interpolation.html

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    <h1 id="interpolation-top" class="title">Interpolation</h1>

    <p>Interpolation is the process of constructing a function that takes on
        specified values at specified points. In engineering and science, one often
        has a number of data points, obtained by sampling or experimentation, which
        represent the values of a function for a limited number of values of the
        independent variable. It is often required to interpolate (i.e. estimate)
        the value of that function for an intermediate value of the independent
        variable.</p>

    <p>Smile package <code>smile.interpolation</code> provides a variety of
        algorithms on 1d and 2d data. These algorithms implement the interface
        <code>Interpolation</code> (or <code>Interpolation2D</code> for 2d data),
        of which the method <code>interpolate</code> takes a value and return
        an interploated value.</p>

    <h2 id="linear" class="title">Piecewise Linear Interpolation</h2>

    <p><code>LinearInterpolation</code> is quick and easy, but it is not very precise. Another disadvantage
        is that the interpolant is not differentiable at the control points.</p>

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        <li class="active"><a href="#java_1" data-toggle="tab">Java</a></li>
        <li><a href="#scala_1" data-toggle="tab">Scala</a></li>
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        <div class="tab-pane" id="scala_1">
            <div class="code" style="text-align: left;">
    <pre class="prettyprint lang-scala"><code>
    val x = Array(0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0)
    val y = Array(0.0, 0.8415, 0.9093, 0.1411, -0.7568, -0.9589, -0.2794)
    val points = (0 until x.length).map(i => Array(x(i), y(i))).toArray

    val linear = new LinearInterpolation(x, y)

    val data = (0 to 60).map { i =>
      val x = i * 0.1
      val y = linear.interpolate(x)
      Array(x, y)
    }.toArray

    val canvas = plot(points, '@', Color.BLACK)
    canvas.add(LinePlot.of(data, Color.RED))
    show(canvas)
    </code></pre>
            </div>
        </div>
        <div class="tab-pane active" id="java_1">
            <div class="code" style="text-align: left;">
          <pre class="prettyprint lang-java"><code>
    double[] x = {0.0, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0};
    double[] y = {0.0, 0.8415, 0.9093, 0.1411, -0.7568, -0.9589, -0.2794};
    double[][] points = new double[x.length][2];
    for (int i = 0; i < x.length; i++) {
      points[i][0] = x[i];
      points[i][1] = y[i];
    }

    var linear = new LinearInterpolation(x, y);
    double[][] data = new double[61][2];
    for (int i = 0; i < data.length; i++) {
      data[i][0] = i * 0.1;
      data[i][1] = linear.interpolate(data[i][0]);
    }

    var canvas = ScatterPlot.of(points, '@', Color.BLACK).canvas();
    canvas.add(LinePlot.of(data, Color.RED));
    canvas.window();
          </code></pre>
            </div>
        </div>
    </div>

    <div style="width: 100%; display: inline-block; text-align: center;">
        <img src="images/linear-interpolation.png" class="enlarge" style="width: 480px;" />
    </div>

    <p><code>BilinearInterpolation</code> is an extension of linear interpolation for
        interpolating functions of two variables on a
        regular grid. The key idea is to perform linear interpolation first in one
        direction, and then again in the other direction.</p>

    <h2 id="cubic" class="title">Cubic Spline Interpolation</h2>

    <p>Spline interpolation uses low-degree polynomials
        in each of the intervals, and chooses the polynomial pieces such that they
        fit smoothly together. The resulting function is called a spline.
        The natural cubic spline is piecewise cubic and twice continuously
        differentiable. Furthermore, its second derivative is zero at the end
        points.</p>

    <p>Like polynomial interpolation, spline interpolation incurs a smaller
        error than linear interpolation and the interpolant is smoother. However,
        the interpolant is easier to evaluate than the high-degree polynomials
        used in polynomial interpolation. It also does not suffer from Runge's
        phenomenon.</p>

