One morning before work, I decided to update my C# implementation of AdaBoost (“adaptive boosting”) regression. AdaBoost regression uses a collection of decision trees. Loosely, each tree is trained on a random selection of the training data, where the probability that an item is included in the current training set is proportional to how poorly predicted it is by previous trees.
More specifically, in very high-level pseudo-code:
train: assign equal weights to all training items create an empty list of decision trees (about 50 or 100) for-each tree create a subset of training data where probability of inclusion is based on item's weight create and train a decision tree assign a confidence weight to new tree (better predictions give higher confidence weight) add newly trained tree to collection update training item weights (worse predictions give higher weight to item) end-loop predict: for-each tree compute predicted y end-loop return median of confidence-weight predictions
The original AdaBoost technique was developed in 1995 and is used for binary classification predictions. The original paper suggested how the technique could be modified to do regression. A 1997 research paper updated the suggestion for regression and it is called AdaBoost.R2 (adaptive boosting for regression version 2). As far as I know, adaptive boosting regression is only implemented using the AdaBoost.R2 algorithm so the terms “adaptive boosting regression”, “AdaBoost”, and “AdaBoost.R2” are all essentially synonymous.
The output of my demo:
Begin AdaBoost.R2 (tree) regression from scratch C# Loading synthetic train (200) and test (40) data First three train X: -0.1660 0.4406 -0.9998 -0.3953 -0.7065 0.0776 -0.1616 0.3704 -0.5911 0.7562 -0.9452 0.3409 -0.1654 0.1174 -0.7192 First three train y: 0.4840 0.1568 0.8054 Setting maxLearners = 100 Setting tree maxDepth = 5 Setting tree minSamples = 2 Setting tree minLeaf = 1 Training AdaBoost.R2 model Done Created 100 learners Accuracy train (within 0.10): 0.8250 Accuracy test (within 0.10): 0.5250 MSE train: 0.0004 MSE test: 0.0022 Predicting for x = -0.1660 0.4406 -0.9998 -0.3953 -0.7065 Predicted y = 0.4890 End demo
The results show that AdaBoost regresssion tends to overfit the data — good accuracy on the training data but not-as-good accuracy on new, previously unseen test data.
I validated my AdaBoost regression implementation by running the demo data through the AdaBoostRegressor module in the scikit-learn Python language library:
Setting n_estimators = 100 Setting max_depth = 5 Setting min_samples_split = 2 Setting min_samples_leaf = 1 Training AdaBoost.R2 model Done Created 100 learners Accuracy train (within 0.10): 0.8200 Accuracy test (within 0.10): 0.5250 MSE train: 0.0002 MSE test: 0.0015
The results are pretty much the same. AdaBoost regression has a random component in the way training subsets are selected for tree, so results will differ for different implementations.
AdaBoost regression is related to gradient boosting regression (specifically via the XGBoost and LightGBM libraries). AdaBoost regression is not used nearly as often as gradient boosting. I suspect that, in practice, gradient boosting regression gives better (less over-fitting) results.
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Demo program. Replace “lt” (less than), “gt”, “lte”, “gte” with Boolean operator symbols (my blog editor often chokes on symbols).
using System;
using System.IO;
using System.Collections.Generic;
namespace AdaBoostRegression // AdaBoost.R2 algorithm
{
internal class AdaBoostRegressionProgram
{
static void Main(string[] args)
{
Console.WriteLine("\nBegin AdaBoost.R2 (tree) " +
"regression from scratch C# ");
// 1. load data
Console.WriteLine("\nLoading synthetic train (200)" +
" and test (40) data");
string trainFile =
"..\\..\\..\\Data\\synthetic_train_200.txt";
int[] colsX = new int[] { 0, 1, 2, 3, 4 };
int colY = 5;
double[][] trainX =
MatLoad(trainFile, colsX, ',', "#");
double[] trainY =
MatToVec(MatLoad(trainFile,
new int[] { colY }, ',', "#"));
string testFile =
"..\\..\\..\\Data\\synthetic_test_40.txt";
double[][] testX =
MatLoad(testFile, colsX, ',', "#");
double[] testY =
MatToVec(MatLoad(testFile,
new int[] { colY }, ',', "#"));
Console.WriteLine("\nFirst three train X: ");
for (int i = 0; i "lt" 3; ++i)
VecShow(trainX[i], 4, 8);
Console.WriteLine("\nFirst three train y: ");
for (int i = 0; i "lt" 3; ++i)
Console.WriteLine(trainY[i].ToString("F4").
PadLeft(8));
// 2. create and train model
int maxLearners = 100;
int maxDepth = 5;
int minSamples = 2;
int minLeaf = 1;
Console.WriteLine("\nSetting maxLearners = " +
maxLearners);
Console.WriteLine("Setting tree maxDepth = " +
maxDepth);
Console.WriteLine("Setting tree minSamples = " +
minSamples);
Console.WriteLine("Setting tree minLeaf = " +
minLeaf);
Console.WriteLine("\nTraining AdaBoost.R2 model ");
AdaBoostRegressor model =
new AdaBoostRegressor(maxLearners, maxDepth,
minSamples, minLeaf, seed: 0); // master seed
model.Train(trainX, trainY);
Console.WriteLine("Done ");
Console.WriteLine("Created " +
model.learners.Count() + " learners ");
// 3. evaluate model
double accTrain = model.Accuracy(trainX, trainY, 0.10);
Console.WriteLine("\nAccuracy train (within 0.10): " +
accTrain.ToString("F4"));
double accTest = model.Accuracy(testX, testY, 0.10);
Console.WriteLine("Accuracy test (within 0.10): " +
accTest.ToString("F4"));
double mseTrain = model.MSE(trainX, trainY);
Console.WriteLine("\nMSE train: " +
mseTrain.ToString("F4"));
double mseTest = model.MSE(testX, testY);
Console.WriteLine("MSE test: " +
mseTest.ToString("F4"));
// 4. use model to make a prediction
double[] x = trainX[0];
Console.WriteLine("\nPredicting for x = ");
VecShow(x, 4, 8);
double yPred = model.Predict(x);
Console.WriteLine("Predicted y = " +
yPred.ToString("F4"));
Console.WriteLine("\nEnd demo ");
Console.ReadLine();
} // Main()
// ------------------------------------------------------
// helpers for Main():
// MatLoad(), MatToVec(), VecShow().
