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Analysis of autocorrelation function of stochastic processes by F-transform of higher degree

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Abstract

The autocorrelation function of a stochastic process is one of the essential mathematical tools in the description of variability that is successfully applied in many scientific fields such as signal processing or financial time series analysis and forecasting. The aim of the paper is to provide an analysis of fuzzy transform of higher degree applied to stochastic processes with a focus on its autocorrelation function. We introduce a higher-degree fuzzy transform of a stochastic process and investigate its basic properties. Further, we introduce a higher-degree fuzzy transform of bivariate functions in the tensor product of polynomial spaces and demonstrate its approximation ability. In addition, we show that the bivariate higher-degree fuzzy transform of multiplicative separable functions is a product of univariate fuzzy transform of the respective functions, which is a desirable property in the processing of functions of higher dimensions. The obtained results are used to demonstrate an interesting identity between the autocorrelation function of the fuzzy transform of a stochastic process and the bivariate fuzzy transform of the autocorrelation function of the stochastic process. This identity provides two ways to determine the autocorrelation function of the fuzzy transform of a stochastic process, which can be useful, especially in a situation where its direct calculation becomes complex or even impossible, for example, the calculation of a sample autocorrelation function.

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Notes

  1. More precisely, in the theory of stochastic processes we distinguish two main approaches to the definition of stationarity of a stochastic process. One approach referring as the strict-sense stationarity requires that the joined distribution function of a stochastic process does not depend on time shift. Nevertheless, for many applications, the strict-sense stationarity is too restrictive, and only a weaker form, referred as the weak-sense stationarity, for stochastic processes is considered requiring that the mean function is constant and the autocorrelation function is independent of time shift (see Sect. 2.3). For more information about stationary stochastic processes, we refer to Jazwinski (2007), Papoulis and Pillai (2002), Yaglom (1962).

  2. For now, let us forget that the mean function is constant and the process is weak-sense stationary in its pure mathematical definition.

  3. Note that the F-transform was originally introduced as a fuzzy transform of zero degree, where the F-transform components are real numbers, therefore, the fuzzy partition, usually determined by a generating function, bandwidth and later also by a shift and central node (see Definition 2), was only one parameter of F-transform.

  4. We consider a “reasonably smooth” function to be a function that has derivatives up to a certain degree at each point of its domain.

  5. For a reader, who is not familiar with the basic concepts of the theory of stochastic processes, we refer to Sect. 2.3 on p. 6.

  6. See Sect. 3.2 on p. 10 for the definition.

  7. Note that recent results on the approximation of stochastic processes by F-transform and filtering out of high frequencies in them can be found in Holčapek and Nguyen (2018), Nguyen and Holčapek (2018).

  8. Note that a bivariate polynomial P of degree n with complex coefficients has the form \(P(t,s)=\sum _{0\le i+j\le n} a_{ij}t^is^j\), while a bivariate polynomial Q from the tensor product of polynomial spaces of degree n over the field of complex numbers has the form \(Q(t,s)=\sum _{0\le i,j\le n} a_{ij}t^is^j\), where \(a_{ij}\) denotes complex coefficients.

  9. See the last paragraph of Sect. 3.2.

  10. A bivariate function f(xy), \(x,y\in {\mathbb {R}}\), is said to be multiplicative separable if there are two univariate functions \(f_1(x)\) and \(f_2(y)\) such that \(f(x,y)=f_1(x)\cdot f_2(y)\) for any \(x,y\in {\mathbb {R}}\). An example of a multiplicative separable function is \(f(x,y)=2\sin (x)\cos (y)\).

  11. Note that \(\psi \) is a non-negative function that is positive on a set of positive Lebesgue measure, in our case, it is easy to see that \(E=\{\mathbf {t}\mid \psi (\mathbf {t})>0\}\) is equal to \(\prod _{i=1}^n (a_i,b_i)\) for suitable \(a_i,b_i\in {\mathbb {R}}\), \(i=1,\ldots ,n\), which is a Lebesgue measurable set. In literature (see, e.g., Cruz-Uribe et al. 2011), such functions are often referred to as a weight or weighted function.

