• maximum Entropy
• spectral Analysis
• bayesian statistics
• stochastic process
• time series

Estimation of parameters of theoretical models is one of the main goals of statistical inference. From a Bayesian perspective, this goal can be achieved by the computation of the posterior distribution for the parameters defining the model under scrutiny.
The posterior distribution is given by the product of the prior distribution and the likelihood function. While the prior distribution is a representation of the state of knowledge of the observer, the likelihood function is entirely defined by the noise statistical properties. For timeseries, the likelihood function is thus entirely defined by its Power spectral density (PSD)-
The problem of the PSD estimation is addressed appealing to Burg's proposed Maximum Entropy Spectral Analysis (MESA) method. Starting from Jaynes' Maximum Entropy Principle, MESA allows to reconstruct the spectrum directly from the observed timeseries making no assumptions on the physical model. The only requirement is the knowledge of the autocorrelation of the data, computed from the timeseries itself.
This thesis begins with an introduction to Burg's method and his recursive algorithm, focusing on the duality with autoregressive processes. In fact, a by-product of MESA are the coefficients for the autoregressive process of order $p$ ($AR(p)$) that better represent the observed process.
Even if Burg's method is theoretically free from prior assumptions, the partial and imperfect knowledge of autocorrelation function requires a choice on the recursive order $p$ that better approximates the true power spectral density.
Since several methods have been proposed in literature, Chapter two is devoted to the study of three of them: Akaike's 'Final Prediction Error' (FPE), Parzen's 'Criterion on autoregressive transfer function (CAT)', Rao's 'Optimum Bayes Decision Rule' (OBD). We characterize them, by studying simulated Gaussian noise with a known PSD and compare the results from each of the above methods with the known PSD.
Having established what method gives a better reproduction of the PSD, in Chapter 3 Burg's method is applied to the simple timeseries of Solar spot's number. We focus on this dataset goes because the link that MESA defines with autoregressive process is explicitly used to perform forecasting and compare the predictions for the future values with the observed data. The result is remarkably accurate for time windows comparable with the autoregressive order, even if no assumptions on the physical model describing the process have been made.
In the last chapter, the problem of parameter estimation for data that are a superposition of signal and noise is addressed. In these cases, the discrimination of noise and signal is a hard task. In general, this problem is studied in a two-step fashion: the noise PSD is characterized in a temporal window where no signal is present. Such an approach requires the restrictive assumption of the noise being stationary for a long period. We relax this hypothesis proposing a new way to perform "on-source" analysis using MESA to characterize the noise PSD simultaneously with the parameters of the deterministic signal of interest.
MESA has the great advantage that the number of the hypotheses to be made is minimized. The PSD estimation is implemented in the Monte Carlo sampling as an additional step, allowing us to characterize data "on-source" and distinguish signal from noise. \\
As a proof-of-principle, this new method is applied to a toy model: a sinusoidal signal superimposed with some Gaussian noise with known PSD, and both parameters and PSD estimates are compared with the true values.
The results are encouraging: parameters are correctly estimated and the posterior on the PSD is consistent with the real PSD.