diff --git a/_sources/gallery/index.rst.txt b/_sources/gallery/index.rst.txt index dfa6275..3582aba 100644 --- a/_sources/gallery/index.rst.txt +++ b/_sources/gallery/index.rst.txt @@ -12,6 +12,17 @@ General-purpose, introductory and illustrative examples. + + + + Calculating User-Specified Functions with QRFs + + Quantile Regression Forests vs. Random Forests - - - - Weighted vs. Unweighted Quantile Runtimes - -
diff --git a/_sources/gallery/plot_predict_custom.rst.txt b/_sources/gallery/plot_predict_custom.rst.txt new file mode 100644 index 0000000..aede31b --- /dev/null +++ b/_sources/gallery/plot_predict_custom.rst.txt @@ -0,0 +1,250 @@ + +:orphan: +:html_theme.sidebar_secondary.remove: + +.. This document is auto-generated by the example-gallery extension. Do not modify directly. + +.. _gallery_plot_predict_custom: + + +Calculating User-Specified Functions with QRFs +============================================== + +An example that demonstrates a way of extracting the empirical distribution +from a quantile regression forest (QRF) for one or more samples in order to +calculate a user-specified function of interest. While a QRF is designed to +estimate quantiles from the empirical distribution calculated for each sample, +in many cases it may be useful to use the empirical distribution to calculate +other quantities of interest. Here, we calculate the ECDF for a test sample. + + +.. altair-plot:: + :remove-code: + + + + from itertools import chain + + import altair as alt + import numpy as np + import pandas as pd + import scipy as sp + from sklearn import datasets + from sklearn.model_selection import train_test_split + + from quantile_forest import RandomForestQuantileRegressor + + np.random.seed(0) + + + def predict(reg, X, quantiles=0.5, what=None): + """Custom prediction method that allows user-specified function. + + Parameters + ---------- + reg : BaseForestQuantileRegressor + Quantile forest model object. + X : array-like, of shape (n_samples, n_features) + New test data. + quantiles : float or list of floats, default=0.5 + Scalar or vector of quantiles for quantile prediction. + what : Any, optional + User-specified function for prediction used instead of quantiles. + Must return a numeric vector. + + Returns + ------- + Output array with the user-specified function applied (if any). + """ + if what is None: + return reg.predict(X, quantiles=quantiles) + + # Get the complete set of proximities for each sample. + proximities = reg.proximity_counts(X) + + # Retrieve the unsorted training responses from the model (stored in sorted order). + reverse_sorter = np.argsort(reg.sorter_, axis=0) + y_train = np.empty_like(reg.forest_.y_train).T + for i in range(y_train.shape[1]): + y_train[:, i] = np.asarray(reg.forest_.y_train)[i, reverse_sorter[:, i]] + + # For each sample, construct an array of the training responses used for prediction. + s = [np.concatenate([np.repeat(y_train[i[0]], i[1]) for i in prox]) for prox in proximities] + + # Apply the user-specified function for each sample. Must return a numeric vector. + y_out = np.array([what(s_i) for s_i in s]) + + return y_out + + + X, y = datasets.load_diabetes(return_X_y=True) + X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=1, random_state=0) + + reg = RandomForestQuantileRegressor().fit(X_train, y_train) + + # Define a user-specified function; here we randomly sample 1000 values with replacement. + func = lambda x: np.random.choice(x, size=1000) + + # Output array with the user-specified function applied to each sample's empirical distribution. + y_out = predict(reg, X_test, what=func) + + # Calculate the ECDF from output array. + y_ecdf = [sp.stats.ecdf(y_i).cdf for y_i in y_out] + + df = pd.DataFrame( + { + "y_value": list(chain.from_iterable([y_i.quantiles for y_i in y_ecdf])), + "y_value2": list(chain.from_iterable([y_i.quantiles for y_i in y_ecdf]))[1:] + [np.nan], + "probability": list(chain.from_iterable([y_i.probabilities for y_i in y_ecdf])), + } + ) + + + def plot_ecdf(df): + tooltip = [ + alt.Tooltip("y_value", title="Response Value"), + alt.Tooltip("probability", title="Probability"), + ] + + circles = ( + alt.Chart(df) + .mark_circle(color="#006aff") + .encode( + x=alt.X("y_value", title="Response Value"), + y=alt.Y("probability", title="Probability"), + tooltip=tooltip, + ) + ) + + lines = ( + alt.Chart(df) + .mark_line(color="#006aff") + .encode( + x=alt.X("y_value", title="Response Value"), + x2=alt.X2("y_value2"), + y=alt.Y("probability", title="Probability"), + tooltip=tooltip, + ) + ) + + chart = (circles + lines).properties(height=400, width=650) + return chart + + + chart = plot_ecdf(df) + chart + + +.. code:: python + + + from itertools import chain + + import altair as alt + import numpy as np + import pandas as pd + import scipy as sp + from sklearn import datasets + from sklearn.model_selection import train_test_split + + from quantile_forest import RandomForestQuantileRegressor + + np.random.seed(0) + + + def predict(reg, X, quantiles=0.5, what=None): + """Custom prediction method that allows user-specified function. + + Parameters + ---------- + reg : BaseForestQuantileRegressor + Quantile forest model object. + X : array-like, of shape (n_samples, n_features) + New test data. + quantiles : float or list of floats, default=0.5 + Scalar or vector of quantiles for quantile prediction. + what : Any, optional + User-specified function for prediction used instead of quantiles. + Must return a numeric vector. + + Returns + ------- + Output array with the user-specified function applied (if any). + """ + if what is None: + return reg.predict(X, quantiles=quantiles) + + # Get the complete set of proximities for each sample. + proximities = reg.proximity_counts(X) + + # Retrieve the unsorted training responses from the model (stored in sorted order). + reverse_sorter = np.argsort(reg.sorter_, axis=0) + y_train = np.empty_like(reg.forest_.y_train).T + for i in range(y_train.shape[1]): + y_train[:, i] = np.asarray(reg.forest_.y_train)[i, reverse_sorter[:, i]] + + # For each sample, construct an array of the training responses used for prediction. + s = [np.concatenate([np.repeat(y_train[i[0]], i[1]) for i in prox]) for prox in proximities] + + # Apply the user-specified function for each sample. Must return a numeric vector. + y_out = np.array([what(s_i) for s_i in s]) + + return y_out + + + X, y = datasets.load_diabetes(return_X_y=True) + X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=1, random_state=0) + + reg = RandomForestQuantileRegressor().fit(X_train, y_train) + + # Define a user-specified function; here we randomly sample 1000 values with replacement. + func = lambda x: np.random.choice(x, size=1000) + + # Output array with the user-specified function applied to each sample's empirical distribution. + y_out = predict(reg, X_test, what=func) + + # Calculate the ECDF from output array. + y_ecdf = [sp.stats.ecdf(y_i).cdf for y_i in y_out] + + df = pd.DataFrame( + { + "y_value": list(chain.from_iterable([y_i.quantiles for y_i in y_ecdf])), + "y_value2": list(chain.from_iterable([y_i.quantiles for y_i in y_ecdf]))[1:] + [np.nan], + "probability": list(chain.from_iterable([y_i.probabilities for y_i in y_ecdf])), + } + ) + + + def plot_ecdf(df): + tooltip = [ + alt.Tooltip("y_value", title="Response Value"), + alt.Tooltip("probability", title="Probability"), + ] + + circles = ( + alt.Chart(df) + .mark_circle(color="#006aff") + .encode( + x=alt.X("y_value", title="Response Value"), + y=alt.Y("probability", title="Probability"), + tooltip=tooltip, + ) + ) + + lines = ( + alt.Chart(df) + .mark_line(color="#006aff") + .encode( + x=alt.X("y_value", title="Response Value"), + x2=alt.X2("y_value2"), + y=alt.Y("probability", title="Probability"), + tooltip=tooltip, + ) + ) + + chart = (circles + lines).properties(height=400, width=650) + return chart + + + chart = plot_ecdf(df) + chart diff --git a/_sources/gallery/plot_quantile_extrapolation.rst.txt b/_sources/gallery/plot_quantile_extrapolation.rst.txt index 8b55306..30afb3c 100644 --- a/_sources/gallery/plot_quantile_extrapolation.rst.txt +++ b/_sources/gallery/plot_quantile_extrapolation.rst.txt @@ -10,11 +10,16 @@ Extrapolation with Quantile Regression Forests ============================================== -An example on a toy dataset that demonstrates that the prediction intervals -produced by a quantile regression forest do not extrapolate outside of the -bounds of the data in the training set, an important limitation of the -approach. Notice that the extrapolated median and interval values fail to -accurately predict values outside of those observed in the training set. +An example on a toy dataset that illustrates the prediction intervals produced +by a quantile regression forest (QRF) on extrapolated data. QRFs do not +intrinsically extrapolate outside of the bounds of the training data, an +important limitation of the approach; notice that the extrapolated interval +with a standard QRF fails to reliably cover values outside of those observed +in the training set. To overcome this limitation, we can use a procedure known +as Xtrapolation that can estimate the extrapolation bounds for samples that +fall outside the range of the training data. Adapted from "Extrapolation-Aware +Nonparametric Statistical Inference" by Niklas Pfister and Peter Bühlmann: +https://arxiv.org/abs/2402.09758. .. altair-plot:: @@ -22,6 +27,8 @@ accurately predict values outside of those observed in the training set. + import math + import altair as alt import numpy as np import pandas as pd @@ -30,13 +37,15 @@ accurately predict values outside of those observed in the training set. np.random.seed(0) - n_samples = 1000 + n_samples = 500 bounds = [0, 15] extrap_frac = 0.25 - func = lambda x: x * np.sin(x) func_str = "f(x) = x sin(x)" + quantiles = [0.025, 0.975, 0.5] + qrf_params = {"max_samples_leaf": None, "min_samples_leaf": 4, "random_state": 0} + def make_func_Xy(func, bounds, n_samples): x = np.linspace(bounds[0], bounds[1], n_samples) @@ -44,49 +53,354 @@ accurately predict values outside of those observed in the training set. std = 0.01 + np.abs(x - 5.0) / 5.0 noise = np.random.normal(scale=std) y = f + noise - return x, y + return np.atleast_2d(x).T, y + + + class Xtrapolation: + """Xtrapolation procedure. + + Performs extrapolation-aware nonparametric statistical inference based on + an existing nonparametric estimate. Adapted from the Python code [1] for + the Xtrapolation procedure introduced in [2]. + + The procedure specifically applies a QRF for generating local polynomials + to estimate derivatives in a single dimension. For multi-dimensional + problems, using the original implementation is strongly encouraged. + + References + ---------- + .. [1] https://github.com/NiklasPfister/ExtrapolationAware-Inference + .. [2] N. Pfister and P. Bühlmann, "Extrapolation-Aware Nonparametric + Statistical Inference", arXiv preprint, 2024. + https://arxiv.org/abs/2402.09758 + """ + + def __init__(self, orders=np.array([1])): + self.orders_ = orders + self.max_order_ = np.max(orders) + + @staticmethod + def _penalized_locpol(fval, v, X, weights, degree, pen=0, penalize_intercept=False): + v = v.reshape(-1, 1) + n = X.shape[0] + dd = degree + 1 + if penalize_intercept: + pen_list = list(range(0, dd)) + else: + pen_list = list(range(1, dd)) + + # Construct design matrices. + DDmat = np.zeros((n * dd, n * dd)) + DYmat = np.zeros((n * dd, 1)) + for i in range(n): + Wi = np.sqrt(weights[i, :].reshape(-1, 1)) + + # Construct DDmat (block-diagonal). + x0v = X[i, :].dot(v) + Di = np.tile((X.dot(v) - x0v).reshape(-1, 1), dd) ** np.arange(dd) * Wi + DDmat[(i * dd) : ((i + 1) * dd), (i * dd) : ((i + 1) * dd)] = (Di.T).dot(Di) + + # Construct DYmat. + DYmat[(i * dd) : ((i + 1) * dd), :] = (Di.T).dot((fval.reshape(-1, 1)) * Wi) + + Z = np.zeros((dd, dd)) + for kk in pen_list: + Z[kk, kk] = math.factorial(kk) + PP = np.kron(np.diag(np.sum(weights, axis=1)) - weights, Z) + penmat = pen * (PP.T).dot(PP) + B = np.linalg.solve(DDmat + penmat, DYmat) + coefs = B.reshape(n, -1) + + # Extract derivatives from coefficients. + deriv_mat = coefs * np.array([math.factorial(k) for k in range(degree + 1)]) + return deriv_mat + + @staticmethod + def _get_tree_weight_matrix(X, Y, X_eval=None, n_trees=100, **kwargs): + """Fit forest and extract weights. + + This implementation extracts the weight matrix from a list of quantile + random forests, each with a single tree fitted on non-bootstrapped + samples. This allows for controlling the bootstrap selection for each + tree and summing the weight matrices across all of the trees. + """ + if "n_estimators" in kwargs: + n_trees = kwargs["n_estimators"] + kwargs["n_estimators"] = 1 + if "random_state" in kwargs: + del kwargs["random_state"] + kwargs["bootstrap"] = False + + trees = [RandomForestQuantileRegressor(random_state=i, **kwargs) for i in range(n_trees)] + + n = X.shape[0] + nn = 0 + if X_eval is not None: + nn = X_eval.shape[0] + X = np.r_[X, X_eval] + weight_mat = np.zeros((n + nn, n + nn)) + + s = 0.5 + bn = int(n * s) + + for tree in trees: + # Draw bootstrap sample. + boot_sample = np.random.choice(np.arange(n), bn, replace=False) + split1 = boot_sample[: int(bn / 2)] + split2 = np.concatenate([boot_sample[int(bn / 2) :], np.arange(nn) + n]) + + # Fit tree. + tree.fit(X[split1, :], Y[split1].flatten()) + + # Extract tree weight matrix. + y_train_leaves = tree._get_y_train_leaves(X[split2, :], 1) + nrows = X[split2, :].shape[0] + matrix = np.zeros((nrows, nrows)) + for leaf in y_train_leaves[0]: + indices = leaf[0] + indices = indices[indices != 0] - 1 + if len(indices) > 0: + matrix[np.ix_(indices, indices)] = 1 + weight_mat[np.ix_(split2, split2)] += matrix + + # Normalize weights (rows correspond to weights - non-symmetric). + weight_mat /= weight_mat.sum(axis=1)[:, None] + + return weight_mat + + def fit_weights(self, X, fval, x0=None, train=False, **kwargs): + """Compute random forest weights for derivative estimation.""" + n, d = X.shape + fval = fval.flatten() + + if train: + d_xtra = d + xtra_features = list(range(d)) + weights = [None] * d_xtra + for jj, var in enumerate(xtra_features): + var_order = list(range(d)) + var_order = np.array([var] + var_order[:var] + var_order[var + 1 :]) + weights[jj] = self._get_tree_weight_matrix(X[:, var_order], fval, x0, **kwargs) + else: + weights = self._get_tree_weight_matrix(X, fval, x0, **kwargs)[n:, :n] + + return weights + + def fit_derivatives(self, X, fval, pen=0.1, **kwargs): + """Estimate derivatives.""" + n, d = X.shape + fval = fval.flatten() + + # Fit weights for local polynomial. + weights = self.fit_weights(X, fval, train=True, **kwargs) + + # Estimate derivatives with local polynomial. + derivatives = np.zeros((self.max_order_ + 1, n, d)) + Xtilde = X[:, list(range(d))] + + # Fit local polynomial. + for jj in range(d): + vv = np.zeros((d, 1)) + vv[jj] = 1 + tmp = self._penalized_locpol( + fval, + vv, + Xtilde, + weights[jj], + degree=self.max_order_ + 1, + pen=pen, + penalize_intercept=False, + ) + for kk in range(self.max_order_ + 1): + derivatives[kk, :, jj] = fval if kk == 0 else tmp[:, kk] + + return derivatives + + def prediction_bounds(self, X, fval, x0, nn=50, **kwargs): + """Compute extrapolation bounds.""" + n, d = X.shape + fval = fval.flatten() + if len(x0.shape) == 1: + x0 = x0.reshape(-1, 1) + n0 = x0.shape[0] + xtra_features = list(range(d)) + + # Fit derivatives. + derivatives = self.fit_derivatives(X, fval, **kwargs) + + # Determine weighting for extrapolation points (using rotation). + mu = derivatives[1].mean(axis=0) + _, D, Vt = np.linalg.svd(derivatives[1] - mu[None, :]) + TT = (Vt.T) * D[None, :] + Xtilde = X[:, xtra_features].dot(TT) + x0tilde = x0[:, xtra_features].dot(TT) + + # Find closest points between rotated points (Euclidean). + weight_x0 = np.zeros((n0, n)) + for ii in range(n0): + xinds = np.argsort(np.sum((x0tilde[None, ii, :] - Xtilde) ** 2, axis=1))[:nn] + weight_x0[ii, xinds] = 1 / nn + + # Precompute factorials. + order_factorials = np.empty(self.max_order_ + 1) + for oo in range(self.max_order_ + 1): + order_factorials[oo] = math.factorial(oo) + + # Iterate over all extrapolation points and average/intersect. + bounds = np.zeros((n0, len(self.orders_), 3)) + for ll, xpt in enumerate(x0): + xinds = np.where(weight_x0[ll, :] != 0)[0] + + # Number of anchor points to check. + f_lower = np.zeros((len(xinds), len(self.orders_))) + f_upper = np.zeros((len(xinds), len(self.orders_))) + f_median = np.zeros((len(xinds), len(self.orders_))) + for ii, xind in enumerate(xinds): + xx = X[xind, :].reshape(1, -1) + vv = (xpt - xx)[:, xtra_features] + vv_norm = np.sqrt(np.sum(vv**2)) + + # Compute directional derivatives. + deriv_mat = np.zeros((n, self.max_order_ + 1)) + deriv_mat[:, 0] = derivatives[0, :, :].mean(axis=1) + if vv_norm > np.finfo(float).eps: + vv_direction = np.array(vv / vv_norm).reshape(-1, 1) + for kk in range(1, self.max_order_ + 1): + deriv_mat[:, kk] = derivatives[kk, :, :].dot(vv_direction**kk).flatten() + + # Select bounds. + deriv_min = np.quantile(deriv_mat, 0, axis=0) + deriv_max = np.quantile(deriv_mat, 1, axis=0) + deriv_median = np.quantile(deriv_mat, 0.5, axis=0) + + # Estimate extrapolation bounds. + mterm = 0 + kk = 0 + for oo in range(self.max_order_ + 1): + if oo in self.orders_: + lo_bdd = deriv_min[oo] * (vv_norm**oo) / order_factorials[oo] + up_bdd = deriv_max[oo] * (vv_norm**oo) / order_factorials[oo] + median_deriv = deriv_median[oo] * (vv_norm**oo) / order_factorials[oo] + f_lower[ii, kk] = mterm + lo_bdd + f_upper[ii, kk] = mterm + up_bdd + f_median[ii, kk] = mterm + median_deriv + kk += 1 + mterm += deriv_mat[xind, oo] * (vv_norm**oo) / order_factorials[oo] + + # Combine bounds over x-indices. + ww = (weight_x0[ll, xinds] / np.sum(weight_x0[ll, :]))[:, None] + f_median = np.sum(f_median * ww, axis=0) + + # Aggregate by optimal-average. + f_lower = np.max(f_lower, axis=0) + f_upper = np.min(f_upper, axis=0) + ind = f_upper < f_lower + average = (f_upper + f_lower) / 2 + f_lower[ind] = average[ind] + f_upper[ind] = average[ind] + + bounds[ll, :, 0] = f_lower + bounds[ll, :, 1] = f_upper + bounds[ll, :, 2] = f_median + + return bounds + + + def train_test_split(train_indices, **kwargs): + """Fit model on training samples and extrapolate on test samples.""" + X_train = X[train_indices, :] + y_train = y[train_indices] + + # Run quantile regression (with forests). + qrf = RandomForestQuantileRegressor(**kwargs) + qrf.fit(X_train, y_train) + qmat = qrf.predict(X, quantiles=quantiles) + + # Xtrapolation. + bounds_list = [None] * len(quantiles) + for i in range(len(quantiles)): + # Run Xtrapolation on quantile. + xtra = Xtrapolation() + bounds_list[i] = xtra.prediction_bounds(X_train, qmat[train_indices, i], X, **kwargs) + + return { + "train_indices": train_indices, + "quantiles": quantiles, + "qmat": qmat, + "bounds_list": bounds_list, + } + + + def prob_randomized_pi(qmat, y, coverage): + """Calculate calibration probability""" + alpha_included = np.mean((qmat[:, 0] <= y) & (y <= qmat[:, 1])) + alpha_excluded = np.mean((qmat[:, 0] < y) & (y < qmat[:, 1])) + if coverage <= alpha_excluded: + prob_si = 1 + elif coverage >= alpha_included: + prob_si = 0 + else: + prob_si = (coverage - alpha_included) / (alpha_excluded - alpha_included) + return prob_si + + def randomized_pi(qmat, prob_si, y): + """Calculate coverage.""" + si_index = np.random.choice([False, True], len(y), replace=True, p=[prob_si, 1 - prob_si]) + included = (qmat[:, 0] < y) & (y < qmat[:, 1]) + boundary = (qmat[:, 0] == y) | (qmat[:, 1] == y) + return included | (boundary & si_index) - def get_train_Xy(X, y, min_idx, max_idx): - X_train = X[min_idx:max_idx] - y_train = y[min_idx:max_idx] - return X_train, y_train + def get_coverage_qrf(qmat, train_indices, test_indices, y_train, level): + """Calculate extrapolation coverage for regular quantile forest.""" + prob_si = prob_randomized_pi(qmat[train_indices, :], y_train, level) + qrf = randomized_pi(qmat, prob_si, y) + return np.mean(qrf[test_indices]) - def get_test_X(X, bounds): - n_samples = len(X) - X_test = np.atleast_2d(np.linspace(*bounds, n_samples)).T - return X_test + + def get_coverage_xtr(bounds_list, train_indices, test_indices, y_train, level): + """Calculate extrapolation coverage for Xtrapolation.""" + bb_low = np.max(bounds_list[0][:, :, 0], axis=1) + bb_upp = np.min(bounds_list[1][:, :, 1], axis=1) + bb_low_train, bb_upp_train = bb_low[train_indices], bb_upp[train_indices] + prob_si = prob_randomized_pi(np.c_[bb_low_train, bb_upp_train], y_train, level) + xtra = randomized_pi(np.c_[bb_low, bb_upp], prob_si, y) + return np.mean(xtra[test_indices]) # Create the full dataset. X, y = make_func_Xy(func, bounds, n_samples) - # Calculate the extrapolation bounds. + # Fit and extrapolate based on train-test split (depending on X). extrap_min_idx = int(n_samples * (extrap_frac / 2)) extrap_max_idx = int(n_samples - (n_samples * (extrap_frac / 2))) + sort_X = np.argsort(X.squeeze()) + train_indices = np.repeat(False, len(y)) + train_indices[sort_X[extrap_min_idx] : sort_X[extrap_max_idx]] = True + res = train_test_split(train_indices, **qrf_params) - # Based on the extrapolation bounds, get the training and test data. - # Training data excludes extrapolated regions; test data includes them. - X_train, y_train = get_train_Xy(X, y, extrap_min_idx, extrap_max_idx) - X_test = get_test_X(X, bounds) - - qrf = RandomForestQuantileRegressor(max_samples_leaf=None, min_samples_leaf=10, random_state=0) - qrf.fit(np.expand_dims(X_train, axis=-1), y_train) - - # Get predictions at 95% prediction intervals and median. - y_pred = qrf.predict(X_test, quantiles=[0.025, 0.5, 0.975]) + # Get coverages on extrapolated samples. + args = (train_indices, ~train_indices, y[train_indices], quantiles[1] - quantiles[0]) + cov_qrf = get_coverage_qrf(res["qmat"], *args) + cov_xtr = get_coverage_xtr(res["bounds_list"], *args) df = pd.DataFrame( { - "X_true": X, - "y_func": func(X), + "X_true": X.squeeze(), + "y_func": func(X.squeeze()), "y_true": y, - "y_pred": y_pred[:, 1], - "y_pred_low": y_pred[:, 0], - "y_pred_upp": y_pred[:, 2], + "y_pred": res["qmat"][:, 2], + "y_pred_low": res["qmat"][:, 0], + "y_pred_upp": res["qmat"][:, 1], + "bb_low": np.max(res["bounds_list"][0][:, :, 0], axis=1), + "bb_upp": np.min(res["bounds_list"][1][:, :, 1], axis=1), + "bb_mid": np.median(res["bounds_list"][2][:, :, :2], axis=(1, 2)), + "train": res["train_indices"], "test_left": [True] * extrap_min_idx + [False] * (len(y) - extrap_min_idx), "test_right": [False] * extrap_max_idx + [True] * (len(y) - extrap_max_idx), + "cov_qrf": cov_qrf, + "cov_xtr": cov_xtr, } ) @@ -132,17 +446,17 @@ accurately predict values outside of those observed in the training set. tooltip=tooltip_true, ) - line_pred = base.mark_line().encode( - x=alt.X("X_true:Q", scale=x_scale, title=""), + line_pred = base.mark_line(clip=True).encode( + x=alt.X("X_true:Q", title="", scale=x_scale), y=alt.Y("y_pred:Q", scale=y_scale), color=alt.condition(alt.datum["extrapolate"], alt.value("red"), alt.value("#006aff")), tooltip=tooltip_pred, ) - bar_pred = base.mark_bar(width=2).encode( - x=alt.X("X_true:Q", scale=x_scale, title=""), - y=alt.Y("y_pred_low:Q", scale=y_scale, title=""), - y2=alt.Y2("y_pred_upp:Q", title=None), + bar_pred = base.mark_bar(clip=True, width=2).encode( + x=alt.X("X_true"), + y=alt.Y("y_pred_low"), + y2=alt.Y2("y_pred_upp"), color=alt.condition(alt.datum["extrapolate"], alt.value("red"), alt.value("#e0f2ff")), opacity=alt.condition(alt.datum["extrapolate"], alt.value(0.05), alt.value(0.8)), tooltip=tooltip_pred, @@ -150,6 +464,25 @@ accurately predict values outside of those observed in the training set. chart = bar_pred + points_true + line_true + line_pred + if "coverage" in df.columns: + text_coverage = ( + base.transform_aggregate(coverage="mean(coverage)") + .transform_calculate( + coverage_text=( + f"'Extrapolated Coverage: '" + f" + format({alt.datum['coverage'] * 100}, '.1f') + '%'" + f" + ' (target = {(quantiles[1] - quantiles[0]) * 100}%)'" + ) + ) + .mark_text(align="left", baseline="top") + .encode( + x=alt.value(5), + y=alt.value(5), + text=alt.Text("coverage_text:N"), + ) + ) + chart += text_coverage + if legend: # For desired legend ordering. data = { @@ -171,27 +504,42 @@ accurately predict values outside of those observed in the training set. chart = chart.resolve_scale(color="independent") chart = chart.properties(height=200, width=300, title=title) - return chart - kwargs = { - "x_domain": [int(np.min(df["X_true"])), int(np.max(df["X_true"]))], - "y_domain": [int(np.min(df["y_true"])), int(np.max(df["y_true"]))], - } + kwargs = {"x_domain": [0, 15], "y_domain": [-15, 20]} + xtra_mapper = {"bb_mid": "y_pred", "bb_low": "y_pred_low", "bb_upp": "y_pred_upp"} - chart1 = plot_extrapolations( - df.query("~(test_left | test_right)"), title="Prediction Intervals on Training Data", **kwargs + chart1 = alt.layer( + plot_extrapolations( + df.query("~(test_left | test_right)").assign(**{"coverage": lambda x: x["cov_qrf"]}), + title="Extrapolation with Standard QRF", + **kwargs, + ).resolve_scale(color="independent"), + plot_extrapolations(df.query("test_left").assign(extrapolate=True), **kwargs), + plot_extrapolations(df.query("test_right").assign(extrapolate=True), **kwargs), ) chart2 = alt.layer( plot_extrapolations( - df.query("~(test_left | test_right)"), - title="Prediction Intervals with Extrapolated Values", + df.query("~(test_left | test_right)").assign(**{"coverage": lambda x: x["cov_xtr"]}), + title="Extrapolation with Xtrapolation Procedure", legend=True, **kwargs, ).resolve_scale(color="independent"), - plot_extrapolations(df.query("test_left").assign(extrapolate=True), **kwargs), - plot_extrapolations(df.query("test_right").assign(extrapolate=True), **kwargs), + plot_extrapolations( + df.query("test_left") + .assign(extrapolate=True) + .drop(columns=["y_pred", "y_pred_low", "y_pred_upp"]) + .rename(xtra_mapper, axis="columns"), + **kwargs, + ), + plot_extrapolations( + df.query("test_right") + .assign(extrapolate=True) + .drop(columns=["y_pred", "y_pred_low", "y_pred_upp"]) + .rename(xtra_mapper, axis="columns"), + **kwargs, + ), ) chart = chart1 | chart2 chart @@ -200,6 +548,8 @@ accurately predict values outside of those observed in the training set. .. code:: python + import math + import altair as alt import numpy as np import pandas as pd @@ -208,13 +558,15 @@ accurately predict values outside of those observed in the training set. np.random.seed(0) - n_samples = 1000 + n_samples = 500 bounds = [0, 15] extrap_frac = 0.25 - func = lambda x: x * np.sin(x) func_str = "f(x) = x sin(x)" + quantiles = [0.025, 0.975, 0.5] + qrf_params = {"max_samples_leaf": None, "min_samples_leaf": 4, "random_state": 0} + def make_func_Xy(func, bounds, n_samples): x = np.linspace(bounds[0], bounds[1], n_samples) @@ -222,49 +574,354 @@ accurately predict values outside of those observed in the training set. std = 0.01 + np.abs(x - 5.0) / 5.0 noise = np.random.normal(scale=std) y = f + noise - return x, y + return np.atleast_2d(x).T, y + + + class Xtrapolation: + """Xtrapolation procedure. + + Performs extrapolation-aware nonparametric statistical inference based on + an existing nonparametric estimate. Adapted from the Python code [1] for + the Xtrapolation procedure introduced in [2]. + + The procedure specifically applies a QRF for generating local polynomials + to estimate derivatives in a single dimension. For multi-dimensional + problems, using the original implementation is strongly encouraged. + + References + ---------- + .. [1] https://github.com/NiklasPfister/ExtrapolationAware-Inference + .. [2] N. Pfister and P. Bühlmann, "Extrapolation-Aware Nonparametric + Statistical Inference", arXiv preprint, 2024. + https://arxiv.org/abs/2402.09758 + """ + + def __init__(self, orders=np.array([1])): + self.orders_ = orders + self.max_order_ = np.max(orders) + + @staticmethod + def _penalized_locpol(fval, v, X, weights, degree, pen=0, penalize_intercept=False): + v = v.reshape(-1, 1) + n = X.shape[0] + dd = degree + 1 + if penalize_intercept: + pen_list = list(range(0, dd)) + else: + pen_list = list(range(1, dd)) + + # Construct design matrices. + DDmat = np.zeros((n * dd, n * dd)) + DYmat = np.zeros((n * dd, 1)) + for i in range(n): + Wi = np.sqrt(weights[i, :].reshape(-1, 1)) + + # Construct DDmat (block-diagonal). + x0v = X[i, :].dot(v) + Di = np.tile((X.dot(v) - x0v).reshape(-1, 1), dd) ** np.arange(dd) * Wi + DDmat[(i * dd) : ((i + 1) * dd), (i * dd) : ((i + 1) * dd)] = (Di.T).dot(Di) + + # Construct DYmat. + DYmat[(i * dd) : ((i + 1) * dd), :] = (Di.T).dot((fval.reshape(-1, 1)) * Wi) + + Z = np.zeros((dd, dd)) + for kk in pen_list: + Z[kk, kk] = math.factorial(kk) + PP = np.kron(np.diag(np.sum(weights, axis=1)) - weights, Z) + penmat = pen * (PP.T).dot(PP) + B = np.linalg.solve(DDmat + penmat, DYmat) + coefs = B.reshape(n, -1) + + # Extract derivatives from coefficients. + deriv_mat = coefs * np.array([math.factorial(k) for k in range(degree + 1)]) + return deriv_mat + + @staticmethod + def _get_tree_weight_matrix(X, Y, X_eval=None, n_trees=100, **kwargs): + """Fit forest and extract weights. + + This implementation extracts the weight matrix from a list of quantile + random forests, each with a single tree fitted on non-bootstrapped + samples. This allows for controlling the bootstrap selection for each + tree and summing the weight matrices across all of the trees. + """ + if "n_estimators" in kwargs: + n_trees = kwargs["n_estimators"] + kwargs["n_estimators"] = 1 + if "random_state" in kwargs: + del kwargs["random_state"] + kwargs["bootstrap"] = False + + trees = [RandomForestQuantileRegressor(random_state=i, **kwargs) for i in range(n_trees)] + + n = X.shape[0] + nn = 0 + if X_eval is not None: + nn = X_eval.shape[0] + X = np.r_[X, X_eval] + weight_mat = np.zeros((n + nn, n + nn)) + + s = 0.5 + bn = int(n * s) + + for tree in trees: + # Draw bootstrap sample. + boot_sample = np.random.choice(np.arange(n), bn, replace=False) + split1 = boot_sample[: int(bn / 2)] + split2 = np.concatenate([boot_sample[int(bn / 2) :], np.arange(nn) + n]) + + # Fit tree. + tree.fit(X[split1, :], Y[split1].flatten()) + + # Extract tree weight matrix. + y_train_leaves = tree._get_y_train_leaves(X[split2, :], 1) + nrows = X[split2, :].shape[0] + matrix = np.zeros((nrows, nrows)) + for leaf in y_train_leaves[0]: + indices = leaf[0] + indices = indices[indices != 0] - 1 + if len(indices) > 0: + matrix[np.ix_(indices, indices)] = 1 + weight_mat[np.ix_(split2, split2)] += matrix + + # Normalize weights (rows correspond to weights - non-symmetric). + weight_mat /= weight_mat.sum(axis=1)[:, None] + + return weight_mat + + def fit_weights(self, X, fval, x0=None, train=False, **kwargs): + """Compute random forest weights for derivative estimation.""" + n, d = X.shape + fval = fval.flatten() + + if train: + d_xtra = d + xtra_features = list(range(d)) + weights = [None] * d_xtra + for jj, var in enumerate(xtra_features): + var_order = list(range(d)) + var_order = np.array([var] + var_order[:var] + var_order[var + 1 :]) + weights[jj] = self._get_tree_weight_matrix(X[:, var_order], fval, x0, **kwargs) + else: + weights = self._get_tree_weight_matrix(X, fval, x0, **kwargs)[n:, :n] + + return weights + + def fit_derivatives(self, X, fval, pen=0.1, **kwargs): + """Estimate derivatives.""" + n, d = X.shape + fval = fval.flatten() + + # Fit weights for local polynomial. + weights = self.fit_weights(X, fval, train=True, **kwargs) + + # Estimate derivatives with local polynomial. + derivatives = np.zeros((self.max_order_ + 1, n, d)) + Xtilde = X[:, list(range(d))] + + # Fit local polynomial. + for jj in range(d): + vv = np.zeros((d, 1)) + vv[jj] = 1 + tmp = self._penalized_locpol( + fval, + vv, + Xtilde, + weights[jj], + degree=self.max_order_ + 1, + pen=pen, + penalize_intercept=False, + ) + for kk in range(self.max_order_ + 1): + derivatives[kk, :, jj] = fval if kk == 0 else tmp[:, kk] + + return derivatives + + def prediction_bounds(self, X, fval, x0, nn=50, **kwargs): + """Compute extrapolation bounds.""" + n, d = X.shape + fval = fval.flatten() + if len(x0.shape) == 1: + x0 = x0.reshape(-1, 1) + n0 = x0.shape[0] + xtra_features = list(range(d)) + + # Fit derivatives. + derivatives = self.fit_derivatives(X, fval, **kwargs) + + # Determine weighting for extrapolation points (using rotation). + mu = derivatives[1].mean(axis=0) + _, D, Vt = np.linalg.svd(derivatives[1] - mu[None, :]) + TT = (Vt.T) * D[None, :] + Xtilde = X[:, xtra_features].dot(TT) + x0tilde = x0[:, xtra_features].dot(TT) + + # Find closest points between rotated points (Euclidean). + weight_x0 = np.zeros((n0, n)) + for ii in range(n0): + xinds = np.argsort(np.sum((x0tilde[None, ii, :] - Xtilde) ** 2, axis=1))[:nn] + weight_x0[ii, xinds] = 1 / nn + + # Precompute factorials. + order_factorials = np.empty(self.max_order_ + 1) + for oo in range(self.max_order_ + 1): + order_factorials[oo] = math.factorial(oo) + + # Iterate over all extrapolation points and average/intersect. + bounds = np.zeros((n0, len(self.orders_), 3)) + for ll, xpt in enumerate(x0): + xinds = np.where(weight_x0[ll, :] != 0)[0] + + # Number of anchor points to check. + f_lower = np.zeros((len(xinds), len(self.orders_))) + f_upper = np.zeros((len(xinds), len(self.orders_))) + f_median = np.zeros((len(xinds), len(self.orders_))) + for ii, xind in enumerate(xinds): + xx = X[xind, :].reshape(1, -1) + vv = (xpt - xx)[:, xtra_features] + vv_norm = np.sqrt(np.sum(vv**2)) + + # Compute directional derivatives. + deriv_mat = np.zeros((n, self.max_order_ + 1)) + deriv_mat[:, 0] = derivatives[0, :, :].mean(axis=1) + if vv_norm > np.finfo(float).eps: + vv_direction = np.array(vv / vv_norm).reshape(-1, 1) + for kk in range(1, self.max_order_ + 1): + deriv_mat[:, kk] = derivatives[kk, :, :].dot(vv_direction**kk).flatten() + + # Select bounds. + deriv_min = np.quantile(deriv_mat, 0, axis=0) + deriv_max = np.quantile(deriv_mat, 1, axis=0) + deriv_median = np.quantile(deriv_mat, 0.5, axis=0) + + # Estimate extrapolation bounds. + mterm = 0 + kk = 0 + for oo in range(self.max_order_ + 1): + if oo in self.orders_: + lo_bdd = deriv_min[oo] * (vv_norm**oo) / order_factorials[oo] + up_bdd = deriv_max[oo] * (vv_norm**oo) / order_factorials[oo] + median_deriv = deriv_median[oo] * (vv_norm**oo) / order_factorials[oo] + f_lower[ii, kk] = mterm + lo_bdd + f_upper[ii, kk] = mterm + up_bdd + f_median[ii, kk] = mterm + median_deriv + kk += 1 + mterm += deriv_mat[xind, oo] * (vv_norm**oo) / order_factorials[oo] + + # Combine bounds over x-indices. + ww = (weight_x0[ll, xinds] / np.sum(weight_x0[ll, :]))[:, None] + f_median = np.sum(f_median * ww, axis=0) + + # Aggregate by optimal-average. + f_lower = np.max(f_lower, axis=0) + f_upper = np.min(f_upper, axis=0) + ind = f_upper < f_lower + average = (f_upper + f_lower) / 2 + f_lower[ind] = average[ind] + f_upper[ind] = average[ind] + + bounds[ll, :, 0] = f_lower + bounds[ll, :, 1] = f_upper + bounds[ll, :, 2] = f_median + + return bounds + + + def train_test_split(train_indices, **kwargs): + """Fit model on training samples and extrapolate on test samples.""" + X_train = X[train_indices, :] + y_train = y[train_indices] + + # Run quantile regression (with forests). + qrf = RandomForestQuantileRegressor(**kwargs) + qrf.fit(X_train, y_train) + qmat = qrf.predict(X, quantiles=quantiles) + + # Xtrapolation. + bounds_list = [None] * len(quantiles) + for i in range(len(quantiles)): + # Run Xtrapolation on quantile. + xtra = Xtrapolation() + bounds_list[i] = xtra.prediction_bounds(X_train, qmat[train_indices, i], X, **kwargs) + + return { + "train_indices": train_indices, + "quantiles": quantiles, + "qmat": qmat, + "bounds_list": bounds_list, + } + + + def prob_randomized_pi(qmat, y, coverage): + """Calculate calibration probability""" + alpha_included = np.mean((qmat[:, 0] <= y) & (y <= qmat[:, 1])) + alpha_excluded = np.mean((qmat[:, 0] < y) & (y < qmat[:, 1])) + if coverage <= alpha_excluded: + prob_si = 1 + elif coverage >= alpha_included: + prob_si = 0 + else: + prob_si = (coverage - alpha_included) / (alpha_excluded - alpha_included) + return prob_si + + def randomized_pi(qmat, prob_si, y): + """Calculate coverage.""" + si_index = np.random.choice([False, True], len(y), replace=True, p=[prob_si, 1 - prob_si]) + included = (qmat[:, 0] < y) & (y < qmat[:, 1]) + boundary = (qmat[:, 0] == y) | (qmat[:, 1] == y) + return included | (boundary & si_index) - def get_train_Xy(X, y, min_idx, max_idx): - X_train = X[min_idx:max_idx] - y_train = y[min_idx:max_idx] - return X_train, y_train + def get_coverage_qrf(qmat, train_indices, test_indices, y_train, level): + """Calculate extrapolation coverage for regular quantile forest.""" + prob_si = prob_randomized_pi(qmat[train_indices, :], y_train, level) + qrf = randomized_pi(qmat, prob_si, y) + return np.mean(qrf[test_indices]) - def get_test_X(X, bounds): - n_samples = len(X) - X_test = np.atleast_2d(np.linspace(*bounds, n_samples)).T - return X_test + + def get_coverage_xtr(bounds_list, train_indices, test_indices, y_train, level): + """Calculate extrapolation coverage for Xtrapolation.""" + bb_low = np.max(bounds_list[0][:, :, 0], axis=1) + bb_upp = np.min(bounds_list[1][:, :, 1], axis=1) + bb_low_train, bb_upp_train = bb_low[train_indices], bb_upp[train_indices] + prob_si = prob_randomized_pi(np.c_[bb_low_train, bb_upp_train], y_train, level) + xtra = randomized_pi(np.c_[bb_low, bb_upp], prob_si, y) + return np.mean(xtra[test_indices]) # Create the full dataset. X, y = make_func_Xy(func, bounds, n_samples) - # Calculate the extrapolation bounds. + # Fit and extrapolate based on train-test split (depending on X). extrap_min_idx = int(n_samples * (extrap_frac / 2)) extrap_max_idx = int(n_samples - (n_samples * (extrap_frac / 2))) + sort_X = np.argsort(X.squeeze()) + train_indices = np.repeat(False, len(y)) + train_indices[sort_X[extrap_min_idx] : sort_X[extrap_max_idx]] = True + res = train_test_split(train_indices, **qrf_params) - # Based on the extrapolation bounds, get the training and test data. - # Training data excludes extrapolated regions; test data includes them. - X_train, y_train = get_train_Xy(X, y, extrap_min_idx, extrap_max_idx) - X_test = get_test_X(X, bounds) - - qrf = RandomForestQuantileRegressor(max_samples_leaf=None, min_samples_leaf=10, random_state=0) - qrf.fit(np.expand_dims(X_train, axis=-1), y_train) - - # Get predictions at 95% prediction intervals and median. - y_pred = qrf.predict(X_test, quantiles=[0.025, 0.5, 0.975]) + # Get coverages on extrapolated samples. + args = (train_indices, ~train_indices, y[train_indices], quantiles[1] - quantiles[0]) + cov_qrf = get_coverage_qrf(res["qmat"], *args) + cov_xtr = get_coverage_xtr(res["bounds_list"], *args) df = pd.DataFrame( { - "X_true": X, - "y_func": func(X), + "X_true": X.squeeze(), + "y_func": func(X.squeeze()), "y_true": y, - "y_pred": y_pred[:, 1], - "y_pred_low": y_pred[:, 0], - "y_pred_upp": y_pred[:, 2], + "y_pred": res["qmat"][:, 2], + "y_pred_low": res["qmat"][:, 0], + "y_pred_upp": res["qmat"][:, 1], + "bb_low": np.max(res["bounds_list"][0][:, :, 0], axis=1), + "bb_upp": np.min(res["bounds_list"][1][:, :, 1], axis=1), + "bb_mid": np.median(res["bounds_list"][2][:, :, :2], axis=(1, 2)), + "train": res["train_indices"], "test_left": [True] * extrap_min_idx + [False] * (len(y) - extrap_min_idx), "test_right": [False] * extrap_max_idx + [True] * (len(y) - extrap_max_idx), + "cov_qrf": cov_qrf, + "cov_xtr": cov_xtr, } ) @@ -310,17 +967,17 @@ accurately predict values outside of those observed in the training set. tooltip=tooltip_true, ) - line_pred = base.mark_line().encode( - x=alt.X("X_true:Q", scale=x_scale, title=""), + line_pred = base.mark_line(clip=True).encode( + x=alt.X("X_true:Q", title="", scale=x_scale), y=alt.Y("y_pred:Q", scale=y_scale), color=alt.condition(alt.datum["extrapolate"], alt.value("red"), alt.value("#006aff")), tooltip=tooltip_pred, ) - bar_pred = base.mark_bar(width=2).encode( - x=alt.X("X_true:Q", scale=x_scale, title=""), - y=alt.Y("y_pred_low:Q", scale=y_scale, title=""), - y2=alt.Y2("y_pred_upp:Q", title=None), + bar_pred = base.mark_bar(clip=True, width=2).encode( + x=alt.X("X_true"), + y=alt.Y("y_pred_low"), + y2=alt.Y2("y_pred_upp"), color=alt.condition(alt.datum["extrapolate"], alt.value("red"), alt.value("#e0f2ff")), opacity=alt.condition(alt.datum["extrapolate"], alt.value(0.05), alt.value(0.8)), tooltip=tooltip_pred, @@ -328,6 +985,25 @@ accurately predict values outside of those observed in the training set. chart = bar_pred + points_true + line_true + line_pred + if "coverage" in df.columns: + text_coverage = ( + base.transform_aggregate(coverage="mean(coverage)") + .transform_calculate( + coverage_text=( + f"'Extrapolated Coverage: '" + f" + format({alt.datum['coverage'] * 100}, '.1f') + '%'" + f" + ' (target = {(quantiles[1] - quantiles[0]) * 100}%)'" + ) + ) + .mark_text(align="left", baseline="top") + .encode( + x=alt.value(5), + y=alt.value(5), + text=alt.Text("coverage_text:N"), + ) + ) + chart += text_coverage + if legend: # For desired legend ordering. data = { @@ -349,27 +1025,42 @@ accurately predict values outside of those observed in the training set. chart = chart.resolve_scale(color="independent") chart = chart.properties(height=200, width=300, title=title) - return chart - kwargs = { - "x_domain": [int(np.min(df["X_true"])), int(np.max(df["X_true"]))], - "y_domain": [int(np.min(df["y_true"])), int(np.max(df["y_true"]))], - } + kwargs = {"x_domain": [0, 15], "y_domain": [-15, 20]} + xtra_mapper = {"bb_mid": "y_pred", "bb_low": "y_pred_low", "bb_upp": "y_pred_upp"} - chart1 = plot_extrapolations( - df.query("~(test_left | test_right)"), title="Prediction Intervals on Training Data", **kwargs + chart1 = alt.layer( + plot_extrapolations( + df.query("~(test_left | test_right)").assign(**{"coverage": lambda x: x["cov_qrf"]}), + title="Extrapolation with Standard QRF", + **kwargs, + ).resolve_scale(color="independent"), + plot_extrapolations(df.query("test_left").assign(extrapolate=True), **kwargs), + plot_extrapolations(df.query("test_right").assign(extrapolate=True), **kwargs), ) chart2 = alt.layer( plot_extrapolations( - df.query("~(test_left | test_right)"), - title="Prediction Intervals with Extrapolated Values", + df.query("~(test_left | test_right)").assign(**{"coverage": lambda x: x["cov_xtr"]}), + title="Extrapolation with Xtrapolation