kzs.md: Spatial Kolmogorov-Zurbenko Spline
In kzs: Kolmogorov-Zurbenko Spatial Smoothing and Applications

Description Usage Arguments Details Value Note Author(s) See Also Examples

The kzs.md function is an extension of the kzs function to d input variables.

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

`y`	a one-dimensional vector of real values representing the response variable to be smoothed.
`x`	a d-dimensional matrix of real values containing the input variables X = (X1, X2, ..., Xd); i.e., each column of the matrix is an input variable.
`smooth`	a real-valued vector of size d in which each element defines the range of smoothing for each corresponding variable in `x`.
`scale`	a real-valued vector of size d in which each element defines a uniform scale along its corresponding input variable.
`k`	an integer specifying the number of iterations `kzs.md` will execute. By default, `k = 1`.
`edges`	a logical indicating whether or not to display the outcome data beyond the ranges of the d input variables in `x`. By default, `edges = TRUE`.

The details for this function are nearly identical to that of kzs, except now extended to d-dimensional space.

a (d+1)-column data frame of the form (x1, x2, ...,xd, yk). See kzs.2d for the general interpretations of these results.

In many applications where input variables can be space, kzs.md can resolve the problem of missing values in time series or or irregularly observed valuesin Geographical Information Systems (GIS) data analysis. For these applications, scale is especially advantageous because it can create a uniform space over a geographic region to which the algorithm will be applied. Additionally, kzs.md can be recommended as a diagnostic tool before applying multiple linear regression analysis due to its capability of displaying nonlinearities of the outcome over the input variables.

There is no graphical output for this function; for two input variables, kzs.2d will produce a 3-dimensional plot. For three input variables, a 4-dimensional movie can be constructed over time.

In general, kzs, kzs.2d and kzs.md are all linear operations, and linear operations are commutative. Thus, for example, the outcome of a kzs.2d operation is equivalent to kzs.1d + kzs.1d; likewise, the outcome of a kzs.3d operation is equivalent to a kzs.2d + kzs.1d, etc...

Derek Cyr cyr.derek@gmail.com and Igor Zurbenko igorg.zurbenko@gmail.com

kzs; For more on the parameter restrictions, see kzs.params

# This example is an extension of the example documented in kzs.2d. We make 
# use of the Sinc function to filter a signal buried in noise over 3-dimensional 
# input variables. See the "Details" section of the "kzs.3d_data" data frame 
# documentation for specific details. 
require(lattice)

# Gridded data for X = (X1, X2) input variables
x1 <- seq(-1.5*pi, 1.5*pi, length = 50)
x2 <- x1	
df <- expand.grid(x1 = x1, x2 = x2)
	
# Time dimension
time <- 1:50

# Change the amplitude of the original function to change from 0 to 1 along time
amplitude <- sort(round(seq(0.02, 1, 0.02), digits = 2))

# Store the time and amplitude together in a data frame
t_amp <- data.frame(cbind(time, amplitude))

# Create the data set of Sinc function outcomes for each amplitude
sinc <- array(0, dim = c(nrow(df), length(amplitude)))
for (i in 1:length(amplitude)) {
	sinc[,i] <- round(amplitude[i]*sin(sqrt(df$x1^2 + df$x2^2)) / 
		    	sqrt(df$x1^2 + df$x2^2)) 
	sinc[,i][is.na(sinc[,i])] <- amplitude[i]
}

# Add noise to distort the signal 
for (j in 1:ncol(sinc)) {
	ez <- rnorm(nrow(sinc), mean = 0, sd = 1)
	sinc[,j] <- sinc[,j] + ez
}

# Change to a data frame and add the gridded input data 
kzs.2d_data <- as.data.frame(cbind(df, sinc)) 

### Movie of the signal buried in noise
grayscale = colorRampPalette(c("white", "gray", "black"))
for (u in 1:50) {
	plot(levelplot(kzs.2d_data[,u+2] ~ x1*x2, kzs.2d_data, 
		col.regions = grayscale, colorkey = FALSE))
}

### Movie of KZS 4-dimensional KZS outcome
data(kzs.3d_data)
bluered = colorRampPalette(c("blue", "cyan2", "green", 
			     "yellow", "red", "firebrick"), space = "rgb")
for (j in 1:50) {
	plot(levelplot(kzs.3d_data[,j+2] ~ x1*x2, kzs.3d_data, 
           at = do.breaks(c(-0.3, 1.0), 100), col.regions = bluered))
}