t-viSNE/modules/pca/pca.js

var PCA = (function () {
    /**
     * The first step is to subtract the mean and center data
     * 
     * @param {Array} matrix - data in an mXn matrix format
     * @returns 
     */
    function computeDeviationMatrix(matrix) {
        var unit = unitSquareMatrix(matrix.length);
        return subtract(matrix, scale(multiply(unit, matrix), 1 / matrix.length));
    }
    /**
     * Computes variance from deviation
     * 
     * @param {Array} deviation - data minus mean as calculated from computeDeviationMatrix
     * @returns 
     */
    function computeDeviationScores(deviation) {
        var devSumOfSquares = multiply(transpose(deviation), deviation);
        return devSumOfSquares;
    }
    /**
     * Calculates the var covar square matrix using either population or sample
     * 
     * @param {Array} devSumOfSquares 
     * @param {boolean} sample - true/false whether data is from sample or not
     * @returns 
     */
    function computeVarianceCovariance(devSumOfSquares, sample) {
        var varianceCovariance;
        if (sample)
            varianceCovariance = scale(devSumOfSquares, 1 / (devSumOfSquares.length - 1));
        else
            varianceCovariance = scale(devSumOfSquares, 1 / (devSumOfSquares.length));
        return varianceCovariance;
    }
    /**
     * Matrix is the deviation sum of squares as computed earlier
     * 
     * @param {Array} matrix - output of computeDeviationScores
     * @returns 
     */
    function computeSVD(matrix) {
        var result = svd(matrix);
        var eigenvectors = result.U;
        var eigenvalues = result.S;
        var results = eigenvalues.map(function (value, i) {
            var obj = {};
            obj.eigenvalue = value;
            obj.vector = eigenvectors.map(function (vector, j) {
                return -1 * vector[i]; //HACK prevent completely negative vectors
            });
            return obj;
        });
        return results;
    }
    /**
     * Get reduced dataset after removing some dimensions
     * 
     * @param {Array} data - initial matrix started out with
     * @param {rest} vectors - eigenvectors selected as part of process
     * @returns 
     */
    function computeAdjustedData(data, ...vectorObjs) {
        //FIXME no need to transpose vectors since they're already in row normal form
        var vectors = vectorObjs.map(function(v){return v.vector});
        var matrixMinusMean = computeDeviationMatrix(data);
        var adjustedData = multiply(vectors, transpose(matrixMinusMean));
        var unit = unitSquareMatrix(data.length);
        var avgData = scale(multiply(unit, data), -1 / data.length); //NOTE get the averages to add back

        var formattedAdjustData = formatData(adjustedData, 2);
        return {
            adjustedData: adjustedData,
            formattedAdjustedData: formattedAdjustData,
            avgData: avgData,
            selectedVectors: vectors
        };
    }

    /**
     * Get original data set from reduced data set (decompress)
     * @param {*} adjustedData = formatted or unformatted adjusted data
     * @param {*} vectors = selectedVectors
     * @param {*} avgData = avgData
     */
    function computeOriginalData(adjustedData, vectors, avgData) {
        var originalWithoutMean = transpose(multiply(transpose(vectors), adjustedData));
        var originalWithMean = subtract(originalWithoutMean, avgData);
        var formattedData = formatData(originalWithMean, 2);
        return {
            originalData: originalWithMean,
            formattedOriginalData: formattedData
        }
    }

    /**
     * Get percentage explained, or loss
     * @param {*} vectors 
     * @param {*} selected 
     */
    function computePercentageExplained(vectors, ...selected) {
        var total = vectors.map(function (v) {
            return v.eigenvalue
        }).reduce(function (a, b) {
            return a + b;
        });
        var explained = selected.map(function (v) {
            return v.eigenvalue
        }).reduce(function (a, b) {
            return a + b;
        });
        return (explained / total);
    }

    function getEigenVectors(data) {
        return computeSVD(computeVarianceCovariance(computeDeviationScores(computeDeviationMatrix(data)), false));
    }

