NORMA {NORMA}

NORMA Optimization

Description

Computes general noise SVR based on NORMA optimization.

Usage

NORMA(x, y, f_0 = 0, beta_0 = 0, lambda = 0, rate = function(t) {     1
  }, kernel = linear_kernel, cost_der = ILF_cost_der,
  cost_name = "ILF_cost_der", gamma = 1, max_iterations = nrow(x),
  stopping_threshold = 0, trace = TRUE, no_beta = TRUE,
  fixed_epsilon = TRUE, ...)

Arguments

x

matrix containing training points. Each row must be a point.

y

numeric containing target for training points x.

f_0

initial hypothesis.

beta_0

initial value for offset b.

lambda

NORMA optimization parameter lambda

rate

learning rate for NORMA optimization. Must be a function with one argument.

kernel

kernel function to use. Must be a function with three arguments such as gaussian_kernel. See also linear_kernel

cost_der

Loss function derivative to use. See also ILF_cost_der. Must be "ILF_cost_der" when ILF derivative is used.

cost_name

character indicating the symbolic name of cost_der.

gamma

gaussian kernel parameter \gamma.

max_iterations

maximum number of NORMA iterations computed.

stopping_threshold

value indicating when to stop NORMA optimization. See also 'Details'.

trace

boolean indicating if information messages should be printed (TRUE) or not (FALSE).

no_beta

boolean indicating if an offset b is used (FALSE) or not (TRUE).

fixed_epsilon

boolean indicating if epsilon should be updated (FALSE) or not (TRUE).

...

additional arguments to be passed to the low level functions.

Details

Optimization will stop when the sum of the differences between all training predicted values of present iteration versus values from previous iteration does not exceeds stopping_threshold.

Value

Returns a list containing:

alpha

matrix representing \alpha parameters of NORMA optimization in each iteration, one per row.

beta

numeric representing \beta parameter of NORMA optimization in each iteration.

n_iterations

Number of NORMA iterations performed.

Author(s)

Jesus Prada, [e-mail]

References

Link to the scientific paper

Kivinen J., Smola A. J., Williamson R.C.: Online learning with kernels. In: IEEE transactions on signal processing, vol. 52, pp. 2165-2176, IEEE (2004).

with theoretical background for NORMA optimization is provided below.

http://realm.sics.se/papers/KivSmoWil04(1).pdf

Examples

NORMA(x=matrix(rnorm(10),nrow=10,ncol=1,byrow=TRUE),y=rnorm(10),kernel=function(x,y,gamma=0){x%*%y},
cost_der=function(phi,sigma_cuad,mu){return((phi-mu)/sigma_cuad)},cost_name="example",
sigma_cuad=1,mu=0)