kzft: Kolmogorov-Zurbenko Fourier Transform

Description

Kolmogorov-Zurbenko Fourier Transform is an iterated Fourier transform.

Usage

kzft(x, m=100, k=1, f=NULL, dim=1)
ff_kzft(xr, xi=NULL, m=100, k=1, f=NULL, dim=1)
transfer_function(m,k,lamda=seq(-0.5,0.5,by=0.01),omega=0)

Arguments

The raw time series

The length of the window size for a regular Fourier transform

The number of iterations for applying the KZFT

The frequency that KZFT is applied at.

dim

A value of 1 will return a vector of the given frequency and a value of 2 will return a matrix (spectra).

The Real component of a complex time series

The Imaginary component of a complex time series

lamda

The frequencies used for the calculating the transfer function

omega

The frequency that KZFT is applied at.

Details

Kolmogorov-Zurbenko Fourier Transform (KZFT) is the Fourier transform applied over every segment of length m iterated k times. If a frequency is not selected then the frequency with the largest amplitude is used. The approach taken here is to iterate fft to create a matrix of frequencies over time, another approach is to use matrices for the KZFT transform (see the KZFT package by Wei Yang in R).

References

I. G. Zurbenko, The spectral Analysis of Time Series. North-Holland, 1986. I. G. Zurbenko, P. S. Porter, Construction of high-resolution wavelets, Signal Processing 65: 315-327, 1998. R. Neagu, I. G. Zurbenko, Tracking and separating non-stationary multi-component chirp signals, J. Franklin Inst., 499-520, 2002. R. H. Shumway, D. S. Stoffer, Time Series Analysis and Its Applications: With R Examples, Springer, 2006. Wei Yang and Igor Zurbenko, kzft: Kolmogorov-Zurbenko Fourier Transform and Applications, R-Project 2007.

Examples

Run this code

# example taken from Wei Yang's KZFT package
# coefficients of kzft(201,5)


# function to calculate polnomial coefficients for kzft
require(polynom)
coeff <- function(m, k)
{
    poly<-polynomial(rep(1,m))
    polyk<-poly^k
    coef<-as.vector(polyk)
    coef<-coef/m^k
    M=(m-1)*k+1
    return(coef[1:M])
}


a<-coeff(201,5);
t<-seq(1:1001)-501;
z<-cos(2*pi*0.025*t);
plot(z*a,type="l",xlab="Time", ylab="Coefficient", main="Coefficients of the kzft");
lines(a);
lines(-1*a);

# example taken from Wei Yang's KZFT package
# transfer function of the kzft(201,5) at frequency 0.025
lamda<-seq(-0.1,0.1,by=0.001)
tf1<-transfer_function(201,1,lamda,0.025)
tf2<-transfer_function(201,5,lamda,0.025)
matplot(lamda,cbind(log(tf1),log(tf2)),type="l",ylim=c(-15,0),
	ylab="Natural log transformation of the coefficients", 
	xlab="Frequency (cycles/time unit)",
    main="Transfer function of kzft(201,5) at frequency 0.025")

# example 
# signal with a frequency of 0.01
t<-1:1000
f<-10/1000
x<-cos(2*pi*f*t) + rnorm(length(t),0,3)

system.time(z1<-kzft(x,200,1)$Complex)
system.time(z2<-kzft(x,200,2)$Complex)
system.time(z3<-kzft(x,200,3)$Complex)

par(mfrow=c(2,2))
plot(x,type="l",main="Original time series",xlab="t", ylab="y")
plot(2*Re(z1)[1:400],type="l",main="kzft(200,1)",xlab="t", ylab="y")
plot(2*Re(z2)[1:400],type="l",main="kzft(200,2)",xlab="t", ylab="y")
plot(2*Re(z3)[1:400],type="l",main="kzft(200,3)",xlab="t", ylab="y")

#
# example ff
# This example uses the ff package for use with large datasets
#
require(ff)
t<-1:1000
f<-10/1000
x<-cos(2*pi*(10/1000)*t)+rnorm(1000,0,3)

#
# ff does not yet handle complex datasets so we need to seperate the complex components
# xr is the real component and xi is assigned the imaginary
# we will simply convert to "ff" format; to read in a dataset from a file, please read the ff documentation.
#
xr<-ff(x)
xi<-ff(vmode="double",length=length(x))

z1<-ff_kzft(xr,xi,200,1)$Re
z2<-ff_kzft(xr,xi,200,2)$Re
z3<-ff_kzft(xr,xi,200,3)$Re

par(mfrow=c(2,2))
plot(x,type="l",main="Original time series",xlab="t", ylab="y")
plot(2*z1[1:400],type="l",main="ff_kzft(200,1)",xlab="t", ylab="y")
plot(2*z2[1:400],type="l",main="ff_kzft(200,2)",xlab="t", ylab="y")
plot(2*z3[1:400],type="l",main="ff_kzft(200,3)",xlab="t", ylab="y")