Edited by: Sergei Pereverzyev, Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austria
Reviewed by: Ran Zhang, Shanghai University of Finance and Economics, China; Frank Filbir, Helmholtz Zentrum München, Germany
This article was submitted to Mathematics of Computation and Data Science, a section of the journal Frontiers in Applied Mathematics and Statistics
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The problem of hidden periodicity of a bivariate exponential sum
Let
The problem of hidden periodicities (the PHP problem) is to find vectors
It is convenient to use the following notations for the exponent vectors
The univariate problem of the parameter estimation has been considered initially by de Prony [
Recently, the problem of parameter estimation, in general, and Prony's method, in particular, have received a lot of attention, and different approaches have been proposed to obtain a solution. On the one hand, various approaches have been developed to stabilize Prony's method (see [
Motivated by Pan and Saff [
The outline of this paper is as follows. In section 2, we recall basic concepts related to bivariate polynomials and the Gröbner basis theory. In section 3, we define bivariate Pronytype polynomials and introduce the method of Pronytype polynomials. Using the new method together with an autocorrelation sequence in section 4, we present an approach that allows more stability in the presence of noise. Numerical results are provided in section 5, where we compare different versions of the method of Pronytype polynomials with the method proposed in Cuyt and WenShin [
We believe that the concept of the Pronytype polynomials can also be extended to the multivariate case. However, first one needs to study in detail properties of such multivariate polynomials, and then to analyze a structure of ideals and varieties they build, which causes certain technical challenges which we hope to overcome in future.
In this subsection based on Cox et al. [
For a pair of nonnegative integers
and for any real number α ∈ ℝ
In contrast to the onedimensional case, dealing with bivariate polynomials naturally requires some fixed order of monomials. Here we stick to the Graded Lexicographic Order. However, we would like to mention that the Graded Reverse Lexicographic Order can be used alternatively (see [
Let
For some
Herewith, there exist the inverse Cantor functions
In the following subsection we summarize some facts about the Gröbner basis theory that help later to deal with common zeros of polynomial systems. For more details we refer to Sturmfels [
We consider a bivariate polynomial
It is wellknown that the ideal generated by the leading terms of the initial set of the polynomials
Suppose
A monomial
Using just the leading terms of the polynomials from
Set
Let
which we call the
The next object we consider is some set of elements from the integer grid
Proof. (A) First of all, we prove that the parameters
Assuming that
Owing to the definition of the exponential sum
Applying multilinearity of determinants to the first row of
Repeating this process up to the penultimate row of the determinant
Among all the determinants
I.: determinants with at least two equal indices
II.: determinants where all indices
Let us consider the determinants of type I. We assume that for some
(B) Now let us show that the set of Pronytype polynomials
Let
Let us start with the number of parameters
Illustration of sets of initial monomials.
Now let for the same
In the same way, for any
In step
Using notation
Proof. From the representation (17) it follows, that as soon as all indices
□
The results from the previous subsection provide the following computational algorithm that allows detecting the parameters and frequency vectors of the
PTP algorithm for 

Input:  
Set up  
Compute  
Output 1:  
Output:  { 
The number of samples of
Proof. Let
Let us start with
(1) Let
(2) Let
(3) Let
Hereby, we identify all
(1) Let
(2) Let
(3) Let
(4) Let
Finally, if
Having analyzed all cases, we can identify all elements of
□
For example,
Sample set
As it was mentioned before, a certain disadvantage of the original Prony method is its instability in case when data are corrupted by noise. Since the Pronytype polynomials are a bivariate generalization of the onedimensional approach, the pure PTPalgorithm also inherits instability in noisy data case. Therefore, in this section we introduce the method based on the Pronytype polynomials and a windowed autocorrelation sequence that allows gaining more stability in case of noise corruption.
In this section, using the onedimensional localized kernel constructed in Filbir et al. [
First of all let us introduce the necessary notations. For a Lebesgue measurable function
Proof. Let us, first of all, consider a trigonometric polynomial of one variable
In this section we discuss the application of the method of Pronytype polynomials to some special type of an
Let
Let us consider a function
The fact that the symmetric exponential sum
Further, we introduce indicators of the exponential sum (27) similar to those used in [
Proof. Since part
Using autocorrelation in the case of noisy data helps us to stabilize the result. Theorem 4.2.1 provides the following algorithm that we call
PTPA algorithm for symmetric 

Input:  
Autocorrelation  
Set up  for 
Compute  
Output 1:  { 
Output:  { 
Proof. To investigate the number of samples of the
Sample sets of PTPA algorithm.
When applying the PTPA algorithm, the structure of the Pronytype polynomials stays the same as for any
see
Since we are shifting the rectangular window sample set
The number of vectors that lack in
and in terms of
As we have seen in the previous section, the PTPA algorithm is applicable for symmetric exponential sums. In this subsection we propose a generalization of this method to nonsymmetric exponential sums.
