A natural cubic smoothing splines module to smooth-out noise and obtain an estimate of the first two derivatives (velocity and acceleration in the case of a particle trajectory).

Various methods have been introduced for the automatic choice of the smoothing parameter.

It can also be used to get an interpolating natural cubic spline.

This code was adapted to python from the Octave splines package created by N.Y. Krakauer (see Refs. below), with algorithmic modifications, mainly to gain performance.

• Krakauer, N. Y. & Fekete, B. M. Are climate model simulations useful for forecasting precipitation trends? hindcast and synthetic-data experiments. Environ. Res. Lett. 9, 024009 (2014).
• Krakauer's original octave splines code http://octave.sourceforge.net/splines/
• E. Afik. Robust and highly performant ring detection algorithm for 3d particle tracking using 2d microscope imaging. Sci. Rep. 5, 13584; doi: 10.1038/srep13584 (2015)
• E. Afik and V. Steinberg. On the role of initial velocities in pair dispersion in a microfluidic chaotic flow. Nature Communications 8, Article number: 468 (2017) doi: 10.1038/s41467-017-00389-8.
• E. Afik and V. Steinberg. A Lagrangian approach to elastic turbulence in a curvilinear microfluidic channel. figshare doi: 10.6084/m9.figshare.5112991 (2017).
• T. Li, A. Yan, N. Bhatia, A. Altinok, E. Afik, P. Durand-Smet, ... & E. M. Meyerowitz. Calcium signals are necessary to establish auxin transporter polarity in a plant stem cell niche. Nature communications 10, Article number: 726 (2019) doi: 10.1038/s41467-019-08575-6.

TODO: render math properly

Carl de Boor (1978), A Practical Guide to Splines, Springer, Chapter XIV

Given noisy data y parametrised by x (with weights $w_i = 1/\epsilon _i^2 $), assuming some underlying $$ \qquad y_i = g(x_i) + \epsilon_i $$ find the natural cubic spline $ f_p(x) $, minimizer of the functional:

$$ \qquad p \sum { w_i | y_i - f_p(x_i) |^2 } + (1-p) \int f_p '' (x)^2 dx $$

** Some notations (following [de Boor]):**

$N$ length of $x$ $$ i = 0,...,N-1 $$ $$ a_i = f_p(x_i) $$ $$ c_i = f_p''(x_i)/2 $$ $$ u = c / 3p $$ $ S[f_p] $ weighted sum of squared errors (first term in the functional above) $$ \Delta x_{i} = x_{i+1}-x_{i} $$ $R$ symmetric tridiagonal matrix of order $N-2$: $$ \qquad \Delta x_{i-1} , 2(\Delta x_{i-1} + \Delta x_i) , \Delta x_i $$ $Q^T$ tridiagonal matrix of order $(N-2) \times N$: $$ \qquad 1/\Delta x_{i-1} , -1/\Delta x_{i-1} - 1/\Delta x_i, 1/\Delta x_i $$ $D$ is the diagonal matrix of the uncertainties on y: $$ \qquad D_{i,i} = 1/w_i $$

The following relations result from the continuity of the first derivative and minimisation the regularised LSQ

expressing $c$ in terms of the $a$ : $$ \qquad R c = 3 Q^T a$$

It should be noted that for natural spline $ c_{0} = c_{N-1} = 0 $, hence in the above equation the first and last elements of $c$ are omitted.

$$ \Rightarrow $$ $$ \qquad { 6(1-p) Q^T D^2 Q + pR } u = Q^T y $$ $$ \qquad a = y - 6(1-p) D^2 Q u $$ $$ \qquad S[f_p] = (y-a)^T D^{-2} (y-a) =|| 6(1-p) D Q u ||^2_2 $$

The ppform (in terms of $u$ and $a$):

$$ f_p(x_i) = a_i $$ $$ f_p'(x_i) = \Delta a_i / \Delta x_i - 3p \Delta x_i ( 2 u_i + u_{i+1} ) $$ $$ f_p''(x_i) = 6p u_i $$ $$ f_p'''(x_i^+) = 6p \Delta u_i / \Delta x_i $$

