New approaches to variablestarsdata processing and interpretation

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New approaches to variable stars data processing
and interpretation

• Zdenek Mikuláek
• Institute for Theoretical Physics and
  Astrophysics, Masaryk University, Brno, Czech
  Republic

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Introduction

• Development from Tsessevichs times in the field
  of variable stars research is large. It has
  arisen
• the number of VS itself - by one or two orders,
  as well as the number of their observers and
  interpreters.
• the volume and common access to high-quality VS
  observing data and computational techniques.
• the number of new efficient statistical
  techniques and methods that are available for
  everybody thanks to wide spread PCs.
• Nevertheless, the methods used for processing of
  data mostly have remained the same as those used
  in Vladimir Platonovichs era.

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• Every astrophysicist likes large quantities and
  better quality of modern observational data, new
  methods of processing are not so popular.
  Majority of them needs a good knowledge of matrix
  calculus.
• A frequent syndrome of VS observers
  Matrixphobia.
• There are exceptions few of mathematically
  erudite theoreticians love new methods and
  matrices so much that they do not use them for
  real observational data.
• Both extremes in the data processing are bad we
  should find our golden mean.
• The contemporary statistics shares inexhaustible
  quantity of methods. It is necessary to select
  several of the most versatile and diverse
  methods, master them and to learn to combine
  them.
• The method of processing must not be unique, bur
  always must be made-to-measure of the set
  problem.

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Advanced Principal Component Analysis

• The majority of VS data processing tasks are
  solved using LSM, strictly speaking linear
  regression (polynomials, harmonic polynomials).
• There are many other methods which are able to
  give us the same or better results. The example
  APCA.
• APCA a combination of LR and standard PCA
  optimal for solving a lot astrophysics problems
• realistic fitting of multicolour light curves
• the determination of the moments of extrema of
  McLC
• modeling of light multicolour curves necessary
  for improvement of ephemerides
• diagnostics of LC secular changes. Classification
  of LCs

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HD 90044 rotating magnetic CP star
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Supersylva extrema of multicolor symmetric LC
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Least square method

• the most popular method among astronomers
  minimalization of the sum of quadrates of
  deflections of y in respect of the before
  established model of observed dependence S. The
  solution of LSM the vector of free parameters
  of the model their uncertainties
• The invention of the scientist an adequate
  modeling of the reality. Consequent steps only
  the technique of solution.
• The finding of real solution is quick if one
  knows a good estimate of the real solution then
  substitution of the S in the space of free
  parameters 1 by a paraboloid
• Then conversion to linear regression solution
  of the systems of k equations with k unknown
  parameters
• Linear regression the model is the linear
  combinations of k functions favorite
  polynomial regression, hpr

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Benefits of orthogonal models

• Linear (linearized) LSM uncertainties of
  parameters.
• Is valid
• No!!!!
• What use is to assign errors of parameters???

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• How to estimate the uncertainty of the prediction?

You must know H. You can transform functions fi
so that form an orthogonal basis e.g. by
Gram-Schmidt orthogonalization procedure. Then H
will be diagonal and the meaning of parameter
uncertainty will have their awaited
sense. Orthogonal polynomials
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Orthogonal model of cubic polynomial
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True weights in LSM

• Canonical weights of VS observers
• visual 1, photographic 3, photoelectric
  10 (20)
• True weights for TW Dra (before 1942)
• faintening 1 visual I 4, vis. II 28
  PEPphotoseries 266!
• True weights should not be stated in advance! It
  should be the result of a preliminary iterative
  analysis.
• The weight is not given only by inner accuracy of
  a particular observational method, but also the
  adequacy of the model. function. If the model is
  wrong, the weights of all type of measurements
  might be nearly equal!

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Robust regression

• Practically all real (untrimmed) astrophysical
  data contain rough errors outliers. They
  devastate LSM method their results are a vagary
  of outliers number and distribution.
• Second problem Observers intending to clean
  their data of outliers occasionally erase also
  non-outliers.
• Both problems can be treated properly by a suited
  robust regression.
• We prefer RR which modifies weights of particular
  measurements by a special function of deflection
  of measured quantity from predicted values. Our
  favorite

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Conclusions

• New methods of variable stars data processing
  enable us better exploit information hidden in
  their observations. Endeavor connected with
  mastering of them will return in new subtle
  discoveries and revealing.
• Matrix calculus, true using of weights, advanced
  principal component analysis, factor analysis,
  robust regression, creation and usage of
  orthogonal models and several other processing
  techniques should appertain to compulsory outfit
  of each variable stars observer of the 21st
  century