``` pyspark.mllib.stat.StatisticschiSqTest(observed, expected=None) :: Experimental :: If `observed` is Vector, conduct Pearson's chi-squared goodness of fit test of the observed data against the expected distribution, or againt the uniform distribution (by default), with each category having an expected frequency of `1 / len(observed)`. (Note: `observed` cannot contain negative values) If `observed` is matrix, conduct Pearson's independence test on the input contingency matrix, which cannot contain negative entries or columns or rows that sum up to 0. If `observed` is an RDD of LabeledPoint, conduct Pearson's independence test for every feature against the label across the input RDD. For each feature, the (feature, label) pairs are converted into a contingency matrix for which the chi-squared statistic is computed. All label and feature values must be categorical. :param observed: it could be a vector containing the observed categorical counts/relative frequencies, or the contingency matrix (containing either counts or relative frequencies), or an RDD of LabeledPoint containing the labeled dataset with categorical features. Real-valued features will be treated as categorical for each distinct value. :param expected: Vector containing the expected categorical counts/relative frequencies. `expected` is rescaled if the `expected` sum differs from the `observed` sum. :return: ChiSquaredTest object containing the test statistic, degrees of freedom, p-value, the method used, and the null hypothesis. ``` Author: Davies Liu <davies@databricks.com> Closes #3091 from davies/his and squashes the following commits: 145d16c [Davies Liu] address comments 0ab0764 [Davies Liu] fix float 5097d54 [Davies Liu] add Hypothesis test Python API
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layout | title | displayTitle |
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global | Basic Statistics - MLlib | <a href="mllib-guide.html">MLlib</a> - Basic Statistics |
- Table of contents {:toc}
\[ \newcommand{\R}{\mathbb{R}} \newcommand{\E}{\mathbb{E}} \newcommand{\x}{\mathbf{x}} \newcommand{\y}{\mathbf{y}} \newcommand{\wv}{\mathbf{w}} \newcommand{\av}{\mathbf{\alpha}} \newcommand{\bv}{\mathbf{b}} \newcommand{\N}{\mathbb{N}} \newcommand{\id}{\mathbf{I}} \newcommand{\ind}{\mathbf{1}} \newcommand{\0}{\mathbf{0}} \newcommand{\unit}{\mathbf{e}} \newcommand{\one}{\mathbf{1}} \newcommand{\zero}{\mathbf{0}} \]
Summary statistics
We provide column summary statistics for RDD[Vector]
through the function colStats
available in Statistics
.
colStats()
returns an instance of
MultivariateStatisticalSummary
,
which contains the column-wise max, min, mean, variance, and number of nonzeros, as well as the
total count.
{% highlight scala %} import org.apache.spark.mllib.linalg.Vector import org.apache.spark.mllib.stat.{MultivariateStatisticalSummary, Statistics}
val observations: RDD[Vector] = ... // an RDD of Vectors
// Compute column summary statistics. val summary: MultivariateStatisticalSummary = Statistics.colStats(observations) println(summary.mean) // a dense vector containing the mean value for each column println(summary.variance) // column-wise variance println(summary.numNonzeros) // number of nonzeros in each column
{% endhighlight %}
colStats()
returns an instance of
MultivariateStatisticalSummary
,
which contains the column-wise max, min, mean, variance, and number of nonzeros, as well as the
total count.
{% highlight java %} import org.apache.spark.api.java.JavaRDD; import org.apache.spark.api.java.JavaSparkContext; import org.apache.spark.mllib.linalg.Vector; import org.apache.spark.mllib.stat.MultivariateStatisticalSummary; import org.apache.spark.mllib.stat.Statistics;
JavaSparkContext jsc = ...
