#region License Information /* HeuristicLab * Copyright (C) 2002-2015 Heuristic and Evolutionary Algorithms Laboratory (HEAL) * and the BEACON Center for the Study of Evolution in Action. * * This file is part of HeuristicLab. * * HeuristicLab is free software: you can redistribute it and/or modify * it under the terms of the GNU General Public License as published by * the Free Software Foundation, either version 3 of the License, or * (at your option) any later version. * * HeuristicLab is distributed in the hope that it will be useful, * but WITHOUT ANY WARRANTY; without even the implied warranty of * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the * GNU General Public License for more details. * * You should have received a copy of the GNU General Public License * along with HeuristicLab. If not, see . */ #endregion using System; using System.Collections; using System.Collections.Generic; using System.Diagnostics; using System.Linq; using HeuristicLab.Core; using HeuristicLab.Problems.DataAnalysis; namespace HeuristicLab.Algorithms.DataAnalysis { // This class implements a greedy decision tree learner which selects splits with the maximum reduction in sum of squared errors. // The tree builder also tracks variable relevance metrics based on the splits and improvement after the split. // The implementation is tuned for gradient boosting where multiple trees have to be calculated for the same training data // each time with a different target vector. Vectors of idx to allow iteration of intput variables in sorted order are // pre-calculated so that optimal thresholds for splits can be calculated in O(n) for each input variable. // After each split the row idx are partitioned in a left an right part. public class RegressionTreeBuilder { private readonly IRandom random; private readonly IRegressionProblemData problemData; private readonly int nCols; private readonly double[][] x; // all training data (original order from problemData), x is constant private double[] y; // training labels (original order from problemData), y can be changed private Dictionary sumImprovements; // for variable relevance calculation private readonly string[] allowedVariables; // all variables in shuffled order private Dictionary varName2Index; // maps the variable names to column indexes private int effectiveVars; // number of variables that are used from allowedVariables private int effectiveRows; // number of rows that are used from private readonly int[][] sortedIdxAll; private readonly int[][] sortedIdx; // random selection from sortedIdxAll (for r < 1.0) private int calls = 0; // helper arrays which are allocated to maximal necessary size only once in the ctor private readonly int[] internalIdx, which, leftTmp, rightTmp; private readonly double[] outx; private readonly int[] outSortedIdx; private RegressionTreeModel.TreeNode[] tree; // tree is represented as a flat array of nodes private int curTreeNodeIdx; // the index where the next tree node is stored private class Partition { public int ParentNodeIdx { get; set; } public int Depth { get; set; } public int StartIdx { get; set; } public int EndIndex { get; set; } public bool Left { get; set; } } private readonly SortedList queue; // prepare and allocate buffer variables in ctor public RegressionTreeBuilder(IRegressionProblemData problemData, IRandom random) { this.problemData = problemData; this.random = random; var rows = problemData.TrainingIndices.Count(); this.nCols = problemData.AllowedInputVariables.Count(); allowedVariables = problemData.AllowedInputVariables.ToArray(); varName2Index = new Dictionary(allowedVariables.Length); for (int i = 0; i < allowedVariables.Length; i++) varName2Index.Add(allowedVariables[i], i); sortedIdxAll = new int[nCols][]; sortedIdx = new int[nCols][]; sumImprovements = new Dictionary(); internalIdx = new int[rows]; which = new int[rows]; leftTmp = new