A Flexible Family of Compactly Supported Covariance Functions Based on Cutoff Polynomials

Abstract

In time series analysis, signal covariance modeling is an essential part of stochastic methods like prediction or filtering. In geodetic applications, covariance functions are rarely treated as true compactly supported functions although large amounts of data would approve such. Covariance models for complex correlation shapes are also rare. Ideally, general families of covariance functions with a large flexibility are desirable to model complex correlations structures like negative correlations. In this paper, we derive isotropic finite covariance functions that are parametrized in a way that positive definiteness is guaranteed. These are based on cutoff polynomials which are derived from operations such as autoconvolution and autocorrelation. Next to the compact support, the resulting autocovariance models share the advantages of (a) positive definiteness by design, (b) extensibility to arbitrary orders and (c) extensive flexibility by employing multiple tunable shape parameters. All these realize various correlation shapes such as negative correlations (the so-called hole effect) and several oscillations. The methodological concepts are derived for homogeneous and isotropic random fields in $\mathbb{R}^d$. The family of covariance functions is then derived for one-dimensional applications. A data example demonstrates the covariance modeling approach using stationary time-series data.

1 Introduction and Related Work

Signal covariance modeling is an important part of many stochastic prediction methods. Within that, finite covariance functions are of important use. The finite support leads to many zero-elements in the covariance matrices which allows the use of sparse data structures and efficient solvers. Both yield advantages in runtime and data allocation and also enable the handling of large tasks. In geostatistical applications, covariance functions are rarely treated as true compactly supported functions although the enormous amount of data would approve such, see e.g. Sansò and Schuh (1987) and Schuh (2016).

In time series analysis simply parametrized covariance functions are helpful for statistical analysis of data. This implies the use of functions depending on a number of tunable shape parameters. Such a parametrization is achieved by a construction using certain operations such as autoconvolution and autocorrelation.

Several covariance functions organized as a family or class exist. Sansò and Schuh (1987), Wendland (1995), Wu (1995), Gaspari and Cohn (1999) and Buhmann (2001) have introduced such classes of finite covariance functions, either given by (a) polynomials, (b) they contain trigonometric expressions, (c) they comprise rational functions or (d) are built from combined expressions of the former. All are defined on a compact support and might be given in a piecewise definition within the finite support.

One well known compactly supported covariance model of interest is the so-called spherical covariance function (Wackernagel 1998; Chilès and Delfiner 1999) which is very often used in geostatistics (Webster and Oliver 2007) and also appears in textbooks on stochastic processes (Priestley 1981). The derivations in this paper use the spherical covariance function

$$\displaystyle \begin{aligned} \gamma\!\left(\tau\right)=\begin{cases} \sigma^2 \,\left(1-\frac{3}{2}\frac{\lvert\tau\rvert}{b} + \frac{1}{2}\frac{\lvert\tau\rvert^3}{b^3}\right) & ,\, \text{for }\lvert\tau\rvert \leq b \ , \\ 0 & ,\,\text{otherwise} \ , \end{cases} {} \end{aligned} $$

(1)

(Matheron 1965, p. 57; Priestley 1981, Eq. (3.7.93); Wackernagel 1998, p. 56; Chilès and Delfiner 1999, p. 81; Webster and Oliver 2007, p. 87) as an initial example and a starting point. Notably, the spherical covariance model can be constructed by a 3D-autoconvolution of a unit sphere but also by a 1D-autocorrelation of a first order monomial cut on the interval $\left [0, 1\right ]$.

This present article uses techniques different from those given in the established families primarily by means of using self-correlation instead of self-convolution. Furthermore, the suggested autocovariance models combine the advantages of (a) guaranteed positive definiteness in $\mathbb{R}^1$, similar to the models of Sansò and Schuh (1987) and Gaspari and Cohn (1999), (b) extensibility to arbitrary orders as e.g. the families of Wendland (1995) and Wu (1995) and (c) extensive flexibility by employing multiple tunable shape parameters as is partially done in Gaspari and Cohn (1999).

Many empirical covariance structures of real-world problems decrease to a minimum below zero, i.e. obtaining negative correlations in a certain interval, cf. e.g. Daley et al. (2015). This phenomenon is called the hole effect (e.g. Journel and Froidevaux 1982; Webster and Oliver 2007) and several globally supported covariance models exist for that purpose, e.g. a damped cosine, see e.g. Gneiting (1999) and Schubert et al. (2020).

The family of covariance functions presented here is able to handle different correlation patterns, such as negative correlations and several oscillations and furthermore provide finite support. Furthermore, we show a way of easily constraining the function to higher classes of continuous differentiability. All in all, this flexibility is needed when the functions are fitted to data-derived empirical autocovariances.

The paper is organized in the following way. Section 2 provides the basic methodology on autocovariance functions. Section 3 introduces the new methodological concepts and the definition of the family of covariance functions which is followed by a data example in Sect. 4. The family of covariance functions is derived only for validity in $\mathbb{R}^1$, but an outlook is given for deriving similar families with validity in $\mathbb{R}^2$ and $\mathbb{R}^3$.

