# find_rho: Estimate local correlation \$rho\$ between two variables. In LAJordanger/localgaussSpec: Local Gaussian Spectral Analysis

## Description

This function use the local likelihood function proposed by Hjort & Jones (1996) to find an estimate of the local correlation \$ρ\$ given \$n\$ bivariate observations and a vector \$b\$ with the specified bandwidths. Note that the code use the simplifying assumption that the the marginals are standard normal, which reduces the number of parameters to estimate from five to one. This function is based on code from H\aakon Otneim.

## Usage

 ```1 2``` ```find_rho(data, grid, b, .kernel = c("normal", "uniform"), return_arguments = FALSE) ```

## Arguments

 `data` A matrix containing the observations, one bivariate observation in each row. `grid` A matrix containing the points where we want to find the local correlations, one bivariate point in each row. `b` A vector containing the two bandwidhts to use. `.kernel` Specification of the kernel to use, either `normal` or `uniform`, with default being the `normal` one if none is selected. Note: The `uniform`-kernel takes much longer to compute (a factor of ten when used on a small sample of size \$n=200\$). `return_arguments` Logical argument, default `FALSE`, that can be used to include the three arguments `data`, `grid` and `b` in the result from this function.

## Details

In the computation, a product kernel that either is based on the standard normal distribution or the uniform distribution is used. The default kernel is the normal one, since it seems to behave better than the uniform kernel with regard to being able to get a proper result (i.e. not `NA`) and since it also seems to compute a bit faster when a computation is possible. Note that the code computes the required densities (related to the normal distribution) directly instead of using `dnorm` and `dmvnorm`, since that gives a speed gain.

## Value

The result of this function is a list. For each grid-point there will be an estimate of the local correlation and the log-density of the corresponding local Gaussian approximation at that point. When `return_arguments` is `TRUE`, the three arguments `data`, `grid` and `b` will also be included.

## References

Hjort, N. L., and Jones, M. C.: "Locally parametric nonparametric density estimation.", The Annals of Statistics (1996): 1619-1647.

LAJordanger/localgaussSpec documentation built on July 28, 2017, 12:15 a.m.