(function (window, factory) {
if (typeof exports === 'object') {
module.exports = factory();
} else if (typeof define === 'function' && define.amd) {
define(factory);
} else {
window.jStat = factory();
}
})(this, function () {
var 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(num) {
return (typeof num === 'number') ? num - num === 0 : false;
}
// 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) {
// 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 (var 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
};
jStat._random_fn = Math.random;
jStat.setRandom = function setRandom(fn) {
if (typeof fn !== 'function')
throw new TypeError('fn is not a function');
jStat._random_fn = fn;
};
// 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 or return a sub matrix by pick some rows
jStat.row = function row(arr, index) {
if (isArray(index)) {
return index.map(function(i) {
return jStat.row(arr, i);
})
}
return arr[index];
};
// return row as array
// rowa([[1,2],[3,4]],0) -> [1,2]
jStat.rowa = function rowa(arr, i) {
return jStat.row(arr, i);
};
// Returns the specified column as a vector or return a sub matrix by pick some
// columns
jStat.col = function col(arr, index) {
if (isArray(index)) {
var submat = jStat.arange(arr.length).map(function() {
return new Array(index.length);
});
index.forEach(function(ind, i){
jStat.arange(arr.length).forEach(function(j) {
submat[j][i] = arr[j][ind];
});
});
return submat;
}
var column = new Array(arr.length);
for (var i = 0; i < arr.length; i++)
column[i] = [arr[i][index]];
return column;
};
// return column as array
// cola([[1,2],[3,4]],0) -> [1,3]
jStat.cola = function cola(arr, i) {
return jStat.col(arr, i).map(function(a){ return a[0] });
};
// 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;
};
// Cumulatively combine the elements of an array or array of arrays using a function.
jStat.cumreduce = function cumreduce(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);
if (ncol > 0)
res[row][0] = arr[row][0];
for (col = 1; col < ncol; col++)
res[row][col] = func(res[row][col-1], arr[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, jStat._random_fn);
};
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 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 < length;
cnt++, current = (min * hival + step * hival * cnt) / hival) {
arr.push((func ? func(current, cnt) : current));
}
return arr;
};
// arange(5) -> [0,1,2,3,4]
// arange(1,5) -> [1,2,3,4]
// arange(5,1,-1) -> [5,4,3,2]
jStat.arange = function arange(start, end, step) {
var rl = [];
var i;
step = step || 1;
if (end === undefined) {
end = start;
start = 0;
}
if (start === end || step === 0) {
return [];
}
if (start < end && step < 0) {
return [];
}
if (start > end && step > 0) {
return [];
}
if (step > 0) {
for (i = start; i < end; i += step) {
rl.push(i);
}
} else {
for (i = start; i > end; i += step) {
rl.push(i);
}
}
return rl;
};
// A=[[1,2,3],[4,5,6],[7,8,9]]
// slice(A,{row:{end:2},col:{start:1}}) -> [[2,3],[5,6]]
// slice(A,1,{start:1}) -> [5,6]
// as numpy code A[:2,1:]
jStat.slice = (function(){
function _slice(list, start, end, step) {
// note it's not equal to range.map mode it's a bug
var i;
var rl = [];
var length = list.length;
if (start === undefined && end === undefined && step === undefined) {
return jStat.copy(list);
}
start = start || 0;
end = end || list.length;
start = start >= 0 ? start : length + start;
end = end >= 0 ? end : length + end;
step = step || 1;
if (start === end || step === 0) {
return [];
}
if (start < end && step < 0) {
return [];
}
if (start > end && step > 0) {
return [];
}
if (step > 0) {
for (i = start; i < end; i += step) {
rl.push(list[i]);
}
} else {
for (i = start; i > end;i += step) {
rl.push(list[i]);
}
}
return rl;
}
function slice(list, rcSlice) {
var colSlice, rowSlice;
rcSlice = rcSlice || {};
if (isNumber(rcSlice.row)) {
if (isNumber(rcSlice.col))
return list[rcSlice.row][rcSlice.col];
var row = jStat.rowa(list, rcSlice.row);
colSlice = rcSlice.col || {};
return _slice(row, colSlice.start, colSlice.end, colSlice.step);
}
if (isNumber(rcSlice.col)) {
var col = jStat.cola(list, rcSlice.col);
rowSlice = rcSlice.row || {};
return _slice(col, rowSlice.start, rowSlice.end, rowSlice.step);
}
rowSlice = rcSlice.row || {};
colSlice = rcSlice.col || {};
var rows = _slice(list, rowSlice.start, rowSlice.end, rowSlice.step);
return rows.map(function(row) {
return _slice(row, colSlice.start, colSlice.end, colSlice.step);
});
}
return slice;
}());
// A=[[1,2,3],[4,5,6],[7,8,9]]
// sliceAssign(A,{row:{start:1},col:{start:1}},[[0,0],[0,0]])
// A=[[1,2,3],[4,0,0],[7,0,0]]
jStat.sliceAssign = function sliceAssign(A, rcSlice, B) {
var nl, ml;
if (isNumber(rcSlice.row)) {
if (isNumber(rcSlice.col))
return A[rcSlice.row][rcSlice.col] = B;
rcSlice.col = rcSlice.col || {};
rcSlice.col.start = rcSlice.col.start || 0;
rcSlice.col.end = rcSlice.col.end || A[0].length;
rcSlice.col.step = rcSlice.col.step || 1;
nl = jStat.arange(rcSlice.col.start,
Math.min(A.length, rcSlice.col.end),
rcSlice.col.step);
var m = rcSlice.row;
nl.forEach(function(n, i) {
A[m][n] = B[i];
});
return A;
}
if (isNumber(rcSlice.col)) {
rcSlice.row = rcSlice.row || {};
rcSlice.row.start = rcSlice.row.start || 0;
rcSlice.row.end = rcSlice.row.end || A.length;
rcSlice.row.step = rcSlice.row.step || 1;
ml = jStat.arange(rcSlice.row.start,
Math.min(A[0].length, rcSlice.row.end),
rcSlice.row.step);
var n = rcSlice.col;
ml.forEach(function(m, j) {
A[m][n] = B[j];
});
return A;
}
if (B[0].length === undefined) {
B = [B];
}
rcSlice.row.start = rcSlice.row.start || 0;
rcSlice.row.end = rcSlice.row.end || A.length;
rcSlice.row.step = rcSlice.row.step || 1;
rcSlice.col.start = rcSlice.col.start || 0;
rcSlice.col.end = rcSlice.col.end || A[0].length;
rcSlice.col.step = rcSlice.col.step || 1;
ml = jStat.arange(rcSlice.row.start,
Math.min(A.length, rcSlice.row.end),
rcSlice.row.step);
nl = jStat.arange(rcSlice.col.start,
Math.min(A[0].length, rcSlice.col.end),
rcSlice.col.step);
ml.forEach(function(m, i) {
nl.forEach(function(n, j) {
A[m][n] = B[i][j];
});
});
return A;
};
// [1,2,3] ->
// [[1,0,0],[0,2,0],[0,0,3]]
jStat.diagonal = function diagonal(diagArray) {
var mat = jStat.zeros(diagArray.length, diagArray.length);
diagArray.forEach(function(t, i) {
mat[i][i] = t;
});
return mat;
};
// return copy of A
jStat.copy = function copy(A) {
return A.map(function(row) {
if (isNumber(row))
return row;
return row.map(function(t) {
return t;
});
});
};
// 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));
};
// Cumulatively combine the elements of a matrix or vector using a function.
