Linear Regression & Empirical Factor Regression Linear Regression SVD Regression D is a diagonal matrix of singular values
Empirical Factor Regression By definition, Regression is now done in factor space using generalized shrinkage (ridge regression) priors on, e.g. RVM Problem of inversion:has many-to-one mapping is canonical “least-norm” inverse
Example: Biscuit Dough Data NIR spectroscopy reflectance values are predictors Response is fat content of dough samples 39 training, 39 testing: data are pooled and testing data responses treated as missing values to be imputed Top 16 factors used, based on size of singular values
Example: Biscuit Dough Data (2) Left: Fitted and predicted vs true values Right: Least-norm inverse of beta ~ 1700 nm range is absorbance region for fat As can be seen, solution is not sparse
Latent Factor Regression Loosen to Under proper constraints on B, this finds common structure in X and isolates idiosyncrasies to noise Now, variation in X has less effect on y The implied prior is When variance, Phi 0, this reverts to empirical linear regression
Sparse Latent Factor Regression WRT gene expression profiling, “multiple biological factors underlie patterns of gene expression variation, so latent factor approaches are natural – we imagine that latent factors reflect individual biological functions… This is a motivating context for sparse models.” Columns of B represents the genes involved in a particular biological factor. Rows of B represent a particular gene’s involvement across biological factors.
Example: Gene Expression Data p = 6128 genes measured using Affymetrix DNA microarrays n = 49 breast cancer tumor samples k = 25 factors Factor 3 separates by red: estrogen receptor positive tumors blue: ER negative
Example: Gene Expression Data Comparison with results obtained using empirical SVD factors
Conclusion Sparse factor regression modeling is a promising framework for dimensionality reduction of predictors. Only those factors that are relevant (e.g. factor 3) are of interest. Therefore, only those genes with non-zero values in that column of B are meaningful.