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/* global module */
// # simple-statistics
//
// A simple, literate statistics system. The code below uses the
// [Javascript module pattern](http://www.adequatelygood.com/2010/3/JavaScript-Module-Pattern-In-Depth),
// eventually assigning `simple-statistics` to `ss` in browsers or the
// `exports` object for node.js
(function() {
var ss = {};
if (typeof module !== 'undefined') {
// Assign the `ss` object to exports, so that you can require
// it in [node.js](http://nodejs.org/)
module.exports = ss;
} else {
// Otherwise, in a browser, we assign `ss` to the window object,
// so you can simply refer to it as `ss`.
this.ss = ss;
}
if (typeof define === 'function' && define.amd) {
define([], function () {
return ss;
});
}
// # [Linear Regression](http://en.wikipedia.org/wiki/Linear_regression)
//
// [Simple linear regression](http://en.wikipedia.org/wiki/Simple_linear_regression)
// is a simple way to find a fitted line
// between a set of coordinates.
function linear_regression() {
var linreg = {},
data = [];
// Assign data to the model. Data is assumed to be an array.
linreg.data = function(x) {
if (!arguments.length) return data;
data = x.slice();
return linreg;
};
// Calculate the slope and y-intercept of the regression line
// by calculating the least sum of squares
linreg.mb = function() {
var m, b;
// Store data length in a local variable to reduce
// repeated object property lookups
var data_length = data.length;
//if there's only one point, arbitrarily choose a slope of 0
//and a y-intercept of whatever the y of the initial point is
if (data_length === 1) {
m = 0;
b = data[0][1];
} else {
// Initialize our sums and scope the `m` and `b`
// variables that define the line.
var sum_x = 0, sum_y = 0,
sum_xx = 0, sum_xy = 0;
// Use local variables to grab point values
// with minimal object property lookups
var point, x, y;
// Gather the sum of all x values, the sum of all
// y values, and the sum of x^2 and (x*y) for each
// value.
//
// In math notation, these would be SS_x, SS_y, SS_xx, and SS_xy
for (var i = 0; i < data_length; i++) {
point = data[i];
x = point[0];
y = point[1];
sum_x += x;
sum_y += y;
sum_xx += x * x;
sum_xy += x * y;
}
// `m` is the slope of the regression line
m = ((data_length * sum_xy) - (sum_x * sum_y)) /
((data_length * sum_xx) - (sum_x * sum_x));
// `b` is the y-intercept of the line.
b = (sum_y / data_length) - ((m * sum_x) / data_length);
}
// Return both values as an object.
return { m: m, b: b };
};
// a shortcut for simply getting the slope of the regression line
linreg.m = function() {
return linreg.mb().m;
};
// a shortcut for simply getting the y-intercept of the regression
// line.
linreg.b = function() {
return linreg.mb().b;
};
// ## Fitting The Regression Line
//
// This is called after `.data()` and returns the
// equation `y = f(x)` which gives the position
// of the regression line at each point in `x`.
linreg.line = function() {
// Get the slope, `m`, and y-intercept, `b`, of the line.
var mb = linreg.mb(),
m = mb.m,
b = mb.b;
// Return a function that computes a `y` value for each
// x value it is given, based on the values of `b` and `a`
// that we just computed.
return function(x) {
return b + (m * x);
};
};
return linreg;
}
// # [R Squared](http://en.wikipedia.org/wiki/Coefficient_of_determination)
//
// The r-squared value of data compared with a function `f`
// is the sum of the squared differences between the prediction
// and the actual value.
function r_squared(data, f) {
if (data.length < 2) return 1;
// Compute the average y value for the actual
// data set in order to compute the
// _total sum of squares_
var sum = 0, average;
for (var i = 0; i < data.length; i++) {
sum += data[i][1];
}
average = sum / data.length;
// Compute the total sum of squares - the
// squared difference between each point
// and the average of all points.
var sum_of_squares = 0;
for (var j = 0; j < data.length; j++) {
sum_of_squares += Math.pow(average - data[j][1], 2);
}
// Finally estimate the error: the squared
// difference between the estimate and the actual data
// value at each point.
var err = 0;
for (var k = 0; k < data.length; k++) {
err += Math.pow(data[k][1] - f(data[k][0]), 2);
}
// As the error grows larger, its ratio to the
// sum of squares increases and the r squared
// value grows lower.
return 1 - (err / sum_of_squares);
}
// # [Bayesian Classifier](http://en.wikipedia.org/wiki/Naive_Bayes_classifier)
//
// This is a naïve bayesian classifier that takes
// singly-nested objects.
function bayesian() {
// The `bayes_model` object is what will be exposed
// by this closure, with all of its extended methods, and will
// have access to all scope variables, like `total_count`.
var bayes_model = {},
// The number of items that are currently
// classified in the model
total_count = 0,
// Every item classified in the model
data = {};
// ## Train
// Train the classifier with a new item, which has a single
// dimension of Javascript literal keys and values.
bayes_model.train = function(item, category) {
// If the data object doesn't have any values
// for this category, create a new object for it.
if (!data[category]) data[category] = {};
// Iterate through each key in the item.
for (var k in item) {
var v = item[k];
// Initialize the nested object `data[category][k][item[k]]`
// with an object of keys that equal 0.
if (data[category][k] === undefined) data[category][k] = {};
if (data[category][k][v] === undefined) data[category][k][v] = 0;
// And increment the key for this key/value combination.
data[category][k][item[k]]++;
}
// Increment the number of items classified
total_count++;
};
// ## Score
// Generate a score of how well this item matches all
// possible categories based on its attributes
bayes_model.score = function(item) {
// Initialize an empty array of odds per category.
var odds = {}, category;
// Iterate through each key in the item,
// then iterate through each category that has been used
// in previous calls to `.train()`
for (var k in item) {
var v = item[k];
for (category in data) {
// Create an empty object for storing key - value combinations
// for this category.
if (odds[category] === undefined) odds[category] = {};
// If this item doesn't even have a property, it counts for nothing,
// but if it does have the property that we're looking for from
// the item to categorize, it counts based on how popular it is
// versus the whole population.
if (data[category][k]) {
odds[category][k + '_' + v] = (data[category][k][v] || 0) / total_count;
} else {
odds[category][k + '_' + v] = 0;
}
}
}
// Set up a new object that will contain sums of these odds by category
var odds_sums = {};
for (category in odds) {
// Tally all of the odds for each category-combination pair -
// the non-existence of a category does not add anything to the
// score.
for (var combination in odds[category]) {
if (odds_sums[category] === undefined) odds_sums[category] = 0;
odds_sums[category] += odds[category][combination];
}
}
return odds_sums;
};
// Return the completed model.
return bayes_model;
}
// # sum
//
// is simply the result of adding all numbers
// together, starting from zero.
//
// This runs on `O(n)`, linear time in respect to the array
function sum(x) {
var value = 0;
for (var i = 0; i < x.length; i++) {
value += x[i];
}
return value;
}
// # mean
//
// is the sum over the number of values
//
// This runs on `O(n)`, linear time in respect to the array
function mean(x) {
// The mean of no numbers is null
if (x.length === 0) return null;
return sum(x) / x.length;
}
// # geometric mean
//
// a mean function that is more useful for numbers in different
// ranges.
