# Source code for econml.grf._base_grftree

#
# This code contains snippets of code from:
# https://github.com/scikit-learn/scikit-learn/blob/master/sklearn/tree/_classes.py
#
# Copyright (c) 2007-2020 The scikit-learn developers.

import numpy as np
from ._criterion import LinearMomentGRFCriterionMSE, LinearMomentGRFCriterion
from ..tree import BaseTree
from sklearn.model_selection import train_test_split
from sklearn.utils import check_array
from sklearn.utils import check_random_state
from sklearn.utils.validation import check_is_fitted
import copy

# =============================================================================
# Types and constants
# =============================================================================

CRITERIA_GRF = {"het": LinearMomentGRFCriterion,
"mse": LinearMomentGRFCriterionMSE}

# =============================================================================
# Base GRF tree
# =============================================================================

[docs]class GRFTree(BaseTree): """A tree of a Generalized Random Forest [grftree1]. This method should be used primarily through the BaseGRF forest class and its derivatives and not as a standalone estimator. It fits a tree that solves the local moment equation problem:: E[ m(Z; theta(x)) | X=x] = 0 For some moment vector function m, that takes as input random samples of a random variable Z and is parameterized by some unknown parameter theta(x). Each node in the tree contains a local estimate of the parameter theta(x), for every region of X that falls within that leaf. Parameters ---------- criterion : {'mse', 'het'}, default='mse' The function to measure the quality of a split. Supported criteria are 'mse' for the mean squared error in a linear moment estimation tree and 'het' for heterogeneity score. These criteria solve any linear moment problem of the form:: E[J * theta(x) - A | X = x] = 0 - The 'mse' criterion finds splits that maximize the score: .. code-block:: sum_{child} weight(child) * theta(child).T @ E[J | X in child] @ theta(child) - In the case of a causal tree, this coincides with minimizing the MSE: .. code-block:: sum_{child} E[(Y - <theta(child), T>)^2 | X=child] weight(child) - In the case of an IV tree, this roughly coincides with minimize the projected MSE:: .. code-block:: sum_{child} E[(Y - <theta(child), E[T|Z]>)^2 | X=child] weight(child) Internally, for the case of more than two treatments or for the case of one treatment with fit_intercept=True then this criterion is approximated by computationally simpler variants for computationaly purposes. In particular, it is replaced by:: sum_{child} weight(child) * rho(child).T @ E[J | X in child] @ rho(child) where: .. code-block:: rho(child) := J(parent)^{-1} E[A - J * theta(parent) | X in child] This can be thought as a heterogeneity inducing score, but putting more weight on scores with a large minimum eigenvalue of the child jacobian E[J | X in child], which leads to smaller variance of the estimate and stronger identification of the parameters. - The 'het' criterion finds splits that maximize the pure parameter heterogeneity score: .. code-block:: sum_{child} weight(child) * rho(child).T @ rho(child) This can be thought as an approximation to the ideal heterogeneity score: .. code-block:: weight(left) * weight(right) || theta(left) - theta(right)||_2^2 / weight(parent)^2 as outlined in [grftree1]_ splitter : {"best"}, default="best" The strategy used to choose the split at each node. Supported strategies are "best" to choose the best split. max_depth : int, default=None The maximum depth of the tree. If None, then nodes are expanded until all leaves are pure or until all leaves contain less than min_samples_split samples. min_samples_split : int or float, default=10 The minimum number of samples required to split an internal node: - If int, then consider min_samples_split as the minimum number. - If float, then min_samples_split is a fraction and ceil(min_samples_split * n_samples) are the minimum number of samples for each split. min_samples_leaf : int or float, default=5 The minimum number of samples required to be at a leaf node. A split point at any depth will only be considered if it leaves at least min_samples_leaf training samples in each of the left and right branches. This may have the effect of smoothing the model, especially in regression. - If int, then consider min_samples_leaf as the minimum number. - If float, then min_samples_leaf is a fraction and ceil(min_samples_leaf * n_samples) are the minimum number of samples for each node. min_weight_fraction_leaf : float, default=0.0 The minimum weighted fraction of the sum total of weights (of all the input samples) required to be at a leaf node. Samples have