Scalar Statistics

var(x::AbstractArray, 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(x::AbstractArray, 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 standard deviation of collection x. If x is an AbstractArray, dim can be specified as a tuple to compute statistics over these dimensions. A weighting vector w can be specified to weight the estimates. 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 collection x. If x is an AbstractArray, dim can be specified as a tuple to compute statistics over these dimensions. 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 a collection, 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 collection x, optionally specifying a precomputed mean m. The coefficient of variation is the ratio of the standard deviation to the mean.

sem(x)

Return the standard error of the mean of collection x, i.e. sqrt(var(x, corrected=true) / length(x)).

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

Compute the median absolute deviation (MAD) of collection x 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 a collection of probabilities p, optionally specifying a real number b such that the entropy is scaled by 1/log(b). Elements with probability 0 or 1 add 0 to the entropy.

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 from q to p, also called the relative entropy of p with respect to q, that is the sum pᵢ * log(pᵢ / qᵢ). Optionally a real number b can be specified such that the divergence is scaled by 1/log(b).

percentile(x, p)

Return the pth percentile of a collection x, i.e. quantile(x, p / 100).

iqr(x)

Compute the interquartile range (IQR) of collection x, i.e. the 75th percentile minus the 25th percentile.

nquantile(x, n::Integer)

Return the n-quantiles of collection x, i.e. the values which partition v into n subsets of nearly equal size.

Equivalent to quantile(x, [0:n]/n). For example, nquantiles(x, 5) returns a vector of quantiles, respectively at [0.0, 0.2, 0.4, 0.6, 0.8, 1.0].

quantile(v, w::AbstractWeights, p)

Compute the weighted quantiles of a vector v at a specified set of probability values p, using weights given by a weight vector w (of type AbstractWeights). Weights must not be negative. The weights and data vectors must have the same length. NaN is returned if x contains any NaN values. An error is raised if w contains any NaN values.

With FrequencyWeights, the function returns the same result as quantile for a vector with repeated values. Weights must be integers.

With non FrequencyWeights, denote $N$ the length of the vector, $w$ the vector of weights, $h = p (\sum_{i<= N} w_i - w_1) + w_1$ the cumulative weight corresponding to the probability $p$ and $S_k = \sum_{i<=k} w_i$ the cumulative weight for each observation, define $v_{k+1}$ the smallest element of v such that $S_{k+1}$ is strictly superior to $h$. The weighted $p$ quantile is given by $v_k + \gamma (v_{k+1} - v_k)$ with $\gamma = (h - S_k)/(S_{k+1} - S_k)$. In particular, when all weights are equal, the function returns the same result as the unweighted quantile.

median(v::RealVector, w::AbstractWeights)

Compute the weighted median of v with weights w (of type AbstractWeights). See the documentation for quantile for more details.

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.