cov(x::UVAL_COLLECTION_TYPES, y::UVAL_COLLECTION_TYPES, n::Int; corrected::Bool = true)

Obtain a distribution on the covariance between two collections of uncertain values.

This is done by repeating the following procedure n times:

1. First, draw a length-L realisation of x by drawing one random number from each uncertain value furnishing the dataset. The draws are independent, so that no element-wise dependencies (e.g. sequential correlations) that are not already present in the data are introduced in the realisation.
2. Second, draw a length-L realisation of y in the same manner.
3. Compute the covariance between the two length-L draws.

This yields n estimates of the covariance between n independent pairs of realisations of x and y. The n-member distribution of covariance estimates is returned as a vector.

If corrected is true (the default) then the sum is scaled with n - 1 for each pair of draws, whereas the sum is scaled with n if corrected is false where n = length(x).

cor(x::UVAL_COLLECTION_TYPES, y::UVAL_COLLECTION_TYPES, n::Int)

Estimate a distribution on Pearson's rank correlation coefficient between two collections of uncertain values.

This is done by repeating the following procedure n times:

1. First, draw a length-L realisation of x by drawing one random number from each uncertain value furnishing the dataset. The draws are independent, so that no element-wise dependencies (e.g. sequential correlations) that are not already present in the data are introduced in the realisation.
2. Second, draw a length-L realisation of y in the same manner.
3. Compute Pearson's rank correlation coefficient between the two length-L draws.

This yields n estimates of Pearson's rank correlation coefficient between n independent pairs of realisations of x and y. The n-member distribution of correlation estimates is returned as a vector.

corkendall(x::UVAL_COLLECTION_TYPES, y::UVAL_COLLECTION_TYPES, n::Int)

Estimate a n-member distribution on Kendalls's rank correlation coefficient between two collections of uncertain values.

This is done by repeating the following procedure n times:

1. First, draw a length-L realisation of x by drawing one random number from each uncertain value furnishing the dataset. The draws are independent, so that no element-wise dependencies (e.g. sequential correlations) that are not already present in the data are introduced in the realisation.
2. Second, draw a length-L realisation of y in the same manner.
3. Compute Kendall's rank correlation coefficient between the two length-L draws.

This yields n computations of Kendall's rank correlation coefficient between n independent pairs of realisations of x and y. The n-member distribution of correlation estimates is returned as a vector.

corspearman(x::UVAL_COLLECTION_TYPES, y::UVAL_COLLECTION_TYPES, n::Int)

Estimate a n-member distribution on Spearman's rank correlation coefficient between two collections of uncertain values.

This is done by repeating the following procedure n times:

1. First, draw a length-L realisation of x by drawing one random number from each uncertain value furnishing the dataset. The draws are independent, so that no element-wise dependencies (e.g. sequential correlations) that are not already present in the data are introduced in the realisation.
2. Second, draw a length-L realisation of y in the same manner.
3. Compute Spearman's rank correlation coefficient between the two length-L draws.

This yields n estimates of Spearman's rank correlation coefficient between n independent pairs of realisations of x and y. The n-member distribution of correlation estimates is returned as a vector.

countne(x::UVAL_COLLECTION_TYPES, y::UVAL_COLLECTION_TYPES, n::Int)

Estimate a n-member distribution on the number of indices at which the elements of two collections of uncertain values are not equal.

This is done by repeating the following procedure n times:

1. Draw a length-L realisation of x by drawing one random number from each uncertain value furnishing the dataset. The draws are independent, so that no element-wise dependencies (e.g. sequential correlations) that are not already present in the data are introduced in the realisation.
2. Draw a length-L realisation of y in the same manner.
3. Count the number of indices at which the elements of the two length-L draws are not equal.

This yields n counts of non-equal values between n pairs of independent realisations of x and y. The n-member distribution of nonequal-value counts is returned as a vector.

counteq(x::UVAL_COLLECTION_TYPES, y::UVAL_COLLECTION_TYPES, n::Int)

Estimate a n-member distribution on the number of indices at which the elements of two collections of uncertain values are equal.

This is done by repeating the following procedure n times:

1. First, draw a length-L realisation of x by drawing one random number from each uncertain value furnishing the dataset. The draws are independent, so that no element-wise dependencies (e.g. sequential correlations) that are not already present in the data are introduced in the realisation.
2. Second, draw a length-L realisation of y in the same manner.
3. Count the number of indices at which the elements of the two length-L draws are equal.

