The min is the lowest number in the array. This runs in O(n), linear time, with respect to the length of the array.

• Error: if the length of x is less than one

min([1, 5, -10, 100, 2]); // => -10

This computes the maximum number in an array.

This runs in O(n), linear time, with respect to the length of the array.

• Error: if the length of x is less than one

max([1, 2, 3, 4]);
// => 4

Our default sum is the Kahan-Babuska algorithm. This method is an improvement over the classical Kahan summation algorithm. It aims at computing the sum of a list of numbers while correcting for floating-point errors. Traditionally, sums are calculated as many successive additions, each one with its own floating-point roundoff. These losses in precision add up as the number of numbers increases. This alternative algorithm is more accurate than the simple way of calculating sums by simple addition.

This runs in O(n), linear time, with respect to the length of the array.

sum([1, 2, 3]); // => 6

The simple sum of an array is the result of adding all numbers together, starting from zero.

This runs in O(n), linear time, with respect to the length of the array.

sumSimple([1, 2, 3]); // => 6

The quantile: this is a population quantile, since we assume to know the entire dataset in this library. This is an implementation of the 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

quantile([3, 6, 7, 8, 8, 9, 10, 13, 15, 16, 20], 0.5); // => 9

The product of an array is the result of multiplying all numbers together, starting using one as the multiplicative identity.

This runs in O(n), linear time, with respect to the length of the array.

product([1, 2, 3, 4]); // => 24

The minimum is the lowest number in the array. With a sorted array, the first element in the array is always the smallest, so this calculation can be done in one step, or constant time.

minSorted([-100, -10, 1, 2, 5]); // => -100

The maximum is the highest number in the array. With a sorted array, the last element in the array is always the largest, so this calculation can be done in one step, or constant time.

maxSorted([-100, -10, 1, 2, 5]); // => 5

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 faster.

• Error: if p ix outside of the range from 0 to 1
• Error: if x is empty

quantileSorted([3, 6, 7, 8, 8, 9, 10, 13, 15, 16, 20], 0.5); // => 9

The mean, also known as average, is the sum of all values over the number of values. This is a measure of central tendency: a method of finding a typical or central value of a set of numbers.

This runs in O(n), linear time, with respect to the length of the array.

• Error: if the length of x is less than one

mean([0, 10]); // => 5

When adding a new value to a list, one does not have to necessary recompute the mean of the list in linear time. They can instead use this function to compute the new mean by providing the current mean, the number of elements in the list that produced it and the new value to add.

addToMean(14, 5, 53); // => 20.5

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 is a measure of central tendency: a method of finding a typical or central value of a set of numbers.

This runs in O(n log(n)) because it needs to sort the array internally before running an O(n) search to find the mode.

mode([0, 0, 1]); // => 0

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 is a measure of central tendency: a method of finding a typical or central value of a set of numbers.

This runs in O(n) because the input is sorted.

• Error: if sorted is empty

modeSorted([0, 0, 1]); // => 0

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.

modeFast uses a Map object to keep track of the mode, instead of the approach used with mode, a sorted array. As a result, it is faster than mode and supports any data type that can be compared with ==. It also requires a JavaScript environment with support for Map, and will throw an error if Map is not available.

This is a measure of central tendency: a method of finding a typical or central value of a set of numbers.

• ReferenceError: if the JavaScript environment doesn't support Map
• Error: if x is empty

modeFast(['rabbits', 'rabbits', 'squirrels']); // => 'rabbits'

The median is the middle number of a list. This is often a good indicator of 'the middle' when there are outliers that skew the mean() value. This is a measure of central tendency: a method of finding a typical or central value of a set of numbers.

The median isn't necessarily one of the elements in the list: the value can be the average of two elements if the list has an even length and the two central values are different.

median([10, 2, 5, 100, 2, 1]); // => 3.5

The median is the middle number of a list. This is often a good indicator of 'the middle' when there are outliers that skew the mean() value. This is a measure of central tendency: a method of finding a typical or central value of a set of numbers.

The median isn't necessarily one of the elements in the list: the value can be the average of two elements if the list has an even length and the two central values are different.

medianSorted([10, 2, 5, 100, 2, 1]); // => 52.5

The Harmonic Mean is a mean function typically used to find the average of rates. This mean is calculated by taking the reciprocal of the arithmetic mean of the reciprocals of the input numbers.