equal weight when sample_weight is not provided. min_var_leaf : None or double in (0, infinity), default=None A constraint on the minimum degree of identification of the parameter of interest. This avoids performing splits where either the variance of the treatment is small or the correlation of the instrument with the treatment is small, or the variance of the instrument is small. Generically for any linear moment problem this translates to conditions on the leaf jacobian matrix J(leaf) that are proxies for a well-conditioned matrix, which leads to smaller variance of the local estimate. The proxy of the well-conditioning is different for different criterion, primarily for computational efficiency reasons. - If criterion='het', then the diagonal entries of J(leaf) are constraint to have absolute value at least min_var_leaf: .. code-block:: for all i in {1, ..., n_outputs}: abs(J(leaf)[i, i]) > min_var_leaf In the context of a causal tree, when residual treatment is passed at fit time, then, this translates to a requirement on Var(T[i]) for every treatment coordinate i. In the context of an IV tree, with residual instruments and residual treatments passed at fit time this translates to Cov(T[i], Z[i]) > min_var_leaf for each coordinate i of the instrument and the treatment. - If criterion='mse', because the criterion stores more information about the leaf jacobian for every candidate split, then we impose further constraints on the pairwise determininants of the leaf jacobian, as they come at small extra computational cost, i.e.:: for all i neq j: sqrt(abs(J(leaf)[i, i] * J(leaf)[j, j] - J(leaf)[i, j] * J(leaf)[j, i])) > min_var_leaf In the context of a causal tree, when residual treatment is passed at fit time, then this translates to a constraint on the pearson correlation coefficient on any two coordinates of the treatment within the leaf, i.e.:: for all i neq j: sqrt( Var(T[i]) * Var(T[j]) * (1 - rho(T[i], T[j])^2) ) ) > min_var_leaf where rho(X, Y) is the Pearson correlation coefficient of two random variables X, Y. Thus this constraint also enforces that no two pairs of treatments be very co-linear within a leaf. This extra constraint primarily has bite in the case of more than two input treatments. min_var_leaf_on_val : bool, default=False Whether the min_var_leaf constraint should also be enforced to hold on the validation set of the honest split too. If min_var_leaf=None then this flag does nothing. Setting this to True should be done with caution, as this partially violates the honesty structure, since parts of the variables other than the X variable (e.g. the variables that go into the jacobian J of the linear model) are used to inform the split structure of the tree. However, this is a benign dependence and for instance in a causal tree or an IV tree does not use the label y. It only uses the treatment T and the instrument Z and their local correlation structures to decide whether a split is feasible. max_features : int, float or {"auto", "sqrt", "log2"}, default=None The number of features to consider when looking for the best split: - If int, then consider max_features features at each split. - If float, then max_features is a fraction and int(max_features * n_features) features are considered at each split. - If "auto", then max_features=n_features. - If "sqrt", then max_features=sqrt(n_features). - If "log2", then max_features=log2(n_features). - If None, then max_features=n_features. Note: the search for a split does not stop until at least one valid partition of the node samples is found, even if it requires to effectively inspect more than max_features features. random_state : int, RandomState instance or None, default=None Controls the randomness of the estimator. The features are always randomly permuted at each split, even if splitter is set to "best". When max_features < n_features, the algorithm will select max_features at random at each split before finding the best split among them. But the best found split may vary across different runs, even if max_features=n_features. That is the case, if the improvement of the criterion is identical for several splits and one split has to be selected at random. To obtain a deterministic behaviour during fitting, random_state has to be fixed to an integer. min_impurity_decrease : float, default=0.0 A node will be split if this split induces a decrease of the impurity greater than or equal to this value. The weighted impurity decrease equation is the following:: N_t / N * (impurity - N_t_R / N_t * right_impurity - N_t_L / N_t * left_impurity) where N is the total number of samples, N_t is the number of samples at the current node, N_t_L is the number of samples in the left child, and N_t_R is the number of samples in