<.
### Affine Transformation
Following the notation from wikipedia if we have samples x, from a normal distribution $ x \sim \mathcal{N}(\mu,\Sigma) $ and we apply an affine transformation $y = c + Bx$ then the resulting values $y$ will be distributed:
$$ y \sim \mathcal{N}(c + B\mu, B \Sigma B^{\rm T}) $$
If we generate samples from the standard normal $x \sim \mathcal{N}(0,1)$ then apply a transformation $y = c + Bx$ we will be left with samples distributed according to:
$$y \sim \mathcal{N}(c, B B^{\rm T}) $$
Thus, we have a means to generate samples from any particular multivariate normal distribution with mean, $\hat{\mu}$, and covariance $\hat{\Sigma}$.
### Cholesky Decomposition
One immediate problem is that the transformation doesn't involve multiplication by $\hat{\Sigma}$ directly. Instead we need to calculate by $B$ where:
$$ B B^{\rm T} = \Sigma $$
In general, we would solve this problem using [*Cholesky decomposition*](http://en.wikipedia.org/wiki/Cholesky_decomposition) which is just a fancy way of finding a particular square root decomposition of a matrix.
While I'm sure there is a handy math library out there we could load to do this efficiently, since I will only be working with 2x2 matrices here I will instead borrow the same line of thinking exhibited in [this blog post](http://metamerist.blogspot.com/2008/03/googlaziness-cholesky-2x2.html).
To summarize, we can set up the equation as follows:
$$B B^{\rm T} = \Sigma $$
$$\begin{bmatrix} a & 0 \\\\ b & c \end{bmatrix}
\begin{bmatrix} a & b \\\\ 0 & c \end{bmatrix} =
\begin{bmatrix} a^2 & ab \\\\ ab & (b^2 + c^2) \end{bmatrix}$$
## Demo
In the following example, I will generate random samples from a distribution $\mathcal{N}(\mu,\Sigma)$ with:
$$\mu = \begin{bmatrix} 0, 0 \end{bmatrix}$$
$$\Sigma = \begin{bmatrix} 3 & -2 \\\\ -2 & 2 \end{bmatrix}$$.
<div id="d3Code"></div>
## D3 Code
var w = 500;
var h = 500;
var numPoints = 1000;
// Define parameters for gaussian
var mu = [0, 0];
var sig = [[3, -2],
[-2, 2]];
//Create SVG element
var svg = d3.select("div#d3Code").append("svg")
.attr("style", "outline: thin solid black;")
.attr("width", w)
.attr("height", h);
//Define plot scale
var xScale = d3.scale.linear()
.domain([-10, 10])
.range([0, w]);
var yScale = d3.scale.linear()
.domain([10, -10])
.range([0, h]);
//Perform cholesky decomposition
var sqrtSig = chol2d(sig);
//Generate random data
var dataset = d3.range(numPoints).map(function() {
return sampleBivariateNormal(mu, sqrtSig);
});
//Draw data on svg element
svg.selectAll("circle")
.data(dataset)
.enter()
.append("circle")
.attr("cx", function(d) {
return xScale(d[0]);
})
.attr("cy", function(d) {
return yScale(d[1]);
})
.attr("r", 2.5);
//2-D Cholesky decomposition of sigma
function chol2d(sig){
var a,b,c;
a = Math.sqrt(sig[0][0]);
b = sig[0][1]/a;
c = Math.sqrt(sig[1][1]-b*b);
return [
[a, 0],
[b, c]
];
}
//Transfrom 2-D standard normal data
function sampleBivariateNormal(mu, sqrtSig){
var stdNorm = d3.random.normal(0, 1);
var x = [stdNorm(), stdNorm()];
var y = [0, 0];
y[0] = mu[0] + sqrtSig[0][0]*x[0] + sqrtSig[0][1]*x[1];
y[1] = mu[1] + sqrtSig[1][0]*x[0] + sqrtSig[1][1]*x[1];
return [y[0], y[1]];
}
---
<div class="container">
<div class="row">
<span class="pull-left">
<a class="btn btn-primary" href="http://Ryan-J-Smith.github.io/D3-Examples">Back to Examples</a></span>
<span class="pull-right">By [Ryan J. Smith](http://www.ryanjsmith.me), 2015</span>
</div>
</div>
]]>