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scinote-web/vendor/assets/javascripts/jstat.js
Jure Grabnar e5b2f29b42 Add libraries for HoT formulas
2016-09-13 09:44:17 +02:00

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this.j$ = this.jStat = (function(Math, undefined) {
// For quick reference.
var concat = Array.prototype.concat;
var slice = Array.prototype.slice;
var toString = Object.prototype.toString;
// Calculate correction for IEEE error
// TODO: This calculation can be improved.
function calcRdx(n, m) {
var val = n > m ? n : m;
return Math.pow(10,
17 - ~~(Math.log(((val > 0) ? val : -val)) * Math.LOG10E));
}
var isArray = Array.isArray || function isArray(arg) {
return toString.call(arg) === '[object Array]';
};
function isFunction(arg) {
return toString.call(arg) === '[object Function]';
}
function isNumber(arg) {
return typeof arg === 'number' && arg === arg;
}
// Converts the jStat matrix to vector.
function toVector(arr) {
return concat.apply([], arr);
}
// The one and only jStat constructor.
function jStat() {
return new jStat._init(arguments);
}
// TODO: Remove after all references in src files have been removed.
jStat.fn = jStat.prototype;
// By separating the initializer from the constructor it's easier to handle
// always returning a new instance whether "new" was used or not.
jStat._init = function _init(args) {
var i;
// If first argument is an array, must be vector or matrix.
if (isArray(args[0])) {
// Check if matrix.
if (isArray(args[0][0])) {
// See if a mapping function was also passed.
if (isFunction(args[1]))
args[0] = jStat.map(args[0], args[1]);
// Iterate over each is faster than this.push.apply(this, args[0].
for (i = 0; i < args[0].length; i++)
this[i] = args[0][i];
this.length = args[0].length;
// Otherwise must be a vector.
} else {
this[0] = isFunction(args[1]) ? jStat.map(args[0], args[1]) : args[0];
this.length = 1;
}
// If first argument is number, assume creation of sequence.
} else if (isNumber(args[0])) {
this[0] = jStat.seq.apply(null, args);
this.length = 1;
// Handle case when jStat object is passed to jStat.
} else if (args[0] instanceof jStat) {
// Duplicate the object and pass it back.
return jStat(args[0].toArray());
// Unexpected argument value, return empty jStat object.
// TODO: This is strange behavior. Shouldn't this throw or some such to let
// the user know they had bad arguments?
} else {
this[0] = [];
this.length = 1;
}
return this;
};
jStat._init.prototype = jStat.prototype;
jStat._init.constructor = jStat;
// Utility functions.
// TODO: for internal use only?
jStat.utils = {
calcRdx: calcRdx,
isArray: isArray,
isFunction: isFunction,
isNumber: isNumber,
toVector: toVector
};
// Easily extend the jStat object.
// TODO: is this seriously necessary?
jStat.extend = function extend(obj) {
var i, j;
if (arguments.length === 1) {
for (j in obj)
jStat[j] = obj[j];
return this;
}
for (i = 1; i < arguments.length; i++) {
for (j in arguments[i])
obj[j] = arguments[i][j];
}
return obj;
};
// Returns the number of rows in the matrix.
jStat.rows = function rows(arr) {
return arr.length || 1;
};
// Returns the number of columns in the matrix.
jStat.cols = function cols(arr) {
return arr[0].length || 1;
};
// Returns the dimensions of the object { rows: i, cols: j }
jStat.dimensions = function dimensions(arr) {
return {
rows: jStat.rows(arr),
cols: jStat.cols(arr)
};
};
// Returns a specified row as a vector
jStat.row = function row(arr, index) {
return arr[index];
};
// Returns the specified column as a vector
jStat.col = function cols(arr, index) {
var column = new Array(arr.length);
for (var i = 0; i < arr.length; i++)
column[i] = [arr[i][index]];
return column;
};
// Returns the diagonal of the matrix
jStat.diag = function diag(arr) {
var nrow = jStat.rows(arr);
var res = new Array(nrow);
for (var row = 0; row < nrow; row++)
res[row] = [arr[row][row]];
return res;
};
// Returns the anti-diagonal of the matrix
jStat.antidiag = function antidiag(arr) {
var nrow = jStat.rows(arr) - 1;
var res = new Array(nrow);
for (var i = 0; nrow >= 0; nrow--, i++)
res[i] = [arr[i][nrow]];
return res;
};
// Transpose a matrix or array.
jStat.transpose = function transpose(arr) {
var obj = [];
var objArr, rows, cols, j, i;
// Make sure arr is in matrix format.
if (!isArray(arr[0]))
arr = [arr];
rows = arr.length;
cols = arr[0].length;
for (i = 0; i < cols; i++) {
objArr = new Array(rows);
for (j = 0; j < rows; j++)
objArr[j] = arr[j][i];
obj.push(objArr);
}
// If obj is vector, return only single array.
return obj.length === 1 ? obj[0] : obj;
};
// Map a function to an array or array of arrays.
// "toAlter" is an internal variable.
jStat.map = function map(arr, func, toAlter) {
var row, nrow, ncol, res, col;
if (!isArray(arr[0]))
arr = [arr];
nrow = arr.length;
ncol = arr[0].length;
res = toAlter ? arr : new Array(nrow);
for (row = 0; row < nrow; row++) {
// if the row doesn't exist, create it
if (!res[row])
res[row] = new Array(ncol);
for (col = 0; col < ncol; col++)
res[row][col] = func(arr[row][col], row, col);
}
return res.length === 1 ? res[0] : res;
};
// Destructively alter an array.
jStat.alter = function alter(arr, func) {
return jStat.map(arr, func, true);
};
// Generate a rows x cols matrix according to the supplied function.
jStat.create = function create(rows, cols, func) {
var res = new Array(rows);
var i, j;
if (isFunction(cols)) {
func = cols;
cols = rows;
}
for (i = 0; i < rows; i++) {
res[i] = new Array(cols);
for (j = 0; j < cols; j++)
res[i][j] = func(i, j);
}
return res;
};
function retZero() { return 0; }
// Generate a rows x cols matrix of zeros.
jStat.zeros = function zeros(rows, cols) {
if (!isNumber(cols))
cols = rows;
return jStat.create(rows, cols, retZero);
};
function retOne() { return 1; }
// Generate a rows x cols matrix of ones.
jStat.ones = function ones(rows, cols) {
if (!isNumber(cols))
cols = rows;
return jStat.create(rows, cols, retOne);
};
// Generate a rows x cols matrix of uniformly random numbers.
jStat.rand = function rand(rows, cols) {
if (!isNumber(cols))
cols = rows;
return jStat.create(rows, cols, Math.random);
};
function retIdent(i, j) { return i === j ? 1 : 0; }
// Generate an identity matrix of size row x cols.
jStat.identity = function identity(rows, cols) {
if (!isNumber(cols))
cols = rows;
return jStat.create(rows, cols, retIdent);
};
// Tests whether a matrix is symmetric
jStat.symmetric = function symmetric(arr) {
var issymmetric = true;
var size = arr.length;
var row, col;
if (arr.length !== arr[0].length)
return false;
for (row = 0; row < size; row++) {
for (col = 0; col < size; col++)
if (arr[col][row] !== arr[row][col])
return false;
}
return true;
};
// Set all values to zero.
jStat.clear = function clear(arr) {
return jStat.alter(arr, retZero);
};
// Generate sequence.
jStat.seq = function seq(min, max, length, func) {
if (!isFunction(func))
func = false;
var arr = [];
var hival = calcRdx(min, max);
var step = (max * hival - min * hival) / ((length - 1) * hival);
var current = min;
var cnt;
// Current is assigned using a technique to compensate for IEEE error.
// TODO: Needs better implementation.
for (cnt = 0;
current <= max;
cnt++, current = (min * hival + step * hival * cnt) / hival) {
arr.push((func ? func(current, cnt) : current));
}
return arr;
};
// TODO: Go over this entire implementation. Seems a tragic waste of resources
// doing all this work. Instead, and while ugly, use new Function() to generate
// a custom function for each static method.
// Quick reference.
var jProto = jStat.prototype;
// Default length.
jProto.length = 0;
// For internal use only.
// TODO: Check if they're actually used, and if they are then rename them
// to _*
jProto.push = Array.prototype.push;
jProto.sort = Array.prototype.sort;
jProto.splice = Array.prototype.splice;
jProto.slice = Array.prototype.slice;
// Return a clean array.
jProto.toArray = function toArray() {
return this.length > 1 ? slice.call(this) : slice.call(this)[0];
};
// Map a function to a matrix or vector.
jProto.map = function map(func, toAlter) {
return jStat(jStat.map(this, func, toAlter));
};
// Destructively alter an array.
jProto.alter = function alter(func) {
jStat.alter(this, func);
return this;
};
// Extend prototype with methods that have no argument.
(function(funcs) {
for (var i = 0; i < funcs.length; i++) (function(passfunc) {
jProto[passfunc] = function(func) {
var self = this,
results;
// Check for callback.
if (func) {
setTimeout(function() {
func.call(self, jProto[passfunc].call(self));
});
return this;
}
results = jStat[passfunc](this);
return isArray(results) ? jStat(results) : results;
};
})(funcs[i]);
})('transpose clear symmetric rows cols dimensions diag antidiag'.split(' '));
// Extend prototype with methods that have one argument.
(function(funcs) {
for (var i = 0; i < funcs.length; i++) (function(passfunc) {
jProto[passfunc] = function(index, func) {
var self = this;
// check for callback
if (func) {
setTimeout(function() {
func.call(self, jProto[passfunc].call(self, index));
});
return this;
}
return jStat(jStat[passfunc](this, index));
};
})(funcs[i]);
})('row col'.split(' '));
// Extend prototype with simple shortcut methods.