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        <li><a href="#scala_2" data-toggle="tab">Scala</a></li>
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        <div class="tab-pane" id="scala_2">
            <div class="code" style="text-align: left;">
          <pre class="prettyprint lang-scala"><code>
    val cubic = new CubicSplineInterpolation1D(x, y)

    val data = (0 to 60).map { i =>
      val x = i * 0.1
      val y = cubic.interpolate(x)
      Array(x, y)
    }.toArray

    val canvas = plot(points, '@', Color.BLACK)
    canvas.add(LinePlot.of(data, Color.BLUE))
    show(canvas)
          </code></pre>
            </div>
        </div>
        <div class="tab-pane active" id="java_2">
            <div class="code" style="text-align: left;">
          <pre class="prettyprint lang-java"><code>
    var cubic = new CubicSplineInterpolation1D(x, y);

    double[][] data = new double[61][2];
    for (int i = 0; i < data.length; i++) {
      data[i][0] = i * 0.1;
      data[i][1] = cubic.interpolate(data[i][0]);
    }

    var canvas = ScatterPlot.of(points, '@', Color.BLACK).canvas();
    canvas.add(LinePlot.of(data, Color.BLUE));
    canvas.window();
          </code></pre>
            </div>
        </div>
    </div>

    <div style="width: 100%; display: inline-block; text-align: center;">
        <img src="images/cubic-interpolation.png" class="enlarge" style="width: 480px;" />
    </div>

    <p>For 2d grid, we provide two implementations: <code>CubicSplineInterpolation2D</code> and
        <code>BicubicInterpolation</code>. <code>CubicSplineInterpolation2D</code> is
        similar to one-dimensional splines as it guarantees the
        continuity of the first and second function derivatives. In contrast,
        <code>BicubicInterpolation</code> guarantees continuity of only
        gradient and cross-derivative. Second derivatives
        could be discontinuous.</p>

    <p>In image processing, bicubic interpolation is often chosen over bilinear
        interpolation or nearest neighbor in image resampling, when speed is not
        an issue. Images resampled with bicubic interpolation are smoother and
        have fewer interpolation artifacts.</p>

    <div style="width: 100%; display: inline-block; text-align: center;">
        <img src="images/grid-interpolation2d.png" class="enlarge" style="width: 480px;" />
    </div>

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        <li class="active"><a href="#java_3" data-toggle="tab">Java</a></li>
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        <div class="tab-pane" id="scala_3">
            <div class="code" style="text-align: left;">
          <pre class="prettyprint lang-scala"><code>
    val x1 = Array(1950.0, 1960, 1970, 1980, 1990)
    val x2 = Array(10.0, 20, 30)
    val y = Array(
            Array(150.697, 199.592, 187.625),
            Array(179.323, 195.072, 250.287),
            Array(203.212, 179.092, 322.767),
            Array(226.505, 153.706, 426.730),
            Array(249.633, 120.281, 598.243)
        )

    val canvas = heatmap(y, Palette.jet(256))
    canvas.setTitle("Original")
    show(canvas)

    val cubic = new CubicSplineInterpolation2D(x1, x2, y)

    val data = Array.ofDim[Double](101, 101)
    for (i <- 0 to 100; j <- 0 to 100)
      data(i)(j) = cubic.interpolate(1950 + i*0.4, 10 + j*0.2)

    val cubicPlot = heatmap(data, Palette.jet(256))
    cubicPlot.setTitle("Cubic")
    show(cubicPlot)

    val bicubic = new BicubicInterpolation(x1, x2, y)

    for (i <- 0 to 100; j <- 0 to 100)
      data(i)(j) = bicubic.interpolate(1950 + i*0.4, 10 + j*0.2)

