# 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

example

````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`.```

example

````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

expand all

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

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`

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`

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.

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`.

Predictor data observation dimension, specified as `'rows'` or `'columns'`.

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. You cannot specify `'ObservationsIn','columns'` for predictor data in a table.

Data Types: `char` | `string`

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

expand all

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

expand all

`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 ```

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 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.

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 $1{0}^{-8}$ through $1{0}^{1}$.

`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`.

expand all

## 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.