layout	title	displayTitle
global	Evaluation Metrics - MLlib	<a href="mllib-guide.html">MLlib</a> - Evaluation Metrics

Table of contents {:toc}

Spark's MLlib comes with a number of machine learning algorithms that can be used to learn from and make predictions on data. When these algorithms are applied to build machine learning models, there is a need to evaluate the performance of the model on some criteria, which depends on the application and its requirements. Spark's MLlib also provides a suite of metrics for the purpose of evaluating the performance of machine learning models.

Specific machine learning algorithms fall under broader types of machine learning applications like classification, regression, clustering, etc. Each of these types have well established metrics for performance evaluation and those metrics that are currently available in Spark's MLlib are detailed in this section.

Classification model evaluation

While there are many different types of classification algorithms, the evaluation of classification models all share similar principles. In a supervised classification problem, there exists a true output and a model-generated predicted output for each data point. For this reason, the results for each data point can be assigned to one of four categories:

True Positive (TP) - label is positive and prediction is also positive
True Negative (TN) - label is negative and prediction is also negative
False Positive (FP) - label is negative but prediction is positive
False Negative (FN) - label is positive but prediction is negative

These four numbers are the building blocks for most classifier evaluation metrics. A fundamental point when considering classifier evaluation is that pure accuracy (i.e. was the prediction correct or incorrect) is not generally a good metric. The reason for this is because a dataset may be highly unbalanced. For example, if a model is designed to predict fraud from a dataset where 95% of the data points are not fraud and 5% of the data points are fraud, then a naive classifier that predicts not fraud, regardless of input, will be 95% accurate. For this reason, metrics like precision and recall are typically used because they take into account the type of error. In most applications there is some desired balance between precision and recall, which can be captured by combining the two into a single metric, called the F-measure.

Binary classification

Binary classifiers are used to separate the elements of a given dataset into one of two possible groups (e.g. fraud or not fraud) and is a special case of multiclass classification. Most binary classification metrics can be generalized to multiclass classification metrics.

Threshold tuning

It is import to understand that many classification models actually output a "score" (often times a probability) for each class, where a higher score indicates higher likelihood. In the binary case, the model may output a probability for each class: $P(Y=1|X)$ and $P(Y=0|X)$. Instead of simply taking the higher probability, there may be some cases where the model might need to be tuned so that it only predicts a class when the probability is very high (e.g. only block a credit card transaction if the model predicts fraud with >90% probability). Therefore, there is a prediction threshold which determines what the predicted class will be based on the probabilities that the model outputs.

Tuning the prediction threshold will change the precision and recall of the model and is an important part of model optimization. In order to visualize how precision, recall, and other metrics change as a function of the threshold it is common practice to plot competing metrics against one another, parameterized by threshold. A P-R curve plots (precision, recall) points for different threshold values, while a receiver operating characteristic, or ROC, curve plots (recall, false positive rate) points.

Available metrics

Metric	Definition
Precision (Postive Predictive Value)	$PPV=\frac{TP}{TP + FP}$
Recall (True Positive Rate)	$TPR=\frac{TP}{P}=\frac{TP}{TP + FN}$
F-measure	$F(\beta) = \left(1 + \beta^2\right) \cdot \left(\frac{PPV \cdot TPR} {\beta^2 \cdot PPV + TPR}\right)$
Receiver Operating Characteristic (ROC)	$FPR(T)=\int^\infty_{T} P_0(T)\,dT \\ TPR(T)=\int^\infty_{T} P_1(T)\,dT$
Area Under ROC Curve	$AUROC=\int^1_{0} \frac{TP}{P} d\left(\frac{FP}{N}\right)$
Area Under Precision-Recall Curve	$AUPRC=\int^1_{0} \frac{TP}{TP+FP} d\left(\frac{TP}{P}\right)$

Examples

The following code snippets illustrate how to load a sample dataset, train a binary classification algorithm on the data, and evaluate the performance of the algorithm by several binary evaluation metrics.

{% highlight scala %} import org.apache.spark.mllib.classification.LogisticRegressionWithLBFGS import org.apache.spark.mllib.evaluation.BinaryClassificationMetrics import org.apache.spark.mllib.regression.LabeledPoint import org.apache.spark.mllib.util.MLUtils

// Load training data in LIBSVM format val data = MLUtils.loadLibSVMFile(sc, "data/mllib/sample_binary_classification_data.txt")

// Split data into training (60%) and test (40%) val Array(training, test) = data.randomSplit(Array(0.6, 0.4), seed = 11L) training.cache()

// Run training algorithm to build the model val model = new LogisticRegressionWithLBFGS() .setNumClasses(2) .run(training)

// Clear the prediction threshold so the model will return probabilities model.clearThreshold

// Compute raw scores on the test set val predictionAndLabels = test.map { case LabeledPoint(label, features) => val prediction = model.predict(features) (prediction, label) }

// Instantiate metrics object val metrics = new BinaryClassificationMetrics(predictionAndLabels)

// Precision by threshold val precision = metrics.precisionByThreshold precision.foreach { case (t, p) => println(s"Threshold: $t, Precision: $p") }

// Recall by threshold val recall = metrics.recallByThreshold recall.foreach { case (t, r) => println(s"Threshold: $t, Recall: $r") }

// Precision-Recall Curve val PRC = metrics.pr

// F-measure val f1Score = metrics.fMeasureByThreshold f1Score.foreach { case (t, f) => println(s"Threshold: $t, F-score: $f, Beta = 1") }

val beta = 0.5 val fScore = metrics.fMeasureByThreshold(beta) f1Score.foreach { case (t, f) => println(s"Threshold: $t, F-score: $f, Beta = 0.5") }

// AUPRC val auPRC = metrics.areaUnderPR println("Area under precision-recall curve = " + auPRC)

// Compute thresholds used in ROC and PR curves val thresholds = precision.map(_._1)

// ROC Curve val roc = metrics.roc

// AUROC val auROC = metrics.areaUnderROC println("Area under ROC = " + auROC)

{% endhighlight %}

{% highlight java %} import scala.Tuple2;

import org.apache.spark.api.java.*; import org.apache.spark.rdd.RDD; import org.apache.spark.api.java.function.Function; import org.apache.spark.mllib.classification.LogisticRegressionModel; import org.apache.spark.mllib.classification.LogisticRegressionWithLBFGS; import org.apache.spark.mllib.evaluation.BinaryClassificationMetrics; import org.apache.spark.mllib.regression.LabeledPoint; import org.apache.spark.mllib.util.MLUtils; import org.apache.spark.SparkConf; import org.apache.spark.SparkContext;

public class BinaryClassification { public static void main(String[] args) { SparkConf conf = new SparkConf().setAppName("Binary Classification Metrics"); SparkContext sc = new SparkContext(conf); String path = "data/mllib/sample_binary_classification_data.txt"; JavaRDD data = MLUtils.loadLibSVMFile(sc, path).toJavaRDD();

