statistics.html

---
layout: layout.njk
permalink: "{{ page.filePathStem }}.html"
---
{% include "toc.njk" %}

<div class="col-md-9 col-md-pull-3">

    <h1 id="statistics-top" class="title">Statistics</h1>

    <p>In Smile, there are many statistical functions to describe and analyze data.</p>

    <h2 id="basic" class="title">Basic Statistic Functions</h2>

    <p>Use the following functions to calculate the descriptive statistics for your data:
        <code>sum</code>, <code>mean</code>,
        <code>median</code>, <code>q1</code>, <code>q3</code>,
        <code>variance</code>, <code>sd</code>, <code>mad</code> (median absolute deviation),
        <code>min</code>, <code>max</code>, <code>whichMin</code>, <code>whichMax</code>, etc.</p>
    <ul class="nav nav-tabs">
        <li class="active"><a href="#scala_1" data-toggle="tab">Scala</a></li>
        <li><a href="#java_1" data-toggle="tab">Java</a></li>
    </ul>
    <div class="tab-content">
        <div class="tab-pane active" id="scala_1">
            <div class="code" style="text-align: left;">
    <pre class="prettyprint lang-scala"><code>
    smile> val x = Array(1.0, 2.0, 3.0, 4.0)
    x: Array[Double] = Array(1.0, 2.0, 3.0, 4.0)

    smile> mean(x)
    res1: Double = 2.5

    smile> sd(x)
    res2: Double = 1.2909944487358054
    </code></pre>
            </div>
        </div>
        <div class="tab-pane" id="java_1">
            <div class="code" style="text-align: left;">
          <pre class="prettyprint lang-java"><code>
    jshell> import static smile.math.MathEx.*

    jshell> import smile.stat.*

    jshell> double[] x = {1.0, 2.0, 3.0, 4.0}
    x ==> double[4] { 1.0, 2.0, 3.0, 4.0 }

    jshell> mean(x)
    $4 ==> 2.5

    jshell> sd(x)
    $5 ==> 1.2909944487358054
          </code></pre>
            </div>
        </div>
    </div>

    <h2 id="distribution" class="title">Distributions</h2>

    <p>Probability distributions are theoretical distributions based on assumptions
        about a source population. The distributions assign probability to the event
        that a random variable has a specific, discrete value, or falls within a
        specified range of continuous values.</p>

    <p>All univariate distributions in Smile implements the interface <code>smile.math.stat.distribution.Distribution</code>.
        We support Bernoulli, beta, binomial, &chi;<sup>2</sup>, exponential, F, gamma, Gaussian, geometric,
        hyper geometric, logistic, log normal, negative binomial, Possion, shift geometric, t, and Weibull distribution.
        In additional, multivariate Gaussian distribution is supported. In fact, we also support finite mixture models
        and can estimate the exponential family mixture models from data.</p>

    <p>A <code>Distribution</code> object can be created with given parameters. Meanwhile they can be created by estimating parameters
        from a given data set. With a <code>Distribution</code> object, we may access its distribution parameter(s), mean,
        variance, standard deviation, entropy, generates a random number following the distribution,
        call its probability density function (the method <code>p</code> or cumulative distribution function (<code>cdf</code>).
        The reverse function of <code>cdf</code> is <code>quantile</code>. We can also calculate the likelihood or
        log likelihood of a sample set.</p>

    <ul class="nav nav-tabs">
        <li class="active"><a href="#scala_2" data-toggle="tab">Scala</a></li>
        <li><a href="#java_2" data-toggle="tab">Java</a></li>
    </ul>
    <div class="tab-content">
        <div class="tab-pane active" id="scala_2">
            <div class="code" style="text-align: left;">
    <pre class="prettyprint lang-scala"><code>
    smile&gt; val e = new ExponentialDistribution(1.0)
    e: smile.stat.distribution.ExponentialDistribution = Exponential Distribution(1.0000)

    smile&gt; e.mean
    res3: Double = 1.0

    smile&gt; e.variance
    res4: Double = 1.0

    smile&gt; e.sd
    res5: Double = 1.0

    smile&gt; e.entropy
    res6: Double = 1.0

    // generate a random number
    smile&gt; e.rand
    res7: Double = 0.3155668608029686

    // PDF
    smile&gt; e.p(2.0)
    res8: Double = 0.1353352832366127

    smile&gt; e.cdf(2.0)
    res9: Double = 0.8646647167633873

    smile&gt; e.quantile(0.1)
    res10: Double = 0.10536051565782628

    smile&gt; e.logLikelihood(Array(1.0, 1.1, 0.9, 1.5))
    res12: Double = -4.5

    // estimate a distribution from data
    smile&gt; val e = ExponentialDistribution.fit(Array(1.0, 1.1, 0.9, 1.5, 1.8, 1.9, 2.0, 0.5))
    e: smile.stat.distribution.ExponentialDistribution = Exponential Distribution(0.7477)
    </code></pre>
            </div>
        </div>
        <div class="tab-pane" id="java_2">
            <div class="code" style="text-align: left;">
          <pre class="prettyprint lang-java"><code>
    jshell> import smile.stat.distribution.*

    jshell> var e = new ExponentialDistribution(1.0)
    e ==> Exponential Distribution(1.0000)