(function(funcs) {
for (var i = 0; i < funcs.length; i++) (function(passfunc) {
jProto[passfunc] = new Function(
'return jStat(jStat.' + passfunc + '.apply(null, arguments));');
})(funcs[i]);
})('create zeros ones rand identity'.split(' '));
// Exposing jStat.
return jStat;
}(Math));
(function(jStat, Math) {
var isFunction = jStat.utils.isFunction;
// Ascending functions for sort
function ascNum(a, b) { return a - b; }
function clip(arg, min, max) {
return Math.max(min, Math.min(arg, max));
}
// sum of an array
jStat.sum = function sum(arr) {
var sum = 0;
var i = arr.length;
var tmp;
while (--i >= 0)
sum += arr[i];
return sum;
};
// sum squared
jStat.sumsqrd = function sumsqrd(arr) {
var sum = 0;
var i = arr.length;
while (--i >= 0)
sum += arr[i] * arr[i];
return sum;
};
// sum of squared errors of prediction (SSE)
jStat.sumsqerr = function sumsqerr(arr) {
var mean = jStat.mean(arr);
var sum = 0;
var i = arr.length;
var tmp;
while (--i >= 0) {
tmp = arr[i] - mean;
sum += tmp * tmp;
}
return sum;
};
// product of an array
jStat.product = function product(arr) {
var prod = 1;
var i = arr.length;
while (--i >= 0)
prod *= arr[i];
return prod;
};
// minimum value of an array
jStat.min = function min(arr) {
var low = arr[0];
var i = 0;
while (++i < arr.length)
if (arr[i] < low)
low = arr[i];
return low;
};
// maximum value of an array
jStat.max = function max(arr) {
var high = arr[0];
var i = 0;
while (++i < arr.length)
if (arr[i] > high)
high = arr[i];
return high;
};
// mean value of an array
jStat.mean = function mean(arr) {
return jStat.sum(arr) / arr.length;
};
// mean squared error (MSE)
jStat.meansqerr = function meansqerr(arr) {
return jStat.sumsqerr(arr) / arr.length;
};
// geometric mean of an array
jStat.geomean = function geomean(arr) {
return Math.pow(jStat.product(arr), 1 / arr.length);
};
// median of an array
jStat.median = function median(arr) {
var arrlen = arr.length;
var _arr = arr.slice().sort(ascNum);
// check if array is even or odd, then return the appropriate
return !(arrlen & 1)
? (_arr[(arrlen / 2) - 1 ] + _arr[(arrlen / 2)]) / 2
: _arr[(arrlen / 2) | 0 ];
};
// cumulative sum of an array
jStat.cumsum = function cumsum(arr) {
var len = arr.length;
var sums = new Array(len);
var i;
sums[0] = arr[0];
for (i = 1; i < len; i++)
sums[i] = sums[i - 1] + arr[i];
return sums;
};
// successive differences of a sequence
jStat.diff = function diff(arr) {
var diffs = [];
var arrLen = arr.length;
var i;
for (i = 1; i < arrLen; i++)
diffs.push(arr[i] - arr[i - 1]);
return diffs;
};
// mode of an array
// if there are multiple modes of an array, return all of them
// is this the appropriate way of handling it?
jStat.mode = function mode(arr) {
var arrLen = arr.length;
var _arr = arr.slice().sort(ascNum);
var count = 1;
var maxCount = 0;
var numMaxCount = 0;
var mode_arr = [];
var i;
for (i = 0; i < arrLen; i++) {
if (_arr[i] === _arr[i + 1]) {
count++;
} else {
if (count > maxCount) {
mode_arr = [_arr[i]];
maxCount = count;
numMaxCount = 0;
}
// are there multiple max counts
else if (count === maxCount) {
mode_arr.push(_arr[i]);
numMaxCount++;
}
// resetting count for new value in array
count = 1;
}
}
return numMaxCount === 0 ? mode_arr[0] : mode_arr;
};
// range of an array
jStat.range = function range(arr) {
return jStat.max(arr) - jStat.min(arr);
};
// variance of an array
// flag indicates population vs sample
jStat.variance = function variance(arr, flag) {
return jStat.sumsqerr(arr) / (arr.length - (flag ? 1 : 0));
};
// standard deviation of an array
// flag indicates population vs sample
jStat.stdev = function stdev(arr, flag) {
return Math.sqrt(jStat.variance(arr, flag));
};
// mean deviation (mean absolute deviation) of an array
jStat.meandev = function meandev(arr) {
var devSum = 0;
var mean = jStat.mean(arr);
var i;
for (i = arr.length - 1; i >= 0; i--)
devSum += Math.abs(arr[i] - mean);
return devSum / arr.length;
};
// median deviation (median absolute deviation) of an array
jStat.meddev = function meddev(arr) {
var devSum = 0;
var median = jStat.median(arr);
var i;
for (i = arr.length - 1; i >= 0; i--)
devSum += Math.abs(arr[i] - median);
return devSum / arr.length;
};
// coefficient of variation
jStat.coeffvar = function coeffvar(arr) {
return jStat.stdev(arr) / jStat.mean(arr);
};
// quartiles of an array
jStat.quartiles = function quartiles(arr) {
var arrlen = arr.length;
var _arr = arr.slice().sort(ascNum);
return [
_arr[ Math.round((arrlen) / 4) - 1 ],
_arr[ Math.round((arrlen) / 2) - 1 ],
_arr[ Math.round((arrlen) * 3 / 4) - 1 ]
];
};
// Arbitary quantiles of an array. Direct port of the scipy.stats
// implementation by Pierre GF Gerard-Marchant.
jStat.quantiles = function quantiles(arr, quantilesArray, alphap, betap) {
var sortedArray = arr.slice().sort(ascNum);
var quantileVals = [quantilesArray.length];
var n = arr.length;
var i, p, m, aleph, k, gamma;
if (typeof alphap === 'undefined')
alphap = 3 / 8;
if (typeof betap === 'undefined')
betap = 3 / 8;
for (i = 0; i < quantilesArray.length; i++) {
p = quantilesArray[i];
m = alphap + p * (1 - alphap - betap);
aleph = n * p + m;
k = Math.floor(clip(aleph, 1, n - 1));
gamma = clip(aleph - k, 0, 1);
quantileVals[i] = (1 - gamma) * sortedArray[k - 1] + gamma * sortedArray[k];
}
return quantileVals;
};
// The percentile rank of score in a given array. Returns the percentage
// of all values in the input array that are less than (kind='strict') or
// less or equal than (kind='weak') score. Default is weak.
jStat.percentileOfScore = function percentileOfScore(arr, score, kind) {
var counter = 0;
var len = arr.length;
var strict = false;
var value, i;
if (kind === 'strict')
strict = true;
for (i = 0; i < len; i++) {
value = arr[i];
if ((strict && value < score) ||
(!strict && value <= score)) {
counter++;
}
}
return counter / len;
};
// covariance of two arrays
jStat.covariance = function covariance(arr1, arr2) {
var u = jStat.mean(arr1);
var v = jStat.mean(arr2);
var arr1Len = arr1.length;
var sq_dev = new Array(arr1Len);
var i;
for (i = 0; i < arr1Len; i++)
sq_dev[i] = (arr1[i] - u) * (arr2[i] - v);
return jStat.sum(sq_dev) / (arr1Len - 1);
};
// (pearson's) population correlation coefficient, rho
jStat.corrcoeff = function corrcoeff(arr1, arr2) {
return jStat.covariance(arr1, arr2) /
jStat.stdev(arr1, 1) /
jStat.stdev(arr2, 1);
};
var jProto = jStat.prototype;
// Extend jProto with method for calculating cumulative sums, as it does not
// run again in case of true.
// If a matrix is passed, automatically assume operation should be done on the
// columns.
jProto.cumsum = function(fullbool, func) {
var arr = [];
var i = 0;
var tmpthis = this;
// Assignment reassignation depending on how parameters were passed in.
if (isFunction(fullbool)) {
func = fullbool;
fullbool = false;
}
// Check if a callback was passed with the function.
if (func) {
setTimeout(function() {
func.call(tmpthis, jProto.cumsum.call(tmpthis, fullbool));
});
return this;
}
// Check if matrix and run calculations.
if (this.length > 1) {
tmpthis = fullbool === true ? this : this.transpose();
for (; i < tmpthis.length; i++)
arr[i] = jStat.cumsum(tmpthis[i]);
return arr;
}
return jStat.cumsum(this[0], fullbool);
};
// Extend jProto with methods which don't require arguments and work on columns.
(function(funcs) {
for (var i = 0; i < funcs.length; i++) (function(passfunc) {
// If a matrix is passed, automatically assume operation should be done on
// the columns.
jProto[passfunc] = function(fullbool, func) {
var arr = [];
var i = 0;
var tmpthis = this;
// Assignment reassignation depending on how parameters were passed in.
if (isFunction(fullbool)) {
func = fullbool;
fullbool = false;
}
// Check if a callback was passed with the function.
if (func) {
setTimeout(function() {
func.call(tmpthis, jProto[passfunc].call(tmpthis, fullbool));
});
return this;
}
// Check if matrix and run calculations.
if (this.length > 1) {
tmpthis = fullbool === true ? this : this.transpose();
for (; i < tmpthis.length; i++)
arr[i] = jStat[passfunc](tmpthis[i]);
return fullbool === true
? jStat[passfunc](jStat.utils.toVector(arr))
: arr;
}
// Pass fullbool if only vector, not a matrix. for variance and stdev.
return jStat[passfunc](this[0], fullbool);
};
})(funcs[i]);
})(('sum sumsqrd sumsqerr product min max mean meansqerr geomean median diff ' +
'mode range variance stdev meandev meddev coeffvar quartiles').split(' '));
// Extend jProto with functions that take arguments. Operations on matrices are
// done on columns.
(function(funcs) {
for (var i = 0; i < funcs.length; i++) (function(passfunc) {
jProto[passfunc] = function() {
var arr = [];
var i = 0;
var tmpthis = this;
var args = Array.prototype.slice.call(arguments);
// If the last argument is a function, we assume it's a callback; we
// strip the callback out and call the function again.
if (isFunction(args[args.length - 1])) {
var callbackFunction = args[args.length - 1];
var argsToPass = args.slice(0, args.length - 1);
setTimeout(function() {
callbackFunction.call(tmpthis,
jProto[passfunc].apply(tmpthis, argsToPass));
});
return this;
// Otherwise we curry the function args and call normally.
} else {
var callbackFunction = undefined;
var curriedFunction = function curriedFunction(vector) {
return jStat[passfunc].apply(tmpthis, [vector].concat(args));
}
}
// If this is a matrix, run column-by-column.
if (this.length > 1) {
tmpthis = tmpthis.transpose();
for (; i < tmpthis.length; i++)
arr[i] = curriedFunction(tmpthis[i]);
return arr;
}
// Otherwise run on the vector.
