## Supervised Learning Workflow and Algorithms

### What Is Supervised Learning?

The aim of supervised, machine learning is to build a model that makes predictions based on evidence in the presence of uncertainty. As adaptive algorithms identify patterns in data, a computer "learns" from the observations. When exposed to more observations, the computer improves its predictive performance.

Specifically, a supervised learning algorithm takes a known set of input data and known responses to the data (output), and trains a model to generate reasonable predictions for the response to new data.

For example, suppose you want to predict whether someone will have a heart attack within a year. You have a set of data on previous patients, including age, weight, height, blood pressure, etc. You know whether the previous patients had heart attacks within a year of their measurements. So, the problem is combining all the existing data into a model that can predict whether a new person will have a heart attack within a year.

You can think of the entire set of input data as a heterogeneous matrix. Rows of the matrix are called observations, examples, or instances, and each contain a set of measurements for a subject (patients in the example). Columns of the matrix are called predictors, attributes, or features, and each are variables representing a measurement taken on every subject (age, weight, height, etc. in the example). You can think of the response data as a column vector where each row contains the output of the corresponding observation in the input data (whether the patient had a heart attack). To fit or train a supervised learning model, choose an appropriate algorithm, and then pass the input and response data to it.

Supervised learning splits into two broad categories: classification and regression.

• In classification, the goal is to assign a class (or label) from a finite set of classes to an observation. That is, responses are categorical variables. Applications include spam filters, advertisement recommendation systems, and image and speech recognition. Predicting whether a patient will have a heart attack within a year is a classification problem, and the possible classes are true and false. Classification algorithms usually apply to nominal response values. However, some algorithms can accommodate ordinal classes (see fitcecoc).

• In regression, the goal is to predict a continuous measurement for an observation. That is, the responses variables are real numbers. Applications include forecasting stock prices, energy consumption, or disease incidence.

Statistics and Machine Learning Toolbox™ supervised learning functionalities comprise a stream-lined, object framework. You can efficiently train a variety of algorithms, combine models into an ensemble, assess model performances, cross-validate, and predict responses for new data.

### Steps in Supervised Learning

While there are many Statistics and Machine Learning Toolbox algorithms for supervised learning, most use the same basic workflow for obtaining a predictor model. (Detailed instruction on the steps for ensemble learning is in Framework for Ensemble Learning.) The steps for supervised learning are:

#### Prepare Data

All supervised learning methods start with an input data matrix, usually called X here. Each row of X represents one observation. Each column of X represents one variable, or predictor. Represent missing entries with NaN values in X. Statistics and Machine Learning Toolbox supervised learning algorithms can handle NaN values, either by ignoring them or by ignoring any row with a NaN value.

You can use various data types for response data Y. Each element in Y represents the response to the corresponding row of X. Observations with missing Y data are ignored.

• For regression, Y must be a numeric vector with the same number of elements as the number of rows of X.

• For classification, Y can be any of these data types. This table also contains the method of including missing entries.

Data TypeMissing Entry
Numeric vectorNaN
Categorical vector<undefined>
Character arrayRow of spaces
String array<missing> or ""
Cell array of character vectors''
Logical vector(Cannot represent)

#### Choose an Algorithm

There are tradeoffs between several characteristics of algorithms, such as:

• Speed of training

• Memory usage

• Predictive accuracy on new data

• Transparency or interpretability, meaning how easily you can understand the reasons an algorithm makes its predictions

Details of the algorithms appear in Characteristics of Classification Algorithms. More detail about ensemble algorithms is in Choose an Applicable Ensemble Aggregation Method.

#### Fit a Model

The fitting function you use depends on the algorithm you choose.

AlgorithmFitting Function for ClassificationFitting Function for Regression
Decision treesfitctreefitrtree
Discriminant analysisfitcdiscrNot applicable
Ensembles (for example, Random Forests [1])fitcensemble, TreeBaggerfitrensemble, TreeBagger
Gaussian kernel modelfitckernel (SVM and logistic regression learners)fitrkernel (SVM and least-squares regression learners)
Gaussian process regression (GPR)Not applicablefitrgp
k-nearest neighborsfitcknnNot applicable
Linear modelfitclinear (SVM and logistic regression)fitrlinear (SVM and least-squares regression)
Multiclass, error-correcting output codes (ECOC) model for SVM or other classifiersfitcecocNot applicable
Naive Bayes modelfitcnbNot applicable
Neural network modelfitcnetfitrnet
Support vector machines (SVM)fitcsvmfitrsvm

For a comparison of these algorithms, see Characteristics of Classification Algorithms.

#### Choose a Validation Method

The three main methods to examine the accuracy of the resulting fitted model are:

• Examine the resubstitution error. For examples, see:

• Examine the cross-validation error. For examples, see:

• Examine the out-of-bag error for bagged decision trees. For examples, see:

#### Examine Fit and Update Until Satisfied

After validating the model, you might want to change it for better accuracy, better speed, or to use less memory.

