SummaryStatistics
but handles n-tuple values instead of scalar values. It can also compute the
full covariance matrix for the input data.
</p>
<p>
Neither <code>DescriptiveStatistics nor SummaryStatistics is
thread-safe. <a href="../apidocs/org/apache/commons/math/stat/descriptive/SynchronizedDescriptiveStatistics.html">
SynchronizedDescriptiveStatistics</a> and
<a href="../apidocs/org/apache/commons/math/stat/descriptive/SynchronizedSummaryStatistics.html">
SynchronizedSummaryStatistics</a>, respectively, provide thread-safe versions for applications that
require concurrent access to statistical aggregates by multiple threads.
<a href="../apidocs/org/apache/commons/math/stat/descriptive/SynchronizedMultiVariateSummaryStatistics.html">
SynchronizedMultivariateSummaryStatistics</a> provides threadsafe MultivariateSummaryStatistics.
</p>
<p>
There is also a utility class,
<a href="../apidocs/org/apache/commons/math/stat/StatUtils.html">
StatUtils</a>, that provides static methods for computing statistics
directly from double[] arrays.
</p>
<p>
Here are some examples showing how to compute Descriptive statistics.
<dl>
<dt>Compute summary statistics for a list of double values
<br><dd>Using the
DescriptiveStatistics aggregate
(values are stored in memory):
<source>
// Get a DescriptiveStatistics instance
DescriptiveStatistics stats = new DescriptiveStatistics();
// Add the data from the array
for( int i = 0; i < inputArray.length; i++) {
stats.addValue(inputArray[i]);
}
// Compute some statistics
double mean = stats.getMean();
double std = stats.getStandardDeviation();
double median = stats.getMedian();
</source>
</dd>
<dd>Using the SummaryStatistics aggregate (values are
<strong>not stored in memory):
<source>
// Get a SummaryStatistics instance
SummaryStatistics stats = new SummaryStatistics();
// Read data from an input stream,
// adding values and updating sums, counters, etc.
while (line != null) {
line = in.readLine();
stats.addValue(Double.parseDouble(line.trim()));
}
in.close();
// Compute the statistics
double mean = stats.getMean();
double std = stats.getStandardDeviation();
//double median = stats.getMedian(); <-- NOT AVAILABLE
</source>
</dd>
<dd>Using the StatUtils utility class:
<source>
// Compute statistics directly from the array
// assume values is a double[] array
double mean = StatUtils.mean(values);
double std = StatUtils.variance(values);
double median = StatUtils.percentile(50);
// Compute the mean of the first three values in the array
mean = StatUtils.mean(values, 0, 3);
</source>
</dd>
<dt>Maintain a "rolling mean" of the most recent 100 values from
an input stream</dt>
<br><dd>Use a
DescriptiveStatistics instance with
window size set to 100
<source>
// Create a DescriptiveStats instance and set the window size to 100
DescriptiveStatistics stats = new DescriptiveStatistics();
stats.setWindowSize(100);
// Read data from an input stream,
// displaying the mean of the most recent 100 observations
// after every 100 observations
long nLines = 0;
while (line != null) {
line = in.readLine();
stats.addValue(Double.parseDouble(line.trim()));
if (nLines == 100) {
nLines = 0;
System.out.println(stats.getMean());
}
}
in.close();
</source>
</dd>
<dt>Compute statistics in a thread-safe manner
<br/>
<dd>Use a SynchronizedDescriptiveStatistics instance
<source>
// Create a SynchronizedDescriptiveStatistics instance and
// use as any other DescriptiveStatistics instance
DescriptiveStatistics stats = new SynchronizedDescriptiveStatistics();
</source>
</dd>
<dt>Compute statistics for multiple samples and overall statistics concurrently
<br/>
<dd>There are two ways to do this using AggregateSummaryStatistics.
The first is to use an <code>AggregateSummaryStatistics instance to accumulate
overall statistics contributed by <code>SummaryStatistics instances created using
<a href="../apidocs/org/apache/commons/math/stat/descriptive/AggregateSummaryStatistics.html#createContributingStatistics()">
AggregateSummaryStatistics.createContributingStatistics()</a>:
<source>
// Create a AggregateSummaryStatistics instance to accumulate the overall statistics
// and AggregatingSummaryStatistics for the subsamples
AggregateSummaryStatistics aggregate = new AggregateSummaryStatistics();
SummaryStatistics setOneStats = aggregate.createContributingStatistics();
SummaryStatistics setTwoStats = aggregate.createContributingStatistics();
// Add values to the subsample aggregates
setOneStats.addValue(2);
setOneStats.addValue(3);
setTwoStats.addValue(2);
setTwoStats.addValue(4);
...