return curriedFunction(this[0]);
};
})(funcs[i]);
})('quantiles percentileOfScore'.split(' '));
}(this.jStat, Math));
// Special functions //
(function(jStat, Math) {
// Log-gamma function
jStat.gammaln = function gammaln(x) {
var j = 0;
var cof = [
76.18009172947146, -86.50532032941677, 24.01409824083091,
-1.231739572450155, 0.1208650973866179e-2, -0.5395239384953e-5
];
var ser = 1.000000000190015;
var xx, y, tmp;
tmp = (y = xx = x) + 5.5;
tmp -= (xx + 0.5) * Math.log(tmp);
for (; j < 6; j++)
ser += cof[j] / ++y;
return Math.log(2.5066282746310005 * ser / xx) - tmp;
};
// gamma of x
jStat.gammafn = function gammafn(x) {
var p = [-1.716185138865495, 24.76565080557592, -379.80425647094563,
629.3311553128184, 866.9662027904133, -31451.272968848367,
-36144.413418691176, 66456.14382024054
];
var q = [-30.8402300119739, 315.35062697960416, -1015.1563674902192,
-3107.771671572311, 22538.118420980151, 4755.8462775278811,
-134659.9598649693, -115132.2596755535];
var fact = false;
var n = 0;
var xden = 0;
var xnum = 0;
var y = x;
var i, z, yi, res, sum, ysq;
if (y <= 0) {
res = y % 1 + 3.6e-16;
if (res) {
fact = (!(y & 1) ? 1 : -1) * Math.PI / Math.sin(Math.PI * res);
y = 1 - y;
} else {
return Infinity;
}
}
yi = y;
if (y < 1) {
z = y++;
} else {
z = (y -= n = (y | 0) - 1) - 1;
}
for (i = 0; i < 8; ++i) {
xnum = (xnum + p[i]) * z;
xden = xden * z + q[i];
}
res = xnum / xden + 1;
if (yi < y) {
res /= yi;
} else if (yi > y) {
for (i = 0; i < n; ++i) {
res *= y;
y++;
}
}
if (fact) {
res = fact / res;
}
return res;
};
// lower incomplete gamma function P(a,x)
jStat.gammap = function gammap(a, x) {
var aln = jStat.gammaln(a);
var ap = a;
var sum = 1 / a;
var del = sum;
var b = x + 1 - a;
var c = 1 / 1.0e-30;
var d = 1 / b;
var h = d;
var i = 1;
// calculate maximum number of itterations required for a
var ITMAX = -~(Math.log((a >= 1) ? a : 1 / a) * 8.5 + a * 0.4 + 17);
var an, endval;
if (x < 0 || a <= 0) {
return NaN;
} else if (x < a + 1) {
for (; i <= ITMAX; i++) {
sum += del *= x / ++ap;
}
return sum * Math.exp(-x + a * Math.log(x) - (aln));
}
for (; i <= ITMAX; i++) {
an = -i * (i - a);
b += 2;
d = an * d + b;
c = b + an / c;
d = 1 / d;
h *= d * c;
}
return 1 - h * Math.exp(-x + a * Math.log(x) - (aln));
};
// natural log factorial of n
jStat.factorialln = function factorialln(n) {
return n < 0 ? NaN : jStat.gammaln(n + 1);
};
// factorial of n
jStat.factorial = function factorial(n) {
return n < 0 ? NaN : jStat.gammafn(n + 1);
};
// combinations of n, m
jStat.combination = function combination(n, m) {
// make sure n or m don't exceed the upper limit of usable values
return (n > 170 || m > 170)
? Math.exp(jStat.combinationln(n, m))
: (jStat.factorial(n) / jStat.factorial(m)) / jStat.factorial(n - m);
};
jStat.combinationln = function combinationln(n, m){
return jStat.factorialln(n) - jStat.factorialln(m) - jStat.factorialln(n - m);
};
// permutations of n, m
jStat.permutation = function permutation(n, m) {
return jStat.factorial(n) / jStat.factorial(n - m);
};
// beta function
jStat.betafn = function betafn(x, y) {
// ensure arguments are positive
if (x <= 0 || y <= 0)
return undefined;
// make sure x + y doesn't exceed the upper limit of usable values
return (x + y > 170)
? Math.exp(jStat.betaln(x, y))
: jStat.gammafn(x) * jStat.gammafn(y) / jStat.gammafn(x + y);
};
// natural logarithm of beta function
jStat.betaln = function betaln(x, y) {
return jStat.gammaln(x) + jStat.gammaln(y) - jStat.gammaln(x + y);
};
// Evaluates the continued fraction for incomplete beta function by modified
// Lentz's method.
jStat.betacf = function betacf(x, a, b) {
var fpmin = 1e-30;
var m = 1;
var qab = a + b;
var qap = a + 1;
var qam = a - 1;
var c = 1;
var d = 1 - qab * x / qap;
var m2, aa, del, h;
// These q's will be used in factors that occur in the coefficients
if (Math.abs(d) < fpmin)
d = fpmin;
d = 1 / d;
h = d;
for (; m <= 100; m++) {
m2 = 2 * m;
aa = m * (b - m) * x / ((qam + m2) * (a + m2));
// One step (the even one) of the recurrence
d = 1 + aa * d;
if (Math.abs(d) < fpmin)
d = fpmin;
c = 1 + aa / c;
if (Math.abs(c) < fpmin)
c = fpmin;
d = 1 / d;
h *= d * c;
aa = -(a + m) * (qab + m) * x / ((a + m2) * (qap + m2));
// Next step of the recurrence (the odd one)
d = 1 + aa * d;
if (Math.abs(d) < fpmin)
d = fpmin;
c = 1 + aa / c;
if (Math.abs(c) < fpmin)
c = fpmin;
d = 1 / d;
del = d * c;
h *= del;
if (Math.abs(del - 1.0) < 3e-7)
break;
}
return h;
};
// Returns the inverse incomplte gamma function
jStat.gammapinv = function gammapinv(p, a) {
var j = 0;
var a1 = a - 1;
var EPS = 1e-8;
var gln = jStat.gammaln(a);
var x, err, t, u, pp, lna1, afac;
if (p >= 1)
return Math.max(100, a + 100 * Math.sqrt(a));
if (p <= 0)
return 0;
if (a > 1) {
lna1 = Math.log(a1);
afac = Math.exp(a1 * (lna1 - 1) - gln);
pp = (p < 0.5) ? p : 1 - p;
t = Math.sqrt(-2 * Math.log(pp));
x = (2.30753 + t * 0.27061) / (1 + t * (0.99229 + t * 0.04481)) - t;
if (p < 0.5)
x = -x;
x = Math.max(1e-3,
a * Math.pow(1 - 1 / (9 * a) - x / (3 * Math.sqrt(a)), 3));
} else {
t = 1 - a * (0.253 + a * 0.12);
if (p < t)
x = Math.pow(p / t, 1 / a);
else
x = 1 - Math.log(1 - (p - t) / (1 - t));
}
for(; j < 12; j++) {
if (x <= 0)
return 0;
err = jStat.gammap(a, x) - p;
if (a > 1)
t = afac * Math.exp(-(x - a1) + a1 * (Math.log(x) - lna1));
else
t = Math.exp(-x + a1 * Math.log(x) - gln);
u = err / t;
x -= (t = u / (1 - 0.5 * Math.min(1, u * ((a - 1) / x - 1))));
if (x <= 0)
x = 0.5 * (x + t);
if (Math.abs(t) < EPS * x)
break;
}
return x;
};
// Returns the error function erf(x)
jStat.erf = function erf(x) {
var cof = [-1.3026537197817094, 6.4196979235649026e-1, 1.9476473204185836e-2,
-9.561514786808631e-3, -9.46595344482036e-4, 3.66839497852761e-4,
4.2523324806907e-5, -2.0278578112534e-5, -1.624290004647e-6,
1.303655835580e-6, 1.5626441722e-8, -8.5238095915e-8,
6.529054439e-9, 5.059343495e-9, -9.91364156e-10,
-2.27365122e-10, 9.6467911e-11, 2.394038e-12,
-6.886027e-12, 8.94487e-13, 3.13092e-13,
-1.12708e-13, 3.81e-16, 7.106e-15,
-1.523e-15, -9.4e-17, 1.21e-16,
-2.8e-17];
var j = cof.length - 1;
var isneg = false;
var d = 0;
var dd = 0;
var t, ty, tmp, res;
if (x < 0) {
x = -x;
isneg = true;
}
t = 2 / (2 + x);
ty = 4 * t - 2;
for(; j > 0; j--) {
tmp = d;
d = ty * d - dd + cof[j];
dd = tmp;
}
res = t * Math.exp(-x * x + 0.5 * (cof[0] + ty * d) - dd);
return isneg ? res - 1 : 1 - res;
};
// Returns the complmentary error function erfc(x)
jStat.erfc = function erfc(x) {
return 1 - jStat.erf(x);
};
// Returns the inverse of the complementary error function
jStat.erfcinv = function erfcinv(p) {
var j = 0;
var x, err, t, pp;
if (p >= 2)
return -100;
if (p <= 0)
return 100;
pp = (p < 1) ? p : 2 - p;
t = Math.sqrt(-2 * Math.log(pp / 2));
x = -0.70711 * ((2.30753 + t * 0.27061) /
(1 + t * (0.99229 + t * 0.04481)) - t);
for (; j < 2; j++) {
err = jStat.erfc(x) - pp;
x += err / (1.12837916709551257 * Math.exp(-x * x) - x * err);
}
return (p < 1) ? x : -x;
};
// Returns the inverse of the incomplete beta function
jStat.ibetainv = function ibetainv(p, a, b) {
var EPS = 1e-8;
var a1 = a - 1;
var b1 = b - 1;
var j = 0;
var lna, lnb, pp, t, u, err, x, al, h, w, afac;
if (p <= 0)
return 0;
if (p >= 1)
return 1;
if (a >= 1 && b >= 1) {
pp = (p < 0.5) ? p : 1 - p;
t = Math.sqrt(-2 * Math.log(pp));
x = (2.30753 + t * 0.27061) / (1 + t* (0.99229 + t * 0.04481)) - t;
if (p < 0.5)
x = -x;
al = (x * x - 3) / 6;
h = 2 / (1 / (2 * a - 1) + 1 / (2 * b - 1));
w = (x * Math.sqrt(al + h) / h) - (1 / (2 * b - 1) - 1 / (2 * a - 1)) *
(al + 5 / 6 - 2 / (3 * h));
x = a / (a + b * Math.exp(2 * w));
} else {
lna = Math.log(a / (a + b));
lnb = Math.log(b / (a + b));
t = Math.exp(a * lna) / a;
u = Math.exp(b * lnb) / b;
w = t + u;
if (p < t / w)
x = Math.pow(a * w * p, 1 / a);
else
x = 1 - Math.pow(b * w * (1 - p), 1 / b);
}
afac = -jStat.gammaln(a) - jStat.gammaln(b) + jStat.gammaln(a + b);
for(; j < 10; j++) {
if (x === 0 || x === 1)
return x;
err = jStat.ibeta(x, a, b) - p;
t = Math.exp(a1 * Math.log(x) + b1 * Math.log(1 - x) + afac);
u = err / t;
x -= (t = u / (1 - 0.5 * Math.min(1, u * (a1 / x - b1 / (1 - x)))));
if (x <= 0)
x = 0.5 * (x + t);
if (x >= 1)
x = 0.5 * (x + t + 1);
if (Math.abs(t) < EPS * x && j > 0)
break;
}
return x;
};
// Returns the incomplete beta function I_x(a,b)
jStat.ibeta = function ibeta(x, a, b) {
// Factors in front of the continued fraction.
var bt = (x === 0 || x === 1) ? 0 :
Math.exp(jStat.gammaln(a + b) - jStat.gammaln(a) -
jStat.gammaln(b) + a * Math.log(x) + b *
Math.log(1 - x));
if (x < 0 || x > 1)
return false;
if (x < (a + 1) / (a + b + 2))
// Use continued fraction directly.
return bt * jStat.betacf(x, a, b) / a;
// else use continued fraction after making the symmetry transformation.
return 1 - bt * jStat.betacf(1 - x, b, a) / b;
};
// Returns a normal deviate (mu=0, sigma=1).