2 Methodology

2.1 Autocovariance Functions and Positive Semi-Definiteness

Autocovariance functions $\gamma (h)$ of a discrete stochastic process have to be positive semi-definite, i.e.,

$$\displaystyle \begin{aligned} \sum_{i,j=1}^N z_i \gamma(t_i-t_j) z_j \geq 0 \text{ for any }z_i,z_j \in \mathbb{R} {} \end{aligned} $$

(2)

with discrete lag $h=\lvert t_i-t_j \rvert =k\,\varDelta t,~k \in \mathbb{Z}$ (cf. Brockwell and Davis 1991, Prop. 1.5.1; Yaglom 1987, Eq. (1.28)).

For analytical covariance functions $\gamma \!\left (\tau \right )$ depending on a lag $\tau$ the positive definiteness requirement (Eq. 2) translates to non-negativity of the Fourier transform of the covariance function, known as Bochner’s theorem (Bochner 1955; Schuh 2016).

In this paper, we will define a family of covariance functions $\gamma _n\!\left (\tau \right )$ where n is the order, which can for instance be the degree of an expansion as a polynomial. Furthermore, for the sake of brevity we define $\tau \geq 0$ and omit the vertical bars.

In this paper, we deal with autocovariance functions $\gamma \!\left (\tau \right )$ of compact support. For reasons of brevity, the common notation of the plus subscript $\left (1-{\tau } \right )_{+}$ which corresponds to $\max \left ({0,\, 1-{\tau }}\right )$, i.e. signifying a cutoff at 1, is extended to $\left (1-{\tau }/{b}\right )_{+}$ indicating a cutoff at the range parameter b as in Eq. (1). However, due to the fact that we deal with non-monotonous functions we use the notation $\left (1-{\tau }/{b}\right )_{\left ( \tau \leq b\right )}$.

2.2 Operations on Covariance Functions

It is well known that operations such as summation, scaling, multiplication and convolution are admissible operations to be applied to positive semi-definite covariance functions, naturally with positive weights to preserve positive definiteness. For instance, covariance tapering is known as the multiplication of an arbitrary covariance function with a finite one to obtain a function of finite support, see e.g. Furrer et al (2006). However, it is a matter of fact that even if one negative definite function is involved in a product the result can nonetheless be positive definite. Hence, it is beneficial to define covariance models that are positive definite by design due to the fact that general models exploit the full parameter space whilst nested models imply restrictions.

This work focuses on the operations autoconvolution and autocorrelation. The former gives rise to the use of B-splines (linear and higher orders) as autocovariance functions, for example. The operations self-convolution (autoconvolution)

$$\displaystyle \begin{aligned} \gamma\!\left(\tau\right) = \left({f} \ast {f} \right)\,&\left(\tau\right) = \int_{\mathbb{R}^d} \, f\!\left(t\right) \, f\!\left(\tau - t\right) \, dt {} \end{aligned} $$

(3)

and self-correlation (autocorrelation)

$$\displaystyle \begin{aligned} \gamma\!\left(\tau\right) = \left({h} \star {h} \right)\,&\left(\tau\right) = \int_{\mathbb{R}^d} \, h\!\left(t\right) \, h\!\left(\tau + t\right) \, dt {} \end{aligned} $$

(4)

of the indicator functions f(t) and h(t) differ only in the sign of t in the second function. Let us use the following notation to introduce, analogous to self-convolution (autoconvolution) $\left ({f} \ast {f} \right )$, the operation self-correlation (autocorrelation)

$$\displaystyle \begin{aligned} \gamma\!\left(\tau,~\sigma,~b,~\vec{d}\right) = \left(h \star h \right)~~&\left(\tau,~\sigma,~b,~\vec{d}\right) \end{aligned} $$

(5)

and denote the involved functions by the autoconvolution kernel ${f}\!\left (t,~{b},~\vec {{d}}\right )$ and autocorrelation kernel $h\!\left (t,~{b},~\vec {{d}}\right )$. ${f}\!\left (t\right )$ and $h\!\left (t\right )$ are also called indicator functions. b is the range parameter signifying the length of support while the $d_i$ in vector $\vec {d}$ are shape parameters. The autocovariance function $\gamma \!\left (\tau \right )$ has a scale parameter given by the variance $\sigma ^2$ such that $\gamma \!\left (0\right )=\sigma ^2$.

2.3 Positive Definiteness in $\mathbb{R}^d$

The application in spatial domains requires positive semi-definiteness of the covariance function in higher dimensions $\mathbb{R}^d$, which is derived here.

In multivariate problems the distance r is taken as the Euclidean distance of the d-dimensional vector $\boldsymbol {h}$ and the isotropic (radial) covariance function reads $\gamma \!\left (r\right ) = \gamma \!\left (\left \lVert \boldsymbol {h}\right \rVert \right ) \text{with } \boldsymbol {h} \in \mathbb{R}^d$. For applications with data in higher dimensions, e.g. spatial data, the reduction to a one-dimensional distance-like norm (e.g. Euclidean) does not guarantee positive definiteness of the covariance function. Instead, the Bochner theorem is generalized to non-negativity $\mathcal {H}_d \! \left \lbrace \gamma \!\left (r\right ) \right \rbrace :={S}\!\left (s\right )\geq 0 \ \forall \, s$ of the Hankel transform $\mathcal {H}_d \! \left \lbrace \gamma \!\left (r\right ) \right \rbrace$ building a spectrum ${S}\!\left (s\right )$ (Bochner 1955; Moritz 1976; Lantuéjoul 2002, p. 25).

