# Smoothing Parameter Selection and Alpha-Stable P-Adic Time Signals

Rachid Sabre, Walid Horrigue

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Rachid Sabre, Walid Horrigue. Smoothing Parameter Selection and Alpha-Stable P-Adic Time Signals. 11th International Conference on Signal Image Processing and Multimedia, May 2023, Vancouver, Canada. pp.1-9, 10.5121/csit.2023.130902 . hal-04158361

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# SMOOTHING PARAMETER SELECTION AND
ALPHA-STABLE P-ADIC TIME SIGNALS

Rachid Sabre¹, and Walid Horrigue²

¹Biogeosciences (UMR CNRS/uB 6282) University of Burgundy/ Institut
Agro, 26, BdDocteurPetitjean Dijon, France.

²Agroécologie, INRAE, Institut Agro, Univ. Bourgogne, Univ. Bourgogne
Franche-Comté, F-21000 Dijon, France

## ABSTRACT

The estimation of the spectral density of stable p-adic signals is already done. Such estimation is based on smoothing the periodogram by using a spectral window. The convergence rate of this estimator depends on bandwidth of spectral window (called the smoothing parameter). The aim of this work is to give a technique for selecting the optimal parameter, i.e. the parameter that achieves the estimation with the best convergence rate.

For that purpose, we were inspired by the cross-validation method of finding the optimal parameter. This method minimizes the integrated square error estimate.

## KEYWORDS

Spectral density, p-adic processes, stable random field.

## 1. INTRODUCTION

In this paper, we are interested in a family of random fields whose energy is infinite, namely the family of alpha stable random fields. These processes are considerably accurate models for many phenomena in several areas such as physics, biology, electronics and electricity, hydrology, economics, communications and radar applications... see [1]-[12]. The estimation of the spectral density of these processes is given in [13] when the time processes is continuous, and in [14]-[15] when the time processes is discrete. The paper [16] has extended this work to stable random fields with p-adic time. More specifically, considers a harmonized p-adic process:

$$X(t_1, t_2) = \int_{Q_p^2} e^{i<t_1\lambda_1 + t_2\lambda_2>} dM(\lambda_1, \lambda_2);$$

$(t_1, t_2) \in Q_p^2$ where $Q_p^2$ is the field of p-adic numbers and M is a alpha stable random measure with a control measure m. The paper [16] studied the case where the measure m is continuous with respect to the Haar measure:

$dm(x_1, x_2) = \Phi(x_1, x_2)d\mathcal{H}(x_1, x_2), \quad (x_1, x_2) \in Q_p^2$. The density function $\Phi$ is called the spectral density of the process X. The paper[16] gives a modified periodogram from the observation of the process on the ball $U_n$. Afterwards, this modified periodogram is smoothed by a spectral window to get an asymptotically unbiased consistent estimate of the spectral density. Thus, the
rate convergence of this estimator depends on the bandwidth of spectral window called the smoothing parameter.

The aim of this work is to give a method of selection for obtaining the optimal smoothing pa- rameter, which allows achieving the best convergence rate of estimator. This method is based on cross validation technique.

The choice of the p-adic numbers is motivated by widespread use in various areas.

In view of the fact that many questions in physics have been solved thanks to p-adic numbers. Moreover, the number of articles dealing with the p-adic numbers definitely proves its im- portance. There are other questions relatedto the string theory (related to the p-adic quantum do- main) and other areas where hierarchically structured fractal behaviors exist (turbulence theory, dynamical systems, statistical physics, biology, see [17]-[21]. The 2-adic number represents a special case useful for computer design see [22]. The quantum mechanical state has been mod- eled by using p-adic statistics [23]-[25] focusing on the probabilities when the number of trials is infinitely large. A new asymptotic of Bernoulli’s classical probabilities has been demonstrated by [26]. The work [27] developed p-adic probability theory for statistical information processes. The paper [28] is interested in symmetric stochastic integrals with respect to p-adic Brownian motion. The works [29]-[32] studied the properties of the trajectories of a p-adic Wiener process. They used the p-adic differentiation operator.