• Change fitting parameters to try to get a more accurate model. For examples, see:

• Change fitting parameters to try to get a smaller model. This sometimes gives a model with more accuracy. For examples, see:

• Try a different algorithm. For applicable choices, see:

When satisfied with a model of some types, you can trim it using the appropriate compact function (compact for classification trees, compact for regression trees, compact for discriminant analysis, compact for naive Bayes, compact for SVM, compact for ECOC models, compact for classification ensembles, and compact for regression ensembles). compact removes training data and other properties not required for prediction, e.g., pruning information for decision trees, from the model to reduce memory consumption. Because kNN classification models require all of the training data to predict labels, you cannot reduce the size of a ClassificationKNN model.

#### Use Fitted Model for Predictions

To predict classification or regression response for most fitted models, use the predict method:

Ypredicted = predict(obj,Xnew)
• obj is the fitted model or fitted compact model.

• Xnew is the new input data.

• Ypredicted is the predicted response, either classification or regression.

### Characteristics of Classification Algorithms

This table shows typical characteristics of the various supervised learning algorithms. The characteristics in any particular case can vary from the listed ones. Use the table as a guide for your initial choice of algorithms. Decide on the tradeoff you want in speed, memory usage, flexibility, and interpretability.

Tip

Try a decision tree or discriminant first, because these classifiers are fast and easy to interpret. If the models are not accurate enough predicting the response, try other classifiers with higher flexibility.

To control flexibility, see the details for each classifier type. To avoid overfitting, look for a model of lower flexibility that provides sufficient accuracy.

ClassifierMulticlass SupportCategorical Predictor SupportPrediction SpeedMemory UsageInterpretability
Decision TreesfitctreeYesYesFastSmallEasy
Discriminant analysisfitcdiscrYesNoFastSmall for linear, large for quadraticEasy
SVMfitcsvmNo.
Combine multiple binary SVM classifiers using fitcecoc.
YesMedium for linear.
Slow for others.
Medium for linear.
All others: medium for multiclass, large for binary.
Easy for linear SVM.
Hard for all other kernel types.
Naive BayesfitcnbYesYesMedium for simple distributions.
Slow for kernel distributions or high-dimensional data
Small for simple distributions.
Medium for kernel distributions or high-dimensional data
Easy
Nearest neighborfitcknnYesYesSlow for cubic.
Medium for others.
MediumHard
Ensemblesfitcensemble and fitrensembleYesYesFast to medium depending on choice of algorithmLow to high depending on choice of algorithm.Hard

The results in this table are based on an analysis of many data sets. The data sets in the study have up to 7000 observations, 80 predictors, and 50 classes. This list defines the terms in the table.

Speed:

• Fast — 0.01 second

• Medium — 1 second

• Slow — 100 seconds

Memory

• Small — 1MB

• Medium — 4MB

• Large — 100MB

Note

The table provides a general guide. Your results depend on your data and the speed of your machine.

#### Categorical Predictor Support

This table describes the data-type support of predictors for each classifier.

ClassifierAll predictors numericAll predictors categoricalSome categorical, some numeric
Decision TreesYesYesYes
Discriminant AnalysisYesNoNo
SVMYesYesYes
Naive BayesYesYesYes
Nearest NeighborEuclidean distance onlyHamming distance onlyNo
EnsemblesYesYes, except subspace ensembles of discriminant analysis classifiersYes, except subspace ensembles

### Misclassification Cost Matrix, Prior Probabilities, and Observation Weights

When you train a classification model, you can specify the misclassification cost matrix, prior probabilities, and observation weights by using the Cost, Prior, and Weights name-value arguments, respectively. Classification learning algorithms use the specified values for cost-sensitive learning and evaluation.

#### Specify Cost, Prior, and Weights Name-Value Arguments

Suppose that you specify Cost as C, Prior as p, and Weights as w. The values C, p, and w have the forms

$\begin{array}{c}C=\left({c}_{ij}\right),\\ p=\left[\begin{array}{cccc}{p}_{1}& {p}_{2}& \cdots & {p}_{K}\end{array}\right],\\ w=\left[\begin{array}{cccc}{w}_{1}& {w}_{2}& \cdots & {w}_{n}\end{array}{\right]}^{\prime }.\end{array}$

• C is a K-by-K numeric matrix, where K is the number of classes. cij = C(i,j) is the cost of classifying an observation into class j when its true class is i.

• wj is the observation weight for observation j, and n is the number of observations.

• p is a 1-by-K numeric vector, where pk is the prior probability of the class k. If you specify Prior as "empirical", then the software sets pk to the sum of observation weights for the observations in class k:

#### Cost, Prior, and W Properties of Classification Model

The software stores the user-specified cost matrix (C) in the Cost property as is, and stores the prior probabilities and observation weights in the Prior and W properties, respectively, after normalization.

