c96244f5ac
## 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.
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| layout | title | displayTitle |
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| global | Classification and regression - spark.ml | Classification and regression - spark.ml |
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Table of Contents
- This will become a table of contents (this text will be scraped). {:toc}
In spark.ml, we implement popular linear methods such as logistic
regression and linear least squares with L_1 or L_2 regularization.
Refer to the linear methods in mllib for
details about implementation and tuning. We also include a DataFrame API for Elastic
net, a hybrid
of L_1 and L_2 regularization proposed in Zou et al, Regularization
and variable selection via the elastic
net.
Mathematically, it is defined as a convex combination of the L_1 and
the L_2 regularization terms:
\[ \alpha \left( \lambda \|\wv\|_1 \right) + (1-\alpha) \left( \frac{\lambda}{2}\|\wv\|_2^2 \right) , \alpha \in [0, 1], \lambda \geq 0 \]
By setting \alpha properly, elastic net contains both L_1 and $L_2$
regularization as special cases. For example, if a linear
regression model is
trained with the elastic net parameter \alpha set to 1, it is
equivalent to a
Lasso model.
On the other hand, if \alpha is set to 0, the trained model reduces
to a ridge
regression model.
We implement Pipelines API for both linear regression and logistic
regression with elastic net regularization.
Classification
Logistic regression
Logistic regression is a popular method to predict a binary response. It is a special case of Generalized Linear models that predicts the probability of the outcome.
For more background and more details about the implementation, refer to the documentation of the logistic regression in spark.mllib.
The current implementation of logistic regression in
spark.mlonly supports binary classes. Support for multiclass regression will be added in the future.
Example
The following example shows how to train a logistic regression model
with elastic net regularization. elasticNetParam corresponds to
\alpha and regParam corresponds to \lambda.
The spark.ml implementation of logistic regression also supports
extracting a summary of the model over the training set. Note that the
predictions and metrics which are stored as DataFrame in
BinaryLogisticRegressionSummary are annotated @transient and hence
only available on the driver.
LogisticRegressionTrainingSummary
provides a summary for a
LogisticRegressionModel.
Currently, only binary classification is supported and the
summary must be explicitly cast to
BinaryLogisticRegressionTrainingSummary.
This will likely change when multiclass classification is supported.
Continuing the earlier example:
{% include_example scala/org/apache/spark/examples/ml/LogisticRegressionSummaryExample.scala %}
Continuing the earlier example:
{% include_example java/org/apache/spark/examples/ml/JavaLogisticRegressionSummaryExample.java %}
Decision tree classifier
Decision trees are a popular family of classification and regression methods.
More information about the spark.ml implementation can be found further in the section on decision trees.
Example
The following examples load a dataset in LibSVM format, split it into training and test sets, train on the first dataset, and then evaluate on the held-out test set.
We use two feature transformers to prepare the data; these help index categories for the label and categorical features, adding metadata to the DataFrame which the Decision Tree algorithm can recognize.
More details on parameters can be found in the Scala API documentation.
{% include_example scala/org/apache/spark/examples/ml/DecisionTreeClassificationExample.scala %}
More details on parameters can be found in the Java API documentation.
{% include_example java/org/apache/spark/examples/ml/JavaDecisionTreeClassificationExample.java %}
More details on parameters can be found in the Python API documentation.
{% include_example python/ml/decision_tree_classification_example.py %}
Random forest classifier
Random forests are a popular family of classification and regression methods.
More information about the spark.ml implementation can be found further in the section on random forests.
Example
The following examples load a dataset in LibSVM format, split it into training and test sets, train on the first dataset, and then evaluate on the held-out test set.
We use two feature transformers to prepare the data; these help index categories for the label and categorical features, adding metadata to the DataFrame which the tree-based algorithms can recognize.
Refer to the Scala API docs for more details.
{% include_example scala/org/apache/spark/examples/ml/RandomForestClassifierExample.scala %}
Refer to the Java API docs for more details.
{% include_example java/org/apache/spark/examples/ml/JavaRandomForestClassifierExample.java %}
Refer to the Python API docs for more details.