The Hankel transform $\mathcal {H}_d \! \left \lbrace \,\right \rbrace$ of order d for an isotropic covariance function $\gamma \!\left (r\right )$ in $\mathbb{R}^d$ is generally defined as (see e.g. Sneddon 1951, Sec. 12; Yaglom 1987, p. 353; Chilès and Delfiner 1999, p. 68; Lantuéjoul 2002, p. 241)

$$\displaystyle \begin{aligned} \begin{aligned} \mathcal{H}_d \! \left\lbrace \gamma\!\left(r\right) \right\rbrace := {S}\!\left(s\right) = &\frac{1}{\left(2\pi\right)^{d/2} \, s^{d/2-1}} ~ \\ & \int_{0}^{\infty} \, \mathrm{J}_{d/2-1}\!\left(s\,r\right) \, r^{d/2} \, \gamma\!\left(r\right) \, dr \end{aligned} {} \end{aligned} $$

(6)

where $\mathrm {J}_{\nu }\!\left (\right )$ is the Bessel function of the first kind (called J-Bessel) and order $\nu$.

Next to the transition to the spectral domain given by the Hankel transform, convolution and correlation theorem also translate to the higher dimensions and their respective transforms. As a result, a self-convolution (Eq. 3) of a function $f(r)$ in $\mathbb{R}^d$ to generate $\gamma \!\left (r\right )$ corresponds to a squaring of $\mathcal {H}_d \! \left \lbrace f(r) \right \rbrace$ (convolution theorem)

$$\displaystyle \begin{aligned} S(s) := \mathcal{H}_d \! \left\lbrace \gamma\!\left(r\right) \right\rbrace = { \mathcal{H}_d \! \left\lbrace f(r) \right\rbrace }^2 \end{aligned} $$

(7)

and hence guaranteed positive definiteness only if $f(r)$ is symmetric.

The crucial difference between self-convolution and self-correlation is that the autocorrelation operation (Eq. 4) translates to the operation of a true absolute value (Euclidean norm) in the (potentially complex-valued) spectral domain (correlation theorem)

$$\displaystyle \begin{aligned} S(s) := \mathcal{H}_d \! \left\lbrace \gamma\!\left(r\right) \right\rbrace = {\lvert \mathcal{H}_d \! \left\lbrace h(r) \right\rbrace \rvert}^2 \end{aligned} $$

(8)

which ensures non-negativity (Bochner’s theorem) for whatever parity of $h(r)$, see e.g. Chilès and Delfiner (1999, Eq. (2.30)). Thus, autocorrelation allows non-symmetric and even one-sided indicator functions $h(t)$, whereas autoconvolution restricts to symmetric indicator functions. A reasonable assumption now is that self-correlation can in general produce more flexible functions due to the non-symmetric nature of the indicator function and guaranteed non-negative spectrum ${S}\!\left (s\right )$.

3 Methodological Advances

Established families for compactly supported rarely exhibit negative correlations, the hole effect. Gneiting (2002, Sec. 2.3) introduces oscillatory compactly supported functions based on the so-called turning bands operator (e.g. Matheron 1973; Lantuéjoul 2002). In fact, the turning bands operator (also TBM, Turning Bands Method) can retrieve the covariance model of the same type which has maximum hole effect and which can build the boundary of the domain of validity, i.e. positive semi-definiteness in a dimension of interest. For details see e.g. the covariance model when applying the TBM operator to the spherical model, which amounts to $\gamma \!\left (\tau \right ) = \left (1-3{\tau } + 2{\tau ^3}\right )_{(\tau \leq 1)}$, see Chilès and Delfiner (1999, Tab. A.2), and builds a limit case for positive definiteness in 1D.

As this only retrieves limit cases and not fully flexible models, allowing polynomial coefficients to vary arbitrarily within the bounds of validity can provide the full flexibility. This is the general idea of this paper and will be derived in the next sections.

3.1 Covariance Functions Given by Cutoff Polynomials

We desire covariance models $\gamma \!\left (\tau \right )$ or $\gamma \!\left (r\right )$ of polynomial type. The general idea is an extension to allow variable polynomial coefficients $a_i$ and define

$$\displaystyle \begin{aligned} \gamma\!\left(\tau\right) &= \begin{cases} \sigma^2 \, \left( 1 + \sum_{i=1}^n a_i \, \left(\frac{\tau}{b}\right)^{i} \right) & , \, \tau \leq b \ ,\\ 0 & , \, \tau > b \ . \end{cases} {} \end{aligned} $$

(9)

The vector includes the polynomial coefficients $a_i$, among which $a_0$ is covered by the variance $\sigma ^2$. With the purpose of defining a family of covariance functions in this section, the family is subscripted by the degree m of the indicator function as $\gamma _{m}\!\left (\tau \right )$ and thus linked to the number of defining shape parameters. This is favorable to a subscript by the polynomial degree n which takes only odd degrees.