Procedure", legend=True, **kwargs, ).resolve_scale(color="independent"), - plot_extrapolations(df.query("test_left").assign(extrapolate=True), **kwargs), - plot_extrapolations(df.query("test_right").assign(extrapolate=True), **kwargs), + plot_extrapolations( + df.query("test_left") + .assign(extrapolate=True) + .drop(columns=["y_pred", "y_pred_low", "y_pred_upp"]) + .rename(xtra_mapper, axis="columns"), + **kwargs, + ), + plot_extrapolations( + df.query("test_right") + .assign(extrapolate=True) + .drop(columns=["y_pred", "y_pred_low", "y_pred_upp"]) + .rename(xtra_mapper, axis="columns"), + **kwargs, + ), ) chart = chart1 | chart2 chart diff --git a/_sources/gallery/plot_quantile_weighting.rst.txt b/_sources/gallery/plot_quantile_weighting.rst.txt deleted file mode 100644 index cd89523..0000000 --- a/_sources/gallery/plot_quantile_weighting.rst.txt +++ /dev/null @@ -1,359 +0,0 @@ - -:orphan: -:html_theme.sidebar_secondary.remove: - -.. This document is auto-generated by the example-gallery extension. Do not modify directly. - -.. _gallery_plot_quantile_weighting: - - -Weighted vs. Unweighted Quantile Runtimes -========================================= - -An example comparison of the prediction runtime when using a quantile -regression forest with weighted and unweighted quantiles to compute the -predicted output values. While weighted and unweighted quantiles produce -identical outputs, the relative runtime of the methods depends on the number -of training samples and the total number of leaf samples across all trees used -to calculate the quantiles. A standard random forest regressor is included for -comparison. - - -.. altair-plot:: - :remove-code: - - - - import time - from contextlib import contextmanager - - import altair as alt - import pandas as pd - from sklearn import datasets - from sklearn.ensemble import RandomForestRegressor - - from quantile_forest import RandomForestQuantileRegressor - - samples = [100, 175, 250, 325, 500] - estimators = [10, 25, 50, 75, 100] - repeats = 3 - - - @contextmanager - def timing(): - t0 = time.process_time() - yield lambda: (t1 - t0) - t1 = time.process_time() - - - legend = { - "RF": "#f2a619", - "QRF Weighted Quantile": "#006aff", - "QRF Unweighted Quantile": "#001751", - } - - dataset = datasets.make_regression( - n_samples=max(samples), n_features=3, n_targets=1, random_state=0 - ) - - # Populate data with timing results over samples and estimators. - data = {"name": [], "n_samples": [], "n_estimators": [], "iteration": [], "runtime": []} - for n_samples in samples: - X = dataset[0][:n_samples, :] - y = dataset[1][:n_samples] - for n_estimators in estimators: - for repeat in range(repeats): - rf = RandomForestRegressor( - n_estimators=n_estimators, - max_depth=7, - random_state=0, - ) - qrf = RandomForestQuantileRegressor( - n_estimators=n_estimators, - max_depth=7, - max_samples_leaf=None, - random_state=0, - ) - - rf.fit(X, y) - qrf.fit(X, y) - - with timing() as rf_time: - _ = rf.predict(X) - with timing() as qrf_weighted_time: - _ = qrf.predict(X, quantiles=0.5, weighted_quantile=True) - with timing() as qrf_unweighted_time: - _ = qrf.predict(X, quantiles=0.5, weighted_quantile=False) - - timings = [rf_time(), qrf_weighted_time(), qrf_unweighted_time()] - - for name, runtime in zip(legend.keys(), timings): - runtime *= 1000 # convert from milliseconds to seconds - - data["name"].extend([name]) - data["n_samples"].extend([n_samples]) - data["n_estimators"].extend([n_estimators]) - data["iteration"].extend([repeat]) - data["runtime"].extend([runtime]) - - df = ( - pd.DataFrame(data) - .groupby(["name", "n_samples", "n_estimators"]) - .agg({"runtime": ["mean", "std"]}) - .pipe(lambda x: x.set_axis(["_".join(map(str, col)) for col in x.columns], axis=1)) - .reset_index() - .assign( - **{ - "ymean": lambda x: x["runtime_mean"], - "ystd": lambda x: x["runtime_std"], - "ymin": lambda x: x["ymean"] - (x["ystd"] / 2), - "ymax": lambda x: x["ymean"] + (x["ystd"] / 2), - } - ) - .drop(columns=["runtime_mean", "runtime_std"]) - ) - - - def plot_timings_by_factor(df, legend, constant, factor, factor_title): - max_value = df[constant].max() - - click = alt.selection_point(fields=["name"], bind="legend") - - color = alt.condition( - click, - alt.Color( - "name:N", - legend=alt.Legend(symbolOpacity=1), - sort=list(legend.keys()), - title=None, - ), - alt.value("lightgray"), - ) - - base = ( - alt.Chart(df) - .transform_filter(alt.datum[constant] == max_value) # hold this factor constant - .transform_joinaggregate(min_runtime="min(ymin):Q", groupby=[constant]) - .transform_calculate(ymean_norm="datum.ymean / datum.min_runtime") - .transform_calculate(ystd_norm="datum.ystd / datum.min_runtime") - .transform_calculate(ymin_norm="datum.ymin / datum.min_runtime") - .transform_calculate(ymax_norm="datum.ymax / datum.min_runtime") - ) - - line = base.mark_line().encode( - x=alt.X(f"{factor}:Q", scale=alt.Scale(nice=False), title=factor_title), - y=alt.Y( - "ymean_norm:Q", - scale=alt.Scale(zero=False), - title="Prediction Runtime (normalized)", - ), - color=color, - ) - - area = base.mark_area(opacity=0.1).encode( - x=alt.X(f"{factor}:Q"), - y=alt.Y("ymin_norm:Q"), - y2=alt.Y2("ymax_norm:Q"), - color=color, - tooltip=[ - alt.Tooltip("name:N", title="Estimator Name"), - alt.Tooltip(f"{factor}:Q", format=",d", title=factor_title), - alt.Tooltip("ymean_norm:Q", format=",.3f", title="Average Runtime"), - alt.Tooltip("ymin_norm:Q", format=",.3f", title="Minimum Runtime"), - alt.Tooltip("ymax_norm:Q", format=",.3f", title="Maximum Runtime"), - ], - ) - - text = ( - base.transform_aggregate(coverage="mean(constant)", groupby=[constant]) - .transform_calculate(text=(f"'{constant} = ' + format({alt.datum[constant]}, ',d')")) - .mark_text(align="left", baseline="top") - .encode( - x=alt.value(5), - y=alt.value(5), - text=alt.Text("text:N"), - ) - ) - - chart = ( - (line + area + text) - .add_params(click) - .properties(height=200, width=300, title=f"Runtime by {factor_title}") - ) - - return chart - - - chart1 = plot_timings_by_factor(df, legend, "n_samples", "n_estimators", "Number of Estimators") - chart2 = plot_timings_by_factor(df, legend, "n_estimators", "n_samples", "Number of Samples") - chart = (chart1 | chart2).configure_range(category=alt.RangeScheme(list(legend.values()))) - chart - - -.. code:: python - - - import time - from contextlib import contextmanager - - import altair as alt - import pandas as pd - from sklearn import datasets - from sklearn.ensemble import RandomForestRegressor - - from quantile_forest import RandomForestQuantileRegressor - - samples = [100, 175, 250, 325, 500] - estimators = [10, 25, 50, 75, 100] - repeats = 3 - - - @contextmanager - def timing(): - t0 = time.process_time() - yield lambda: (t1 - t0) - t1 = time.process_time() - - - legend = { - "RF": "#f2a619", - "QRF Weighted Quantile": "#006aff", - "QRF Unweighted Quantile": "#001751", - } - - dataset = datasets.make_regression( - n_samples=max(samples), n_features=3, n_targets=1, random_state=0 - ) - - # Populate data with timing results over samples and estimators. - data = {"name": [], "n_samples": [], "n_estimators": [], "iteration": [], "runtime": []} - for n_samples in samples: - X = dataset[0][:n_samples, :] - y = dataset[1][:n_samples] - for n_estimators in estimators: - for repeat in range(repeats): - rf = RandomForestRegressor( - n_estimators=n_estimators, - max_depth=7, - random_state=0, - ) - qrf = RandomForestQuantileRegressor( - n_estimators=n_estimators, - max_depth=7, - max_samples_leaf=None, - random_state=0, - ) - - rf.fit(X, y) - qrf.fit(X, y) - - with timing() as rf_time: - _ = rf.predict(X) - with timing() as qrf_weighted_time: - _ = qrf.predict(X, quantiles=0.5, weighted_quantile=True) - with timing() as qrf_unweighted_time: - _ = qrf.predict(X, quantiles=0.5, weighted_quantile=False) - - timings = [rf_time(), qrf_weighted_time(), qrf_unweighted_time()] - - for name, runtime in zip(legend.keys(), timings): - runtime *= 1000 # convert from milliseconds to seconds - - data["name"].extend([name]) - data["n_samples"].extend([n_samples]) - data["n_estimators"].extend([n_estimators]) - data["iteration"].extend([repeat]) - data["runtime"].extend([runtime]) - - df = ( - pd.DataFrame(data) - .groupby(["name", "n_samples", "n_estimators"]) - .agg({"runtime": ["mean", "std"]}) - .pipe(lambda x: x.set_axis(["_".join(map(str, col)) for col in x.columns], axis=1)) - .reset_index() - .assign( - **{ - "ymean": lambda x: x["runtime_mean"], - "ystd": lambda x: x["runtime_std"], - "ymin": lambda x: x["ymean"] - (x["ystd"] / 2), - "ymax": lambda x: x["ymean"] + (x["ystd"] / 2), - } - ) - .drop(columns=["runtime_mean", "runtime_std"]) - ) - - - def plot_timings_by_factor(df, legend, constant, factor, factor_title): - max_value = df[constant].max() - - click = alt.selection_point(fields=["name"], bind="legend") - - color = alt.condition( - click, - alt.Color( - "name:N", - legend=alt.Legend(symbolOpacity=1), - sort=list(legend.keys()), - title=None, - ), - alt.value("lightgray"), - ) - - base = ( - alt.Chart(df) - .transform_filter(alt.datum[constant] == max_value) # hold this factor constant - .transform_joinaggregate(min_runtime="min(ymin):Q", groupby=[constant]) - .transform_calculate(ymean_norm="datum.ymean / datum.min_runtime") - .transform_calculate(ystd_norm="datum.ystd / datum.min_runtime") - .transform_calculate(ymin_norm="datum.ymin / datum.min_runtime") - .transform_calculate(ymax_norm="datum.ymax / datum.min_runtime") - ) - - line = base.mark_line().encode( - x=alt.X(f"{factor}:Q", scale=alt.Scale(nice=False), title=factor_title), - y=alt.Y( - "ymean_norm:Q", - scale=alt.Scale(zero=False), - title="Prediction Runtime (normalized)", - ), - color=color, - ) - - area = base.mark_area(opacity=0.1).encode( - x=alt.X(f"{factor}:Q"), - y=alt.Y("ymin_norm:Q"), - y2=alt.Y2("ymax_norm:Q"), - color=color, - tooltip=[ - alt.Tooltip("name:N", title="Estimator Name"), - alt.Tooltip(f"{factor}:Q", format=",d", title=factor_title), - alt.Tooltip("ymean_norm:Q", format=",.3f", title="Average Runtime"), - alt.Tooltip("ymin_norm:Q", format=",.3f", title="Minimum Runtime"), - alt.Tooltip("ymax_norm:Q", format=",.3f", title="Maximum Runtime"), - ], - ) - - text = ( - base.transform_aggregate(coverage="mean(constant)", groupby=[constant]) - .transform_calculate(text=(f"'{constant} = ' + format({alt.datum[constant]}, ',d')")) - .mark_text(align="left", baseline="top") - .encode( - x=alt.value(5), - y=alt.value(5), - text=alt.Text("text:N"), - ) - ) - - chart = ( - (line + area + text) - .add_params(click) - .properties(height=200, width=300, title=f"Runtime by {factor_title}") - ) - - return chart - - - chart1 = plot_timings_by_factor(df, legend, "n_samples", "n_estimators", "Number of Estimators") - chart2 = plot_timings_by_factor(df, legend, "n_estimators", "n_samples", "Number of Samples") - chart = (chart1 | chart2).configure_range(category=alt.RangeScheme(list(legend.values()))) - chart diff --git a/_sources/user_guide/fit_predict.rst.txt b/_sources/user_guide/fit_predict.rst.txt index bbaed97..687038e 100644 --- a/_sources/user_guide/fit_predict.rst.txt +++ b/_sources/user_guide/fit_predict.rst.txt @@ -140,3 +140,8 @@ The predictions of a standard random forest can also be recovered from a quantil >>> y_pred_qrf = qrf.predict(X_test, **kwargs) >>> np.allclose(y_pred_rf, y_pred_qrf) True + +Predicting with User-Specified Functions +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +While a QRF is designed to estimate quantiles from the empirical distribution calculated for each sample, in many cases it may be useful to use the empirical distribution to calculate other quantities of interest. For more details, see :ref:`gallery_plot_predict_custom`. diff --git a/_static/_image_hashes.json b/_static/_image_hashes.json index a38d695..6977a59 100644 --- a/_static/_image_hashes.json +++ b/_static/_image_hashes.json @@ -1 +1 @@ -{"plot_quantile_interpolation.png": "66b159ef696d268916cfe5e7569cdaec", "plot_quantile_extrapolation.png": "4d6b6eeaf92d2184349ab64ae0b5b940", "plot_quantile_multioutput.png": "beca1cd595c390681c067c879aa556c9", "plot_quantile_example.png": "0bb47ee3c7146a61ae1bc808328f59b9", "plot_quantile_conformalized.png": "dadc075cb7747a31f4af804dc7caf752", "plot_quantile_intervals.png": "fb626d604bb32d19534d32c94c024e4c", "plot_quantile_vs_standard.png": "cfd72ec618746421589c8e18ffde31e8", "plot_quantile_weighting.png": "02304471ac013a3ec6a78f6fd2c34d29"} \ No newline at end of file +{"plot_predict_custom.png": "e4925268c2e493b63423fc025c567e35", "plot_quantile_interpolation.png": "66b159ef696d268916cfe5e7569cdaec", "plot_quantile_extrapolation.png": "fa93cb2fef869563a944c6b1ad5b5f34", "plot_quantile_multioutput.png": "beca1cd595c390681c067c879aa556c9", "plot_quantile_example.png": "0bb47ee3c7146a61ae1bc808328f59b9", "plot_quantile_conformalized.png": "dadc075cb7747a31f4af804dc7caf752", "plot_quantile_intervals.png": "fb626d604bb32d19534d32c94c024e4c", "plot_quantile_vs_standard.png": "cfd72ec618746421589c8e18ffde31e8"} \ No newline at end of file diff --git a/_static/plot_predict_custom-thumb.png b/_static/plot_predict_custom-thumb.png new file mode 100644 index 0000000..18bb2ae Binary files /dev/null and b/_static/plot_predict_custom-thumb.png differ diff --git a/_static/plot_predict_custom.png b/_static/plot_predict_custom.png new file mode 100644 index 0000000..e2fcab3 Binary files /dev/null and b/_static/plot_predict_custom.png differ diff --git a/_static/plot_quantile_extrapolation-thumb.png b/_static/plot_quantile_extrapolation-thumb.png index 2eea967..47456e8 100644 Binary files a/_static/plot_quantile_extrapolation-thumb.png and b/_static/plot_quantile_extrapolation-thumb.png differ diff --git a/_static/plot_quantile_extrapolation.png b/_static/plot_quantile_extrapolation.png index 1a2b1c0..2b6071e 100644 Binary files a/_static/plot_quantile_extrapolation.png and b/_static/plot_quantile_extrapolation.png differ diff --git a/_static/plot_quantile_weighting-thumb.png b/_static/plot_quantile_weighting-thumb.png deleted file mode 100644 index 88a8adf..0000000 Binary files a/_static/plot_quantile_weighting-thumb.png and /dev/null differ diff --git a/_static/plot_quantile_weighting.png b/_static/plot_quantile_weighting.png deleted file mode 100644 index bfde97a..0000000 Binary files a/_static/plot_quantile_weighting.png and /dev/null differ diff --git a/gallery/index.html b/gallery/index.html index a2931e0..e5a0541 100644 --- a/gallery/index.html +++ b/gallery/index.html @@ -422,6 +422,17 @@

