Transcript occular`s successors - Asteroid Occultation Updates

OCCULAR’S SUCCESSORS:
OCCULTATION TIME EXTRACTOR (OTE)
AND LIGHT CURVE STATISTICAL
ANALYZER (LCSA)
T. GEORGE, B. ANDERSON, H. PAVLOV
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 2009. Change
include ability to find
‘penumbral’ light curves,
sub-frame timing,
asymmetrical transitions
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
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
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
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
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
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
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 yi (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.
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( yi | 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:
Key Elements of BI Analysis of
Occultations (continued)
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 σi (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.
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.
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.
Key Elements of BI Analysis of
Occultations – Sample Output
Key Elements of BI Analysis of
Occultations – Sample Output OTE
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
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 6months to a year to complete
Advisory Panel
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David Dunham *
Dave Herald *
Steve Preston *
Tony George *
Kazuhisa Miyashita
Mitsuru Soma
Brad Timerson *
John Talbot
Eric Frapa
* confirmed