DescriptiveStatistics.md

[hide]
#I "../../out/lib/net40"
#r "MathNet.Numerics.dll"
#r "MathNet.Numerics.FSharp.dll"
open System.Numerics
open MathNet.Numerics
open MathNet.Numerics.Statistics

Descriptive Statistics

Initialization

We need to reference Math.NET Numerics and open the statistics namespace:

[lang=csharp]
using MathNet.Numerics.Statistics;

Univariate Statistical Analysis

The primary class for statistical analysis is Statistics which provides common descriptive statics as static extension methods to IEnumerable<double> sequences. However, various statistics can be computed much more efficiently if the data source has known properties or structure, that's why the following classes provide specialized static implementations:

• ArrayStatistics provides routines optimized for single-dimensional arrays. Some of these routines end with the Inplace suffix, indicating that they reorder the input array slightly towards being sorted during execution - without fully sorting them, which could be expensive.
• SortedArrayStatistics provides routines optimized for an array sorting ascendingly. Especially order-statistics are very efficient this way, some even with constant time complexity.
• StreamingStatistics processes large amounts of data without keeping them in memory. Useful if data larger than local memory is streamed directly from a disk or network.

Another alternative, in case you need to gather a whole set of statistical characteristics in one pass, is provided by the DescriptiveStatistics class:

[lang=csharp]
var samples = new ChiSquare(5).Samples().Take(1000);
var statistics = new DescriptiveStatistics(samples);

var largestElement = statistics.Maximum;
var smallestElement = statistics.Minimum;
var median = statistics.Median;

var mean = statistics.Mean;
var variance = statistics.Variance;
var stdDev = statistics.StandardDeviation;

var kurtosis = statistics.Kurtosis;
var skewness = statistics.Skewness;

Minimum & Maximum

The minimum and maximum values of a sample set can be evaluated with the Minimum and Maximum functions of all four classes: Statistics, ArrayStatistics, SortedArrayStatistics and StreamingStatistics. The one in SortedArrayStatistics is the fastest with constant time complexity, but expects the array to be sorted ascendingly.

Both min and max are directly affected by outliers and are therefore no robust statistics at all. For a more robust alternative, consider using Quantiles instead.

[lang=csharp]
var samples = new ChiSquare(5).Samples().Take(1000).ToArray();
var largestElement = samples.Maximum();
var smallestElement = samples.Minimum();

Mean

The arithmetic mean or average of the provided samples. In statistics, the sample mean is a measure of the central tendency and estimates the expected value of the distribution. The mean is affected by outliers, so if you need a more robust estimate consider to use the Median instead.

Statistics.Mean(data) StreamingStatistics.Mean(stream) ArrayStatistics.Mean(data)

$$$ \overline{x} = \frac{1}{N}\sum_{i=1}^N x_i

[lang=fsharp]
let whiteNoise = Generate.Normal(1000, mean=10.0, standardDeviation=2.0)
// [fsi:val samples : float [] = [|12.90021939; 9.631515037; 7.810008046; 14.13301053; ...|] ]
Statistics.Mean whiteNoise
// [fsi:val it : float = 10.02162347]

let wave = Generate.Sinusoidal(1000, samplingRate=100., frequency=5., amplitude=0.5)
Statistics.Mean wave
// [fsi:val it : float = -4.133520783e-17]

Variance and Standard Deviation

Variance ${\sigma }^2$ and the Standard Deviation $\sigma$ are measures of how far the samples are spread out.

If the whole population is available, the functions with the Population-prefix will evaluate the respective measures with an $N$ normalizer for a population of size $N$.

Statistics.PopulationVariance(population) Statistics.PopulationStandardDeviation(population)

$$$ \sigma^2 = \frac{1}{N}\sum_{i=1}^N (x_i - \mu)^2

On the other hand, if only a sample of the full population is available, the functions without the Population-prefix will estimate unbiased population measures by applying Bessel's correction with an $N-1$ normalizer to a sample set of size $N$.