    jshell> e.mean()
    $8 ==> 1.0

    jshell> e.variance()
    $9 ==> 1.0

    jshell> e.sd()
    $10 ==> 1.0

    jshell> e.entropy()
    $11 ==> 1.0

    jshell> e.rand()
    [main] INFO smile.math.MathEx - Set RNG seed 19650218 for thread main
    $12 ==> 0.38422274023788616

    jshell> e.p(2)
    $13 ==> 0.1353352832366127

    jshell> e.cdf(2)
    $14 ==> 0.8646647167633873

    jshell> e.quantile(0.1)
    $15 ==> 0.10536051565782628

    jshell> double[] samples = {1.0, 1.1, 0.9, 1.5}
    samples ==> double[4] { 1.0, 1.1, 0.9, 1.5 }

    jshell> e.logLikelihood(samples)
    $17 ==> -4.5

    jshell> double[] data = {1.0, 1.1, 0.9, 1.5, 1.8, 1.9, 2.0, 0.5}
    data ==> double[8] { 1.0, 1.1, 0.9, 1.5, 1.8, 1.9, 2.0, 0.5 }

    jshell> var d = ExponentialDistribution.fit(data)
    d ==> Exponential Distribution(0.7477)
          </code></pre>
            </div>
        </div>
    </div>

    <p>The below is a more advanced example of estimating a mixture model of Gaussian, exponential and gamma distribution.
        The result is quite accurate for this complicated case.</p>
    <ul class="nav nav-tabs">
        <li class="active"><a href="#scala_3" data-toggle="tab">Scala</a></li>
        <li><a href="#java_3" data-toggle="tab">Java</a></li>
    </ul>
    <div class="tab-content">
        <div class="tab-pane active" id="scala_3">
            <div class="code" style="text-align: left;">
    <pre class="prettyprint lang-scala"><code>
    smile&gt; val gaussian = new GaussianDistribution(-2.0, 1.0)
    smile&gt; val exp = new ExponentialDistribution(0.8)
    smile&gt; val gamma = new GammaDistribution(2.0, 3.0)

    // generate the samples
    smile&gt; val data = Array.fill(500)(gaussian.rand()) ++ Array.fill(500)(exp.rand()) ++ Array.fill(1000)(gamma.rand())

    // define the initial guess of the components in the mixture model
    smile&gt; val a = new Mixture.Component(0.3, new GaussianDistribution(0.0, 1.0))
    smile&gt; val b = new Mixture.Component(0.3, new ExponentialDistribution(1.0))
    smile&gt; val c = new Mixture.Component(0.4, new GammaDistribution(1.0, 2.0))

    // estimate the model
    smile&gt; val mixture = ExponentialFamilyMixture.fit(data, a, b, c)
    mixture: smile.stat.distribution.ExponentialFamilyMixture = Mixture[3]:{ (Gaussian Distribution(-2.0135, 0.9953):0.2478) (Exponential Distribution(0.7676):0.2882) (Gamma Distribution(2.7008, 2.4051):0.4640)}
    </code></pre>
            </div>
        </div>
        <div class="tab-pane" id="java_3">
            <div class="code" style="text-align: left;">
          <pre class="prettyprint lang-java"><code>
    jshell> var gaussian = new GaussianDistribution(-2.0, 1.0)
    gaussian ==> Gaussian Distribution(-2.0000, 1.0000)

    jshell> var exp = new ExponentialDistribution(0.8)
    exp ==> Exponential Distribution(0.8000)

    jshell> var gamma = new GammaDistribution(2.0, 3.0)
    gamma ==> Gamma Distribution(3.0000, 2.0000)

    jshell>     import java.util.stream.*

    jshell> var data = DoubleStream.concat(
       ...>     DoubleStream.concat(
       ...>         DoubleStream.generate(gaussian::rand).limit(500),
       ...>         DoubleStream.generate(exp::rand).limit(500)),
       ...>     DoubleStream.generate(gamma::rand).limit(1000)).toArray()
    data ==> double[2000] { -2.396693222610913, -3.11796309434 ... 0928, 2.995037488374675, 1

    jshell> var a = new Mixture.Component(0.3, new GaussianDistribution(0.0, 1.0))
    a ==> smile.stat.distribution.Mixture$Component@32a068d1

    jshell> var b = new Mixture.Component(0.3, new ExponentialDistribution(1.0))
    b ==> smile.stat.distribution.Mixture$Component@365c30cc

    jshell> var c = new Mixture.Component(0.4, new GammaDistribution(1.0, 2.0))
    c ==> smile.stat.distribution.Mixture$Component@4148db48

    jshell> var mixture = ExponentialFamilyMixture.fit(data, a, b, c)
    mixture ==> Mixture(3)[0.31 x Gaussian Distribution(-1.3630, 1.5056) + 0.17 x Exponential Distribution(0.5566) + 0.52 x Gamma Distribution(3.7170, 1.5014)]
          </code></pre>
            </div>
        </div>
    </div>

    <p>If the distribution family is not known, nonparametric methods such as
        kernel density estimation can be used. Kernel density estimation
        is a fundamental data smoothing problem where inferences about the population
        are made, based on a finite data sample. It is also known as the
        Parzen window method.</p>

