jStat/dist/jstat.js

(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) {

console.warn('The npm package jStat is no longer maintained. Instead use jstat (lowercase).');
console.warn('Visit https://www.npmjs.com/package/jstat for more information.');

// For quick reference.
var concat = Array.prototype.concat;
var slice = Array.prototype.slice;
var toString = Object.prototype.toString;

// Calculate correction for IEEE error
// TODO: This calculation can be improved.
function calcRdx(n, m) {
  var val = n > m ? n : m;
  return Math.pow(10,
                  17 - ~~(Math.log(((val > 0) ? val : -val)) * Math.LOG10E));
}


var isArray = Array.isArray || function isArray(arg) {
  return toString.call(arg) === '[object Array]';
};


function isFunction(arg) {
  return toString.call(arg) === '[object Function]';
}


function isNumber(arg) {
  return typeof arg === 'number' && arg === arg;
}


// Converts the jStat matrix to vector.
function toVector(arr) {
  return concat.apply([], arr);
}


// The one and only jStat constructor.
function jStat() {
  return new jStat._init(arguments);
}


// TODO: Remove after all references in src files have been removed.
jStat.fn = jStat.prototype;


// By separating the initializer from the constructor it's easier to handle
// always returning a new instance whether "new" was used or not.
jStat._init = function _init(args) {
  // 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) {
  return Math.pow(jStat.product(arr), 1 / arr.length);
};


// median of an array
jStat.median = function median(arr) {
  var arrlen = arr.length;
  var _arr = arr.slice().sort(ascNum);
  // check if array is even or odd, then return the appropriate
  return !(arrlen & 1)
    ? (_arr[(arrlen / 2) - 1 ] + _arr[(arrlen / 2)]) / 2
    : _arr[(arrlen / 2) | 0 ];
};


// cumulative sum of an array
jStat.cumsum = function cumsum(arr) {
  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 arrlen = arr.length;
  var sorted = arr.slice().sort(ascNum);
  var ranks = new Array(arrlen);
  var val;
  for (var i = 0; i < arrlen; i++) {
    var first = sorted.indexOf(arr[i]);
    var last = sorted.lastIndexOf(arr[i]);
    if (first === last) {
      val = first;
    } else {
      val = (first + last) / 2;
    }
    ranks[i] = val + 1;
  }
  return ranks;
};


// 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;
};


// 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 (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(a, b, c) {
      if (!(this instanceof arguments.callee))
        return new arguments.callee(a, b, c);
      this._a = a;
      this._b = b;
      this._c = c;
      return this;
    };
    // distribution method to be used on a jStat instance
    jStat.fn[func] = function(a, b, c) {
      var newthis = jStat[func](a, b, c);
      newthis.data = this;
      return newthis;
    };
    // sample instance method
    jStat[func].prototype.sample = function(arr) {
      var a = this._a;
      var b = this._b;
      var c = this._c;
      if (arr)
        return jStat.alter(arr, function() {
          return jStat[func].sample(a, b, c);
        });
      else
        return jStat[func].sample(a, b, c);
    };
    // generate the pdf, cdf and inv instance methods
    (function(vals) {
      for (var i = 0; i < vals.length; i++) (function(fnfunc) {
        jStat[func].prototype[fnfunc] = function(x) {
          var a = this._a;
          var b = this._b;
          var c = this._c;
          if (!x && x !== 0)
            x = this.data;
          if (typeof x !== 'number') {
            return jStat.fn.map.call(x, function(x) {
              return jStat[func][fnfunc](x, a, b, c);
            });
          }
          return jStat[func][fnfunc](x, a, b, c);
        };
      })(vals[i]);
    })('pdf cdf inv'.split(' '));
    // generate the mean, median, mode and variance instance methods
    (function(vals) {
      for (var i = 0; i < vals.length; i++) (function(fnfunc) {
        jStat[func].prototype[fnfunc] = function() {
          return jStat[func][fnfunc](this._a, this._b, this._c);
        };
      })(vals[i]);
    })('mean median mode variance'.split(' '));
  })(list[i]);
})((
  'beta centralF cauchy chisquare exponential gamma invgamma kumaraswamy ' +
  '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;
  },

  sample: function sample(l) {
    var p = 1, k = 0, L = Math.exp(-l);
    do {
      k++;
      p *= jStat._random_fn();
    } while (p > L);
    return k - 1;
  }
});

// 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) {
    var alen = a.length,
    alend = alen * 2,
    vals = new Array(alend),
    rowshift = alen - 1,
    colshift = alend - 1,
    mrow = rowshift - alen + 1,
    mcol = colshift,
    i = 0,
    result = 0,
    j;
    // check for special 2x2 case
    if (alen === 2) {
      return a[0][0] * a[1][1] - a[0][1] * a[1][0];
    }
    for (; i < alend; i++) {
      vals[i] = 1;
    }
    for (i = 0; i < alen; i++) {
      for (j = 0; j < alen; j++) {
        vals[(mrow < 0) ? mrow + alen : mrow ] *= a[i][j];
        vals[(mcol < alen) ? mcol + alen : mcol ] *= a[i][j];
        mrow++;
        mcol--;
      }
      mrow = --rowshift - alen + 1;
      mcol = --colshift;
    }
    for (i = 0; i < alen; i++) {
      result += vals[i];
    }
    for (; i < alend; i++) {
      result -= vals[i];
    }
    return result;
  },

  gauss_elimination: function gauss_elimination(a, b) {
    var i = 0,
    j = 0,
    n = a.length,
    m = a[0].length,
    factor = 1,
    sum = 0,
    x = [],
    maug, pivot, temp, k;
    a = jStat.aug(a, b);
    maug = a[0].length;
    for(i = 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;
});