//
// this is the nth root of the input numbers multiplied by each other
//
// This runs on `O(n)`, linear time in respect to the array
function geometric_mean(x) {
// The mean of no numbers is null
if (x.length === 0) return null;
// the starting value.
var value = 1;
for (var i = 0; i < x.length; i++) {
// the geometric mean is only valid for positive numbers
if (x[i] <= 0) return null;
// repeatedly multiply the value by each number
value *= x[i];
}
return Math.pow(value, 1 / x.length);
}
// # harmonic mean
//
// a mean function typically used to find the average of rates
//
// this is the reciprocal of the arithmetic mean of the reciprocals
// of the input numbers
//
// This runs on `O(n)`, linear time in respect to the array
function harmonic_mean(x) {
// The mean of no numbers is null
if (x.length === 0) return null;
var reciprocal_sum = 0;
for (var i = 0; i < x.length; i++) {
// the harmonic mean is only valid for positive numbers
if (x[i] <= 0) return null;
reciprocal_sum += 1 / x[i];
}
// divide n by the the reciprocal sum
return x.length / reciprocal_sum;
}
// root mean square (RMS)
//
// a mean function used as a measure of the magnitude of a set
// of numbers, regardless of their sign
//
// this is the square root of the mean of the squares of the
// input numbers
//
// This runs on `O(n)`, linear time in respect to the array
function root_mean_square(x) {
if (x.length === 0) return null;
var sum_of_squares = 0;
for (var i = 0; i < x.length; i++) {
sum_of_squares += Math.pow(x[i], 2);
}
return Math.sqrt(sum_of_squares / x.length);
}
// # min
//
// This is simply the minimum number in the set.
//
// This runs on `O(n)`, linear time in respect to the array
function min(x) {
var value;
for (var i = 0; i < x.length; i++) {
// On the first iteration of this loop, min is
// undefined and is thus made the minimum element in the array
if (x[i] < value || value === undefined) value = x[i];
}
return value;
}
// # max
//
// This is simply the maximum number in the set.
//
// This runs on `O(n)`, linear time in respect to the array
function max(x) {
var value;
for (var i = 0; i < x.length; i++) {
// On the first iteration of this loop, max is
// undefined and is thus made the maximum element in the array
if (x[i] > value || value === undefined) value = x[i];
}
return value;
}
// # [variance](http://en.wikipedia.org/wiki/Variance)
//
// is the sum of squared deviations from the mean
//
// depends on `mean()`
function variance(x) {
// The variance of no numbers is null
if (x.length === 0) return null;
var mean_value = mean(x),
deviations = [];
// Make a list of squared deviations from the mean.
for (var i = 0; i < x.length; i++) {
deviations.push(Math.pow(x[i] - mean_value, 2));
}
// Find the mean value of that list
return mean(deviations);
}
// # [standard deviation](http://en.wikipedia.org/wiki/Standard_deviation)
//
// is just the square root of the variance.
//
// depends on `variance()`
function standard_deviation(x) {
// The standard deviation of no numbers is null
if (x.length === 0) return null;
return Math.sqrt(variance(x));
}
// The sum of deviations to the Nth power.
// When n=2 it's the sum of squared deviations.
// When n=3 it's the sum of cubed deviations.
//
// depends on `mean()`
function sum_nth_power_deviations(x, n) {
var mean_value = mean(x),
sum = 0;
for (var i = 0; i < x.length; i++) {
sum += Math.pow(x[i] - mean_value, n);
}
return sum;
}
// # [variance](http://en.wikipedia.org/wiki/Variance)
//
// is the sum of squared deviations from the mean
//
// depends on `sum_nth_power_deviations`
function sample_variance(x) {
// The variance of no numbers is null
if (x.length <= 1) return null;
var sum_squared_deviations_value = sum_nth_power_deviations(x, 2);
// Find the mean value of that list
return sum_squared_deviations_value / (x.length - 1);
}
// # [standard deviation](http://en.wikipedia.org/wiki/Standard_deviation)
//
// is just the square root of the variance.
//
// depends on `sample_variance()`
function sample_standard_deviation(x) {
// The standard deviation of no numbers is null
if (x.length <= 1) return null;
return Math.sqrt(sample_variance(x));
}
// # [covariance](http://en.wikipedia.org/wiki/Covariance)
//
// sample covariance of two datasets:
// how much do the two datasets move together?
// x and y are two datasets, represented as arrays of numbers.
//
// depends on `mean()`
function sample_covariance(x, y) {
// The two datasets must have the same length which must be more than 1
if (x.length <= 1 || x.length != y.length){
return null;
}
// determine the mean of each dataset so that we can judge each
// value of the dataset fairly as the difference from the mean. this
// way, if one dataset is [1, 2, 3] and [2, 3, 4], their covariance
// does not suffer because of the difference in absolute values
var xmean = mean(x),
ymean = mean(y),
sum = 0;
// for each pair of values, the covariance increases when their
// difference from the mean is associated - if both are well above
// or if both are well below
// the mean, the covariance increases significantly.
for (var i = 0; i < x.length; i++){
sum += (x[i] - xmean) * (y[i] - ymean);
}
// the covariance is weighted by the length of the datasets.
return sum / (x.length - 1);
}
// # [correlation](http://en.wikipedia.org/wiki/Correlation_and_dependence)
//
// Gets a measure of how correlated two datasets are, between -1 and 1
//
// depends on `sample_standard_deviation()` and `sample_covariance()`
function sample_correlation(x, y) {
var cov = sample_covariance(x, y),
xstd = sample_standard_deviation(x),
ystd = sample_standard_deviation(y);
if (cov === null || xstd === null || ystd === null) {
return null;
}
return cov / xstd / ystd;
}
// # [median](http://en.wikipedia.org/wiki/Median)
//
// The middle number of a list. This is often a good indicator of 'the middle'
// when there are outliers that skew the `mean()` value.
function median(x) {
// The median of an empty list is null
if (x.length === 0) return null;
// Sorting the array makes it easy to find the center, but
// use `.slice()` to ensure the original array `x` is not modified
var sorted = x.slice().sort(function (a, b) { return a - b; });
// If the length of the list is odd, it's the central number
if (sorted.length % 2 === 1) {
return sorted[(sorted.length - 1) / 2];
// Otherwise, the median is the average of the two numbers
// at the center of the list
} else {
var a = sorted[(sorted.length / 2) - 1];
var b = sorted[(sorted.length / 2)];
return (a + b) / 2;
}
}
// # [mode](http://bit.ly/W5K4Yt)
//
// The mode is the number that appears in a list the highest number of times.
// There can be multiple modes in a list: in the event of a tie, this
// algorithm will return the most recently seen mode.
//
// This implementation is inspired by [science.js](https://github.com/jasondavies/science.js/blob/master/src/stats/mode.js)
//
// This runs on `O(n)`, linear time in respect to the array
function mode(x) {
// Handle edge cases:
// The median of an empty list is null
if (x.length === 0) return null;
else if (x.length === 1) return x[0];
// Sorting the array lets us iterate through it below and be sure
// that every time we see a new number it's new and we'll never
// see the same number twice
var sorted = x.slice().sort(function (a, b) { return a - b; });
// This assumes it is dealing with an array of size > 1, since size
// 0 and 1 are handled immediately. Hence it starts at index 1 in the
// array.
var last = sorted[0],
// store the mode as we find new modes
value,
// store how many times we've seen the mode
max_seen = 0,
// how many times the current candidate for the mode
// has been seen
seen_this = 1;
// end at sorted.length + 1 to fix the case in which the mode is
// the highest number that occurs in the sequence. the last iteration
// compares sorted[i], which is undefined, to the highest number
// in the series
for (var i = 1; i < sorted.length + 1; i++) {
// we're seeing a new number pass by
if (sorted[i] !== last) {
// the last number is the new mode since we saw it more
// often than the old one
if (seen_this > max_seen) {
max_seen = seen_this;
value = last;
}
seen_this = 1;
last = sorted[i];
// if this isn't a new number, it's one more occurrence of
// the potential mode
} else { seen_this++; }
}
return value;
}
// # [t-test](http://en.wikipedia.org/wiki/Student's_t-test)
//
// This is to compute a one-sample t-test, comparing the mean
// of a sample to a known value, x.
