kriging.js

kriging.js is a Javascript library providing spatial prediction and mapping capabilities via the ordinary kriging algorithm.

Kriging is a type of gaussian process where 2-dimensional coordinates are mapped to some target variable using kernel regression. This algorithm has been specifically designed to accurately model smaller data sets by assigning a prior to the variogram parameters.

Fitting a Model

The first step is to link kriging.js to your html code and assign your coordinate and target variables to 3 separate arrays.

<script src="kriging.js" type="text/javascript"></script>
<script type="text/javascript">
	var t = [ /* Target variable */ ];
	var x = [ /* X-axis coordinates */ ];
	var y = [ /* Y-axis coordinates */ ];
	var model = "exponential";
	var sigma2 = 0, alpha = 100;
	var variogram = kriging.train(t, x, y, model, sigma2, alpha);
</script>

The train method in the kriging object fits your input to whatever variogram model you specify - gaussian, exponential or spherical - and returns a variogram object.

Error and Bayesian Prior

Notice the σ² (sigma2) and α (alpha) variables, these correspond to the variance parameters of the gaussian process and the prior of the variogram model, respectively. A diffuse α prior is typically used; a formal mathematical definition of the model is provided below.

Predicting New Values

Values can be predicted for new coordinate pairs by using the predict method in the kriging object.

  var xnew, ynew /* Pair of new coordinates to predict */;
  var tpredicted = kriging.predict(xnew, ynew, variogram);
  

Creating a Map

Variogram and Probability Model

The various variogram models can be interpreted as kernel functions for 2-dimensional coordinates a, b and parameters nugget, range, sill and A. Reparameterized as a linear function, with w = [nugget, (sill-nugget)/range], this becomes:

• Gaussian: k(a,b) = w[0] + w[1] * ( 1 - exp{ -( ||a-b|| / range )² / A } )
• Exponential: k(a,b) = w[0] + w[1] * ( 1 - exp{ -( ||a-b|| / range ) / A } )
• Spherical: k(a,b) = w[0] + w[1] * ( 1.5 * ( ||a-b|| / range ) - 0.5 * ( ||a-b|| / range )³ )

The variance parameter α of the prior distribution for w should be manually set, according to:

• w ~ N(w|0, αI)

Using the fitted kernel function hyperparameters and setting K as the Gram matrix, the prior and likelihood for the gaussian process become:

• y ~ N(y|0, K)
• t|y ~ N(t|y, σ²I)

The variance parameter σ² of the likelihood reflects the error in the gaussian process and should be manually set.