// Split initial RDD into two... [60% training data, 40% testing data].
JavaRDD<LabeledPoint>[] splits = data.randomSplit(new double[] {0.6, 0.4}, 11L);
JavaRDD<LabeledPoint> training = splits[0].cache();
JavaRDD<LabeledPoint> test = splits[1];

// Run training algorithm to build the model.
final LogisticRegressionModel model = new LogisticRegressionWithLBFGS()
  .setNumClasses(2)
  .run(training.rdd());

// Clear the prediction threshold so the model will return probabilities
model.clearThreshold();

// Compute raw scores on the test set.
JavaRDD<Tuple2<Object, Object>> predictionAndLabels = test.map(
  new Function<LabeledPoint, Tuple2<Object, Object>>() {
    public Tuple2<Object, Object> call(LabeledPoint p) {
      Double prediction = model.predict(p.features());
      return new Tuple2<Object, Object>(prediction, p.label());
    }
  }
);

// Get evaluation metrics.
BinaryClassificationMetrics metrics = new BinaryClassificationMetrics(predictionAndLabels.rdd());

// Precision by threshold
JavaRDD<Tuple2<Object, Object>> precision = metrics.precisionByThreshold().toJavaRDD();
System.out.println("Precision by threshold: " + precision.toArray());

// Recall by threshold
JavaRDD<Tuple2<Object, Object>> recall = metrics.recallByThreshold().toJavaRDD();
System.out.println("Recall by threshold: " + recall.toArray());

// F Score by threshold
JavaRDD<Tuple2<Object, Object>> f1Score = metrics.fMeasureByThreshold().toJavaRDD();
System.out.println("F1 Score by threshold: " + f1Score.toArray());

JavaRDD<Tuple2<Object, Object>> f2Score = metrics.fMeasureByThreshold(2.0).toJavaRDD();
System.out.println("F2 Score by threshold: " + f2Score.toArray());

// Precision-recall curve
JavaRDD<Tuple2<Object, Object>> prc = metrics.pr().toJavaRDD();
System.out.println("Precision-recall curve: " + prc.toArray());

// Thresholds
JavaRDD<Double> thresholds = precision.map(
  new Function<Tuple2<Object, Object>, Double>() {
    public Double call (Tuple2<Object, Object> t) {
      return new Double(t._1().toString());
    }
  }
);

// ROC Curve
JavaRDD<Tuple2<Object, Object>> roc = metrics.roc().toJavaRDD();
System.out.println("ROC curve: " + roc.toArray());

// AUPRC
System.out.println("Area under precision-recall curve = " + metrics.areaUnderPR());

// AUROC
System.out.println("Area under ROC = " + metrics.areaUnderROC());

// Save and load model
model.save(sc, "myModelPath");
LogisticRegressionModel sameModel = LogisticRegressionModel.load(sc, "myModelPath");

} }

{% endhighlight %}

{% highlight python %} from pyspark.mllib.classification import LogisticRegressionWithLBFGS from pyspark.mllib.evaluation import BinaryClassificationMetrics from pyspark.mllib.regression import LabeledPoint from pyspark.mllib.util import MLUtils

Several of the methods available in scala are currently missing from pyspark

Load training data in LIBSVM format

data = MLUtils.loadLibSVMFile(sc, "data/mllib/sample_binary_classification_data.txt")

Split data into training (60%) and test (40%)

training, test = data.randomSplit([0.6, 0.4], seed = 11L) training.cache()

Run training algorithm to build the model

model = LogisticRegressionWithLBFGS.train(training)

Compute raw scores on the test set

predictionAndLabels = test.map(lambda lp: (float(model.predict(lp.features)), lp.label))

Instantiate metrics object

metrics = BinaryClassificationMetrics(predictionAndLabels)

Area under precision-recall curve

print("Area under PR = %s" % metrics.areaUnderPR)

Area under ROC curve

print("Area under ROC = %s" % metrics.areaUnderROC)

{% endhighlight %}

Multiclass classification

A multiclass classification describes a classification problem where there are $M \gt 2$ possible labels for each data point (the case where $M=2$ is the binary classification problem). For example, classifying handwriting samples to the digits 0 to 9, having 10 possible classes.

For multiclass metrics, the notion of positives and negatives is slightly different. Predictions and labels can still be positive or negative, but they must be considered under the context of a particular class. Each label and prediction take on the value of one of the multiple classes and so they are said to be positive for their particular class and negative for all other classes. So, a true positive occurs whenever the prediction and the label match, while a true negative occurs when neither the prediction nor the label take on the value of a given class. By this convention, there can be multiple true negatives for a given data sample. The extension of false negatives and false positives from the former definitions of positive and negative labels is straightforward.

Label based metrics

Opposed to binary classification where there are only two possible labels, multiclass classification problems have many possible labels and so the concept of label-based metrics is introduced. Overall precision measures precision across all labels - the number of times any class was predicted correctly (true positives) normalized by the number of data points. Precision by label considers only one class, and measures the number of time a specific label was predicted correctly normalized by the number of times that label appears in the output.

Available metrics

Define the class, or label, set as

$$L = {\ell_0, \ell_1, \ldots, \ell_{M-1} } $$

The true output vector $\mathbf{y}$ consists of $N$ elements

$$\mathbf{y}_0, \mathbf{y}1, \ldots, \mathbf{y}{N-1} \in L $$

A multiclass prediction algorithm generates a prediction vector $\hat{\mathbf{y}}$ of $N$ elements

$$\hat{\mathbf{y}}_0, \hat{\mathbf{y}}1, \ldots, \hat{\mathbf{y}}{N-1} \in L $$

For this section, a modified delta function $\hat{\delta}(x)$ will prove useful

$$\hat{\delta}(x) = \begin{cases}1 & \text{if $x = 0$}, \ 0 & \text{otherwise}.\end{cases}$$

