Merge pull request #1110 from diluculo/makima

Add Makima (Modified Akima) Interpolation to CubicSpline

Author: cdrnet

src/Numerics.Tests/InterpolationTests/MakimaSplineTest.cs

@@ -0,0 +1,109 @@
+// <copyright file="MakimaSplineTest.cs" company="Math.NET">
+// Math.NET Numerics, part of the Math.NET Project
+// https://numerics.mathdotnet.com
+// https://github.com/mathnet/mathnet-numerics
+//
+// Copyright (c) 2009-$CURRENT_YEAR$ Math.NET
+//
+// Permission is hereby granted, free of charge, to any person
+// obtaining a copy of this software and associated documentation
+// files (the "Software"), to deal in the Software without
+// restriction, including without limitation the rights to use,
+// copy, modify, merge, publish, distribute, sublicense, and/or sell
+// copies of the Software, and to permit persons to whom the
+// Software is furnished to do so, subject to the following
+// conditions:
+//
+// The above copyright notice and this permission notice shall be
+// included in all copies or substantial portions of the Software.
+//
+// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
+// EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES
+// OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
+// NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT
+// HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY,
+// WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
+// FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR
+// OTHER DEALINGS IN THE SOFTWARE.
+// </copyright>
+
+using MathNet.Numerics.Interpolation;
+using NUnit.Framework;
+
+namespace MathNet.Numerics.Tests.InterpolationTests
+{
+    [TestFixture, Category("Interpolation")]
+    public class MakimaSplineTest
+    {
+        // Test sample data
+        readonly double[] _t = { -2.0, -1.0, 0.0, 1.0, 2.0 };
+        readonly double[] _y = { 1.0, 2.0, -1.0, 0.0, 1.0 };
+
+        /// <summary>
+        /// Verifies that the Makima interpolation exactly fits the sample points.
+        /// </summary>
+        [Test]
+        public void FitsAtSamplePoints()
+        {
+            IInterpolation it = CubicSpline.InterpolateMakima(_t, _y);
+            for (var i = 0; i < _y.Length; i++)
+            {
+                Assert.AreEqual(_y[i], it.Interpolate(_t[i]), 1e-15, "Exact point " + i);
+            }
+        }
+
+        /// <summary>
+        /// Verifies that at arbitrary points, the interpolation matches the reference values.
+        /// (Note: Replace the expected values with verified reference values.)
+        /// </summary>
+        /// <param name="t">The x-value to interpolate.</param>
+        /// <param name="expected">The expected interpolated value.</param>
+        /// <param name="maxAbsoluteError">Maximum allowed absolute error.</param>
+        [TestCase(-2.4, 0.14266666666666683, 1e-10)]
+        [TestCase(-0.9, 1.805, 1e-10)]
+        [TestCase(-0.5, 0.29166666666666674, 1e-10)]
+        [TestCase(-0.1, -0.955, 1e-10)]
+        [TestCase(0.1, -0.954, 1e-10)]
+        [TestCase(0.4, -0.696, 1e-10)]
+        [TestCase(1.2, 0.2, 1e-10)]
+        [TestCase(10.0, 9.0, 1e-10)]
+        [TestCase(-10.0, 527.0, 1e-10)]
+        public void FitsAtArbitraryPoints(double t, double expected, double maxAbsoluteError)
+        {
+            IInterpolation it = CubicSpline.InterpolateMakima(_t, _y);
+            Assert.AreEqual(expected, it.Interpolate(t), maxAbsoluteError, "Interpolation at {0}", t);
+        }
+
+        /// <summary>
+        /// Verifies that Makima interpolation handles linear data correctly.
+        /// </summary>
+        /// <param name="samples">The number of sample points.</param>
+        [TestCase(5)]
+        [TestCase(7)]
+        [TestCase(15)]
+        public void SupportsLinearCase(int samples)
+        {
+            double[] x, y, xtest, ytest;
+            // LinearInterpolationCase.Build is a utility to generate linear test data.
+            LinearInterpolationCase.Build(out x, out y, out xtest, out ytest, samples);
+            IInterpolation it = CubicSpline.InterpolateMakima(x, y);
+            for (var i = 0; i < xtest.Length; i++)
+            {
+                Assert.AreEqual(ytest[i], it.Interpolate(xtest[i]), 1e-15, "Linear case with {0} samples, sample {1}", samples, i);
+            }
+        }
+
+        /// <summary>
+        /// Verifies that the interpolation throws an exception when provided with too few samples.
+        /// </summary>
+        [Test]
+        public void FewSamples()
+        {
+            Assert.That(() => CubicSpline.InterpolateMakima(new double[0], new double[0]), Throws.ArgumentException);
+            Assert.That(() => CubicSpline.InterpolateMakima(new double[4], new double[4]), Throws.ArgumentException);
+            Assert.That(CubicSpline.InterpolateMakima(new[] { 1.0, 2.0, 3.0, 4.0, 5.0 },
+                                                      new[] { 2.0, 2.0, 2.0, 2.0, 2.0 })
+                          .Interpolate(1.0), Is.EqualTo(2.0));
+        }
+    }
+}

src/Numerics/Interpolation/CubicSpline.cs

@@ -210,6 +210,126 @@ public static CubicSpline InterpolateAkima(IEnumerable<double> x, IEnumerable<do
             return InterpolateAkimaInplace(x.ToArray(), y.ToArray());
         }
 