    <ul class="nav nav-tabs">
        <li class="active"><a href="#scala_4" data-toggle="tab">Scala</a></li>
        <li><a href="#java_4" data-toggle="tab">Java</a></li>
    </ul>
    <div class="tab-content">
        <div class="tab-pane active" id="scala_4">
            <div class="code" style="text-align: left;">
    <pre class="prettyprint lang-scala"><code>
    smile&gt; val k = new KernelDensity(data)
    k: smile.stat.distribution.KernelDensity = smile.stat.distribution.KernelDensity@69724abb

    smile&gt; k.p(1.0)
    res2: Double = 0.11397721599552492

    smile&gt; mixture.p(1.0)
    res3: Double = 0.1272572973513569
    </code></pre>
            </div>
        </div>
        <div class="tab-pane" id="java_4">
            <div class="code" style="text-align: left;">
          <pre class="prettyprint lang-java"><code>
    jshell> var k = new KernelDensity(data)
    k ==> smile.stat.distribution.KernelDensity@146044d7

    jshell> k.p(1)
    $32 ==> 0.11955905354604122

    jshell> mixture.p(1)
    $33 ==> 0.14009430199392497
          </code></pre>
            </div>
        </div>
    </div>

    <h2 id="test" class="title">Hypothesis Test</h2>

    <p>A statistical hypothesis test is a method of making decisions using data,
        whether from a controlled experiment or an observational study (not controlled).
        In statistics, a result is called statistically significant if it is unlikely
        to have occurred by chance alone, according to a pre-determined threshold
        probability, the significance level.</p>

    <h3 id="chisq-test" class="title">&chi;<sup>2</sup> Test</h3>

    <h4 class="title">One-Sample Test</h4>
    <p>Given the array x containing the observed numbers of events,
        and an array prob containing the expected probabilities of events, and given
        the number of constraints (normally one), a small value of p-value
        indicates a significant difference between the distributions.</p>

    <ul class="nav nav-tabs">
        <li class="active"><a href="#scala_5" data-toggle="tab">Scala</a></li>
        <li><a href="#java_5" data-toggle="tab">Java</a></li>
    </ul>
    <div class="tab-content">
        <div class="tab-pane active" id="scala_5">
            <div class="code" style="text-align: left;">
    <pre class="prettyprint lang-scala"><code>
    smile> val bins = Array(20, 22, 13, 22, 10, 13)
    bins: Array[Int] = Array(20, 22, 13, 22, 10, 13)

    smile> val prob = Array(1.0/6, 1.0/6, 1.0/6, 1.0/6, 1.0/6, 1.0/6)
    prob: Array[Double] = Array(0.16666666666666666, 0.16666666666666666, 0.16666666666666666, 0.16666666666666666, 0.16666666666666666, 0.16666666666666666)

    smile> chisqtest(bins, prob)
    res8: stat.hypothesis.ChiSqTest = One Sample Chi-squared Test(t = 8.3600, df = 5.000, p-value = 0.137480)
    </code></pre>
            </div>
        </div>
        <div class="tab-pane" id="java_5">
            <div class="code" style="text-align: left;">
          <pre class="prettyprint lang-java"><code>
    jshell> import smile.stat.hypothesis.*

    jshell> int[] bins = {20, 22, 13, 22, 10, 13}
    bins ==> int[6] { 20, 22, 13, 22, 10, 13 }

    jshell> double[] prob = {1.0/6, 1.0/6, 1.0/6, 1.0/6, 1.0/6, 1.0/6}
    prob ==> double[6] { 0.16666666666666666, 0.16666666666666 ... 666, 0.16666666666666666 }

    jshell> ChiSqTest.test(bins, prob)
    $37 ==> One Sample Chi-squared Test(t = 8.3600, df = 5.000, p-value = 0.137480)
          </code></pre>
            </div>
        </div>
    </div>

    <h4 class="title">Two-Sample Test</h4>
    <p>Two-sample chisq test. Given the arrays x and y, containing two
        sets of binned data, and given one constraint, a small value of
        p-value indicates a significant difference between two distributions.</p>
    <ul class="nav nav-tabs">
        <li class="active"><a href="#scala_6" data-toggle="tab">Scala</a></li>
        <li><a href="#java_6" data-toggle="tab">Java</a></li>
    </ul>
    <div class="tab-content">
        <div class="tab-pane active" id="scala_6">
            <div class="code" style="text-align: left;">
          <pre class="prettyprint lang-scala"><code>
    smile> val bins1 = Array(8, 13, 16, 10, 3)
    bins1: Array[Int] = Array(8, 13, 16, 10, 3)

    smile> val bins2 = Array(4,  9, 14, 16, 7)
    bins2: Array[Int] = Array(4, 9, 14, 16, 7)

    smile> chisqtest2(bins1, bins2)
    res11: stat.hypothesis.ChiSqTest = Two Sample Chi-squared Test(t = 5.1786, df = 4.000, p-value = 0.269462)
          </code></pre>
            </div>
        </div>
        <div class="tab-pane" id="java_6">
            <div class="code" style="text-align: left;">
          <pre class="prettyprint lang-java"><code>
    jshell> int[] bins1 = {8, 13, 16, 10, 3}
    bins1 ==> int[5] { 8, 13, 16, 10, 3 }

    jshell> int[] bins2 = {4,  9, 14, 16, 7}
    bins2 ==> int[5] { 4, 9, 14, 16, 7 }

    jshell> ChiSqTest.test(bins1, bins2)
    $40 ==> Two Sample Chi-squared Test(t = 5.1786, df = 4.000, p-value = 0.269462)
          </code></pre>
            </div>
        </div>
    </div>