//
// in this case, we're trying to determine whether the
// population mean is equal to the value that we know, which is `x`
// here. usually the results here are used to look up a
// [p-value](http://en.wikipedia.org/wiki/P-value), which, for
// a certain level of significance, will let you determine that the
// null hypothesis can or cannot be rejected.
//
// Depends on `standard_deviation()` and `mean()`
function t_test(sample, x) {
// The mean of the sample
var sample_mean = mean(sample);
// The standard deviation of the sample
var sd = standard_deviation(sample);
// Square root the length of the sample
var rootN = Math.sqrt(sample.length);
// Compute the known value against the sample,
// returning the t value
return (sample_mean - x) / (sd / rootN);
}
// # [2-sample t-test](http://en.wikipedia.org/wiki/Student's_t-test)
//
// This is to compute two sample t-test.
// Tests whether "mean(X)-mean(Y) = difference", (
// in the most common case, we often have `difference == 0` to test if two samples
// are likely to be taken from populations with the same mean value) with
// no prior knowledge on standard deviations of both samples
// other than the fact that they have the same standard deviation.
//
// Usually the results here are used to look up a
// [p-value](http://en.wikipedia.org/wiki/P-value), which, for
// a certain level of significance, will let you determine that the
// null hypothesis can or cannot be rejected.
//
// `diff` can be omitted if it equals 0.
//
// [This is used to confirm or deny](http://www.monarchlab.org/Lab/Research/Stats/2SampleT.aspx)
// a null hypothesis that the two populations that have been sampled into
// `sample_x` and `sample_y` are equal to each other.
//
// Depends on `sample_variance()` and `mean()`
function t_test_two_sample(sample_x, sample_y, difference) {
var n = sample_x.length,
m = sample_y.length;
// If either sample doesn't actually have any values, we can't
// compute this at all, so we return `null`.
if (!n || !m) return null ;
// default difference (mu) is zero
if (!difference) difference = 0;
var meanX = mean(sample_x),
meanY = mean(sample_y);
var weightedVariance = ((n - 1) * sample_variance(sample_x) +
(m - 1) * sample_variance(sample_y)) / (n + m - 2);
return (meanX - meanY - difference) /
Math.sqrt(weightedVariance * (1 / n + 1 / m));
}
// # chunk
//
// Split an array into chunks of a specified size. This function
// has the same behavior as [PHP's array_chunk](http://php.net/manual/en/function.array-chunk.php)
// function, and thus will insert smaller-sized chunks at the end if
// the input size is not divisible by the chunk size.
//
// `sample` is expected to be an array, and `chunkSize` a number.
// The `sample` array can contain any kind of data.
function chunk(sample, chunkSize) {
// a list of result chunks, as arrays in an array
var output = [];
// `chunkSize` must be zero or higher - otherwise the loop below,
// in which we call `start += chunkSize`, will loop infinitely.
// So, we'll detect and return null in that case to indicate
// invalid input.
if (chunkSize <= 0) {
return null;
}
// `start` is the index at which `.slice` will start selecting
// new array elements
for (var start = 0; start < sample.length; start += chunkSize) {
// for each chunk, slice that part of the array and add it
// to the output. The `.slice` function does not change
// the original array.
output.push(sample.slice(start, start + chunkSize));
}
return output;
}
// # shuffle_in_place
//
// A [Fisher-Yates shuffle](http://en.wikipedia.org/wiki/Fisher%E2%80%93Yates_shuffle)
// in-place - which means that it will change the order of the original
// array by reference.
function shuffle_in_place(sample, randomSource) {
// a custom random number source can be provided if you want to use
// a fixed seed or another random number generator, like
// [random-js](https://www.npmjs.org/package/random-js)
randomSource = randomSource || Math.random;
// store the current length of the sample to determine
// when no elements remain to shuffle.
var length = sample.length;
// temporary is used to hold an item when it is being
// swapped between indices.
var temporary;
// The index to swap at each stage.
var index;
// While there are still items to shuffle
while (length > 0) {
// chose a random index within the subset of the array
// that is not yet shuffled
index = Math.floor(randomSource() * length--);
// store the value that we'll move temporarily
temporary = sample[length];
// swap the value at `sample[length]` with `sample[index]`
sample[length] = sample[index];
sample[index] = temporary;
}
return sample;
}
// # shuffle
//
// A [Fisher-Yates shuffle](http://en.wikipedia.org/wiki/Fisher%E2%80%93Yates_shuffle)
// is a fast way to create a random permutation of a finite set.
function shuffle(sample, randomSource) {
// slice the original array so that it is not modified
sample = sample.slice();
// and then shuffle that shallow-copied array, in place
return shuffle_in_place(sample.slice(), randomSource);
}
// # sample
//
// Create a [simple random sample](http://en.wikipedia.org/wiki/Simple_random_sample)
// from a given array of `n` elements.
function sample(array, n, randomSource) {
// shuffle the original array using a fisher-yates shuffle
var shuffled = shuffle(array, randomSource);
// and then return a subset of it - the first `n` elements.
return shuffled.slice(0, n);
}
// # quantile
//
// This is a population quantile, since we assume to know the entire
// dataset in this library. Thus I'm trying to follow the
// [Quantiles of a Population](http://en.wikipedia.org/wiki/Quantile#Quantiles_of_a_population)
// algorithm from wikipedia.
//
// Sample is a one-dimensional array of numbers,
// and p is either a decimal number from 0 to 1 or an array of decimal
// numbers from 0 to 1.
// In terms of a k/q quantile, p = k/q - it's just dealing with fractions or dealing
// with decimal values.
// When p is an array, the result of the function is also an array containing the appropriate
// quantiles in input order
function quantile(sample, p) {
// We can't derive quantiles from an empty list
if (sample.length === 0) return null;
// Sort a copy of the array. We'll need a sorted array to index
// the values in sorted order.
var sorted = sample.slice().sort(function (a, b) { return a - b; });
if (p.length) {
// Initialize the result array
var results = [];
// For each requested quantile
for (var i = 0; i < p.length; i++) {
results[i] = quantile_sorted(sorted, p[i]);
}
return results;
} else {
return quantile_sorted(sorted, p);
}
}
// # quantile
//
// This is the internal implementation of quantiles: when you know
// that the order is sorted, you don't need to re-sort it, and the computations
// are much faster.
function quantile_sorted(sample, p) {
var idx = (sample.length) * p;
if (p < 0 || p > 1) {
return null;
} else if (p === 1) {
// If p is 1, directly return the last element
return sample[sample.length - 1];
} else if (p === 0) {
// If p is 0, directly return the first element
return sample[0];
} else if (idx % 1 !== 0) {
// If p is not integer, return the next element in array
return sample[Math.ceil(idx) - 1];
} else if (sample.length % 2 === 0) {
// If the list has even-length, we'll take the average of this number
// and the next value, if there is one
return (sample[idx - 1] + sample[idx]) / 2;
} else {
// Finally, in the simple case of an integer value
// with an odd-length list, return the sample value at the index.