// Full sample data is reported by the aggregate
double totalSampleSum = aggregate.getSum();
</source>
The above approach has the disadvantages that the <code>addValue calls must be synchronized on the
<code>SummaryStatistics instance maintained by the aggregate and each value addition updates the
aggregate as well as the subsample. For applications that can wait to do the aggregation until all values
have been added, a static
<a href="../apidocs/org/apache/commons/math/stat/descriptive/AggregateSummaryStatistics.html#aggregate(java.util.Collection)">
aggregate</a> method is available, as shown in the following example.
This method should be used when aggregation needs to be done across threads.
<source>
// Create SummaryStatistics instances for the subsample data
SummaryStatistics setOneStats = new SummaryStatistics();
SummaryStatistics setTwoStats = new SummaryStatistics();
// Add values to the subsample SummaryStatistics instances
setOneStats.addValue(2);
setOneStats.addValue(3);
setTwoStats.addValue(2);
setTwoStats.addValue(4);
...
// Aggregate the subsample statistics
Collection<SummaryStatistics> aggregate = new ArrayList<SummaryStatistics>();
aggregate.add(setOneStats);
aggregate.add(setTwoStats);
StatisticalSummary aggregatedStats = AggregateSummaryStatistics.aggregate(aggregate);
// Full sample data is reported by aggregatedStats
double totalSampleSum = aggregatedStats.getSum();
</source>
</dd>
</dl>
</p>
</subsection>
<subsection name="1.3 Frequency distributions">
<p>
<a href="../apidocs/org/apache/commons/math/stat/Frequency.html">
org.apache.commons.math.stat.descriptive.Frequency</a>
provides a simple interface for maintaining counts and percentages of discrete
values.
</p>
<p>
Strings, integers, longs and chars are all supported as value types,
as well as instances of any class that implements <code>Comparable.
The ordering of values used in computing cumulative frequencies is by
default the <i>natural ordering, but this can be overriden by supplying a
<code>Comparator to the constructor. Adding values that are not
comparable to those that have already been added results in an
<code>IllegalArgumentException.
</p>
<p>
Here are some examples.
<dl>
<dt>Compute a frequency distribution based on integer values
<br><dd>Mixing integers, longs, Integers and Longs: <source> Frequency f = new Frequency(); f.addValue(1); f.addValue(new Integer(1)); f.addValue(new Long(1)); f.addValue(2); f.addValue(new Integer(-1)); System.out.prinltn(f.getCount(1)); // displays 3 System.out.println(f.getCumPct(0)); // displays 0.2 System.out.println(f.getPct(new Integer(1))); // displays 0.6 System.out.println(f.getCumPct(-2)); // displays 0 System.out.println(f.getCumPct(10)); // displays 1 </source> </dd> <dt>Count string frequencies <br>
<dd>Using case-sensitive comparison, alpha sort order (natural comparator): <source> Frequency f = new Frequency(); f.addValue("one"); f.addValue("One"); f.addValue("oNe"); f.addValue("Z"); System.out.println(f.getCount("one")); // displays 1 System.out.println(f.getCumPct("Z")); // displays 0.5 System.out.println(f.getCumPct("Ot")); // displays 0.25 </source> </dd> <dd>Using case-insensitive comparator: <source> Frequency f = new Frequency(String.CASE_INSENSITIVE_ORDER); f.addValue("one"); f.addValue("One"); f.addValue("oNe"); f.addValue("Z"); System.out.println(f.getCount("one")); // displays 3 System.out.println(f.getCumPct("z")); // displays 1 </source> </dd> </dl> </p> </subsection> <subsection name="1.4 Simple regression"> <p> <a href="../apidocs/org/apache/commons/math/stat/regression/SimpleRegression.html"> org.apache.commons.math.stat.regression.SimpleRegression</a> provides ordinary least squares regression with one independent variable, estimating the linear model: </p> <p> <code> y = intercept + slope * x </p> <p> Standard errors for <code>intercept and
slope are
available as well as ANOVA, r-square and Pearson's r statistics.
</p>
<p>
Observations (x,y pairs) can be added to the model one at a time or they
can be provided in a 2-dimensional array. The observations are not stored
in memory, so there is no limit to the number of observations that can be
added to the model.
