# Classification and regression

2020-01-21

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This page covers algorithms for Classification and Regression. It also includes sections discussing specific classes of algorithms, such as linear methods, trees, and ensembles.

# Classification

## Logistic regression

Logistic regression is a popular method to predict a categorical response. It is a special case of Generalized Linear models that predicts the probability of the outcomes. In spark.ml logistic regression can be used to predict a binary outcome by using binomial logistic regression, or it can be used to predict a multiclass outcome by using multinomial logistic regression. Use the family parameter to select between these two algorithms, or leave it unset and Spark will infer the correct variant.

> Multinomial logistic regression can be used for binary classification by setting the family param to "multinomial". It will produce two sets of coefficients and two intercepts.

> When fitting LogisticRegressionModel without intercept on dataset with constant nonzero column, Spark MLlib outputs zero coefficients for constant nonzero columns. This behavior is the same as R glmnet but different from LIBSVM.

### Binomial logistic regression

For more background and more details about the implementation of binomial logistic regression, refer to the documentation of logistic regression in spark.mllib.

Examples

The following example shows how to train binomial and multinomial logistic regression models for binary classification with elastic net regularization. elasticNetParam corresponds to $\alpha$ and regParam corresponds to $\lambda$.

More details on parameters can be found in the [Scala API documentation](api/scala/index.html#org.apache.spark.ml.classification.LogisticRegression). {% include_example scala/org/apache/spark/examples/ml/LogisticRegressionWithElasticNetExample.scala %}
More details on parameters can be found in the [Java API documentation](api/java/org/apache/spark/ml/classification/LogisticRegression.html). {% include_example java/org/apache/spark/examples/ml/JavaLogisticRegressionWithElasticNetExample.java %}
More details on parameters can be found in the [Python API documentation](api/python/pyspark.ml.html#pyspark.ml.classification.LogisticRegression). {% include_example python/ml/logistic_regression_with_elastic_net.py %}
More details on parameters can be found in the [R API documentation](api/R/spark.logit.html). {% include_example binomial r/ml/logit.R %}

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 LogisticRegressionSummary are annotated @transient and hence only available on the driver.

[LogisticRegressionTrainingSummary](api/scala/index.html#org.apache.spark.ml.classification.LogisticRegressionTrainingSummary) provides a summary for a [LogisticRegressionModel](api/scala/index.html#org.apache.spark.ml.classification.LogisticRegressionModel). In the case of binary classification, certain additional metrics are available, e.g. ROC curve. The binary summary can be accessed via the binarySummary method. See [BinaryLogisticRegressionTrainingSummary](api/scala/index.html#org.apache.spark.ml.classification.BinaryLogisticRegressionTrainingSummary). Continuing the earlier example: {% include_example scala/org/apache/spark/examples/ml/LogisticRegressionSummaryExample.scala %}
[LogisticRegressionTrainingSummary](api/java/org/apache/spark/ml/classification/LogisticRegressionTrainingSummary.html) provides a summary for a [LogisticRegressionModel](api/java/org/apache/spark/ml/classification/LogisticRegressionModel.html). In the case of binary classification, certain additional metrics are available, e.g. ROC curve. The binary summary can be accessed via the binarySummary method. See [BinaryLogisticRegressionTrainingSummary](api/java/org/apache/spark/ml/classification/BinaryLogisticRegressionTrainingSummary.html). Continuing the earlier example: {% include_example java/org/apache/spark/examples/ml/JavaLogisticRegressionSummaryExample.java %}
[LogisticRegressionTrainingSummary](api/python/pyspark.ml.html#pyspark.ml.classification.LogisticRegressionSummary) provides a summary for a [LogisticRegressionModel](api/python/pyspark.ml.html#pyspark.ml.classification.LogisticRegressionModel). In the case of binary classification, certain additional metrics are available, e.g. ROC curve. See [BinaryLogisticRegressionTrainingSummary](api/python/pyspark.ml.html#pyspark.ml.classification.BinaryLogisticRegressionTrainingSummary). Continuing the earlier example: {% include_example python/ml/logistic_regression_summary_example.py %}

### Multinomial logistic regression

Multiclass classification is supported via multinomial logistic (softmax) regression. In multinomial logistic regression, the algorithm produces $K$ sets of coefficients, or a matrix of dimension $K \times J$ where $K$ is the number of outcome classes and $J$ is the number of features. If the algorithm is fit with an intercept term then a length $K$ vector of intercepts is available.