    function analyseTopResult(data) {
        var eigenVectors = getEigenVectors(data);
        var sorted = eigenVectors.sort(function (a, b) {
            return b.eigenvalue - a.eigenvalue;
        });
        console.log('Sorted Vectors', sorted);
        var selected = sorted[0].vector;
        return computeAdjustedData(data, selected);
    }

    function formatData(data, precision) {
        var TEN = Math.pow(10, precision || 2);
        return data.map(function (d, i) {
            return d.map(function (n) {
                return Math.round(n * TEN) / TEN;
            })
        })
    }
    /**
     * Multiplies AxB, where A and B are matrices of nXm and mXn dimensions
     * @param {} a 
     * @param {*} b 
     */
    function multiply(a, b) {
        if (!a[0] || !b[0] || !a.length || !b.length) {
            throw new Error('Both A and B should be matrices');
        }

        if (b.length !== a[0].length) {
            throw new Error('Columns in A should be the same as the number of rows in B');
        }
        var product = [];

        for (var i = 0; i < a.length; i++) {
            product[i] = []; //initialize a new row
            for (var j = 0; j < b[0].length; j++) {
                for (var k = 0; k < a[0].length; k++) {
                    (product[i])[j] = !!(product[i])[j] ? (product[i])[j] + (a[i])[k] * (b[k])[j] : (a[i])[k] * (b[k])[j];
                }
            }
        }
        return product;
    }
    /**
     * Utility function to subtract matrix b from a
     * 
     * @param {any} a 
     * @param {any} b 
     * @returns 
     */
    function subtract(a, b) {
        if (!(a.length === b.length && a[0].length === b[0].length))
            throw new Error('Both A and B should have the same dimensions');
        var result = [];
        for (var i = 0; i < a.length; i++) {
            result[i] = [];
            for (var j = 0; j < b[0].length; j++) {
                (result[i])[j] = (a[i])[j] - (b[i])[j];
            }
        }
        return result;
    }
    /**
     * Multiplies a matrix into a factor
     * 
     * @param {any} matrix 
     * @param {any} factor 
     * @returns 
     */
    function scale(matrix, factor) {
        var result = [];
        for (var i = 0; i < matrix.length; i++) {
            result[i] = [];
            for (var j = 0; j < matrix[0].length; j++) {
                (result[i])[j] = (matrix[i])[j] * factor;
            }
        }
        return result;
    }

    /**
     * Generates a unit square matrix
     * @param {*} rows = number of rows to fill
     */
    function unitSquareMatrix(rows) {
        var result = [];
        for (var i = 0; i < rows; i++) {
            result[i] = [];
            for (var j = 0; j < rows; j++) {
                (result[i])[j] = 1;
            }
        }
        return result;
    }
    /**
     * Transposes a matrix, converts rows to columns
     * @param {*} matrix 
     */
    function transpose(matrix) {
        var operated = clone(matrix);
        return operated[0].map(function (m, c) {
            return matrix.map(function (r) {
                return r[c];
            });
        });
    }
    /**
     * Deep Clones a matrix
     * @param {*} arr 
     */
    function clone(arr) {
        var string = JSON.stringify(arr);
        var result = JSON.parse(string);
        return result;
    }

    /**
     * Compute the thin SVD from G. H. Golub and C. Reinsch, Numer. Math. 14, 403-420 (1970)
     * From the Numeric JS Implementation Copyright (C) 2011 by Sébastien Loisel
     * The C implementation from which this has been taken may be found here: http://www.public.iastate.edu/~dicook/JSS/paper/code/svd.c
     * @param {*} A = m*n matrix
     */
    function svd(A) {
        var temp;
        var prec = Math.pow(2, -52) // assumes double prec
        var tolerance = 1.e-64 / prec;
        var itmax = 50;
        var c = 0;
        var i = 0;
        var j = 0;
        var k = 0;
        var l = 0;
        var u = clone(A);
        var m = u.length;
        var n = u[0].length;

        if (m < n) throw "Need more rows than columns"

        var e = new Array(n); //vector1
        var q = new Array(n); //vector2
        for (i = 0; i < n; i++) e[i] = q[i] = 0.0;
        var v = rep([n, n], 0);

        function pythag(a, b) {
            a = Math.abs(a)
            b = Math.abs(b)
            if (a > b)
                return a * Math.sqrt(1.0 + (b * b / a / a))
            else if (b == 0.0)
                return a
            return b * Math.sqrt(1.0 + (a * a / b / b))
        }

