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but nothing +> else can serve as the foundation stone --as the first step." +> +> --- John Tukey + + + + +## El papel de la exploración en el análisis de datos {.unnumbered} + +El estándar científico para contestar preguntas o tomar decisiones es +uno que se basa en el análisis de datos. Es decir, en primer lugar se +deben reunir todos los datos que puedan contener o sugerir alguna guía +para entender mejor la pregunta o la decisión a la que nos enfrentamos. +Esta recopilación de datos ---que pueden ser cualitativos, +cuantitativos, o una mezcla de los dos--- debe entonces ser analizada +para extraer información relevante para nuestro problema. + +En análisis de datos existen dos distintos tipos de trabajo: + +- El trabajo **exploratorio** o de **detective**: ¿cuáles son los + aspectos importantes de estos datos? ¿qué indicaciones generales + muestran los datos? ¿qué tareas de análisis debemos empezar + haciendo? ¿cuáles son los caminos generales para formular con + precisión y contestar algunas preguntas que nos interesen? + +- El trabajo **inferencial**, **confirmatorio**, o de **juez**: ¿cómo + evaluar el peso de la evidencia de los descubrimientos del paso + anterior? ¿qué tan bien soportadas están las respuestas y + conclusiones por nuestro conjunto de datos? + +## Preguntas y datos {-} + +Cuando observamos un conjunto de datos, independientemente de su tamaño, el paso inicial más importante es entender bajo qué proceso se generan los datos. + +A grandes rasgos, cuanto más sepamos de este proceso, mejor podemos contestar preguntas de interés. +En muchos casos, tendremos que hacer algunos supuestos de cómo se generan estos datos para dar respuestas (condicionales a esos supuestos). + +## Algunos conceptos básicos {.unnumbered} + +Empezamos explicando algunas ideas que no serán útiles más adelante. El primer concepto se refiere a entender cómo se distribuyen los datos a los largo de su escala de medición. Comenzamos con un ejemplo: los siguientes datos fueron registrados en un restaurante durante cuatro días consecutivos. + + +```{.r .fold-hide} +library(tidyverse) +library(patchwork) +source("R/funciones_auxiliares.R") +# usamos los datos tips del paquete reshape2 +propinas <- read_csv("./data/propinas.csv") +``` + +Y vemos una muestra + + +``` r +sample_n(propinas, 10) |> formatear_tabla() +``` + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
cuenta_total propina fumador dia momento num_personas
24.71 5.85 No Jue Comida 2
19.77 2.00 No Dom Cena 4
16.00 2.00 Si Jue Comida 2
15.06 3.00 No Sab Cena 2
17.07 3.00 No Sab Cena 3
13.13 2.00 No Dom Cena 2
44.30 2.50 Si Sab Cena 3
10.34 2.00 Si Jue Comida 2
14.78 3.23 No Dom Cena 2
25.28 5.00 Si Sab Cena 2
+ +Aquí la unidad de observación es una cuenta particular. Tenemos tres +mediciones numéricas de cada cuenta: cúanto fue la cuenta total, la +propina, y el número de personas asociadas a la cuenta. Los datos están +separados según se fumó o no en la mesa, y temporalmente en dos partes: +el día (Jueves, Viernes, Sábado o Domingo), cada uno separado por Cena y +Comida. + +
+

Denotamos por \(x\) el valor de +medición de una unidad de observación. Usualmente utilizamos +sub-índices para identificar entre diferentes puntos de datos +(observaciones), por ejemplo, \(x_n\) +para la \(n-\)ésima observación. De tal +forma que una colección de \(N\) +observaciones la escribimos como \[\begin{align} +\{x_1, \ldots, x_N\}. +\end{align}\]

+
+ +El primer tipo de comparaciones que nos interesa hacer es para una +medición: ¿Varían mucho o poco los datos de un tipo de medición? ¿Cuáles +son valores típicos o centrales? ¿Existen valores atípicos? + +Supongamos entonces que consideramos simplemente la variable de +`cuenta_total`. Podemos comenzar por **ordenar los datos**, y ver cuáles +datos están en los extremos y cuáles están en los lugares centrales: + + +``` r +propinas <- propinas |> + mutate(orden_cuenta = rank(cuenta_total, ties.method = "first"), + f = (orden_cuenta - 0.5) / n()) +cuenta <- propinas |> select(orden_cuenta, f, cuenta_total) |> arrange(f) +bind_rows(head(cuenta), tail(cuenta)) |> formatear_tabla() +``` + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
orden_cuenta f cuenta_total
1 0.0020492 3.07
2 0.0061475 5.75
3 0.0102459 7.25
4 0.0143443 7.25
5 0.0184426 7.51
6 0.0225410 7.56
239 0.9774590 44.30
240 0.9815574 45.35
241 0.9856557 48.17
242 0.9897541 48.27
243 0.9938525 48.33
244 0.9979508 50.81
+ +También podemos graficar los datos en orden, interpolando valores +consecutivos. + + + +A esta función le llamamos la **función de cuantiles** para la variable +`cuenta_total`. Nos sirve para comparar directamente los distintos +valores que observamos los datos según el orden que ocupan. En +particular, podemos estudiar la **dispersión y valores centrales** de +los datos observados: + +- El **rango** de datos va de unos 3 dólares hasta 50 dólares +- Los **valores centrales**, por ejemplo el 50% de los valores más + centrales, están entre unos 13 y 25 dólares. +- El valor que divide en dos mitades iguales a los datos es de + alrededor de 18 dólares. + +
+

El cuantil \(f\), +que denotamos por \(q(f)\) es valor a +lo largo de la escala de medición de los datos tal que aproximadamente +una fracción \(f\) de los datos son +menores o iguales a \(q(f)\).

+ +
+ +En nuestro ejemplo: + +- Los **valores centrales** ---del cuantil 0.25 al 0.75, por decir un + ejemplo--- están entre unos 13 y 25 dólares. Estos dos cuantiles se + llaman **cuartil inferior** y **cuartil superior** respectivamente +- El cuantil 0.5 (o también conocido como **mediana**) está alrededor + de 18 dólares. + +Éste último puede ser utilizado para dar un valor *central* de la +distribución de valores para `cuenta_total`. Asimismo podemos dar +resúmenes más refinados si es necesario. Por ejemplo, podemos reportar +que: + +- El cuantil 0.95 es de unos 35 dólares --- sólo 5% de las cuentas son + de más de 35 dólares +- El cuantil 0.05 es de unos 8 dólares --- sólo 5% de las cuentas son + de 8 dólares o menos. + +Finalmente, la forma de la gráfica se interpreta usando su pendiente +(tasa de cambio) haciendo comparaciones en diferentes partes de la +gráfica: + +- La distribución de valores tiene asimetría: el 10% de las cuentas + más altas tiene considerablemente más dispersión que el 10% de las + cuentas más bajas. + +- Entre los cuantiles 0.2 y 0.7 es donde existe *mayor* densidad de + datos: la pendiente (tasa de cambio) es baja, lo que significa que + al avanzar en los valores observados, los cuantiles (el porcentaje + de casos) aumenta rápidamente. + +- Cuando la pendiente alta, quiere decir que los datos tienen más + dispersión local o están más separados. + +**Observación**: Hay varias maneras de definir los cuantiles (ver +[@cleveland93]): Supongamos que queremos definir $q(f)$, y denotamos los +datos ordenados como $x_{(1)}, x_{(2)}, \ldots, x_{(N)}$, de forma que +$x_{(1)}$ es el dato más chico y $x_{(N)}$ es el dato más grande. Para +cada $x_{(i)}$ definimos\ +$$f_i = i / N$$ entonces definimos el cuantil $q(f_i)=x_{(i)}$. Para +cualquier $f$ entre 0 y 1, podemos definir $q(f)$ como sigue: si $f$ +está entre $f_i$ y $f_{i+1}$ interpolamos linealmente los valores +correspondientes $x_{(i)}$ y $x_{(i-1)}$. + +En la práctica, es más conveniente usar $f_i= \frac{i - 0.5}{N}$. La +gráfica de cuantiles no cambia mucho comparado con la difinición +anterior, y esto nos permitirá comparar de mejor manera con +distribuciones teóricas que no tienen definido su cuantil 0 y el 1, pues +tienen soporte en los números reales (como la distribución normal, por +ejemplo). + +------------------------------------------------------------------------ + +Asociada a la función de cuantiles $q$ tenemos la **distribución +acumulada empírica de los datos**, que es aproximadamente inversa de la +función de cuantiles, y se define como: + +$$\hat{F}(x) = i/N$$ si $x_{(i)} \leq x < x_{(i+1)}$. + +Nótese que +$\hat{F}(q(f_i)) = i/N = f_i$ (demuéstralo). + + +``` r +acum_cuenta <- ecdf(cuenta$cuenta_total) +cuenta <- cuenta |> + mutate(dea_cuenta_total = acum_cuenta(cuenta_total)) +g_acum <- ggplot(cuenta, aes(x = cuenta_total, y = dea_cuenta_total)) + geom_point() + + labs(subtitle = "Distribución acum empírica de cuenta total") + xlab("") +g_cuantiles + g_acum +``` + + + +La función de distribución acumulada empírica es otra forma de graficar la dispersión de los datos. En su gráfica vemos que proporción de los datos que son iguales o están por debajo de cada valor en el eje horizontal. + +Nota: + +* En análisis de datos, es más frecuente utilizar la función de cuantiles pues existen versiones más generales que son útiles, por ejemplo, para evaluar ajuste de modelos probabilísticos + +* En la teoría, generalmente es más común utilizar la fda empírica, que tiene una única definición que veremos coincide con definiciones teóricas. + + +### Histogramas {.unnumbered} + +En algunos casos, es más natural hacer un **histograma**, donde +dividimos el rango de la variable en cubetas o intervalos (en este caso +de igual longitud), y graficamos por medio de barras cuántos datos caen +en cada cubeta: + + + +Es una gráfica más popular, pero perdemos cierto nivel de detalle, y +distintas particiones resaltan distintos aspectos de los datos. + +
+

¿Cómo se ve la gráfica de cuantiles de las propinas? ¿Cómo crees que +esta gráfica se compara con distintos histogramas?

+
+ + +``` r +g_1 <- ggplot(propinas, aes(sample = propina)) + + geom_qq(distribution = stats::qunif) + xlab("f") + ylab("propina") +g_1 +``` + + + +Finalmente, una gráfica más compacta que resume la gráfica de cuantiles +o el histograma es el **diagrama de caja y brazos**. Mostramos dos +versiones, la clásica de Tukey (T) y otra versión menos común de +Spear/Tufte (ST): + + +``` r +library(ggthemes) +cuartiles <- quantile(cuenta$cuenta_total) +t(cuartiles) |> formatear_tabla() +``` + + + + + + + + + + + + + + + + + + + + +
0% 25% 50% 75% 100%
3.07 13.3475 17.795 24.1275 50.81
+ +``` r +g_1 <- ggplot(cuenta, aes(x = f, y = cuenta_total)) + + labs(subtitle = "Gráfica de cuantiles: Cuenta total") + + geom_hline(yintercept = cuartiles[2], colour = "gray") + + geom_hline(yintercept = cuartiles[3], colour = "gray") + + geom_hline(yintercept = cuartiles[4], colour = "gray") + + geom_point(alpha = 0.5) + geom_line() +g_2 <- ggplot(cuenta, aes(x = factor("ST", levels =c("ST")), y = cuenta_total)) + + geom_tufteboxplot() + + labs(subtitle = " ") + xlab("") + ylab("") +g_3 <- ggplot(cuenta, aes(x = factor("T"), y = cuenta_total)) + + geom_boxplot() + + labs(subtitle = " ") + xlab("") + ylab("") +g_4 <- ggplot(cuenta, aes(x = factor("P"), y = cuenta_total)) + + geom_jitter(height = 0, width =0.2, alpha = 0.5) + + labs(subtitle = " ") + xlab("") + ylab("") +g_5 <- ggplot(cuenta, aes(x = factor("V"), y = cuenta_total)) + + geom_violin() + + labs(subtitle = " ") + xlab("") + ylab("") +g_1 + g_2 + g_3 + g_4 + + plot_layout(widths = c(8, 2, 2, 2)) +``` + + + +::: {.cols data-latex=""} +::: {.col data-latex="{0.50\\textwidth}"} +El diagrama de la derecha explica los elementos de la versión típica del +diagrama de caja y brazos (*boxplot*). *RIC* se refiere al **Rango +Intercuantílico**, definido por la diferencia entre los cuantiles 25% y +75%. +::: + +::: {.col data-latex="{0.10\\textwidth}"} +  +::: + +::: {.col data-latex="{0.4\\textwidth}"} + + +Figura: [Jumanbar / CC +BY-SA](https://creativecommons.org/licenses/by-sa/3.0) +::: +::: + +**Ventajas en el análisis inicial** + +En un principio del análisis, estos resúmenes (cuantiles) pueden ser más +útiles que utilizar medias y varianzas, por ejemplo. La razón es que los +cuantiles: + +- Son cantidades más fácilmente interpretables +- Los cuantiles centrales son más resistentes a valores atípicos que + medias o varianzas +- Permiten identificar valores extremos +- Es fácil comparar cuantiles de distintos bonches de datos en la + misma escala + +**Nota:** Existen diferentes definiciones para calcular cuantiles de una muestra de +datos, puedes leer más en [este artículo](https://www.amherst.edu/media/view/129116/original/Sample+Quantiles.pdf). + +### Media y desviación estándar {.unnumbered} + +Las medidas más comunes de localización y dispersión para un conjunto de +datos son la media muestral y la [desviación estándar +muestral](https://es.wikipedia.org/wiki/Desviación_t%C3%ADpica). + +En general, no son muy apropiadas para iniciar el análisis exploratorio, +pues: + +- Son medidas más difíciles de interpretar y explicar que los + cuantiles. En este sentido, son medidas especializadas. Por ejemplo, + compara una explicación intuitiva de la mediana contra una + explicación intuitiva de la media. +- No son resistentes a valores atípicos. Su falta de resistencia los + vuelve poco útiles en las primeras etapas de limpieza y descripción + y en resúmenes deficientes para distribuciones irregulares (con + colas largas por ejemplo). + +
+

La media, o promedio, se denota por \(\bar +x\) y se define como \[\begin{align} +\bar x = \frac1N \sum_{n = 1}^N x_n. +\end{align}\] La desviación estándar muestral se define como +\[\begin{align} +\text{std}(x) = \sqrt{\frac1{N-1} \sum_{n = 1}^N (x_n - \bar x)^2}. +\end{align}\]

+
+ +**Observación**: Si $N$ no es muy chica, no importa mucho si dividimos +por $N$ o por $N-1$ en la fórmula de la desviación estándar. La razón de +que típicamente se usa $N-1$ la veremos más adelante, en la parte de +estimación. + +Por otro lado, ventajas de estas medidas de centralidad y dispersión +son: + +- La media y desviación estándar son computacionalmente convenientes. Por lo tanto regresaremos a estas medidas una vez que estudiemos modelos de probabilidad básicos. + +- En muchas ocasiones conviene usar estas medidas pues permite hacer + comparaciones históricas o tradicionales ---pues análisis anteriores + pudieran estar basados en éstas. + + + +
+
    +
  1. Considera el caso de tener \(N\) +observaciones y asume que ya tienes calculado el promedio para dichas +observaciones. Este promedio lo denotaremos por \(\bar x_N\). Ahora, considera que has +obtenido \(M\) observaciones más. +Escribe una fórmula recursiva para la media del conjunto total de datos +\(\bar x_{N+M}\) en función de lo que +ya tenías precalculado \(\bar +x_N.\)

  2. +
  3. ¿En qué situaciones esta propiedad puede ser +conveniente?

  4. +
+
+ +## Ejemplos {.unnumbered} + +### Precios de casas {.unnumbered} + +En este ejemplo consideremos los [datos de precios de ventas de la +ciudad de Ames, +Iowa](https://www.kaggle.com/prevek18/ames-housing-dataset). En +particular nos interesa entender la variación del precio de las casas. + + + +Por este motivo calculamos los cuantiles que corresponden al 25%, 50% y +75% (**cuartiles**), así como el mínimo y máximo de los precios de las +casas: + + +``` r +quantile(casas |> pull(precio_miles)) +``` + +``` +## 0% 25% 50% 75% 100% +## 37.9 132.0 165.0 215.0 755.0 +``` + +
+

Comprueba que el mínimo y máximo están asociados a los cuantiles 0% y +100%, respectivamente.

