PRIM for multivariate data

PRIM for multivariate data.

Usage

prim.box(x, y, box.init=NULL, peel.alpha=0.05, paste.alpha=0.01,
     mass.min=0.05, threshold, pasting=TRUE, verbose=FALSE,
     threshold.type=0, y.fun=mean)

prim.hdr(prim, threshold, threshold.type, y.fun=mean)
prim.combine(prim1, prim2, y.fun=mean)

Arguments

x

matrix of data values

y

vector of response values

y.fun

function applied to response y. Default is mean.

box.init

initial covering box

peel.alpha

peeling quantile tuning parameter

paste.alpha

pasting quantile tuning parameter

mass.min

minimum mass tuning parameter

threshold

threshold tuning parameter(s)

threshold.type

threshold direction indicator: 1 = ">= threshold", -1 = "<= threshold", 0 = ">= threshold[1] & <= threshold[2]"

pasting

flag for pasting

verbose

flag for printing output during execution

prim,prim1,prim2

objects of type prim

Details

The data are \((\bold{X}_1, Y_1), \dots, (\bold{X}_n, Y_n)\) where \(\bold{X}_i\) is d-dimensional and \(Y_i\) is a scalar response. PRIM finds modal (and/or anti-modal) regions in the conditional expectation \(m(\bold{x}) = \bold{E} (Y | \bold{x}).\)

In general, \(Y_i\) can be real-valued. See vignette("prim"). Here, we focus on the special case for binary \(Y_i\). Let \(Y_i\) = 1 when \(\bold{X}_i \sim F^+\); and \(Y_i\) = -1 when \(\bold{X}_i \sim F^-\) where \(F^+\) and \(F^-\) are different distribution functions. In this set-up, PRIM finds the regions where \(F^+\) and \(F^-\) are most different.

The tuning parameters peel.alpha and paste.alpha control the `patience' of PRIM. Smaller values involve more patience. Larger values less patience. The peeling steps remove data from a box till either the box mean is smaller than threshold or the box mass is less than mass.min. Pasting is optional, and is used to correct any possible over-peeling. The default values for peel.alpha, paste.alpha and mass.min are taken from Friedman & Fisher (1999).

The type of PRIM estimate is controlled threshold and threshold.type:

threshold.type=1

search for {\(m(\bold{x}) \geq\) threshold}.

threshold.type=-1

search for {\(m(\bold{x}) \leq\) threshold}.

threshold.type=0

search for both {\(m(\bold{x}) \geq\) threshold[1]} and {\(m(\bold{x}) \leq\) threshold[2]}.

There are two ways of using PRIM. One is prim.box with pre-specified threshold(s). This is appropriate when the threshold(s) are known to produce good estimates.

On the other hand, if the user doesn't provide threshold values then prim.box computes box sequences which cover the data range. These can then be pruned at a later stage. prim.hdr allows the user to specify many different threshold values in an efficient manner, without having to recomputing the entire PRIM box sequence. prim.combine can be used to join the regions computed from prim.hdr. See the examples below.

Value

– prim.box produces a PRIM estimate, an object of type prim, which is a list with 8 fields:

x

list of data matrices

y

list of response variable vectors

y.mean

list of vectors of box mean for y

box

list of matrices of box limits (first row = minima, second row = maxima)

mass

vector of box masses (proportion of points inside a box)

num.class

total number of PRIM boxes

num.hdr.class

total number of PRIM boxes which form the HDR

ind

threshold direction indicator: 1 = ">= threshold", -1 = "<=threshold"

The above lists have num.class fields, one for each box.

– prim.hdr takes a prim object and prunes it using different threshold values. Returns another prim object. This is much faster for experimenting with different threshold values than calling prim.box each time.

– prim.combine combines two prim objects into a single prim object. Usually used in conjunction with prim.hdr. See examples below.

Examples

data(quasiflow)
qf <- quasiflow[1:1000,1:2]
qf.label <- quasiflow[1:1000,4]

## using only one command
thr <- c(0.25, -0.3)
qf.prim1 <- prim.box(x=qf, y=qf.label, threshold=thr, threshold.type=0)

## alternative - requires more commands but allows more control
## in intermediate stages
qf.primp <- prim.box(x=qf, y=qf.label, threshold.type=1)
   ## default threshold too low, try higher one

qf.primp.hdr <- prim.hdr(prim=qf.primp, threshold=0.25, threshold.type=1)
qf.primn <- prim.box(x=qf, y=qf.label, threshold=-0.3, threshold.type=-1)
qf.prim2 <- prim.combine(qf.primp.hdr, qf.primn)

plot(qf.prim1, alpha=0.2)   ## orange=x1>x2, blue x2<x1