> Multinomial coefficients are available as coefficientMatrix and intercepts are available as interceptVector.

> coefficients and intercept methods on a logistic regression model trained with multinomial family are not supported. Use coefficientMatrix and interceptVector instead.

The conditional probabilities of the outcome classes $k \in {1, 2, ..., K}$ are modeled using the softmax function.

$P(Y=k|\mathbf{X}, \boldsymbol{\beta}_k, \beta_{0k}) = \frac{e^{\boldsymbol{\beta}_k \cdot \mathbf{X} + \beta_{0k}}}{\sum_{k'=0}^{K-1} e^{\boldsymbol{\beta}_{k'} \cdot \mathbf{X} + \beta_{0k'}}}$

We minimize the weighted negative log-likelihood, using a multinomial response model, with elastic-net penalty to control for overfitting.

$\min_{\beta, \beta_0} -\left[\sum_{i=1}^L w_i \cdot \log P(Y = y_i|\mathbf{x}_i)\right] + \lambda \left[\frac{1}{2}\left(1 - \alpha\right)||\boldsymbol{\beta}||_2^2 + \alpha ||\boldsymbol{\beta}||_1\right]$

For a detailed derivation please see here.

Examples

The following example shows how to train a multiclass logistic regression model with elastic net regularization, as well as extract the multiclass training summary for evaluating the model.

{% include_example scala/org/apache/spark/examples/ml/MulticlassLogisticRegressionWithElasticNetExample.scala %}
{% include_example java/org/apache/spark/examples/ml/JavaMulticlassLogisticRegressionWithElasticNetExample.java %}
{% include_example python/ml/multiclass_logistic_regression_with_elastic_net.py %}
More details on parameters can be found in the [R API documentation](api/R/spark.logit.html). {% include_example multinomial r/ml/logit.R %}

## 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.

Examples

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](api/scala/index.html#org.apache.spark.ml.classification.DecisionTreeClassifier). {% include_example scala/org/apache/spark/examples/ml/DecisionTreeClassificationExample.scala %}
More details on parameters can be found in the [Java API documentation](api/java/org/apache/spark/ml/classification/DecisionTreeClassifier.html). {% include_example java/org/apache/spark/examples/ml/JavaDecisionTreeClassificationExample.java %}
More details on parameters can be found in the [Python API documentation](api/python/pyspark.ml.html#pyspark.ml.classification.DecisionTreeClassifier). {% include_example python/ml/decision_tree_classification_example.py %}
Refer to the [R API docs](api/R/spark.decisionTree.html) for more details. {% include_example classification r/ml/decisionTree.R %}

## 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.

Examples

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](api/scala/index.html#org.apache.spark.ml.classification.RandomForestClassifier) for more details. {% include_example scala/org/apache/spark/examples/ml/RandomForestClassifierExample.scala %}
Refer to the [Java API docs](api/java/org/apache/spark/ml/classification/RandomForestClassifier.html) for more details. {% include_example java/org/apache/spark/examples/ml/JavaRandomForestClassifierExample.java %}
Refer to the [Python API docs](api/python/pyspark.ml.html#pyspark.ml.classification.RandomForestClassifier) for more details. {% include_example python/ml/random_forest_classifier_example.py %}
Refer to the [R API docs](api/R/spark.randomForest.html) for more details. {% include_example classification r/ml/randomForest.R %}

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.

