Fit binary decision tree for multiclass classification
returns a fitted binary classification decision tree based on the input
variables (also known as predictors, features, or attributes) contained in the
table tree
= fitctree(Tbl
,ResponseVarName
)Tbl
and output (response or labels) contained in
Tbl.ResponseVarName
. The returned binary tree splits
branching nodes based on the values of a column of
Tbl
.
fits a tree with additional options specified by one or more name-value pair
arguments, using any of the previous syntaxes. For example, you can specify the
algorithm used to find the best split on a categorical predictor, grow a
cross-validated tree, or hold out a fraction of the input data for
validation.tree
= fitctree(___,Name,Value
)
Grow a classification tree using the ionosphere
data set.
load ionosphere
tc = fitctree(X,Y)
tc = ClassificationTree ResponseName: 'Y' CategoricalPredictors: [] ClassNames: {'b' 'g'} ScoreTransform: 'none' NumObservations: 351 Properties, Methods
You can control the depth of the trees using the MaxNumSplits
, MinLeafSize
, or MinParentSize
name-value pair parameters. fitctree
grows deep decision trees by default. You can grow shallower trees to reduce model complexity or computation time.
Load the ionosphere
data set.
load ionosphere
The default values of the tree depth controllers for growing classification trees are:
n - 1
for MaxNumSplits
. n
is the training sample size.
1
for MinLeafSize
.
10
for MinParentSize
.
These default values tend to grow deep trees for large training sample sizes.
Train a classification tree using the default values for tree depth control. Cross-validate the model by using 10-fold cross-validation.
rng(1); % For reproducibility MdlDefault = fitctree(X,Y,'CrossVal','on');
Draw a histogram of the number of imposed splits on the trees. Also, view one of the trees.
numBranches = @(x)sum(x.IsBranch); mdlDefaultNumSplits = cellfun(numBranches, MdlDefault.Trained); figure; histogram(mdlDefaultNumSplits)
view(MdlDefault.Trained{1},'Mode','graph')
The average number of splits is around 15.
Suppose that you want a classification tree that is not as complex (deep) as the ones trained using the default number of splits. Train another classification tree, but set the maximum number of splits at 7, which is about half the mean number of splits from the default classification tree. Cross-validate the model by using 10-fold cross-validation.
Mdl7 = fitctree(X,Y,'MaxNumSplits',7,'CrossVal','on'); view(Mdl7.Trained{1},'Mode','graph')
Compare the cross-validation classification errors of the models.
classErrorDefault = kfoldLoss(MdlDefault)
classErrorDefault = 0.1168
classError7 = kfoldLoss(Mdl7)
classError7 = 0.1311
Mdl7
is much less complex and performs only slightly worse than MdlDefault
.
This example shows how to optimize hyperparameters automatically using fitctree
. The example uses Fisher's iris data.
Load Fisher's iris data.
load fisheriris
Optimize the cross-validation loss of the classifier, using the data in meas
to predict the response in species
.
X = meas; Y = species; Mdl = fitctree(X,Y,'OptimizeHyperparameters','auto')
|======================================================================================| | Iter | Eval | Objective | Objective | BestSoFar | BestSoFar | MinLeafSize | | | result | | runtime | (observed) | (estim.) | | |======================================================================================| | 1 | Best | 0.066667 | 0.83006 | 0.066667 | 0.066667 | 31 | | 2 | Accept | 0.066667 | 0.21725 | 0.066667 | 0.066667 | 12 | | 3 | Best | 0.04 | 0.14417 | 0.04 | 0.040003 | 2 | | 4 | Accept | 0.66667 | 0.12419 | 0.04 | 0.15796 | 73 | | 5 | Accept | 0.04 | 0.14783 | 0.04 | 0.040009 | 2 | | 6 | Accept | 0.66667 | 0.15588 | 0.04 | 0.040012 | 74 | | 7 | Accept | 0.066667 | 0.15864 | 0.04 | 0.040012 | 20 | | 8 | Accept | 0.04 | 0.10172 | 0.04 | 0.040008 | 4 | | 9 | Best | 0.033333 | 0.11429 | 0.033333 | 0.03335 | 1 | | 10 | Accept | 0.066667 | 0.078155 | 0.033333 | 0.03335 | 7 | | 11 | Accept | 0.04 | 0.13669 | 0.033333 | 0.033348 | 3 | | 12 | Accept | 0.066667 | 0.089281 | 0.033333 | 0.033348 | 26 | | 13 | Accept | 0.046667 | 0.12558 | 0.033333 | 0.033347 | 5 | | 14 | Accept | 0.033333 | 0.12133 | 0.033333 | 0.03334 | 1 | | 15 | Accept | 0.066667 | 0.10213 | 0.033333 | 0.033339 | 15 | | 16 | Accept | 0.033333 | 0.25367 | 0.033333 | 0.033337 | 1 | | 17 | Accept | 0.033333 | 0.088846 | 0.033333 | 0.033336 | 1 | | 18 | Accept | 0.33333 | 0.092997 | 0.033333 | 0.033336 | 43 | | 19 | Accept | 0.066667 | 0.11667 | 0.033333 | 0.033336 | 9 | | 20 | Accept | 0.066667 | 0.1473 | 0.033333 | 0.033336 | 6 | |======================================================================================| | Iter | Eval | Objective | Objective | BestSoFar | BestSoFar | MinLeafSize | | | result | | runtime | (observed) | (estim.) | | |======================================================================================| | 21 | Accept | 0.066667 | 0.099125 | 0.033333 | 0.033336 | 17 | | 22 | Accept | 0.066667 | 0.13919 | 0.033333 | 0.033336 | 10 | | 23 | Accept | 0.066667 | 0.1234 | 0.033333 | 0.033336 | 36 | | 24 | Accept | 0.33333 | 0.11026 | 0.033333 | 0.034075 | 54 | | 25 | Accept | 0.04 | 0.23708 | 0.033333 | 0.034054 | 2 | | 26 | Accept | 0.04 | 0.14081 | 0.033333 | 0.034022 | 3 | | 27 | Accept | 0.04 | 0.093676 | 0.033333 | 0.033997 | 4 | | 28 | Accept | 0.066667 | 0.091611 | 0.033333 | 0.033973 | 23 | | 29 | Accept | 0.066667 | 0.15805 | 0.033333 | 0.033946 | 8 | | 30 | Accept | 0.066667 | 0.14283 | 0.033333 | 0.033922 | 13 |
__________________________________________________________ Optimization completed. MaxObjectiveEvaluations of 30 reached. Total function evaluations: 30 Total elapsed time: 43.4069 seconds Total objective function evaluation time: 4.6827 Best observed feasible point: MinLeafSize ___________ 1 Observed objective function value = 0.033333 Estimated objective function value = 0.033922 Function evaluation time = 0.11429 Best estimated feasible point (according to models): MinLeafSize ___________ 1 Estimated objective function value = 0.033922 Estimated function evaluation time = 0.13541
Mdl = ClassificationTree ResponseName: 'Y' CategoricalPredictors: [] ClassNames: {'setosa' 'versicolor' 'virginica'} ScoreTransform: 'none' NumObservations: 150 HyperparameterOptimizationResults: [1x1 BayesianOptimization] Properties, Methods
Load the census1994
data set. Consider a model that predicts a person's salary category given their age, working class, education level, martial status, race, sex, capital gain and loss, and number of working hours per week.
load census1994 X = adultdata(:,{'age','workClass','education_num','marital_status','race',... 'sex','capital_gain','capital_loss','hours_per_week','salary'});
Display the number of categories represented in the categorical variables using summary
.
summary(X)
Variables: age: 32561x1 double Values: Min 17 Median 37 Max 90 workClass: 32561x1 categorical Values: Federal-gov 960 Local-gov 2093 Never-worked 7 Private 22696 Self-emp-inc 1116 Self-emp-not-inc 2541 State-gov 1298 Without-pay 14 NumMissing 1836 education_num: 32561x1 double Values: Min 1 Median 10 Max 16 marital_status: 32561x1 categorical Values: Divorced 4443 Married-AF-spouse 23 Married-civ-spouse 14976 Married-spouse-absent 418 Never-married 10683 Separated 1025 Widowed 993 race: 32561x1 categorical Values: Amer-Indian-Eskimo 311 Asian-Pac-Islander 1039 Black 3124 Other 271 White 27816 sex: 32561x1 categorical Values: Female 10771 Male 21790 capital_gain: 32561x1 double Values: Min 0 Median 0 Max 99999 capital_loss: 32561x1 double Values: Min 0 Median 0 Max 4356 hours_per_week: 32561x1 double Values: Min 1 Median 40 Max 99 salary: 32561x1 categorical Values: <=50K 24720 >50K 7841
Because there are few categories represented in the categorical variables compared to levels in the continuous variables, the standard CART, predictor-splitting algorithm prefers splitting a continuous predictor over the categorical variables.
Train a classification tree using the entire data set. To grow unbiased trees, specify usage of the curvature test for splitting predictors. Because there are missing observations in the data, specify usage of surrogate splits.
Mdl = fitctree(X,'salary','PredictorSelection','curvature',... 'Surrogate','on');
Estimate predictor importance values by summing changes in the risk due to splits on every predictor and dividing the sum by the number of branch nodes. Compare the estimates using a bar graph.
imp = predictorImportance(Mdl); figure; bar(imp); title('Predictor Importance Estimates'); ylabel('Estimates'); xlabel('Predictors'); h = gca; h.XTickLabel = Mdl.PredictorNames; h.XTickLabelRotation = 45; h.TickLabelInterpreter = 'none';
In this case, capital_gain
is the most important predictor, followed by education_num
.
This example shows how to optimize hyperparameters of a classification tree automatically using a tall array. The sample data set airlinesmall.csv
is a large data set that contains a tabular file of airline flight data. This example creates a tall table containing the data and uses it to run the optimization procedure.
When you perform calculations on tall arrays, MATLAB® uses either a parallel pool (default if you have Parallel Computing Toolbox™) or the local MATLAB session. If you want to run the example using the local MATLAB session when you have Parallel Computing Toolbox, you can change the global execution environment by using the mapreducer
function.
Create a datastore that references the folder location with the data. Select a subset of the variables to work with, and treat 'NA'
values as missing data so that datastore
replaces them with NaN
values. Create a tall table that contains the data in the datastore.
ds = datastore('airlinesmall.csv'); ds.SelectedVariableNames = {'Month','DayofMonth','DayOfWeek',... 'DepTime','ArrDelay','Distance','DepDelay'}; ds.TreatAsMissing = 'NA'; tt = tall(ds) % Tall table
Starting parallel pool (parpool) using the 'local' profile ... Connected to the parallel pool (number of workers: 6). tt = M×7 tall table Month DayofMonth DayOfWeek DepTime ArrDelay Distance DepDelay _____ __________ _________ _______ ________ ________ ________ 10 21 3 642 8 308 12 10 26 1 1021 8 296 1 10 23 5 2055 21 480 20 10 23 5 1332 13 296 12 10 22 4 629 4 373 -1 10 28 3 1446 59 308 63 10 8 4 928 3 447 -2 10 10 6 859 11 954 -1 : : : : : : : : : : : : : :
Determine the flights that are late by 10 minutes or more by defining a logical variable that is true for a late flight. This variable contains the class labels. A preview of this variable includes the first few rows.
Y = tt.DepDelay > 10 % Class labels
Y = M×1 tall logical array 1 0 1 1 0 1 0 0 : :
Create a tall array for the predictor data.
X = tt{:,1:end-1} % Predictor data
X = M×6 tall double matrix 10 21 3 642 8 308 10 26 1 1021 8 296 10 23 5 2055 21 480 10 23 5 1332 13 296 10 22 4 629 4 373 10 28 3 1446 59 308 10 8 4 928 3 447 10 10 6 859 11 954 : : : : : : : : : : : :
Remove rows in X
and Y
that contain missing data.
R = rmmissing([X Y]); % Data with missing entries removed
X = R(:,1:end-1);
Y = R(:,end);
Standardize the predictor variables.