General-purpose, introductory and illustrative examples.

+ + + + Calculating User-Specified Functions with QRFs + + Quantile Regression Forests vs. Random Forests - - - - Weighted vs. Unweighted Quantile Runtimes - -
diff --git a/gallery/plot_predict_custom.html b/gallery/plot_predict_custom.html new file mode 100644 index 0000000..37f3e83 --- /dev/null +++ b/gallery/plot_predict_custom.html @@ -0,0 +1,611 @@ + + + + + + + + + + + Calculating User-Specified Functions with QRFs — quantile-forest 1.3.4 documentation + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
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quantile-forest

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Calculating User-Specified Functions with QRFs#

+

An example that demonstrates a way of extracting the empirical distribution +from a quantile regression forest (QRF) for one or more samples in order to +calculate a user-specified function of interest. While a QRF is designed to +estimate quantiles from the empirical distribution calculated for each sample, +in many cases it may be useful to use the empirical distribution to calculate +other quantities of interest. Here, we calculate the ECDF for a test sample.

+ +
+ +
from itertools import chain
+
+import altair as alt
+import numpy as np
+import pandas as pd
+import scipy as sp
+from sklearn import datasets
+from sklearn.model_selection import train_test_split
+
+from quantile_forest import RandomForestQuantileRegressor
+
+np.random.seed(0)
+
+
+def predict(reg, X, quantiles=0.5, what=None):
+    """Custom prediction method that allows user-specified function.
+
+    Parameters
+    ----------
+    reg : BaseForestQuantileRegressor
+        Quantile forest model object.
+    X : array-like, of shape (n_samples, n_features)
+        New test data.
+    quantiles : float or list of floats, default=0.5
+        Scalar or vector of quantiles for quantile prediction.
+    what : Any, optional
+        User-specified function for prediction used instead of quantiles.
+        Must return a numeric vector.
+
+    Returns
+    -------
+    Output array with the user-specified function applied (if any).
+    """
+    if what is None:
+        return reg.predict(X, quantiles=quantiles)
+
+    # Get the complete set of proximities for each sample.
+    proximities = reg.proximity_counts(X)
+
+    # Retrieve the unsorted training responses from the model (stored in sorted order).
+    reverse_sorter = np.argsort(reg.sorter_, axis=0)
+    y_train = np.empty_like(reg.forest_.y_train).T
+    for i in range(y_train.shape[1]):
+        y_train[:, i] = np.asarray(reg.forest_.y_train)[i, reverse_sorter[:, i]]
+
+    # For each sample, construct an array of the training responses used for prediction.
+    s = [np.concatenate([np.repeat(y_train[i[0]], i[1]) for i in prox]) for prox in proximities]
+
+    # Apply the user-specified function for each sample. Must return a numeric vector.
+    y_out = np.array([what(s_i) for s_i in s])
+
+    return y_out
+
+
+X, y = datasets.load_diabetes(return_X_y=True)
+X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=1, random_state=0)
+
+reg = RandomForestQuantileRegressor().fit(X_train, y_train)
+
+# Define a user-specified function; here we randomly sample 1000 values with replacement.
+func = lambda x: np.random.choice(x, size=1000)
+
+# Output array with the user-specified function applied to each sample's empirical distribution.
+y_out = predict(reg, X_test, what=func)
+
+# Calculate the ECDF from output array.
+y_ecdf = [sp.stats.ecdf(y_i).cdf for y_i in y_out]
+
+df = pd.DataFrame(
+    {
+        "y_value": list(chain.from_iterable([y_i.quantiles for y_i in y_ecdf])),
+        "y_value2": list(chain.from_iterable([y_i.quantiles for y_i in y_ecdf]))[1:] + [np.nan],
+        "probability": list(chain.from_iterable([y_i.probabilities for y_i in y_ecdf])),
+    }
+)
+
+
+def plot_ecdf(df):
+    tooltip = [
+        alt.Tooltip("y_value", title="Response Value"),
+        alt.Tooltip("probability", title="Probability"),
+    ]
+
+    circles = (
+        alt.Chart(df)
+        .mark_circle(color="#006aff")
+        .encode(
+            x=alt.X("y_value", title="Response Value"),
+            y=alt.Y("probability", title="Probability"),
+            tooltip=tooltip,
+        )
+    )
+
+    lines = (
+        alt.Chart(df)
+        .mark_line(color="#006aff")
+        .encode(
+            x=alt.X("y_value", title="Response Value"),
+            x2=alt.X2("y_value2"),
+            y=alt.Y("probability", title="Probability"),
+            tooltip=tooltip,
+        )
+    )
+
+    chart = (circles + lines).properties(height=400, width=650)
+    return chart
+
+
+chart = plot_ecdf(df)
+chart
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-402,18 +402,23 @@

Extrapolation with Quantile Regression Forests#

-

An example on a toy dataset that demonstrates that the prediction intervals -produced by a quantile regression forest do not extrapolate outside of the -bounds of the data in the training set, an important limitation of the -approach. Notice that the extrapolated median and interval values fail to -accurately predict values outside of those observed in the training set.

+

An example on a toy dataset that illustrates the prediction intervals produced +by a quantile regression forest (QRF) on extrapolated data. QRFs do not +intrinsically extrapolate outside of the bounds of the training data, an +important limitation of the approach; notice that the extrapolated interval +with a standard QRF fails to reliably cover values outside of those observed +in the training set. To overcome this limitation, we can use a procedure known +as Xtrapolation that can estimate the extrapolation bounds for samples that +fall outside the range of the training data. Adapted from “Extrapolation-Aware +Nonparametric Statistical Inference” by Niklas Pfister and Peter Bühlmann: +https://arxiv.org/abs/2402.09758.