return sample[idx];
}
}
// # [Interquartile range](http://en.wikipedia.org/wiki/Interquartile_range)
//
// A measure of statistical dispersion, or how scattered, spread, or
// concentrated a distribution is. It's computed as the difference between
// the third quartile and first quartile.
function iqr(sample) {
// We can't derive quantiles from an empty list
if (sample.length === 0) return null;
// Interquartile range is the span between the upper quartile,
// at `0.75`, and lower quartile, `0.25`
return quantile(sample, 0.75) - quantile(sample, 0.25);
}
// # [Median Absolute Deviation](http://en.wikipedia.org/wiki/Median_absolute_deviation)
//
// The Median Absolute Deviation (MAD) is a robust measure of statistical
// dispersion. It is more resilient to outliers than the standard deviation.
function mad(x) {
// The mad of nothing is null
if (!x || x.length === 0) return null;
var median_value = median(x),
median_absolute_deviations = [];
// Make a list of absolute deviations from the median
for (var i = 0; i < x.length; i++) {
median_absolute_deviations.push(Math.abs(x[i] - median_value));
}
// Find the median value of that list
return median(median_absolute_deviations);
}
// ## Compute Matrices for Jenks
//
// Compute the matrices required for Jenks breaks. These matrices
// can be used for any classing of data with `classes <= n_classes`
function jenksMatrices(data, n_classes) {
// in the original implementation, these matrices are referred to
// as `LC` and `OP`
//
// * lower_class_limits (LC): optimal lower class limits
// * variance_combinations (OP): optimal variance combinations for all classes
var lower_class_limits = [],
variance_combinations = [],
// loop counters
i, j,
// the variance, as computed at each step in the calculation
variance = 0;
// Initialize and fill each matrix with zeroes
for (i = 0; i < data.length + 1; i++) {
var tmp1 = [], tmp2 = [];
// despite these arrays having the same values, we need
// to keep them separate so that changing one does not change
// the other
for (j = 0; j < n_classes + 1; j++) {
tmp1.push(0);
tmp2.push(0);
}
lower_class_limits.push(tmp1);
variance_combinations.push(tmp2);
}
for (i = 1; i < n_classes + 1; i++) {
lower_class_limits[1][i] = 1;
variance_combinations[1][i] = 0;
// in the original implementation, 9999999 is used but
// since Javascript has `Infinity`, we use that.
for (j = 2; j < data.length + 1; j++) {
variance_combinations[j][i] = Infinity;
}
}
for (var l = 2; l < data.length + 1; l++) {
// `SZ` originally. this is the sum of the values seen thus
// far when calculating variance.
var sum = 0,
// `ZSQ` originally. the sum of squares of values seen
// thus far
sum_squares = 0,
// `WT` originally. This is the number of
w = 0,
// `IV` originally
i4 = 0;
// in several instances, you could say `Math.pow(x, 2)`
// instead of `x * x`, but this is slower in some browsers
// introduces an unnecessary concept.
for (var m = 1; m < l + 1; m++) {
// `III` originally
var lower_class_limit = l - m + 1,
val = data[lower_class_limit - 1];
// here we're estimating variance for each potential classing
// of the data, for each potential number of classes. `w`
// is the number of data points considered so far.
w++;
// increase the current sum and sum-of-squares
sum += val;
sum_squares += val * val;
// the variance at this point in the sequence is the difference
// between the sum of squares and the total x 2, over the number
// of samples.
variance = sum_squares - (sum * sum) / w;
i4 = lower_class_limit - 1;
if (i4 !== 0) {
for (j = 2; j < n_classes + 1; j++) {
// if adding this element to an existing class
// will increase its variance beyond the limit, break
// the class at this point, setting the `lower_class_limit`
// at this point.
if (variance_combinations[l][j] >=
(variance + variance_combinations[i4][j - 1])) {
lower_class_limits[l][j] = lower_class_limit;
variance_combinations[l][j] = variance +
variance_combinations[i4][j - 1];
}
}
}
}
lower_class_limits[l][1] = 1;
variance_combinations[l][1] = variance;
}
// return the two matrices. for just providing breaks, only
// `lower_class_limits` is needed, but variances can be useful to
// evaluate goodness of fit.
return {
lower_class_limits: lower_class_limits,
variance_combinations: variance_combinations
};
}
// ## Pull Breaks Values for Jenks
//
// the second part of the jenks recipe: take the calculated matrices
// and derive an array of n breaks.
function jenksBreaks(data, lower_class_limits, n_classes) {
var k = data.length - 1,
kclass = [],
countNum = n_classes;
// the calculation of classes will never include the upper and
// lower bounds, so we need to explicitly set them
kclass[n_classes] = data[data.length - 1];
kclass[0] = data[0];
// the lower_class_limits matrix is used as indices into itself
// here: the `k` variable is reused in each iteration.
while (countNum > 1) {
kclass[countNum - 1] = data[lower_class_limits[k][countNum] - 2];
k = lower_class_limits[k][countNum] - 1;
countNum--;
}
return kclass;
}
// # [Jenks natural breaks optimization](http://en.wikipedia.org/wiki/Jenks_natural_breaks_optimization)
//
// Implementations: [1](http://danieljlewis.org/files/2010/06/Jenks.pdf) (python),
// [2](https://github.com/vvoovv/djeo-jenks/blob/master/main.js) (buggy),
// [3](https://github.com/simogeo/geostats/blob/master/lib/geostats.js#L407) (works)
//
// Depends on `jenksBreaks()` and `jenksMatrices()`
function jenks(data, n_classes) {
if (n_classes > data.length) return null;
// sort data in numerical order, since this is expected
// by the matrices function
data = data.slice().sort(function (a, b) { return a - b; });
// get our basic matrices
var matrices = jenksMatrices(data, n_classes),
// we only need lower class limits here
lower_class_limits = matrices.lower_class_limits;
// extract n_classes out of the computed matrices
return jenksBreaks(data, lower_class_limits, n_classes);
}
// # [Skewness](http://en.wikipedia.org/wiki/Skewness)
//
// A measure of the extent to which a probability distribution of a
// real-valued random variable "leans" to one side of the mean.
// The skewness value can be positive or negative, or even undefined.
//
// Implementation is based on the adjusted Fisher-Pearson standardized
// moment coefficient, which is the version found in Excel and several
// statistical packages including Minitab, SAS and SPSS.
//
// Depends on `sum_nth_power_deviations()` and `sample_standard_deviation`
function sample_skewness(x) {
// The skewness of less than three arguments is null
if (x.length < 3) return null;
var n = x.length,
cubed_s = Math.pow(sample_standard_deviation(x), 3),
sum_cubed_deviations = sum_nth_power_deviations(x, 3);
return n * sum_cubed_deviations / ((n - 1) * (n - 2) * cubed_s);
}
// # Standard Normal Table
// A standard normal table, also called the unit normal table or Z table,
// is a mathematical table for the values of Φ (phi), which are the values of
// the cumulative distribution function of the normal distribution.
// It is used to find the probability that a statistic is observed below,
// above, or between values on the standard normal distribution, and by
// extension, any normal distribution.