Metric	Definition
Confusion Matrix	$C_{ij} = \sum_{k=0}^{N-1} \hat{\delta}(\mathbf{y}_k-\ell_i) \cdot \hat{\delta}(\hat{\mathbf{y}}_k - \ell_j)\\ \\ \left( \begin{array}{ccc} \sum_{k=0}^{N-1} \hat{\delta}(\mathbf{y}_k-\ell_1) \cdot \hat{\delta}(\hat{\mathbf{y}}_k - \ell_1) & \ldots & \sum_{k=0}^{N-1} \hat{\delta}(\mathbf{y}_k-\ell_1) \cdot \hat{\delta}(\hat{\mathbf{y}}_k - \ell_N) \\ \vdots & \ddots & \vdots \\ \sum_{k=0}^{N-1} \hat{\delta}(\mathbf{y}_k-\ell_N) \cdot \hat{\delta}(\hat{\mathbf{y}}_k - \ell_1) & \ldots & \sum_{k=0}^{N-1} \hat{\delta}(\mathbf{y}_k-\ell_N) \cdot \hat{\delta}(\hat{\mathbf{y}}_k - \ell_N) \end{array} \right)$
Overall Precision	$PPV = \frac{TP}{TP + FP} = \frac{1}{N}\sum_{i=0}^{N-1} \hat{\delta}\left(\hat{\mathbf{y}}_i - \mathbf{y}_i\right)$
Overall Recall	$TPR = \frac{TP}{TP + FN} = \frac{1}{N}\sum_{i=0}^{N-1} \hat{\delta}\left(\hat{\mathbf{y}}_i - \mathbf{y}_i\right)$
Overall F1-measure	$F1 = 2 \cdot \left(\frac{PPV \cdot TPR} {PPV + TPR}\right)$
Precision by label	$PPV(\ell) = \frac{TP}{TP + FP} = \frac{\sum_{i=0}^{N-1} \hat{\delta}(\hat{\mathbf{y}}_i - \ell) \cdot \hat{\delta}(\mathbf{y}_i - \ell)} {\sum_{i=0}^{N-1} \hat{\delta}(\hat{\mathbf{y}}_i - \ell)}$
Recall by label	$TPR(\ell)=\frac{TP}{P} = \frac{\sum_{i=0}^{N-1} \hat{\delta}(\hat{\mathbf{y}}_i - \ell) \cdot \hat{\delta}(\mathbf{y}_i - \ell)} {\sum_{i=0}^{N-1} \hat{\delta}(\mathbf{y}_i - \ell)}$
F-measure by label	$F(\beta, \ell) = \left(1 + \beta^2\right) \cdot \left(\frac{PPV(\ell) \cdot TPR(\ell)} {\beta^2 \cdot PPV(\ell) + TPR(\ell)}\right)$
Weighted precision	$PPV_{w}= \frac{1}{N} \sum\nolimits_{\ell \in L} PPV(\ell) \cdot \sum_{i=0}^{N-1} \hat{\delta}(\mathbf{y}_i-\ell)$
Weighted recall	$TPR_{w}= \frac{1}{N} \sum\nolimits_{\ell \in L} TPR(\ell) \cdot \sum_{i=0}^{N-1} \hat{\delta}(\mathbf{y}_i-\ell)$
Weighted F-measure	$F_{w}(\beta)= \frac{1}{N} \sum\nolimits_{\ell \in L} F(\beta, \ell) \cdot \sum_{i=0}^{N-1} \hat{\delta}(\mathbf{y}_i-\ell)$

Examples

The following code snippets illustrate how to load a sample dataset, train a multiclass classification algorithm on the data, and evaluate the performance of the algorithm by several multiclass classification evaluation metrics.

{% highlight scala %} import org.apache.spark.mllib.classification.LogisticRegressionWithLBFGS import org.apache.spark.mllib.evaluation.MulticlassMetrics import org.apache.spark.mllib.regression.LabeledPoint import org.apache.spark.mllib.util.MLUtils

// Load training data in LIBSVM format val data = MLUtils.loadLibSVMFile(sc, "data/mllib/sample_multiclass_classification_data.txt")

// Split data into training (60%) and test (40%) val Array(training, test) = data.randomSplit(Array(0.6, 0.4), seed = 11L) training.cache()

// Run training algorithm to build the model val model = new LogisticRegressionWithLBFGS() .setNumClasses(3) .run(training)

// Compute raw scores on the test set val predictionAndLabels = test.map { case LabeledPoint(label, features) => val prediction = model.predict(features) (prediction, label) }

// Instantiate metrics object val metrics = new MulticlassMetrics(predictionAndLabels)

// Confusion matrix println("Confusion matrix:") println(metrics.confusionMatrix)

// Overall Statistics val precision = metrics.precision val recall = metrics.recall // same as true positive rate val f1Score = metrics.fMeasure println("Summary Statistics") println(s"Precision = $precision") println(s"Recall = $recall") println(s"F1 Score = $f1Score")

// Precision by label val labels = metrics.labels labels.foreach { l => println(s"Precision($l) = " + metrics.precision(l)) }

// Recall by label labels.foreach { l => println(s"Recall($l) = " + metrics.recall(l)) }

// False positive rate by label labels.foreach { l => println(s"FPR($l) = " + metrics.falsePositiveRate(l)) }

// F-measure by label labels.foreach { l => println(s"F1-Score($l) = " + metrics.fMeasure(l)) }

// Weighted stats println(s"Weighted precision: ${metrics.weightedPrecision}") println(s"Weighted recall: ${metrics.weightedRecall}") println(s"Weighted F1 score: ${metrics.weightedFMeasure}") println(s"Weighted false positive rate: ${metrics.weightedFalsePositiveRate}")

{% endhighlight %}

{% highlight java %} import scala.Tuple2;

import org.apache.spark.api.java.*; import org.apache.spark.rdd.RDD; import org.apache.spark.api.java.function.Function; import org.apache.spark.mllib.classification.LogisticRegressionModel; import org.apache.spark.mllib.classification.LogisticRegressionWithLBFGS; import org.apache.spark.mllib.evaluation.MulticlassMetrics; import org.apache.spark.mllib.regression.LabeledPoint; import org.apache.spark.mllib.util.MLUtils; import org.apache.spark.mllib.linalg.Matrix; import org.apache.spark.SparkConf; import org.apache.spark.SparkContext;

public class MulticlassClassification { public static void main(String[] args) { SparkConf conf = new SparkConf().setAppName("Multiclass Classification Metrics"); SparkContext sc = new SparkContext(conf); String path = "data/mllib/sample_multiclass_classification_data.txt"; JavaRDD data = MLUtils.loadLibSVMFile(sc, path).toJavaRDD();

// Split initial RDD into two... [60% training data, 40% testing data].
JavaRDD<LabeledPoint>[] splits = data.randomSplit(new double[] {0.6, 0.4}, 11L);
JavaRDD<LabeledPoint> training = splits[0].cache();
JavaRDD<LabeledPoint> test = splits[1];

// Run training algorithm to build the model.
final LogisticRegressionModel model = new LogisticRegressionWithLBFGS()
  .setNumClasses(3)
  .run(training.rdd());

// Compute raw scores on the test set.
JavaRDD<Tuple2<Object, Object>> predictionAndLabels = test.map(
  new Function<LabeledPoint, Tuple2<Object, Object>>() {
    public Tuple2<Object, Object> call(LabeledPoint p) {
      Double prediction = model.predict(p.features());
      return new Tuple2<Object, Object>(prediction, p.label());
    }
  }
);

// Get evaluation metrics.
MulticlassMetrics metrics = new MulticlassMetrics(predictionAndLabels.rdd());

// Confusion matrix
Matrix confusion = metrics.confusionMatrix();
System.out.println("Confusion matrix: \n" + confusion);

// Overall statistics
System.out.println("Precision = " + metrics.precision());
System.out.println("Recall = " + metrics.recall());
System.out.println("F1 Score = " + metrics.fMeasure());