This yields n counts of non-equal values between n pairs of independent realisations of x and y. The n-member distribution of equal-value counts is returned as a vector.

maxad(x::UVAL_COLLECTION_TYPES, y::UVAL_COLLECTION_TYPES, n::Int)

Obtain a distribution over the maximum absolute deviation between two collections of uncertain values.

This is done by repeating the following procedure n times:

1. First, draw a length-L realisation of x by drawing one random number from each uncertain value furnishing the dataset. The draws are independent, so that no element-wise dependencies (e.g. sequential correlations) that are not already present in the data are introduced in the realisation.
2. Second, draw a length-L realisation of y in the same manner.
3. Compute the maximum absolute deviation between the two length-L draws.

This yields n estimates of the maximum absolute deviation between n independent pairs of realisations of x and y. The n-member distribution of maximum absolute deviation estimates is returned as a vector.

meanad(x::UVAL_COLLECTION_TYPES, y::UVAL_COLLECTION_TYPES, n::Int)

Obtain a distribution over the mean absolute deviation between two collections of uncertain values.

This is done by repeating the following procedure n times:

1. First, draw a length-L realisation of x by drawing one random number from each uncertain value furnishing the dataset. The draws are independent, so that no element-wise dependencies (e.g. sequential correlations) that are not already present in the data are introduced in the realisation.
2. Second, draw a length-L realisation of y in the same manner.
3. Compute the mean absolute deviation between the two length-L draws.

This yields n estimates of the mean absolute deviation between n independent pairs of realisations of x and y. The n-member distribution of mean absolute deviation estimates is returned as a vector.

msd(x::UVAL_COLLECTION_TYPES, y::UVAL_COLLECTION_TYPES, n::Int)

Obtain a distribution over the mean squared deviation between two collections of uncertain values.

This is done by repeating the following procedure n times:

1. First, draw a length-L realisation of x by drawing one random number from each uncertain value furnishing the dataset. The draws are independent, so that no element-wise dependencies (e.g. sequential correlations) that are not already present in the data are introduced in the realisation.
2. Second, draw a length-L realisation of y in the same manner.
3. Compute the mean squared deviation between the two length-L draws.

This yields n estimates of the mean squared deviation between n independent pairs of realisations of x and y. The n-member distribution of mean squared deviation estimates is returned as a vector.

psnr(x::UVAL_COLLECTION_TYPES, y::UVAL_COLLECTION_TYPES, maxv, n::Int)

Obtain a distribution over the peak signal-to-noise ratio (PSNR) between two collections of uncertain values.

This is done by repeating the following procedure n times:

1. First, draw a length-L realisation of x by drawing one random number from each uncertain value furnishing the dataset. The draws are independent, so that no element-wise dependencies (e.g. sequential correlations) that are not already present in the data are introduced in the realisation.
2. Second, draw a length-L realisation of y in the same manner.
3. Compute the PSNR between the two length-L draws.

This yields n estimates of the peak signal-to-noise ratio between n independent pairs of realisations of x and y. The n-member distribution of PSNR estimates is returned as a vector.

The PSNR is computed as 10 * log10(maxv^2 / msd(x_draw, y_draw)), where maxv is the maximum possible value x or y can take

rmsd(x::UVAL_COLLECTION_TYPES, y::UVAL_COLLECTION_TYPES, n::Int, normalize = false)

Obtain a distribution over the root mean squared deviation between two collections of uncertain values.

This is done by repeating the following procedure n times:

1. First, draw a length-L realisation of x by drawing one random number from each uncertain value furnishing the dataset. The draws are independent, so that no element-wise dependencies (e.g. sequential correlations) that are not already present in the data are introduced in the realisation.
2. Second, draw a length-L realisation of y in the same manner.
3. Compute the root mean squared deviation between the two length-L draws.

This yields n estimates of the root mean squared deviation between n independent pairs of realisations of x and y. The n-member distribution of root mean squared deviation estimates is returned as a vector.

The root mean squared deviation is computed as sqrt(msd(x_draw, y_draw)) at each iteration. Optionally, x_draw and y_draw may be normalised.

sqL2dist(x::UVAL_COLLECTION_TYPES, y::UVAL_COLLECTION_TYPES, n::Int)

Obtain a distribution over the squared L2 distance between two collections of uncertain values.

