# gausskernel: Gaussian Kernel Distance Computation In KRLS: Kernel-Based Regularized Least Squares

## Description

Given a N by D numeric data matrix, this function computes the N by N distance matrix with the pairwise distances between the rows of the data matrix as measured by a Gaussian Kernel.

## Usage

 `1` ```gausskernel(X = NULL, sigma = NULL) ```

## Arguments

 `X` N by N numeric data matrix. `sigma` Positive scalar that specifies the bandwidth of the Gaussian kernel (see details).

## Details

Given two D dimensional vectors x_i and x_j. The Gaussian kernel is defined as

k(x_i,x_j)=exp(-|| x_i - x_j ||^2 / sigma^2)

where ||x_i - x_j|| is the Euclidean distance given by

||x_i - x_j||=((x_i1-x_j1)^2 + (x_i2-x_j2)^2 + ... + (x_iD-x_jD)^2)^.5

and sigma^2 is the bandwidth of the kernel.

Note that the Gaussian kernel is a measure of similarity between x_i and x_j. It evalues to 1 if the x_i and x_j are identical, and approaches 0 as x_i and x_j move further apart.

The function relies on the `dist` function in the stats package for an initial estimate of the euclidean distance.

## Value

An N by N numeric distance matrix that contains the pairwise distances between the rows in X.

## Author(s)

Jens Hainmueller (Stanford) and Chad Hazlett (MIT)

`dist` function in the stats package.

## Examples

 ```1 2``` ```X <- matrix(rnorm(6),ncol=2) gausskernel(X=X,sigma=1) ```

### Example output

```## KRLS Package for Kernel-based Regularized Least Squares.

## See Hainmueller and Hazlett (2014) for details.

1         2         3
1 1.0000000 0.1034381 0.1863869
2 0.1034381 1.0000000 0.5184201
3 0.1863869 0.5184201 1.0000000
```

KRLS documentation built on May 2, 2019, 5:51 a.m.