This is a measure of central tendency: a method of finding a typical or central value of a set of numbers.

This runs in O(n), linear time, with respect to the length of the array.

• Error: if x is empty
• Error: if x contains a negative number

harmonicMean([2, 3]).toFixed(2) // => '2.40'

The Geometric Mean is 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.

The geometric mean is often useful for proportional growth: given growth rates for multiple years, like 80%, 16.66% and 42.85%, a simple mean will incorrectly estimate an average growth rate, whereas a geometric mean will correctly estimate a growth rate that, over those years, will yield the same end value.

This runs in O(n), linear time, with respect to the length of the array.

• Error: if x is empty
• Error: if x contains a negative number

var growthRates = [1.80, 1.166666, 1.428571];
var averageGrowth = ss.geometricMean(growthRates);
var averageGrowthRates = [averageGrowth, averageGrowth, averageGrowth];
var startingValue = 10;
var startingValueMean = 10;
growthRates.forEach(function(rate) {
  startingValue *= rate;
});
averageGrowthRates.forEach(function(rate) {
  startingValueMean *= rate;
});
startingValueMean === startingValue;

The Root Mean Square (RMS) is 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 in O(n), linear time, with respect to the length of the array.

• Error: if x is empty

rootMeanSquare([-1, 1, -1, 1]); // => 1

Skewness is 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.

• Error: if x has length less than 3

sampleSkewness([2, 4, 6, 3, 1]); // => 0.590128656384365

The variance is the sum of squared deviations from the mean.

This is an implementation of variance, not sample variance: see the sampleVariance method if you want a sample measure.

• Error: if x's length is 0

variance([1, 2, 3, 4, 5, 6]); // => 2.9166666666666665

The sample variance is the sum of squared deviations from the mean. The sample variance is distinguished from the variance by the usage of Bessel's Correction: instead of dividing the sum of squared deviations by the length of the input, it is divided by the length minus one. This corrects the bias in estimating a value from a set that you don't know if full.

References:

• Wolfram MathWorld on Sample Variance

• Error: if the length of x is less than 2

sampleVariance([1, 2, 3, 4, 5]); // => 2.5

The standard deviation is the square root of the variance. This is also known as the population standard deviation. It's useful for measuring the amount of variation or dispersion in a set of values.

Standard deviation is only appropriate for full-population knowledge: for samples of a population, sampleStandardDeviation is more appropriate.

variance([2, 4, 4, 4, 5, 5, 7, 9]); // => 4
standardDeviation([2, 4, 4, 4, 5, 5, 7, 9]); // => 2

The sample standard deviation is the square root of the sample variance.

sampleStandardDeviation([2, 4, 4, 4, 5, 5, 7, 9]).toFixed(2);
// => '2.14'

The Median Absolute Deviation is a robust measure of statistical dispersion. It is more resilient to outliers than the standard deviation.

medianAbsoluteDeviation([1, 1, 2, 2, 4, 6, 9]); // => 1

The Interquartile range is 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.

interquartileRange([0, 1, 2, 3]); // => 2

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.

var input = [1, 2, 3];
// since the variance of a set is the mean squared
// deviations, we can calculate that with sumNthPowerDeviations:
sumNthPowerDeviations(input, 2) / input.length;

The Z-Score, or 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.

zScore(78, 80, 5); // => -0.4

The correlation is a measure of how correlated two datasets are, between -1 and 1

sampleCorrelation([1, 2, 3, 4, 5, 6], [2, 2, 3, 4, 5, 60]).toFixed(2);
// => '0.69'

Sample covariance of two datasets: how much do the two datasets move together? x and y are two datasets, represented as arrays of numbers.

• Error: if x and y do not have equal lengths
• Error: if x or y have length of one or less

sampleCovariance([1, 2, 3, 4, 5, 6], [6, 5, 4, 3, 2, 1]); // => -3.5

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.

var samples = [[0, 0], [1, 1]];
var regressionLine = linearRegressionLine(linearRegression(samples));
rSquared(samples, regressionLine); // = 1 this line is a perfect fit

Simple linear regression is a simple way to find a fitted line between a set of coordinates. This algorithm finds the slope and y-intercept of a regression line using the least sum of squares.