JavaRDD mat = ... // an RDD of Vectors
// Compute column summary statistics. MultivariateStatisticalSummary summary = Statistics.colStats(mat.rdd()); System.out.println(summary.mean()); // a dense vector containing the mean value for each column System.out.println(summary.variance()); // column-wise variance System.out.println(summary.numNonzeros()); // number of nonzeros in each column
{% endhighlight %}
{% highlight python %} from pyspark.mllib.stat import Statistics
sc = ... # SparkContext
mat = ... # an RDD of Vectors
Compute column summary statistics.
summary = Statistics.colStats(mat) print summary.mean() print summary.variance() print summary.numNonzeros()
{% endhighlight %}
Correlations
Calculating the correlation between two series of data is a common operation in Statistics. In MLlib we provide the flexibility to calculate pairwise correlations among many series. The supported correlation methods are currently Pearson's and Spearman's correlation.
{% highlight scala %} import org.apache.spark.SparkContext import org.apache.spark.mllib.linalg._ import org.apache.spark.mllib.stat.Statistics
val sc: SparkContext = ...
val seriesX: RDD[Double] = ... // a series val seriesY: RDD[Double] = ... // must have the same number of partitions and cardinality as seriesX
// compute the correlation using Pearson's method. Enter "spearman" for Spearman's method. If a // method is not specified, Pearson's method will be used by default. val correlation: Double = Statistics.corr(seriesX, seriesY, "pearson")
val data: RDD[Vector] = ... // note that each Vector is a row and not a column
// calculate the correlation matrix using Pearson's method. Use "spearman" for Spearman's method. // If a method is not specified, Pearson's method will be used by default. val correlMatrix: Matrix = Statistics.corr(data, "pearson")
{% endhighlight %}
{% highlight java %} import org.apache.spark.api.java.JavaDoubleRDD; import org.apache.spark.api.java.JavaSparkContext; import org.apache.spark.mllib.linalg.*; import org.apache.spark.mllib.stat.Statistics;
JavaSparkContext jsc = ...
JavaDoubleRDD seriesX = ... // a series JavaDoubleRDD seriesY = ... // must have the same number of partitions and cardinality as seriesX
// compute the correlation using Pearson's method. Enter "spearman" for Spearman's method. If a // method is not specified, Pearson's method will be used by default. Double correlation = Statistics.corr(seriesX.srdd(), seriesY.srdd(), "pearson");
JavaRDD data = ... // note that each Vector is a row and not a column
// calculate the correlation matrix using Pearson's method. Use "spearman" for Spearman's method. // If a method is not specified, Pearson's method will be used by default. Matrix correlMatrix = Statistics.corr(data.rdd(), "pearson");
{% endhighlight %}
{% highlight python %} from pyspark.mllib.stat import Statistics
sc = ... # SparkContext
seriesX = ... # a series seriesY = ... # must have the same number of partitions and cardinality as seriesX
Compute the correlation using Pearson's method. Enter "spearman" for Spearman's method. If a
method is not specified, Pearson's method will be used by default.
print Statistics.corr(seriesX, seriesY, method="pearson")
data = ... # an RDD of Vectors
calculate the correlation matrix using Pearson's method. Use "spearman" for Spearman's method.
If a method is not specified, Pearson's method will be used by default.
print Statistics.corr(data, method="pearson")
{% endhighlight %}
Stratified sampling
Unlike the other statistics functions, which reside in MLlib, stratified sampling methods,
sampleByKey
and sampleByKeyExact
, can be performed on RDD's of key-value pairs. For stratified
sampling, the keys can be thought of as a label and the value as a specific attribute. For example
the key can be man or woman, or document ids, and the respective values can be the list of ages
of the people in the population or the list of words in the documents. The sampleByKey
method
will flip a coin to decide whether an observation will be sampled or not, therefore requires one
pass over the data, and provides an expected sample size. sampleByKeyExact
requires significant
more resources than the per-stratum simple random sampling used in sampleByKey
, but will provide
the exact sampling size with 99.99% confidence. sampleByKeyExact
is currently not supported in
python.