+
+ +Una posible comparación es considerar los precios y sus variación en +función de zona de la ciudad en que se encuentra una vivienda. Podemos +usar diagramas de caja y brazos para hacer una **comparación burda** de +los precios en distintas zonas de la ciudad: + + +``` r +ggplot(casas, aes(x = nombre_zona, y = precio_miles)) + + geom_boxplot() + + coord_flip() +``` + + + +La primera pregunta que nos hacemos es cómo pueden variar las +características de las casas dentro de cada zona. Para esto, podemos +considerar el área de las casas. En lugar de graficar el precio, +graficamos el precio por metro cuadrado, por ejemplo: + + + + +``` r +ggplot(casas, aes(x = nombre_zona, y = precio_m2)) + + geom_boxplot() + + coord_flip() +``` + + + +Podemos cuantificar la variación que observamos de zona a zona y la +variación que hay dentro de cada una de las zonas. Una primera +aproximación es observar las variación del precio al calcular la mediana +dentro de cada zona, y después cuantificar por medio de cuantiles cómo +varía la mediana entre zonas: + + +``` r +casas |> + group_by(nombre_zona) |> + summarise(mediana_zona = median(precio_m2), .groups = "drop") |> + pull(mediana_zona) |> + quantile() |> + round() +``` + +``` +## 0% 25% 50% 75% 100% +## 963 1219 1298 1420 1725 +``` + +Por otro lado, las variaciones con respecto a las medianas **dentro** de +cada zona, por grupo, se resume como: + + +``` r +quantile(casas |> group_by(nombre_zona) |> + mutate(residual = precio_m2 - median(precio_m2)) |> + pull(residual)) |> + round() +``` + +``` +## 0% 25% 50% 75% 100% +## -765 -166 0 172 1314 +``` + +Nótese que este último paso tiene sentido pues la variación dentro de +las zonas, en términos de precio por metro cuadrado, es similar. Esto no +lo podríamos haber hecho de manera efectiva si se hubiera utilizado el +precio de las casas sin ajustar por su tamaño. + +Podemos resumir este primer análisis con un par de gráficas de cuantiles +(@cleveland93): + + +``` r +mediana <- median(casas$precio_m2) +resumen <- casas |> + group_by(nombre_zona) |> + mutate(mediana_zona = median(precio_m2)) |> + mutate(residual = precio_m2 - mediana_zona) |> + ungroup() |> + mutate(mediana_zona = mediana_zona - mediana) |> + select(nombre_zona, mediana_zona, residual) |> + pivot_longer(mediana_zona:residual, names_to = "tipo", values_to = "valor") +ggplot(resumen, aes(sample = valor)) + + geom_qq(distribution = stats::qunif) + + facet_wrap(~ tipo) + + ylab("Precio por m2") + xlab("f") + + labs(subtitle = "Precio por m2 por zona", + caption = paste0("Mediana total de ", round(mediana))) +``` + + + +Vemos que la mayor parte de la variación del precio por metro cuadrado +ocurre dentro de cada zona, una vez que controlamos por el tamaño de las +casas. La variación dentro de cada zona es aproximadamente simétrica, +aunque la cola derecha es ligeramente más larga con algunos valores +extremos. + +Podemos seguir con otro indicador importante: la calificación de calidad +de los terminados de las casas. Como primer intento podríamos hacer: + + + +Lo que indica que las calificaciones de calidad están distribuidas de +manera muy distinta a lo largo de las zonas, y que probablemente no va +ser simple desentrañar qué variación del precio se debe a la zona y cuál +se debe a la calidad. + +### Prueba Enlace {.unnumbered} + +Consideremos la prueba Enlace (2011) de matemáticas para primarias. Una +primera pregunta que alguien podría hacerse es: ¿cuáles escuelas son +mejores en este rubro, las privadas o las públicas? + + + + +``` r +enlace_tbl <- enlace |> group_by(tipo) |> + summarise(n_escuelas = n(), + cuantiles = list(cuantil(mate_6, c(0.05, 0.25, 0.5, 0.75, 0.95)))) |> + unnest(cols = cuantiles) |> mutate(valor = round(valor)) +enlace_tbl |> + spread(cuantil, valor) |> + formatear_tabla() +``` + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
tipo n_escuelas 0.05 0.25 0.5 0.75 0.95
Indígena/Conafe 13599 304 358 412 478 588
General 60166 380 454 502 548 631
Particular 6816 479 551 593 634 703
+ +Para un análisis exploratorio podemos utilizar distintas gráficas. Por +ejemplo, podemos utilizar nuevamente las gráficas de caja y brazos, así +como graficar los percentiles. Nótese que en la gráfica 1 se utilizan +los cuantiles 0.05, 0.25, 0.5, 0.75 y 0.95: + + +``` +## Warning: Using `size` aesthetic for lines was deprecated in ggplot2 3.4.0. +## ℹ Please use `linewidth` instead. +## This warning is displayed once every 8 hours. +## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was +## generated. +``` + + + +Se puede discutir qué tan apropiada es cada gráfica con el objetivo de +realizar comparaciones. Sin duda, graficar más cuantiles es más útil +para hacer comparaciones. Por ejemplo, en la Gráfica 1 podemos ver que +la mediana de las escuelas generales está cercana al cuantil 5% de las +escuelas particulares. Por otro lado, el diagrama de caja y brazos +muestra también valores "atípicos". + +Antes de contestar prematuramente la pregunta: ¿cuáles son las mejores +escuelas? busquemos mejorar la interpretabilidad de nuestras +comparaciones. Podemos comenzar por agregar, por ejemplo, el nivel del +marginación del municipio donde se encuentra la escuela. + + + +Para este objetivo, podemos usar páneles (pequeños múltiplos útiles para +hacer comparaciones) y graficar: + + + +Esta gráfica pone en contexto la pregunta inicial, y permite evidenciar +la dificultad de contestarla. En particular: + +1. Señala que la pregunta no sólo debe concentarse en el tipo de + "sistema": pública, privada, etc. Por ejemplo, las escuelas públicas + en zonas de marginación baja no tienen una distribución de + calificaciones muy distinta a las privadas en zonas de marginación + alta. +2. El contexto de la escuela es importante. +3. Debemos de pensar qué factores --por ejemplo, el entorno familiar de + los estudiantes-- puede resultar en comparaciones que favorecen a + las escuelas privadas. Un ejemplo de esto es considerar si los + estudiantes tienen que trabajar o no. A su vez, esto puede o no ser + reflejo de la calidad del sistema educativo. +4. Si esto es cierto, entonces la pregunta inicial es demasiado vaga y + mal planteada. Quizá deberíamos intentar entender cuánto "aporta" + cada escuela a cada estudiante, como medida de qué tan buena es cada + escuela. + +### Estados y calificaciones en SAT {.unnumbered} + +¿Cómo se relaciona el gasto por alumno, a nivel estatal, con sus +resultados académicos? Hay trabajo considerable en definir estos +términos, pero supongamos que tenemos el [siguiente conjunto de +datos](http://jse.amstat.org/datasets/sat.txt) [@Guber], que son datos +oficiales agregados por `estado` de Estados Unidos. Consideremos el +subconjunto de variables `sat`, que es la calificación promedio de los +alumnos en cada estado (para 1997) y `expend`, que es el gasto en miles +de dólares por estudiante en (1994-1995). + + +``` r +sat <- read_csv("data/sat.csv") +sat_tbl <- sat |> select(state, expend, sat) |> + gather(variable, valor, expend:sat) |> + group_by(variable) |> + summarise(cuantiles = list(cuantil(valor))) |> + unnest(cols = c(cuantiles)) |> + mutate(valor = round(valor, 1)) |> + spread(cuantil, valor) +sat_tbl |> formatear_tabla() +``` + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
variable 0 0.25 0.5 0.75 1
expend 3.7 4.9 5.8 6.4 9.8
sat 844.0 897.2 945.5 1032.0 1107.0
+ +Esta variación es considerable para promedios del SAT: el percentil 75 +es alrededor de 1050 puntos, mientras que el percentil 25 corresponde a +alrededor de 800. Igualmente, hay diferencias considerables de gasto por +alumno (miles de dólares) a lo largo de los estados. + +Ahora hacemos nuestro primer ejercico de comparación: ¿Cómo se ven las +calificaciones para estados en distintos niveles de gasto? Podemos usar +una gráfica de dispersión: + + +``` r +library(ggrepel) +ggplot(sat, aes(x = expend, y = sat, label = state)) + + geom_point(colour = "red", size = 2) + geom_text_repel(colour = "gray50") + + xlab("Gasto por alumno (miles de dólares)") + + ylab("Calificación promedio en SAT") +``` + + + +Estas comparaciones no son de alta calidad, solo estamos usando 2 +variables ---que son muy pocas--- y no hay mucho que podamos decir en +cuanto explicación. Sin duda nos hace falta una imagen más completa. +Necesitaríamos entender la correlación que existe entre las demás +características de nuestras unidades de estudio. + +**Las unidades que estamos comparando pueden diferir fuertemente en +otras propiedades importantes (o dimensiones), lo cual no permite +interpretar la gráfica de manera sencilla.** + +Una variable que tenemos es el porcentaje de alumnos de cada estado que +toma el SAT. Podemos agregar como sigue: + + +``` r +ggplot(sat, aes(x = expend, y = math, label=state, colour = frac)) + + geom_point() + geom_text_repel() + + xlab("Gasto por alumno (miles de dólares)") + + ylab("Calificación en matemáticas") +``` + + + +Esto nos permite entender por qué nuestra comparación inicial es +relativamente pobre. Los estados con mejores resultados promedio en el +SAT son aquellos donde una fracción relativamente baja de los +estudiantes toma el examen. La diferencia es considerable. + +En este punto podemos hacer varias cosas. Una primera idea es intentar +comparar estados más similares en cuanto a la población de alumnos que +asiste. Podríamos hacer grupos como sigue: + + +``` r +set.seed(991) +k_medias_sat <- kmeans(sat |> select(frac), centers = 4, nstart = 100, iter.max = 100) +sat$clase <- k_medias_sat$cluster +sat <- sat |> group_by(clase) |> + mutate(clase_media = round(mean(frac))) |> + ungroup() |> + mutate(clase_media = factor(clase_media)) +sat <- sat |> + mutate(rank_p = rank(frac, ties= "first") / length(frac)) +ggplot(sat, aes(x = rank_p, y = frac, label = state, + colour = clase_media)) + + geom_point(size = 2) +``` + + + +Estos resultados indican que es más probable que buenos alumnos decidan +hacer el SAT. Lo interesante es que esto ocurre de manera diferente en +cada estado. Por ejemplo, en algunos estados era más común otro examen: +el ACT. + +Si hacemos *clusters* de estados según el % de alumnos, empezamos a ver +otra historia. Para esto, ajustemos rectas de mínimos cuadrados como +referencia: + + + +Esto da una imagen muy diferente a la que originalmente planteamos. Nota +que dependiendo de cómo categorizamos, esta gráfica puede variar (puedes +intentar con más o menos grupos, por ejemplo). + +### Tablas de conteos {.unnumbered} + +Consideremos los siguientes datos de tomadores de té (del paquete +FactoMineR [@factominer]): + + +``` r +tea <- read_csv("data/tea.csv") +# nombres y códigos +te <- tea |> select(how, price, sugar) |> + rename(presentacion = how, precio = price, azucar = sugar) |> + mutate( + presentacion = fct_recode(presentacion, + suelto = "unpackaged", bolsas = "tea bag", mixto = "tea bag+unpackaged"), + precio = fct_recode(precio, + marca = "p_branded", variable = "p_variable", barato = "p_cheap", + marca_propia = "p_private label", desconocido = "p_unknown", fino = "p_upscale"), + azucar = fct_recode(azucar, + sin_azúcar = "No.sugar", con_azúcar = "sugar")) +``` + + +``` r +sample_n(te, 10) +``` + +``` +## # A tibble: 10 × 3 +## presentacion precio azucar +## +## 1 bolsas marca sin_azúcar +## 2 bolsas variable sin_azúcar +## 3 bolsas marca con_azúcar +## 4 bolsas fino con_azúcar +## 5 mixto variable con_azúcar +## 6 mixto fino con_azúcar +## 7 bolsas marca sin_azúcar +## 8 bolsas fino sin_azúcar +## 9 mixto variable con_azúcar +## 10 mixto variable sin_azúcar +``` + +Nos interesa ver qué personas compran té suelto, y de qué tipo. +Empezamos por ver las proporciones que compran té según su empaque (en +bolsita o suelto): + + +``` r +precio <- te |> + count(precio) |> + mutate(prop = round(100 * n / sum(n))) |> + select(-n) +tipo <- te |> + count(presentacion) |> + mutate(pct = round(100 * n / sum(n))) +tipo |> formatear_tabla() +``` + + + + + + + + + + + + + + + + + + + + + + + + + + +
presentacion n pct
bolsas 170 57
mixto 94 31
suelto 36 12
+ +La mayor parte de las personas toma té en bolsas. Sin embargo, el tipo +de té (en términos de precio o marca) que compran es muy distinto +dependiendo de la presentación: + + +``` r +tipo <- tipo |> select(presentacion, prop_presentacion = pct) +tabla_cruzada <- te |> + count(presentacion, precio) |> + # porcentajes por presentación + group_by(presentacion) |> + mutate(prop = round(100 * n / sum(n))) |> + select(-n) +tabla_cruzada |> + pivot_wider(names_from = presentacion, values_from = prop, + values_fill = list(prop = 0)) |> + formatear_tabla() +``` + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
precio bolsas mixto suelto
marca 41 21 14
barato 3 1 3
marca_propia 9 4 3
desconocido 6 1 0
fino 8 20 56
variable 32 52 25
+ +Estos datos podemos examinarlos un rato y llegar a conclusiones, pero +esta tabla no necesariamente es la mejor manera de mostrar patrones en +los datos. Tampoco son muy útiles gráficas como la siguiente: + + +``` r +ggplot(tabla_cruzada |> ungroup() |> + mutate(price = fct_reorder(precio, prop)), + aes(x = precio, y = prop, group = presentacion, colour = presentacion)) + + geom_point() + coord_flip() + geom_line() +``` + + + +En lugar de eso, calcularemos *perfiles columna*. Esto es, comparamos +cada una de las columnas con la columna marginal (en la tabla de tipo de +estilo de té): + + +``` r +num_grupos <- n_distinct(te |> select(presentacion)) +tabla <- te |> + count(presentacion, precio) |> + group_by(presentacion) |> + mutate(prop_precio = (100 * n / sum(n))) |> + group_by(precio) |> + mutate(prom_prop = sum(prop_precio)/num_grupos) |> + mutate(perfil = 100 * (prop_precio / prom_prop - 1)) +tabla +``` + +``` +## # A tibble: 17 × 6 +## # Groups: precio [6] +## presentacion precio n prop_precio prom_prop perfil +## +## 1 bolsas marca 70 41.2 25.4 61.8 +## 2 bolsas barato 5 2.94 2.26 30.1 +## 3 bolsas marca_propia 16 9.41 5.48 71.7 +## 4 bolsas desconocido 11 6.47 2.51 158. +## 5 bolsas fino 14 8.24 28.0 -70.6 +## 6 bolsas variable 54 31.8 36.3 -12.5 +## 7 mixto marca 20 21.3 25.4 -16.4 +## 8 mixto barato 1 1.06 2.26 -52.9 +## 9 mixto marca_propia 4 4.26 5.48 -22.4 +## 10 mixto desconocido 1 1.06 2.51 -57.6 +## 11 mixto fino 19 20.2 28.0 -27.8 +## 12 mixto variable 49 52.1 36.3 43.6 +## 13 suelto marca 5 13.9 25.4 -45.4 +## 14 suelto barato 1 2.78 2.26 22.9 +## 15 suelto marca_propia 1 2.78 5.48 -49.3 +## 16 suelto fino 20 55.6 28.0 98.4 +## 17 suelto variable 9 25 36.3 -31.1 +``` + + +``` r +tabla_perfil <- tabla |> + select(presentacion, precio, perfil, pct = prom_prop) |> + pivot_wider(names_from = presentacion, values_from = perfil, + values_fill = list(perfil = -100.0)) +if_profile <- function(x){ + any(x < 0) & any(x > 0) +} +marcar <- marcar_tabla_fun(25, "red", "black") +tab_out <- tabla_perfil |> + arrange(desc(bolsas)) |> + select(-pct, everything()) |> + mutate(across(where(is.numeric), round, 0)) |> + mutate(across(where(if_profile), marcar)) |> + knitr::kable(format_table_salida(), escape = FALSE, digits = 0, booktabs = T) |> + kableExtra::kable_styling(latex_options = c("striped", "scale_down"), + bootstrap_options = c( "hover", "condensed"), + full_width = FALSE) +``` + +``` +## Warning: There was 1 warning in `mutate()`. +## ℹ In argument: `across(where(is.numeric), round, 0)`. +## ℹ In group 1: `precio = marca`. +## Caused by warning: +## ! The `...` argument of `across()` is deprecated as of dplyr 1.1.0. +## Supply arguments directly to `.fns` through an anonymous function instead. +## +## # Previously +## across(a:b, mean, na.rm = TRUE) +## +## # Now +## across(a:b, \(x) mean(x, na.rm = TRUE)) +``` + +``` r +if (knitr::is_latex_output()) { + gsub("marca_propia", "marca-propia", tab_out) +} else { + tab_out +} +``` + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
precio bolsas mixto suelto pct
desconocido 158 -58 -100 3
marca_propia 72 -22 -49 5
marca 62 -16 -45 25
barato 30 -53 23 2
variable -12 44 -31 36
fino -71 -28 98 28
+ +Leemos esta tabla como sigue: por ejemplo, los compradores de té suelto +compran té *fino* a una tasa casi el doble (98%) que el promedio. + +También podemos graficar como: + + +``` r +tabla_graf <- tabla_perfil |> + ungroup() |> + mutate(precio = fct_reorder(precio, bolsas)) |> + select(-pct) |> + pivot_longer(cols = -precio, names_to = "presentacion", values_to = "perfil") +g_perfil <- ggplot(tabla_graf, + aes(x = precio, xend = precio, y = perfil, yend = 0, group = presentacion)) + + geom_point() + geom_segment() + facet_wrap(~presentacion) + + geom_hline(yintercept = 0 , colour = "gray")+ coord_flip() +g_perfil +``` + + + +**Observación**: hay dos maneras de construir la columna promedio: +tomando los porcentajes sobre todos los datos, o promediando los +porcentajes de las columnas. Si los grupos de las columnas están +desbalanceados, estos promedios son diferentes. + +- Cuando usamos porcentajes sobre la población, perfiles columna y + renglón dan el mismo resultado +- Sin embargo, cuando hay un grupo considerablemente más grande que + otros, las comparaciones se vuelven vs este grupo particular. No + siempre queremos hacer esto. + +#### Interpretación {.unnumbered} + +En el último ejemplo de tomadores de té utilizamos una muestra de +personas, no toda la población de tomadores de té. Eso quiere decir que +tenemos cierta incertidumbre de cómo se generalizan o no los resultados +que obtuvimos en nuestro análisis a la población general. + +Nuestra respuesta depende de cómo se extrajo la muestra que estamos +considerando. Si el mecanismo de extracción incluye algún proceso +probabilístico, entonces es posible en principio entender qué tan bien +generalizan los resultados de nuestro análisis a la población general, y +entender esto depende de entender qué tanta variación hay de muestra a +muestra, de todas las posibles muestras que pudimos haber extraido. + +En las siguiente secciones discutiremos estos aspectos, en los cuales +pasamos del trabajo de "detective" al trabajo de "juez" en nuestro +trabajo analítico. + +## Suavizamiento loess {.unnumbered} + +Las gráficas de dispersión son la herramienta básica para describir la +relación entre dos variables cuantitativas, y como vimos en ejemplo +anteriores, muchas veces podemos apreciar mejor la relación entre ellas +si agregamos una curva `loess` que suavice. + +Los siguientes datos muestran los premios ofrecidos y las ventas totales +de una lotería a lo largo de 53 sorteos (las unidades son cantidades de +dinero indexadas). Graficamos en escalas logarítmicas y agregamos una +curva `loess`. + + + +``` r +# cargamos los datos +load(here::here("data", "ventas_sorteo.Rdata")) + +ggplot(ventas.sorteo, aes(x = premio, y = ventas.tot.1)) + + geom_point() + + geom_smooth(method = "loess", alpha = 0.5, degree = 1, se = FALSE) + + scale_x_log10(breaks = c(20000, 40000, 80000)) + + scale_y_log10(breaks = c(10000, 15000, 22000, 33000)) +``` + + + +El patrón no era difícil de ver en los datos originales, sin embargo, la +curva lo hace más claro, el logaritmo de las ventas tiene una relación +no lineal con el logaritmo del premio: para premios no muy grandes no +parece haber gran diferencia, pero cuando los premios empiezan a crecer +por encima de 20,000, las ventas crecen más rápidamente que para premios +menores. Este efecto se conoce como *bola de nieve*, y es frecuente en +este tipo de loterías. + +Antes de adentrarnos a los modelos `loess` comenzamos explicando cómo se +ajustan familias paramétricas de curvas a conjuntos de datos dados. El +modelo de *regresion lineal* ajusta una recta a un conjunto de datos. +Por ejemplo, consideremos la familia $$f_{a,b}(x) = a x + b,$$ para un +conjunto de datos bivariados $\{ (x_1, y_1), \ldots, (x_N, y_N)\}$. +Buscamos encontrar $a$ y $b$ tales que $f_{a,b}$ de un ajuste *óptimo* a +los datos. Para esto, se minimiza la suma de errores cuadráticos + +$$\frac1N \sum_{n = 1}^N ( y_n - a x_n - b)^2.$$ + +En este caso, las constantes $a$ y $b$ se pueden encontrar diferenciando +la función de mínimos cuadrados. Nótese que podemos repetir el argumento +con otras familias de funciones (por ejemplo cuadráticas). + + +``` r +ggplot(ventas.sorteo, aes(x = premio, y = ventas.tot.1)) + + geom_point() + + geom_smooth(method = "lm", alpha = 0.5, se = FALSE) + + scale_x_log10(breaks = c(20000, 40000, 80000)) + + scale_y_log10(breaks = c(10000, 15000, 22000, 33000)) +``` + + + +Si observamos la gráfica notamos que este modelo lineal (en los +logaritmos) no resumen adecuadamente estos datos. Podríamos experimentar +con otras familias (por ejemplo, una cuadrática o cúbica, potencias, +exponenciales, etc.); sin embargo, en la etapa exploratoria es mejor +tomar una ruta de ajuste más flexible y robusta. Regresión local nos +provee de un método con estas características: + +**Curvas loess (regresión local):** Una manera de mejorar la +flexibilidad de los modelos lineales es considerar rectas de manera +local. Es decir, en cada $x$ posible consideramos cuál es la recta +que mejor ajusta a los datos, considerando solamente valores de $x_n$ +que están cercanos a $x$. + +La siguiente gráfica muestra qué recta se ajusta alrededor de cada +punto, y cómo queda el suavizador completo, con distintos valores de +suavizamiento. El tono de los puntos indican en cada paso que ventana de datos es +considerada: + + + +**Escogiendo de los parámetros.** El parámetro de suavizamiento se +encuentra por ensayo y error. La idea general es que debemos encontrar +una curva que explique patrones importantes en los datos (que *ajuste* +los datos) pero que no muestre variaciones a escalas más chicas +difíciles de explicar (que pueden ser el resultado de influencias de +otras variables, variación muestral, ruido o errores de redondeo, por +ejemplo). En el proceso de prueba y error iteramos el ajuste y en cada +paso hacemos análisis de residuales, con el fin de seleccionar un +suavizamiento adecuado. + +En lugar de usar ajustes locales lineales, podemos usar ajustes locales +cuadráticos que nos permiten capturar formas locales cuadráticas sin +tener que suavizar demasiado poco: + + + +### Opcional: cálculo del suavizador {.unnumbered} + +La idea es producir ajustes locales de rectas o funciones lineales o +cuadráticas. Consideremos especificar dos parámetros: + +- Parámetro de suavizamiento $\alpha$: cuando $\alpha$ es más grande, + la curva ajustada es más suave. + +- Grado de los polinomios locales que ajustamos $\lambda$: + generalmente se toma $\lambda=1,2$. + +Entonces, supongamos que los datos están dados por +$(x_1,y_1), \ldots, (x_N, y_N)$, y sean $\alpha$ un parámetro de +suavizamiento fijo, y $\lambda=1$. Denotamos como $\hat{g}(x)$ la curva +`loess` ajustada, y como $w_n(x)$ a una función de peso (que depende de +x) para la observación $(x_n, y_n)$. + +Para poder calcular $w_n(x)$ debemos comenzar calculando +$q=\lfloor{N\alpha}\rfloor$ que suponemos mayor que uno, esta $q$ es el número +de puntos que se utilizan en cada ajuste local. Ahora definimos +la función *tricubo*: \begin{align} +T(u)=\begin{cases} +(1-|u|^3)^3, & \text{para $|u| < 1$}.\\ +0, & \text{en otro caso}. +\end{cases} +\end{align} + +entonces, para el punto $x$ definimos el peso correspondiente al dato +$(x_n,y_n)$, denotado por $w_n(x)$ como: + +$$w_n(x)=T\bigg(\frac{|x-x_n|}{d_q(x)}\bigg)$$ + +donde $d_q(x)$ es el valor de la $q$-ésima distancia más chica (la más +grande entre las $q$ más chicas) entre los valores $|x-x_j|$, +$j=1,\ldots,N$. De esta forma, las observaciones $x_n$ reciben más peso +cuanto más cerca estén de $x$. + +En palabras, de $x_1,...,x_N$ tomamos los $q$ datos más cercanos a $x$, +que denotamos $x_{i_1}(x) \leq x_{i_2}(x) \leq \cdots \leq x_{i_q}(x)$. +Los re-escalamos a $[0,1]$ haciendo corresponder $x$ a $0$ y el punto +más alejado de $x$ (que es $x_{i_q}$) a 1. + +Aplicamos el tricubo (gráfica de abajo), para encontrar los pesos de +cada punto. Los puntos que están a una distancia mayor a $d_q(x)$ +reciben un peso de cero, y los más cercanos un peso que depende de que +tan cercanos están a $x$. Nótese que $x$ es el punto ancla en dónde +estamos ajustando la regresión local. + + +``` r +tricubo <- function(x) { + ifelse(abs(x) < 1, (1 - abs(x) ^ 3) ^ 3, 0) +} +curve(tricubo, from = -1.5, to = 1.5) +``` + + + +Finalmente, para cada valor de $x_k$ que está en el conjunto de datos +$\{x_1,...,x_n\}$, ajustamos una recta de mínimos cuadrados ponderados +por los pesos $w_n(x)$, es decir, minimizamos (en el caso lineal): + +$$\sum_{i=1}^nw_n(x_k)(y_i-ax_n-b)^2.$$ + +**Observaciones:** + +1. Cualquier función (continua y quizás diferenciable) con la forma de + flan del tricubo que se desvanece fuera de $(-1,1),$ es creciente en + $(-1,0)$ y decreciente en $(0, 1)$ es un buen candidato para usarse + en lugar del tricubo. La razón por la que escogemos precisamente + esta forma algebráica no tiene que ver con el análisis exploratorio, + sino con las ventajas teóricas adicionales que tiene en la + inferencia. + +2. El caso $\lambda=2$ es similar. La única diferencia es en el paso de + ajuste, donde usamos funciones cuadráticas, y obtendríamos entonces + tres familias de parámetros $a(x_k), b_1(x_k), b_2(x_k),$ para cada + $k \in \{1, \ldots, N\}$. + +## Caso de estudio: nacimientos en México {.unnumbered} + +Podemos usar el suavizamiento `loess` para entender y describir el +comportamiento de series de tiempo, en las cuáles intentamos entender la +dependencia de una serie de mediciones indexadas por el tiempo. +Típicamente es necesario utilizar distintas componentes para describir +exitosamente una serie de tiempo, y para esto usamos distintos tipos de +suavizamientos. Veremos que distintas componentes varían en distintas +escalas de tiempo (unas muy lentas, como la tendencia, otras más +rapidamente, como variación quincenal, etc.). + +Este caso de estudio esta basado en un análisis propuesto por [A. +Vehtari y A. +Gelman](https://statmodeling.stat.columbia.edu/2016/05/18/birthday-analysis-friday-the-13th-update/), +junto con un análisis de serie de tiempo de @cleveland93. + +En nuestro caso, usaremos los datos de nacimientos registrados por día +en México desde 1999. Los usaremos para contestar las preguntas: ¿cuáles +son los cumpleaños más frecuentes? y ¿en qué mes del año hay más +nacimientos? + +Podríamos utilizar una gráfica popular (ver por ejemplo [esta +visualización](http://thedailyviz.com/2016/09/17/how-common-is-your-birthday-dailyviz/)) +como: + + + +Sin embargo, ¿cómo criticarías este análisis desde el punto de vista de +los tres primeros principios del diseño analítico? ¿Las comparaciones +son útiles? ¿Hay aspectos multivariados? ¿Qué tan bien explica o sugiere +estructura, mecanismos o causalidad? + +### Datos de natalidad para México {.unnumbered} + + +``` r +library(lubridate) +library(ggthemes) +theme_set(theme_minimal(base_size = 14)) +natalidad <- read_rds("./data/nacimientos/natalidad.rds") |> + mutate(dia_semana = weekdays(fecha)) |> + mutate(dia_año = yday(fecha)) |> + mutate(año = year(fecha)) |> + mutate(mes = month(fecha)) |> + ungroup() |> + mutate(dia_semana = recode(dia_semana, Monday = "Lunes", Tuesday = "Martes", Wednesday = "Miércoles", + Thursday = "Jueves", Friday = "Viernes", Saturday = "Sábado", Sunday = "Domingo")) |> + #necesario pues el LOCALE puede cambiar + mutate(dia_semana = recode(dia_semana, lunes = "Lunes", martes = "Martes", miércoles = "Miércoles", + jueves = "Jueves", viernes = "Viernes", sábado = "Sábado", domingo = "Domingo")) |> + mutate(dia_semana = fct_relevel(dia_semana, c("Lunes", "Martes", "Miércoles", + "Jueves", "Viernes", "Sábado", "Domingo"))) +``` + +Consideremos los *datos agregados* del número de nacimientos +(registrados) por día desde 1999 hasta 2016. Un primer intento podría +ser hacer una gráfica de la serie de tiempo. Sin embargo, vemos que no +es muy útil: + + + +Hay varias características que notamos. Primero, parece haber una +tendencia ligeramente decreciente del número de nacimientos a lo largo +de los años. Segundo, la gráfica sugiere un patrón anual. Y por último, +encontramos que hay dispersión producida por los días de la semana. + +Sólo estas características hacen que la comparación entre días sea +difícil de realizar. Supongamos que comparamos el número de nacimientos +de dos miércoles dados. Esa comparación será diferente dependiendo: del +año donde ocurrieron, el mes donde ocurrieron, si semana santa ocurrió +en algunos de los miércoles, y así sucesivamente. + +Como en nuestros ejemplos anteriores, la idea del siguiente análisis es +aislar las componentes que observamos en la serie de tiempo: extraemos +componentes ajustadas, y luego examinamos los residuales. + +En este caso particular, asumiremos una **descomposición aditiva** de la +serie de tiempo [@cleveland93]. + +
+