{% include_example python/ml/random_forest_classifier_example.py %}
Gradient-boosted tree classifier
Gradient-boosted trees (GBTs) are a popular classification and regression method using ensembles of decision trees.
More information about the spark.ml implementation can be found further in the section on GBTs.
Example
The following examples load a dataset in LibSVM format, split it into training and test sets, train on the first dataset, and then evaluate on the held-out test set.
We use two feature transformers to prepare the data; these help index categories for the label and categorical features, adding metadata to the DataFrame which the tree-based algorithms can recognize.
Refer to the Scala API docs for more details.
{% include_example scala/org/apache/spark/examples/ml/GradientBoostedTreeClassifierExample.scala %}
Refer to the Java API docs for more details.
{% include_example java/org/apache/spark/examples/ml/JavaGradientBoostedTreeClassifierExample.java %}
Refer to the Python API docs for more details.
{% include_example python/ml/gradient_boosted_tree_classifier_example.py %}
Multilayer perceptron classifier
Multilayer perceptron classifier (MLPC) is a classifier based on the feedforward artificial neural network.
MLPC consists of multiple layers of nodes.
Each layer is fully connected to the next layer in the network. Nodes in the input layer represent the input data. All other nodes map inputs to outputs
by a linear combination of the inputs with the node's weights $\wv$ and bias $\bv$ and applying an activation function.
This can be written in matrix form for MLPC with $K+1$ layers as follows:
\[ \mathrm{y}(\x) = \mathrm{f_K}(...\mathrm{f_2}(\wv_2^T\mathrm{f_1}(\wv_1^T \x+b_1)+b_2)...+b_K) \]
Nodes in intermediate layers use sigmoid (logistic) function:
\[ \mathrm{f}(z_i) = \frac{1}{1 + e^{-z_i}} \]
Nodes in the output layer use softmax function:
\[ \mathrm{f}(z_i) = \frac{e^{z_i}}{\sum_{k=1}^N e^{z_k}} \]
The number of nodes $N$ in the output layer corresponds to the number of classes.
MLPC employs backpropagation for learning the model. We use the logistic loss function for optimization and L-BFGS as an optimization routine.
Example
One-vs-Rest classifier (a.k.a. One-vs-All)
OneVsRest is an example of a machine learning reduction for performing multiclass classification given a base classifier that can perform binary classification efficiently. It is also known as "One-vs-All."
OneVsRest is implemented as an Estimator. For the base classifier it takes instances of Classifier and creates a binary classification problem for each of the k classes. The classifier for class i is trained to predict whether the label is i or not, distinguishing class i from all other classes.
Predictions are done by evaluating each binary classifier and the index of the most confident classifier is output as label.
Example
The example below demonstrates how to load the
Iris dataset, parse it as a DataFrame and perform multiclass classification using OneVsRest. The test error is calculated to measure the algorithm accuracy.
Refer to the Scala API docs for more details.
{% include_example scala/org/apache/spark/examples/ml/OneVsRestExample.scala %}
Refer to the Java API docs for more details.
{% include_example java/org/apache/spark/examples/ml/JavaOneVsRestExample.java %}
Refer to the Python API docs for more details.
{% include_example python/ml/one_vs_rest_example.py %}
Naive Bayes
Naive Bayes classifiers are a family of simple
probabilistic classifiers based on applying Bayes' theorem with strong (naive) independence
assumptions between the features. The spark.ml implementation currently supports both multinomial
naive Bayes
and Bernoulli naive Bayes.
More information can be found in the section on Naive Bayes in MLlib.
Example
Refer to the Scala API docs for more details.
{% include_example scala/org/apache/spark/examples/ml/NaiveBayesExample.scala %}
Refer to the Java API docs for more details.
{% include_example java/org/apache/spark/examples/ml/JavaNaiveBayesExample.java %}
Refer to the Python API docs for more details.
{% include_example python/ml/naive_bayes_example.py %}
Regression
Linear regression
The interface for working with linear regression models and model summaries is similar to the logistic regression case.
Example
The following example demonstrates training an elastic net regularized linear regression model and extracting model summary statistics.
Generalized linear regression
Contrasted with linear regression where the output is assumed to follow a Gaussian
distribution, generalized linear models (GLMs) are specifications of linear models where the response variable Y_i follows some
distribution from the exponential family of distributions.
