samQL {SAM}

Training function of Sparse Additive Models

Description

The regression model is learned using training data.

Usage

samQL(
  X,
  y,
  p = 3,
  lambda = NULL,
  nlambda = NULL,
  lambda.min.ratio = 0.005,
  thol = 1e-05,
  max.ite = 1e+05,
  regfunc = "L1"
)

Arguments

X

The n by d design matrix of the training set, where n is sample size and d is dimension.

y

The n-dimensional response vector of the training set, where n is sample size.

p

The number of basis spline functions. The default value is 3.

lambda

A user supplied lambda sequence. Typical usage is to have the program compute its own lambda sequence based on nlambda and lambda.min.ratio. Supplying a value of lambda overrides this. WARNING: use with care. Do not supply a single value for lambda. Supply instead a decreasing sequence of lambda values. samQL relies on its warms starts for speed, and its often faster to fit a whole path than compute a single fit.

nlambda

The number of lambda values. The default value is 30.

lambda.min.ratio

Smallest value for lambda, as a fraction of lambda.max, the (data derived) entry value (i.e. the smallest value for which all coefficients are zero). The default is 5e-3.

thol

Stopping precision. The default value is 1e-5.

max.ite

The number of maximum iterations. The default value is 1e5.

regfunc

A string indicating the regularizer. The default value is "L1". You can also assign "MCP" or "SCAD" to it.

Details

We adopt various computational algorithms including the block coordinate descent, fast iterative soft-thresholding algorithm, and newton method. The computation is further accelerated by "warm-start" and "active-set" tricks.

Value

p

The number of basis spline functions used in training.

X.min

A vector with each entry corresponding to the minimum of each input variable. (Used for rescaling in testing)

X.ran

A vector with each entry corresponding to the range of each input variable. (Used for rescaling in testing)

lambda

A sequence of regularization parameter used in training.

w

The solution path matrix (d*p by length of lambda) with each column corresponding to a regularization parameter. Since we use the basis expansion, the length of each column is d*p+1.

intercept

The solution path of the intercept.

df

The degree of freedom of the solution path (The number of non-zero component function)

knots

The p-1 by d matrix. Each column contains the knots applied to the corresponding variable.

Boundary.knots

The 2 by d matrix. Each column contains the boundary points applied to the corresponding variable.

func_norm

The functional norm matrix (d by length of lambda) with each column corresponds to a regularization parameter. Since we have d input variables, the length of each column is d.

sse

Sums of square errors of the solution path.

See Also

SAM,plot.samQL,print.samQL,predict.samQL

Examples

## generating training data
n = 100
d = 500
X = 0.5*matrix(runif(n*d),n,d) + matrix(rep(0.5*runif(n),d),n,d)

## generating response
y = -2*sin(X[,1]) + X[,2]^2-1/3 + X[,3]-1/2 + exp(-X[,4])+exp(-1)-1

## Training
out.trn = samQL(X,y)
out.trn

## plotting solution path
plot(out.trn)

## generating testing data
nt = 1000
Xt = 0.5*matrix(runif(nt*d),nt,d) + matrix(rep(0.5*runif(nt),d),nt,d)

yt = -2*sin(Xt[,1]) + Xt[,2]^2-1/3 + Xt[,3]-1/2 + exp(-Xt[,4])+exp(-1)-1

## predicting response
out.tst = predict(out.trn,Xt)