A classification model trained by the fitcdiscr, fitcgam, fitcknn, fitcnb, or fitcnet function uses the Cost property for prediction, but the functions do not use Cost for training. Therefore, the Cost property of the model is not read-only; you can change the property value by using dot notation after creating the trained model. For models that use Cost for training, the property is read-only.

The software normalizes the prior probabilities to sum to 1 and normalizes observation weights to sum up to the value of the prior probability in the respective class.

#### Cost-Sensitive Learning

These classification models support cost-sensitive learning:

• Classification decision tree, trained by fitctree

• Classification ensemble, trained by fitcensemble or TreeBagger

• Gaussian kernel classification with SVM and logistic regression learners, trained by fitckernel

• Multiclass, error-correcting output codes (ECOC) model, trained by fitcecoc

• Linear classification for SVM and logistic regression, trained by fitclinear

• SVM classification, trained by fitcsvm

The fitting functions use the misclassification cost matrix specified by the Cost name-value argument for model training. Approaches to cost-sensitive learning vary from one classifier to another.

• fitcecoc converts the specified cost matrix and prior probability values for multiclass classification into the values for binary classification for each binary learner. For more information, see Prior Probabilities and Misclassification Cost.

• fitctree applies average cost correction for growing a tree.

• fitcensemble, TreeBagger, fitckernel, fitclinear, and fitcsvm adjust prior probabilities and observation weights for the specified cost matrix.

• fitcensemble and TreeBagger generate in-bag samples by oversampling classes with large misclassification costs and undersampling classes with small misclassification costs. Consequently, out-of-bag samples have fewer observations from classes with large misclassification costs and more observations from classes with small misclassification costs. If you train a classification ensemble using a small data set and a highly skewed cost matrix, then the number of out-of-bag observations per class might be very low. Therefore, the estimated out-of-bag error can have a large variance and might be difficult to interpret. The same phenomenon can occur for classes with large prior probabilities.

Adjust Prior Probabilities and Observation Weights for Misclassification Cost Matrix.  For model training, the fitcensemble, TreeBagger, fitckernel, fitclinear, and fitcsvm functions update the class prior probabilities p to p* and the observation weights w to w* to incorporate the penalties described in the cost matrix C.

For a binary classification model, the software completes these steps:

1. Update p to incorporate the cost matrix C.

$\begin{array}{l}{\stackrel{^}{p}}_{1}={p}_{1}{c}_{12},\\ {\stackrel{^}{p}}_{2}={p}_{2}{c}_{21}.\end{array}$

2. Normalize $\stackrel{^}{p}$ so that the updated prior probabilities sum to 1.

${p}^{*}=\frac{1}{{\stackrel{^}{p}}_{1}+{\stackrel{^}{p}}_{2}}\stackrel{^}{p}.$

3. Remove observations from the training data corresponding to classes with zero prior probability.

4. Normalize the observation weights wj to sum up to the updated prior probability of the class to which the observation belongs. That is, the normalized weight for observation j in class k is

5. Remove observations that have zero weight.

If you have three or more classes for an ensemble model, trained by fitcensemble or TreeBagger, the software also adjusts prior probabilities for the misclassification cost matrix. This conversion is more complex. First, the software attempts to solve a matrix equation described in Zhou and Liu [2]. If the software fails to find a solution, it applies the “average cost” adjustment described in Breiman et al. [3]. For more information, see Zadrozny et al. [4].

#### Cost-Sensitive Evaluation

You can account for the cost imbalance in classification models and data sets by conducting a cost-sensitive analysis:

• Perform a cost-sensitive test by using the compareHoldout or testcholdout function. Both functions statistically compare the predictive performance of two classification models by including a cost matrix in the analysis. For details, see Cost-Sensitive Testing.

• Compare observed misclassification costs, returned by the object functions loss, resubLoss, and kfoldLoss of classification models. Specify the LossFun name-value argument as "classifcost". The functions return a weighted average misclassification cost of the input data, training data, and data for cross-validation, respectively. For details, see the object function reference page of any classification model object. For example, see Classification Loss.

For an example of cost-sensitive evaluation, see Conduct Cost-Sensitive Comparison of Two Classification Models.

## References

[1] Breiman, L. "Random Forests." Machine Learning 45, 2001, pp. 5–32.

[2] Zhou, Z.-H., and X.-Y. Liu. “On Multi-Class Cost-Sensitive Learning.” Computational Intelligence. Vol. 26, Issue 3, 2010, pp. 232–257 CiteSeerX.

[3] Breiman, L., J. H. Friedman, R. A. Olshen, and C. J. Stone. Classification and Regression Trees. Boca Raton, FL: Chapman & Hall, 1984.

[4] Zadrozny, B., J. Langford, and N. Abe. “Cost-Sensitive Learning by Cost-Proportionate Example Weighting.” Third IEEE International Conference on Data Mining, 435–442. 2003.