.. ****************************************************************************** .. * Copyright 2020-2022 Intel Corporation .. * .. * Licensed under the Apache License, Version 2.0 (the "License"); .. * you may not use this file except in compliance with the License. .. * You may obtain a copy of the License at .. * .. * http://www.apache.org/licenses/LICENSE-2.0 .. * .. * Unless required by applicable law or agreed to in writing, software .. * distributed under the License is distributed on an "AS IS" BASIS, .. * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. .. * See the License for the specific language governing permissions and .. * limitations under the License. .. *******************************************************************************/ .. default-domain:: cpp .. _alg_df: ================================================== Decision Forest Classification and Regression (DF) ================================================== .. include:: ../../../includes/ensembles/df-introduction.rst ------------------------ Mathematical formulation ------------------------ .. _df_t_math: Training -------- Given :math:n feature vectors :math:X=\{x_1=(x_{11},\ldots,x_{1p}),\ldots,x_n=(x_{n1},\ldots,x_{np})\} of size :math:p, their non-negative observation weights :math:W=\{w_1,\ldots,w_n\} and :math:n responses :math:Y=\{y_1,\ldots,y_n\}, .. tabs:: .. group-tab:: Classification - :math:y_i \in \{0, \ldots, C-1\}, where :math:C is the number of classes .. group-tab:: Regression - :math:y_i \in \mathbb{R} the problem is to build a decision forest classification or regression model. The library uses the following algorithmic framework for the training stage. Let :math:S = (X, Y) be the set of observations. Given positive integer parameters, such as the number of trees :math:B, the bootstrap parameter :math:N = f*n, where :math:f is a fraction of observations used for a training of each tree in the forest, and the number of features per node :math:m, the algorithm does the following for :math:b = 1, \ldots, B: - Selects randomly with replacement the set :math:D_b of :math:N vectors from the set :math:S. The set :math:D_b is called a *bootstrap* set. - Trains a :ref:decision tree
classifier :math:T_b on :math:D_b using parameter :math:m for each tree. :ref:Decision tree
:math:T is trained using the training set :math:D of size :math:N. Each node :math:t in the tree corresponds to the subset :math:D_t of the training set :math:D, with the root node being :math:D itself. Each internal node :math:t represents a binary test (split) that divides the subset :math:X_t in two subsets, :math:X_{t_L} and :math:X_{t_R}, corresponding to their children, :math:t_L and :math:t_R. .. _df_t_math_dense: Training method: *Dense* ++++++++++++++++++++++++ In *dense* training method, all possible splits for each feature are taken from the subset of selected features for the current node and evaluated for best split computation. .. _df_t_math_hist: Training method: *Hist* +++++++++++++++++++++++ In *hist* training method, only a selected subset of splits is considered for best split computation. This subset of splits is computed for each feature at the initialization stage of the algorithm. After computing the subset of splits, each value from the initially provided data is substituted with the value of the corresponding bin. Bins are continuous intervals between selected splits. Split Criteria ++++++++++++++ The metric for measuring the best split is called *impurity*, :math:i(t). It generally reflects the homogeneity of responses within the subset :math:D_t in the node :math:t. .. tabs:: .. group-tab:: Classification *Gini index* is an impurity metric for classification, calculated as follows: .. math:: {I}_{Gini}\left(D\right)=1-\sum _{i=0}^{C-1}{p}_{i}^{2} where - :math:D is a set of observations that reach the node; - :math:p_i is specified in the table below: .. tabularcolumns:: |\Y{0.2}|\Y{0.8}| .. list-table:: Decision Forest Split Criteria Calculation :widths: 10 10 :header-rows: 1 :align: left :class: longtable * - Without sample weights - With sample weights * - :math:p_i is the observed fraction of observations that belong to class :math:i in :math:D - :math:p_i is the observed weighted fraction of observations that belong to class :math:i in :math:D: .. math:: p_i = \frac{\sum_{d \in \{d \in D | y_d = i \}} W_d}{\sum_{d \in D} W_d} .. group-tab:: Regression *MSE* is an impurity metric for regression, calculated as follows: .. tabularcolumns:: |\Y{0.2}|\Y{0.8}| .. list-table:: MSE Impurity Metric :widths: 10 10 :header-rows: 1 :align: left :class: longtable * - Without sample weights - With sample weights * - :math:I_{\mathrm{MSE}}\left(D\right) = \frac{1}{W(D)} \sum _{i=1}^{W(D)}{\left(y_i - \frac{1}{W(D)} \sum _{j=1}^{W(D)} y_j \right)}^{2} - :math:I_{\mathrm{MSE}}\left(D\right) = \frac{1}{W(D)} \sum _{i \in D}{w_i \left(y_i - \frac{1}{W(D)} \sum _{j \in D} w_j y_j \right)}^{2} * - :math:W(S) = \sum_{s \in S} 1, which is equivalent to the number of elements in :math:S - :math:W(S) = \sum_{s \in S} w_s Let the *impurity decrease* in the node :math:t be .. math:: \Delta i\left(t\right)=i\left(t\right)–\frac{|{D}_{t}{}_{{}_{L}}|}{|{D}_{t}|}i\left({t}_{L}\right)–\frac{|{D}_{t}{}_{{}_{R}}|}{|{D}_{t}|}i\left({t}_{R}\right).