//
// The probabilities are taken from http://en.wikipedia.org/wiki/Standard_normal_table
// The table used is the cumulative, and not cumulative from 0 to mean
// (even though the latter has 5 digits precision, instead of 4).
var standard_normal_table = [
/* z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 */
/* 0.0 */
0.5000, 0.5040, 0.5080, 0.5120, 0.5160, 0.5199, 0.5239, 0.5279, 0.5319, 0.5359,
/* 0.1 */
0.5398, 0.5438, 0.5478, 0.5517, 0.5557, 0.5596, 0.5636, 0.5675, 0.5714, 0.5753,
/* 0.2 */
0.5793, 0.5832, 0.5871, 0.5910, 0.5948, 0.5987, 0.6026, 0.6064, 0.6103, 0.6141,
/* 0.3 */
0.6179, 0.6217, 0.6255, 0.6293, 0.6331, 0.6368, 0.6406, 0.6443, 0.6480, 0.6517,
/* 0.4 */
0.6554, 0.6591, 0.6628, 0.6664, 0.6700, 0.6736, 0.6772, 0.6808, 0.6844, 0.6879,
/* 0.5 */
0.6915, 0.6950, 0.6985, 0.7019, 0.7054, 0.7088, 0.7123, 0.7157, 0.7190, 0.7224,
/* 0.6 */
0.7257, 0.7291, 0.7324, 0.7357, 0.7389, 0.7422, 0.7454, 0.7486, 0.7517, 0.7549,
/* 0.7 */
0.7580, 0.7611, 0.7642, 0.7673, 0.7704, 0.7734, 0.7764, 0.7794, 0.7823, 0.7852,
/* 0.8 */
0.7881, 0.7910, 0.7939, 0.7967, 0.7995, 0.8023, 0.8051, 0.8078, 0.8106, 0.8133,
/* 0.9 */
0.8159, 0.8186, 0.8212, 0.8238, 0.8264, 0.8289, 0.8315, 0.8340, 0.8365, 0.8389,
/* 1.0 */
0.8413, 0.8438, 0.8461, 0.8485, 0.8508, 0.8531, 0.8554, 0.8577, 0.8599, 0.8621,
/* 1.1 */
0.8643, 0.8665, 0.8686, 0.8708, 0.8729, 0.8749, 0.8770, 0.8790, 0.8810, 0.8830,
/* 1.2 */
0.8849, 0.8869, 0.8888, 0.8907, 0.8925, 0.8944, 0.8962, 0.8980, 0.8997, 0.9015,
/* 1.3 */
0.9032, 0.9049, 0.9066, 0.9082, 0.9099, 0.9115, 0.9131, 0.9147, 0.9162, 0.9177,
/* 1.4 */
0.9192, 0.9207, 0.9222, 0.9236, 0.9251, 0.9265, 0.9279, 0.9292, 0.9306, 0.9319,
/* 1.5 */
0.9332, 0.9345, 0.9357, 0.9370, 0.9382, 0.9394, 0.9406, 0.9418, 0.9429, 0.9441,
/* 1.6 */
0.9452, 0.9463, 0.9474, 0.9484, 0.9495, 0.9505, 0.9515, 0.9525, 0.9535, 0.9545,
/* 1.7 */
0.9554, 0.9564, 0.9573, 0.9582, 0.9591, 0.9599, 0.9608, 0.9616, 0.9625, 0.9633,
/* 1.8 */
0.9641, 0.9649, 0.9656, 0.9664, 0.9671, 0.9678, 0.9686, 0.9693, 0.9699, 0.9706,
/* 1.9 */
0.9713, 0.9719, 0.9726, 0.9732, 0.9738, 0.9744, 0.9750, 0.9756, 0.9761, 0.9767,
/* 2.0 */
0.9772, 0.9778, 0.9783, 0.9788, 0.9793, 0.9798, 0.9803, 0.9808, 0.9812, 0.9817,
/* 2.1 */
0.9821, 0.9826, 0.9830, 0.9834, 0.9838, 0.9842, 0.9846, 0.9850, 0.9854, 0.9857,
/* 2.2 */
0.9861, 0.9864, 0.9868, 0.9871, 0.9875, 0.9878, 0.9881, 0.9884, 0.9887, 0.9890,
/* 2.3 */
0.9893, 0.9896, 0.9898, 0.9901, 0.9904, 0.9906, 0.9909, 0.9911, 0.9913, 0.9916,
/* 2.4 */
0.9918, 0.9920, 0.9922, 0.9925, 0.9927, 0.9929, 0.9931, 0.9932, 0.9934, 0.9936,
/* 2.5 */
0.9938, 0.9940, 0.9941, 0.9943, 0.9945, 0.9946, 0.9948, 0.9949, 0.9951, 0.9952,
/* 2.6 */
0.9953, 0.9955, 0.9956, 0.9957, 0.9959, 0.9960, 0.9961, 0.9962, 0.9963, 0.9964,
/* 2.7 */
0.9965, 0.9966, 0.9967, 0.9968, 0.9969, 0.9970, 0.9971, 0.9972, 0.9973, 0.9974,
/* 2.8 */
0.9974, 0.9975, 0.9976, 0.9977, 0.9977, 0.9978, 0.9979, 0.9979, 0.9980, 0.9981,
/* 2.9 */
0.9981, 0.9982, 0.9982, 0.9983, 0.9984, 0.9984, 0.9985, 0.9985, 0.9986, 0.9986,
/* 3.0 */
0.9987, 0.9987, 0.9987, 0.9988, 0.9988, 0.9989, 0.9989, 0.9989, 0.9990, 0.9990
];
// # [Cumulative Standard Normal Probability](http://en.wikipedia.org/wiki/Standard_normal_table)
//
// Since probability tables cannot be
// printed for every normal distribution, as there are an infinite variety
// of normal distributions, it is common practice to convert a normal to a
// standard normal and then use the standard normal table to find probabilities
function cumulative_std_normal_probability(z) {
// Calculate the position of this value.
var absZ = Math.abs(z),
// Each row begins with a different
// significant digit: 0.5, 0.6, 0.7, and so on. So the row is simply
// this value's significant digit: 0.567 will be in row 0, so row=0,
// 0.643 will be in row 1, so row=10.
row = Math.floor(absZ * 10),
column = 10 * (Math.floor(absZ * 100) / 10 - Math.floor(absZ * 100 / 10)),
index = Math.min((row * 10) + column, standard_normal_table.length - 1);
// The index we calculate must be in the table as a positive value,
// but we still pay attention to whether the input is positive
// or negative, and flip the output value as a last step.
if (z >= 0) {
return standard_normal_table[index];
} else {
// due to floating-point arithmetic, values in the table with
// 4 significant figures can nevertheless end up as repeating
// fractions when they're computed here.
return +(1 - standard_normal_table[index]).toFixed(4);
}
}
// # [Z-Score, or Standard Score](http://en.wikipedia.org/wiki/Standard_score)
//
// The standard score is the number of standard deviations an observation
// or datum is above or below the mean. Thus, a positive standard score
// represents a datum above the mean, while a negative standard score
// represents a datum below the mean. It is a dimensionless quantity
// obtained by subtracting the population mean from an individual raw
// score and then dividing the difference by the population standard
// deviation.
