glm
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Generalized Linear Models
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JavaScript
/******************************************************************\
For an m-by-n matrix A with m >= n, the Thin Singular Value Decomposition
(See: http://en.wikipedia.org/wiki/Singular_value_decomposition#Thin_SVD )
returns:
- an m-by-n matrix U with orthogonal columns,
- an n-by-n diagonal matrix S,
- and an n-by-n orthogonal matrix V
such that A = U*S*V'. (here V' indicates the transposed of V)
The function thinsvd(A), for m >=n, returns an array containing:
- an m x n bidimensional array U with orthogonal columns
- an m-sized unidimensional array containing the n singular values, sorted in descending order
- an n x n bidimensional orthogonal array containing V (NOTE: _not_ V' as NumPy's linalg.svd(A,full_matrices=0) does!!)
If m < n, it returns an array containing:
- an m x m bidimensional array containing U
- an m-sized unidimensional array containing the n singular values, sorted in descending order
- an m x n bidimensional array V with orthogonal columns
The Singular Values s[] allow to compute the following values of the input argument:
- Two norm l2n = s[0]
- Condition number cn = l2n / s[Math.min(m,n)-1]
- Rank r = number of singular values larger than cn * eps,
(eps being 2.22E-16 i.e. Math.pow(2.0,-52.0)
With square random matrices, complexity grows as O(n^3)
Decomposing a 100x100 matrix typically takes, with a
single-core 1.7GHz Pentium M:
- Chrome 3.0.195.27: 880 ms
- Firefox 3.5.4: 950 ms
- Safari 4.0.3: 1,360 ms
- MSIE 8.0: 9,554 ms (and three "slow script" warnings)
\******************************************************************/
var hypot = function(a, b) {
var at = Math.abs(a);
var bt = Math.abs(b);
var q;
if( at > bt ) {
q = bt / at;
return at * Math.sqrt( 1.0 + q*q );
} else {
if ( bt > 0.0 ){
q = at / bt;
return bt * Math.sqrt( 1.0 + q*q );
} else {
return 0.0;
}
}
};
exports.GLM.thinsvd = function (A) {
// Derived from JAMA public domain code: http://math.nist.gov/javanumerics/jama/
var i, j, k, t, f, g, cs, sn;
var lowrise = (A.length < A[0].length); // if true, then rows < columns. in that case, transpose A and exchange U and V on return
// make a copy, so the original matrix will be preserved
var AT = [];
if(lowrise) {
for(i=0; i<A[0].length; i++) {
AT[i] = [];
for(j=0; j<A.length; j++) {
AT[i][j] = A[j][i]; // swap rows with columns
}
}
} else {
for(i=0; i<A.length; i++) {
AT[i] = [];
for(j=0; j<A[0].length; j++) {
AT[i][j] = A[i][j]; // swap rows with columns
}
}
}
A = AT;
var m = A.length;
var n = A[0].length;
var nu = Math.min(m,n);
var s = [];
var U = [];
for(i=0; i<m; i++) {
U[i] = [];
for(j=0; j<n; j++) {
U[i][j] = 0.;
}
}
var V = [];
for(i=0; i<n; i++) {
V[i] = [];
}
var e = [];
var work = [];
var wantu = true;
var wantv = true;
// Reduce A to bidiagonal form, storing the diagonal elements
// in s and the super-diagonal elements in e.
var nct = Math.min(m-1,n);
var nrt = Math.max(0,Math.min(n-2,m));
for (k = 0; k < Math.max(nct,nrt); k++) {
if (k < nct) {
// Compute the transformation for the k-th column and
// place the k-th diagonal in s[k].
// Compute 2-norm of k-th column without under/overflow.
s[k] = 0;
for (i = k; i < m; i++) {
s[k] = hypot(s[k],A[i][k]);
}
if (s[k] !== 0.0) {
if (A[k][k] < 0.0) {
s[k] = -s[k];
}
for (i = k; i < m; i++) {
A[i][k] /= s[k];
}
A[k][k] += 1.0;
}
s[k] = -s[k];
}
for (j = k+1; j < n; j++) {
if ((k < nct) && (s[k] !== 0.0)) {
// Apply the transformation.
t = 0;
for (i = k; i < m; i++) {
t += A[i][k]*A[i][j];
}
t = -t/A[k][k];
for (i = k; i < m; i++) {
A[i][j] += t*A[i][k];
}
}
// Place the k-th row of A into e for the
// subsequent calculation of the row transformation.
e[j] = A[k][j];
}
if (wantu && (k < nct)) {
// Place the transformation in U for subsequent back
// multiplication.
for (i = k; i < m; i++) {
U[i][k] = A[i][k];
}
}
if (k < nrt) {
// Compute the k-th row transformation and place the
// k-th super-diagonal in e[k].
