# econml.grf._base_grftree.GRFTree

class econml.grf._base_grftree.GRFTree(*, criterion='mse', splitter='best', max_depth=None, min_samples_split=10, min_samples_leaf=5, min_weight_fraction_leaf=0.0, min_var_leaf=None, min_var_leaf_on_val=False, max_features=None, random_state=None, min_impurity_decrease=0.0, min_balancedness_tol=0.45, honest=True)[source]

Bases: `econml.tree._tree_classes.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:

```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:

```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:

```sum_{child} weight(child) * rho(child).T @ rho(child)
```

This can be thought as an approximation to the ideal heterogeneity score:

```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:

```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, {“auto”, “sqrt”, “log2”}, or None, 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.

feature_importances_

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.

Type

ndarray of shape (n_features,)

max_features_

The inferred value of max_features.

Type

int

n_features_

The number of features when `fit` is performed.

Type

int

n_outputs_

The number of outputs when `fit` is performed.

Type

int

n_relevant_outputs_

The first n_relevant_outputs_ where the ones we cared about when `fit` was performed.

Type

int

n_y_

The raw label dimension when `fit` is performed.

Type

int

n_samples_

The number of training samples when `fit` is performed.

Type

int

honest_

Whether honesty was enabled when `fit` was performed

Type

int

tree_

The underlying Tree object. Please refer to `help(econml.tree._tree.Tree)` for attributes of Tree object.

Type

Tree instance

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

__init__(*, criterion='mse', splitter='best', max_depth=None, min_samples_split=10, min_samples_leaf=5, min_weight_fraction_leaf=0.0, min_var_leaf=None, min_var_leaf_on_val=False, max_features=None, random_state=None, min_impurity_decrease=0.0, min_balancedness_tol=0.45, honest=True)[source]

Methods

 `__init__`(*[, criterion, splitter, ...]) `apply`(X[, check_input]) Return the index of the leaf that each sample is predicted as. `decision_path`(X[, check_input]) Return the decision path in the tree. `feature_importances`([max_depth, ...]) 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::. `fit`(X, y, n_y, n_outputs, n_relevant_outputs) Fit the tree from the data Return the depth of the decision tree. Return the number of leaves of the decision tree. `get_params`([deep]) Get parameters for this estimator. Regenerate the train_test_split of input sample indices that was used for the training and the evaluation split of the honest tree construction structure. This method should be called before fit. `predict`(X[, check_input]) Return the prefix of relevant fitted local parameters for each X, i.e. theta(X). `predict_alpha_and_jac`(X[, check_input]) Predict the local jacobian `E[J | X=x]` and the local alpha `E[A | X=x]` of a linear moment equation. `predict_full`(X[, check_input]) Return the fitted local parameters for each X, i.e. theta(X). `predict_moment`(X, parameter[, check_input]) Predict the local moment value for each sample and at the given parameter. `set_params`(**params) Set the parameters of this estimator.

Attributes

apply(X, check_input=True)

Return the index of the leaf that each sample is predicted as.

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

X_leaves – For each datapoint x in X, return the index of the leaf x ends up in. Leaves are numbered within `[0; self.tree_.node_count)`, possibly with gaps in the numbering.

Return type

array_like of shape (n_samples,)

decision_path(X, check_input=True)

Return the decision path in the tree.

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

indicator – Return a node indicator CSR matrix where non zero elements indicates that the samples goes through the nodes.

Return type

sparse matrix of shape (n_samples, n_nodes)

feature_importances(max_depth=4, depth_decay_exponent=2.0)[source]

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_ – Normalized total parameter heterogeneity inducing importance of each feature

Return type

ndarray of shape (n_features,)

fit(X, y, n_y, n_outputs, n_relevant_outputs, sample_weight=None, check_input=True)[source]

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.

get_depth()

Return the depth of the decision tree. The depth of a tree is the maximum distance between the root and any leaf.

Returns

self.tree_.max_depth – The maximum depth of the tree.

Return type

int

get_n_leaves()

Return the number of leaves of the decision tree.

Returns

self.tree_.n_leaves – Number of leaves.

Return type

int

get_params(deep=True)

Get parameters for this estimator.

Parameters

deep (bool, default=True) – If True, will return the parameters for this estimator and contained subobjects that are estimators.

Returns

params – Parameter names mapped to their values.

Return type

dict

get_train_test_split_inds()

Regenerate the train_test_split of input sample indices that was used for the training and the evaluation split of the honest tree construction structure. Uses the same random seed that was used at `fit` time and re-generates the indices.

init()[source]

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.

predict(X, check_input=True)[source]

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)[ – The estimated relevant parameters for each row of X

Return type

n_relevant_outputs] : array_like of shape (n_samples, n_relevant_outputs)

predict_alpha_and_jac(X, check_input=True)[source]

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

predict_full(X, check_input=True)[source]

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) – All the estimated parameters for each row of X

Return type

array_like of shape (n_samples, n_outputs)

predict_moment(X, parameter, check_input=True)[source]

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 – The local moment E[J | X=x] theta(x) - E[A | X=x] for each sample x

Return type

array_like of shape (n_samples, n_outputs)

set_params(**params)

Set the parameters of this estimator.

The method works on simple estimators as well as on nested objects (such as `Pipeline`). The latter have parameters of the form `<component>__<parameter>` so that it’s possible to update each component of a nested object.

Parameters

**params (dict) – Estimator parameters.

Returns

self – Estimator instance.

Return type

estimator instance