The AstroStat Slog » Fisher information

A test for global maximum

hlee — Wed, 02 Jul 2008 02:10:09 +0000

If getting the first derivative (score function) and the second derivative (empirical Fisher information) of a (pseudo) likelihood function is feasible and checking regularity conditions is viable, a test for global maximum (Li and Jiang, JASA, 1999, Vol. 94, pp. 847-854) seems to be a useful reference for verifying the best fit solution.

I didn’t see any ways to confirm that best fit results from XSPEC or Sherpa are a global maximum without searching whole parameter space. My little understanding tells that many fitting algorithms do not guarantee a global maximum. By checking that the best fit solution is the global maximum and subsequently, the obtained error bar is expected to have the nominal coverage, we could save efforts of searching whole parameter space.

[ArXiv] 1st week, May 2008

hlee — Mon, 12 May 2008 02:42:54 +0000

I think I have to review spatial statistics in astronomy, focusing on tessellation (void structure), point process (expanding 2 (3) point correlation function), and marked point process (spatial distribution of hardness ratios of X-ray distant sources, different types of galaxies -not only morphological differences but other marks such as absolute magnitudes and existence of particular features). When? Someday…

In addition to Bayesian methodologies, like this week’s astro-ph, studies on characterizing empirical spatial distributions of voids and galaxies frequently appear, which I believe can be enriched further with the ideas from stochastic geometry and spatial statistics. Click for what was appeared in arXiv this week.

[astro-ph:0805.0156]R. D’Abrusco, G. Longo, N. A. Walton
Quasar candidates selection in the Virtual Observatory era
[astro-ph:0805.0201] S. Vegetti& L.V.E. Koopmans
Bayesian Strong Gravitational-Lens Modelling on Adaptive Grids: Objective Detection of Mass Substructure in Galaxies (many like to see this paper: nest sampling implemented, discusses penalty function and tessllation)
[astro-ph:0805.0238] J. A. Carter et al.
Analytic Approximations for Transit Light Curve Observables, Uncertainties, and Covariances
[astro-ph:0805.0269] S.M.Leach et al.
Component separation methods for the Planck mission
[astro-ph:0805.0276] M. Grossi et al.
The mass density field in simulated non-Gaussian scenarios
[astro-ph:0805.0790] Ceccarelli, Padilla, & Lambas
Large-scale modulation of star formation in void walls
[astro-ph:0805.0797] Ceccarelli et al.
Voids in the 2dFGRS and LCDM simulations: spatial and dynamical properties
[astro-ph:0805.0875] S. Basilakos and L. Perivolaropoulos
Testing GRBs as Standard Candles
[astro-ph:0805.0968] A. A. Stanislavsky et al.
Statistical Modeling of Solar Flare Activity from Empirical Time Series of Soft X-ray Solar Emission

[ArXiv] 2nd week, Mar. 2008

hlee — Fri, 14 Mar 2008 19:44:34 +0000

Warning! The list is long this week but diverse. Some are of CHASC’s obvious interest.

[astro-ph:0803.0997] V. Smolcic et.al.
A new method to separate star forming from AGN galaxies at intermediate redshift: The submillijansky radio population in the VLA-COSMOS survey
[astro-ph:0803.1048] T.A. Carroll and M. Kopf
Zeeman-Tomography of the Solar Photosphere — 3-Dimensional Surface Structures Retrieved from Hinode Observations
[astro-ph:0803.1066] M. Beasley et.al.
A 2dF spectroscopic study of globular clusters in NGC 5128: Probing the formation history of the nearest giant Elliptical
[astro-ph:0803.1098] Z. Lorenzo
A new luminosity function for galaxies as given by the mass-luminosity relationship
[astro-ph:0803.1199] D. Coe et.al.
LensPerfect: Gravitational Lens Massmap Reconstructions Yielding Exact Reproduction of All Multiple Images (could it be related to GREAT08 Challenge?)
[astro-ph:0803.1213] H.Y.Wang et.al.
Reconstructing the cosmic density field with the distribution of dark matter halos
[astro-ph:0803.1420] E. Lantz et.al.
Multi-imaging and Bayesian estimation for photon counting with EMCCD’s
[astro-ph:0803.1491] Wu, Rozo, & Wechsler
The Effect of Halo Assembly Bias on Self Calibration in Galaxy Cluster Surveys
[astro-ph:0803.1616] P. Mukherjee et.al.
Planck priors for dark energy surveys (some CHASCians would like to check!)
[astro-ph:0803.1738] P. Mukherjee and A. R. Liddle
Planck and reionization history: a model selection view
[astro-ph:0803.1814] J. Cardoso et.al.
Component separation with flexible models. Application to the separation of astrophysical emissions
[astro-ph:0803.1851] A. R. Marble et.al.
The Flux Auto- and Cross-Correlation of the Lyman-alpha Forest. I. Spectroscopy of QSO Pairs with Arcminute Separations and Similar Redshifts
[astro-ph:0803.1857] R. Marble et.al.
The Flux Auto- and Cross-Correlation of the Lyman-alpha Forest. II. Modelling Anisotropies with Cosmological Hydrodynamic Simulations

[ArXiv] 1st week, Nov. 2007

hlee — Fri, 02 Nov 2007 21:59:08 +0000

To be exact, the title of this posting should contain 5th week, Oct, which seems to be the week of EGRET. In addition to astro-ph papers, although they are not directly related to astrostatistics, I include a few statistics papers which may be profitable for astronomical data analysis.