Let us consider the
Let us denote by
Test for coefficients  
Input:  
Form  
Compute  
Find  
Output:  { 
In order to find the exact parameters of the exponential sum
Test for parameters  
Input:  
Form  
Compute  
Find  
Output:  z_{1} = z_{1, p}, …, z_{N} = z_{N, p} 
In general, we have the algorithm to find the frequency vectors of the exponential sum
PTPAS algorithm  
Input:  
Set up  
Compute  
Output 1  { 
Compute  { 

Test for coefficients 

Test for parameters 
Output 1:  { 
Output:  { 
In this section we present some numerical results related to the stability of the suggested methods in case of noise corruption.
We have implemented the PTP, PTPA, and PTPAS algorithms in Mathematica with a working precision of 50 digits. For numerical computation we use the following numerical method, which we call an
Using the data of one hundred randomly generated collections of coefficients and the separated frequency vectors
The obtained numerical results of Experiment 1 are shown in
Results of Experiment 1.
10^{−2}  20  97  97 
10^{−3}  61  100  100 
10^{−4}  90  100  100 
10^{−5}  100  100  100 
Results of Experiment 1.
10^{−2}  2.59  8.96 × 10^{−3}  2.46  2.44 
10^{−3}  1.93  1.40 × 10^{−2}  0.285  0.404 
10^{−4}  1.31  4.53 × 10^{−3}  0.150  0.150 
10^{−5}  0.47  1.09 × 10^{−3}  8.67 × 10^{−6}  1.16 × 10^{−6} 
10^{−6}  0.22  1.77 × 10^{−4}  1.69 × 10^{−6}  1.02 × 10^{−7} 
10^{−7}  0.01  1.10 × 10^{−5}  1.97 × 10^{−7}  1.37 × 10^{−8} 
10^{−8}  7.42 × 10^{−4}  9.93 × 10^{−7}  1.45 × 10^{−8}  5.48 × 10^{−10} 
10^{−9}  9.76 × 10^{−5}  1.27 × 10^{−7}  1.03 × 10^{−9}  1.40 × 10^{−10} 
10^{−10}  1.00 × 10^{−5}  1.01 × 10^{−8}  1.62 × 10^{−10}  1.04 × 10^{−11} 
10^{−11}  1.07 × 10^{−6}  1.14 × 10^{−9}  2.52 × 10^{−11}  1.39 × 10^{−12} 
10^{−12}  1.38 × 10^{−7}  1.54 × 10^{−10}  2.48 × 10^{−12}  7.89 × 10^{−14} 
10^{−13}  9.42 × 10^{−9}  1.36 × 10^{−11}  1.12 × 10^{−13}  1.60 × 10^{−14} 
10^{−14}  9.06 × 10^{−10}  1.43 × 10^{−12}  1.93 × 10^{−14}  1.11 × 10^{−15} 
10^{−15}  7.46 × 10^{−11}  9.03 × 10^{−14}  1.68 × 10^{−15}  1.22 × 10^{−16} 
10^{−16}  1.41 × 10^{−11}  1.10 × 10^{−14}  2.13 × 10^{−16}  2.11 × 10^{−17} 
10^{−17}  1.13 × 10^{−12}  1.18 × 10^{−15}  1.37 × 10^{−17}  1.32 × 10^{−18} 
10^{−18}  9.32 × 10^{−14}  1.10 × 10^{−16}  1.94 × 10^{−18}  1.41 × 10^{−19} 
10^{−19}  1.33 × 10^{−14}  1.09 × 10^{−17}  8.40 × 10^{−20}  2.03 × 10^{−20} 
10^{−20}  1.11 × 10^{−15}  1.45 × 10^{−18}  1.23 × 10^{−20}  9.17 × 10^{−22} 
10^{−21}  9.43 × 10^{−17}  1.20 × 10^{−19}  2.84 × 10^{−21}  1.04 × 10^{−22} 
10^{−22}  1.12 × 10^{−17}  1.18 × 10^{−20}  2.86 × 10^{−22}  1.17 × 10^{−23} 
10^{−23}  8.34 × 10^{−19}  1.33 × 10^{−21}  1.32 × 10^{−23}  1.90 × 10^{−24}, 
10^{−24}  9.23 × 10^{−20}  1.19 × 10^{−22}  8.55 × 10^{−25}  1.77 × 10^{−25} 
10^{−25}  1.20 × 10^{−20}  1.22 × 10^{−23}  1.92 × 10^{−25}  2.46 × 10^{−26} 
10^{−26}  1.06 × 10^{−21}  1.23 × 10^{−24}  2.24 × 10^{−26}  8.73 × 10^{−28} 
10^{−27}  1.03 × 10^{−22}  1.23 × 10^{−25}  1.31 × 10^{−27}  6.47 × 10^{−29} 
10^{−28}  9.42 × 10^{−24}  1.04 × 10^{−26}  2.36 × 10^{−28}  1.33 × 10^{−29} 
10^{−29}  6.92 × 10^{−25}  1.21 × 10^{−27}  6.95 × 10^{−30}  1.48 × 10^{−30} 
10^{−30}  1.14 × 10^{−25}  6.06 × 10^{−29}  1.15 × 10^{−30}  1.18 × 10^{−31} 
Results of Experiment 1.