Examples

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](api/scala/index.html#org.apache.spark.ml.classification.GBTClassifier) for more details. {% include_example scala/org/apache/spark/examples/ml/GradientBoostedTreeClassifierExample.scala %}
Refer to the [Java API docs](api/java/org/apache/spark/ml/classification/GBTClassifier.html) for more details. {% include_example java/org/apache/spark/examples/ml/JavaGradientBoostedTreeClassifierExample.java %}
Refer to the [Python API docs](api/python/pyspark.ml.html#pyspark.ml.classification.GBTClassifier) for more details. {% include_example python/ml/gradient_boosted_tree_classifier_example.py %}
Refer to the [R API docs](api/R/spark.gbt.html) for more details. {% include_example classification r/ml/gbt.R %}

## 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.

Examples

Refer to the [Scala API docs](api/scala/index.html#org.apache.spark.ml.classification.MultilayerPerceptronClassifier) for more details. {% include_example scala/org/apache/spark/examples/ml/MultilayerPerceptronClassifierExample.scala %}
Refer to the [Java API docs](api/java/org/apache/spark/ml/classification/MultilayerPerceptronClassifier.html) for more details. {% include_example java/org/apache/spark/examples/ml/JavaMultilayerPerceptronClassifierExample.java %}
Refer to the [Python API docs](api/python/pyspark.ml.html#pyspark.ml.classification.MultilayerPerceptronClassifier) for more details. {% include_example python/ml/multilayer_perceptron_classification.py %}
Refer to the [R API docs](api/R/spark.mlp.html) for more details. {% include_example r/ml/mlp.R %}

## Linear Support Vector Machine

A support vector machine constructs a hyperplane or set of hyperplanes in a high- or infinite-dimensional space, which can be used for classification, regression, or other tasks. Intuitively, a good separation is achieved by the hyperplane that has the largest distance to the nearest training-data points of any class (so-called functional margin), since in general the larger the margin the lower the generalization error of the classifier. LinearSVC in Spark ML supports binary classification with linear SVM. Internally, it optimizes the Hinge Loss using OWLQN optimizer.

Examples

Refer to the [Scala API docs](api/scala/index.html#org.apache.spark.ml.classification.LinearSVC) for more details. {% include_example scala/org/apache/spark/examples/ml/LinearSVCExample.scala %}
Refer to the [Java API docs](api/java/org/apache/spark/ml/classification/LinearSVC.html) for more details. {% include_example java/org/apache/spark/examples/ml/JavaLinearSVCExample.java %}
Refer to the [Python API docs](api/python/pyspark.ml.html#pyspark.ml.classification.LinearSVC) for more details. {% include_example python/ml/linearsvc.py %}
Refer to the [R API docs](api/R/spark.svmLinear.html) for more details. {% include_example r/ml/svmLinear.R %}

## 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.

Examples

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](api/scala/index.html#org.apache.spark.ml.classification.OneVsRest) for more details. {% include_example scala/org/apache/spark/examples/ml/OneVsRestExample.scala %}
Refer to the [Java API docs](api/java/org/apache/spark/ml/classification/OneVsRest.html) for more details. {% include_example java/org/apache/spark/examples/ml/JavaOneVsRestExample.java %}
Refer to the [Python API docs](api/python/pyspark.ml.html#pyspark.ml.classification.OneVsRest) for more details. {% include_example python/ml/one_vs_rest_example.py %}

## Naive Bayes

Naive Bayes classifiers are a family of simple probabilistic, multiclass classifiers based on applying Bayes' theorem with strong (naive) independence assumptions between every pair of features.

Naive Bayes can be trained very efficiently. With a single pass over the training data, it computes the conditional probability distribution of each feature given each label. For prediction, it applies Bayes' theorem to compute the conditional probability distribution of each label given an observation.