En el estudio de series de tiempo una estructura +común es considerar el efecto de diversos factores como tendencia, +estacionalidad, ciclicidad e irregularidades de manera aditiva. Esto es, +consideramos la descomposición \[\begin{align} +y(t) = f_{t}(t) + f_{e}(t) + f_{c}(t) + \varepsilon. +\end{align}\] Una estrategia de ajuste, como veremos más +adelante, es proceder de manera modular. Es decir, se ajustan +los componentes de manera secuencial considerando los residuales de los +anteriores.

+
+ +### Tendencia {.unnumbered} + +Comenzamos por extraer la tendencia, haciendo promedios `loess` +[@cleveland1979robust] con vecindades relativamente grandes. Quizá +preferiríamos suavizar menos para capturar más variación lenta, pero si +hacemos esto en este punto empezamos a absorber parte de la componente +anual: + + +``` r +mod_1 <- loess(n ~ as.numeric(fecha), data = natalidad, span = 0.2, degree = 1) +datos_dia <- natalidad |> + mutate(ajuste_1 = fitted(mod_1)) |> + mutate(res_1 = n - ajuste_1) +``` + + + +Notemos que a principios de 2000 el suavizador está en niveles de +alrededor de 7000 nacimientos diarios, hacia 2015 ese número es más +cercano a unos 6000. + +### Componente anual {.unnumbered} + +Al obtener la tendencia podemos aislar el efecto a largo plazo y +proceder a realizar mejores comparaciones (por ejemplo, comparar un día +de 2000 y de 2015 tendria más sentido). Ahora, ajustamos **los +residuales del suavizado anterior**, pero con menos suavizamiento. Así +evitamos capturar tendencia: + + +``` r +mod_anual <- loess(res_1 ~ as.numeric(fecha), data = datos_dia, degree = 2, span = 0.005) +datos_dia <- datos_dia |> mutate(ajuste_2 = fitted(mod_anual)) |> + mutate(res_2 = res_1 - ajuste_2) +``` + + + +### Día de la semana {.unnumbered} + +Hasta ahora, hemos aislado los efectos por plazos largos de tiempo +(tendencia) y hemos incorporado las variaciones estacionales (componente +anual) de nuestra serie de tiempo. Ahora, veremos cómo capturar el +efecto por día de la semana. En este caso, podemos hacer suavizamiento +`loess` para cada serie de manera independiente + + +``` r +datos_dia <- datos_dia |> + group_by(dia_semana) |> + nest() |> + mutate(ajuste_mod = + map(data, ~ loess(res_2 ~ as.numeric(fecha), data = .x, span = 0.1, degree = 1))) |> + mutate(ajuste_3 = map(ajuste_mod, fitted)) |> + select(-ajuste_mod) |> + unnest(cols = c(data, ajuste_3)) |> + mutate(res_3 = res_2 - ajuste_3) |> + ungroup() +``` + + + +### Residuales {.unnumbered} + +Por último, examinamos los residuales finales quitando los efectos +ajustados: + + +``` +## `geom_smooth()` using formula = 'y ~ x' +``` + + + +**Observación**: nótese que la distribución de estos residuales presenta +irregularidades interesantes. La distribución es de *colas largas*, y no +se debe a unos cuantos datos atípicos. Esto generalmente es indicación +que hay factores importantes que hay que examinar mas a detalle en los +residuales: + + + +### Reestimación {.unnumbered} + +Cuando hacemos este proceso secuencial de llevar el ajuste a los +residual, a veces conviene iterarlo. La razón es que en una segunda o +tercera pasada podemos hacer mejores estimaciones de cada componente, y +es posible suavizar menos sin capturar *componentes de más alta +frecuencia.* + +Así que podemos regresar a la serie original para hacer mejores +estimaciones, más suavizadas: + + +``` r +# Quitamos componente anual y efecto de día de la semana +datos_dia <- datos_dia |> mutate(n_1 = n - ajuste_2 - ajuste_3) +# Reajustamos +mod_1 <- loess(n_1 ~ as.numeric(fecha), data = datos_dia, span = 0.02, degree = 2, + family = "symmetric") +``` + + + + + + + +Y ahora repetimos con la componente de día de la semana: + + + +### Análisis de componentes {.unnumbered} + +Ahora comparamos las componentes estimadas y los residuales en una misma +gráfica. Por definición, la suma de todas estas componentes da los datos +originales. + + + +Este último paso nos permite diversas comparaciones que explican la +variación que vimos en los datos. Una gran parte de los residuales está +entre $\pm 250$ nacimientos por día. Sin embargo, vemos que las colas +tienen una dispersión mucho mayor: + + +``` r +quantile(datos_dia$res_6, c(00, .01,0.05, 0.10, 0.90, 0.95, 0.99, 1)) |> round() +``` + +``` +## 0% 1% 5% 10% 90% 95% 99% 100% +## -2238 -1134 -315 -202 188 268 516 2521 +``` + +¿A qué se deben estas colas tan largas? + + + +#### Viernes 13? {.unnumbered} + +Podemos empezar con una curosidad. Los días Viernes o Martes 13, ¿nacen +menos niños? + + + +Nótese que fue útil agregar el indicador de Semana santa por el Viernes +13 de Semana Santa que se ve como un atípico en el panel de los viernes +13. + +### Residuales: antes y después de 2006 {.unnumbered} + +Veamos primero una agregación sobre los años de los residuales. Lo +primero es observar un cambio que sucedió repentinamente en 2006: + + + +La razón es un cambio en la ley acerca de cuándo pueden entrar los niños +a la primaria. Antes era por edad y había poco margen. Ese exceso de +nacimientos son reportes falsos para que los niños no tuvieran que +esperar un año completo por haber nacido unos cuantos días después de la +fecha límite. + +Otras características que debemos investigar: + +- Efectos de Año Nuevo, Navidad, Septiembre 16 y otros días feriados + como Febrero 14. +- Semana santa: como la fecha cambia, vemos que los residuales + negativos tienden a ocurrir dispersos alrededor del día 100 del año. + +### Otros días especiales: más de residuales {.unnumbered} + +Ahora promediamos residuales (es posible agregar barras para indicar +dispersión a lo largo de los años) para cada día del año. Podemos +identificar ahora los residuales más grandes: se deben, por ejemplo, a +días feriados, con consecuencias adicionales que tienen en días ajuntos +(excesos de nacimientos): + + + +### Semana santa {.unnumbered} + +Para Semana Santa tenemos que hacer unos cálculos. Si alineamos los +datos por días antes de Domingo de Pascua, obtenemos un patrón de caída +fuerte de nacimientos el Viernes de Semana Santa, y la característica +forma de "valle con hombros" en días anteriores y posteriores estos +Viernes. ¿Por qué ocurre este patrón? + + + +Nótese un defecto de nuestro modelo: el patrón de "hombros" alrededor +del Viernes Santo no es suficientemente fuerte para equilibrar los +nacimientos faltantes. ¿Cómo podríamos mejorar nuestra descomposición? diff --git a/01-exploratorio_files/figure-html/fig-casas-1.png b/01-exploratorio_files/figure-html/fig-casas-1.png new file mode 100644 index 0000000000000000000000000000000000000000..c317c51ce45fee7f81c8368a39cdc0ae618dd586 GIT binary patch literal 24217 zcmeFZbyU<}6fcUyASEp&E!`rBIMh%g4PsHEG*SWsNIM9k0wN(DgGeYP-J#M5jC3PC zbk|VtJMdG%`_^6S-u2d7@4fZ@Fy=ewyU#wm&pw|suQk;bi3lzcU}0eqDczF4jfI5+ z#liwj;9&z#Cbtb6u&}VPG*#~?03XktI|l-Ru(7ehU@#62&iV7_adC0+@bK{Q@d*eB z2nh)u&`Xde3_M%m5q&!ot>S7gM*WklZ%Uso12@5hliJ!myeH+ zpPyeqKtNDX@XD1dLPA1UuU-`v78Vf^xpwWEsHo`m>(|A^#BSWU0f9io#lhZEfv4ckbN1d-vYGd-w0(fAHXej*gD5uCAV*p1!`mfq}uphYt-6 z4Ie#v^!V{(BO@bYV`CE&6H`-DGcz-Db8`y|3rkDOCr_SOSy@?ITie*!*xK6K+1WjP z`qbXu-oe4a(a{kGgE=`lIXgSMxVSuf_RQ7Q)y>V#-QC^8!^6|l)62`t+uQs3^XG6l z+{ee~#fuldzP^5belK6X^!N9F_3G8@*RS8ac@q#25EvL36cqIK?c3ns;CJudg@lBJ zhK7cPg@uQQzkmNeA|fI(GBPSEDmpqkCMG5}Ha0FUE7PD*`uzEGMn*@~E32xis;jGOYHGfI{rc_Ox9{J-*Vfk7 z)z#J4*EcjYG&VLiH8nLiH@CF3{P^*swY9aat*yPi9f?H#{Q0w^qocF4v#YDCySuxm zr>D2Kx390SzrTNAU|?`?aA;^~czAeZWMp)7^w+OnV`F3E&&}Sj<>T@-lax z#Vy9;KEK#~&@jr_@KgPbPjkon)DItww(cZl{vM4EIV@&o zQZ6hkF!+Bm_!b;o7FA~<*|_7Fm2Z0Ymk;eln( zZ`JFCl339E4^Khy9>J}fX|P9`+f+;$(^`mj*jw&H7fRQG%UFmTT4L~e{t{d1dV#gv zCll?`cCB^qZoB6D=2B2eI@PrWsx`M$ox^H<^Np};-zWF6AFTpKpwWWap8W(~{oy^- zS1hb9xpVLs&j*&qMyXz&qUh1vqw+7pTdlKG)uha=pTM4|;fu?95k~jl>+XZGVm-f| z&Wub7?B|pT=)b!DsWM}9PO z&95K3Uz6FFTv>R)pJvyf>@02LWbJnpcf%pAWB9Fm(N{1QS0D(KWz*s|5FFt`_25lz z#d>vu)}U{8Kv7B6U6>d0QYSmoyg1kZ-zm1-T*c1OIPa3qShqlFv{;JuL(76D_Kdxo zoYQ(5hIM_#xl|2b7EOzwILZT&d7LkACR(hgN>(pUKATUD9cabDx=4-#cEdMT><_N9 zw#`^^^4|NopF(KNc{uet03zgES^0~l-nnY0X)n0wf^OtRlw;ugxYb{B=HXY0YP;7# zJp#2a4e~ASxzyLY6{=Ws$Z%WLsXs(dPi5((i;Eu^^0txWE@m2x=S1*l1c;_De|_0v zowU%xMdBBCUwJSx&!ErNrSjk#O+C7Q{rVNrl~nfjp)2olrOl^?&tpx#lqH09iigPs zO4?{p<&S^A-cK=;J8r(%(EZ|pxK(0)8Ry`lxVsmzF$pJFS31hcX{-0)Xu{&sfL8wY zLF(XmzVxtnL$ZO9Fn;E6yVYWYRp#C0FPvsqx3bH>Rt65U75Z__yML*gVqDBx3?8tS zw)cP~7L(8|{J6Janp`(}8w)z3H74uk{B~Vmh!@@ErEl;};8hWt#5gi$nm@2|@<)%; z<1{ufT605e5Edb*yk$L`wJmBsH-B%zBG`Ggm5tjJn*rPVYoF#!SRCbk+ERPr<1yt= z{POW)mln~lLr^Ci>i5e>U!vZ4CVy3gL^W3|mpd@Mu5kvkA(fZc2i-moYi-8CK&9ad9to^4!^))4U&+tfe zE(=pxEN-9d0(;_^>c;{>)6V34Lk^sQo$prLZ_R-SqO6T_L{`ZE*x?ZWhF;r@I4LmuPVyy*MZ zkQ7NWly@dWd$qWDhPey^iz~URD@3&fkTt~vQZ}+%+E7NFr$AF9wGyfu%WyniF|O3&7i zB9mmAnvs9^*A6{#(fyZo)SHhMXmWrl7+Q1<-E`u_*^0i{a9M5sxGuzVdyoRI8?gOs zKY6Nn`5WZR2h&*3s(h^H`$-?5t&wiE*^eQDkZOb)%nS9T(@8i)FCe+ycKMK}BCFWD z-j+?ev^gXG3v}DT@WL-h#8#zW;xNN-;y{8Ifwbh-oZul{faf417J^Ngplj&qY63&? 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zj6*}3yPHEb4K?QXnm|OIUUF*)>;wG#cl4&WL?ICIf2|*hD6z!{(bvP5dLY2R3yNxI J(@z_F{V)BFk=g(N literal 0 HcmV?d00001 diff --git a/404.html b/404.html new file mode 100644 index 0000000..1f7980d --- /dev/null +++ b/404.html @@ -0,0 +1,328 @@ + + + + + + + Page not found | Fundamentos de Estadística con Remuestreo + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/88-visualizacion.md b/88-visualizacion.md new file mode 100644 index 0000000..c26c3d8 --- /dev/null +++ b/88-visualizacion.md @@ -0,0 +1,392 @@ +# Apéndice: Principios de visualizacion {-} + +> "The simple graph has brought more information to the data analyst's +> mind than any other device." +> +> --- John Tukey + +## El cuarteto de Anscombe {-} + +En 1971 un estadístico llamado Frank Anscombe (fundador del departamento de +Estadística de la Universidad de Yale) publicó cuatro conjuntos de dato. Cada +uno consiste de 11 observaciones. La peculariedad de estos conjuntos es +que tienen las mismas propiedades estadísticas. + +Sin embargo, cuando analizamos los datos de manera gráfica en un histograma +encontramos rápidamente que los conjuntos de datos son muy distintos. + +
      + + + +
      + +
      +
      + + +Media de $x$: 9 +Varianza muestral de $x$: 11 +Media de $y$: 7.50 +Varianza muestral de $y$: 4.12 +Correlación entre $x$ y $y$: 0.816 +Línea de regresión lineal: $y = 3.00 + 0.500x$ + +
      + +En la gráfica del primer conjunto de datos, se ve clara una +relación lineal simple con un modelo que cumple los supuestos de +normalidad. La segunda gráfica (arriba a la derecha) muestra unos datos que +tienen una asociación pero definitivamente no es lineal. En la tercera gráfica +(abajo a la izquierda) están puntos alineados perfectamente en una línea recta, +excepto por uno de ellos. En la última gráfica podemos ver un ejemplo en el cual +basta tener una observación atípica para que se produzca un coeficiente de +correlación alto aún cuando en realidad no existe una asociación lineal entre +las dos variables. + +
      + + +
      + + + +![](images/datasaurus.gif) + + + +
      + +El cuarteto de Ascombe inspiró una técnica reciente para crear datos que comparten las +mismas propiedades estadísticas al igual que en el cuarteto, pero que producen gráficas +muy distintas ([Matejka, +Fitzmaurice](https://www.autodeskresearch.com/publications/samestats)). + +
      + + +## Introducción {-} + +La visualización de datos no trata de hacer gráficas “bonitas” o “divertidas”, +ni de simplificar lo complejo o ayudar a una persona “que no entiende mucho” a +entender ideas complejas. Más bien, trata de aprovechar nuestra gran capacidad +de procesamiento visual para exhibir de manera clara aspectos importantes de los +datos. + +El siguiente ejemplo de [@tufte06], ilustra claramente la diferencia entre estos +dos enfoques. A la izquierda están gráficas (más o menos típicas de Powerpoint) +basadas en la filosofía de simplificar, de intentar no “ahogar” al lector con +datos. El resultado es una colección incoherente, de bajo contenido, que no +tiene mucho qué decir y que es, “indeferente al contenido y la evidencia”. +A la derecha está una variación del rediseño de Tufte en forma de tabla, que en +este caso particular es una manera eficiente de mostrar claramente los patrones +que hay en este conjunto simple de datos. + +¿Qué principios son los que soportan la efectividad de esta tabla sobre la +gráfica de la derecha? Veremos que hay dos conjuntos de principios importantes: +unos relacionados con el diseño y otros con la naturaleza del análisis de datos, +independientemente del método de visualización. + + + + + +## Visualización popular de datos {-} + +Publicaciones populares (periódicos, revistas, sitios internet) muchas veces +incluyen visualización de datos como parte de sus artículos o reportajes. En +general siguen el mismo patrón que en la visión tradicionalista de la +estadística: sirven más para divertir que para explicar, tienden a explicar +ideas simples y conjuntos chicos de datos, y se consideran como una “ayuda” +para los “lectores menos sofisticados”. Casi siempre se trata de gráficas +triviales (muchas veces con errores graves) que no aportan mucho a artículos que +tienen un nivel de complejidad mucho mayor (es la filosofía: lo escrito para el +adulto, lo graficado para el niño). + + + +## Teoría de visualización de datos {-} + +Existe teoría fundamentada acerca de la visualización. Después del trabajo +pionero de Tukey, los principios e indicadores de Tufte se basan en un estudio +de la historia de la graficación y ejercicios de muestreo de la práctica gráfica +a lo largo de varias disciplinas (¿cuáles son las mejores gráficas? ¿por qué?) +El trabajo de Cleveland es orientado a la práctica del análisis de datos +(¿cuáles gráficas nos han ayudado a mostrar claramente los resultados del +análisis?), por una parte, y a algunos estudios de percepción visual. + +En resumen, hablaremos de las siguientes guías: + +### Principios generales del diseño analítico {-} + +Aplicables a una presentación o análisis completos, y como guía para construir +nuevas visualizaciones [@tufte06]. + +
      +