For functions $\gamma \!\left (r\right )$ with finite support the integration Eq. (6) is done up to a fixed upper limit which leads to a particular form of a Riemann-Liouville integral. The Hankel transform of compactly supported monomials $\mathcal {H}_d \! \left \lbrace r_{(r\leq 1)}^k \right \rbrace$ is given by algebraic combinations of rational, trigonometric, J-Bessel, Struve and Lommel functions (cf. Gradshteyn and Ryzhik 2000, Eqs. (6.561)) but has a compact notation given by (cf. Gradshteyn and Ryzhik 2000, Eq. (6.569); Erdélyi 1954, Eq. (13.1.56), p. 193)

(10)

where is one particular form of the generalized hypergeometric function. Eventually, linear combinations of Eq. (10), given by the weighted sum using the polynomial coefficients $a_i$, provide the result for the Hankel transform of a general compactly supported polynomial and thus an evaluation of the positive definiteness in various dimensions $\mathbb{R}^d$. Eq. (10) is independent of the support range b and the transforms $\mathcal {H}_d \! \left \lbrace r_{(r\leq 1)}^k \right \rbrace$ are solely weighted by the $a_i$.

The spectra $S(s)$ for covariance models given by combinations of Eq. (10) may follow typical spectra for shaping filters in the sense that designated extrema in the spectrum are modelled, corresponding to the oscillating behavior of the autocovariance function. Beyond that however, they are of oscillating and slowly attenuating nature. It should be noted here that the validation of $S(s)\geq 0$ can be cumbersome as the spectrum might asymptotically reach a negative value. The use of an indicator function bypasses the problem and can guarantee positive semi-definiteness in $\mathbb{R}^d$. This will be done in the next section, however only for $\mathbb{R}^1$.

Note that Hristopulos (2015) uses compactly supported polynomials in the spectral domain and achieves a linear combination of Eq. (10) as a family of globally supported covariance functions in time domain, called the Bessel-Lommel covariance functions.

3.2 Idea of Parametrization

From this point on the paper will deal only with univariate autocorrelation leading to a family of covariance functions valid in $\mathbb{R}^1$.

The idea is that the autocovariance function is generated from an analytical (univariate) self-correlation

$$\displaystyle \begin{aligned} \gamma\!\left(\tau\right) = \left(h \star h \right)\ \left(\tau\right) = \int_{-\infty}^{\infty} h\!\left(t\right) \, h\!\left(t + \tau\right) \, dt {} \end{aligned} $$

(11)

(see e.g. Yaglom 1987, Eq. (2.45); Chilès and Delfiner 1999, Eq. (2.30); Eq. (7.22); Lantuéjoul 2002, p. 190; Schlather 2012, Eq. (2.12) and Iske 2018, Cor. 8.12) of an indicator function $h\!\left (t\right )$. In order to achieve compactly supported covariance functions one has to use a compactly supported indicator function.

In general, it proves beneficial to assess the function as generated from autocorrelation operation such that several properties of covariance functions are automatically satisfied and the covariance functions are positive definite by design.

In order to achieve valid covariance models for 1D, we first restrict ourselves to covariance functions generated from one-dimensional (univariate) autocorrelation operations. The resulting functions can nonetheless suit as isotropic covariance functions in $\mathbb{R}^d$ only if positive definiteness in the higher dimension is ensured additionally.¹ On the other hand, covariance models valid in a certain dimensions can always be used in a lower dimension.

In contrast to autoconvolution, autocorrelation allows non-symmetric and even one-sided indicator functions. What is not demonstrated here, self-correlation has the advantages of enabling to model one more lobe with each order, which is not possible with self-convolution. Furthermore, the maximum hole effect can only be achieved by self-correlation. As a result, we acknowledge the assumption that self-correlation can in general produce more flexible functions. In addition, the indicator functions are restricted without loss of generality to one-sided functions.

Clearly, the number of intervals in the piece-wise definition of the indicator function plays a role. A desirable property of the self-correlation is the possibility to achieve an autocovariance function consisting of just one defined interval (apart from the symmetric counterpart and the zero outside the support range b) by an arbitrary indicator function that shares this property. Hence, we restrict to one-sided indicator functions defined by a polynomial in a single interval, although multi-interval covariance models are also common, see Gaspari and Cohn (1999).

3.3 Parameterizing Polynomials by Univariate Self-Correlation

In order to achieve valid covariance functions for 1D the idea is that the autocovariance function is generated from a univariate self-correlation (Eq. 11). Doing that using a one-sided indicator function $h\!\left (t\right )$ of cutoff polynomial type of degree m

$$\displaystyle \begin{aligned} h\!\left(t\right) &= \begin{cases} \alpha \, \sum_{i=0}^m d_i \, \left(\frac{t}{b}\right)^{i} & ,\, 0\leq t \leq b \ ,\\ 0 & ,\, t <0 ,\ t > b \ , \end{cases} {} \end{aligned} $$

(12)

constructs an autocovariance function by

$$\displaystyle \begin{aligned} \gamma\!\left(\tau\right) = \int_{0}^{b-\tau} h\!\left(t\right) \, h\!\left(t + \tau\right) \, dt {} \end{aligned} $$

(13)

which automatically fulfills the symmetry $\gamma \!\left (\tau \right ) = \gamma \!\left (-\tau \right )$.