the right child. N, N_t, N_t_R and N_t_L all refer to the weighted sum, if sample_weight is passed. min_balancedness_tol: float in [0, .5], default=.45 How imbalanced a split we can tolerate. This enforces that each split leaves at least (.5 - min_balancedness_tol) fraction of samples on each side of the split; or fraction of the total weight of samples, when sample_weight is not None. Default value, ensures that at least 5% of the parent node weight falls in each side of the split. Set it to 0.0 for no balancedness and to .5 for perfectly balanced splits. For the formal inference theory to be valid, this has to be any positive constant bounded away from zero. honest: bool, default=True Whether the data should be split in two equally sized samples, such that the one half-sample is used to determine the optimal split at each node and the other sample is used to determine the value of every node. Attributes ---------- feature_importances_ : ndarray of shape (n_features,) The feature importances based on the amount of parameter heterogeneity they create. The higher, the more important the feature. The importance of a feature is computed as the (normalized) total heterogeneity that the feature creates. Each split that the feature was chosen adds:: parent_weight * (left_weight * right_weight) * mean((value_left[k] - value_right[k])**2) / parent_weight**2 to the importance of the feature. Each such quantity is also weighted by the depth of the split. By default splits below max_depth=4 are not used in this calculation and also each split at depth depth, is re-weighted by 1 / (1 + depth)**2.0. See the method feature_importances for a method that allows one to change these defaults. max_features_ : int The inferred value of max_features. n_features_ : int The number of features when fit is performed. n_outputs_ : int The number of outputs when fit is performed. n_relevant_outputs_ : int The first n_relevant_outputs_ where the ones we cared about when fit was performed. n_y_ : int The raw label dimension when fit is performed. n_samples_ : int The number of training samples when fit is performed. honest_ : int Whether honesty was enabled when fit was performed tree_ : Tree instance The underlying Tree object. Please refer to help(econml.tree._tree.Tree) for attributes of Tree object. References ---------- .. [grftree1] Athey, Susan, Julie Tibshirani, and Stefan Wager. "Generalized random forests." The Annals of Statistics 47.2 (2019): 1148-1178 https://arxiv.org/pdf/1610.01271.pdf """
[docs] def __init__(self, *, criterion="mse", splitter="best", max_depth=None, min_samples_split=10, min_samples_leaf=5, min_weight_fraction_leaf=0., min_var_leaf=None, min_var_leaf_on_val=False, max_features=None, random_state=None, min_impurity_decrease=0., min_balancedness_tol=0.45, honest=True): super().__init__(criterion=criterion, splitter=splitter, max_depth=max_depth, min_samples_split=min_samples_split, min_samples_leaf=min_samples_leaf, min_weight_fraction_leaf=min_weight_fraction_leaf, min_var_leaf=min_var_leaf, min_var_leaf_on_val=min_var_leaf_on_val, max_features=max_features, random_state=random_state, min_impurity_decrease=min_impurity_decrease, min_balancedness_tol=min_balancedness_tol, honest=honest)
def _get_valid_criteria(self): return CRITERIA_GRF def _get_valid_min_var_leaf_criteria(self): return (LinearMomentGRFCriterion,) def _get_store_jac(self): return True
[docs] def init(self,): """ This method should be called before fit. We added this pre-fit step so that this step can be executed without parallelism as it contains code that holds the gil and can hinder parallel execution. We also did not merge this step to __init__ as we want __init__ to just be storing the parameters for easy cloning. We also don't want to directly pass a RandomState object as random_state, as we want to keep the starting seed to be able to replicate the randomness of the object outside the object. """ self.random_seed_ = self.random_state self.random_state_ = check_random_state(self.random_seed_) return self
[docs] def fit(self, X, y, n_y, n_outputs, n_relevant_outputs, sample_weight=None, check_input=True): """ Fit the tree from the data Parameters ---------- X : (n, d) array The features to split on y : (n, m) array All the variables required to calculate the criterion function, evaluate splits and estimate local values, i.e. all the values that go into the moment function except X. n_y, n_outputs, n_relevant_outputs : auxiliary info passed to the criterion objects that help the object parse the variable y into each separate variable components. - In the case when isinstance(criterion, LinearMomentGRFCriterion), then the first n_y columns of y are the raw outputs, the next n_outputs columns contain the A part of the moment and the next n_outputs * n_outputs columnts contain the J part of the moment in row contiguous format. The first n_relevant_outputs parameters of the linear moment are the ones that we care about. The rest are nuisance parameters. sample_weight : (n,) array, default=None The sample weights check_input : bool, defaul=True Whether to check the input parameters for validity. Should be set to False to improve running time in parallel execution, if the variables have already been checked by the forest class that spawned this tree. """ return super().fit(X, y, n_y, n_outputs, n_relevant_outputs, sample_weight=sample_weight, check_input=check_input)
