kriging.js
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Javascript library for geospatial prediction and mapping via ordinary kriging http://oeo4b.github.io
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JavaScript
// Extend the Array class
Array.prototype.max = function() {
return Math.max.apply(null, this);
};
Array.prototype.min = function() {
return Math.min.apply(null, this);
};
Array.prototype.mean = function() {
var i, sum;
for(i=0,sum=0;i<this.length;i++)
sum += this[i];
return sum / this.length;
};
Array.prototype.pip = function(x, y) {
var i, j, c = false;
for(i=0,j=this.length-1;i<this.length;j=i++) {
if( ((this[i][1]>y) != (this[j][1]>y)) &&
(x<(this[j][0]-this[i][0]) * (y-this[i][1]) / (this[j][1]-this[i][1]) + this[i][0]) ) {
c = !c;
}
}
return c;
}
var kriging = function() {
var kriging = {};
var createArrayWithValues = function(value, n) {
var array = [];
for ( var i = 0; i < n; i++) {
array.push(value);
}
return array;
};
// Matrix algebra
kriging_matrix_diag = function(c, n) {
var Z = createArrayWithValues(0, n * n);
for(i=0;i<n;i++) Z[i*n+i] = c;
return Z;
};
kriging_matrix_transpose = function(X, n, m) {
var i, j, Z = Array(m*n);
for(i=0;i<n;i++)
for(j=0;j<m;j++)
Z[j*n+i] = X[i*m+j];
return Z;
};
kriging_matrix_scale = function(X, c, n, m) {
var i, j;
for(i=0;i<n;i++)
for(j=0;j<m;j++)
X[i*m+j] *= c;
};
kriging_matrix_add = function(X, Y, n, m) {
var i, j, Z = Array(n*m);
for(i=0;i<n;i++)
for(j=0;j<m;j++)
Z[i*m+j] = X[i*m+j] + Y[i*m+j];
return Z;
};
// Naive matrix multiplication
kriging_matrix_multiply = function(X, Y, n, m, p) {
var i, j, k, Z = Array(n*p);
for(i=0;i<n;i++) {
for(j=0;j<p;j++) {
Z[i*p+j] = 0;
for(k=0;k<m;k++)
Z[i*p+j] += X[i*m+k]*Y[k*p+j];
}
}
return Z;
};
// Cholesky decomposition
kriging_matrix_chol = function(X, n) {
var i, j, k, sum, p = Array(n);
for(i=0;i<n;i++) p[i] = X[i*n+i];
for(i=0;i<n;i++) {
for(j=0;j<i;j++)
p[i] -= X[i*n+j]*X[i*n+j];
if(p[i]<=0) return false;
p[i] = Math.sqrt(p[i]);
for(j=i+1;j<n;j++) {
for(k=0;k<i;k++)
X[j*n+i] -= X[j*n+k]*X[i*n+k];
X[j*n+i] /= p[i];
}
}
for(i=0;i<n;i++) X[i*n+i] = p[i];
return true;
};
// Inversion of cholesky decomposition
kriging_matrix_chol2inv = function(X, n) {
var i, j, k, sum;
for(i=0;i<n;i++) {
X[i*n+i] = 1/X[i*n+i];
for(j=i+1;j<n;j++) {
sum = 0;
for(k=i;k<j;k++)
sum -= X[j*n+k]*X[k*n+i];
X[j*n+i] = sum/X[j*n+j];
}
}
for(i=0;i<n;i++)
for(j=i+1;j<n;j++)
X[i*n+j] = 0;
for(i=0;i<n;i++) {
X[i*n+i] *= X[i*n+i];
for(k=i+1;k<n;k++)
X[i*n+i] += X[k*n+i]*X[k*n+i];
for(j=i+1;j<n;j++)
for(k=j;k<n;k++)
X[i*n+j] += X[k*n+i]*X[k*n+j];
}
for(i=0;i<n;i++)
for(j=0;j<i;j++)
X[i*n+j] = X[j*n+i];
};
// Inversion via gauss-jordan elimination
kriging_matrix_solve = function(X, n) {
var m = n;
var b = Array(n*n);
var indxc = Array(n);
var indxr = Array(n);
var ipiv = Array(n);
var i, icol, irow, j, k, l, ll;
var big, dum, pivinv, temp;
for(i=0;i<n;i++)