// ------------------------------------------------------
static double[][] MatLoad(string fn, int[] usecols,
char sep, string comment)
{
List"lt"double[]"gt" result =
new List"lt"double[]"gt"();
string line = "";
FileStream ifs = new FileStream(fn, FileMode.Open);
StreamReader sr = new StreamReader(ifs);
while ((line = sr.ReadLine()) != null)
{
if (line.StartsWith(comment) == true)
continue;
string[] tokens = line.Split(sep);
List"lt"double"gt" lst = new List"lt"double"gt"();
for (int j = 0; j "lt" usecols.Length; ++j)
lst.Add(double.Parse(tokens[usecols[j]]));
double[] row = lst.ToArray();
result.Add(row);
}
sr.Close(); ifs.Close();
return result.ToArray();
}
static double[] MatToVec(double[][] M)
{
int nRows = M.Length;
int nCols = M[0].Length;
double[] result = new double[nRows * nCols];
int k = 0;
for (int i = 0; i "lt" nRows; ++i)
for (int j = 0; j "lt" nCols; ++j)
result[k++] = M[i][j];
return result;
}
static void VecShow(double[] vec, int dec, int wid)
{
for (int i = 0; i "lt" vec.Length; ++i)
Console.Write(vec[i].ToString("F" + dec).
PadLeft(wid));
Console.WriteLine("");
}
} // class Program
// ========================================================
class AdaBoostRegressor
{
public int maxLearners;
public int maxDepth;
public int minSamples;
public int minLeaf;
private Random rnd;
private int seed; // master also passed to each tree
public List"lt"DecisionTreeRegressor"gt" learners;
public List"lt"double"gt" betas;
public AdaBoostRegressor(int maxLearners, int maxDepth,
int minSamples, int minLeaf, int seed)
{
this.maxLearners = maxLearners;
this.maxDepth = maxDepth;
this.minSamples = minSamples;
this.minLeaf = minLeaf;
this.rnd = new Random(seed);
this.seed = seed;
this.learners = new List"lt"DecisionTreeRegressor"gt"();
this.betas = new List"lt"double"gt"();
// each learner has a "model weight" derived
// from its beta-confidence weight
} // ctor
public void Train(double[][] trainX, double[] trainY)
{
int N = trainX.Length;
double[] weights = new double[N];
for (int i = 0; i "lt" N; ++i)
weights[i] = 1.0 / N;
for (int t = 0; t "lt" this.maxLearners; ++t)
{
// 1. use weights to select a sample
int[] sampleIdxs = this.SelectIdxs(weights, N);
double[][] sampleX = ExtractRows(trainX,
sampleIdxs);
double[] sampleY = ExtractVals(trainY,
sampleIdxs);
// 2. construct regression machine t
DecisionTreeRegressor lrnr =
new DecisionTreeRegressor(this.maxDepth,
this.minSamples, this.minLeaf, seed:this.seed);
lrnr.Train(sampleX, sampleY);
// 3. compute predicteds for all training items
double[] predY = new double[N];
for (int i = 0; i "lt" N; ++i)
predY[i] = lrnr.Predict(trainX[i]);
// 4. calculate loss each sample
double[] absDiffs = new double[N];
for (int i = 0; i "lt" N; ++i)
{
absDiffs[i] = Math.Abs(predY[i] - trainY[i]);
}
double D = 0.0; // largest abs diff
for (int i = 0; i "lt" N; ++i)
{
if (absDiffs[i] "gt" D)
D = absDiffs[i];
}
double[] L = new double[N];
for (int i = 0; i "lt" N; ++i)
L[i] = absDiffs[i] / D; // normalized
// 5. calculate average loss
double Lbar = 0.0;
for (int i = 0; i "lt" N; ++i)
Lbar += L[i] * weights[i];
if (Lbar "gte" 0.5) // bad learner
continue;
// 6. compute beta (low beta = high confidence)
this.learners.Add(lrnr); // lrnr is good
double beta = Lbar / (1.0 - Lbar);
this.betas.Add(beta); // each learner has a beta
// 7. update training weights
double sum = 0.0;
for (int i = 0; i "lt" N; ++i)
{
weights[i] *= Math.Pow(beta, (1.0 - Lbar));
sum += weights[i];
}
for (int i = 0; i "lt" N; ++i)
weights[i] /= sum;
} // t
} // Train()
public double Predict(double[] x)
{
// each learner makes a prediction
double[] preds = new double[this.learners.Count];
for (int i = 0; i "lt" this.learners.Count; ++i)
{
preds[i] = this.learners[i].Predict(x);
}
// each learner has a "model weight" derived
// from its beta-confidence weight
double[] modelWts = new double[this.learners.Count];
for (int i = 0; i "lt" this.learners.Count; ++i)
modelWts[i] = Math.Log(1.0 / this.betas[i]);
// final prediction is weighted median of
// the predictions
double result = WeightedMedian(preds, modelWts);
//double result = WeightedMean(preds, modelWts);
return result;
} // Predict()
public double Accuracy(double[][] dataX,
double[] dataY, double pctClose)
{
int nCorrect = 0; int nWrong = 0;
for (int i = 0; i "lt" dataX.Length; ++i)
{
double predY = this.Predict(dataX[i]);
double actuaY = dataY[i];
if (Math.Abs(predY - actuaY) "lt"
Math.Abs(pctClose * actuaY))
++nCorrect;
else
++nWrong;
}
return (nCorrect * 1.0) / (nCorrect + nWrong);
}