      Principio 1. Muestra comparaciones, contrastes, +diferencias.
      +Principio 2. Muestra causalidad, mecanismo, +explicación, estructura sistemática.
      +Principio 3. Muestra datos multivariados, es decir, más +de una o dos variables.
      +Principio 4. Integra palabras, números, imágenes y +diagramas.
      +Principio 5. Describe la totalidad de la evidencia. +Muestra fuentes usadas y problemas relevantes.
      +Principio 6. Las presentaciones analíticas, a fin de +cuentas, se sostienen o caen dependiendo de la calidad, relevancia e +integridad de su contenido.

      +
      + +### Técnicas de visualización {-} +Esta categoría incluye técnicas específicas que dependen de la forma de nuestros +datos y el tipo de pregunta que queremos investigar (@tukey77, @cleveland93, +@cleveland94, @tufte06). + +
      +

      Tipos de gráficas: cuantiles, histogramas, caja y +brazos, gráficas de dispersión, puntos/barras/ líneas, series de +tiempo.
      +Técnicas para mejorar gráficas: Transformación de +datos, transparencia, vibración, banking 45, suavizamiento y bandas de +confianza.
      +Pequeños múltiplos

      +
      + +### Indicadores de calidad gráfica {-} +Aplicables a cualquier gráfica en particular. Estas son guías concretas y +relativamente objetivas para evaluar la calidad de una gráfica [@tufte86]. + +
      +

      Integridad Gráfica. El factor de engaño, es decir, +la distorsión gráfica de las cantidades representadas, debe ser +mínimo.
      +Chartjunk. Minimizar el uso de decoración gráfica que +interfiera con la interpretación de los datos: 3D, rejillas, rellenos +con patrones.
      +Tinta de datos. Maximizar la proporción de tinta de +datos vs. tinta total de la gráfica. For non-data- ink, less is +more. For data-ink, less is a bore.
      +Densidad de datos. Las mejores gráficas tienen mayor +densidad de datos, que es la razón entre el tamaño del conjunto de datos +y el área de la gráfica. Las gráficas se pueden encoger mucho. +Percepción visual. Algunas tareas son más fáciles para el ojo humano que +otras [@cleveland94].

      +
      + +### Factor de engaño y Chartjunk {-} + +
      +![](images/pies.jpg) +
      + + +El **factor de engaño** es el cociente entre el efecto mostrado en una gráfica y +el efecto correspondiente en los datos. Idealmente, el factor de engaño debe ser +1 (ninguna distorsión). + +El **chartjunk** son aquellos elementos gráficos que no corresponden a variación +de datos, o que entorpecen la interpretación de una gráfica. +Estos son los indicadores de calidad más fáciles de entender y aplicar, y +afortunadamente cada vez son menos comunes. + +Un diseño popular que califica como chartjunk y además introduce factores de +engaño es el *pie* de 3D. En la gráfica de la derecha, podemos ver como la +rebanada C se ve más grande que la rebanada A, aunque claramente ese no es el +caso (factor de engaño). La razón es la variación en la perspectiva que no +corresponde a variación en los datos (chartjunk). + +#### Crítica gráfica: Gráfica de *pie* {-} + +Todavía elementos que pueden mejorar la comprensión de nuestra +gráfica de *pie*: se trata de la +decodificiación que hay que hacer categoría - color - cuantificación. Podemos +agregar las etiquetas como se muestra en la serie de la derecha, pero entonces: +¿por qué no mostrar simplemente la tabla de datos? ¿qué agrega el *pie* a la +interpretación? + +La deficiencias en el *pie* se pueden ver claramente al intentar graficar más +categorías (13) . En el primer *pie* no podemos distinguir realmente cuáles son +las categorías grandes y cuáles las chicas, y es muy difícil tener una imagen +mental clara de estos datos. Agregar los porcentajes ayuda, pero entonces, otra +vez, preguntamos cuál es el propósito del pie. La tabla de la izquierda hace +todo el trabajo (una vez que ordenamos las categrías de la más grande a la más +chica). Es posible hacer una gráfica de barras como la de abajo a la izquierda. + + +
      + +
      +![](images/barras_pie.jpg) +
      + + +Hay otros tipos de **chartjunk** comunes: uno es la textura de barras, por ejemplo. +El efecto es la producción de un efecto moiré que es desagradable y quita la +atención de los datos, como en la gráfica de barras de abajo. Otro común son las +rejillas, como mostramos en las gráficas de la izquierda. Nótese como en estos +casos hay efectos ópticos no planeados que degradan la percepción de los +patrones en los datos. + + + + + + +### Pequeños múltiplos y densidad gráfica {-} + +La densidad de una gráfica es el tamaño del conjunto de datos que se grafica +comparado con el área total de la gráfica. En el siguiente ejemplo, graficamos +en logaritmo-10 de millones de cabezas de ganado en Francia (cerdos, res, ovejas y +caballos). La gráfica de la izquierda es pobre en densidad pues sólo representa +4 datos. La manera más fácil de mejorar la densidad es hacer más chica la +gráfica: + + + + +La razón de este encogimiento es una que tiene qué ver con las oportunidades perdidas de una gráfica grande. Si repetimos este mismo patrón (misma escala, mismos tipos de ganado) para distintos países obtenemos la siguiente gráfica: + +
      +![](images/europe_plot.jpg) + +
      + +Esta es una gráfica de puntos. Es útil como sustituto de una gráfica de barras, +y es superior en el sentido de que una mayor proporción de la tinta que se usa +es tinta de datos. Otra vez, mayor proporción de tinta de datos representa más +oportunidades que se pueden capitalizar. + +
      + + +#### Más pequeños múltiplos {-} + +Los pequeños múltiplos presentan oportunidades para mostrar más acerca +de nuestro problema de interés. Consideramos por ejemplo la relación +de radiación solar y niveles de ozono: + + + +En el ejemplo anterior incluyendo +una variable adicional (velocidad del viento) podemos entender más +acerca de la relación de radiación solar y niveles de ozono: + + + + +### Tinta de datos {-} + +Maximizar la proporción de tinta de datos en nuestras gráficas tiene beneficios +inmediatos. La regla es: si hay tinta que no representa variación en los datos, +o la eliminación de esa tinta no representa pérdidas de significado, esa tinta +debe ser eliminada. El ejemplo más claro es el de las rejillas en gráficas y +tablas: + + + + + + + +¿Por qué usar grises en lugar de negros? La respuesta tiene qué ver con el +principio de tinta de datos: si marcamos las diferencias sutil pero claramente, +tenemos más oportunidades abiertas para hacer énfasis en lo que nos interesa: a +una gráfica o tabla saturada no se le puede hacer más - es difícil agregar +elementos adicionales que ayuden a la comprensión. Si comenzamos marcando con +sutileza, entonces se puede hacer más. Los mapas geográficos son un buen ejemplo +de este principio. + +El espacio en blanco es suficientemente bueno para indicar las fronteras en una +tabla, y facilita la lectura: + + + +Para un ejemplo del proceso de rediseño de una tabla, ver [aquí](https://www.darkhorseanalytics.com/blog/clear-off-the-table). +Finalmente, podemos ver un ejemplo que intenta incorporar +los elementos del diseño analítico, incluyendo pequeños múltiplos: + + + + +### Decoración {-} + + + + +### Percepción de escala {-} + +Entre la percepción visual y la interpretación de una gráfica están implícitas +tareas visuales específicas que las personas debemos realizar para ver +correctamente la gráfica. En la década de los ochenta, William S. Cleveland y +Robert McGill realizaron algunos experimentos identificando y clasificando estas +tareas para diferentes tipos de gráficos [@cleveland84]. +En estos, se le pregunta a la persona que compare dos valores dentro de una +gráfica, por ejemplo, en dos barras en una gráfica de barras, o dos rebanadas de +una gráfica de pie. + + + +Los resultados de Cleveland y McGill fueron replicados por Heer y Bostock en +2010 y los resultados se muestran en las gráficas de la derecha: + + + +## Ejemplo: gráfica de Minard {-} + +Concluimos esta sección con una gráfica que, aunque poco común, ejemplifica +los principios de una buena gráfica, y es reconocida como una de las mejores +visualizaciones de la historia. + +> Una gráfica excelente, presenta datos interesantes de forma bien +diseñada: es una cuestión de fondo, de diseño, y estadística... [Se] compone de +ideas complejas comunicadas con claridad, precisión y eficiencia. ... [Es] lo +que da al espectador la mayor cantidad de ideas, en el menor tiempo, con +la menor cantidad de tinta, y en el espacio más pequeño. ... Es casi siempre +multivariado. ... Una excelente gráfica debe decir la verdad acerca de los +datos. (Tufte, 1983) + +La [famosa visualización](https://upload.wikimedia.org/wikipedia/commons/2/29/Minard.png) de Charles Joseph Minard de la marcha de Napoleón sobre +Moscú, ilustra los principios de una buena gráfica. Tufte señala que esta imagen +"bien podría ser el mejor gráfico estadístico jamás dibujado", y sostiene que +"cuenta una historia rica y coherente con sus datos multivariados, mucho más +esclarecedora que un solo número que rebota en el tiempo". Se representan seis +variables: el tamaño del ejército, su ubicación en una superficie bidimensional, +la dirección del movimiento del ejército y la temperatura en varias fechas +durante la retirada de Moscú". + + + + +Hoy en día Minard es reconocido como uno de los principales contribuyentes a la +teoría de análisis de datos y creación de **infografías** con un fundamento +estadístico. + +Se grafican 6 variables: el número de tropas de Napoleón, la distancia, la +temperatura, la latitud y la longitud, la dirección en que viajaban las tropas +y la localización relativa a fechas específicas. + +La gráfica de Minard, como la describe E.J. Marey, parece "desafiar la pluma del +historiador con su brutal elocuencia", la combinación de datos del mapa, y la +serie de tiempo, dibujados en 1869, "retratan una secuencia de pérdidas +devastadoras que sufrieron las tropas de Napoleón en 1812". Comienza en la +izquierda, en la frontera de Polonia y Rusia, cerca del río Niemen. La línea +gruesa dorada muestra el tamaño de la Gran Armada (422,000) en el momento en que +invadía Rusia en junio de 1812. + +El ancho de esta banda indica el tamaño de la armada en cada punto del mapa. En +septiembre, la armada llegó a Moscú, que ya había sido saqueada y dejada +desértica, con sólo 100,000 hombres. + +El camino del retiro de Napoleón desde Moscú está representado por la línea +oscura (gris) que está en la parte inferior, que está relacionada a su vez con +la temperatura y las fechas en el diagrama de abajo. Fue un invierno muy frío, +y muchos se congelaron en su salida de Rusia. Como se muestra en el mapa, cruzar +el río Berezina fue un desastre, y el ejército de Napoleón logró regresar a +Polonia con tan sólo 10,000 hombres. + +También se muestran los movimientos de las tropas auxiliaries, que buscaban +proteger por atrás y por la delantera mientras la armada avanzaba hacia Moscú. +La gráfica de Minard cuenta una historia rica y cohesiva, coherente con datos +multivariados y con los hechos históricos, y que puede ser más ilustrativa que +tan sólo representar un número rebotando a lo largo del tiempo. diff --git a/88-visualizacion_files/figure-html/unnamed-chunk-1-1.png b/88-visualizacion_files/figure-html/unnamed-chunk-1-1.png new file mode 100644 index 0000000000000000000000000000000000000000..8a7884ef0a62786be85666c7aa2131fe0eb2b74a GIT binary patch literal 52261 zcmdSBWmwdE)HXVxqA0>f0RX{Dq=x|QzkR#du6gaHKUlI}2&2I&sz27#gD ztYPo{#Cx9e;e0sfz23R>Vi;!r@r$+Yb>H_Ie>oYk>sLvxA`pn{;?IR&AP|=X5r_*7 zS1{ofw~N9d@B`cWxtbjUanBe1A0u?`j{yR4A0aOMRM9zMb=>JCfy3$9pFPWGuFpsw 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+ + + + + +
      + + + +
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      + + +
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      + +
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      +

      Sección 1 Análisis exploratorio

      +
      +

      “Exploratory data analysis can never be the whole story, but nothing +else can serve as the foundation stone –as the first step.”