Z = zscore(X);
Optimize hyperparameters automatically using the 'OptimizeHyperparameters'
name-value pair argument. Find the optimal 'MinLeafSize'
value that minimizes holdout cross-validation loss. (Specifying 'auto'
uses 'MinLeafSize'
.) For reproducibility, use the 'expected-improvement-plus'
acquisition function and set the seeds of the random number generators using rng
and tallrng
. The results can vary depending on the number of workers and the execution environment for the tall arrays. For details, see Control Where Your Code Runs.
rng('default') tallrng('default') [Mdl,FitInfo,HyperparameterOptimizationResults] = fitctree(Z,Y,... 'OptimizeHyperparameters','auto',... 'HyperparameterOptimizationOptions',struct('Holdout',0.3,... 'AcquisitionFunctionName','expected-improvement-plus'))
Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 3: Completed in 5.6 sec - Pass 2 of 3: Completed in 2.1 sec - Pass 3 of 3: Completed in 3.4 sec Evaluation completed in 13 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 1: Completed in 0.73 sec Evaluation completed in 0.9 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 1: Completed in 1.2 sec Evaluation completed in 1.5 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.64 sec - Pass 2 of 4: Completed in 1.1 sec - Pass 3 of 4: Completed in 0.72 sec - Pass 4 of 4: Completed in 1.1 sec Evaluation completed in 5.9 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.53 sec - Pass 2 of 4: Completed in 0.73 sec - Pass 3 of 4: Completed in 0.58 sec - Pass 4 of 4: Completed in 0.8 sec Evaluation completed in 3.5 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.54 sec - Pass 2 of 4: Completed in 0.72 sec - Pass 3 of 4: Completed in 0.54 sec - Pass 4 of 4: Completed in 0.79 sec Evaluation completed in 3.5 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.51 sec - Pass 2 of 4: Completed in 0.72 sec - Pass 3 of 4: Completed in 0.59 sec - Pass 4 of 4: Completed in 0.87 sec Evaluation completed in 3.3 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.56 sec - Pass 2 of 4: Completed in 0.8 sec - Pass 3 of 4: Completed in 0.54 sec - Pass 4 of 4: Completed in 0.89 sec Evaluation completed in 3.4 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.57 sec - Pass 2 of 4: Completed in 0.74 sec - Pass 3 of 4: Completed in 0.58 sec - Pass 4 of 4: Completed in 1 sec Evaluation completed in 3.5 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.54 sec - Pass 2 of 4: Completed in 1.3 sec - Pass 3 of 4: Completed in 0.68 sec - Pass 4 of 4: Completed in 1.3 sec Evaluation completed in 4.3 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.65 sec - Pass 2 of 4: Completed in 0.73 sec - Pass 3 of 4: Completed in 0.65 sec - Pass 4 of 4: Completed in 1.7 sec Evaluation completed in 4.4 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.63 sec - Pass 2 of 4: Completed in 0.85 sec - Pass 3 of 4: Completed in 0.58 sec - Pass 4 of 4: Completed in 2.2 sec Evaluation completed in 4.9 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 1.2 sec - Pass 2 of 4: Completed in 0.88 sec - Pass 3 of 4: Completed in 0.58 sec - Pass 4 of 4: Completed in 3 sec Evaluation completed in 6.9 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.72 sec - Pass 2 of 4: Completed in 0.96 sec - Pass 3 of 4: Completed in 0.59 sec - Pass 4 of 4: Completed in 4.2 sec Evaluation completed in 7.2 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.77 sec - Pass 2 of 4: Completed in 0.95 sec - Pass 3 of 4: Completed in 0.65 sec - Pass 4 of 4: Completed in 4.8 sec Evaluation completed in 7.9 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.79 sec - Pass 2 of 4: Completed in 1 sec - Pass 3 of 4: Completed in 0.61 sec - Pass 4 of 4: Completed in 5.1 sec Evaluation completed in 8.4 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.89 sec - Pass 2 of 4: Completed in 1.1 sec - Pass 3 of 4: Completed in 0.59 sec - Pass 4 of 4: Completed in 5.8 sec Evaluation completed in 9.3 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 1 sec - Pass 2 of 4: Completed in 1.2 sec - Pass 3 of 4: Completed in 0.63 sec - Pass 4 of 4: Completed in 5.2 sec Evaluation completed in 8.9 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 1.6 sec - Pass 2 of 4: Completed in 1.3 sec - Pass 3 of 4: Completed in 0.74 sec - Pass 4 of 4: Completed in 4.8 sec Evaluation completed in 9.2 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 1 sec - Pass 2 of 4: Completed in 1.3 sec - Pass 3 of 4: Completed in 0.68 sec - Pass 4 of 4: Completed in 3.9 sec Evaluation completed in 7.7 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 1.6 sec - Pass 2 of 4: Completed in 1.3 sec - Pass 3 of 4: Completed in 0.7 sec - Pass 4 of 4: Completed in 3 sec Evaluation completed in 7.3 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 1.1 sec - Pass 2 of 4: Completed in 1.3 sec - Pass 3 of 4: Completed in 0.66 sec - Pass 4 of 4: Completed in 2.5 sec Evaluation completed in 6.1 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 1.1 sec - Pass 2 of 4: Completed in 1.3 sec - Pass 3 of 4: Completed in 0.66 sec - Pass 4 of 4: Completed in 2.2 sec Evaluation completed in 5.9 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 1.1 sec - Pass 2 of 4: Completed in 1.3 sec - Pass 3 of 4: Completed in 0.69 sec - Pass 4 of 4: Completed in 1.9 sec Evaluation completed in 5.5 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 1.2 sec - Pass 2 of 4: Completed in 1.4 sec - Pass 3 of 4: Completed in 0.67 sec - Pass 4 of 4: Completed in 1.6 sec Evaluation completed in 5.5 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 1.3 sec - Pass 2 of 4: Completed in 1.4 sec - Pass 3 of 4: Completed in 0.65 sec - Pass 4 of 4: Completed in 1.5 sec Evaluation completed in 5.4 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 1.1 sec - Pass 2 of 4: Completed in 1.3 sec - Pass 3 of 4: Completed in 0.67 sec - Pass 4 of 4: Completed in 1.3 sec Evaluation completed in 5 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 1.2 sec - Pass 2 of 4: Completed in 1.3 sec - Pass 3 of 4: Completed in 0.73 sec - Pass 4 of 4: Completed in 1.3 sec Evaluation completed in 5.1 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 1.1 sec - Pass 2 of 4: Completed in 1.3 sec - Pass 3 of 4: Completed in 0.65 sec - Pass 4 of 4: Completed in 1.4 sec Evaluation completed in 5.1 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 1: Completed in 2.4 sec Evaluation completed in 2.6 sec |======================================================================================| | Iter | Eval | Objective | Objective | BestSoFar | BestSoFar | MinLeafSize | | | result | | runtime | (observed) | (estim.) | | |======================================================================================| | 1 | Best | 0.11572 | 197.12 | 0.11572 | 0.11572 | 10 | Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 1: Completed in 0.4 sec Evaluation completed in 0.56 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 1: Completed in 0.93 sec Evaluation completed in 1.1 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.5 sec - Pass 2 of 4: Completed in 0.7 sec - Pass 3 of 4: Completed in 1.1 sec - Pass 4 of 4: Completed in 0.84 sec Evaluation completed in 3.7 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 1: Completed in 1.4 sec Evaluation completed in 1.6 sec | 2 | Accept | 0.19635 | 10.496 | 0.11572 | 0.12008 | 48298 | Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 1: Completed in 0.33 sec Evaluation completed in 0.47 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 1: Completed in 0.83 sec Evaluation completed in 0.99 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.49 sec - Pass 2 of 4: Completed in 0.68 sec - Pass 3 of 4: Completed in 0.52 sec - Pass 4 of 4: Completed in 0.74 sec Evaluation completed in 3 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.48 sec - Pass 2 of 4: Completed in 0.69 sec - Pass 3 of 4: Completed in 0.51 sec - Pass 4 of 4: Completed in 0.73 sec Evaluation completed in 3 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.51 sec - Pass 2 of 4: Completed in 0.71 sec - Pass 3 of 4: Completed in 0.68 sec - Pass 4 of 4: Completed in 0.77 sec Evaluation completed in 3.2 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.48 sec - Pass 2 of 4: Completed in 0.7 sec - Pass 3 of 4: Completed in 0.55 sec - Pass 4 of 4: Completed in 0.86 sec Evaluation completed in 3.1 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.5 sec - Pass 2 of 4: Completed in 0.69 sec - Pass 3 of 4: Completed in 0.55 sec - Pass 4 of 4: Completed in 0.76 sec Evaluation completed in 3.1 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.52 sec - Pass 2 of 4: Completed in 0.7 sec - Pass 3 of 4: Completed in 0.5 sec - Pass 4 of 4: Completed in 0.76 sec Evaluation completed in 3 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.54 sec - Pass 2 of 4: Completed in 0.75 sec - Pass 3 of 4: Completed in 0.55 sec - Pass 4 of 4: Completed in 1.3 sec Evaluation completed in 4 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.53 sec - Pass 2 of 4: Completed in 0.74 sec - Pass 3 of 4: Completed in 0.55 sec - Pass 4 of 4: Completed in 0.78 sec Evaluation completed in 3.1 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.56 sec - Pass 2 of 4: Completed in 0.76 sec - Pass 3 of 4: Completed in 0.56 sec - Pass 4 of 4: Completed in 0.78 sec Evaluation completed in 3.2 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 1: Completed in 0.75 sec Evaluation completed in 0.87 sec | 3 | Best | 0.1048 | 44.614 | 0.1048 | 0.11431 | 3166 | Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 1: Completed in 0.3 sec Evaluation completed in 0.45 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 1: Completed in 0.83 sec Evaluation completed in 0.97 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.99 sec - Pass 2 of 4: Completed in 0.68 sec - Pass 3 of 4: Completed in 0.52 sec - Pass 4 of 4: Completed in 0.73 sec Evaluation completed in 3.5 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.47 sec - Pass 2 of 4: Completed in 0.76 sec - Pass 3 of 4: Completed in 0.54 sec - Pass 4 of 4: Completed in 0.82 sec Evaluation completed in 3.2 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.52 sec - Pass 2 of 4: Completed in 0.74 sec - Pass 3 of 4: Completed in 0.54 sec - Pass 4 of 4: Completed in 0.81 sec Evaluation completed in 3.2 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.55 sec - Pass 2 of 4: Completed in 0.7 sec - Pass 3 of 4: Completed in 0.53 sec - Pass 4 of 4: Completed in 0.81 sec Evaluation completed in 3.1 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.52 sec - Pass 2 of 4: Completed in 0.77 sec - Pass 3 of 4: Completed in 0.58 sec - Pass 4 of 4: Completed in 0.89 sec Evaluation completed in 3.3 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.51 sec - Pass 2 of 4: Completed in 0.74 sec - Pass 3 of 4: Completed in 0.6 sec - Pass 4 of 4: Completed in 1.5 sec Evaluation completed in 3.9 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 1 sec - Pass 2 of 4: Completed in 0.75 sec - Pass 3 of 4: Completed in 0.55 sec - Pass 4 of 4: Completed in 1.1 sec Evaluation completed in 4.1 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.52 sec - Pass 2 of 4: Completed in 0.78 sec - Pass 3 of 4: Completed in 0.5 sec - Pass 4 of 4: Completed in 1.3 sec Evaluation completed in 3.6 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.63 sec - Pass 2 of 4: Completed in 1.3 sec - Pass 3 of 4: Completed in 0.61 sec - Pass 4 of 4: Completed in 1.4 sec Evaluation completed in 4.5 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.59 sec - Pass 2 of 4: Completed in 0.8 sec - Pass 3 of 4: Completed in 0.57 sec - Pass 4 of 4: Completed in 1.5 sec Evaluation completed in 4.1 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.66 sec - Pass 2 of 4: Completed in 0.81 sec - Pass 3 of 4: Completed in 0.52 sec - Pass 4 of 4: Completed in 1.5 sec Evaluation completed in 4.2 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.62 sec - Pass 2 of 4: Completed in 0.75 sec - Pass 3 of 4: Completed in 0.61 sec - Pass 4 of 4: Completed in 1.4 sec Evaluation completed in 3.9 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.61 sec - Pass 2 of 4: Completed in 0.78 sec - Pass 3 of 4: Completed in 1.1 sec - Pass 4 of 4: Completed in 1.6 sec Evaluation completed in 4.7 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.6 sec - Pass 2 of 4: Completed in 0.81 sec - Pass 3 of 4: Completed in 0.55 sec - Pass 4 of 4: Completed in 1.5 sec Evaluation completed in 4 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.61 sec - Pass 2 of 4: Completed in 0.78 sec - Pass 3 of 4: Completed in 0.55 sec - Pass 4 of 4: Completed in 0.9 sec Evaluation completed in 3.4 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.59 sec - Pass 2 of 4: Completed in 0.81 sec - Pass 3 of 4: Completed in 0.53 sec - Pass 4 of 4: Completed in 0.81 sec Evaluation completed in 3.3 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.59 sec - Pass 2 of 4: Completed in 0.78 sec - Pass 3 of 4: Completed in 1 sec - Pass 4 of 4: Completed in 1.3 sec Evaluation completed in 4.3 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 1: Completed in 0.8 sec Evaluation completed in 0.94 sec | 4 | Best | 0.10094 | 91.723 | 0.10094 | 0.10574 | 180 | Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 1: Completed in 0.3 sec Evaluation completed in 0.42 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 1: Completed in 0.93 sec Evaluation completed in 1.1 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.55 sec - Pass 2 of 4: Completed in 0.66 sec - Pass 3 of 4: Completed in 0.55 sec - Pass 4 of 4: Completed in 0.83 sec Evaluation completed in 3.1 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.49 sec - Pass 2 of 4: Completed in 0.71 sec - Pass 3 of 4: Completed in 0.54 sec - Pass 4 of 4: Completed in 0.76 sec Evaluation completed in 3.1 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.5 