// Compute 2-norm without under/overflow.
e[k] = 0;
for (i = k+1; i < n; i++) {
e[k] = hypot(e[k],e[i]);
}
if (e[k] !== 0.0) {
if (e[k+1] < 0.0) {
e[k] = -e[k];
}
for (i = k+1; i < n; i++) {
e[i] /= e[k];
}
e[k+1] += 1.0;
}
e[k] = -e[k];
if ((k+1 < m) && (e[k] !== 0.0)) {
// Apply the transformation.
for (i = k+1; i < m; i++) {
work[i] = 0.0;
}
for (j = k+1; j < n; j++) {
for (i = k+1; i < m; i++) {
work[i] += e[j]*A[i][j];
}
}
for (j = k+1; j < n; j++) {
t = -e[j]/e[k+1];
for (i = k+1; i < m; i++) {
A[i][j] += t*work[i];
}
}
}
if (wantv) {
// Place the transformation in V for subsequent
// back multiplication.
for (i = k+1; i < n; i++) {
V[i][k] = e[i];
}
}
}
}
// Set up the final bidiagonal matrix or order p.
var p = Math.min(n,m+1);
if (nct < n) {
s[nct] = A[nct][nct];
}
if (m < p) {
s[p-1] = 0.0;
}
if (nrt+1 < p) {
e[nrt] = A[nrt][p-1];
}
e[p-1] = 0.0;
// If required, generate U.
if (wantu) {
for (j = nct; j < nu; j++) {
for (i = 0; i < m; i++) {
U[i][j] = 0.0;
}
U[j][j] = 1.0;
}
for (k = nct-1; k >= 0; k--) {
if (s[k] !== 0.0) {
for (j = k+1; j < nu; j++) {
t = 0;
for (i = k; i < m; i++) {
t += U[i][k]*U[i][j];
}
t = -t/U[k][k];
for (i = k; i < m; i++) {
U[i][j] += t*U[i][k];
}
}
for (i = k; i < m; i++ ) {
U[i][k] = -U[i][k];
}
U[k][k] = 1.0 + U[k][k];
for (i = 0; i < k-1; i++) {
U[i][k] = 0.0;
}
} else {
for (i = 0; i < m; i++) {
U[i][k] = 0.0;
}
U[k][k] = 1.0;
}
}
}
// If required, generate V.
if (wantv) {
for (k = n-1; k >= 0; k--) {
if ((k < nrt) && (e[k] !== 0.0)) {
for (j = k+1; j < nu; j++) {
t = 0;
for (i = k+1; i < n; i++) {
t += V[i][k]*V[i][j];
}
t = -t/V[k+1][k];
for (i = k+1; i < n; i++) {
V[i][j] += t*V[i][k];
}
}
}
for (i = 0; i < n; i++) {
V[i][k] = 0.0;
}
V[k][k] = 1.0;
}
}
// Main iteration loop for the singular values.
var pp = p-1;
var iter = 0;
var totiter = 0;
var eps = 2.2205E-16; // Math.pow(2.0,-52.0);
var tiny = 1.6034E-291; // Math.pow(2.0,-966.0);
while (p > 0) {
var kase;
// Here is where a test for too many iterations would go.
// This section of the program inspects for
// negligible elements in the s and e arrays. On
// completion the variables kase and k are set as follows.
// kase = 1 if s(p) and e[k-1] are negligible and k<p
// kase = 2 if s(k) is negligible and k<p
// kase = 3 if e[k-1] is negligible, k<p, and
// s(k), ..., s(p) are not negligible (qr step).
// kase = 4 if e(p-1) is negligible (convergence).
for (k = p-2; k >= -1; k--) {
if (k == -1) {
break;
}
if (Math.abs(e[k]) <= tiny + eps*(Math.abs(s[k]) + Math.abs(s[k+1]))) {
e[k] = 0.0;
break;
}
}
if (k == p-2) {
kase = 4;
} else {
var ks;
for (ks = p-1; ks >= k; ks--) {
if (ks == k) {
break;
}
t = (ks != p ? Math.abs(e[ks]) : 0.0) + (ks != k+1 ? Math.abs(e[ks-1]) : 0.0);
if (Math.abs(s[ks]) <= tiny + eps*t) {
s[ks] = 0.0;
break;
}
}
if (ks == k) {
kase = 3;
} else if (ks == p-1) {
kase = 1;
} else {
kase = 2;
k = ks;
}
}
k++;
// Perform the task indicated by kase.
