Analytical and Numerical Investigations of Probabilistic Monochromatic Problem

A probabilistic formalism, relying on Bayes’ theorem and linear Gaussian inversion, is adapted, so that a monochromatic problem can be investigated. The formalism enables an objective test in probabilistic terms of the quantities and model concepts involved in the problem at hand. With this formalism, an amplitude (linear parameter), a frequency (non-linear parameter) and a hyperparameter of the Gaussian amplitude prior are inferred jointly given simulated data sets with Gaussian noise contributions. For the amplitude, an analytical normal posterior follows which is conditional on the frequency and the hyperparameter. The remaining posterior estimates the frequency with an uncertainty of MHz, while the convolution of a standard approach would achieve an uncertainty of some GHz. This improvement in the estimation is investigated analytically and numerically, revealing for instance the positive effect of a high signal-to-noise ratio and/or a large number of data points. As a fixed choice of the hyperparameter imposes certain results on the amplitude and frequency, this parameter is estimated and, thus, tested for plausibility as well. From abstract point of view, the model posterior is investigated as well.


Introduction
Fourier transformation tools are used to obtain information about spectra for a given data set.As any data has an uncertainty, Fourier transformation techniques can be supported by probabilistic theory captured by Bayes' theorem [1] to improve scientific results and conclusions.The works [2] [3] demonstrate some advantages of the probabilistic ansatz over a conventional approach.
A common and misleading assumption in the field is that the Nyquist theorem determines the spectral band limitation.However, as the Nyquist frequency follows from finite data sampling, it can only be an upper limit but cannot be an estimator for the limit caused actually by a source and/or diagnostic throughput.A similar reasoning applies to the lower spectral limit.
After a basic formalism has been derived, it could be shown in Ref. [4] that the band limitation can be well inferred from experimental data.Thereby, a linear Gaussian inversion technique was used to infer spectral amplitudes.Furthermore, a settings posterior was introduced which estimates non-linear parameters and hyperparameters of a problem.For instance, the spectral band limits and the uncertainty on the derived quasi-continuous spectrum, originating in non-probed Fourier coefficients, have been inferred jointly.
For many applied analysis schemes, certain model assumptions and their implications are not tested but assumed to be valid with infinite trust.From scientific point of view, any analysis scheme should be tested given simulated noisy data for which all model assumptions are clear.Then, analysis results and model assumptions can be investigated which is achieved objectively by a probabilistic ansatz.If this can be carried out analytically, valuable information is available when only actual measured data are given for a scientific problem.
The problem of a monochromatic source is a good example to show the powerfulness of Bayesian inference and to test model assumptions.In this work, the formalism derived in Ref. [4] is adapted to a monochromatic problem in Section 2 and applied to simulated data with different noise levels in Section 3.
In addition, analytical and numerical investigations are carried out, so that the probabilistic findings can be understood in more detail.For example, a better signal-to-noise ratio improves the estimation of the frequency, while more data points compensate a worsened signal-to-noise ratio.The conclusion section can be found at the end.

Adapted Probabilistic Formalism
The formalism derived in Ref. [4] is adapted below to a monochromatic problem, involving one frequency parameter 1 f and one amplitude parameter 1 S .In addition, a hyperparameter is at hand, entering in the normal prior for 1 S .
Abstractly, Bayes' theorem reads then with the joint posterior on the left-hand, the likelihood ( ) by the evidence, the amplitude prior

S Pr p S σ
the prior for the hyperparameter, and the prior for the frequency ( ) The amplitude 1 S maps linearly to the data domain via the vector ( )  ( ) with the covariance matrix . The amplitude prior takes the form ( ) ( ) for vanishing prior mean and variance  )

S S S S p S f
conditional on 1 f and 1 , S Pr σ .In the above equation, the posterior variance is given by ( ) , and the posterior mean reads From the remaining terms, the so-called settings posterior is derived which can be interpreted as Ockham's razor with respect to 1 S .This settings posterior has no general analytical solution, and, thus, the normalisation constant K ′ cannot be stated further.For the remainder of the paper, ( ) Finally, the evidence can be identified as The constant

Monochromatic Problem
For a monochromatic even source, data sets with different noise levels are simulated, and the formalism derived in the previous section is applied.To explain the results found, analytical and numerical investigations are carried out.
While this problem is treated abstractly in the following, the main results will be presented for two examples with low and high noise contribution to ease the presentation.
(A) Simulated Data: Two Examples Data sets are modelled for the real-world interferometer found in Ref. [5] which achieves an optical path difference to obtain an interferogram.Formally, the interferogram ( ) are generated with the noise levels D σ at 0.05 and 1.0 (see Figure 1).