sec - Pass 2 of 4: Completed in 0.7 sec - Pass 3 of 4: Completed in 0.56 sec - Pass 4 of 4: Completed in 0.78 sec Evaluation completed in 3.1 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.56 sec - Pass 2 of 4: Completed in 0.72 sec - Pass 3 of 4: Completed in 0.51 sec - Pass 4 of 4: Completed in 0.81 sec Evaluation completed in 3.2 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.52 sec - Pass 2 of 4: Completed in 1.3 sec - Pass 3 of 4: Completed in 1.1 sec - Pass 4 of 4: Completed in 0.88 sec Evaluation completed in 4.3 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.52 sec - Pass 2 of 4: Completed in 0.7 sec - Pass 3 of 4: Completed in 0.5 sec - Pass 4 of 4: Completed in 0.98 sec Evaluation completed in 3.3 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.56 sec - Pass 2 of 4: Completed in 0.74 sec - Pass 3 of 4: Completed in 0.5 sec - Pass 4 of 4: Completed in 1.1 sec Evaluation completed in 3.5 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.61 sec - Pass 2 of 4: Completed in 0.81 sec - Pass 3 of 4: Completed in 0.56 sec - Pass 4 of 4: Completed in 1.2 sec Evaluation completed in 3.8 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.59 sec - Pass 2 of 4: Completed in 0.81 sec - Pass 3 of 4: Completed in 0.7 sec - Pass 4 of 4: Completed in 1.4 sec Evaluation completed in 4.1 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.56 sec - Pass 2 of 4: Completed in 0.73 sec - Pass 3 of 4: Completed in 0.59 sec - Pass 4 of 4: Completed in 1.4 sec Evaluation completed in 3.8 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.58 sec - Pass 2 of 4: Completed in 0.8 sec - Pass 3 of 4: Completed in 0.52 sec - Pass 4 of 4: Completed in 1.2 sec Evaluation completed in 3.7 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.64 sec - Pass 2 of 4: Completed in 0.79 sec - Pass 3 of 4: Completed in 0.54 sec - Pass 4 of 4: Completed in 1.1 sec Evaluation completed in 3.6 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.57 sec - Pass 2 of 4: Completed in 0.78 sec - Pass 3 of 4: Completed in 0.56 sec - Pass 4 of 4: Completed in 0.97 sec Evaluation completed in 3.5 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.55 sec - Pass 2 of 4: Completed in 0.75 sec - Pass 3 of 4: Completed in 0.55 sec - Pass 4 of 4: Completed in 0.89 sec Evaluation completed in 3.3 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.6 sec - Pass 2 of 4: Completed in 1.3 sec - Pass 3 of 4: Completed in 0.61 sec - Pass 4 of 4: Completed in 0.85 sec Evaluation completed in 3.9 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.6 sec - Pass 2 of 4: Completed in 0.82 sec - Pass 3 of 4: Completed in 0.56 sec - Pass 4 of 4: Completed in 0.79 sec Evaluation completed in 3.4 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 1: Completed in 1.3 sec Evaluation completed in 1.4 sec | 5 | Best | 0.10087 | 82.84 | 0.10087 | 0.10085 | 219 | Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 1: Completed in 0.32 sec Evaluation completed in 0.45 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 1: Completed in 0.87 sec Evaluation completed in 1 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.5 sec - Pass 2 of 4: Completed in 0.7 sec - Pass 3 of 4: Completed in 0.56 sec - Pass 4 of 4: Completed in 0.76 sec Evaluation completed in 3.1 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.53 sec - Pass 2 of 4: Completed in 0.74 sec - Pass 3 of 4: Completed in 0.54 sec - Pass 4 of 4: Completed in 0.79 sec Evaluation completed in 3.2 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.49 sec - Pass 2 of 4: Completed in 0.66 sec - Pass 3 of 4: Completed in 0.5 sec - Pass 4 of 4: Completed in 0.78 sec Evaluation completed in 3 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.49 sec - Pass 2 of 4: Completed in 0.68 sec - Pass 3 of 4: Completed in 0.51 sec - Pass 4 of 4: Completed in 0.81 sec Evaluation completed in 3.1 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.55 sec - Pass 2 of 4: Completed in 0.68 sec - Pass 3 of 4: Completed in 0.54 sec - Pass 4 of 4: Completed in 0.86 sec Evaluation completed in 3.2 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.51 sec - Pass 2 of 4: Completed in 0.71 sec - Pass 3 of 4: Completed in 0.53 sec - Pass 4 of 4: Completed in 1 sec Evaluation completed in 3.4 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.59 sec - Pass 2 of 4: Completed in 1.2 sec - Pass 3 of 4: Completed in 0.55 sec - Pass 4 of 4: Completed in 0.85 sec Evaluation completed in 3.8 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.6 sec - Pass 2 of 4: Completed in 0.74 sec - Pass 3 of 4: Completed in 0.6 sec - Pass 4 of 4: Completed in 0.84 sec Evaluation completed in 3.3 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.51 sec - Pass 2 of 4: Completed in 0.77 sec - Pass 3 of 4: Completed in 0.58 sec - Pass 4 of 4: Completed in 0.87 sec Evaluation completed in 3.3 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.53 sec - Pass 2 of 4: Completed in 0.78 sec - Pass 3 of 4: Completed in 1.1 sec - Pass 4 of 4: Completed in 0.92 sec Evaluation completed in 3.9 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.54 sec - Pass 2 of 4: Completed in 0.81 sec - Pass 3 of 4: Completed in 0.59 sec - Pass 4 of 4: Completed in 0.77 sec Evaluation completed in 3.2 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.53 sec - Pass 2 of 4: Completed in 1.2 sec - Pass 3 of 4: Completed in 0.68 sec - Pass 4 of 4: Completed in 0.86 sec Evaluation completed in 3.9 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 1: Completed in 0.77 sec Evaluation completed in 0.93 sec | 6 | Accept | 0.10155 | 61.043 | 0.10087 | 0.10089 | 1089 | Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 1: Completed in 0.33 sec Evaluation completed in 0.46 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 1: Completed in 0.89 sec Evaluation completed in 1 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.56 sec - Pass 2 of 4: Completed in 0.72 sec - Pass 3 of 4: Completed in 0.56 sec - Pass 4 of 4: Completed in 0.8 sec Evaluation completed in 3.2 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.53 sec - Pass 2 of 4: Completed in 0.69 sec - Pass 3 of 4: Completed in 0.54 sec - Pass 4 of 4: Completed in 0.85 sec Evaluation completed in 3.1 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.51 sec - Pass 2 of 4: Completed in 1.2 sec - Pass 3 of 4: Completed in 0.59 sec - Pass 4 of 4: Completed in 0.83 sec Evaluation completed in 3.8 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 1 sec - Pass 2 of 4: Completed in 0.76 sec - Pass 3 of 4: Completed in 0.56 sec - Pass 4 of 4: Completed in 0.87 sec Evaluation completed in 4 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 1 sec - Pass 2 of 4: Completed in 0.77 sec - Pass 3 of 4: Completed in 0.53 sec - Pass 4 of 4: Completed in 0.9 sec Evaluation completed in 3.8 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 1 sec - Pass 2 of 4: Completed in 0.77 sec - Pass 3 of 4: Completed in 0.51 sec - Pass 4 of 4: Completed in 0.98 sec Evaluation completed in 3.9 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 1 sec - Pass 2 of 4: Completed in 0.78 sec - Pass 3 of 4: Completed in 0.62 sec - Pass 4 of 4: Completed in 1.1 sec Evaluation completed in 4.2 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.52 sec - Pass 2 of 4: Completed in 0.71 sec - Pass 3 of 4: Completed in 0.53 sec - Pass 4 of 4: Completed in 1.5 sec Evaluation completed in 3.9 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.59 sec - Pass 2 of 4: Completed in 0.72 sec - Pass 3 of 4: Completed in 0.59 sec - Pass 4 of 4: Completed in 2 sec Evaluation completed in 4.5 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.59 sec - Pass 2 of 4: Completed in 0.81 sec - Pass 3 of 4: Completed in 0.57 sec - Pass 4 of 4: Completed in 2.7 sec Evaluation completed in 5.4 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.64 sec - Pass 2 of 4: Completed in 0.87 sec - Pass 3 of 4: Completed in 1.2 sec - Pass 4 of 4: Completed in 3.7 sec Evaluation completed in 7.1 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.73 sec - Pass 2 of 4: Completed in 0.92 sec - Pass 3 of 4: Completed in 0.6 sec - Pass 4 of 4: Completed in 4.4 sec Evaluation completed in 7.3 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.86 sec - Pass 2 of 4: Completed in 1.5 sec - Pass 3 of 4: Completed in 0.64 sec - Pass 4 of 4: Completed in 4.8 sec Evaluation completed in 8.6 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.9 sec - Pass 2 of 4: Completed in 1.1 sec - Pass 3 of 4: Completed in 0.65 sec - Pass 4 of 4: Completed in 5.2 sec Evaluation completed in 8.5 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 1 sec - Pass 2 of 4: Completed in 1.3 sec - Pass 3 of 4: Completed in 0.73 sec - Pass 4 of 4: Completed in 5.6 sec Evaluation completed in 9.4 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 1.5 sec - Pass 2 of 4: Completed in 1.6 sec - Pass 3 of 4: Completed in 0.75 sec - Pass 4 of 4: Completed in 5.8 sec Evaluation completed in 10 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 1.3 sec - Pass 2 of 4: Completed in 1.4 sec - Pass 3 of 4: Completed in 1.2 sec - Pass 4 of 4: Completed in 5.1 sec Evaluation completed in 9.9 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 1.4 sec - Pass 2 of 4: Completed in 1.5 sec - Pass 3 of 4: Completed in 0.7 sec - Pass 4 of 4: Completed in 4.1 sec Evaluation completed in 8.5 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 1.4 sec - Pass 2 of 4: Completed in 1.6 sec - Pass 3 of 4: Completed in 0.71 sec - Pass 4 of 4: Completed in 3.6 sec Evaluation completed in 7.9 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 1.5 sec - Pass 2 of 4: Completed in 1.8 sec - Pass 3 of 4: Completed in 0.74 sec - Pass 4 of 4: Completed in 3.2 sec Evaluation completed in 7.9 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 1.4 sec - Pass 2 of 4: Completed in 1.7 sec - Pass 3 of 4: Completed in 0.73 sec - Pass 4 of 4: Completed in 2.8 sec Evaluation completed in 7.3 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 1.5 sec - Pass 2 of 4: Completed in 1.7 sec - Pass 3 of 4: Completed in 0.82 sec - Pass 4 of 4: Completed in 2.4 sec Evaluation completed in 7 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 2 sec - Pass 2 of 4: Completed in 1.9 sec - Pass 3 of 4: Completed in 0.79 sec - Pass 4 of 4: Completed in 2.3 sec Evaluation completed in 7.6 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 1.6 sec - Pass 2 of 4: Completed in 1.8 sec - Pass 3 of 4: Completed in 0.73 sec - Pass 4 of 4: Completed in 2.2 sec Evaluation completed in 6.9 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 1.6 sec - Pass 2 of 4: Completed in 1.7 sec - Pass 3 of 4: Completed in 0.79 sec - Pass 4 of 4: Completed in 2.3 sec Evaluation completed in 7 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 1.7 sec - Pass 2 of 4: Completed in 1.9 sec - Pass 3 of 4: Completed in 0.8 sec - Pass 4 of 4: Completed in 1.8 sec Evaluation completed in 6.8 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 1.7 sec - Pass 2 of 4: Completed in 1.8 sec - Pass 3 of 4: Completed in 0.77 sec - Pass 4 of 4: Completed in 1.8 sec Evaluation completed in 6.6 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 1.4 sec - Pass 2 of 4: Completed in 1.6 sec - Pass 3 of 4: Completed in 0.73 sec - Pass 4 of 4: Completed in 1.8 sec Evaluation completed in 6.1 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 1.5 sec - Pass 2 of 4: Completed in 1.7 sec - Pass 3 of 4: Completed in 1.3 sec - Pass 4 of 4: Completed in 1.7 sec Evaluation completed in 6.7 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 1.5 sec - Pass 2 of 4: Completed in 1.7 sec - Pass 3 of 4: Completed in 0.73 sec - Pass 4 of 4: Completed in 1.8 sec Evaluation completed in 6.3 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 1: Completed in 1.3 sec Evaluation completed in 1.5 sec | 7 | Accept | 0.13495 | 241.76 | 0.10087 | 0.10089 | 1 | Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 1: Completed in 0.31 sec Evaluation completed in 0.44 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 1: Completed in 0.87 sec Evaluation completed in 1 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.47 sec - Pass 2 of 4: Completed in 0.67 sec - Pass 3 of 4: Completed in 0.54 sec - Pass 4 of 4: Completed in 0.74 sec Evaluation completed in 3 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.51 sec - Pass 2 of 4: Completed in 0.79 sec - Pass 3 of 4: Completed in 0.56 sec - Pass 4 of 4: Completed in 0.76 sec Evaluation completed in 3.2 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.49 sec - Pass 2 of 4: Completed in 0.69 sec - Pass 3 of 4: Completed in 1.1 sec - Pass 4 of 4: Completed in 0.78 sec Evaluation completed in 3.6 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 1 sec - Pass 2 of 4: Completed in 0.78 sec - Pass 3 of 4: Completed in 0.53 sec - Pass 4 of 4: Completed in 0.85 sec Evaluation completed in 3.8 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.54 sec - Pass 2 of 4: Completed in 0.77 sec - Pass 3 of 4: Completed in 0.52 sec - Pass 4 of 4: Completed in 0.89 sec Evaluation completed in 3.3 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.51 sec - Pass 2 of 4: Completed in 0.73 sec - Pass 3 of 4: Completed in 0.6 sec - Pass 4 of 4: Completed in 1.1 sec Evaluation completed in 3.5 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.53 sec - Pass 2 of 4: Completed in 0.77 sec - Pass 3 of 4: Completed in 0.54 sec - Pass 4 of 4: Completed in 1.3 sec Evaluation completed in 3.7 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.62 sec - Pass 2 of 4: Completed in 0.8 sec - Pass 3 of 4: Completed in 0.61 sec - Pass 4 of 4: Completed in 1.6 sec Evaluation completed in 4.4 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 1.2 sec - Pass 2 of 4: Completed in 1.5 sec - Pass 3 of 4: Completed in 1.1 sec - Pass 4 of 4: Completed in 