[astro-ph:0710.4966]
Uncertainties of the antiproton flux from Dark Matter annihilation in comparison to the EGRET excess of diffuse gamma rays by Iris Gebauer
[astro-ph:0710.5106]
The dark connection between the Canis Major dwarf, the Monoceros ring, the gas flaring, the rotation curve and the EGRET excess of diffuse Galactic Gamma Rays by W. de Boer et.al.
[astro-ph:0710.5119]
Determination of the Dark Matter profile from the EGRET excess of diffuse Galactic gamma radiation by Markus Weber
[astro-ph:0710.5171]
Systematic Bias in Cosmic Shear: Beyond the Fisher Matrix by A.Amara and A. Refregier
[astro-ph:0710.5560]
Principal Component Analysis of the Time- and Position-Dependent Point Spread Function of the Advanced Camera for Surveys by M.J. Jee et.al.
[astro-ph:0710.5637]
A method of open cluster membership determination by G. Javakhishvili et.al.
[stat.CO:0710.5670]
An Elegant Method for Generating Multivariate Poisson Data by I. Yahav and G.Shmueli
[astro-ph:0710.5788]
Variations in Stellar Clustering with Environment: Dispersed Star Formation and the Origin of Faint Fuzzies by B. G. Elmegreen
[math.ST:0710.5749]
On the Laplace transform of some quadratic forms and the exact distribution of the sample variance from a gamma or uniform parent distribution by T.Royen
[math.ST:0710.5797]
The Distribution of Maxima of Approximately Gaussian Random Fields by Y. Nardi, D.Siegmund and B.Yakir
[astro-ph:0711.0177]
Maximum Likelihood Method for Cross Correlations with Astrophysical Sources by R.Jansson and G. R. Farrar
[stat.ME:0711.0198]
A Geometric Approach to Confidence Sets for Ratios: Fieller’s Theorem, Generalizations, and Bootstrap by U. von Luxburg and V. H. Franz

Cross-validation for model selection

hlee — Mon, 20 Aug 2007 03:35:48 +0000

One of the most frequently cited papers in model selection would be An Asymptotic Equivalence of Choice of Model by Cross-Validation and Akaike’s Criterion by M. Stone, Journal of the Royal Statistical Society. Series B (Methodological), Vol. 39, No. 1 (1977), pp. 44-47.
(Akaike’s 1974 paper, introducing Akaike Information Criterion (AIC), is the most often cited paper in the subject of model selection).

The popularity of AIC comes from its simplicity. By penalizing the log of maximum likelihood with the number of model parameters (p), one can choose the best model that describes/generates the data. Nonetheless, we know that AIC has its shortcoming: all candidate models are nested each other and come from the same parametric family. For an exponential family, the trace of multiplication of score function and Fisher information becomes equivalent to the number of parameters, where you can easily raise a question, “what happens when the trace cannot be obtained analytically?”

The general form of AIC is called TIC (Takeuchi’s information criterion, Takeuchi, 1976), where the penalized term is written as the trace of multiplication of score function and Fisher information. Still, I haven’t answered to the question above.

I personally think that a trick to avoid such dilemma is the key content of Stone (1974), using cross-validation. Stone proved that computing the log likelihood by cross-validation is equivalent to AIC, without computing the score function and Fisher information or getting an exact estimate of the number of parameters. Cross-validation enables to obtain the penalized maximum log likelihoods across models (penalizing is necessary due to estimating the parameters) so that comparison among models for selection becomes feasible while it elevates worries of getting the proper number of parameters (penalization).

Numerous tactics are available for the purpose of model selection. Although variable selection (candidate models are generally nested) is a very hot topic in statistics these days and tones of publication could be found, when it comes to applying resampling methods to model selection, there are not many works. As Stone proved, cross-validation relieves any difficulties of calculating the score function and Fisher information of a model. I was working on non-nested model selection (selecting a best model from different parametric families) with Jackknife with Prof. Babu and Prof. Rao at Penn State until last year (paper hasn’t submitted yet) based on finding that the Jackknife enables to get the unbiased maximum likelihood. Even though high cost of computation compared to cross-validation and the jackknife, the bootstrap has occasionally appeared for model selection.

I’m not sure cross-validation or the jackknife is a feasible approach to be implemented in astronomical softwares, when they compute statistics. Certainly it has advantages when it comes to calculating likelihoods, like Cash statistics.