The operation yields covariance models $\gamma (\tau )$ of polynomial type (Eq. 9) of degree $n=2m+1$ with coefficients $a_i$ depending on a set of defining parameters $d_i$. The scalar $\alpha$ signifies a scaling by the total overlapping area in the integration and is applied to realize $\gamma \!\left (0\right ) = \sigma ^2$. It depends on all parameters $d_i$ but is not specified further, as it will be replaced by $\sigma ^2$.

We exemplarily look at the self-correlation of a finite straight-line indicator function, i.e. $m=1$. For this case only, the operation requires only one² defining parameter $d_1$ and the result is a cubic polynomial with only constant, linear and cubic term given by

$$\displaystyle \begin{aligned} \begin{aligned} \gamma_{m=1}\!\left(\tau\right) = &\sigma^2 \, \Bigg( 1-\left(\frac{{d_{1}}^2}{2\,{d_{1}}^2+6\,d_{1}+6}+1\right) \, \frac{\tau}{b} \\ & + \left(\frac{{d_{1}}^2}{2\,{d_{1}}^2+6\,d_{1}+6}\right) \, \left(\frac{\tau}{b}\right)^3 \Bigg)_{(\tau\leq b)} \ . \end{aligned} {} \end{aligned} $$

(14)

For this case, the expressions for $a_1$ and $a_3$ fulfill the linear condition $a_1+a_3=-1$ such that

$$\displaystyle \begin{aligned} \vec{a} = \begin{bmatrix} 1,~ & -c_0 - 1 ,~ & 0,~ & c_0 \end{bmatrix} {} \end{aligned} $$

(15)

can be given as an equivalent expression for Eq. (14) using a different defining parameter $c_0$. However, the rational expression for $a_3$ in Eq. (14) leads to $c_0\in \left [0,~2\right ]$. These parameter bounds need to be additionally enforced when using Eq. (15), whilst they are guaranteed by the parametrization (14). As a result, the parametrization using $d_i$ and Eq. (12) is favored despite its non-linear nature.

Figure 1 shows this family of functions for a selection of defining parameters $d_1$ over its full range of values. The triangular and spherical model as well as the model with extreme hole effect (TBM operator) are part of this family. Naturally, all functions of this family are at least valid in $\mathbb{R}^1$. For dimensions $d=2$ and $d=3$, only the spherical model (Eq. 1) ($d_1=-1$, $c_0=0.5$) is valid which can be shown by evaluation of Eq. (10) as well as by the geometrical considerations of convolving the unit sphere in 3D.

Fig. 1

Covariance models corresponding to the flexible family $\gamma _{m=1}\! \left (\tau \right )$ (Eq. 14). The colors are used for visual separability. The spherical model (dashed line) is the subcase for $d_1=-1$ or $c_0=0.5$

With the generation by Eqs. (13) and (12) we introduce a family of covariance functions that is extensible to arbitrary orders m which yields lengthy, non-linear expressions for the polynomial coefficients $a_i$ similar to Gaspari et al (2006, Eqs. (33), (C.1) and (C.2)). Despite being long and non-linear the formulas are manageable and converge when fitting the tuning parameters. The equations for $m=2\text{ to }5$ are given in the Appendix Eqs. (16) to (19). There, similar to Eq. (15), a formulation using linear constraint relations among shape parameters can be achieved, if $c_{10}$ to $c_{16}$ are considered as the defining tuning parameters, whose bounds are not materialized though.

Higher order models of this family naturally provide more flexibility than for $m=1$ and realize covariance functions with a bigger hole effect compared to Fig. 1.

So in general, nonlinear expressions are obtained. The big advantage however is that the covariance function is well-shaped and, by virtue of the univariate self-correlation, positive semi-definiteness in $\mathbb{R}^1$ is also guaranteed.

In general, the number of defining parameters is $m+1$, see Eq. (12), but can collapse to less as in the case of Eq. (14) which does not involve $d_0$. The covariance functions are polynomials of degree n where in general $m = \left (n-1\right )/2$ holds. The variance $\sigma ^2$ takes the role of $a_0$ and a factor to all other $a_i$ and is also a parameter to be estimated. In addition there is the finite range b. In total, the number of parameters to estimate is $m+3$.

Clearly, the continuity properties of the indicator function define the continuity class of the covariance function. With no specifications to the continuity of $h\!\left (t\right )$, $\gamma \!\left (\tau \right )$ from Eq. (13) will in general only be $\mathcal C^0$ at $\tau =b$. The covariance function’s property of continuous differentiability at $\tau =b$ can be easily increased to higher classes, e.g. $\mathcal C^1$. This can be done by setting the parameters $d_0$, $d_1$, etc. to exactly zero which result in $\mathcal C^{1}$, $\mathcal C^{2}$, etc., respectively.