Input data: These Multinomial, Complement and Bernoulli models are typically used for document classification. Within that context, each observation is a document and each feature represents a term. A feature's value is the frequency of the term (in Multinomial or Complement Naive Bayes) or a zero or one indicating whether the term was found in the document (in Bernoulli Naive Bayes). Feature values for Multinomial and Bernoulli models must be non-negative. The model type is selected with an optional parameter "multinomial", "complement", "bernoulli" or "gaussian", with "multinomial" as the default. For document classification, the input feature vectors should usually be sparse vectors. Since the training data is only used once, it is not necessary to cache it.

Additive smoothing can be used by setting the parameter $\lambda$ (default to $1.0$).

Examples

Refer to the [Scala API docs](api/scala/index.html#org.apache.spark.ml.classification.NaiveBayes) for more details. {% include_example scala/org/apache/spark/examples/ml/NaiveBayesExample.scala %}
Refer to the [Java API docs](api/java/org/apache/spark/ml/classification/NaiveBayes.html) for more details. {% include_example java/org/apache/spark/examples/ml/JavaNaiveBayesExample.java %}
Refer to the [Python API docs](api/python/pyspark.ml.html#pyspark.ml.classification.NaiveBayes) for more details. {% include_example python/ml/naive_bayes_example.py %}
Refer to the [R API docs](api/R/spark.naiveBayes.html) for more details. {% include_example r/ml/naiveBayes.R %}

## Factorization machines classifier

For more background and more details about the implementation of factorization machines, refer to the Factorization Machines section.

Examples

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 scale features to be between 0 and 1 to prevent the exploding gradient problem.

Refer to the [Scala API docs](api/scala/index.html#org.apache.spark.ml.classification.FMClassifier) for more details. {% include_example scala/org/apache/spark/examples/ml/FMClassifierExample.scala %}
Refer to the [Java API docs](api/java/org/apache/spark/ml/classification/FMClassifier.html) for more details. {% include_example java/org/apache/spark/examples/ml/JavaFMClassifierExample.java %}
Refer to the [Python API docs](api/python/pyspark.ml.html#pyspark.ml.classification.FMClassifier) for more details. {% include_example python/ml/fm_classifier_example.py %}

# Regression

## Linear regression

The interface for working with linear regression models and model summaries is similar to the logistic regression case.

> When fitting LinearRegressionModel without intercept on dataset with constant nonzero column by "l-bfgs" solver, Spark MLlib outputs zero coefficients for constant nonzero columns. This behavior is the same as R glmnet but different from LIBSVM.

Examples

The following example demonstrates training an elastic net regularized linear regression model and extracting model summary statistics.

More details on parameters can be found in the [Scala API documentation](api/scala/index.html#org.apache.spark.ml.regression.LinearRegression). {% include_example scala/org/apache/spark/examples/ml/LinearRegressionWithElasticNetExample.scala %}
More details on parameters can be found in the [Java API documentation](api/java/org/apache/spark/ml/regression/LinearRegression.html). {% include_example java/org/apache/spark/examples/ml/JavaLinearRegressionWithElasticNetExample.java %}
More details on parameters can be found in the [Python API documentation](api/python/pyspark.ml.html#pyspark.ml.regression.LinearRegression). {% include_example python/ml/linear_regression_with_elastic_net.py %}

## 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

Gaussian Continuous Identity*, Log, Inverse
Binomial Binary Logit*, Probit, CLogLog
Poisson Count Log*, Identity, Sqrt
Gamma Continuous Inverse*, Identity, Log
Tweedie Zero-inflated continuous Power link function

Examples

The following example demonstrates training a GLM with a Gaussian response and identity link function and extracting model summary statistics.