The spectral theory and Fourier transforms of p-adic time stationary processes were developed in [34] where the process X(t) ∈ Qp where Qp is in time in the field of p-adic numbers. It also gave an estimate of the spectral density via the construction of a periodogram inspired by real- time stationary processes. Another estimator was proposed in [35] from discrete-time observa- tions X(τk)k∈Z where (τk)k∈Z where the instant observations are taken from Qp, associated with a Poisson process.

The paper is organized as follows: Section 2 recalls the asymptotically unbiased and consistent estimate (smoothing the periodogram) and the results (propositions 2.1-2.5) shown in [34]. Sec- tion 3 shows that the estimator which tends to the spectral density in probability convergence mode (proposition 2.6), gives the cross validation criterion and shows that criterion gives the optimum smoothing parameter (theorems 3.1-3.3). Section 4 contains the concluding remarks, the potential applications and the open research problems.

2. PERIODOGRAM AND SPECTRAL DENSITY ESTIMATION

Consider a process X = {X$_{t}$$_{1}$$_{,t}$$_{2}$ / (t$_{1}$, t$_{2}$ ) ∈ Q 2 p } where Q 2 p is the field of p-adic numbers having the following integral representation

$$
X_{t_1, t_2} = \int_{Q_p^2} e^{i <t_1 \lambda_1 + t_2 \lambda_2>} dM(\lambda_1, \lambda_2)(1)
$$

∀ ( t$_{1}$, t$_{2}$ ) ∈ Q 2 p where M is a symmetric α stable SαS random measure with independent and iso- tropic increments. There exists a control measure m that is defined by:

$$
m(A \times B) = [M(A \times B), M(A \times B)]_{\alpha}^{1/\alpha}.
$$

Assume that the measure m is absolutely continuous with respect to Haar measure: dm = Φ(x$_{1}$, x$_{2}$ )dH (x$_{1}$, x$_{2}$ ) where H is Haar measure.
This density is estimated in [13] when the process has continuous real time, in [14]-[15] when the process and the random field have discrete time. The paper [16] studied the process when the time is p-adic. They give an asymptotically and consistent estimate of the density $\Phi$ called the spectral density of the process $X = \{X_{t_1,t_2}/(t_1, t_2) \in Q_p^2\}$. For that, they take the ball $U_n = \{(x, y) \in Q_{p;:}^2 | (x, y)|_p \le p^n\}$ as the observation of the process. They consider the following periodogram: For $(\lambda_1, \lambda_2) \in Q_p^2$

$$d_n(\lambda_1, \lambda_2) = A_n \mathrm{Re} \int_{U_n} e^{-i\langle t_1 \lambda_1 + t_2 \lambda_2 \rangle} p^{-n} h(t_1 p^n, t_2 p^n) X(t_1, t_2) d\mathcal{H}((t_1, t_2)) \quad (2)$$

$$H(\lambda_1, \lambda_2) = \int_{Z_p^2} h(t_1, t_2) e^{-i\langle t_1 \lambda_1 + t_2 \lambda_2 \rangle} d\mathcal{H}(t_1, t_2)$$

$$B_{\alpha} = \int_{Q_p^2} |\mathcal{H}(\lambda_1, \lambda_2)|^{\alpha} d\mathcal{H}(\lambda_1, \lambda_2) < +\infty$$

$$H_n(\lambda_1, \lambda_2) = \left(\frac{p^{2n}}{B_{\alpha}}\right)^{\frac{1}{\alpha}} H(p^{-n}\lambda_1, p^{-n}\lambda_2) = A_n H(p^{-n}\lambda_1, p^{-n}\lambda_2)$$

Therefore, $A_n = \left(\frac{p^{2n}}{B_{\alpha}}\right)^{\frac{1}{\alpha}}$.

$$\int_{Q_p^2} |H_n(\lambda_1, \lambda_2)|^\alpha d\mathcal{H}(\lambda_1, \lambda_2) =$$

$$\int_{Q_p^2} \frac{p^{2n}}{B_{\alpha}} |H(p^{-n}\lambda_1 p^{-n}\lambda_2)|^{\alpha} d\mathcal{H}(\lambda_1, \lambda_2) = \frac{p^{2n}}{B_{\alpha}} p^{-2n} \int_{Q_p^2} |H(v_1, v_2)|^{\alpha} d\mathcal{H}(v_1, v_2)$$

