LPTrans: This function computes m specially-designed LP orthonormal...

In LPTime: LP Nonparametric Approach to Non-Gaussian Non-Linear Time Series Modelling

Description

Computes LP Score functions for a given random variable X.

Usage

LPTrans(x, m)

Arguments

x	Observation from random variable X.
m	The number of LP transformations to be computed.

Details

For random variable X(either discrete or continuous) construct the LP transformed series by Gram Schmidt orthonormalization of the powers of

\mbox{T}_{1}[X] = \frac{F^{\scriptsize\mbox{mid}}(X) - 0.5}{σ [ F^{\scriptsize\mbox{mid}}(X)]}

where F^{\scriptsize\mbox{mid}}(x; \, X) = F(x; X) - 0.5p(x; \, X), \; p(x;\, X) = \mbox{Pr}[X = x],\; F(x;\, X) = \mbox{Pr}[X ≤q x],
and σ(X) denotes the standard deviation of the random variable X.
For X continuous, \mbox{T}_{j}[X] = \mbox{Leg}_{j}[F(X)], where \mbox{Leg}_j denotes jth shifted orthonormal Legendre Polynomial \mbox{Leg}_j(u), \; 0 < u < 1. Now define the UNIT LP basis function as follows:

\mbox{S}_{j}(u; \, X) = \mbox{T}_{j}[Q(u; \, X)], \; 0 < u < 1.

Our score functions are custom constructed (non-parametrically designed data-adaptive score functions) for each random variable X which can be discrete or continuous.

Value

A matrix of order n \times m where n is the number of observations on X. Each column of the matrix is an orthonormal LP score function.

Author(s)

Subhadeep Mukhopadhyay

References

Mukhopadhyay, S. and Parzen, E. (2014). LP approach to statistical modeling.arXiv:1405.2601.

Mukhopadhyay, S. and Parzen, E. (2013). Nonlinear time series modeling by LPTime,nonparametric empirical learning. arXiv:1308.0642.

Parzen, E. and Mukhopadhyay, S. (2013b). United Statistical Algorithms, LP comoment,Copula Density, Nonparametric Modeling. 59th ISI World Statistics Congress (WSC), Hong Kong.

Examples

library(lattice)
#Example from Eye Trajectory data
data(EyeTrack.sample)
x.coords <- EyeTrack.sample[,1]

x.diff <- diff(x.coords) #Differenced x-coordinate series
trans.x.diff <- LPTrans(x.diff, m = 4)
head(trans.x.diff)

x.diff.std <- (x.diff - mean(x.diff))/sd(x.diff)

x.series <- cbind(x.diff.std, ts(LPTrans(x.diff, m = 4)))
colnames(x.series) <- c("Difference of X",paste("LPTrans(diff(X)) [,",1:4,"]", sep = ""))
xyplot(x.series,outer = TRUE,
       main = "Plot of differenced x-coordinates
and its LP-transformations over time"
       )

LPTime documentation built on May 2, 2019, 7:18 a.m.