-
import altair as alt
+
import math
+
+import altair as alt
 import numpy as np
 import pandas as pd
 
@@ -430,13 +437,15 @@
 
 np.random.seed(0)
 
-n_samples = 1000
+n_samples = 500
 bounds = [0, 15]
 extrap_frac = 0.25
-
 func = lambda x: x * np.sin(x)
 func_str = "f(x) = x sin(x)"
 
+quantiles = [0.025, 0.975, 0.5]
+qrf_params = {"max_samples_leaf": None, "min_samples_leaf": 4, "random_state": 0}
+
 
 def make_func_Xy(func, bounds, n_samples):
     x = np.linspace(bounds[0], bounds[1], n_samples)
@@ -444,49 +453,354 @@
     std = 0.01 + np.abs(x - 5.0) / 5.0
     noise = np.random.normal(scale=std)
     y = f + noise
-    return x, y
+    return np.atleast_2d(x).T, y
+
+
+class Xtrapolation:
+    """Xtrapolation procedure.
+
+    Performs extrapolation-aware nonparametric statistical inference based on
+    an existing nonparametric estimate. Adapted from the Python code [1] for
+    the Xtrapolation procedure introduced in [2].
+
+    The procedure specifically applies a QRF for generating local polynomials
+    to estimate derivatives in a single dimension. For multi-dimensional
+    problems, using the original implementation is strongly encouraged.
+
+    References
+    ----------
+    .. [1] https://github.com/NiklasPfister/ExtrapolationAware-Inference
+    .. [2] N. Pfister and P. Bühlmann, "Extrapolation-Aware Nonparametric
+           Statistical Inference", arXiv preprint, 2024.
+           https://arxiv.org/abs/2402.09758
+    """
+
+    def __init__(self, orders=np.array([1])):
+        self.orders_ = orders
+        self.max_order_ = np.max(orders)
+
+    @staticmethod
+    def _penalized_locpol(fval, v, X, weights, degree, pen=0, penalize_intercept=False):
+        v = v.reshape(-1, 1)
+        n = X.shape[0]
+        dd = degree + 1
+        if penalize_intercept:
+            pen_list = list(range(0, dd))
+        else:
+            pen_list = list(range(1, dd))
+
+        # Construct design matrices.
+        DDmat = np.zeros((n * dd, n * dd))
+        DYmat = np.zeros((n * dd, 1))
+        for i in range(n):
+            Wi = np.sqrt(weights[i, :].reshape(-1, 1))
+
+            # Construct DDmat (block-diagonal).
+            x0v = X[i, :].dot(v)
+            Di = np.tile((X.dot(v) - x0v).reshape(-1, 1), dd) ** np.arange(dd) * Wi
+            DDmat[(i * dd) : ((i + 1) * dd), (i * dd) : ((i + 1) * dd)] = (Di.T).dot(Di)
+
+            # Construct DYmat.
+            DYmat[(i * dd) : ((i + 1) * dd), :] = (Di.T).dot((fval.reshape(-1, 1)) * Wi)
+
+        Z = np.zeros((dd, dd))
+        for kk in pen_list:
+            Z[kk, kk] = math.factorial(kk)
+        PP = np.kron(np.diag(np.sum(weights, axis=1)) - weights, Z)
+        penmat = pen * (PP.T).dot(PP)
+        B = np.linalg.solve(DDmat + penmat, DYmat)
+        coefs = B.reshape(n, -1)
+
+        # Extract derivatives from coefficients.
+        deriv_mat = coefs * np.array([math.factorial(k) for k in range(degree + 1)])
+        return deriv_mat
+
+    @staticmethod
+    def _get_tree_weight_matrix(X, Y, X_eval=None, n_trees=100, **kwargs):
+        """Fit forest and extract weights.
+
+        This implementation extracts the weight matrix from a list of quantile
+        random forests, each with a single tree fitted on non-bootstrapped
+        samples. This allows for controlling the bootstrap selection for each
+        tree and summing the weight matrices across all of the trees.
+        """
+        if "n_estimators" in kwargs:
+            n_trees = kwargs["n_estimators"]
+        kwargs["n_estimators"] = 1
+        if "random_state" in kwargs:
+            del kwargs["random_state"]
+        kwargs["bootstrap"] = False
+
+        trees = [RandomForestQuantileRegressor(random_state=i, **kwargs) for i in range(n_trees)]
+
+        n = X.shape[0]
+        nn = 0
+        if X_eval is not None:
+            nn = X_eval.shape[0]
+            X = np.r_[X, X_eval]
+        weight_mat = np.zeros((n + nn, n + nn))
+
+        s = 0.5
+        bn = int(n * s)
+
+        for tree in trees:
+            # Draw bootstrap sample.
+            boot_sample = np.random.choice(np.arange(n), bn, replace=False)
+            split1 = boot_sample[: int(bn / 2)]
+            split2 = np.concatenate([boot_sample[int(bn / 2) :], np.arange(nn) + n])
+
+            # Fit tree.
+            tree.fit(X[split1, :], Y[split1].flatten())
+
+            # Extract tree weight matrix.
+            y_train_leaves = tree._get_y_train_leaves(X[split2, :], 1)
+            nrows = X[split2, :].shape[0]
+            matrix = np.zeros((nrows, nrows))
+            for leaf in y_train_leaves[0]:
+                indices = leaf[0]
+                indices = indices[indices != 0] - 1
+                if len(indices) > 0:
+                    matrix[np.ix_(indices, indices)] = 1
+            weight_mat[np.ix_(split2, split2)] += matrix
+
+        # Normalize weights (rows correspond to weights - non-symmetric).
+        weight_mat /= weight_mat.sum(axis=1)[:, None]
+
+        return weight_mat
+
+    def fit_weights(self, X, fval, x0=None, train=False, **kwargs):
+        """Compute random forest weights for derivative estimation."""
+        n, d = X.shape
+        fval = fval.flatten()
+
+        if train:
+            d_xtra = d
+            xtra_features = list(range(d))
+            weights = [None] * d_xtra
+            for jj, var in enumerate(xtra_features):
+                var_order = list(range(d))
+                var_order = np.array([var] + var_order[:var] + var_order[var + 1 :])
+                weights[jj] = self._get_tree_weight_matrix(X[:, var_order], fval, x0, **kwargs)
+        else:
+            weights = self._get_tree_weight_matrix(X, fval, x0, **kwargs)[n:, :n]
+
+        return weights
+
+    def fit_derivatives(self, X, fval, pen=0.1, **kwargs):
+        """Estimate derivatives."""
+        n, d = X.shape
+        fval = fval.flatten()
+
+        # Fit weights for local polynomial.
+        weights = self.fit_weights(X, fval, train=True, **kwargs)
+
+        # Estimate derivatives with local polynomial.
+        derivatives = np.zeros((self.max_order_ + 1, n, d))
+        Xtilde = X[:, list(range(d))]
+
+        # Fit local polynomial.
+        for jj in range(d):
+            vv = np.zeros((d, 1))
+            vv[jj] = 1
+            tmp = self._penalized_locpol(
+                fval,
+                vv,
+                Xtilde,
+                weights[jj],
+                degree=self.max_order_ + 1,
+                pen=pen,
+                penalize_intercept=False,
+            )
+            for kk in range(self.max_order_ + 1):
+                derivatives[kk, :, jj] = fval if kk == 0 else tmp[:, kk]
+
+        return derivatives
+
+    def prediction_bounds(self, X, fval, x0, nn=50, **kwargs):
+        """Compute extrapolation bounds."""
+        n, d = X.shape
+        fval = fval.flatten()
+        if len(x0.shape) == 1:
+            x0 = x0.reshape(-1, 1)
+        n0 = x0.shape[0]
+        xtra_features = list(range(d))
+
+        # Fit derivatives.
+        derivatives = self.fit_derivatives(X, fval, **kwargs)
+
+        # Determine weighting for extrapolation points (using rotation).
+        mu = derivatives[1].mean(axis=0)
+        _, D, Vt = np.linalg.svd(derivatives[1] - mu[None, :])
+        TT = (Vt.T) * D[None, :]
+        Xtilde = X[:, xtra_features].dot(TT)
+        x0tilde = x0[:, xtra_features].dot(TT)
+
+        # Find closest points between rotated points (Euclidean).
+        weight_x0 = np.zeros((n0, n))
+        for ii in range(n0):
+            xinds = np.argsort(np.sum((x0tilde[None, ii, :] - Xtilde) ** 2, axis=1))[:nn]
+            weight_x0[ii, xinds] = 1 / nn
+
+        # Precompute factorials.
+        order_factorials = np.empty(self.max_order_ + 1)
+        for oo in range(self.max_order_ + 1):
+            order_factorials[oo] = math.factorial(oo)
+
+        # Iterate over all extrapolation points and average/intersect.
+        bounds = np.zeros((n0, len(self.orders_), 3))
+        for ll, xpt in enumerate(x0):
+            xinds = np.where(weight_x0[ll, :] != 0)[0]
+
+            # Number of anchor points to check.
+            f_lower = np.zeros((len(xinds), len(self.orders_)))
+            f_upper = np.zeros((len(xinds), len(self.orders_)))
+            f_median = np.zeros((len(xinds), len(self.orders_)))
+            for ii, xind in enumerate(xinds):
+                xx = X[xind, :].reshape(1, -1)
+                vv = (xpt - xx)[:, xtra_features]
+                vv_norm = np.sqrt(np.sum(vv**2))
+
+                # Compute directional derivatives.
+                deriv_mat = np.zeros((n, self.max_order_ + 1))
+                deriv_mat[:, 0] = derivatives[0, :, :].mean(axis=1)
+                if vv_norm > np.finfo(float).eps:
+                    vv_direction = np.array(vv / vv_norm).reshape(-1, 1)
+                    for kk in range(1, self.max_order_ + 1):
+                        deriv_mat[:, kk] = derivatives[kk, :, :].dot(vv_direction**kk).flatten()
+
+                # Select bounds.
+                deriv_min = np.quantile(deriv_mat, 0, axis=0)
+                deriv_max = np.quantile(deriv_mat, 1, axis=0)
+                deriv_median = np.quantile(deriv_mat, 0.5, axis=0)
+
+                # Estimate extrapolation bounds.
+                mterm = 0
+                kk = 0
+                for oo in range(self.max_order_ + 1):
+                    if oo in self.orders_:
+                        lo_bdd = deriv_min[oo] * (vv_norm**oo) / order_factorials[oo]
+                        up_bdd = deriv_max[oo] * (vv_norm**oo) / order_factorials[oo]
+                        median_deriv = deriv_median[oo] * (vv_norm**oo) / order_factorials[oo]
+                        f_lower[ii, kk] = mterm + lo_bdd
+                        f_upper[ii, kk] = mterm + up_bdd
+                        f_median[ii, kk] = mterm + median_deriv
+                        kk += 1
+                    mterm += deriv_mat[xind, oo] * (vv_norm**oo) / order_factorials[oo]
+
+            # Combine bounds over x-indices.
+            ww = (weight_x0[ll, xinds] / np.sum(weight_x0[ll, :]))[:, None]
+            f_median = np.sum(f_median * ww, axis=0)
+
+            # Aggregate by optimal-average.
+            f_lower = np.max(f_lower, axis=0)
+            f_upper = np.min(f_upper, axis=0)
+            ind = f_upper < f_lower
+            average = (f_upper + f_lower) / 2
+            f_lower[ind] = average[ind]
+            f_upper[ind] = average[ind]
+
+            bounds[ll, :, 0] = f_lower
+            bounds[ll, :, 1] = f_upper
+            bounds[ll, :, 2] = f_median
+
+        return bounds
+
+
+def train_test_split(train_indices, **kwargs):
+    """Fit model on training samples and extrapolate on test samples."""
+    X_train = X[train_indices, :]
+    y_train = y[train_indices]
+
+    # Run quantile regression (with forests).
+    qrf = RandomForestQuantileRegressor(**kwargs)
+    qrf.fit(X_train, y_train)
+    qmat = qrf.predict(X, quantiles=quantiles)
+
+    # Xtrapolation.
+    bounds_list = [None] * len(quantiles)
+    for i in range(len(quantiles)):
+        # Run Xtrapolation on quantile.
+        xtra = Xtrapolation()
+        bounds_list[i] = xtra.prediction_bounds(X_train, qmat[train_indices, i], X, **kwargs)
+
+    return {
+        "train_indices": train_indices,
+        "quantiles": quantiles,
+        "qmat": qmat,
+        "bounds_list": bounds_list,
+    }
+
+
+def prob_randomized_pi(qmat, y, coverage):
+    """Calculate calibration probability"""
+    alpha_included = np.mean((qmat[:, 0] <= y) & (y <= qmat[:, 1]))
+    alpha_excluded = np.mean((qmat[:, 0] < y) & (y < qmat[:, 1]))
+    if coverage <= alpha_excluded:
+        prob_si = 1
+    elif coverage >= alpha_included:
+        prob_si = 0
+    else:
+        prob_si = (coverage - alpha_included) / (alpha_excluded - alpha_included)
+    return prob_si
 
 
-def get_train_Xy(X, y, min_idx, max_idx):
-    X_train = X[min_idx:max_idx]
-    y_train = y[min_idx:max_idx]
-    return X_train, y_train
+def randomized_pi(qmat, prob_si, y):
+    """Calculate coverage."""
+    si_index = np.random.choice([False, True], len(y), replace=True, p=[prob_si, 1 - prob_si])
+    included = (qmat[:, 0] < y) & (y < qmat[:, 1])
+    boundary = (qmat[:, 0] == y) | (qmat[:, 1] == y)
+    return included | (boundary & si_index)
 
 
-def get_test_X(X, bounds):
-    n_samples = len(X)
-    X_test = np.atleast_2d(np.linspace(*bounds, n_samples)).T
-    return X_test
+def get_coverage_qrf(qmat, train_indices, test_indices, y_train, level):
+    """Calculate extrapolation coverage for regular quantile forest."""
+    prob_si = prob_randomized_pi(qmat[train_indices, :], y_train, level)
+    qrf = randomized_pi(qmat, prob_si, y)
+    return np.mean(qrf[test_indices])
+
+
+def get_coverage_xtr(bounds_list, train_indices, test_indices, y_train, level):
+    """Calculate extrapolation coverage for Xtrapolation."""
+    bb_low = np.max(bounds_list[0][:, :, 0], axis=1)
+    bb_upp = np.min(bounds_list[1][:, :, 1], axis=1)
+    bb_low_train, bb_upp_train = bb_low[train_indices], bb_upp[train_indices]
+    prob_si = prob_randomized_pi(np.c_[bb_low_train, bb_upp_train], y_train, level)
+    xtra = randomized_pi(np.c_[bb_low, bb_upp], prob_si, y)
+    return np.mean(xtra[test_indices])
 