Since $|p^{-n}|_p = p^{+n}$, they obtain

$$\int_{Q_p^2} |H_n(\lambda_1, \lambda_2)|^{\alpha} d\mathcal{H}(\lambda_1, \lambda_2) = 1$$

The following propositions 2.1-2.5 are proved in [16]

**Proposition 2.1** Let

$$\Psi_n(\lambda_1, \lambda_2) = \Delta \int_{Q_p^2} |H_n(\lambda_1 - u_1, \lambda_2 - u_2)|^\alpha \Phi(u_1, u_2) d\mathcal{H}(u_1, u_2). \text{If } \Phi \text{ is a continuous and bounded function, then } B_\alpha(\Psi_n(\lambda_1, \lambda_2) - \Phi(\lambda_1, \lambda_2)) \text{ converges to zero as } n \text{ tends to infinity.}$$

**Proposition 2.2** Let $(\lambda_1, \lambda_2) \in Q_p^2$ the characteristic function of $d_n(\lambda_1, \lambda_2)$, $E\exp\{\mathrm{ird}_n(\lambda_1, \lambda_2)\}$, converges to $\exp\{-C_\alpha |r|^\alpha \Phi(\lambda_1, \lambda_2)\}$.

The periodogram is modified as follows:

$$I_n(\lambda_1, \lambda_2) = C_{q,\alpha} |d_n(\lambda_1, \lambda_2)|^q, \quad (3)$$

where $0<q<2$ and the normalization constant is given by $C_{(q,\alpha)} = \frac{D_q}{F_{q,\alpha} C_{\alpha}^{q/\alpha}}$ where $D_q = \int \frac{1-\cos(u)}{|u|^{1+q}} du$ and $F_{q,\alpha} = \int \frac{1-\exp(-|u|^\alpha)}{|u|^{1+q}} du$ $C_\alpha = (\alpha\pi)^{-1} \int_0^\pi |\cos(\theta)|^\alpha d\theta$,

**Propositions 2.3** Let $(\lambda_1, \lambda_2) \in Q_p^2$, then $EI_n(\lambda_1, \lambda_2) = (\Psi_n(\lambda_1, \lambda_2))^{\frac{q}{\alpha}}$ and $I_n(\lambda_1, \lambda_2)$ is an asymptotically unbiased estimator of the spectral density but not consistent
$EI_n(\lambda_1, \lambda_2) - (\Phi(\lambda_1, \lambda_2))^{q/\alpha} = o(1)$ and $VarI_n(\lambda_1, \lambda_2) - V_{\alpha,q}(\Phi(\lambda_1, \lambda_2))^2$, with $V_{\alpha,q} = \frac{c_{q,\alpha}^2}{c_{2q,\alpha}} - 1$.

In order to have an asymptotically and consistent estimate, we smooth the periodogram that was
modified using a spectral window.

$$
f_n(\lambda_1, \lambda_2) = \int_{Q_p^2} W_n(\lambda_1 - u_1, \lambda_2 - u_2) I_n(u_1, u_2) d\mathcal{H}(u_1, u_2)
$$

Where $W_n(x_1, x_2) = |M_n|_p W(x_1 M_n, x_2 M_n)$ such that

$$
M_n \to \infty ; \frac{M_n}{n} \to 0; |M_n|_p \to 0 \text{ and } \frac{|M_n|_p}{|p^n|_p} \to \infty. \quad (4)
$$

The function W is an even nonnegative function vanishing outside [-1,1]²and
∫Qp² W(v₁, v₂)dH(v₁; v₂) = 1.

**Proposition 2.4** Let $(\lambda_1, \lambda_2) \in Q_p^2$ and $Bias(f_n(\lambda_1, \lambda_2)) = E[f_n(\lambda_1, \lambda_2)] - (\Phi(\lambda_1, \lambda_2))^{p/\alpha}$, then $Bias(f_n(\lambda_1, \lambda_2)) = o(1)$. Moreover, if $\Phi$ verifies $|\Phi(x_1, x_2) - \Phi(y_1, y_2)| \le cste|(x_1 - y_1, x_2 - y_2)|_p^{-k}$, then, $Bias(f_n(\lambda_1, \lambda_2)) = O\left(\frac{1}{|M_n|_p^{\frac{kp}{\alpha}}}\right)$.