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 %}
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 %}
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 %}
Refer to the [R API docs](api/R/spark.glm.html) for more details. {% include_example r/ml/glm.R %}

## 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.

Examples

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](api/scala/index.html#org.apache.spark.ml.regression.DecisionTreeRegressor). {% include_example scala/org/apache/spark/examples/ml/DecisionTreeRegressionExample.scala %}
More details on parameters can be found in the [Java API documentation](api/java/org/apache/spark/ml/regression/DecisionTreeRegressor.html). {% include_example java/org/apache/spark/examples/ml/JavaDecisionTreeRegressionExample.java %}
More details on parameters can be found in the [Python API documentation](api/python/pyspark.ml.html#pyspark.ml.regression.DecisionTreeRegressor). {% include_example python/ml/decision_tree_regression_example.py %}
Refer to the [R API docs](api/R/spark.decisionTree.html) for more details. {% include_example regression r/ml/decisionTree.R %}

## 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.

Examples

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](api/scala/index.html#org.apache.spark.ml.regression.RandomForestRegressor) for more details. {% include_example scala/org/apache/spark/examples/ml/RandomForestRegressorExample.scala %}
Refer to the [Java API docs](api/java/org/apache/spark/ml/regression/RandomForestRegressor.html) for more details. {% include_example java/org/apache/spark/examples/ml/JavaRandomForestRegressorExample.java %}
Refer to the [Python API docs](api/python/pyspark.ml.html#pyspark.ml.regression.RandomForestRegressor) for more details. {% include_example python/ml/random_forest_regressor_example.py %}
Refer to the [R API docs](api/R/spark.randomForest.html) for more details. {% include_example regression r/ml/randomForest.R %}

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.

Examples

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](api/scala/index.html#org.apache.spark.ml.regression.GBTRegressor) for more details. {% include_example scala/org/apache/spark/examples/ml/GradientBoostedTreeRegressorExample.scala %}
Refer to the [Java API docs](api/java/org/apache/spark/ml/regression/GBTRegressor.html) for more details. {% include_example java/org/apache/spark/examples/ml/JavaGradientBoostedTreeRegressorExample.java %}
Refer to the [Python API docs](api/python/pyspark.ml.html#pyspark.ml.regression.GBTRegressor) for more details. {% include_example python/ml/gradient_boosted_tree_regressor_example.py %}
Refer to the [R API docs](api/R/spark.gbt.html) for more details. {% include_example regression r/ml/gbt.R %}

## 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

> When fitting AFTSurvivalRegressionModel without intercept on dataset with constant nonzero column, Spark MLlib outputs zero coefficients for constant nonzero columns. This behavior is different from R survival::survreg.

Examples

Refer to the [Scala API docs](api/scala/index.html#org.apache.spark.ml.regression.AFTSurvivalRegression) for more details. {% include_example scala/org/apache/spark/examples/ml/AFTSurvivalRegressionExample.scala %}
Refer to the [Java API docs](api/java/org/apache/spark/ml/regression/AFTSurvivalRegression.html) for more details. {% include_example java/org/apache/spark/examples/ml/JavaAFTSurvivalRegressionExample.java %}
Refer to the [Python API docs](api/python/pyspark.ml.html#pyspark.ml.regression.AFTSurvivalRegression) for more details. {% include_example python/ml/aft_survival_regression.py %}
Refer to the [R API docs](api/R/spark.survreg.html) for more details. {% include_example r/ml/survreg.R %}

## Isotonic regression

Isotonic regression belongs to the family of regression algorithms. Formally isotonic regression is a problem where given a finite set of real numbers $Y = {y_1, y_2, ..., y_n}$ representing observed responses and $X = {x_1, x_2, ..., x_n}$ the unknown response values to be fitted finding a function that minimizes

$$f(x) = \sum_{i=1}^n w_i (y_i - x_i)^2$$

with respect to complete order subject to $x_1\le x_2\le ...\le x_n$ where $w_i$ are positive weights. The resulting function is called isotonic regression and it is unique. It can be viewed as least squares problem under order restriction. Essentially isotonic regression is a monotonic function best fitting the original data points.

We implement a pool adjacent violators algorithm which uses an approach to parallelizing isotonic regression. The training input is a DataFrame which contains three columns label, features and weight. Additionally, IsotonicRegression algorithm has one optional parameter called $isotonic$ defaulting to true. This argument specifies if the isotonic regression is isotonic (monotonically increasing) or antitonic (monotonically decreasing).

