# ClassificationKernel

Gaussian kernel classification model using random feature expansion

## Description

`ClassificationKernel` is a trained model object for a binary Gaussian kernel classification model using random feature expansion. `ClassificationKernel` is more practical for big data applications that have large training sets but can also be applied to smaller data sets that fit in memory.

Unlike other classification models, and for economical memory usage, `ClassificationKernel` model objects do not store the training data. However, they do store information such as the number of dimensions of the expanded space, the kernel scale parameter, prior-class probabilities, and the regularization strength.

You can use trained `ClassificationKernel` models to continue training using the training data and to predict labels or classification scores for new data. For details, see `resume` and `predict`.

## Creation

Create a `ClassificationKernel` object using the `fitckernel` function. This function maps data in a low-dimensional space into a high-dimensional space, then fits a linear model in the high-dimensional space by minimizing the regularized objective function. The linear model in the high-dimensional space is equivalent to the model with a Gaussian kernel in the low-dimensional space. Available linear classification models include regularized support vector machine (SVM) and logistic regression models.

## Properties

expand all

### Kernel Classification Properties

Linear classification model type, specified as `'logistic'` or `'svm'`.

In the following table, $f\left(x\right)=T\left(x\right)\beta +b.$

• x is an observation (row vector) from p predictor variables.

• $T\left(·\right)$ is a transformation of an observation (row vector) for feature expansion. T(x) maps x in ${ℝ}^{p}$ to a high-dimensional space (${ℝ}^{m}$).

• β is a vector of m coefficients.

• b is the scalar bias.

ValueAlgorithmLoss Function`FittedLoss` Value
`'logistic'`Logistic regressionDeviance (logistic): $\ell \left[y,f\left(x\right)\right]=\mathrm{log}\left\{1+\mathrm{exp}\left[-yf\left(x\right)\right]\right\}$`'logit'`
`'svm'`Support vector machineHinge: $\ell \left[y,f\left(x\right)\right]=\mathrm{max}\left[0,1-yf\left(x\right)\right]$`'hinge'`

Number of dimensions of the expanded space, specified as a positive integer.

Data Types: `single` | `double`

Kernel scale parameter, specified as a positive scalar.

Data Types: `char` | `single` | `double`

Box constraint, specified as a positive scalar.

Data Types: `double` | `single`

Regularization term strength, specified as a nonnegative scalar.

Data Types: `single` | `double`

Loss function used to fit the linear model, specified as `'hinge'` or `'logit'`.

ValueAlgorithmLoss Function`Learner` Value
`'hinge'`Support vector machineHinge: $\ell \left[y,f\left(x\right)\right]=\mathrm{max}\left[0,1-yf\left(x\right)\right]$`'svm'`
`'logit'`Logistic regressionDeviance (logistic): $\ell \left[y,f\left(x\right)\right]=\mathrm{log}\left\{1+\mathrm{exp}\left[-yf\left(x\right)\right]\right\}$`'logistic'`

Complexity penalty type, which is always ```'ridge (L2)'```.

The software composes the objective function for minimization from the sum of the average loss function (see `FittedLoss`) and the regularization term, ridge (L2) penalty.

The ridge (L2) penalty is

`$\frac{\lambda }{2}\sum _{j=1}^{p}{\beta }_{j}^{2}$`

where λ specifies the regularization term strength (see `Lambda`). The software excludes the bias term (β0) from the regularization penalty.

### Other Classification Properties

Categorical predictor indices, specified as a vector of positive integers. `CategoricalPredictors` contains index values corresponding to the columns of the predictor data that contain categorical predictors. If none of the predictors are categorical, then this property is empty (`[]`).

Data Types: `single` | `double`

Unique class labels used in training, specified as a categorical or character array, logical or numeric vector, or cell array of character vectors. `ClassNames` has the same data type as the class labels `Y`. (The software treats string arrays as cell arrays of character vectors.) `ClassNames` also determines the class order.

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

Misclassification costs, specified as a square numeric matrix. `Cost` has K rows and columns, where K is the number of classes.

`Cost(i,j)` is the cost of classifying a point into class `j` if its true class is `i`. The order of the rows and columns of `Cost` corresponds to the order of the classes in `ClassNames`.

Data Types: `double`

Parameters used for training the `ClassificationKernel` model, specified as a structure.