      +

      — John Tukey

      +
      +
      +

      El papel de la exploración en el análisis de datos

      +

      El estándar científico para contestar preguntas o tomar decisiones es +uno que se basa en el análisis de datos. Es decir, en primer lugar se +deben reunir todos los datos que puedan contener o sugerir alguna guía +para entender mejor la pregunta o la decisión a la que nos enfrentamos. +Esta recopilación de datos —que pueden ser cualitativos, +cuantitativos, o una mezcla de los dos— debe entonces ser analizada +para extraer información relevante para nuestro problema.

      +

      En análisis de datos existen dos distintos tipos de trabajo:

      +
        +
      • El trabajo exploratorio o de detective: ¿cuáles son los +aspectos importantes de estos datos? ¿qué indicaciones generales +muestran los datos? ¿qué tareas de análisis debemos empezar +haciendo? ¿cuáles son los caminos generales para formular con +precisión y contestar algunas preguntas que nos interesen?

      • +
      • El trabajo inferencial, confirmatorio, o de juez: ¿cómo +evaluar el peso de la evidencia de los descubrimientos del paso +anterior? ¿qué tan bien soportadas están las respuestas y +conclusiones por nuestro conjunto de datos?

      • +
      +
      +
      +

      Preguntas y datos

      +

      Cuando observamos un conjunto de datos, independientemente de su tamaño, el paso inicial más importante es entender bajo qué proceso se generan los datos.

      +

      A grandes rasgos, cuanto más sepamos de este proceso, mejor podemos contestar preguntas de interés. +En muchos casos, tendremos que hacer algunos supuestos de cómo se generan estos datos para dar respuestas (condicionales a esos supuestos).

      +
      +
      +

      Algunos conceptos básicos

      +

      Empezamos explicando algunas ideas que no serán útiles más adelante. El primer concepto se refiere a entender cómo se distribuyen los datos a los largo de su escala de medición. Comenzamos con un ejemplo: los siguientes datos fueron registrados en un restaurante durante cuatro días consecutivos.

      +
      library(tidyverse)
      +library(patchwork)
      +source("R/funciones_auxiliares.R")
      +# usamos los datos tips del paquete reshape2
      +propinas <- read_csv("./data/propinas.csv")
      +

      Y vemos una muestra

      +
      sample_n(propinas, 10) |> formatear_tabla()
      + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
      +cuenta_total + +propina + +fumador + +dia + +momento + +num_personas +
      +24.71 + +5.85 + +No + +Jue + +Comida + +2 +
      +19.77 + +2.00 + +No + +Dom + +Cena + +4 +
      +16.00 + +2.00 + +Si + +Jue + +Comida + +2 +
      +15.06 + +3.00 + +No + +Sab + +Cena + +2 +
      +17.07 + +3.00 + +No + +Sab + +Cena + +3 +
      +13.13 + +2.00 + +No + +Dom + +Cena + +2 +
      +44.30 + +2.50 + +Si + +Sab + +Cena + +3 +
      +10.34 + +2.00 + +Si + +Jue + +Comida + +2 +
      +14.78 + +3.23 + +No + +Dom + +Cena + +2 +
      +25.28 + +5.00 + +Si + +Sab + +Cena + +2 +
      +

      Aquí la unidad de observación es una cuenta particular. Tenemos tres +mediciones numéricas de cada cuenta: cúanto fue la cuenta total, la +propina, y el número de personas asociadas a la cuenta. Los datos están +separados según se fumó o no en la mesa, y temporalmente en dos partes: +el día (Jueves, Viernes, Sábado o Domingo), cada uno separado por Cena y +Comida.

      +
      +

      +Denotamos por \(x\) el valor de +medición de una unidad de observación. Usualmente utilizamos +sub-índices para identificar entre diferentes puntos de datos +(observaciones), por ejemplo, \(x_n\) +para la \(n-\)ésima observación. De tal +forma que una colección de \(N\) +observaciones la escribimos como \[\begin{align} +\{x_1, \ldots, x_N\}. +\end{align}\] +

      +
      +

      El primer tipo de comparaciones que nos interesa hacer es para una +medición: ¿Varían mucho o poco los datos de un tipo de medición? ¿Cuáles +son valores típicos o centrales? ¿Existen valores atípicos?

      +

      Supongamos entonces que consideramos simplemente la variable de +cuenta_total. Podemos comenzar por ordenar los datos, y ver cuáles +datos están en los extremos y cuáles están en los lugares centrales:

      +
      propinas <- propinas |>
      +  mutate(orden_cuenta = rank(cuenta_total, ties.method = "first"),
      +         f = (orden_cuenta - 0.5) / n())
      +cuenta <- propinas |> select(orden_cuenta, f, cuenta_total) |> arrange(f)
      +bind_rows(head(cuenta), tail(cuenta)) |> formatear_tabla()
      + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
      +orden_cuenta + +f + +cuenta_total +
      +1 + +0.0020492 + +3.07 +
      +2 + +0.0061475 + +5.75 +
      +3 + +0.0102459 + +7.25 +
      +4 + +0.0143443 + +7.25 +
      +5 + +0.0184426 + +7.51 +
      +6 + +0.0225410 + +7.56 +
      +239 + +0.9774590 + +44.30 +
      +240 + +0.9815574 + +45.35 +
      +241 + +0.9856557 + +48.17 +
      +242 + +0.9897541 + +48.27 +
      +243 + +0.9938525 + +48.33 +
      +244 + +0.9979508 + +50.81 +
      +

      También podemos graficar los datos en orden, interpolando valores +consecutivos.

      +

      +

      A esta función le llamamos la función de cuantiles para la variable +cuenta_total. Nos sirve para comparar directamente los distintos +valores que observamos los datos según el orden que ocupan. En +particular, podemos estudiar la dispersión y valores centrales de +los datos observados:

      +
        +
      • El rango de datos va de unos 3 dólares hasta 50 dólares
      • +
      • Los valores centrales, por ejemplo el 50% de los valores más +centrales, están entre unos 13 y 25 dólares.
      • +
      • El valor que divide en dos mitades iguales a los datos es de +alrededor de 18 dólares.
      • +
      +
      +

      +El cuantil \(f\), +que denotamos por \(q(f)\) es valor a +lo largo de la escala de medición de los datos tal que aproximadamente +una fracción \(f\) de los datos son +menores o iguales a \(q(f)\). +

      +
        +
      • +Al cuantil \(f=0.5\) le llamamos la +mediana.
        +
      • +
      • +A los cuantiles \(f=0.25\) y \(f=0.75\) les llamamos cuartiles inferior y +superior. +
      • +
      +
      +

      En nuestro ejemplo:

      +
        +
      • Los valores centrales —del cuantil 0.25 al 0.75, por decir un +ejemplo— están entre unos 13 y 25 dólares. Estos dos cuantiles se +llaman cuartil inferior y cuartil superior respectivamente
      • +
      • El cuantil 0.5 (o también conocido como mediana) está alrededor +de 18 dólares.
      • +
      +

      Éste último puede ser utilizado para dar un valor central de la +distribución de valores para cuenta_total. Asimismo podemos dar +resúmenes más refinados si es necesario. Por ejemplo, podemos reportar +que:

      +
        +
      • El cuantil 0.95 es de unos 35 dólares — sólo 5% de las cuentas son +de más de 35 dólares
      • +
      • El cuantil 0.05 es de unos 8 dólares — sólo 5% de las cuentas son +de 8 dólares o menos.
      • +
      +

      Finalmente, la forma de la gráfica se interpreta usando su pendiente +(tasa de cambio) haciendo comparaciones en diferentes partes de la +gráfica:

      +
        +
      • La distribución de valores tiene asimetría: el 10% de las cuentas +más altas tiene considerablemente más dispersión que el 10% de las +cuentas más bajas.

      • +
      • Entre los cuantiles 0.2 y 0.7 es donde existe mayor densidad de +datos: la pendiente (tasa de cambio) es baja, lo que significa que +al avanzar en los valores observados, los cuantiles (el porcentaje +de casos) aumenta rápidamente.

      • +
      • Cuando la pendiente alta, quiere decir que los datos tienen más +dispersión local o están más separados.

      • +
      +

      Observación: Hay varias maneras de definir los cuantiles (ver +(William S. Cleveland 1993)): Supongamos que queremos definir \(q(f)\), y denotamos los +datos ordenados como \(x_{(1)}, x_{(2)}, \ldots, x_{(N)}\), de forma que +\(x_{(1)}\) es el dato más chico y \(x_{(N)}\) es el dato más grande. Para +cada \(x_{(i)}\) definimos
      +\[f_i = i / N\] entonces definimos el cuantil \(q(f_i)=x_{(i)}\). Para +cualquier \(f\) entre 0 y 1, podemos definir \(q(f)\) como sigue: si \(f\) +está entre \(f_i\) y \(f_{i+1}\) interpolamos linealmente los valores +correspondientes \(x_{(i)}\) y \(x_{(i-1)}\).

      +

      En la práctica, es más conveniente usar \(f_i= \frac{i - 0.5}{N}\). La +gráfica de cuantiles no cambia mucho comparado con la difinición +anterior, y esto nos permitirá comparar de mejor manera con +distribuciones teóricas que no tienen definido su cuantil 0 y el 1, pues +tienen soporte en los números reales (como la distribución normal, por +ejemplo).

      +
      +

      Asociada a la función de cuantiles \(q\) tenemos la distribución +acumulada empírica de los datos, que es aproximadamente inversa de la +función de cuantiles, y se define como:

      +

      \[\hat{F}(x) = i/N\] si \(x_{(i)} \leq x < x_{(i+1)}\).

      +

      Nótese que +\(\hat{F}(q(f_i)) = i/N = f_i\) (demuéstralo).

      +
      acum_cuenta <- ecdf(cuenta$cuenta_total)
      +cuenta <- cuenta |>
      +  mutate(dea_cuenta_total = acum_cuenta(cuenta_total))
      +g_acum <- ggplot(cuenta, aes(x = cuenta_total, y = dea_cuenta_total)) + geom_point() +
      +  labs(subtitle = "Distribución acum empírica de cuenta total") + xlab("")
      +g_cuantiles + g_acum
      +

      +

      La función de distribución acumulada empírica es otra forma de graficar la dispersión de los datos. En su gráfica vemos que proporción de los datos que son iguales o están por debajo de cada valor en el eje horizontal.

      +

      Nota:

      +
        +
      • En análisis de datos, es más frecuente utilizar la función de cuantiles pues existen versiones más generales que son útiles, por ejemplo, para evaluar ajuste de modelos probabilísticos

      • +
      • En la teoría, generalmente es más común utilizar la fda empírica, que tiene una única definición que veremos coincide con definiciones teóricas.

      • +
      +
      +

      Histogramas

      +

      En algunos casos, es más natural hacer un histograma, donde +dividimos el rango de la variable en cubetas o intervalos (en este caso +de igual longitud), y graficamos por medio de barras cuántos datos caen +en cada cubeta:

      +

      +

      Es una gráfica más popular, pero perdemos cierto nivel de detalle, y +distintas particiones resaltan distintos aspectos de los datos.

      +
      +

      +¿Cómo se ve la gráfica de cuantiles de las propinas? ¿Cómo crees que +esta gráfica se compara con distintos histogramas? +

      +
      +
      g_1 <- ggplot(propinas, aes(sample = propina)) +
      +  geom_qq(distribution = stats::qunif) + xlab("f") + ylab("propina")
      +g_1
      +

      +

      Finalmente, una gráfica más compacta que resume la gráfica de cuantiles +o el histograma es el diagrama de caja y brazos. Mostramos dos +versiones, la clásica de Tukey (T) y otra versión menos común de +Spear/Tufte (ST):

      +
      library(ggthemes)
      +cuartiles <- quantile(cuenta$cuenta_total)
      +t(cuartiles) |>  formatear_tabla()
      + + + + + + + + + + + + + + + + + + + +
      +0% + +25% + +50% + +75% + +100% +
      +3.07 + +13.3475 + +17.795 + +24.1275 + +50.81 +
      +
      g_1 <- ggplot(cuenta, aes(x = f, y = cuenta_total)) +
      +  labs(subtitle = "Gráfica de cuantiles: Cuenta total") +
      +  geom_hline(yintercept = cuartiles[2], colour = "gray") +
      +  geom_hline(yintercept = cuartiles[3], colour = "gray") +
      +  geom_hline(yintercept = cuartiles[4], colour = "gray") +
      +  geom_point(alpha = 0.5) + geom_line()
      +g_2 <- ggplot(cuenta, aes(x = factor("ST", levels =c("ST")), y = cuenta_total)) +
      +  geom_tufteboxplot() +
      +  labs(subtitle = " ") +  xlab("") + ylab("")
      +g_3 <- ggplot(cuenta, aes(x = factor("T"), y = cuenta_total)) +
      +  geom_boxplot() +
      +  labs(subtitle = " ") +  xlab("") + ylab("")
      +g_4 <- ggplot(cuenta, aes(x = factor("P"), y = cuenta_total)) +
      +  geom_jitter(height = 0, width =0.2, alpha = 0.5) +
      +  labs(subtitle = " ") +  xlab("") + ylab("")
      +g_5 <- ggplot(cuenta, aes(x = factor("V"), y = cuenta_total)) +
      +  geom_violin() +
      +  labs(subtitle = " ") +  xlab("") + ylab("")
      +g_1 + g_2 + g_3 + g_4 +
      +  plot_layout(widths = c(8, 2, 2, 2))
      +

      +
      +
      +

      El diagrama de la derecha explica los elementos de la versión típica del +diagrama de caja y brazos (boxplot). RIC se refiere al Rango +Intercuantílico, definido por la diferencia entre los cuantiles 25% y +75%.

      +
      +
      +

       

      +
      +
      +

      +

      Figura: Jumanbar / CC +BY-SA

      +
      +
      +

      Ventajas en el análisis inicial

      +

      En un principio del análisis, estos resúmenes (cuantiles) pueden ser más +útiles que utilizar medias y varianzas, por ejemplo. La razón es que los +cuantiles:

      +
        +
      • Son cantidades más fácilmente interpretables
      • +
      • Los cuantiles centrales son más resistentes a valores atípicos que +medias o varianzas
      • +
      • Permiten identificar valores extremos
      • +
      • Es fácil comparar cuantiles de distintos bonches de datos en la +misma escala
      • +
      +

      Nota: Existen diferentes definiciones para calcular cuantiles de una muestra de +datos, puedes leer más en este artículo.

      +
      +
      +

      Media y desviación estándar

      +

      Las medidas más comunes de localización y dispersión para un conjunto de +datos son la media muestral y la desviación estándar +muestral.

      +

      En general, no son muy apropiadas para iniciar el análisis exploratorio, +pues:

      +
        +
      • Son medidas más difíciles de interpretar y explicar que los +cuantiles. En este sentido, son medidas especializadas. Por ejemplo, +compara una explicación intuitiva de la mediana contra una +explicación intuitiva de la media.
      • +
      • No son resistentes a valores atípicos. Su falta de resistencia los +vuelve poco útiles en las primeras etapas de limpieza y descripción +y en resúmenes deficientes para distribuciones irregulares (con +colas largas por ejemplo).
      • +
      +
      +

      +La media, o promedio, se denota por \(\bar x\) y se define como \[\begin{align} +\bar x = \frac1N \sum_{n = 1}^N x_n. +\end{align}\] La desviación estándar muestral se define como +\[\begin{align} +\text{std}(x) = \sqrt{\frac1{N-1} \sum_{n = 1}^N (x_n - \bar x)^2}. +\end{align}\] +

      +
      +

      Observación: Si \(N\) no es muy chica, no importa mucho si dividimos +por \(N\) o por \(N-1\) en la fórmula de la desviación estándar. La razón de +que típicamente se usa \(N-1\) la veremos más adelante, en la parte de +estimación.

      +

      Por otro lado, ventajas de estas medidas de centralidad y dispersión +son:

      +
        +
      • La media y desviación estándar son computacionalmente convenientes. Por lo tanto regresaremos a estas medidas una vez que estudiemos modelos de probabilidad básicos.

      • +
      • En muchas ocasiones conviene usar estas medidas pues permite hacer +comparaciones históricas o tradicionales —pues análisis anteriores +pudieran estar basados en éstas.

      • +
      +
      +
        +
      1. +

        +Considera el caso de tener \(N\) +observaciones y asume que ya tienes calculado el promedio para dichas +observaciones. Este promedio lo denotaremos por \(\bar x_N\). Ahora, considera que has +obtenido \(M\) observaciones más. +Escribe una fórmula recursiva para la media del conjunto total de datos +\(\bar x_{N+M}\) en función de lo que +ya tenías precalculado \(\bar x_N.\) +

        +
      2. +
      3. +

        +¿En qué situaciones esta propiedad puede ser +conveniente? +

        +
      4. +
      +
      +
      +
      +
      +

      Ejemplos

      +
      +

      Precios de casas

      +

      En este ejemplo consideremos los datos de precios de ventas de la +ciudad de Ames, +Iowa. En +particular nos interesa entender la variación del precio de las casas.

      +

      Por este motivo calculamos los cuantiles que corresponden al 25%, 50% y +75% (cuartiles), así como el mínimo y máximo de los precios de las +casas:

      +
      quantile(casas |> pull(precio_miles))
      +
      ##    0%   25%   50%   75%  100% 
      +##  37.9 132.0 165.0 215.0 755.0
      +
      +

      +Comprueba que el mínimo y máximo están asociados a los cuantiles 0% y +100%, respectivamente. +

      +
      +

      Una posible comparación es considerar los precios y sus variación en +función de zona de la ciudad en que se encuentra una vivienda. Podemos +usar diagramas de caja y brazos para hacer una comparación burda de +los precios en distintas zonas de la ciudad:

      +
      ggplot(casas, aes(x = nombre_zona, y = precio_miles)) +
      +  geom_boxplot() +
      +  coord_flip()
      +

      +

      La primera pregunta que nos hacemos es cómo pueden variar las +características de las casas dentro de cada zona. Para esto, podemos +considerar el área de las casas. En lugar de graficar el precio, +graficamos el precio por metro cuadrado, por ejemplo:

      +
      ggplot(casas, aes(x = nombre_zona, y = precio_m2)) +
      +  geom_boxplot() +
      +  coord_flip()
      +

      +

      Podemos cuantificar la variación que observamos de zona a zona y la +variación que hay dentro de cada una de las zonas. Una primera +aproximación es observar las variación del precio al calcular la mediana +dentro de cada zona, y después cuantificar por medio de cuantiles cómo +varía la mediana entre zonas:

      +
      casas |>
      +  group_by(nombre_zona) |>
      +  summarise(mediana_zona = median(precio_m2), .groups = "drop") |>
      +  pull(mediana_zona) |>
      +  quantile() |>
      +  round()
      +
      ##   0%  25%  50%  75% 100% 
      +##  963 1219 1298 1420 1725
      +

      Por otro lado, las variaciones con respecto a las medianas dentro de +cada zona, por grupo, se resume como:

      +
      quantile(casas |> group_by(nombre_zona) |>
      +           mutate(residual = precio_m2 - median(precio_m2)) |>
      +           pull(residual)) |>
      +  round()
      +
      ##   0%  25%  50%  75% 100% 
      +## -765 -166    0  172 1314
      +

      Nótese que este último paso tiene sentido pues la variación dentro de +las zonas, en términos de precio por metro cuadrado, es similar. Esto no +lo podríamos haber hecho de manera efectiva si se hubiera utilizado el +precio de las casas sin ajustar por su tamaño.