int[rows]; rightTmp = new int[rows]; outx = new double[rows]; outSortedIdx = new int[rows]; queue = new SortedList(); x = new double[nCols][]; y = problemData.Dataset.GetDoubleValues(problemData.TargetVariable, problemData.TrainingIndices).ToArray(); int col = 0; foreach (var inputVariable in problemData.AllowedInputVariables) { x[col] = problemData.Dataset.GetDoubleValues(inputVariable, problemData.TrainingIndices).ToArray(); sortedIdxAll[col] = Enumerable.Range(0, rows).OrderBy(r => x[col][r]).ToArray(); sortedIdx[col] = new int[rows]; col++; } } // simple API produces a single regression tree optimizing sum of squared errors // this can be used if only a simple regression tree should be produced // for a set of trees use the method CreateRegressionTreeForGradientBoosting below // // r and m work in the same way as for alglib random forest // r is fraction of rows to use for training // m is fraction of variables to use for training public IRegressionModel CreateRegressionTree(int maxDepth, double r = 0.5, double m = 0.5) { // subtract mean of y first var yAvg = y.Average(); for (int i = 0; i < y.Length; i++) y[i] -= yAvg; var seLoss = new SquaredErrorLoss(); var zeros = Enumerable.Repeat(0.0, y.Length); var ones = Enumerable.Repeat(1.0, y.Length); var model = CreateRegressionTreeForGradientBoosting(y, maxDepth, problemData.TrainingIndices.ToArray(), seLoss.GetLineSearchFunc(y, zeros, ones), r, m); return new GradientBoostedTreesModel(new[] { new ConstantRegressionModel(yAvg), model }, new[] { 1.0, 1.0 }); } // specific interface that allows to specify the target labels and the training rows which is necessary when for gradient boosted trees public IRegressionModel CreateRegressionTreeForGradientBoosting(double[] y, int maxDepth, int[] idx, LineSearchFunc lineSearch, double r = 0.5, double m = 0.5) { Debug.Assert(maxDepth > 0); Debug.Assert(r > 0); Debug.Assert(r <= 1.0); Debug.Assert(y.Count() == this.y.Length); Debug.Assert(m > 0); Debug.Assert(m <= 1.0); this.y = y; // y is changed in gradient boosting // shuffle row idx HeuristicLab.Random.ListExtensions.ShuffleInPlace(idx, random); int nRows = idx.Count(); // shuffle variable idx HeuristicLab.Random.ListExtensions.ShuffleInPlace(allowedVariables, random); // only select a part of the rows and columns randomly effectiveRows = (int)Math.Ceiling(nRows * r); effectiveVars = (int)Math.Ceiling(nCols * m); // the which array is used for partining row idxs Array.Clear(which, 0, which.Length); // mark selected rows for (int row = 0; row < effectiveRows; row++) { which[idx[row]] = 1; // we use the which vector as a temporary variable here internalIdx[row] = idx[row]; } for (int col = 0; col < nCols; col++) { int i = 0; for (int row = 0; row < nRows; row++) { if (which[sortedIdxAll[col][row]] > 0) { Debug.Assert(i < effectiveRows); sortedIdx[col][i] = sortedIdxAll[col][row]; i++; } } } // prepare array for the tree nodes (a tree of maxDepth=1 has 1 node, a tree of maxDepth=d has 2^d - 1 nodes) int numNodes = (int)Math.Pow(2, maxDepth) - 1; this.tree = new RegressionTreeModel.TreeNode[numNodes]; this.curTreeNodeIdx = 0; // start and end idx are inclusive queue.Add(calls++, new Partition() { ParentNodeIdx = -1, Depth = maxDepth, StartIdx = 0, EndIndex = effectiveRows - 1 }); CreateRegressionTreeForIdx(lineSearch); return new RegressionTreeModel(tree); } private void CreateRegressionTreeForIdx(LineSearchFunc lineSearch) { while (queue.Any()) { var f = queue.First().Value; // actually a stack queue.RemoveAt(0); var depth = f.Depth; var startIdx = f.StartIdx; var endIdx = f.EndIndex; Debug.Assert(endIdx - startIdx >= 0); Debug.Assert(startIdx >= 0); Debug.Assert(endIdx < internalIdx.Length); double threshold; string bestVariableName; // stop when only one row is left or no split is possible if (depth <= 1 || endIdx - startIdx == 0 || !FindBestVariableAndThreshold(startIdx, endIdx, out threshold, out