**Proposition 2.5** Let$(\lambda_1, \lambda_2)$ be in $Q_p^2$. Assume that $\Phi \in L_{Q_p^2}$.
Then $Var(f_n(\lambda_1, \lambda_2)) = O(p^{-n}M_n^{-3n})$.

From propositions 2.4 and 2.5, we show in the following proposition that $(f_n(\lambda_1, \lambda_2))^{\frac{\alpha}{q}}$ converges to $\Phi(\lambda_1, \lambda_2)$ in probability.

**Proposition 2.6** Let $\lambda_1, \lambda_2$ p-adic numbers such that $\phi(\lambda_1, \lambda_2) > 0$. Then, $(f_n(\lambda_1, \lambda_2))^{\frac{\alpha}{q}}$ converges in probability to $\phi(\lambda_1, \lambda_2)$.

**Proof** We show that $f_n(\lambda_1, \lambda_2)$ converges in mean quadratic to $\phi(\lambda_1, \lambda_2)^{\frac{q}{\alpha}}$. $E|f_n(\lambda_1, \lambda_2) - \phi(\lambda_1, \lambda_2)^{\frac{q}{\alpha}}|^2 = E f_n(\lambda_1, \lambda_2) - \phi(\lambda_1, \lambda_2)^{\frac{q}{\alpha}})^2 + Var(f_n(\lambda_1, \lambda_2)$. Then, from proposition 2.3, $E|f_n(\lambda_1, \lambda_2) - \phi(\lambda_1, \lambda_2)^{\frac{q}{\alpha}}|^2$ converges to zero. Thus, $(f_n(\lambda_1, \lambda_2))^{\frac{\alpha}{q}}$ converges to $\phi(\lambda_1, \lambda_2)$ in probability.

It is clear that the choice of $M_n$ has an important role since the convergence speeds depend on this smoothing parameter. The papers [35]-[36] gave a criterion of choice of $h$ in the one-dimensional case, they were restricted to the parametric case. The objective of this work is to give a criterion for the selection of these parameters by non-parametric methods. Let's note by $f(x_1, x_2) = (\Phi(x_1, x_2))^{\frac{q}{\alpha}}$ and $h = \frac{1}{M_n}$ the width of the two spectral windows. We are therefore looking for a criterion $CV(h)$ allowing us to select $h$ to minimize the mean integrated square error (MISE), where
$$ MISE(h) = \iint E[f_n(x_1, x_2) - f(x_1, x_2)]^2 \rho(x_1, x_2) dx_1 dx_2, \quad (5) $$

$\rho$ being a weight function that is assumed to be known and null outside of $[0,2\pi] \times [0,2\pi]$. Although $MISE(h)$ it is a good measure of the quality of $f_n$, it can not help us to choose $h$, since it depends on the unknown function $f$. We will therefore try to estimate it. For this, we adopt the method of cross validation that has been proposed in [35]. Indeed, consider the integrated square error (ISE) defined by:

$$ \begin{align*} ISE(h) &= \iint [f_n(x_1, x_2) - f(x_1, x_2)]^2 \rho(x_1, x_2) dx_1 dx_2 = A - 2C + B \\ \text{where } A &= \int_0^{2\pi} \int_0^{2\pi} f_n^2(x_1, x_2) \rho(x_1, x_2) dx_1 dx_2 \\ C &= \int_0^{2\pi} \int_0^{2\pi} f_n(x_1, x_2) f(x_1, x_2) \rho(x_1, x_2) dx_1 dx_2 \\ B &= \int_0^{2\pi} \int_0^{2\pi} f^2(x_1, x_2) \rho(x_1, x_2) dx_1 dx_2. \end{align*} $$

Since $B$ is Independent of $h$, to choose $h$ minimizing $ISE(h)$ is to choose $h$ minimizing $A - 2C$. The term $A$ is calculable since we know $f_n$, whereas, in the term $C$, we have $f$ unknown. Weproceed by the principle of “leave-out- $I$”.

