resubLoss

Class: RegressionSVM

Resubstitution loss for support vector machine regression model

Syntax

L = resubLoss(mdl)
L = resubLoss(mdl,Name,Value)


Description

L = resubLoss(mdl) returns the resubstitution loss for the support vector machine (SVM) regression model mdl, using the training data stored in mdl.X and corresponding response values stored in mdl.Y.

L = resubLoss(mdl,Name,Value) returns the resubstitution loss with additional options specified by one or more Name,Value pair arguments. For example, you can specify the loss function or observation weights.

Input Arguments

mdl — Full, trained SVM regression model

RegressionSVM model

Full, trained SVM regression model, specified as a RegressionSVM model returned by fitrsvm.

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.

'LossFun' — Loss function

'mse' (default) | 'epsiloninsensitive' | function handle

Loss function, specified as the comma-separated pair consisting of 'LossFun' and 'mse', 'epsiloninsensitive', or a function handle.

• The following table lists the available loss functions. Specify one using its corresponding value.

  Value	Loss Function
  'mse'	Mean Squared Error
  'epsiloninsensitive'	Epsilon-Insensitive Loss Function

• Specify your own function using function handle notation.

  Suppose that n = size(X,1) is the sample size. Your function must have the signature lossvalue = lossfun(Y,Yfit,W), where:

  • The output argument lossvalue is a numeric value.

  • You choose the function name (lossfun).

  • Y is an n-by-1 numeric vector of observed response values.

  • Yfit is an n-by-1 numeric vector of predicted response values, calculated using the corresponding predictor values in X (similar to the output of predict).

  • W is an n-by-1 numeric vector of observation weights.

  Specify your function using 'LossFun',@lossfun.

Example: 'LossFun','epsiloninsensitive'

Data Types: char | string | function_handle

'Weights' — Observation weights

ones(size(X,1),1) (default) | numeric vector

Observation weights, specified as the comma-separated pair consisting of 'Weights' and a numeric vector. Weights must be the same length as the number of rows in X. The software weighs the observations in each row of X using the corresponding weight value in Weights.

Data Types: single | double

Output Arguments

L — Resubstitution loss

scalar value

Resubstitution loss, returned as a scalar value.

The resubstitution loss is the loss calculated between the response training data and the model’s predicted response values based on the input training data.

Resubstitution loss can be an overly optimistic estimate of the predictive error on new data. If the resubstitution loss is high, the model’s predictions are not likely to be very good. However, having a low resubstitution loss does not guarantee good predictions for new data.

To better assess the predictive accuracy of your model, cross validate the model using crossval.

Examples

Resubstitution Loss for SVM Regression Model

This example shows how to train an SVM regression model, then calculate the resubstitution loss using mean square error (MSE) and epsilon-insensitive loss.

This example uses the abalone data from the UCI Machine Learning Repository. Download the data and save it in your current directory with the name 'abalone.data'.

Read the data into a table.

tbl = readtable('abalone.data','Filetype','text','ReadVariableNames',false);
rng default  % for reproducibility

The sample data contains 4177 observations. All of the predictor variables are continuous except for sex, which is a categorical variable with possible values 'M' (for males), 'F' (for females), and 'I' (for infants). The goal is to predict the number of rings on the abalone, and thereby determine its age, using physical measurements.

Train an SVM regression model to the data, using a Gaussian kernel function with an automatic kernel scale. Standardize the data.

mdl = fitrsvm(tbl,'Var9','KernelFunction','gaussian','KernelScale','auto','Standardize',true);

Calculate the resubstitution loss using mean square error (MSE).

mse_loss = resubLoss(mdl)

mse_loss =

    4.0603

Calculate the epsilon-insensitive loss.

eps_loss = resubLoss(mdl,'LossFun','epsiloninsensitive')

eps_loss =

    1.1027

More About

Mean Squared Error

The weighted mean squared error is calculated as follows:

$$\text{mse}=\frac{{\displaystyle \sum _{j=1}^n{w}_j{\left(f\left(x_j\right)-y_j\right)}^2}}{{\displaystyle \sum _{j=1}^n{w}_j}},$$

where:

• n is the number of rows of data

• x_j is the jth row of data

• y_j is the true response to x_j

• f(x_j) is the response prediction of the SVM regression model mdl to x_j

• w is the vector of weights.

The weights in w are all equal to one by default. You can specify different values for weights using the 'Weights' name-value pair argument. If you specify weights, each value is divided by the sum of all weights, such that the normalized weights add to one.

Epsilon-Insensitive Loss Function

The epsilon-insensitive loss function ignores errors that are within the distance epsilon (ε) of the function value. It is formally described as:

$$Los{s}_{\epsilon }=\{\begin{array}{c}0,if\left|y-f\left(x\right)\right|\le \epsilon \\ \left|y-f\left(x\right)\right|-\epsilon ,otherwise.\end{array}$$

The mean epsilon-insensitive loss is calculated as follows:

$$Loss=\frac{{\displaystyle \sum _{j=1}^n{w}_j\max \left(0,\left|y_j-f\left(x_j\right)\right|-\epsilon \right)}}{{\displaystyle \sum _{j=1}^n{w}_j}},$$

References

[1] Nash, W.J., T. L. Sellers, S. R. Talbot, A. J. Cawthorn, and W. B. Ford. "The Population Biology of Abalone (Haliotis species) in Tasmania. I. Blacklip Abalone (H. rubra) from the North Coast and Islands of Bass Strait." Sea Fisheries Division, Technical Report No. 48, 1994.

[2] Waugh, S. "Extending and Benchmarking Cascade-Correlation: Extensions to the Cascade-Correlation Architecture and Benchmarking of Feed-forward Supervised Artificial Neural Networks." University of Tasmania Department of Computer Science thesis, 1995.

[3] Clark, D., Z. Schreter, A. Adams. "A Quantitative Comparison of Dystal and Backpropagation." submitted to the Australian Conference on Neural Networks, 1996.

[4] Lichman, M. UCI Machine Learning Repository, [http://archive.ics.uci.edu/ml]. Irvine, CA: University of California, School of Information and Computer Science.