      +

      Podemos resumir este primer análisis con un par de gráficas de cuantiles +(William S. Cleveland (1993)):

      +
      mediana <- median(casas$precio_m2)
      +resumen <- casas |>
      +  group_by(nombre_zona) |>
      +  mutate(mediana_zona = median(precio_m2)) |>
      +  mutate(residual = precio_m2 - mediana_zona) |>
      +  ungroup() |>
      +  mutate(mediana_zona = mediana_zona - mediana) |>
      +  select(nombre_zona, mediana_zona, residual) |>
      +  pivot_longer(mediana_zona:residual, names_to = "tipo", values_to = "valor")
      +ggplot(resumen, aes(sample = valor)) +
      +  geom_qq(distribution = stats::qunif) +
      +  facet_wrap(~ tipo) +
      +  ylab("Precio por m2") + xlab("f") +
      +  labs(subtitle = "Precio por m2 por zona",
      +       caption = paste0("Mediana total de ", round(mediana)))
      +

      +

      Vemos que la mayor parte de la variación del precio por metro cuadrado +ocurre dentro de cada zona, una vez que controlamos por el tamaño de las +casas. La variación dentro de cada zona es aproximadamente simétrica, +aunque la cola derecha es ligeramente más larga con algunos valores +extremos.

      +

      Podemos seguir con otro indicador importante: la calificación de calidad +de los terminados de las casas. Como primer intento podríamos hacer:

      +

      +

      Lo que indica que las calificaciones de calidad están distribuidas de +manera muy distinta a lo largo de las zonas, y que probablemente no va +ser simple desentrañar qué variación del precio se debe a la zona y cuál +se debe a la calidad.

      +
      +
      +

      Prueba Enlace

      +

      Consideremos la prueba Enlace (2011) de matemáticas para primarias. Una +primera pregunta que alguien podría hacerse es: ¿cuáles escuelas son +mejores en este rubro, las privadas o las públicas?

      +
      enlace_tbl <- enlace |> group_by(tipo) |>
      +  summarise(n_escuelas = n(),
      +            cuantiles = list(cuantil(mate_6, c(0.05, 0.25, 0.5, 0.75, 0.95)))) |>
      +  unnest(cols = cuantiles) |> mutate(valor = round(valor))
      +enlace_tbl |>
      +  spread(cuantil, valor) |>
      +  formatear_tabla()
      + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
      +tipo + +n_escuelas + +0.05 + +0.25 + +0.5 + +0.75 + +0.95 +
      +Indígena/Conafe + +13599 + +304 + +358 + +412 + +478 + +588 +
      +General + +60166 + +380 + +454 + +502 + +548 + +631 +
      +Particular + +6816 + +479 + +551 + +593 + +634 + +703 +
      +

      Para un análisis exploratorio podemos utilizar distintas gráficas. Por +ejemplo, podemos utilizar nuevamente las gráficas de caja y brazos, así +como graficar los percentiles. Nótese que en la gráfica 1 se utilizan +los cuantiles 0.05, 0.25, 0.5, 0.75 y 0.95:

      +
      ## Warning: Using `size` aesthetic for lines was deprecated in ggplot2 3.4.0.
      +## ℹ Please use `linewidth` instead.
      +## This warning is displayed once every 8 hours.
      +## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
      +## generated.
      +

      +

      Se puede discutir qué tan apropiada es cada gráfica con el objetivo de +realizar comparaciones. Sin duda, graficar más cuantiles es más útil +para hacer comparaciones. Por ejemplo, en la Gráfica 1 podemos ver que +la mediana de las escuelas generales está cercana al cuantil 5% de las +escuelas particulares. Por otro lado, el diagrama de caja y brazos +muestra también valores “atípicos”.

      +

      Antes de contestar prematuramente la pregunta: ¿cuáles son las mejores +escuelas? busquemos mejorar la interpretabilidad de nuestras +comparaciones. Podemos comenzar por agregar, por ejemplo, el nivel del +marginación del municipio donde se encuentra la escuela.

      +

      Para este objetivo, podemos usar páneles (pequeños múltiplos útiles para +hacer comparaciones) y graficar:

      +

      +

      Esta gráfica pone en contexto la pregunta inicial, y permite evidenciar +la dificultad de contestarla. En particular:

      +
        +
      1. Señala que la pregunta no sólo debe concentarse en el tipo de +“sistema”: pública, privada, etc. Por ejemplo, las escuelas públicas +en zonas de marginación baja no tienen una distribución de +calificaciones muy distinta a las privadas en zonas de marginación +alta.
      2. +
      3. El contexto de la escuela es importante.
      4. +
      5. Debemos de pensar qué factores –por ejemplo, el entorno familiar de +los estudiantes– puede resultar en comparaciones que favorecen a +las escuelas privadas. Un ejemplo de esto es considerar si los +estudiantes tienen que trabajar o no. A su vez, esto puede o no ser +reflejo de la calidad del sistema educativo.
      6. +
      7. Si esto es cierto, entonces la pregunta inicial es demasiado vaga y +mal planteada. Quizá deberíamos intentar entender cuánto “aporta” +cada escuela a cada estudiante, como medida de qué tan buena es cada +escuela.
      8. +
      +
      +
      +

      Estados y calificaciones en SAT

      +

      ¿Cómo se relaciona el gasto por alumno, a nivel estatal, con sus +resultados académicos? Hay trabajo considerable en definir estos +términos, pero supongamos que tenemos el siguiente conjunto de +datos (Guber 1999), que son datos +oficiales agregados por estado de Estados Unidos. Consideremos el +subconjunto de variables sat, que es la calificación promedio de los +alumnos en cada estado (para 1997) y expend, que es el gasto en miles +de dólares por estudiante en (1994-1995).

      +
      sat <- read_csv("data/sat.csv")
      +sat_tbl <- sat |> select(state, expend, sat) |>
      +  gather(variable, valor, expend:sat) |>
      +  group_by(variable) |>
      +  summarise(cuantiles = list(cuantil(valor))) |>
      +  unnest(cols = c(cuantiles)) |>
      +  mutate(valor = round(valor, 1)) |>
      +  spread(cuantil, valor)
      +sat_tbl |> formatear_tabla()
      + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
      +variable + +0 + +0.25 + +0.5 + +0.75 + +1 +
      +expend + +3.7 + +4.9 + +5.8 + +6.4 + +9.8 +
      +sat + +844.0 + +897.2 + +945.5 + +1032.0 + +1107.0 +
      +

      Esta variación es considerable para promedios del SAT: el percentil 75 +es alrededor de 1050 puntos, mientras que el percentil 25 corresponde a +alrededor de 800. Igualmente, hay diferencias considerables de gasto por +alumno (miles de dólares) a lo largo de los estados.

      +

      Ahora hacemos nuestro primer ejercico de comparación: ¿Cómo se ven las +calificaciones para estados en distintos niveles de gasto? Podemos usar +una gráfica de dispersión:

      +
      library(ggrepel)
      +ggplot(sat, aes(x = expend, y = sat, label = state)) +
      +  geom_point(colour = "red", size = 2) + geom_text_repel(colour = "gray50") +
      +  xlab("Gasto por alumno (miles de dólares)") +
      +  ylab("Calificación promedio en SAT")
      +

      +

      Estas comparaciones no son de alta calidad, solo estamos usando 2 +variables —que son muy pocas— y no hay mucho que podamos decir en +cuanto explicación. Sin duda nos hace falta una imagen más completa. +Necesitaríamos entender la correlación que existe entre las demás +características de nuestras unidades de estudio.

      +

      Las unidades que estamos comparando pueden diferir fuertemente en +otras propiedades importantes (o dimensiones), lo cual no permite +interpretar la gráfica de manera sencilla.

      +

      Una variable que tenemos es el porcentaje de alumnos de cada estado que +toma el SAT. Podemos agregar como sigue:

      +
      ggplot(sat, aes(x = expend, y = math, label=state, colour = frac)) +
      +  geom_point() + geom_text_repel() +
      +  xlab("Gasto por alumno (miles de dólares)") +
      +  ylab("Calificación en matemáticas")
      +

      +

      Esto nos permite entender por qué nuestra comparación inicial es +relativamente pobre. Los estados con mejores resultados promedio en el +SAT son aquellos donde una fracción relativamente baja de los +estudiantes toma el examen. La diferencia es considerable.

      +

      En este punto podemos hacer varias cosas. Una primera idea es intentar +comparar estados más similares en cuanto a la población de alumnos que +asiste. Podríamos hacer grupos como sigue:

      +
      set.seed(991)
      +k_medias_sat <- kmeans(sat |> select(frac), centers = 4,  nstart = 100, iter.max = 100)
      +sat$clase <- k_medias_sat$cluster
      +sat <- sat |> group_by(clase) |>
      +  mutate(clase_media = round(mean(frac))) |>
      +  ungroup() |>
      +  mutate(clase_media = factor(clase_media))
      +sat <- sat |>
      +  mutate(rank_p = rank(frac, ties= "first") / length(frac))
      +ggplot(sat, aes(x = rank_p, y = frac, label = state,
      +                colour = clase_media)) +
      +  geom_point(size = 2)
      +

      +

      Estos resultados indican que es más probable que buenos alumnos decidan +hacer el SAT. Lo interesante es que esto ocurre de manera diferente en +cada estado. Por ejemplo, en algunos estados era más común otro examen: +el ACT.

      +

      Si hacemos clusters de estados según el % de alumnos, empezamos a ver +otra historia. Para esto, ajustemos rectas de mínimos cuadrados como +referencia:

      +

      +

      Esto da una imagen muy diferente a la que originalmente planteamos. Nota +que dependiendo de cómo categorizamos, esta gráfica puede variar (puedes +intentar con más o menos grupos, por ejemplo).

      +
      +
      +

      Tablas de conteos

      +

      Consideremos los siguientes datos de tomadores de té (del paquete +FactoMineR (Lê et al. 2008)):

      +
      tea <- read_csv("data/tea.csv")
      +# nombres y códigos
      +te <- tea |> select(how, price, sugar) |>
      +  rename(presentacion = how, precio = price, azucar = sugar) |>
      +  mutate(
      +    presentacion = fct_recode(presentacion,
      +                              suelto = "unpackaged", bolsas = "tea bag", mixto = "tea bag+unpackaged"),
      +    precio = fct_recode(precio,
      +                        marca = "p_branded", variable = "p_variable", barato = "p_cheap",
      +                        marca_propia = "p_private label", desconocido = "p_unknown", fino = "p_upscale"),
      +    azucar = fct_recode(azucar,
      +                        sin_azúcar = "No.sugar", con_azúcar = "sugar"))
      +
      sample_n(te, 10)
      +
      ## # A tibble: 10 × 3
      +##    presentacion precio   azucar    
      +##    <fct>        <fct>    <fct>     
      +##  1 bolsas       marca    sin_azúcar
      +##  2 bolsas       variable sin_azúcar
      +##  3 bolsas       marca    con_azúcar
      +##  4 bolsas       fino     con_azúcar
      +##  5 mixto        variable con_azúcar
      +##  6 mixto        fino     con_azúcar
      +##  7 bolsas       marca    sin_azúcar
      +##  8 bolsas       fino     sin_azúcar
      +##  9 mixto        variable con_azúcar
      +## 10 mixto        variable sin_azúcar
      +

      Nos interesa ver qué personas compran té suelto, y de qué tipo. +Empezamos por ver las proporciones que compran té según su empaque (en +bolsita o suelto):

      +
      precio <- te |>
      +  count(precio) |>
      +  mutate(prop = round(100 * n / sum(n))) |>
      +  select(-n)
      +tipo <- te |>
      +  count(presentacion) |>
      +  mutate(pct = round(100 * n / sum(n)))
      +tipo |> formatear_tabla()
      + + + + + + + + + + + + + + + + + + + + + + + + + +
      +presentacion + +n + +pct +
      +bolsas + +170 + +57 +
      +mixto + +94 + +31 +
      +suelto + +36 + +12 +
      +

      La mayor parte de las personas toma té en bolsas. Sin embargo, el tipo +de té (en términos de precio o marca) que compran es muy distinto +dependiendo de la presentación:

      +
      tipo <- tipo |> select(presentacion, prop_presentacion = pct)
      +tabla_cruzada <- te |>
      +  count(presentacion, precio) |>
      +  # porcentajes por presentación
      +  group_by(presentacion) |>
      +  mutate(prop = round(100 * n / sum(n))) |>
      +  select(-n)
      +tabla_cruzada |>
      +  pivot_wider(names_from = presentacion, values_from = prop,
      +              values_fill = list(prop = 0)) |>
      +  formatear_tabla()
      + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
      +precio + +bolsas + +mixto + +suelto +
      +marca + +41 + +21 + +14 +
      +barato + +3 + +1 + +3 +
      +marca_propia + +9 + +4 + +3 +
      +desconocido + +6 + +1 + +0 +
      +fino + +8 + +20 + +56 +
      +variable + +32 + +52 + +25 +
      +

      Estos datos podemos examinarlos un rato y llegar a conclusiones, pero +esta tabla no necesariamente es la mejor manera de mostrar patrones en +los datos. Tampoco son muy útiles gráficas como la siguiente:

      +
      ggplot(tabla_cruzada |> ungroup() |>
      +         mutate(price = fct_reorder(precio, prop)),
      +       aes(x = precio, y = prop, group = presentacion, colour = presentacion)) +
      +  geom_point() + coord_flip() + geom_line()
      +

      +

      En lugar de eso, calcularemos perfiles columna. Esto es, comparamos +cada una de las columnas con la columna marginal (en la tabla de tipo de +estilo de té):

      +
      num_grupos <- n_distinct(te |> select(presentacion))
      +tabla <- te |>
      +  count(presentacion, precio) |>
      +  group_by(presentacion) |>
      +  mutate(prop_precio = (100 * n / sum(n))) |>
      +  group_by(precio) |>
      +  mutate(prom_prop = sum(prop_precio)/num_grupos) |>
      +  mutate(perfil = 100 * (prop_precio / prom_prop - 1))
      +tabla
      +
      ## # A tibble: 17 × 6
      +## # Groups:   precio [6]
      +##    presentacion precio           n prop_precio prom_prop perfil
      +##    <fct>        <fct>        <int>       <dbl>     <dbl>  <dbl>
      +##  1 bolsas       marca           70       41.2      25.4    61.8
      +##  2 bolsas       barato           5        2.94      2.26   30.1
      +##  3 bolsas       marca_propia    16        9.41      5.48   71.7
      +##  4 bolsas       desconocido     11        6.47      2.51  158. 
      +##  5 bolsas       fino            14        8.24     28.0   -70.6
      +##  6 bolsas       variable        54       31.8      36.3   -12.5
      +##  7 mixto        marca           20       21.3      25.4   -16.4
      +##  8 mixto        barato           1        1.06      2.26  -52.9
      +##  9 mixto        marca_propia     4        4.26      5.48  -22.4
      +## 10 mixto        desconocido      1        1.06      2.51  -57.6
      +## 11 mixto        fino            19       20.2      28.0   -27.8
      +## 12 mixto        variable        49       52.1      36.3    43.6
      +## 13 suelto       marca            5       13.9      25.4   -45.4
      +## 14 suelto       barato           1        2.78      2.26   22.9
      +## 15 suelto       marca_propia     1        2.78      5.48  -49.3
      +## 16 suelto       fino            20       55.6      28.0    98.4
      +## 17 suelto       variable         9       25        36.3   -31.1
      +
      tabla_perfil <- tabla |>
      +  select(presentacion, precio, perfil, pct = prom_prop) |>
      +  pivot_wider(names_from = presentacion, values_from = perfil,
      +              values_fill = list(perfil = -100.0))
      +if_profile <- function(x){
      +  any(x < 0) & any(x > 0)
      +}
      +marcar <- marcar_tabla_fun(25, "red", "black")
      +tab_out <- tabla_perfil |>
      +  arrange(desc(bolsas)) |>
      +  select(-pct, everything()) |>
      +  mutate(across(where(is.numeric), round, 0)) |>
      +  mutate(across(where(if_profile), marcar)) |>
      +  knitr::kable(format_table_salida(), escape = FALSE, digits = 0, booktabs = T) |>
      +  kableExtra::kable_styling(latex_options = c("striped", "scale_down"),
      +                            bootstrap_options = c( "hover", "condensed"),
      +                            full_width = FALSE)
      +
      ## Warning: There was 1 warning in `mutate()`.
      +## ℹ In argument: `across(where(is.numeric), round, 0)`.
      +## ℹ In group 1: `precio = marca`.
      +## Caused by warning:
      +## ! The `...` argument of `across()` is deprecated as of dplyr 1.1.0.
      +## Supply arguments directly to `.fns` through an anonymous function instead.
      +## 
      +##   # Previously
      +##   across(a:b, mean, na.rm = TRUE)
      +## 
      +##   # Now
      +##   across(a:b, \(x) mean(x, na.rm = TRUE))
      +
      if (knitr::is_latex_output()) {
      +  gsub("marca_propia", "marca-propia", tab_out)
      +} else {
      +  tab_out
      +}
      + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
      +precio + +bolsas + +mixto + +suelto + +pct +
      +desconocido + +158 + +-58 + +-100 + +3 +
      +marca_propia + +72 + +-22 + +-49 + +5 +
      +marca + +62 + +-16 + +-45 + +25 +
      +barato + +30 + +-53 + +23 + +2 +
      +variable + +-12 + +44 + +-31 + +36 +
      +fino + +-71 + +-28 + +98 + +28 +
      +

      Leemos esta tabla como sigue: por ejemplo, los compradores de té suelto +compran té fino a una tasa casi el doble (98%) que el promedio.