### 3. CONSTRUCTION OF THE CROSS-VALIDATION ESTIMATOR

In this section, we will define the estimator and give some results in the form of a proposition or theorem, the proof of which is appended in the last section. Let $j, j' \in \{0,1, ..., n-1\}$ such that $\frac{2\pi j}{n} \in U_n$ and $\frac{2\pi j'}{n} \in U_n$. The construction of “leave-out- $I$” consists of finding an estimator $f_n^{j,j'}(\omega_j, \omega_{j'})$ that replace $f(w_j, w_{j'})$ in the expression of $C$ and such that $I_n(\omega_j, \omega_{j'})$ and $f_n^{j,j'}(\omega_j, \omega_{j'})$ are asymptotically independent. Thus, we can estimate $C$ by: $\frac{1}{n^2} \sum_{j \in A_n} \sum_{j' \in A_n} f_n^{j,j'}(w_j, w_{j'}) I_n(w_j, w_{j'}) \rho(w_j, w_{j'})$ where $w_j = \frac{2\pi j}{n}, w_{j'} = \frac{2\pi j'}{n}, \bar{n} = \lfloor \frac{n-1}{2} \rfloor$ and $A_n = \{j \in \{0,1, ..., n-1\} \text{ such that } \frac{2\pi j}{n} \in U_n\}$

$$ \begin{align*} f_n^{j,j'}(x_1, x_2) &= \int_{U_n^2} I_n^{j,j'}(u_1, u_2) W_n(x_1 - u_1, x_2 - u_2) du_1 du_2, \text{ where} \\ & \quad I_n^{j,j'}(u_1, u_2) = I_n(u_1, u_2) \text{ if } (u_1, u_2) \notin B_{j,j'} \\ & \quad I_n^{j,j'}(u_1, u_2) = \theta_1(u_1, u_2) I_n(w_{j-1}, w_{j'-1}) + \\ & \quad \theta_2(u_1, u_2) I_n(w_{j+1}, w_{j'-1}) + \\ & \quad \theta_3(u_1, u_2) I_n(w_{j-1}, w_{j'+1}) + \\ & \quad \theta_4(u_1, u_2) I_n(w_{j+1}, w_{j'+1}) \quad \text{otherwise} \end{align*} $$

$B_{j,j'} = [w_{j-1}, w_{j+1}] \times [w_{j'-1}, w_{j'+1}]$. The construction of $I_n^{j,j'}(u_1, u_2)$ where $(u_1, u_2) \in A_{j,j'}$ is done as if $I_n$ was bi-linear. In this case,

$$ \theta_1(u_1, u_2) = \alpha\beta; \theta_2(u_1, u_2) = (1-\alpha)\beta; \theta_3(u_1, u_2) = \alpha(1-\beta) \text{ and} \\ \theta_4(u_1, u_2) = (1-\alpha)(1-\beta) \text{ where } \alpha = \frac{u_1-w_{j+1}}{w_{j-1}-w_{j+1}} \text{ and } \beta = \frac{u_2-w_{j'+1}}{w_{j'-1}-w_{j'+1}}. $$
The following proposition shows that $f_n^{j,j'}$ is an estimator asymptotically unbiased of the function $f$.

**Theorem 3.1** For all $(x_1, x_2) \in Q_p^2$

$$E[f_n^{j,j'}(x_1, x_2) - f_n(x_1, x_2)] = O\left(\frac{1}{n^2}\right).$$

From this result, we establish our criterion, noted CV "cross validation", defined by:

$$CV(h) = CV_1(h) + \int_{U_n^2} f^2(u_1, u_2) \rho(u_1, u_2) du_1 du_2$$

$$\text{where } CV_1(h) = \int_{U_n^2} f_n^2(u_1, u_2) \rho(u_1, u_2) du_1 du_2 -$$

$$\frac{2}{n^2} \sum_{j \in A_n} \sum_{j' \in A_n} f_n^{j,j'}(w_j, w_{j'}) I_n(w_j, w_{j'}) \rho(w_j, w_{j'})$$

