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fitctree

Fit binary classification decision tree for multiclass classification

Syntax

tree = fitctree(Tbl,ResponseVarName)
tree = fitctree(Tbl,formula)
tree = fitctree(Tbl,Y)
tree = fitctree(X,Y)
tree = fitctree(___,Name,Value)

Description

tree = fitctree(Tbl,ResponseVarName) returns a fitted binary classification decision tree based on the input variables (also known as predictors, features, or attributes) contained in the table Tbl and output (response or labels) contained in ResponseVarName. The returned binary tree splits branching nodes based on the values of a column of Tbl.

tree = fitctree(Tbl,formula) returns a fitted binary classification decision tree based on the input variables contained in the table Tbl. formula is an explanatory model of the response and a subset of predictor variables in Tbl used to fit tree.

tree = fitctree(Tbl,Y) returns a fitted binary classification decision tree based on the input variables contained in the table Tbl and output in vector Y.

example

tree = fitctree(X,Y) returns a fitted binary classification decision tree based on the input variables contained in matrix X and output Y. The returned binary tree splits branching nodes based on the values of a column of X.

example

tree = fitctree(___,Name,Value) 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.

Examples

collapse all

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 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 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 = 0.1140
classError7 = kfoldLoss(Mdl7)
classError7 = 0.1254

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 |     0.52658 |    0.066667 |    0.066667 |           31 |
|    2 | Accept |    0.066667 |     0.31459 |    0.066667 |    0.066667 |           12 |
|    3 | Best   |        0.04 |     0.15141 |        0.04 |    0.040003 |            2 |
|    4 | Accept |     0.66667 |     0.22534 |        0.04 |     0.15796 |           73 |
|    5 | Accept |        0.04 |     0.28039 |        0.04 |    0.040009 |            2 |
|    6 | Accept |     0.66667 |     0.11729 |        0.04 |    0.040012 |           74 |
|    7 | Accept |    0.066667 |     0.12783 |        0.04 |    0.040012 |           20 |
|    8 | Accept |        0.04 |     0.23242 |        0.04 |    0.040008 |            4 |
|    9 | Best   |    0.033333 |     0.32521 |    0.033333 |     0.03335 |            1 |
|   10 | Accept |    0.066667 |     0.16289 |    0.033333 |     0.03335 |            7 |
|   11 | Accept |        0.04 |     0.23017 |    0.033333 |    0.033348 |            3 |
|   12 | Accept |    0.066667 |     0.13759 |    0.033333 |    0.033348 |           26 |
|   13 | Accept |    0.046667 |     0.15854 |    0.033333 |    0.033347 |            5 |
|   14 | Accept |    0.033333 |     0.13087 |    0.033333 |     0.03334 |            1 |
|   15 | Accept |    0.066667 |     0.16629 |    0.033333 |    0.033339 |           15 |
|   16 | Accept |    0.033333 |     0.25347 |    0.033333 |    0.033337 |            1 |
|   17 | Accept |    0.033333 |      0.2882 |    0.033333 |    0.033336 |            1 |
|   18 | Accept |     0.33333 |     0.14085 |    0.033333 |    0.033336 |           43 |
|   19 | Accept |    0.066667 |     0.24593 |    0.033333 |    0.033336 |            9 |
|   20 | Accept |    0.066667 |     0.19255 |    0.033333 |    0.033336 |            6 |
|======================================================================================|
| Iter | Eval   | Objective   | Objective   | BestSoFar   | BestSoFar   |  MinLeafSize |
|      | result |             | runtime     | (observed)  | (estim.)    |              |
|======================================================================================|
|   21 | Accept |    0.066667 |     0.16676 |    0.033333 |    0.033336 |           17 |
|   22 | Accept |    0.066667 |     0.16223 |    0.033333 |    0.033336 |           10 |
|   23 | Accept |    0.066667 |     0.21164 |    0.033333 |    0.033336 |           36 |
|   24 | Accept |     0.33333 |     0.18707 |    0.033333 |    0.034075 |           54 |
|   25 | Accept |        0.04 |     0.23593 |    0.033333 |    0.034054 |            2 |
|   26 | Accept |        0.04 |     0.16931 |    0.033333 |    0.034022 |            3 |
|   27 | Accept |        0.04 |     0.26958 |    0.033333 |    0.033997 |            4 |
|   28 | Accept |    0.066667 |     0.14697 |    0.033333 |    0.033973 |           23 |
|   29 | Accept |    0.066667 |     0.14113 |    0.033333 |    0.033946 |            8 |
|   30 | Accept |    0.066667 |     0.19308 |    0.033333 |    0.033922 |           13 |

__________________________________________________________
Optimization completed.
MaxObjectiveEvaluations of 30 reached.
Total function evaluations: 30
Total elapsed time: 65.7631 seconds.
Total objective function evaluation time: 6.2921

Best observed feasible point:
    MinLeafSize
    ___________

         1     

Observed objective function value = 0.033333
Estimated objective function value = 0.033922
Function evaluation time = 0.32521

Best estimated feasible point (according to models):
    MinLeafSize
    ___________

         1     

Estimated objective function value = 0.033922
Estimated function evaluation time = 0.19735
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.

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
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   
      :          :             :           :          :           :           :
      :          :             :           :          :           :           :

When you execute calculations on tall arrays, the default execution environment uses either the local MATLAB session or a local parallel pool (if you have Parallel Computing Toolbox™). You can use the mapreducer function to change the execution environment.