4 An Example: Milan Cathedral Deformation Time Series

This example is the well known deformation time series of Milan Cathedral (Sansò 1985). The time series measurements are levelling heights of a pillar in the period of 1965 to 1977. For details see also Schubert et al. (2020). The time series is detrended using a linear function and the remaining residuals define the stochastic signal. Based on the detrended time series, the biased estimator is used to determine the empirical covariances.

The fitting of the analytical covariance function is done using the MATLAB function fmincon (MathWorks 2022) with the optimization with respect to the parameters $\sigma$, b and $d_0$ to $d_m$ formulated in the least-squares sense. If needed the support range b and variance $\sigma ^2$ can be constrained by lower and upper bounds. The other shape parameters $d_i$ can be left unconstrained unless higher continuity classes should be achieved.

A first fitted covariance model of order $m=2$ obtains a compact support of $b={6.04}^{years}$ where the transition is of $\mathcal {C}^0$, see Fig. 2.

Fig. 2

Compactly supported covariance models fitted to the empirical covariances of the Milan Cathedral deformation time series

During the model fitting it became apparent that a model with two lobes requires $\mathcal {C}^1$ continuity at $\tau =b$. Hence, a linear equality constraint is applied directly to the parameter $d_0$, i.e. $d_0=0$.

The fitted covariance model is of degree $m=5$ and exhibits two lobes up to the support range of $b={10.24}^{years}$. The shape is comparable to a damped oscillatory behavior, as would result from a modelling using AR and ARMA-processes (see Schubert et al. 2020), but it has finite support.

Performing the collocation prediction of the pillar deformation, the residual sum of squares (RSS) in a leave-one-out cross-validation (LOOCV) shows a reduction from 0.5314 mm2 for the first model to 0.5011 mm2 for the second model. This demonstrates the better representation of the stochastic behavior by the higher order model.

5 Conclusions

We introduced a family of compactly supported autocovariance functions based on cutoff polynomials. The functions are parameterized by a set of defining parameters which build the polynomial coefficients in a non-linear fashion resulting from the construction by self-correlation. These covariance models define a general family with a large flexibility and the ability to model various oscillatory shapes.

In summary, we have introduced versatile compactly supported functions suited for geodetic applications with large amounts of data. The resulting autocovariance models are flexible due to multiple tunable shape parameters and share the advantages of being positive definite by design, extensible to arbitrary orders and easy to constrain to different continuity classes. As an extension to this paper, families of covariance functions that are guaranteed to be positive definite in 2D, 3D and on the sphere have been derived and will be published soon.

Appendix A: Higher Order Covariance Models of the 1D Family

These families are given for $\sigma ^2=1$ and for a support range of $b=1$. For arbitrary b replace $\tau$ by ${\tau }/{b}$. The expressions involve auxiliary variables $c_l$ and defining parameters $d_i$.

For $m=2$:

$$\displaystyle \begin{aligned} &c_0 = {d_{2}}^2 \ , \\ &c_1 = -5\,{d_{1}}^2+{d_{2}}^2+20\,d_{0}\,d_{2} \ , \\ &c_2 = 5\,{d_{1}}^2+15\,d_{1}\,d_{2}+9\,{d_{2}}^2+10\,d_{0}\,d_{2} \ , \\ &c_3 = 30\,{d_{0}}^2+30\,d_{0}\,d_{1}+20\,d_{0}\,d_{2}+10\,{d_{1}}^2+15\,d_{1}\,d_{2}+6\,{d_{2}}^2 \ , \\ &c_{10} = \frac{c_0}{c_3} \ , ~~ c_{11} = \frac{c_1}{c_3} \ , ~~ c_{12} = \frac{c_2}{c_3} \ , \\ & \gamma_{m=2}\!\left(\tau\right)= \big( -c_{10}\,\tau ^5+\left(c_{10}-c_{11}\right)\,\tau ^3+\left(c_{11}+c_{12}\right)\,\tau ^2+\left(-c_{12}-1\right)\,\tau +1 \big)_{\left(\tau \leq 1 \right)} {} \end{aligned} $$

(A.1)

For $m=3$:

$$\displaystyle \begin{aligned} &c_0 = 3\,{d_{3}}^2 \ , \\ &c_1 = -14\,{d_{2}}^2+3\,{d_{3}}^2+42\,d_{1}\,d_{3} \ , \\ &c_2 = -70\,{d_{1}}^2+168\,d_{1}\,d_{3}+14\,{d_{2}}^2+140\,d_{2}\,d_{3}+280\,d_{0}\,d_{2}+102\,{d_{3}}^2+420\,d_{0}\,d_{3} \ , \\ &c_3 = 70\,{d_{1}}^2+210\,d_{1}\,d_{2}+252\,d_{1}\,d_{3}+126\,{d_{2}}^2+280\,d_{2}\,d_{3}+140\,d_{0}\,d_{2}+150\,{d_{3}}^2+210\,d_{0}\,d_{3} \ , \\ &c_4 = 420\,{d_{0}}^2+420\,d_{0}\,d_{1}+280\,d_{0}\,d_{2}+210\,d_{0}\,d_{3}+140\,{d_{1}}^2+210\,d_{1}\,d_{2}+168\,d_{1}\,d_{3}+84\,{d_{2}}^2+140\,d_{2}\,d_{3}+60\,{d_{3}}^2 \ , \\ &c_{10} = \frac{c_0}{c_4} \ , ~~ c_{11} = \frac{c_1}{c_4} \ , ~~ c_{12} = \frac{c_2}{c_4} \ , ~~ c_{13} = \frac{c_3}{c_4} \ , \\ & \gamma_{m=3}\!\left(\tau\right)= \big( c_{10}\,\tau ^7+\left(c_{11}-c_{10}\right)\,\tau ^5+\left(-c_{11}-c_{12}\right)\,\tau ^3 +\left(c_{12}+c_{13}\right)\,\tau ^2+\left(-c_{13}-1\right)\,\tau +1 \big)_{\left(\tau \leq 1 \right)} {} \end{aligned} $$

