### 28.5 Polynomial Interpolation

Octave comes with good support for various kinds of interpolation, most of which are described in Interpolation. One simple alternative to the functions described in the aforementioned chapter, is to fit a single polynomial, or a piecewise polynomial (spline) to some given data points. To avoid a highly fluctuating polynomial, one most often wants to fit a low-order polynomial to data. This usually means that it is necessary to fit the polynomial in a least-squares sense, which just is what the `polyfit` function does.

Function File: p = polyfit (x, y, n)
Function File: [p, s] = polyfit (x, y, n)
Function File: [p, s, mu] = polyfit (x, y, n)

  Return the coefficients of a polynomial p(x) of degree n that minimizes the least-squares-error of the fit to the points `[x, y]`. If n is a logical vector, it is used as a mask to selectively force the corresponding polynomial coefficients to be used or ignored.

  The polynomial coefficients are returned in a row vector.

  The optional output s is a structure containing the following fields:

  ‘`R`’

    Triangular factor R from the QR decomposition.

  ‘`X`’

    The Vandermonde matrix used to compute the polynomial coefficients.

  ‘`C`’

    The unscaled covariance matrix, formally equal to the inverse of x’*x, but computed in a way minimizing roundoff error propagation.

  ‘`df`’

    The degrees of freedom.

  ‘`normr`’

    The norm of the residuals.

  ‘`yf`’

    The values of the polynomial for each value of x.

  The second output may be used by `polyval` to calculate the statistical error limits of the predicted values. In particular, the standard deviation of p coefficients is given by
  `sqrt (diag (s.C)/s.df)*s.normr`.

  When the third output, mu, is present the coefficients, p, are associated with a polynomial in xhat = (x-mu(1))/mu(2). Where mu(1) = mean (x), and mu(2) = std (x). This linear transformation of x improves the numerical stability of the fit.

  See also: polyval, polyaffine, roots, vander, zscore.

In situations where a single polynomial isn’t good enough, a solution is to use several polynomials pieced together. The function `splinefit` fits a peicewise polynomial (spline) to a set of data.

Function File: pp = splinefit (x, y, breaks)
Function File: pp = splinefit (x, y, p)
Function File: pp = splinefit (…, "periodic", periodic)
Function File: pp = splinefit (…, "robust", robust)
Function File: pp = splinefit (…, "beta", beta)
Function File: pp = splinefit (…, "order", order)
Function File: pp = splinefit (…, "constraints", constraints)

  Fit a piecewise cubic spline with breaks (knots) breaks to the noisy data, x and y. x is a vector, and y is a vector or N-D array. If y is an N-D array, then x(j) is matched to y(:,…,:,j).

  The fitted spline is returned as a piecewise polynomial, pp, and may be evaluated using `ppval`.

  p is a positive integer defining the number of intervals along x, and p+1 is the number of breaks. The number of points in each interval differ by no more than 1.

  The optional property periodic is a logical value which specifies whether a periodic boundary condition is applied to the spline. The length of the period is `max (breaks) - min (breaks)`. The default value is `false`.

  The optional property robust is a logical value which specifies if robust fitting is to be applied to reduce the influence of outlying data points. Three iterations of weighted least squares are performed. Weights are computed from previous residuals. The sensitivity of outlier identification is controlled by the property beta. The value of beta is stricted to the range, 0 < beta < 1. The default value is beta = 1/2. Values close to 0 give all data equal weighting. Increasing values of beta reduce the influence of outlying data. Values close to unity may cause instability or rank deficiency.

  The splines are constructed of polynomials with degree order. The default is a cubic, order=3. A spline with P pieces has P+order degrees of freedom. With periodic boundary conditions the degrees of freedom are reduced to P.

  The optional property, constaints, is a structure specifying linear constraints on the fit. The structure has three fields, `"xc"`, `"yc"`, and `"cc"`.

  `"xc"`

    Vector of the x-locations of the constraints.

  `"yc"`

    Constraining values at the locations xc. The default is an array of zeros.

  `"cc"`

    Coefficients (matrix). The default is an array of ones. The number of rows is limited to the order of the piecewise polynomials, order.

  Constraints are linear combinations of derivatives of order 0 to order-1 according to

      cc(1,j) * y(xc(j)) + cc(2,j) * y'(xc(j)) + ... = yc(:,...,:,j).

  See also: interp1, unmkpp, ppval, spline, pchip, ppder, ppint, ppjumps.