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, set the random seed for tall arrays and use the 'expected-improvement-plus' acquisition function.

rng('default') % For reproducibility
[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 4: Completed in 0.61 sec
- Pass 2 of 4: Completed in 0.86 sec
- Pass 3 of 4: Completed in 1.2 sec
- Pass 4 of 4: Completed in 1.2 sec
Evaluation completed in 4.8 sec

Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.1 sec
Evaluation completed in 1.4 sec
Evaluating tall expression using the Parallel Pool 'local':
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Evaluating tall expression using the Parallel Pool 'local':
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Evaluation completed in 6.7 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1.4 sec
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Evaluation completed in 6.8 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1.4 sec
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- Pass 3 of 4: Completed in 1.4 sec
- Pass 4 of 4: Completed in 1.9 sec
Evaluation completed in 6.7 sec
Evaluating tall expression using the Parallel Pool 'local':
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Evaluation completed in 7.1 sec
Evaluating tall expression using the Parallel Pool 'local':
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Evaluation completed in 6.6 sec
Evaluating tall expression using the Parallel Pool 'local':
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Evaluation completed in 7.4 sec
Evaluating tall expression using the Parallel Pool 'local':
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Evaluation completed in 7.3 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1.4 sec
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Evaluation completed in 7.5 sec
Evaluating tall expression using the Parallel Pool 'local':
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- Pass 3 of 4: Completed in 0.99 sec
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Evaluation completed in 6.5 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1.5 sec
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Evaluation completed in 7.8 sec
Evaluating tall expression using the Parallel Pool 'local':
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Evaluation completed in 7.2 sec
Evaluating tall expression using the Parallel Pool 'local':
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Evaluation completed in 8.6 sec
Evaluating tall expression using the Parallel Pool 'local':
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Evaluation completed in 8.3 sec
Evaluating tall expression using the Parallel Pool 'local':
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Evaluation completed in 9.1 sec
Evaluating tall expression using the Parallel Pool 'local':
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Evaluation completed in 8.7 sec
Evaluating tall expression using the Parallel Pool 'local':
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Evaluation completed in 9.5 sec
Evaluating tall expression using the Parallel Pool 'local':
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Evaluation completed in 8.7 sec
Evaluating tall expression using the Parallel Pool 'local':
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Evaluation completed in 8.6 sec
Evaluating tall expression using the Parallel Pool 'local':
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Evaluation completed in 8.5 sec
Evaluating tall expression using the Parallel Pool 'local':
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Evaluation completed in 9 sec
Evaluating tall expression using the Parallel Pool 'local':
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Evaluation completed in 8.1 sec
Evaluating tall expression using the Parallel Pool 'local':
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Evaluation completed in 8.4 sec
Evaluating tall expression using the Parallel Pool 'local':
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Evaluation completed in 8.4 sec
Evaluating tall expression using the Parallel Pool 'local':
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Evaluation completed in 8.3 sec
Evaluating tall expression using the Parallel Pool 'local':
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Evaluation completed in 9.1 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 2 sec
Evaluation completed in 2.2 sec
|======================================================================================|
| Iter | Eval   | Objective   | Objective   | BestSoFar   | BestSoFar   |  MinLeafSize |
|      | result |             | runtime     | (observed)  | (estim.)    |              |
|======================================================================================|
|    1 | Best   |     0.11755 |      252.16 |     0.11755 |     0.11755 |           10 |
Evaluating tall expression using the Parallel Pool 'local':
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Evaluation completed in 0.75 sec
Evaluating tall expression using the Parallel Pool 'local':
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Evaluating tall expression using the Parallel Pool 'local':
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Evaluation completed in 7.1 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.7 sec
Evaluation completed in 1.9 sec
|    2 | Accept |     0.19549 |      15.739 |     0.11755 |     0.12176 |        48298 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.1 sec
Evaluation completed in 1.3 sec
Evaluating tall expression using the Parallel Pool 'local':
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Evaluating tall expression using the Parallel Pool 'local':
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Evaluation completed in 7.2 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.88 sec
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Evaluation completed in 5.4 sec
Evaluating tall expression using the Parallel Pool 'local':
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Evaluation completed in 7.1 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1.4 sec
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Evaluation completed in 6.5 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.91 sec
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Evaluation completed in 6.7 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1.4 sec
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Evaluation completed in 7 sec
Evaluating tall expression using the Parallel Pool 'local':
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Evaluation completed in 6.9 sec
Evaluating tall expression using the Parallel Pool 'local':
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Evaluation completed in 7.2 sec
Evaluating tall expression using the Parallel Pool 'local':
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Evaluation completed in 6.6 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1.5 sec
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Evaluation completed in 7.2 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.8 sec
Evaluation completed in 2 sec
|    3 | Best   |     0.10411 |      92.721 |     0.10411 |     0.11109 |         3166 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.1 sec
Evaluation completed in 1.3 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.7 sec
Evaluation completed in 1.9 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1.4 sec
- Pass 2 of 4: Completed in 1.2 sec
- Pass 3 of 4: Completed in 1.5 sec
- Pass 4 of 4: Completed in 1.7 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 1.7 sec
- Pass 3 of 4: Completed in 1.5 sec