linearRegression([[0, 0], [1, 1]]); // => { m: 1, b: 0 }

Given the output of linearRegression: an object with m and b values indicating slope and intercept, respectively, generate a line function that translates x values into y values.

var l = linearRegressionLine(linearRegression([[0, 0], [1, 1]]));
l(0) // = 0
l(2) // = 2
linearRegressionLine({ b: 0, m: 1 })(1); // => 1
linearRegressionLine({ b: 1, m: 1 })(1); // => 2

A Fisher-Yates shuffle is a fast way to create a random permutation of a finite set. This is a function around shuffle_in_place that adds the guarantee that it will not modify its input.

var shuffled = shuffle([1, 2, 3, 4]);
shuffled; // = [2, 3, 1, 4] or any other random permutation

A Fisher-Yates shuffle in-place - which means that it will change the order of the original array by reference.

This is an algorithm that generates a random permutation of a set.

var x = [1, 2, 3, 4];
shuffleInPlace(x);
// x is shuffled to a value like [2, 1, 4, 3]

Sampling with replacement is a type of sampling that allows the same item to be picked out of a population more than once.

var values = [1, 2, 3, 4];
sampleWithReplacement(values, 2); // returns 2 random values, like [2, 4];

Create a simple random sample from a given array of n elements.

The sampled values will be in any order, not necessarily the order they appear in the input.

var values = [1, 2, 4, 5, 6, 7, 8, 9];
sample(values, 3); // returns 3 random values, like [2, 5, 8];

Bayesian Classifier

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

var bayes = new BayesianClassifier();
bayes.train({
  species: 'Cat'
}, 'animal');
var result = bayes.score({
  species: 'Cat'
})
// result
// {
//   animal: 1
// }

This is a single-layer Perceptron Classifier that takes arrays of numbers and predicts whether they should be classified as either 0 or 1 (negative or positive examples).

// Create the model
var p = new PerceptronModel();
// Train the model with input with a diagonal boundary.
for (var i = 0; i < 5; i++) {
    p.train([1, 1], 1);
    p.train([0, 1], 0);
    p.train([1, 0], 0);
    p.train([0, 0], 0);
}
p.predict([0, 0]); // 0
p.predict([0, 1]); // 0
p.predict([1, 0]); // 0
p.predict([1, 1]); // 1

The 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.

• Error: if p is outside 0 and 1

bernoulliDistribution(0.3); // => [0.7, 0.3]

The 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.

The 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.

Percentage Points of the χ2 (Chi-Squared) Distribution

The χ2 (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).

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.

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.

tTest([1, 2, 3, 4, 5, 6], 3.385).toFixed(2); // => '0.16'

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, 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 reject a null hypothesis that the two populations that have been sampled into sampleX and sampleY are equal to each other.

tTestTwoSample([1, 2, 3, 4], [3, 4, 5, 6], 0); // => -2.1908902300206643

Cumulative Standard Normal Probability

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.

You can use .5 + .5 * errorFunction(x / Math.sqrt(2)) to calculate the probability instead of looking it up in a table.

Kernel density estimation is a useful tool for, among other things, estimating the shape of the underlying probability distribution from a sample.

Gaussian error function

The errorFunction(x/(sd * Math.sqrt(2))) is the probability that a value in a normal distribution with standard deviation sd is within x of the mean.

This function returns a numerical approximation to the exact value. It uses Horner's method to evaluate the polynomial of τ (tau).

errorFunction(1).toFixed(2); // => '0.84'

The Inverse Gaussian error function returns a numerical approximation to the value that would have caused errorFunction() to return x.

The Probit is the inverse of cumulativeStdNormalProbability(), and is also known as the normal quantile function.

It returns the number of standard deviations from the mean where the p'th quantile of values can be found in a normal distribution. So, for example, probit(0.5 + 0.6827/2) ≈ 1 because 68.27% of values are normally found within 1 standard deviation above or below the mean.

Ckmeans clustering is an improvement on heuristic-based clustering approaches like Jenks. The algorithm was developed in Haizhou Wang and Mingzhou Song as a dynamic programming approach to the problem of clustering numeric data into groups with the least within-group sum-of-squared-deviations.

Minimizing the difference within groups - what Wang & Song refer to as withinss, or within sum-of-squares, means that groups are optimally homogenous within and the data is split into representative groups. This is very useful for visualization, where you may want to represent a continuous variable in discrete color or style groups. This function can provide groups that emphasize differences between data.

