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wiki:astronomy:observational_astronomy:snr [2024/11/04 19:10] – created Roy Proutywiki:astronomy:observational_astronomy:snr [2024/11/04 21:06] (current) – Roy Prouty
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-# Hypothesis Testing +====== Signal to Noise ======
-For the purposes of hypothesis testing in the context of the sciences, there are two buckets of hypotheses. Here, the main goal of any good scientific experiment is to discern the likelihood measurements of the experiment are drawn from a statistical model generated by either of the two hypotheses.+
  
-These two hypotheses are +===== Hypothesis Testing =====
-1. Null Hypothesis +
-2. Alternate Hypothesis+
  
-## Null Hypothesis +Hypothesis Testing For the purposes of hypothesis testing in the context of the sciences, there are two buckets of hypotheses. Here, the main goal of any good scientific experiment is to discern the likelihood measurements of the experiment are drawn from a statistical model generated by either of the two hypotheses.
-The Null Hypothesis is the supposition that the interrogated phenomenon does not occur or that the measurements resulting from the experiment are consistent with random fluctuations in the measurement apparatus.+
  
-## Alternate Hypothesis +These two hypotheses are 1. Null Hypothesis 2. Alternate Hypothesis
-The Alternate Hypothesis is the supposition that the interrogated phenomenon does occur.+
  
-# Experiments! +==== Null Hypothesis ====
-The goal of any good science experiment is to discern the likelihood that measurements generated by the experiment are generated by one hypothesis or the other. In most cases, we can never truly *know*, but we can be *damn sure*.+
  
-## Generating Statistical Models Based on Hypotheses + The Null Hypothesis is the supposition that the interrogated phenomenon does not occur or that the measurements resulting from the experiment are consistent with random fluctuations in the measurement apparatus.
-In the context of photometry and other measures derived from photometric methods (e.g., variation in light curves), the Null Hypothesis is the supposition that the observer only observed noise and not any astrophysical source. The Alternate Hypothesis is therefore the supposition that the astrophysical source was observed.+
  
-# Telescope Systemics Equation +==== Alternate Hypothesis ==== 
-For any given CCD-like detector, the light frame reported by the detector is the direct sum of a few disparate sources of counts.+ 
 + The Alternate Hypothesis is the supposition that the interrogated phenomenon does occur. 
 + 
 +==== Experiments! ==== 
 + 
 + The goal of any good science experiment is to discern the likelihood that measurements generated by the experiment are generated by one hypothesis or the other. In most cases, we can never truly *know*, but we can be *damn sure*. 
 + 
 +---- 
 + 
 +==== Generating Statistical Models Based on Hypotheses ==== 
 + 
 + In the context of photometry and other measures derived from photometric methods (e.g., variation in light curves), the Null Hypothesis is the supposition that the observer only observed noise and not any astrophysical source. The Alternate Hypothesis is therefore the supposition that the astrophysical source was observed. We must endeavor to generate statistical models of each if we are to have any hope of testing them with data. Let's discuss the provenance of the sources of detector signals and noise in the context of photometry all while remembering them as random processes. Then we'll do some arm-wavy math to justify some nice and *normal*-looking statistical models for each hypothesis. 
 + 
 +=== Telescope Systemics Equation === 
 + 
 +As discussed in [[wiki:astronomy:observational_astronomy:data_reduction_telescope|Data Reduction I:Top of Telescope]]:  For any given CCD-like detector, the light frame reported by the detector is the direct sum of a few disparate sources of counts.
  
 $$L_{ij}(\lambda, t) = \biggl(\epsilon_{ij}(\lambda)I_{ij}(\lambda) + T_{ij}\biggr)t + B_{ij}$$ $$L_{ij}(\lambda, t) = \biggl(\epsilon_{ij}(\lambda)I_{ij}(\lambda) + T_{ij}\biggr)t + B_{ij}$$
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 The Light frame ($L_{ij}(\lambda)$; measured in counts) is the sum of: The Light frame ($L_{ij}(\lambda)$; measured in counts) is the sum of:
  