\text{ } Termination Criteria ++++++++++++++++++++ The library supports the following termination criteria of decision forest training: Minimal number of observations in a leaf node Node :math:t is not processed if :math:|D_t| is smaller than the predefined value. Splits that produce nodes with the number of observations smaller than that value are not allowed. Minimal number of observations in a split node Node :math:t is not processed if :math:|D_t| is smaller than the predefined value. Splits that produce nodes with the number of observations smaller than that value are not allowed. Minimum weighted fraction of the sum total of weights of all the input observations required to be at a leaf node Node :math:t is not processed if :math:|D_t| is smaller than the predefined value. Splits that produce nodes with the number of observations smaller than that value are not allowed. Maximal tree depth Node :math:t is not processed if its depth in the tree reached the predefined value. Impurity threshold Node :math:t is not processed if its impurity is smaller than the predefined threshold. Maximal number of leaf nodes Grow trees with positive maximal number of leaf nodes in a :ref:best-first  fashion. Best nodes are defined by relative reduction in impurity. If maximal number of leaf nodes equals zero, then this criterion does not limit the number of leaf nodes, and trees grow in a :ref:depth-first  fashion. Tree Building Strategies ++++++++++++++++++++++++ Maximal number of leaf nodes defines the strategy of tree building: :ref:depth-first  or :ref:best-first . .. _df_t_depth_first_strategy: Depth-first Strategy ~~~~~~~~~~~~~~~~~~~~ If maximal number of leaf nodes equals zero, a :ref:decision tree
is built using depth-first strategy. In each terminal node :math:t, the following recursive procedure is applied: - Stop if the termination criteria are met. - Choose randomly without replacement :math:m feature indices :math:J_t \in \{0, 1, \ldots, p-1\}. - For each :math:j \in J_t, find the best split :math:s_{j,t} that partitions subset :math:D_t and maximizes impurity decrease :math:\Delta i(t). - A node is a split if this split induces a decrease of the impurity greater than or equal to the predefined value. Get the best split :math:s_t that maximizes impurity decrease :math:\Delta i in all :math:s_{j,t} splits. - Apply this procedure recursively to :math:t_L and :math:t_R. .. _df_t_best_first_strategy: Best-first Strategy ~~~~~~~~~~~~~~~~~~~ If maximal number of leaf nodes is positive, a :ref:decision tree
is built using best-first strategy. In each terminal node :math:t, the following steps are applied: - Stop if the termination criteria are met. - Choose randomly without replacement :math:m feature indices :math:J_t \in \{0, 1, \ldots, p-1\}. - For each :math:j \in J_t, find the best split :math:s_{j,t} that partitions subset :math:D_t and maximizes impurity decrease :math:\Delta i(t). - A node is a split if this split induces a decrease of the impurity greater than or equal to the predefined value and the number of split nodes is less or equal to :math:\mathrm{maxLeafNodes} – 1. Get the best split :math:s_t that maximizes impurity decrease :math:\Delta i in all :math:s_{j,t} splits. - Put a node into a sorted array, where sort criterion is the improvement in impurity :math:\Delta i(t)|D_t|. The node with maximal improvement is the first in the array. For a leaf node, the improvement in impurity is zero. - Apply this procedure to :math:t_L and :math:t_R and grow a tree one by one getting the first element from the array until the array is empty. .. _df_i_math: Inference --------- Given decision forest classification or regression model and vectors :math:x_1, \ldots, x_r, the problem is to calculate the responses for those vectors. .. _df_i_math_dense_hist: Inference methods: *Dense* and *Hist* ------------------------------------- *Dense* and *hist* inference methods perform prediction in the same way. To solve the problem for