// If n and m are specified it returns a object of normal deviates.
jStat.randn = function randn(n, m) {
var u, v, x, y, q, mat;
if (!m)
m = n;
if (n)
return jStat.create(n, m, function() { return jStat.randn(); });
do {
u = Math.random();
v = 1.7156 * (Math.random() - 0.5);
x = u - 0.449871;
y = Math.abs(v) + 0.386595;
q = x * x + y * (0.19600 * y - 0.25472 * x);
} while (q > 0.27597 && (q > 0.27846 || v * v > -4 * Math.log(u) * u * u));
return v / u;
};
// Returns a gamma deviate by the method of Marsaglia and Tsang.
jStat.randg = function randg(shape, n, m) {
var oalph = shape;
var a1, a2, u, v, x, mat;
if (!m)
m = n;
if (!shape)
shape = 1;
if (n) {
mat = jStat.zeros(n,m);
mat.alter(function() { return jStat.randg(shape); });
return mat;
}
if (shape < 1)
shape += 1;
a1 = shape - 1 / 3;
a2 = 1 / Math.sqrt(9 * a1);
do {
do {
x = jStat.randn();
v = 1 + a2 * x;
} while(v <= 0);
v = v * v * v;
u = Math.random();
} while(u > 1 - 0.331 * Math.pow(x, 4) &&
Math.log(u) > 0.5 * x*x + a1 * (1 - v + Math.log(v)));
// alpha > 1
if (shape == oalph)
return a1 * v;
// alpha < 1
do {
u = Math.random();
} while(u === 0);
return Math.pow(u, 1 / oalph) * a1 * v;
};
// making use of static methods on the instance
(function(funcs) {
for (var i = 0; i < funcs.length; i++) (function(passfunc) {
jStat.fn[passfunc] = function() {
return jStat(
jStat.map(this, function(value) { return jStat[passfunc](value); }));
}
})(funcs[i]);
})('gammaln gammafn factorial factorialln'.split(' '));
(function(funcs) {
for (var i = 0; i < funcs.length; i++) (function(passfunc) {
jStat.fn[passfunc] = function() {
return jStat(jStat[passfunc].apply(null, arguments));
};
})(funcs[i]);
})('randn'.split(' '));
}(this.jStat, Math));
(function(jStat, Math) {
// generate all distribution instance methods
(function(list) {
for (var i = 0; i < list.length; i++) (function(func) {
// distribution instance method
jStat[func] = function(a, b, c) {
if (!(this instanceof arguments.callee))
return new arguments.callee(a, b, c);
this._a = a;
this._b = b;
this._c = c;
return this;
};
// distribution method to be used on a jStat instance
jStat.fn[func] = function(a, b, c) {
var newthis = jStat[func](a, b, c);
newthis.data = this;
return newthis;
};
// sample instance method
jStat[func].prototype.sample = function(arr) {
var a = this._a;
var b = this._b;
var c = this._c;
if (arr)
return jStat.alter(arr, function() {
return jStat[func].sample(a, b, c);
});
else
return jStat[func].sample(a, b, c);
};
// generate the pdf, cdf and inv instance methods
(function(vals) {
for (var i = 0; i < vals.length; i++) (function(fnfunc) {
jStat[func].prototype[fnfunc] = function(x) {
var a = this._a;
var b = this._b;
var c = this._c;
if (!x && x !== 0)
x = this.data;
if (typeof x !== 'number') {
return jStat.fn.map.call(x, function(x) {
return jStat[func][fnfunc](x, a, b, c);
});
}
return jStat[func][fnfunc](x, a, b, c);
};
})(vals[i]);
})('pdf cdf inv'.split(' '));
// generate the mean, median, mode and variance instance methods
(function(vals) {
for (var i = 0; i < vals.length; i++) (function(fnfunc) {
jStat[func].prototype[fnfunc] = function() {
return jStat[func][fnfunc](this._a, this._b, this._c);
};
})(vals[i]);
})('mean median mode variance'.split(' '));
})(list[i]);
})((
'beta centralF cauchy chisquare exponential gamma invgamma kumaraswamy ' +
'lognormal normal pareto studentt weibull uniform binomial negbin hypgeom ' +
'poisson triangular'
).split(' '));
// extend beta function with static methods
jStat.extend(jStat.beta, {
pdf: function pdf(x, alpha, beta) {
// PDF is zero outside the support
if (x > 1 || x < 0)
return 0;
// PDF is one for the uniform case
if (alpha == 1 && beta == 1)
return 1;
if (alpha < 512 || beta < 512) {
return (Math.pow(x, alpha - 1) * Math.pow(1 - x, beta - 1)) /
jStat.betafn(alpha, beta);
} else {
return Math.exp((alpha - 1) * Math.log(x) +
(beta - 1) * Math.log(1 - x) -
jStat.betaln(alpha, beta));
}
},
cdf: function cdf(x, alpha, beta) {
return (x > 1 || x < 0) ? (x > 1) * 1 : jStat.ibeta(x, alpha, beta);
},
inv: function inv(x, alpha, beta) {
return jStat.ibetainv(x, alpha, beta);
},
mean: function mean(alpha, beta) {
return alpha / (alpha + beta);
},
median: function median(alpha, beta) {
throw new Error('median not yet implemented');
},
mode: function mode(alpha, beta) {
return (alpha * beta) / (Math.pow(alpha + beta, 2) * (alpha + beta + 1));
},
// return a random sample
sample: function sample(alpha, beta) {
var u = jStat.randg(alpha);
return u / (u + jStat.randg(beta));
},
variance: function variance(alpha, beta) {
return (alpha * beta) / (Math.pow(alpha + beta, 2) * (alpha + beta + 1));
}
});
// extend F function with static methods
jStat.extend(jStat.centralF, {
pdf: function pdf(x, df1, df2) {
if (x < 0)
return undefined;
return Math.sqrt((Math.pow(df1 * x, df1) * Math.pow(df2, df2)) /
(Math.pow(df1 * x + df2, df1 + df2))) /
(x * jStat.betafn(df1/2, df2/2));
},
cdf: function cdf(x, df1, df2) {
return jStat.ibeta((df1 * x) / (df1 * x + df2), df1 / 2, df2 / 2);
},
inv: function inv(x, df1, df2) {
return df2 / (df1 * (1 / jStat.ibetainv(x, df1 / 2, df2 / 2) - 1));
},
mean: function mean(df1, df2) {
return (df2 > 2) ? df2 / (df2 - 2) : undefined;
},
mode: function mode(df1, df2) {
return (df1 > 2) ? (df2 * (df1 - 2)) / (df1 * (df2 + 2)) : undefined;
},
// return a random sample
sample: function sample(df1, df2) {
var x1 = jStat.randg(df1 / 2) * 2;
var x2 = jStat.randg(df2 / 2) * 2;
return (x1 / df1) / (x2 / df2);
},
variance: function variance(df1, df2) {
if (df2 <= 4)
return undefined;
return 2 * df2 * df2 * (df1 + df2 - 2) /
(df1 * (df2 - 2) * (df2 - 2) * (df2 - 4));
}
});
// extend cauchy function with static methods
jStat.extend(jStat.cauchy, {
pdf: function pdf(x, local, scale) {
return (scale / (Math.pow(x - local, 2) + Math.pow(scale, 2))) / Math.PI;
},
cdf: function cdf(x, local, scale) {
return Math.atan((x - local) / scale) / Math.PI + 0.5;
},
inv: function(p, local, scale) {
return local + scale * Math.tan(Math.PI * (p - 0.5));
},
median: function median(local, scale) {
return local;
},
mode: function mode(local, scale) {
return local;
},
sample: function sample(local, scale) {
return jStat.randn() *
Math.sqrt(1 / (2 * jStat.randg(0.5))) * scale + local;
}
});
// extend chisquare function with static methods
jStat.extend(jStat.chisquare, {
pdf: function pdf(x, dof) {
return Math.exp((dof / 2 - 1) * Math.log(x) - x / 2 - (dof / 2) *
Math.log(2) - jStat.gammaln(dof / 2));
},
cdf: function cdf(x, dof) {
return jStat.gammap(dof / 2, x / 2);
},
inv: function(p, dof) {
return 2 * jStat.gammapinv(p, 0.5 * dof);
},
mean : function(dof) {
return dof;
},
// TODO: this is an approximation (is there a better way?)
median: function median(dof) {
return dof * Math.pow(1 - (2 / (9 * dof)), 3);
},
mode: function mode(dof) {
return (dof - 2 > 0) ? dof - 2 : 0;
},
sample: function sample(dof) {
return jStat.randg(dof / 2) * 2;
},
variance: function variance(dof) {
return 2 * dof;
}
});
// extend exponential function with static methods
jStat.extend(jStat.exponential, {
pdf: function pdf(x, rate) {
return x < 0 ? 0 : rate * Math.exp(-rate * x);
},
cdf: function cdf(x, rate) {
return x < 0 ? 0 : 1 - Math.exp(-rate * x);
},
inv: function(p, rate) {
return -Math.log(1 - p) / rate;
},
mean : function(rate) {
return 1 / rate;
},
median: function (rate) {
return (1 / rate) * Math.log(2);
},
mode: function mode(rate) {
return 0;
},
sample: function sample(rate) {
return -1 / rate * Math.log(Math.random());
},
variance : function(rate) {
return Math.pow(rate, -2);
}
});
// extend gamma function with static methods
jStat.extend(jStat.gamma, {
pdf: function pdf(x, shape, scale) {
return Math.exp((shape - 1) * Math.log(x) - x / scale -
jStat.gammaln(shape) - shape * Math.log(scale));
},
cdf: function cdf(x, shape, scale) {
return jStat.gammap(shape, x / scale);
},
inv: function(p, shape, scale) {
return jStat.gammapinv(p, shape) * scale;
},
mean : function(shape, scale) {
return shape * scale;
},
mode: function mode(shape, scale) {
if(shape > 1) return (shape - 1) * scale;
return undefined;
},
sample: function sample(shape, scale) {
return jStat.randg(shape) * scale;
},
variance: function variance(shape, scale) {
return shape * scale * scale;
}
});
// extend inverse gamma function with static methods
jStat.extend(jStat.invgamma, {
pdf: function pdf(x, shape, scale) {
return Math.exp(-(shape + 1) * Math.log(x) - scale / x -
jStat.gammaln(shape) + shape * Math.log(scale));
},
cdf: function cdf(x, shape, scale) {
return 1 - jStat.gammap(shape, scale / x);
},
inv: function(p, shape, scale) {
return scale / jStat.gammapinv(1 - p, shape);
},
mean : function(shape, scale) {
return (shape > 1) ? scale / (shape - 1) : undefined;
},
mode: function mode(shape, scale) {
return scale / (shape + 1);
},
sample: function sample(shape, scale) {
return scale / jStat.randg(shape);
},
variance: function variance(shape, scale) {
if (shape <= 2)
return undefined;
return scale * scale / ((shape - 1) * (shape - 1) * (shape - 2));
}
});
// extend kumaraswamy function with static methods
jStat.extend(jStat.kumaraswamy, {
pdf: function pdf(x, alpha, beta) {
return Math.exp(Math.log(alpha) + Math.log(beta) + (alpha - 1) *
Math.log(x) + (beta - 1) *
Math.log(1 - Math.pow(x, alpha)));
},
cdf: function cdf(x, alpha, beta) {
return (1 - Math.pow(1 - Math.pow(x, alpha), beta));
},
mean : function(alpha, beta) {
return (beta * jStat.gammafn(1 + 1 / alpha) *
jStat.gammafn(beta)) / (jStat.gammafn(1 + 1 / alpha + beta));
},
median: function median(alpha, beta) {
return Math.pow(1 - Math.pow(2, -1 / beta), 1 / alpha);
},
mode: function mode(alpha, beta) {
if (!(alpha >= 1 && beta >= 1 && (alpha !== 1 && beta !== 1)))