1.9 sec Evaluation completed in 6.3 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.69 sec - Pass 2 of 4: Completed in 0.88 sec - Pass 3 of 4: Completed in 0.75 sec - Pass 4 of 4: Completed in 2.1 sec Evaluation completed in 5.1 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.61 sec - Pass 2 of 4: Completed in 0.79 sec - Pass 3 of 4: Completed in 0.54 sec - Pass 4 of 4: Completed in 2.2 sec Evaluation completed in 4.8 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.61 sec - Pass 2 of 4: Completed in 0.84 sec - Pass 3 of 4: Completed in 0.58 sec - Pass 4 of 4: Completed in 2.2 sec Evaluation completed in 4.9 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.68 sec - Pass 2 of 4: Completed in 0.85 sec - Pass 3 of 4: Completed in 0.59 sec - Pass 4 of 4: Completed in 2.2 sec Evaluation completed in 4.9 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.68 sec - Pass 2 of 4: Completed in 0.91 sec - Pass 3 of 4: Completed in 0.58 sec - Pass 4 of 4: Completed in 2.4 sec Evaluation completed in 5.3 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.92 sec - Pass 2 of 4: Completed in 0.86 sec - Pass 3 of 4: Completed in 0.57 sec - Pass 4 of 4: Completed in 1.6 sec Evaluation completed in 4.7 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.69 sec - Pass 2 of 4: Completed in 0.91 sec - Pass 3 of 4: Completed in 0.63 sec - Pass 4 of 4: Completed in 1.3 sec Evaluation completed in 4.1 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.67 sec - Pass 2 of 4: Completed in 0.86 sec - Pass 3 of 4: Completed in 0.56 sec - Pass 4 of 4: Completed in 0.99 sec Evaluation completed in 3.6 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 1.2 sec - Pass 2 of 4: Completed in 0.9 sec - Pass 3 of 4: Completed in 0.57 sec - Pass 4 of 4: Completed in 0.95 sec Evaluation completed in 4.2 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.73 sec - Pass 2 of 4: Completed in 0.91 sec - Pass 3 of 4: Completed in 0.57 sec - Pass 4 of 4: Completed in 0.91 sec Evaluation completed in 3.7 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.76 sec - Pass 2 of 4: Completed in 0.93 sec - Pass 3 of 4: Completed in 0.57 sec - Pass 4 of 4: Completed in 0.9 sec Evaluation completed in 3.7 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 1: Completed in 0.91 sec Evaluation completed in 1.1 sec | 8 | Accept | 0.10246 | 115.31 | 0.10087 | 0.10089 | 58 | Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 1: Completed in 0.34 sec Evaluation completed in 0.49 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 1: Completed in 0.87 sec Evaluation completed in 1 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.55 sec - Pass 2 of 4: Completed in 0.72 sec - Pass 3 of 4: Completed in 0.57 sec - Pass 4 of 4: Completed in 0.8 sec Evaluation completed in 3.2 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.48 sec - Pass 2 of 4: Completed in 0.7 sec - Pass 3 of 4: Completed in 0.52 sec - Pass 4 of 4: Completed in 0.76 sec Evaluation completed in 3 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.51 sec - Pass 2 of 4: Completed in 0.69 sec - Pass 3 of 4: Completed in 0.54 sec - Pass 4 of 4: Completed in 0.79 sec Evaluation completed in 3.1 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.53 sec - Pass 2 of 4: Completed in 0.72 sec - Pass 3 of 4: Completed in 0.56 sec - Pass 4 of 4: Completed in 0.81 sec Evaluation completed in 3.2 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 1 sec - Pass 2 of 4: Completed in 0.75 sec - Pass 3 of 4: Completed in 0.56 sec - Pass 4 of 4: Completed in 1.4 sec Evaluation completed in 4.3 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.55 sec - Pass 2 of 4: Completed in 0.77 sec - Pass 3 of 4: Completed in 0.6 sec - Pass 4 of 4: Completed in 0.9 sec Evaluation completed in 3.3 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.5 sec - Pass 2 of 4: Completed in 0.73 sec - Pass 3 of 4: Completed in 1.1 sec - Pass 4 of 4: Completed in 1.1 sec Evaluation completed in 4 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.56 sec - Pass 2 of 4: Completed in 0.75 sec - Pass 3 of 4: Completed in 0.53 sec - Pass 4 of 4: Completed in 1.2 sec Evaluation completed in 3.5 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.61 sec - Pass 2 of 4: Completed in 0.76 sec - Pass 3 of 4: Completed in 1.1 sec - Pass 4 of 4: Completed in 1.1 sec Evaluation completed in 4.1 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.6 sec - Pass 2 of 4: Completed in 0.74 sec - Pass 3 of 4: Completed in 0.53 sec - Pass 4 of 4: Completed in 0.95 sec Evaluation completed in 3.4 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.55 sec - Pass 2 of 4: Completed in 0.76 sec - Pass 3 of 4: Completed in 0.57 sec - Pass 4 of 4: Completed in 0.94 sec Evaluation completed in 3.4 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.58 sec - Pass 2 of 4: Completed in 0.77 sec - Pass 3 of 4: Completed in 0.54 sec - Pass 4 of 4: Completed in 0.83 sec Evaluation completed in 3.3 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.59 sec - Pass 2 of 4: Completed in 0.78 sec - Pass 3 of 4: Completed in 0.57 sec - Pass 4 of 4: Completed in 0.83 sec Evaluation completed in 3.4 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.6 sec - Pass 2 of 4: Completed in 0.76 sec - Pass 3 of 4: Completed in 0.55 sec - Pass 4 of 4: Completed in 0.77 sec Evaluation completed in 3.3 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.72 sec - Pass 2 of 4: Completed in 0.81 sec - Pass 3 of 4: Completed in 0.6 sec - Pass 4 of 4: Completed in 0.76 sec Evaluation completed in 3.6 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 1: Completed in 1.3 sec Evaluation completed in 1.4 sec | 9 | Accept | 0.10173 | 77.229 | 0.10087 | 0.10086 | 418 | Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 1: Completed in 0.32 sec Evaluation completed in 0.46 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 1: Completed in 0.84 sec Evaluation completed in 1 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.5 sec - Pass 2 of 4: Completed in 0.72 sec - Pass 3 of 4: Completed in 0.57 sec - Pass 4 of 4: Completed in 0.75 sec Evaluation completed in 3.1 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.49 sec - Pass 2 of 4: Completed in 0.68 sec - Pass 3 of 4: Completed in 0.55 sec - Pass 4 of 4: Completed in 0.76 sec Evaluation completed in 3 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.52 sec - Pass 2 of 4: Completed in 0.71 sec - Pass 3 of 4: Completed in 0.54 sec - Pass 4 of 4: Completed in 0.91 sec Evaluation completed in 3.3 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.53 sec - Pass 2 of 4: Completed in 0.69 sec - Pass 3 of 4: Completed in 0.52 sec - Pass 4 of 4: Completed in 0.82 sec Evaluation completed in 3.1 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.51 sec - Pass 2 of 4: Completed in 0.7 sec - Pass 3 of 4: Completed in 0.61 sec - Pass 4 of 4: Completed in 0.82 sec Evaluation completed in 3.2 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 1.1 sec - Pass 2 of 4: Completed in 0.78 sec - Pass 3 of 4: Completed in 0.54 sec - Pass 4 of 4: Completed in 0.95 sec Evaluation completed in 3.9 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.55 sec - Pass 2 of 4: Completed in 0.71 sec - Pass 3 of 4: Completed in 0.53 sec - Pass 4 of 4: Completed in 1.1 sec Evaluation completed in 3.5 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.54 sec - Pass 2 of 4: Completed in 0.7 sec - Pass 3 of 4: Completed in 0.58 sec - Pass 4 of 4: Completed in 1.3 sec Evaluation completed in 3.7 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.55 sec - Pass 2 of 4: Completed in 0.73 sec - Pass 3 of 4: Completed in 0.59 sec - Pass 4 of 4: Completed in 1.7 sec Evaluation completed in 4.2 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.58 sec - Pass 2 of 4: Completed in 0.79 sec - Pass 3 of 4: Completed in 0.56 sec - Pass 4 of 4: Completed in 1.7 sec Evaluation completed in 4.2 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.71 sec - Pass 2 of 4: Completed in 1.7 sec - Pass 3 of 4: Completed in 0.59 sec - Pass 4 of 4: Completed in 1.7 sec Evaluation completed in 5.2 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.65 sec - Pass 2 of 4: Completed in 0.83 sec - Pass 3 of 4: Completed in 0.61 sec - Pass 4 of 4: Completed in 1.4 sec Evaluation completed in 4.1 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.67 sec - Pass 2 of 4: Completed in 0.87 sec - Pass 3 of 4: Completed in 0.58 sec - Pass 4 of 4: Completed in 1.4 sec Evaluation completed in 4.1 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.61 sec - Pass 2 of 4: Completed in 0.82 sec - Pass 3 of 4: Completed in 0.55 sec - Pass 4 of 4: Completed in 1.1 sec Evaluation completed in 3.7 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.65 sec - Pass 2 of 4: Completed in 0.84 sec - Pass 3 of 4: Completed in 0.62 sec - Pass 4 of 4: Completed in 0.89 sec Evaluation completed in 3.6 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.64 sec - Pass 2 of 4: Completed in 0.81 sec - Pass 3 of 4: Completed in 0.56 sec - Pass 4 of 4: Completed in 0.88 sec Evaluation completed in 3.6 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.62 sec - Pass 2 of 4: Completed in 0.9 sec - Pass 3 of 4: Completed in 0.55 sec - Pass 4 of 4: Completed in 0.86 sec Evaluation completed in 3.5 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.65 sec - Pass 2 of 4: Completed in 0.81 sec - Pass 3 of 4: Completed in 0.56 sec - Pass 4 of 4: Completed in 0.8 sec Evaluation completed in 3.3 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 1: Completed in 0.77 sec Evaluation completed in 0.89 sec | 10 | Accept | 0.10114 | 94.532 | 0.10087 | 0.10091 | 123 | Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 1: Completed in 0.86 sec Evaluation completed in 1 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 1: Completed in 0.83 sec Evaluation completed in 0.99 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.48 sec - Pass 2 of 4: Completed in 0.7 sec - Pass 3 of 4: Completed in 0.54 sec - Pass 4 of 4: Completed in 0.8 sec Evaluation completed in 3 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.5 sec - Pass 2 of 4: Completed in 0.72 sec - Pass 3 of 4: Completed in 0.53 sec - Pass 4 of 4: Completed in 0.79 sec Evaluation completed in 3.1 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.56 sec - Pass 2 of 4: Completed in 0.73 sec - Pass 3 of 4: Completed in 0.54 sec - Pass 4 of 4: Completed in 0.85 sec Evaluation completed in 3.3 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.5 sec - Pass 2 of 4: Completed in 0.69 sec - Pass 3 of 4: Completed in 0.55 sec - Pass 4 of 4: Completed in 0.81 sec Evaluation completed in 3.2 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.55 sec - Pass 2 of 4: Completed in 0.82 sec - Pass 3 of 4: Completed in 0.64 sec - Pass 4 of 4: Completed in 0.94 sec Evaluation completed in 3.5 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.49 sec - Pass 2 of 4: Completed in 0.77 sec - Pass 3 of 4: Completed in 0.53 sec - Pass 4 of 4: Completed in 0.97 sec Evaluation completed in 3.3 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.56 sec - Pass 2 of 4: Completed in 0.78 sec - Pass 3 of 4: Completed in 0.58 sec - Pass 4 of 4: Completed in 1.3 sec Evaluation completed in 3.8 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.55 sec - Pass 2 of 4: Completed in 0.81 sec - Pass 3 of 4: Completed in 0.56 sec - Pass 4 of 4: Completed in 1.3 sec Evaluation completed in 3.8 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.55 sec - Pass 2 of 4: Completed in 0.76 sec - Pass 3 of 4: Completed in 0.56 sec - Pass 4 of 4: Completed in 1.5 sec Evaluation completed in 3.9 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.59 sec - Pass 2 of 4: Completed in 0.76 sec - Pass 3 of 4: Completed in 0.55 sec - Pass 4 of 4: Completed in 1.5 sec Evaluation completed in 4 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.59 sec - Pass 2 of 4: Completed in 0.8 sec - Pass 3 of 4: Completed in 0.59 sec - Pass 4 of 4: Completed in 1.4 sec Evaluation completed in 3.9 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.59 sec - Pass 2 of 4: Completed in 0.77 sec - Pass 3 of 4: Completed in 0.55 sec - Pass 4 of 4: Completed in 1.2 sec Evaluation completed in 3.7 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 1.1 sec - Pass 2 of 4: Completed in 0.8 sec - Pass 3 of 4: Completed in 0.58 sec - Pass 4 of 4: Completed in 1.1 sec Evaluation completed in 4.2 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.65 sec - Pass 2 of 4: Completed in 0.84 sec - Pass 3 of 4: Completed in 1.1 sec - Pass 4 of 4: Completed in 1 sec Evaluation completed in 4.3 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.63 sec - Pass 2 of 4: Completed in 0.84 sec - Pass 3 of 4: Completed in 0.59 sec - Pass 4 of 4: Completed in 0.9 sec Evaluation completed in 3.5 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 1.2 sec - Pass 2 of 4: Completed in 0.83 sec - Pass 3 of 4: Completed in 0.56 sec - Pass 4 of 4: Completed in 0.81 sec Evaluation completed in 4 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.65 sec - Pass 2 of 4: Completed in 0.79 sec - Pass 3 of 4: Completed in 0.59 sec - Pass 4 of 4: Completed in 0.8 sec Evaluation completed in 3.4 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 1: Completed in 0.77 sec Evaluation completed in 0.89 sec | 11 | Best | 0.1008 | 90.637 | 0.1008 | 0.10088 | 178 | Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 1: Completed in 0.38 sec Evaluation completed in 0.52 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 1: Completed in 0.88 sec Evaluation completed in 1 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.49 sec - Pass 2 of 4: Completed in 0.69 sec - Pass 3 of 4: Completed in 0.51 sec - Pass 4 of 4: Completed in 0.78 sec Evaluation completed in 3 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.59 sec - Pass 2 of 4: Completed in 0.72 sec - Pass 3 of 4: Completed in 0.53 sec - Pass 4 of 4: Completed in 0.79 sec Evaluation completed in 3.1 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.58 sec - Pass 2 of 4: Completed in 0.74 sec - Pass 3 of 4: Completed in 0.55 sec - Pass 4 of 4: Completed in 0.93 sec Evaluation completed in 3.4 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.57 sec - Pass 2 of 4: Completed in 0.79 sec - Pass 3 of 4: Completed in 0.58 sec - Pass 4 of 4: Completed in 0.83 sec Evaluation completed in 3.3 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.53 sec - Pass 2 of 4: Completed in 1.2 sec - Pass 3 of 4: Completed in 0.59 sec - Pass 4 of 4: Completed in 0.91 sec Evaluation completed in 3.8 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.58 sec - Pass 2 of 4: Completed in 0.85 sec - Pass 3 of 4: Completed in 0.58 sec - Pass 4 of 4: Completed in 1 sec Evaluation completed in 3.6 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.56 sec - Pass 2 of 4: Completed in 0.77 sec - Pass 3 of 4: Completed in 0.55 sec - Pass 4 of 4: Completed in 1.2 sec Evaluation completed in 3.6 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 1.1 sec - Pass 2 of 4: Completed in 0.81 sec - Pass 3 of 4: Completed in 0.52 sec - Pass 4 of 4: Completed in 1.4 sec Evaluation completed in 4.4 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.54 sec - Pass 2 of 4: Completed in 0.74 sec - Pass 3 of 4: Completed in 0.54 sec - Pass 4 of 4: Completed in 1.5 sec Evaluation completed in 3.9 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.61 sec - Pass 2 of 4: Completed in 0.79 sec - Pass 3 of 4: Completed in 0.58 sec - Pass 4 of 4: Completed in 1.5 sec Evaluation completed in 4 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.61 sec - Pass 2 of 4: Completed in 0.82 sec - Pass 3 of 4: Completed in 0.61 sec - Pass 4 of 4: Completed in 1.4 sec Evaluation completed in 4 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.66 sec - Pass 2 of 4: Completed in 0.77 sec - Pass 3 of 4: Completed in 0.54 sec - Pass 4 of 4: Completed in 1.2 sec Evaluation completed in 3.8 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.61 sec - Pass 2 of 4: Completed in 0.79 sec - Pass 3 of 4: Completed in 0.56 sec - Pass 4 of 4: Completed in 1.2 sec Evaluation completed in 3.7 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.62 sec - Pass 2 of 4: Completed in 0.85 sec - Pass 3 of 4: Completed in 0.56 sec - Pass 4 of 4: Completed in 1 sec Evaluation completed in 3.6 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.61 sec - Pass 2 of 4: Completed in 0.86 sec - Pass 3 of 4: Completed in 1.1 sec - Pass 4 of 4: Completed in 0.96 sec Evaluation completed in 4.1 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.65 sec - Pass 2 of 4: Completed in 0.8 sec - Pass 3 of 4: Completed in 0.59 sec - Pass 4 of 4: Completed in 0.86 sec Evaluation completed in 3.5 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.69 sec - Pass 2 of 4: Completed in 0.84 sec - Pass 3 of 4: Completed in 0.53 sec - Pass 4 of 4: Completed in 0.83 sec Evaluation completed in 3.5 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 1: Completed in 0.76 sec Evaluation completed in 0.89 sec | 12 | Accept | 0.1008 | 90.267 | 0.1008 | 0.10086 | 179 | Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 1: Completed in 0.32 sec Evaluation completed in 0.45 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 1: Completed in 0.9 sec Evaluation completed in 1.1 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.58 sec - Pass 2 of 4: Completed in 0.71 sec - Pass 3 of 4: Completed in 0.53 sec - Pass 4 of 4: Completed in 0.77 sec Evaluation completed in 3.2 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.49 sec - Pass 2 of 4: Completed in 0.69 sec - Pass 3 of 4: Completed in 0.58 sec - Pass 4 of 4: Completed in 0.77 sec Evaluation completed in 3.1 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.52 sec - Pass 2 of 4: Completed in 0.71 sec - Pass 3 of 4: Completed in 0.51 sec - Pass 4 of 4: Completed in 0.78 sec Evaluation completed in 3.1 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.54 sec - Pass 2 of 4: Completed in 0.7 sec - Pass 3 of 4: Completed in 0.54 sec - Pass 4 of 4: Completed in 0.72 sec Evaluation completed in 3.1 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.54 sec - Pass 2 of 4: Completed in 0.74 sec - Pass 3 of 4: Completed in 0.51 sec - Pass 4 of 4: Completed in 1.3 sec Evaluation completed in 3.6 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.54 sec - Pass 2 of 4: Completed in 0.78 sec - Pass 3 of 4: Completed in 0.59 sec - Pass 4 of 4: Completed in 0.74 sec Evaluation completed in 3.2 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 1: Completed in 0.83 sec Evaluation completed in 0.97 sec | 13 | Accept | 0.11126 | 32.134 | 0.1008 | 0.10084 | 10251 | Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 1: Completed in 0.32 sec Evaluation completed in 0.45 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 1: Completed in 0.85 sec Evaluation completed in 0.99 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.51 sec - Pass 2 of 4: Completed in 0.75 sec - Pass 3 of 4: Completed in 0.55 sec - Pass 4 of 4: Completed in 0.74 sec Evaluation completed in 3.1 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.5 sec - Pass 2 of 4: Completed in 0.7 sec - Pass 3 of 4: Completed in 0.57 sec - Pass 4 of 4: Completed in 0.78 sec Evaluation completed in 3.1 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.5 sec - Pass 2 of 4: Completed in 0.68 sec - Pass 3 of 4: Completed in 0.53 sec - Pass 4 of 4: Completed in 0.79 sec Evaluation completed in 3 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.5 sec - Pass 2 of 4: Completed in 1.3 sec - Pass 3 of 4: Completed in 0.54 sec - Pass 4 of 4: Completed in 0.91 sec Evaluation completed in 3.7 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.53 sec - Pass 2 of 4: Completed in 1.2 sec - Pass 3 of 4: Completed in 0.59 sec - Pass 4 of 4: Completed in 0.86 sec Evaluation completed in 3.7 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.51 sec - Pass 2 of 4: Completed in 0.71 sec - Pass 3 of 4: Completed in 0.59 sec - Pass 4 of 4: Completed in 1 sec Evaluation completed in 3.4 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.56 sec - Pass 2 of 4: Completed in 0.71 sec - Pass 3 of 4: Completed in 0.64 sec - Pass 4 of 4: Completed in 0.99 sec Evaluation completed in 3.4 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.54 sec - Pass 2 of 4: Completed in 1.2 sec - Pass 3 of 4: Completed in 0.58 sec - Pass 4 of 4: Completed in 0.94 sec Evaluation completed in 3.8 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.51 sec - Pass 2 of 4: Completed in 0.77 sec - Pass 3 of 4: Completed in 0.59 sec - Pass 4 of 4: Completed in 0.9 sec Evaluation completed in 3.3 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.56 sec - Pass 2 of 4: Completed in 0.78 sec - Pass 3 of 4: Completed in 0.56 sec - Pass 4 of 4: Completed in 0.9 sec Evaluation completed in 3.3 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.55 sec - Pass 2 of 4: Completed in 0.72 sec - Pass 3 of 4: Completed in 0.52 sec - Pass 4 of 4: Completed in 0.89 sec Evaluation completed in 3.2 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.58 sec - Pass 2 of 4: Completed in 0.76 sec - Pass 3 of 4: Completed in 0.54 sec - Pass 4 of 4: Completed in 0.8 sec Evaluation completed in 3.3 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.56 sec - Pass 2 of 4: Completed in 1.3 sec - Pass 3 of 4: Completed in 0.61 sec - Pass 4 of 4: Completed in 0.76 sec Evaluation completed in 3.8 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 1: Completed in 0.83 sec Evaluation completed in 0.97 sec | 14 | Accept | 0.10154 | 66.262 | 0.1008 | 0.10085 | 736 | Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 1: Completed in 0.36 sec Evaluation completed in 0.5 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 1: Completed in 0.89 sec Evaluation completed in 1 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.53 sec - Pass 2 of 4: Completed in 0.69 sec - Pass 3 of 4: Completed in 0.56 sec - Pass 4 of 4: Completed in 0.74 sec Evaluation completed in 3.1 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.5 sec - Pass 2 of 4: Completed in 0.67 sec - Pass 3 of 4: Completed in 0.56 sec - Pass 4 of 4: Completed in 0.78 sec Evaluation completed in 3.1 sec Evaluating tall expression using the Parallel Pool 'local': - Pass 1 of 4: Completed in 0.52 sec - Pass 2 of 4: Completed in 0.69 sec - Pass 3 of 4: Completed in 0.54 sec - Pass 4 of 4: Completed in 0.84 sec Evaluation completed in 3.1 sec Evaluating tall expression using the Parallel Pool 'local': Evaluation 0% ...
Mdl = CompactClassificationTree ResponseName: 'Y' CategoricalPredictors: [] ClassNames: [0 1] ScoreTransform: 'none' Properties, Methods
FitInfo = struct with no fields.
HyperparameterOptimizationResults = BayesianOptimization with properties: ObjectiveFcn: @createObjFcn/tallObjFcn VariableDescriptions: [4×1 optimizableVariable] Options: [1×1 struct] MinObjective: 0.1004 XAtMinObjective: [1×1 table] MinEstimatedObjective: 0.1008 XAtMinEstimatedObjective: [1×1 table] NumObjectiveEvaluations: 30 TotalElapsedTime: 3.0367e+03 NextPoint: [1×1 table] XTrace: [30×1 table] ObjectiveTrace: [30×1 double] ConstraintsTrace: [] UserDataTrace: {30×1 cell} ObjectiveEvaluationTimeTrace: [30×1 double] IterationTimeTrace: [30×1 double] ErrorTrace: [30×1 double] FeasibilityTrace: [30×1 logical] FeasibilityProbabilityTrace: [30×1 double] IndexOfMinimumTrace: [30×1 double] ObjectiveMinimumTrace: [30×1 double] EstimatedObjectiveMinimumTrace: [30×1 double]
Tbl
— Sample dataSample data used to train the model, specified as a table. Each row of Tbl
corresponds to one observation, and each column corresponds to one predictor variable.
Optionally, Tbl
can contain one additional column for the response
variable. Multicolumn variables and cell arrays other than cell arrays of character
vectors are not allowed.
If Tbl
contains the
response variable, and you want to use all
remaining variables in Tbl
as
predictors, then specify the response variable by
using ResponseVarName
.
If Tbl
contains the
response variable, and you want to use only a
subset of the remaining variables in
Tbl
as predictors, then specify
a formula by using
formula
.
If Tbl
does not contain the
response variable, then specify a response
variable by using Y
. The
length of the response variable and the number of
rows in Tbl
must be
equal.
Data Types: table
ResponseVarName
— Response variable nameTbl
Response variable name, specified as the name of a variable in
Tbl
.
You must specify ResponseVarName
as a character vector or string scalar.
For example, if the response variable Y
is
stored as Tbl.Y
, then specify it as
'Y'
. Otherwise, the software
treats all columns of Tbl
, including
Y
, as predictors when training
the model.
The response variable must be a categorical, character, or string array; a logical or numeric
vector; or a cell array of character vectors. If
Y
is a character array, then each
element of the response variable must correspond to one row of
the array.
A good practice is to specify the order of the classes by using the
ClassNames
name-value
argument.
Data Types: char
| string
formula
— Explanatory model of response variable and subset of predictor variablesExplanatory model of the response variable and a subset of the predictor variables,
specified as a character vector or string scalar in the form
'Y~x1+x2+x3'
. In this form, Y
represents the
response variable, and x1
, x2
, and
x3
represent the predictor variables.
To specify a subset of variables in Tbl
as predictors for
training the model, use a formula. If you specify a formula, then the software does not
use any variables in Tbl
that do not appear in
formula
.
The variable names in the formula must be both variable names in Tbl
(Tbl.Properties.VariableNames
) and valid MATLAB® identifiers. You can verify the variable names in Tbl
by
using the isvarname
function. If the variable names
are not valid, then you can convert them by using the matlab.lang.makeValidName
function.
Data Types: char
| string
Y
— Class labelsClass labels, specified as a numeric vector, categorical vector, logical vector, character
array, string array, or cell array of character vectors. Each row of
Y
represents the classification of the corresponding row of
X
.
When fitting the tree, fitctree
considers NaN
,
''
(empty character vector), ""
(empty
string), <missing>
, and <undefined>
values in Y
to be missing values. fitctree
does not use observations with missing values for Y
in the
fit.
For numeric Y
, consider fitting a regression
tree using fitrtree
instead.
Data Types: single
| double
| categorical
| logical
| char
| string
| cell
X
— Predictor dataPredictor data, specified as a numeric matrix. Each row of X
corresponds
to one observation, and each column corresponds to one predictor variable.
fitctree
considers NaN
values
in X
as missing values. fitctree
does
not use observations with all missing values for X
in
the fit. fitctree
uses observations with some
missing values for X
to find splits on variables
for which these observations have valid values.
Data Types: single
| double
Specify optional
comma-separated pairs of Name,Value
arguments. Name
is
the argument name and Value
is the corresponding value.
Name
must appear inside quotes. You can specify several name and value
pair arguments in any order as
Name1,Value1,...,NameN,ValueN
.
'CrossVal','on','MinLeafSize',40
specifies a
cross-validated classification tree with a minimum of 40 observations per
leaf.Note
You cannot use any cross-validation name-value pair argument along with the
'OptimizeHyperparameters'
name-value pair argument. You can modify
the cross-validation for 'OptimizeHyperparameters'
only by using the
'HyperparameterOptimizationOptions'
name-value pair
argument.