- Pass 4 of 4: Completed in 1.8 sec
Evaluation completed in 7 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1.4 sec
- Pass 2 of 4: Completed in 1.2 sec
- Pass 3 of 4: Completed in 1.4 sec
- Pass 4 of 4: Completed in 1.8 sec
Evaluation completed in 6.5 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1.4 sec
- Pass 2 of 4: Completed in 1.7 sec
- Pass 3 of 4: Completed in 0.99 sec
- Pass 4 of 4: Completed in 1.8 sec
Evaluation completed in 6.7 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 1.5 sec
- Pass 4 of 4: Completed in 1.8 sec
Evaluation completed in 7.2 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1 sec
- Pass 2 of 4: Completed in 1.7 sec
- Pass 3 of 4: Completed in 1.5 sec
- Pass 4 of 4: Completed in 1.9 sec
Evaluation completed in 6.8 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 1.5 sec
- Pass 4 of 4: Completed in 1.8 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 1 sec
- Pass 4 of 4: Completed in 1.8 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 1.8 sec
- Pass 3 of 4: Completed in 1.5 sec
- Pass 4 of 4: Completed in 2 sec
Evaluation completed in 7.5 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1.5 sec
- Pass 2 of 4: Completed in 1.2 sec
- Pass 3 of 4: Completed in 1.4 sec
- Pass 4 of 4: Completed in 1.9 sec
Evaluation completed in 6.8 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.93 sec
- Pass 2 of 4: Completed in 1.7 sec
- Pass 3 of 4: Completed in 1 sec
- Pass 4 of 4: Completed in 1.9 sec
Evaluation completed in 6.2 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 1.5 sec
- Pass 4 of 4: Completed in 1.8 sec
Evaluation completed in 7.4 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1.5 sec
- Pass 2 of 4: Completed in 1.8 sec
- Pass 3 of 4: Completed in 1.5 sec
- Pass 4 of 4: Completed in 1.9 sec
Evaluation completed in 7.3 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.98 sec
- Pass 2 of 4: Completed in 1.7 sec
- Pass 3 of 4: Completed in 1.5 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 1.5 sec
- Pass 2 of 4: Completed in 1.7 sec
- Pass 3 of 4: Completed in 1.5 sec
- Pass 4 of 4: Completed in 1.8 sec
Evaluation completed in 7.2 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1.6 sec
- Pass 2 of 4: Completed in 1.9 sec
- Pass 3 of 4: Completed in 1.5 sec
- Pass 4 of 4: Completed in 1.8 sec
Evaluation completed in 7.4 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1.5 sec
- Pass 2 of 4: Completed in 1.8 sec
- Pass 3 of 4: Completed in 1.5 sec
- Pass 4 of 4: Completed in 1.8 sec
Evaluation completed in 7.2 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.8 sec
Evaluation completed in 1.9 sec
|    4 | Best   |     0.10005 |      154.39 |     0.10005 |     0.10428 |          180 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.1 sec
Evaluation completed in 1.3 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.2 sec
Evaluation completed in 1.4 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 1.5 sec
- Pass 4 of 4: Completed in 1.8 sec
Evaluation completed in 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 1.5 sec
- Pass 4 of 4: Completed in 1.8 sec
Evaluation completed in 7.1 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1.4 sec
- Pass 2 of 4: Completed in 1.8 sec
- Pass 3 of 4: Completed in 1.5 sec
- Pass 4 of 4: Completed in 1.7 sec
Evaluation completed in 7 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 1.5 sec
- Pass 4 of 4: Completed in 1.8 sec
Evaluation completed in 7 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 1.5 sec
- Pass 4 of 4: Completed in 1.8 sec
Evaluation completed in 7 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 1.4 sec
- Pass 4 of 4: Completed in 1.7 sec
Evaluation completed in 7 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 1.5 sec
- Pass 4 of 4: Completed in 1.8 sec
Evaluation completed in 7.1 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 1.5 sec
- Pass 4 of 4: Completed in 1.8 sec
Evaluation completed in 7.6 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1.4 sec
- Pass 2 of 4: Completed in 1.7 sec
- Pass 3 of 4: Completed in 1.5 sec
- Pass 4 of 4: Completed in 1.9 sec
Evaluation completed in 7.3 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 1 sec
- Pass 4 of 4: Completed in 1.9 sec
Evaluation completed in 6.8 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.98 sec
- Pass 2 of 4: Completed in 1.7 sec
- Pass 3 of 4: Completed in 1.5 sec
- Pass 4 of 4: Completed in 1.9 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 1.5 sec
- Pass 4 of 4: Completed in 1.9 sec
Evaluation completed in 7.2 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 1.5 sec
- Pass 4 of 4: Completed in 1.9 sec
Evaluation completed in 7.2 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 1.5 sec
- Pass 4 of 4: Completed in 1.8 sec
Evaluation completed in 7.2 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 1.5 sec
- Pass 4 of 4: Completed in 1.8 sec
Evaluation completed in 7.2 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 1.5 sec
- Pass 4 of 4: Completed in 1.8 sec
Evaluation completed in 7.2 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 1.5 sec
- Pass 4 of 4: Completed in 1.8 sec
Evaluation completed in 7.5 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.8 sec
Evaluation completed in 1.9 sec
|    5 | Accept |     0.10047 |      156.46 |     0.10005 |     0.10003 |          146 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.1 sec
Evaluation completed in 1.3 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.7 sec
Evaluation completed in 1.9 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1.4 sec
- Pass 2 of 4: Completed in 1.8 sec
- Pass 3 of 4: Completed in 1.5 sec
- Pass 4 of 4: Completed in 1.8 sec
Evaluation completed in 7.1 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 1.5 sec
- Pass 4 of 4: Completed in 1.8 sec
Evaluation completed in 7.1 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 1.5 sec
- Pass 4 of 4: Completed in 1.7 sec
Evaluation completed in 6.9 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1.4 sec
- Pass 2 of 4: Completed in 1.6 sec
- Pass 3 of 4: Completed in 1.5 sec
- Pass 4 of 4: Completed in 1.7 sec
Evaluation completed in 6.8 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 1.5 sec
- Pass 4 of 4: Completed in 1.8 sec
Evaluation completed in 7 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 1.5 sec
- Pass 4 of 4: Completed in 1.8 sec
Evaluation completed in 7 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 1.5 sec
- Pass 4 of 4: Completed in 1.7 sec
Evaluation completed in 7 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 1.5 sec
- Pass 4 of 4: Completed in 1.7 sec
Evaluation completed in 6.9 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 1.5 sec
- Pass 4 of 4: Completed in 1.7 sec
Evaluation completed in 6.9 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 1.5 sec
- Pass 4 of 4: Completed in 1.7 sec
Evaluation completed in 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 1.5 sec
- Pass 4 of 4: Completed in 1.7 sec
Evaluation completed in 6.9 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 1.5 sec
- Pass 4 of 4: Completed in 1.8 sec
Evaluation completed in 7 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 1.5 sec
- Pass 4 of 4: Completed in 1.7 sec
Evaluation completed in 6.9 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.8 sec
Evaluation completed in 1.9 sec
|    6 | Accept |     0.10113 |      119.54 |     0.10005 |     0.10008 |          991 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.1 sec
Evaluation completed in 1.2 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.2 sec
Evaluation completed in 1.3 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 1.5 sec
