Is the kidney data Gaussian

or how can we model

the kidney data distribution ? for high dimensions d = 264 and low sample n = 36 do PCA to reduce data to 7 dimensions analyse data in PC directions: data could be normal - QQ−plots some outliers ? sphere or whiten PC data: normalise with covariance matrix this uncorrelates data and makes all directions same length Independent Component Analysis - ICA decomposes data into independent non-Gaussian directions first direction: most non-Gaussian has large skewness and kurtosis second direction: ’more’ Gaussian etc ICA does not give unique directions does not converge if ICA cannot find non-Gaussian directions compare first IC direction with data simulated from Gaussian calculate p-value ICA finds 4 outliers outliers not in a cluster try: removing these outliers ICA finds 3 new ones ...

ICA gives clear indication that data non-Gaussian try: transform data – use sphered data power transformations of distance to origin re-sphere and compare to Gaussian power transform with α = 0.55 works best how Gaussian is this transformed data? do ICA on it and compare to original data transformation has moved data closer to Gaussian but transformed data clearly non-Gaussian how can we simulate this data? calculate EDF of real data want simulated data which envelops EDF simulated Gaussian data does not cover EDF of data apply ’un-sphering’ and power transform to simulated Gaussian compare to EDF of data best for β = 2 with β = 1/α averages of such data very close to real data, but power transformed simulated data has large variability so better to take averages of such data can adequately model kidney data by average of un-sphered and power transformed Gaussians