Here, we have used the data of one hundred randomly generated collections of coefficients and the well separated frequency vectors
Results of Experiment 2.
10^{−2}  23  100  100 
10^{−3}  61  100  100 
10^{−4}  90  100  100 
10^{−5}  100  100  100 
Results of Experiment 2.
10^{−2}  5.12  0.154  1.53 × 10^{−3}  3.14 × 10^{−4} 
10^{−3}  5.36  0.187  1.68 × 10^{−4}  2.39 × 10^{−5} 
10^{−4}  4.51  0.0674  249 × 10^{−5}  2.66 × 10^{−6} 
10^{−5}  3.62  2.65 × 10^{−3}  2.10 × 10^{−6}  2.47 × 10^{−7} 
10^{−6}  3.28  3.15 × 10^{−4}  2.83 × 10^{−7}  2.33 × 10^{−8} 
10^{−7}  1.95  2.50 × 10^{−5}  1.70 × 10^{−8}  2.18 × 10^{−9} 
10^{−8}  1.74  2.63 × 10^{−6}  3.22 × 10^{−9}  2.27 × 10^{−10} 
10^{−9}  1.00  2.95 × 10^{−7}  2.39 × 10^{−10}  2.43 × 10^{−11} 
10^{−10}  0.52  3.12 × 10^{−8}  2.43 × 10^{−11}  2.82 × 10^{−12} 
10^{−11}  0.29  2.81 × 10^{−9}  1.94 × 10^{−12}  2.55 × 10^{−13} 
10^{−12}  0.42  3.46 × 10^{−10}  2.64 × 10^{−13}  1.76 × 10^{−14} 
10^{−13}  0.34  2.26 × 10^{−11}  2.85 × 10^{−14}  2.42 × 10^{−15} 
10^{−14}  0.12  3.04 × 10^{−12}  2.46 × 10^{−15}  2.28 × 10^{−16} 
10^{−15}  0.061  3.11 × 10^{−13}  2.15 × 10^{−16}  2.69 × 10^{−17} 
10^{−16}  5.17 × 10^{−4}  3.17 × 10^{−14}  2.22 × 10^{−17}  2.04 × 10^{−18} 
10^{−17}  5.16 × 10^{−5}  2.56 × 10^{−15}  2.49 × 10^{−18}  2.52 × 10^{−19} 
10^{−18}  7.94 × 10^{−6}  2.77 × 10^{−16}  2.27 × 10^{−19}  2.22 × 10^{−20} 
10^{−19}  5.77 × 10^{−7}  2.69 × 10^{−17}  3.33 × 10^{−20}  3.10 × 10^{−21} 
10^{−20}  7.21 × 10^{−8}  3.07 × 10^{−18}  1.87 × 10^{−21}  2.43 × 10^{−22} 
10^{−21}  2.38 × 10^{−9}  3.30 × 10^{−19}  2.09 × 10^{−22}  2.45 × 10^{−23} 
10^{−22}  2.68 × 10^{−10}  2.96 × 10^{−20}  2.50 × 10^{−23}  2.74 × 10^{−24} 
10^{−23}  2.98 × 10^{−11}  3.45 × 10^{−21}  2.15 × 10^{−24}  2.46 × 10^{−25} 
10^{−24}  9.00 × 10^{−12}  3.10 × 10^{−22}  2.07 × 10^{−25}  2.65 × 10^{−26} 
10^{−25}  6.67 × 10^{−13}  3.32 × 10^{−23}  2.94 × 10^{−26}  2.10 × 10^{−27} 
10^{−26}  5.66 × 10^{−14}  2.92 × 10^{−24}  2.29 × 10^{−27}  2.31 × 10^{−28} 
10^{−27}  5.78 × 10^{−15}  3.07 × 10^{−25}  2.53 × 10^{−28}  2.27 × 10^{−29} 
10^{−28}  4.18 × 10^{−16}  3.08 × 10^{−26}  2.17 × 10^{−29}  2.73 × 10^{−30} 
10^{−29}  1.97 × 10^{−17}  2.98 × 10^{−27}  2.26 × 10^{−30}  2.27 × 10^{−31} 
10^{−30}  1.26 × 10^{−18}  3.49 × 10^{−28}  1.89 × 10^{−31}  2.36 × 10^{−32} 
Results of Experiment 2.
As numerical computations show, the methods of the PTP type stay more stable in the case of noise corruption. Moreover, the PTPA algorithm has a good performance, even if the level of noise is of the order 10^{−2}. Of course, in this case we need to ask for more samples (see Lemma 4.2.1) of the exponential sum.
All datasets generated and analyzed for this study are included in the article/supplementary material.
All authors listed have made a substantial, direct and intellectual contribution to the work, and approved it for publication.
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
First of all, the authors would like to thank Prof. Stefan Kunis for his valuable comments and remarks to improve this paper. The authors are most grateful to Frederic Schoppert for his numerical computations and the discussions about the number of samples required by the method of Pronytype polynomials. Furthermore, the authors are thankful to Prof. Anne FrühbisKrüger for fruitful discussions.