simple_statistics.js

simple-statistics

A simple, literate statistics system. The code below uses the Javascript module pattern, 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

        exports = 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;
    }

Linear Regression

Simple linear regression is a simple way to find a fitted line between a set of coordinates.

    ss.linear_regression = function() {
        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;
        };

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() {

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,
                    m, b;

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 SSx, SSy, SSxx, and SSxy

                for (var i = 0; i < data.length; i++) {
                    sum_x += data[i][0];
                    sum_y += data[i][1];

                    sum_xx += data[i][0] * data[i][0];
                    sum_xy += data[i][0] * data[i][1];
                }

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 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

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.

    ss.r_squared = function(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, it's ratio to the sum of squares increases and the r squared value grows lower.

        return 1 - (err / sum_of_squares);
    };

Bayesian Classifier

This is a naïve bayesian classifier that takes singly-nested objects.

    ss.bayesian = function() {

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

    ss.sum = function(x) {
        var sum = 0;
        for (var i = 0; i < x.length; i++) {
            sum += x[i];
        }
        return sum;
    };

mean

is the sum over the number of values

This runs on O(n), linear time in respect to the array

    ss.mean = function(x) {

The mean of no numbers is null

        if (x.length === 0) return null;

        return ss.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 multipled by each other

This runs on O(n), linear time in respect to the array

    ss.geometric_mean = function(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);
    };

Alias this into its common name

    ss.average = ss.mean;

min

This is simply the minimum number in the set.

This runs on O(n), linear time in respect to the array

    ss.min = function(x) {
        var min;
        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] < min || min === undefined) min = x[i];
        }
        return min;
    };

max

This is simply the maximum number in the set.

This runs on O(n), linear time in respect to the array

    ss.max = function(x) {
        var max;
        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] > max || max === undefined) max = x[i];
        }
        return max;
    };

variance

is the sum of squared deviations from the mean

    ss.variance = function(x) {

The variance of no numbers is null

        if (x.length === 0) return null;

        var mean = ss.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, 2));
        }

Find the mean value of that list

        return ss.mean(deviations);
    };

standard deviation

is just the square root of the variance.

    ss.standard_deviation = function(x) {

The standard deviation of no numbers is null

        if (x.length === 0) return null;

        return Math.sqrt(ss.variance(x));
    };

     ss.sum_squared_deviations = function(x) {

The variance of no numbers is null

        if (x.length <= 1) return null;

        var mean = ss.mean(x),
            sum = 0;

Make a list of squared deviations from the mean.

        for (var i = 0; i < x.length; i++) {
            sum += Math.pow(x[i] - mean, 2);
        }

        return sum;
     };

variance

is the sum of squared deviations from the mean

    ss.sample_variance = function(x) {
        var sum_squared_deviations = ss.sum_squared_deviations(x);
        if (sum_squared_deviations === null) return null;

Find the mean value of that list

        return sum_squared_deviations / (x.length - 1);
    };

standard deviation

is just the square root of the variance.

    ss.sample_standard_deviation = function(x) {

The standard deviation of no numbers is null

        if (x.length <= 1) return null;

        return Math.sqrt(ss.sample_variance(x));
    };

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.

    ss.sample_covariance = function(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 = ss.mean(x),
            ymean = ss.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

Gets a measure of how correlated two datasets are, between -1 and 1

    ss.sample_correlation = function(x, y) {
        var cov = ss.sample_covariance(x, y),
            xstd = ss.sample_standard_deviation(x),
            ystd = ss.sample_standard_deviation(y);

        if (cov === null || xstd === null || ystd === null) {
            return null;
        }

        return cov / xstd / ystd;
    };

median

    ss.median = function(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

This implementation is inspired by science.js

    ss.mode = function(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

            mode,

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;
                    seen_this = 1;
                    mode = last;
                }
                last = sorted[i];

if this isn't a new number, it's one more occurrence of the potential mode

            } else { seen_this++; }
        }
        return mode;
    };

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, which, for a certain level of significance, will let you determine that the null hypothesis can or cannot be rejected.

    ss.t_test = function(sample, x) {

The mean of the sample

      var sample_mean = ss.mean(sample);

The standard deviation of the sample

      var sd = ss.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);
    };

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 algorithm from wikipedia.

Sample is a one-dimensional array of numbers, and p is a decimal number 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.

    ss.quantile = function(sample, p) {

We can't derive quantiles from an empty list

        if (sample.length === 0) return null;

invalid bounds. Microsoft Excel accepts 0 and 1, but we won't.

        if (p >= 1 || p <= 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; });

Find a potential index in the list. In Wikipedia's terms, this is Ip.

        var idx = (sorted.length) * p;

If this isn't an integer, we'll round up to the next value in the list.

        if (idx % 1 !== 0) {
            return sorted[Math.ceil(idx) - 1];
        } else if (sample.length % 2 === 0) {

If the list has even-length and we had an integer in the first place, we'll take the average of this number and the next value, if there is one

            return (sorted[idx - 1] + sorted[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 sorted[idx];
        }
    };

Compute the matrices required for Jenks breaks. These matrices can be used for any classing of data with classes <= n_classes

    ss.jenksMatrices = function(data, n_classes) {

in the original implementation, these matrices are referred to as LC and OP

  • lowerclasslimits (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 = [];
            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 lowerclasslimit 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 evaluage goodness of fit.

        return {
            lower_class_limits: lower_class_limits,
            variance_combinations: variance_combinations
        };
    };

the second part of the jenks recipe: take the calculated matrices and derive an array of n breaks.

    ss.jenksBreaks = function(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 lowerclasslimits matrix is used as indexes 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

Implementations: 1 (python), 2 (buggy), 3 (works)

    ss.jenks = function(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 = ss.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 ss.jenksBreaks(data, lower_class_limits, n_classes);

    };

Mixin

Mixin simplestatistics to the Array native object. This is an optional feature that lets you treat simplestatistics as a native feature of Javascript.

    ss.mixin = function() {
        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',
            'mean', 'min', 'max', 'quantile', 'geometric_mean'];

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);
            };
        }

for each array function, define a function off of the Array prototype which automatically gets the array as the first argument. We use 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(Array.prototype, arrayMethods[i], {
                value: wrap(arrayMethods[i]),
                configurable: true,
                enumerable: false,
                writable: true
            });
        }
    };

})(this);