Fit the logistic-normal by transforming, then fitting a normal

The alr is a bijection onto R^{D-1}, so the MLE on the simplex is the multivariate-normal MLE of the transformed data; no optimisation is needed. The parameter-free Jacobian is included in loglik so that value is comparable with aitlnp.

Usage

Lgtnfit(X, ddof = 1)

Arguments

X

One composition per row; strictly positive.

ddof

Divisor correction: 1 for the unbiased covariance, 0 for the MLE.

Value

List with mu, Sigma, center, loglik, n, D.

Details

Formula: Y = alr(X); muhat = mean(Y); Sigmahat = sum (Y_k - muhat)(Y_k - muhat)' / (n - ddof)

References

Aitchison (1986), The Statistical Analysis of Compositional Data, Chapter 7.

Examples

D <- data.frame(x = c(1, 2, 3, 4), y = c(2, 4, 5, 9))
Lgtnfit(D)
#> $mu
#>          x 
#> -0.6770126 
#> 
#> $Sigma
#>            [,1]
#> [1,] 0.01535756
#> 
#> $center
#>         x           
#> 0.3369284 0.6630716 
#> 
#> $loglik
#> [1] -5.887618
#> 
#> $n
#> [1] 4
#> 
#> $D
#> [1] 2
#> 
#> $method
#> [1] "Logistic-normal MLE via the alr transform"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"