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.
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 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. 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 for a more comprehensive review of GLMs and their applications.
Available families
| Family | Response Type | Supported Links | |
|---|---|---|---|
| Gaussian | Continuous | Identity*, Log, Inverse | |
| Binomial | Binary | Logit*, Probit, CLogLog | |
| Poisson | Count | Log*, Identity, Sqrt | |
| Gamma | Continuous | Inverse*, Idenity, Log | |
| * Canonical Link | |||
Example
The following example demonstrates training a GLM with a Gaussian response and identity link function and extracting model summary statistics.
{% include_example scala/org/apache/spark/examples/ml/GeneralizedLinearRegressionExample.scala %}
{% include_example java/org/apache/spark/examples/ml/JavaGeneralizedLinearRegressionExample.java %}
{% include_example python/ml/generalized_linear_regression_example.py %}
Decision tree regression
Decision trees are a popular family of classification and regression methods.
More information about the spark.ml implementation can be found further in the section on decision trees.
Example
The following examples load a dataset in LibSVM format, split it into training and test sets, train on the first dataset, and then evaluate on the held-out test set.
We use a feature transformer to index categorical features, adding metadata to the DataFrame which the Decision Tree algorithm can recognize.
More details on parameters can be found in the Scala API documentation.
{% include_example scala/org/apache/spark/examples/ml/DecisionTreeRegressionExample.scala %}
More details on parameters can be found in the Java API documentation.
{% include_example java/org/apache/spark/examples/ml/JavaDecisionTreeRegressionExample.java %}
More details on parameters can be found in the Python API documentation.
{% include_example python/ml/decision_tree_regression_example.py %}
Random forest regression
Random forests are a popular family of classification and regression methods.
More information about the spark.ml implementation can be found further in the section on random forests.
Example
The following examples load a dataset in LibSVM format, split it into training and test sets, train on the first dataset, and then evaluate on the held-out test set.
We use a feature transformer to index categorical features, adding metadata to the DataFrame which the tree-based algorithms can recognize.
Refer to the Scala API docs for more details.
{% include_example scala/org/apache/spark/examples/ml/RandomForestRegressorExample.scala %}
Refer to the Java API docs for more details.
{% include_example java/org/apache/spark/examples/ml/JavaRandomForestRegressorExample.java %}
Refer to the Python API docs for more details.
{% include_example python/ml/random_forest_regressor_example.py %}
Gradient-boosted tree regression
Gradient-boosted trees (GBTs) are a popular regression method using ensembles of decision trees.
More information about the spark.ml implementation can be found further in the section on GBTs.
Example
Note: For this example dataset, GBTRegressor actually only needs 1 iteration, but that will not
be true in general.
Refer to the Scala API docs for more details.
{% include_example scala/org/apache/spark/examples/ml/GradientBoostedTreeRegressorExample.scala %}
Refer to the Java API docs for more details.
{% include_example java/org/apache/spark/examples/ml/JavaGradientBoostedTreeRegressorExample.java %}
Refer to the Python API docs for more details.
{% include_example python/ml/gradient_boosted_tree_regressor_example.py %}
Survival regression
In spark.ml, we implement the Accelerated failure time (AFT)
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 a
log-linear model for survival analysis. Different from a
Proportional hazards model
designed for the same purpose, the AFT model is easier to parallelize
because each instance contributes 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 the corresponding density function.
The most commonly used AFT model is based on the Weibull distribution of the survival time.
The Weibull distribution for lifetime corresponds to the extreme value distribution for the
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 a 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 on the 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
Example
Decision trees
Decision trees and their ensembles are popular methods for the machine learning tasks of classification and regression. Decision trees are widely used since they are easy to interpret, handle categorical features, extend to the multiclass classification setting, do not require feature scaling, and are able to capture non-linearities and feature interactions. Tree ensemble algorithms such as random forests and boosting are among the top performers for classification and regression tasks.
The spark.ml implementation supports decision trees for binary and multiclass classification and for regression,
using both continuous and categorical features. The implementation partitions data by rows,
allowing distributed training with millions or even billions of instances.