    val bicubicPlot = heatmap(data, Palette.jet(256))
    bicubicPlot.setTitle("Bicubic")
    show(bicubicPlot)
          </code></pre>
            </div>
        </div>
        <div class="tab-pane active" id="java_3">
            <div class="code" style="text-align: left;">
          <pre class="prettyprint lang-java"><code>
    double[] x1 = {1950.0, 1960, 1970, 1980, 1990};
    double[] x2 = {10.0, 20, 30};
    double[][] y = {
            {150.697, 199.592, 187.625},
            {179.323, 195.072, 250.287},
            {203.212, 179.092, 322.767},
            {226.505, 153.706, 426.730},
            {249.633, 120.281, 598.243}
        };

    var canvas = Heatmap.of(y, Palette.jet(256)).canvas();
    canvas.setTitle("Original");
    canvas.window();

    var cubic = new CubicSplineInterpolation2D(x1, x2, y);

    double[][] data = new double[101][101];
    for (int i = 0; i <= 100; i++) {
        for (int j = 0; j <= 100; j++) {
            data[i][j] = cubic.interpolate(1950 + i*0.4, 10 + j*0.2);
        }
    }

    var cubicPlot = Heatmap.of(data, Palette.jet(256)).canvas();
    cubicPlot.setTitle("Cubic");
    cubicPlot.window();

    var bicubic = new BicubicInterpolation(x1, x2, y);

    for (int i = 0; i <= 100; i++) {
        for (int j = 0; j <= 100; j++) {
            data[i][j] = cubic.interpolate(1950 + i*0.4, 10 + j*0.2);
        }
    }

    var bicubicPlot = Heatmap.of(data, Palette.jet(256)).canvas();
    bicubicPlot.setTitle("Bicubic");
    bicubicPlot.window();
          </code></pre>
            </div>
        </div>
    </div>

    <h2 id="kriging" class="title">Kriging Interpolation</h2>

    <p>Kriging interpolation can be used for the data points irregularly distributed in space.
        Kriging belongs to the family of linear least squares estimation algorithms,
        also known as Gauss-Markov estimation or Gaussian process regression.
        We implement ordinary kriging for interpolation with power variogram.</p>

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    </ul>
    <div class="tab-content">
        <div class="tab-pane" id="scala_4">
            <div class="code" style="text-align: left;">
          <pre class="prettyprint lang-scala"><code>
    val kriging = new KrigingInterpolation1D(x, y)

    val data = Array.ofDim[Double](61, 2)
    for (i <- 0 to 60) {
      data(i)(0) = i * 0.1
      data(i)(1) = kriging.interpolate(data(i)(0))
    }

    val canvas = plot(points, '@', Color.BLACK)
    canvas.add(LinePlot.of(data, Color.CYAN))
    show(canvas)
          </code></pre>
            </div>
        </div>
        <div class="tab-pane active" id="java_4">
            <div class="code" style="text-align: left;">
          <pre class="prettyprint lang-java"><code>
    var kriging = new KrigingInterpolation1D(x, y);

    double[][] data = new double[61][2];
    for (int i = 0; i < data.length; i++) {
      data[i][0] = i * 0.1;
      data[i][1] = kriging.interpolate(data[i][0]);
    }

    var canvas = ScatterPlot.of(points, '@', Color.BLACK).canvas();
    canvas.add(LinePlot.of(data, Color.CYAN));
    canvas.window();
          </code></pre>
            </div>
        </div>
    </div>

    <div style="width: 100%; display: inline-block; text-align: center;">
        <img src="images/kriging-interpolation.png" class="enlarge" style="width: 480px;" />
    </div>

    <h2 id="rbf" class="title">RBF Interpolation</h2>

    <p>Radial basis function interpolation is a popular method for the data points are
        irregularly distributed in space. In its basic form, radial basis function
        interpolation is in the form
        <code>y(x) = &Sigma; w<sub>i</sub> &phi;(||x-c<sub>i</sub>||)</code>,
        where the approximating function y(x) is represented as a sum of N radial
        basis functions &phi;, each associated with a different center c<sub>i</sub>, and weighted
        by an appropriate coefficient w<sub>i</sub>. For distance, one usually chooses
        Euclidean distance. The weights w<sub>i</sub> can
        be estimated using the matrix methods of linear least squares, because
        the approximating function is linear in the weights.</p>

