7.6.0
The min is the lowest number in the array.
This runs in O(n)
, linear time, with respect to the length of the array.
number
:
minimum value
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.
number
:
maximum value
max([1, 2, 3, 4]);
// => 4
This computes the minimum & maximum number in an array.
This runs in O(n)
, linear time, with respect to the length of the array.
Array<number>
:
minimum & maximum value
extent([1, 2, 3, 4]);
// => [1, 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.
number
:
sum of all input numbers
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.
number
:
sum of all input numbers
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
number
:
quantile
quantile([3, 6, 7, 8, 8, 9, 10, 13, 15, 16, 20], 0.5); // => 9
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.
number
:
value value
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
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.
number
:
product of all input numbers
product([1, 2, 3, 4]); // => 24
These are special versions of methods that assume your input is sorted. This assumptions lets them run a lot faster, usually in O(1).
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.
number
:
minimum value
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.
number
:
maximum value
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.
(number)
desired quantile: a number between 0 to 1, inclusive
number
:
quantile value
quantileSorted([3, 6, 7, 8, 8, 9, 10, 13, 15, 16, 20], 0.5); // => 9
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.
number
:
value value
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
These are different ways to identifying centers or locations of a distribution.
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.
number
:
mean
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.
number
:
the new mean
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.
number
:
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.
number
:
mode
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.
(Array<any>)
a sample of one or more data points
any?
:
mode
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.
number
:
median value
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.
number
:
median value
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.
number
:
harmonic mean
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.
number
:
geometric mean
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.
number
:
root mean square
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.
number
:
sample skewness
sampleSkewness([2, 4, 6, 3, 1]); // => 0.590128656384365
These are different ways of determining how spread out a distribution is.
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.
number
:
variance: a value greater than or equal to zero.
zero indicates that all values are identical.
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:
number
:
sample variance
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.
number
:
standard deviation
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.
number
:
sample standard deviation
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.
number
:
median absolute 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.
number
:
interquartile range: the span between lower and upper quartile,
0.25 and 0.75
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.
number
:
sum of nth power 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.
number
:
z score
zScore(78, 80, 5); // => -0.4
The correlation is a measure of how correlated two datasets are, between -1 and 1
number
:
sample correlation
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.
number
:
sample covariance
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.
number
:
r-squared 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.
Object
:
object containing slope and intersect of regression line
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.
Function
:
method that computes y-value at any given
x-value on the line.
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.
(Array)
sample of 0 or more numbers
(Function
= Math.random
)
an optional entropy source that
returns numbers between 0 inclusive and 1 exclusive: the range [0, 1)
Array
:
shuffled version of 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.
(Array)
sample of one or more numbers
(Function
= Math.random
)
an optional entropy source that
returns numbers between 0 inclusive and 1 exclusive: the range [0, 1)
Array
:
x
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.
(Array<any>)
an array of any kind of value
(number)
count of how many elements to take
(Function
= Math.random
)
an optional entropy source that
returns numbers between 0 inclusive and 1 exclusive: the range [0, 1)
Array
:
n sampled items from the population
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.
(Array<any>)
input array. can contain any type
(number)
count of how many elements to take
(Function
= Math.random
)
an optional entropy source that
returns numbers between 0 inclusive and 1 exclusive: the range [0, 1)
Array
:
subset of n elements in original array
var values = [1, 2, 4, 5, 6, 7, 8, 9];
sample(values, 3); // returns 3 random values, like [2, 5, 8];
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
Train the classifier with a new example, which is a numeric array of features and a 0 or 1 label.
PerceptronModel
:
this
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.
(number)
input value, between 0 and 1 inclusive
Array<number>
:
values of bernoulli distribution at this point
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.
Array<number>
:
output
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
.
(number)
location poisson distribution
Array<number>
:
values of poisson distribution at that point
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.
(number)
expected value of the population mean
number
:
value
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.
(number
= 0
)
(number | null)
:
test result
tTestTwoSample([1, 2, 3, 4], [3, 4, 5, 6], 0); // => -2.1908902300206643
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:
(string)
alternative hypothesis, either 'two_sided' (default), 'greater', or 'less'
(number)
number of values in permutation distribution.
(Function
= Math.random
)
an optional entropy source
number
:
p-value The probability of observing the difference between groups (as or more extreme than what we did), assuming the null hypothesis.
var control = [2, 5, 3, 6, 7, 2, 5];
var treatment = [20, 5, 13, 12, 7, 2, 2];
permutationTest(control, treatment); // ~0.1324
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.
(number)
number
:
cumulative standard normal probability
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).
(number)
input
number
:
error estimation
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.
(number)
value of error function
number
:
estimated inverted value
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.
(number)
number
:
probit
Breaks methods split datasets into chunks. Often these are used for segmentation or visualization of a dataset. A method of computing breaks that splits data evenly can make for a better choropleth map, for instance, because each color will be represented equally.
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.
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
(number)
number of desired classes. This cannot be
greater than the number of values in the data array.
Array<Array<number>>
:
clustered input
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.