        //rep function
        function rep(s, v, k) {
            if (typeof k === "undefined") {
                k = 0;
            }
            var n = s[k],
                ret = Array(n),
                i;
            if (k === s.length - 1) {
                for (i = n - 2; i >= 0; i -= 2) {
                    ret[i + 1] = v;
                    ret[i] = v;
                }
                if (i === -1) {
                    ret[0] = v;
                }
                return ret;
            }
            for (i = n - 1; i >= 0; i--) {
                ret[i] = rep(s, v, k + 1);
            }
            return ret;
        }

        //Householder's reduction to bidiagonal form

        var f = 0.0;
        var g = 0.0;
        var h = 0.0;
        var x = 0.0;
        var y = 0.0;
        var z = 0.0;
        var s = 0.0;

        for (i = 0; i < n; i++) {
            e[i] = g; //vector
            s = 0.0; //sum
            l = i + 1; //stays i+1
            for (j = i; j < m; j++)
                s += (u[j][i] * u[j][i]);
            if (s <= tolerance)
                g = 0.0;
            else {
                f = u[i][i];
                g = Math.sqrt(s);
                if (f >= 0.0) g = -g;
                h = f * g - s
                u[i][i] = f - g;
                for (j = l; j < n; j++) {
                    s = 0.0
                    for (k = i; k < m; k++)
                        s += u[k][i] * u[k][j]
                    f = s / h
                    for (k = i; k < m; k++)
                        u[k][j] += f * u[k][i]
                }
            }
            q[i] = g
            s = 0.0
            for (j = l; j < n; j++)
                s = s + u[i][j] * u[i][j]
            if (s <= tolerance)
                g = 0.0
            else {
                f = u[i][i + 1]
                g = Math.sqrt(s)
                if (f >= 0.0) g = -g
                h = f * g - s
                u[i][i + 1] = f - g;
                for (j = l; j < n; j++) e[j] = u[i][j] / h
                for (j = l; j < m; j++) {
                    s = 0.0
                    for (k = l; k < n; k++)
                        s += (u[j][k] * u[i][k])
                    for (k = l; k < n; k++)
                        u[j][k] += s * e[k]
                }
            }
            y = Math.abs(q[i]) + Math.abs(e[i])
            if (y > x)
                x = y
        }

        // accumulation of right hand transformations
        for (i = n - 1; i != -1; i += -1) {
            if (g != 0.0) {
                h = g * u[i][i + 1]
                for (j = l; j < n; j++)
                    v[j][i] = u[i][j] / h //u is array, v is square of columns
                for (j = l; j < n; j++) {
                    s = 0.0
                    for (k = l; k < n; k++)
                        s += u[i][k] * v[k][j]
                    for (k = l; k < n; k++)
                        v[k][j] += (s * v[k][i])
                }
            }
            for (j = l; j < n; j++) {
                v[i][j] = 0;
                v[j][i] = 0;
            }
            v[i][i] = 1;
            g = e[i]
            l = i
        }

        // accumulation of left hand transformations
        for (i = n - 1; i != -1; i += -1) {
            l = i + 1
            g = q[i]
            for (j = l; j < n; j++)
                u[i][j] = 0;
            if (g != 0.0) {
                h = u[i][i] * g
                for (j = l; j < n; j++) {
                    s = 0.0
                    for (k = l; k < m; k++) s += u[k][i] * u[k][j];
                    f = s / h
                    for (k = i; k < m; k++) u[k][j] += f * u[k][i];
                }
                for (j = i; j < m; j++) u[j][i] = u[j][i] / g;
            } else
                for (j = i; j < m; j++) u[j][i] = 0;
            u[i][i] += 1;
        }