The number of breaks (or knots) used to construct the piecewise polynomial is a significant factor in suppressing the noise present in the input data, x and y. This is demonstrated by the example below.

    x = 2 * pi * rand (1, 200);
    y = sin (x) + sin (2 * x) + 0.2 * randn (size (x));
    ## Uniform breaks
    breaks = linspace (0, 2 * pi, 41); % 41 breaks, 40 pieces
    pp1 = splinefit (x, y, breaks);
    ## Breaks interpolated from data
    pp2 = splinefit (x, y, 10);  % 11 breaks, 10 pieces
    ## Plot
    xx = linspace (0, 2 * pi, 400);
    y1 = ppval (pp1, xx);
    y2 = ppval (pp2, xx);
    plot (x, y, ".", xx, [y1; y2])
    axis tight
    ylim auto
    legend ({"data", "41 breaks, 40 pieces", "11 breaks, 10 pieces"})

The result of which can be seen in Figure 28.1.

Figure 28.1: Comparison of a fitting a piecewise polynomial with 41 breaks to one with 11 breaks. The fit with the large number of breaks exhibits a fast ripple that is not present in the underlying function.

The piecewise polynomial fit, provided by `splinefit`, has continuous derivatives up to the order-1. For example, a cubic fit has continuous first and second derivatives. This is demonstrated by the code

    ## Data (200 points)
    x = 2 * pi * rand (1, 200);
    y = sin (x) + sin (2 * x) + 0.1 * randn (size (x));
    ## Piecewise constant
    pp1 = splinefit (x, y, 8, "order", 0);
    ## Piecewise linear
    pp2 = splinefit (x, y, 8, "order", 1);
    ## Piecewise quadratic
    pp3 = splinefit (x, y, 8, "order", 2);
    ## Piecewise cubic
    pp4 = splinefit (x, y, 8, "order", 3);
    ## Piecewise quartic
    pp5 = splinefit (x, y, 8, "order", 4);
    ## Plot
    xx = linspace (0, 2 * pi, 400);
    y1 = ppval (pp1, xx);
    y2 = ppval (pp2, xx);
    y3 = ppval (pp3, xx);
    y4 = ppval (pp4, xx);
    y5 = ppval (pp5, xx);
    plot (x, y, ".", xx, [y1; y2; y3; y4; y5])
    axis tight
    ylim auto
    legend ({"data", "order 0", "order 1", "order 2", "order 3", "order 4"})

The result of which can be seen in Figure 28.2.

Figure 28.2: Comparison of a piecewise constant, linear, quadratic, cubic, and quartic polynomials with 8 breaks to noisy data. The higher order solutions more accurately represent the underlying function, but come with the expense of computational complexity.

When the underlying function to provide a fit to is periodic, `splinefit` is able to apply the boundary conditions needed to manifest a periodic fit. This is demonstrated by the code below.

    ## Data (100 points)
    x = 2 * pi * [0, (rand (1, 98)), 1];
    y = sin (x) - cos (2 * x) + 0.2 * randn (size (x));
    ## No constraints
    pp1 = splinefit (x, y, 10, "order", 5);
    ## Periodic boundaries
    pp2 = splinefit (x, y, 10, "order", 5, "periodic", true);
    ## Plot
    xx = linspace (0, 2 * pi, 400);
    y1 = ppval (pp1, xx);
    y2 = ppval (pp2, xx);
    plot (x, y, ".", xx, [y1; y2])
    axis tight
    ylim auto
    legend ({"data", "no constraints", "periodic"})

The result of which can be seen in Figure 28.3.

Figure 28.3: Comparison of piecewise polynomial fits to a noisy periodic function with, and without, periodic boundary conditions.

More complex constraints may be added as well. For example, the code below illustrates a periodic fit with values that have been clamped at the endpoints, and a second periodic fit which is hinged at the endpoints.

    ## Data (200 points)
    x = 2 * pi * rand (1, 200);
    y = sin (2 * x) + 0.1 * randn (size (x));
    ## Breaks
    breaks = linspace (0, 2 * pi, 10);
    ## Clamped endpoints, y = y' = 0
    xc = [0, 0, 2*pi, 2*pi];
    cc = [(eye (2)), (eye (2))];
    con = struct ("xc", xc, "cc", cc);
    pp1 = splinefit (x, y, breaks, "constraints", con);
    ## Hinged periodic endpoints, y = 0
    con = struct ("xc", 0);
    pp2 = splinefit (x, y, breaks, "constraints", con, "periodic", true);
    ## Plot
    xx = linspace (0, 2 * pi, 400);
    y1 = ppval (pp1, xx);
    y2 = ppval (pp2, xx);
    plot (x, y, ".", xx, [y1; y2])
    axis tight
    ylim auto
    legend ({"data", "clamped", "hinged periodic"})

The result of which can be seen in Figure 28.4.

Figure 28.4: Comparison of two periodic piecewise cubic fits to a noisy periodic signal. One fit has its endpoints clamped and the second has its endpoints hinged.