This is done by repeating the following procedure n times:

1. First, draw a length-L realisation of x by drawing one random number from each uncertain value furnishing the dataset. The draws are independent, so that no element-wise dependencies (e.g. sequential correlations) that are not already present in the data are introduced in the realisation.
2. Second, draw a length-L realisation of y in the same manner.
3. Compute the squared L2 distance between the two length-L draws.

This yields n estimates of the squared L2 distance between n independent pairs of realisations of x and y. The n-member distribution of squared L2 distance estimates is returned as a vector.

The squared L2 distance is computed as $\sum_{i=1}^n |x_i - y_i|^2$.

crosscor(x::UVAL_COLLECTION_TYPES, y::UVAL_COLLECTION_TYPES, [lags], n::Int; demean = true)

Obtain a distribution over the cross correlation between two collections of uncertain values.

This is done by repeating the following procedure n times:

1. First, draw a length-L realisation of x by drawing one random number from each uncertain value furnishing the dataset. The draws are independent, so that no element-wise dependencies (e.g. sequential correlations) that are not already present in the data are introduced in the realisation.
2. Second, draw a length-L realisation of y in the same manner.
3. Compute the cross correlation between the two length-L draws.

This yields n estimates of the cross correlation between n independent pairs of realisations of x and y. The n-member distribution of cross correlation estimates is returned as a vector.

demean specifies whether, at each iteration, the respective means of the draws should be subtracted from them before computing their cross correlation.

When left unspecified, the lags used are -min(n-1, 10*log10(n)) to min(n, 10*log10(n)).

The output is normalized by sqrt(var(x_draw)*var(y_draw)). See crosscov for the unnormalized form.

crosscov(x::UVAL_COLLECTION_TYPES, y::UVAL_COLLECTION_TYPES, [lags], n::Int; demean = true)

Obtain a distribution over the cross covariance function (CCF) between two collections of uncertain values.

This is done by repeating the following procedure n times:

1. First, draw a length-L realisation of x by drawing one random number from each uncertain value furnishing the dataset. The draws are independent, so that no element-wise dependencies (e.g. sequential correlations) that are not already present in the data are introduced in the realisation.
2. Second, draw a length-L realisation of y in the same manner.
3. Compute the CCF between the two length-L draws.

This yields n estimates of the CCF between n independent pairs of realisations of x and y. The n-member distribution of CCF estimates is returned as a vector.

demean specifies whether, at each iteration, the respective means of the draws should be subtracted from them before computing their CCF.

When left unspecified, the lags used are -min(n-1, 10*log10(n)) to min(n, 10*log10(n)).

The output is not normalized. See crosscor for a function with normalization.

gkldiv(x::UVAL_COLLECTION_TYPES, y::UVAL_COLLECTION_TYPES, n::Int)

Obtain a distribution over the generalized Kullback-Leibler divergence between two collections of uncertain values.

This is done by repeating the following procedure n times:

1. First, draw a length-L realisation of x by drawing one random number from each uncertain value furnishing the dataset. The draws are independent, so that no element-wise dependencies (e.g. sequential correlations) that are not already present in the data are introduced in the realisation.
2. Second, draw a length-L realisation of y in the same manner.
3. Compute the generalized Kullback-Leibler divergence between the two length-L draws.

This yields n estimates of the generalized Kullback-Leibler divergence between n independent pairs of realisations of x and y. The n-member distribution of generalized Kullback-Leibler divergence estimates is returned as a vector.

kldivergence(x::UVAL_COLLECTION_TYPES, y::UVAL_COLLECTION_TYPES, [b], n::Int)

Obtain a distribution over the Kullback-Leibler divergence between two collections of uncertain values.

This is done by repeating the following procedure n times:

1. First, draw a length-L realisation of x by drawing one random number from each uncertain value furnishing the dataset. The draws are independent, so that no element-wise dependencies (e.g. sequential correlations) that are not already present in the data are introduced in the realisation.
2. Second, draw a length-L realisation of y in the same manner.
3. Compute the Kullback-Leibler divergence between the two length-L draws.

This yields n estimates of the Kullback-Leibler divergence between n independent pairs of realisations of x and y. The n-member distribution of Kullback-Leibler divergence estimates is returned as a vector.

Optionally a real number b can be specified such that the divergence is scaled by 1/log(b).