// ------------------------------------------------------
public double MSE(double[][] dataX,
double[] dataY)
{
int n = dataX.Length;
double sum = 0.0;
for (int i = 0; i "lt" n; ++i)
{
double actualY = dataY[i];
double predY = this.Predict(dataX[i]);
sum += (actualY - predY) * (actualY - predY);
}
return sum / n;
}
// ------------------------------------------------------
// helper functions for AdaBoostRegressor
// ------------------------------------------------------
//private static double WeightedMean(double[] values,
// double[] weights)
//{
// int n = values.Length;
// double sumWts = 0.0;
// double sum = 0.0;
// for (int i = 0; i "lt" n; ++i)
// {
// sumWts += weights[i];
// sum += values[i] * weights[i];
// }
// return sum / sumWts;
//}
// ------------------------------------------------------
private static double WeightedMedian(double[] values,
double[] weights)
{
// no interpolation for even n
// don't assume weights sum to 1.0
int n = values.Length;
double sumWts = 0.0;
for (int i = 0; i "lt" n; ++i)
sumWts += weights[i];
double thresh = sumWts / 2;
int[] sortedIdxs = ArgSort(values);
double accum = 0.0;
for (int j = 0; j "lt" n; ++j)
{
accum += weights[sortedIdxs[j]];
if (accum "gte" thresh)
return values[sortedIdxs[j]];
}
return values[sortedIdxs[n - 1]];
}
// helper for WeightedMedian()
private static int[] ArgSort(double[] values)
{
int n = values.Length;
double[] copy = new double[n];
int[] indices = new int[n];
for (int i = 0; i "lt" n; ++i)
{
copy[i] = values[i];
indices[i] = i;
}
Array.Sort(copy, indices); // in parallel
return indices;
}
private static double[][] MatMake(int nRows, int nCols)
{
double[][] result = new double[nRows][];
for (int i = 0; i "lt" nRows; ++i)
result[i] = new double[nCols];
return result;
}
private int SelectIdx(double[] probs)
{
// roulette wheel selection
double p = this.rnd.NextDouble();
double accum = 0.0;
for (int i = 0; i "lt" probs.Length; ++i)
{
accum += probs[i];
if (p "lt" accum)
return i;
}
return probs.Length - 1;
}
private int[] SelectIdxs(double[] probs, int n)
{
// with replacement
int[] result = new int[n];
for (int i = 0; i "lt" n; ++i)
result[i] = this.SelectIdx(probs);
return result;
}
private static double[][] ExtractRows(double[][] mat,
int[] idxs)
{
int nRows = idxs.Length; // of the result
int nCols = mat[0].Length; // source and result
double[][] result = MatMake(nRows, nCols);
for (int i = 0; i "lt" nRows; ++i)
{
for (int j = 0; j "lt" nCols; ++j)
{
int srcRow = idxs[i];
result[i][j] = mat[srcRow][j];
}
}
return result;
}
private static double[] ExtractVals(double[] vec,
int[] idxs)
{
int nVals = idxs.Length;
double[] result = new double[nVals];
for (int i = 0; i "lt" nVals; ++i)
{
int srcIdx = idxs[i];
result[i] = vec[srcIdx];
}
return result;
}
} // class AdaBoostRegressor
// ========================================================
public class DecisionTreeRegressor
{
public int maxDepth;
public int minSamples; // aka min_samples_split
public int minLeaf; // min number of values in a leaf
public int numSplitCols; // mostly for random forest
public List"lt"Node"gt" tree = new List"lt"Node"gt"();
public Random rnd; // order in which cols are searched
public double[][] trainX; // store data by ref
public double[] trainY;
// ------------------------------------------------------
public class Node
{
public int id;
public int colIdx; // aka featureIdx
public double thresh;
public int left; // index into List
public int right;
public double value;
public bool isLeaf;
public List"lt"int"gt" rows; // rows in train data
public Node()
{
this.id = -1;
this.colIdx = -1;
this.thresh = 0.0; // aka split value
this.left = -1;
this.right = -1;
this.value = 0.0; // aka pred y
this.isLeaf = false;
this.rows = null;
}
} // class Node
// --------------------------------------------
public DecisionTreeRegressor(int maxDepth = 2,
int minSamples = 2, int minLeaf = 1,
int numSplitCols = -1, int seed = 0)
{
// if maxDepth = 0, tree has just a root node
// if maxDepth = 1, at most 3 nodes (root, l, r)
// if maxDepth = n, at most 2^(n+1) - 1 nodes
this.maxDepth = maxDepth;
this.minSamples = minSamples;
this.minLeaf = minLeaf;
this.numSplitCols = numSplitCols; // for ran. forest
// create full tree List with null nodes
int numNodes = (int)Math.Pow(2, (maxDepth + 1)) - 1;
for (int i = 0; i "lt" numNodes; ++i)
{
this.tree.Add(null); // empty nodes
}
this.rnd = new Random(seed);
}
// ------------------------------------------------------
// public: Train(), Predict().
// helpers: MakeTree(), BestSplit(), TreeTargetMean(),
// TreeTargetVariance().