</p>
<p>
<strong>Usage Notes: -
<li> When there are fewer than two observations in the model, or when
there is no variation in the x values (i.e. all x values are the same)
all statistics return <code>NaN. At least two observations with
different x coordinates are requred to estimate a bivariate regression
model.</li>
<li> getters for the statistics always compute values based on the current
set of observations -- i.e., you can get statistics, then add more data
and get updated statistics without using a new instance. There is no
"compute" method that updates all statistics. Each of the getters performs
the necessary computations to return the requested statistic.</li>
</ul>
</p>
<p>
<strong>Implementation Notes:
-
<li> As observations are added to the model, the sum of x values, y values,
cross products (x times y), and squared deviations of x and y from their
respective means are updated using updating formulas defined in
"Algorithms for Computing the Sample Variance: Analysis and
Recommendations", Chan, T.F., Golub, G.H., and LeVeque, R.J.
1983, American Statistician, vol. 37, pp. 242-247, referenced in
Weisberg, S. "Applied Linear Regression". 2nd Ed. 1985. All regression
statistics are computed from these sums.</li>
<li> Inference statistics (confidence intervals, parameter significance levels)
are based on on the assumption that the observations included in the model are
drawn from a <a href="http://mathworld.wolfram.com/BivariateNormalDistribution.html">
Bivariate Normal Distribution</a>
</ul>
</p>
<p>
Here are some examples.
<dl>
<dt>Estimate a model based on observations added one at a time
<br>
<dd>Instantiate a regression instance and add data points <source> regression = new SimpleRegression(); regression.addData(1d, 2d); // At this point, with only one observation, // all regression statistics will return NaN regression.addData(3d, 3d); // With only two observations, // slope and intercept can be computed // but inference statistics will return NaN regression.addData(3d, 3d); // Now all statistics are defined. </source> </dd> <dd>Compute some statistics based on observations added so far <source> System.out.println(regression.getIntercept()); // displays intercept of regression line System.out.println(regression.getSlope()); // displays slope of regression line System.out.println(regression.getSlopeStdErr()); // displays slope standard error </source> </dd> <dd>Use the regression model to predict the y value for a new x value <source> System.out.println(regression.predict(1.5d) // displays predicted y value for x = 1.5 </source> More data points can be added and subsequent getXxx calls will incorporate additional data in statistics. </dd> <dt>Estimate a model from a double[][] array of data points <br>
<dd>Instantiate a regression object and load dataset <source> double[][] data = { { 1, 3 }, {2, 5 }, {3, 7 }, {4, 14 }, {5, 11 }}; SimpleRegression regression = new SimpleRegression(); regression.addData(data); </source> </dd> <dd>Estimate regression model based on data <source> System.out.println(regression.getIntercept()); // displays intercept of regression line System.out.println(regression.getSlope()); // displays slope of regression line System.out.println(regression.getSlopeStdErr()); // displays slope standard error </source> More data points -- even another double[][] array -- can be added and subsequent getXxx calls will incorporate additional data in statistics. </dd> </dl> </p> </subsection> <subsection name="1.5 Multiple linear regression"> <p> <a href="../apidocs/org/apache/commons/math/stat/regression/MultipleLinearRegression.html"> org.apache.commons.math.stat.regression.MultipleLinearRegression</a> provides ordinary least squares regression with a generic multiple variable linear model, which in matrix notation can be expressed as: </p> <p> <code> y=X*b+u </p> <p> where y is an <code>n-vector regressand, X is a
[n,k] matrix whose k columns are called
<b>regressors, b is k-vector of regression parameters and u is an n-vector
of <b>error terms or residuals. The notation is quite standard in literature,
cf eg <a href="http://www.econ.queensu.ca/ETM">Davidson and MacKinnon, Econometrics Theory and Methods, 2004.
</p>
<p>
Two implementations are provided: <a href="../apidocs/org/apache/commons/math/stat/regression/OLSMultipleLinearRegression.html">
org.apache.commons.math.stat.regression.OLSMultipleLinearRegression</a> and
<a href="../apidocs/org/apache/commons/math/stat/regression/GLSMultipleLinearRegression.html">
org.apache.commons.math.stat.regression.GLSMultipleLinearRegression</a>
</p>
<p>
Observations (x,y and covariance data matrices) can be added to the model via the <code>addData(double[] y, double[][] x, double[][] covariance) method.
The observations are stored in memory until the next time the addData method is invoked.
</p>
<p>
<strong>Usage Notes: -
<li> Data is validated when invoking the
addData(double[] y, double[][] x, double[][] covariance) method and
<code>IllegalArgumentException is thrown when inappropriate.
</li>
<li> Only the GLS regressions require the covariance matrix, so in the OLS regression it is ignored and can be safely
inputted as <code>null.
</ul>
</p>
<p>
Here are some examples.