Access fields of `ModelParameters` using dot notation. For example, access the relative tolerance on the linear coefficients and the bias term by using `Mdl.ModelParameters.BetaTolerance`.

Data Types: `struct`

Predictor names in order of their appearance in the predictor data, specified as a cell array of character vectors. The length of `PredictorNames` is equal to the number of columns used as predictor variables in the training data `X` or `Tbl`.

Data Types: `cell`

Expanded predictor names, specified as a cell array of character vectors.

If the model uses encoding for categorical variables, then `ExpandedPredictorNames` includes the names that describe the expanded variables. Otherwise, `ExpandedPredictorNames` is the same as `PredictorNames`.

Data Types: `cell`

Prior class probabilities, specified as a numeric vector. `Prior` has as many elements as classes in `ClassNames`, and the order of the elements corresponds to the elements of `ClassNames`.

Data Types: `double`

Response variable name, specified as a character vector.

Data Types: `char`

Score transformation function to apply to predicted scores, specified as a function name or function handle.

For kernel classification models and before the score transformation, the predicted classification score for the observation x (row vector) is $f\left(x\right)=T\left(x\right)\beta +b.$

• $T\left(·\right)$ is a transformation of an observation for feature expansion.

• β is the estimated column vector of coefficients.

• b is the estimated scalar bias.

To change the score transformation function to `function`, for example, use dot notation.

• For a built-in function, enter this code and replace `function` with a value from the table.

`Mdl.ScoreTransform = 'function';`

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 that you define, enter its function handle.

`Mdl.ScoreTransform = @function;`

`function` must accept a matrix of the original scores for each class, and then return a matrix of the same size representing the transformed scores for each class.

Data Types: `char` | `function_handle`

## Object Functions

 `edge` Classification edge for Gaussian kernel classification model `loss` Classification loss for Gaussian kernel classification model `margin` Classification margins for Gaussian kernel classification model `partialDependence` Compute partial dependence `plotPartialDependence` Create partial dependence plot (PDP) and individual conditional expectation (ICE) plots `predict` Predict labels for Gaussian kernel classification model `resume` Resume training of Gaussian kernel classification model

## Examples

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Train a binary kernel classification model using SVM.

Load the `ionosphere` data set. This data set has 34 predictors and 351 binary responses for radar returns, either bad (`'b'`) or good (`'g'`).

```load ionosphere [n,p] = size(X)```
```n = 351 ```
```p = 34 ```
`resp = unique(Y)`
```resp = 2x1 cell {'b'} {'g'} ```

Train a binary kernel classification model that identifies whether the radar return is bad (`'b'`) or good (`'g'`). Extract a fit summary to determine how well the optimization algorithm fits the model to the data.

```rng('default') % For reproducibility [Mdl,FitInfo] = fitckernel(X,Y)```
```Mdl = ClassificationKernel ResponseName: 'Y' ClassNames: {'b' 'g'} Learner: 'svm' NumExpansionDimensions: 2048 KernelScale: 1 Lambda: 0.0028 BoxConstraint: 1 Properties, Methods ```
```FitInfo = struct with fields: Solver: 'LBFGS-fast' LossFunction: 'hinge' Lambda: 0.0028 BetaTolerance: 1.0000e-04 GradientTolerance: 1.0000e-06 ObjectiveValue: 0.2604 GradientMagnitude: 0.0028 RelativeChangeInBeta: 8.2512e-05 FitTime: 0.1155 History: [] ```

`Mdl` is a `ClassificationKernel` model. To inspect the in-sample classification error, you can pass `Mdl` and the training data or new data to the `loss` function. Or, you can pass `Mdl` and new predictor data to the `predict` function to predict class labels for new observations. You can also pass `Mdl` and the training data to the `resume` function to continue training.

`FitInfo` is a structure array containing optimization information. Use `FitInfo` to determine whether optimization termination measurements are satisfactory.

For better accuracy, you can increase the maximum number of optimization iterations (`'IterationLimit'`) and decrease the tolerance values (`'BetaTolerance'` and `'GradientTolerance'`) by using the name-value pair arguments. Doing so can improve measures like `ObjectiveValue` and `RelativeChangeInBeta` in `FitInfo`. You can also optimize model parameters by using the `'OptimizeHyperparameters'` name-value pair argument.

Load the `ionosphere` data set. This data set has 34 predictors and 351 binary responses for radar returns, either bad (`'b'`) or good (`'g'`).

`load ionosphere`

Partition the data set into training and test sets. Specify a 20% holdout sample for the test set.