{% highlight scala %} import org.apache.spark.SparkContext import org.apache.spark.SparkContext._ import org.apache.spark.rdd.PairRDDFunctions
val sc: SparkContext = ...
val data = ... // an RDD[(K, V)] of any key value pairs val fractions: Map[K, Double] = ... // specify the exact fraction desired from each key
// Get an exact sample from each stratum val approxSample = data.sampleByKey(withReplacement = false, fractions) val exactSample = data.sampleByKeyExact(withReplacement = false, fractions)
{% endhighlight %}
{% highlight java %} import java.util.Map;
import org.apache.spark.api.java.JavaPairRDD; import org.apache.spark.api.java.JavaSparkContext;
JavaSparkContext jsc = ...
JavaPairRDD<K, V> data = ... // an RDD of any key value pairs Map<K, Object> fractions = ... // specify the exact fraction desired from each key
// Get an exact sample from each stratum JavaPairRDD<K, V> approxSample = data.sampleByKey(false, fractions); JavaPairRDD<K, V> exactSample = data.sampleByKeyExact(false, fractions);
{% endhighlight %}
Note: sampleByKeyExact()
is currently not supported in Python.
{% highlight python %}
sc = ... # SparkContext
data = ... # an RDD of any key value pairs fractions = ... # specify the exact fraction desired from each key as a dictionary
approxSample = data.sampleByKey(False, fractions);
{% endhighlight %}
Hypothesis testing
Hypothesis testing is a powerful tool in statistics to determine whether a result is statistically
significant, whether this result occurred by chance or not. MLlib currently supports Pearson's
chi-squared ( $\chi^2$) tests for goodness of fit and independence. The input data types determine
whether the goodness of fit or the independence test is conducted. The goodness of fit test requires
an input type of Vector
, whereas the independence test requires a Matrix
as input.
MLlib also supports the input type RDD[LabeledPoint]
to enable feature selection via chi-squared
independence tests.
{% highlight scala %} import org.apache.spark.SparkContext import org.apache.spark.mllib.linalg._ import org.apache.spark.mllib.regression.LabeledPoint import org.apache.spark.mllib.stat.Statistics._
val sc: SparkContext = ...
val vec: Vector = ... // a vector composed of the frequencies of events
// compute the goodness of fit. If a second vector to test against is not supplied as a parameter,
// the test runs against a uniform distribution.
val goodnessOfFitTestResult = Statistics.chiSqTest(vec)
println(goodnessOfFitTestResult) // summary of the test including the p-value, degrees of freedom,
// test statistic, the method used, and the null hypothesis.
val mat: Matrix = ... // a contingency matrix
// conduct Pearson's independence test on the input contingency matrix val independenceTestResult = Statistics.chiSqTest(mat) println(independenceTestResult) // summary of the test including the p-value, degrees of freedom...
val obs: RDD[LabeledPoint] = ... // (feature, label) pairs.
// The contingency table is constructed from the raw (feature, label) pairs and used to conduct // the independence test. Returns an array containing the ChiSquaredTestResult for every feature // against the label. val featureTestResults: Array[ChiSqTestResult] = Statistics.chiSqTest(obs) var i = 1 featureTestResults.foreach { result => println(s"Column $i:\n$result") i += 1 } // summary of the test
{% endhighlight %}
{% highlight java %} import org.apache.spark.api.java.JavaRDD; import org.apache.spark.api.java.JavaSparkContext; import org.apache.spark.mllib.linalg.*; import org.apache.spark.mllib.regression.LabeledPoint; import org.apache.spark.mllib.stat.Statistics; import org.apache.spark.mllib.stat.test.ChiSqTestResult;
JavaSparkContext jsc = ...
Vector vec = ... // a vector composed of the frequencies of events
// compute the goodness of fit. If a second vector to test against is not supplied as a parameter,
// the test runs against a uniform distribution.
ChiSqTestResult goodnessOfFitTestResult = Statistics.chiSqTest(vec);
// summary of the test including the p-value, degrees of freedom, test statistic, the method used,
// and the null hypothesis.