    <h4 class="title">Independence Test</h4>

    <p>Independence test on a two-dimensional contingency table in the form of an array of
        integers. The rows of contingency table
        are labels by the values of one nominal variable, the columns are labels
        by the values of the other nominal variable, and whose entries are
        non-negative integers giving the number of observed events for each
        combination of row and column. Continuity correction
        will be applied when computing the test statistic for 2x2 tables: one half
        is subtracted from all |O-E| differences. The correlation coefficient is
        calculated as Cramer's V.</p>

    <ul class="nav nav-tabs">
        <li class="active"><a href="#scala_7" data-toggle="tab">Scala</a></li>
        <li><a href="#java_7" data-toggle="tab">Java</a></li>
    </ul>
    <div class="tab-content">
        <div class="tab-pane active" id="scala_7">
            <div class="code" style="text-align: left;">
          <pre class="prettyprint lang-scala"><code>
    smile> val x = Array(Array(12, 7), Array(5, 7))
    x: Array[Array[Int]] = Array(Array(12, 7), Array(5, 7))

    smile> chisqtest(x)
    res13: stat.hypothesis.ChiSqTest = Pearson's Chi-squared Test(t = 0.6411, df = 1.000, p-value = 0.423305)
          </code></pre>
            </div>
        </div>
        <div class="tab-pane" id="java_7">
            <div class="code" style="text-align: left;">
          <pre class="prettyprint lang-java"><code>
    jshell> int[][] x = { {12, 7}, {5, 7} }
    x ==> int[2][] { int[2] { 12, 7 }, int[2] { 5, 7 } }

    jshell> ChiSqTest.test(x)
    $42 ==> Pearson's Chi-squared Test(t = 0.6411, df = 1.000, p-value = 0.423305)
          </code></pre>
            </div>
        </div>
    </div>

    <h3 id="f-test" class="title">F Test</h3>

    <p>Test if the arrays x and y have significantly different variances.
        Small values of p-value indicate that the two arrays have significantly
        different variances.</p>
    <ul class="nav nav-tabs">
        <li class="active"><a href="#scala_8" data-toggle="tab">Scala</a></li>
        <li><a href="#java_8" data-toggle="tab">Java</a></li>
    </ul>
    <div class="tab-content">
        <div class="tab-pane active" id="scala_8">
            <div class="code" style="text-align: left;">
          <pre class="prettyprint lang-scala"><code>
    smile> val x = Array(0.48074284, -0.52975023, 1.28590721, 0.63456079, -0.41761197, 2.76072411,
                   1.30321095, -1.16454533, 2.27210509, 1.46394553, -0.31713164, 1.26247543,
                   2.65886430, 0.40773450, 1.18055440, -0.39611251, 2.13557687, 0.40878860,
                   1.28461394, -0.02906355)
    x: Array[Double] = Array(
      0.48074284,
      -0.52975023,
      1.28590721,
      0.63456079,
      -0.41761197,
      2.76072411,
      1.30321095,
      -1.16454533,
      2.27210509,
      1.46394553,
      -0.31713164,
      1.26247543,
      2.6588643,
      0.4077345,
      1.1805544,
      -0.39611251,
      2.13557687,
      0.4087886,
      1.28461394,
      -0.02906355
    )

    smile> val y = Array(1.7495879, 1.9359727, 3.1294928, 0.0861894, 2.1643415, 0.1913219,
                   -0.3947444, 1.6910837, 1.1548294, 0.2763955, 0.4794719, 3.1805501,
                   1.5700497, 2.6860190, -0.4410879, 1.8900183, 1.3422381, -0.1701592)
    y: Array[Double] = Array(
      1.7495879,
      1.9359727,
      3.1294928,
      0.0861894,
      2.1643415,
      0.1913219,
      -0.3947444,
      1.6910837,
      1.1548294,
      0.2763955,
      0.4794719,
      3.1805501,
      1.5700497,
      2.686019,
      -0.4410879,
      1.8900183,
      1.3422381,
      -0.1701592
    )

    smile> ftest(x, y)
    res16: stat.hypothesis.FTest = F-test(f = 1.0958, df1 = 17, df2 = 19, p-value = 0.841464)

    smile> val z = Array(0.6621329, 0.4688975, -0.1553013, 0.4564548, 2.2776146, 2.1543678,
                   2.8555142, 1.5852899, 0.9091290, 1.6060025, 1.0111968, 1.2479493,
                   0.9407034, 1.7167572, 0.5380608, 2.1290007, 1.8695506, 1.2139096)
    z: Array[Double] = Array(
      0.6621329,
      0.4688975,
      -0.1553013,
      0.4564548,
      2.2776146,
      2.1543678,
      2.8555142,
      1.5852899,
      0.909129,
      1.6060025,
      1.0111968,
      1.2479493,
      0.9407034,
      1.7167572,
      0.5380608,
      2.1290007,
      1.8695506,
      1.2139096
    )

