predict

Predict responses using neighborhood component analysis (NCA) classifier

Syntax

[labels,postprobs,classnames] = predict(mdl,X)

Description

[labels,postprobs,classnames] = predict(mdl,X) computes the predicted labels, labels, corresponding to the rows of X, using the model mdl.

Examples

Tune NCA Model for Classification

Load the sample data.

load("twodimclassdata.mat")

This data set is simulated using the scheme described in [1]. This is a two-class classification problem in two dimensions. Data from the first class (class –1) are drawn from two bivariate normal distributions $$N({\mu }_1,\Sigma )$$ or $$N({\mu }_2,\Sigma )$$ with equal probability, where $${\mu }_1=[-0.75,-1.5]$$, $${\mu }_2=[0.75,1.5]$$, and $$\Sigma =I_2$$. Similarly, data from the second class (class 1) are drawn from two bivariate normal distributions $$N({\mu }_3,\Sigma )$$ or $$N({\mu }_4,\Sigma )$$ with equal probability, where $${\mu }_3=[1.5,-1.5]$$, $${\mu }_4=[-1.5,1.5]$$, and $$\Sigma =I_2$$. The normal distribution parameters used to create this data set result in tighter clusters in data than the data used in [1].

Create a scatter plot of the data grouped by the class.

gscatter(X(:,1),X(:,2),y)
xlabel("x1")
ylabel("x2")

Add 100 irrelevant features to $$X$$. First generate data from a Normal distribution with a mean of 0 and a variance of 20.

n = size(X,1);
rng("default")
XwithBadFeatures = [X,randn(n,100)*sqrt(20)];

Normalize the data so that all points are between 0 and 1.

XwithBadFeatures = (XwithBadFeatures-min(XwithBadFeatures,[],1))./ ...
    range(XwithBadFeatures,1);
X = XwithBadFeatures;

Fit a neighborhood component analysis (NCA) model to the data using the default Lambda (regularization parameter, $$\lambda$$) value. Use the LBFGS solver and display the convergence information.

ncaMdl = fscnca(X,y,FitMethod="exact",Verbose=1, ...
    Solver="lbfgs");

 o Solver = LBFGS, HessianHistorySize = 15, LineSearchMethod = weakwolfe

|====================================================================================================|
|   ITER   |   FUN VALUE   |  NORM GRAD  |  NORM STEP  |  CURV  |    GAMMA    |    ALPHA    | ACCEPT |
|====================================================================================================|
|        0 |  9.519258e-03 |   1.494e-02 |   0.000e+00 |        |   4.015e+01 |   0.000e+00 |   YES  |
|        1 | -3.093574e-01 |   7.186e-03 |   4.018e+00 |    OK  |   8.956e+01 |   1.000e+00 |   YES  |
|        2 | -4.809455e-01 |   4.444e-03 |   7.123e+00 |    OK  |   9.943e+01 |   1.000e+00 |   YES  |
|        3 | -4.938877e-01 |   3.544e-03 |   1.464e+00 |    OK  |   9.366e+01 |   1.000e+00 |   YES  |
|        4 | -4.964759e-01 |   2.901e-03 |   6.084e-01 |    OK  |   1.554e+02 |   1.000e+00 |   YES  |
|        5 | -4.972077e-01 |   1.323e-03 |   6.129e-01 |    OK  |   1.195e+02 |   5.000e-01 |   YES  |
|        6 | -4.974743e-01 |   1.569e-04 |   2.155e-01 |    OK  |   1.003e+02 |   1.000e+00 |   YES  |
|        7 | -4.974868e-01 |   3.844e-05 |   4.161e-02 |    OK  |   9.835e+01 |   1.000e+00 |   YES  |
|        8 | -4.974874e-01 |   1.417e-05 |   1.073e-02 |    OK  |   1.043e+02 |   1.000e+00 |   YES  |
|        9 | -4.974874e-01 |   4.893e-06 |   1.781e-03 |    OK  |   1.530e+02 |   1.000e+00 |   YES  |
|       10 | -4.974874e-01 |   9.404e-08 |   8.947e-04 |    OK  |   1.670e+02 |   1.000e+00 |   YES  |

         Infinity norm of the final gradient = 9.404e-08
              Two norm of the final step     = 8.947e-04, TolX   = 1.000e-06
Relative infinity norm of the final gradient = 9.404e-08, TolFun = 1.000e-06
EXIT: Local minimum found.

Plot the feature weights. The weights of the irrelevant features should be very close to zero.

semilogx(ncaMdl.FeatureWeights,"o")
xlabel("Feature index")
ylabel("Feature weight")    
grid on

Predict the classes using the NCA model and compute the confusion matrix.

ypred = predict(ncaMdl,X);
confusionchart(y,ypred)

The confusion matrix shows that 40 of the data that are in class –1 are predicted as belonging to class –1, and 60 of the data from class –1 are predicted to be in class 1. Similarly, 94 of the data from class 1 are predicted to be from class 1, and 6 of them are predicted to be from class –1. The prediction accuracy for class –1 is not good.