// Stats by labels
for (int i = 0; i < metrics.labels().length; i++) {
    System.out.format("Class %f precision = %f\n", metrics.labels()[i], metrics.precision(metrics.labels()[i]));
    System.out.format("Class %f recall = %f\n", metrics.labels()[i], metrics.recall(metrics.labels()[i]));
    System.out.format("Class %f F1 score = %f\n", metrics.labels()[i], metrics.fMeasure(metrics.labels()[i]));
}

//Weighted stats
System.out.format("Weighted precision = %f\n", metrics.weightedPrecision());
System.out.format("Weighted recall = %f\n", metrics.weightedRecall());
System.out.format("Weighted F1 score = %f\n", metrics.weightedFMeasure());
System.out.format("Weighted false positive rate = %f\n", metrics.weightedFalsePositiveRate());

// Save and load model
model.save(sc, "myModelPath");
LogisticRegressionModel sameModel = LogisticRegressionModel.load(sc, "myModelPath");

} }

{% endhighlight %}

{% highlight python %} from pyspark.mllib.classification import LogisticRegressionWithLBFGS from pyspark.mllib.util import MLUtils from pyspark.mllib.evaluation import MulticlassMetrics

Load training data in LIBSVM format

data = MLUtils.loadLibSVMFile(sc, "data/mllib/sample_multiclass_classification_data.txt")

Split data into training (60%) and test (40%)

training, test = data.randomSplit([0.6, 0.4], seed = 11L) training.cache()

Run training algorithm to build the model

model = LogisticRegressionWithLBFGS.train(training, numClasses=3)

Compute raw scores on the test set

predictionAndLabels = test.map(lambda lp: (float(model.predict(lp.features)), lp.label))

Instantiate metrics object

metrics = MulticlassMetrics(predictionAndLabels)

Overall statistics

precision = metrics.precision() recall = metrics.recall() f1Score = metrics.fMeasure() print("Summary Stats") print("Precision = %s" % precision) print("Recall = %s" % recall) print("F1 Score = %s" % f1Score)

Statistics by class

labels = data.map(lambda lp: lp.label).distinct().collect() for label in sorted(labels): print("Class %s precision = %s" % (label, metrics.precision(label))) print("Class %s recall = %s" % (label, metrics.recall(label))) print("Class %s F1 Measure = %s" % (label, metrics.fMeasure(label, beta=1.0)))

Weighted stats

print("Weighted recall = %s" % metrics.weightedRecall) print("Weighted precision = %s" % metrics.weightedPrecision) print("Weighted F(1) Score = %s" % metrics.weightedFMeasure()) print("Weighted F(0.5) Score = %s" % metrics.weightedFMeasure(beta=0.5)) print("Weighted false positive rate = %s" % metrics.weightedFalsePositiveRate) {% endhighlight %}

Multilabel classification

A multilabel classification problem involves mapping each sample in a dataset to a set of class labels. In this type of classification problem, the labels are not mutually exclusive. For example, when classifying a set of news articles into topics, a single article might be both science and politics.

Because the labels are not mutually exclusive, the predictions and true labels are now vectors of label sets, rather than vectors of labels. Multilabel metrics, therefore, extend the fundamental ideas of precision, recall, etc. to operations on sets. For example, a true positive for a given class now occurs when that class exists in the predicted set and it exists in the true label set, for a specific data point.

Available metrics

Here we define a set $D$ of $N$ documents

$$D = \left{d_0, d_1, ..., d_{N-1}\right}$$

Define $L_0, L_1, ..., L_{N-1}$ to be a family of label sets and $P_0, P_1, ..., P_{N-1}$ to be a family of prediction sets where $L_i$ and $P_i$ are the label set and prediction set, respectively, that correspond to document $d_i$.

The set of all unique labels is given by

$$L = \bigcup_{k=0}^{N-1} L_k$$

The following definition of indicator function $I_A(x)$ on a set $A$ will be necessary

$$I_A(x) = \begin{cases}1 & \text{if $x \in A$}, \ 0 & \text{otherwise}.\end{cases}$$

Metric	Definition
Precision	$\frac{1}{N} \sum_{i=0}^{N-1} \frac{\left\|P_i \cap L_i\right\|}{\left\|P_i\right\|}$
Recall	$\frac{1}{N} \sum_{i=0}^{N-1} \frac{\left\|L_i \cap P_i\right\|}{\left\|L_i\right\|}$
Accuracy	$\frac{1}{N} \sum_{i=0}^{N - 1} \frac{\left\|L_i \cap P_i \right\|} {\left\|L_i\right\| + \left\|P_i\right\| - \left\|L_i \cap P_i \right\|}$
Precision by label	$PPV(\ell)=\frac{TP}{TP + FP}= \frac{\sum_{i=0}^{N-1} I_{P_i}(\ell) \cdot I_{L_i}(\ell)} {\sum_{i=0}^{N-1} I_{P_i}(\ell)}$
Recall by label	$TPR(\ell)=\frac{TP}{P}= \frac{\sum_{i=0}^{N-1} I_{P_i}(\ell) \cdot I_{L_i}(\ell)} {\sum_{i=0}^{N-1} I_{L_i}(\ell)}$
F1-measure by label	$F1(\ell) = 2 \cdot \left(\frac{PPV(\ell) \cdot TPR(\ell)} {PPV(\ell) + TPR(\ell)}\right)$
Hamming Loss	$\frac{1}{N \cdot \left\|L\right\|} \sum_{i=0}^{N - 1} \left\|L_i\right\| + \left\|P_i\right\| - 2\left\|L_i \cap P_i\right\|$
Subset Accuracy	$\frac{1}{N} \sum_{i=0}^{N-1} I_{\{L_i\}}(P_i)$
F1 Measure	$\frac{1}{N} \sum_{i=0}^{N-1} 2 \frac{\left\|P_i \cap L_i\right\|}{\left\|P_i\right\| \cdot \left\|L_i\right\|}$
Micro precision	$\frac{TP}{TP + FP}=\frac{\sum_{i=0}^{N-1} \left\|P_i \cap L_i\right\|} {\sum_{i=0}^{N-1} \left\|P_i \cap L_i\right\| + \sum_{i=0}^{N-1} \left\|P_i - L_i\right\|}$
Micro recall	$\frac{TP}{TP + FN}=\frac{\sum_{i=0}^{N-1} \left\|P_i \cap L_i\right\|} {\sum_{i=0}^{N-1} \left\|P_i \cap L_i\right\| + \sum_{i=0}^{N-1} \left\|L_i - P_i\right\|}$
Micro F1 Measure	$2 \cdot \frac{TP}{2 \cdot TP + FP + FN}=2 \cdot \frac{\sum_{i=0}^{N-1} \left\|P_i \cap L_i\right\|}{2 \cdot \sum_{i=0}^{N-1} \left\|P_i \cap L_i\right\| + \sum_{i=0}^{N-1} \left\|L_i - P_i\right\| + \sum_{i=0}^{N-1} \left\|P_i - L_i\right\|}$

Examples

The following code snippets illustrate how to evaluate the performance of a multilabel classifer. The examples use the fake prediction and label data for multilabel classification that is shown below.