The widths of spectral windows will be chosen at the points $\hat{h}$ minimizing the criterion $CV(h)$:

$$\hat{h}_1 = \underset{h}{\operatorname{argmin}} CV(h) = \underset{h}{\operatorname{argmin}} CV_1(h) \quad (6)$$

Subsequently, to facilitate writing and to generalise without losing generality, we will take

$$\rho(u_1, u_2) = \frac{1}{2\pi} \text{ on } U_n^2 \text{ and null outside.}$$

Now, we establish results similar to those given in [37], concerning the estimation of the intensities of a punctual process. It is to show that, on average, when $n$ is large enough, the criterion $CV(h)$ is approximately equal to the integrated quadratic error $ISE(h)$ and that the variance of $CV(h)$ is asymptotically zero. This allows us to confirm that the parameters $\hat{h}$ minimizing the criterion $CV(h)$ are close to those that minimize the integral squared error (ISE) when $n$ is large enough. These results are stated in the following theorem:

**Theorem 3.2** We have

$$|E\{CV(h) - ISE(h)\}| = O\left(\frac{1}{n^2}\right).$$

$$\operatorname{var}\{CV(h)\} = O\left(\frac{1}{n^2h^2}\right)$$

Thus, since

$$E\{[CV(h) - MISE(h)]^2\} = \\ \operatorname{var}\{CV(h)\} + [E\{CV(h) - MISE(h)\}]^2 = O\left(\frac{1}{n^2h^2}\right).$$

The widths of the spectral windows $\hat{h}_1$ and $\hat{h}_2$ obtained by cross validation, defined in (6), are asymptotically optimal, i.e. the integrated square error at $\hat{h}$ converges in probability to the small integrated square error.

**Theorem 3.3** The width of the spectral windows $\hat{h}$ obtained by cross validation are asymptotically optimal:

$$\frac{ISE(\hat{h})}{ISE(\hat{\hat{h}})} \to 1 \quad \text{in probability, where}$$
$$ \hat{h} = \underset{h}{\operatorname{argmin}} CV(h) \quad \text{and} \quad (\hat{h}) = \underset{h}{\operatorname{argmin}} ISE(h). $$

To show this result, we use the similar technique used in [38].

## 4. CONCLUSION

In this work, we propose a method to find the smoothing parameter that optimizes the estimation of the spectral density for a p-adic time alpha-sand process. The approach is based on the cross-validation technique, which has already given very good results in other fields. This kind of process finds its applications to model phenomena, which have an infinite energy such as:

Dynamic images of an agricultural field taken by drones, to identify weeds. These images can have a great variance, in particular when the images are disturbed by climatic conditions. The amount of certain microorganisms in the soil varies in a significant way that can be modelled by alpha stable random field. The measurement of microorganisms can be taken with a constant error and with some jumps when encountering rocks in the ground. This work could be completed by the study of a more general case when the measurement is mixed and when the observations are taken with a constant error.

## ACKNOWLEDGEMENTS

I would like to thank the referees for their interest in this paper and their valuable comments and suggestions. I also thank TaouakoulBrahim for proof reading.

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## AUTHORS

**Rachid Sabre:** received the PhD degree in statistics from the University of Rouen, Rouen, France, in 1993 and Habilitation (HdR) from the University of Burgundy, Dijon, France, in 2003. He joined Agrosup Dijon, Dijon, France, in 1995, where he is an Associate Professor. From 1998 through 2010, he served as a member of Institut de Mathématiques de Bourgogne, France. He was a member of the Scientific Council AgroSup Dijon from 2009 to 2013. Since 2012, he has been a member of Laboratoire Electronique, Informatique, et Image (Le2i), France. He is author/co-author of numerous papers in scientific and technical journals and conference proceedings. His research interests lie in areas of statistical process and spectral analysis for signal and image processing.

**Walid Horrigue** received the PhD degree in statistics from the University of Lille, France, in 2012. He joined UMR "Agroécologie Equipe Biocom" INRA (National Institute for Agricultural Research) Dijon, France, in 2013, as a researcher where he develops research on modeling the microbiological organism in agricultural soil. He is author/co-author of some papers in scientific and technical journals and conference proceedings. His research interests lie in areas of nonparametric statistic "functional estimate with censored data"

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