# Spatial Kolmogorov-Zurbenko Spline

### Description

The `kzs.2d`

function is a spatial extension of the `kzs`

function for two input variables.

### Usage

1 |

### Arguments

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

`x` |
a two-dimensional matrix of real values containing the input variables X = (X1, X2). Each column represents an input variable. |

`smooth` |
a vector of size two that defines the width of the smoothing window along each input variable. |

`scale` |
a vector of size two in which each element will define a uniformly spaced scale along its respective input variable. |

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

`edges` |
a logical indicating whether or not to display the outcome data beyond the rectangular range of the two
input variables. By default, |

`plot` |
a logical indicating whether or not to produce a 3-dimensional plot of the |

### Details

The details for this function are nearly identical to that of `kzs`

, except now extended to
three dimensional space. The only difference is that the `kzs.2d`

function averages all `y`

that
are contained within a rectangular window made up of sides `smooth[1]`

and `smooth[2]`

.

### Value

a three column data frame of the form `(x1, x2, yk)`

:

`x1 ` |
the |

`x2 ` |
the |

`yk ` |
the smoothed response values resulting from |

### Note

Data set (Y, X1, X2) must be provided, usually as 3-dimensional observations that occur in time or
space; `kzs.2d`

is designed for the general situation, including time series data. In many applications
where an input variable can be time, `kzs.2d`

can resolve the problem of missing values in time series or
or irregularly observed values in Geographical Information Systems (GIS) data analysis. The name of this
function, `kzs.2d`

, simply means that there are two input variables required for use.

The graphical output of `kzs.2d`

is a result of the `wireframe()`

function within the lattice package.

### Author(s)

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

### See Also

`kzs`

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

### Examples

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 | ```
# EXAMPLE - Estimating the Sinc function in the interval (-3pi, 3pi)
# Load the LATTICE package
# Gridded data for X = (x1, x2) input variables
x1 <- seq(-3*pi, 3*pi, length = 60)
x2 <- x1
df <- expand.grid(x1 = x1, x2 = x2)
# Apply the Sinc function to the (x1, x2) coordinates
df$z <- sin(sqrt(df$x1^2 + df$x2^2)) / sqrt(df$x1^2 + df$x2^2)
df$z[is.na(df$z)] <- 1
# Any point outside the circle of radius 3pi is set to 0. This provides
# a better picture of the outcome solely for the purposes of this example.
dst <- sqrt((df$x1 - 0)^2 + (df$x2 - 0)^2)
df$dist <- dst
df$z[df$dist > 3*pi] <- 0
# Add noise to distort the signal
ez <- rnorm(length(df$z), mean = 0, sd = 1) * 1/4
df$zn <- ez + df$z
### (1) 3D plot of the signal to be estimated by kzs.2d()
wireframe(z ~ x1 * x2, df, main = "Signal to be estimated", drape = TRUE,
colorkey = TRUE, scales = list(arrows = FALSE))
### (2) 3D plot of the signal buried in noise
wireframe(zn ~ x1 * x2, df, main = "Signal buried in noise", drape = TRUE,
colorkey = TRUE, scales = list(arrows = FALSE))
### (3) Execute kzs.2d()
# kzs.2d() may take time to run; k = 1 iteration is used here, but k = 2
# will provide a smoother outcome.
sw <- c(1, 1)
sc <- c(0.2, 0.2)
kzs.2d(y = df[,5], x = df[,1:2], smooth = sw, scale = sc, k = 1, edges = TRUE,
plot = TRUE)
``` |