// ------------------------------------------------------
public void Train(double[][] trainX, double[] trainY)
{
this.trainX = trainX; //
this.trainY = trainY;
this.MakeTree();
// optionally delete rows in each Node to save space
// when tree is part of an ensemble
for (int i = 0; i "lt" this.tree.Count; ++i)
if (this.tree[i] != null)
this.tree[i].rows = null;
}
// ------------------------------------------------------
public double Predict(double[] x)
{
int p = 0;
Node currNode = this.tree[p];
while (currNode != null &&
currNode.isLeaf == false &&
p "lt" this.tree.Count)
{
if (x[currNode.colIdx] "lte" currNode.thresh)
p = currNode.left;
else
p = currNode.right;
currNode = this.tree[p];
}
return this.tree[p].value;
}
// ------------------------------------------------------
// helpers: MakeTree(), BestSplit(),
// TreeTargetMean(), TreeTargetVariance()
private void MakeTree()
{
// no recursion, no pointers, List storage, no stack
if (this.numSplitCols == -1) // use all cols
this.numSplitCols = this.trainX[0].Length;
// prepare root node
List"lt"int"gt" allRows = new List"lt"int"gt"();
for (int i = 0; i "lt" this.trainX.Length; ++i)
allRows.Add(i);
double grandMean = this.TreeTargetMean(allRows);
// wait to supply colIdx and thresh in loop
Node root = new Node();
root.id = 0;
root.left = 1;
root.right = 2;
root.value = grandMean;
root.isLeaf = false; // already set
root.rows = allRows;
this.tree[0] = root;
for (int i = 0; i "lt" this.tree.Count; ++i)
{
Node currNode = this.tree[i];
// curr node has values except colIdx and thresh
// curr node too deep to have children OR
// curr node not enough rows to split then
// leave both children as null
if (currNode == null ||
currNode.rows.Count == 0) { continue; }
// if parent cannot be split, make parent a leaf
if (currNode.id "gte" (int)Math.Pow(2,
(this.maxDepth)) - 1 ||
currNode.rows.Count "lt" this.minSamples)
{
currNode.isLeaf = true;
continue;
}
// parent has enough rows to try to split
double[] splitInfo = this.BestSplit(currNode.rows);
int colIdx = (int)splitInfo[0];
double splitVal = splitInfo[1]; //split value
if (colIdx == -1) // unable to split, parent leaf
{
currNode.isLeaf = true;
continue;
}
// complete the fields for curr node
currNode.colIdx = colIdx;
currNode.thresh = splitVal;
// construct the children, except colIdx and thresh
Node leftNode = new Node();
Node rightNode = new Node();
// construct the children rows using split info
// all info except colIdx and thresh
List"lt"int"gt" leftIdxs = new List"lt"int"gt"();
List"lt"int"gt" rightIdxs = new List"lt"int"gt"();
for (int k = 0; k "lt" currNode.rows.Count; ++k)
{
int r = currNode.rows[k];
if (this.trainX[r][colIdx] "lte" splitVal)
leftIdxs.Add(r);
else
rightIdxs.Add(r);
}
// assign -1 to children if out of range
leftNode.id = currNode.id * 2 + 1;
if (leftNode.id "gt" (int)Math.Pow(2,
(maxDepth + 1)) - 2) leftNode.id = -1;
leftNode.left = leftNode.id * 2 + 1;
if (leftNode.left "gt" (int)Math.Pow(2,
(maxDepth + 1)) - 2) leftNode.left = -1;
leftNode.right = leftNode.id * 2 + 2;
if (leftNode.right "gt" (int)Math.Pow(2,
(maxDepth + 1)) - 2) leftNode.right = -1;
leftNode.rows = leftIdxs;
leftNode.value =
this.TreeTargetMean(leftNode.rows);
this.tree[leftNode.id] = leftNode;
rightNode.id = currNode.id * 2 + 2;
if (rightNode.id "gt" (int)Math.Pow(2,
(maxDepth + 1)) - 2) rightNode.id = -1;
rightNode.left = rightNode.id * 2 + 1;
if (rightNode.left "gt" (int)Math.Pow(2,
(maxDepth + 1)) - 2) rightNode.left = -1;
rightNode.right = rightNode.id * 2 + 2;
if (rightNode.right "gt" (int)Math.Pow(2,
(maxDepth + 1)) - 2) rightNode.right = -1;
rightNode.rows = rightIdxs;
rightNode.value =
this.TreeTargetMean(rightNode.rows);
this.tree[rightNode.id] = rightNode;
} // i
return;
}