- Pass 4 of 4: Completed in 1.7 sec
Evaluation completed in 6.9 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 1.5 sec
- Pass 4 of 4: Completed in 1.7 sec
Evaluation completed in 7 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 1.5 sec
- Pass 4 of 4: Completed in 1.7 sec
Evaluation completed in 6.9 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 1.5 sec
- Pass 4 of 4: Completed in 1.8 sec
Evaluation completed in 6.9 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 1.6 sec
- Pass 4 of 4: Completed in 1.8 sec
Evaluation completed in 7.1 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 1.6 sec
- Pass 4 of 4: Completed in 1.8 sec
Evaluation completed in 7.1 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 1.5 sec
- Pass 4 of 4: Completed in 1.8 sec
Evaluation completed in 7.1 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1.4 sec
- Pass 2 of 4: Completed in 1.8 sec
- Pass 3 of 4: Completed in 1.5 sec
- Pass 4 of 4: Completed in 1.9 sec
Evaluation completed in 7.4 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1.5 sec
- Pass 2 of 4: Completed in 1.9 sec
- Pass 3 of 4: Completed in 1.5 sec
- Pass 4 of 4: Completed in 2.1 sec
Evaluation completed in 7.6 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 1.5 sec
- Pass 4 of 4: Completed in 2.2 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 1.8 sec
- Pass 3 of 4: Completed in 1.5 sec
- Pass 4 of 4: Completed in 2.4 sec
Evaluation completed in 8.2 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1.7 sec
- Pass 2 of 4: Completed in 2 sec
- Pass 3 of 4: Completed in 1.5 sec
- Pass 4 of 4: Completed in 2.7 sec
Evaluation completed in 8.8 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1.7 sec
- Pass 2 of 4: Completed in 2 sec
- Pass 3 of 4: Completed in 1.5 sec
- Pass 4 of 4: Completed in 2.8 sec
Evaluation completed in 9 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1.9 sec
- Pass 2 of 4: Completed in 2.1 sec
- Pass 3 of 4: Completed in 1.6 sec
- Pass 4 of 4: Completed in 2.9 sec
Evaluation completed in 9.3 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1.9 sec
- Pass 2 of 4: Completed in 2.2 sec
- Pass 3 of 4: Completed in 1.3 sec
- Pass 4 of 4: Completed in 3 sec
Evaluation completed in 9.3 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 2.1 sec
- Pass 2 of 4: Completed in 2.4 sec
- Pass 3 of 4: Completed in 1.7 sec
- Pass 4 of 4: Completed in 3.1 sec
Evaluation completed in 10 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 2.2 sec
- Pass 2 of 4: Completed in 2.4 sec
- Pass 3 of 4: Completed in 1.6 sec
- Pass 4 of 4: Completed in 3.1 sec
Evaluation completed in 10 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1.7 sec
- Pass 2 of 4: Completed in 2.5 sec
- Pass 3 of 4: Completed in 1.6 sec
- Pass 4 of 4: Completed in 3 sec
Evaluation completed in 9.7 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 2.3 sec
- Pass 2 of 4: Completed in 3 sec
- Pass 3 of 4: Completed in 1.8 sec
- Pass 4 of 4: Completed in 2.9 sec
Evaluation completed in 11 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 2.4 sec
- Pass 2 of 4: Completed in 2.6 sec
- Pass 3 of 4: Completed in 1.8 sec
- Pass 4 of 4: Completed in 3 sec
Evaluation completed in 10 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 2.4 sec
- Pass 2 of 4: Completed in 2.8 sec
- Pass 3 of 4: Completed in 1.7 sec
- Pass 4 of 4: Completed in 2.9 sec
Evaluation completed in 10 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 2.4 sec
- Pass 2 of 4: Completed in 2.7 sec
- Pass 3 of 4: Completed in 1.7 sec
- Pass 4 of 4: Completed in 3.1 sec
Evaluation completed in 11 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 2.5 sec
- Pass 2 of 4: Completed in 2.7 sec
- Pass 3 of 4: Completed in 1.7 sec
- Pass 4 of 4: Completed in 2.8 sec
Evaluation completed in 10 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 2.5 sec
- Pass 2 of 4: Completed in 2.7 sec
- Pass 3 of 4: Completed in 1.7 sec
- Pass 4 of 4: Completed in 2.8 sec
Evaluation completed in 10 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 2.4 sec
- Pass 2 of 4: Completed in 2.8 sec
- Pass 3 of 4: Completed in 1.7 sec
- Pass 4 of 4: Completed in 2.7 sec
Evaluation completed in 10 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 2.4 sec
- Pass 2 of 4: Completed in 2.7 sec
- Pass 3 of 4: Completed in 1.7 sec
- Pass 4 of 4: Completed in 2.7 sec
Evaluation completed in 10 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 2.5 sec
- Pass 2 of 4: Completed in 2.7 sec
- Pass 3 of 4: Completed in 1.7 sec
- Pass 4 of 4: Completed in 2.8 sec
Evaluation completed in 10 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 2.5 sec
- Pass 2 of 4: Completed in 2.7 sec
- Pass 3 of 4: Completed in 1.7 sec
- Pass 4 of 4: Completed in 2.7 sec
Evaluation completed in 10 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 2.5 sec
- Pass 2 of 4: Completed in 2.8 sec
- Pass 3 of 4: Completed in 1.7 sec
- Pass 4 of 4: Completed in 2.9 sec
Evaluation completed in 11 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 2.4 sec
- Pass 2 of 4: Completed in 2.7 sec
- Pass 3 of 4: Completed in 1.7 sec
- Pass 4 of 4: Completed in 2.8 sec
Evaluation completed in 10 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.8 sec
Evaluation completed in 1.9 sec
|    7 | Accept |     0.13372 |      329.49 |     0.10005 |     0.10007 |            1 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 0.56 sec
Evaluation completed in 0.7 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.7 sec
Evaluation completed in 1.9 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1.4 sec
- Pass 2 of 4: Completed in 1.6 sec
- Pass 3 of 4: Completed in 1.5 sec
- Pass 4 of 4: Completed in 1.8 sec
Evaluation completed in 7 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 1.5 sec
- Pass 4 of 4: Completed in 1.7 sec
Evaluation completed in 6.9 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 1.5 sec
- Pass 4 of 4: Completed in 1.8 sec
Evaluation completed in 6.9 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1.4 sec
- Pass 2 of 4: Completed in 1.8 sec
- Pass 3 of 4: Completed in 1.5 sec
- Pass 4 of 4: Completed in 1.8 sec
Evaluation completed in 7.2 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1.4 sec
- Pass 2 of 4: Completed in 2 sec
- Pass 3 of 4: Completed in 1.5 sec
- Pass 4 of 4: Completed in 1.8 sec
Evaluation completed in 7.5 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1.4 sec
- Pass 2 of 4: Completed in 1.7 sec
- Pass 3 of 4: Completed in 1.5 sec
- Pass 4 of 4: Completed in 1.8 sec
Evaluation completed in 7 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 1.5 sec
- Pass 4 of 4: Completed in 1.8 sec
Evaluation completed in 7 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1.4 sec
- Pass 2 of 4: Completed in 1.2 sec
- Pass 3 of 4: Completed in 1.5 sec
- Pass 4 of 4: Completed in 1.9 sec
Evaluation completed in 6.7 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.96 sec
- Pass 2 of 4: Completed in 1.7 sec
- Pass 3 of 4: Completed in 1.5 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 1.7 sec
- Pass 3 of 4: Completed in 1.5 sec
- Pass 4 of 4: Completed in 1.8 sec
Evaluation completed in 7.1 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1.5 sec
- Pass 2 of 4: Completed in 1.8 sec
- Pass 3 of 4: Completed in 1.5 sec
- Pass 4 of 4: Completed in 1.8 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 1.8 sec
- Pass 3 of 4: Completed in 1.5 sec
- Pass 4 of 4: Completed in 1.8 sec
Evaluation completed in 7.2 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 1.5 sec
- Pass 4 of 4: Completed in 1.8 sec
Evaluation completed in 7.2 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 1.5 sec