 
 # Create the full dataset.
 X, y = make_func_Xy(func, bounds, n_samples)
 
-# Calculate the extrapolation bounds.
+# Fit and extrapolate based on train-test split (depending on X).
 extrap_min_idx = int(n_samples * (extrap_frac / 2))
 extrap_max_idx = int(n_samples - (n_samples * (extrap_frac / 2)))
+sort_X = np.argsort(X.squeeze())
+train_indices = np.repeat(False, len(y))
+train_indices[sort_X[extrap_min_idx] : sort_X[extrap_max_idx]] = True
+res = train_test_split(train_indices, **qrf_params)
 
-# Based on the extrapolation bounds, get the training and test data.
-# Training data excludes extrapolated regions; test data includes them.
-X_train, y_train = get_train_Xy(X, y, extrap_min_idx, extrap_max_idx)
-X_test = get_test_X(X, bounds)
-
-qrf = RandomForestQuantileRegressor(max_samples_leaf=None, min_samples_leaf=10, random_state=0)
-qrf.fit(np.expand_dims(X_train, axis=-1), y_train)
-
-# Get predictions at 95% prediction intervals and median.
-y_pred = qrf.predict(X_test, quantiles=[0.025, 0.5, 0.975])
+# Get coverages on extrapolated samples.
+args = (train_indices, ~train_indices, y[train_indices], quantiles[1] - quantiles[0])
+cov_qrf = get_coverage_qrf(res["qmat"], *args)
+cov_xtr = get_coverage_xtr(res["bounds_list"], *args)
 
 df = pd.DataFrame(
     {
-        "X_true": X,
-        "y_func": func(X),
+        "X_true": X.squeeze(),
+        "y_func": func(X.squeeze()),
         "y_true": y,
-        "y_pred": y_pred[:, 1],
-        "y_pred_low": y_pred[:, 0],
-        "y_pred_upp": y_pred[:, 2],
+        "y_pred": res["qmat"][:, 2],
+        "y_pred_low": res["qmat"][:, 0],
+        "y_pred_upp": res["qmat"][:, 1],
+        "bb_low": np.max(res["bounds_list"][0][:, :, 0], axis=1),
+        "bb_upp": np.min(res["bounds_list"][1][:, :, 1], axis=1),
+        "bb_mid": np.median(res["bounds_list"][2][:, :, :2], axis=(1, 2)),
+        "train": res["train_indices"],
         "test_left": [True] * extrap_min_idx + [False] * (len(y) - extrap_min_idx),
         "test_right": [False] * extrap_max_idx + [True] * (len(y) - extrap_max_idx),
+        "cov_qrf": cov_qrf,
+        "cov_xtr": cov_xtr,
     }
 )
 
@@ -532,17 +846,17 @@
         tooltip=tooltip_true,
     )
 
-    line_pred = base.mark_line().encode(
-        x=alt.X("X_true:Q", scale=x_scale, title=""),
+    line_pred = base.mark_line(clip=True).encode(
+        x=alt.X("X_true:Q", title="", scale=x_scale),
         y=alt.Y("y_pred:Q", scale=y_scale),
         color=alt.condition(alt.datum["extrapolate"], alt.value("red"), alt.value("#006aff")),
         tooltip=tooltip_pred,
     )
 
-    bar_pred = base.mark_bar(width=2).encode(
-        x=alt.X("X_true:Q", scale=x_scale, title=""),
-        y=alt.Y("y_pred_low:Q", scale=y_scale, title=""),
-        y2=alt.Y2("y_pred_upp:Q", title=None),
+    bar_pred = base.mark_bar(clip=True, width=2).encode(
+        x=alt.X("X_true"),
+        y=alt.Y("y_pred_low"),
+        y2=alt.Y2("y_pred_upp"),
         color=alt.condition(alt.datum["extrapolate"], alt.value("red"), alt.value("#e0f2ff")),
         opacity=alt.condition(alt.datum["extrapolate"], alt.value(0.05), alt.value(0.8)),
         tooltip=tooltip_pred,
@@ -550,6 +864,25 @@
 
     chart = bar_pred + points_true + line_true + line_pred
 
+    if "coverage" in df.columns:
+        text_coverage = (
+            base.transform_aggregate(coverage="mean(coverage)")
+            .transform_calculate(
+                coverage_text=(
+                    f"'Extrapolated Coverage: '"
+                    f" + format({alt.datum['coverage'] * 100}, '.1f') + '%'"
+                    f" + ' (target = {(quantiles[1] - quantiles[0]) * 100}%)'"
+                )
+            )
+            .mark_text(align="left", baseline="top")
+            .encode(
+                x=alt.value(5),
+                y=alt.value(5),
+                text=alt.Text("coverage_text:N"),
+            )
+        )
+        chart += text_coverage
+
     if legend:
         # For desired legend ordering.
         data = {
@@ -571,27 +904,42 @@
         chart = chart.resolve_scale(color="independent")
 
     chart = chart.properties(height=200, width=300, title=title)
-
     return chart
 
 
-kwargs = {
-    "x_domain": [int(np.min(df["X_true"])), int(np.max(df["X_true"]))],
-    "y_domain": [int(np.min(df["y_true"])), int(np.max(df["y_true"]))],
-}
+kwargs = {"x_domain": [0, 15], "y_domain": [-15, 20]}
+xtra_mapper = {"bb_mid": "y_pred", "bb_low": "y_pred_low", "bb_upp": "y_pred_upp"}
 
-chart1 = plot_extrapolations(
-    df.query("~(test_left | test_right)"), title="Prediction Intervals on Training Data", **kwargs
+chart1 = alt.layer(
+    plot_extrapolations(
+        df.query("~(test_left | test_right)").assign(**{"coverage": lambda x: x["cov_qrf"]}),
+        title="Extrapolation with Standard QRF",
+        **kwargs,
+    ).resolve_scale(color="independent"),
+    plot_extrapolations(df.query("test_left").assign(extrapolate=True), **kwargs),
+    plot_extrapolations(df.query("test_right").assign(extrapolate=True), **kwargs),
 )
 chart2 = alt.layer(
     plot_extrapolations(
-        df.query("~(test_left | test_right)"),
-        title="Prediction Intervals with Extrapolated Values",
+        df.query("~(test_left | test_right)").assign(**{"coverage": lambda x: x["cov_xtr"]}),
+        title="Extrapolation with Xtrapolation Procedure",
         legend=True,
         **kwargs,
     ).resolve_scale(color="independent"),
-    plot_extrapolations(df.query("test_left").assign(extrapolate=True), **kwargs),
-    plot_extrapolations(df.query("test_right").assign(extrapolate=True), **kwargs),
+    plot_extrapolations(
+        df.query("test_left")
+        .assign(extrapolate=True)
+        .drop(columns=["y_pred", "y_pred_low", "y_pred_upp"])
+        .rename(xtra_mapper, axis="columns"),
+        **kwargs,
+    ),
+    plot_extrapolations(
+        df.query("test_right")
+        .assign(extrapolate=True)
+        .drop(columns=["y_pred", "y_pred_low", "y_pred_upp"])
+        .rename(xtra_mapper, axis="columns"),
+        **kwargs,
+    ),
 )
 chart = chart1 | chart2
 chart
diff --git a/gallery/plot_quantile_interpolation.html b/gallery/plot_quantile_interpolation.html
index 2577aee..722e80d 100644
--- a/gallery/plot_quantile_interpolation.html
+++ b/gallery/plot_quantile_interpolation.html
@@ -415,7 +415,7 @@
   // embed when document is loaded, to ensure vega library is available
   // this works on all modern browsers, except IE8 and older
   document.addEventListener("DOMContentLoaded", function(event) {
-      var spec = {"config": {"view": {"continuousWidth": 300, "continuousHeight": 300, "stroke": null}, "facet": {"spacing": 15}, "range": {"category": ["#000000", "#006aff", "#ffd237", "#0d4599", "#f2a619", "#a6e5ff"]}, "scale": {"bandPaddingInner": 0.9}}, "data": {"name": "data-8357d4f1f0593893b30cb552bbc9af18"}, "facet": {"column": {"field": "X", "header": {"labelOrient": "bottom", "titleOrient": "bottom"}, "title": "Samples (Feature Values)", "type": "nominal"}}, "spec": {"layer": [{"mark": {"type": "bar"}, "encoding": {"color": {"condition": {"param": "param_10", "field": "method", "sort": ["Actual", "Linear", "Lower", "Higher", "Midpoint", "Nearest"], "title": null, "type": "nominal"}, "value": "lightgray"}, "tooltip": [{"field": "method", "title": "Method", "type": "nominal"}, {"field": "X", "title": "X Values", "type": "nominal"}, {"field": "y_pred", "format": ".3f", "title": "Median Y Value", "type": "nominal"}, {"field": "y_pred_low", "format": ".3f", "title": "Lower Y Value", "type": "nominal"}, {"field": "y_pred_upp", "format": ".3f", "title": "Upper Y Value", "type": "nominal"}], "x": {"axis": {"labels": false, "tickSize": 0}, "field": "method", "sort": ["Actual", "Linear", "Lower", "Higher", "Midpoint", "Nearest"], "title": null, "type": "nominal"}, "y": {"field": "y_pred_low", "title": "", "type": "quantitative"}, "y2": {"field": "y_pred_upp", "title": null}}, "name": "view_9"}, {"mark": {"type": "circle", "opacity": 1, "size": 75}, "encoding": {"color": {"condition": {"param": "param_10", "field": "method", "sort": ["Actual", "Linear", "Lower", "Higher", "Midpoint", "Nearest"], "title": null, "type": "nominal"}, "value": "lightgray"}, "tooltip": [{"field": "method", "title": "Method", "type": "nominal"}, {"field": "X", "title": "X Values", "type": "nominal"}, {"field": "y_pred", "format": ".3f", "title": "Median Y Value", "type": "nominal"}, {"field": "y_pred_low", "format": ".3f", "title": "Lower Y Value", "type": "nominal"}, {"field": "y_pred_upp", "format": ".3f", "title": "Upper Y Value", "type": "nominal"}], "x": {"axis": {"labels": false, "tickSize": 0}, "field": "method", "sort": ["Actual", "Linear", "Lower", "Higher", "Midpoint", "Nearest"], "title": null, "type": "nominal"}, "y": {"field": "y_pred", "title": "Actual and Predicted Values", "type": "quantitative"}}}], "height": 400, "width": {"step": 20}}, "params": [{"name": "param_10", "select": {"type": "point", "fields": ["method"]}, "bind": "legend", "views": ["view_9"]}], "$schema": "https://vega.github.io/schema/vega-lite/v5.16.3.json", "datasets": {"data-8357d4f1f0593893b30cb552bbc9af18": [{"method": "Actual", "X": "Sample 1 ([-1, -1])", "y_pred": -2.0, "y_pred_low": -2.0, "y_pred_upp": -2.0}, {"method": "Actual", "X": "Sample 2 ([-1, -1])", "y_pred": -1.0, "y_pred_low": -1.0, "y_pred_upp": -1.0}, {"method": "Actual", "X": "Sample 3 ([-1, -1])", "y_pred": 0.0, "y_pred_low": 0.0, "y_pred_upp": 0.0}, {"method": "Actual", "X": "Sample 4 ([1, 1])", "y_pred": 1.0, "y_pred_low": 1.0, "y_pred_upp": 1.0}, {"method": "Actual", "X": "Sample 5 ([1, 1])", "y_pred": 2.0, "y_pred_low": 2.0, "y_pred_upp": 2.0}, {"method": "Linear", "X": "Sample 1 ([-1, -1])", "y_pred": -1.0, "y_pred_low": -1.95, "y_pred_upp": -0.050000000000000044}, {"method": "Linear", "X": "Sample 2 ([-1, -1])", "y_pred": -1.0, "y_pred_low": -1.95, "y_pred_upp": -0.050000000000000044}, {"method": "Linear", "X": "Sample 3 ([-1, -1])", "y_pred": -1.0, "y_pred_low": -1.95, "y_pred_upp": -0.050000000000000044}, {"method": "Linear", "X": "Sample 4 ([1, 1])", "y_pred": 1.5, "y_pred_low": 1.025, "y_pred_upp": 1.975}, {"method": "Linear", "X": "Sample 5 ([1, 1])", "y_pred": 1.5, "y_pred_low": 1.025, "y_pred_upp": 1.975}, {"method": "Lower", "X": "Sample 1 ([-1, -1])", "y_pred": -1.0, "y_pred_low": -2.0, "y_pred_upp": -1.0}, {"method": "Lower", "X": "Sample 2 ([-1, -1])", "y_pred": -1.0, "y_pred_low": -2.0, "y_pred_upp": -1.0}, {"method": "Lower", "X": "Sample 3 ([-1, -1])", 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diff --git a/gallery/plot_quantile_weighting.html b/gallery/plot_quantile_weighting.html
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Weighted vs. Unweighted Quantile Runtimes#

-
An example comparison of the prediction runtime when using a quantile
-regression forest with weighted and unweighted quantiles to compute the
-predicted output values. While weighted and unweighted quantiles produce
-identical outputs, the relative runtime of the methods depends on the number
-of training samples and the total number of leaf samples across all trees used
-to calculate the quantiles. A standard random forest regressor is included for
-comparison.