-1. The counts resulting from the counts generated by photoelectrons across a heterogeous detector and optical system and over some integration time ($t$) $\epsilon_{ij}(\lambda)I_{ij}(\lambda)t$ +  - The counts resulting from the counts generated by photoelectrons across a heterogeneous detector and optical system and over some integration time ($t$) $\epsilon_{ij}(\lambda)I_{ij}(\lambda)t$ 
-2. The counts resulting from the detector and optical system even when no light is incident on the optical system. With no light, the optical system is said to be in-the-dark and so any counts generated from such a system constitute the so-called Dark counts. $D_{ij}(t) = T_{ij}t + B_{ij}$. The Dark counts is the result of thermally agitated electrons finding themselves in the read-out circuitry as well as the relatively constant base-level of electronic noise owing to a variety of sources.+  - The counts resulting from the detector and optical system even when no light is incident on the optical system. With no light, the optical system is said to be in-the-dark and so any counts generated from such a system constitute the so-called Dark counts. $D_{ij}(t) = T_{ij}t + B_{ij}$. The Dark counts is the result of thermally agitated electrons finding themselves in the read-out circuitry as well as the relatively constant base-level of electronic noise owing to a variety of sources. 
 + 
 +What's worse is that there is no guarantee that each pixel must abide the same random distribution as its neighbors. There may be pixel-level or optical system-level defects ensuring this. So each pixel in each frame is a sampling of some unknown probability distribution. 
 + 
 +=== Stacks of Signals === 
 + 
 +The measurement of counts in the pixels of each light frame are clearly owing to a few separate phenomenon occurring in unison at the time of observation. The generation of photoelectrons from incident light, the liberation of electrons owing to their temperature, and the liberation of electrons due to electronic effects are all inherently random, [[wiki:maths:poisson|Possionian]] processes.\\ \\ 
 + 
 +These processes can be isolated and sampled repeatedly, allowing us to build-out a rough distribution of measured counts. For each the Light and the Dark, we can take (separately) an ensemble average and recover what is called the "signal" from each of these sources of counts.\\ \\ 
 + 
 +Therefore, a sequence of $N$ Light frames,  $\{L_{ij}^0(\lambda), ~L_{ij}^1(\lambda), ~L_{ij}^2(\lambda), \cdots , ~L_{ij}^N(\lambda)\}$ can be averaged to yield the average $\langle L_{ij}^0(\lambda)\rangle$.\\ \\ 
 + 
 +Being more clear, consider a the small array of pixels constituting a single frame read-out from a detector with pixels labeled $i$ & $j$. {{ :wiki:astronomy:observational_astronomy:pixelarray.png?400 }}\\ \\ 
 + 
 +We can imagine rotating this array onto its side ... {{ :wiki:astronomy:observational_astronomy:pixelarrayonside.png?400 }}\\ \\ 
 + 
 +And finally, considering that each pixel of each frame captured is a sampling of a probability distribution, we should sample that distribution $N$ times to generate a stack of frames. {{ :wiki:astronomy:observational_astronomy:pixelarraystack.png?400 }}\\ \\ 
 + 
 +If each pixel value (the counts) from the $i,j$-th pixel is a random sampling of a [[wiki:maths:poisson|Poisson Distribution]], then $L^n_{ij}\thicksim\mathcal{Pois(\lambda)} \rightarrow \mathcal{Norm(\lambda, \sqrt{\lambda})}$ if $\lambda$ is large enough (and it is for these purposes!). \\ \\ 
 + 
 +So we can treat each "stack" of pixels as its own ensemble of randomly sampled events (identically and independently distributed random events). Each stack of pixels should therefore constitute its own $\mathcal{Pois(\lambda_{ij})}\rightarrow \mathcal{Norm(\lambda_{ij}, \sqrt{\lambda_{ij}})}$. **Perhaps this is the best demonstration yet of why we need to take multiple frames of the same type.** 
 + 
 +---- 
 + 
 +==== Noise ==== 
 + 
 +For a Normal Distribution, we can call the $\sigma$ (i.e., the uncertainty or the standard of deviation) the noise in the signal ($\mu$--the average value measured in the stack). So then, any light frame pixel can be said to have an average value with some signal ($\mu$) along with some noise ($\sigma$). If there is high confidence in the measured signal (meaning low uncertainty), then $\mu \gg \sigma$. 
 + 
 +Further, for the sources of unwanted signal, they each come with their own signal and noise -- trusted to be Normally Distributed. So in the construction of the calibrated frame, only the *signal* from unwanted signal sources can be removed. The uncertainty associated with these signals cannot be separated -- they were measured at the same time! All one can hope to do here is either (a) not collect unwanted signal & noise in the first place or (b) acknowledge the uncertainty in the final result. 
 + 
 +{{ :wiki:astronomy:observational_astronomy:lightframe.png?600 |}} 
 + 
 +{{ :wiki:astronomy:observational_astronomy:calibratedframe.png?600 |}} 
 +In the above, despite the average Dark Signal being removed, the uncertainty associated with the collection of the Dark cannot be disentangled from the uncertainty in the source signal -- so it remains. The distribution of dark signal is therefore centered on $0$, but with some spread about $0$. 
 + 
 +==== Signal to Noise Ratio ==== 
 +In returning to the ultimate goal of claiming one hypothesis true over another ... The Null Hypothesis is that the signal is indistinguishable from the noise. The Alternate Hypothesis is that the signal is distinguishable from the noise. We can construct the statistical model of the Alternate Hypothesis with $\mathcal{Norm}(\mu, \sqrt{\mu^2 + \mu_D^2 + \mu_U^2 + ... })$. And the Null Hypothesis with $\mathcal{Norm}(0, \sqrt{\mu_D^2 + \mu_U^2 + ... })$ -- where the uncertainties are just the sum of the uncertainties in quadrature. \\ \\ 
 + 
 +So the figure of merit here is the Signal-to-Noise Ratio (SNR). In short, this quantity is the number of standard deviations the average calibrated signal is from the uncertainty in that average signal *collected*.\\ \\ 
 + 
 +**At the UMBC Observatory, we aim for an SNR > 10 to confidently reject the Null Hypothesis.**