- Pass 4 of 4: Completed in 1.9 sec
Evaluation completed in 7.3 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 1.5 sec
- Pass 4 of 4: Completed in 1.7 sec
Evaluation completed in 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 1.5 sec
- Pass 4 of 4: Completed in 1.8 sec
Evaluation completed in 7.1 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.7 sec
Evaluation completed in 1.8 sec
|    8 | Accept |     0.10116 |      145.82 |     0.10005 |     0.10015 |          357 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.1 sec
Evaluation completed in 1.2 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.7 sec
Evaluation completed in 1.9 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 1.5 sec
- Pass 4 of 4: Completed in 1.8 sec
Evaluation completed in 7.1 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 1.5 sec
- Pass 4 of 4: Completed in 1.7 sec
Evaluation completed in 6.9 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.92 sec
- Pass 2 of 4: Completed in 1.7 sec
- Pass 3 of 4: Completed in 1.5 sec
- Pass 4 of 4: Completed in 1.8 sec
Evaluation completed in 6.5 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1.4 sec
- Pass 2 of 4: Completed in 1.7 sec
- Pass 3 of 4: Completed in 1.5 sec
- Pass 4 of 4: Completed in 1.8 sec
Evaluation completed in 7.1 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 1.5 sec
- Pass 4 of 4: Completed in 1.8 sec
Evaluation completed in 7 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1.4 sec
- Pass 2 of 4: Completed in 1.8 sec
- Pass 3 of 4: Completed in 1.5 sec
- Pass 4 of 4: Completed in 1.8 sec
Evaluation completed in 7.1 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 1.5 sec
- Pass 4 of 4: Completed in 1.7 sec
Evaluation completed in 6.9 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 1.5 sec
- Pass 4 of 4: Completed in 1.8 sec
Evaluation completed in 7.2 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 1 sec
- Pass 4 of 4: Completed in 1.7 sec
Evaluation completed in 6.6 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.92 sec
- Pass 2 of 4: Completed in 1.7 sec
- Pass 3 of 4: Completed in 1.5 sec
- Pass 4 of 4: Completed in 1.7 sec
Evaluation completed in 6.4 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 1.5 sec
- Pass 4 of 4: Completed in 1.7 sec
Evaluation completed in 7 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.7 sec
Evaluation completed in 1.9 sec
|    9 | Accept |     0.10212 |      101.69 |     0.10005 |     0.10018 |         1688 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 0.58 sec
Evaluation completed in 0.73 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.7 sec
Evaluation completed in 1.9 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 1.5 sec
- Pass 4 of 4: Completed in 1.8 sec
Evaluation completed in 6.9 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 1.6 sec
- Pass 4 of 4: Completed in 1.8 sec
Evaluation completed in 7.2 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1.6 sec
- Pass 2 of 4: Completed in 1.6 sec
- Pass 3 of 4: Completed in 1.5 sec
- Pass 4 of 4: Completed in 1.7 sec
Evaluation completed in 7.1 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 1.4 sec
- Pass 4 of 4: Completed in 1.7 sec
Evaluation completed in 6.9 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 0.97 sec
- Pass 4 of 4: Completed in 1.8 sec
Evaluation completed in 6.5 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1.4 sec
- Pass 2 of 4: Completed in 1.7 sec
- Pass 3 of 4: Completed in 1.5 sec
- Pass 4 of 4: Completed in 1.9 sec
Evaluation completed in 7.1 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 1.4 sec
- Pass 4 of 4: Completed in 1.8 sec
Evaluation completed in 7.1 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 1.5 sec
- Pass 4 of 4: Completed in 1.9 sec
Evaluation completed in 7.2 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 1.5 sec
- Pass 4 of 4: Completed in 2 sec
Evaluation completed in 7.4 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1.5 sec
- Pass 2 of 4: Completed in 1.8 sec
- Pass 3 of 4: Completed in 1.5 sec
- Pass 4 of 4: Completed in 2.1 sec
Evaluation completed in 7.5 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 1.5 sec
- Pass 4 of 4: Completed in 2 sec
Evaluation completed in 7.5 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 1.5 sec
- Pass 4 of 4: Completed in 2.1 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 1.9 sec
- Pass 3 of 4: Completed in 1.5 sec
- Pass 4 of 4: Completed in 2.1 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 1.9 sec
- Pass 3 of 4: Completed in 1.5 sec
- Pass 4 of 4: Completed in 2 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 1.9 sec
- Pass 3 of 4: Completed in 1.5 sec
- Pass 4 of 4: Completed in 2 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 1.9 sec
- Pass 3 of 4: Completed in 1.6 sec
- Pass 4 of 4: Completed in 1.9 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 1.9 sec
- Pass 3 of 4: Completed in 1.5 sec
- Pass 4 of 4: Completed in 1.9 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 2 sec
- Pass 3 of 4: Completed in 1.5 sec
- Pass 4 of 4: Completed in 1.9 sec
Evaluation completed in 7.6 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.7 sec
Evaluation completed in 1.9 sec
|   10 | Accept |     0.10215 |      168.67 |     0.10005 |     0.10029 |           60 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.1 sec
Evaluation completed in 1.2 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.7 sec
Evaluation completed in 1.9 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 1.5 sec
- Pass 4 of 4: Completed in 1.7 sec
Evaluation completed in 6.9 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 1.5 sec
- Pass 4 of 4: Completed in 1.8 sec
Evaluation completed in 7.1 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1.5 sec
- Pass 2 of 4: Completed in 1.8 sec
- Pass 3 of 4: Completed in 1.5 sec
- Pass 4 of 4: Completed in 1.7 sec
Evaluation completed in 7.2 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 1.5 sec
- Pass 4 of 4: Completed in 1.7 sec
Evaluation completed in 7.1 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 1.5 sec
- Pass 4 of 4: Completed in 1.7 sec
Evaluation completed in 7 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 1.5 sec
- Pass 4 of 4: Completed in 1.8 sec
Evaluation completed in 7.1 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1.4 sec
- Pass 2 of 4: Completed in 1.8 sec
- Pass 3 of 4: Completed in 1.5 sec
- Pass 4 of 4: Completed in 1.9 sec
Evaluation completed in 7.2 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 1.5 sec
- Pass 4 of 4: Completed in 1.9 sec
Evaluation completed in 7.3 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1.4 sec
- Pass 2 of 4: Completed in 1.8 sec
- Pass 3 of 4: Completed in 1.5 sec
- Pass 4 of 4: Completed in 1.9 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 1.8 sec
- Pass 3 of 4: Completed in 1.5 sec
- Pass 4 of 4: Completed in 2 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 1.8 sec
- Pass 3 of 4: Completed in 1.5 sec
- Pass 4 of 4: Completed in 1.9 sec
Evaluation completed in 7.4 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1.5 sec
- Pass 2 of 4: Completed in 1.8 sec
- Pass 3 of 4: Completed in 1 sec
- Pass 4 of 4: Completed in 1.9 sec
Evaluation completed in 6.9 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1 sec
- Pass 2 of 4: Completed in 1.8 sec
- Pass 3 of 4: Completed in 1.5 sec
- Pass 4 of 4: Completed in 1.9 sec
Evaluation completed in 6.9 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1.5 sec
- Pass 2 of 4: Completed in 1.9 sec
- Pass 3 of 4: Completed in 1.5 sec
- Pass 4 of 4: Completed in 1.8 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 1.8 sec