C.M. Hurvich, J.S. Simonoff, C-L Tsai (1998), Smoothing parameter selection in nonparametric regression using an improved Akaike information criterion, J. Royal Statistical Society, 60B:271-293

P. Craven and G. Wahba (1978), Smoothing noisy data with spline functions, Numerische Mathematik, 31:377-403

M. F. Hutchinson and F. R. de Hoog (1985), Smoothing noisy data with spline functions, Numerische Mathematik, 47:99-106

V. Cherkassky and F. Mulier (2007), Learning from Data: Concepts, Theory, and Methods. Wiley

Nir Krakauer, octave-splines 1.2.4

L. Wasserman (2004), All of Nonparametric Statistics

Following Hurvich et al. (1998): "Classical" methods to choose the smoothing parameter are based on the minimisation of an approximately unbiased estimator of either the mean average squared error: $$ \qquad MASE = \frac{1}{N} E { || g(x_i) - f_p(x_i) ||^2 } $$ (as the case of GCV) or the expected Kullback-Leibler discrepency (as the case for the AIC).

The smoothing parameter is the the minimizer of $$ \qquad \log \hat \sigma ^2 + \psi (H_p / N) $$ where $$ \qquad \hat \sigma ^2 = \frac{1}{N} \sum { y_i - f_p(x_i) }^2 = \frac{1}{N} || (I-H_p)y ||^2 $$ and $\psi$ is the penalty function designed to decrease with increasing smoothness of $f_p$.

The Hat matrix $H_p$ (also referred to as the Smoothing matrix $L$ [Wasserman (2004)] or the Influence matrix $A_p$ [Hutchinson & de Hoog (1985)]), is the matrix transforming the (noisy) data to the estimator: $$ \qquad a = H_p y $$

Denote $h=tr(H_p)/N$; some possiblities for the penalty function: $$ \qquad \psi_{GCV}(H) = -2\log [1-h] $$ $$ \qquad \psi_{AIC}(H) = 2 h $$ $$ \qquad \psi_{AIC_C}(H) = \frac{1+h}{1-(h+2/N)} = 1 + 2 \frac{tr(H) +1}{N- tr(H)-2} \qquad if \quad tr(H)+2>N \quad \psi_{AIC_C}=\infty$$ $$ \qquad \psi_{T}(H) = -\log [1 - 2 h] $$

Cherkassky and Mulier (2007) [p.129]

Vapnik-Chervonenkis penalization factor or Vapnik's measure: $$ \qquad \psi_{VM} = -\log \left[ 1 - \sqrt{ h-h \log h +\log N / 2N } \right] $$

All require the estimation of $I-H_p$ and $Tr{H_p}$. Using de Boor's notation $$\qquad I - H_p = \frac{1}{6(1-p)} D^2 Q { Q^T D^2 Q + \frac{p}{6(1-p)} R }^{-1} Q^T = \dots$$ Following Craven & Wahba (1978), denote $F = DQR^{-1/2}$ , for $R^{-1/2}$ is the (symmetric?) square-root of the inverse of $R$. Rewritte in terms of $F$: $$ \qquad \ldots = \frac{1}{6(1-p)} DF {F^T F +\frac{p}{6(1-p)} I }^{-1} F^T D^{-1} = \ldots$$ Let the signular value decomposition of $F$ be $ F = U \Sigma V^T $ with non-zero singluar values $\sigma_i$ and then: $$\qquad \ldots = D U \left( \frac{\sigma_i^2}{6(1-p)\sigma_i^2 + p} \right) U^T D^{-1}$$ and so finally Craven & Wahba (1978) suggest to choose by $$ \qquad \min_p \frac{\hat \sigma^2} { [Tr(I-H_p)/N]^2} = \min_p N \frac{ || \Lambda U^T y ||^2}{\left(Tr(\Lambda) \right)^2}$$ where $ \Lambda $ denotes the diagonal matrix with the elements $\frac{\sigma_i^2}{6(1-p)\sigma_i^2 + p} $ on the diagonal.