jProto.cumreduce = function cumreduce(func, toAlter) {
return jStat(jStat.cumreduce(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] = 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;
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;
};
// sum of an array in each row
jStat.sumrow = function sumrow(arr) {
var sum = 0;
var i = arr.length;
while (--i >= 0)
sum += arr[i];
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;
};
// unique values of an array
jStat.unique = function unique(arr) {
var hash = {}, _arr = [];
for(var i = 0; i < arr.length; i++) {
if (!hash[arr[i]]) {
hash[arr[i]] = true;
_arr.push(arr[i]);
}
}
return _arr;
};
// 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) {
var logs = arr.map(Math.log)
var meanOfLogs = jStat.mean(logs)
return Math.exp(meanOfLogs)
};
// 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) {
return jStat.cumreduce(arr, function (a, b) { return a + b; });
};
// cumulative product of an array
jStat.cumprod = function cumprod(arr) {
return jStat.cumreduce(arr, function (a, b) { return a * b; });
};
// 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;
};
// ranks of an array
jStat.rank = function (arr) {
var i;
var distinctNumbers = [];
var numberCounts = {};
for (i = 0; i < arr.length; i++) {
var number = arr[i];
if (numberCounts[number]) {
numberCounts[number]++;
} else {
numberCounts[number] = 1;
distinctNumbers.push(number);
}
}
var sortedDistinctNumbers = distinctNumbers.sort(ascNum);
var numberRanks = {};
var currentRank = 1;
for (i = 0; i < sortedDistinctNumbers.length; i++) {
var number = sortedDistinctNumbers[i];
var count = numberCounts[number];
var first = currentRank;
var last = currentRank + count - 1;
var rank = (first + last) / 2;
numberRanks[number] = rank;
currentRank += count;
}
return arr.map(function (number) {
return numberRanks[number];
});
};
// 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 = true indicates sample instead of population
jStat.variance = function variance(arr, flag) {
return jStat.sumsqerr(arr) / (arr.length - (flag ? 1 : 0));
};
// pooled variance of an array of arrays
jStat.pooledvariance = function pooledvariance(arr) {
var sumsqerr = arr.reduce(function (a, samples) {return a + jStat.sumsqerr(samples);}, 0);
var count = arr.reduce(function (a, samples) {return a + samples.length;}, 0);
return sumsqerr / (count - arr.length);
};
// deviation of an array
jStat.deviation = function (arr) {
var mean = jStat.mean(arr);
var arrlen = arr.length;
var dev = new Array(arrlen);
for (var i = 0; i < arrlen; i++) {
dev[i] = arr[i] - mean;
}
return dev;
};
// standard deviation of an array
// flag = true indicates sample instead of population
jStat.stdev = function stdev(arr, flag) {
return Math.sqrt(jStat.variance(arr, flag));
};
// pooled standard deviation of an array of arrays
jStat.pooledstdev = function pooledstdev(arr) {
return Math.sqrt(jStat.pooledvariance(arr));
};
// mean deviation (mean absolute deviation) of an array
jStat.meandev = function meandev(arr) {
var mean = jStat.mean(arr);
var a = [];
for (var i = arr.length - 1; i >= 0; i--) {
a.push(Math.abs(arr[i] - mean));
}
return jStat.mean(a);
};
// median deviation (median absolute deviation) of an array
jStat.meddev = function meddev(arr) {
var median = jStat.median(arr);
var a = [];
for (var i = arr.length - 1; i >= 0; i--) {
a.push(Math.abs(arr[i] - median));
}
return jStat.median(a);
};
// 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;
};
// Return the k-th percentile of values in a range, where k is in the range 0..1, inclusive.
// Passing true for the exclusive parameter excludes both endpoints of the range.
jStat.percentile = function percentile(arr, k, exclusive) {
var _arr = arr.slice().sort(ascNum);
var realIndex = k * (_arr.length + (exclusive ? 1 : -1)) + (exclusive ? 0 : 1);
var index = parseInt(realIndex);
var frac = realIndex - index;
if (index + 1 < _arr.length) {
return _arr[index - 1] + frac * (_arr[index] - _arr[index - 1]);
} else {
return _arr[index - 1];
}
}
// 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;
};
// Histogram (bin count) data
jStat.histogram = function histogram(arr, binCnt) {
binCnt = binCnt || 4;
var first = jStat.min(arr);
var binWidth = (jStat.max(arr) - first) / binCnt;
var len = arr.length;
var bins = [];
var i;
for (i = 0; i < binCnt; i++)
bins[i] = 0;
for (i = 0; i < len; i++)
bins[Math.min(Math.floor(((arr[i] - first) / binWidth)), binCnt - 1)] += 1;
return bins;
};
// 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);
};
// (spearman's) rank correlation coefficient, sp
jStat.spearmancoeff = function (arr1, arr2) {
arr1 = jStat.rank(arr1);
arr2 = jStat.rank(arr2);
//return pearson's correlation of the ranks:
return jStat.corrcoeff(arr1, arr2);
}
// statistical standardized moments (general form of skew/kurt)
jStat.stanMoment = function stanMoment(arr, n) {
var mu = jStat.mean(arr);
var sigma = jStat.stdev(arr);
var len = arr.length;
var skewSum = 0;
for (var i = 0; i < len; i++)
skewSum += Math.pow((arr[i] - mu) / sigma, n);
return skewSum / arr.length;
};
// (pearson's) moment coefficient of skewness
jStat.skewness = function skewness(arr) {
return jStat.stanMoment(arr, 3);
};
// (pearson's) (excess) kurtosis
jStat.kurtosis = function kurtosis(arr) {
return jStat.stanMoment(arr, 4) - 3;
};
var jProto = jStat.prototype;
// Extend jProto with method for calculating cumulative sums and products.
// This differs from the similar extension below as cumsum and cumprod should
// not be run again in the case fullbool === true.
// If a matrix is passed, automatically assume operation should be done on the
// 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 arr;
}
// Pass fullbool if only vector, not a matrix. for variance and stdev.
return jStat[passfunc](this[0], fullbool);
};
})(funcs[i]);
})(('cumsum cumprod').split(' '));
// 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) {
if (passfunc !== 'sumrow')
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 sumrow product min max unique mean meansqerr ' +
'geomean median diff rank mode range variance deviation stdev meandev ' +
'meddev coeffvar quartiles histogram skewness kurtosis').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);
var callbackFunction;
// 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])) {
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 {
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(' '));
}(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;
};
/*
* log-gamma function to support poisson distribution sampling. The
* algorithm comes from SPECFUN by Shanjie Zhang and Jianming Jin and their
* book "Computation of Special Functions", 1996, John Wiley & Sons, Inc.
