edge

Class: ClassificationLinear

Classification edge for linear classification models

Syntax

e = edge(Mdl,X,Y)

e = edge(Mdl,Tbl,ResponseVarName)

e = edge(Mdl,Tbl,Y)

e = edge(___,Name,Value)

Description

e = edge(Mdl,X,Y) returns the classification edges for the binary, linear classification model Mdl using predictor data in X and corresponding class labels in Y. e contains a classification edge for each regularization strength in Mdl.

e = edge(Mdl,Tbl,ResponseVarName) returns the classification edges for the trained linear classifier Mdl using the predictor data in Tbl and the class labels in Tbl.ResponseVarName.

e = edge(Mdl,Tbl,Y) returns the classification edges for the classifier Mdl using the predictor data in table Tbl and the class labels in vector Y.

e = edge(___,Name,Value) specifies options using one or more name-value pair arguments in addition to any of the input argument combinations in previous syntaxes. For example, you can specify that columns in the predictor data correspond to observations or supply observation weights.

Input Arguments

Mdl — Binary, linear classification model

ClassificationLinear model object

Binary, linear classification model, specified as a ClassificationLinear model object. You can create a ClassificationLinear model object using fitclinear.

X — Predictor data

full matrix | sparse matrix

Predictor data, specified as an n-by-p full or sparse matrix. This orientation of X indicates that rows correspond to individual observations, and columns correspond to individual predictor variables.

Note

If you orient your predictor matrix so that observations correspond to columns and specify 'ObservationsIn','columns', then you might experience a significant reduction in computation time.

The length of Y and the number of observations in X must be equal.

Data Types: single | double

Y — Class labels

categorical array | character array | string array | logical vector | numeric vector | cell array of character vectors

Class labels, specified as a categorical, character, or string array, logical or numeric vector, or cell array of character vectors.

• The data type of Y must be the same as the data type of Mdl.ClassNames. (The software treats string arrays as cell arrays of character vectors.)

• The distinct classes in Y must be a subset of Mdl.ClassNames.

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

• The length of Y must be equal to the number of observations in X or Tbl.

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

Tbl — Sample data

table

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 additional columns for the response variable and observation weights. Tbl must contain all the predictors used to train Mdl. Multicolumn variables and cell arrays other than cell arrays of character vectors are not allowed.

If Tbl contains the response variable used to train Mdl, then you do not need to specify ResponseVarName or Y.

If you train Mdl using sample data contained in a table, then the input data for edge must also be in a table.

ResponseVarName — Response variable name

name of variable in Tbl

Response variable name, specified as the name of a variable in Tbl. If Tbl contains the response variable used to train Mdl, then you do not need to specify ResponseVarName.

If you specify ResponseVarName, then you must specify it as a character vector or string scalar. For example, if the response variable is stored as Tbl.Y, then specify ResponseVarName as 'Y'. Otherwise, the software treats all columns of Tbl, including Tbl.Y, as predictors.

The response variable must be a categorical, character, or string array, a logical or numeric vector, or a cell array of character vectors. If the response variable is a character array, then each element must correspond to one row of the array.

Data Types: char | string

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.

'ObservationsIn' — Predictor data observation dimension

'rows' (default) | 'columns'

Predictor data observation dimension, specified as the comma-separated pair consisting of 'ObservationsIn' and 'columns' or 'rows'.

Note

If you orient your predictor matrix so that observations correspond to columns and specify 'ObservationsIn','columns', then you might experience a significant reduction in optimization execution time. You cannot specify 'ObservationsIn','columns' for predictor data in a table.

'Weights' — Observation weights

ones(size(X,1),1) (default) | numeric vector | name of variable in Tbl

Observation weights, specified as the comma-separated pair consisting of 'Weights' and a numeric vector or the name of a variable in Tbl.

• If you specify Weights as a numeric vector, then the size of Weights must be equal to the number of observations in X or Tbl.

• If you specify Weights as the name of a variable in Tbl, then the name must be a character vector or string scalar. For example, if the weights are stored as Tbl.W, then specify Weights as 'W'. Otherwise, the software treats all columns of Tbl, including Tbl.W, as predictors.

If you supply weights, then for each regularization strength, edge computes the weighted classification edge and normalizes weights to sum up to the value of the prior probability in the respective class.

