# harmonic

##### Basis for Harmonic Functions

Evaluates a basis for the harmonic polynomials in $x$ and $y$ of degree less than or equal to $n$.

##### Usage

`harmonic(x, y, n)`

##### Arguments

- x
- Vector of $x$ coordinates
- y
- Vector of $y$ coordinates
- n
- Maximum degree of polynomial

##### Details

This function computes a basis for the harmonic polynomials
in two variables $x$ and $y$ up to a given degree $n$
and evaluates them at given $x,y$ locations.
It can be used in model formulas (for example in
the model-fitting functions
`lm,glm,gam`

and `ppm`

) to specify a
linear predictor which is a harmonic function.

A function $f(x,y)$ is harmonic if $$\frac{\partial^2}{\partial x^2} f + \frac{\partial^2}{\partial y^2}f = 0.$$ The harmonic polynomials of degree less than or equal to $n$ have a basis consisting of $2 n$ functions.

This function was implemented on a suggestion of P. McCullagh for fitting nonstationary spatial trend to point process models.

##### Value

- A data frame with
`2 * n`

columns giving the values of the basis functions at the coordinates. Each column is labelled by an algebraic expression for the corresponding basis function.

##### See Also

##### Examples

```
data(longleaf)
X <- unmark(longleaf)
# inhomogeneous point pattern
<testonly># smaller dataset
longleaf <- longleaf[seq(1,longleaf$n, by=50)]</testonly>
# fit Poisson point process with log-cubic intensity
fit.3 <- ppm(X, ~ polynom(x,y,3), Poisson())
# fit Poisson process with log-cubic-harmonic intensity
fit.h <- ppm(X, ~ harmonic(x,y,3), Poisson())
# Likelihood ratio test
lrts <- 2 * (fit.3$maxlogpl - fit.h$maxlogpl)
x <- X$x
y <- X$y
df <- ncol(polynom(x,y,3)) - ncol(harmonic(x,y,3))
pval <- 1 - pchisq(lrts, df=df)
```

*Documentation reproduced from package spatstat, version 1.13-1, License: GPL (>= 2)*