+        /// <summary>
+        /// Creates a modified Akima (makima) cubic spline interpolation from a set of (x,y) value pairs,
+        /// sorted ascendingly by x.
+        /// </summary>
+        /// <param name="x">Sample points (must be sorted ascendingly)</param>
+        /// <param name="y">Sample values corresponding to x</param>
+        /// <returns>A CubicSpline instance using the makima method</returns>
+        public static CubicSpline InterpolateMakimaSorted(double[] x, double[] y)
+        {
+            if (x.Length != y.Length)
+            {
+                throw new ArgumentException("All vectors must have the same dimensionality.");
+            }
+            if (x.Length < 5)
+            {
+                throw new ArgumentException("The given array is too small. It must be at least 5 long.", nameof(x));
+            }
+
+            var n = x.Length;
+            var dx = new double[n - 1];
+            for (var i = 0; i < n - 1; i++)
+            {
+                dx[i] = x[i + 1] - x[i];
+            }
+
+            // Allocate m with padding: length = n + 3
+            var mLen = n + 3;
+            var m = new double[mLen];
+            // m[2] .. m[n+1] : slopes between adjacent points.
+            for (var i = 0; i < n - 1; i++)
+            {
+                m[i + 2] = (y[i + 1] - y[i]) / dx[i];
+            }
+
+            // Extrapolate two additional points on the left.
+            m[1] = 2 * m[2] - m[3];
+            m[0] = 2 * m[1] - m[2];
+
+            // Extrapolate two additional points on the right.
+            m[n + 1] = 2 * m[n] - m[n - 1];
+            m[n + 2] = 2 * m[n + 1] - m[n];
+
+            // Compute differences (dm) and sums (pm) for makima weights.
+            var dmLen = mLen - 1;
+            var dm = new double[dmLen];
+            var pm = new double[dmLen];
+            for (var i = 0; i < dmLen; i++)
+            {
+                dm[i] = Math.Abs(m[i + 1] - m[i]);
+                pm[i] = Math.Abs(m[i + 1] + m[i]);
+            }
+
+            // Compute modified weights: f1 and f2 for indices 0 .. n-1.
+            var f1 = new double[n];
+            var f2 = new double[n];
+            for (var i = 0; i < n; i++)
+            {
+                f1[i] = dm[i + 2] + 0.5 * pm[i + 2];
+                f2[i] = dm[i] + 0.5 * pm[i];
+            }
+            var f12 = new double[n];
+            for (var i = 0; i < n; i++)
+            {
+                f12[i] = f1[i] + f2[i];
+            }
+
+            // Initial derivative estimates: t[i] = 0.5*(m[i+3] + m[i])
+            var t = new double[n];
+            for (var i = 0; i < n; i++)
+            {
+                t[i] = 0.5 * (m[i + 3] + m[i]);
+            }
+
+            // Determine threshold from maximum of f12.
+            var maxF12 = 0.0;
+            for (var i = 0; i < n; i++)
+            {
+                if (f12[i] > maxF12)
+                    maxF12 = f12[i];
+            }
+            var threshold = 1e-9 * maxF12;
+
+            // For indices where f12 is significant, update t[i] using weighted average.
+            for (var i = 0; i < n; i++)
+            {
+                if (f12[i] > threshold)
+                {
+                    t[i] = (f1[i] * m[i + 1] + f2[i] * m[i + 2]) / f12[i];
+                }
+            }
+
+            // Construct the Hermite cubic spline using the computed derivatives.
+            return InterpolateHermiteSorted(x, y, t);
+        }
+
+        /// <summary>
+        /// Creates a modified Akima (makima) cubic spline interpolation from an unsorted set of (x,y) value pairs.
+        /// WARNING: This method works in-place and will sort the input arrays.
+        /// </summary>
+        /// <param name="x">Sample points</param>
+        /// <param name="y">Sample values corresponding to x</param>
+        /// <returns>A CubicSpline instance using the makima method</returns>
+        public static CubicSpline InterpolateMakimaInplace(double[] x, double[] y)
+        {
+            // Sorting.Sort is assumed to sort x and apply the same permutation to y.
+            Sorting.Sort(x, y);
+            return InterpolateMakimaSorted(x, y);
+        }
+
+        /// <summary>
+        /// Creates a modified Akima (makima) cubic spline interpolation from an unsorted set of (x,y) value pairs.
+        /// </summary>
+        /// <param name="x">Sample points as an IEnumerable</param>
+        /// <param name="y">Sample values corresponding to x as an IEnumerable</param>
+        /// <returns>A CubicSpline instance using the makima method</returns>
+        public static CubicSpline InterpolateMakima(IEnumerable<double> x, IEnumerable<double> y)
+        {
+            return InterpolateMakimaInplace(x.ToArray(), y.ToArray());
+        }
+
         /// <summary>
         /// Create a piecewise cubic Hermite interpolating polynomial from an unsorted set of (x,y) value pairs.
         /// Monotone-preserving interpolation with continuous first derivative.