kza (version 2.03)

kzft: Kolmogorov-Zurbenko Fourier Transform

Description

Kolmogorov-Zurbenko Fourier Transform is an iterated Fourier transform.

Usage

kzft(x, m = NULL, k = 1, f = NULL, dim = 1, index = NULL, trim = FALSE) 
Rkzft(x, m = NULL, k = 1, f = NULL, dim = 2)
coeff(m, k)
max_freq(x, m, start = 1, t = NULL)
transfer_function(m, k, lamda = seq(-0.5,0.5,by=0.01), omega = 0 )

Arguments

x

The raw time series

m

The window size for transform

k

The number of iterations for applying the KZFT

f

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).

index

An indexing set

trim

Remove k*m from edges of array.

start

Search for max amplitude begining at start.

t

An indexing set

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). Rkzft is an implementation written completely in R. This function uses the coeff function to generate the coefficients. The frequency_kzft function returns the frequency with the largest variation.

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. Igor G. Zurbenko, Amy L. Potrzeba, Tidal Waves in Atmosphere and Their Effects, Acta Geophysica Volume 58, Number 2, 356-373

See Also

kzp, kztp,

Examples

# 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 with missing values
period=101
f<-1/period
t<-1:2000
s<-1*sin(2*pi*f*t)
x<-s
noise<-3*rnorm(length(t))
x<-s+noise
m=101

rand_idx <- sample(t,100,replace=FALSE)
x[rand_idx]<-NA
t[rand_idx]<-NA
x<-as.vector(na.omit(x))
t<-as.vector(na.omit(t))

system.time(z1<-kzft(x, m=m, k=1, f=f, dim=1, index=t))
system.time(z2<-kzft(x, m=m, k=2, f=f, dim=1, index=t))
system.time(z3<-kzft(x, m=m, k=3, f=f, dim=1, index=t))

par(mfrow=c(2,2))
plot(x,type="l",main="Original time series",xlab="t", ylab="y")
lines(s,col="blue")
plot(2*Re(z1),type="l",main="kzft(101,1)",xlab="t", ylab="y", ylim=c(-6,6))
lines(s,col="blue")
plot(2*Re(z2),type="l",main="kzft(101,2)",xlab="t", ylab="y", ylim=c(-6,6))
lines(s,col="blue")
plot(2*Re(z3),type="l",main="kzft(101,3)",xlab="t", ylab="y", ylim=c(-6,6))
lines(s,col="blue")
par(mfrow=c(1,1))