```rng('default') % For reproducibility Partition = cvpartition(Y,'Holdout',0.20); trainingInds = training(Partition); % Indices for the training set XTrain = X(trainingInds,:); YTrain = Y(trainingInds); testInds = test(Partition); % Indices for the test set XTest = X(testInds,:); YTest = Y(testInds);```

Train a binary kernel classification model that identifies whether the radar return is bad (`'b'`) or good (`'g'`).

`Mdl = fitckernel(XTrain,YTrain,'IterationLimit',5,'Verbose',1);`
```|=================================================================================================================| | Solver | Pass | Iteration | Objective | Step | Gradient | Relative | sum(beta~=0) | | | | | | | magnitude | change in Beta | | |=================================================================================================================| | LBFGS | 1 | 0 | 1.000000e+00 | 0.000000e+00 | 2.811388e-01 | | 0 | | LBFGS | 1 | 1 | 7.585395e-01 | 4.000000e+00 | 3.594306e-01 | 1.000000e+00 | 2048 | | LBFGS | 1 | 2 | 7.160994e-01 | 1.000000e+00 | 2.028470e-01 | 6.923988e-01 | 2048 | | LBFGS | 1 | 3 | 6.825272e-01 | 1.000000e+00 | 2.846975e-02 | 2.388909e-01 | 2048 | | LBFGS | 1 | 4 | 6.699435e-01 | 1.000000e+00 | 1.779359e-02 | 1.325304e-01 | 2048 | | LBFGS | 1 | 5 | 6.535619e-01 | 1.000000e+00 | 2.669039e-01 | 4.112952e-01 | 2048 | |=================================================================================================================| ```

`Mdl` is a `ClassificationKernel` model.

Predict the test-set labels, construct a confusion matrix for the test set, and estimate the classification error for the test set.