Being a dynamic approach, this algorithm is based on two matrices that store incrementally-computed values for squared deviations and backtracking indexes.

This implementation is based on Ckmeans 3.4.6, which introduced a new divide and conquer approach that improved runtime from O(kn^2) to O(kn log(n)).

Unlike the original implementation, this implementation does not include any code to automatically determine the optimal number of clusters: this information needs to be explicitly provided.

References

Ckmeans.1d.dp: Optimal k-means Clustering in One Dimension by Dynamic Programming Haizhou Wang and Mingzhou Song ISSN 2073-4859

from The R Journal Vol. 3/2, December 2011

• Error: if the number of requested clusters is higher than the size of the data

ckmeans([-1, 2, -1, 2, 4, 5, 6, -1, 2, -1], 3);
// The input, clustered into groups of similar numbers.
//= [[-1, -1, -1, -1], [2, 2, 2], [4, 5, 6]]);

Given an array of x, this will find the extent of the x and return an array of breaks that can be used to categorize the x into a number of classes. The returned array will always be 1 longer than the number of classes because it includes the minimum value.

equalIntervalBreaks([1, 2, 3, 4, 5, 6], 4); // => [1, 2.25, 3.5, 4.75, 6]

Split an array into chunks of a specified size. This function has the same behavior as PHP's array_chunk function, and thus will insert smaller-sized chunks at the end if the input size is not divisible by the chunk size.

x is expected to be an array, and chunkSize a number. The x array can contain any kind of data.

• Error: if chunk size is less than 1 or not an integer

chunk([1, 2, 3, 4, 5, 6], 2);
// => [[1, 2], [3, 4], [5, 6]]

The χ2 (Chi-Squared) Goodness-of-Fit 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, chiSquared, 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.

// Data from Poisson goodness-of-fit example 10-19 in William W. Hines & Douglas C. Montgomery,
// "Probability and Statistics in Engineering and Management Science", Wiley (1980).
var data1019 = [
    0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
    0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
    1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
    2, 2, 2, 2, 2, 2, 2, 2, 2,
    3, 3, 3, 3
];
ss.chiSquaredGoodnessOfFit(data1019, ss.poissonDistribution, 0.05); //= false

We use ε, epsilon, as a stopping criterion when we want to iterate until we're "close enough". Epsilon is a very small number: for simple statistics, that number is 0.0001

This is used in calculations like the binomialDistribution, in which the process of finding a value is iterative: it progresses until it is close enough.

Below is an example of using epsilon in gradient descent, where we're trying to find a local minimum of a function's derivative, given by the fDerivative method.

Type: number

// From calculation, we expect that the local minimum occurs at x=9/4
var x_old = 0;
// The algorithm starts at x=6
var x_new = 6;
var stepSize = 0.01;

function fDerivative(x) {
  return 4 * Math.pow(x, 3) - 9 * Math.pow(x, 2);
}

// The loop runs until the difference between the previous
// value and the current value is smaller than epsilon - a rough
// meaure of 'close enough'
while (Math.abs(x_new - x_old) > ss.epsilon) {
  x_old = x_new;
  x_new = x_old - stepSize * fDerivative(x_old);
}

console.log('Local minimum occurs at', x_new);

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.

• Error: if n is less than 0 or not an integer

factorial(5); // => 120

Compute the gamma function of a value using Nemes' approximation. The gamma of n is equivalent to (n-1)!, but unlike the factorial function, gamma is defined for all real n except zero and negative integers (where NaN is returned). Note, the gamma function is also well-defined for complex numbers, though this implementation currently does not handle complex numbers as input values. Nemes' approximation is defined here as Theorem 2.2. Negative values use Euler's reflection formula for computation.

gamma(11.5); // 11899423.084037038
gamma(-11.5); // 2.29575810481609e-8
gamma(5); // 24

For a sorted input, counting the number of unique values is possible in constant time and constant memory. This is a simple implementation of the algorithm.

Values are compared with ===, so objects and non-primitive objects are not handled in any special way.

uniqueCountSorted([1, 2, 3]); // => 3
uniqueCountSorted([1, 1, 1]); // => 1

Approximate equality.

Bisection method is a root-finding method that repeatedly bisects an interval to find the root.

This function returns a numerical approximation to the exact value.

