Schema-based Program Synthesis and the AutoBayes System Part III

1. Schema-based Program Synthesis and the AutoBayes SystemPart III Johann Schumann SGT, NASA Ames

2. Extending AutoBayes • some extensions are straight-forward: add text-book formulas • additional symbolic simplification rules might be required • adding schemas requires substantial work • “hard-coded” schema as first step • applicability constraints and control • functional mechanisms to handle scalar/vector/matrix cases are available • support for documentation generation • no schema language, Prolog syntax used

3. Non-Gaussian PDF • Data characteristics are modeled using probability density functions (PDFs) • Example: Gaussians, exponential, ... • AB contains a number of built-in PDFs, which can be extended (hands-on demo) • Having multiple PDFs adds a lot of power over libraries

4. Exercise 1: • For clustering, often Gaussian distribution of data is used. • How about angles: 0 == 360 • you get 5 clusters • A different distribution (vonMises-Fisher) automatically solves this problem • In AutoBayes: just replace the “gauss” by “vonmises1” -- no programming required • multiple PDFs in one spec

5. Exercise 2: • Take the “estimate foot” example (norm.ab) • try to generate multiple solutions • pragma schema_control_arbitrary_init_values=true • enables numerical algorithm • pragma schema_control_use_generic_optimize=true • allows AB to use the generic “optimize(...)” statement

6. Exercise 3: • Take the “estimate foot” example (norm.ab) and modify it to work with different probability densities • examples: • vonmises1 • exponential • poisson • cauchy

7. Excercise 4 • generate multiple programs for a simple clustering example: mog.ab • autobayes -maxprog 20 mog.ab

8. Exercise 5 • Add the pareto distribution to AutoBayes • must modify the file • synth/distribution.pl and • interface/symbols.pl

9. AutoBayes as a Prolog Program • AutoBayes is a pretty large program • ~180 prolog files, 100,000LoC (with AutoFilter) • Heavy use of • meta-programming (call, etc.) • rewriting (using an engine implemented in Prolog) • functional programming elements for all sorts of list/vector/array handling • backtracking and backtrackable global data structures • procedural (non-logical) elements, e.g., file I/O, flags, etc. • no use of modules but naming conventions • everything SWI Prolog + few C extensions to handle backtrackable global counters and flags

10. AutoBayes Weak Points • The input parser is very inflexible (uses Prolog operators) • Very bad error messages–often just “no” • no “schema language”: AutoBayes extension only by union of Prolog/domain specialist • Only primitive control of schema selection: need for a schema-selection mechanism • not all schemas are fully documented • large code-base, which needs to be maintained

11. Summary • AutoBayes suitable for a wide range of data analysis tasks • AutoBayes generated customized algorithms • AutoBayes schema-based program synthesis + symbolic • logic + functional + procedural elements used • AutoBayes extension: easy to very hard • AutoBayes debugging: a pain, but explanations and LaTeX output very helpful • AutoBayes is NASA OpenSource: bugfixes/extensions always welcome • AutoBayes has a 160+ pages Users manual • AutoBayes useful for classroom projects to PhD projects

12. AutoBayes in Air Traffic Control • The US Airspace is very crowded and extreme growth rates are expected over the next years • Air Traffic Control (ATC) is still mostly done manually • Next Generation Air Traffic Systems (NGATS) are • highly computerized • researched/developed at NASA http://www.youtube.com/watch?v=LPFzYyNzT40 The statistical analysis of air traffic radar data (position, speed, altitude, etc. for each aircraft every 12 seconds) is important for the development, testing, and assessment of air traffic control algorithms. Of particular interest: separation assurance and trajectory prediction

13. altitude CAS-Mach Transition • most climb profiles • start with a segment of constant CAS (calibrated air speed) • followed by a segment of constant mach (speed relative to the speed of sound, depends on altitude) • transition altitude is not published

14. Transition altitude How to detect the transition? mach • aircraft data contain mach, CAS, altitude • data are very noisy • task: determine the most likely point, where • mach goes from increase to constant • CAS goes from constant to decrease • get the altitude at this point CAS alt

15. Finding the Transitions I 1 const nat N. const double sigma_sq 2 double m_level, m_rate. nat t_0. 3 data double mach(0 .. N-1). 4 mach(t) ~ N( if(t < t_0)? 4.1 m_level + m_rate*(t-t_0): 4.2 m_level, 4.3 sigma_sq). 5 max pr(mach|{t_0, m_level, m_rate}) for {t_0, m_level, m_rate}. • Declare all variables, • unknown parameters, and • data. • The mach data is Gaussian distributed: • Before transition: grows linearly • After transition: mean mach number is constant • Variance is given • Ask AutoBayes to estimate the best values for the unknowns a = 0.69428 b = 0.0072091 c = 0.44532 t_0 = 35 mach Noise! Red: actual trajectory Blue: estimated profile

16. How about the CAS transition? 1 const nat N. const double sigma_sq 2 double m_level, m_rate. nat t_0. 3 double cas_level, cas_rate. 4 data double mach(0..N-1). 5 data double cas(0..N-1). 6 mach(t) ~ N( if(t < t_0)? 6- m_level + m_rate*(t-t_0): m_level, sigma_sq). 7 cas(t) ~ N( if(t < t_0)? 7- cas_level : cas_level + cas_rate*(t-t_0), sigma_sq). 8 max pr({cas, mach}|{t_0, m_level, m_rate, cas_level, cas_rate}) for {t_0, m_level, m_rate, cas_level, cas_rate}. A small modification to the AutoBayes model allows the generation of an entirely new algorithm for finding the best transition in Mach and CAS NOW: 682LoC

17. B737 Transition Points B737 ~32,000ft altitude B737 ~26,000ft ~280kn CAS mach mach Likelihood of transition 1 week ZOA_SFO data 423 of 1645 climb scenarios

18. Different AC Types B733 A320 B737 B733 A320 B737