// ------------------------------------------------------
private double[] BestSplit(List"lt"int"gt" rows)
{
// implicit params numSplitCols, minLeaf, numsplitCols
// result[0] = best col idx (as double)
// result[1] = best split value
rows.Sort();
int bestColIdx = -1; // indicates bad split
double bestThresh = 0.0;
double bestVar = double.MaxValue; // smaller better
int nRows = rows.Count; // or dataY.Length
int nCols = this.trainX[0].Length;
if (nRows == 0)
{
throw new Exception("empty data in BestSplit()");
}
// process cols in scrambled order
int[] colIndices = new int[nCols];
for (int k = 0; k "lt" nCols; ++k)
colIndices[k] = k;
// shuffle, inline Fisher-Yates
int n = colIndices.Length;
for (int i = 0; i "lt" n; ++i)
{
int ri = rnd.Next(i, n); // be careful
int tmp = colIndices[i];
colIndices[i] = colIndices[ri];
colIndices[ri] = tmp;
}
// numSplitCols is usually all columns (-1)
for (int j = 0; j "lt" this.numSplitCols; ++j)
{
int colIdx = colIndices[j];
HashSet"lt"double"gt" examineds =
new HashSet"lt"double"gt"();
for (int i = 0; i "lt" nRows; ++i) // each row
{
// if curr thresh been seen, skip it
double thresh = this.trainX[rows[i]][colIdx];
if (examineds.Contains(thresh)) continue;
examineds.Add(thresh);
// get row idxs where x is lte, gt thresh
List"lt"int"gt" leftIdxs = new List"lt"int"gt"();
List"lt"int"gt" rightIdxs = new List"lt"int"gt"();
for (int k = 0; k "lt" nRows; ++k)
{
if (this.trainX[rows[k]][colIdx] "lte" thresh)
leftIdxs.Add(rows[k]);
else
rightIdxs.Add(rows[k]);
}
// Check if proposed split has too few values
if (leftIdxs.Count "lt" this.minLeaf ||
rightIdxs.Count "lt" this.minLeaf)
continue; // to next row
double leftVar =
this.TreeTargetVariance(leftIdxs);
double rightVar =
this.TreeTargetVariance(rightIdxs);
double weightedVar = (leftIdxs.Count * leftVar +
rightIdxs.Count * rightVar) / nRows;
if (weightedVar "lt" bestVar)
{
// if this never happens, bestColIdx remains -1
// which means a bad split. used in MakeTree()
bestColIdx = colIdx;
bestThresh = thresh;
bestVar = weightedVar;
}
} // each row
} // j each col
double[] result = new double[2]; // out params ugly
result[0] = 1.0 * bestColIdx;
result[1] = bestThresh;
return result;
} // BestSplit()
// ------------------------------------------------------
private double TreeTargetMean(List"lt"int"gt" rows)
{
// mean of rows items in trainY: for node prediction
double sum = 0.0;
for (int i = 0; i "lt" rows.Count; ++i)
{
int r = rows[i];
sum += this.trainY[r];
}
return sum / rows.Count;
}
// ------------------------------------------------------
private double TreeTargetVariance(List"lt"int"gt" rows)
{
double mean = this.TreeTargetMean(rows);
double sum = 0.0;
for (int i = 0; i "lt" rows.Count; ++i)
{
int r = rows[i];
sum += (this.trainY[r] - mean) *
(this.trainY[r] - mean);
}
return sum / rows.Count;
}
// ------------------------------------------------------
} // class DecisionTreeRegressor
// ========================================================
} // ns
Training data:
# synthetic_train_200.txt # -0.1660, 0.4406, -0.9998, -0.3953, -0.7065, 0.4840 0.0776, -0.1616, 0.3704, -0.5911, 0.7562, 0.1568 -0.9452, 0.3409, -0.1654, 0.1174, -0.7192, 0.8054 0.9365, -0.3732, 0.3846, 0.7528, 0.7892, 0.1345 -0.8299, -0.9219, -0.6603, 0.7563, -0.8033, 0.7955 0.0663, 0.3838, -0.3690, 0.3730, 0.6693, 0.3206 -0.9634, 0.5003, 0.9777, 0.4963, -0.4391, 0.7377 -0.1042, 0.8172, -0.4128, -0.4244, -0.7399, 0.4801 -0.9613, 0.3577, -0.5767, -0.4689, -0.0169, 0.6861 -0.7065, 0.1786, 0.3995, -0.7953, -0.1719, 0.5569 0.3888, -0.1716, -0.9001, 0.0718, 0.3276, 0.2500 0.1731, 0.8068, -0.7251, -0.7214, 0.6148, 0.3297 -0.2046, -0.6693, 0.8550, -0.3045, 0.5016, 0.2129 0.2473, 0.5019, -0.3022, -0.4601, 0.7918, 0.2613 -0.1438, 0.9297, 0.3269, 0.2434, -0.7705, 0.5171 