for(j=0;j<n;j++) {
if(i==j) b[i*n+j] = 1;
else b[i*n+j] = 0;
}
for(j=0;j<n;j++) ipiv[j] = 0;
for(i=0;i<n;i++) {
big = 0;
for(j=0;j<n;j++) {
if(ipiv[j]!=1) {
for(k=0;k<n;k++) {
if(ipiv[k]==0) {
if(Math.abs(X[j*n+k])>=big) {
big = Math.abs(X[j*n+k]);
irow = j;
icol = k;
}
}
}
}
}
++(ipiv[icol]);
if(irow!=icol) {
for(l=0;l<n;l++) {
temp = X[irow*n+l];
X[irow*n+l] = X[icol*n+l];
X[icol*n+l] = temp;
}
for(l=0;l<m;l++) {
temp = b[irow*n+l];
b[irow*n+l] = b[icol*n+l];
b[icol*n+l] = temp;
}
}
indxr[i] = irow;
indxc[i] = icol;
if(X[icol*n+icol]==0) return false; // Singular
pivinv = 1 / X[icol*n+icol];
X[icol*n+icol] = 1;
for(l=0;l<n;l++) X[icol*n+l] *= pivinv;
for(l=0;l<m;l++) b[icol*n+l] *= pivinv;
for(ll=0;ll<n;ll++) {
if(ll!=icol) {
dum = X[ll*n+icol];
X[ll*n+icol] = 0;
for(l=0;l<n;l++) X[ll*n+l] -= X[icol*n+l]*dum;
for(l=0;l<m;l++) b[ll*n+l] -= b[icol*n+l]*dum;
}
}
}
for(l=(n-1);l>=0;l--)
if(indxr[l]!=indxc[l]) {
for(k=0;k<n;k++) {
temp = X[k*n+indxr[l]];
X[k*n+indxr[l]] = X[k*n+indxc[l]];
X[k*n+indxc[l]] = temp;
}
}
return true;
}
// Variogram models
kriging_variogram_gaussian = function(h, nugget, range, sill, A) {
return nugget + ((sill-nugget)/range)*
( 1.0 - Math.exp(-(1.0/A)*Math.pow(h/range, 2)) );
};
kriging_variogram_exponential = function(h, nugget, range, sill, A) {
return nugget + ((sill-nugget)/range)*
( 1.0 - Math.exp(-(1.0/A) * (h/range)) );
};
kriging_variogram_spherical = function(h, nugget, range, sill, A) {
if(h>range) return nugget + (sill-nugget)/range;
return nugget + ((sill-nugget)/range)*
( 1.5*(h/range) - 0.5*Math.pow(h/range, 3) );
};
// Train using gaussian processes with bayesian priors
kriging.train = function(t, x, y, model, sigma2, alpha) {
var variogram = {
t : t,
x : x,
y : y,
nugget : 0.0,
range : 0.0,
sill : 0.0,
A : 1/3,
n : 0
};
switch(model) {
case "gaussian":
variogram.model = kriging_variogram_gaussian;
break;
case "exponential":
variogram.model = kriging_variogram_exponential;
break;
case "spherical":
variogram.model = kriging_variogram_spherical;
break;
};
// Lag distance/semivariance
var i, j, k, l, n = t.length;
var distance = Array((n*n-n)/2);
for(i=0,k=0;i<n;i++)
for(j=0;j<i;j++,k++) {
distance[k] = Array(2);
distance[k][0] = Math.pow(
Math.pow(x[i]-x[j], 2)+
Math.pow(y[i]-y[j], 2), 0.5);
distance[k][1] = Math.abs(t[i]-t[j]);
}
distance.sort(function(a, b) { return a[0] - b[0]; });
variogram.range = distance[(n*n-n)/2-1][0];
// Bin lag distance
var lags = ((n*n-n)/2)>30?30:(n*n-n)/2;
var tolerance = variogram.range/lags;
var lag = createArrayWithValues(0,lags);
var semi = createArrayWithValues(0,lags);
if(lags<30) {
for(l=0;l<lags;l++) {
lag[l] = distance[l][0];
semi[l] = distance[l][1];
}
}
else {
for(i=0,j=0,k=0,l=0;i<lags&&j<((n*n-n)/2);i++,k=0) {
while( distance[j][0]<=((i+1)*tolerance) ) {
lag[l] += distance[j][0];
semi[l] += distance[j][1];
j++;k++;
if(j>=((n*n-n)/2)) break;
}
if(k>0) {
