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See the License for the specific language governing permissions and limitations under the License. \end{comment} \subsection{Stepwise Linear Regression} \noindent{\bf Description} \smallskip Our stepwise linear regression script selects a linear model based on the Akaike information criterion (AIC): the model that gives rise to the lowest AIC is computed. \\ \smallskip \noindent{\bf Usage} \smallskip {\hangindent=\parindent\noindent\it% {\tt{}-f }path/\/{\tt{}StepLinearRegDS.dml} {\tt{} -nvargs} {\tt{} X=}path/file {\tt{} Y=}path/file {\tt{} B=}path/file {\tt{} S=}path/file {\tt{} O=}path/file {\tt{} icpt=}int {\tt{} thr=}double {\tt{} fmt=}format } \smallskip \noindent{\bf Arguments} \begin{Description} \item[{\tt X}:] Location (on HDFS) to read the matrix of feature vectors, each row contains one feature vector. \item[{\tt Y}:] Location (on HDFS) to read the 1-column matrix of response values \item[{\tt B}:] Location (on HDFS) to store the estimated regression parameters (the $\beta_j$'s), with the intercept parameter~$\beta_0$ at position {\tt B[}$m\,{+}\,1$, {\tt 1]} if available \item[{\tt S}:] (default:\mbox{ }{\tt " "}) Location (on HDFS) to store the selected feature-ids in the order as computed by the algorithm; by default the selected feature-ids are forwarded to the standard output. \item[{\tt O}:] (default:\mbox{ }{\tt " "}) Location (on HDFS) to store the CSV-file of summary statistics defined in Table~\ref{table:linreg:stats}; by default the summary statistics are forwarded to the standard output. \item[{\tt icpt}:] (default:\mbox{ }{\tt 0}) Intercept presence and shifting/rescaling the features in~$X$:\\ {\tt 0} = no intercept (hence no~$\beta_0$), no shifting or rescaling of the features;\\ {\tt 1} = add intercept, but do not shift/rescale the features in~$X$;\\ {\tt 2} = add intercept, shift/rescale the features in~$X$ to mean~0, variance~1 \item[{\tt thr}:] (default:\mbox{ }{\tt 0.01}) Threshold to stop the algorithm: if the decrease in the value of the AIC falls below {\tt thr} no further features are being checked and the algorithm stops. \item[{\tt fmt}:] (default:\mbox{ }{\tt "text"}) Matrix file output format, such as {\tt text}, {\tt mm}, or {\tt csv}; see read/write functions in SystemML Language Reference for details. \end{Description} \noindent{\bf Details} \smallskip Stepwise linear regression iteratively selects predictive variables in an automated procedure. Currently, our implementation supports forward selection: starting from an empty model (without any variable) the algorithm examines the addition of each variable based on the AIC as a model comparison criterion. The AIC is defined as \begin{equation} AIC = -2 \log{L} + 2 edf,\label{eq:AIC} \end{equation} where $L$ denotes the likelihood of the fitted model and $edf$ is the equivalent degrees of freedom, i.e., the number of estimated parameters. This procedure is repeated until including no additional variable improves the model by a certain threshold specified in the input parameter {\tt thr}. For fitting a model in each iteration we use the direct solve'' method as in the script {\tt LinearRegDS.dml} discussed in Section~\ref{sec:LinReg}. \smallskip \noindent{\bf Returns} \smallskip Similar to the outputs from {\tt LinearRegDS.dml} the stepwise linear regression script computes the estimated regression coefficients and stores them in matrix $B$ on HDFS. The format of matrix $B$ is identical to the one produced by the scripts for linear regression (see Section~\ref{sec:LinReg}). Additionally, {\tt StepLinearRegDS.dml} outputs the variable indices (stored in the 1-column matrix $S$) in the order they have been selected by the algorithm, i.e., $i$th entry in matrix $S$ corresponds to the variable which improves the AIC the most in $i$th iteration. If the model with the lowest AIC includes no variables matrix $S$ will be empty (contains one 0). Moreover, the estimated summary statistics as defined in Table~\ref{table:linreg:stats} are printed out or stored in a file (if requested). In the case where an empty model achieves the best AIC these statistics will not be produced. \smallskip \noindent{\bf Examples} \smallskip {\hangindent=\parindent\noindent\tt \hml -f StepLinearRegDS.dml -nvargs X=/user/biadmin/X.mtx Y=/user/biadmin/Y.mtx B=/user/biadmin/B.mtx S=/user/biadmin/selected.csv O=/user/biadmin/stats.csv icpt=2 thr=0.05 fmt=csv }