Scalar Statistics

var(x, w::AbstractWeights, [dim]; mean=nothing, corrected=false)

Compute the variance of a real-valued array x, optionally over a dimension dim. Observations in x are weighted using weight vector w. The uncorrected (when corrected=false) sample variance is defined as:

where $n$ is the length of the input and $μ$ is the mean. The unbiased estimate (when corrected=true) of the population variance is computed by replacing $\frac{1}{\sum{w}}$ with a factor dependent on the type of weights used:

• AnalyticWeights: $\frac{1}{\sum w - \sum {w^2} / \sum w}$

• FrequencyWeights: $\frac{1}{\sum{w} - 1}$

• ProbabilityWeights: $\frac{n}{(n - 1) \sum w}$ where $n$ equals count(!iszero, w)

• Weights: ArgumentError (bias correction not supported)

std(v, w::AbstractWeights, [dim]; mean=nothing, corrected=false)

Compute the standard deviation of a real-valued array x, optionally over a dimension dim. Observations in x are weighted using weight vector w. The uncorrected (when corrected=false) sample standard deviation is defined as:

where $n$ is the length of the input and $μ$ is the mean. The unbiased estimate (when corrected=true) of the population standard deviation is computed by replacing $\frac{1}{\sum{w}}$ with a factor dependent on the type of weights used:

• AnalyticWeights: $\frac{1}{\sum w - \sum {w^2} / \sum w}$

• FrequencyWeights: $\frac{1}{\sum{w} - 1}$

• ProbabilityWeights: $\frac{n}{(n - 1) \sum w}$ where $n$ equals count(!iszero, w)

• Weights: ArgumentError (bias correction not supported)

mean_and_var(x, [w::AbstractWeights], [dim]; corrected=false) -> (mean, var)

Return the mean and variance of a real-valued array x, optionally over a dimension dim, as a tuple. Observations in x can be weighted using weight vector w. Finally, bias correction is be applied to the variance calculation if corrected=true. See var documentation for more details.

mean_and_std(x, [w::AbstractWeights], [dim]; corrected=false) -> (mean, std)

Return the mean and standard deviation of a real-valued array x, optionally over a dimension dim, as a tuple. A weighting vector w can be specified to weight the estimates. Finally, bias correction is applied to the standard deviation calculation if corrected=true. See std documentation for more details.

skewness(v, [wv::AbstractWeights], m=mean(v))

Compute the standardized skewness of a real-valued array v, optionally specifying a weighting vector wv and a center m.

kurtosis(v, [wv::AbstractWeights], m=mean(v))

Compute the excess kurtosis of a real-valued array v, optionally specifying a weighting vector wv and a center m.

moment(v, k, [wv::AbstractWeights], m=mean(v))

Return the kth order central moment of a real-valued array v, optionally specifying a weighting vector wv and a center m.

span(x)

Return the span of an integer array, i.e. the range minimum(x):maximum(x). The minimum and maximum of x are computed in one-pass using extrema.

variation(x, m=mean(x))

Return the coefficient of variation of an array x, optionally specifying a precomputed mean m. The coefficient of variation is the ratio of the standard deviation to the mean.

sem(a)

Return the standard error of the mean of a, i.e. sqrt(var(a) / length(a)).

mad(v; center=median(v), normalize=true)

Compute the median absolute deviation (MAD) of v around center (by default, around the median).

If normalize is set to true, the MAD is multiplied by 1 / quantile(Normal(), 3/4) ≈ 1.4826, in order to obtain a consistent estimator of the standard deviation under the assumption that the data is normally distributed.

zscore(X, [μ, σ])

Compute the z-scores of X, optionally specifying a precomputed mean μ and standard deviation σ. z-scores are the signed number of standard deviations above the mean that an observation lies, i.e. $(x - μ) / σ$.

μ and σ should be both scalars or both arrays. The computation is broadcasting. In particular, when μ and σ are arrays, they should have the same size, and size(μ, i) == 1 || size(μ, i) == size(X, i) for each dimension.

zscore!([Z], X, μ, σ)

Compute the z-scores of an array X with mean μ and standard deviation σ. z-scores are the signed number of standard deviations above the mean that an observation lies, i.e. $(x - μ) / σ$.

If a destination array Z is provided, the scores are stored in Z and it must have the same shape as X. Otherwise X is overwritten.

entropy(p, [b])

Compute the entropy of an array p, optionally specifying a real number b such that the entropy is scaled by 1/log(b).

renyientropy(p, α)

Compute the Rényi (generalized) entropy of order α of an array p.

crossentropy(p, q, [b])

Compute the cross entropy between p and q, optionally specifying a real number b such that the result is scaled by 1/log(b).

kldivergence(p, q, [b])

Compute the Kullback-Leibler divergence of q from p, optionally specifying a real number b such that the divergence is scaled by 1/log(b).

mode(a, [r])

Return the mode (most common number) of an array, optionally over a specified range r. If several modes exist, the first one (in order of appearance) is returned.

modes(a, [r])::Vector

Return all modes (most common numbers) of an array, optionally over a specified range r.

summarystats(a)

Compute summary statistics for a real-valued array a. Returns a SummaryStats object containing the mean, minimum, 25th percentile, median, 75th percentile, and maxmimum.

describe(a)

Pretty-print the summary statistics provided by summarystats: the mean, minimum, 25th percentile, median, 75th percentile, and maximum.