  12. Recall that the support of a real-valued function \(\psi \) is defined as the subset of the domain containing those elements which are not mapped to zero or the topological closure of this set, if a topological structure is assumed. In this paper, we consider the closed supports, i.e., \(\mathrm {Supp}(\psi )=\mathop {\mathrm {cl}}{\{\mathbf {t}\mid \mathbf {t}\in {\mathbb {R}}^n,\, \psi (\mathbf {t})\ne 0\}}\), where \(\mathrm {cl}\) denotes the standard topological closure on \({\mathbb {R}}^n\). Then, the support of \(\psi \) is the n-orthotope \(\prod _{i=1}^n[t_{k_i}-h_i,t_{k_i}+h_i]\), where \(t_{k_i}=k_i r_i\) and \(h_i\) is the ith component of the bandwidth \(\mathbf {h}\).

  13. For more information on weighted function spaces, see Cruz-Uribe et al. (2011).

  14. Note that the Cauchy–Schwarz inequality expressed using the inner product has the form \(|\langle \psi ,\zeta \rangle |^2\le \langle \psi ,\psi \rangle \langle \zeta ,\zeta \rangle \).

  15. Obviously, we have \(|\alpha +\beta |^2=(\alpha +\beta )({\bar{\alpha }}+{\bar{\beta }})=\alpha {\bar{\alpha }}+\alpha {\bar{\beta }}+{\bar{\alpha }}\beta +\beta {\bar{\beta }}\) and \(|\alpha -\beta |^2=(\alpha -\beta )({\bar{\alpha }}-{\bar{\beta }})=\alpha {\bar{\alpha }}-\alpha {\bar{\beta }}-{\bar{\alpha }}\beta +\beta {\bar{\beta }}\ge 0\). From the second inequality, we find that \(\alpha {\bar{\alpha }}+\beta {\bar{\beta }}\ge \alpha {\bar{\beta }}+{\bar{\alpha }}\beta \), from which we obtain \(|\alpha +\beta |^2\le 2\alpha {\bar{\alpha }}+2\beta {\bar{\beta }}=2|\alpha |^2+2|\beta |^2.\)

  16. Note that \( \int _c^d\int _c^d f(t)f(s)\, \mathrm{d}t \mathrm{d}s=\int _c^d f(t)\,\mathrm{d}t\cdot \int _c^d f(s)\,\mathrm{d}s\), which implies the Riemann integrability of f(t)f(s) on \([c,d]\times [c,d]\). Moreover, it is well known that the product of two Riemann integrable functions defined on a rectangle \([a,b]\times [c,d]\) is again a Riemann integrable function (see, e.g., Theorem 6.13 in Rudin 1976). In our case, we consider the product of functions \(\varGamma _\xi (t,s)\) and \(F(t,s)=f(t)f(s)\).

  17. The first attempt to introduce the \(\hbox {F}^m\)-transform of a stochastic process can be found in Holčapek et al. (2013b), nevertheless, no proper definition has been given here.

  18. A complex-valued discrete time stochastic process \(\xi \) is a discrete function that assigns a complex-valued random variable \(\xi (t)\) for any time \(t\in {\mathbb {Z}}\). The concept of weak-sense stationarity for discrete time stochastic processes is defined in the same way as for continuous time stochastic processes, where \({\mathbb {R}}\) is replaced by \({\mathbb {Z}}\) in the definition.

  19. Note that \(\varGamma _\xi (t,s)={\mathbb {E}}[\xi (t)\xi (s)]\), \(\varGamma _\xi ^*(\tau )=\varGamma _\xi (0,\tau )\) and \(\varGamma ^*_\xi (\tau )=\varGamma _\xi ^*(-\tau )\) for any \(t,s,\tau \in {\mathbb {R}}\).

  20. For \(n,m\in {\mathbb {N}}\), we use \(\mathrm {mod}(n,m)\) to denote the remainder after dividing n by m. For example, \(\mathrm {mod}(11,4)=3\), because \(10=4\cdot 2+3\) and \(0\le 3<4\) (the remainder 3 has to be less than divisor 4).

  21. This follows from \(\int _a^b\int _c^d |f(t,s)|^2 \,\mathrm{d}t\mathrm{d}s=\int _a^b |f_1(t)|^2\,\mathrm{d}t\cdot \int _c^d |f_2(s)|^2\,\mathrm{d}s\).

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Acknowledgements

We would like to thank the editors and anonymous referees for their valuable comments that significantly helped us to improve the paper. This work was supported by the ERDF/ESF project AI-Met4AI (No. CZ.02.1.01/0.0/0.0/17_049/0008414). The additional support was also provided by the Czech Science Foundation through the Project No. 18-13951S.

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Holčapek, M., Nguyen, L. Analysis of autocorrelation function of stochastic processes by F-transform of higher degree. Soft Comput 25, 7707–7730 (2021). https://doi.org/10.1007/s00500-020-05543-x

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