      +

      También podemos graficar como:

      +
      tabla_graf <- tabla_perfil |>
      +  ungroup() |>
      +  mutate(precio = fct_reorder(precio, bolsas)) |>
      +  select(-pct) |>
      +  pivot_longer(cols = -precio, names_to = "presentacion", values_to = "perfil")
      +g_perfil <- ggplot(tabla_graf,
      +                   aes(x = precio, xend = precio, y = perfil, yend = 0, group = presentacion)) +
      +  geom_point() + geom_segment() + facet_wrap(~presentacion) +
      +  geom_hline(yintercept = 0 , colour = "gray")+ coord_flip()
      +g_perfil
      +

      +

      Observación: hay dos maneras de construir la columna promedio: +tomando los porcentajes sobre todos los datos, o promediando los +porcentajes de las columnas. Si los grupos de las columnas están +desbalanceados, estos promedios son diferentes.

      +
        +
      • Cuando usamos porcentajes sobre la población, perfiles columna y +renglón dan el mismo resultado
      • +
      • Sin embargo, cuando hay un grupo considerablemente más grande que +otros, las comparaciones se vuelven vs este grupo particular. No +siempre queremos hacer esto.
      • +
      +
      +

      Interpretación

      +

      En el último ejemplo de tomadores de té utilizamos una muestra de +personas, no toda la población de tomadores de té. Eso quiere decir que +tenemos cierta incertidumbre de cómo se generalizan o no los resultados +que obtuvimos en nuestro análisis a la población general.

      +

      Nuestra respuesta depende de cómo se extrajo la muestra que estamos +considerando. Si el mecanismo de extracción incluye algún proceso +probabilístico, entonces es posible en principio entender qué tan bien +generalizan los resultados de nuestro análisis a la población general, y +entender esto depende de entender qué tanta variación hay de muestra a +muestra, de todas las posibles muestras que pudimos haber extraido.

      +

      En las siguiente secciones discutiremos estos aspectos, en los cuales +pasamos del trabajo de “detective” al trabajo de “juez” en nuestro +trabajo analítico.

      +
      +
      +
      +
      +

      Suavizamiento loess

      +

      Las gráficas de dispersión son la herramienta básica para describir la +relación entre dos variables cuantitativas, y como vimos en ejemplo +anteriores, muchas veces podemos apreciar mejor la relación entre ellas +si agregamos una curva loess que suavice.

      +

      Los siguientes datos muestran los premios ofrecidos y las ventas totales +de una lotería a lo largo de 53 sorteos (las unidades son cantidades de +dinero indexadas). Graficamos en escalas logarítmicas y agregamos una +curva loess.

      +
      # cargamos los datos
      +load(here::here("data", "ventas_sorteo.Rdata"))
      +
      +ggplot(ventas.sorteo, aes(x = premio, y = ventas.tot.1)) +
      +  geom_point() +
      +  geom_smooth(method = "loess", alpha = 0.5, degree = 1, se = FALSE) +
      +  scale_x_log10(breaks = c(20000, 40000, 80000)) +
      +  scale_y_log10(breaks = c(10000, 15000, 22000, 33000))
      +

      +

      El patrón no era difícil de ver en los datos originales, sin embargo, la +curva lo hace más claro, el logaritmo de las ventas tiene una relación +no lineal con el logaritmo del premio: para premios no muy grandes no +parece haber gran diferencia, pero cuando los premios empiezan a crecer +por encima de 20,000, las ventas crecen más rápidamente que para premios +menores. Este efecto se conoce como bola de nieve, y es frecuente en +este tipo de loterías.

      +

      Antes de adentrarnos a los modelos loess comenzamos explicando cómo se +ajustan familias paramétricas de curvas a conjuntos de datos dados. El +modelo de regresion lineal ajusta una recta a un conjunto de datos. +Por ejemplo, consideremos la familia \[f_{a,b}(x) = a x + b,\] para un +conjunto de datos bivariados \(\{ (x_1, y_1), \ldots, (x_N, y_N)\}\). +Buscamos encontrar \(a\) y \(b\) tales que \(f_{a,b}\) de un ajuste óptimo a +los datos. Para esto, se minimiza la suma de errores cuadráticos

      +

      \[\frac1N \sum_{n = 1}^N ( y_n - a x_n - b)^2.\]

      +

      En este caso, las constantes \(a\) y \(b\) se pueden encontrar diferenciando +la función de mínimos cuadrados. Nótese que podemos repetir el argumento +con otras familias de funciones (por ejemplo cuadráticas).

      +
      ggplot(ventas.sorteo, aes(x = premio, y = ventas.tot.1)) +
      +  geom_point() +
      +  geom_smooth(method = "lm", alpha = 0.5, se = FALSE) +
      +  scale_x_log10(breaks = c(20000, 40000, 80000)) +
      +  scale_y_log10(breaks = c(10000, 15000, 22000, 33000))
      +

      +

      Si observamos la gráfica notamos que este modelo lineal (en los +logaritmos) no resumen adecuadamente estos datos. Podríamos experimentar +con otras familias (por ejemplo, una cuadrática o cúbica, potencias, +exponenciales, etc.); sin embargo, en la etapa exploratoria es mejor +tomar una ruta de ajuste más flexible y robusta. Regresión local nos +provee de un método con estas características:

      +

      Curvas loess (regresión local): Una manera de mejorar la +flexibilidad de los modelos lineales es considerar rectas de manera +local. Es decir, en cada \(x\) posible consideramos cuál es la recta +que mejor ajusta a los datos, considerando solamente valores de \(x_n\) +que están cercanos a \(x\).

      +

      La siguiente gráfica muestra qué recta se ajusta alrededor de cada +punto, y cómo queda el suavizador completo, con distintos valores de +suavizamiento. El tono de los puntos indican en cada paso que ventana de datos es +considerada:

      +

      +

      Escogiendo de los parámetros. El parámetro de suavizamiento se +encuentra por ensayo y error. La idea general es que debemos encontrar +una curva que explique patrones importantes en los datos (que ajuste +los datos) pero que no muestre variaciones a escalas más chicas +difíciles de explicar (que pueden ser el resultado de influencias de +otras variables, variación muestral, ruido o errores de redondeo, por +ejemplo). En el proceso de prueba y error iteramos el ajuste y en cada +paso hacemos análisis de residuales, con el fin de seleccionar un +suavizamiento adecuado.

      +

      En lugar de usar ajustes locales lineales, podemos usar ajustes locales +cuadráticos que nos permiten capturar formas locales cuadráticas sin +tener que suavizar demasiado poco:

      +

      +
      +

      Opcional: cálculo del suavizador

      +

      La idea es producir ajustes locales de rectas o funciones lineales o +cuadráticas. Consideremos especificar dos parámetros:

      +
        +
      • Parámetro de suavizamiento \(\alpha\): cuando \(\alpha\) es más grande, +la curva ajustada es más suave.

      • +
      • Grado de los polinomios locales que ajustamos \(\lambda\): +generalmente se toma \(\lambda=1,2\).

      • +
      +

      Entonces, supongamos que los datos están dados por +\((x_1,y_1), \ldots, (x_N, y_N)\), y sean \(\alpha\) un parámetro de +suavizamiento fijo, y \(\lambda=1\). Denotamos como \(\hat{g}(x)\) la curva +loess ajustada, y como \(w_n(x)\) a una función de peso (que depende de +x) para la observación \((x_n, y_n)\).

      +

      Para poder calcular \(w_n(x)\) debemos comenzar calculando +\(q=\lfloor{N\alpha}\rfloor\) que suponemos mayor que uno, esta \(q\) es el número +de puntos que se utilizan en cada ajuste local. Ahora definimos +la función tricubo: \[\begin{align} +T(u)=\begin{cases} +(1-|u|^3)^3, & \text{para $|u| < 1$}.\\ +0, & \text{en otro caso}. +\end{cases} +\end{align}\]

      +

      entonces, para el punto \(x\) definimos el peso correspondiente al dato +\((x_n,y_n)\), denotado por \(w_n(x)\) como:

      +

      \[w_n(x)=T\bigg(\frac{|x-x_n|}{d_q(x)}\bigg)\]

      +

      donde \(d_q(x)\) es el valor de la \(q\)-ésima distancia más chica (la más +grande entre las \(q\) más chicas) entre los valores \(|x-x_j|\), +\(j=1,\ldots,N\). De esta forma, las observaciones \(x_n\) reciben más peso +cuanto más cerca estén de \(x\).

      +

      En palabras, de \(x_1,...,x_N\) tomamos los \(q\) datos más cercanos a \(x\), +que denotamos \(x_{i_1}(x) \leq x_{i_2}(x) \leq \cdots \leq x_{i_q}(x)\). +Los re-escalamos a \([0,1]\) haciendo corresponder \(x\) a \(0\) y el punto +más alejado de \(x\) (que es \(x_{i_q}\)) a 1.

      +

      Aplicamos el tricubo (gráfica de abajo), para encontrar los pesos de +cada punto. Los puntos que están a una distancia mayor a \(d_q(x)\) +reciben un peso de cero, y los más cercanos un peso que depende de que +tan cercanos están a \(x\). Nótese que \(x\) es el punto ancla en dónde +estamos ajustando la regresión local.

      +
      tricubo <- function(x) {
      +  ifelse(abs(x) < 1, (1 - abs(x) ^ 3) ^ 3, 0)
      +}
      +curve(tricubo, from = -1.5, to = 1.5)
      +

      +

      Finalmente, para cada valor de \(x_k\) que está en el conjunto de datos +\(\{x_1,...,x_n\}\), ajustamos una recta de mínimos cuadrados ponderados +por los pesos \(w_n(x)\), es decir, minimizamos (en el caso lineal):

      +

      \[\sum_{i=1}^nw_n(x_k)(y_i-ax_n-b)^2.\]

      +

      Observaciones:

      +
        +
      1. Cualquier función (continua y quizás diferenciable) con la forma de +flan del tricubo que se desvanece fuera de \((-1,1),\) es creciente en +\((-1,0)\) y decreciente en \((0, 1)\) es un buen candidato para usarse +en lugar del tricubo. La razón por la que escogemos precisamente +esta forma algebráica no tiene que ver con el análisis exploratorio, +sino con las ventajas teóricas adicionales que tiene en la +inferencia.

      2. +
      3. El caso \(\lambda=2\) es similar. La única diferencia es en el paso de +ajuste, donde usamos funciones cuadráticas, y obtendríamos entonces +tres familias de parámetros \(a(x_k), b_1(x_k), b_2(x_k),\) para cada +\(k \in \{1, \ldots, N\}\).

      4. +
      +
      +
      +
      +

      Caso de estudio: nacimientos en México

      +

      Podemos usar el suavizamiento loess para entender y describir el +comportamiento de series de tiempo, en las cuáles intentamos entender la +dependencia de una serie de mediciones indexadas por el tiempo. +Típicamente es necesario utilizar distintas componentes para describir +exitosamente una serie de tiempo, y para esto usamos distintos tipos de +suavizamientos. Veremos que distintas componentes varían en distintas +escalas de tiempo (unas muy lentas, como la tendencia, otras más +rapidamente, como variación quincenal, etc.).

      +

      Este caso de estudio esta basado en un análisis propuesto por A. +Vehtari y A. +Gelman, +junto con un análisis de serie de tiempo de William S. Cleveland (1993).

      +

      En nuestro caso, usaremos los datos de nacimientos registrados por día +en México desde 1999. Los usaremos para contestar las preguntas: ¿cuáles +son los cumpleaños más frecuentes? y ¿en qué mes del año hay más +nacimientos?

      +

      Podríamos utilizar una gráfica popular (ver por ejemplo esta +visualización) +como:

      +

      +

      Sin embargo, ¿cómo criticarías este análisis desde el punto de vista de +los tres primeros principios del diseño analítico? ¿Las comparaciones +son útiles? ¿Hay aspectos multivariados? ¿Qué tan bien explica o sugiere +estructura, mecanismos o causalidad?

      +
      +

      Datos de natalidad para México

      +
      library(lubridate)
      +library(ggthemes)
      +theme_set(theme_minimal(base_size = 14))
      +natalidad <- read_rds("./data/nacimientos/natalidad.rds") |>
      +  mutate(dia_semana = weekdays(fecha)) |>
      +  mutate(dia_año = yday(fecha)) |>
      +  mutate(año = year(fecha)) |>
      +  mutate(mes = month(fecha)) |>
      +  ungroup() |>
      +  mutate(dia_semana = recode(dia_semana, Monday = "Lunes", Tuesday = "Martes", Wednesday = "Miércoles",
      +                             Thursday = "Jueves", Friday = "Viernes", Saturday = "Sábado", Sunday = "Domingo")) |>
      +  #necesario pues el LOCALE puede cambiar
      +  mutate(dia_semana = recode(dia_semana, lunes = "Lunes", martes = "Martes", miércoles = "Miércoles",
      +                             jueves = "Jueves", viernes = "Viernes", sábado = "Sábado", domingo = "Domingo")) |>
      +  mutate(dia_semana = fct_relevel(dia_semana, c("Lunes", "Martes", "Miércoles",
      +                                                "Jueves", "Viernes", "Sábado", "Domingo")))
      +

      Consideremos los datos agregados del número de nacimientos +(registrados) por día desde 1999 hasta 2016. Un primer intento podría +ser hacer una gráfica de la serie de tiempo. Sin embargo, vemos que no +es muy útil:

      +

      +

      Hay varias características que notamos. Primero, parece haber una +tendencia ligeramente decreciente del número de nacimientos a lo largo +de los años. Segundo, la gráfica sugiere un patrón anual. Y por último, +encontramos que hay dispersión producida por los días de la semana.

      +

      Sólo estas características hacen que la comparación entre días sea +difícil de realizar. Supongamos que comparamos el número de nacimientos +de dos miércoles dados. Esa comparación será diferente dependiendo: del +año donde ocurrieron, el mes donde ocurrieron, si semana santa ocurrió +en algunos de los miércoles, y así sucesivamente.

      +

      Como en nuestros ejemplos anteriores, la idea del siguiente análisis es +aislar las componentes que observamos en la serie de tiempo: extraemos +componentes ajustadas, y luego examinamos los residuales.

      +

      En este caso particular, asumiremos una descomposición aditiva de la +serie de tiempo (William S. Cleveland 1993).

      +
      +

      +En el estudio de series de tiempo una estructura +común es considerar el efecto de diversos factores como tendencia, +estacionalidad, ciclicidad e irregularidades de manera aditiva. Esto es, +consideramos la descomposición \[\begin{align} +y(t) = f_{t}(t) + f_{e}(t) + f_{c}(t) + \varepsilon. +\end{align}\] Una estrategia de ajuste, como veremos más +adelante, es proceder de manera modular. Es decir, se ajustan +los componentes de manera secuencial considerando los residuales de los +anteriores. +

      +
      +
      +
      +

      Tendencia

      +

      Comenzamos por extraer la tendencia, haciendo promedios loess +(William S. Cleveland 1979) con vecindades relativamente grandes. Quizá +preferiríamos suavizar menos para capturar más variación lenta, pero si +hacemos esto en este punto empezamos a absorber parte de la componente +anual:

      +
      mod_1 <- loess(n ~ as.numeric(fecha), data = natalidad, span = 0.2, degree = 1)
      +datos_dia <- natalidad |> 
      +  mutate(ajuste_1 = fitted(mod_1)) |>
      +  mutate(res_1 = n - ajuste_1)
      +

      +

      Notemos que a principios de 2000 el suavizador está en niveles de +alrededor de 7000 nacimientos diarios, hacia 2015 ese número es más +cercano a unos 6000.

      +
      +
      +

      Componente anual

      +

      Al obtener la tendencia podemos aislar el efecto a largo plazo y +proceder a realizar mejores comparaciones (por ejemplo, comparar un día +de 2000 y de 2015 tendria más sentido). Ahora, ajustamos los +residuales del suavizado anterior, pero con menos suavizamiento. Así +evitamos capturar tendencia:

      +
      mod_anual <- loess(res_1 ~ as.numeric(fecha), data = datos_dia, degree = 2, span = 0.005)
      +datos_dia <- datos_dia |> mutate(ajuste_2 = fitted(mod_anual)) |>
      +  mutate(res_2 = res_1 - ajuste_2)
      +

      +
      +
      +

      Día de la semana

      +

      Hasta ahora, hemos aislado los efectos por plazos largos de tiempo +(tendencia) y hemos incorporado las variaciones estacionales (componente +anual) de nuestra serie de tiempo. Ahora, veremos cómo capturar el +efecto por día de la semana. En este caso, podemos hacer suavizamiento +loess para cada serie de manera independiente

      +
      datos_dia <- datos_dia |>
      +  group_by(dia_semana) |>
      +  nest() |>
      +  mutate(ajuste_mod =
      +           map(data, ~ loess(res_2 ~ as.numeric(fecha), data = .x, span = 0.1, degree = 1))) |>
      +  mutate(ajuste_3 =  map(ajuste_mod, fitted)) |>
      +  select(-ajuste_mod) |>
      +  unnest(cols = c(data, ajuste_3)) |>
      +  mutate(res_3 = res_2 - ajuste_3) |>
      +  ungroup()
      +

      +
      +
      +

      Residuales

      +

      Por último, examinamos los residuales finales quitando los efectos +ajustados:

      +
      ## `geom_smooth()` using formula = 'y ~ x'
      +

      +

      Observación: nótese que la distribución de estos residuales presenta +irregularidades interesantes. La distribución es de colas largas, y no +se debe a unos cuantos datos atípicos. Esto generalmente es indicación +que hay factores importantes que hay que examinar mas a detalle en los +residuales:

      +

      +
      +
      +

      Reestimación

      +

      Cuando hacemos este proceso secuencial de llevar el ajuste a los +residual, a veces conviene iterarlo. La razón es que en una segunda o +tercera pasada podemos hacer mejores estimaciones de cada componente, y +es posible suavizar menos sin capturar componentes de más alta +frecuencia.

      +

      Así que podemos regresar a la serie original para hacer mejores +estimaciones, más suavizadas:

      +
      # Quitamos componente anual y efecto de día de la semana
      +datos_dia <- datos_dia |> mutate(n_1 = n - ajuste_2 - ajuste_3)
      +# Reajustamos
      +mod_1 <- loess(n_1 ~ as.numeric(fecha), data = datos_dia, span = 0.02, degree = 2,
      +               family = "symmetric")
      +

      +

      +

      Y ahora repetimos con la componente de día de la semana:

      +

      +
      +
      +

      Análisis de componentes

      +

      Ahora comparamos las componentes estimadas y los residuales en una misma +gráfica. Por definición, la suma de todas estas componentes da los datos +originales.

      +

      +

      Este último paso nos permite diversas comparaciones que explican la +variación que vimos en los datos. Una gran parte de los residuales está +entre \(\pm 250\) nacimientos por día. Sin embargo, vemos que las colas +tienen una dispersión mucho mayor:

      +
      quantile(datos_dia$res_6, c(00, .01,0.05, 0.10, 0.90, 0.95, 0.99, 1)) |> round()
      +
      ##    0%    1%    5%   10%   90%   95%   99%  100% 
      +## -2238 -1134  -315  -202   188   268   516  2521
      +

      ¿A qué se deben estas colas tan largas?

      +
      +

      Viernes 13?

      +

      Podemos empezar con una curosidad. Los días Viernes o Martes 13, ¿nacen +menos niños?

      +

      +

      Nótese que fue útil agregar el indicador de Semana santa por el Viernes +13 de Semana Santa que se ve como un atípico en el panel de los viernes +13.