(A.2)

For $m=4$:

$$\displaystyle \begin{aligned} &c_0 = 2\,{d_{4}}^2 \ , \\ &c_1 = -9\,{d_{3}}^2+2\,{d_{4}}^2+24\,d_{2}\,d_{4} \ , \\ &c_2 = 42\,{d_{2}}^2+504\,d_{0}\,d_{4}-126\,d_{1}\,d_{3} \ , \\ &c_3 = 756\,d_{0}\,d_{4}+126\,d_{1}\,d_{3}+630\,d_{1}\,d_{4}+396\,d_{2}\,d_{4}+315\,d_{3}\,d_{4}-42\,{d_{2}}^2+9\,{d_{3}}^2+250\,{d_{4}}^2 \ , \\ &\begin{array}{l}\displaystyle c_4 = -210\,{d_{1}}^2+504\,d_{1}\,d_{3}+1050\,d_{1}\,d_{4}+42\,{d_{2}}^2+420\,d_{2}\,d_{3}+864\,d_{2}\,d_{4}+840\,d_{0}\,d_{2}+306\,{d_{3}}^2+945\,d_{3}\,d_{4} \\\displaystyle +1260\,d_{0}\,d_{3}+590\,{d_{4}}^2+1764\,d_{0}\,d_{4} \ , \end{array} \\ &\begin{array}{l}\displaystyle c_5 = 210\,{d_{1}}^2+630\,d_{1}\,d_{2}+756\,d_{1}\,d_{3}+840\,d_{1}\,d_{4}+378\,{d_{2}}^2+840\,d_{2}\,d_{3}+900\,d_{2}\,d_{4}+420\,d_{0}\,d_{2} \\\displaystyle +450\,{d_{3}}^2+945\,d_{3}\,d_{4}+630\,d_{0}\,d_{3}+490\,{d_{4}}^2+756\,d_{0}\,d_{4} \ , \end{array} \\ {} \end{aligned} $$

(A.3)

$$\displaystyle \begin{aligned} &\hspace{-50pt} \begin{array}{l}\displaystyle c_6 = 1260\,{d_{0}}^2+1260\,d_{0}\,d_{1}+840\,d_{0}\,d_{2}+630\,d_{0}\,d_{3}+504\,d_{0}\,d_{4}+420\,{d_{1}}^2+630\,d_{1}\,d_{2}+504\,d_{1}\,d_{3}\\\displaystyle +420\,d_{1}\,d_{4}+252\,{d_{2}}^2+420\,d_{2}\,d_{3} +360\,d_{2}\,d_{4}+180\,{d_{3}}^2+315\,d_{3}\,d_{4}+140\,{d_{4}}^2 \ , \end{array} \\ &\hspace{-50pt}c_{10} = \frac{c_0}{c_6} \ , ~~ c_{11} = \frac{c_1}{c_6} \ , ~~ c_{12} = \frac{c_1+c_2}{c_6} \ , ~~ c_{13} = \frac{c_3}{c_6} \ , ~~ c_{14} = \frac{c_4}{c_6} \ , ~~ c_{15} = \frac{c_5}{c_6} \ , \\ &\begin{array}{l}\displaystyle \gamma_{m=4}\!\left(\tau\right)=\big(-c_{10}\,\tau ^9+\left(c_{10}-c_{11}\right)\,\tau ^7+\left(c_{11}-c_{12}\right)\,\tau ^5+\left(c_{12}+c_{13}\right)\,\tau ^4+\left(-c_{13}-c_{14}\right)\,\tau ^3 \\\displaystyle +\left(c_{14}+c_{15}\right)\,\tau ^2 +\left(-c_{15}-1\right)\,\tau +1 \big)_{\left(\tau \leq 1 \right)} \end{array} \end{aligned} $$