0.1568, -0.1837, -0.5259, 0.8068, 0.1474, 0.3307 -0.9943, 0.2343, -0.3467, 0.0541, 0.7719, 0.5581 0.2467, -0.9684, 0.8589, 0.3818, 0.9946, 0.1092 -0.6553, -0.7257, 0.8652, 0.3936, -0.8680, 0.7018 0.8460, 0.4230, -0.7515, -0.9602, -0.9476, 0.1996 -0.9434, -0.5076, 0.7201, 0.0777, 0.1056, 0.5664 0.9392, 0.1221, -0.9627, 0.6013, -0.5341, 0.1533 0.6142, -0.2243, 0.7271, 0.4942, 0.1125, 0.1661 0.4260, 0.1194, -0.9749, -0.8561, 0.9346, 0.2230 0.1362, -0.5934, -0.4953, 0.4877, -0.6091, 0.3810 0.6937, -0.5203, -0.0125, 0.2399, 0.6580, 0.1460 -0.6864, -0.9628, -0.8600, -0.0273, 0.2127, 0.5387 0.9772, 0.1595, -0.2397, 0.1019, 0.4907, 0.1611 0.3385, -0.4702, -0.8673, -0.2598, 0.2594, 0.2270 -0.8669, -0.4794, 0.6095, -0.6131, 0.2789, 0.4700 0.0493, 0.8496, -0.4734, -0.8681, 0.4701, 0.3516 0.8639, -0.9721, -0.5313, 0.2336, 0.8980, 0.1412 0.9004, 0.1133, 0.8312, 0.2831, -0.2200, 0.1782 0.0991, 0.8524, 0.8375, -0.2102, 0.9265, 0.2150 -0.6521, -0.7473, -0.7298, 0.0113, -0.9570, 0.7422 0.6190, -0.3105, 0.8802, 0.1640, 0.7577, 0.1056 0.6895, 0.8108, -0.0802, 0.0927, 0.5972, 0.2214 0.1982, -0.9689, 0.1870, -0.1326, 0.6147, 0.1310 -0.3695, 0.7858, 0.1557, -0.6320, 0.5759, 0.3773 -0.1596, 0.3581, 0.8372, -0.9992, 0.9535, 0.2071 -0.2468, 0.9476, 0.2094, 0.6577, 0.1494, 0.4132 0.1737, 0.5000, 0.7166, 0.5102, 0.3961, 0.2611 0.7290, -0.3546, 0.3416, -0.0983, -0.2358, 0.1332 -0.3652, 0.2438, -0.1395, 0.9476, 0.3556, 0.4170 -0.6029, -0.1466, -0.3133, 0.5953, 0.7600, 0.4334 -0.4596, -0.4953, 0.7098, 0.0554, 0.6043, 0.2775 0.1450, 0.4663, 0.0380, 0.5418, 0.1377, 0.2931 -0.8636, -0.2442, -0.8407, 0.9656, -0.6368, 0.7429 0.6237, 0.7499, 0.3768, 0.1390, -0.6781, 0.2185 -0.5499, 0.1850, -0.3755, 0.8326, 0.8193, 0.4399 -0.4858, -0.7782, -0.6141, -0.0008, 0.4572, 0.4197 0.7033, -0.1683, 0.2334, -0.5327, -0.7961, 0.1776 0.0317, -0.0457, -0.6947, 0.2436, 0.0880, 0.3345 0.5031, -0.5559, 0.0387, 0.5706, -0.9553, 0.3107 -0.3513, 0.7458, 0.6894, 0.0769, 0.7332, 0.3170 0.2205, 0.5992, -0.9309, 0.5405, 0.4635, 0.3532 -0.4806, -0.4859, 0.2646, -0.3094, 0.5932, 0.3202 0.9809, -0.3995, -0.7140, 0.8026, 0.0831, 0.1600 0.9495, 0.2732, 0.9878, 0.0921, 0.0529, 0.1289 -0.9476, -0.6792, 0.4913, -0.9392, -0.2669, 0.5966 0.7247, 0.3854, 0.3819, -0.6227, -0.1162, 0.1550 -0.5922, -0.5045, -0.4757, 0.5003, -0.0860, 0.5863 -0.8861, 0.0170, -0.5761, 0.5972, -0.4053, 0.7301 0.6877, -0.2380, 0.4997, 0.0223, 0.0819, 0.1404 0.9189, 0.6079, -0.9354, 0.4188, -0.0700, 0.1907 -0.1428, -0.7820, 0.2676, 0.6059, 0.3936, 0.2790 0.5324, -0.3151, 0.6917, -0.1425, 0.6480, 0.1071 -0.8432, -0.9633, -0.8666, -0.0828, -0.7733, 0.7784 -0.9444, 0.5097, -0.2103, 0.4939, -0.0952, 0.6787 -0.0520, 0.6063, -0.1952, 0.8094, -0.9259, 0.4836 0.5477, -0.7487, 0.2370, -0.9793, 0.0773, 0.1241 0.2450, 0.8116, 0.9799, 0.4222, 0.4636, 0.2355 0.8186, -0.1983, -0.5003, -0.6531, -0.7611, 0.1511 -0.4714, 0.6382, -0.3788, 0.9648, -0.4667, 0.5950 0.0673, -0.3711, 0.8215, -0.2669, -0.1328, 0.2677 -0.9381, 0.4338, 0.7820, -0.9454, 0.0441, 0.5518 -0.3480, 0.7190, 0.1170, 0.3805, -0.0943, 0.4724 -0.9813, 0.1535, -0.3771, 0.0345, 0.8328, 0.5438 -0.1471, -0.5052, -0.2574, 0.8637, 0.8737, 0.3042 -0.5454, -0.3712, -0.6505, 0.2142, -0.1728, 0.5783 0.6327, -0.6297, 0.4038, -0.5193, 0.1484, 0.1153 -0.5424, 0.3282, -0.0055, 0.0380, -0.6506, 0.6613 0.1414, 0.9935, 0.6337, 0.1887, 0.9520, 0.2540 -0.9351, -0.8128, -0.8693, -0.0965, -0.2491, 0.7353 0.9507, -0.6640, 0.9456, 0.5349, 0.6485, 0.1059 -0.0462, -0.9737, -0.2940, -0.0159, 0.4602, 0.2606 -0.0627, -0.0852, -0.7247, -0.9782, 0.5166, 0.2977 0.0478, 0.5098, -0.0723, -0.7504, -0.3750, 0.3335 0.0090, 0.3477, 0.5403, -0.7393, -0.9542, 0.4415 -0.9748, 0.3449, 0.3736, -0.1015, 0.8296, 0.4358 0.2887, -0.9895, -0.0311, 0.7186, 0.6608, 0.2057 0.1570, -0.4518, 0.1211, 