• TypeError: Argument func must be a function

bisect(Math.cos,0,4,100,0.003); // => 1.572265625

Thecoefficient of variation_ is the ratio of the standard deviation to the mean. .._coefficient of variation: https://en.wikipedia.org/wiki/Coefficient_of_variation

coefficientOfVariation([1, 2, 3, 4]).toFixed(3); // => 0.516
coefficientOfVariation([1, 2, 3, 4, 5]).toFixed(3); // => 0.527
coefficientOfVariation([-1, 0, 1, 2, 3, 4]).toFixed(3); // => 1.247

Implementation of Combinations with replacement Combinations are unique subsets of a collection - in this case, k x from a collection at a time. 'With replacement' means that a given element can be chosen multiple times. Unlike permutation, order doesn't matter for combinations.

combinationsReplacement([1, 2], 2); // => [[1, 1], [1, 2], [2, 2]]

Implementation of Combinations Combinations are unique subsets of a collection - in this case, k x from a collection at a time. https://en.wikipedia.org/wiki/Combination

combinations([1, 2, 3], 2); // => [[1,2], [1,3], [2,3]]

When combining two lists of values for which one already knows the means, one does not have to necessary recompute the mean of the combined lists in linear time. They can instead use this function to compute the combined mean by providing the mean & number of values of the first list and the mean & number of values of the second list.

combineMeans(5, 3, 4, 3); // => 4.5

When combining two lists of values for which one already knows the variances, one does not have to necessary recompute the variance of the combined lists in linear time. They can instead use this function to compute the combined variance by providing the variance, mean & number of values of the first list and the variance, mean & number of values of the second list.

combineVariances(14 / 3, 5, 3, 8 / 3, 4, 3); // => 47 / 12

Logistic Cumulative Distribution Function

Calculate Euclidean distance between two points.

The extent is the lowest & highest number in the array. With a sorted array, the first element in the array is always the lowest while the last element is always the largest, so this calculation can be done in one step, or constant time.

extentSorted([-100, -10, 1, 2, 5]); // => [-100, 5]

This computes the minimum & maximum number in an array.

This runs in O(n), linear time, with respect to the length of the array.

• Error: if the length of x is less than one

extent([1, 2, 3, 4]);
// => [1, 4]

Compute the logarithm of the gamma function of a value using Lanczos' approximation. This function takes as input any real-value n greater than 0. This function is useful for values of n too large for the normal gamma function (n > 165). The code is based on Lanczo's Gamma approximation, defined here.

gammaln(500); // 2605.1158503617335
gammaln(2.4); // 0.21685932244884043

The jenks natural breaks optimization is an algorithm commonly used in cartography and visualization to decide upon groupings of data values that minimize variance within themselves and maximize variation between themselves.

For instance, cartographers often use jenks in order to choose which values are assigned to which colors in a choropleth map.

Type: Object

Perform k-means clustering.

• Error: If any centroids wind up friendless (i.e., without associated points).

kMeansCluster([[0.0, 0.5], [1.0, 0.5]], 2); // => {labels: [0, 1], centroids: [[0.0, 0.5], [1.0 0.5]]}

The log average is an equivalent way of computing the geometric mean of an array suitable for large or small products.

It's found by calculating the average logarithm of the elements and exponentiating.

• Error: if x is empty
• Error: if x contains a negative number

The Logit is the inverse of cumulativeStdLogisticProbability, and is also known as the logistic quantile function.

The mean, also known as average, is the sum of all values over the number of values. This is a measure of central tendency: a method of finding a typical or central value of a set of numbers.

The simple mean uses the successive addition method internally to calculate it's result. Errors in floating-point addition are not accounted for, so if precision is required, the standard mean method should be used instead.

This runs in O(n), linear time, with respect to the length of the array.

• Error: if the length of x is less than one

mean([0, 10]); // => 5

Conducts a permutation test to determine if two data sets are significantly different from each other, using the difference of means between the groups as the test statistic. The function allows for the following hypotheses:

• two_tail = Null hypothesis: the two distributions are equal.
• greater = Null hypothesis: observations from sampleX tend to be smaller than those from sampleY.
• less = Null hypothesis: observations from sampleX tend to be greater than those from sampleY. Learn more about one-tail vs two-tail tests.

var control = [2, 5, 3, 6, 7, 2, 5];
var treatment = [20, 5, 13, 12, 7, 2, 2];
permutationTest(control, treatment); // ~0.1324

Implementation of Heap's Algorithm for generating permutations.

