From d50a66cc04bfa1c483f04daffe465322316c745e Mon Sep 17 00:00:00 2001
From: Yanbo Liang <ybliang8@gmail.com>
Date: Mon, 9 Nov 2015 08:57:29 -0800
Subject: [SPARK-10689][ML][DOC] User guide and example code for
 AFTSurvivalRegression

Add user guide and example code for ```AFTSurvivalRegression```.

Author: Yanbo Liang <ybliang8@gmail.com>

Closes #9491 from yanboliang/spark-10689.
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+---
+layout: global
+title: Survival Regression - ML
+displayTitle: <a href="ml-guide.html">ML</a> - Survival Regression
+---
+
+
+`\[
+\newcommand{\R}{\mathbb{R}}
+\newcommand{\E}{\mathbb{E}}
+\newcommand{\x}{\mathbf{x}}
+\newcommand{\y}{\mathbf{y}}
+\newcommand{\wv}{\mathbf{w}}
+\newcommand{\av}{\mathbf{\alpha}}
+\newcommand{\bv}{\mathbf{b}}
+\newcommand{\N}{\mathbb{N}}
+\newcommand{\id}{\mathbf{I}}
+\newcommand{\ind}{\mathbf{1}}
+\newcommand{\0}{\mathbf{0}}
+\newcommand{\unit}{\mathbf{e}}
+\newcommand{\one}{\mathbf{1}}
+\newcommand{\zero}{\mathbf{0}}
+\]`
+
+
+In `spark.ml`, we implement the [Accelerated failure time (AFT)](https://en.wikipedia.org/wiki/Accelerated_failure_time_model) 
+model which is a parametric survival regression model for censored data. 
+It describes a model for the log of survival time, so it's often called 
+log-linear model for survival analysis. Different from 
+[Proportional hazards](https://en.wikipedia.org/wiki/Proportional_hazards_model) model
+designed for the same purpose, the AFT model is more easily to parallelize 
+because each instance contribute to the objective function independently.
+
+Given the values of the covariates $x^{'}$, for random lifetime $t_{i}$ of 
+subjects i = 1, ..., n, with possible right-censoring, 
+the likelihood function under the AFT model is given as:
+`\[
+L(\beta,\sigma)=\prod_{i=1}^n[\frac{1}{\sigma}f_{0}(\frac{\log{t_{i}}-x^{'}\beta}{\sigma})]^{\delta_{i}}S_{0}(\frac{\log{t_{i}}-x^{'}\beta}{\sigma})^{1-\delta_{i}}
+\]`
+Where $\delta_{i}$ is the indicator of the event has occurred i.e. uncensored or not.
+Using $\epsilon_{i}=\frac{\log{t_{i}}-x^{'}\beta}{\sigma}$, the log-likelihood function
+assumes the form:
+`\[
+\iota(\beta,\sigma)=\sum_{i=1}^{n}[-\delta_{i}\log\sigma+\delta_{i}\log{f_{0}}(\epsilon_{i})+(1-\delta_{i})\log{S_{0}(\epsilon_{i})}]
+\]`
+Where $S_{0}(\epsilon_{i})$ is the baseline survivor function,
+and $f_{0}(\epsilon_{i})$ is corresponding density function.
+
+The most commonly used AFT model is based on the Weibull distribution of the survival time. 
+The Weibull distribution for lifetime corresponding to extreme value distribution for 
+log of the lifetime, and the $S_{0}(\epsilon)$ function is:
+`\[   
+S_{0}(\epsilon_{i})=\exp(-e^{\epsilon_{i}})
+\]`
+the $f_{0}(\epsilon_{i})$ function is:
+`\[
+f_{0}(\epsilon_{i})=e^{\epsilon_{i}}\exp(-e^{\epsilon_{i}})
+\]`
+The log-likelihood function for AFT model with Weibull distribution of lifetime is:
+`\[
+\iota(\beta,\sigma)= -\sum_{i=1}^n[\delta_{i}\log\sigma-\delta_{i}\epsilon_{i}+e^{\epsilon_{i}}]
+\]`
+Due to minimizing the negative log-likelihood equivalent to maximum a posteriori probability,
+the loss function we use to optimize is $-\iota(\beta,\sigma)$.
+The gradient functions for $\beta$ and $\log\sigma$ respectively are:
+`\[   
+\frac{\partial (-\iota)}{\partial \beta}=\sum_{1=1}^{n}[\delta_{i}-e^{\epsilon_{i}}]\frac{x_{i}}{\sigma}
+\]`
+`\[ 
+\frac{\partial (-\iota)}{\partial (\log\sigma)}=\sum_{i=1}^{n}[\delta_{i}+(\delta_{i}-e^{\epsilon_{i}})\epsilon_{i}]
+\]`
+
+The AFT model can be formulated as a convex optimization problem, 
+i.e. the task of finding a minimizer of a convex function $-\iota(\beta,\sigma)$ 
+that depends coefficients vector $\beta$ and the log of scale parameter $\log\sigma$.
+The optimization algorithm underlying the implementation is L-BFGS.
+The implementation matches the result from R's survival function 
+[survreg](https://stat.ethz.ch/R-manual/R-devel/library/survival/html/survreg.html)
+
+## Example:
+
+<div class="codetabs">
+
+<div data-lang="scala" markdown="1">
+{% include_example scala/org/apache/spark/examples/ml/AFTSurvivalRegressionExample.scala %}
+</div>
+
+<div data-lang="java" markdown="1">
+{% include_example java/org/apache/spark/examples/ml/JavaAFTSurvivalRegressionExample.java %}
+</div>
+
+<div data-lang="python" markdown="1">
+{% include_example python/ml/aft_survival_regression.py %}
+</div>
+
+</div>
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