All weights are very close to zero, which indicates that the value of $$\lambda$$ used in training the model is too large. When $$\lambda \to \infty$$, all features weights approach to zero. Hence, it is important to tune the regularization parameter in most cases to detect the relevant features.

Use five-fold cross-validation to tune $$\lambda$$ for feature selection by using fscnca. Tuning $$\lambda$$ means finding the $$\lambda$$ value that will produce the minimum classification loss. To tune $$\lambda$$ using cross-validation:

1. Partition the data into five folds. For each fold, cvpartition assigns four-fifths of the data as a training set and one-fifth of the data as a test set. Again for each fold, cvpartition creates a stratified partition, where each partition has roughly the same proportion of classes.

cvp = cvpartition(y,"KFold",5);
numtestsets = cvp.NumTestSets;
lambdavalues = linspace(0,2,20)/length(y); 
lossvalues = zeros(length(lambdavalues),numtestsets);

2. Train the neighborhood component analysis (NCA) model for each $$\lambda$$ value using the training set in each fold.

3. Compute the classification loss for the corresponding test set in the fold using the NCA model. Record the loss value.

4. Repeat this process for all folds and all $$\lambda$$ values.

for i = 1:length(lambdavalues)                
    for k = 1:numtestsets
        
        % Extract the training set from the partition object
        Xtrain = X(cvp.training(k),:);
        ytrain = y(cvp.training(k),:);
        
        % Extract the test set from the partition object
        Xtest  = X(cvp.test(k),:);
        ytest  = y(cvp.test(k),:);
        
        % Train an NCA model for classification using the training set
        ncaMdl = fscnca(Xtrain,ytrain,FitMethod="exact", ...
            Solver="lbfgs",Lambda=lambdavalues(i));
        
        % Compute the classification loss for the test set using the NCA
        % model
        lossvalues(i,k) = loss(ncaMdl,Xtest,ytest, ...
            LossFunction="quadratic");   
   
    end                          
end

Plot the average loss values of the folds versus the $$\lambda$$ values. If the $$\lambda$$ value that corresponds to the minimum loss falls on the boundary of the tested $$\lambda$$ values, the range of $$\lambda$$ values should be reconsidered.

plot(lambdavalues,mean(lossvalues,2),"o-")
xlabel("Lambda values")
ylabel("Loss values")
grid on

Find the $$\lambda$$ value that corresponds to the minimum average loss.

[~,idx] = min(mean(lossvalues,2)); % Find the index
bestlambda = lambdavalues(idx) % Find the best lambda value

bestlambda = 
0.0037

Fit the NCA model to all of the data using the best $$\lambda$$ value. Use the LBFGS solver and display the convergence information.

ncaMdl = fscnca(X,y,FitMethod="exact",Verbose=1, ...
    Solver="lbfgs",Lambda=bestlambda);

 o Solver = LBFGS, HessianHistorySize = 15, LineSearchMethod = weakwolfe

|====================================================================================================|
|   ITER   |   FUN VALUE   |  NORM GRAD  |  NORM STEP  |  CURV  |    GAMMA    |    ALPHA    | ACCEPT |
|====================================================================================================|
|        0 | -1.246913e-01 |   1.231e-02 |   0.000e+00 |        |   4.873e+01 |   0.000e+00 |   YES  |
|        1 | -3.411330e-01 |   5.717e-03 |   3.618e+00 |    OK  |   1.068e+02 |   1.000e+00 |   YES  |
|        2 | -5.226111e-01 |   3.763e-02 |   8.252e+00 |    OK  |   7.825e+01 |   1.000e+00 |   YES  |
|        3 | -5.817731e-01 |   8.496e-03 |   2.340e+00 |    OK  |   5.591e+01 |   5.000e-01 |   YES  |
|        4 | -6.132632e-01 |   6.863e-03 |   2.526e+00 |    OK  |   8.228e+01 |   1.000e+00 |   YES  |
|        5 | -6.135264e-01 |   9.373e-03 |   7.341e-01 |    OK  |   3.244e+01 |   1.000e+00 |   YES  |
|        6 | -6.147894e-01 |   1.182e-03 |   2.933e-01 |    OK  |   2.447e+01 |   1.000e+00 |   YES  |
|        7 | -6.148714e-01 |   6.392e-04 |   6.688e-02 |    OK  |   3.195e+01 |   1.000e+00 |   YES  |
|        8 | -6.149524e-01 |   6.521e-04 |   9.934e-02 |    OK  |   1.236e+02 |   1.000e+00 |   YES  |
|        9 | -6.149972e-01 |   1.154e-04 |   1.191e-01 |    OK  |   1.171e+02 |   1.000e+00 |   YES  |
|       10 | -6.149990e-01 |   2.922e-05 |   1.983e-02 |    OK  |   7.365e+01 |   1.000e+00 |   YES  |
|       11 | -6.149993e-01 |   1.556e-05 |   8.354e-03 |    OK  |   1.288e+02 |   1.000e+00 |   YES  |
|       12 | -6.149994e-01 |   1.147e-05 |   7.256e-03 |    OK  |   2.332e+02 |   1.000e+00 |   YES  |
|       13 | -6.149995e-01 |   1.040e-05 |   6.781e-03 |    OK  |   2.287e+02 |   1.000e+00 |   YES  |
|       14 | -6.149996e-01 |   9.015e-06 |   6.265e-03 |    OK  |   9.974e+01 |   1.000e+00 |   YES  |
|       15 | -6.149996e-01 |   7.763e-06 |   5.206e-03 |    OK  |   2.919e+02 |   1.000e+00 |   YES  |
|       16 | -6.149997e-01 |   8.374e-06 |   1.679e-02 |    OK  |   6.878e+02 |   1.000e+00 |   YES  |
|       17 | -6.149997e-01 |   9.387e-06 |   9.542e-03 |    OK  |   1.284e+02 |   5.000e-01 |   YES  |
|       18 | -6.149997e-01 |   3.250e-06 |   5.114e-03 |    OK  |   1.225e+02 |   1.000e+00 |   YES  |
|       19 | -6.149997e-01 |   1.574e-06 |   1.275e-03 |    OK  |   1.808e+02 |   1.000e+00 |   YES  |