Statistics.Variance(samples) Statistics.StandardDeviation(samples)

$$$ s^2 = \frac{1}{N-1}\sum_{i=1}^N (x_i - \overline{x})^2

[lang=fsharp]
Statistics.Variance whiteNoise
// [fsi:val it : float = 3.819436094]
Statistics.StandardDeviation whiteNoise
// [fsi:val it : float = 1.954337764]

Statistics.Variance wave
// [fsi:val it : float = 0.1251251251]

Combined Routines

Since mean and variance are often needed together, there are routines that evaluate both in a single pass:

Statistics.MeanVariance(samples) ArrayStatistics.MeanVariance(samples) StreamingStatistics.MeanVariance(samples)

[lang=fsharp]
Statistics.MeanVariance whiteNoise
// [fsi:val it : float * float = (10.02162347, 3.819436094)]

Covariance

The sample covariance is an estimation of the Covariance, a measure of how much two random variables change together. Similarly to the variance above, there are two versions in order to apply Bessel's correction to bias in case of sample data.

Statistics.Covariance(samples1, samples2)

$$$ q = \frac{1}{N-1}\sum_{i=1}^N (x_i - \overline{x})(y_i - \overline{y})

Statistics.PopulationCovariance(population1, population2)

$$$ q = \frac{1}{N}\sum_{i=1}^N (x_i - \mu_x)(y_i - \mu_y)

[lang=fsharp]
Statistics.Covariance(whiteNoise, whiteNoise)
// [fsi:val it : float = 3.819436094]
Statistics.Covariance(whiteNoise, wave)
// [fsi:val it : float = 0.04397985084]

Order Statistics

Order Statistic

The k-th order statistic of a sample set is the k-th smallest value. Note that, as an exception to most of Math.NET Numerics, the order k is one-based, meaning the smallest value is the order statistic of order 1 (there is no order 0).

Statistics.OrderStatistic(data, order) SortedArrayStatistics.OrderStatistic(data, order)

If the samples are sorted ascendingly, this is trivial and can be evaluated in constant time, which is what the SortedArrayStatistics implementation does.

If you have the samples in an array which is not (guaranteed to be) sorted, but if it is fine if the array does incrementally get sorted over multiple calls, you can also use the following in-place implementation. It is usually faster than fully sorting the array, unless you need to compute it for more than a handful orders.

ArrayStatistics.OrderStatisticInplace(data, order)

For convenience there's also an option that returns a function Func<int, double>, mapping from order to the resulting order statistic. Internally it sorts a copy of the provided data and then on each invocation uses efficient sorted algorithms:

Statistics.OrderStatisticFunc(data)

Such Inplace and Func variants are a common pattern throughout the Statistics class and also the rest of the library.

[lang=fsharp]
Statistics.OrderStatistic(whiteNoise, 1)
// [fsi:val it : float = 3.633070184]
Statistics.OrderStatistic(whiteNoise, 1000)
// [fsi:val it : float = 16.65183566]

let os = Statistics.orderStatisticFunc whiteNoise
os 250
// [fsi:val it : float = 8.645491746]
os 500
// [fsi:val it : float = 10.11872428]
os 750
// [fsi:val it : float = 11.33170746]

Median

Median is a robust indicator of central tendency and much less affected by outliers than the sample mean. The median is estimated by the value exactly in the middle of the sorted set of samples and thus separating the higher half of the data from the lower half.

Statistics.Median(data) SortedArrayStatistics.Median(data) ArrayStatistics.MedianInplace(data)

The median is only unique if the sample size is odd. This implementation internally uses the default quantile definition, which is equivalent to mode 8 in R and is approximately median-unbiased regardless of the sample distribution. If you need another convention, use QuantileCustom instead, see below for details.

[lang=fsharp]
Statistics.Median whiteNoise
// [fsi:val it : float = 10.11872428]
Statistics.Median wave
// [fsi:val it : float = -2.452600839e-16]

Quartiles and the 5-number summary

Quartiles group the ascendingly sorted data into four equal groups, where each group represents a quarter of the data. The lower quartile is estimated by the middle number between the first two groups and the upper quartile by the middle number between the remaining two groups. The middle number between the two middle groups estimates the median as discussed above.

Statistics.LowerQuartile(data) Statistics.UpperQuartile(data) SortedArrayStatistics.LowerQuartile(data) SortedArrayStatistics.UpperQuartile(data) ArrayStatistics.LowerQuartileInplace(data) ArrayStatistics.UpperQuartileInplace(data)

[lang=fsharp]
Statistics.LowerQuartile whiteNoise
// [fsi:val it : float = 8.645491746]
Statistics.UpperQuartile whiteNoise
// [fsi:val it : float = 11.33213732]

Using that data we can provide a useful set of indicators usually named 5-number summary, which consists of the minimum value, the lower quartile, the median, the upper quartile and the maximum value. All these values can be visualized in the popular box plot diagrams.