<dl>
<dt>OLS regression
<br><dd>Instantiate an OLS regression object and load dataset <source> MultipleLinearRegression regression = new OLSMultipleLinearRegression(); double[] y = new double[]{11.0, 12.0, 13.0, 14.0, 15.0, 16.0}; double[] x = new double[6][]; x[0] = new double[]{1.0, 0, 0, 0, 0, 0}; x[1] = new double[]{1.0, 2.0, 0, 0, 0, 0}; x[2] = new double[]{1.0, 0, 3.0, 0, 0, 0}; x[3] = new double[]{1.0, 0, 0, 4.0, 0, 0}; x[4] = new double[]{1.0, 0, 0, 0, 5.0, 0}; x[5] = new double[]{1.0, 0, 0, 0, 0, 6.0}; regression.addData(y, x, null); // we don't need covariance </source> </dd> <dd>Estimate of regression values honours the
MultipleLinearRegression interface:
<source>
double[] beta = regression.estimateRegressionParameters();
double[] residuals = regression.estimateResiduals();
double[][] parametersVariance = regression.estimateRegressionParametersVariance();
double regressandVariance = regression.estimateRegressandVariance();
</source>
</dd>
<dt>GLS regression
<br><dd>Instantiate an GLS regression object and load dataset <source> MultipleLinearRegression regression = new GLSMultipleLinearRegression(); double[] y = new double[]{11.0, 12.0, 13.0, 14.0, 15.0, 16.0}; double[] x = new double[6][]; x[0] = new double[]{1.0, 0, 0, 0, 0, 0}; x[1] = new double[]{1.0, 2.0, 0, 0, 0, 0}; x[2] = new double[]{1.0, 0, 3.0, 0, 0, 0}; x[3] = new double[]{1.0, 0, 0, 4.0, 0, 0}; x[4] = new double[]{1.0, 0, 0, 0, 5.0, 0}; x[5] = new double[]{1.0, 0, 0, 0, 0, 6.0}; double[][] omega = new double[6][]; omega[0] = new double[]{1.1, 0, 0, 0, 0, 0}; omega[1] = new double[]{0, 2.2, 0, 0, 0, 0}; omega[2] = new double[]{0, 0, 3.3, 0, 0, 0}; omega[3] = new double[]{0, 0, 0, 4.4, 0, 0}; omega[4] = new double[]{0, 0, 0, 0, 5.5, 0}; omega[5] = new double[]{0, 0, 0, 0, 0, 6.6}; regression.addData(y, x, omega); // we do need covariance </source> </dd> <dd>Estimate of regression values honours the same
MultipleLinearRegression interface as
the OLS regression.
</dd>
</dl>
</p>
</subsection>
<subsection name="1.6 Rank transformations">
<p>
Some statistical algorithms require that input data be replaced by ranks.
The <a href="../apidocs/org/apache/commons/math/stat/ranking/package-summary.html">
org.apache.commons.math.stat.ranking</a> package provides rank transformation.
<a href="../apidocs/org/apache/commons/math/stat/ranking/RankingAlgorithm.html">
RankingAlgorithm</a> defines the interface for ranking.
<a href="../apidocs/org/apache/commons/math/stat/ranking/NaturalRanking.html">
NaturalRanking</a> provides an implementation that has two configuration options.
<ul>
<li>
Ties strategy</a> deterimines how ties in the source data are handled by the ranking
<li>
NaN strategy</a> determines how NaN values in the source data are handled.
</ul>
</p>
<p>
Examples:
<source>
NaturalRanking ranking = new NaturalRanking(NaNStrategy.MINIMAL,
TiesStrategy.MAXIMUM);
double[] data = { 20, 17, 30, 42.3, 17, 50,
Double.NaN, Double.NEGATIVE_INFINITY, 17 };
double[] ranks = ranking.rank(exampleData);
</source>
results in <code>ranks containing {6, 5, 7, 8, 5, 9, 2, 2, 5}.
<source>
new NaturalRanking(NaNStrategy.REMOVED,TiesStrategy.SEQUENTIAL).rank(exampleData);
</source>
returns <code>{5, 2, 6, 7, 3, 8, 1, 4}.
</p>
<p>
The default <code>NaNStrategy is NaNStrategy.MAXIMAL. This makes NaN
values larger than any other value (including <code>Double.POSITIVE_INFINITY). The
default <code>TiesStrategy is TiesStrategy.AVERAGE, which assigns tied
values the average of the ranks applicable to the sequence of ties. See the
<a href="../apidocs/org/apache/commons/math/stat/ranking/NaturalRanking.html">
NaturalRanking</a> for more examples and
TiesStrategy</a> and NaNStrategy
for details on these configuration options.