'AlgorithmForCategorical'
— Algorithm for best categorical predictor split'Exact'
| 'PullLeft'
| 'PCA'
| 'OVAbyClass'
Algorithm to find the best split on a categorical predictor with
C categories for data and K ≥
3 classes, specified as the comma-separated pair consisting of
'AlgorithmForCategorical'
and one of the
following values.
Value | Description |
---|---|
'Exact' | Consider all 2C–1 – 1 combinations. |
'PullLeft' | Start with all C categories on the right branch. Consider moving each category to the left branch as it achieves the minimum impurity for the K classes among the remaining categories. From this sequence, choose the split that has the lowest impurity. |
'PCA' | Compute a score for each category using the inner product between the first principal component of a weighted covariance matrix (of the centered class probability matrix) and the vector of class probabilities for that category. Sort the scores in ascending order, and consider all C – 1 splits. |
'OVAbyClass' | Start with all C categories on the right branch. For each class, order the categories based on their probability for that class. For the first class, consider moving each category to the left branch in order, recording the impurity criterion at each move. Repeat for the remaining classes. From this sequence, choose the split that has the minimum impurity. |
fitctree
automatically
selects the optimal subset of algorithms for each split using the known
number of classes and levels of a categorical predictor. For
K = 2 classes, fitctree
always performs
the exact search. To specify a particular algorithm, use the
'AlgorithmForCategorical'
name-value pair
argument.
For more details, see Splitting Categorical Predictors in Classification Trees.
Example: 'AlgorithmForCategorical','PCA'
'CategoricalPredictors'
— Categorical predictors list'all'
Categorical predictors list, specified as one of the values in this table.
Value | Description |
---|---|
Vector of positive integers |
Each entry in the vector is an index value corresponding to the column of the predictor data that contains a categorical variable. The index values are between 1 and If |
Logical vector |
A |
Character matrix | Each row of the matrix is the name of a predictor variable. The names must match the entries in PredictorNames . Pad the names with extra blanks so each row of the character matrix has the same length. |
String array or cell array of character vectors | Each element in the array is the name of a predictor variable. The names must match the entries in PredictorNames . |
'all' | All predictors are categorical. |
By default, if the predictor data is in a table
(Tbl
), fitctree
assumes that a variable is
categorical if it is a logical vector, unordered categorical vector, character array, string
array, or cell array of character vectors. If the predictor data is a matrix
(X
), fitctree
assumes that all predictors are
continuous. To identify any other predictors as categorical predictors, specify them by using
the 'CategoricalPredictors'
name-value argument.
Example: 'CategoricalPredictors','all'
Data Types: single
| double
| logical
| char
| string
| cell
'ClassNames'
— Names of classes to use for trainingNames of classes to use for training, specified as a categorical, character, or string array;
a logical or numeric vector; or a cell array of character vectors.
ClassNames
must have the same data type as the response variable
in Tbl
or Y
.
If ClassNames
is a character array, then each element must correspond to
one row of the array.
Use ClassNames
to:
Specify the order of the classes during training.
Specify the order of any input or output argument dimension that
corresponds to the class order. For example, use
ClassNames
to specify the order of the dimensions of
Cost
or the column order of classification scores
returned by predict
.
Select a subset of classes for training. For example, suppose that the set
of all distinct class names in Y
is
{'a','b','c'}
. To train the model using observations
from classes 'a'
and 'c'
only, specify
'ClassNames',{'a','c'}
.
The default value for ClassNames
is the set of all distinct class names in
the response variable in Tbl
or Y
.
Example: 'ClassNames',{'b','g'}
Data Types: categorical
| char
| string
| logical
| single
| double
| cell
'Cost'
— Cost of misclassificationCost of misclassification of a point, specified as the comma-separated
pair consisting of 'Cost'
and one of the
following:
Square matrix, where Cost(i,j)
is the cost
of classifying a point into class j
if its
true class is i
(i.e., the rows correspond to
the true class and the columns correspond to the predicted
class). To specify the class order for the corresponding rows
and columns of Cost
, also specify the
ClassNames
name-value pair
argument.
Structure S
having two fields:
S.ClassNames
containing the group names
as a variable of the same data type as Y
,
and S.ClassificationCosts
containing the cost
matrix.
The default is Cost(i,j)=1
if
i~=j
, and Cost(i,j)=0
if
i=j
.
Data Types: single
| double
| struct
'MaxDepth'
— Maximum tree depthMaximum tree depth, specified as the comma-separated pair consisting
of 'MaxDepth'
and a positive integer. Specify a value
for this argument to return a tree that has fewer levels and requires
fewer passes through the tall array to compute. Generally, the algorithm
of fitctree
takes one pass through the data and an
additional pass for each tree level. The function does not set a maximum
tree depth, by default.
Note
This option applies only when you use
fitctree
on tall arrays. See Tall Arrays for more information.
'MaxNumCategories'
— Maximum category levels10
(default) | nonnegative scalar valueMaximum category levels, specified as the comma-separated pair
consisting of 'MaxNumCategories'
and a nonnegative
scalar value. fitctree
splits a
categorical predictor using the exact search algorithm if the predictor
has at most MaxNumCategories
levels in the split
node. Otherwise, fitctree
finds the best
categorical split using one of the inexact algorithms.
Passing a small value can lead to loss of accuracy and passing a large value can increase computation time and memory overload.
Example: 'MaxNumCategories',8
'MaxNumSplits'
— Maximal number of decision splitssize(X,1) - 1
(default) | positive integerMaximal number of decision splits (or branch nodes), specified as the
comma-separated pair consisting of 'MaxNumSplits'
and
a positive integer. fitctree
splits
MaxNumSplits
or fewer branch nodes. For more
details on splitting behavior, see Algorithms.
Example: 'MaxNumSplits',5
Data Types: single
| double
'MergeLeaves'
— Leaf merge flag'on'
(default) | 'off'
Leaf merge flag, specified as the comma-separated pair consisting of
'MergeLeaves'
and 'on'
or
'off'
.
If MergeLeaves
is 'on'
, then
fitctree
:
Merges leaves that originate from the same parent node, and that yields a sum of risk values greater or equal to the risk associated with the parent node
Estimates the optimal sequence of pruned subtrees, but does not prune the classification tree
Otherwise, fitctree
does not merge
leaves.
Example: 'MergeLeaves','off'
'MinLeafSize'
— Minimum number of leaf node observations1
(default) | positive integer valueMinimum number of leaf node observations, specified as the
comma-separated pair consisting of 'MinLeafSize'
and
a positive integer value. Each leaf has at least
MinLeafSize
observations per tree leaf. If you
supply both MinParentSize
and
MinLeafSize
, fitctree
uses
the setting that gives larger leaves: MinParentSize =
max(MinParentSize,2*MinLeafSize)
.
Example: 'MinLeafSize',3
Data Types: single
| double
'MinParentSize'
— Minimum number of branch node observations10
(default) | positive integer valueMinimum number of branch node observations, specified as the
comma-separated pair consisting of 'MinParentSize'
and a positive integer value. Each branch node in the tree has at least
MinParentSize
observations. If you supply both
MinParentSize
and MinLeafSize
,
fitctree
uses the setting
that gives larger leaves: MinParentSize =
max(MinParentSize,2*MinLeafSize)
.
Example: 'MinParentSize',8
Data Types: single
| double
'NumBins'
— Number of bins for numeric predictors[]
(empty) (default) | positive integer scalarNumber of bins for numeric predictors, specified as the comma-separated pair
consisting of 'NumBins'
and a positive integer scalar.
If the 'NumBins'
value is empty (default), then
fitctree
does not bin any predictors.
If you specify the 'NumBins'
value as a positive integer scalar
(numBins
), then fitctree
bins every
numeric predictor into at most numBins
equiprobable bins, and
then grows trees on the bin indices instead of the original data.
The number of bins can be less than numBins
if a
predictor has fewer than numBins
unique
values.
fitctree
does not bin categorical
predictors.
When you use a large training data set, this binning option speeds up training but might cause
a potential decrease in accuracy. You can try 'NumBins',50
first, and
then change the value depending on the accuracy and training speed.
A trained model stores the bin edges in the BinEdges
property.
Example: 'NumBins',50
Data Types: single
| double
'NumVariablesToSample'
— Number of predictors to select at random for each split'all'
(default) | positive integer valueNumber of predictors to select at random for each split, specified as the comma-separated pair consisting of 'NumVariablesToSample'
and a positive integer value. Alternatively, you can specify 'all'
to use all available predictors.
If the training data includes many predictors and you want to analyze predictor
importance, then specify 'NumVariablesToSample'
as
'all'
. Otherwise, the software might not select some predictors,
underestimating their importance.
To reproduce the random selections, you must set the seed of the random number generator by using rng
and specify 'Reproducible',true
.
Example: 'NumVariablesToSample',3
Data Types: char
| string
| single
| double
'PredictorNames'
— Predictor variable namesPredictor variable names, specified as a string array of unique names or cell array of unique
character vectors. The functionality of PredictorNames
depends on the
way you supply the training data.
If you supply X
and Y
, then you
can use PredictorNames
to assign names to the predictor
variables in X
.
The order of the names in PredictorNames
must correspond to the column order of X
.
That is, PredictorNames{1}
is the name of
X(:,1)
,
PredictorNames{2}
is the name of
X(:,2)
, and so on. Also,
size(X,2)
and
numel(PredictorNames)
must be
equal.
By default, PredictorNames
is
{'x1','x2',...}
.
If you supply Tbl
, then you can use
PredictorNames
to choose which predictor variables to
use in training. That is, fitctree
uses only the
predictor variables in PredictorNames
and the response
variable during training.
PredictorNames
must be a subset of
Tbl.Properties.VariableNames
and cannot
include the name of the response variable.
By default, PredictorNames
contains the
names of all predictor variables.
A good practice is to specify the predictors for training
using either 'PredictorNames'
or
formula
, but not both.
Example: 'PredictorNames',{'SepalLength','SepalWidth','PetalLength','PetalWidth'}
Data Types: string
| cell
'PredictorSelection'
— Algorithm used to select the best split predictor'allsplits'
(default) | 'curvature'
| 'interaction-curvature'
Algorithm used to select the best split predictor at each node,
specified as the comma-separated pair consisting of
'PredictorSelection'
and a value in this
table.
Value | Description |
---|---|
'allsplits' | Standard CART — Selects the split predictor that maximizes the split-criterion gain over all possible splits of all predictors [1]. |
'curvature' | Curvature test — Selects the split predictor that minimizes the p-value of chi-square tests of independence between each predictor and the response [4]. Training speed is similar to standard CART. |
'interaction-curvature' | Interaction test — Chooses the split predictor that minimizes the p-value of chi-square tests of independence between each predictor and the response, and that minimizes the p-value of a chi-square test of independence between each pair of predictors and response [3]. Training speed can be slower than standard CART. |
For 'curvature'
and
'interaction-curvature'
, if all tests yield
p-values greater than 0.05, then
fitctree
stops splitting nodes.
Tip
Standard CART tends to select split predictors containing many distinct values, e.g., continuous variables, over those containing few distinct values, e.g., categorical variables [4]. Consider specifying the curvature or interaction test if any of the following are true:
If there are predictors that have relatively fewer distinct values than other predictors, for example, if the predictor data set is heterogeneous.
If an analysis of predictor importance is
your goal. For more on predictor importance
estimation, see predictorImportance
and Introduction to Feature Selection.
Trees grown using standard CART are not sensitive to predictor variable interactions. Also, such trees are less likely to identify important variables in the presence of many irrelevant predictors than the application of the interaction test. Therefore, to account for predictor interactions and identify importance variables in the presence of many irrelevant variables, specify the interaction test [3].
Prediction speed is unaffected by the value of
'PredictorSelection'
.
For details on how fitctree
selects split
predictors, see Node Splitting Rules and Choose Split Predictor Selection Technique.
Example: 'PredictorSelection','curvature'
'Prior'
— Prior probabilities'empirical'
(default) | 'uniform'
| vector of scalar values | structurePrior probabilities for each class, specified as one of the following:
Character vector or string scalar.
Vector (one scalar value for each class). To specify the class order for the
corresponding elements of 'Prior'
, set the
'ClassNames'
name-value argument.
Structure S
with two fields.
S.ClassNames
contains the class names as a variable
of the same type as the response variable in Y
or
Tbl
.
S.ClassProbs
contains a vector of corresponding
probabilities.
fitctree
normalizes the weights in each class
('Weights'
) to add up to the value of the prior probability of
the respective class.
Example: 'Prior','uniform'
Data Types: char
| string
| single
| double
| struct
'Prune'
— Flag to estimate optimal sequence of pruned subtrees'on'
(default) | 'off'
Flag to estimate the optimal sequence of pruned subtrees, specified as
the comma-separated pair consisting of 'Prune'
and
'on'
or 'off'
.
If Prune
is 'on'
, then
fitctree
grows the classification tree
without pruning it, but estimates the optimal sequence of pruned
subtrees. Otherwise, fitctree
grows the
classification tree without estimating the optimal sequence of pruned
subtrees.
To prune a trained ClassificationTree
model,
pass it to prune
.
Example: 'Prune','off'
'PruneCriterion'
— Pruning criterion'error'
(default) | 'impurity'
Pruning criterion, specified as the comma-separated pair consisting of
'PruneCriterion'
and 'error'
or 'impurity'
.
If you specify 'impurity'
, then
fitctree
uses the impurity measure specified
by the 'SplitCriterion'
name-value pair
argument.
For details, see Impurity and Node Error.