return undefined;
return Math.pow((alpha - 1) / (alpha * beta - 1), 1 / alpha);
},
variance: function variance(alpha, beta) {
throw new Error('variance not yet implemented');
// TODO: complete this
}
});
// extend lognormal function with static methods
jStat.extend(jStat.lognormal, {
pdf: function pdf(x, mu, sigma) {
return Math.exp(-Math.log(x) - 0.5 * Math.log(2 * Math.PI) -
Math.log(sigma) - Math.pow(Math.log(x) - mu, 2) /
(2 * sigma * sigma));
},
cdf: function cdf(x, mu, sigma) {
return 0.5 +
(0.5 * jStat.erf((Math.log(x) - mu) / Math.sqrt(2 * sigma * sigma)));
},
inv: function(p, mu, sigma) {
return Math.exp(-1.41421356237309505 * sigma * jStat.erfcinv(2 * p) + mu);
},
mean: function mean(mu, sigma) {
return Math.exp(mu + sigma * sigma / 2);
},
median: function median(mu, sigma) {
return Math.exp(mu);
},
mode: function mode(mu, sigma) {
return Math.exp(mu - sigma * sigma);
},
sample: function sample(mu, sigma) {
return Math.exp(jStat.randn() * sigma + mu);
},
variance: function variance(mu, sigma) {
return (Math.exp(sigma * sigma) - 1) * Math.exp(2 * mu + sigma * sigma);
}
});
// extend normal function with static methods
jStat.extend(jStat.normal, {
pdf: function pdf(x, mean, std) {
return Math.exp(-0.5 * Math.log(2 * Math.PI) -
Math.log(std) - Math.pow(x - mean, 2) / (2 * std * std));
},
cdf: function cdf(x, mean, std) {
return 0.5 * (1 + jStat.erf((x - mean) / Math.sqrt(2 * std * std)));
},
inv: function(p, mean, std) {
return -1.41421356237309505 * std * jStat.erfcinv(2 * p) + mean;
},
mean : function(mean, std) {
return mean;
},
median: function median(mean, std) {
return mean;
},
mode: function (mean, std) {
return mean;
},
sample: function sample(mean, std) {
return jStat.randn() * std + mean;
},
variance : function(mean, std) {
return std * std;
}
});
// extend pareto function with static methods
jStat.extend(jStat.pareto, {
pdf: function pdf(x, scale, shape) {
if (x <= scale)
return undefined;
return (shape * Math.pow(scale, shape)) / Math.pow(x, shape + 1);
},
cdf: function cdf(x, scale, shape) {
return 1 - Math.pow(scale / x, shape);
},
mean: function mean(scale, shape) {
if (shape <= 1)
return undefined;
return (shape * Math.pow(scale, shape)) / (shape - 1);
},
median: function median(scale, shape) {
return scale * (shape * Math.SQRT2);
},
mode: function mode(scale, shape) {
return scale;
},
variance : function(scale, shape) {
if (shape <= 2)
return undefined;
return (scale*scale * shape) / (Math.pow(shape - 1, 2) * (shape - 2));
}
});
// extend studentt function with static methods
jStat.extend(jStat.studentt, {
pdf: function pdf(x, dof) {
return (jStat.gammafn((dof + 1) / 2) / (Math.sqrt(dof * Math.PI) *
jStat.gammafn(dof / 2))) *
Math.pow(1 + ((x * x) / dof), -((dof + 1) / 2));
},
cdf: function cdf(x, dof) {
var dof2 = dof / 2;
return jStat.ibeta((x + Math.sqrt(x * x + dof)) /
(2 * Math.sqrt(x * x + dof)), dof2, dof2);
},
inv: function(p, dof) {
var x = jStat.ibetainv(2 * Math.min(p, 1 - p), 0.5 * dof, 0.5);
x = Math.sqrt(dof * (1 - x) / x);
return (p > 0) ? x : -x;
},
mean: function mean(dof) {
return (dof > 1) ? 0 : undefined;
},
median: function median(dof) {
return 0;
},
mode: function mode(dof) {
return 0;
},
sample: function sample(dof) {
return jStat.randn() * Math.sqrt(dof / (2 * jStat.randg(dof / 2)));
},
variance: function variance(dof) {
return (dof > 2) ? dof / (dof - 2) : (dof > 1) ? Infinity : undefined;
}
});
// extend weibull function with static methods
jStat.extend(jStat.weibull, {
pdf: function pdf(x, scale, shape) {
if (x < 0)
return 0;
return (shape / scale) * Math.pow((x / scale), (shape - 1)) *
Math.exp(-(Math.pow((x / scale), shape)));
},
cdf: function cdf(x, scale, shape) {
return x < 0 ? 0 : 1 - Math.exp(-Math.pow((x / scale), shape));
},
inv: function(p, scale, shape) {
return scale * Math.pow(-Math.log(1 - p), 1 / shape);
},
mean : function(scale, shape) {
return scale * jStat.gammafn(1 + 1 / shape);
},
median: function median(scale, shape) {
return scale * Math.pow(Math.log(2), 1 / shape);
},
mode: function mode(scale, shape) {
if (shape <= 1)
return undefined;
return scale * Math.pow((shape - 1) / shape, 1 / shape);
},
sample: function sample(scale, shape) {
return scale * Math.pow(-Math.log(Math.random()), 1 / shape);
},
variance: function variance(scale, shape) {
return scale * scale * jStat.gammafn(1 + 2 / shape) -
Math.pow(this.mean(scale, shape), 2);
}
});
// extend uniform function with static methods
jStat.extend(jStat.uniform, {
pdf: function pdf(x, a, b) {
return (x < a || x > b) ? 0 : 1 / (b - a);
},
cdf: function cdf(x, a, b) {
if (x < a)
return 0;
else if (x < b)
return (x - a) / (b - a);
return 1;
},
mean: function mean(a, b) {
return 0.5 * (a + b);
},
median: function median(a, b) {
return jStat.mean(a, b);
},
mode: function mode(a, b) {
throw new Error('mode is not yet implemented');
},
sample: function sample(a, b) {
return (a / 2 + b / 2) + (b / 2 - a / 2) * (2 * Math.random() - 1);
},
variance: function variance(a, b) {
return Math.pow(b - a, 2) / 12;
}
});
// extend uniform function with static methods
jStat.extend(jStat.binomial, {
pdf: function pdf(k, n, p) {
return (p === 0 || p === 1) ?
((n * p) === k ? 1 : 0) :
jStat.combination(n, k) * Math.pow(p, k) * Math.pow(1 - p, n - k);
},
cdf: function cdf(x, n, p) {
var binomarr = [],
k = 0;
if (x < 0) {
return 0;
}
if (x < n) {
for (; k <= x; k++) {
binomarr[ k ] = jStat.binomial.pdf(k, n, p);
}
return jStat.sum(binomarr);
}
return 1;
}
});
// extend uniform function with static methods
jStat.extend(jStat.negbin, {
pdf: function pdf(k, r, p) {
return k !== k | 0
? false
: k < 0
? 0
: jStat.combination(k + r - 1, r - 1) * Math.pow(1 - p, k) * Math.pow(p, r);
},
cdf: function cdf(x, r, p) {
var sum = 0,
k = 0;
if (x < 0) return 0;
for (; k <= x; k++) {
sum += jStat.negbin.pdf(k, r, p);
}
return sum;
}
});
// extend uniform function with static methods
jStat.extend(jStat.hypgeom, {
pdf: function pdf(k, N, m, n) {
// Hypergeometric PDF.
// A simplification of the CDF algorithm below.
// k = number of successes drawn
// N = population size
// m = number of successes in population
// n = number of items drawn from population
if(k !== k | 0) {
return false;
} else if(k < 0 || k < m - (N - n)) {
// It's impossible to have this few successes drawn.
return 0;
} else if(k > n || k > m) {
// It's impossible to have this many successes drawn.
return 0;
} else if (m * 2 > N) {
// More than half the population is successes.
if(n * 2 > N) {
// More than half the population is sampled.
return jStat.hypgeom.pdf(N - m - n + k, N, N - m, N - n)
} else {
// Half or less of the population is sampled.
return jStat.hypgeom.pdf(n - k, N, N - m, n);
}
} else if(n * 2 > N) {
// Half or less is successes.
return jStat.hypgeom.pdf(m - k, N, m, N - n);
} else if(m < n) {
// We want to have the number of things sampled to be less than the
// successes available. So swap the definitions of successful and sampled.
return jStat.hypgeom.pdf(k, N, n, m);
} else {
// If we get here, half or less of the population was sampled, half or
// less of it was successes, and we had fewer sampled things than
// successes. Now we can do this complicated iterative algorithm in an
// efficient way.
// The basic premise of the algorithm is that we partially normalize our
// intermediate product to keep it in a numerically good region, and then
// finish the normalization at the end.
// This variable holds the scaled probability of the current number of
// successes.
var scaledPDF = 1;
// This keeps track of how much we have normalized.
var samplesDone = 0;
for(var i = 0; i < k; i++) {
// For every possible number of successes up to that observed...
while(scaledPDF > 1 && samplesDone < n) {
// Intermediate result is growing too big. Apply some of the
// normalization to shrink everything.
scaledPDF *= 1 - (m / (N - samplesDone));
// Say we've normalized by this sample already.
samplesDone++;
}
// Work out the partially-normalized hypergeometric PDF for the next
// number of successes
scaledPDF *= (n - i) * (m - i) / ((i + 1) * (N - m - n + i + 1));
}
for(; samplesDone < n; samplesDone++) {
// Apply all the rest of the normalization
scaledPDF *= 1 - (m / (N - samplesDone));
}
// Bound answer sanely before returning.
return Math.min(1, Math.max(0, scaledPDF));
}
},
cdf: function cdf(x, N, m, n) {
// Hypergeometric CDF.
// This algorithm is due to Prof. Thomas S. Ferguson, <tom@math.ucla.edu>,
// and comes from his hypergeometric test calculator at
// <http://www.math.ucla.edu/~tom/distributions/Hypergeometric.html>.
// x = number of successes drawn
// N = population size
// m = number of successes in population
// n = number of items drawn from population
if(x < 0 || x < m - (N - n)) {
// It's impossible to have this few successes drawn or fewer.
return 0;
} else if(x >= n || x >= m) {
// We will always have this many successes or fewer.
return 1;
} else if (m * 2 > N) {
// More than half the population is successes.
if(n * 2 > N) {
// More than half the population is sampled.
return jStat.hypgeom.cdf(N - m - n + x, N, N - m, N - n)
} else {
// Half or less of the population is sampled.
return 1 - jStat.hypgeom.cdf(n - x - 1, N, N - m, n);
}
} else if(n * 2 > N) {
// Half or less is successes.
return 1 - jStat.hypgeom.cdf(m - x - 1, N, m, N - n);
} else if(m < n) {
// We want to have the number of things sampled to be less than the
// successes available. So swap the definitions of successful and sampled.
return jStat.hypgeom.cdf(x, N, n, m);
} else {
// If we get here, half or less of the population was sampled, half or
// less of it was successes, and we had fewer sampled things than
// successes. Now we can do this complicated iterative algorithm in an
// efficient way.