(B) Application of Formalism
The linear mapping of the amplitude parameter is given by ( ) M .For the two simulated data sets, Figure 2 shows ( )    The settings posterior ( )

| , ,
which takes very large values even on logarithmic scale and, thus, ln K p × is used in the following.For the two cases, ln K p × is shown in Figure 3(a) and .Identifying this global maximum by ( ) takes reasonable numbers and is presented for the

(C) Analytical and Numerical Investigations
To have further inside in the results, the conditional amplitude posterior and K p × are analytically investigated.In order to do so, the noise contribution in the data is neglected, meaning that i D is replaced with the noise-less interferogram ( )   4)).Formally, one finds for the inverse S Po D S Pr As long as the spatial sampling is well enough, one can use the approximation as shown by Equation (A1), to get the posterior variance ( ) This variance increases quadratically with the noise level and reduces with the number of data samples.Thus, more data points per spatial unit can compensate the noise contribution, at least partly.However, this assumes that the noise is independent of the data sampling.
The modulated sinc function in the denominator of Equation (10) depends on For the posterior mean, ( ) ( ) follows, leaving aside the noise contribution but the uncertainty on the mean is still captured by

σ
. With the approximation (A1) and the trigonometric identity (B1), one can resolve ( ) Indeed, one finds the original amplitude there is a peaking of ( ) with the FWHM determined by both trigonometric arguments.Away from f, the oscillating amplitude drops like ( ) the sinc function which can be seen in Figure 2.
2) Conditional Posterior for f 1 Now the information is available to investigate ( ) the noise, the triple product in the exponent of the exponential in The global maximum is close to 1 f f = , and, assuming a sufficiently large frequency f, one gets the approximation when the Taylor series expansions ( ) remains, and the term which is independent on and square root of the variance In fact,

, S Po
σ changes with 1 f on a GHz, but the peaking in 1 f is of the order of MHz.Hence, the factorisation ( ) ( ) ( ) and  is obtained when the ratio of the exponent to its maximum at 1,Po f reads

3) Posterior for
Pr After the factorisation, one combines Equations ( 16) and ( 18) like ( ) ( ) with the important term Journal of Applied Mathematics and Physics The exponent of P becomes 1 cos 2π sinc 2π 2 which is monotonically rising in are at hand as could be seen in Figure 3(a) and Figure 3(b).
As the exponent vanishes for supposed a large 1,Po f .Interestingly, D N reduces the noise impact drastically, so that even for elevated noise levels D S σ ≈ , the maximum is still given by S with minor corrections.
The uniform prior ( ) 1 , S Pr p σ may be finite for the domain and the maximum of P in this domain is denoted by Max P .Then, the peaking can be made more obvious by after some algebra for a large 1,Po f .With the normalisation of the above equation ( ) .

S Pr Max
In case, the prior is finite for the domain of Equation ( 23).With the above expression and Equation ( 19), the constant follows with proper unit Hz.K depends in a complicated way on the noise level, the signal, the number of data points, the spatial domain for which the data is acquired, and limits to handle large numerical values.Thereby,