*/
jStat.loggam = function loggam(x) {
var x0, x2, xp, gl, gl0;
var k, n;
var a = [8.333333333333333e-02, -2.777777777777778e-03,
7.936507936507937e-04, -5.952380952380952e-04,
8.417508417508418e-04, -1.917526917526918e-03,
6.410256410256410e-03, -2.955065359477124e-02,
1.796443723688307e-01, -1.39243221690590e+00];
x0 = x;
n = 0;
if ((x == 1.0) || (x == 2.0)) {
return 0.0;
}
if (x <= 7.0) {
n = Math.floor(7 - x);
x0 = x + n;
}
x2 = 1.0 / (x0 * x0);
xp = 2 * Math.PI;
gl0 = a[9];
for (k = 8; k >= 0; k--) {
gl0 *= x2;
gl0 += a[k];
}
gl = gl0 / x0 + 0.5 * Math.log(xp) + (x0 - 0.5) * Math.log(x0) - x0;
if (x <= 7.0) {
for (k = 1; k <= n; k++) {
gl -= Math.log(x0 - 1.0);
x0 -= 1.0;
}
}
return gl;
}
// 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;
if (x > 171.6243769536076) {
return Infinity;
}
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, which is usually typeset with a
// lower-case greek gamma as the function symbol
jStat.gammap = function gammap(a, x) {
return jStat.lowRegGamma(a, x) * jStat.gammafn(a);
};
// The lower regularized incomplete gamma function, usually written P(a,x)
jStat.lowRegGamma = function lowRegGamma(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;
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 of the lower regularized inomplete 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.lowRegGamma(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;
if (!m)
m = n;
if (n)
return jStat.create(n, m, function() { return jStat.randn(); });
do {
u = jStat._random_fn();
v = 1.7156 * (jStat._random_fn() - 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 = jStat._random_fn();
} 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 = jStat._random_fn();
} 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(' '));
}(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 f(a, b, c) {
if (!(this instanceof f))
return new f(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 ' +
'laplace lognormal noncentralt normal pareto studentt weibull uniform ' +
'binomial negbin hypgeom poisson triangular tukey arcsine'
).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) {
return jStat.ibetainv(0.5, alpha, beta);
},
mode: function mode(alpha, beta) {
return (alpha - 1 ) / ( alpha + beta - 2);
},
// 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, {
// This implementation of the pdf function avoids float overflow
// See the way that R calculates this value:
// https://svn.r-project.org/R/trunk/src/nmath/df.c
pdf: function pdf(x, df1, df2) {
var p, q, f;
if (x < 0)
return 0;
if (df1 <= 2) {
if (x === 0 && df1 < 2) {
return Infinity;
}
if (x === 0 && df1 === 2) {
return 1;
}
return (1 / jStat.betafn(df1 / 2, df2 / 2)) *
Math.pow(df1 / df2, df1 / 2) *
Math.pow(x, (df1/2) - 1) *
Math.pow((1 + (df1 / df2) * x), -(df1 + df2) / 2);
}
p = (df1 * x) / (df2 + x * df1);
q = df2 / (df2 + x * df1);
f = df1 * q / 2.0;
return f * jStat.binomial.pdf((df1 - 2) / 2, (df1 + df2 - 2) / 2, p);
},
cdf: function cdf(x, df1, df2) {
if (x < 0)
return 0;
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) {
if (scale < 0) { return 0; }
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) {
if (x < 0)
return 0;
return (x === 0 && dof === 2) ? 0.5 :
Math.exp((dof / 2 - 1) * Math.log(x) - x / 2 - (dof / 2) *
Math.log(2) - jStat.gammaln(dof / 2));
},
cdf: function cdf(x, dof) {
if (x < 0)
return 0;
return jStat.lowRegGamma(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(jStat._random_fn());
},
variance : function(rate) {
return Math.pow(rate, -2);
}
});
// extend gamma function with static methods
jStat.extend(jStat.gamma, {
pdf: function pdf(x, shape, scale) {
if (x < 0)
return 0;
return (x === 0 && shape === 1) ? 1 / scale :
Math.exp((shape - 1) * Math.log(x) - x / scale -
jStat.gammaln(shape) - shape * Math.log(scale));
},
cdf: function cdf(x, shape, scale) {
if (x < 0)
return 0;
return jStat.lowRegGamma(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) {
if (x <= 0)
return 0;
return Math.exp(-(shape + 1) * Math.log(x) - scale / x -
jStat.gammaln(shape) + shape * Math.log(scale));
},
cdf: function cdf(x, shape, scale) {
if (x <= 0)
return 0;
return 1 - jStat.lowRegGamma(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) {
if (x === 0 && alpha === 1)
return beta;
else if (x === 1 && beta === 1)
return alpha;
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) {
if (x < 0)
return 0;
else if (x > 1)
return 1;
return (1 - Math.pow(1 - Math.pow(x, alpha), beta));
},
inv: function inv(p, alpha, beta) {
return Math.pow(1 - Math.pow(1 - p, 1 / beta), 1 / alpha);
},
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) {
if (x <= 0)
return 0;
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) {
if (x < 0)
return 0;
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 noncentralt function with static methods
jStat.extend(jStat.noncentralt, {
pdf: function pdf(x, dof, ncp) {
var tol = 1e-14;
if (Math.abs(ncp) < tol) // ncp approx 0; use student-t
return jStat.studentt.pdf(x, dof)
if (Math.abs(x) < tol) { // different formula for x == 0
return Math.exp(jStat.gammaln((dof + 1) / 2) - ncp * ncp / 2 -
0.5 * Math.log(Math.PI * dof) - jStat.gammaln(dof / 2));
}
// formula for x != 0
return dof / x *
(jStat.noncentralt.cdf(x * Math.sqrt(1 + 2 / dof), dof+2, ncp) -
jStat.noncentralt.cdf(x, dof, ncp));
},
cdf: function cdf(x, dof, ncp) {
var tol = 1e-14;
var min_iterations = 200;
if (Math.abs(ncp) < tol) // ncp approx 0; use student-t
return jStat.studentt.cdf(x, dof);
// turn negative x into positive and flip result afterwards
var flip = false;
if (x < 0) {
flip = true;
ncp = -ncp;
}
var prob = jStat.normal.cdf(-ncp, 0, 1);
var value = tol + 1;
// use value at last two steps to determine convergence
var lastvalue = value;
var y = x * x / (x * x + dof);
var j = 0;
var p = Math.exp(-ncp * ncp / 2);
var q = Math.exp(-ncp * ncp / 2 - 0.5 * Math.log(2) -
jStat.gammaln(3 / 2)) * ncp;
while (j < min_iterations || lastvalue > tol || value > tol) {
lastvalue = value;
if (j > 0) {
p *= (ncp * ncp) / (2 * j);
q *= (ncp * ncp) / (2 * (j + 1 / 2));
}
value = p * jStat.beta.cdf(y, j + 0.5, dof / 2) +
q * jStat.beta.cdf(y, j+1, dof/2);
prob += 0.5 * value;
j++;
}
return flip ? (1 - prob) : prob;
}
});
// 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 0;
return (shape * Math.pow(scale, shape)) / Math.pow(x, shape + 1);
},
cdf: function cdf(x, scale, shape) {
if (x < scale)
return 0;
return 1 - Math.pow(scale / x, shape);
},
inv: function inv(p, scale, shape) {
return scale / Math.pow(1 - p, 1 / 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) {
dof = dof > 1e100 ? 1e100 : dof;
return (1/(Math.sqrt(dof) * jStat.betafn(0.5, 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 > 0.5) ? 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 || scale < 0 || shape < 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 0;
return scale * Math.pow((shape - 1) / shape, 1 / shape);
},
sample: function sample(scale, shape) {
