Compositional Scalar-on-Function Regression with Application to Sediment Particle Size Distributions

Abstract

The chemical composition of sediments is controlled predominantly by the sediment grain size, and thus evaluating their relationship is an important task in sedimentary geochemistry. The grain size is characterized by the respective particle size distribution, which can be expressed as a probability density function. Because of the relative character of densities, the Bayes space methodology was employed to build a regression model between a real response and a density function as a covariate, here the chemical composition and the particle size density. For practical computations, density functions were expressed in the standard $L^2$ space using the centred logratio transformation and spline approximation of the input discretized densities was utilized by respecting the induced zero-integral constraint. After a concise simulation study, supporting the relevance of the proposed regression model, the new methodology was applied to examine the relationship between sediment grain size and geochemical composition, with samples being obtained in the Czech Republic in the Skalka Reservoir and in the Ohře River floodplain upstream of the reservoir, to reveal proper grain size proxies. The Al/Si and Zr/Rb logratios in the sediments that were studied showed grain-size control, which makes them suitable for this purpose.

Appendix: Computation of ZB-Spline Coefficients

In this appendix, the computation of ZB-spline coefficients for a smoothing spline from the $L_0^2$ space is briefly recalled; for more details see Machalová et al. (2021). Let data $(t_{j},f_{j})$, $a\le t_{j}\le b$, weights $w_{j}>0$, $j=1,\ldots ,n$, $n\ge g+1$ and a parameter $\alpha \in (0,1)$ be given. The task is to find a spline $s_{k}(t)\in {{{\mathscr {Z}}}}_{k}^{\varDelta \lambda }[a,b]\subset L_0^2(I)$, which minimizes the functional

$$\begin{aligned} J_{l}(s_k) = (1-\alpha )\int _{a}^{b}\left[ s_{k}^{(2)}(t)\right] ^{2}\,\mathrm{d}t + \alpha \sum \limits _{j=1}^{n} w_{j}\left[ f_{j}-s_{k}(t_{j})\right] ^{2}. \end{aligned}$$

(64)

As proved in Machalová et al. (2021), the resulting smoothing spline is given by the formula

$$\begin{aligned} s_{k}(t) = \sum \limits _{j=-k}^{g-1}z_{j}^{*}Z_{j}^{k+1}(t), \end{aligned}$$

where $Z_{j}^{k+1} \in L_0^2, j=-k,\ldots ,g-1$ are ZB-spline basis functions defined in (33), and the vector ${\mathbf {z}}^{*}=\left( z_{-k}^{*},\ldots ,z_{g-1}^{*}\right) ^{'}$ is obtained by

$$\begin{aligned} \mathbf{z}^{*}=\left\{ \mathbf{K}^{'} \mathbf{D }\left[ (1-\alpha ){\mathbf{S}_2}^{'}{} \mathbf{M}^{(2)}_{B}{} \mathbf{S}_2+\alpha \mathbf{B}_{k+1}^{'}(\mathbf{t})\mathbf{W}{} \mathbf{B}_{k+1}(\mathbf{t})\right] \mathbf{DK}\right\} ^{-1} \alpha \mathbf{K}^{'}{} \mathbf{D}{} \mathbf{B}_{k+1}^{'}(\mathbf{t})\mathbf{W f}. \end{aligned}$$

(65)

Here, $\mathbf{W} = \hbox {diag}(\mathbf{w }), \mathbf{w }=(w_1,\ldots ,w_n)', \mathbf{f }=(f_1,\ldots ,f_n)'$,

$$\begin{aligned} \mathbf{B}_{k+1}(\mathbf{t}) = \left( \begin{matrix} B^{k+1}_{-k}({t_1}) &{} \ldots &{} B^{k+1}_{g}({t_1}) \\ \vdots &{} \ddots &{} \vdots \\ B^{k+1}_{-k}({t_n}) &{} \ldots &{} B^{k+1}_{-g}({t_n}) \end{matrix}\right) \in {\mathbb {R}}^{n,g+k+1}, \end{aligned}$$

is the collocation matrix, $\mathbf{D }, \mathbf{L }, \mathbf{M}^{(2)}_{B}$ are given in (36), (37), (47), and $\mathbf{S }_2 = \mathbf{D }_2 \mathbf{L }_2 \mathbf{D }_1 \mathbf{L }_1 \in {\mathbb {R}}^{g+k+1-2,g+k+1}$ with $\mathbf{D }_j$, $\mathbf{L }_j$, $j=1,2$ are given in (48), (49).