When having many pairs of variables that demands numerous scatter plots, one possibility is using parallel coordinates and a matrix of correlation coefficients. If Gaussian distribution is assumed, which seems to be almost all cases, particularly when parametrizing measurement errors or fitting models of physics, then error bars of these coefficients also can be reported in a matrix form. If one considers more complex relationships with multiple tiers of data sets, then one might want to check ANCOVA (ANalysis of COVAriance) to find out how statisticians structure observations and their uncertainties into a model to extract useful information.

I’m not saying those simple examples from wikipedia, wikiversity, or publicly available tutorials on ANCOVA are directly applicable to statistical modeling for astronomical data. Most likely not. Astrophysics generally handles complicated nonlinear models of physics. However, identifying dependent variables, independent variables, latent variables, covariates, response variables, predictors, to name some jargon in statistical model, and defining their relationships in a rather comprehensive way as used in ANCOVA, instead of pairing variables for scatter plots, would help to quantify relationships appropriately and to remove artificial correlations. Those spurious correlations appear frequently because of data projection. For example, datum points on a circle on the XY plane of the 3D space centered at zero, when seen horizontally, look like that they form a bar, not a circle, producing a perfect correlation.

As a matter of fact, astronomers are aware of removing these unnecessary correlations via some corrections. For example, fitting a straight line or a 2nd order polynomial for extinction correction. However, I rarely satisfy with such linear shifts of data with uncertainty because of changes in the uncertainty structure. Consider what happens when subtracting background leading negative values, a unrealistic consequence. Unless probabilistically guaranteed, linear operation requires lots of care. We do not know whether residuals y-E(Y|X=x) are perfectly normal only if μ and σs in the gaussian density function can be operated linearly (about Gaussian distribution, please see the post why Gaussianity? and the reference therein). An alternative to the subtraction is linear approximation or nonparametric model fitting as we saw through applications of principle component analysis (PCA). PCA is used for whitening and approximating nonlinear functional data (curves and images). Taking the sources of uncertainty and their hierarchical structure properly is not an easy problem both astronomically and statistically. Nevertheless, identifying properties of the observed both from physics and statistics and putting into a comprehensive and structured model could help to find out the causality^[2] and the significance of correlation, better than throwing numerous scatter plots with lines from simple regression analysis.

In order to understand why statisticians studied ANCOVA or, in general, ANOVA (ANalysis Of VAriance) in addition to the material in wiki:ANCOVA, you might want to check this page^[3] and to utilize your search engine with keywords of interest on top of ANCOVA to narrow down results.

From the linear model perspective, if a response is considered to be a function of redshift (z), then z becomes a covariate. The significance of this covariate in addition to other factors in the model, can be tested later when one fully fit/analyze the statistical model. If one wants to design a model, say rotation speed (indicator of dark matter occupation) as a function of redshift, the degrees of spirality, and the number of companions – this is a very hypothetical proposal due to my lack of knowledge in observational cosmology. I only want to point that the model fitting problem can be seen from statistical modeling like ANCOVA by identifying covariates and relationships – because the covariate z is continuous, and the degrees are fixed effect (0 to 7, 8 tuples), and the number of companions are random effect (cluster size is random), the comprehensive model could be described by ANCOVA. To my knowledge, scatter plots and simple linear regression are marginalizing all additional contributing factors and information which can be the main contributors of correlations, although it may seem Y and X are highly correlated in the scatter plot. At some points, we must marginalize over unknowns. Nonetheless, we still have some nuisance parameters and latent variables that can be factored into the model, different from ignoring them, to obtain advanced insights and knowledge from observations in many measures/dimensions.

Something, I think, can be done with a small/ergonomic chart/table via hypothesis testing, multivariate regression, model selection, variable selection, dimension reduction, projection pursuit, or names of some state of the art statistical methods, is done in astronomy with numerous scatter plots, with colors, symbols, and lines to account all possible relationships within pairs whose correlation can be artificial. I also feel that trees, electricity, or efforts can be saved from producing nice looking scatter plots. Fitting/Analyzing more comprehensive models put into a statistical fashion helps to identify independent variables or covariates causing strong correlation, to find collinear variables, and to drop redundant or uncorrelated predictors. Bayes factors or p-values can be assessed for comparing models, testing significance their variables, and computing error bars appropriately, not the way that the null hypothesis probability is interpreted.

Lastly, ANCOVA is a complete [MADS].