- Pass 3 of 4: Completed in 1.5 sec
- Pass 4 of 4: Completed in 1.8 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 1.6 sec
- Pass 4 of 4: Completed in 1.8 sec
Evaluation completed in 7.2 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 1.5 sec
- Pass 4 of 4: Completed in 1.8 sec
Evaluation completed in 7.2 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 1.5 sec
- Pass 4 of 4: Completed in 1.8 sec
Evaluation completed in 7.2 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.8 sec
Evaluation completed in 2 sec
|   11 | Accept |     0.10095 |      166.59 |     0.10005 |     0.10022 |          132 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.1 sec
Evaluation completed in 1.2 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.7 sec
Evaluation completed in 1.9 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1.4 sec
- Pass 2 of 4: Completed in 1.6 sec
- Pass 3 of 4: Completed in 1.5 sec
- Pass 4 of 4: Completed in 1.8 sec
Evaluation completed in 6.9 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 1.5 sec
- Pass 4 of 4: Completed in 1.7 sec
Evaluation completed in 6.9 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 1.5 sec
- Pass 4 of 4: Completed in 1.7 sec
Evaluation completed in 6.9 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 1.5 sec
- Pass 4 of 4: Completed in 1.8 sec
Evaluation completed in 7.1 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 1.5 sec
- Pass 4 of 4: Completed in 1.8 sec
Evaluation completed in 7.1 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 1.5 sec
- Pass 4 of 4: Completed in 1.8 sec
Evaluation completed in 7 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 1.5 sec
- Pass 4 of 4: Completed in 1.8 sec
Evaluation completed in 7.1 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 0.97 sec
- Pass 4 of 4: Completed in 1.8 sec
Evaluation completed in 6.5 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 1.5 sec
- Pass 4 of 4: Completed in 1.9 sec
Evaluation completed in 7.2 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 1.5 sec
- Pass 4 of 4: Completed in 1.9 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 1.5 sec
- Pass 4 of 4: Completed in 2.1 sec
Evaluation completed in 7.5 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 1.5 sec
- Pass 4 of 4: Completed in 1.8 sec
Evaluation completed in 7.1 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1.5 sec
- Pass 2 of 4: Completed in 1.8 sec
- Pass 3 of 4: Completed in 1.5 sec
- Pass 4 of 4: Completed in 1.8 sec
Evaluation completed in 7.2 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 1.5 sec
- Pass 4 of 4: Completed in 1.8 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 1.8 sec
- Pass 3 of 4: Completed in 1.5 sec
- Pass 4 of 4: Completed in 1.8 sec
Evaluation completed in 7.2 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1.5 sec
- Pass 2 of 4: Completed in 1.2 sec
- Pass 3 of 4: Completed in 1.5 sec
- Pass 4 of 4: Completed in 1.7 sec
Evaluation completed in 6.6 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 1.5 sec
- Pass 4 of 4: Completed in 1.7 sec
Evaluation completed in 7.1 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.8 sec
Evaluation completed in 1.9 sec
|   12 | Accept |     0.10147 |      155.66 |     0.10005 |     0.10064 |          207 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 0.57 sec
Evaluation completed in 0.72 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.7 sec
Evaluation completed in 1.9 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 1.5 sec
- Pass 4 of 4: Completed in 1.7 sec
Evaluation completed in 6.9 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1.5 sec
- Pass 2 of 4: Completed in 1.6 sec
- Pass 3 of 4: Completed in 1.5 sec
- Pass 4 of 4: Completed in 1.8 sec
Evaluation completed in 7.1 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 1.5 sec
- Pass 4 of 4: Completed in 1.8 sec
Evaluation completed in 7 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1.4 sec
- Pass 2 of 4: Completed in 1.8 sec
- Pass 3 of 4: Completed in 1.5 sec
- Pass 4 of 4: Completed in 1.8 sec
Evaluation completed in 7.1 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 1.5 sec
- Pass 4 of 4: Completed in 1.7 sec
Evaluation completed in 7.1 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 1.6 sec
- Pass 4 of 4: Completed in 1.9 sec
Evaluation completed in 7.4 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 1.5 sec
- Pass 4 of 4: Completed in 1.8 sec
Evaluation completed in 7.2 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1.4 sec
- Pass 2 of 4: Completed in 1.8 sec
- Pass 3 of 4: Completed in 1.5 sec
- Pass 4 of 4: Completed in 1.9 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 1.5 sec
- Pass 4 of 4: Completed in 2 sec
Evaluation completed in 7.4 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1.5 sec
- Pass 2 of 4: Completed in 1.8 sec
- Pass 3 of 4: Completed in 1.5 sec
- Pass 4 of 4: Completed in 2 sec
Evaluation completed in 7.5 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 1.5 sec
- Pass 4 of 4: Completed in 1.9 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 1.8 sec
- Pass 3 of 4: Completed in 1.5 sec
- Pass 4 of 4: Completed in 1.9 sec
Evaluation completed in 7.4 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1.5 sec
- Pass 2 of 4: Completed in 1.8 sec
- Pass 3 of 4: Completed in 1.5 sec
- Pass 4 of 4: Completed in 1.9 sec
Evaluation completed in 7.4 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 1.5 sec
- Pass 4 of 4: Completed in 1.9 sec
Evaluation completed in 7.5 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 1.5 sec
- Pass 4 of 4: Completed in 1.9 sec
Evaluation completed in 7.4 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 1.5 sec
- Pass 2 of 4: Completed in 1.8 sec
- Pass 3 of 4: Completed in 1.5 sec
- Pass 4 of 4: Completed in 1.8 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 1.8 sec
- Pass 3 of 4: Completed in 1.5 sec
- Pass 4 of 4: Completed in 1.8 sec
Evaluation completed in 7.3 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.8 sec
Evaluation completed in 2 sec
|   13 | Accept |     0.10063 |      158.96 |     0.10005 |     0.10063 |          121 |
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.1 sec
Evaluation completed in 1.2 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 1.7 sec
Evaluation completed in 1.9 sec
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 0.91 sec
- Pass 2 of 4: Completed in 1.7 sec
- Pass 3 of 4: Completed in 1.5 sec
- Pass 4 of 4: Completed in 1.8 sec
Evaluation completed in 6.4 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 1.5 sec
- Pass 4 of 4: Completed in 1.7 sec
Evaluation completed in 6.9 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 1.5 sec
- Pass 4 of 4: Completed in 1.7 sec
Evaluation completed in 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 1.5 sec
- Pass 4 of 4: Completed in 1.7 sec
Evaluation completed in 7 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 1.5 sec
- Pass 4 of 4: Completed in 1.8 sec
Evaluation completed in 7 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 1.5 sec
- Pass 4 of 4: Completed in 1.8 sec
Evaluation completed in 7.1 sec
Evaluating tall expression using the Parallel Pool 'local':
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Evalu...
Mdl = 
  classreg.learning.classif.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: 0.1001
                   XAtMinObjective: [1×1 table]
             MinEstimatedObjective: 0.1006
          XAtMinEstimatedObjective: [1×1 table]
           NumObjectiveEvaluations: 30
                  TotalElapsedTime: 4.8903e+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]