        // diagonalization of the bidiagonal form
        prec = prec * x
        for (k = n - 1; k != -1; k += -1) {
            for (var iteration = 0; iteration < itmax; iteration++) { // test f splitting
                var test_convergence = false
                for (l = k; l != -1; l += -1) {
                    if (Math.abs(e[l]) <= prec) {
                        test_convergence = true
                        break
                    }
                    if (Math.abs(q[l - 1]) <= prec)
                        break
                }
                if (!test_convergence) { // cancellation of e[l] if l>0
                    c = 0.0
                    s = 1.0
                    var l1 = l - 1
                    for (i = l; i < k + 1; i++) {
                        f = s * e[i]
                        e[i] = c * e[i]
                        if (Math.abs(f) <= prec)
                            break
                        g = q[i]
                        h = pythag(f, g)
                        q[i] = h
                        c = g / h
                        s = -f / h
                        for (j = 0; j < m; j++) {
                            y = u[j][l1]
                            z = u[j][i]
                            u[j][l1] = y * c + (z * s)
                            u[j][i] = -y * s + (z * c)
                        }
                    }
                }
                // test f convergence
                z = q[k]
                if (l == k) { //convergence
                    if (z < 0.0) { //q[k] is made non-negative
                        q[k] = -z
                        for (j = 0; j < n; j++)
                            v[j][k] = -v[j][k]
                    }
                    break //break out of iteration loop and move on to next k value
                }
                if (iteration >= itmax - 1)
                    throw 'Error: no convergence.'
                // shift from bottom 2x2 minor
                x = q[l]
                y = q[k - 1]
                g = e[k - 1]
                h = e[k]
                f = ((y - z) * (y + z) + (g - h) * (g + h)) / (2.0 * h * y)
                g = pythag(f, 1.0)
                if (f < 0.0)
                    f = ((x - z) * (x + z) + h * (y / (f - g) - h)) / x
                else
                    f = ((x - z) * (x + z) + h * (y / (f + g) - h)) / x
                // next QR transformation
                c = 1.0
                s = 1.0
                for (i = l + 1; i < k + 1; i++) {
                    g = e[i]
                    y = q[i]
                    h = s * g
                    g = c * g
                    z = pythag(f, h)
                    e[i - 1] = z
                    c = f / z
                    s = h / z
                    f = x * c + g * s
                    g = -x * s + g * c
                    h = y * s
                    y = y * c
                    for (j = 0; j < n; j++) {
                        x = v[j][i - 1]
                        z = v[j][i]
                        v[j][i - 1] = x * c + z * s
                        v[j][i] = -x * s + z * c
                    }
                    z = pythag(f, h)
                    q[i - 1] = z
                    c = f / z
                    s = h / z
                    f = c * g + s * y
                    x = -s * g + c * y
                    for (j = 0; j < m; j++) {
                        y = u[j][i - 1]
                        z = u[j][i]
                        u[j][i - 1] = y * c + z * s
                        u[j][i] = -y * s + z * c
                    }
                }
                e[l] = 0.0
                e[k] = f
                q[k] = x
            }
        }

        for (i = 0; i < q.length; i++)
            if (q[i] < prec) q[i] = 0

        //sort eigenvalues	
        for (i = 0; i < n; i++) {
            for (j = i - 1; j >= 0; j--) {
                if (q[j] < q[i]) {
                    c = q[j]
                    q[j] = q[i]
                    q[i] = c
                    for (k = 0; k < u.length; k++) {
                        temp = u[k][i];
                        u[k][i] = u[k][j];
                        u[k][j] = temp;
                    }
                    for (k = 0; k < v.length; k++) {
                        temp = v[k][i];
                        v[k][i] = v[k][j];
                        v[k][j] = temp;
                    }
                    i = j
                }
            }
        }

        return {
            U: u,
            S: q,
            V: v
        }
    }

    return {
        computeDeviationScores: computeDeviationScores,
        computeDeviationMatrix: computeDeviationMatrix,
        computeSVD: computeSVD,
        computePercentageExplained: computePercentageExplained,
        computeOriginalData: computeOriginalData,
        computeVarianceCovariance: computeVarianceCovariance,
        computeAdjustedData: computeAdjustedData,
        getEigenVectors: getEigenVectors,
        analyseTopResult: analyseTopResult,
        transpose: transpose,
        multiply: multiply,
        clone: clone,
        scale: scale
    }
})();

if(typeof module !== 'undefined')
module.exports = PCA;