Statistics.FiveNumberSummary(data) SortedArrayStatistics.FiveNumberSummary(data) ArrayStatistics.FiveNumberSummaryInplace(data)

[lang=fsharp]
Statistics.FiveNumberSummary whiteNoise
// [fsi:val it : float [] = [|3.633070184; 8.645937823; 10.12165054; 11.33213732; 16.65183566|] ]
Statistics.FiveNumberSummary wave
// [fsi:val it : float [] = [|-0.5; -0.3584185509; -2.452600839e-16; 0.3584185509; 0.5|] ]

The difference between the upper and the lower quartile is called inter-quartile range (IQR) and is a robust indicator of spread. In box plots the IQR is the total height of the box.

Statistics.InterquartileRange(data) SortedArrayStatistics.InterquartileRange(data) ArrayStatistics.InterquartileRangeInplace(data)

Just like median, quartiles use the default R8 quantile definition internally.

[lang=fsharp]
Statistics.InterquartileRange whiteNoise
// [fsi:val it : float = 2.686199498]

Percentiles

Percentiles extend the concept further by grouping the sorted values into 100 equal groups and looking at the 101 places (0,1,..,100) between and around them. The 0-percentile represents the minimum value, 25 the first quartile, 50 the median, 75 the upper quartile and 100 the maximum value.

Statistics.Percentile(data, p) Statistics.PercentileFunc(data) SortedArrayStatistics.Percentile(data, p) ArrayStatistics.PercentileInplace(data, p)

Just like median, percentiles use the default R8 quantile definition internally.

[lang=fsharp]
Statistics.Percentile(whiteNoise, 5)
// [fsi:val it : float = 6.693373507]
Statistics.Percentile(whiteNoise, 98)
// [fsi:val it : float = 13.97580653]

Quantiles

Instead of grouping into 4 or 100 boxes, quantiles generalize the concept to an infinite number of boxes and thus to arbitrary real numbers $\tau$ between 0.0 and 1.0, where 0.0 represents the minimum value, 0.5 the median and 1.0 the maximum value. Quantiles are closely related to the inverse cumulative distribution function of the sample distribution.

Statistics.Quantile(data, tau) Statistics.QuantileFunc(data) SortedArrayStatistics.Quantile(data, tau) ArrayStatistics.QuantileInplace(data, tau)

[lang=fsharp]
Statistics.Quantile(whiteNoise, 0.98)
// [fsi:val it : float = 13.97580653]

Quantile Conventions and Compatibility

Remember that all these descriptive statistics do not compute but merely estimate statistical indicators of the value distribution. In the case of quantiles, there is usually not a single number between the two groups specified by $\tau$. There are multiple ways to deal with this: the R project supports 9 modes and Mathematica and SciPy have their own way to parametrize the behavior.

The QuantileCustom functions support all 9 modes from the R-project, which includes the one used by Microsoft Excel, and also the 4-parameter variant of Mathematica:

Statistics.QuantileCustom(data, tau, definition) Statistics.QuantileCustomFunc(data, definition) SortedArrayStatistics.QuantileCustom(data, tau, a, b, c, d) SortedArrayStatistics.QuantileCustom(data, tau, definition) ArrayStatistics.QuantileCustomInplace(data, tau, a, b, c, d) ArrayStatistics.QuantileCustomInplace(data, tau, definition)

The QuantileDefinition enumeration has the following options:

• R1, SAS3, EmpiricalInvCDF

• R2, SAS5, EmpiricalInvCDFAverage

• R3, SAS2, Nearest

• R4, SAS1, California

• R5, Hydrology, Hazen

• R6, SAS4, Nist, Weibull, SPSS

• R7, Excel, Mode, S

• R8, Median, Default

• R9, Normal

  [lang=fsharp] Statistics.QuantileCustom(whiteNoise, 0.98, QuantileDefinition.R3) // [fsi:val it : float = 13.97113209] Statistics.QuantileCustom(whiteNoise, 0.98, QuantileDefinition.Excel) // [fsi:val it : float = 13.97127374]