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-
import time
-from contextlib import contextmanager
-
-import altair as alt
-import pandas as pd
-from sklearn import datasets
-from sklearn.ensemble import RandomForestRegressor
-
-from quantile_forest import RandomForestQuantileRegressor
-
-samples = [100, 175, 250, 325, 500]
-estimators = [10, 25, 50, 75, 100]
-repeats = 3
-
-
-@contextmanager
-def timing():
-    t0 = time.process_time()
-    yield lambda: (t1 - t0)
-    t1 = time.process_time()
-
-
-legend = {
-    "RF": "#f2a619",
-    "QRF Weighted Quantile": "#006aff",
-    "QRF Unweighted Quantile": "#001751",
-}
-
-dataset = datasets.make_regression(
-    n_samples=max(samples), n_features=3, n_targets=1, random_state=0
-)
-
-# Populate data with timing results over samples and estimators.
-data = {"name": [], "n_samples": [], "n_estimators": [], "iteration": [], "runtime": []}
-for n_samples in samples:
-    X = dataset[0][:n_samples, :]
-    y = dataset[1][:n_samples]
-    for n_estimators in estimators:
-        for repeat in range(repeats):
-            rf = RandomForestRegressor(
-                n_estimators=n_estimators,
-                max_depth=7,
-                random_state=0,
-            )
-            qrf = RandomForestQuantileRegressor(
-                n_estimators=n_estimators,
-                max_depth=7,
-                max_samples_leaf=None,
-                random_state=0,
-            )
-
-            rf.fit(X, y)
-            qrf.fit(X, y)
-
-            with timing() as rf_time:
-                _ = rf.predict(X)
-            with timing() as qrf_weighted_time:
-                _ = qrf.predict(X, quantiles=0.5, weighted_quantile=True)
-            with timing() as qrf_unweighted_time:
-                _ = qrf.predict(X, quantiles=0.5, weighted_quantile=False)
-
-            timings = [rf_time(), qrf_weighted_time(), qrf_unweighted_time()]
-
-            for name, runtime in zip(legend.keys(), timings):
-                runtime *= 1000  # convert from milliseconds to seconds
-
-                data["name"].extend([name])
-                data["n_samples"].extend([n_samples])
-                data["n_estimators"].extend([n_estimators])
-                data["iteration"].extend([repeat])
-                data["runtime"].extend([runtime])
-
-df = (
-    pd.DataFrame(data)
-    .groupby(["name", "n_samples", "n_estimators"])
-    .agg({"runtime": ["mean", "std"]})
-    .pipe(lambda x: x.set_axis(["_".join(map(str, col)) for col in x.columns], axis=1))
-    .reset_index()
-    .assign(
-        **{
-            "ymean": lambda x: x["runtime_mean"],
-            "ystd": lambda x: x["runtime_std"],
-            "ymin": lambda x: x["ymean"] - (x["ystd"] / 2),
-            "ymax": lambda x: x["ymean"] + (x["ystd"] / 2),
-        }
-    )
-    .drop(columns=["runtime_mean", "runtime_std"])
-)
-
-
-def plot_timings_by_factor(df, legend, constant, factor, factor_title):
-    max_value = df[constant].max()
-
-    click = alt.selection_point(fields=["name"], bind="legend")
-
-    color = alt.condition(
-        click,
-        alt.Color(
-            "name:N",
-            legend=alt.Legend(symbolOpacity=1),
-            sort=list(legend.keys()),
-            title=None,
-        ),
-        alt.value("lightgray"),
-    )
-
-    base = (
-        alt.Chart(df)
-        .transform_filter(alt.datum[constant] == max_value)  # hold this factor constant
-        .transform_joinaggregate(min_runtime="min(ymin):Q", groupby=[constant])
-        .transform_calculate(ymean_norm="datum.ymean / datum.min_runtime")
-        .transform_calculate(ystd_norm="datum.ystd / datum.min_runtime")
-        .transform_calculate(ymin_norm="datum.ymin / datum.min_runtime")
-        .transform_calculate(ymax_norm="datum.ymax / datum.min_runtime")
-    )
-
-    line = base.mark_line().encode(
-        x=alt.X(f"{factor}:Q", scale=alt.Scale(nice=False), title=factor_title),
-        y=alt.Y(
-            "ymean_norm:Q",
-            scale=alt.Scale(zero=False),
-            title="Prediction Runtime (normalized)",
-        ),
-        color=color,
-    )
-
-    area = base.mark_area(opacity=0.1).encode(
-        x=alt.X(f"{factor}:Q"),
-        y=alt.Y("ymin_norm:Q"),
-        y2=alt.Y2("ymax_norm:Q"),
-        color=color,
-        tooltip=[
-            alt.Tooltip("name:N", title="Estimator Name"),
-            alt.Tooltip(f"{factor}:Q", format=",d", title=factor_title),
-            alt.Tooltip("ymean_norm:Q", format=",.3f", title="Average Runtime"),
-            alt.Tooltip("ymin_norm:Q", format=",.3f", title="Minimum Runtime"),
-            alt.Tooltip("ymax_norm:Q", format=",.3f", title="Maximum Runtime"),
-        ],
-    )
-
-    text = (
-        base.transform_aggregate(coverage="mean(constant)", groupby=[constant])
-        .transform_calculate(text=(f"'{constant} = ' + format({alt.datum[constant]}, ',d')"))
-        .mark_text(align="left", baseline="top")
-        .encode(
-            x=alt.value(5),
-            y=alt.value(5),
-            text=alt.Text("text:N"),
-        )
-    )
-
-    chart = (
-        (line + area + text)
-        .add_params(click)
-        .properties(height=200, width=300, title=f"Runtime by {factor_title}")
-    )
-
-    return chart
-
-
-chart1 = plot_timings_by_factor(df, legend, "n_samples", "n_estimators", "Number of Estimators")
-chart2 = plot_timings_by_factor(df, legend, "n_estimators", "n_samples", "Number of Samples")
-chart = (chart1 | chart2).configure_range(category=alt.RangeScheme(list(legend.values())))
-chart
-

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-      © Copyright 2022-2024, Zillow Group.
-      

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-    Created using Sphinx 7.2.6.
-    

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diff --git a/objects.inv b/objects.inv
index 1854e08..03e8f44 100644
Binary files a/objects.inv and b/objects.inv differ
diff --git a/searchindex.js b/searchindex.js
index 7724c0e..dc8060a 100644
--- a/searchindex.js
+++ b/searchindex.js
@@ -1 +1 @@
-Search.setIndex({"docnames": ["api", "gallery/index", "gallery/plot_quantile_conformalized", "gallery/plot_quantile_example", "gallery/plot_quantile_extrapolation", "gallery/plot_quantile_interpolation", "gallery/plot_quantile_intervals", "gallery/plot_quantile_multioutput", "gallery/plot_quantile_vs_standard", "gallery/plot_quantile_weighting", "generated/quantile_forest.ExtraTreesQuantileRegressor", "generated/quantile_forest.RandomForestQuantileRegressor", "getting_started/developers", "getting_started/installation", "index", "references", "releases/changes", "user_guide/fit_predict", "user_guide/introduction", "user_guide/proximities", "user_guide/quantile_ranks"], "filenames": ["api.rst", "gallery/index.rst", "gallery/plot_quantile_conformalized.rst", "gallery/plot_quantile_example.rst", "gallery/plot_quantile_extrapolation.rst", "gallery/plot_quantile_interpolation.rst", "gallery/plot_quantile_intervals.rst", "gallery/plot_quantile_multioutput.rst", "gallery/plot_quantile_vs_standard.rst", "gallery/plot_quantile_weighting.rst", "generated/quantile_forest.ExtraTreesQuantileRegressor.rst", "generated/quantile_forest.RandomForestQuantileRegressor.rst", "getting_started/developers.rst", "getting_started/installation.rst", "index.rst", "references.rst", "releases/changes.rst", "user_guide/fit_predict.rst", "user_guide/introduction.rst", "user_guide/proximities.rst", "user_guide/quantile_ranks.rst"], "titles": ["API Reference", "General Examples", "QRFs for Conformalized Quantile Regression", "Predicting with Quantile Regression Forests", "Extrapolation with Quantile Regression Forests", "Comparing Quantile Interpolation Methods", "Quantile Regression Forests Prediction Intervals", "Multiple-Output Quantile Regression with QRFs", "Quantile Regression Forests vs. Random Forests", "Weighted vs. Unweighted Quantile Runtimes", "ExtraTreesQuantileRegressor", "RandomForestQuantileRegressor", "Developer\u2019s Guide", "Getting Started", "quantile-forest", "References", "Release Notes", "Fitting and Predicting", "Introduction", "Proximity Counts", "Quantile Ranks"], "terms": {"thi": [0, 2, 5, 7, 9, 10, 11, 14, 16, 17, 18], "i": [0, 2, 4, 7, 8, 9, 10, 11, 12, 14, 17, 18, 19, 20], "full": [0, 4, 17, 18], "document": [0, 16], "packag": [0, 12, 13, 14, 17, 18], "purpos": 1, "introductori": [1, 14], "illustr": [1, 3, 5, 8], "compar": [1, 3], "quantil": [1, 10, 11, 13, 15, 16], "interpol": [1, 10, 11, 16], "method": [1, 9, 10, 11, 14, 17, 18, 19], "extrapol": 1, "regress": [1, 9, 10, 11, 14, 15, 16, 17], "forest": [1, 2, 5, 9, 10, 11, 13, 15, 16, 20], "multipl": [1, 10, 11, 16], "output": [1, 2, 9, 10, 11, 16, 17, 18, 19, 20], "qrf": [1, 3, 4, 6, 8, 9, 10, 11, 14, 17], "predict": [1, 2, 4, 5, 7, 8, 9, 10, 11, 14, 16, 18], "conform": [1, 16], "interv": [1, 2, 3, 4, 5, 7, 10, 11, 18], "v": [1, 4, 10, 11, 12], "random": [1, 3, 4, 7, 9, 10, 11, 18], "weight": [1, 10, 11, 16, 18], "unweight": [1, 10, 11, 17, 18], "runtim": 1, 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"Version 1.0.0 (released Mar 23, 2022)": [[16, "version-1-0-0-released-mar-23-2022"]], "Fitting and Predicting": [[17, "fitting-and-predicting"]], "Fitting a Model": [[17, "fitting-a-model"]], "Making Predictions": [[17, "making-predictions"]], "Quantile Weighting": [[17, "quantile-weighting"]], "Out-of-Bag Estimation": [[17, "out-of-bag-estimation"]], "Random Forest Predictions": [[17, "random-forest-predictions"]], "Predicting with User-Specified Functions": [[17, "predicting-with-user-specified-functions"]], "Introduction": [[18, "introduction"]], "Quantile Regression Forests": [[18, "quantile-regression-forests"]], "Proximity Counts": [[19, "proximity-counts"]], "Quantile Ranks": [[20, "quantile-ranks"]]}, "indexentries": {"extratreesquantileregressor (class in quantile_forest)": [[10, "quantile_forest.ExtraTreesQuantileRegressor"]], "apply() (quantile_forest.extratreesquantileregressor method)": [[10, "quantile_forest.ExtraTreesQuantileRegressor.apply"]], "decision_path() (quantile_forest.extratreesquantileregressor method)": [[10, "quantile_forest.ExtraTreesQuantileRegressor.decision_path"]], "estimators_samples_ (quantile_forest.extratreesquantileregressor property)": [[10, "quantile_forest.ExtraTreesQuantileRegressor.estimators_samples_"]], "feature_importances_ (quantile_forest.extratreesquantileregressor property)": [[10, "quantile_forest.ExtraTreesQuantileRegressor.feature_importances_"]], "fit() (quantile_forest.extratreesquantileregressor method)": [[10, "quantile_forest.ExtraTreesQuantileRegressor.fit"]], "get_metadata_routing() (quantile_forest.extratreesquantileregressor method)": [[10, "quantile_forest.ExtraTreesQuantileRegressor.get_metadata_routing"]], "get_params() (quantile_forest.extratreesquantileregressor method)": [[10, "quantile_forest.ExtraTreesQuantileRegressor.get_params"]], "predict() (quantile_forest.extratreesquantileregressor method)": [[10, "quantile_forest.ExtraTreesQuantileRegressor.predict"]], "proximity_counts() 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"randomforestquantileregressor (class in quantile_forest)": [[11, "quantile_forest.RandomForestQuantileRegressor"]], "apply() (quantile_forest.randomforestquantileregressor method)": [[11, "quantile_forest.RandomForestQuantileRegressor.apply"]], "decision_path() (quantile_forest.randomforestquantileregressor method)": [[11, "quantile_forest.RandomForestQuantileRegressor.decision_path"]], "estimators_samples_ (quantile_forest.randomforestquantileregressor property)": [[11, "quantile_forest.RandomForestQuantileRegressor.estimators_samples_"]], "feature_importances_ (quantile_forest.randomforestquantileregressor property)": [[11, "quantile_forest.RandomForestQuantileRegressor.feature_importances_"]], "fit() (quantile_forest.randomforestquantileregressor method)": [[11, "quantile_forest.RandomForestQuantileRegressor.fit"]], "get_metadata_routing() (quantile_forest.randomforestquantileregressor method)": [[11, "quantile_forest.RandomForestQuantileRegressor.get_metadata_routing"]], "get_params() (quantile_forest.randomforestquantileregressor method)": [[11, "quantile_forest.RandomForestQuantileRegressor.get_params"]], "predict() (quantile_forest.randomforestquantileregressor method)": [[11, "quantile_forest.RandomForestQuantileRegressor.predict"]], "proximity_counts() (quantile_forest.randomforestquantileregressor method)": [[11, "quantile_forest.RandomForestQuantileRegressor.proximity_counts"]], "quantile_ranks() (quantile_forest.randomforestquantileregressor method)": [[11, "quantile_forest.RandomForestQuantileRegressor.quantile_ranks"]], "score() (quantile_forest.randomforestquantileregressor method)": [[11, "quantile_forest.RandomForestQuantileRegressor.score"]], "set_fit_request() (quantile_forest.randomforestquantileregressor method)": [[11, "quantile_forest.RandomForestQuantileRegressor.set_fit_request"]], "set_params() (quantile_forest.randomforestquantileregressor method)": [[11, "quantile_forest.RandomForestQuantileRegressor.set_params"]], "set_predict_request() (quantile_forest.randomforestquantileregressor method)": [[11, "quantile_forest.RandomForestQuantileRegressor.set_predict_request"]], "set_score_request() (quantile_forest.randomforestquantileregressor method)": [[11, "quantile_forest.RandomForestQuantileRegressor.set_score_request"]]}})
\ No newline at end of file
diff --git a/user_guide/fit_predict.html b/user_guide/fit_predict.html
index 86aec34..55195b9 100644
--- a/user_guide/fit_predict.html
+++ b/user_guide/fit_predict.html
@@ -560,6 +560,10 @@ 
Random Forest Predictions
+
Predicting with User-Specified Functions#

+
While a QRF is designed to estimate quantiles from the empirical distribution calculated for each sample, in many cases it may be useful to use the empirical distribution to calculate other quantities of interest. For more details, see Calculating User-Specified Functions with QRFs.

+
@@ -613,6 +617,7 @@

Random Forest PredictionsQuantile Weighting
  • Out-of-Bag Estimation
  • Random Forest Predictions
    • +
  • Predicting with User-Specified Functions