Document predictions:

doc 0 - predict 0, 1 - class 0, 2
doc 1 - predict 0, 2 - class 0, 1
doc 2 - predict none - class 0
doc 3 - predict 2 - class 2
doc 4 - predict 2, 0 - class 2, 0
doc 5 - predict 0, 1, 2 - class 0, 1
doc 6 - predict 1 - class 1, 2

Predicted classes:

class 0 - doc 0, 1, 4, 5 (total 4)
class 1 - doc 0, 5, 6 (total 3)
class 2 - doc 1, 3, 4, 5 (total 4)

True classes:

class 0 - doc 0, 1, 2, 4, 5 (total 5)
class 1 - doc 1, 5, 6 (total 3)
class 2 - doc 0, 3, 4, 6 (total 4)

{% highlight scala %} import org.apache.spark.mllib.evaluation.MultilabelMetrics import org.apache.spark.rdd.RDD;

val scoreAndLabels: RDD[(Array[Double], Array[Double])] = sc.parallelize( Seq((Array(0.0, 1.0), Array(0.0, 2.0)), (Array(0.0, 2.0), Array(0.0, 1.0)), (Array(), Array(0.0)), (Array(2.0), Array(2.0)), (Array(2.0, 0.0), Array(2.0, 0.0)), (Array(0.0, 1.0, 2.0), Array(0.0, 1.0)), (Array(1.0), Array(1.0, 2.0))), 2)

// Instantiate metrics object val metrics = new MultilabelMetrics(scoreAndLabels)

// Summary stats println(s"Recall = ${metrics.recall}") println(s"Precision = ${metrics.precision}") println(s"F1 measure = ${metrics.f1Measure}") println(s"Accuracy = ${metrics.accuracy}")

// Individual label stats metrics.labels.foreach(label => println(s"Class $label precision = ${metrics.precision(label)}")) metrics.labels.foreach(label => println(s"Class $label recall = ${metrics.recall(label)}")) metrics.labels.foreach(label => println(s"Class $label F1-score = ${metrics.f1Measure(label)}"))

// Micro stats println(s"Micro recall = ${metrics.microRecall}") println(s"Micro precision = ${metrics.microPrecision}") println(s"Micro F1 measure = ${metrics.microF1Measure}")

// Hamming loss println(s"Hamming loss = ${metrics.hammingLoss}")

// Subset accuracy println(s"Subset accuracy = ${metrics.subsetAccuracy}")

{% endhighlight %}

{% highlight java %} import scala.Tuple2;

import org.apache.spark.api.java.*; import org.apache.spark.rdd.RDD; import org.apache.spark.mllib.evaluation.MultilabelMetrics; import org.apache.spark.SparkConf; import java.util.Arrays; import java.util.List;

public class MultilabelClassification { public static void main(String[] args) { SparkConf conf = new SparkConf().setAppName("Multilabel Classification Metrics"); JavaSparkContext sc = new JavaSparkContext(conf);

List<Tuple2<double[], double[]>> data = Arrays.asList(
    new Tuple2<double[], double[]>(new double[]{0.0, 1.0}, new double[]{0.0, 2.0}),
    new Tuple2<double[], double[]>(new double[]{0.0, 2.0}, new double[]{0.0, 1.0}),
    new Tuple2<double[], double[]>(new double[]{}, new double[]{0.0}),
    new Tuple2<double[], double[]>(new double[]{2.0}, new double[]{2.0}),
    new Tuple2<double[], double[]>(new double[]{2.0, 0.0}, new double[]{2.0, 0.0}),
    new Tuple2<double[], double[]>(new double[]{0.0, 1.0, 2.0}, new double[]{0.0, 1.0}),
    new Tuple2<double[], double[]>(new double[]{1.0}, new double[]{1.0, 2.0})
    );
JavaRDD<Tuple2<double[], double[]>> scoreAndLabels = sc.parallelize(data);

// Instantiate metrics object
MultilabelMetrics metrics = new MultilabelMetrics(scoreAndLabels.rdd());

// Summary stats
System.out.format("Recall = %f\n", metrics.recall());
System.out.format("Precision = %f\n", metrics.precision());
System.out.format("F1 measure = %f\n", metrics.f1Measure());
System.out.format("Accuracy = %f\n", metrics.accuracy());

// Stats by labels
for (int i = 0; i < metrics.labels().length - 1; i++) {
    System.out.format("Class %1.1f precision = %f\n", metrics.labels()[i], metrics.precision(metrics.labels()[i]));
    System.out.format("Class %1.1f recall = %f\n", metrics.labels()[i], metrics.recall(metrics.labels()[i]));
    System.out.format("Class %1.1f F1 score = %f\n", metrics.labels()[i], metrics.f1Measure(metrics.labels()[i]));
}

// Micro stats
System.out.format("Micro recall = %f\n", metrics.microRecall());
System.out.format("Micro precision = %f\n", metrics.microPrecision());
System.out.format("Micro F1 measure = %f\n", metrics.microF1Measure());

// Hamming loss
System.out.format("Hamming loss = %f\n", metrics.hammingLoss());

// Subset accuracy
System.out.format("Subset accuracy = %f\n", metrics.subsetAccuracy());

} }

{% endhighlight %}

{% highlight python %} from pyspark.mllib.evaluation import MultilabelMetrics

scoreAndLabels = sc.parallelize([ ([0.0, 1.0], [0.0, 2.0]), ([0.0, 2.0], [0.0, 1.0]), ([], [0.0]), ([2.0], [2.0]), ([2.0, 0.0], [2.0, 0.0]), ([0.0, 1.0, 2.0], [0.0, 1.0]), ([1.0], [1.0, 2.0])])

Instantiate metrics object

metrics = MultilabelMetrics(scoreAndLabels)

Summary stats

print("Recall = %s" % metrics.recall()) print("Precision = %s" % metrics.precision()) print("F1 measure = %s" % metrics.f1Measure()) print("Accuracy = %s" % metrics.accuracy)

Individual label stats

labels = scoreAndLabels.flatMap(lambda x: x[1]).distinct().collect() for label in labels: print("Class %s precision = %s" % (label, metrics.precision(label))) print("Class %s recall = %s" % (label, metrics.recall(label))) print("Class %s F1 Measure = %s" % (label, metrics.f1Measure(label)))

Micro stats

print("Micro precision = %s" % metrics.microPrecision) print("Micro recall = %s" % metrics.microRecall) print("Micro F1 measure = %s" % metrics.microF1Measure)

Hamming loss

print("Hamming loss = %s" % metrics.hammingLoss)

Subset accuracy

print("Subset accuracy = %s" % metrics.subsetAccuracy)

{% endhighlight %}

Ranking systems

The role of a ranking algorithm (often thought of as a recommender system) is to return to the user a set of relevant items or documents based on some training data. The definition of relevance may vary and is usually application specific. Ranking system metrics aim to quantify the effectiveness of these rankings or recommendations in various contexts. Some metrics compare a set of recommended documents to a ground truth set of relevant documents, while other metrics may incorporate numerical ratings explicitly.