Data Types: double | single

Output Arguments

e — Classification edges

numeric scalar | numeric row vector

Classification edges, returned as a numeric scalar or row vector.

e is the same size as Mdl.Lambda. e(j) is the classification edge of the linear classification model trained using the regularization strength Mdl.Lambda(j).

Examples

Estimate Test-Sample Edge

Load the NLP data set.

load nlpdata

X is a sparse matrix of predictor data, and Y is a categorical vector of class labels. There are more than two classes in the data.

The models should identify whether the word counts in a web page are from the Statistics and Machine Learning Toolbox™ documentation. So, identify the labels that correspond to the Statistics and Machine Learning Toolbox™ documentation web pages.

Ystats = Y == 'stats';

Train a binary, linear classification model that can identify whether the word counts in a documentation web page are from the Statistics and Machine Learning Toolbox™ documentation. Specify to holdout 30% of the observations. Optimize the objective function using SpaRSA.

rng(1); % For reproducibility 
CVMdl = fitclinear(X,Ystats,'Solver','sparsa','Holdout',0.30);
CMdl = CVMdl.Trained{1};

CVMdl is a ClassificationPartitionedLinear model. It contains the property Trained, which is a 1-by-1 cell array holding a ClassificationLinear model that the software trained using the training set.

Extract the training and test data from the partition definition.

trainIdx = training(CVMdl.Partition);
testIdx = test(CVMdl.Partition);

Estimate the training- and test-sample edges.

eTrain = edge(CMdl,X(trainIdx,:),Ystats(trainIdx))

eTrain = 15.6660

eTest = edge(CMdl,X(testIdx,:),Ystats(testIdx))

eTest = 15.4767

Feature Selection Using Test-Sample Edges

One way to perform feature selection is to compare test-sample edges from multiple models. Based solely on this criterion, the classifier with the highest edge is the best classifier.

Load the NLP data set.

load nlpdata

X is a sparse matrix of predictor data, and Y is a categorical vector of class labels. There are more than two classes in the data.

The models should identify whether the word counts in a web page are from the Statistics and Machine Learning Toolbox™ documentation. So, identify the labels that correspond to the Statistics and Machine Learning Toolbox™ documentation web pages. For quicker execution time, orient the predictor data so that individual observations correspond to columns.

Ystats = Y == 'stats';
X = X';
rng(1); % For reproducibility

Create a data partition which holds out 30% of the observations for testing.

Partition = cvpartition(Ystats,'Holdout',0.30);
testIdx = test(Partition); % Test-set indices
XTest = X(:,testIdx);     
YTest = Ystats(testIdx);

Partition is a cvpartition object that defines the data set partition.

Randomly choose half of the predictor variables.

p = size(X,1); % Number of predictors
idxPart = randsample(p,ceil(0.5*p));

Train two binary, linear classification models: one that uses the all of the predictors and one that uses half of the predictors. Optimize the objective function using SpaRSA, and indicate that observations correspond to columns.

CVMdl = fitclinear(X,Ystats,'CVPartition',Partition,'Solver','sparsa',...
    'ObservationsIn','columns');
PCVMdl = fitclinear(X(idxPart,:),Ystats,'CVPartition',Partition,'Solver','sparsa',...
    'ObservationsIn','columns');

CVMdl and PCVMdl are ClassificationPartitionedLinear models.

Extract the trained ClassificationLinear models from the cross-validated models.

CMdl = CVMdl.Trained{1};
PCMdl = PCVMdl.Trained{1};

Estimate the test sample edge for each classifier.

fullEdge = edge(CMdl,XTest,YTest,'ObservationsIn','columns')

fullEdge = 15.4767

partEdge = edge(PCMdl,XTest(idxPart,:),YTest,'ObservationsIn','columns')

partEdge = 13.4458

Based on the test-sample edges, the classifier that uses all of the predictors is the better model.

Find Good Lasso Penalty Using Edge

To determine a good lasso-penalty strength for a linear classification model that uses a logistic regression learner, compare test-sample edges.

Load the NLP data set. Preprocess the data as in Feature Selection Using Test-Sample Edges.

load nlpdata
Ystats = Y == 'stats';
X = X'; 

Partition = cvpartition(Ystats,'Holdout',0.30);
testIdx = test(Partition);
XTest = X(:,testIdx);
YTest = Ystats(testIdx);

Create a set of 11 logarithmically-spaced regularization strengths from $$10^{-8}$$ through $$10^1$$.

