whittaker2: Implementation notes for Whittaker smoothing and...

In albatross: PARAFAC Analysis of Fluorescence Excitation-Emission Matrices

Implementation notes for Whittaker smoothing and interpolation of surfaces

Description

Smooth Eilers2003 or estimate the baseline Eilers2004-PTW of a surface measured on an arbitrary grid by minimising a sum of penalties. Combined difference orders and two different methods of preventing negative values in the output are supported.

This is not a public interface. Subject to change without further notice. Please do not call from outside albatross.

Usage

  whittaker2(x, y, z, lambda, d, p, logscale, nonneg)
  diffmat(x, y, d)
  vandermonde(x0)

Arguments

x	Grid values along the rows of z.
y	Grid values along the columns of z.
z	Matrix containing the surface values to smooth or NAs to interpolate.
lambda	A vector of smoothness penalties, one for every difference order. Must be of the same length as d.
d	whittaker2 A vector of difference orders corresponding to elements of lambda. diffmat Difference order, an integer scalar.
p	If not missing, use the asymmetric penalty method Eilers2004-PTW to estimate the baseline by penalising the differences with weight p if 3{\hat{z} < z

z.hat < zz&#x0302; &lt; z and 1 - p otherwise. Typically, values around 3{10^{-3}

Details

Finite difference approximation

How to differentiate a function tabulated on a fixed, potentially nonuniform grid before you even know its values? Use its Taylor series.

First derivative is special because it's possible to use central differences and get a second-order accurate result, even on a non-uniform grid, by carefully choosing the points where the derivative is calculated. Let x + \frac{h}{2} and x - \frac{h}{2} be a pair of adjacent points from the grid. Here's the Taylor series expansion for f around x, with the Lagrange form of the reminder:

3{ f \left(x + \frac{h}{2}\right) = f(x) + \frac{h}{2} f'(x) + \frac{h^2}{8} f''(x) + \frac{h^3}{48} f'''(\zeta)

f(x + h/2) = f(x) + h/2 f'(x) + h^2/8 f”(x) + h^3/48 f”'(\zeta) f (x + ^h/₂) = f(x) + ^h/₂ f'(x) + ^h²/₈ f”(x) + ^h³/₄₈ f”'(&zeta;)

3{ f \left(x - \frac{h}{2}\right) = f(x) - \frac{h}{2} f'(x) + \frac{h^2}{8} f''(x) - \frac{h^3}{48} f'''(\eta)

Value

whittaker2	A matrix of the same shape as z, with values smoothed, interpolated, or baseline estimated.
diffmat	A difference matrix 3{\mathbf D

D<b>D</b> that, when multiplied by 3{\mathrm{vec}(\mathbf{Z})

References

\bibliography

See Also

feemscatter

Examples

  data(feems)
  z <- feemscatter(feems$a, rep(25, 4), 'omit')
  str(albatross:::whittaker2(
    attr(z, 'emission'), attr(z, 'excitation'), z,
    c(1, 1e-3), 1:2, logscale = NA, nonneg = 1
  ))

albatross documentation built on May 29, 2024, 9:10 a.m.