Input Arguments

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Sample 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

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, logical or numeric vector, or 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.

It is a good practice to specify the order of the classes by using the ClassNames name-value pair argument.

Data Types: char | string

Explanatory model of the response and a subset of the predictor variables, specified as a character vector or string scalar in the form of 'Y~X1+X2+X3'. In this form, Y represents the response variable, and X1, X2, and X3 represent the predictor variables. The variables must be variable names in Tbl (Tbl.Properties.VariableNames).

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.

Data Types: char | string

Class 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

Predictor 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

Name-Value Pair Arguments

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.

Example: '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.

Model Parameters

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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.

ValueDescription
'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.

Example: 'AlgorithmForCategorical','PCA'

Categorical predictors list, specified as the comma-separated pair consisting of 'CategoricalPredictors' and one of the values in this table.

ValueDescription
Vector of positive integersAn entry in the vector is the index value corresponding to the column of the predictor data (X or Tbl) that contains a categorical variable.
Logical vectorA true entry means that the corresponding column of predictor data (X or Tbl) is a categorical variable.
Character matrixEach 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 vectorsEach 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 contains logical values, categorical values, a string array, or a cell array of character vectors. If the predictor data is a matrix (X), fitctree assumes all predictors are continuous. To identify any categorical predictors when the data is a matrix, use the 'CategoricalPredictors' name-value pair argument.

Example: 'CategoricalPredictors','all'

Data Types: single | double | logical | char | string | cell

Names of classes to use for training, specified as the comma-separated pair consisting of 'ClassNames' and 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 Y.

If ClassNames is a character array, then each element must correspond to one row of the array.

Use ClassNames to:

  • Order 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 Y.

Example: 'ClassNames',{'b','g'}

Data Types: categorical | char | string | logical | single | double | cell

Cost 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

Maximum tree depth, specified as a positive integer. Specify a value for this parameter to return a tree with fewer levels that 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. There is no maximum tree depth by default.

Note

This option only applies when using fitctree on tall arrays. See Extended Capabilities for more information.

Maximum 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

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'

Minimum 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

Predictor variable names, specified as the comma-separated pair consisting of 'PredictorNames' and 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 give the predictor variables in X names.

    • 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 in 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.

    • It is a good practice to specify the predictors for training using either 'PredictorNames' or formula only.

Example: 'PredictorNames',{'SepalLength','SepalWidth','PetalLength','PetalWidth'}

Data Types: string | cell

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.

ValueDescription
'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.

  • 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.

Example: 'PredictorSelection','curvature'

Prior probabilities for each class, specified as the comma-separated pair consisting of 'Prior' and one of the following.

  • A character vector or string scalar:

    • 'empirical' determines class probabilities from class frequencies in Y. If you pass observation weights, fitctree uses the weights to compute the class probabilities.

    • 'uniform' sets all class probabilities equal.

  • A vector (one scalar value for each class). To specify the class order for the corresponding elements of Prior, also specify the ClassNames name-value pair argument.

  • A structure S with two fields:

    • S.ClassNames containing the class names as a variable of the same type as Y

    • S.ClassProbs containing a vector of corresponding probabilities

If you set values for both weights and prior, the weights are renormalized to add up to the value of the prior probability in the respective class.

Example: 'Prior','uniform'

Data Types: char | string | single | double | struct

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'

Pruning criterion, specified as the comma-separated pair consisting of 'PruneCriterion' and 'error' or 'impurity'.

