LPTime (version 1.0-2)

LPTrans: This function computes $m$ specially-designed LP orthonormal basis functions of a random variable

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.

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.

Details

For random variable $X$(either discrete or continuous) construct the LP transformed series by Gram Schmidt orthonormalization of the powers of $$\mbox{T}_{1}[X] = \frac{F^{\scriptsize\mbox{mid}}(X) - 0.5}{\sigma [ F^{\scriptsize\mbox{mid}}(X)]}$$ where $F^{\scriptsize\mbox{mid}}(x; \, X) = F(x; X) - 0.5p(x; \, X), \; p(x;\, X) = \mbox{Pr}[X = x],\; F(x;\, X) = \mbox{Pr}[X \leq x]$, and $\sigma(X)$ 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.

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"
       )