System.out.println(goodnessOfFitTestResult);
Matrix mat = ... // a contingency matrix
// conduct Pearson's independence test on the input contingency matrix ChiSqTestResult independenceTestResult = Statistics.chiSqTest(mat); // summary of the test including the p-value, degrees of freedom... System.out.println(independenceTestResult);
JavaRDD obs = ... // an RDD of labeled points
// The contingency table is constructed from the raw (feature, label) pairs and used to conduct // the independence test. Returns an array containing the ChiSquaredTestResult for every feature // against the label. ChiSqTestResult[] featureTestResults = Statistics.chiSqTest(obs.rdd()); int i = 1; for (ChiSqTestResult result : featureTestResults) { System.out.println("Column " + i + ":"); System.out.println(result); // summary of the test i++; }
{% endhighlight %}
{% highlight python %} from pyspark import SparkContext from pyspark.mllib.linalg import Vectors, Matrices from pyspark.mllib.regresssion import LabeledPoint from pyspark.mllib.stat import Statistics
sc = SparkContext()
vec = Vectors.dense(...) # a vector composed of the frequencies of events
compute the goodness of fit. If a second vector to test against is not supplied as a parameter,
the test runs against a uniform distribution.
goodnessOfFitTestResult = Statistics.chiSqTest(vec) print goodnessOfFitTestResult # summary of the test including the p-value, degrees of freedom, # test statistic, the method used, and the null hypothesis.
mat = Matrices.dense(...) # a contingency matrix
conduct Pearson's independence test on the input contingency matrix
independenceTestResult = Statistics.chiSqTest(mat) print independenceTestResult # summary of the test including the p-value, degrees of freedom...
obs = sc.parallelize(...) # LabeledPoint(feature, label) .
The contingency table is constructed from an RDD of LabeledPoint and used to conduct
the independence test. Returns an array containing the ChiSquaredTestResult for every feature
against the label.
featureTestResults = Statistics.chiSqTest(obs)
for i, result in enumerate(featureTestResults): print "Column $d:" % (i + 1) print result {% endhighlight %}
Random data generation
Random data generation is useful for randomized algorithms, prototyping, and performance testing. MLlib supports generating random RDDs with i.i.d. values drawn from a given distribution: uniform, standard normal, or Poisson.
{% highlight scala %} import org.apache.spark.SparkContext import org.apache.spark.mllib.random.RandomRDDs._
val sc: SparkContext = ...
// Generate a random double RDD that contains 1 million i.i.d. values drawn from the
// standard normal distribution N(0, 1)
, evenly distributed in 10 partitions.
val u = normalRDD(sc, 1000000L, 10)
// Apply a transform to get a random double RDD following N(1, 4)
.
val v = u.map(x => 1.0 + 2.0 * x)
{% endhighlight %}
{% highlight java %} import org.apache.spark.SparkContext; import org.apache.spark.api.JavaDoubleRDD; import static org.apache.spark.mllib.random.RandomRDDs.*;
JavaSparkContext jsc = ...
// Generate a random double RDD that contains 1 million i.i.d. values drawn from the
// standard normal distribution N(0, 1)
, evenly distributed in 10 partitions.
JavaDoubleRDD u = normalJavaRDD(jsc, 1000000L, 10);
// Apply a transform to get a random double RDD following N(1, 4)
.
JavaDoubleRDD v = u.map(
new Function<Double, Double>() {
public Double call(Double x) {
return 1.0 + 2.0 * x;
}
});
{% endhighlight %}
{% highlight python %} from pyspark.mllib.random import RandomRDDs
sc = ... # SparkContext
Generate a random double RDD that contains 1 million i.i.d. values drawn from the
standard normal distribution N(0, 1)
, evenly distributed in 10 partitions.
u = RandomRDDs.uniformRDD(sc, 1000000L, 10)
Apply a transform to get a random double RDD following N(1, 4)
.
v = u.map(lambda x: 1.0 + 2.0 * x) {% endhighlight %}