Available metrics

A ranking system usually deals with a set of $M$ users

$$U = \left{u_0, u_1, ..., u_{M-1}\right}$$

Each user ($u_i$) having a set of $N$ ground truth relevant documents

$$D_i = \left{d_0, d_1, ..., d_{N-1}\right}$$

And a list of $Q$ recommended documents, in order of decreasing relevance

$$R_i = \left[r_0, r_1, ..., r_{Q-1}\right]$$

The goal of the ranking system is to produce the most relevant set of documents for each user. The relevance of the sets and the effectiveness of the algorithms can be measured using the metrics listed below.

It is necessary to define a function which, provided a recommended document and a set of ground truth relevant documents, returns a relevance score for the recommended document.

$$rel_D(r) = \begin{cases}1 & \text{if $r \in D$}, \ 0 & \text{otherwise}.\end{cases}$$

Metric	Definition	Notes
Precision at k	$p(k)=\frac{1}{M} \sum_{i=0}^{M-1} {\frac{1}{k} \sum_{j=0}^{\text{min}(\left\|D\right\|, k) - 1} rel_{D_i}(R_i(j))}$	Precision at k is a measure of how many of the first k recommended documents are in the set of true relevant documents averaged across all users. In this metric, the order of the recommendations is not taken into account.
Mean Average Precision	$MAP=\frac{1}{M} \sum_{i=0}^{M-1} {\frac{1}{\left\|D_i\right\|} \sum_{j=0}^{Q-1} \frac{rel_{D_i}(R_i(j))}{j + 1}}$	MAP is a measure of how many of the recommended documents are in the set of true relevant documents, where the order of the recommendations is taken into account (i.e. penalty for highly relevant documents is higher).
Normalized Discounted Cumulative Gain	$NDCG(k)=\frac{1}{M} \sum_{i=0}^{M-1} {\frac{1}{IDCG(D_i, k)}\sum_{j=0}^{n-1} \frac{rel_{D_i}(R_i(j))}{\text{ln}(j+1)}} \\ \text{Where} \\ \hspace{5 mm} n = \text{min}\left(\text{max}\left(\|R_i\|,\|D_i\|\right),k\right) \\ \hspace{5 mm} IDCG(D, k) = \sum_{j=0}^{\text{min}(\left\|D\right\|, k) - 1} \frac{1}{\text{ln}(j+1)}$	NDCG at k is a measure of how many of the first k recommended documents are in the set of true relevant documents averaged across all users. In contrast to precision at k, this metric takes into account the order of the recommendations (documents are assumed to be in order of decreasing relevance).

Examples

The following code snippets illustrate how to load a sample dataset, train an alternating least squares recommendation model on the data, and evaluate the performance of the recommender by several ranking metrics. A brief summary of the methodology is provided below.

MovieLens ratings are on a scale of 1-5:

5: Must see
4: Will enjoy
3: It's okay
2: Fairly bad
1: Awful

So we should not recommend a movie if the predicted rating is less than 3. To map ratings to confidence scores, we use:

5 -> 2.5
4 -> 1.5
3 -> 0.5
2 -> -0.5
1 -> -1.5.

This mappings means unobserved entries are generally between It's okay and Fairly bad. The semantics of 0 in this expanded world of non-positive weights are "the same as never having interacted at all."

{% highlight scala %} import org.apache.spark.mllib.evaluation.{RegressionMetrics, RankingMetrics} import org.apache.spark.mllib.recommendation.{ALS, Rating}

// Read in the ratings data val ratings = sc.textFile("data/mllib/sample_movielens_data.txt").map { line => val fields = line.split("::") Rating(fields(0).toInt, fields(1).toInt, fields(2).toDouble - 2.5) }.cache()

// Map ratings to 1 or 0, 1 indicating a movie that should be recommended val binarizedRatings = ratings.map(r => Rating(r.user, r.product, if (r.rating > 0) 1.0 else 0.0)).cache()

// Summarize ratings val numRatings = ratings.count() val numUsers = ratings.map(.user).distinct().count() val numMovies = ratings.map(.product).distinct().count() println(s"Got $numRatings ratings from $numUsers users on $numMovies movies.")

// Build the model val numIterations = 10 val rank = 10 val lambda = 0.01 val model = ALS.train(ratings, rank, numIterations, lambda)

// Define a function to scale ratings from 0 to 1 def scaledRating(r: Rating): Rating = { val scaledRating = math.max(math.min(r.rating, 1.0), 0.0) Rating(r.user, r.product, scaledRating) }

// Get sorted top ten predictions for each user and then scale from [0, 1] val userRecommended = model.recommendProductsForUsers(10).map{ case (user, recs) => (user, recs.map(scaledRating)) }

// Assume that any movie a user rated 3 or higher (which maps to a 1) is a relevant document // Compare with top ten most relevant documents val userMovies = binarizedRatings.groupBy(.user) val relevantDocuments = userMovies.join(userRecommended).map{ case (user, (actual, predictions)) => (predictions.map(.product), actual.filter(.rating > 0.0).map(.product).toArray) }

// Instantiate metrics object val metrics = new RankingMetrics(relevantDocuments)

// Precision at K Array(1, 3, 5).foreach{ k => println(s"Precision at $k = ${metrics.precisionAt(k)}") }

// Mean average precision println(s"Mean average precision = ${metrics.meanAveragePrecision}")

// Normalized discounted cumulative gain Array(1, 3, 5).foreach{ k => println(s"NDCG at $k = ${metrics.ndcgAt(k)}") }

// Get predictions for each data point val allPredictions = model.predict(ratings.map(r => (r.user, r.product))).map(r => ((r.user, r.product), r.rating)) val allRatings = ratings.map(r => ((r.user, r.product), r.rating)) val predictionsAndLabels = allPredictions.join(allRatings).map{ case ((user, product), (predicted, actual)) => (predicted, actual) }

// Get the RMSE using regression metrics val regressionMetrics = new RegressionMetrics(predictionsAndLabels) println(s"RMSE = ${regressionMetrics.rootMeanSquaredError}")

// R-squared println(s"R-squared = ${regressionMetrics.r2}")

{% endhighlight %}

{% highlight java %} import scala.Tuple2;

import org.apache.spark.api.java.; import org.apache.spark.rdd.RDD; import org.apache.spark.mllib.recommendation.MatrixFactorizationModel; import org.apache.spark.SparkConf; import org.apache.spark.api.java.function.Function; import java.util.; import org.apache.spark.mllib.evaluation.RegressionMetrics; import org.apache.spark.mllib.evaluation.RankingMetrics; import org.apache.spark.mllib.recommendation.ALS; import org.apache.spark.mllib.recommendation.Rating;

