# GPois: Fit a tilted Gaussian density via a Poisson GAM In Hanchao-Zhang/ProDenICA: Product Density Estimation for ICA using tilted Gaussian density estimates

## Description

This is a contrast method for `ProDenICA`. It fits a tilted Gaussian density estimate by multiplying the Gaussian density by an exponential tilt function using a cubic smoothing spline

## Usage

 `1` ```GPois(x, df = 6, B = 500, order = 1, widen = 1.2, density.return = FALSE, ...) ```

## Arguments

 `x` vector of real values `df` degrees of freedom for the smoothing-spline fit; default is 6 `B` number of grid points for density estimate; default is 500 `order` A robustness parameter to avoid responding to outliers in `x`. The range of `x` is estimated by the `order`th and `n-order+1`th order statistics. Default is `order=1` `widen` an expansion factor to widen the range of `x`; default is `widen=1.2` `density.return` logical variable, with default `FALSE`. If `density.return=TRUE`, the estimated density is returned `...` additional arguments to GAM; typically not used

## Details

See Section 14.7.4 of 'Elements of Statistical Learning (Hastie, Tibshirani and Friedman, 2009, 2nd Edition)' for details

## Value

a list with components

 `Gs` estimated contrast function, which is the log of the tilting function, evaluated at the original values of `x`. `mean(Gs)` is measure of negentropy `gs` estimated first derivative of `Gs` at `x` `gps` estimated second derivative of `Gs` at `x` `density` if `density.return=TRUE`, a list with components `\$x` the grid of B values of `x`, and `\$y` the estimated density.

## Author(s)

Trevor Hastie and Rob Tibshirani

## References

Hastie, T. and Tibshirani, R. (2003) Independent Component Analysis through Product Density Estimation in Advances in Neural Information Processing Systems 15 (Becker, S. and Obermayer, K., eds), MIT Press, Cambridge, MA. pp 649-656
Hastie, T., Tibshirani, R. and Friedman, J. (2009) Elements of Statistical Learning (2nd edition), Springer.
http://www-stat.stanford.edu/~hastie/Papers/ESLII.pdf

`ProDenICA`, `G1` and `G0`
 ``` 1 2 3 4 5 6 7 8 9 10``` ```p=2 ### Can use letters a-r below for dist dist="n" N=1024 A0<-mixmat(p) s<-scale(cbind(rjordan(dist,N),rjordan(dist,N))) x <- s %*% A0 fit=ProDenICA(x,Gfunc=GPois, whiten=TRUE, density=TRUE) par(mfrow=c(2,1)) plot(fit) ```