Machine Learning for Survival Analysis: A Survey <s> Regularized Cox models. <s> SUMMARY We propose a new method for estimation in linear models. The 'lasso' minimizes the residual sum of squares subject to the sum of the absolute value of the coefficients being less than a constant. Because of the nature of this constraint it tends to produce some coefficients that are exactly 0 and hence gives interpretable models. Our simulation studies suggest that the lasso enjoys some of the favourable properties of both subset selection and ridge regression. It produces interpretable models like subset selection and exhibits the stability of ridge regression. There is also an interesting relationship with recent work in adaptive function estimation by Donoho and Johnstone. The lasso idea is quite general and can be applied in a variety of statistical models: extensions to generalized regression models and tree-based models are briefly described. <s> BIB001 </s> Machine Learning for Survival Analysis: A Survey <s> Regularized Cox models. <s> I propose a new method for variable selection and shrinkage in Cox's proportional hazards model. My proposal minimizes the log partial likelihood subject to the sum of the absolute values of the parameters being bounded by a constant. Because of the nature of this constraint, it shrinks coefficients and produces some coefficients that are exactly zero. As a result it reduces the estimation variance while providing an interpretable final model. The method is a variation of the 'lasso' proposal of Tibshirani, designed for the linear regression context. Simulations indicate that the lasso can be more accurate than stepwise selection in this setting. <s> BIB002
With the continuing development of data collection and detection techniques, most real-world domains tend to accumulate high-dimensional data. In some cases, the number of variables (P) in the given data will almost equal, or even exceed, the number of instances (N ). It is therefore challenging to build a good prediction model that takes into account all the available features, and, in some cases, the model may provide inaccurate results because of the overfitting problem (van Houwelingen and Putter 2011). This encourages the use of sparsity norms to select vital features in high-dimensional data under the assumption that most of the features are not significant  . In order to identify the features that are most relevant to the outcome variable among what can be tens of thousands of features, a number of different penalty functions, including lasso, group lasso, fused lasso, and graph lasso, have been developed for prediction models using sparse learning methods. The family of -norm penalty functions γ : γ , γ > 0 are the most commonly used of these penalty functions. The smaller the value of γ , the sparser the solution, but when 0 ≤ γ < 1, the penalty is nonconvex, which makes the optimization problem more challenging to solve. The regularizers for the most commonly used regularized Cox models are summarized in Table 3 . Lasso-Cox: Lasso BIB001 is an 1 -norm regularizer that performs feature selection and estimates the regression coefficients simultaneously. In BIB002 , the 1 -norm penalty was incorporated into the log-partial likelihood shown in Equation (14) to obtain the Lasso-Cox algorithm, which inherits the properties of 1 -norm in feature selection. 