// The basic premise of the algorithm is that we partially normalize our
// intermediate sum to keep it in a numerically good region, and then
// finish the normalization at the end.
// Holds the intermediate, scaled total CDF.
var scaledCDF = 1;
// This variable holds the scaled probability of the current number of
// successes.
var scaledPDF = 1;
// This keeps track of how much we have normalized.
var samplesDone = 0;
for(var i = 0; i < x; i++) {
// For every possible number of successes up to that observed...
while(scaledCDF > 1 && samplesDone < n) {
// Intermediate result is growing too big. Apply some of the
// normalization to shrink everything.
var factor = 1 - (m / (N - samplesDone));
scaledPDF *= factor;
scaledCDF *= factor;
// Say we've normalized by this sample already.
samplesDone++;
}
// Work out the partially-normalized hypergeometric PDF for the next
// number of successes
scaledPDF *= (n - i) * (m - i) / ((i + 1) * (N - m - n + i + 1));
// Add to the CDF answer.
scaledCDF += scaledPDF;
}
for(; samplesDone < n; samplesDone++) {
// Apply all the rest of the normalization
scaledCDF *= 1 - (m / (N - samplesDone));
}
// Bound answer sanely before returning.
return Math.min(1, Math.max(0, scaledCDF));
}
}
});
// extend uniform function with static methods
jStat.extend(jStat.poisson, {
pdf: function pdf(k, l) {
return Math.pow(l, k) * Math.exp(-l) / jStat.factorial(k);
},
cdf: function cdf(x, l) {
var sumarr = [],
k = 0;
if (x < 0) return 0;
for (; k <= x; k++) {
sumarr.push(jStat.poisson.pdf(k, l));
}
return jStat.sum(sumarr);
},
mean : function(l) {
return l;
},
variance : function(l) {
return l;
},
sample: function sample(l) {
var p = 1, k = 0, L = Math.exp(-l);
do {
k++;
p *= Math.random();
} while (p > L);
return k - 1;
}
});
// extend triangular function with static methods
jStat.extend(jStat.triangular, {
pdf: function pdf(x, a, b, c) {
return (b <= a || c < a || c > b)
? undefined
: (x < a || x > b)
? 0
: (x <= c)
? (2 * (x - a)) / ((b - a) * (c - a))
: (2 * (b - x)) / ((b - a) * (b - c));
},
cdf: function cdf(x, a, b, c) {
if (b <= a || c < a || c > b)
return undefined;
if (x < a) {
return 0;
} else {
if (x <= c)
return Math.pow(x - a, 2) / ((b - a) * (c - a));
return 1 - Math.pow(b - x, 2) / ((b - a) * (b - c));
}
// never reach this
return 1;
},
mean: function mean(a, b, c) {
return (a + b + c) / 3;
},
median: function median(a, b, c) {
if (c <= (a + b) / 2) {
return b - Math.sqrt((b - a) * (b - c)) / Math.sqrt(2);
} else if (c > (a + b) / 2) {
return a + Math.sqrt((b - a) * (c - a)) / Math.sqrt(2);
}
},
mode: function mode(a, b, c) {
return c;
},
sample: function sample(a, b, c) {
var u = Math.random();
if (u < ((c - a) / (b - a)))
return a + Math.sqrt(u * (b - a) * (c - a))
return b - Math.sqrt((1 - u) * (b - a) * (b - c));
},
variance: function variance(a, b, c) {
return (a * a + b * b + c * c - a * b - a * c - b * c) / 18;
}
});
}(this.jStat, Math));
/* Provides functions for the solution of linear system of equations, integration, extrapolation,
* interpolation, eigenvalue problems, differential equations and PCA analysis. */
(function(jStat, Math) {
var push = Array.prototype.push;
var isArray = jStat.utils.isArray;
jStat.extend({
// add a vector/matrix to a vector/matrix or scalar
add: function add(arr, arg) {
// check if arg is a vector or scalar
if (isArray(arg)) {
if (!isArray(arg[0])) arg = [ arg ];
return jStat.map(arr, function(value, row, col) {
return value + arg[row][col];
});
}
return jStat.map(arr, function(value) { return value + arg; });
},
// subtract a vector or scalar from the vector
subtract: function subtract(arr, arg) {
// check if arg is a vector or scalar
if (isArray(arg)) {
if (!isArray(arg[0])) arg = [ arg ];
return jStat.map(arr, function(value, row, col) {
return value - arg[row][col] || 0;
});
}
return jStat.map(arr, function(value) { return value - arg; });
},
// matrix division
divide: function divide(arr, arg) {
if (isArray(arg)) {
if (!isArray(arg[0])) arg = [ arg ];
return jStat.multiply(arr, jStat.inv(arg));
}
return jStat.map(arr, function(value) { return value / arg; });
},
// matrix multiplication
multiply: function multiply(arr, arg) {
var row, col, nrescols, sum,
nrow = arr.length,
ncol = arr[0].length,
res = jStat.zeros(nrow, nrescols = (isArray(arg)) ? arg[0].length : ncol),
rescols = 0;
if (isArray(arg)) {
for (; rescols < nrescols; rescols++) {
for (row = 0; row < nrow; row++) {
sum = 0;
for (col = 0; col < ncol; col++)
sum += arr[row][col] * arg[col][rescols];
res[row][rescols] = sum;
}
}
return (nrow === 1 && rescols === 1) ? res[0][0] : res;
}
return jStat.map(arr, function(value) { return value * arg; });
},
// Returns the dot product of two matricies
dot: function dot(arr, arg) {
if (!isArray(arr[0])) arr = [ arr ];
if (!isArray(arg[0])) arg = [ arg ];
// convert column to row vector
var left = (arr[0].length === 1 && arr.length !== 1) ? jStat.transpose(arr) : arr,
right = (arg[0].length === 1 && arg.length !== 1) ? jStat.transpose(arg) : arg,
res = [],
row = 0,
nrow = left.length,
ncol = left[0].length,
sum, col;
for (; row < nrow; row++) {
res[row] = [];
sum = 0;
for (col = 0; col < ncol; col++)
sum += left[row][col] * right[row][col];
res[row] = sum;
}
return (res.length === 1) ? res[0] : res;
},
// raise every element by a scalar
pow: function pow(arr, arg) {
return jStat.map(arr, function(value) { return Math.pow(value, arg); });
},
// generate the absolute values of the vector
abs: function abs(arr) {
return jStat.map(arr, function(value) { return Math.abs(value); });
},
// TODO: make compatible with matrices
// computes the p-norm of the vector
norm: function norm(arr, p) {
var nnorm = 0,
i = 0;
// check the p-value of the norm, and set for most common case
if (isNaN(p)) p = 2;
// check if multi-dimensional array, and make vector correction
if (isArray(arr[0])) arr = arr[0];
// vector norm
for (; i < arr.length; i++) {
nnorm += Math.pow(Math.abs(arr[i]), p);
}
return Math.pow(nnorm, 1 / p);
},
// TODO: make compatible with matrices
// computes the angle between two vectors in rads
angle: function angle(arr, arg) {
return Math.acos(jStat.dot(arr, arg) / (jStat.norm(arr) * jStat.norm(arg)));
},
// augment one matrix by another
aug: function aug(a, b) {
var newarr = a.slice(),
i = 0;
for (; i < newarr.length; i++) {
push.apply(newarr[i], b[i]);
}
return newarr;
},
inv: function inv(a) {
var rows = a.length,
cols = a[0].length,
b = jStat.identity(rows, cols),
c = jStat.gauss_jordan(a, b),
obj = [],
i = 0,
j;
for (; i < rows; i++) {
obj[i] = [];
for (j = cols - 1; j < c[0].length; j++)
obj[i][j - cols] = c[i][j];
}
return obj;
},
// calculate the determinant of a matrix
det: function det(a) {
var alen = a.length,
alend = alen * 2,
vals = new Array(alend),
rowshift = alen - 1,
colshift = alend - 1,
mrow = rowshift - alen + 1,
mcol = colshift,
i = 0,
result = 0,
j;
// check for special 2x2 case
if (alen === 2) {
return a[0][0] * a[1][1] - a[0][1] * a[1][0];
}
for (; i < alend; i++) {
vals[i] = 1;
}
for (i = 0; i < alen; i++) {
for (j = 0; j < alen; j++) {
vals[(mrow < 0) ? mrow + alen : mrow ] *= a[i][j];
vals[(mcol < alen) ? mcol + alen : mcol ] *= a[i][j];
mrow++;
mcol--;
}
mrow = --rowshift - alen + 1;
mcol = --colshift;
}
for (i = 0; i < alen; i++) {
result += vals[i];
}
for (; i < alend; i++) {
result -= vals[i];
}
return result;
},
gauss_elimination: function gauss_elimination(a, b) {
var i = 0,
j = 0,
n = a.length,
m = a[0].length,
factor = 1,
sum = 0,
x = [],
maug, pivot, temp, k;
a = jStat.aug(a, b);
maug = a[0].length;
for(; i < n; i++) {
pivot = a[i][i];
j = i;
for (k = i + 1; k < m; k++) {
if (pivot < Math.abs(a[k][i])) {
pivot = a[k][i];
j = k;
}
}
if (j != i) {
for(k = 0; k < maug; k++) {
temp = a[i][k];
a[i][k] = a[j][k];
a[j][k] = temp;
}
}
for (j = i + 1; j < n; j++) {
factor = a[j][i] / a[i][i];
for(k = i; k < maug; k++) {
a[j][k] = a[j][k] - factor * a[i][k];
}
}
}
for (i = n - 1; i >= 0; i--) {
sum = 0;
for (j = i + 1; j<= n - 1; j++) {
sum = x[j] * a[i][j];
}
x[i] =(a[i][maug - 1] - sum) / a[i][i];
}
return x;
},
gauss_jordan: function gauss_jordan(a, b) {
var m = jStat.aug(a, b),
h = m.length,
w = m[0].length;
// find max pivot
for (var y = 0; y < h; y++) {
var maxrow = y;
for (var y2 = y+1; y2 < h; y2++) {
if (Math.abs(m[y2][y]) > Math.abs(m[maxrow][y]))
maxrow = y2;