</p>
</subsection>
<subsection name="1.7 Covariance and correlation">
<p>
The <a href="../apidocs/org/apache/commons/math/stat/correlation/package-summary.html">
org.apache.commons.math.stat.correlation</a> package computes covariances
and correlations for pairs of arrays or columns of a matrix.
<a href="../apidocs/org/apache/commons/math/stat/correlation/Covariance.html">
Covariance</a> computes covariances,
<a href="../apidocs/org/apache/commons/math/stat/correlation/PearsonsCorrelation.html">
PearsonsCorrelation</a> provides Pearson's Product-Moment correlation coefficients and
<a href="../apidocs/org/apache/commons/math/stat/correlation/SpearmansCorrelation.html">
SpearmansCorrelation</a> computes Spearman's rank correlation.
</p>
<p>
<strong>Implementation Notes
<ul>
<li>
Unbiased covariances are given by the formula <br><code>cov(X, Y) = sum [(xi - E(X))(yi - E(Y))] / (n - 1) where <code>E(X) is the mean of
X and E(Y)
is the mean of the <code>Y values. Non-bias-corrected estimates use
<code>n in place of n - 1. Whether or not covariances are
bias-corrected is determined by the optional parameter, "biasCorrected," which
defaults to <code>true.
</li>
<li>
<a href="../apidocs/org/apache/commons/math/stat/correlation/PearsonsCorrelation.html">
PearsonsCorrelation</a> computes correlations defined by the formula <code>cor(X, Y) = sum[(xi - E(X))(yi - E(Y))] / [(n - 1)s(X)s(Y)]
where <code>E(X) and
E(Y) are means of X and Y
and <code>s(X), s(Y) are standard deviations.
</li>
<li>
<a href="../apidocs/org/apache/commons/math/stat/correlation/SpearmansCorrelation.html">
SpearmansCorrelation</a> applies a rank transformation to the input data and computes Pearson's
correlation on the ranked data. The ranking algorithm is configurable. By default,
<a href="../apidocs/org/apache/commons/math/stat/ranking/NaturalRanking.html">
NaturalRanking</a> with default strategies for handling ties and NaN values is used.
</li>
</ul>
</p>
<p>
<strong>Examples:
<dl>
<dt>Covariance of 2 arrays
<br><dd>To compute the unbiased covariance between 2 double arrays, <code>x and
y, use:
<source>
new Covariance().covariance(x, y)
</source>
For non-bias-corrected covariances, use
<source>
covariance(x, y, false)
</source>
</dd>
<br><dt>Covariance matrix <br>
<dd> A covariance matrix over the columns of a source matrix
data
can be computed using
<source>
new Covariance().computeCovarianceMatrix(data)
</source>
The i-jth entry of the returned matrix is the unbiased covariance of the ith and jth
columns of <code>data. As above, to get non-bias-corrected covariances,
use
<source>
computeCovarianceMatrix(data, false)
</source>
</dd>
<br><dt>Pearson's correlation of 2 arrays <br>
<dd>To compute the Pearson's product-moment correlation between two double arrays <code>x and
y, use:
<source>
new PearsonsCorrelation().correlation(x, y)
</source>
</dd>
<br><dt>Pearson's correlation matrix <br>
<dd> A (Pearson's) correlation matrix over the columns of a source matrix
data
can be computed using
<source>
new PearsonsCorrelation().computeCorrelationMatrix(data)
</source>
The i-jth entry of the returned matrix is the Pearson's product-moment correlation between the
ith and jth columns of <code>data.
</dd>
<br><dt>Pearson's correlation significance and standard errors <br>
<dd> To compute standard errors and/or significances of correlation coefficients associated with Pearson's correlation coefficients, start by creating a <code>PearsonsCorrelation instance <source> PearsonsCorrelation correlation = new PearsonsCorrelation(data); </source> where <code>data is either a rectangular array or a
RealMatrix.
Then the matrix of standard errors is
<source>
correlation.getCorrelationStandardErrors();
</source>
The formula used to compute the standard error is <br/>
<code>SEr = ((1 - r2) / (n - 2))1/2where <code>r is the estimated correlation coefficient and <code>n is the number of observations in the source dataset.
<strong>p-values for the (2-sided) null hypotheses that elements of a correlation matrix are zero populate the RealMatrix returned by <source> correlation.getCorrelationPValues() </source> <code>getCorrelationPValues().getEntry(i,j) is the probability that a random variable distributed as <code>tn-2 takes a value with absolute value greater than or equal to <br>
<code>|rij|((n - 2) / (1 - rij2))1/2, where <code>rij is the estimated correlation between the ith and jth columns of the source array or RealMatrix. This is sometimes referred to as the <i>significance of the coefficient.
