# corFunctions: Fast implementations of (robust) correlation estimators In ccaPP: (Robust) Canonical Correlation Analysis via Projection Pursuit

## Description

Estimate the correlation of two vectors via fast C++ implementations, with a focus on robust and nonparametric methods.

## Usage

 ``` 1 2 3 4 5 6 7 8 9 10``` ```corPearson(x, y) corSpearman(x, y, consistent = FALSE) corKendall(x, y, consistent = FALSE) corQuadrant(x, y, consistent = FALSE) corM(x, y, prob = 0.9, initial = c("quadrant", "spearman", "kendall", "pearson"), tol = 1e-06) ```

## Arguments

 `x, y` numeric vectors. `consistent` a logical indicating whether a consistent estimate at the bivariate normal distribution should be returned (defaults to `FALSE`). `prob` numeric; probability for the quantile of the chi-squared distribution to be used for tuning the Huber loss function (defaults to 0.9). `initial` a character string specifying the starting values for the Huber M-estimator. For `"quadrant"` (the default), `"spearman"` or `"kendall"`, the consistent version of the respecive correlation measure is used together with the medians and MAD's. For `"pearson"`, the Pearson correlation is used together with the means and standard deviations. `tol` a small positive numeric value to be used for determining convergence.

## Details

`corPearson` estimates the classical Pearson correlation. `corSpearman`, `corKendall` and `corQuadrant` estimate the Spearman, Kendall and quadrant correlation, respectively, which are nonparametric correlation measures that are somewhat more robust. `corM` estimates the correlation based on a bivariate M-estimator of location and scatter with a Huber loss function, which is sufficiently robust in the bivariate case, but loses robustness with increasing dimension.

The nonparametric correlation measures do not estimate the same population quantities as the Pearson correlation, the latter of which is consistent at the bivariate normal model. Let rho denote the population correlation at the normal model. Then the Spearman correlation estimates (6/pi) arcsin(rho/2), while the Kendall and quadrant correlation estimate (2/pi) arcsin(rho). Consistent estimates are thus easily obtained by taking the corresponding inverse expressions.

The Huber M-estimator, on the other hand, is consistent at the bivariate normal model.

## Value

The respective correlation estimate.

## Note

The Kendall correlation uses a naive n^2 implementation if n < 30 and a fast O(n log(n)) implementation for larger values, where n denotes the number of observations.

Functionality for removing observations with missing values is currently not implemented.

## Author(s)

Andreas Alfons, O(n log(n)) implementation of the Kendall correlation by David Simcha

`ccaGrid`, `ccaProj`, `cor`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27``` ```## generate data library("mvtnorm") set.seed(1234) # for reproducibility sigma <- matrix(c(1, 0.6, 0.6, 1), 2, 2) xy <- rmvnorm(100, sigma=sigma) x <- xy[, 1] y <- xy[, 2] ## compute correlations # Pearson correlation corPearson(x, y) # Spearman correlation corSpearman(x, y) corSpearman(x, y, consistent=TRUE) # Kendall correlation corKendall(x, y) corKendall(x, y, consistent=TRUE) # quadrant correlation corQuadrant(x, y) corQuadrant(x, y, consistent=TRUE) # Huber M-estimator corM(x, y) ```