Lambda = logspace(-8,1,11);

Train binary, linear classification models that use each of the regularization strengths. Optimize the objective function using SpaRSA. Lower the tolerance on the gradient of the objective function to 1e-8.

rng(10); % For reproducibility
CVMdl = fitclinear(X,Ystats,'ObservationsIn','columns',...
    'CVPartition',Partition,'Learner','logistic','Solver','sparsa',...
    'Regularization','lasso','Lambda',Lambda,'GradientTolerance',1e-8)

CVMdl = 
  ClassificationPartitionedLinear
    CrossValidatedModel: 'Linear'
           ResponseName: 'Y'
        NumObservations: 31572
                  KFold: 1
              Partition: [1x1 cvpartition]
             ClassNames: [0 1]
         ScoreTransform: 'none'


  Properties, Methods

Extract the trained linear classification model.

Mdl = CVMdl.Trained{1}

Mdl = 
  ClassificationLinear
      ResponseName: 'Y'
        ClassNames: [0 1]
    ScoreTransform: 'logit'
              Beta: [34023x11 double]
              Bias: [1x11 double]
            Lambda: [1x11 double]
           Learner: 'logistic'


  Properties, Methods

Mdl is a ClassificationLinear model object. Because Lambda is a sequence of regularization strengths, you can think of Mdl as 11 models, one for each regularization strength in Lambda.

Estimate the test-sample edges.

e = edge(Mdl,X(:,testIdx),Ystats(testIdx),'ObservationsIn','columns')

e = 1×11

    0.9986    0.9986    0.9986    0.9986    0.9986    0.9932    0.9765    0.9205    0.8332    0.8128    0.8128

Because there are 11 regularization strengths, e is a 1-by-11 vector of edges.

Plot the test-sample edges for each regularization strength. Identify the regularization strength that maximizes the edges over the grid.

figure;
plot(log10(Lambda),log10(e),'-o')
[~, maxEIdx] = max(e);
maxLambda = Lambda(maxEIdx);
hold on
plot(log10(maxLambda),log10(e(maxEIdx)),'ro');
ylabel('log_{10} test-sample edge')
xlabel('log_{10} Lambda')
legend('Edge','Max edge')
hold off

Several values of Lambda yield similarly high edges. Higher values of lambda lead to predictor variable sparsity, which is a good quality of a classifier.

Choose the regularization strength that occurs just before the edge starts decreasing.

LambdaFinal = Lambda(5);

Train a linear classification model using the entire data set and specify the regularization strength yielding the maximal edge.

MdlFinal = fitclinear(X,Ystats,'ObservationsIn','columns',...
    'Learner','logistic','Solver','sparsa','Regularization','lasso',...
    'Lambda',LambdaFinal);

To estimate labels for new observations, pass MdlFinal and the new data to predict.

More About

Classification Edge

The classification edge is the weighted mean of the classification margins.

One way to choose among multiple classifiers, for example to perform feature selection, is to choose the classifier that yields the greatest edge.

Classification Margin

The classification margin for binary classification is, for each observation, the difference between the classification score for the true class and the classification score for the false class.

The software defines the classification margin for binary classification as

$$m=2yf\left(x\right).$$

x is an observation. If the true label of x is the positive class, then y is 1, and –1 otherwise. f(x) is the positive-class classification score for the observation x. The classification margin is commonly defined as m = yf(x).

If the margins are on the same scale, then they serve as a classification confidence measure. Among multiple classifiers, those that yield greater margins are better.

Classification Score

For linear classification models, the raw classification score for classifying the observation x, a row vector, into the positive class is defined by

$$f_j(x)=x{\beta }_j+b_j.$$

For the model with regularization strength j, $${\beta }_j$$ is the estimated column vector of coefficients (the model property Beta(:,j)) and $$b_j$$ is the estimated, scalar bias (the model property Bias(j)).

The raw classification score for classifying x into the negative class is –f(x). The software classifies observations into the class that yields the positive score.

If the linear classification model consists of logistic regression learners, then the software applies the 'logit' score transformation to the raw classification scores (see ScoreTransform).

Algorithms

By default, observation weights are prior class probabilities. If you supply weights using Weights, then the software normalizes them to sum to the prior probabilities in the respective classes. The software uses the normalized weights to estimate the weighted edge.

Extended Capabilities

Tall Arrays

Calculate with arrays that have more rows than fit in memory.

Usage notes and limitations:

• edge does not support tall table data.

For more information, see Tall Arrays.