For $m=5$:

$$\displaystyle \begin{aligned} &t_0 = 5\,{d_{5}}^2 \ , \\ &t_1 = -22\,{d_{4}}^2+5\,{d_{5}}^2+55\,d_{3}\,d_{5} \ , \\ &t_2 = 99\,{d_{3}}^2+660\,d_{1}\,d_{5}-264\,d_{2}\,d_{4} \ , \\ &\begin{array}{l}\displaystyle t_3 = 5544\,d_{0}\,d_{4}-1386\,d_{1}\,d_{3}+13860\,d_{0}\,d_{5}+6270\,d_{1}\,d_{5}+264\,d_{2}\,d_{4}+4620\,d_{2}\,d_{5}+3410\,d_{3}\,d_{5}+2772\,d_{4}\,d_{5} \\\displaystyle +462\,{d_{2}}^2-99\,{d_{3}}^2+22\,{d_{4}}^2+2305\,{d_{5}}^2 \ , \end{array} \\ &\begin{array}{l}\displaystyle t_4 = 8316\,d_{0}\,d_{4}+1386\,d_{1}\,d_{3}+20790\,d_{0}\,d_{5}+6930\,d_{1}\,d_{4}+16830\,d_{1}\,d_{5}+4356\,d_{2}\,d_{4}+12705\,d_{2}\,d_{5}+3465\,d_{3}\,d_{4}\\\displaystyle +10450\,d_{3}\,d_{5}+11088\,d_{4}\,d_{5}-462\,{d_{2}}^2+99\,{d_{3}}^2+2750\,{d_{4}}^2+7595\,{d_{5}}^2 \ , \end{array} \\ &\begin{array}{l}\displaystyle t_5 = -2310\,{d_{1}}^2+5544\,d_{1}\,d_{3}+11550\,d_{1}\,d_{4}+17820\,d_{1}\,d_{5}+462\,{d_{2}}^2+4620\,d_{2}\,d_{3}+9504\,d_{2}\,d_{4}+15015\,d_{2}\,d_{5}\\\displaystyle +9240\,d_{0}\,d_{2}+3366\,{d_{3}}^2+10395\,d_{3}\,d_{4}+14960\,d_{3}\,d_{5}+13860\,d_{0}\,d_{3}+6490\,{d_{4}}^2+16632\,d_{4}\,d_{5}+19404\,d_{0}\,d_{4}\\\displaystyle +9730\,{d_{5}}^2+25410\,d_{0}\,d_{5} \ , \end{array} \\ & \begin{array}{l}\displaystyle t_6 = 2310\,{d_{1}}^2+6930\,d_{1}\,d_{2}+8316\,d_{1}\,d_{3}+9240\,d_{1}\,d_{4}+9900\,d_{1}\,d_{5}+4158\,{d_{2}}^2+9240\,d_{2}\,d_{3}+9900\,d_{2}\,d_{4} \\\displaystyle +10395\,d_{2}\,d_{5}+4620\,d_{0}\,d_{2}+4950\,{d_{3}}^2+10395\,d_{3}\,d_{4}+10780\,d_{3}\,d_{5}+6930\,d_{0}\,d_{3}+5390\,{d_{4}}^2\\\displaystyle +11088\,d_{4}\,d_{5}+8316\,d_{0}\,d_{4}+5670\,{d_{5}}^2+9240\,d_{0}\,d_{5} \ , \end{array} \\ & \begin{array}{l}\displaystyle t_7 = 13860\,{d_{0}}^2+13860\,d_{0}\,d_{1}+9240\,d_{0}\,d_{2}+6930\,d_{0}\,d_{3}+5544\,d_{0}\,d_{4}+4620\,d_{0}\,d_{5}+4620\,{d_{1}}^2+6930\,d_{1}\,d_{2} \\\displaystyle +5544\,d_{1}\,d_{3}+4620\,d_{1}\,d_{4}+3960\,d_{1}\,d_{5}+2772\,{d_{2}}^2+4620\,d_{2}\,d_{3}+3960\,d_{2}\,d_{4}+3465\,d_{2}\,d_{5}+1980\,{d_{3}}^2\\\displaystyle +3465\,d_{3}\,d_{4}+3080\,d_{3}\,d_{5}+1540\,{d_{4}}^2+2772\,d_{4}\,d_{5}+1260\,{d_{5}}^2 \ , \end{array} \\ &t_{10} = \frac{t_0}{t_7} \ , ~~ t_{11} = \frac{t_1}{t_7} \ , ~~ t_{12} = \frac{t_1+t_2}{t_7} \ , ~~ t_{13} = \frac{t_3}{t_7} \ , ~~ t_{14} = \frac{t_4}{t_7} \ , ~~ t_{15} = \frac{t_5}{t_7} \ , ~~ t_{16} = \frac{t_6}{t_7} \ , \\ &\begin{array}{l}\displaystyle \gamma( \tau)= \big( t_{10}\,\tau ^{11}+\left(t_{11}-t_{10}\right)\,\tau ^9+\left(t_{12}-t_{11}\right)\,\tau ^7+\left(-t_{12}-t_{13}\right)\,\tau ^5+\left(t_{13}+t_{14}\right)\,\tau ^4+\left(-t_{14}-t_{15}\right)\,\tau ^3 \\\displaystyle +\left(t_{15}+t_{16}\right)\,\tau ^2+\left(-t_{16}-1\right)\,\tau +1 \big)_{\left(\tau \leq 1 \right)} \end{array} {} \end{aligned} $$

(A.4)