Table 1. Summary table of the various regression methods explored in this thesis Method OLS PLS LASSO ELASTIC NET SPLS Summary Chooses subspaces from explanatory matrix, X, sequentially projects response vector, y, onto the subspaces of the column spaces of X to determine the model variable coefficients. Shrinks some coefficients and sets others equal to zero in accordance with shrinkage parameter, t. Provides a sparse model that can be used for both feature selection and composition predictions. Generalizes the lasso. Shrinks some coefficients and sets others equal to zero; averages highly correlated features and shrinks averages. Provides a sparse model that has more terms than the lasso and can be used for feature selection and composition predictions. Adds an L1-constraint to the PLS algorithm to impose sparsity. Performs feature selection yet retains stable properties of PLS. Penalty None , Tuning Parameters None k, # of dimensions t t, α κ, t1, t2, K Advantage(s) Provides a stable model. Provides an interpretable model, selects subset of predictors with the strongest effects on the response variable. Can be used for feature selection when less data are available. Performs well in the p >> N case. Provides an interpretable model that is more stable than the lasso. Useful for feature selection. Performs well in the p >> N case. Disadvantage(s) Does not provide a stable model when the variables are correlated; predictions suffer from wild variance so they are unreliable. Provides a complex model in which all coefficients are linear combinations of the original channels. Involves a complex optimization problem with no simple, closed-form representation. Does not perform as well in the p >> N case. Arbitrarily chooses one covariate from a group of highly collinear covariates to use in the model and discards the rest. Cannot be used for feature selection in situations when less data are available because it overwhelms the data with too many model variables. Shrinkage properties not yet well understood.
model RMSEP values using Student’s t-test reveal that there is no statistical difference in the prediction accuracies of the four models for this suite of 100 igneous rocks. These results are shown in Table 7, which gives both the p-values and the absolute differences in RMSEP for the model comparisons. MODEL SELECTION Partial Least Squares The number of components in the PLS model was constrained to 15 to prevent overfitting. The selected models for the ten major elemental oxides all used fewer than 15 components, as shown in Table 2. MgO had the most complex model with 14 components, while Na2O had the most parsimonious model with 3 components. The other elements had components numbering from 6 to 12, with SiO2 and Al2O3 on the lower end of that range, and TiO2 and Fe2O3 on the upper end. Lasso The tuning parameter, t, was constrained to 0.1 ≤ t ≤ 0.9 using the CRAN R package caret. Larger values of t would have permitted models with more coefficients. Caret constrains to a maximum value of 0.9 because larger values do not improve the predictions sufficiently to compensate for the lost model simplicity. These t values, shown in Table 3, minimize the number of components in the model and the model RMSEP values according to the one SE heuristic.
Elastic Net For the elastic net models, the tuning parameters were constrained such that 0.1 ≤ t ≤ 3.0 and α = {0.1, 1.0} using the CRAN R package caret. Models were selected according to the one SE heuristic. Most elemental oxide models selected α = 0.1, which represents 10% L1-penalty and 90% L2-penalty. The exception was SiO2, which selected α = 1.0. This is equivalent to a lasso penalty because it is 100% L1-penalty. The MnO model had the smallest t value (t = 0.1); Al2O3, Fe2O3, MgO, and CaO had the largest t value (t = 3.0). The latter four models had the most coefficients. Sparse Partial Least Squares For the SPLS models, tuning parameters were constrained using caret: η = {0.1, 0.3, 0.5, 0.7, 0.9} and K = {1, 2, 3, 4, 5}. The η parameter is a combination of the t1 and t2 parameters; it controls the sparsity of the model. The sparser models are for Al2O3, TiO2, and MgO. TiO2 has the largest number of hidden components (K = 4). The third parameter was κ = 0.5, which simply dictates the starting point for the SPLS algorithm.
availability may lead to more accurate models. Thus, all the models could perform better for spectra obtained on Mars because they will have more emission lines from which to build their models. The ultimate goal of this thesis is to determine the best method(s) for determining the elemental compositions of Mars rocks from LIBS spectral data. SPLS, lasso, and elastic net not only provide models with accuracies comparable to PLS model accuracies, but they also give interpretable models. These methods can improve researchers’ understanding of how Mars evolved by painting a clearer picture of elemental compositions and interactions in rocks. FUTURE WORK DATA PREPROCESSING: AVERAGED VERSUS UNAVERAGED SPECTRA Spectra from each of the 50 shots for each sample were averaged in this thesis, which resulted in 100 averaged spectra that were used for analysis. This was done to average out noise. This is permissible because the samples are homogenous (rock powders). However, the spectra from MSL will be from rocks, which by their nature are heterogeneous. Each shot could contain distinct compositional information. In the context of this application, information about