The `kzs.md`

function is an extension of the `kzs`

function to *d* input variables.

1 |

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

`x` |
a |

`smooth` |
a real-valued vector of size |

`scale` |
a real-valued vector of size |

`k` |
an integer specifying the number of iterations |

`edges` |
a logical indicating whether or not to display the outcome data beyond the ranges of the |

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`

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 | ```
# 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))
}
``` |

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