}
var tmp = m[y];
m[y] = m[maxrow];
m[maxrow] = tmp
for (var y2 = y+1; y2 < h; y2++) {
c = m[y2][y] / m[y][y];
for (var x = y; x < w; x++) {
m[y2][x] -= m[y][x] * c;
}
}
}
// backsubstitute
for (var y = h-1; y >= 0; y--) {
c = m[y][y];
for (var y2 = 0; y2 < y; y2++) {
for (var x = w-1; x > y-1; x--) {
m[y2][x] -= m[y][x] * m[y2][y] / c;
}
}
m[y][y] /= c;
for (var x = h; x < w; x++) {
m[y][x] /= c;
}
}
return m;
},
lu: function lu(a, b) {
throw new Error('lu not yet implemented');
},
cholesky: function cholesky(a, b) {
throw new Error('cholesky not yet implemented');
},
gauss_jacobi: function gauss_jacobi(a, b, x, r) {
var i = 0;
var j = 0;
var n = a.length;
var l = [];
var u = [];
var d = [];
var xv, c, h, xk;
for (; i < n; i++) {
l[i] = [];
u[i] = [];
d[i] = [];
for (j = 0; j < n; j++) {
if (i > j) {
l[i][j] = a[i][j];
u[i][j] = d[i][j] = 0;
} else if (i < j) {
u[i][j] = a[i][j];
l[i][j] = d[i][j] = 0;
} else {
d[i][j] = a[i][j];
l[i][j] = u[i][j] = 0;
}
}
}
h = jStat.multiply(jStat.multiply(jStat.inv(d), jStat.add(l, u)), -1);
c = jStat.multiply(jStat.inv(d), b);
xv = x;
xk = jStat.add(jStat.multiply(h, x), c);
i = 2;
while (Math.abs(jStat.norm(jStat.subtract(xk,xv))) > r) {
xv = xk;
xk = jStat.add(jStat.multiply(h, xv), c);
i++;
}
return xk;
},
gauss_seidel: function gauss_seidel(a, b, x, r) {
var i = 0;
var n = a.length;
var l = [];
var u = [];
var d = [];
var j, xv, c, h, xk;
for (; i < n; i++) {
l[i] = [];
u[i] = [];
d[i] = [];
for (j = 0; j < n; j++) {
if (i > j) {
l[i][j] = a[i][j];
u[i][j] = d[i][j] = 0;
} else if (i < j) {
u[i][j] = a[i][j];
l[i][j] = d[i][j] = 0;
} else {
d[i][j] = a[i][j];
l[i][j] = u[i][j] = 0;
}
}
}
h = jStat.multiply(jStat.multiply(jStat.inv(jStat.add(d, l)), u), -1);
c = jStat.multiply(jStat.inv(jStat.add(d, l)), b);
xv = x;
xk = jStat.add(jStat.multiply(h, x), c);
i = 2;
while (Math.abs(jStat.norm(jStat.subtract(xk, xv))) > r) {
xv = xk;
xk = jStat.add(jStat.multiply(h, xv), c);
i = i + 1;
}
return xk;
},
SOR: function SOR(a, b, x, r, w) {
var i = 0;
var n = a.length;
var l = [];
var u = [];
var d = [];
var j, xv, c, h, xk;
for (; i < n; i++) {
l[i] = [];
u[i] = [];
d[i] = [];
for (j = 0; j < n; j++) {
if (i > j) {
l[i][j] = a[i][j];
u[i][j] = d[i][j] = 0;
} else if (i < j) {
u[i][j] = a[i][j];
l[i][j] = d[i][j] = 0;
} else {
d[i][j] = a[i][j];
l[i][j] = u[i][j] = 0;
}
}
}
h = jStat.multiply(jStat.inv(jStat.add(d, jStat.multiply(l, w))),
jStat.subtract(jStat.multiply(d, 1 - w),
jStat.multiply(u, w)));
c = jStat.multiply(jStat.multiply(jStat.inv(jStat.add(d,
jStat.multiply(l, w))), b), w);
xv = x;
xk = jStat.add(jStat.multiply(h, x), c);
i = 2;
while (Math.abs(jStat.norm(jStat.subtract(xk, xv))) > r) {
xv = xk;
xk = jStat.add(jStat.multiply(h, xv), c);
i++;
}
return xk;
},
householder: function householder(a) {
var m = a.length;
var n = a[0].length;
var i = 0;
var w = [];
var p = [];
var alpha, r, k, j, factor;
for (; i < m - 1; i++) {
alpha = 0;
for (j = i + 1; j < n; j++)
alpha += (a[j][i] * a[j][i]);
factor = (a[i + 1][i] > 0) ? -1 : 1;
alpha = factor * Math.sqrt(alpha);
r = Math.sqrt((((alpha * alpha) - a[i + 1][i] * alpha) / 2));
w = jStat.zeros(m, 1);
w[i + 1][0] = (a[i + 1][i] - alpha) / (2 * r);
for (k = i + 2; k < m; k++) w[k][0] = a[k][i] / (2 * r);
p = jStat.subtract(jStat.identity(m, n),
jStat.multiply(jStat.multiply(w, jStat.transpose(w)), 2));
a = jStat.multiply(p, jStat.multiply(a, p));
}
return a;
},
// TODO: not working properly.
QR: function QR(a, b) {
var m = a.length;
var n = a[0].length;
var i = 0;
var w = [];
var p = [];
var x = [];
var j, alpha, r, k, factor, sum;
for (; i < m - 1; i++) {
alpha = 0;
for (j = i + 1; j < n; j++)
alpha += (a[j][i] * a[j][i]);
factor = (a[i + 1][i] > 0) ? -1 : 1;
alpha = factor * Math.sqrt(alpha);
r = Math.sqrt((((alpha * alpha) - a[i + 1][i] * alpha) / 2));
w = jStat.zeros(m, 1);
w[i + 1][0] = (a[i + 1][i] - alpha) / (2 * r);
for (k = i + 2; k < m; k++)
w[k][0] = a[k][i] / (2 * r);
p = jStat.subtract(jStat.identity(m, n),
jStat.multiply(jStat.multiply(w, jStat.transpose(w)), 2));
a = jStat.multiply(p, a);
b = jStat.multiply(p, b);
}
for (i = m - 1; i >= 0; i--) {
sum = 0;
for (j = i + 1; j <= n - 1; j++)
sum = x[j] * a[i][j];
x[i] = b[i][0] / a[i][i];
}
return x;
},
jacobi: function jacobi(a) {
var condition = 1;
var count = 0;
var n = a.length;
var e = jStat.identity(n, n);
var ev = [];
var b, i, j, p, q, maxim, theta, s;
// condition === 1 only if tolerance is not reached
while (condition === 1) {
count++;
maxim = a[0][1];
p = 0;
q = 1;
for (i = 0; i < n; i++) {
for (j = 0; j < n; j++) {
if (i != j) {
if (maxim < Math.abs(a[i][j])) {
maxim = Math.abs(a[i][j]);
p = i;
q = j;
}
}
}
}
if (a[p][p] === a[q][q])
theta = (a[p][q] > 0) ? Math.PI / 4 : -Math.PI / 4;
else
theta = Math.atan(2 * a[p][q] / (a[p][p] - a[q][q])) / 2;
s = jStat.identity(n, n);
s[p][p] = Math.cos(theta);
s[p][q] = -Math.sin(theta);
s[q][p] = Math.sin(theta);
s[q][q] = Math.cos(theta);
// eigen vector matrix
e = jStat.multiply(e, s);
b = jStat.multiply(jStat.multiply(jStat.inv(s), a), s);
a = b;
condition = 0;
for (i = 1; i < n; i++) {
for (j = 1; j < n; j++) {
if (i != j && Math.abs(a[i][j]) > 0.001) {
condition = 1;
}
}
}
}
for (i = 0; i < n; i++) ev.push(a[i][i]);
//returns both the eigenvalue and eigenmatrix
return [e, ev];
},
rungekutta: function rungekutta(f, h, p, t_j, u_j, order) {
var k1, k2, u_j1, k3, k4;
if (order === 2) {
while (t_j <= p) {
k1 = h * f(t_j, u_j);
k2 = h * f(t_j + h, u_j + k1);
u_j1 = u_j + (k1 + k2) / 2;
u_j = u_j1;
t_j = t_j + h;
}
}
if (order === 4) {
while (t_j <= p) {
k1 = h * f(t_j, u_j);
k2 = h * f(t_j + h / 2, u_j + k1 / 2);
k3 = h * f(t_j + h / 2, u_j + k2 / 2);
k4 = h * f(t_j +h, u_j + k3);
u_j1 = u_j + (k1 + 2 * k2 + 2 * k3 + k4) / 6;
u_j = u_j1;
t_j = t_j + h;
}
}
return u_j;
},
romberg: function romberg(f, a, b, order) {
var i = 0;
var h = (b - a) / 2;
var x = [];
var h1 = [];
var g = [];
var m, a1, j, k, I, d;
while (i < order / 2) {
I = f(a);
for (j = a, k = 0; j <= b; j = j + h, k++) x[k] = j;
m = x.length;
for (j = 1; j < m - 1; j++) {
I += (((j % 2) !== 0) ? 4 : 2) * f(x[j]);
}
I = (h / 3) * (I + f(b));
g[i] = I;
h /= 2;
i++;
}
a1 = g.length;
m = 1;
while (a1 !== 1) {
for (j = 0; j < a1 - 1; j++)
h1[j] = ((Math.pow(4, m)) * g[j + 1] - g[j]) / (Math.pow(4, m) - 1);
a1 = h1.length;
g = h1;
h1 = [];
m++;
}
return g;
},
richardson: function richardson(X, f, x, h) {
function pos(X, x) {
var i = 0;
var n = X.length;
var p;
for (; i < n; i++)
if (X[i] === x) p = i;
return p;
}
var n = X.length,
h_min = Math.abs(x - X[pos(X, x) + 1]),
i = 0,
g = [],
h1 = [],
y1, y2, m, a, j;
while (h >= h_min) {
y1 = pos(X, x + h);
y2 = pos(X, x);
g[i] = (f[y1] - 2 * f[y2] + f[2 * y2 - y1]) / (h * h);
h /= 2;
i++;
}
a = g.length;
m = 1;
while (a != 1) {
for (j = 0; j < a - 1; j++)
h1[j] = ((Math.pow(4, m)) * g[j + 1] - g[j]) / (Math.pow(4, m) - 1);
a = h1.length;
g = h1;
h1 = [];
m++;
}
return g;
},
simpson: function simpson(f, a, b, n) {
var h = (b - a) / n;
var I = f(a);
var x = [];
var j = a;
var k = 0;
var i = 1;
var m;
for (; j <= b; j = j + h, k++)
x[k] = j;
m = x.length;
for (; i < m - 1; i++) {
I += ((i % 2 !== 0) ? 4 : 2) * f(x[i]);
}
return (h / 3) * (I + f(b));
},
hermite: function hermite(X, F, dF, value) {
var n = X.length;
var p = 0;
var i = 0;
var l = [];
var dl = [];
var A = [];
var B = [];
var j;
for (; i < n; i++) {
l[i] = 1;
for (j = 0; j < n; j++) {
if (i != j) l[i] *= (value - X[j]) / (X[i] - X[j]);
}
dl[i] = 0;
for (j = 0; j < n; j++) {
if (i != j) dl[i] += 1 / (X [i] - X[j]);
}
A[i] = (1 - 2 * (value - X[i]) * dl[i]) * (l[i] * l[i]);
B[i] = (value - X[i]) * (l[i] * l[i]);
p += (A[i] * F[i] + B[i] * dF[i]);
}
return p;
},
lagrange: function lagrange(X, F, value) {
var p = 0;
var i = 0;
var j, l;
var n = X.length;
for (; i < n; i++) {
l = F[i];
for (j = 0; j < n; j++) {
// calculating the lagrange polynomial L_i
if (i != j) l *= (value - X[j]) / (X[i] - X[j]);
}
// adding the lagrange polynomials found above
p += l;
}
return p;
},
cubic_spline: function cubic_spline(X, F, value) {
var n = X.length;
var i = 0, j;
var A = [];
var B = [];
var alpha = [];
var c = [];
var h = [];
var b = [];
var d = [];
for (; i < n - 1; i++)
h[i] = X[i + 1] - X[i];
alpha[0] = 0;
for (i = 1; i < n - 1; i++) {
alpha[i] = (3 / h[i]) * (F[i + 1] - F[i]) -
(3 / h[i-1]) * (F[i] - F[i-1]);