    smile> ftest(x, z)
    res18: stat.hypothesis.FTest = F-test(f = 2.0460, df1 = 19, df2 = 17, p-value = 0.143778)
          </code></pre>
            </div>
        </div>
        <div class="tab-pane" id="java_8">
            <div class="code" style="text-align: left;">
          <pre class="prettyprint lang-java"><code>
    jshell> double[] x = {0.48074284, -0.52975023, 1.28590721, 0.63456079, -0.41761197, 2.76072411,
       ...>             1.30321095, -1.16454533, 2.27210509, 1.46394553, -0.31713164, 1.26247543,
       ...>             2.65886430, 0.40773450, 1.18055440, -0.39611251, 2.13557687, 0.40878860,
       ...>             1.28461394, -0.02906355}
    x ==> double[20] { 0.48074284, -0.52975023, 1.28590721, ...  1.28461394, -0.02906355 }

    jshell> double[] y = {1.7495879, 1.9359727, 3.1294928, 0.0861894, 2.1643415, 0.1913219,
       ...>             -0.3947444, 1.6910837, 1.1548294, 0.2763955, 0.4794719, 3.1805501,
       ...>             1.5700497, 2.6860190, -0.4410879, 1.8900183, 1.3422381, -0.1701592}
    y ==> double[18] { 1.7495879, 1.9359727, 3.1294928, 0.0 ... 3, 1.3422381, -0.1701592 }

    jshell> FTest.test(x, y)
    $45 ==> F-test(f = 1.0958, df1 = 17, df2 = 19, p-value = 0.841464)

    jshell> double[] z = {0.6621329, 0.4688975, -0.1553013, 0.4564548, 2.2776146, 2.1543678,
       ...>             2.8555142, 1.5852899, 0.9091290, 1.6060025, 1.0111968, 1.2479493,
       ...>             0.9407034, 1.7167572, 0.5380608, 2.1290007, 1.8695506, 1.2139096}
    z ==> double[18] { 0.6621329, 0.4688975, -0.1553013, 0. ... 07, 1.8695506, 1.2139096 }

    jshell> FTest.test(x, z)
    $47 ==> F-test(f = 2.0460, df1 = 19, df2 = 17, p-value = 0.143778)
          </code></pre>
            </div>
        </div>
    </div>

    <h3 id="t-test" class="title">t Test</h3>

    <h4 class="title">One-Sample Test</h4>
    <p>Independent one-sample t-test whether the mean of a normally distributed
        population has a value specified in a null hypothesis. Small values of
        p-value indicate that the array has significantly different mean.</p>
    <ul class="nav nav-tabs">
        <li class="active"><a href="#scala_9" data-toggle="tab">Scala</a></li>
        <li><a href="#java_9" data-toggle="tab">Java</a></li>
    </ul>
    <div class="tab-content">
        <div class="tab-pane active" id="scala_9">
            <div class="code" style="text-align: left;">
          <pre class="prettyprint lang-scala"><code>
    smile> ttest(x, 1.0)
    res19: stat.hypothesis.TTest = One Sample t-test(t = -0.6641, df = 19.000, p-value = 0.514609)

    smile> ttest(x, 1.1)
    res20: stat.hypothesis.TTest = One Sample t-test(t = -1.0648, df = 19.000, p-value = 0.300300)
          </code></pre>
            </div>
        </div>
        <div class="tab-pane" id="java_9">
            <div class="code" style="text-align: left;">
          <pre class="prettyprint lang-java"><code>
    jshell> TTest.test(x, 1.0)
    $48 ==> One Sample t-test(t = -0.6641, df = 19.000, p-value = 0.514609)

    jshell> TTest.test(x, 1.1)
    $49 ==> One Sample t-test(t = -1.0648, df = 19.000, p-value = 0.300300)
          </code></pre>
            </div>
        </div>
    </div>

    <h4 class="title">Paired Two-Sample Test</h4>
    <p>Given the paired arrays x and y, test if they have significantly
        different means. Small values of p-value indicate that the two arrays
        have significantly different means.</p>
    <ul class="nav nav-tabs">
        <li class="active"><a href="#scala_10" data-toggle="tab">Scala</a></li>
        <li><a href="#java_10" data-toggle="tab">Java</a></li>
    </ul>
    <div class="tab-content">
        <div class="tab-pane active" id="scala_10">
            <div class="code" style="text-align: left;">
          <pre class="prettyprint lang-scala"><code>
    smile> ttest(y, z)
    res21: stat.hypothesis.TTest = Paired t-test(t = -0.1502, df = 17.000, p-value = 0.882382)
          </code></pre>
            </div>
        </div>
        <div class="tab-pane" id="java_10">
            <div class="code" style="text-align: left;">
          <pre class="prettyprint lang-java"><code>
    jshell> TTest.testPaired(y, z)
    $53 ==> Paired t-test(t = -0.1502, df = 17.000, p-value = 0.882382)
          </code></pre>
            </div>
        </div>
    </div>