[docs] def predict(self, X, check_input=True): """Return the prefix of relevant fitted local parameters for each X, i.e. theta(X). Parameters ---------- X : {array-like} of shape (n_samples, n_features) The input samples. Internally, it will be converted to dtype=np.float64. check_input : bool, default=True Allow to bypass several input checking. Don't use this parameter unless you know what you do. Returns ------- theta(X)[:n_relevant_outputs] : array-like of shape (n_samples, n_relevant_outputs) The estimated relevant parameters for each row of X """ check_is_fitted(self) X = self._validate_X_predict(X, check_input) pred = self.tree_.predict(X) return pred
[docs] def predict_full(self, X, check_input=True): """Return the fitted local parameters for each X, i.e. theta(X). Parameters ---------- X : {array-like} of shape (n_samples, n_features) The input samples. Internally, it will be converted to dtype=np.float64. check_input : bool, default=True Allow to bypass several input checking. Don't use this parameter unless you know what you do. Returns ------- theta(X) : array-like of shape (n_samples, n_outputs) All the estimated parameters for each row of X """ check_is_fitted(self) X = self._validate_X_predict(X, check_input) pred = self.tree_.predict_full(X) return pred
[docs] def predict_alpha_and_jac(self, X, check_input=True): """Predict the local jacobian E[J | X=x] and the local alpha E[A | X=x] of a linear moment equation. Parameters ---------- X : {array-like} of shape (n_samples, n_features) The input samples. Internally, it will be converted to dtype=np.float64 check_input : bool, default=True Allow to bypass several input checking. Don't use this parameter unless you know what you do. Returns ------- alpha : array-like of shape (n_samples, n_outputs) The local alpha E[A | X=x] for each sample x jac : array-like of shape (n_samples, n_outputs * n_outputs) The local jacobian E[J | X=x] flattened in a C contiguous format """ check_is_fitted(self) X = self._validate_X_predict(X, check_input) return self.tree_.predict_precond_and_jac(X)
[docs] def predict_moment(self, X, parameter, check_input=True): """ Predict the local moment value for each sample and at the given parameter:: E[J | X=x] theta(x) - E[A | X=x] Parameters ---------- X : {array-like} of shape (n_samples, n_features) The input samples. Internally, it will be converted to dtype=np.float64 parameter : {array-like} of shape (n_samples, n_outputs) A parameter estimate for each sample check_input : bool, default=True Allow to bypass several input checking. Don't use this parameter unless you know what you do. Returns ------- moment : array-like of shape (n_samples, n_outputs) The local moment E[J | X=x] theta(x) - E[A | X=x] for each sample x """ alpha, jac = self.predict_alpha_and_jac(X) return alpha - np.einsum('ijk,ik->ij', jac.reshape((-1, self.n_outputs_, self.n_outputs_)), parameter)
[docs] def feature_importances(self, max_depth=4, depth_decay_exponent=2.0): """The feature importances based on the amount of parameter heterogeneity they create. The higher, the more important the feature. The importance of a feature is computed as the (normalized) total heterogeneity that the feature creates. Each split that the feature was chosen adds:: parent_weight * (left_weight * right_weight) * mean((value_left[k] - value_right[k])**2) / parent_weight**2 to the importance of the feature. Each such quantity is also weighted by the depth of the split. Parameters ---------- max_depth : int, default=4 Splits of depth larger than max_depth are not used in this calculation depth_decay_exponent: double, default=2.0 The contribution of each split to the total score is re-weighted by 1 / (1 + depth)**2.0. Returns ------- feature_importances_ : ndarray of shape (n_features,) Normalized total parameter heterogeneity inducing importance of each feature """ check_is_fitted(self) return self.tree_.compute_feature_heterogeneity_importances(normalize=True, max_depth=max_depth, depth_decay=depth_decay_exponent)
@property def feature_importances_(self): return self.feature_importances()