//
// The z-score is only defined if one knows the population parameters;
// if one only has a sample set, then the analogous computation with
// sample mean and sample standard deviation yields the
// Student's t-statistic.
function z_score(x, mean, standard_deviation) {
return (x - mean) / standard_deviation;
}
// We use `ε`, epsilon, as a stopping criterion when we want to iterate
// until we're "close enough".
var epsilon = 0.0001;
// # [Factorial](https://en.wikipedia.org/wiki/Factorial)
//
// A factorial, usually written n!, is the product of all positive
// integers less than or equal to n. Often factorial is implemented
// recursively, but this iterative approach is significantly faster
// and simpler.
function factorial(n) {
// factorial is mathematically undefined for negative numbers
if (n < 0 ) { return null; }
// typically you'll expand the factorial function going down, like
// 5! = 5 * 4 * 3 * 2 * 1. This is going in the opposite direction,
// counting from 2 up to the number in question, and since anything
// multiplied by 1 is itself, the loop only needs to start at 2.
var accumulator = 1;
for (var i = 2; i <= n; i++) {
// for each number up to and including the number `n`, multiply
// the accumulator my that number.
accumulator *= i;
}
return accumulator;
}
// # Bernoulli Distribution
//
// The [Bernoulli distribution](http://en.wikipedia.org/wiki/Bernoulli_distribution)
// is the probability discrete
// distribution of a random variable which takes value 1 with success
// probability `p` and value 0 with failure
// probability `q` = 1 - `p`. It can be used, for example, to represent the
// toss of a coin, where "1" is defined to mean "heads" and "0" is defined
// to mean "tails" (or vice versa). It is
// a special case of a Binomial Distribution
// where `n` = 1.
function bernoulli_distribution(p) {
// Check that `p` is a valid probability (0 ≤ p ≤ 1)
if (p < 0 || p > 1 ) { return null; }
return binomial_distribution(1, p);
}
// # Binomial Distribution
//
// The [Binomial Distribution](http://en.wikipedia.org/wiki/Binomial_distribution) is the discrete probability
// distribution of the number of successes in a sequence of n independent yes/no experiments, each of which yields
// success with probability `probability`. Such a success/failure experiment is also called a Bernoulli experiment or
// Bernoulli trial; when trials = 1, the Binomial Distribution is a Bernoulli Distribution.
function binomial_distribution(trials, probability) {
// Check that `p` is a valid probability (0 ≤ p ≤ 1),
// that `n` is an integer, strictly positive.
if (probability < 0 || probability > 1 ||
trials <= 0 || trials % 1 !== 0) {
return null;
}
// a [probability mass function](https://en.wikipedia.org/wiki/Probability_mass_function)
function probability_mass(x, trials, probability) {
return factorial(trials) /
(factorial(x) * factorial(trials - x)) *
(Math.pow(probability, x) * Math.pow(1 - probability, trials - x));
}
// We initialize `x`, the random variable, and `accumulator`, an accumulator
// for the cumulative distribution function to 0. `distribution_functions`
// is the object we'll return with the `probability_of_x` and the
// `cumulative_probability_of_x`, as well as the calculated mean &
// variance. We iterate until the `cumulative_probability_of_x` is
// within `epsilon` of 1.0.
var x = 0,
cumulative_probability = 0,
cells = {};
// This algorithm iterates through each potential outcome,
// until the `cumulative_probability` is very close to 1, at
// which point we've defined the vast majority of outcomes
do {
cells[x] = probability_mass(x, trials, probability);
cumulative_probability += cells[x];
x++;
// when the cumulative_probability is nearly 1, we've calculated
// the useful range of this distribution
} while (cumulative_probability < 1 - epsilon);
return cells;
}
// # Poisson Distribution
//
// The [Poisson Distribution](http://en.wikipedia.org/wiki/Poisson_distribution)
// is a discrete probability distribution that expresses the probability
// of a given number of events occurring in a fixed interval of time
// and/or space if these events occur with a known average rate and
// independently of the time since the last event.
//
// The Poisson Distribution is characterized by the strictly positive
// mean arrival or occurrence rate, `λ`.
function poisson_distribution(lambda) {
// Check that lambda is strictly positive
if (lambda <= 0) { return null; }
// our current place in the distribution
var x = 0,
// and we keep track of the current cumulative probability, in
// order to know when to stop calculating chances.
cumulative_probability = 0,
// the calculated cells to be returned
cells = {};
// a [probability mass function](https://en.wikipedia.org/wiki/Probability_mass_function)
function probability_mass(x, lambda) {
return (Math.pow(Math.E, -lambda) * Math.pow(lambda, x)) /
factorial(x);
}
// This algorithm iterates through each potential outcome,
// until the `cumulative_probability` is very close to 1, at
// which point we've defined the vast majority of outcomes
do {
cells[x] = probability_mass(x, lambda);
cumulative_probability += cells[x];
x++;
// when the cumulative_probability is nearly 1, we've calculated
// the useful range of this distribution
} while (cumulative_probability < 1 - epsilon);
return cells;
}
// # Percentage Points of the χ2 (Chi-Squared) Distribution
// The [χ2 (Chi-Squared) Distribution](http://en.wikipedia.org/wiki/Chi-squared_distribution) is used in the common
// chi-squared tests for goodness of fit of an observed distribution to a theoretical one, the independence of two
// criteria of classification of qualitative data, and in confidence interval estimation for a population standard
// deviation of a normal distribution from a sample standard deviation.
//
// Values from Appendix 1, Table III of William W. Hines & Douglas C. Montgomery, "Probability and Statistics in
// Engineering and Management Science", Wiley (1980).
var chi_squared_distribution_table = {
1: { 0.995: 0.00, 0.99: 0.00, 0.975: 0.00, 0.95: 0.00, 0.9: 0.02, 0.5: 0.45, 0.1: 2.71, 0.05: 3.84, 0.025: 5.02, 0.01: 6.63, 0.005: 7.88 },
2: { 0.995: 0.01, 0.99: 0.02, 0.975: 0.05, 0.95: 0.10, 0.9: 0.21, 0.5: 1.39, 0.1: 4.61, 0.05: 5.99, 0.025: 7.38, 0.01: 9.21, 0.005: 10.60 },