Example: 'PruneCriterion','impurity'

Response 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

Score transformation, specified as the comma-separated pair consisting of 'ScoreTransform' and a character vector, string scalar, or function handle.

This table summarizes the available character vectors and string scalars.

ValueDescription
'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 + ex)
'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 + ex) – 1

For a MATLAB® function or a function you define, use its function handle for 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

Surrogate 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

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 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 class.

Data Types: single | double | char | string

Cross-Validation

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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'

Partition 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.

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

Number 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:

  1. Randomly partitions the data into k sets

  2. For each set, reserves the set as validation data, and trains the model using the other k – 1 sets

  3. 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

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'

Hyperparameter Optimization

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Maximal 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

Minimum 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

Number of predictors to select at random for each split, specified as the comma-separated pair consisting of 'NumVariablesToSample' and a positive integer value. You can also specify 'all' to use all available predictors.

Example: 'NumVariablesToSample',3

Data Types: char | string | single | double

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).

Example: 'SplitCriterion','deviance'

Hyperparameter Optimization

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Parameters 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:

  • MaxNumSplitsfitctree searches among integers, by default log-scaled in the range [1,max(2,NumObservations-1)].

  • MinLeafSizefitctree 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'.

  • NumVariablesToSamplefitctree 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'

Options 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 NameValuesDefault
Optimizer
  • 'bayesopt' — Use Bayesian optimization. Internally, this setting calls bayesopt.

  • 'gridsearch' — Use grid search with NumGridDivisions values per dimension.

  • 'randomsearch' — Search at random among MaxObjectiveEvaluations points.

'gridsearch' searches in a random order, using uniform sampling without replacement from the grid. After optimization, you can get a table in grid order by using the command sortrows(Mdl.HyperparameterOptimizationResults).

'bayesopt'
AcquisitionFunctionName

  • 'expected-improvement-per-second-plus'

  • 'expected-improvement'

  • 'expected-improvement-plus'

  • 'expected-improvement-per-second'

  • 'lower-confidence-bound'

  • 'probability-of-improvement'

For details, see the bayesopt AcquisitionFunctionName name-value pair argument, or Acquisition Function Types.

'expected-improvement-per-second-plus'
MaxObjectiveEvaluationsMaximum 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 tic and toc. Run time can exceed MaxTime because MaxTime does not interrupt function evaluations.

Inf
NumGridDivisionsFor '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
ShowPlotsLogical 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
SaveIntermediateResultsLogical 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.

  • 0 — No iterative display

  • 1 — Iterative display

  • 2 — Iterative display with extra information

For details, see the bayesopt Verbose name-value pair argument.

1
UseParallelLogical value indicating whether to run Bayesian optimization in parallel, which requires Parallel Computing Toolbox™. For details, see Parallel Bayesian Optimization.false
Repartition

Logical value indicating whether to repartition the cross-validation at every iteration. If false, the optimizer uses a single partition for the optimization.

true usually gives the most robust results because this setting takes partitioning noise into account. However, for good results, true requires at least twice as many function evaluations.

false
Use no more than one of the following three field names.
CVPartitionA cvpartition object, as created by cvpartition.'Kfold',5 if you do not specify any cross-validation field
HoldoutA scalar in the range (0,1) representing the holdout fraction.
KfoldAn integer greater than 1.

Example: 'HyperparameterOptimizationOptions',struct('MaxObjectiveEvaluations',60)

Data Types: struct

Output Arguments

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Classification 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.

More About

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Curvature Test

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.

  1. 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.

  2. 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

    π^jk=i=1nI{yi=k}wi.

    wi is the weight of observation i, wi=1, I is the indicator function, and n is the sample size. If all observations have the same weight, then π^jk=njkn, where njk is the number of observations in level j of the predictor that are in class k.

  3. Compute the test statistic

    t=nk=1Kj=1J(π^jkπ^j+π^+k)2π^j+π^+k

    π^j+=kπ^jk, that is, the marginal probability of observing the predictor at level j. π^+k=jπ^jk, 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.

  4. 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].

Impurity and Node Error

ClassificationTree 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

    1ip2(i),

    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

    ip(i)log2p(i).

    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

    P(L)P(R)(i|L(i)R(i)|)2,

    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 hence similar to the parent node, and so 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).

Interaction Test

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.

  1. 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.

  2. 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.

  3. 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].

Predictive Measure of Association

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 jk. At node t, the predictive measure of association between the optimal split xj < u and a surrogate split xk < v is

λjk=min(PL,PR)(1PLjLkPRjRk)min(PL,PR).

  • 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 xju. The subscript R stands for the right child of node t.

  • PLjLk is the proportion of observations at node t, such that xj < u and xk < v.

  • PRjRk is the proportion of observations at node t, such that xju and xkv.

  • 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.

Surrogate Decision Splits

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.

Tips

  • 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.

Algorithms

collapse all

Node Splitting Rules

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:

    1. fitctree computes the weighted impurity of node t, it. For supported impurity measures, see SplitCriterion.

    2. fitctree estimates the probability that an observation is in node t using

      P(T)=jTwj.

      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.

    3. 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.

    4. 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:

      1. fitctree splits the observations in node t into left and right child nodes (tL and tR, respectively).

      2. 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

          ΔI=P(T)itP(TL)itLP(TR)itR.

        • If xi contains missing values then, assuming that the observations are missing at random, the impurity gain is

          ΔIU=P(TTU)itP(TL)itLP(TR)itR.

          TTU is the set of all observation indices in node t that are not missing.

        • If you use surrogate decision splits, then:

          1. fitctree computes the predictive measures of association between the decision split xj < u and all possible decision splits xk < v, jk.

          2. 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.

          3. 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.

          4. 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.

      3. 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'):

    1. 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.

    2. 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' ):

    1. 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.

    2. If fitctree chooses a split predictor, then it uses standard CART to choose the cut point (see step 4 in the standard CART process).

Tree Depth Control

  • 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:

    1. Determine how many branch nodes in the current layer must be unsplit so that there are at most MaxNumSplits branch nodes.

    2. Sort the branch nodes by their impurity gains.

    3. Unsplit the number of least successful branches.

    4. 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.

Parallelization

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.

References

[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.

Extended Capabilities

Introduced in R2014a