lag[l] /= k;
semi[l] /= k;
l++;
}
}
if(l<2) return variogram; // Error: Not enough points
}
// Feature transformation
n = l;
variogram.range = lag[n-1]-lag[0];
var X = createArrayWithValues(1,2 * n);
var Y = Array(n);
var A = variogram.A;
for(i=0;i<n;i++) {
switch(model) {
case "gaussian":
X[i*2+1] = 1.0-Math.exp(-(1.0/A)*Math.pow(lag[i]/variogram.range, 2));
break;
case "exponential":
X[i*2+1] = 1.0-Math.exp(-(1.0/A)*lag[i]/variogram.range);
break;
case "spherical":
X[i*2+1] = 1.5*(lag[i]/variogram.range)-
0.5*Math.pow(lag[i]/variogram.range, 3);
break;
};
Y[i] = semi[i];
}
// Least squares
var Xt = kriging_matrix_transpose(X, n, 2);
var Z = kriging_matrix_multiply(Xt, X, 2, n, 2);
Z = kriging_matrix_add(Z, kriging_matrix_diag(1/alpha, 2), 2, 2);
var cloneZ = Z.slice(0);
if(kriging_matrix_chol(Z, 2))
kriging_matrix_chol2inv(Z, 2);
else {
kriging_matrix_solve(cloneZ, 2);
Z = cloneZ;
}
var W = kriging_matrix_multiply(kriging_matrix_multiply(Z, Xt, 2, 2, n), Y, 2, n, 1);
// Variogram parameters
variogram.nugget = W[0];
variogram.sill = W[1]*variogram.range+variogram.nugget;
variogram.n = x.length;
// Gram matrix with prior
n = x.length;
var K = Array(n*n);
for(i=0;i<n;i++) {
for(j=0;j<i;j++) {
K[i*n+j] = variogram.model(Math.pow(Math.pow(x[i]-x[j], 2)+
Math.pow(y[i]-y[j], 2), 0.5),
variogram.nugget,
variogram.range,
variogram.sill,
variogram.A);
K[j*n+i] = K[i*n+j];
}
K[i*n+i] = variogram.model(0, variogram.nugget,
variogram.range,
variogram.sill,
variogram.A);
}
// Inverse penalized Gram matrix projected to target vector
var C = kriging_matrix_add(K, kriging_matrix_diag(sigma2, n), n, n);
var cloneC = C.slice(0);
if(kriging_matrix_chol(C, n))
kriging_matrix_chol2inv(C, n);
else {
kriging_matrix_solve(cloneC, n);
C = cloneC;
}
// Copy unprojected inverted matrix as K
var K = C.slice(0);
var M = kriging_matrix_multiply(C, t, n, n, 1);
variogram.K = K;
variogram.M = M;
return variogram;
};
// Model prediction
kriging.predict = function(x, y, variogram) {
var i, k = Array(variogram.n);
for(i=0;i<variogram.n;i++)
k[i] = variogram.model(Math.pow(Math.pow(x-variogram.x[i], 2)+
Math.pow(y-variogram.y[i], 2), 0.5),
variogram.nugget, variogram.range,
variogram.sill, variogram.A);
return kriging_matrix_multiply(k, variogram.M, 1, variogram.n, 1)[0];
};
kriging.variance = function(x, y, variogram) {
var i, k = Array(variogram.n);
for(i=0;i<variogram.n;i++)
k[i] = variogram.model(Math.pow(Math.pow(x-variogram.x[i], 2)+
Math.pow(y-variogram.y[i], 2), 0.5),
variogram.nugget, variogram.range,
variogram.sill, variogram.A);
return variogram.model(0, variogram.nugget, variogram.range,
variogram.sill, variogram.A)+
kriging_matrix_multiply(kriging_matrix_multiply(k, variogram.K,
1, variogram.n, variogram.n),
k, 1, variogram.n, 1)[0];
};
// Gridded matrices or contour paths
kriging.grid = function(polygons, variogram, width) {
var i, j, k, n = polygons.length;