    <h4 class="title">Independent (Unpaired) Two-Sample Test</h4>
    <p>Test if the arrays x and y have significantly different means. Small
     values of p-value indicate that the two arrays have significantly
      different means. If the parameter equalVariance is true, the data arrays are assumed to be
      drawn from populations with the same true variance. Otherwise, The data
      arrays are allowed to be drawn from populations with unequal variances.</p>
    <ul class="nav nav-tabs">
        <li class="active"><a href="#scala_11" data-toggle="tab">Scala</a></li>
        <li><a href="#java_11" data-toggle="tab">Java</a></li>
    </ul>
    <div class="tab-content">
        <div class="tab-pane active" id="scala_11">
            <div class="code" style="text-align: left;">
          <pre class="prettyprint lang-scala"><code>
    smile> ttest2(x, y)
    res22: stat.hypothesis.TTest = Unequal Variance Two Sample t-test(t = -1.1219, df = 35.167, p-value = 0.269491)

    smile> ttest2(x, y, true)
    res23: stat.hypothesis.TTest = Equal Variance Two Sample t-test(t = -1.1247, df = 36.000, p-value = 0.268153)

    smile> ttest2(x, z)
    res24: stat.hypothesis.TTest = Unequal Variance Two Sample t-test(t = -1.5180, df = 34.025, p-value = 0.138243)

    smile> ttest2(x, z, true)
    res25: stat.hypothesis.TTest = Equal Variance Two Sample t-test(t = -1.4901, df = 36.000, p-value = 0.144906)
          </code></pre>
            </div>
        </div>
        <div class="tab-pane" id="java_11">
            <div class="code" style="text-align: left;">
          <pre class="prettyprint lang-java"><code>
    jshell> TTest.test(x, y, false)
    $54 ==> Unequal Variance Two Sample t-test(t = -1.1219, df = 35.167, p-value = 0.269491)

    jshell> TTest.test(x, y, true)
    $55 ==> Equal Variance Two Sample t-test(t = -1.1247, df = 36.000, p-value = 0.268153)

    jshell> TTest.test(x, z, false)
    $56 ==> Unequal Variance Two Sample t-test(t = -1.5180, df = 34.025, p-value = 0.138243)

    jshell> TTest.test(x, z, true)
    $57 ==> Equal Variance Two Sample t-test(t = -1.4901, df = 36.000, p-value = 0.144906)
          </code></pre>
            </div>
        </div>
    </div>

    <h3 id="ks-test" class="title">Kolmogorov–Smirnov Test</h3>

    <h4 class="title">One-Sample Test</h4>
    <p>The one-sample K-S test for the null hypothesis that the data set x
        is drawn from the given distribution. Small values of p-value show that
        the cumulative distribution function of x is significantly different from
        the given distribution. The array x is modified by being sorted into
        ascending order.</p>
    <ul class="nav nav-tabs">
        <li class="active"><a href="#scala_12" data-toggle="tab">Scala</a></li>
        <li><a href="#java_12" data-toggle="tab">Java</a></li>
    </ul>
    <div class="tab-content">
        <div class="tab-pane active" id="scala_12">
            <div class="code" style="text-align: left;">
          <pre class="prettyprint lang-scala"><code>
    smile> val x = Array(
                   0.53236606, -1.36750258, -1.47239199, -0.12517888, -1.24040594, 1.90357309,
                   -0.54429527, 2.22084140, -1.17209146, -0.68824211, -1.75068914, 0.48505896,
                   2.75342248, -0.90675303, -1.05971929, 0.49922388, -1.23214498, 0.79284888,
                   0.85309580, 0.17903487, 0.39894754, -0.52744720, 0.08516943, -1.93817962,
                   0.25042913, -0.56311389, -1.08608388, 0.11912253, 2.87961007, -0.72674865,
                   1.11510699, 0.39970074, 0.50060532, -0.82531807, 0.14715616, -0.96133601,
                   -0.95699473, -0.71471097, -0.50443258, 0.31690224, 0.04325009, 0.85316056,
                   0.83602606, 1.46678847, 0.46891827, 0.69968175, 0.97864326, 0.66985742,
                   -0.20922486, -0.15265994)
    x: Array[Double] = Array(
      0.53236606,
      -1.36750258,
      -1.47239199,
      -0.12517888,
      -1.24040594,
      1.90357309,
      -0.54429527,
      2.2208414,
      -1.17209146,
      -0.68824211,
      -1.75068914,
      0.48505896,
      2.75342248,
      -0.90675303,
      -1.05971929,
      0.49922388,
      -1.23214498,
      0.79284888,
      0.8530958,
      0.17903487,
      0.39894754,
      -0.5274472,
      0.08516943,
      -1.93817962,
    ...

