kzs (version 1.4)

kzs: Kolmogorov-Zurbenko Spline

Description

This is a one-dimensional iterative smoothing algorithm based on convolutions of rectangular kernels.

Usage

kzs(y, x, smooth, scale, k = 1, edges = TRUE, plot = TRUE)

Arguments

y

a one-dimensional vector of real values representing the response variable to be smoothed.

x

a one-dimensional vector of real values representing the input variable.

smooth

a real number defining the width of the smoothing window, i.e., the width of the rectangular kernel.

scale

for an irregularly spaced x, scale is a positive real number that will define a uniform scale along x.

k

an integer specifying the number of iterations kzs will execute; k may also be interpreted as the order of smoothness (as a polynomial of degree k-1). By default, k = 1.

edges

a logical indicating whether or not to display the outcome data beyond the initial range of x. By default, edges = TRUE.

plot

a logical indicating whether or not to produce a plot of the kzs outcome. This is TRUE by default.

Value

• a two-column data frame of paired values (xk, yk):
• xk
  x values in increments of scale
• yk
  smoothed response values resulting from k iterations of kzs

Details

The relation between variables Y and X as a function of a current value of X = x [namely, Y(x)] is often desired as a result of practical research. Usually we search for some simple function, Y(x), when given a data set of pairs (Xi, Yi). When plotted, these pairs frequently resemble a noisy plot, and thus Y(x) is desired to be a smooth outcome that captures patterns or long-term trends in the original data, while suppressing the noise. The kzs function is based on convolutions of the rectangular kernel, which is equilvalent to repeated applications of a moving average. According to the Central Limit Theorem, repeated convolutions with rectangular kernels will converge to the Gaussian kernel; the resulting kernel will have finite support equal to smooth*k, which will result in a smooth outcome with diminished noise leakage, which is a feature that the standard Gaussian kernel does not exhibit.

References

Zurbenko, I.G. (1986). The Spectral Analysis of Time Series. North Holland Series in Statistics and Probability, Elsevier Science, Amsterdam.

See Also

kzs.params, kzs.2d, kzs.md

Examples

# Total time t
t <- seq(from = -round(400*pi), to = round(400*pi), by = .25) 

# Construct the signal over time
ts <- 0.5*sin(sqrt((2*pi*abs(t))/200))
signal <- ifelse(t < 0, -ts, ts)

# Bury the signal in noise [randomly, from N(0, 1)]
et <- rnorm(length(t), mean = 0, sd = 1)
yt <- et + signal

# Data frame of (t, yt) 
pts <- data.frame(cbind(t, yt))


### EXAMPLE 1 - Apply kzs to the signal buried in noise                 

# Plot of the true signal
plot(signal ~ t, xlab = "t", ylab = "Signal", main = "True Signal",
type = "l")

# Plot of signal + noise
plot(yt ~ t, ylab = "yt", main = "Signal buried in noise", type = "p")

# Apply 3 iterations of kzs
kzs(y = pts[,2], x = pts[,1], smooth = 80, scale = .2, k = 3, edges = TRUE,
plot = TRUE)
lines(signal ~ t, col = "red")
title(main = "kzs(smooth = 80, scale = .2, k = 3, edges = TRUE)")
legend("topright", c("True signal","kzs estimate"), cex = 0.8,
col = c("red", "black"), lty = 1:1, lwd = 2, bty = "n")

### EXAMPLE 2 - Irregularly observed data over time

# Cancel a random 20 percent of (t, yt) leaving irregularly observed time points
obs <- seq(1:length(t))
t20 <- sample(obs, size = length(obs)/5)
pts20 <- pts[-t20,]        

# Plot of (t,yt) with 20 percent of the data removed
plot(pts20$yt ~ pts20$t, main = "Signal buried in noise
20 percent of 
(t, yt) deleted", xlab = "t", ylab = "yt", type = "p")

# Apply 3 iterations of kzs
kzs(y = pts20[,2], x = pts20[,1], smooth = 80, scale = .2, k = 3, edges = TRUE, 
plot = TRUE)
lines(signal ~ t, col = "red")
title(main = "kzs(smooth = 80, scale = .2, k = 3, edges = TRUE)")
legend("topright", c("True signal","kzs estimate"), cex = 0.8, 
col = c("red", "black"), lty = 1:1, lwd = 2, bty = "n")