      +
      +
      +
      +

      Residuales: antes y después de 2006

      +

      Veamos primero una agregación sobre los años de los residuales. Lo +primero es observar un cambio que sucedió repentinamente en 2006:

      +

      +

      La razón es un cambio en la ley acerca de cuándo pueden entrar los niños +a la primaria. Antes era por edad y había poco margen. Ese exceso de +nacimientos son reportes falsos para que los niños no tuvieran que +esperar un año completo por haber nacido unos cuantos días después de la +fecha límite.

      +

      Otras características que debemos investigar:

      +
        +
      • Efectos de Año Nuevo, Navidad, Septiembre 16 y otros días feriados +como Febrero 14.
      • +
      • Semana santa: como la fecha cambia, vemos que los residuales +negativos tienden a ocurrir dispersos alrededor del día 100 del año.
      • +
      +
      +
      +

      Otros días especiales: más de residuales

      +

      Ahora promediamos residuales (es posible agregar barras para indicar +dispersión a lo largo de los años) para cada día del año. Podemos +identificar ahora los residuales más grandes: se deben, por ejemplo, a +días feriados, con consecuencias adicionales que tienen en días ajuntos +(excesos de nacimientos):

      +

      +
      +
      +

      Semana santa

      +

      Para Semana Santa tenemos que hacer unos cálculos. Si alineamos los +datos por días antes de Domingo de Pascua, obtenemos un patrón de caída +fuerte de nacimientos el Viernes de Semana Santa, y la característica +forma de “valle con hombros” en días anteriores y posteriores estos +Viernes. ¿Por qué ocurre este patrón?

      +

      +

      Nótese un defecto de nuestro modelo: el patrón de “hombros” alrededor +del Viernes Santo no es suficientemente fuerte para equilibrar los +nacimientos faltantes. ¿Cómo podríamos mejorar nuestra descomposición?

      + +
      +
      +
      +

      References

      +
      +
      +Cleveland, William S. 1979. “Robust Locally Weighted Regression and Smoothing Scatterplots.” Journal of the American Statistical Association 74 (368): 829–36. +
      +
      +Cleveland, William S. 1993. Visualizing Data. Hobart Press. +
      +
      +Guber, D. 1999. “Getting What You Pay for: The Debate over Equity in Public School Expenditures.” Journal of Statistics Education 7 (2). +
      +
      +Lê, Sébastien, Julie Josse, François Husson, et al. 2008. “FactoMineR: An r Package for Multivariate Analysis.” Journal of Statistical Software 25 (1): 1–18. +
      +
      +
      + +
      +
      +
      + + +
      +
      + + + + + + + + + + + + + + + diff --git "a/ap\303\251ndice-principios-de-visualizacion.html" "b/ap\303\251ndice-principios-de-visualizacion.html" new file mode 100644 index 0000000..989fbde --- /dev/null +++ "b/ap\303\251ndice-principios-de-visualizacion.html" @@ -0,0 +1,677 @@ + + + + + + + Apéndice: Principios de visualizacion | Fundamentos de Estadística con Remuestreo + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
      + + + +
      +
      + + +
      +
      + +
      +
      +

      Apéndice: Principios de visualizacion

      +
      +

      “The simple graph has brought more information to the data analyst’s +mind than any other device.”

      +

      — John Tukey

      +
      +
      +

      El cuarteto de Anscombe

      +

      En 1971 un estadístico llamado Frank Anscombe (fundador del departamento de +Estadística de la Universidad de Yale) publicó cuatro conjuntos de dato. Cada +uno consiste de 11 observaciones. La peculariedad de estos conjuntos es +que tienen las mismas propiedades estadísticas.

      +

      Sin embargo, cuando analizamos los datos de manera gráfica en un histograma +encontramos rápidamente que los conjuntos de datos son muy distintos.

      +
      +

      +
      +


      +

      +

      Media de \(x\): 9
      +Varianza muestral de \(x\): 11
      +Media de \(y\): 7.50
      +Varianza muestral de \(y\): 4.12
      +Correlación entre \(x\) y \(y\): 0.816
      +Línea de regresión lineal: \(y = 3.00 + 0.500x\)

      +
      + +
      +

      En la gráfica del primer conjunto de datos, se ve clara una +relación lineal simple con un modelo que cumple los supuestos de +normalidad. La segunda gráfica (arriba a la derecha) muestra unos datos que +tienen una asociación pero definitivamente no es lineal. En la tercera gráfica +(abajo a la izquierda) están puntos alineados perfectamente en una línea recta, +excepto por uno de ellos. En la última gráfica podemos ver un ejemplo en el cual +basta tener una observación atípica para que se produzca un coeficiente de +correlación alto aún cuando en realidad no existe una asociación lineal entre +las dos variables.

      +
      + +
      +
      +

      +
      +

      El cuarteto de Ascombe inspiró una técnica reciente para crear datos que comparten las +mismas propiedades estadísticas al igual que en el cuarteto, pero que producen gráficas +muy distintas (Matejka, +Fitzmaurice).

      +
      + +
      +
      +
      +

      Introducción

      +

      La visualización de datos no trata de hacer gráficas “bonitas” o “divertidas”, +ni de simplificar lo complejo o ayudar a una persona “que no entiende mucho” a +entender ideas complejas. Más bien, trata de aprovechar nuestra gran capacidad +de procesamiento visual para exhibir de manera clara aspectos importantes de los +datos.

      +

      El siguiente ejemplo de (Tufte 2006), ilustra claramente la diferencia entre estos +dos enfoques. A la izquierda están gráficas (más o menos típicas de Powerpoint) +basadas en la filosofía de simplificar, de intentar no “ahogar” al lector con +datos. El resultado es una colección incoherente, de bajo contenido, que no +tiene mucho qué decir y que es, “indeferente al contenido y la evidencia”. +A la derecha está una variación del rediseño de Tufte en forma de tabla, que en +este caso particular es una manera eficiente de mostrar claramente los patrones +que hay en este conjunto simple de datos.

      +

      ¿Qué principios son los que soportan la efectividad de esta tabla sobre la +gráfica de la derecha? Veremos que hay dos conjuntos de principios importantes: +unos relacionados con el diseño y otros con la naturaleza del análisis de datos, +independientemente del método de visualización.

      +

      +
      + +
      +

      Teoría de visualización de datos

      +

      Existe teoría fundamentada acerca de la visualización. Después del trabajo +pionero de Tukey, los principios e indicadores de Tufte se basan en un estudio +de la historia de la graficación y ejercicios de muestreo de la práctica gráfica +a lo largo de varias disciplinas (¿cuáles son las mejores gráficas? ¿por qué?) +El trabajo de Cleveland es orientado a la práctica del análisis de datos +(¿cuáles gráficas nos han ayudado a mostrar claramente los resultados del +análisis?), por una parte, y a algunos estudios de percepción visual.

      +

      En resumen, hablaremos de las siguientes guías:

      +
      +

      Principios generales del diseño analítico

      +

      Aplicables a una presentación o análisis completos, y como guía para construir +nuevas visualizaciones (Tufte 2006).

      +
      +

      +Principio 1. Muestra comparaciones, contrastes, +diferencias.
      +Principio 2. Muestra causalidad, mecanismo, +explicación, estructura sistemática.
      +Principio 3. Muestra datos multivariados, es decir, más +de una o dos variables.
      +Principio 4. Integra palabras, números, imágenes y +diagramas.
      +Principio 5. Describe la totalidad de la evidencia. +Muestra fuentes usadas y problemas relevantes.
      +Principio 6. Las presentaciones analíticas, a fin de +cuentas, se sostienen o caen dependiendo de la calidad, relevancia e +integridad de su contenido. +

      +
      +
      +
      +

      Técnicas de visualización

      +

      Esta categoría incluye técnicas específicas que dependen de la forma de nuestros +datos y el tipo de pregunta que queremos investigar (Tukey (1977), William S. Cleveland (1993), +W. S. Cleveland (1994), Tufte (2006)).

      +
      +

      +Tipos de gráficas: cuantiles, histogramas, caja y +brazos, gráficas de dispersión, puntos/barras/ líneas, series de +tiempo.
      +Técnicas para mejorar gráficas: Transformación de +datos, transparencia, vibración, banking 45, suavizamiento y bandas de +confianza.
      +Pequeños múltiplos +

      +
      +
      +
      +

      Indicadores de calidad gráfica

      +

      Aplicables a cualquier gráfica en particular. Estas son guías concretas y +relativamente objetivas para evaluar la calidad de una gráfica (Tufte 1986).

      +
      +

      +Integridad Gráfica. El factor de engaño, es decir, +la distorsión gráfica de las cantidades representadas, debe ser +mínimo.
      +Chartjunk. Minimizar el uso de decoración gráfica que +interfiera con la interpretación de los datos: 3D, rejillas, rellenos +con patrones.
      +Tinta de datos. Maximizar la proporción de tinta de +datos vs. tinta total de la gráfica. For non-data- ink, less is +more. For data-ink, less is a bore.
      +Densidad de datos. Las mejores gráficas tienen mayor +densidad de datos, que es la razón entre el tamaño del conjunto de datos +y el área de la gráfica. Las gráficas se pueden encoger mucho. +Percepción visual. Algunas tareas son más fáciles para el ojo humano que +otras (W. S. Cleveland 1994). +

      +
      +
      +
      +

      Factor de engaño y Chartjunk

      +
      +

      +
      +

      El factor de engaño es el cociente entre el efecto mostrado en una gráfica y +el efecto correspondiente en los datos. Idealmente, el factor de engaño debe ser +1 (ninguna distorsión).

      +

      El chartjunk son aquellos elementos gráficos que no corresponden a variación +de datos, o que entorpecen la interpretación de una gráfica.
      +Estos son los indicadores de calidad más fáciles de entender y aplicar, y +afortunadamente cada vez son menos comunes.

      +

      Un diseño popular que califica como chartjunk y además introduce factores de +engaño es el pie de 3D. En la gráfica de la derecha, podemos ver como la +rebanada C se ve más grande que la rebanada A, aunque claramente ese no es el +caso (factor de engaño). La razón es la variación en la perspectiva que no +corresponde a variación en los datos (chartjunk).

      +
      +

      Crítica gráfica: Gráfica de pie

      +

      Todavía elementos que pueden mejorar la comprensión de nuestra +gráfica de pie: se trata de la +decodificiación que hay que hacer categoría - color - cuantificación. Podemos +agregar las etiquetas como se muestra en la serie de la derecha, pero entonces: +¿por qué no mostrar simplemente la tabla de datos? ¿qué agrega el pie a la +interpretación?

      +

      La deficiencias en el pie se pueden ver claramente al intentar graficar más +categorías (13) . En el primer pie no podemos distinguir realmente cuáles son +las categorías grandes y cuáles las chicas, y es muy difícil tener una imagen +mental clara de estos datos. Agregar los porcentajes ayuda, pero entonces, otra +vez, preguntamos cuál es el propósito del pie. La tabla de la izquierda hace +todo el trabajo (una vez que ordenamos las categrías de la más grande a la más +chica). Es posible hacer una gráfica de barras como la de abajo a la izquierda.

      +
      + +
      +
      +

      +
      +

      Hay otros tipos de chartjunk comunes: uno es la textura de barras, por ejemplo. +El efecto es la producción de un efecto moiré que es desagradable y quita la +atención de los datos, como en la gráfica de barras de abajo. Otro común son las +rejillas, como mostramos en las gráficas de la izquierda. Nótese como en estos +casos hay efectos ópticos no planeados que degradan la percepción de los +patrones en los datos.

      +

      +
      +
      +
      +

      Pequeños múltiplos y densidad gráfica

      +

      La densidad de una gráfica es el tamaño del conjunto de datos que se grafica +comparado con el área total de la gráfica. En el siguiente ejemplo, graficamos +en logaritmo-10 de millones de cabezas de ganado en Francia (cerdos, res, ovejas y +caballos). La gráfica de la izquierda es pobre en densidad pues sólo representa +4 datos. La manera más fácil de mejorar la densidad es hacer más chica la +gráfica:

      +

      +

      La razón de este encogimiento es una que tiene qué ver con las oportunidades perdidas de una gráfica grande. Si repetimos este mismo patrón (misma escala, mismos tipos de ganado) para distintos países obtenemos la siguiente gráfica:

      +
      +

      +
      +

      Esta es una gráfica de puntos. Es útil como sustituto de una gráfica de barras, +y es superior en el sentido de que una mayor proporción de la tinta que se usa +es tinta de datos. Otra vez, mayor proporción de tinta de datos representa más +oportunidades que se pueden capitalizar.

      +
      + +
      +
      +

      Más pequeños múltiplos

      +

      Los pequeños múltiplos presentan oportunidades para mostrar más acerca +de nuestro problema de interés. Consideramos por ejemplo la relación +de radiación solar y niveles de ozono:

      +

      +

      En el ejemplo anterior incluyendo +una variable adicional (velocidad del viento) podemos entender más +acerca de la relación de radiación solar y niveles de ozono:

      +

      +
      +
      +
      +

      Tinta de datos

      +

      Maximizar la proporción de tinta de datos en nuestras gráficas tiene beneficios +inmediatos. La regla es: si hay tinta que no representa variación en los datos, +o la eliminación de esa tinta no representa pérdidas de significado, esa tinta +debe ser eliminada. El ejemplo más claro es el de las rejillas en gráficas y +tablas:

      +

      +

      +

      ¿Por qué usar grises en lugar de negros? La respuesta tiene qué ver con el +principio de tinta de datos: si marcamos las diferencias sutil pero claramente, +tenemos más oportunidades abiertas para hacer énfasis en lo que nos interesa: a +una gráfica o tabla saturada no se le puede hacer más - es difícil agregar +elementos adicionales que ayuden a la comprensión. Si comenzamos marcando con +sutileza, entonces se puede hacer más. Los mapas geográficos son un buen ejemplo +de este principio.

      +

      El espacio en blanco es suficientemente bueno para indicar las fronteras en una +tabla, y facilita la lectura:

      +

      +

      Para un ejemplo del proceso de rediseño de una tabla, ver aquí. +Finalmente, podemos ver un ejemplo que intenta incorporar +los elementos del diseño analítico, incluyendo pequeños múltiplos:

      +

      +
      +
      +

      Decoración

      +

      +
      +
      +

      Percepción de escala

      +

      Entre la percepción visual y la interpretación de una gráfica están implícitas +tareas visuales específicas que las personas debemos realizar para ver +correctamente la gráfica. En la década de los ochenta, William S. Cleveland y +Robert McGill realizaron algunos experimentos identificando y clasificando estas +tareas para diferentes tipos de gráficos (William S. Cleveland and McGill 1984). +En estos, se le pregunta a la persona que compare dos valores dentro de una +gráfica, por ejemplo, en dos barras en una gráfica de barras, o dos rebanadas de +una gráfica de pie.

      +

      +

      Los resultados de Cleveland y McGill fueron replicados por Heer y Bostock en +2010 y los resultados se muestran en las gráficas de la derecha:

      +

      +
      +
      +
      +

      Ejemplo: gráfica de Minard

      +

      Concluimos esta sección con una gráfica que, aunque poco común, ejemplifica +los principios de una buena gráfica, y es reconocida como una de las mejores +visualizaciones de la historia.

      +
      +

      Una gráfica excelente, presenta datos interesantes de forma bien +diseñada: es una cuestión de fondo, de diseño, y estadística… [Se] compone de +ideas complejas comunicadas con claridad, precisión y eficiencia. … [Es] lo +que da al espectador la mayor cantidad de ideas, en el menor tiempo, con +la menor cantidad de tinta, y en el espacio más pequeño. … Es casi siempre +multivariado. … Una excelente gráfica debe decir la verdad acerca de los +datos. (Tufte, 1983)

      +
      +

      La famosa visualización de Charles Joseph Minard de la marcha de Napoleón sobre +Moscú, ilustra los principios de una buena gráfica. Tufte señala que esta imagen +“bien podría ser el mejor gráfico estadístico jamás dibujado”, y sostiene que +“cuenta una historia rica y coherente con sus datos multivariados, mucho más +esclarecedora que un solo número que rebota en el tiempo”. Se representan seis +variables: el tamaño del ejército, su ubicación en una superficie bidimensional, +la dirección del movimiento del ejército y la temperatura en varias fechas +durante la retirada de Moscú”.

      +

      +

      Hoy en día Minard es reconocido como uno de los principales contribuyentes a la +teoría de análisis de datos y creación de infografías con un fundamento +estadístico.

      +

      Se grafican 6 variables: el número de tropas de Napoleón, la distancia, la +temperatura, la latitud y la longitud, la dirección en que viajaban las tropas +y la localización relativa a fechas específicas.

      +

      La gráfica de Minard, como la describe E.J. Marey, parece “desafiar la pluma del +historiador con su brutal elocuencia”, la combinación de datos del mapa, y la +serie de tiempo, dibujados en 1869, “retratan una secuencia de pérdidas +devastadoras que sufrieron las tropas de Napoleón en 1812”. Comienza en la +izquierda, en la frontera de Polonia y Rusia, cerca del río Niemen. La línea +gruesa dorada muestra el tamaño de la Gran Armada (422,000) en el momento en que +invadía Rusia en junio de 1812.

      +

      El ancho de esta banda indica el tamaño de la armada en cada punto del mapa. En +septiembre, la armada llegó a Moscú, que ya había sido saqueada y dejada +desértica, con sólo 100,000 hombres.

      +

      El camino del retiro de Napoleón desde Moscú está representado por la línea +oscura (gris) que está en la parte inferior, que está relacionada a su vez con +la temperatura y las fechas en el diagrama de abajo. Fue un invierno muy frío, +y muchos se congelaron en su salida de Rusia. Como se muestra en el mapa, cruzar +el río Berezina fue un desastre, y el ejército de Napoleón logró regresar a +Polonia con tan sólo 10,000 hombres.

      +

      También se muestran los movimientos de las tropas auxiliaries, que buscaban +proteger por atrás y por la delantera mientras la armada avanzaba hacia Moscú. +La gráfica de Minard cuenta una historia rica y cohesiva, coherente con datos +multivariados y con los hechos históricos, y que puede ser más ilustrativa que +tan sólo representar un número rebotando a lo largo del tiempo.

      + +
      +
      +

      References

      +
      +
      +Cleveland, W. S. 1994. The Elements of Graphing Data. AT&T Bell Laboratories. https://books.google.com.mx/books?id=KMsZAQAAIAAJ. +
      +
      +Cleveland, William S. 1993. Visualizing Data. Hobart Press. +
      +
      +Cleveland, William S., and Robert McGill. 1984. “Graphical Perception: Theory, Experimentation, and Application to the Development of Graphical Methods.” Journal of the American Statistical Association 79 (387): pp. 531–554. http://www.jstor.org/stable/2288400. +
      +
      +Tufte, Edward R. 1986. The Visual Display of Quantitative Information. Cheshire, CT, USA: Graphics Press. +
      +
      +———. 2006. Beautiful Evidence. Cheshire, CT: Graphics Press. +
      +
      +Tukey, J. W. 1977. Exploratory Data Analysis. Addison-Wesley Series in Behavioral Science. Addison-Wesley Publishing Company. https://books.google.com.mx/books?id=UT9dAAAAIAAJ. +
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      + +
      +
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      + + +
      +
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