This function returns the quantile in which one would find the given value in the given array. With a sorted array, leveraging binary search, we can find this information in logarithmic time.

quantileRankSorted([1, 2, 3, 4], 3); // => 0.75
quantileRankSorted([1, 2, 3, 3, 4], 3); // => 0.7
quantileRankSorted([1, 2, 3, 4], 6); // => 1
quantileRankSorted([1, 2, 3, 3, 5], 4); // => 0.8

This function returns the quantile in which one would find the given value in the given array. It will copy and sort your array before each run, so if you know your array is already sorted, you should use quantileRankSorted instead.

quantileRank([4, 3, 1, 2], 3); // => 0.75
quantileRank([4, 3, 2, 3, 1], 3); // => 0.7
quantileRank([2, 4, 1, 3], 6); // => 1
quantileRank([5, 3, 1, 2, 3], 4); // => 0.8

Rearrange items in arr so that all items in [left, k] range are the smallest. The k-th element will have the (k - left + 1)-th smallest value in [left, right].

Implements Floyd-Rivest selection algorithm https://en.wikipedia.org/wiki/Floyd-Rivest_algorithm

var arr = [65, 28, 59, 33, 21, 56, 22, 95, 50, 12, 90, 53, 28, 77, 39];
quickselect(arr, 8);
// = [39, 28, 28, 33, 21, 12, 22, 50, 53, 56, 59, 65, 90, 77, 95]

Relative error.

This is more difficult to calculate than it first appears [1,2]. The usual formula for the relative error between an actual value A and an expected value E is |(A-E)/E|, but:

1. If the expected value is 0, any other value has infinite relative error, which is counter-intuitive: if the expected voltage is 0, getting 1/10th of a volt doesn't feel like an infinitely large error.

2. This formula does not satisfy the mathematical definition of a metric [3]. [4] solved this problem by defining the relative error as |ln(|A/E|)|, but that formula only works if all values are positive: for example, it reports the relative error of -10 and 10 as 0.

Our implementation sticks with convention and returns:

• 0 if the actual and expected values are both zero
• Infinity if the actual value is non-zero and the expected value is zero
• |(A-E)/E| in all other cases

[1] https://math.stackexchange.com/questions/677852/how-to-calculate-relative-error-when-true-value-is-zero [2] https://en.wikipedia.org/wiki/Relative_change_and_difference [3] https://en.wikipedia.org/wiki/Metric_(mathematics)#Definition [4] F.W.J. Olver: "A New Approach to Error Arithmetic." SIAM Journal on Numerical Analysis, 15(2), 1978, 10.1137/0715024.

Kurtosis is a measure of the heaviness of a distribution's tails relative to its variance. The kurtosis value can be positive or negative, or even undefined.

Implementation is based on Fisher's excess kurtosis definition and uses unbiased moment estimators. This is the version found in Excel and available in several statistical packages, including SAS and SciPy.

• Error: if x has length less than 4

sampleKurtosis([1, 2, 2, 3, 5]); // => 1.4555765595463122

The rank correlation is a measure of the strength of monotonic relationship between two arrays

Calculate the silhouette metric for a set of N-dimensional points arranged in groups. The metric is the largest individual silhouette value for the data.

silhouetteMetric([[0.25], [0.75]], [0, 0]); // => 1.0

Calculate the silhouette values for clustered data.

silhouette([[0.25], [0.75]], [0, 0]); // => [1.0, 1.0]

When removing a value from a list, one does not have to necessary recompute the mean of the list in linear time. They can instead use this function to compute the new mean by providing the current mean, the number of elements in the list that produced it and the value to remove.

subtractFromMean(20.5, 6, 53); // => 14

This function calculates the Wilcoxon rank sum statistic for the first sample with respect to the second. The Wilcoxon rank sum test is a non-parametric alternative to the t-test which is equivalent to the Mann-Whitney U test. The statistic is calculated by pooling all the observations together, ranking them, and then summing the ranks associated with one of the samples. If this rank sum is sufficiently large or small we reject the hypothesis that the two samples come from the same distribution in favor of the alternative that one is shifted with respect to the other.

wilcoxonRankSum([1, 4, 8], [9, 12, 15]); // => 6