This function computes m specially-designed LP orthonormal basis functions of a random variable
Computes LP Score functions for a given random variable X.
LPTrans(x, m)
x |
Observation from random variable X. |
m |
The number of LP transformations to be computed. |
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.
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.
Subhadeep Mukhopadhyay
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.
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"
)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.