For example, if <code>data is a RealMatrix with 2 columns and 10 rows, then <source> new PearsonsCorrelation(data).getCorrelationPValues().getEntry(0,1) </source> is the significance of the Pearson's correlation coefficient between the two columns of <code>data. If this value is less than .01, we can say that the correlation between the two columns of data is significant at the 99% level. </dd> <br>
<dt>Spearman's rank correlation coefficient <br>
<dd>To compute the Spearman's rank-moment correlation between two double arrays <code>x and
y:
<source>
new SpearmansCorrelation().correlation(x, y)
</source>
This is equivalent to
<source>
RankingAlgorithm ranking = new NaturalRanking();
new PearsonsCorrelation().correlation(ranking.rank(x), ranking.rank(y))
</source>
</dd>
<br></dl> </p> </subsection> <subsection name="1.8 Statistical tests"> <p> The interfaces and implementations in the <a href="../apidocs/org/apache/commons/math/stat/inference/"> org.apache.commons.math.stat.inference</a> package provide <a href="http://www.itl.nist.gov/div898/handbook/prc/section2/prc22.htm"> Student's t</a>, <a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda35f.htm"> Chi-Square</a> and <a href="http://www.itl.nist.gov/div898/handbook/prc/section4/prc43.htm"> One-Way ANOVA</a> test statistics as well as <a href="http://www.cas.lancs.ac.uk/glossary_v1.1/hyptest.html#pvalue"> p-values</a> associated with
t-,
<code>Chi-Square and One-Way ANOVA tests. The
interfaces are
<a href="../apidocs/org/apache/commons/math/stat/inference/TTest.html">
TTest</a>,
<a href="../apidocs/org/apache/commons/math/stat/inference/ChiSquareTest.html">
ChiSquareTest</a>, and
<a href="../apidocs/org/apache/commons/math/stat/inference/OneWayAnova.html">
OneWayAnova</a> with provided implementations
<a href="../apidocs/org/apache/commons/math/stat/inference/TTestImpl.html">
TTestImpl</a>,
<a href="../apidocs/org/apache/commons/math/stat/inference/ChiSquareTestImpl.html">
ChiSquareTestImpl</a> and
<a href="../apidocs/org/apache/commons/math/stat/inference/OneWayAnovaImpl.html">
OneWayAnovaImpl</a>, respectively.
The
<a href="../apidocs/org/apache/commons/math/stat/inference/TestUtils.html">
TestUtils</a> class provides static methods to get test instances or
to compute test statistics directly. The examples below all use the
static methods in <code>TestUtils to execute tests. To get
test object instances, either use e.g.,
<code>TestUtils.getTTest() or use the implementation constructors
directly, e.g.,
<code>new TTestImpl().
</p>
<p>
<strong>Implementation Notes
<ul>
<li>Both one- and two-sample t-tests are supported. Two sample tests
can be either paired or unpaired and the unpaired two-sample tests can
be conducted under the assumption of equal subpopulation variances or
without this assumption. When equal variances is assumed, a pooled
variance estimate is used to compute the t-statistic and the degrees
of freedom used in the t-test equals the sum of the sample sizes minus 2.
When equal variances is not assumed, the t-statistic uses both sample
variances and the
<a href="http://www.itl.nist.gov/div898/handbook/prc/section3/gifs/nu3.gif">
Welch-Satterwaite approximation</a> is used to compute the degrees
of freedom. Methods to return t-statistics and p-values are provided in each
case, as well as boolean-valued methods to perform fixed significance
level tests. The names of methods or methods that assume equal
subpopulation variances always start with "homoscedastic." Test or
test-statistic methods that just start with "t" do not assume equal
variances. See the examples below and the API documentation for
more details.</li>
<li>The validity of the p-values returned by the t-test depends on the
assumptions of the parametric t-test procedure, as discussed
<a href="http://www.basic.nwu.edu/statguidefiles/ttest_unpaired_ass_viol.html">
here</a>
<li>p-values returned by t-, chi-square and Anova tests are exact, based
on numerical approximations to the t-, chi-square and F distributions in the
<code>distributions package.