3: { 0.995: 0.07, 0.99: 0.11, 0.975: 0.22, 0.95: 0.35, 0.9: 0.58, 0.5: 2.37, 0.1: 6.25, 0.05: 7.81, 0.025: 9.35, 0.01: 11.34, 0.005: 12.84 },
4: { 0.995: 0.21, 0.99: 0.30, 0.975: 0.48, 0.95: 0.71, 0.9: 1.06, 0.5: 3.36, 0.1: 7.78, 0.05: 9.49, 0.025: 11.14, 0.01: 13.28, 0.005: 14.86 },
5: { 0.995: 0.41, 0.99: 0.55, 0.975: 0.83, 0.95: 1.15, 0.9: 1.61, 0.5: 4.35, 0.1: 9.24, 0.05: 11.07, 0.025: 12.83, 0.01: 15.09, 0.005: 16.75 },
6: { 0.995: 0.68, 0.99: 0.87, 0.975: 1.24, 0.95: 1.64, 0.9: 2.20, 0.5: 5.35, 0.1: 10.65, 0.05: 12.59, 0.025: 14.45, 0.01: 16.81, 0.005: 18.55 },
7: { 0.995: 0.99, 0.99: 1.25, 0.975: 1.69, 0.95: 2.17, 0.9: 2.83, 0.5: 6.35, 0.1: 12.02, 0.05: 14.07, 0.025: 16.01, 0.01: 18.48, 0.005: 20.28 },
8: { 0.995: 1.34, 0.99: 1.65, 0.975: 2.18, 0.95: 2.73, 0.9: 3.49, 0.5: 7.34, 0.1: 13.36, 0.05: 15.51, 0.025: 17.53, 0.01: 20.09, 0.005: 21.96 },
9: { 0.995: 1.73, 0.99: 2.09, 0.975: 2.70, 0.95: 3.33, 0.9: 4.17, 0.5: 8.34, 0.1: 14.68, 0.05: 16.92, 0.025: 19.02, 0.01: 21.67, 0.005: 23.59 },
10: { 0.995: 2.16, 0.99: 2.56, 0.975: 3.25, 0.95: 3.94, 0.9: 4.87, 0.5: 9.34, 0.1: 15.99, 0.05: 18.31, 0.025: 20.48, 0.01: 23.21, 0.005: 25.19 },
11: { 0.995: 2.60, 0.99: 3.05, 0.975: 3.82, 0.95: 4.57, 0.9: 5.58, 0.5: 10.34, 0.1: 17.28, 0.05: 19.68, 0.025: 21.92, 0.01: 24.72, 0.005: 26.76 },
12: { 0.995: 3.07, 0.99: 3.57, 0.975: 4.40, 0.95: 5.23, 0.9: 6.30, 0.5: 11.34, 0.1: 18.55, 0.05: 21.03, 0.025: 23.34, 0.01: 26.22, 0.005: 28.30 },
13: { 0.995: 3.57, 0.99: 4.11, 0.975: 5.01, 0.95: 5.89, 0.9: 7.04, 0.5: 12.34, 0.1: 19.81, 0.05: 22.36, 0.025: 24.74, 0.01: 27.69, 0.005: 29.82 },
14: { 0.995: 4.07, 0.99: 4.66, 0.975: 5.63, 0.95: 6.57, 0.9: 7.79, 0.5: 13.34, 0.1: 21.06, 0.05: 23.68, 0.025: 26.12, 0.01: 29.14, 0.005: 31.32 },
15: { 0.995: 4.60, 0.99: 5.23, 0.975: 6.27, 0.95: 7.26, 0.9: 8.55, 0.5: 14.34, 0.1: 22.31, 0.05: 25.00, 0.025: 27.49, 0.01: 30.58, 0.005: 32.80 },
16: { 0.995: 5.14, 0.99: 5.81, 0.975: 6.91, 0.95: 7.96, 0.9: 9.31, 0.5: 15.34, 0.1: 23.54, 0.05: 26.30, 0.025: 28.85, 0.01: 32.00, 0.005: 34.27 },
17: { 0.995: 5.70, 0.99: 6.41, 0.975: 7.56, 0.95: 8.67, 0.9: 10.09, 0.5: 16.34, 0.1: 24.77, 0.05: 27.59, 0.025: 30.19, 0.01: 33.41, 0.005: 35.72 },
18: { 0.995: 6.26, 0.99: 7.01, 0.975: 8.23, 0.95: 9.39, 0.9: 10.87, 0.5: 17.34, 0.1: 25.99, 0.05: 28.87, 0.025: 31.53, 0.01: 34.81, 0.005: 37.16 },
19: { 0.995: 6.84, 0.99: 7.63, 0.975: 8.91, 0.95: 10.12, 0.9: 11.65, 0.5: 18.34, 0.1: 27.20, 0.05: 30.14, 0.025: 32.85, 0.01: 36.19, 0.005: 38.58 },
20: { 0.995: 7.43, 0.99: 8.26, 0.975: 9.59, 0.95: 10.85, 0.9: 12.44, 0.5: 19.34, 0.1: 28.41, 0.05: 31.41, 0.025: 34.17, 0.01: 37.57, 0.005: 40.00 },
21: { 0.995: 8.03, 0.99: 8.90, 0.975: 10.28, 0.95: 11.59, 0.9: 13.24, 0.5: 20.34, 0.1: 29.62, 0.05: 32.67, 0.025: 35.48, 0.01: 38.93, 0.005: 41.40 },
22: { 0.995: 8.64, 0.99: 9.54, 0.975: 10.98, 0.95: 12.34, 0.9: 14.04, 0.5: 21.34, 0.1: 30.81, 0.05: 33.92, 0.025: 36.78, 0.01: 40.29, 0.005: 42.80 },
23: { 0.995: 9.26, 0.99: 10.20, 0.975: 11.69, 0.95: 13.09, 0.9: 14.85, 0.5: 22.34, 0.1: 32.01, 0.05: 35.17, 0.025: 38.08, 0.01: 41.64, 0.005: 44.18 },
24: { 0.995: 9.89, 0.99: 10.86, 0.975: 12.40, 0.95: 13.85, 0.9: 15.66, 0.5: 23.34, 0.1: 33.20, 0.05: 36.42, 0.025: 39.36, 0.01: 42.98, 0.005: 45.56 },
25: { 0.995: 10.52, 0.99: 11.52, 0.975: 13.12, 0.95: 14.61, 0.9: 16.47, 0.5: 24.34, 0.1: 34.28, 0.05: 37.65, 0.025: 40.65, 0.01: 44.31, 0.005: 46.93 },
26: { 0.995: 11.16, 0.99: 12.20, 0.975: 13.84, 0.95: 15.38, 0.9: 17.29, 0.5: 25.34, 0.1: 35.56, 0.05: 38.89, 0.025: 41.92, 0.01: 45.64, 0.005: 48.29 },
27: { 0.995: 11.81, 0.99: 12.88, 0.975: 14.57, 0.95: 16.15, 0.9: 18.11, 0.5: 26.34, 0.1: 36.74, 0.05: 40.11, 0.025: 43.19, 0.01: 46.96, 0.005: 49.65 },
28: { 0.995: 12.46, 0.99: 13.57, 0.975: 15.31, 0.95: 16.93, 0.9: 18.94, 0.5: 27.34, 0.1: 37.92, 0.05: 41.34, 0.025: 44.46, 0.01: 48.28, 0.005: 50.99 },
29: { 0.995: 13.12, 0.99: 14.26, 0.975: 16.05, 0.95: 17.71, 0.9: 19.77, 0.5: 28.34, 0.1: 39.09, 0.05: 42.56, 0.025: 45.72, 0.01: 49.59, 0.005: 52.34 },
30: { 0.995: 13.79, 0.99: 14.95, 0.975: 16.79, 0.95: 18.49, 0.9: 20.60, 0.5: 29.34, 0.1: 40.26, 0.05: 43.77, 0.025: 46.98, 0.01: 50.89, 0.005: 53.67 },
40: { 0.995: 20.71, 0.99: 22.16, 0.975: 24.43, 0.95: 26.51, 0.9: 29.05, 0.5: 39.34, 0.1: 51.81, 0.05: 55.76, 0.025: 59.34, 0.01: 63.69, 0.005: 66.77 },
50: { 0.995: 27.99, 0.99: 29.71, 0.975: 32.36, 0.95: 34.76, 0.9: 37.69, 0.5: 49.33, 0.1: 63.17, 0.05: 67.50, 0.025: 71.42, 0.01: 76.15, 0.005: 79.49 },
60: { 0.995: 35.53, 0.99: 37.48, 0.975: 40.48, 0.95: 43.19, 0.9: 46.46, 0.5: 59.33, 0.1: 74.40, 0.05: 79.08, 0.025: 83.30, 0.01: 88.38, 0.005: 91.95 },
70: { 0.995: 43.28, 0.99: 45.44, 0.975: 48.76, 0.95: 51.74, 0.9: 55.33, 0.5: 69.33, 0.1: 85.53, 0.05: 90.53, 0.025: 95.02, 0.01: 100.42, 0.005: 104.22 },
80: { 0.995: 51.17, 0.99: 53.54, 0.975: 57.15, 0.95: 60.39, 0.9: 64.28, 0.5: 79.33, 0.1: 96.58, 0.05: 101.88, 0.025: 106.63, 0.01: 112.33, 0.005: 116.32 },
90: { 0.995: 59.20, 0.99: 61.75, 0.975: 65.65, 0.95: 69.13, 0.9: 73.29, 0.5: 89.33, 0.1: 107.57, 0.05: 113.14, 0.025: 118.14, 0.01: 124.12, 0.005: 128.30 },
100: { 0.995: 67.33, 0.99: 70.06, 0.975: 74.22, 0.95: 77.93, 0.9: 82.36, 0.5: 99.33, 0.1: 118.50, 0.05: 124.34, 0.025: 129.56, 0.01: 135.81, 0.005: 140.17 }
};
// # χ2 (Chi-Squared) Goodness-of-Fit Test
//
// The [χ2 (Chi-Squared) Goodness-of-Fit Test](http://en.wikipedia.org/wiki/Goodness_of_fit#Pearson.27s_chi-squared_test)
// uses a measure of goodness of fit which is the sum of differences between observed and expected outcome frequencies
// (that is, counts of observations), each squared and divided by the number of observations expected given the
// hypothesized distribution. The resulting χ2 statistic, `chi_squared`, can be compared to the chi-squared distribution
// to determine the goodness of fit. In order to determine the degrees of freedom of the chi-squared distribution, one
// takes the total number of observed frequencies and subtracts the number of estimated parameters. The test statistic
// follows, approximately, a chi-square distribution with (k − c) degrees of freedom where `k` is the number of non-empty
// cells and `c` is the number of estimated parameters for the distribution.