Rank Statistics

Ranks

Rank statistics are the counterpart to order statistics. The Ranks function evaluates the rank of each sample and returns them as an array of doubles. The return type is double instead of int in order to deal with ties, if one of the values appears multiple times. Similar to QuantileDefinition, the RankDefinition enumeration controls how ties should be handled:

• Average, Default: Replace ties with their mean (causing non-integer ranks).
• Min, Sports: Replace ties with their minimum, as typical in sports ranking.
• Max: Replace ties with their maximum.
• First: Permutation with increasing values at each index of ties.
• EmpiricalCDF

Statistics.Ranks(data, definition) SortedArrayStatistics.Ranks(data, definition) ArrayStatistics.RanksInplace(data, definition)

[lang=fsharp]
Statistics.Ranks(whiteNoise)
// [fsi:val it : float [] = [|634.0; 736.0; 405.0; 395.0; 197.0; 167.0; 722.0; 44.0; ...|] ]
Statistics.Ranks([| 13.0; 14.0; 11.0; 12.0; 13.0 |], RankDefinition.Average)
// [fsi:val it : float [] = [|3.5; 5.0; 1.0; 2.0; 3.5|] ]
Statistics.Ranks([| 13.0; 14.0; 11.0; 12.0; 13.0 |], RankDefinition.Sports)
// [fsi:val it : float [] = [|3.0; 5.0; 1.0; 2.0; 3.0|] ]

Quantile Rank

Counterpart of the Quantile function, estimates $\tau$ of the provided $\tau$-quantile value $x$ from the provided samples. The $\tau$-quantile is the data value where the cumulative distribution function crosses $\tau$.

Statistics.QuantileRank(data, x, definition) Statistics.QuantileRankFunc(data, definition) SortedArrayStatistics.QuantileRank(data, x, definition)

[lang=fsharp]
Statistics.QuantileRank(whiteNoise, 13.0)
// [fsi:val it : float = 0.9370045563]
Statistics.QuantileRank(whiteNoise, 6.7, RankDefinition.Average)
// [fsi:val it : float = 0.04960610389]

Empirical Distribution Functions

Statistics.EmpiricalCDF(data, x) Statistics.EmpiricalCDFFunc(data) Statistics.EmpiricalInvCDF(data, tau) Statistics.EmpiricalInvCDFFunc(data) SortedArrayStatistics.EmpiricalCDF(data, x)

[lang=fsharp]
let ecdf = Statistics.EmpiricalCDFFunc whiteNoise
Generate.LinearSpacedMap(20, start=3.0, stop=17.0, map=ecdf)
// [fsi:val it : float [] =]
// [fsi:    [|0.0; 0.001; 0.002; 0.005; 0.022; 0.05; 0.094; 0.172; 0.278; 0.423; 0.555; ]
// [fsi:      0.705; 0.843; 0.921; 0.944; 0.983; 0.992; 0.997; 0.999; 1.0|] ]

let eicdf = Statistics.empiricalInvCDFFunc whiteNoise
[ for tau in 0.0..0.05..1.0 -> eicdf tau ]
// [fsi:val it : float [] =]
// [fsi:    [3.633070184; 6.682142043; 7.520000817; 8.040513497; 8.347587493; ]
// [fsi:     8.645491746; 9.02681611; 9.298987151; 9.522627142; 9.819352699; 10.11872428; ]
// [fsi:     10.35991046; 10.57530906; 10.8259542; 11.08605473; 11.33170746; 11.54356436; ]
// [fsi:     11.90973541; 12.4294346; 13.36889423; 16.65183566] ]

Histograms

A histogram can be computed using the Histogram class. Its constructor takes the samples enumerable, the number of buckets to create, plus optionally the range (minimum, maximum) of the sample data if available.

[lang=csharp]
var histogram = new Histogram(samples, 10);
var bucket3count = histogram[2].Count;

Correlation

The Correlation class supports computing Pearson's product-momentum and Spearman's ranked correlation coefficient, as well as their correlation matrix for a set of vectors.

Code Sample: Computing the correlation coefficient of 1000 samples of f(x) = 2x and g(x) = x^2:

[lang=csharp]
double[] dataF = Generate.LinearSpacedMap(1000, 0, 100, x => 2*x);
double[] dataG = Generate.LinearSpacedMap(1000, 0, 100, x => x*x);
double correlation = Correlation.Pearson(dataF, dataG);