SVM

Description

This module performs support vector regression on a set of training points in order to determine the generating function. Currently least squares support vector regression is implemented. The optimal solution to the Langrangian is found by a conjugate gradient algorithm (CGA).

Synopsis

Documentation

data DataSet Source

Each data set is a list of vectors and values which are training points of the form f(x) = y forall {x,y}.

Constructors

DataSet
Fields points :: Array Int [Double] values :: DoubleArray

data SVMSolution Source

The solution contains the dual weights, the support vectors and the bias.

Constructors

SVMSolution
Fields alpha :: DoubleArray sv :: Array Int [Double] bias :: Double

newtype KernelFunction Source

Every kernel function represents an inner product in feature space. The parameters are:

A list of kernel parameters that can be interpreted differently by each kernel function.
The first point in the inner product.
The second point in the inner product.

Constructors

KernelFunction ([Double] -> [Double] -> [Double] -> Double)

class SVM a whereSource

A support vector machine (SVM) can estimate a function based upon some training data. Instances of this class need only implement the dual cost and the kernel function. Default implementations are given for finding the SVM solution, for simulating a function and for creating a kernel matrix from a set of training points. All SVMs should return a solution which contains a list of the support vectors and their dual weigths. dcost represents the coefficient of the dual cost function. This term gets added to the diagonal elements of the kernel matrix and may be different for each type of SVM.

Methods

createKernelMatrix :: a -> Array Int [Double] -> KernelMatrix Source

Creates a KernelMatrix from the training points in the DataSet. If kf is the KernelFunction then the elements of the kernel matrix are given by K[i,j] = kf x[i] x[j], where the x[i] are taken from the training points. The kernel matrix is symmetric and positive semi-definite.Only the bottom half of the kernel matrix is stored.

dcost :: a -> Double Source

The derivative of the cost function is added to the diagonal elements of the kernel matrix. This places a cost on the norm of the solution, which helps prevent overfitting of the training data.

evalKernel :: a -> [Double] -> [Double] -> Double Source

This function provides access to the KernelFunction used by the SVM.

simulate :: a -> SVMSolution -> Array Int [Double] -> [Double]Source

This function takes an SVMSolution produced by the SVM passed in, and a list of points in the space, and it returns a list of valuues y = f(x), where f is the generating function represented by the support vector solution.

solve :: a -> DataSet -> Double -> Int -> SVMSolution Source

This function takes a DataSet and feeds it to the SVM. Then it returns the SVMSolution which is the support vector solution for the function which generated the points in the training set. The function also takes values for epsilon and the max iterations, which are used as stopping criteria in the conjugate gradient algorithm.

Instances

SVM LSSVM

data LSSVM Source

A least squares support vector machine. The cost represents the relative expense of missing a training versus a more complicated generating function. The higher this number the better the fit of the training set, but at a cost of poorer generalization. The LSSVM uses every training point in the solution and performs least squares regression on the dual of the problem.

Constructors

LSSVM
Fields kf :: KernelFunction The kernel function defines the feature space. cost :: Double The cost coefficient in the Lagrangian. params :: [Double] Any parameters needed by the `KernelFunction`.

Instances

SVM LSSVM

newtype KernelMatrix Source

The kernel matrix has been implemented as an unboxed array for performance reasons.

Constructors

KernelMatrix DoubleArray

reciprocalKernelFunction :: [Double] -> [Double] -> [Double] -> Double Source

The reciprocal kernel is the result of exponential basis functions, exp(-k*(x+a)). The inner product is an integral over all k >= 0.

radialKernelFunction :: [Double] -> [Double] -> [Double] -> Double Source

This is the kernel when radial basis functions are used.

linearKernelFunction :: [Double] -> [Double] -> [Double] -> Double Source

This is a simple dot product between the two data points, corresponding to a featureless space.

splineKernelFunction :: [Double] -> [Double] -> [Double] -> Double Source

polyKernelFunction :: [Double] -> [Double] -> [Double] -> Double Source

mlpKernelFunction :: [Double] -> [Double] -> [Double] -> Double Source

Provides a solution similar to neural net.