Checking CHIPS Assumptions

Question 1 / Assumption 1

How do we know \(x_{\tau}\) is a Random Variable?

We start our journey with the Sun, a dynamic celestial entity. By considering the Sun as a source of data, we treat it as a Random Value Generator. The surface processes of the Sun, akin to a Stochastic Process, infuse a sense of randomness into any measurement we take of its surface.

As we preprocess solar images, the pixel values on the resulting .fits file transform into a Random Variable/RV, ranging from 0 to \(\infty\). This RV, resembling an exponential distribution (truncated), such as the Gamma distribution, implies that the parameter \(\tau\) is also an RV.

Enter the Logistic function \(y_\tau\), a Surjective function fitting onto \(\tau\) and producing values in the [0-1] range. This, in turn, makes \(y_\tau\) another RV. And behold, \(x_\tau\) is simply the normalized version of \(y_\tau\), maintaining its status as a Random Variable.

Therefore, we deduce that \(\tau\), \(y_\tau\), and \(x_\tau\) can be effectively modeled using Probability Density Functions (PDFs) of exponential distributions. This foundational understanding assures us that \(x_{\tau}\) is not just a static value but a dynamic entity with inherent randomness.

Figure 01: Example showing PDFs that can fit through three parameters discussed in previous paragraph.

Question 2 / Assumption 2

How do we know \(\mathcal{F}(I>I_{th},x_{\tau_th})\) is an Random Variable and can be described using PDF?

To show that the integration of a PDF results in a uniform distribution, we use the Cumulative Distribution Function (CDF). The CDF is the integral of the PDF.

Let \(f_x\) be the PDF of a Random Variable/RV x and \(F_X\) be the corresponding CDF,

\[\begin{equation} F_X(x) = \int_{-\infty}^x f_x dx \end{equation}\]

We understand \(F_X\) values lies between [0-1] and monotonically increasing. We define \(Z=F_X(X)\), then,

\[\begin{eqnarray} F_Z(X) &=& Porb(F_X(X)\leq x)\\ F_Z(X) &=& Prob(X\leq F^{-1}_X(x))\\ F_Z(X) &=& F_X(F^{-1}_X(x))\\ F_Z(X) &=& x \end{eqnarray}\]

The derivative of \(F_Z(X)\) with respect to \(x\) is a constant, so Z is uniformly distributed with a PDF \(\mathcal{U}(0,1)\).

Question 3 / Assumption 3

Can we compare the probabilities[\(\theta\)], i.e., \(\mathcal{F}(I>I_{th},x_{\tau_th})\)?

We can efficiently compare the probabilities, denoted by \(\theta\), derived within a single solar image. The reference intensities, labeled as \(I_s\), for all the computed \(\mathcal{F}(I>I_{th},x_{\tau_th})\) values within a solar image remain constant. This constancy allows for straightforward comparisons of the obtained \(\theta\) values against each other.

When comparing \(\theta\) values from two distinct solar images, slight variations may arise. This discrepancy stems from the fact that the reference intensities \(I_s\) differ from image to image, influenced by solar cycle variations. However, in adherence to our model assumption, where the Sun is considered the source of data and treated as a Random Value Generator, all images and datasets are generated from the Sun. Therefore, from a statistical standpoint, we should be able to compare them effectively, particularly when using the same threshold intensities \(I_{th}\), specifically for \(\tau\) parameter.