resonance spectra have highly correlated spectral channels, poor signal-to-noise ratios, and fall into the >> case. The goal was to find a reliable automated p N method to render MRS results accessible to radiologists who lacked the training to interpret the spectra themselves. PLS and ridge regression were also investigated. These regression techniques worked well for this application (Menze 2004). et al., The lasso has also been used to predict elemental compositions for geological samples, as in the context of this thesis (Dyar 2012). et al., Generalizing the Ridge and Lasso The ridge and lasso penalties can be generalized as follows . (6) This is known as an penalty. For = 0, this reduces to subset selection, where Lq q the penalty restricts the number of non-zero parameters. For = 1, this reduces to q the lasso; = 2 corresponds to the ridge penalty. These are the lasso penalty q L1 and the ridge penalty, respectively. Note that for 1, the constraint region is L2 q ≤ not convex and the prior is not uniform in direction. This makes solving the optimization problem more difficult (Hastie, 2009). et al., Given this generalization, it may seem reasonable to estimate based on q the data. Other values of may work better than = 0, 1, or 2. However, extra q q variance is generated during this estimation process. The improvement in model
resonance spectra have highly correlated spectral channels, poor signal-to-noise ratios, and fall into the p >> N case. The goal was to find a reliable automated method to render MRS results accessible to radiologists who lacked the training to interpret the spectra themselves. PLS and ridge regression were also investigated. These regression techniques worked well for this application (Menze et al., 2004). The lasso has also been used to predict elemental compositions for geological samples, as in the context of this thesis (Dyar et al., 2012). Generalizing the Ridge and Lasso The ridge and lasso penalties can be generalized as follows . (6) This is known as an Lq penalty. For q = 0, this reduces to subset selection, where the penalty restricts the number of non-zero parameters. For q = 1, this reduces to the lasso; q = 2 corresponds to the ridge penalty. These are the L1 lasso penalty and the L2 ridge penalty, respectively. Note that for q ≤ 1, the constraint region is not convex and the prior is not uniform in direction. This makes solving the optimization problem more difficult (Hastie, et al., 2009). Given this generalization, it may seem reasonable to estimate q based on the data. Other values of q may work better than q = 0, 1, or 2. However, extra variance is generated during this estimation process. The improvement in model