return scale * Math.pow(-Math.log(jStat._random_fn()), 1 / shape);
},
variance: function variance(scale, shape) {
return scale * scale * jStat.gammafn(1 + 2 / shape) -
Math.pow(jStat.weibull.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;
},
inv: function(p, a, b) {
return a + (p * (b - a));
},
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 * jStat._random_fn() - 1);
},
variance: function variance(a, b) {
return Math.pow(b - a, 2) / 12;
}
});
// Got this from http://www.math.ucla.edu/~tom/distributions/binomial.html
function betinc(x, a, b, eps) {
var a0 = 0;
var b0 = 1;
var a1 = 1;
var b1 = 1;
var m9 = 0;
var a2 = 0;
var c9;
while (Math.abs((a1 - a2) / a1) > eps) {
a2 = a1;
c9 = -(a + m9) * (a + b + m9) * x / (a + 2 * m9) / (a + 2 * m9 + 1);
a0 = a1 + c9 * a0;
b0 = b1 + c9 * b0;
m9 = m9 + 1;
c9 = m9 * (b - m9) * x / (a + 2 * m9 - 1) / (a + 2 * m9);
a1 = a0 + c9 * a1;
b1 = b0 + c9 * b1;
a0 = a0 / b1;
b0 = b0 / b1;
a1 = a1 / b1;
b1 = 1;
}
return a1 / a;
}
// 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 betacdf;
var eps = 1e-10;
if (x < 0)
return 0;
if (x >= n)
return 1;
if (p < 0 || p > 1 || n <= 0)
return NaN;
x = Math.floor(x);
var z = p;
var a = x + 1;
var b = n - x;
var s = a + b;
var bt = Math.exp(jStat.gammaln(s) - jStat.gammaln(b) -
jStat.gammaln(a) + a * Math.log(z) + b * Math.log(1 - z));
if (z < (a + 1) / (s + 2))
betacdf = bt * betinc(z, a, b, eps);
else
betacdf = 1 - bt * betinc(1 - z, b, a, eps);
return Math.round((1 - betacdf) * (1 / eps)) / (1 / eps);
}
});
// extend uniform function with static methods
jStat.extend(jStat.negbin, {
pdf: function pdf(k, r, p) {
if (k !== k >>> 0)
return false;
if (k < 0)
return 0;
return 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) {
if (l < 0 || (k % 1) !== 0 || k < 0) {
return 0;
}
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;
},
sampleSmall: function sampleSmall(l) {
var p = 1, k = 0, L = Math.exp(-l);
do {
k++;
p *= jStat._random_fn();
} while (p > L);
return k - 1;
},
sampleLarge: function sampleLarge(l) {
var lam = l;
var k;
var U, V, slam, loglam, a, b, invalpha, vr, us;
slam = Math.sqrt(lam);
loglam = Math.log(lam);
b = 0.931 + 2.53 * slam;
a = -0.059 + 0.02483 * b;
invalpha = 1.1239 + 1.1328 / (b - 3.4);
vr = 0.9277 - 3.6224 / (b - 2);
while (1) {
U = Math.random() - 0.5;
V = Math.random();
us = 0.5 - Math.abs(U);
k = Math.floor((2 * a / us + b) * U + lam + 0.43);
if ((us >= 0.07) && (V <= vr)) {
return k;
}
if ((k < 0) || ((us < 0.013) && (V > us))) {
continue;
}
/* log(V) == log(0.0) ok here */
/* if U==0.0 so that us==0.0, log is ok since always returns */
if ((Math.log(V) + Math.log(invalpha) - Math.log(a / (us * us) + b)) <= (-lam + k * loglam - jStat.loggam(k + 1))) {
return k;
}
}
},
sample: function sample(l) {
if (l < 10)
return this.sampleSmall(l);
else
return this.sampleLarge(l);
}
});
// extend triangular function with static methods
jStat.extend(jStat.triangular, {
pdf: function pdf(x, a, b, c) {
if (b <= a || c < a || c > b) {
return NaN;
} else {
if (x < a || x > b) {
return 0;
} else if (x < c) {
return (2 * (x - a)) / ((b - a) * (c - a));
} else if (x === c) {
return (2 / (b - a));
} else { // x > c
return (2 * (b - x)) / ((b - a) * (b - c));
}
}
},
cdf: function cdf(x, a, b, c) {
if (b <= a || c < a || c > b)
return NaN;
if (x <= a)
return 0;
else if (x >= b)
return 1;
if (x <= c)
return Math.pow(x - a, 2) / ((b - a) * (c - a));
else // x > c
return 1 - Math.pow(b - x, 2) / ((b - a) * (b - c));
},
inv: function inv(p, a, b, c) {
if (b <= a || c < a || c > b) {
return NaN;
} else {
if (p <= ((c - a) / (b - a))) {
return a + (b - a) * Math.sqrt(p * ((c - a) / (b - a)));
} else { // p > ((c - a) / (b - a))
return a + (b - a) * (1 - Math.sqrt((1 - p) * (1 - ((c - a) / (b - a)))));
}
}
},
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 = jStat._random_fn();
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;
}
});
// extend arcsine function with static methods
jStat.extend(jStat.arcsine, {
pdf: function pdf(x, a, b) {
if (b <= a) return NaN;
return (x <= a || x >= b) ? 0 :
(2 / Math.PI) *
Math.pow(Math.pow(b - a, 2) -
Math.pow(2 * x - a - b, 2), -0.5);
},
cdf: function cdf(x, a, b) {
if (x < a)
return 0;
else if (x < b)
return (2 / Math.PI) * Math.asin(Math.sqrt((x - a)/(b - a)));
return 1;
},
inv: function(p, a, b) {
return a + (0.5 - 0.5 * Math.cos(Math.PI * p)) * (b - a);
},
mean: function mean(a, b) {
if (b <= a) return NaN;
return (a + b) / 2;
},
median: function median(a, b) {
if (b <= a) return NaN;
return (a + b) / 2;
},
mode: function mode(/*a, b*/) {
throw new Error('mode is not yet implemented');
},
sample: function sample(a, b) {
return ((a + b) / 2) + ((b - a) / 2) *
Math.sin(2 * Math.PI * jStat.uniform.sample(0, 1));
},
variance: function variance(a, b) {
if (b <= a) return NaN;
return Math.pow(b - a, 2) / 8;
}
});
function laplaceSign(x) { return x / Math.abs(x); }
jStat.extend(jStat.laplace, {
pdf: function pdf(x, mu, b) {
return (b <= 0) ? 0 : (Math.exp(-Math.abs(x - mu) / b)) / (2 * b);
},
cdf: function cdf(x, mu, b) {
if (b <= 0) { return 0; }
if(x < mu) {
return 0.5 * Math.exp((x - mu) / b);
} else {
return 1 - 0.5 * Math.exp(- (x - mu) / b);
}
},
mean: function(mu/*, b*/) {
return mu;
},
median: function(mu/*, b*/) {
return mu;
},
mode: function(mu/*, b*/) {
return mu;
},
variance: function(mu, b) {
return 2 * b * b;
},
sample: function sample(mu, b) {
var u = jStat._random_fn() - 0.5;
return mu - (b * laplaceSign(u) * Math.log(1 - (2 * Math.abs(u))));
}
});
function tukeyWprob(w, rr, cc) {
var nleg = 12;
var ihalf = 6;
var C1 = -30;
var C2 = -50;
var C3 = 60;
var bb = 8;
var wlar = 3;
var wincr1 = 2;
var wincr2 = 3;
var xleg = [
0.981560634246719250690549090149,
0.904117256370474856678465866119,
0.769902674194304687036893833213,
0.587317954286617447296702418941,
0.367831498998180193752691536644,
0.125233408511468915472441369464
];
var aleg = [
0.047175336386511827194615961485,
0.106939325995318430960254718194,
0.160078328543346226334652529543,
0.203167426723065921749064455810,
0.233492536538354808760849898925,
0.249147045813402785000562436043
];
var qsqz = w * 0.5;
// if w >= 16 then the integral lower bound (occurs for c=20)
// is 0.99999999999995 so return a value of 1.
if (qsqz >= bb)
return 1.0;
// find (f(w/2) - 1) ^ cc
// (first term in integral of hartley's form).
var pr_w = 2 * jStat.normal.cdf(qsqz, 0, 1, 1, 0) - 1; // erf(qsqz / M_SQRT2)
// if pr_w ^ cc < 2e-22 then set pr_w = 0
if (pr_w >= Math.exp(C2 / cc))
pr_w = Math.pow(pr_w, cc);
else
pr_w = 0.0;
// if w is large then the second component of the
// integral is small, so fewer intervals are needed.
var wincr;
if (w > wlar)
wincr = wincr1;
else
wincr = wincr2;
// find the integral of second term of hartley's form
// for the integral of the range for equal-length
// intervals using legendre quadrature. limits of
// integration are from (w/2, 8). two or three
// equal-length intervals are used.
// blb and bub are lower and upper limits of integration.
