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OCCULARS SUCCESSORS: OCCULTATION TIME EXTRACTOR (OTE) AND LIGHT CURVE STATISTICAL ANALYZER (LCSA) T. GEORGE, B. ANDERSON, H. PAVLOV

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History of Occular First released July 2007 as version 2.07 Originally designed to find simple square wave occultation signals in noisy data Upgraded as version 4.0 February Change include ability to find penumbral light curves, sub-frame timing, asymmetrical transitions

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Pros of Occular Worked with variety of input formats, Tangra, Limovie or any data in csv format Analyzed data reasonable quickly Provided output graphs and reports that determined D and R times and error bars Transition times derived could be used to estimate stellar diameters

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Cons of Occular Occular will always find a signal, even if one does not exist. User judgment was always a factor in evaluating results D and R error bars were based on a Monte Carlo simulations – multiple runs made with simulated noise equal to the noise in the original data D and R error bars were not statistically valid, more of an estimate than a scientific measurement D error bars were not independent of R error bars Discrimination between suspected real signals and false signals was based on Occular Confidence Level – a semi-statistical parameter that was based on the Monte Carlo simulations – not statistically based

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BInOccular – a Successor to Occular? Bob Anderson pursued basic research on applying Bayesian Inference (BI) statistical techniques to the analysis of occultations BI advantage, if input data is normally distributed, then output results will also be normally distributed – error bars will be statistically valid. D error bars can be independent of R error bars

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BInOccular stalls … Bob Anderson converts his computing equipment to Mac environment BI analysis proceeds well Bob decides that such a major program upgrade should be programmed by someone within IOTA and who can support the program over the years ahead. BInOccular is stalled

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Hristo Pavlov joins the team … With BInOccular stalled, Tony George surveys select IOTA members to see if we can identify someone to take over the machine code programming for Bob Anderson Hristo Pavlov was contacted, since he had previously had an interest in incorporating Occular into Tangra Hristo agrees to take over the writing of the computer code

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New Project is Born Hristo suggests splitting the project in two phases: Occultation Timing Extractor – a program to extract simple square wave occultations from data with high signal-to-noise ratios. These would be data where the occultation is relatively easily seen in the data Light Curve Signal Analyzer – a program to extract more complex occultation light curves. This can include: – Extract light curve from data where the occultation is not readily apparent. – Analysis of occultations of large diameter stars and irregular asteroid limb angles. – Automatic discrimination of negative event data from positive event data

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Key Elements of BI Analysis of Occultations (see Occular Successor Statistical Paper by Bob Anderson for details) In BI analysis, We start with a parameterized model of a light curve where xi is the time of the reading and θ1 … θn are the parameters of the light curve. For example, a star disk intersecting an asteroid disk model will have up to 7 parameters in the solution, such as star and asteroid diameter, asteroid shape, asteroid speed, track offset, magnitude drop, etc. Given an occultation observation y i (i=1 to m), we want to determine the values for θ1 … θn that best 'explains/fits' the observed data using the selected light curve model. In order to solve this problem using the equivalent approaches of Bayesian Inference (BI) we must also select a noise model. For star/asteroid occultations it is reasonable to assume that the readings are affected by noise that has Gaussian distribution.

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Key Elements of BI Analysis of Occultations (continued) The probability of a series of independent measurements is simply the product of the individual probabilities, so the conditional probability of the complete observation can be calculated as: Note: p( y i | θ1 … θn ) is the usual notation for conditional probability and is verbalized as 'the probability of yi given θ1 … θn'. Given two explicit models (theoretical light curve and noise), we can now calculate the probability of each observation point relative to the theoretical light curve as follows:

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Key Elements of BI Analysis of Occultations (continued) Because the noise in the data is Gaussian, the resulting calculations of the probability distribution of solutions of the light curve model are also Gaussian. This aspect of the BI approach gives us the additional important information about the parameter distributions that allows us to confidently compute error bars that are statistically valid. The right hand side of the above equation is simply the Gaussian probability density function. All we are saying is that the observed data points differ from the theoretical value given by our light curve model by the addition of Gaussian noise characterized by σ i (the noise at that point) and furthermore that points that lie off the expected light curve are less probable than those that lie on or near to it.

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Key Elements of BI Analysis of Occultations – Methods Used MLE – Maximum Likelihood Estimation a method of estimating the parameters of a statistical model when applied to a data set. Maximum-likelihood estimation provides estimates for the model's parameters (D and R times for example). MCMC – Markov Chain Monte Carlo a method of sampling from probability distributions that has the desired distribution as its equilibrium distribution. The state of the chain after a large number of steps is then used as a sample of the desired distribution. The quality of the sample improves as a function of the number of steps. AIC – Akaike Information Criterion The Akaike information criterion is a measure of the relative goodness of fit of a statistical model. It can be used to decide which model (square wave, penumbral, straight line) best fits the data. This method would be used in the LCSA.

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Key Elements of BI Analysis of Occultations – Sample Output

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Key Elements of BI Analysis of Occultations – Sample Output OTE

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Project Responsibilities Hristo Pavlov – program designer and code programmer Bob Anderson – Bayesian Inference methods and implementation consultant Tony George – OTE beta tester. LCSA project coordinator Advisory Panels – review OTE and LCSA and provide input and guidance

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Project Timing OTE – start immediately – will be worked on after Tangra2 is released – may be done in 6 months LCSA – start in several months – may take 6- months to a year to complete

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Advisory Panel David Dunham * Dave Herald * Steve Preston * Tony George * Kazuhisa Miyashita Mitsuru Soma Brad Timerson * John Talbot Eric Frapa * confirmed

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