Training returns an IsotonicRegressionModel that can be used to predict labels for both known and unknown features. The result of isotonic regression is treated as piecewise linear function. The rules for prediction therefore are:

• If the prediction input exactly matches a training feature then associated prediction is returned. In case there are multiple predictions with the same feature then one of them is returned. Which one is undefined (same as java.util.Arrays.binarySearch).
• If the prediction input is lower or higher than all training features then prediction with lowest or highest feature is returned respectively. In case there are multiple predictions with the same feature then the lowest or highest is returned respectively.
• If the prediction input falls between two training features then prediction is treated as piecewise linear function and interpolated value is calculated from the predictions of the two closest features. In case there are multiple values with the same feature then the same rules as in previous point are used.

Examples

Refer to the [IsotonicRegression Scala docs](api/scala/index.html#org.apache.spark.ml.regression.IsotonicRegression) for details on the API. {% include_example scala/org/apache/spark/examples/ml/IsotonicRegressionExample.scala %}
Refer to the [IsotonicRegression Java docs](api/java/org/apache/spark/ml/regression/IsotonicRegression.html) for details on the API. {% include_example java/org/apache/spark/examples/ml/JavaIsotonicRegressionExample.java %}
Refer to the [IsotonicRegression Python docs](api/python/pyspark.ml.html#pyspark.ml.regression.IsotonicRegression) for more details on the API. {% include_example python/ml/isotonic_regression_example.py %}
Refer to the [IsotonicRegression R API docs](api/R/spark.isoreg.html) for more details on the API. {% include_example r/ml/isoreg.R %}

## Factorization machines regressor

For more background and more details about the implementation of factorization machines, refer to the Factorization Machines section.

Examples

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 scale features to be between 0 and 1 to prevent the exploding gradient problem.

Refer to the [Scala API docs](api/scala/index.html#org.apache.spark.ml.regression.FMRegressor) for more details. {% include_example scala/org/apache/spark/examples/ml/FMRegressorExample.scala %}
Refer to the [Java API docs](api/java/org/apache/spark/ml/regression/FMRegressor.html) for more details. {% include_example java/org/apache/spark/examples/ml/JavaFMRegressorExample.java %}
Refer to the [Python API docs](api/python/pyspark.ml.html#pyspark.ml.regression.FMRegressor) for more details. {% include_example python/ml/fm_regressor_example.py %}

# Linear methods

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 guide for the RDD-based API for details about implementation and tuning; this information is still relevant.

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.

# Factorization Machines

Factorization Machines are able to estimate interactions between features even in problems with huge sparsity (like advertising and recommendation system). The spark.ml implementation supports factorization machines for binary classification and for regression.

Factorization machines formula is:

$$\hat{y} = w_0 + \sum\limits^n_{i-1} w_i x_i + \sum\limits^n_{i=1} \sum\limits^n_{j=i+1} \langle v_i, v_j \rangle x_i x_j$$

The first two terms denote intercept and linear term (same as in linear regression), and the last term denotes pairwise interactions term. $$v_i$$ describes the i-th variable with k factors.

FM can be used for regression and optimization criterion is mean square error. FM also can be used for binary classification through sigmoid function. The optimization criterion is logistic loss.

The pairwise interactions can be reformulated:

$$\sum\limits^n_{i=1} \sum\limits^n_{j=i+1} \langle v_i, v_j \rangle x_i x_j = \frac{1}{2}\sum\limits^k_{f=1} \left(\left( \sum\limits^n_{i=1}v_{i,f}x_i \right)^2 - \sum\limits^n_{i=1}v_{i,f}^2x_i^2 \right)$$

This equation has only linear complexity in both k and n - i.e. its computation is in $$O(kn)$$.

In general, in order to prevent the exploding gradient problem, it is best to scale continuous features to be between 0 and 1, or bin the continuous features and one-hot encode them.

# 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.

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) 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.