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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>
<ul class="nav nav-tabs">
<li class="active"><a href="#java_1" data-toggle="tab">Java</a></li>
<li><a href="#scala_1" data-toggle="tab">Scala</a></li>
</ul>
<div class="tab-content">
<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>
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</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 class="active"><a href="#java_2" data-toggle="tab">Java</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>
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</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>
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</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>
<li><a href="#scala_3" data-toggle="tab">Scala</a></li>
</ul>
<div class="tab-content">
<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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<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) = Σ w<sub>i</sub> φ(||x-c<sub>i</sub>||)</code>,
where the approximating function y(x) is represented as a sum of N radial
basis functions φ, 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 φ 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> > 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="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>
<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 φ(r) goes to infinity as r → 0, and is
finite for r > 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 φ
is φ(r) = r<sup>-p</sup> with (typically) 1 < p ≤ 3.</p>
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<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>
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</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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<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>
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</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>
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</div>
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<img src="images/laplace-interpolation.png" class="enlarge" style="width: 480px;" />
</div>
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