    <p>The points c<sub>i</sub> often called the centers or collocation points
        of the RBF interpolant. Note also that the centers c<sub>i</sub> can be
        located at arbitrary points in the domain, and do not require a grid.
        For certain RBF exponential convergence has been shown. Radial basis
        functions were successfully applied to problems as diverse as computer
        graphics, neural networks, for the solution of differential equations
        via collocation methods and many other problems.</p>

    <p>Other popular choices for &phi; comprise the Gaussian function and the
        so-called thin plate splines. Thin plate splines result from the solution of
        a variational problem. The advantage of the thin plate splines is that
        their conditioning is invariant under scalings. Gaussians, multi-quadrics
        and inverse multi-quadrics are infinitely smooth and involve a scale
        or shape parameter, r<sub><small>0</small></sub> &gt; 0.
        Decreasing r<sub><small>0</small></sub> tends to
        flatten the basis function. For a given function, the quality of
        approximation may strongly depend on this parameter. In particular,
        increasing r<sub><small>0</small></sub> has the effect of better conditioning
        (the separation distance  of the scaled points increases).</p>

    <p>A variant on RBF interpolation is normalized radial basis function (NRBF)
        interpolation, in which we require the sum of the basis functions to be unity.
        NRBF arises more naturally from a Bayesian statistical perspective. However,
        there is no evidence that either the NRBF method is consistently superior
        to the RBF method, or vice versa.</p>

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        <div class="tab-pane" id="scala_5">
            <div class="code" style="text-align: left;">
          <pre class="prettyprint lang-scala"><code>
    val rbf = new RBFInterpolation1D(x, y, new smile.math.rbf.GaussianRadialBasis())

    val data = Array.ofDim[Double](61, 2)
    for (i <- 0 to 60) {
      data(i)(0) = i * 0.1
      data(i)(1) = rbf.interpolate(data(i)(0))
    }

    val canvas = plot(points, '@', Color.BLACK)
    canvas.add(LinePlot.of(data, Color.GREEN))
    show(canvas)
          </code></pre>
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        <div class="tab-pane active" id="java_5">
            <div class="code" style="text-align: left;">
          <pre class="prettyprint lang-java"><code>
    var rbf = new RBFInterpolation1D(x, y, new smile.math.rbf.GaussianRadialBasis());

    double[][] data = new double[61][2];
    for (int i = 0; i < data.length; i++) {
      data[i][0] = i * 0.1;
      data[i][1] = rbf.interpolate(data[i][0]);
    }

    var canvas = ScatterPlot.of(points, '@', Color.BLACK).canvas();
    canvas.add(LinePlot.of(data, Color.GREEN));
    canvas.window();
          </code></pre>
            </div>
        </div>
    </div>

    <div style="width: 100%; display: inline-block; text-align: center;">
        <img src="images/rbf-interpolation.png" class="enlarge" style="width: 480px;" />
    </div>

    <h2 id="shepard" class="title">Shepard Interpolation</h2>
    <p>Shepard's interpolation is a special case of normalized radial basis function
        interpolation if the function &phi;(r) goes to infinity as r &rarr; 0, and is
        finite for r &gt; 0. In this case, the weights w<sub>i</sub> are just equal to
        the respective function values y<sub>i</sub>. So we need not solve linear
        equations, and thus it works for very large N. An example of such &phi;
        is &phi;(r) = r<sup>-p</sup> with (typically) 1 &lt; p &le; 3.</p>