The `splinefit` function also provides the convenience of a robust fitting, where the effect of outlying data is reduced. In the example below, three different fits are provided. Two with differing levels of outlier suppression and a third illustrating the non-robust solution.

    ## Data
    x = linspace (0, 2*pi, 200);
    y = sin (x) + sin (2 * x) + 0.05 * randn (size (x));
    ## Add outliers
    x = [x, linspace(0,2*pi,60)];
    y = [y, -ones(1,60)];
    ## Fit splines with hinged conditions
    con = struct ("xc", [0, 2*pi]);
    ## Robust fitting, beta = 0.25
    pp1 = splinefit (x, y, 8, "constraints", con, "beta", 0.25);
    ## Robust fitting, beta = 0.75
    pp2 = splinefit (x, y, 8, "constraints", con, "beta", 0.75);
    ## No robust fitting
    pp3 = splinefit (x, y, 8, "constraints", con);
    ## Plot
    xx = linspace (0, 2*pi, 400);
    y1 = ppval (pp1, xx);
    y2 = ppval (pp2, xx);
    y3 = ppval (pp3, xx);
    plot (x, y, ".", xx, [y1; y2; y3])
    legend ({"data with outliers","robust, beta = 0.25", ...
             "robust, beta = 0.75", "no robust fitting"})
    axis tight
    ylim auto

The result of which can be seen in Figure 28.5.

Figure 28.5: Comparison of two different levels of robust fitting (beta = 0.25 and 0.75) to noisy data combined with outlying data. A conventional fit, without robust fitting (beta = 0) is also included.

The function, `ppval`, evaluates the piecewise polynomials, created by `mkpp` or other means, and `unmkpp` returns detailed information about the piecewise polynomial.

The following example shows how to combine two linear functions and a quadratic into one function. Each of these functions is expressed on adjoined intervals.

    x = [-2, -1, 1, 2];
    p = [ 0,  1, 0;
          1, -2, 1;
          0, -1, 1 ];
    pp = mkpp (x, p);
    xi = linspace (-2, 2, 50);
    yi = ppval (pp, xi);
    plot (xi, yi);

Function File: pp = mkpp (breaks, coefs)
Function File: pp = mkpp (breaks, coefs, d)

  Construct a piecewise polynomial (pp) structure from sample points breaks and coefficients coefs. breaks must be a vector of strictly increasing values. The number of intervals is given by `ni = length (breaks) - 1`. When m is the polynomial order coefs must be of size: ni x m + 1.

  The i-th row of coefs, `coefs (i,:)`, contains the coefficients for the polynomial over the i-th interval, ordered from highest (m) to lowest (0).

  coefs may also be a multi-dimensional array, specifying a vector-valued or array-valued polynomial. In that case the polynomial order is defined by the length of the last dimension of coefs. The size of first dimension(s) are given by the scalar or vector d. If d is not given it is set to `1`. In any case coefs is reshaped to a 2-D matrix of size `[ni*prod(d m)]`

  See also: unmkpp, ppval, spline, pchip, ppder, ppint, ppjumps.

Function File: [x, p, n, k, d] = unmkpp (pp)

  Extract the components of a piecewise polynomial structure pp. The components are:

  x

    Sample points.

  p

    Polynomial coefficients for points in sample interval. `p (i, :)` contains the coefficients for the polynomial over interval i ordered from highest to lowest. If `d > 1`, `p (r, i, :)` contains the coefficients for the r-th polynomial defined on interval i.

  n

    Number of polynomial pieces.

  k

    Order of the polynomial plus 1.

  d

    Number of polynomials defined for each interval.

  See also: mkpp, ppval, spline, pchip.

Function File: yi = ppval (pp, xi)

  Evaluate the piecewise polynomial structure pp at the points xi. If pp describes a scalar polynomial function, the result is an array of the same shape as xi. Otherwise, the size of the result is `[pp.dim, length(xi)]` if xi is a vector, or `[pp.dim, size(xi)]` if it is a multi-dimensional array.

  See also: mkpp, unmkpp, spline, pchip.

Function File: ppd = ppder (pp)
Function File: ppd = ppder (pp, m)

  Compute the piecewise m-th derivative of a piecewise polynomial struct pp. If m is omitted the first derivative is calculated.

  See also: mkpp, ppval, ppint.

Function File: ppi = ppint (pp)
Function File: ppi = ppint (pp, c)

  Compute the integral of the piecewise polynomial struct pp. c, if given, is the constant of integration.

  See also: mkpp, ppval, ppder.

Function File: jumps = ppjumps (pp)

  Evaluate the boundary jumps of a piecewise polynomial. If there are n intervals, and the dimensionality of pp is d, the resulting array has dimensions `[d, n-1]`.

  See also: mkpp.