function chi_squared_goodness_of_fit(data, distribution_type, significance) {
// Estimate from the sample data, a weighted mean.
var input_mean = mean(data),
// Calculated value of the χ2 statistic.
chi_squared = 0,
// Degrees of freedom, calculated as (number of class intervals -
// number of hypothesized distribution parameters estimated - 1)
degrees_of_freedom,
// Number of hypothesized distribution parameters estimated, expected to be supplied in the distribution test.
// Lose one degree of freedom for estimating `lambda` from the sample data.
c = 1,
// The hypothesized distribution.
// Generate the hypothesized distribution.
hypothesized_distribution = distribution_type(input_mean),
observed_frequencies = [],
expected_frequencies = [],
k;
// Create an array holding a histogram from the sample data, of
// the form `{ value: numberOfOcurrences }`
for (var i = 0; i < data.length; i++) {
if (observed_frequencies[data[i]] === undefined) {
observed_frequencies[data[i]] = 0;
}
observed_frequencies[data[i]]++;
}
// The histogram we created might be sparse - there might be gaps
// between values. So we iterate through the histogram, making
// sure that instead of undefined, gaps have 0 values.
for (i = 0; i < observed_frequencies.length; i++) {
if (observed_frequencies[i] === undefined) {
observed_frequencies[i] = 0;
}
}
// Create an array holding a histogram of expected data given the
// sample size and hypothesized distribution.
for (k in hypothesized_distribution) {
if (k in observed_frequencies) {
expected_frequencies[k] = hypothesized_distribution[k] * data.length;
}
}
// Working backward through the expected frequencies, collapse classes
// if less than three observations are expected for a class.
// This transformation is applied to the observed frequencies as well.
for (k = expected_frequencies.length - 1; k >= 0; k--) {
if (expected_frequencies[k] < 3) {
expected_frequencies[k - 1] += expected_frequencies[k];
expected_frequencies.pop();
observed_frequencies[k - 1] += observed_frequencies[k];
observed_frequencies.pop();
}
}
// Iterate through the squared differences between observed & expected
// frequencies, accumulating the `chi_squared` statistic.
for (k = 0; k < observed_frequencies.length; k++) {
chi_squared += Math.pow(
observed_frequencies[k] - expected_frequencies[k], 2) /
expected_frequencies[k];
}
// Calculate degrees of freedom for this test and look it up in the
// `chi_squared_distribution_table` in order to
// accept or reject the goodness-of-fit of the hypothesized distribution.
degrees_of_freedom = observed_frequencies.length - c - 1;
return chi_squared_distribution_table[degrees_of_freedom][significance] < chi_squared;
}
// # Mixin
//
// Mixin simple_statistics to a single Array instance if provided
// or the Array native object if not. This is an optional
// feature that lets you treat simple_statistics as a native feature
// of Javascript.
function mixin(array) {
var support = !!(Object.defineProperty && Object.defineProperties);
if (!support) throw new Error('without defineProperty, simple-statistics cannot be mixed in');
// only methods which work on basic arrays in a single step
// are supported
var arrayMethods = ['median', 'standard_deviation', 'sum',
'sample_skewness',
'mean', 'min', 'max', 'quantile', 'geometric_mean',
'harmonic_mean', 'root_mean_square'];
// create a closure with a method name so that a reference
// like `arrayMethods[i]` doesn't follow the loop increment
function wrap(method) {
return function() {
// cast any arguments into an array, since they're
// natively objects
var args = Array.prototype.slice.apply(arguments);
// make the first argument the array itself
args.unshift(this);
// return the result of the ss method
return ss[method].apply(ss, args);
};
}
// select object to extend
var extending;
if (array) {
// create a shallow copy of the array so that our internal
// operations do not change it by reference
extending = array.slice();
} else {
extending = Array.prototype;
}
// for each array function, define a function that gets
// the array as the first argument.
// We use [defineProperty](https://developer.mozilla.org/en-US/docs/JavaScript/Reference/Global_Objects/Object/defineProperty)
// because it allows these properties to be non-enumerable:
// `for (var in x)` loops will not run into problems with this
// implementation.
for (var i = 0; i < arrayMethods.length; i++) {
Object.defineProperty(extending, arrayMethods[i], {
value: wrap(arrayMethods[i]),
configurable: true,
enumerable: false,
writable: true
});
}
return extending;
}
ss.linear_regression = linear_regression;
ss.standard_deviation = standard_deviation;
ss.r_squared = r_squared;
ss.median = median;
ss.mean = mean;
ss.mode = mode;
ss.min = min;
ss.max = max;
ss.sum = sum;
ss.quantile = quantile;
ss.quantile_sorted = quantile_sorted;
ss.iqr = iqr;
ss.mad = mad;
ss.chunk = chunk;
ss.shuffle = shuffle;
ss.shuffle_in_place = shuffle_in_place;
ss.sample = sample;
ss.sample_covariance = sample_covariance;
ss.sample_correlation = sample_correlation;
ss.sample_variance = sample_variance;
ss.sample_standard_deviation = sample_standard_deviation;
ss.sample_skewness = sample_skewness;
ss.geometric_mean = geometric_mean;
ss.harmonic_mean = harmonic_mean;
ss.root_mean_square = root_mean_square;
ss.variance = variance;
ss.t_test = t_test;
ss.t_test_two_sample = t_test_two_sample;
// jenks
ss.jenksMatrices = jenksMatrices;
ss.jenksBreaks = jenksBreaks;
ss.jenks = jenks;
ss.bayesian = bayesian;
// Distribution-related methods
ss.epsilon = epsilon; // We make ε available to the test suite.
ss.factorial = factorial;
ss.bernoulli_distribution = bernoulli_distribution;
ss.binomial_distribution = binomial_distribution;
ss.poisson_distribution = poisson_distribution;
ss.chi_squared_goodness_of_fit = chi_squared_goodness_of_fit;
// Normal distribution
ss.z_score = z_score;
ss.cumulative_std_normal_probability = cumulative_std_normal_probability;
ss.standard_normal_table = standard_normal_table;
// Alias this into its common name
ss.average = mean;
ss.interquartile_range = iqr;
ss.mixin = mixin;
ss.median_absolute_deviation = mad;
ss.rms = root_mean_square;
})(this);
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