}
for (i = 1; i < n - 1; i++) {
A[i] = [];
B[i] = [];
A[i][i-1] = h[i-1];
A[i][i] = 2 * (h[i - 1] + h[i]);
A[i][i+1] = h[i];
B[i][0] = alpha[i];
}
c = jStat.multiply(jStat.inv(A), B);
for (j = 0; j < n - 1; j++) {
b[j] = (F[j + 1] - F[j]) / h[j] - h[j] * (c[j + 1][0] + 2 * c[j][0]) / 3;
d[j] = (c[j + 1][0] - c[j][0]) / (3 * h[j]);
}
for (j = 0; j < n; j++) {
if (X[j] > value) break;
}
j -= 1;
return F[j] + (value - X[j]) * b[j] + jStat.sq(value-X[j]) *
c[j] + (value - X[j]) * jStat.sq(value - X[j]) * d[j];
},
gauss_quadrature: function gauss_quadrature() {
throw new Error('gauss_quadrature not yet implemented');
},
PCA: function PCA(X) {
var m = X.length;
var n = X[0].length;
var flag = false;
var i = 0;
var j, temp1;
var u = [];
var D = [];
var result = [];
var temp2 = [];
var Y = [];
var Bt = [];
var B = [];
var C = [];
var V = [];
var Vt = [];
for (i = 0; i < m; i++) {
u[i] = jStat.sum(X[i]) / n;
}
for (i = 0; i < n; i++) {
B[i] = [];
for(j = 0; j < m; j++) {
B[i][j] = X[j][i] - u[j];
}
}
B = jStat.transpose(B);
for (i = 0; i < m; i++) {
C[i] = [];
for (j = 0; j < m; j++) {
C[i][j] = (jStat.dot([B[i]], [B[j]])) / (n - 1);
}
}
result = jStat.jacobi(C);
V = result[0];
D = result[1];
Vt = jStat.transpose(V);
for (i = 0; i < D.length; i++) {
for (j = i; j < D.length; j++) {
if(D[i] < D[j]) {
temp1 = D[i];
D[i] = D[j];
D[j] = temp1;
temp2 = Vt[i];
Vt[i] = Vt[j];
Vt[j] = temp2;
}
}
}
Bt = jStat.transpose(B);
for (i = 0; i < m; i++) {
Y[i] = [];
for (j = 0; j < Bt.length; j++) {
Y[i][j] = jStat.dot([Vt[i]], [Bt[j]]);
}
}
return [X, D, Vt, Y];
}
});
// extend jStat.fn with methods that require one argument
(function(funcs) {
for (var i = 0; i < funcs.length; i++) (function(passfunc) {
jStat.fn[passfunc] = function(arg, func) {
var tmpthis = this;
// check for callback
if (func) {
setTimeout(function() {
func.call(tmpthis, jStat.fn[passfunc].call(tmpthis, arg));
}, 15);
return this;
}
return jStat(jStat[passfunc](this, arg));
};
}(funcs[i]));
}('add divide multiply subtract dot pow abs norm angle'.split(' ')));
}(this.jStat, Math));
(function(jStat, Math) {
var slice = [].slice;
var isNumber = jStat.utils.isNumber;
// flag==true denotes use of sample standard deviation
// Z Statistics
jStat.extend({
// 2 different parameter lists:
// (value, mean, sd)
// (value, array, flag)
zscore: function zscore() {
var args = slice.call(arguments);
if (isNumber(args[1])) {
return (args[0] - args[1]) / args[2];
}
return (args[0] - jStat.mean(args[1])) / jStat.stdev(args[1], args[2]);
},
// 3 different paramter lists:
// (value, mean, sd, sides)
// (zscore, sides)
// (value, array, sides, flag)
ztest: function ztest() {
var args = slice.call(arguments);
if (args.length === 4) {
if(isNumber(args[1])) {
var z = jStat.zscore(args[0],args[1],args[2])
return (args[3] === 1) ?
(jStat.normal.cdf(-Math.abs(z),0,1)) :
(jStat.normal.cdf(-Math.abs(z),0,1)* 2);
}
var z = args[0]
return (args[2] === 1) ?
(jStat.normal.cdf(-Math.abs(z),0,1)) :
(jStat.normal.cdf(-Math.abs(z),0,1)*2);
}
var z = jStat.zscore(args[0],args[1],args[3])
return (args[1] === 1) ?
(jStat.normal.cdf(-Math.abs(z), 0, 1)) :
(jStat.normal.cdf(-Math.abs(z), 0, 1)*2);
}
});
jStat.extend(jStat.fn, {
zscore: function zscore(value, flag) {
return (value - this.mean()) / this.stdev(flag);
},
ztest: function ztest(value, sides, flag) {
var zscore = Math.abs(this.zscore(value, flag));
return (sides === 1) ?
(jStat.normal.cdf(-zscore, 0, 1)) :
(jStat.normal.cdf(-zscore, 0, 1) * 2);
}
});
// T Statistics
jStat.extend({
// 2 parameter lists
// (value, mean, sd, n)
// (value, array)
tscore: function tscore() {
var args = slice.call(arguments);
return (args.length === 4) ?
((args[0] - args[1]) / (args[2] / Math.sqrt(args[3]))) :
((args[0] - jStat.mean(args[1])) /
(jStat.stdev(args[1], true) / Math.sqrt(args[1].length)));
},
// 3 different paramter lists:
// (value, mean, sd, n, sides)
// (tscore, n, sides)
// (value, array, sides)
ttest: function ttest() {
var args = slice.call(arguments);
var tscore;
if (args.length === 5) {
tscore = Math.abs(jStat.tscore(args[0], args[1], args[2], args[3]));
return (args[4] === 1) ?
(jStat.studentt.cdf(-tscore, args[3]-1)) :
(jStat.studentt.cdf(-tscore, args[3]-1)*2);
}
if (isNumber(args[1])) {
tscore = Math.abs(args[0])
return (args[2] == 1) ?
(jStat.studentt.cdf(-tscore, args[1]-1)) :
(jStat.studentt.cdf(-tscore, args[1]-1) * 2);
}
tscore = Math.abs(jStat.tscore(args[0], args[1]))
return (args[2] == 1) ?
(jStat.studentt.cdf(-tscore, args[1].length-1)) :
(jStat.studentt.cdf(-tscore, args[1].length-1) * 2);
}
});
jStat.extend(jStat.fn, {
tscore: function tscore(value) {
return (value - this.mean()) / (this.stdev(true) / Math.sqrt(this.cols()));
},
ttest: function ttest(value, sides) {
return (sides === 1) ?
(1 - jStat.studentt.cdf(Math.abs(this.tscore(value)), this.cols()-1)) :
(jStat.studentt.cdf(-Math.abs(this.tscore(value)), this.cols()-1)*2);
}
});
// F Statistics
jStat.extend({
// Paramter list is as follows:
// (array1, array2, array3, ...)
// or it is an array of arrays
// array of arrays conversion
anovafscore: function anovafscore() {
var args = slice.call(arguments),
expVar, sample, sampMean, sampSampMean, tmpargs, unexpVar, i, j;
if (args.length === 1) {
tmpargs = new Array(args[0].length);
for (i = 0; i < args[0].length; i++) {
tmpargs[i] = args[0][i];
}
args = tmpargs;
}
// 2 sample case
if (args.length === 2) {
return jStat.variance(args[0]) / jStat.variance(args[1]);
}
// Builds sample array
sample = new Array();
for (i = 0; i < args.length; i++) {
sample = sample.concat(args[i]);
}
sampMean = jStat.mean(sample);
// Computes the explained variance
expVar = 0;
for (i = 0; i < args.length; i++) {
expVar = expVar + args[i].length * Math.pow(jStat.mean(args[i]) - sampMean, 2);
}
expVar /= (args.length - 1);
// Computes unexplained variance
unexpVar = 0;
for (i = 0; i < args.length; i++) {
sampSampMean = jStat.mean(args[i]);
for (j = 0; j < args[i].length; j++) {
unexpVar += Math.pow(args[i][j] - sampSampMean, 2);
}
}
unexpVar /= (sample.length - args.length);
return expVar / unexpVar;
},
// 2 different paramter setups
// (array1, array2, array3, ...)
// (anovafscore, df1, df2)
anovaftest: function anovaftest() {
var args = slice.call(arguments),
df1, df2, n, i;
if (isNumber(args[0])) {
return 1 - jStat.centralF.cdf(args[0], args[1], args[2]);
}
anovafscore = jStat.anovafscore(args);
df1 = args.length - 1;
n = 0;
for (i = 0; i < args.length; i++) {
n = n + args[i].length;
}
df2 = n - df1 - 1;
return 1 - jStat.centralF.cdf(anovafscore, df1, df2);
},
ftest: function ftest(fscore, df1, df2) {
return 1 - jStat.centralF.cdf(fscore, df1, df2);
}
});
jStat.extend(jStat.fn, {
anovafscore: function anovafscore() {
return jStat.anovafscore(this.toArray());
},
anovaftes: function anovaftes() {
var n = 0;
var i;
for (i = 0; i < this.length; i++) {
n = n + this[i].length;
}
return jStat.ftest(this.anovafscore(), this.length - 1, n - this.length);
}
});
// Error Bounds
jStat.extend({
// 2 different parameter setups
// (value, alpha, sd, n)
// (value, alpha, array)
normalci: function normalci() {
var args = slice.call(arguments),
ans = new Array(2),
change;
if (args.length === 4) {
change = Math.abs(jStat.normal.inv(args[1] / 2, 0, 1) *
args[2] / Math.sqrt(args[3]));
} else {
change = Math.abs(jStat.normal.inv(args[1] / 2, 0, 1) *
jStat.stdev(args[2]) / Math.sqrt(args[2].length));
}
ans[0] = args[0] - change;
ans[1] = args[0] + change;
return ans;
},
// 2 different parameter setups
// (value, alpha, sd, n)
// (value, alpha, array)
tci: function tci() {
var args = slice.call(arguments),
ans = new Array(2),
change;
if (args.length === 4) {
change = Math.abs(jStat.studentt.inv(args[1] / 2, args[3] - 1) *
args[2] / Math.sqrt(args[3]));
} else {
change = Math.abs(jStat.studentt.inv(args[1] / 2, args[2].length - 1) *
jStat.stdev(args[2], true) / Math.sqrt(args[2].length));
}
ans[0] = args[0] - change;
ans[1] = args[0] + change;
return ans;
},
significant: function significant(pvalue, alpha) {
return pvalue < alpha;
}
});
jStat.extend(jStat.fn, {
normalci: function normalci(value, alpha) {
return jStat.normalci(value, alpha, this.toArray());
},
tci: function tci(value, alpha) {
return jStat.tci(value, alpha, this.toArray());
}
});
}(this.jStat, Math));
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