// Read in the ratings data public class Ranking { public static void main(String[] args) { SparkConf conf = new SparkConf().setAppName("Ranking Metrics"); JavaSparkContext sc = new JavaSparkContext(conf); String path = "data/mllib/sample_movielens_data.txt"; JavaRDD data = sc.textFile(path); JavaRDD ratings = data.map( new Function<String, Rating>() { public Rating call(String line) { String[] parts = line.split("::"); return new Rating(Integer.parseInt(parts[0]), Integer.parseInt(parts[1]), Double.parseDouble(parts[2]) - 2.5); } } ); ratings.cache();

// Train an ALS model
final MatrixFactorizationModel model = ALS.train(JavaRDD.toRDD(ratings), 10, 10, 0.01);

// Get top 10 recommendations for every user and scale ratings from 0 to 1
JavaRDD<Tuple2<Object, Rating[]>> userRecs = model.recommendProductsForUsers(10).toJavaRDD();
JavaRDD<Tuple2<Object, Rating[]>> userRecsScaled = userRecs.map(
  new Function<Tuple2<Object, Rating[]>, Tuple2<Object, Rating[]>>() {
    public Tuple2<Object, Rating[]> call(Tuple2<Object, Rating[]> t) {
      Rating[] scaledRatings = new Rating[t._2().length];
      for (int i = 0; i < scaledRatings.length; i++) {
        double newRating = Math.max(Math.min(t._2()[i].rating(), 1.0), 0.0);
        scaledRatings[i] = new Rating(t._2()[i].user(), t._2()[i].product(), newRating);
      }
      return new Tuple2<Object, Rating[]>(t._1(), scaledRatings);
    }
  }
);
JavaPairRDD<Object, Rating[]> userRecommended = JavaPairRDD.fromJavaRDD(userRecsScaled);

// Map ratings to 1 or 0, 1 indicating a movie that should be recommended
JavaRDD<Rating> binarizedRatings = ratings.map(
  new Function<Rating, Rating>() {
    public Rating call(Rating r) {
      double binaryRating;
      if (r.rating() > 0.0) {
        binaryRating = 1.0;
      }
      else {
        binaryRating = 0.0;
      }
      return new Rating(r.user(), r.product(), binaryRating);
    }
  }
);

// Group ratings by common user
JavaPairRDD<Object, Iterable<Rating>> userMovies = binarizedRatings.groupBy(
  new Function<Rating, Object>() {
    public Object call(Rating r) {
      return r.user();
    }
  }
);

// Get true relevant documents from all user ratings
JavaPairRDD<Object, List<Integer>> userMoviesList = userMovies.mapValues(
  new Function<Iterable<Rating>, List<Integer>>() {
    public List<Integer> call(Iterable<Rating> docs) {
      List<Integer> products = new ArrayList<Integer>();
      for (Rating r : docs) {
        if (r.rating() > 0.0) {
          products.add(r.product());
        }
      }
      return products;
    }
  }
);

// Extract the product id from each recommendation
JavaPairRDD<Object, List<Integer>> userRecommendedList = userRecommended.mapValues(
  new Function<Rating[], List<Integer>>() {
    public List<Integer> call(Rating[] docs) {
      List<Integer> products = new ArrayList<Integer>();
      for (Rating r : docs) {
        products.add(r.product());
      }
      return products;
    }
  }
);
JavaRDD<Tuple2<List<Integer>, List<Integer>>> relevantDocs = userMoviesList.join(userRecommendedList).values();

// Instantiate the metrics object
RankingMetrics metrics = RankingMetrics.of(relevantDocs);

// Precision and NDCG at k
Integer[] kVector = {1, 3, 5};
for (Integer k : kVector) {
  System.out.format("Precision at %d = %f\n", k, metrics.precisionAt(k));
  System.out.format("NDCG at %d = %f\n", k, metrics.ndcgAt(k));
}

// Mean average precision
System.out.format("Mean average precision = %f\n", metrics.meanAveragePrecision());

// Evaluate the model using numerical ratings and regression metrics
JavaRDD<Tuple2<Object, Object>> userProducts = ratings.map(
  new Function<Rating, Tuple2<Object, Object>>() {
    public Tuple2<Object, Object> call(Rating r) {
      return new Tuple2<Object, Object>(r.user(), r.product());
    }
  }
);
JavaPairRDD<Tuple2<Integer, Integer>, Object> predictions = JavaPairRDD.fromJavaRDD(
  model.predict(JavaRDD.toRDD(userProducts)).toJavaRDD().map(
    new Function<Rating, Tuple2<Tuple2<Integer, Integer>, Object>>() {
      public Tuple2<Tuple2<Integer, Integer>, Object> call(Rating r){
        return new Tuple2<Tuple2<Integer, Integer>, Object>(
          new Tuple2<Integer, Integer>(r.user(), r.product()), r.rating());
      }
    }
));
JavaRDD<Tuple2<Object, Object>> ratesAndPreds =
  JavaPairRDD.fromJavaRDD(ratings.map(
    new Function<Rating, Tuple2<Tuple2<Integer, Integer>, Object>>() {
      public Tuple2<Tuple2<Integer, Integer>, Object> call(Rating r){
        return new Tuple2<Tuple2<Integer, Integer>, Object>(
          new Tuple2<Integer, Integer>(r.user(), r.product()), r.rating());
      }
    }
)).join(predictions).values();

// Create regression metrics object
RegressionMetrics regressionMetrics = new RegressionMetrics(ratesAndPreds.rdd());

// Root mean squared error
System.out.format("RMSE = %f\n", regressionMetrics.rootMeanSquaredError());

// R-squared
System.out.format("R-squared = %f\n", regressionMetrics.r2());

} }

{% endhighlight %}

{% highlight python %} from pyspark.mllib.recommendation import ALS, Rating from pyspark.mllib.evaluation import RegressionMetrics, RankingMetrics

Read in the ratings data

lines = sc.textFile("data/mllib/sample_movielens_data.txt")

def parseLine(line): fields = line.split("::") return Rating(int(fields[0]), int(fields[1]), float(fields[2]) - 2.5) ratings = lines.map(lambda r: parseLine(r))

Train a model on to predict user-product ratings

model = ALS.train(ratings, 10, 10, 0.01)

Get predicted ratings on all existing user-product pairs

testData = ratings.map(lambda p: (p.user, p.product)) predictions = model.predictAll(testData).map(lambda r: ((r.user, r.product), r.rating))

ratingsTuple = ratings.map(lambda r: ((r.user, r.product), r.rating)) scoreAndLabels = predictions.join(ratingsTuple).map(lambda tup: tup[1])

Instantiate regression metrics to compare predicted and actual ratings

metrics = RegressionMetrics(scoreAndLabels)

Root mean sqaured error

print("RMSE = %s" % metrics.rootMeanSquaredError)

R-squared

print("R-squared = %s" % metrics.r2)

{% endhighlight %}

Regression model evaluation

Regression analysis is used when predicting a continuous output variable from a number of independent variables.