    smile> kstest(x, new GaussianDistribution(0, 1))
    res27: stat.hypothesis.KSTest = Gaussian Distribution(0.0000, 1.0000) Kolmogorov-Smirnov Test(d = 0.0930, p-value = 0.759824)
          </code></pre>
            </div>
        </div>
        <div class="tab-pane" id="java_12">
            <div class="code" style="text-align: left;">
          <pre class="prettyprint lang-java"><code>
    jshell> double[] x = {0.53236606, -1.36750258, -1.47239199, -0.12517888, -1.24040594, 1.90357309,
       ...>             -0.54429527, 2.22084140, -1.17209146, -0.68824211, -1.75068914, 0.48505896,
       ...>             2.75342248, -0.90675303, -1.05971929, 0.49922388, -1.23214498, 0.79284888,
       ...>             0.85309580, 0.17903487, 0.39894754, -0.52744720, 0.08516943, -1.93817962,
       ...>             0.25042913, -0.56311389, -1.08608388, 0.11912253, 2.87961007, -0.72674865,
       ...>             1.11510699, 0.39970074, 0.50060532, -0.82531807, 0.14715616, -0.96133601,
       ...>             -0.95699473, -0.71471097, -0.50443258, 0.31690224, 0.04325009, 0.85316056,
       ...>             0.83602606, 1.46678847, 0.46891827, 0.69968175, 0.97864326, 0.66985742,
       ...>             -0.20922486, -0.15265994}
    x ==> double[50] { 0.53236606, -1.36750258, -1.47239199 ... -0.20922486, -0.15265994 }

    jshell> KSTest.test(x, new GaussianDistribution(0, 1))
    $59 ==> Gaussian Distribution(0.0000, 1.0000) Kolmogorov-Smirnov Test(d = 0.0930, p-value = 0.759824)
          </code></pre>
            </div>
        </div>
    </div>

    <h4 class="title">Two-Sample Test</h4>
    <p>The two-sample K–S for the null hypothesis that the data sets
        are drawn from the same distribution. Small values of p-value show that
        the cumulative distribution function of x is significantly different from
        that of y. The arrays x and y are modified by being sorted into
        ascending order.</p>

    <ul class="nav nav-tabs">
        <li class="active"><a href="#scala_13" data-toggle="tab">Scala</a></li>
        <li><a href="#java_13" data-toggle="tab">Java</a></li>
    </ul>
    <div class="tab-content">
        <div class="tab-pane active" id="scala_13">
            <div class="code" style="text-align: left;">
          <pre class="prettyprint lang-scala"><code>
    smile> val y = Array(
                   0.95791391, 0.16203847, 0.56622013, 0.39252941, 0.99126354, 0.65639108,
                   0.07903248, 0.84124582, 0.76718719, 0.80756577, 0.12263981, 0.84733360,
                   0.85190907, 0.77896244, 0.84915723, 0.78225903, 0.95788055, 0.01849366,
                   0.21000365, 0.97951772, 0.60078520, 0.80534223, 0.77144013, 0.28495121,
                   0.41300867, 0.51547517, 0.78775718, 0.07564151, 0.82871088, 0.83988694)
    y: Array[Double] = Array(
      0.95791391,
      0.16203847,
      0.56622013,
      0.39252941,
      0.99126354,
      0.65639108,
      0.07903248,
      0.84124582,
      0.76718719,
      0.80756577,
      0.12263981,
      0.8473336,
      0.85190907,
      0.77896244,
      0.84915723,
      0.78225903,
      0.95788055,
      0.01849366,
      0.21000365,
      0.97951772,
      0.6007852,
      0.80534223,
      0.77144013,
      0.28495121,
    ...

    smile> kstest(x, y)
    res29: stat.hypothesis.KSTest = Two Sample Kolmogorov-Smirnov Test(d = 0.4600, p-value = 0.000416466)
          </code></pre>
            </div>
        </div>
        <div class="tab-pane" id="java_13">
            <div class="code" style="text-align: left;">
          <pre class="prettyprint lang-java"><code>
    jshell> KSTest.test(x, new GaussianDistribution(0, 1))
    $59 ==> Gaussian Distribution(0.0000, 1.0000) Kolmogorov-Smirnov Test(d = 0.0930, p-value = 0.759824)

    jshell> double[] y = {0.95791391, 0.16203847, 0.56622013, 0.39252941, 0.99126354, 0.65639108,
       ...>               0.07903248, 0.84124582, 0.76718719, 0.80756577, 0.12263981, 0.84733360,
       ...>               0.85190907, 0.77896244, 0.84915723, 0.78225903, 0.95788055, 0.01849366,
       ...>               0.21000365, 0.97951772, 0.60078520, 0.80534223, 0.77144013, 0.28495121,
       ...>               0.41300867, 0.51547517, 0.78775718, 0.07564151, 0.82871088, 0.83988694}
    y ==> double[30] { 0.95791391, 0.16203847, 0.56622013,  ... , 0.82871088, 0.83988694 }

    jshell> KSTest.test(x, y)
    $61 ==> Two Sample Kolmogorov-Smirnov Test(d = 0.4600, p-value = 0.000416466)
          </code></pre>
            </div>
        </div>
    </div>

    <h3 id="correlation-test" class="title">Correlation Test</h3>

    <h4 class="title">Pearson Correlation</h4>

    <p>The t-test is used to establish if the correlation coefficient is
        significantly different from zero, and, hence that there is evidence
        of an association between the two variables. There is then the
        underlying assumption that the data is from a normal distribution
        sampled randomly. If this is not true, then it is better to use
        Spearman's coefficient of rank correlation (for non-parametric variables).</p>
    <ul class="nav nav-tabs">
        <li class="active"><a href="#scala_14" data-toggle="tab">Scala</a></li>
        <li><a href="#java_14" data-toggle="tab">Java</a></li>
    </ul>
    <div class="tab-content">
        <div class="tab-pane active" id="scala_14">
            <div class="code" style="text-align: left;">
          <pre class="prettyprint lang-scala"><code>
    smile> val x = Array(44.4, 45.9, 41.9, 53.3, 44.7, 44.1, 50.7, 45.2, 60.1)
    x: Array[Double] = Array(44.4, 45.9, 41.9, 53.3, 44.7, 44.1, 50.7, 45.2, 60.1)