Example: 'PruneCriterion','impurity'
'Reproducible'
— Flag to enforce reproducibilityfalse
(logical 0
) (default) | true
(logical 1
)Flag to enforce reproducibility over repeated runs of training a model, specified as the
comma-separated pair consisting of 'Reproducible'
and either
false
or true
.
If 'NumVariablesToSample'
is not 'all'
, then the
software selects predictors at random for each split. To reproduce the random
selections, you must specify 'Reproducible',true
and set the seed of
the random number generator by using rng
. Note that setting 'Reproducible'
to
true
can slow down training.
Example: 'Reproducible',true
Data Types: logical
'ResponseName'
— Response variable name'Y'
(default) | character vector | string scalarResponse variable name, specified as the comma-separated pair
consisting of 'ResponseName'
and a character vector
or string scalar representing the name of the response variable.
This name-value pair is not valid when using the
ResponseVarName
or formula
input arguments.
Example: 'ResponseName','IrisType'
Data Types: char
| string
'ScoreTransform'
— Score transformation'none'
(default) | 'doublelogit'
| 'invlogit'
| 'ismax'
| 'logit'
| function handle | ...Score transformation, specified as a character vector, string scalar, or function handle.
This table summarizes the available character vectors and string scalars.
Value | Description |
---|---|
'doublelogit' | 1/(1 + e–2x) |
'invlogit' | log(x / (1 – x)) |
'ismax' | Sets the score for the class with the largest score to 1, and sets the scores for all other classes to 0 |
'logit' | 1/(1 + e–x) |
'none' or 'identity' | x (no transformation) |
'sign' | –1 for x < 0 0 for x = 0 1 for x > 0 |
'symmetric' | 2x – 1 |
'symmetricismax' | Sets the score for the class with the largest score to 1, and sets the scores for all other classes to –1 |
'symmetriclogit' | 2/(1 + e–x) – 1 |
For a MATLAB function or a function you define, use its function handle for the score transform. The function handle must accept a matrix (the original scores) and return a matrix of the same size (the transformed scores).
Example: 'ScoreTransform','logit'
Data Types: char
| string
| function_handle
'SplitCriterion'
— Split criterion'gdi'
(default) | 'twoing'
| 'deviance'
Split criterion, specified as the comma-separated pair consisting of
'SplitCriterion'
and 'gdi'
(Gini's diversity index), 'twoing'
for the twoing
rule, or 'deviance'
for maximum deviance reduction
(also known as cross entropy).
For details, see Impurity and Node Error.
Example: 'SplitCriterion','deviance'
'Surrogate'
— Surrogate decision splits flag'off'
(default) | 'on'
| 'all'
| positive integer valueSurrogate decision
splits flag, specified as the comma-separated pair consisting
of 'Surrogate'
and 'on'
,
'off'
, 'all'
, or a positive
integer value.
When set to 'on'
, fitctree
finds
at most 10 surrogate splits at each branch node.
When set to 'all'
, fitctree
finds
all surrogate splits at each branch node. The
'all'
setting can use considerable
time and memory.
When set to a positive integer value, fitctree
finds
at most the specified number of surrogate splits at each
branch node.
Use surrogate splits to improve the accuracy of predictions for data with missing values. The setting also lets you compute measures of predictive association between predictors. For more details, see Node Splitting Rules.
Example: 'Surrogate','on'
Data Types: single
| double
| char
| string
'Weights'
— Observation weightsones(size(X,1),1)
(default) | vector of scalar values | name of variable in Tbl
Observation weights, specified as the comma-separated pair consisting of 'Weights'
and a vector of scalar values or the name of a variable in Tbl
. The software weights the observations in each row of X
or Tbl
with the corresponding value in Weights
. The size of Weights
must equal the number of rows in X
or Tbl
.
If you specify the input data as a table Tbl
, then
Weights
can be the name of a variable in Tbl
that contains a numeric vector. In this case, you must specify
Weights
as a character vector or string scalar. For example, if
the weights vector W
is stored as Tbl.W
, then
specify it as 'W'
. Otherwise, the software treats all columns of
Tbl
, including W
, as predictors when training
the model.
fitctree
normalizes the weights in each class to add up to the value
of the prior probability of the respective class.
Data Types: single
| double
| char
| string
'CrossVal'
— Flag to grow cross-validated decision tree'off'
(default) | 'on'
Flag to grow a cross-validated decision tree, specified as the
comma-separated pair consisting of 'CrossVal'
and
'on'
or 'off'
.
If 'on'
, fitctree
grows a
cross-validated decision tree with 10 folds. You can override this
cross-validation setting using one of the 'KFold'
,
'Holdout'
, 'Leaveout'
, or
'CVPartition'
name-value pair arguments. You can
only use one of these four arguments at a time when creating a
cross-validated tree.
Alternatively, cross-validate tree
later using
the crossval
method.
Example: 'CrossVal','on'
'CVPartition'
— Partition for cross-validated treecvpartition
objectPartition to use in a cross-validated tree, specified as the
comma-separated pair consisting of 'CVPartition'
and
an object created using cvpartition
.
If you use 'CVPartition'
, you cannot use any of the
'KFold'
, 'Holdout'
, or
'Leaveout'
name-value pair arguments.
'Holdout'
— Fraction of data for holdout validation0
(default) | scalar value in the range [0,1]
Fraction of data used for holdout validation, specified as the
comma-separated pair consisting of 'Holdout'
and a
scalar value in the range [0,1]
. Holdout validation
tests the specified fraction of the data, and uses the rest of the data
for training.
If you use 'Holdout'
, you cannot use any of the
'CVPartition'
, 'KFold'
, or
'Leaveout'
name-value pair arguments.
Example: 'Holdout',0.1
Data Types: single
| double
'KFold'
— Number of folds10
(default) | positive integer value greater than 1Number of folds to use in a cross-validated classifier, specified
as the comma-separated pair consisting of 'KFold'
and
a positive integer value greater than 1. If you specify, e.g., 'KFold',k
,
then the software:
Randomly partitions the data into k sets
For each set, reserves the set as validation data, and trains the model using the other k – 1 sets
Stores the k
compact, trained
models in the cells of a k
-by-1 cell vector
in the Trained
property of the cross-validated
model.
To create a cross-validated model, you can use one of these
four options only: CVPartition
, Holdout
, KFold
,
or Leaveout
.
Example: 'KFold',8
Data Types: single
| double
'Leaveout'
— Leave-one-out cross-validation flag'off'
(default) | 'on'
Leave-one-out cross-validation flag, specified as the comma-separated
pair consisting of 'Leaveout'
and
'on'
or 'off'
. Specify
'on'
to use leave-one-out
cross-validation.
If you use 'Leaveout'
, you cannot use any of the
'CVPartition'
, 'Holdout'
, or
'KFold'
name-value pair arguments.
Example: 'Leaveout','on'
'OptimizeHyperparameters'
— Parameters to optimize'none'
(default) | 'auto'
| 'all'
| string array or cell array of eligible parameter names | vector of optimizableVariable
objectsParameters to optimize, specified as the comma-separated pair
consisting of 'OptimizeHyperparameters'
and one of
the following:
'none'
— Do not optimize.
'auto'
— Use
{'MinLeafSize'}
'all'
— Optimize all eligible
parameters.
String array or cell array of eligible parameter names
Vector of optimizableVariable
objects,
typically the output of hyperparameters
The optimization attempts to minimize the cross-validation loss
(error) for fitctree
by varying the parameters. For
information about cross-validation loss (albeit in a different context),
see Classification Loss. To control the
cross-validation type and other aspects of the optimization, use the
HyperparameterOptimizationOptions
name-value
pair.
Note
'OptimizeHyperparameters'
values override any values you set using
other name-value pair arguments. For example, setting
'OptimizeHyperparameters'
to 'auto'
causes the
'auto'
values to apply.
The eligible parameters for fitctree
are:
MaxNumSplits
—
fitctree
searches among integers, by
default log-scaled in the range
[1,max(2,NumObservations-1)]
.
MinLeafSize
—
fitctree
searches among integers, by
default log-scaled in the range
[1,max(2,floor(NumObservations/2))]
.
SplitCriterion
— For two classes,
fitctree
searches among
'gdi'
and 'deviance'
.
For three or more classes, fitctree
also
searches among 'twoing'
.
NumVariablesToSample
—
fitctree
does not optimize over this
hyperparameter. If you pass
NumVariablesToSample
as a parameter name,
fitctree
simply uses the full number of
predictors. However, fitcensemble
does
optimize over this hyperparameter.
Set nondefault parameters by passing a vector of
optimizableVariable
objects that have nondefault
values. For example,
load fisheriris params = hyperparameters('fitctree',meas,species); params(1).Range = [1,30];
Pass params
as the value of
OptimizeHyperparameters
.
By default, iterative display appears at the command line, and
plots appear according to the number of hyperparameters in the optimization. For the
optimization and plots, the objective function is log(1 + cross-validation loss) for regression and the misclassification rate for classification. To control
the iterative display, set the Verbose
field of the
'HyperparameterOptimizationOptions'
name-value pair argument. To
control the plots, set the ShowPlots
field of the
'HyperparameterOptimizationOptions'
name-value pair argument.
For an example, see Optimize Classification Tree.
Example: 'auto'
'HyperparameterOptimizationOptions'
— Options for optimizationOptions for optimization, specified as the comma-separated pair consisting of
'HyperparameterOptimizationOptions'
and a structure. This
argument modifies the effect of the OptimizeHyperparameters
name-value pair argument. All fields in the structure are optional.
Field Name | Values | Default |
---|---|---|
Optimizer |
| 'bayesopt' |
AcquisitionFunctionName |
Acquisition functions whose names include
| 'expected-improvement-per-second-plus' |
MaxObjectiveEvaluations | Maximum number of objective function evaluations. | 30 for 'bayesopt' or 'randomsearch' , and the entire grid for 'gridsearch' |
MaxTime | Time limit, specified as a positive real. The time limit is in seconds, as measured by | Inf |
NumGridDivisions | For 'gridsearch' , the number of values in each dimension. The value can be
a vector of positive integers giving the number of
values for each dimension, or a scalar that
applies to all dimensions. This field is ignored
for categorical variables. | 10 |
ShowPlots | Logical value indicating whether to show plots. If true , this field plots
the best objective function value against the
iteration number. If there are one or two
optimization parameters, and if
Optimizer is
'bayesopt' , then
ShowPlots also plots a model of
the objective function against the
parameters. | true |
SaveIntermediateResults | Logical value indicating whether to save results when Optimizer is
'bayesopt' . If
true , this field overwrites a
workspace variable named
'BayesoptResults' at each
iteration. The variable is a BayesianOptimization object. | false |
Verbose | Display to the command line.
For details, see the
| 1 |
UseParallel | Logical value indicating whether to run Bayesian optimization in parallel, which requires Parallel Computing Toolbox™. Due to the nonreproducibility of parallel timing, parallel Bayesian optimization does not necessarily yield reproducible results. For details, see Parallel Bayesian Optimization. | false |
Repartition | Logical value indicating whether to repartition the cross-validation at every iteration. If
| false |
Use no more than one of the following three field names. | ||
CVPartition | A cvpartition object, as created by cvpartition . | 'Kfold',5 if you do not specify any cross-validation
field |
Holdout | A scalar in the range (0,1) representing the holdout fraction. | |
Kfold | An integer greater than 1. |
Example: 'HyperparameterOptimizationOptions',struct('MaxObjectiveEvaluations',60)
Data Types: struct
tree
— Classification treeClassification tree, returned as a classification tree object.
Using the 'CrossVal'
, 'KFold'
,
'Holdout'
, 'Leaveout'
, or
'CVPartition'
options results in a tree of class
ClassificationPartitionedModel
.
You cannot use a partitioned tree for prediction, so this kind of tree does
not have a predict
method. Instead, use
kfoldPredict
to predict
responses for observations not used for training.
Otherwise, tree
is of class ClassificationTree
, and you can
use the predict
method to make
predictions.
The curvature test is a statistical test assessing the null hypothesis that two variables are unassociated.
The curvature test between predictor variable x and y is conducted using this process.
If x is continuous, then partition it into its quartiles. Create a nominal variable that bins observations according to which section of the partition they occupy. If there are missing values, then create an extra bin for them.
For each level in the partitioned predictor j = 1...J and class in the response k = 1,...,K, compute the weighted proportion of observations in class k
wi is the weight of observation i, , I is the indicator function, and n is the sample size. If all observations have the same weight, then , where njk is the number of observations in level j of the predictor that are in class k.
Compute the test statistic
, that is, the marginal probability of observing the predictor at level j. , that is the marginal probability of observing class k. If n is large enough, then t is distributed as a χ2 with (K – 1)(J – 1) degrees of freedom.
If the p-value for the test is less than 0.05, then reject the null hypothesis that there is no association between x and y.
When determining the best split predictor at each node, the standard CART algorithm prefers to select continuous predictors that have many levels. Sometimes, such a selection can be spurious and can also mask more important predictors that have fewer levels, such as categorical predictors.
The curvature test can be applied instead of standard CART to determine the best split predictor at each node. In that case, the best split predictor variable is the one that minimizes the significant p-values (those less than 0.05) of curvature tests between each predictor and the response variable. Such a selection is robust to the number of levels in individual predictors.
Note
If levels of a predictor are pure for a particular class, then
fitctree
merges those levels. Therefore, in step 3
of the algorithm, J can be less than the actual number of
levels in the predictor. For example, if x has 4 levels, and
all observations in bins 1 and 2 belong to class 1, then those levels are pure
for class 1. Consequently, fitctree
merges the
observations in bins 1 and 2, and J reduces to 3.
For more details on how the curvature test applies to growing classification trees, see Node Splitting Rules and [4].