Array<number>
:
array of class break positions
equalIntervalBreaks([1, 2, 3, 4, 5, 6], 4); // => [1, 2.25, 3.5, 4.75, 6]
Perform k-means clustering.
(number)
How many clusters to create.
(Function
= Math.random
)
An optional entropy source that generates uniform values in [0, 1).
kMeansReturn
:
Labels (same length as data) and centroids (same length as numCluster).
kMeansCluster([[0.0, 0.5], [1.0, 0.5]], 2); // => {labels: [0, 1], centroids: [[0.0, 0.5], [1.0 0.5]]}
Calculate the silhouette values for clustered data.
Array<number>
:
The silhouette value for each point.
silhouette([[0.25], [0.75]], [0, 0]); // => [1.0, 1.0]
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.
number
:
The silhouette metric for the groupings.
silhouetteMetric([[0.25], [0.75]], [0, 0]); // => 1.0
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.
Array<Array>
:
a chunked array
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.
(Function)
a function that returns a point in a distribution:
for instance, binomial, bernoulli, or poisson
(number)
number
:
chi squared goodness of fit
// 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.
(number)
input, must be an integer number 1 or greater
number
:
factorial: n!
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.
(number)
Any real number except for zero and negative integers.
number
:
The gamma of the input value.
gamma(11.5); // 11899423.084037038
gamma(-11.5); // 2.29575810481609e-8
gamma(5); // 24
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.
(number)
Any real number greater than zero.
number
:
The logarithm of gamma of the input value.
gammaln(500); // 2605.1158503617335
gammaln(2.4); // 0.21685932244884043
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.
(Array<any>)
an array of any kind of value
number
:
count of unique values
uniqueCountSorted([1, 2, 3]); // => 3
uniqueCountSorted([1, 1, 1]); // => 1
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.
Array<number>
:
minimum & maximum value
extentSorted([-100, -10, 1, 2, 5]); // => [-100, 5]
The rank correlation is a measure of the strength of monotonic relationship between two arrays
number
:
sample rank correlation
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.
number
:
sample kurtosis
sampleKurtosis([1, 2, 2, 3, 5]); // => 1.4555765595463122
Implementation of Heap's Algorithm for generating permutations.
(Array)
any type of data
Array<Array>
:
array of permutations
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
(Array)
any type of data
(int)
the number of objects in each group (without replacement)
Array<Array>
:
array of permutations
combinations([1, 2, 3], 2); // => [[1,2], [1,3], [2,3]]
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.
(Array)
any type of data
(int)
the number of objects in each group (without replacement)
Array<Array>
:
array of permutations
combinationsReplacement([1, 2], 2); // => [[1, 1], [1, 2], [2, 2]]
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.
(number)
mean of the first list
(number)
number of items in the first list
(number)
mean of the second list
(number)
number of items in the second list
number
:
the combined mean
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.
(number)
variance of the first list
(number)
mean of the first list
(number)
number of items in the first list
(number)
variance of the second list
(number)
mean of the second list
(number)
number of items in the second list
number
:
the combined mean
combineVariances(14 / 3, 5, 3, 8 / 3, 4, 3); // => 47 / 12
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.
number
:
geometric mean
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.
number
:
mean
mean([0, 10]); // => 5
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.
(number)
current mean
(number)
number of items in the list
(number)
the value to remove
number
:
the new mean
subtractFromMean(20.5, 6, 53); // => 14
Thecoefficient of variation
is the ratio of the standard deviation to the mean.
..coefficient of variation
: https://en.wikipedia.org/wiki/Coefficient_of_variation
(Array)
input
number
:
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
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.
number
:
rank sum for sampleX
wilcoxonRankSum([1, 4, 8], [9, 12, 15]); // => 6
Kernel density estimation is a useful tool for, among other things, estimating the shape of the underlying probability distribution from a sample.
(any)
sample values
(any)
The kernel function to use. If a function is provided, it should return non-negative values and integrate to 1. Defaults to 'gaussian'.
(any)
The "bandwidth selection" method to use, or a fixed bandwidth value. Defaults to "nrd", the commonly-used
"normal reference distribution" rule-of-thumb
.
Function
:
An estimated
probability density function
for the given sample. The returned function runs in
O(X.length)
.
Logistic Cumulative Distribution Function
(number)
number
:
cumulative standard logistic probability
The Logit is the inverse of cumulativeStdLogisticProbability, and is also known as the logistic quantile function.
(number)
number
:
logit
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.
(Function)
input function
(number)
start of interval
(number)
end of interval
(number)
the maximum number of iterations
(number)
the error tolerance
number
:
estimated root value
bisect(Math.cos,0,4,100,0.003); // => 1.572265625
Type: Object
Calculate Euclidean distance between two points.
number
:
Distance.
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
void
:
mutates input array
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:
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.
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:
|(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.
number
:
The relative error.
Approximate equality.
(number)
The value to be tested.
(number)
The reference value.
(number
= epsilon
)
The acceptable relative difference.
boolean
:
Whether numbers are within tolerance.