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NuSVR

NuSVR

A model type for constructing a ν-support vector regressor, based on LIBSVM.jl, and implementing the MLJ model interface.

From MLJ, the type can be imported using

NuSVR = @load NuSVR pkg=LIBSVM

Do model = NuSVR() to construct an instance with default hyper-parameters. Provide keyword arguments to override hyper-parameter defaults, as in NuSVR(kernel=...).

Reference for algorithm and core C-library: C.-C. Chang and C.-J. Lin (2011): "LIBSVM: a library for support vector machines." ACM Transactions on Intelligent Systems and Technology, 2(3):27:1–27:27. Updated at https://www.csie.ntu.edu.tw/~cjlin/papers/libsvm.pdf.

This model is a re-parameterization of EpsilonSVR in which the epsilon hyper-parameter is replaced with a new parameter nu (denoted $ν$ in the cited reference) which attempts to control the number of support vectors directly.

Training data

In MLJ or MLJBase, bind an instance model to data with:

mach = machine(model, X, y)

where

  • X: any table of input features (eg, a DataFrame) whose columns each have Continuous element scitype; check column scitypes with schema(X)
  • y: is the target, which can be any AbstractVector whose element scitype is Continuous; check the scitype with scitype(y)

Train the machine using fit!(mach, rows=...).

Hyper-parameters

    • kernel=LIBSVM.Kernel.RadialBasis: either an object that can be

    called, as in kernel(x1, x2), or one of the built-in kernels from the LIBSVM.jl package listed below. Here x1 and x2 are vectors whose lengths match the number of columns of the training data X (see "Examples" below).

    • LIBSVM.Kernel.Linear: (x1, x2) -> x1'*x2
    • LIBSVM.Kernel.Polynomial: (x1, x2) -> gamma*x1'*x2 + coef0)^degree
    • LIBSVM.Kernel.RadialBasis: (x1, x2) -> (exp(-gamma*norm(x1 - x2)^2))
    • LIBSVM.Kernel.Sigmoid: (x1, x2) - > tanh(gamma*x1'*x2 + coef0)

    Here gamma, coef0, degree are other hyper-parameters. Serialization of models with user-defined kernels comes with some restrictions. See LIVSVM.jl issue91

  • gamma = 0.0: kernel parameter (see above); if gamma==-1.0 then gamma = 1/nfeatures is used in training, where nfeatures is the number of features (columns of X). If gamma==0.0 then gamma = 1/(var(Tables.matrix(X))*nfeatures) is used. Actual value used appears in the report (see below).

  • coef0 = 0.0: kernel parameter (see above)

  • degree::Int32 = Int32(3): degree in polynomial kernel (see above)

  • cost=1.0 (range (0, Inf)): the parameter denoted $C$ in the cited reference; for greater regularization, decrease cost

  • nu=0.5 (range (0, 1]): An upper bound on the fraction of training errors and a lower bound of the fraction of support vectors. Denoted $ν$ in the cited paper. Changing nu changes the thickness of some neighborhood of the graph of the prediction function ("tube" or "slab") and a training error is said to occur when a data point (x, y) lies outside of that neighborhood.

  • cachesize=200.0 cache memory size in MB

  • tolerance=0.001: tolerance for the stopping criterion

  • shrinking=true: whether to use shrinking heuristics

Operations

  • predict(mach, Xnew): return predictions of the target given features Xnew having the same scitype as X above.

Fitted parameters

The fields of fitted_params(mach) are:

  • libsvm_model: the trained model object created by the LIBSVM.jl package

Report

The fields of report(mach) are:

  • gamma: actual value of the kernel parameter gamma used in training

Examples

Using a built-in kernel

using MLJ
import LIBSVM

NuSVR = @load NuSVR pkg=LIBSVM                 ## model type
model = NuSVR(kernel=LIBSVM.Kernel.Polynomial) ## instance

X, y = make_regression(rng=123) ## table, vector
mach = machine(model, X, y) |> fit!

Xnew, _ = make_regression(3, rng=123)

julia> yhat = predict(mach, Xnew)
3-element Vector{Float64}:
  0.2008156459920009
  0.1131520519131709
 -0.2076156254934889

User-defined kernels

k(x1, x2) = x1'*x2 ## equivalent to `LIBSVM.Kernel.Linear`
model = NuSVR(kernel=k)
mach = machine(model, X, y) |> fit!

julia> yhat = predict(mach, Xnew)
3-element Vector{Float64}:
  1.1211558175964662
  0.06677125944808422
 -0.6817578942749346

See also EpsilonSVR, LIVSVM.jl and the original C implementation documentation.