<li>p-values returned by t-tests are for two-sided tests and the boolean-valued
methods supporting fixed significance level tests assume that the hypotheses
are two-sided. One sided tests can be performed by dividing returned p-values
(resp. critical values) by 2.</li>
<li>Degrees of freedom for chi-square tests are integral values, based on the
number of observed or expected counts (number of observed counts - 1)
for the goodness-of-fit tests and (number of columns -1) * (number of rows - 1)
for independence tests.</li>
</ul>
</p>
<p>
<strong>Examples:
<dl>
<dt>One-sample t tests
<br><dd>To compare the mean of a double[] array to a fixed value: <source> double[] observed = {1d, 2d, 3d}; double mu = 2.5d; System.out.println(TestUtils.t(mu, observed)); </source> The code above will display the t-statisitic associated with a one-sample t-test comparing the mean of the <code>observed values against <code>mu. </dd> <dd>To compare the mean of a dataset described by a <a href="../apidocs/org/apache/commons/math/stat/descriptive/StatisticalSummary.html"> org.apache.commons.math.stat.descriptive.StatisticalSummary</a> to a fixed value: <source> double[] observed ={1d, 2d, 3d}; double mu = 2.5d; SummaryStatistics sampleStats = new SummaryStatistics(); for (int i = 0; i < observed.length; i++) { sampleStats.addValue(observed[i]); } System.out.println(TestUtils.t(mu, observed)); </source> </dd> <dd>To compute the p-value associated with the null hypothesis that the mean of a set of values equals a point estimate, against the two-sided alternative that the mean is different from the target value: <source> double[] observed = {1d, 2d, 3d}; double mu = 2.5d; System.out.println(TestUtils.tTest(mu, observed)); </source> The snippet above will display the p-value associated with the null hypothesis that the mean of the population from which the <code>observed values are drawn equals
mu.
</dd>
<dd>To perform the test using a fixed significance level, use:
<source>
TestUtils.tTest(mu, observed, alpha);
</source>
where <code>0 < alpha < 0.5 is the significance level of
the test. The boolean value returned will be <code>true iff the
null hypothesis can be rejected with confidence <code>1 - alpha.
To test, for example at the 95% level of confidence, use
<code>alpha = 0.05
</dd>
<br><dt>Two-Sample t-tests <br>
<dd>Example 1: Paired test evaluating the null hypothesis that the mean difference between corresponding (paired) elements of the <code>double[] arrays <code>sample1 and
sample2 is zero.
<p>
To compute the t-statistic:
<source>
TestUtils.pairedT(sample1, sample2);
</source>
</p>
<p>
To compute the p-value:
<source>
TestUtils.pairedTTest(sample1, sample2);
</source>
</p>
<p>
To perform a fixed significance level test with alpha = .05:
<source>
TestUtils.pairedTTest(sample1, sample2, .05);
</source>
</p>
The last example will return <code>true iff the p-value
returned by <code>TestUtils.pairedTTest(sample1, sample2)
is less than <code>.05
</dd>
<dd>Example 2: unpaired, two-sided, two-sample t-test using
<code>StatisticalSummary instances, without assuming that
subpopulation variances are equal.
<p>
First create the <code>StatisticalSummary instances. Both
<code>DescriptiveStatistics and SummaryStatistics
implement this interface. Assume that <code>summary1 and
<code>summary2 are SummaryStatistics instances,
each of which has had at least 2 values added to the (virtual) dataset that
it describes. The sample sizes do not have to be the same -- all that is required
is that both samples have at least 2 elements.
</p>
<p>Note: The SummaryStatistics class does
not store the dataset that it describes in memory, but it does compute all
statistics necessary to perform t-tests, so this method can be used to
conduct t-tests with very large samples. One-sample tests can also be
performed this way.
(See <a href="#1.2 Descriptive statistics">Descriptive statistics for details
on the <code>SummaryStatistics class.)
</p>
<p>
To compute the t-statistic:
<source>
TestUtils.t(summary1, summary2);
</source>
</p>
<p>
To compute the p-value:
<source>
TestUtils.tTest(sample1, sample2);
</source>
</p>
<p>
To perform a fixed significance level test with alpha = .05:
<source>
TestUtils.tTest(sample1, sample2, .05);
</source>
</p>
<p>
In each case above, the test does not assume that the subpopulation
variances are equal. To perform the tests under this assumption,
replace "t" at the beginning of the method name with "homoscedasticT"
</p>
</dd>
<br><dt>Chi-square tests <br>
<dd>To compute a chi-square statistic measuring the agreement between a <code>long[] array of observed counts and a
double[]
array of expected counts, use:
<source>
long[] observed = {10, 9, 11};
double[] expected = {10.1, 9.8, 10.3};
System.out.println(TestUtils.chiSquare(expected, observed));
</source>
the value displayed will be
<code>sum((expected[i] - observed[i])^2 / expected[i])
</dd>
<dd> To get the p-value associated with the null hypothesis that
<code>observed conforms to expected use:
<source>
TestUtils.chiSquareTest(expected, observed);
</source>
</dd>
<dd> To test the null hypothesis that observed conforms to
<code>expected with alpha siginficance level
(equiv. <code>100 * (1-alpha)% confidence) where
0 < alpha < 1 </code> use:
<source>
TestUtils.chiSquareTest(expected, observed, alpha);
</source>
The boolean value returned will be <code>true iff the null hypothesis
can be rejected with confidence <code>1 - alpha.