each given query vector :math:x_i, the algorithm does the following: .. tabs:: .. group-tab:: Classification For each tree in the forest, it finds the leaf node that gives :math:x_i its label. The label :math:y that the majority of trees in the forest vote for is chosen as the predicted label for the query vector :math:x_i. .. group-tab:: Regression For each tree in the forest, it finds the leaf node that gives :math:x_i the response as the mean of dependent variables. The mean of responses from all trees in the forest is the predicted response for the query vector :math:x_i. Additional Characteristics Calculated by the Decision Forest ------------------------------------------------------------ Decision forests can produce additional characteristics, such as an estimate of generalization error and an importance measure (relative decisive power) of each of p features (variables). Out-of-bag Error ---------------- The estimate of the generalization error based on the training data can be obtained and calculated as follows: .. tabs:: .. group-tab:: Classification - For each vector :math:x_i in the dataset :math:X, predict its label :math:\hat{y_i} by having the majority of votes from the trees that contain :math:x_i in their OOB set, and vote for that label. - Calculate the OOB error of the decision forest :math:T as the average of misclassifications: .. math:: OOB(T) = \frac{1}{|{D}^{\text{'}}|}\sum _{y_i \in {D}^{\text{'}}}I\{y_i \ne \hat{y_i}\}\text{,where }{D}^{\text{'}}={\bigcup }_{b=1}^{B}\overline{D_b}. - If OOB error value per each observation is required, then calculate the prediction error for :math:x_i: :math:OOB(x_i) = I\{{y}_{i}\ne \hat{{y}_{i}}\} .. group-tab:: Regression - For each vector :math:x_i in the dataset :math:X, predict its response :math:\hat{y_i} as the mean of prediction from the trees that contain :math:x_i in their OOB set: :math:\hat{y_i} = \frac{1}{{|B}_{i}|}\sum _{b=1}^{|B_i|}\hat{y_{ib}}, where :math:B_i= \bigcup{T_b}: x_i \in \overline{D_b} and :math:\hat{y_{ib}} is the result of prediction :math:x_i by :math:T_b. - Calculate the OOB error of the decision forest :math:T as the Mean-Square Error (MSE): .. math:: OOB(T) = \frac{1}{|{D}^{\text{'}}|}\sum _{{y}_{i} \in {D}^{\text{'}}}\sum {(y_i-\hat{y_i})}^{2}, \text{where } {D}^{\text{'}}={\bigcup}_{b=1}^{B}\overline{{D}_{b}} - If OOB error value per each observation is required, then calculate the prediction error for :math:x_i: .. math:: OOB(x_i) = {(y_i-\hat{y_i})}^{2} Variable Importance ------------------- There are two main types of variable importance measures: - *Mean Decrease Impurity* importance (MDI) Importance of the :math:j-th variable for predicting :math:Y is the sum of weighted impurity decreases :math:p(t) \Delta i(s_t, t) for all nodes :math:t that use :math:x_j, averaged over all :math:B trees in the forest: .. math:: MDI\left(j\right)=\frac{1}{B}\sum _{b=1}^{B} \sum _{t\in {T}_{b}:v\left({s}_{t}\right)=j}p\left(t\right)\Delta i\left({s}_{t},t\right), where :math:p\left(t\right)=\frac{|{D}_{t}|}{|D|} is the fraction of observations reaching node :math:t in the tree :math:T_b, and :math:v(s_t) is the index of the variable used in split :math:s_t. - *Mean Decrease Accuracy* (MDA) Importance of the :math:j-th variable for predicting :math:Y is the average increase in the OOB error over all trees in the forest when the values of the :math:j-th variable are randomly permuted in the OOB set. For that reason, this latter measure is also known as *permutation importance*. In more details, the library calculates MDA importance as follows: - Let :math:\pi (X,j) be the set of feature vectors where the :math:j-th variable is randomly permuted over all vectors in the set. - Let :math:E_b be the OOB error calculated for :math:T_b: on its out-of-bag dataset :math:\overline{D_b}. - Let :math:E_{b,j} be the OOB error calculated for :math:T_b: using :math:\pi \left(\overline{{X}_{b}},j\right), and its out-of-bag dataset :math:\overline{D_b} is permuted on the :math:j-th variable. Then * :math:{\delta }_{b,j}={E}_{b}-{E}_{b,j} is the OOB error increase for the tree :math:T_b. * :math:Raw MDA\left(j\right)=\frac{1}{B}\sum _{b=1}^{B}{\delta }_{b,j} is MDA importance. * :math:Scaled MDA\left(j\right)=\frac{Raw MDA\left({x}_{j}\right)}{\frac{{\sigma }_{j}}{\sqrt{B}}}, where :math:{\sigma }_{j}^{2} is the variance of :math:D_{b,j} --------------------- Programming Interface --------------------- Refer to :ref:API Reference: Decision Forest Classification and Regression . ---------------- Distributed mode ---------------- The algorithm supports distributed execution in SMPD mode (only on GPU).