Feature PDF (Spectral Histogram)

For a small number of sinusiods, the total number of parameters for the SINAR model is small, so the feature PDF can be approximated by Gaussian mixtures, etc. The function software/module_sinar_test.m implements the acid using Gaussian mixtures test with method='gmm'. However, for more sinusoids, a PDF estimation method more suited for SINAR should be used. The fixed parameters for the SINAR model of AR order $P$ and $k$ sinusoids include the AR parameter estimates $\sigma^2$, ${\bf a}=[a_1,a_2 \ldots a_P]$ , the vector of $k$ frequencies ${\bf f}$, and the $2k$-dimensional vector of real and imaginary amplitudes ${\bf d}$. Let $H_m$ be an arbitrary (real data) hypothesis and assume we want to estimate $p(\sigma^2, {\bf a}, {\bf f}, {\bf d} \vert H_m)$. We make the simplifying assumption that the AR and sinusoidal parameters are independent:

$$p(\sigma^2, {\bf a}, {\bf f}, {\bf d} \vert H_m) = p(\sigma^2, {\bf a} \vert H_m) \; p({\bf f}, {\bf d} \vert H_m).$$

If we expand

$$p({\bf f}, {\bf d} \vert H_m) = p({\bf d}\vert {\bf f} , H_m) \; p({\bf f}\vert H_m),$$

we can implement $p({\bf f}\vert H_m)$ as a histogram. If we assume the individual frequencies are independent, we have

$$p({\bf f}\vert H_m) = \prod_{i=1}^k h(f_i\vert H_m),$$

where $h(f\vert H_m)$ is the frequency histogram for class $H_m$. In addition, we can assume a standard exponential distribution for the squared magnitudes,

$$p({\bf d}\vert H_m) = \prod_{i=1}^k \frac{1}{\mu(f_i)} e^{-d_i/\mu(f_i)},$$

where $\mu(f)$ can be constant or can depend on the frequency. In the second case, a means of estimating $\mu(f)$ should be used. These ideas are implemented by the function software/spect_hist.m. The function software/module_sinar_test.m implements the acid using histogram with method='hist'.