0.3435, -0.2951, 0.3244 0.7117, -0.6099, 0.4946, -0.4208, 0.5476, 0.1096 -0.2929, -0.5726, 0.5346, -0.3827, 0.4665, 0.2465 0.4889, -0.5572, -0.5718, -0.6021, -0.7150, 0.2163 -0.7782, 0.3491, 0.5996, -0.8389, -0.5366, 0.6516 -0.5847, 0.8347, 0.4226, 0.1078, -0.3910, 0.6134 0.8469, 0.4121, -0.0439, -0.7476, 0.9521, 0.1571 -0.6803, -0.5948, -0.1376, -0.1916, -0.7065, 0.7156 0.2878, 0.5086, -0.5785, 0.2019, 0.4979, 0.2980 0.2764, 0.1943, -0.4090, 0.4632, 0.8906, 0.2960 -0.8877, 0.6705, -0.6155, -0.2098, -0.3998, 0.7107 -0.8398, 0.8093, -0.2597, 0.0614, -0.0118, 0.6502 -0.8476, 0.0158, -0.4769, -0.2859, -0.7839, 0.7715 0.5751, -0.7868, 0.9714, -0.6457, 0.1448, 0.1175 0.4802, -0.7001, 0.1022, -0.5668, 0.5184, 0.1090 0.4458, -0.6469, 0.7239, -0.9604, 0.7205, 0.0779 0.5175, 0.4339, 0.9747, -0.4438, -0.9924, 0.2879 0.8678, 0.7158, 0.4577, 0.0334, 0.4139, 0.1678 0.5406, 0.5012, 0.2264, -0.1963, 0.3946, 0.2088 -0.9938, 0.5498, 0.7928, -0.5214, -0.7585, 0.7687 0.7661, 0.0863, -0.4266, -0.7233, -0.4197, 0.1466 0.2277, -0.3517, -0.0853, -0.1118, 0.6563, 0.1767 0.3499, -0.5570, -0.0655, -0.3705, 0.2537, 0.1632 0.7547, -0.1046, 0.5689, -0.0861, 0.3125, 0.1257 0.8186, 0.2110, 0.5335, 0.0094, -0.0039, 0.1391 0.6858, -0.8644, 0.1465, 0.8855, 0.0357, 0.1845 -0.4967, 0.4015, 0.0805, 0.8977, 0.2487, 0.4663 0.6760, -0.9841, 0.9787, -0.8446, -0.3557, 0.1509 -0.1203, -0.4885, 0.6054, -0.0443, -0.7313, 0.4854 0.8557, 0.7919, -0.0169, 0.7134, -0.1628, 0.2002 0.0115, -0.6209, 0.9300, -0.4116, -0.7931, 0.4052 -0.7114, -0.9718, 0.4319, 0.1290, 0.5892, 0.3661 0.3915, 0.5557, -0.1870, 0.2955, -0.6404, 0.2954 -0.3564, -0.6548, -0.1827, -0.5172, -0.1862, 0.4622 0.2392, -0.4959, 0.5857, -0.1341, -0.2850, 0.2470 -0.3394, 0.3947, -0.4627, 0.6166, -0.4094, 0.5325 0.7107, 0.7768, -0.6312, 0.1707, 0.7964, 0.2757 -0.1078, 0.8437, -0.4420, 0.2177, 0.3649, 0.4028 -0.3139, 0.5595, -0.6505, -0.3161, -0.7108, 0.5546 0.4335, 0.3986, 0.3770, -0.4932, 0.3847, 0.1810 -0.2562, -0.2894, -0.8847, 0.2633, 0.4146, 0.4036 0.2272, 0.2966, -0.6601, -0.7011, 0.0284, 0.2778 -0.0743, -0.1421, -0.0054, -0.6770, -0.3151, 0.3597 -0.4762, 0.6891, 0.6007, -0.1467, 0.2140, 0.4266 -0.4061, 0.7193, 0.3432, 0.2669, -0.7505, 0.6147 -0.0588, 0.9731, 0.8966, 0.2902, -0.6966, 0.4955 -0.0627, -0.1439, 0.1985, 0.6999, 0.5022, 0.3077 0.1587, 0.8494, -0.8705, 0.9827, -0.8940, 0.4263 -0.7850, 0.2473, -0.9040, -0.4308, -0.8779, 0.7199 0.4070, 0.3369, -0.2428, -0.6236, 0.4940, 0.2215 -0.0242, 0.0513, -0.9430, 0.2885, -0.2987, 0.3947 -0.5416, -0.1322, -0.2351, -0.0604, 0.9590, 0.3683 0.1055, 0.7783, -0.2901, -0.5090, 0.8220, 0.2984 -0.9129, 0.9015, 0.1128, -0.2473, 0.9901, 0.4776 -0.9378, 0.1424, -0.6391, 0.2619, 0.9618, 0.5368 0.7498, -0.0963, 0.4169, 0.5549, -0.0103, 0.1614 -0.2612, -0.7156, 0.4538, -0.0460, -0.1022, 0.3717 0.7720, 0.0552, -0.1818, -0.4622, -0.8560, 0.1685 -0.4177, 0.0070, 0.9319, -0.7812, 0.3461, 0.3052 -0.0001, 0.5542, -0.7128, -0.8336, -0.2016, 0.3803 0.5356, -0.4194, -0.5662, -0.9666, -0.2027, 0.1776 -0.2378, 0.3187, -0.8582, -0.6948, -0.9668, 0.5474 -0.1947, -0.3579, 0.1158, 0.9869, 0.6690, 0.2992 0.3992, 0.8365, -0.9205, -0.8593, -0.0520, 0.3154 -0.0209, 0.0793, 0.7905, -0.1067, 0.7541, 0.1864 -0.4928, -0.4524, -0.3433, 0.0951, -0.5597, 0.6261 -0.8118, 0.7404, -0.5263, -0.2280, 0.1431, 0.6349 0.0516, -0.8480, 0.7483, 0.9023, 0.6250, 0.1959 -0.3212, 0.1093, 0.9488, -0.3766, 0.3376, 0.2735 -0.3481, 0.5490, -0.3484, 0.7797, 0.5034, 0.4379 -0.5785, -0.9170, -0.3563, -0.9258, 0.3877, 0.4121 0.3407, -0.1391, 0.5356, 0.0720, -0.9203, 0.3458 -0.3287, -0.8954, 0.2102, 0.0241, 0.2349, 0.3247 -0.1353, 0.6954, -0.0919, -0.9692, 0.7461, 0.3338 0.9036, -0.8982, -0.5299, -0.8733, -0.1567, 0.1187 0.7277, -0.8368, -0.0538, -0.7489, 0.5458, 0.0830 0.9049, 0.8878, 0.2279, 