|====================================================================================================|
|   ITER   |   FUN VALUE   |  NORM GRAD  |  NORM STEP  |  CURV  |    GAMMA    |    ALPHA    | ACCEPT |
|====================================================================================================|
|       20 | -6.149997e-01 |   5.764e-07 |   6.765e-04 |    OK  |   2.905e+02 |   1.000e+00 |   YES  |

         Infinity norm of the final gradient = 5.764e-07
              Two norm of the final step     = 6.765e-04, TolX   = 1.000e-06
Relative infinity norm of the final gradient = 5.764e-07, TolFun = 1.000e-06
EXIT: Local minimum found.

Plot the feature weights.

semilogx(ncaMdl.FeatureWeights,"o")
xlabel("Feature index")
ylabel("Feature weight")    
grid on

fscnca correctly figures out that the first two features are relevant and that the rest are not. The first two features are not individually informative, but when taken together result in an accurate classification model.

Predict the classes using the new model and compute the accuracy.

ypred = predict(ncaMdl,X);
confusionchart(y,ypred)

Confusion matrix shows that prediction accuracy for class –1 has improved. 88 of the data from class –1 are predicted to be from –1, and 12 of them are predicted to be from class 1. Additionally, 92 of the data from class 1 are predicted to be from class 1, and 8 of them are predicted to be from class –1.

References

[1] Yang, W., K. Wang, W. Zuo. "Neighborhood Component Feature Selection for High-Dimensional Data." Journal of Computers. Vol. 7, Number 1, January, 2012.

Input Arguments

mdl — Neighborhood component analysis model for classification

FeatureSelectionNCAClassification object

Neighborhood component analysis model for classification, specified as a FeatureSelectionNCAClassification object.

X — Predictor variable values

table | n-by-p matrix

Predictor variable values, specified as a table or an n-by-p matrix, where n is the number of observations and p is the number of predictor variables used to train mdl.

By default, each row of X corresponds to one observation, and each column corresponds to one variable.

For a numeric matrix:

• The variables in the columns of X must have the same order as the predictor variables that trained mdl.

• If you train mdl using a table (for example, Tbl), and Tbl contains only numeric predictor variables, then X can be a numeric matrix. To treat numeric predictors in Tbl as categorical during training, identify categorical predictors by using the CategoricalPredictors name-value argument of fscnca. If Tbl contains heterogeneous predictor variables (for example, numeric and categorical data types), and X is a numeric matrix, then predict throws an error.

For a table:

• predict does not support multicolumn variables or cell arrays other than cell arrays of character vectors.

• If you train mdl using a table (for example, Tbl), then all predictor variables in X must have the same variable names and data types as the variables that trained mdl (stored in mdl.PredictorNames). However, the column order of X does not need to correspond to the column order of Tbl. Also, Tbl and X can contain additional variables (response variables, observation weights, and so on), but predict ignores them.

• If you train mdl using a numeric matrix, then the predictor names in mdl.PredictorNames must be the same as the corresponding predictor variable names in X. To specify predictor names during training, use the CategoricalPredictors name-value argument of fscnca. All predictor variables in X must be numeric vectors. X can contain additional variables (response variables, observation weights, and so on), but predict ignores them.

Data Types: table | single | double

Output Arguments

labels — Predicted class labels

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

Predicted class labels corresponding to the rows of X, returned as a categorical, logical, or numeric vector, a cell array of character vectors of length n, or a character array with n rows. n is the number of observations. The type of labels is the same as for ResponseName or Y used in training.

postprobs — Posterior probabilities

n-by-c matrix

Posterior probabilities, returned as an n-by-c matrix, where n is the number of observations and c is the number of classes. A posterior probability, postprobs(i,:), represents the membership of an observation in X(i,:) in classes 1 through c.

classnames — Class names

cell array of character vectors

Class names corresponding to posterior probabilities, returned as a cell array of character vectors. Each character vector is the class name corresponding to a column of postprobs.

Version History

Introduced in R2016b