```label = predict(Mdl,XTest); ConfusionTest = confusionchart(YTest,label);```

`L = loss(Mdl,XTest,YTest)`
```L = 0.3594 ```

`Mdl` misclassifies all bad radar returns as good returns.

Continue training by using `resume`. This function continues training with the same options used for training `Mdl`.

`UpdatedMdl = resume(Mdl,XTrain,YTrain);`
```|=================================================================================================================| | Solver | Pass | Iteration | Objective | Step | Gradient | Relative | sum(beta~=0) | | | | | | | magnitude | change in Beta | | |=================================================================================================================| | LBFGS | 1 | 0 | 6.535619e-01 | 0.000000e+00 | 2.669039e-01 | | 2048 | | LBFGS | 1 | 1 | 6.132547e-01 | 1.000000e+00 | 6.355537e-03 | 1.522092e-01 | 2048 | | LBFGS | 1 | 2 | 5.938316e-01 | 4.000000e+00 | 3.202847e-02 | 1.498036e-01 | 2048 | | LBFGS | 1 | 3 | 4.169274e-01 | 1.000000e+00 | 1.530249e-01 | 7.234253e-01 | 2048 | | LBFGS | 1 | 4 | 3.679212e-01 | 5.000000e-01 | 2.740214e-01 | 2.495886e-01 | 2048 | | LBFGS | 1 | 5 | 3.332261e-01 | 1.000000e+00 | 1.423488e-02 | 9.558680e-02 | 2048 | | LBFGS | 1 | 6 | 3.235335e-01 | 1.000000e+00 | 7.117438e-03 | 7.137260e-02 | 2048 | | LBFGS | 1 | 7 | 3.112331e-01 | 1.000000e+00 | 6.049822e-02 | 1.252157e-01 | 2048 | | LBFGS | 1 | 8 | 2.972144e-01 | 1.000000e+00 | 7.117438e-03 | 5.796240e-02 | 2048 | | LBFGS | 1 | 9 | 2.837450e-01 | 1.000000e+00 | 8.185053e-02 | 1.484733e-01 | 2048 | | LBFGS | 1 | 10 | 2.797642e-01 | 1.000000e+00 | 3.558719e-02 | 5.856842e-02 | 2048 | | LBFGS | 1 | 11 | 2.771280e-01 | 1.000000e+00 | 2.846975e-02 | 2.349433e-02 | 2048 | | LBFGS | 1 | 12 | 2.741570e-01 | 1.000000e+00 | 3.914591e-02 | 3.113194e-02 | 2048 | | LBFGS | 1 | 13 | 2.725701e-01 | 5.000000e-01 | 1.067616e-01 | 8.729821e-02 | 2048 | | LBFGS | 1 | 14 | 2.667147e-01 | 1.000000e+00 | 3.914591e-02 | 3.491723e-02 | 2048 | | LBFGS | 1 | 15 | 2.621152e-01 | 1.000000e+00 | 7.117438e-03 | 5.104726e-02 | 2048 | | LBFGS | 1 | 16 | 2.601652e-01 | 1.000000e+00 | 3.558719e-02 | 3.764904e-02 | 2048 | | LBFGS | 1 | 17 | 2.589052e-01 | 1.000000e+00 | 3.202847e-02 | 3.655744e-02 | 2048 | | LBFGS | 1 | 18 | 2.583185e-01 | 1.000000e+00 | 7.117438e-03 | 6.490571e-02 | 2048 | | LBFGS | 1 | 19 | 2.556482e-01 | 1.000000e+00 | 9.252669e-02 | 4.601390e-02 | 2048 | | LBFGS | 1 | 20 | 2.542643e-01 | 1.000000e+00 | 7.117438e-02 | 4.141838e-02 | 2048 | |=================================================================================================================| | Solver | Pass | Iteration | Objective | Step | Gradient | Relative | sum(beta~=0) | | | | | | | magnitude | change in Beta | | |=================================================================================================================| | LBFGS | 1 | 21 | 2.532117e-01 | 1.000000e+00 | 1.067616e-02 | 1.661720e-02 | 2048 | | LBFGS | 1 | 22 | 2.529890e-01 | 1.000000e+00 | 2.135231e-02 | 1.231678e-02 | 2048 | | LBFGS | 1 | 23 | 2.523232e-01 | 1.000000e+00 | 3.202847e-02 | 1.958586e-02 | 2048 | | LBFGS | 1 | 24 | 2.506736e-01 | 1.000000e+00 | 1.779359e-02 | 2.474613e-02 | 2048 | | LBFGS | 1 | 25 | 2.501995e-01 | 1.000000e+00 | 1.779359e-02 | 2.514352e-02 | 2048 | | LBFGS | 1 | 26 | 2.488242e-01 | 1.000000e+00 | 3.558719e-03 | 1.531810e-02 | 2048 | | LBFGS | 1 | 27 | 2.485295e-01 | 5.000000e-01 | 3.202847e-02 | 1.229760e-02 | 2048 | | LBFGS | 1 | 28 | 2.482244e-01 | 1.000000e+00 | 4.270463e-02 | 8.970983e-03 | 2048 | | LBFGS | 1 | 29 | 2.479714e-01 | 1.000000e+00 | 3.558719e-03 | 7.393900e-03 | 2048 | | LBFGS | 1 | 30 | 2.477316e-01 | 1.000000e+00 | 3.202847e-02 | 3.268087e-03 | 2048 | | LBFGS | 1 | 31 | 2.476178e-01 | 2.500000e-01 | 3.202847e-02 | 5.445890e-03 | 2048 | | LBFGS | 1 | 32 | 2.474874e-01 | 1.000000e+00 | 1.779359e-02 | 3.535903e-03 | 2048 | | LBFGS | 1 | 33 | 2.473980e-01 | 1.000000e+00 | 7.117438e-03 | 2.821725e-03 | 2048 | | LBFGS | 1 | 34 | 2.472935e-01 | 1.000000e+00 | 3.558719e-03 | 2.699880e-03 | 2048 | | LBFGS | 1 | 35 | 2.471418e-01 | 1.000000e+00 | 3.558719e-03 | 1.242523e-02 | 2048 | | LBFGS | 1 | 36 | 2.469862e-01 | 1.000000e+00 | 2.846975e-02 | 7.895605e-03 | 2048 | | LBFGS | 1 | 37 | 2.469598e-01 | 1.000000e+00 | 2.135231e-02 | 6.657676e-03 | 2048 | | LBFGS | 1 | 38 | 2.466941e-01 | 1.000000e+00 | 3.558719e-02 | 4.654690e-03 | 2048 | | LBFGS | 1 | 39 | 2.466660e-01 | 5.000000e-01 | 1.423488e-02 | 2.885769e-03 | 2048 | | LBFGS | 1 | 40 | 2.465605e-01 | 1.000000e+00 | 3.558719e-03 | 4.562565e-03 | 2048 | |=================================================================================================================| | Solver | Pass | Iteration | Objective | Step | Gradient | Relative | sum(beta~=0) | | | | | | | magnitude | change in Beta | | |=================================================================================================================| | LBFGS | 1 | 41 | 2.465362e-01 | 1.000000e+00 | 1.423488e-02 | 5.652180e-03 | 2048 | | LBFGS | 1 | 42 | 2.463528e-01 | 1.000000e+00 | 3.558719e-03 | 2.389759e-03 | 2048 | | LBFGS | 1 | 43 | 2.463207e-01 | 1.000000e+00 | 1.511170e-03 | 3.738286e-03 | 2048 | | LBFGS | 1 | 44 | 2.462585e-01 | 5.000000e-01 | 7.117438e-02 | 2.321693e-03 | 2048 | | LBFGS | 1 | 45 | 2.461742e-01 | 1.000000e+00 | 7.117438e-03 | 2.599725e-03 | 2048 | | LBFGS | 1 | 46 | 2.461434e-01 | 1.000000e+00 | 3.202847e-02 | 3.186923e-03 | 2048 | | LBFGS | 1 | 47 | 2.461115e-01 | 1.000000e+00 | 7.117438e-03 | 1.530711e-03 | 2048 | | LBFGS | 1 | 48 | 2.460814e-01 | 1.000000e+00 | 1.067616e-02 | 1.811714e-03 | 2048 | | LBFGS | 1 | 49 | 2.460533e-01 | 5.000000e-01 | 1.423488e-02 | 1.012252e-03 | 2048 | | LBFGS | 1 | 50 | 2.460111e-01 | 1.000000e+00 | 1.423488e-02 | 4.166762e-03 | 2048 | | LBFGS | 1 | 51 | 2.459414e-01 | 1.000000e+00 | 1.067616e-02 | 3.271946e-03 | 2048 | | LBFGS | 1 | 52 | 2.458809e-01 | 1.000000e+00 | 1.423488e-02 | 1.846440e-03 | 2048 | | LBFGS | 1 | 53 | 2.458479e-01 | 1.000000e+00 | 1.067616e-02 | 1.180871e-03 | 2048 | | LBFGS | 1 | 54 | 2.458146e-01 | 1.000000e+00 | 1.455008e-03 | 1.422954e-03 | 2048 | | LBFGS | 1 | 55 | 2.457878e-01 | 1.000000e+00 | 7.117438e-03 | 1.880892e-03 | 2048 | | LBFGS | 1 | 56 | 2.457519e-01 | 1.000000e+00 | 2.491103e-02 | 1.074764e-03 | 2048 | | LBFGS | 1 | 57 | 2.457420e-01 | 1.000000e+00 | 7.473310e-02 | 9.511878e-04 | 2048 | | LBFGS | 1 | 58 | 2.457212e-01 | 1.000000e+00 | 3.558719e-03 | 3.718564e-04 | 2048 | | LBFGS | 1 | 59 | 2.457089e-01 | 1.000000e+00 | 4.270463e-02 | 