4) Model Posterior
The posterior of the model with uniform priors for    This joint posterior peaks close to S and f used to simulate the data, and its width increases with the noise level.However, S and f are usually located inside the peak, and, hence, one can trust the results within the posterior uncertainties.
The marginal posterior for 1 S is obtained by the marginalisation ( ) ( )  data domain.If the measurements have a normal noise contribution (Gaussian likelihood) and the amplitude prior is chosen to be Gaussian, an analytical linear inversion technique is enabled.The amplitude posterior, being conditional on the frequency and prior information, has similiarities with the result of a conventional Fourier transformation analysis approach.However, the estimation of the frequency reveals a well localised domain which is in accordance with the measured data.The corresponding posterior for the frequency is orders of magnitude narrower than the width of the convolution function used by conventional approaches to estimate the spectral resolution.The prior information captured by one hyperparameter is tested as well, and its posterior peaks over a one order of magnitude and is very robust for significantly different noise levels on the data.
The findings for the monochromatic problem are investigated analytically.
This reveals, for instance, the influence of the signal-to-noise ratio, the amount of data points and the spatial domains covered by the measurements on the results.The presented approach can be followed to examine more complex problems which involve more than one frequency and one amplitude and other diagnostic imperfections like a variable offset.Since this offset has a linear correspondence to the data domain, it would enter in the conditional amplitude posterior.Any non-linear parameter and additional hyperparameter would be estimated by the joint settings posterior specific for a certain problem.In doing so, a profound understanding of model implications could be established in analytical terms for applications relying on Fourier transformations.This is essential for designing a diagnostic for a given problem with somewhat known signal-to-noise ratio and hardware limitations, or for comparing results of different models in an objective way when a data set is given and, for example, the number of contributing frequencies is unknown.
is approximated by an integral over the spatial domain using the trigonometric identities (B1) and (B2).The approximation reveals two modulated sinc functions with a spatial and a spectral dependence.The spectral dependence is due to the sum and difference frequencies the frequency in a non-linear manner.The data D S. Schmuck, J. Svensson DOI: 10.4236/jamp.2019.74054795 Journal of Applied Mathematics and Physics with D N entries are assumed to be acquired independently, and the measurement uncertainty of each data point follows a normal distribution with standard deviation D σ .With these assumptions, the Gaussian likelihood becomes

.
After some analytical operations, one obtaines the Gaussian amplitude posterior

(
. The peak has the full width at half maximum (FWHM) of about 8 GHz which is roughly twice as large as the classical spectral resolution stated by
resembles a sample of V , assuming a Gaussian noise component with variance 2 D σ .Furthermore, the noise is chosen to be independent of x.(a) 0.05 D σ = (low noise); (b) 1.0 D σ = (high noise).Journal of Applied Mathematics and Physics

Figure 3 (
Figure 3(b) versus the spectral parameter and for three prior values of 1 , S Pr σ .All three cross-sections look similar, but close to 50 GHz the peak is highest for values near 1 ,

D
(see Equation (

1 f
, L and SS L for which the interferogram is available.The sinc function becomes unity and vanishes for small and large frequencies, respectively.For instance, since both data examples share the same spatial domain, the sinc function is close to 0 above about 20 GHz.The amplitude prior influences larger than 1 which is obtained for a large noise level, a small value of 1 , S Pr σ and/or few data points.
numerical values and is treated in more detail in the next subsection.The remaining term can be rewritten by a quadratic exponent of a Gaussian with posterior mean 1,Po f f =

1 f
the noise but decreases with the square root of the number of data points per spatial unit, the signal level and the spatial domain covered.For the data set examples, one inserts 31.56L = mm, as shown in Figure 3(c) and Figure 3(d).

σ,
the frequency at which the maximum of the exponent occurs is identified as the posterior mean data sets, the ratios are shown in Figure 4(a) and Figure 4(b) in the vicinity of the peak.While the uncertainty of 1,Po f and, hence, the width of the Gaussian augment with the noise level (see also Figure 5(a) and Figure 5(b)) as described by Equation (16), f remains included in the 2-Figure 5(a) and Figure 5(b)).

Figure 4 . 1 D= 1 (
Figure 4. Ratio of exponent to its maximum dependent on frequency 1 f and keeping constant prior standard deviation 1 , S Pr σ the vicinity of S (see Figure5(c) and Figure5(d)) and is in agreement with expression (22).For the different noise levels, ( ) posterior weighs the constants obtained with respect to the spectral and amplitude domains.Hence, ( ) | p H D increases with the constants in the numerator and smaller domains in the denominator.When one investigates Journal of Applied Mathematics and Physics

Figure 3 (
Figure 3(a) and Figure 3(b), it is obvious that ( ) | p H D increases, when can be evaluated and is shown for both data sets in Figure7(a) and Figure 7(b).
Figure 7(c) and Figure 7(d) for both data sets.Another way to achieve the marginalisation is to draw samples for 1 f from ( ) 1 1 , , | , S Pr GMax p f σ

Figure 6 . 1 D
Figure 6.Model posterior ( ) | p H D .The stated model uses a uniform prior for the domain which is present on side only.The spatial dependence of the sinc function is given by the length L of the total domain covered.
This allows the tracking of the dependencies of the Journal of Applied Mathematics and Physics posteriors on the original amplitude S and frequency f, noise level D