if(n==0) return;
// Boundaries of polygons space
var xlim = [polygons[0][0][0], polygons[0][0][0]];
var ylim = [polygons[0][0][1], polygons[0][0][1]];
for(i=0;i<n;i++) // Polygons
for(j=0;j<polygons[i].length;j++) { // Vertices
if(polygons[i][j][0]<xlim[0])
xlim[0] = polygons[i][j][0];
if(polygons[i][j][0]>xlim[1])
xlim[1] = polygons[i][j][0];
if(polygons[i][j][1]<ylim[0])
ylim[0] = polygons[i][j][1];
if(polygons[i][j][1]>ylim[1])
ylim[1] = polygons[i][j][1];
}
// Alloc for O(n^2) space
var xtarget, ytarget;
var a = Array(2), b = Array(2);
var lxlim = Array(2); // Local dimensions
var lylim = Array(2); // Local dimensions
var x = Math.ceil((xlim[1]-xlim[0])/width);
var y = Math.ceil((ylim[1]-ylim[0])/width);
var A = Array(x+1);
for(i=0;i<=x;i++) A[i] = Array(y+1);
for(i=0;i<n;i++) {
// Range for polygons[i]
lxlim[0] = polygons[i][0][0];
lxlim[1] = lxlim[0];
lylim[0] = polygons[i][0][1];
lylim[1] = lylim[0];
for(j=1;j<polygons[i].length;j++) { // Vertices
if(polygons[i][j][0]<lxlim[0])
lxlim[0] = polygons[i][j][0];
if(polygons[i][j][0]>lxlim[1])
lxlim[1] = polygons[i][j][0];
if(polygons[i][j][1]<lylim[0])
lylim[0] = polygons[i][j][1];
if(polygons[i][j][1]>lylim[1])
lylim[1] = polygons[i][j][1];
}
// Loop through polygon subspace
a[0] = Math.floor(((lxlim[0]-((lxlim[0]-xlim[0])%width)) - xlim[0])/width);
a[1] = Math.ceil(((lxlim[1]-((lxlim[1]-xlim[1])%width)) - xlim[0])/width);
b[0] = Math.floor(((lylim[0]-((lylim[0]-ylim[0])%width)) - ylim[0])/width);
b[1] = Math.ceil(((lylim[1]-((lylim[1]-ylim[1])%width)) - ylim[0])/width);
for(j=a[0];j<=a[1];j++)
for(k=b[0];k<=b[1];k++) {
xtarget = xlim[0] + j*width;
ytarget = ylim[0] + k*width;
if(polygons[i].pip(xtarget, ytarget))
A[j][k] = kriging.predict(xtarget,
ytarget,
variogram);
}
}
A.xlim = xlim;
A.ylim = ylim;
A.zlim = [variogram.t.min(), variogram.t.max()];
A.width = width;
return A;
};
kriging.contour = function(value, polygons, variogram) {
};
// Plotting on the DOM
kriging.plot = function(canvas, grid, xlim, ylim, colors) {
// Clear screen
var ctx = canvas.getContext("2d");
ctx.clearRect(0, 0, canvas.width, canvas.height);
// Starting boundaries
var range = [xlim[1]-xlim[0], ylim[1]-ylim[0], grid.zlim[1]-grid.zlim[0]];
var i, j, x, y, z;
var n = grid.length;
var m = grid[0].length;
var wx = Math.ceil(grid.width*canvas.width/(xlim[1]-xlim[0]));
var wy = Math.ceil(grid.width*canvas.height/(ylim[1]-ylim[0]));
for(i=0;i<n;i++)
for(j=0;j<m;j++) {
if(grid[i][j]==undefined) continue;
x = canvas.width*(i*grid.width+grid.xlim[0]-xlim[0])/range[0];
y = canvas.height*(1-(j*grid.width+grid.ylim[0]-ylim[0])/range[1]);
z = (grid[i][j]-grid.zlim[0])/range[2];
if(z<0.0) z = 0.0;
if(z>1.0) z = 1.0;
ctx.fillStyle = colors[Math.floor((colors.length-1)*z)];
ctx.fillRect(Math.round(x-wx/2), Math.round(y-wy/2), wx, wy);
}
};
return kriging;
}();
if (module && module.exports){
module.exports = kriging;
}