Classification

Standard classification problems

In a classification problem, we would typically have some input vectors x and some desired output labels y. Let's consider then a simple classification problem called the yin-yang problem. In this problem, we have two classes of elements. Elements belonging to the positive class, shown in blue; and elements belonging to the negative class, shown in red.

This data can be downloaded in Excel format here. In order to load this data into an application, let's use the ExcelReader class together with some extensions methods from the Accord.Math namespace. Add the following using namespace clauses on top of your source file:

using Accord.Controls;
using Accord.IO;
using Accord.Math;

Then, let's write the following code:

// Read the Excel worksheet into a DataTable
DataTable table = new ExcelReader("examples.xls").GetWorksheet("Sheet1");

// Convert the DataTable to input and output vectors
double[][] inputs = table.ToArray<double>("X", "Y");
int[] outputs = table.Columns["G"].ToArray<int>();

// Plot the data
ScatterplotBox.Show("Yin-Yang", inputs, outputs).Hold();

After we run and execute this code, we will get the following scatter plot shown on the screen:

Models

Naive Bayes

Naive Bayes classifiers is a simple probabilistic classifiers based on Bayes' theorem with strong independence assumptions between the features.

p(A|B) = p(B|A) * p(A) / p(B)
p = probability | = given that '*' = multiplied by
B is a consequent of some antecedent A

// In our problem, we have 2 classes (samples can be either
// positive or negative), and 2 inputs (x and y coordinates).

var nb = new NaiveBayes<NormalDistribution>(classes: 2,
    inputs: 2, prior: new NormalDistribution());

// The Naive Bayes expects the class labels to 
// range from 0 to k, so we convert -1 to be 0:
//
outputs = outputs.Apply(x => x < 0 ? 0 : x);

// Estimate the Naive Bayes
double error = nb.Estimate(inputs, outputs);

// Classify the samples using the model
int[] answers = inputs.Apply(nb.Compute);

// Plot the results
ScatterplotBox.Show("Expected results", inputs, outputs);
ScatterplotBox.Show("Naive Bayes results", inputs, answers)
    .Hold();

Support Vector Machines

SVMs are supervised learning models with associated learning algorithms that analyze data and recognize patterns, used for classification and regression analysis.

Linear

In the Linear SVM the ideia is design a hyperplane that classifies the training vectors in two classes.

// Create a linear binary machine with 2 inputs
var svm = new SupportVectorMachine(inputs: 2);

// Create a L2-regularized L2-loss optimization algorithm for
// the dual form of the learning problem. This is *exactly* the
// same method used by LIBLINEAR when specifying -s 1 in the 
// command line (i.e. L2R_L2LOSS_SVC_DUAL).
//
var teacher = new LinearCoordinateDescent(svm, inputs, outputs);

// Teach the vector machine
double error = teacher.Run();

// Classify the samples using the model
int[] answers = inputs.Apply(svm.Compute).Apply(System.Math.Sign);

// Plot the results
ScatterplotBox.Show("Expected results", inputs, outputs);
ScatterplotBox.Show("LinearSVM results", inputs, answers);

Kernel

Kernel methods enable them to operate in high-dimensional, implicit feature space without ever computing the coordinates of the data in that space.

// Estimate the kernel from the data
var gaussian = Gaussian.Estimate(inputs);

// Create a Gaussian binary support machine with 2 inputs
var svm = new KernelSupportVectorMachine(gaussian, inputs: 2);

// Create a new Sequential Minimal Optimization (SMO) learning 
// algorithm and estimate the complexity parameter C from data
var teacher = new SequentialMinimalOptimization(svm, inputs, outputs)
{
    UseComplexityHeuristic = true
};

// Teach the vector machine
double error = teacher.Run();

// Classify the samples using the model
int[] answers = inputs.Apply(svm.Compute).Apply(System.Math.Sign);

// Plot the results
ScatterplotBox.Show("Expected results", inputs, outputs);
ScatterplotBox.Show("GaussianSVM results", inputs, answers);

``` // Grab the index of multipliers higher than 0 int[] idx = teacher.Lagrange.Find(x => x > 0);

// Select the input vectors for those double[][] sv = inputs.Submatrix(idx);

// Plot the support vectors selected by the machine ScatterplotBox.Show("Support vectors", sv).Hold();

<img src="http://accord-framework.net/images/guides/tutorials/results-ksvm-sv.png" width="400" />

## Decision Trees
The goal is to create a model that predicts the value of a target variable by learning simple decision rules inferred from the data features.

```csharp
// In our problem, we have 2 classes (samples can be either
// positive or negative), and 2 continuous-valued inputs.
DecisionTree tree = new DecisionTree(attributes: new[] 
{
    DecisionVariable.Continuous("X"),
    DecisionVariable.Continuous("Y")
}, outputClasses: 2);

C45Learning teacher = new C45Learning(tree);

// The C4.5 algorithm expects the class labels to
// range from 0 to k, so we convert -1 to be zero:
//
outputs = outputs.Apply(x => x < 0 ? 0 : x);

double error = teacher.Run(inputs, outputs);

// Classify the samples using the model
int[] answers = inputs.Apply(tree.Compute);

// Plot the results
ScatterplotBox.Show("Expected results", inputs, outputs);
ScatterplotBox.Show("Decision Tree results", inputs, answers)
    .Hold();

Neural Networks

<< pick example from the Tutorial's network method >>

See Resilient Backpropagation.

Logistic Regression

Logistic regression measures the relationship between the categorical dependent variable and one or more independent variables by estimating probabilities using a logistic function.

// In our problem, we have 2 inputs (x, y pairs)
var logistic = new LogisticRegression(inputs: 2);

// Create a iterative re-weighted least squares algorithm
var teacher = new IterativeReweightedLeastSquares(logistic);

// Logistic Regression expects the output labels 
// to range from 0 to k, so we convert -1 to be 0:
//
outputs = outputs.Apply(x => x < 0 ? 0 : x);

// Iterate until stop criteria is met
double error = double.PositiveInfinity;
double previous;

do
{
   previous = error;

   // Compute one learning iteration
   error = teacher.Run(inputs, outputs);

} while (Math.Abs(previous - error) < 1e-10 * previous);

// Classify the samples using the model
int[] answers = inputs.Apply(logistic.Compute).Apply(Math.Round).ToInt32();

// Plot the results
ScatterplotBox.Show("Expected results", inputs, outputs);
ScatterplotBox.Show("Logistic Regression results", inputs, answers)
 .Hold();

See Logistic Regression.

Variations

Multi-label problems

In some problems, samples can belong to more than one single class at a time. Those problems are denoted multiple label classification problems and can be solved in different manners. One way to attack a multi-label problem is by using a 1-vs-all support vector machine.

See Multi-label SVM.

Sequence classification

A sequence classification problem is a classification problem where input vectors can have varying length. Those problems can be attacked in multiple ways. One of them is to use a classifier that has been specifically designed to work with sequences. The other one is to extract a fixed number of features from those varying length vectors, and then use them with any standard classification algorithms, such as support vector machines.

For an example on how to transform sequences into fixed length vectors, see Dynamic Time Warp Support Vector Machine.

For examples of sequence classifiers, see Hidden Markov Classifier Learning and Hidden Conditional Random Field Learning.