Available metrics

Metric	Definition
Mean Squared Error (MSE)	$MSE = \frac{\sum_{i=0}^{N-1} (\mathbf{y}_i - \hat{\mathbf{y}}_i)^2}{N}$
Root Mean Squared Error (RMSE)	$RMSE = \sqrt{\frac{\sum_{i=0}^{N-1} (\mathbf{y}_i - \hat{\mathbf{y}}_i)^2}{N}}$
Mean Absoloute Error (MAE)	$MAE=\sum_{i=0}^{N-1} \left\|\mathbf{y}_i - \hat{\mathbf{y}}_i\right\|$
Coefficient of Determination $(R^2)$	$R^2=1 - \frac{MSE}{\text{VAR}(\mathbf{y}) \cdot (N-1)}=1-\frac{\sum_{i=0}^{N-1} (\mathbf{y}_i - \hat{\mathbf{y}}_i)^2}{\sum_{i=0}^{N-1}(\mathbf{y}_i-\bar{\mathbf{y}})^2}$
Explained Variance	$1 - \frac{\text{VAR}(\mathbf{y} - \mathbf{\hat{y}})}{\text{VAR}(\mathbf{y})}$

Examples

The following code snippets illustrate how to load a sample dataset, train a linear regression algorithm on the data, and evaluate the performance of the algorithm by several regression metrics.

{% highlight scala %} import org.apache.spark.mllib.regression.LabeledPoint import org.apache.spark.mllib.regression.LinearRegressionModel import org.apache.spark.mllib.regression.LinearRegressionWithSGD import org.apache.spark.mllib.linalg.Vectors import org.apache.spark.mllib.evaluation.RegressionMetrics import org.apache.spark.mllib.util.MLUtils

// Load the data val data = MLUtils.loadLibSVMFile(sc, "data/mllib/sample_linear_regression_data.txt").cache()

// Build the model val numIterations = 100 val model = LinearRegressionWithSGD.train(data, numIterations)

// Get predictions val valuesAndPreds = data.map{ point => val prediction = model.predict(point.features) (prediction, point.label) }

// Instantiate metrics object val metrics = new RegressionMetrics(valuesAndPreds)

// Squared error println(s"MSE = ${metrics.meanSquaredError}") println(s"RMSE = ${metrics.rootMeanSquaredError}")

// R-squared println(s"R-squared = ${metrics.r2}")

// Mean absolute error println(s"MAE = ${metrics.meanAbsoluteError}")

// Explained variance println(s"Explained variance = ${metrics.explainedVariance}")

{% endhighlight %}

{% highlight java %} import scala.Tuple2;

import org.apache.spark.api.java.*; import org.apache.spark.api.java.function.Function; import org.apache.spark.mllib.linalg.Vectors; import org.apache.spark.mllib.regression.LabeledPoint; import org.apache.spark.mllib.regression.LinearRegressionModel; import org.apache.spark.mllib.regression.LinearRegressionWithSGD; import org.apache.spark.mllib.evaluation.RegressionMetrics; import org.apache.spark.SparkConf;

public class LinearRegression { public static void main(String[] args) { SparkConf conf = new SparkConf().setAppName("Linear Regression Example"); JavaSparkContext sc = new JavaSparkContext(conf);

// Load and parse the data
String path = "data/mllib/sample_linear_regression_data.txt";
JavaRDD<String> data = sc.textFile(path);
JavaRDD<LabeledPoint> parsedData = data.map(
  new Function<String, LabeledPoint>() {
    public LabeledPoint call(String line) {
      String[] parts = line.split(" ");
      double[] v = new double[parts.length - 1];
      for (int i = 1; i < parts.length - 1; i++)
        v[i - 1] = Double.parseDouble(parts[i].split(":")[1]);
      return new LabeledPoint(Double.parseDouble(parts[0]), Vectors.dense(v));
    }
  }
);
parsedData.cache();

// Building the model
int numIterations = 100;
final LinearRegressionModel model =
  LinearRegressionWithSGD.train(JavaRDD.toRDD(parsedData), numIterations);

// Evaluate model on training examples and compute training error
JavaRDD<Tuple2<Object, Object>> valuesAndPreds = parsedData.map(
  new Function<LabeledPoint, Tuple2<Object, Object>>() {
    public Tuple2<Object, Object> call(LabeledPoint point) {
      double prediction = model.predict(point.features());
      return new Tuple2<Object, Object>(prediction, point.label());
    }
  }
);

// Instantiate metrics object
RegressionMetrics metrics = new RegressionMetrics(valuesAndPreds.rdd());

// Squared error
System.out.format("MSE = %f\n", metrics.meanSquaredError());
System.out.format("RMSE = %f\n", metrics.rootMeanSquaredError());

// R-squared
System.out.format("R Squared = %f\n", metrics.r2());

// Mean absolute error
System.out.format("MAE = %f\n", metrics.meanAbsoluteError());

// Explained variance
System.out.format("Explained Variance = %f\n", metrics.explainedVariance());

// Save and load model
model.save(sc.sc(), "myModelPath");
LinearRegressionModel sameModel = LinearRegressionModel.load(sc.sc(), "myModelPath");

} }

{% endhighlight %}

{% highlight python %} from pyspark.mllib.regression import LabeledPoint, LinearRegressionWithSGD from pyspark.mllib.evaluation import RegressionMetrics from pyspark.mllib.linalg import DenseVector

Load and parse the data

def parsePoint(line): values = line.split() return LabeledPoint(float(values[0]), DenseVector([float(x.split(':')[1]) for x in values[1:]]))

data = sc.textFile("data/mllib/sample_linear_regression_data.txt") parsedData = data.map(parsePoint)

Build the model

model = LinearRegressionWithSGD.train(parsedData)

Get predictions

valuesAndPreds = parsedData.map(lambda p: (float(model.predict(p.features)), p.label))

Instantiate metrics object

metrics = RegressionMetrics(valuesAndPreds)

Squared Error

print("MSE = %s" % metrics.meanSquaredError) print("RMSE = %s" % metrics.rootMeanSquaredError)

Files

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File metadata and controls

Classification model evaluation

Binary classification

Threshold tuning

Several of the methods available in scala are currently missing from pyspark

Load training data in LIBSVM format

Split data into training (60%) and test (40%)

Run training algorithm to build the model

Compute raw scores on the test set

Instantiate metrics object

Area under precision-recall curve

Area under ROC curve

Multiclass classification

Label based metrics

Load training data in LIBSVM format

Split data into training (60%) and test (40%)

Run training algorithm to build the model

Compute raw scores on the test set

Instantiate metrics object

Overall statistics

Statistics by class

Weighted stats

Multilabel classification

Instantiate metrics object

Summary stats

Individual label stats

Micro stats

Hamming loss

Subset accuracy

Ranking systems

Read in the ratings data

Train a model on to predict user-product ratings

Get predicted ratings on all existing user-product pairs

Instantiate regression metrics to compare predicted and actual ratings

Root mean sqaured error

R-squared

Regression model evaluation

Load and parse the data

Build the model

Get predictions

Instantiate metrics object

Squared Error

R-squared

Mean absolute error

Explained variance