Users can find more information about the decision tree algorithm in the MLlib Decision Tree guide. The main differences between this API and the original MLlib Decision Tree API are:
- support for ML Pipelines
- separation of Decision Trees for classification vs. regression
- use of DataFrame metadata to distinguish continuous and categorical features
The Pipelines API for Decision Trees offers a bit more functionality than the original API.
In particular, for classification, users can get the predicted probability of each class (a.k.a. class conditional probabilities);
for regression, users can get the biased sample variance of prediction.
Ensembles of trees (Random Forests and Gradient-Boosted Trees) are described below in the Tree ensembles section.
Inputs and Outputs
We list the input and output (prediction) column types here. All output columns are optional; to exclude an output column, set its corresponding Param to an empty string.
Input Columns
| Param name | Type(s) | Default | Description |
|---|---|---|---|
| labelCol | Double | "label" | Label to predict |
| featuresCol | Vector | "features" | Feature vector |
Output Columns
| Param name | Type(s) | Default | Description | Notes |
|---|---|---|---|---|
| predictionCol | Double | "prediction" | Predicted label | |
| rawPredictionCol | Vector | "rawPrediction" | Vector of length # classes, with the counts of training instance labels at the tree node which makes the prediction | Classification only |
| probabilityCol | Vector | "probability" | Vector of length # classes equal to rawPrediction normalized to a multinomial distribution | Classification only |
| varianceCol | Double | The biased sample variance of prediction | Regression only |
Tree Ensembles
The DataFrame API supports two major tree ensemble algorithms: Random Forests and Gradient-Boosted Trees (GBTs).
Both use spark.ml decision trees as their base models.
Users can find more information about ensemble algorithms in the MLlib Ensemble guide.
In this section, we demonstrate the DataFrame API for ensembles.
The main differences between this API and the original MLlib ensembles API are:
- support for DataFrames and ML Pipelines
- separation of classification vs. regression
- use of DataFrame metadata to distinguish continuous and categorical features
- more functionality for random forests: estimates of feature importance, as well as the predicted probability of each class (a.k.a. class conditional probabilities) for classification.
Random Forests
Random forests
are ensembles of decision trees.
Random forests combine many decision trees in order to reduce the risk of overfitting.
The spark.ml implementation supports random forests for binary and multiclass classification and for regression,
using both continuous and categorical features.
For more information on the algorithm itself, please see the spark.mllib documentation on random forests.
Inputs and Outputs
We list the input and output (prediction) column types here. All output columns are optional; to exclude an output column, set its corresponding Param to an empty string.
Input Columns
| Param name | Type(s) | Default | Description |
|---|---|---|---|
| labelCol | Double | "label" | Label to predict |
| featuresCol | Vector | "features" | Feature vector |
Output Columns (Predictions)
| Param name | Type(s) | Default | Description | Notes |
|---|---|---|---|---|
| predictionCol | Double | "prediction" | Predicted label | |
| rawPredictionCol | Vector | "rawPrediction" | Vector of length # classes, with the counts of training instance labels at the tree node which makes the prediction | Classification only |
| probabilityCol | Vector | "probability" | Vector of length # classes equal to rawPrediction normalized to a multinomial distribution | Classification only |
Gradient-Boosted Trees (GBTs)
Gradient-Boosted Trees (GBTs)
are ensembles of decision trees.
GBTs iteratively train decision trees in order to minimize a loss function.
The spark.ml implementation supports GBTs for binary classification and for regression,
using both continuous and categorical features.
For more information on the algorithm itself, please see the spark.mllib documentation on GBTs.
Inputs and Outputs
We list the input and output (prediction) column types here. All output columns are optional; to exclude an output column, set its corresponding Param to an empty string.
Input Columns
| Param name | Type(s) | Default | Description |
|---|---|---|---|
| labelCol | Double | "label" | Label to predict |
| featuresCol | Vector | "features" | Feature vector |
Note that GBTClassifier currently only supports binary labels.
Output Columns (Predictions)
| Param name | Type(s) | Default | Description | Notes |
|---|---|---|---|---|
| predictionCol | Double | "prediction" | Predicted label |
In the future, GBTClassifier will also output columns for rawPrediction and probability, just as RandomForestClassifier does.