</dd>
<dd>To compute a chi-square statistic statistic associated with a
<a href="http://www.itl.nist.gov/div898/handbook/prc/section4/prc45.htm">
chi-square test of independence</a> based on a two-dimensional (long[][])
<code>counts array viewed as a two-way table, use:
<source>
TestUtils.chiSquareTest(counts);
</source>
The rows of the 2-way table are
<code>count[0], ... , count[count.length - 1]. The chi-square statistic returned is <code>sum((counts[i][j] - expected[i][j])^2/expected[i][j]) where the sum is taken over all table entries and <code>expected[i][j] is the product of the row and column sums at row <code>i, column
j divided by the total count.
</dd>
<dd>To compute the p-value associated with the null hypothesis that
the classifications represented by the counts in the columns of the input 2-way
table are independent of the rows, use:
<source>
TestUtils.chiSquareTest(counts);
</source>
</dd>
<dd>To perform a chi-square test of independence with alpha
siginficance level (equiv. <code>100 * (1-alpha)% confidence)
where <code>0 < alpha < 1 use:
<source>
TestUtils.chiSquareTest(counts, alpha);
</source>
The boolean value returned will be <code>true iff the null
hypothesis can be rejected with confidence <code>1 - alpha.
</dd>
<br><dt>One-Way Anova tests <br>
<dd>To conduct a One-Way Analysis of Variance (ANOVA) to evaluate the null hypothesis that the means of a collection of univariate datasets are the same, start by loading the datasets into a collection, e.g. <source> double[] classA = {93.0, 103.0, 95.0, 101.0, 91.0, 105.0, 96.0, 94.0, 101.0 }; double[] classB = {99.0, 92.0, 102.0, 100.0, 102.0, 89.0 }; double[] classC = {110.0, 115.0, 111.0, 117.0, 128.0, 117.0 }; List classes = new ArrayList(); classes.add(classA); classes.add(classB); classes.add(classC); </source> Then you can compute ANOVA F- or p-values associated with the null hypothesis that the class means are all the same using a <code>OneWayAnova instance or
TestUtils
methods:
<source>
double fStatistic = TestUtils.oneWayAnovaFValue(classes); // F-value
double pValue = TestUtils.oneWayAnovaPValue(classes); // P-value
</source>
To test perform a One-Way Anova test with signficance level set at 0.01
(so the test will, assuming assumptions are met, reject the null
hypothesis incorrectly only about one in 100 times), use
<source>
TestUtils.oneWayAnovaTest(classes, 0.01); // returns a boolean
// true means reject null hypothesis
</source>
</dd>
</dl>
</p>
</subsection>
</section>
</body>
</document>
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Commons Math example source code file (stat.xml)
The Commons Math stat.xml source code<?xml version="1.0"?>
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The ASF licenses this file to You under the Apache License, Version 2.0
(the "License"); you may not use this file except in compliance with
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<?xml-stylesheet type="text/xsl" href="./xdoc.xsl"?>
<!-- $Revision: 925895 $ $Date: 2010-03-21 17:05:20 -0400 (Sun, 21 Mar 2010) $ -->
<document url="stat.html">
<properties>
<title>The Commons Math User Guide - Statistics
</properties>
<body>
<section name="1 Statistics">
<subsection name="1.1 Overview">
<p>
The statistics package provides frameworks and implementations for
basic Descriptive statistics, frequency distributions, bivariate regression,
and t-, chi-square and ANOVA test statistics.
</p>
<p>
<a href="#a1.2_Descriptive_statistics">Descriptive statistics | Aggregate | Statistics Included | Values stored? | |
|---|---|---|---|---|
| <a href="../apidocs/org/apache/commons/math/stat/descriptive/DescriptiveStatistics.html"> DescriptiveStatistics</a> | min, max, mean, geometric mean, n, sum, sum of squares, standard deviation, variance, percentiles, skewness, kurtosis, median</td> | Yes | Yes | |
| <a href="../apidocs/org/apache/commons/math/stat/descriptive/SummaryStatistics.html"> SummaryStatistics</a> | min, max, mean, geometric mean, n, sum, sum of squares, standard deviation, variance</td> | No | No |