if(kase == 1) {
// Deflate negligible s(p).
f = e[p-2];
e[p-2] = 0.0;
for (j = p-2; j >= k; j--) {
t = hypot(s[j],f);
cs = s[j]/t;
sn = f/t;
s[j] = t;
if (j != k) {
f = -sn*e[j-1];
e[j-1] = cs*e[j-1];
}
if (wantv) {
for (i = 0; i < n; i++) {
t = cs*V[i][j] + sn*V[i][p-1];
V[i][p-1] = -sn*V[i][j] + cs*V[i][p-1];
V[i][j] = t;
}
}
}
} else if (kase == 2) {
f = e[k-1];
e[k-1] = 0.0;
for (j = k; j < p; j++) {
t = hypot(s[j],f);
cs = s[j]/t;
sn = f/t;
s[j] = t;
f = -sn*e[j];
e[j] = cs*e[j];
if (wantu) {
for (i = 0; i < m; i++) {
t = cs*U[i][j] + sn*U[i][k-1];
U[i][k-1] = -sn*U[i][j] + cs*U[i][k-1];
U[i][j] = t;
}
}
}
} else if (kase == 3) {
// Calculate the shift.
var scale = Math.max(Math.max(Math.max(Math.max(
Math.abs(s[p-1]),Math.abs(s[p-2])),Math.abs(e[p-2])),
Math.abs(s[k])),Math.abs(e[k]));
var sp = s[p-1]/scale;
var spm1 = s[p-2]/scale;
var epm1 = e[p-2]/scale;
var sk = s[k]/scale;
var ek = e[k]/scale;
var b = ((spm1 + sp)*(spm1 - sp) + epm1*epm1)/2.0;
var c = (sp*epm1)*(sp*epm1);
var shift = 0.0;
if ((b !== 0.0) || (c !== 0.0)) {
shift = Math.sqrt(b*b + c);
if (b < 0.0) {
shift = -shift;
}
shift = c/(b + shift);
}
f = (sk + sp)*(sk - sp) + shift;
g = sk*ek;
// Chase zeros.
for (j = k; j < p-1; j++) {
t = hypot(f,g);
cs = f/t;
sn = g/t;
if (j != k) {
e[j-1] = t;
}
f = cs*s[j] + sn*e[j];
e[j] = cs*e[j] - sn*s[j];
g = sn*s[j+1];
s[j+1] = cs*s[j+1];
if (wantv) {
for (i = 0; i < n; i++) {
t = cs*V[i][j] + sn*V[i][j+1];
V[i][j+1] = -sn*V[i][j] + cs*V[i][j+1];
V[i][j] = t;
}
}
t = hypot(f,g);
cs = f/t;
sn = g/t;
s[j] = t;
f = cs*e[j] + sn*s[j+1];
s[j+1] = -sn*e[j] + cs*s[j+1];
g = sn*e[j+1];
e[j+1] = cs*e[j+1];
if (wantu && (j < m-1)) {
for (i = 0; i < m; i++) {
t = cs*U[i][j] + sn*U[i][j+1];
U[i][j+1] = -sn*U[i][j] + cs*U[i][j+1];
U[i][j] = t;
}
}
}
e[p-2] = f;
iter = iter + 1;
} else if(kase == 4) {
// Convergence.
// Make the singular values positive.
if (s[k] <= 0.0) {
s[k] = (s[k] < 0.0 ? -s[k] : 0.0);
if (wantv) {
for (i = 0; i <= pp; i++) {
V[i][k] = -V[i][k];
}
}
}
// Order the singular values.
while (k < pp) {
if (s[k] >= s[k+1]) {
break;
}
t = s[k];
s[k] = s[k+1];
s[k+1] = t;
if (wantv && (k < n-1)) {
for (i = 0; i < n; i++) {
t = V[i][k+1]; V[i][k+1] = V[i][k]; V[i][k] = t;
}
}
if (wantu && (k < m-1)) {
for (i = 0; i < m; i++) {
t = U[i][k+1]; U[i][k+1] = U[i][k]; U[i][k] = t;
}
}
k++;
}
totiter += iter;
iter = 0;
p--;
}
}
if(lowrise) {
return [V,s,U,totiter];
} else {
return [U,s,V,totiter];
}
/*
Two norm: s[0]
Two norm condition number: s[0]/s[Math.min(m,n)-1]
Rank:
function rank (s) {
var eps = 2.22E-16; // Math.pow(2.0,-52.0);
tol = Math.max(m,n)*s[0]*eps;
var r = 0;
for (i = 0; i < s.length; i++) {
if (s[i] > tol) {
r++;
}
}
return r;
}
*/
}