A decision tree splits nodes based on either impurity or node error.
Impurity means one of several things, depending on your choice
of the SplitCriterion
name-value pair argument:
Gini's Diversity Index (gdi
) —
The Gini index of a node is
where the sum is over the classes i at the
node, and p(i) is the observed
fraction of classes with class i that reach the
node. A node with just one class (a pure node)
has Gini index 0
; otherwise the Gini index is positive.
So the Gini index is a measure of node impurity.
Deviance ('deviance'
) —
With p(i) defined the same as
for the Gini index, the deviance of a node is
A pure node has deviance 0
; otherwise, the
deviance is positive.
Twoing rule ('twoing'
) —
Twoing is not a purity measure of a node, but is a different measure
for deciding how to split a node. Let L(i)
denote the fraction of members of class i in the
left child node after a split, and R(i)
denote the fraction of members of class i in the
right child node after a split. Choose the split criterion to maximize
where P(L) and P(R) are the fractions of observations that split to the left and right respectively. If the expression is large, the split made each child node purer. Similarly, if the expression is small, the split made each child node similar to each other, and therefore similar to the parent node. The split did not increase node purity.
Node error — The node error is the fraction of misclassified classes at a node. If j is the class with the largest number of training samples at a node, the node error is
1 – p(j).
The interaction test is a statistical test that assesses the null hypothesis that there is no interaction between a pair of predictor variables and the response variable.
The interaction test assessing the association between predictor variables x1 and x2 with respect to y is conducted using this process.
If x1 or x2 is continuous, then partition that variable into its quartiles. Create a nominal variable that bins observations according to which section of the partition they occupy. If there are missing values, then create an extra bin for them.
Create the nominal variable z with J = J1J2 levels that assigns an index to observation i according to which levels of x1 and x2 it belongs. Remove any levels of z that do not correspond to any observations.
Conduct a curvature test between z and y.
When growing decision trees, if there are important interactions between pairs of predictors, but there are also many other less important predictors in the data, then standard CART tends to miss the important interactions. However, conducting curvature and interaction tests for predictor selection instead can improve detection of important interactions, which can yield more accurate decision trees.
For more details on how the interaction test applies to growing decision trees, see Curvature Test, Node Splitting Rules and [3].
The predictive measure of association is a value that indicates the similarity between decision rules that split observations. Among all possible decision splits that are compared to the optimal split (found by growing the tree), the best surrogate decision split yields the maximum predictive measure of association. The second-best surrogate split has the second-largest predictive measure of association.
Suppose xj and xk are predictor variables j and k, respectively, and j ≠ k. At node t, the predictive measure of association between the optimal split xj < u and a surrogate split xk < v is
PL is the proportion of observations in node t, such that xj < u. The subscript L stands for the left child of node t.
PR is the proportion of observations in node t, such that xj ≥ u. The subscript R stands for the right child of node t.
is the proportion of observations at node t, such that xj < u and xk < v.
is the proportion of observations at node t, such that xj ≥ u and xk ≥ v.
Observations with missing values for xj or xk do not contribute to the proportion calculations.
λjk is a value in (–∞,1]. If λjk > 0, then xk < v is a worthwhile surrogate split for xj < u.
A surrogate decision split is an alternative to the optimal decision split at a given node in a decision tree. The optimal split is found by growing the tree; the surrogate split uses a similar or correlated predictor variable and split criterion.
When the value of the optimal split predictor for an observation is missing, the observation is sent to the left or right child node using the best surrogate predictor. When the value of the best surrogate split predictor for the observation is also missing, the observation is sent to the left or right child node using the second-best surrogate predictor, and so on. Candidate splits are sorted in descending order by their predictive measure of association.
By default, Prune
is 'on'
. However,
this specification does not prune the classification tree. To prune a trained
classification tree, pass the classification tree to prune
.
After training a model, you can generate C/C++ code that predicts labels for new data. Generating C/C++ code requires MATLAB Coder™. For details, see Introduction to Code Generation.
fitctree
uses these processes to determine how to split
node t.
For standard CART (that is, if PredictorSelection
is
'allpairs'
) and for all predictors
xi, i =
1,...,p:
fitctree
computes the weighted
impurity of node t,
it. For
supported impurity measures, see SplitCriterion
.
fitctree
estimates the probability
that an observation is in node t using
wj
is the weight of observation j, and
T is the set of all observation indices
in node t. If you do not specify
Prior
or Weights
,
then wj =
1/n, where n is the
sample size.
fitctree
sorts
xi in
ascending order. Each element of the sorted predictor is a
splitting candidate or cut point.
fitctree
stores any indices
corresponding to missing values in the set
TU, which is
the unsplit set.
fitctree
determines the best way to
split node t using
xi by
maximizing the impurity gain (ΔI) over all
splitting candidates. That is, for all splitting candidates in
xi:
fitctree
splits the
observations in node t into left
and right child nodes
(tL and
tR,
respectively).
fitctree
computes
ΔI. Suppose that for a
particular splitting candidate,
tL and
tR
contain observation indices in the sets
TL and
TR, respectively.
If xi does not contain any missing values, then the impurity gain for the current splitting candidate is
If xi contains missing values then, assuming that the observations are missing at random, the impurity gain is
T – TU is the set of all observation indices in node t that are not missing.
If you use surrogate decision splits, then:
fitctree
computes
the predictive measures of association between
the decision split
xj <
u and all possible decision
splits
xk <
v, j ≠
k.
fitctree
sorts the
possible alternative decision splits in descending
order by their predictive measure of association
with the optimal split. The surrogate split is the
decision split yielding the largest
measure.
fitctree
decides the
child node assignments for observations with a
missing value for
xi using
the surrogate split. If the surrogate predictor
also contains a missing value, then
fitctree
uses the decision
split with the second largest measure, and so on,
until there are no other surrogates. It is
possible for fitctree
to
split two different observations at node
t using two different surrogate
splits. For example, suppose the predictors
x1 and
x2 are
the best and second best surrogates, respectively,
for the predictor
xi,
i ∉ {1,2}, at node
t. If observation
m of predictor
xi is
missing (i.e.,
xmi is
missing), but
xm1
is not missing, then
x1 is
the surrogate predictor for observation
xmi. If
observations
x(m
+ 1),i and
x(m +
1),1 are missing, but
x(m
+ 1),2 is not missing, then
x2 is
the surrogate predictor for observation
m + 1.
fitctree
uses the
appropriate impurity gain formula. That is, if
fitctree
fails to assign
all missing observations in node
t to children nodes using
surrogate splits, then the impurity gain is
ΔIU.
Otherwise, fitctree
uses
ΔI for the impurity
gain.
fitctree
chooses the
candidate that yields the largest impurity
gain.
fitctree
splits the predictor variable
at the cut point that maximizes the impurity gain.
For the curvature test (that is, if
PredictorSelection
is
'curvature'
):
fitctree
conducts curvature
tests between each predictor and the response for
observations in node t.
If all p-values are at least
0.05, then fitctree
does not
split node t.
If there is a minimal p-value,
then fitctree
chooses the
corresponding predictor to split node
t.
If more than one p-value is
zero due to underflow, then
fitctree
applies standard
CART to the corresponding predictors to choose the
split predictor.
If fitctree
chooses a split predictor,
then it uses standard CART to choose the cut point (see step 4
in the standard CART process).
For the interaction test (that is, if
PredictorSelection
is
'interaction-curvature'
):
For observations in node t,
fitctree
conducts curvature
tests between each predictor and the response and
interaction
tests between each pair of predictors and the response.
If all p-values are at least
0.05, then fitctree
does not
split node t.
If there is a minimal p-value
and it is the result of a curvature test, then
fitctree
chooses the
corresponding predictor to split node
t.
If there is a minimal p-value
and it is the result of an interaction test, then
fitctree
chooses the split
predictor using standard CART on the corresponding
pair of predictors.
If more than one p-value is
zero due to underflow, then
fitctree
applies standard
CART to the corresponding predictors to choose the
split predictor.
If fitctree
chooses a split predictor,
then it uses standard CART to choose the cut point (see step 4
in the standard CART process).
If MergeLeaves
is 'on'
and
PruneCriterion
is 'error'
(which
are the default values for these name-value pair arguments), then the
software applies pruning only to the leaves and by using classification
error. This specification amounts to merging leaves that share the most
popular class per leaf.
To accommodate MaxNumSplits
,
fitctree
splits all nodes in the current
layer, and then counts the number of branch
nodes. A layer is the set of nodes that are equidistant from the root node.
If the number of branch nodes exceeds MaxNumSplits
,
fitctree
follows this procedure:
Determine how many branch nodes in the current layer must be
unsplit so that there are at most
MaxNumSplits
branch nodes.
Sort the branch nodes by their impurity gains.
Unsplit the number of least successful branches.
Return the decision tree grown so far.
This procedure produces maximally balanced trees.
The software splits branch nodes layer by layer until at least one of these events occurs:
There are MaxNumSplits
branch nodes.
A proposed split causes the number of observations in at least one
branch node to be fewer than
MinParentSize
.
A proposed split causes the number of observations in at least one
leaf node to be fewer than MinLeafSize
.
The algorithm cannot find a good split within a layer (i.e., the
pruning criterion (see PruneCriterion
), does
not improve for all proposed splits in a layer). A special case is
when all nodes are pure (i.e., all observations in the node have the
same class).
For values 'curvature'
or
'interaction-curvature'
of
PredictorSelection
, all tests yield
p-values greater than 0.05.
MaxNumSplits
and MinLeafSize
do not
affect splitting at their default values. Therefore, if you set
'MaxNumSplits'
, splitting might stop due to the value
of MinParentSize
, before MaxNumSplits
splits occur.
For dual-core systems and above, fitctree
parallelizes
training decision trees using Intel® Threading Building Blocks (TBB). For details on Intel TBB, see https://software.intel.com/en-us/intel-tbb.
[1] Breiman, L., J. Friedman, R. Olshen, and C. Stone. Classification and Regression Trees. Boca Raton, FL: CRC Press, 1984.
[2] Coppersmith, D., S. J. Hong, and J. R. M. Hosking. “Partitioning Nominal Attributes in Decision Trees.” Data Mining and Knowledge Discovery, Vol. 3, 1999, pp. 197–217.
[3] Loh, W.Y. “Regression Trees with Unbiased Variable Selection and Interaction Detection.” Statistica Sinica, Vol. 12, 2002, pp. 361–386.
[4] Loh, W.Y. and Y.S. Shih. “Split Selection Methods for Classification Trees.” Statistica Sinica, Vol. 7, 1997, pp. 815–840.
Usage notes and limitations:
Supported syntaxes are:
tree = fitctree(Tbl,Y)
tree = fitctree(X,Y)
tree =
fitctree(___,Name,Value)
[tree,FitInfo,HyperparameterOptimizationResults]
= fitctree(___,Name,Value)
—
fitctree
returns the
additional output arguments
FitInfo
and
HyperparameterOptimizationResults
when you specify the
'OptimizeHyperparameters'
name-value pair argument.
The FitInfo
output argument is an empty structure array currently
reserved for possible future use.
The HyperparameterOptimizationResults
output argument is a BayesianOptimization
object or a
table of hyperparameters with associated values that describe the cross-validation
optimization of hyperparameters.
'HyperparameterOptimizationResults'
is nonempty when the
'OptimizeHyperparameters'
name-value pair argument is
nonempty at the time you create the model. The values in
'HyperparameterOptimizationResults'
depend on the value
you specify for the 'HyperparameterOptimizationOptions'
name-value pair argument when you create the model.
If you specify 'bayesopt'
(default), then
HyperparameterOptimizationResults
is an object of class BayesianOptimization
.
If you specify 'gridsearch'
or
'randomsearch'
, then
HyperparameterOptimizationResults
is a table of the hyperparameters used, observed objective
function values (cross-validation loss), and rank of
observations from lowest (best) to highest (worst).
Supported name-value pair arguments, and any differences, are:
'AlgorithmForCategorical'
'CategoricalPredictors'
'ClassNames'
'Cost'
'HyperparameterOptimizationOptions'
— For
cross-validation, tall optimization supports only 'Holdout'
validation. By
default, the software selects and reserves 20% of the data as holdout validation data, and
trains the model using the rest of the data. You can specify a different value for the holdout
fraction by using this argument. For example, specify
'HyperparameterOptimizationOptions',struct('Holdout',0.3)
to reserve 30%
of the data as validation data.
'MaxNumCategories'
'MaxNumSplits'
— for tall
optimization, fitctree
searches among integers, by default log-scaled in
the range
[1,max(2,min(10000,NumObservations-1))]
.
'MergeLeaves'
'MinLeafSize'
'MinParentSize'
'NumVariablesToSample'
'OptimizeHyperparameters'
'PredictorNames'
'Prior'
'ResponseName'
'ScoreTransform'
'SplitCriterion'
'Weights'
This additional name-value pair argument is specific to tall arrays:
'MaxDepth'
— A positive integer
specifying the maximum depth of the output tree.
Specify a value for this argument to return a tree
that has fewer levels and requires fewer passes
through the tall array to compute. Generally, the
algorithm of fitctree
takes
one pass through the data and an additional pass
for each tree level. The function does not set a
maximum tree depth, by default.
For more information, see Tall Arrays.
To perform parallel hyperparameter optimization, use the
'HyperparameterOptimizationOptions', struct('UseParallel',true)
name-value argument in the call to this function.
For more information on parallel hyperparameter optimization, see Parallel Bayesian Optimization.
For general information about parallel computing, see Run MATLAB Functions with Automatic Parallel Support (Parallel Computing Toolbox).
ClassificationPartitionedModel
| ClassificationTree
| kfoldPredict
| predict
| prune
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