    smile> val y  = Array(2.6,  3.1,  2.5,  5.0,  3.6,  4.0,  5.2,  2.8,  3.8)
    y: Array[Double] = Array(2.6, 3.1, 2.5, 5.0, 3.6, 4.0, 5.2, 2.8, 3.8)

    smile> pearsontest(x, y)
    res32: stat.hypothesis.CorTest = Pearson Correlation Test(cor = 0.57, t = 1.8411, df = 7.000, p-value = 0.108173)
          </code></pre>
            </div>
        </div>
        <div class="tab-pane" id="java_14">
            <div class="code" style="text-align: left;">
          <pre class="prettyprint lang-java"><code>
    jshell> double[] x = {44.4, 45.9, 41.9, 53.3, 44.7, 44.1, 50.7, 45.2, 60.1}
    x ==> double[9] { 44.4, 45.9, 41.9, 53.3, 44.7, 44.1, 50.7, 45.2, 60.1 }

    jshell> double[] y = {2.6,  3.1,  2.5,  5.0,  3.6,  4.0,  5.2,  2.8,  3.8}
    y ==> double[9] { 2.6, 3.1, 2.5, 5.0, 3.6, 4.0, 5.2, 2.8, 3.8 }

    jshell> CorTest.pearson(x, y)
    $64 ==> Pearson Correlation Test(cor = 0.57, t = 1.8411, df = 7.000, p-value = 0.108173)
          </code></pre>
            </div>
        </div>
    </div>

    <h4 id="spearman-test" class="title">Spearman Rank Correlation</h4>

    <p>The Spearman Rank Correlation
        Coefficient is a form of the Pearson coefficient with the data converted
        to rankings (i.e. when variables are ordinal). It can be used when there
        is non-parametric data and hence Pearson cannot be used.</p>

    <p>The raw scores are converted to ranks and the differences between
        the ranks of each observation on the two variables are calculated.</p>

    <p>The p-value is calculated by approximation, which is good for n &gt; 10.</p>

    <ul class="nav nav-tabs">
        <li class="active"><a href="#scala_15" data-toggle="tab">Scala</a></li>
        <li><a href="#java_15" data-toggle="tab">Java</a></li>
    </ul>
    <div class="tab-content">
        <div class="tab-pane active" id="scala_15">
            <div class="code" style="text-align: left;">
          <pre class="prettyprint lang-scala"><code>
    smile> spearmantest(x, y)
    res33: stat.hypothesis.CorTest = Spearman Correlation Test(cor = 0.60, t = 1.9843, df = 7.000, p-value = 0.0876228)
          </code></pre>
            </div>
        </div>
        <div class="tab-pane" id="java_15">
            <div class="code" style="text-align: left;">
          <pre class="prettyprint lang-java"><code>
    jshell> CorTest.spearman(x, y)
    $65 ==> Spearman Correlation Test(cor = 0.60, t = 1.9843, df = 7.000, p-value = 0.0876228)
          </code></pre>
            </div>
        </div>
    </div>

    <h4 id="kendall-test" class="title">Kendall Rank Correlation</h4>

    <p>The Kendall Tau Rank Correlation
        Coefficient is used to measure the degree of correspondence
        between sets of rankings where the measures are not equidistant.
        It is used with non-parametric data. The p-value is calculated by
        approximation, which is good for n &gt; 10.</p>

    <ul class="nav nav-tabs">
        <li class="active"><a href="#scala_16" data-toggle="tab">Scala</a></li>
        <li><a href="#java_16" data-toggle="tab">Java</a></li>
    </ul>
    <div class="tab-content">
        <div class="tab-pane active" id="scala_16">
            <div class="code" style="text-align: left;">
          <pre class="prettyprint lang-scala"><code>
    smile> kendalltest(x, y)
    res34: stat.hypothesis.CorTest = Kendall Correlation Test(cor = 0.44, t = 1.6681, df = 0.000, p-value = 0.0952928)
          </code></pre>
            </div>
        </div>
        <div class="tab-pane" id="java_16">
            <div class="code" style="text-align: left;">
          <pre class="prettyprint lang-java"><code>
    jshell> CorTest.kendall(x, y)
    $66 ==> Kendall Correlation Test(cor = 0.44, t = 1.6681, df = 0.000, p-value = 0.0952928)
          </code></pre>
            </div>
        </div>
    </div>

    <div id="btnv">
        <span class="btn-arrow-left">&larr; &nbsp;</span>
        <a class="btn-prev-text" href="linear-algebra.html" title="Previous Section: Linear Algebra"><span>Linear Algebra</span></a>
        <a class="btn-next-text" href="wavelet.html" title="Next Section: Wavelet"><span>Wavelet</span></a>
        <span class="btn-arrow-right">&nbsp;&rarr;</span>
    </div>
</div>

<script type="text/javascript">
    $('#toc').toc({exclude: 'h1, h5, h6', context: '', autoId: true, numerate: false});
</script>