var blb = qsqz;
var binc = (bb - qsqz) / wincr;
var bub = blb + binc;
var einsum = 0.0;
// integrate over each interval
var cc1 = cc - 1.0;
for (var wi = 1; wi <= wincr; wi++) {
var elsum = 0.0;
var a = 0.5 * (bub + blb);
// legendre quadrature with order = nleg
var b = 0.5 * (bub - blb);
for (var jj = 1; jj <= nleg; jj++) {
var j, xx;
if (ihalf < jj) {
j = (nleg - jj) + 1;
xx = xleg[j-1];
} else {
j = jj;
xx = -xleg[j-1];
}
var c = b * xx;
var ac = a + c;
// if exp(-qexpo/2) < 9e-14,
// then doesn't contribute to integral
var qexpo = ac * ac;
if (qexpo > C3)
break;
var pplus = 2 * jStat.normal.cdf(ac, 0, 1, 1, 0);
var pminus= 2 * jStat.normal.cdf(ac, w, 1, 1, 0);
// if rinsum ^ (cc-1) < 9e-14,
// then doesn't contribute to integral
var rinsum = (pplus * 0.5) - (pminus * 0.5);
if (rinsum >= Math.exp(C1 / cc1)) {
rinsum = (aleg[j-1] * Math.exp(-(0.5 * qexpo))) * Math.pow(rinsum, cc1);
elsum += rinsum;
}
}
elsum *= (((2.0 * b) * cc) / Math.sqrt(2 * Math.PI));
einsum += elsum;
blb = bub;
bub += binc;
}
// if pr_w ^ rr < 9e-14, then return 0
pr_w += einsum;
if (pr_w <= Math.exp(C1 / rr))
return 0;
pr_w = Math.pow(pr_w, rr);
if (pr_w >= 1) // 1 was iMax was eps
return 1;
return pr_w;
}
function tukeyQinv(p, c, v) {
var p0 = 0.322232421088;
var q0 = 0.993484626060e-01;
var p1 = -1.0;
var q1 = 0.588581570495;
var p2 = -0.342242088547;
var q2 = 0.531103462366;
var p3 = -0.204231210125;
var q3 = 0.103537752850;
var p4 = -0.453642210148e-04;
var q4 = 0.38560700634e-02;
var c1 = 0.8832;
var c2 = 0.2368;
var c3 = 1.214;
var c4 = 1.208;
var c5 = 1.4142;
var vmax = 120.0;
var ps = 0.5 - 0.5 * p;
var yi = Math.sqrt(Math.log(1.0 / (ps * ps)));
var t = yi + (((( yi * p4 + p3) * yi + p2) * yi + p1) * yi + p0)
/ (((( yi * q4 + q3) * yi + q2) * yi + q1) * yi + q0);
if (v < vmax) t += (t * t * t + t) / v / 4.0;
var q = c1 - c2 * t;
if (v < vmax) q += -c3 / v + c4 * t / v;
return t * (q * Math.log(c - 1.0) + c5);
}
jStat.extend(jStat.tukey, {
cdf: function cdf(q, nmeans, df) {
// Identical implementation as the R ptukey() function as of commit 68947
var rr = 1;
var cc = nmeans;
var nlegq = 16;
var ihalfq = 8;
var eps1 = -30.0;
var eps2 = 1.0e-14;
var dhaf = 100.0;
var dquar = 800.0;
var deigh = 5000.0;
var dlarg = 25000.0;
var ulen1 = 1.0;
var ulen2 = 0.5;
var ulen3 = 0.25;
var ulen4 = 0.125;
var xlegq = [
0.989400934991649932596154173450,
0.944575023073232576077988415535,
0.865631202387831743880467897712,
0.755404408355003033895101194847,
0.617876244402643748446671764049,
0.458016777657227386342419442984,
0.281603550779258913230460501460,
0.950125098376374401853193354250e-1
];
var alegq = [
0.271524594117540948517805724560e-1,
0.622535239386478928628438369944e-1,
0.951585116824927848099251076022e-1,
0.124628971255533872052476282192,
0.149595988816576732081501730547,
0.169156519395002538189312079030,
0.182603415044923588866763667969,
0.189450610455068496285396723208
];
if (q <= 0)
return 0;
// df must be > 1
// there must be at least two values
if (df < 2 || rr < 1 || cc < 2) return NaN;
if (!Number.isFinite(q))
return 1;
if (df > dlarg)
return tukeyWprob(q, rr, cc);
// calculate leading constant
var f2 = df * 0.5;
var f2lf = ((f2 * Math.log(df)) - (df * Math.log(2))) - jStat.gammaln(f2);
var f21 = f2 - 1.0;
// integral is divided into unit, half-unit, quarter-unit, or
// eighth-unit length intervals depending on the value of the
// degrees of freedom.
var ff4 = df * 0.25;
var ulen;
if (df <= dhaf) ulen = ulen1;
else if (df <= dquar) ulen = ulen2;
else if (df <= deigh) ulen = ulen3;
else ulen = ulen4;
f2lf += Math.log(ulen);
// integrate over each subinterval
var ans = 0.0;
for (var i = 1; i <= 50; i++) {
var otsum = 0.0;
// legendre quadrature with order = nlegq
// nodes (stored in xlegq) are symmetric around zero.
var twa1 = (2 * i - 1) * ulen;
for (var jj = 1; jj <= nlegq; jj++) {
var j, t1;
if (ihalfq < jj) {
j = jj - ihalfq - 1;
t1 = (f2lf + (f21 * Math.log(twa1 + (xlegq[j] * ulen))))
- (((xlegq[j] * ulen) + twa1) * ff4);
} else {
j = jj - 1;
t1 = (f2lf + (f21 * Math.log(twa1 - (xlegq[j] * ulen))))
+ (((xlegq[j] * ulen) - twa1) * ff4);
}
// if exp(t1) < 9e-14, then doesn't contribute to integral
var qsqz;
if (t1 >= eps1) {
if (ihalfq < jj) {
qsqz = q * Math.sqrt(((xlegq[j] * ulen) + twa1) * 0.5);
} else {
qsqz = q * Math.sqrt(((-(xlegq[j] * ulen)) + twa1) * 0.5);
}
// call wprob to find integral of range portion
var wprb = tukeyWprob(qsqz, rr, cc);
var rotsum = (wprb * alegq[j]) * Math.exp(t1);
otsum += rotsum;
}
// end legendre integral for interval i
// L200:
}
// if integral for interval i < 1e-14, then stop.
// However, in order to avoid small area under left tail,
// at least 1 / ulen intervals are calculated.
if (i * ulen >= 1.0 && otsum <= eps2)
break;
// end of interval i
// L330:
ans += otsum;
}
if (otsum > eps2) { // not converged
throw new Error('tukey.cdf failed to converge');
}
if (ans > 1)
ans = 1;
return ans;
},
inv: function(p, nmeans, df) {
// Identical implementation as the R qtukey() function as of commit 68947
var rr = 1;
var cc = nmeans;
var eps = 0.0001;
var maxiter = 50;
// df must be > 1 ; there must be at least two values
if (df < 2 || rr < 1 || cc < 2) return NaN;
if (p < 0 || p > 1) return NaN;
if (p === 0) return 0;
if (p === 1) return Infinity;
// Initial value
var x0 = tukeyQinv(p, cc, df);
// Find prob(value < x0)
var valx0 = jStat.tukey.cdf(x0, nmeans, df) - p;
// Find the second iterate and prob(value < x1).
// If the first iterate has probability value
// exceeding p then second iterate is 1 less than
// first iterate; otherwise it is 1 greater.
var x1;
if (valx0 > 0.0)
x1 = Math.max(0.0, x0 - 1.0);
else
x1 = x0 + 1.0;
var valx1 = jStat.tukey.cdf(x1, nmeans, df) - p;
// Find new iterate
var ans;
for(var iter = 1; iter < maxiter; iter++) {
ans = x1 - ((valx1 * (x1 - x0)) / (valx1 - valx0));
valx0 = valx1;
// New iterate must be >= 0
x0 = x1;
if (ans < 0.0) {
ans = 0.0;
valx1 = -p;
}
// Find prob(value < new iterate)
valx1 = jStat.tukey.cdf(ans, nmeans, df) - p;
x1 = ans;
// If the difference between two successive
// iterates is less than eps, stop
var xabs = Math.abs(x1 - x0);
if (xabs < eps)
return ans;
}
throw new Error('tukey.inv failed to converge');
}
});
}(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;
function isUsable(arg) {
return isArray(arg) || arg instanceof jStat;
}
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 (isUsable(arg)) {
if (!isUsable(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 (isUsable(arg)) {
if (!isUsable(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 (isUsable(arg)) {
if (!isUsable(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, ncol, res, rescols;
// eg: arr = 2 arg = 3 -> 6 for res[0][0] statement closure
if (arr.length === undefined && arg.length === undefined) {
return arr * arg;
}
nrow = arr.length,
ncol = arr[0].length,
res = jStat.zeros(nrow, nrescols = (isUsable(arg)) ? arg[0].length : ncol),
rescols = 0;
if (isUsable(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; });
},
// outer([1,2,3],[4,5,6])
// ===
// [[1],[2],[3]] times [[4,5,6]]
// ->
// [[4,5,6],[8,10,12],[12,15,18]]
outer:function outer(A, B) {
return jStat.multiply(A.map(function(t){ return [t] }), [B]);
},
// Returns the dot product of two matricies
dot: function dot(arr, arg) {
if (!isUsable(arr[0])) arr = [ arr ];
if (!isUsable(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); });
},
// exponentiate every element
exp: function exp(arr) {
return jStat.map(arr, function(value) { return Math.exp(value); });
},
// generate the natural log of every element
log: function exp(arr) {
return jStat.map(arr, function(value) { return Math.log(value); });
},
// generate the absolute values of the vector
abs: function abs(arr) {
return jStat.map(arr, function(value) { return Math.abs(value); });
},
// computes the p-norm of the vector
// In the case that a matrix is passed, uses the first row as 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 (isUsable(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);
},
// computes the angle between two vectors in rads
// In case a matrix is passed, this uses the first row as the vector
angle: function angle(arr, arg) {
return Math.acos(jStat.dot(arr, arg) / (jStat.norm(arr) * jStat.norm(arg)));
},
// augment one matrix by another
// Note: this function returns a matrix, not a jStat object
aug: function aug(a, b) {
var newarr = [];
var i;
for (i = 0; i < a.length; i++) {
newarr.push(a[i].slice());
}
for (i = 0; i < newarr.length; i++) {
push.apply(newarr[i], b[i]);
}
return newarr;
},
// The inv() function calculates the inverse of a matrix
// Create the inverse by augmenting the matrix by the identity matrix of the
// appropriate size, and then use G-J elimination on the augmented matrix.
