[SPARK-15186][ML][DOCS] Add user guide for generalized linear regression

## What changes were proposed in this pull request?

This patch adds a user guide section for generalized linear regression and includes the examples from [#12754](https://github.com/apache/spark/pull/12754).

## How was this patch tested?

Documentation only, no tests required.

## Approach

In general, it is a bit unclear what level of detail ought to be included in the user guide since there is a lot of variability within the current user guide. I tried to give a fairly brief mathematical introduction to GLMs, and cover what types of problems they could be used for. Additionally, I included a brief blurb on the IRLS solver. The input/output columns are given in a table as is found elsewhere in the docs (though, again, these appear rather intermittently in the current docs), as well as a table providing the supported families and their link functions.

Author: sethah <seth.hendrickson16@gmail.com>

Closes #13139 from sethah/SPARK-15186.
This commit is contained in:
sethah 2016-05-27 12:55:48 -07:00 committed by Joseph K. Bradley
parent a96e4151a9
commit c96244f5ac

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@ -374,6 +374,138 @@ regression model and extracting model summary statistics.
</div> </div>
## Generalized linear regression
Contrasted with linear regression where the output is assumed to follow a Gaussian
distribution, [generalized linear models](https://en.wikipedia.org/wiki/Generalized_linear_model) (GLMs) are specifications of linear models where the response variable $Y_i$ follows some
distribution from the [exponential family of distributions](https://en.wikipedia.org/wiki/Exponential_family).
Spark's `GeneralizedLinearRegression` interface
allows for flexible specification of GLMs which can be used for various types of
prediction problems including linear regression, Poisson regression, logistic regression, and others.
Currently in `spark.ml`, only a subset of the exponential family distributions are supported and they are listed
[below](#available-families).
**NOTE**: Spark currently only supports up to 4096 features through its `GeneralizedLinearRegression`
interface, and will throw an exception if this constraint is exceeded. See the [advanced section](ml-advanced) for more details.
Still, for linear and logistic regression, models with an increased number of features can be trained
using the `LinearRegression` and `LogisticRegression` estimators.
GLMs require exponential family distributions that can be written in their "canonical" or "natural" form, aka
[natural exponential family distributions](https://en.wikipedia.org/wiki/Natural_exponential_family). The form of a natural exponential family distribution is given as:
$$
f_Y(y|\theta, \tau) = h(y, \tau)\exp{\left( \frac{\theta \cdot y - A(\theta)}{d(\tau)} \right)}
$$
where $\theta$ is the parameter of interest and $\tau$ is a dispersion parameter. In a GLM the response variable $Y_i$ is assumed to be drawn from a natural exponential family distribution:
$$
Y_i \sim f\left(\cdot|\theta_i, \tau \right)
$$
where the parameter of interest $\theta_i$ is related to the expected value of the response variable $\mu_i$ by
$$
\mu_i = A'(\theta_i)
$$
Here, $A'(\theta_i)$ is defined by the form of the distribution selected. GLMs also allow specification
of a link function, which defines the relationship between the expected value of the response variable $\mu_i$
and the so called _linear predictor_ $\eta_i$:
$$
g(\mu_i) = \eta_i = \vec{x_i}^T \cdot \vec{\beta}
$$
Often, the link function is chosen such that $A' = g^{-1}$, which yields a simplified relationship
between the parameter of interest $\theta$ and the linear predictor $\eta$. In this case, the link
function $g(\mu)$ is said to be the "canonical" link function.
$$
\theta_i = A'^{-1}(\mu_i) = g(g^{-1}(\eta_i)) = \eta_i
$$
A GLM finds the regression coefficients $\vec{\beta}$ which maximize the likelihood function.
$$
\max_{\vec{\beta}} \mathcal{L}(\vec{\theta}|\vec{y},X) =
\prod_{i=1}^{N} h(y_i, \tau) \exp{\left(\frac{y_i\theta_i - A(\theta_i)}{d(\tau)}\right)}
$$
where the parameter of interest $\theta_i$ is related to the regression coefficients $\vec{\beta}$
by
$$
\theta_i = A'^{-1}(g^{-1}(\vec{x_i} \cdot \vec{\beta}))
$$
Spark's generalized linear regression interface also provides summary statistics for diagnosing the
fit of GLM models, including residuals, p-values, deviances, the Akaike information criterion, and
others.
[See here](http://data.princeton.edu/wws509/notes/) for a more comprehensive review of GLMs and their applications.
### Available families
<table class="table">
<thead>
<tr>
<th>Family</th>
<th>Response Type</th>
<th>Supported Links</th></tr>
</thead>
<tbody>
<tr>
<td>Gaussian</td>
<td>Continuous</td>
<td>Identity*, Log, Inverse</td>
</tr>
<tr>
<td>Binomial</td>
<td>Binary</td>
<td>Logit*, Probit, CLogLog</td>
</tr>
<tr>
<td>Poisson</td>
<td>Count</td>
<td>Log*, Identity, Sqrt</td>
</tr>
<tr>
<td>Gamma</td>
<td>Continuous</td>
<td>Inverse*, Idenity, Log</td>
</tr>
<tfoot><tr><td colspan="4">* Canonical Link</td></tr></tfoot>
</tbody>
</table>
**Example**
The following example demonstrates training a GLM with a Gaussian response and identity link
function and extracting model summary statistics.
<div class="codetabs">
<div data-lang="scala" markdown="1">
Refer to the [Scala API docs](api/scala/index.html#org.apache.spark.ml.regression.GeneralizedLinearRegression) for more details.
{% include_example scala/org/apache/spark/examples/ml/GeneralizedLinearRegressionExample.scala %}
</div>
<div data-lang="java" markdown="1">
Refer to the [Java API docs](api/java/org/apache/spark/ml/regression/GeneralizedLinearRegression.html) for more details.
{% include_example java/org/apache/spark/examples/ml/JavaGeneralizedLinearRegressionExample.java %}
</div>
<div data-lang="python" markdown="1">
Refer to the [Python API docs](api/python/pyspark.ml.html#pyspark.ml.regression.GeneralizedLinearRegression) for more details.
{% include_example python/ml/generalized_linear_regression_example.py %}
</div>
</div>
## Decision tree regression ## Decision tree regression