6.237270e-04 | 2048 | | LBFGS | 1 | 60 | 2.457047e-01 | 5.000000e-01 | 1.423488e-02 | 3.647573e-04 | 2048 | |=================================================================================================================| | Solver | Pass | Iteration | Objective | Step | Gradient | Relative | sum(beta~=0) | | | | | | | magnitude | change in Beta | | |=================================================================================================================| | LBFGS | 1 | 61 | 2.456991e-01 | 1.000000e+00 | 1.423488e-02 | 5.666884e-04 | 2048 | | LBFGS | 1 | 62 | 2.456898e-01 | 1.000000e+00 | 1.779359e-02 | 4.697056e-04 | 2048 | | LBFGS | 1 | 63 | 2.456792e-01 | 1.000000e+00 | 1.779359e-02 | 5.984927e-04 | 2048 | | LBFGS | 1 | 64 | 2.456603e-01 | 1.000000e+00 | 1.403782e-03 | 5.414985e-04 | 2048 | | LBFGS | 1 | 65 | 2.456482e-01 | 1.000000e+00 | 3.558719e-03 | 6.506293e-04 | 2048 | | LBFGS | 1 | 66 | 2.456358e-01 | 1.000000e+00 | 1.476262e-03 | 1.284139e-03 | 2048 | | LBFGS | 1 | 67 | 2.456124e-01 | 1.000000e+00 | 3.558719e-03 | 8.636596e-04 | 2048 | | LBFGS | 1 | 68 | 2.455980e-01 | 1.000000e+00 | 1.067616e-02 | 9.861527e-04 | 2048 | | LBFGS | 1 | 69 | 2.455780e-01 | 1.000000e+00 | 1.067616e-02 | 5.102487e-04 | 2048 | | LBFGS | 1 | 70 | 2.455633e-01 | 1.000000e+00 | 3.558719e-03 | 1.228077e-03 | 2048 | | LBFGS | 1 | 71 | 2.455449e-01 | 1.000000e+00 | 1.423488e-02 | 7.864590e-04 | 2048 | | LBFGS | 1 | 72 | 2.455261e-01 | 1.000000e+00 | 3.558719e-02 | 1.090815e-03 | 2048 | | LBFGS | 1 | 73 | 2.455142e-01 | 1.000000e+00 | 1.067616e-02 | 1.701506e-03 | 2048 | | LBFGS | 1 | 74 | 2.455075e-01 | 1.000000e+00 | 1.779359e-02 | 1.504577e-03 | 2048 | | LBFGS | 1 | 75 | 2.455008e-01 | 1.000000e+00 | 3.914591e-02 | 1.144021e-03 | 2048 | | LBFGS | 1 | 76 | 2.454943e-01 | 1.000000e+00 | 2.491103e-02 | 3.015254e-04 | 2048 | | LBFGS | 1 | 77 | 2.454918e-01 | 5.000000e-01 | 3.202847e-02 | 9.837523e-04 | 2048 | | LBFGS | 1 | 78 | 2.454870e-01 | 1.000000e+00 | 1.779359e-02 | 4.328953e-04 | 2048 | | LBFGS | 1 | 79 | 2.454865e-01 | 5.000000e-01 | 3.558719e-03 | 7.126815e-04 | 2048 | | LBFGS | 1 | 80 | 2.454775e-01 | 1.000000e+00 | 5.693950e-02 | 8.992562e-04 | 2048 | |=================================================================================================================| | Solver | Pass | Iteration | Objective | Step | Gradient | Relative | sum(beta~=0) | | | | | | | magnitude | change in Beta | | |=================================================================================================================| | LBFGS | 1 | 81 | 2.454686e-01 | 1.000000e+00 | 1.183730e-03 | 1.590246e-04 | 2048 | | LBFGS | 1 | 82 | 2.454612e-01 | 1.000000e+00 | 2.135231e-02 | 1.389570e-04 | 2048 | | LBFGS | 1 | 83 | 2.454506e-01 | 1.000000e+00 | 3.558719e-03 | 6.162089e-04 | 2048 | | LBFGS | 1 | 84 | 2.454436e-01 | 1.000000e+00 | 1.423488e-02 | 1.877414e-03 | 2048 | | LBFGS | 1 | 85 | 2.454378e-01 | 1.000000e+00 | 1.423488e-02 | 3.370852e-04 | 2048 | | LBFGS | 1 | 86 | 2.454249e-01 | 1.000000e+00 | 1.423488e-02 | 8.133615e-04 | 2048 | | LBFGS | 1 | 87 | 2.454101e-01 | 1.000000e+00 | 1.067616e-02 | 3.872088e-04 | 2048 | | LBFGS | 1 | 88 | 2.453963e-01 | 1.000000e+00 | 1.779359e-02 | 5.670260e-04 | 2048 | | LBFGS | 1 | 89 | 2.453866e-01 | 1.000000e+00 | 1.067616e-02 | 1.444984e-03 | 2048 | | LBFGS | 1 | 90 | 2.453821e-01 | 1.000000e+00 | 7.117438e-03 | 2.457270e-03 | 2048 | | LBFGS | 1 | 91 | 2.453790e-01 | 5.000000e-01 | 6.761566e-02 | 8.228766e-04 | 2048 | | LBFGS | 1 | 92 | 2.453603e-01 | 1.000000e+00 | 2.135231e-02 | 1.084233e-03 | 2048 | | LBFGS | 1 | 93 | 2.453540e-01 | 1.000000e+00 | 2.135231e-02 | 2.060005e-04 | 2048 | | LBFGS | 1 | 94 | 2.453482e-01 | 1.000000e+00 | 1.779359e-02 | 1.560883e-04 | 2048 | | LBFGS | 1 | 95 | 2.453461e-01 | 1.000000e+00 | 1.779359e-02 | 1.614693e-03 | 2048 | | LBFGS | 1 | 96 | 2.453371e-01 | 1.000000e+00 | 3.558719e-02 | 2.145835e-04 | 2048 | | LBFGS | 1 | 97 | 2.453305e-01 | 1.000000e+00 | 4.270463e-02 | 7.602088e-04 | 2048 | | LBFGS | 1 | 98 | 2.453283e-01 | 2.500000e-01 | 2.135231e-02 | 3.422253e-04 | 2048 | | LBFGS | 1 | 99 | 2.453246e-01 | 1.000000e+00 | 3.558719e-03 | 3.872561e-04 | 2048 | | LBFGS | 1 | 100 | 2.453214e-01 | 1.000000e+00 | 3.202847e-02 | 1.732237e-04 | 2048 | |=================================================================================================================| | Solver | Pass | Iteration | Objective | Step | Gradient | Relative | sum(beta~=0) | | | | | | | magnitude | change in Beta | | |=================================================================================================================| | LBFGS | 1 | 101 | 2.453168e-01 | 1.000000e+00 | 1.067616e-02 | 3.065286e-04 | 2048 | | LBFGS | 1 | 102 | 2.453155e-01 | 5.000000e-01 | 4.626335e-02 | 3.402368e-04 | 2048 | | LBFGS | 1 | 103 | 2.453136e-01 | 1.000000e+00 | 1.779359e-02 | 2.215029e-04 | 2048 | | LBFGS | 1 | 104 | 2.453119e-01 | 1.000000e+00 | 3.202847e-02 | 4.142355e-04 | 2048 | | LBFGS | 1 | 105 | 2.453093e-01 | 1.000000e+00 | 1.423488e-02 | 2.186007e-04 | 2048 | | LBFGS | 1 | 106 | 2.453090e-01 | 1.000000e+00 | 2.846975e-02 | 1.338602e-03 | 2048 | | LBFGS | 1 | 107 | 2.453048e-01 | 1.000000e+00 | 1.423488e-02 | 3.208296e-04 | 2048 | | LBFGS | 1 | 108 | 2.453040e-01 | 1.000000e+00 | 3.558719e-02 | 1.294488e-03 | 2048 | | LBFGS | 1 | 109 | 2.452977e-01 | 1.000000e+00 | 1.423488e-02 | 8.328380e-04 | 2048 | | LBFGS | 1 | 110 | 2.452934e-01 | 1.000000e+00 | 2.135231e-02 | 5.149259e-04 | 2048 | | LBFGS | 1 | 111 | 2.452886e-01 | 1.000000e+00 | 1.779359e-02 | 3.650664e-04 | 2048 | | LBFGS | 1 | 112 | 2.452854e-01 | 1.000000e+00 | 1.067616e-02 | 2.633981e-04 | 2048 | | LBFGS | 1 | 113 | 2.452836e-01 | 1.000000e+00 | 1.067616e-02 | 1.804300e-04 | 2048 | | LBFGS | 1 | 114 | 2.452817e-01 | 1.000000e+00 | 7.117438e-03 | 4.251642e-04 | 2048 | | LBFGS | 1 | 115 | 2.452741e-01 | 1.000000e+00 | 1.779359e-02 | 9.018440e-04 | 2048 | | LBFGS | 1 | 116 | 2.452691e-01 | 1.000000e+00 | 2.135231e-02 | 9.941716e-05 | 2048 | |=================================================================================================================| ```

Predict the test-set labels, construct a confusion matrix for the test set, and estimate the classification error for the test set.

```UpdatedLabel = predict(UpdatedMdl,XTest); UpdatedConfusionTest = confusionchart(YTest,UpdatedLabel);```

`UpdatedL = loss(UpdatedMdl,XTest,YTest)`
```UpdatedL = 0.1284 ```

The classification error decreases after `resume` updates the classification model with more iterations.