inv: function inv(a) {
var rows = a.length;
var cols = a[0].length;
var b = jStat.identity(rows, cols);
var c = jStat.gauss_jordan(a, b);
var result = [];
var i = 0;
var j;
//We need to copy the inverse portion to a new matrix to rid G-J artifacts
for (; i < rows; i++) {
result[i] = [];
for (j = cols; j < c[0].length; j++)
result[i][j - cols] = c[i][j];
}
return result;
},
// calculate the determinant of a matrix
det: function det(a) {
if (a.length === 2) {
return a[0][0] * a[1][1] - a[0][1] * a[1][0];
}
var determinant = 0;
for (var i = 0; i < a.length; i++) {
// build a sub matrix without column `i`
var submatrix = [];
for (var row = 1; row < a.length; row++) {
submatrix[row - 1] = [];
for (var col = 0; col < a.length; col++) {
if (col < i) {
submatrix[row - 1][col] = a[row][col];
} else if (col > i) {
submatrix[row - 1][col - 1] = a[row][col];
}
}
}
// alternate between + and - between determinants
var sign = i % 2 ? -1 : 1;
determinant += det(submatrix) * a[0][i] * sign;
}
return determinant
},
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 = 0; 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 = 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);
var h = m.length;
var w = m[0].length;
var c = 0;
var x, y, y2;
// find max pivot
for (y = 0; y < h; y++) {
var maxrow = y;
for (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 (y2 = y+1; y2 < h; y2++) {
c = m[y2][y] / m[y][y];
for (x = y; x < w; x++) {
m[y2][x] -= m[y][x] * c;
}
}
}
// backsubstitute
for (y = h-1; y >= 0; y--) {
c = m[y][y];
for (y2 = 0; y2 < y; y2++) {
for (x = w-1; x > y-1; x--) {
m[y2][x] -= m[y][x] * m[y2][y] / c;
}
}
m[y][y] /= c;
for (x = h; x < w; x++) {
m[y][x] /= c;
}
}
return m;
},
// solve equation
// Ax=b
// A is upper triangular matrix
// A=[[1,2,3],[0,4,5],[0,6,7]]
// b=[1,2,3]
// triaUpSolve(A,b) // -> [2.666,0.1666,1.666]
// if you use matrix style
// A=[[1,2,3],[0,4,5],[0,6,7]]
// b=[[1],[2],[3]]
// will return [[2.666],[0.1666],[1.666]]
triaUpSolve: function triaUpSolve(A, b) {
var size = A[0].length;
var x = jStat.zeros(1, size)[0];
var parts;
var matrix_mode = false;
if (b[0].length != undefined) {
b = b.map(function(i){ return i[0] });
matrix_mode = true;
}
jStat.arange(size - 1, -1, -1).forEach(function(i) {
parts = jStat.arange(i + 1, size).map(function(j) {
return x[j] * A[i][j];
});
x[i] = (b[i] - jStat.sum(parts)) / A[i][i];
});
if (matrix_mode)
return x.map(function(i){ return [i] });
return x;
},
triaLowSolve: function triaLowSolve(A, b) {
// like to triaUpSolve but A is lower triangular matrix
var size = A[0].length;
var x = jStat.zeros(1, size)[0];
var parts;
var matrix_mode=false;
if (b[0].length != undefined) {
b = b.map(function(i){ return i[0] });
matrix_mode = true;
}
jStat.arange(size).forEach(function(i) {
parts = jStat.arange(i).map(function(j) {
return A[i][j] * x[j];
});
x[i] = (b[i] - jStat.sum(parts)) / A[i][i];
})
if (matrix_mode)
return x.map(function(i){ return [i] });
return x;
},
// A -> [L,U]
// A=LU
// L is lower triangular matrix
// U is upper triangular matrix
lu: function lu(A) {
var size = A.length;
//var L=jStat.diagonal(jStat.ones(1,size)[0]);
var L = jStat.identity(size);
var R = jStat.zeros(A.length, A[0].length);
var parts;
jStat.arange(size).forEach(function(t) {
R[0][t] = A[0][t];
});
jStat.arange(1, size).forEach(function(l) {
jStat.arange(l).forEach(function(i) {
parts = jStat.arange(i).map(function(jj) {
return L[l][jj] * R[jj][i];
});
L[l][i] = (A[l][i] - jStat.sum(parts)) / R[i][i];
});
jStat.arange(l, size).forEach(function(j) {
parts = jStat.arange(l).map(function(jj) {
return L[l][jj] * R[jj][j];
});
R[l][j] = A[parts.length][j] - jStat.sum(parts);
});
});
return [L, R];
},
// A -> T
// A=TT'
// T is lower triangular matrix
cholesky: function cholesky(A) {
var size = A.length;
var T = jStat.zeros(A.length, A[0].length);
var parts;
jStat.arange(size).forEach(function(i) {
parts = jStat.arange(i).map(function(t) {
return Math.pow(T[i][t],2);
});
T[i][i] = Math.sqrt(A[i][i] - jStat.sum(parts));
jStat.arange(i + 1, size).forEach(function(j) {
parts = jStat.arange(i).map(function(t) {
return T[i][t] * T[j][t];
});
T[j][i] = (A[i][j] - jStat.sum(parts)) / T[i][i];
});
});
return T;
},
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;
},
// A -> [Q,R]
// Q is orthogonal matrix
// R is upper triangular
QR: (function() {
// x -> Q
// find a orthogonal matrix Q st.
// Qx=y
// y is [||x||,0,0,...]