bestVariableName)) { CreateLeafNode(startIdx, endIdx, lineSearch); if (f.ParentNodeIdx >= 0) if (f.Left) { tree[f.ParentNodeIdx].leftIdx = curTreeNodeIdx; } else { tree[f.ParentNodeIdx].rightIdx = curTreeNodeIdx; } curTreeNodeIdx++; } else { int splitIdx; CreateInternalNode(f.StartIdx, f.EndIndex, bestVariableName, threshold, out splitIdx); // connect to parent tree if (f.ParentNodeIdx >= 0) if (f.Left) { tree[f.ParentNodeIdx].leftIdx = curTreeNodeIdx; } else { tree[f.ParentNodeIdx].rightIdx = curTreeNodeIdx; } Debug.Assert(splitIdx + 1 <= endIdx); Debug.Assert(startIdx <= splitIdx); queue.Add(calls++, new Partition() { ParentNodeIdx = curTreeNodeIdx, Left = true, Depth = depth - 1, StartIdx = startIdx, EndIndex = splitIdx }); // left part before right part (stack organization) queue.Add(calls++, new Partition() { ParentNodeIdx = curTreeNodeIdx, Left = false, Depth = depth - 1, StartIdx = splitIdx + 1, EndIndex = endIdx }); curTreeNodeIdx++; } } } private void CreateLeafNode(int startIdx, int endIdx, LineSearchFunc lineSearch) { // max depth reached or only one element tree[curTreeNodeIdx].varName = RegressionTreeModel.TreeNode.NO_VARIABLE; tree[curTreeNodeIdx].val = lineSearch(internalIdx, startIdx, endIdx); } // routine for building the tree for the row idx stored in internalIdx between startIdx and endIdx // the lineSearch function calculates the optimal prediction value for tree leaf nodes // (in the case of squared errors it is the average of target values for the rows represented by the node) // startIdx and endIdx are inclusive private void CreateInternalNode(int startIdx, int endIdx, string splittingVar, double threshold, out int splitIdx) { int bestVarIdx = varName2Index[splittingVar]; // split - two pass // store which index goes where for (int k = startIdx; k <= endIdx; k++) { if (x[bestVarIdx][internalIdx[k]] <= threshold) which[internalIdx[k]] = -1; // left partition else which[internalIdx[k]] = 1; // right partition } // partition sortedIdx for each variable int i; int j; for (int col = 0; col < nCols; col++) { i = 0; j = 0; int k; for (k = startIdx; k <= endIdx; k++) { Debug.Assert(Math.Abs(which[sortedIdx[col][k]]) == 1); if (which[sortedIdx[col][k]] < 0) { leftTmp[i++] = sortedIdx[col][k]; } else { rightTmp[j++] = sortedIdx[col][k]; } } Debug.Assert(i > 0); // at least on element in the left partition Debug.Assert(j > 0); // at least one element in the right partition Debug.Assert(i + j == endIdx - startIdx + 1); k = startIdx; for (int l = 0; l < i; l++) sortedIdx[col][k++] = leftTmp[l]; for (int l = 0; l < j; l++) sortedIdx[col][k++] = rightTmp[l]; } // partition row indices i = startIdx; j = endIdx; while (i <= j) { Debug.Assert(Math.Abs(which[internalIdx[i]]) == 1); Debug.Assert(Math.Abs(which[internalIdx[j]]) == 1); if (which[internalIdx[i]] < 0) i++; else if (which[internalIdx[j]] > 0) j--; else { Debug.Assert(which[internalIdx[i]] > 0); Debug.Assert(which[internalIdx[j]] < 0); // swap int tmp = internalIdx[i]; internalIdx[i] = internalIdx[j]; internalIdx[j] = tmp; i++; j--; } } Debug.Assert(j + 1 == i); Debug.Assert(i <= endIdx); Debug.Assert(startIdx <= j); tree[curTreeNodeIdx].varName = splittingVar; tree[curTreeNodeIdx].val = threshold; splitIdx = j; } private bool FindBestVariableAndThreshold(int startIdx, int endIdx, out double threshold, out string bestVar) { Debug.Assert(startIdx < endIdx + 1); // at least 2 elements int rows = endIdx - startIdx + 1; Debug.Assert(rows >= 2); double sumY = 0.0; for (int i = startIdx; i <= endIdx; i++) { sumY += y[internalIdx[i]]; } double bestImprovement = 1.0 / rows * sumY * sumY; double bestThreshold = double.PositiveInfinity; bestVar = RegressionTreeModel.TreeNode.NO_VARIABLE; for (int col = 0; col < effectiveVars; col++) { // sort values for variable to prepare for threshold selection var curVariable = allowedVariables[col]; var curVariableIdx = varName2Index[curVariable]; for (int i = startIdx; i <= endIdx; i++) { var sortedI = sortedIdx[curVariableIdx][i]; outSortedIdx[i - startIdx] = sortedI; outx[i - startIdx] = x[curVariableIdx][sortedI]; } double curImprovement; double curThreshold; FindBestThreshold(outx, outSortedIdx, rows, y, sumY, out curThreshold, out curImprovement); if (curImprovement > bestImprovement) { bestImprovement = curImprovement; bestThreshold = curThreshold; bestVar = allowedVariables[col]; } } if (bestVar == RegressionTreeModel.TreeNode.NO_VARIABLE) { threshold = bestThreshold; return false; } else { UpdateVariableRelevance(bestVar, sumY, bestImprovement, rows); threshold = bestThreshold; return true; } } // TODO: assumption is that the Average(y) = 0 private void UpdateVariableRelevance(string bestVar, double sumY, double bestImprovement, int rows) { if (string.IsNullOrEmpty(bestVar)) return; // update variable relevance double err = sumY * sumY / rows; double errAfterSplit = bestImprovement; double delta = (errAfterSplit - err); // relative reduction in squared error double v; if (!sumImprovements.TryGetValue(bestVar, out v)) { sumImprovements[bestVar] = delta; } sumImprovements[bestVar] = v + delta; } // x [0..N-1] contains rows sorted values in the range from [0..rows-1] // sortedIdx [0..N-1] contains the idx of the values in x in the original dataset in the range from [0..rows-1] // rows specifies the number of valid entries in x and sortedIdx // y [0..N-1] contains the target values in original sorting order // sumY is y.Sum() // // the routine returns the best threshold (x[i] + x[i+1]) / 2 for i = [0 .. rows-2] by calculating the reduction in squared error // additionally the reduction in squared error is returned in bestImprovement // if all elements of x are equal the routing fails to produce a threshold private static void FindBestThreshold(double[] x, int[] sortedIdx, int rows, double[] y, double sumY, out double bestThreshold, out double bestImprovement) { Debug.Assert(rows >= 2); double sl = 0.0; double sr = sumY; double nl = 0.0; double nr = rows; bestImprovement = 1.0 / rows * sumY * sumY; bestThreshold = double.NegativeInfinity; // for all thresholds // if we have n rows there are n-1 possible splits for (int i = 0; i < rows - 1; i++) { sl += y[sortedIdx[i]]; sr -= y[sortedIdx[i]]; nl++; nr--; Debug.Assert(nl > 0); Debug.Assert(nr > 0); if (x[i] < x[i + 1]) { // don't try to split when two elements are equal // goal is to find the split with leading to minimal total variance of left and right parts // without partitioning the variance is var(y) = E(y²) - E(y)² // = 1/n * sum(y²) - (1/n * sum(y))² // ------------- // if we split into right and left part the overall variance is the weigthed combination nl/n * var(y_l) + nr/n * var(y_r) // = nl/n * (1/nl * sum(y_l²) - (1/nl * sum(y_l))²) + nr/n * (1/nr * sum(y_r²) - (1/nr * sum(y_r))²) // = 1/n * sum(y_l²) - 1/nl * 1/n * sum(y_l)² + 1/n * sum(y_r²) - 1/nr * 1/n * sum(y_r)² // = 1/n * (sum(y_l²) + sum(y_r²)) - 1/n * (sum(y_l)² / nl + sum(y_r)² / nr) // = 1/n * sum(y²) - 1/n * (sum(y_l)² / nl + sum(y_r)² / nr) // ------------- // not changed by split (and the same for total variance without partitioning) // // therefore we need to find the maximum value (sum(y_l)² / nl + sum(y_r)² / nr) (ignoring the factor 1/n) // and this value must be larger than 1/n * sum(y)² to be an improvement over no split double curQuality = sl * sl / nl + sr * sr / nr; if (curQuality > bestImprovement) { bestThreshold = (x[i] + x[i + 1]) / 2.0; bestImprovement = curQuality; } } } // if all elements where the same then no split can be found } public IEnumerable> GetVariableRelevance() { // values are scaled: the most important variable has relevance = 100 double scaling = 100 / sumImprovements.Max(t => t.Value); return sumImprovements .Select(t => new KeyValuePair(t.Key, t.Value * scaling)) .OrderByDescending(t => t.Value); } } }