    <ul class="nav nav-tabs">
        <li class="active"><a href="#java_6" data-toggle="tab">Java</a></li>
        <li><a href="#scala_6" data-toggle="tab">Scala</a></li>
    </ul>
    <div class="tab-content">
        <div class="tab-pane" id="scala_6">
            <div class="code" style="text-align: left;">
          <pre class="prettyprint lang-scala"><code>
    val shepard = new ShepardInterpolation1D(x, y, 3)

    val data = Array.ofDim[Double](61, 2)
    for (i <- 0 to 60) {
      data(i)(0) = i * 0.1
      data(i)(1) = shepard.interpolate(data(i)(0))
    }

    val canvas = plot(points, '@', Color.BLACK)
    canvas.add(LinePlot.of(data, Color.PINK))
    show(canvas)
          </code></pre>
            </div>
        </div>
        <div class="tab-pane active" id="java_6">
            <div class="code" style="text-align: left;">
          <pre class="prettyprint lang-java"><code>
    var shepard = new ShepardInterpolation1D(x, y, 3);

    double[][] data = new double[61][2];
    for (int i = 0; i < data.length; i++) {
      data[i][0] = i * 0.1;
      data[i][1] = shepard.interpolate(data[i][0]);
    }

    var canvas = ScatterPlot.of(points, '@', Color.BLACK).canvas();
    canvas.add(LinePlot.of(data, Color.PINK));
    canvas.window();
          </code></pre>
            </div>
        </div>
    </div>

    <div style="width: 100%; display: inline-block; text-align: center;">
        <img src="images/shepard-interpolation.png" class="enlarge" style="width: 480px;" />
    </div>

    <p>Shepard's interpolation is rarely as accurate as the well-tuned application of
        other radial basis functions. However, it is simple, fast, and often sufficient
        for quick and dirty applications.</p>

    <div style="width: 100%; display: inline-block; text-align: center;">
        <img src="images/irregular-interpolation2d.png" class="enlarge" style="width: 480px;" />
    </div>

    <h2 id="laplace" class="title">Laplace Interpolation</h2>

    <p>Laplace's interpolation can restore missing or unmeasured values on a 2-dimensional
        evenly spaced regular grid. In some sense, Laplace interpolation
        produces the smoothest possible interpolant, which are obtained by solving
        a very sparse linear equations with biconjugate gradient method.</p>

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        <li class="active"><a href="#java_7" data-toggle="tab">Java</a></li>
        <li><a href="#scala_7" data-toggle="tab">Scala</a></li>
    </ul>
    <div class="tab-content">
        <div class="tab-pane" id="scala_7">
            <div class="code" style="text-align: left;">
          <pre class="prettyprint lang-scala"><code>
    val zz = Array.ofDim[Double](101, 101)
    val ww = Array.ofDim[Double](101, 101)
    for (i <- 0 to 100; j <- 0 to 100) {
      zz(i)(j) = if (java.lang.Math.random() < 0.2) Double.NaN else data(i)(j)
      ww(i)(j) = zz(i)(j)
    }

    val canvas = heatmap(ww, Palette.jet(256))
    canvas.setTitle("Missing Values")
    show(canvas)

    LaplaceInterpolation.interpolate(zz)
    val canvas = heatmap(zz, Palette.jet(256))
    canvas.setTitle("Laplace")
    show(canvas)
          </code></pre>
            </div>
        </div>
        <div class="tab-pane active" id="java_7">
            <div class="code" style="text-align: left;">
          <pre class="prettyprint lang-java"><code>
    double[][] zz = new double[101][101];
    double[][] ww = new double[101][101];
    for (int i = 0; i <= 100; i++) {
        for (int j = 0; j <= 100; j++) {
            zz[i][j] = Math.random() < 0.2 ? Double.NaN : data[i][j];
            ww[i][j] = zz[i][j];
        }
    }

    var canvas = Heatmap.of(ww, Palette.jet(256)).canvas();
    canvas.setTitle("Missing Values");
    canvas.window();

    LaplaceInterpolation.interpolate(zz);
    var canvas = Heatmap.of(zz, Palette.jet(256)).canvas();
    canvas.setTitle("Laplace");
    canvas.window();
          </code></pre>
            </div>
        </div>
    </div>

    <div style="width: 100%; display: inline-block; text-align: center;">
        <img src="images/laplace-interpolation.png" class="enlarge" style="width: 480px;" />
    </div>

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