// quick ref
var sum = jStat.sum;
var range = jStat.arange;
function qr2(x) {
// quick impletation
// https://www.stat.wisc.edu/~larget/math496/qr.html
var n = x.length;
var p = x[0].length;
var r = jStat.zeros(p, p);
x = jStat.copy(x);
var i,j,k;
for(j = 0; j < p; j++){
r[j][j] = Math.sqrt(sum(range(n).map(function(i){
return x[i][j] * x[i][j];
})));
for(i = 0; i < n; i++){
x[i][j] = x[i][j] / r[j][j];
}
for(k = j+1; k < p; k++){
r[j][k] = sum(range(n).map(function(i){
return x[i][j] * x[i][k];
}));
for(i = 0; i < n; i++){
x[i][k] = x[i][k] - x[i][j]*r[j][k];
}
}
}
return [x, r];
}
return qr2;
}()),
lstsq: (function() {
// solve least squard problem for Ax=b as QR decomposition way if b is
// [[b1],[b2],[b3]] form will return [[x1],[x2],[x3]] array form solution
// else b is [b1,b2,b3] form will return [x1,x2,x3] array form solution
function R_I(A) {
A = jStat.copy(A);
var size = A.length;
var I = jStat.identity(size);
jStat.arange(size - 1, -1, -1).forEach(function(i) {
jStat.sliceAssign(
I, { row: i }, jStat.divide(jStat.slice(I, { row: i }), A[i][i]));
jStat.sliceAssign(
A, { row: i }, jStat.divide(jStat.slice(A, { row: i }), A[i][i]));
jStat.arange(i).forEach(function(j) {
var c = jStat.multiply(A[j][i], -1);
var Aj = jStat.slice(A, { row: j });
var cAi = jStat.multiply(jStat.slice(A, { row: i }), c);
jStat.sliceAssign(A, { row: j }, jStat.add(Aj, cAi));
var Ij = jStat.slice(I, { row: j });
var cIi = jStat.multiply(jStat.slice(I, { row: i }), c);
jStat.sliceAssign(I, { row: j }, jStat.add(Ij, cIi));
})
});
return I;
}
function qr_solve(A, b){
var array_mode = false;
if (b[0].length === undefined) {
// [c1,c2,c3] mode
b = b.map(function(x){ return [x] });
array_mode = true;
}
var QR = jStat.QR(A);
var Q = QR[0];
var R = QR[1];
var attrs = A[0].length;
var Q1 = jStat.slice(Q,{col:{end:attrs}});
var R1 = jStat.slice(R,{row:{end:attrs}});
var RI = R_I(R1);
var Q2 = jStat.transpose(Q1);
if(Q2[0].length === undefined){
Q2 = [Q2]; // The confusing jStat.multifly implementation threat nature process again.
}
var x = jStat.multiply(jStat.multiply(RI, Q2), b);
if(x.length === undefined){
x = [[x]]; // The confusing jStat.multifly implementation threat nature process again.
}
if (array_mode)
return x.map(function(i){ return i[0] });
return x;
}
return qr_solve;
}()),
jacobi: function jacobi(a) {
var condition = 1;
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) {
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;
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 h_min = Math.abs(x - X[pos(X, x) + 1]);
var i = 0;
var g = [];
var h1 = [];
var 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 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;
}
if (typeof jStat[passfunc](this, arg) === 'number')
return jStat[passfunc](this, arg);
else
return jStat(jStat[passfunc](this, arg));
};
}(funcs[i]));
}('add divide multiply subtract dot pow exp log abs norm angle'.split(' ')));
}(jStat, Math));
(function(jStat, Math) {
var slice = [].slice;
var isNumber = jStat.utils.isNumber;
var isArray = jStat.utils.isArray;
// 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);
var z;
if (isArray(args[1])) {
// (value, array, sides, flag)
z = jStat.zscore(args[0],args[1],args[3]);
return (args[2] === 1) ?
(jStat.normal.cdf(-Math.abs(z), 0, 1)) :
(jStat.normal.cdf(-Math.abs(z), 0, 1)*2);
} else {
if (args.length > 2) {
// (value, mean, sd, sides)
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);
} else {
// (zscore, sides)
z = args[0];
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;
}
// 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]);
}
var 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);
}
});
// Tukey's range test
jStat.extend({
// 2 parameter lists
// (mean1, mean2, n1, n2, sd)
// (array1, array2, sd)
qscore: function qscore() {
var args = slice.call(arguments);
var mean1, mean2, n1, n2, sd;
if (isNumber(args[0])) {
mean1 = args[0];
mean2 = args[1];
n1 = args[2];
n2 = args[3];
sd = args[4];
} else {
mean1 = jStat.mean(args[0]);
mean2 = jStat.mean(args[1]);
n1 = args[0].length;
n2 = args[1].length;
sd = args[2];
}
return Math.abs(mean1 - mean2) / (sd * Math.sqrt((1 / n1 + 1 / n2) / 2));
},
// 3 different parameter lists:
// (qscore, n, k)
// (mean1, mean2, n1, n2, sd, n, k)
// (array1, array2, sd, n, k)
qtest: function qtest() {
var args = slice.call(arguments);
var qscore;
if (args.length === 3) {
qscore = args[0];
args = args.slice(1);
} else if (args.length === 7) {
qscore = jStat.qscore(args[0], args[1], args[2], args[3], args[4]);
args = args.slice(5);
} else {
qscore = jStat.qscore(args[0], args[1], args[2]);
args = args.slice(3);
}
var n = args[0];
var k = args[1];
return 1 - jStat.tukey.cdf(qscore, k, n - k);
},
tukeyhsd: function tukeyhsd(arrays) {
var sd = jStat.pooledstdev(arrays);
var means = arrays.map(function (arr) {return jStat.mean(arr);});
var n = arrays.reduce(function (n, arr) {return n + arr.length;}, 0);
var results = [];
for (var i = 0; i < arrays.length; ++i) {
for (var j = i + 1; j < arrays.length; ++j) {
var p = jStat.qtest(means[i], means[j], arrays[i].length, arrays[j].length, sd, n, arrays.length);
results.push([[i, j], p]);
}
}
return results;
}
});
// 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());
}
});
// internal method for calculating the z-score for a difference of proportions test
function differenceOfProportions(p1, n1, p2, n2) {
if (p1 > 1 || p2 > 1 || p1 <= 0 || p2 <= 0) {
throw new Error("Proportions should be greater than 0 and less than 1")
}
var pooled = (p1 * n1 + p2 * n2) / (n1 + n2);
var se = Math.sqrt(pooled * (1 - pooled) * ((1/n1) + (1/n2)));
return (p1 - p2) / se;
}
// Difference of Proportions
jStat.extend(jStat.fn, {
oneSidedDifferenceOfProportions: function oneSidedDifferenceOfProportions(p1, n1, p2, n2) {
var z = differenceOfProportions(p1, n1, p2, n2);
return jStat.ztest(z, 1);
},
twoSidedDifferenceOfProportions: function twoSidedDifferenceOfProportions(p1, n1, p2, n2) {
var z = differenceOfProportions(p1, n1, p2, n2);
return jStat.ztest(z, 2);
}
});
}(jStat, Math));
jStat.models = (function(){
function sub_regress(exog) {
var var_count = exog[0].length;
var modelList = jStat.arange(var_count).map(function(endog_index) {
var exog_index =
jStat.arange(var_count).filter(function(i){return i!==endog_index});
return ols(jStat.col(exog, endog_index).map(function(x){ return x[0] }),
jStat.col(exog, exog_index))
});
return modelList;
}
// do OLS model regress
// exog have include const columns ,it will not generate it .In fact, exog is
// "design matrix" look at
//https://en.wikipedia.org/wiki/Design_matrix
function ols(endog, exog) {
var nobs = endog.length;
var df_model = exog[0].length - 1;
var df_resid = nobs-df_model - 1;
var coef = jStat.lstsq(exog, endog);
var predict =
jStat.multiply(exog, coef.map(function(x) { return [x] }))
.map(function(p) { return p[0] });
var resid = jStat.subtract(endog, predict);
var ybar = jStat.mean(endog);
// constant cause problem
// var SST = jStat.sum(endog.map(function(y) {
// return Math.pow(y-ybar,2);
// }));
var SSE = jStat.sum(predict.map(function(f) {
return Math.pow(f - ybar, 2);
}));
var SSR = jStat.sum(endog.map(function(y, i) {
return Math.pow(y - predict[i], 2);
}));
var SST = SSE + SSR;
var R2 = (SSE / SST);
return {
exog:exog,
endog:endog,
nobs:nobs,
df_model:df_model,
df_resid:df_resid,
coef:coef,
predict:predict,
resid:resid,
ybar:ybar,
SST:SST,
SSE:SSE,
SSR:SSR,
R2:R2
};
}
// H0: b_I=0
// H1: b_I!=0
function t_test(model) {
var subModelList = sub_regress(model.exog);
//var sigmaHat=jStat.stdev(model.resid);
var sigmaHat = Math.sqrt(model.SSR / (model.df_resid));
var seBetaHat = subModelList.map(function(mod) {
var SST = mod.SST;
var R2 = mod.R2;
return sigmaHat / Math.sqrt(SST * (1 - R2));
});
var tStatistic = model.coef.map(function(coef, i) {
return (coef - 0) / seBetaHat[i];
});
var pValue = tStatistic.map(function(t) {
var leftppf = jStat.studentt.cdf(t, model.df_resid);
return (leftppf > 0.5 ? 1 - leftppf : leftppf) * 2;
});
var c = jStat.studentt.inv(0.975, model.df_resid);
var interval95 = model.coef.map(function(coef, i) {
var d = c * seBetaHat[i];
return [coef - d, coef + d];
})
return {
se: seBetaHat,
t: tStatistic,
p: pValue,
sigmaHat: sigmaHat,
interval95: interval95
};
}
function F_test(model) {
var F_statistic =
(model.R2 / model.df_model) / ((1 - model.R2) / model.df_resid);
var fcdf = function(x, n1, n2) {
return jStat.beta.cdf(x / (n2 / n1 + x), n1 / 2, n2 / 2)
}
var pvalue = 1 - fcdf(F_statistic, model.df_model, model.df_resid);
return { F_statistic: F_statistic, pvalue: pvalue };
}
function ols_wrap(endog, exog) {
var model = ols(endog,exog);
var ttest = t_test(model);
var ftest = F_test(model);
// Provide the Wherry / Ezekiel / McNemar / Cohen Adjusted R^2
// Which matches the 'adjusted R^2' provided by R's lm package
var adjust_R2 =
1 - (1 - model.R2) * ((model.nobs - 1) / (model.df_resid));
model.t = ttest;
model.f = ftest;
model.adjust_R2 = adjust_R2;
return model;
}
return { ols: ols_wrap };
})();
//To regress, simply build X matrix
//(append column of 1's) using
//buildxmatrix and build the Y
//matrix using buildymatrix
//(simply the transpose)
//and run regress.