0.9470, -0.3103, 0.2194 0.7957, -0.1308, -0.5284, 0.8817, 0.3684, 0.2172 0.4647, -0.4931, 0.2010, 0.6292, -0.8918, 0.3371 -0.7390, 0.6849, 0.2367, 0.0626, -0.5034, 0.7039 -0.1567, -0.8711, 0.7940, -0.5932, 0.6525, 0.1710 0.7635, -0.0265, 0.1969, 0.0545, 0.2496, 0.1445 0.7675, 0.1354, -0.7698, -0.5460, 0.1920, 0.1728 -0.5211, -0.7372, -0.6763, 0.6897, 0.2044, 0.5217 0.1913, 0.1980, 0.2314, -0.8816, 0.5006, 0.1998 0.8964, 0.0694, -0.6149, 0.5059, -0.9854, 0.1825 0.1767, 0.7104, 0.2093, 0.6452, 0.7590, 0.2832 -0.3580, -0.7541, 0.4426, -0.1193, -0.7465, 0.5657 -0.5996, 0.5766, -0.9758, -0.3933, -0.9572, 0.6800 0.9950, 0.1641, -0.4132, 0.8579, 0.0142, 0.2003 -0.4717, -0.3894, -0.2567, -0.5111, 0.1691, 0.4266 0.3917, -0.8561, 0.9422, 0.5061, 0.6123, 0.1212 -0.0366, -0.1087, 0.3449, -0.1025, 0.4086, 0.2475 0.3633, 0.3943, 0.2372, -0.6980, 0.5216, 0.1925 -0.5325, -0.6466, -0.2178, -0.3589, 0.6310, 0.3568 0.2271, 0.5200, -0.1447, -0.8011, -0.7699, 0.3128 0.6415, 0.1993, 0.3777, -0.0178, -0.8237, 0.2181 -0.5298, -0.0768, -0.6028, -0.9490, 0.4588, 0.4356 0.6870, -0.1431, 0.7294, 0.3141, 0.1621, 0.1632 -0.5985, 0.0591, 0.7889, -0.3900, 0.7419, 0.2945 0.3661, 0.7984, -0.8486, 0.7572, -0.6183, 0.3449 0.6995, 0.3342, -0.3113, -0.6972, 0.2707, 0.1712 0.2565, 0.9126, 0.1798, -0.6043, -0.1413, 0.2893 -0.3265, 0.9839, -0.2395, 0.9854, 0.0376, 0.4770 0.2690, -0.1722, 0.9818, 0.8599, -0.7015, 0.3954 -0.2102, -0.0768, 0.1219, 0.5607, -0.0256, 0.3949 0.8216, -0.9555, 0.6422, -0.6231, 0.3715, 0.0801 -0.2896, 0.9484, -0.7545, -0.6249, 0.7789, 0.4370 -0.9985, -0.5448, -0.7092, -0.5931, 0.7926, 0.5402
Test data:
# synthetic_test_40.txt # 0.7462, 0.4006, -0.0590, 0.6543, -0.0083, 0.1935 0.8495, -0.2260, -0.0142, -0.4911, 0.7699, 0.1078 -0.2335, -0.4049, 0.4352, -0.6183, -0.7636, 0.5088 0.1810, -0.5142, 0.2465, 0.2767, -0.3449, 0.3136 -0.8650, 0.7611, -0.0801, 0.5277, -0.4922, 0.7140 -0.2358, -0.7466, -0.5115, -0.8413, -0.3943, 0.4533 0.4834, 0.2300, 0.3448, -0.9832, 0.3568, 0.1360 -0.6502, -0.6300, 0.6885, 0.9652, 0.8275, 0.3046 -0.3053, 0.5604, 0.0929, 0.6329, -0.0325, 0.4756 -0.7995, 0.0740, -0.2680, 0.2086, 0.9176, 0.4565 -0.2144, -0.2141, 0.5813, 0.2902, -0.2122, 0.4119 -0.7278, -0.0987, -0.3312, -0.5641, 0.8515, 0.4438 0.3793, 0.1976, 0.4933, 0.0839, 0.4011, 0.1905 -0.8568, 0.9573, -0.5272, 0.3212, -0.8207, 0.7415 -0.5785, 0.0056, -0.7901, -0.2223, 0.0760, 0.5551 0.0735, -0.2188, 0.3925, 0.3570, 0.3746, 0.2191 0.1230, -0.2838, 0.2262, 0.8715, 0.1938, 0.2878 0.4792, -0.9248, 0.5295, 0.0366, -0.9894, 0.3149 -0.4456, 0.0697, 0.5359, -0.8938, 0.0981, 0.3879 0.8629, -0.8505, -0.4464, 0.8385, 0.5300, 0.1769 0.1995, 0.6659, 0.7921, 0.9454, 0.9970, 0.2330 -0.0249, -0.3066, -0.2927, -0.4923, 0.8220, 0.2437 0.4513, -0.9481, -0.0770, -0.4374, -0.9421, 0.2879 -0.3405, 0.5931, -0.3507, -0.3842, 0.8562, 0.3987 0.9538, 0.0471, 0.9039, 0.7760, 0.0361, 0.1706 -0.0887, 0.2104, 0.9808, 0.5478, -0.3314, 0.4128 -0.8220, -0.6302, 0.0537, -0.1658, 0.6013, 0.4306 -0.4123, -0.2880, 0.9074, -0.0461, -0.4435, 0.5144 0.0060, 0.2867, -0.7775, 0.5161, 0.7039, 0.3599 -0.7968, -0.5484, 0.9426, -0.4308, 0.8148, 0.2979 0.7811, 0.8450, -0.6877, 0.7594, 0.2640, 0.2362 -0.6802, -0.1113, -0.8325, -0.6694, -0.6056, 0.6544 0.3821, 0.1476, 0.7466, -0.5107, 0.2592, 0.1648 0.7265, 0.9683, -0.9803, -0.4943, -0.5523, 0.2454 -0.9049, -0.9797, -0.0196, -0.9090, -0.4433, 0.6447 -0.4607, 0.1811, -0.2389, 0.4050, -0.0078, 0.5229 0.2664, -0.2932, -0.4259, -0.7336, 0.8742, 0.1834 -0.4507, 0.1029, -0.6294, -0.1158, -0.6294, 0.6081 0.8948, -0.0124, 0.9278, 0.2899, -0.0314, 0.1534 -0.1323, -0.8813, -0.0146, -0.0697, 0.6135, 0.2386
.NET Test Automation Recipes
Software Testing
SciPy Programming Succinctly
Keras Succinctly
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