//Regressions
jStat.extend({
buildxmatrix: function buildxmatrix(){
//Parameters will be passed in as such
//(array1,array2,array3,...)
//as (x1,x2,x3,...)
//needs to be (1,x1,x2,x3,...)
var matrixRows = new Array(arguments.length);
for(var i=0;i<arguments.length;i++){
var array = [1];
matrixRows[i]= array.concat(arguments[i]);
}
return jStat(matrixRows);
},
builddxmatrix: function builddxmatrix() {
//Paramters will be passed in as such
//([array1,array2,...]
var matrixRows = new Array(arguments[0].length);
for(var i=0;i<arguments[0].length;i++){
var array = [1]
matrixRows[i]= array.concat(arguments[0][i]);
}
return jStat(matrixRows);
},
buildjxmatrix: function buildjxmatrix(jMat) {
//Builds from jStat Matrix
var pass = new Array(jMat.length)
for(var i=0;i<jMat.length;i++){
pass[i] = jMat[i];
}
return jStat.builddxmatrix(pass);
},
buildymatrix: function buildymatrix(array){
return jStat(array).transpose();
},
buildjymatrix: function buildjymatrix(jMat){
return jMat.transpose();
},
matrixmult: function matrixmult(A,B){
var i, j, k, result, sum;
if (A.cols() == B.rows()) {
if(B.rows()>1){
result = [];
for (i = 0; i < A.rows(); i++) {
result[i] = [];
for (j = 0; j < B.cols(); j++) {
sum = 0;
for (k = 0; k < A.cols(); k++) {
sum += A.toArray()[i][k] * B.toArray()[k][j];
}
result[i][j] = sum;
}
}
return jStat(result);
}
result = [];
for (i = 0; i < A.rows(); i++) {
result[i] = [];
for (j = 0; j < B.cols(); j++) {
sum = 0;
for (k = 0; k < A.cols(); k++) {
sum += A.toArray()[i][k] * B.toArray()[j];
}
result[i][j] = sum;
}
}
return jStat(result);
}
},
//regress and regresst to be fixed
regress: function regress(jMatX,jMatY){
//print("regressin!");
//print(jMatX.toArray());
var innerinv = jStat.xtranspxinv(jMatX);
//print(innerinv);
var xtransp = jMatX.transpose();
var next = jStat.matrixmult(jStat(innerinv),xtransp);
return jStat.matrixmult(next,jMatY);
},
regresst: function regresst(jMatX,jMatY,sides){
var beta = jStat.regress(jMatX,jMatY);
var compile = {};
compile.anova = {};
var jMatYBar = jStat.jMatYBar(jMatX, beta);
compile.yBar = jMatYBar;
var yAverage = jMatY.mean();
compile.anova.residuals = jStat.residuals(jMatY, jMatYBar);
compile.anova.ssr = jStat.ssr(jMatYBar, yAverage);
compile.anova.msr = compile.anova.ssr / (jMatX[0].length - 1);
compile.anova.sse = jStat.sse(jMatY, jMatYBar);
compile.anova.mse =
compile.anova.sse / (jMatY.length - (jMatX[0].length - 1) - 1);
compile.anova.sst = jStat.sst(jMatY, yAverage);
compile.anova.mst = compile.anova.sst / (jMatY.length - 1);
compile.anova.r2 = 1 - (compile.anova.sse / compile.anova.sst);
if (compile.anova.r2 < 0) compile.anova.r2 = 0;
compile.anova.fratio = compile.anova.msr / compile.anova.mse;
compile.anova.pvalue =
jStat.anovaftest(compile.anova.fratio,
jMatX[0].length - 1,
jMatY.length - (jMatX[0].length - 1) - 1);
compile.anova.rmse = Math.sqrt(compile.anova.mse);
compile.anova.r2adj = 1 - (compile.anova.mse / compile.anova.mst);
if (compile.anova.r2adj < 0) compile.anova.r2adj = 0;
compile.stats = new Array(jMatX[0].length);
var covar = jStat.xtranspxinv(jMatX);
var sds, ts, ps;
for(var i=0; i<beta.length;i++){
sds=Math.sqrt(compile.anova.mse * Math.abs(covar[i][i]));
ts= Math.abs(beta[i] / sds);
ps= jStat.ttest(ts, jMatY.length - jMatX[0].length - 1, sides);
compile.stats[i]=[beta[i], sds, ts, ps];
}
compile.regress = beta;
return compile;
},
xtranspx: function xtranspx(jMatX){
return jStat.matrixmult(jMatX.transpose(),jMatX);
},
xtranspxinv: function xtranspxinv(jMatX){
var inner = jStat.matrixmult(jMatX.transpose(),jMatX);
var innerinv = jStat.inv(inner);
return innerinv;
},
jMatYBar: function jMatYBar(jMatX, beta) {
var yBar = jStat.matrixmult(jMatX, beta);
return new jStat(yBar);
},
residuals: function residuals(jMatY, jMatYBar) {
return jStat.matrixsubtract(jMatY, jMatYBar);
},
ssr: function ssr(jMatYBar, yAverage) {
var ssr = 0;
for(var i = 0; i < jMatYBar.length; i++) {
ssr += Math.pow(jMatYBar[i] - yAverage, 2);
}
return ssr;
},
sse: function sse(jMatY, jMatYBar) {
var sse = 0;
for(var i = 0; i < jMatY.length; i++) {
sse += Math.pow(jMatY[i] - jMatYBar[i], 2);
}
return sse;
},
sst: function sst(jMatY, yAverage) {
var sst = 0;
for(var i = 0; i < jMatY.length; i++) {
sst += Math.pow(jMatY[i] - yAverage, 2);
}
return sst;
},
matrixsubtract: function matrixsubtract(A,B){
var ans = new Array(A.length);
for(var i=0;i<A.length;i++){
ans[i] = new Array(A[i].length);
for(var j=0;j<A[i].length;j++){
ans[i][j]=A[i][j]-B[i][j];
}
}
return jStat(ans);
}
});
// Make it compatible with previous version.
jStat.jStat = jStat;
return jStat;
});