1. This is not an assuring absolute statement but a personal impression after reading articles of various fields in addition to astronomy. My readings of other fields tell that many rely on correlation statistics but less scatter plots by adding straight lines going through data sets for the purpose of imposing relationships within variable pairs
2. the way that chi-square fitting is done and the goodness-of-fit test is carried out is understood by the notion that X causes Y and by the practice that the objective function, the sum of (Y-E[Y|X])^2/σ^2 is minimized
3. It’s a website of Vassar college, that had a first female faculty in astronomy, Maria Mitchell. It is said that the first building constructed is the Vassar College Observatory, which is now a national historic landmark. This historical factor is the only reason of pointing this website to drag some astronomers attentions beyond statistics.

The need of source separation methods in astronomy has led various adaptations of decomposition methods available. It is not difficult to locate those applications from journals of various fields including astronomical journals. However, they are most likely soliciting one dimension reduction method of their choice over others to emphasize that their strategy works better. I rarely come up with a paper which gathered and summarized component separation methods applicable to astronomical data. In that regards, the following paper seems useful to overview methods of reducing dimensionality for astronomers.

[arxiv:0805.0269]
Component separation methods for the Planck mission
S.M.Leach et al.
Check its appendix for method description.

Various library/modules are available through software/data analysis system so that one can try various dimension reduction methods conveniently. The only concern I have is the challenge of interpretation after these computational/mathematical/statistical analysis, how to assign physics interpretation to images/spectra produced by decomposition. I think this is a big open question.

First, a simple definition of the Mahalanobis distance:
$$D^2(X_i)=(X_i-\bar{X})^T\hat{\Sigma}^{-1}(X_i-\bar{X})$$
It can be seen as an Euclidean distance after strandardizing multivariate data. In one way or the other, scatter plots and regression analyses (includes all sorts of fancy correlation studies) reflect the notion of this distance.

To my knowledge, the Mahalanobis distance is employed for exploring multivariate data when one is up to finding, diagnosing, or justifying removal of outliers, identical to astronomers’ 3 or 5 σ in univariate cases. Classical text books on multivariate data analysis or classification/clustering have detail information. One would like to check the wiki site here.

Surprisingly, despite its popularity, the terminology itself is under-represented in ADS. A single paper in ApJS, none in other major astronomical journals, was found with the name Mahalanobis in their abstracts.

ApJS, v.176, pp.276-292 (2008): The Palomar Testbed Interferometer Calibrator Catalog (van Belle et al)

Unfortunately their description and the usage of the Mahalanobis distance is quite ambiguous. See the quote:

The Mahalanobis distance (MD) is a multivariate generalization of one-dimensional Euclidean distance

They only showed MD for measuring distance among uncorrelated variables/metrics and didn’t introduce that the generalization is obtained from the covariance matrix.

Since standardization (generalization or normalization) is a pretty common practice, the lack of appearance in abstracts does not mean it’s not used in astronomy. So I did the full text search among A&A, ApJ, AJ, and MNRAS, which lead me four more publications containing the Mahalanobis distance. Quite less than I expected.

1. A&A, 330, 215 (1998): Study of an unbiased sample of B stars observed with Hipparcos: the discovery of a large amount of new slowly pulsating B stars (Waelkens et al.)
2. MNRAS, 334, 20 (2002): UBV(RI)C photometry of Hipparcos red stars (Koen et al.)
3. AJ, 99, 1108 (1990): Kinematics and composition of H II regions in spiral galaxies. II – M51, M101 and NGC 2403 (Zaritsky, Elston, and Hill)
4. A&A, 343, 496 (1999): An analysis of the incidence of the VEGA phenomenon among main-sequence and POST main-sequence stars (Plets and Vynckier )

The last two papers have the definition in the way how I know (p.1114 from Zaritsky, Elston, and Hill’s and p. 504 from Plets and Vynckier’s).

Including the usage given in these papers, the Mahalanobis distance is popularly used in exploratory data analysis (EDA): 1. measuring distance, 2. outlier removal, 3. checking normality of multivariate data. Due to its requirement of estimating the (inverse) covariance matrix, it shares tools with principal component analysis (PCA), linear discriminant analysis (LDA), and other methods requiring Wishart distributions.

By the way, Wishart distribution is also underrepresented in ADS. Only one paper appeared via abstract text search.
[2006MNRAS.372.1104P] Likelihood techniques for the combined analysis of CMB temperature and polarization power spectra (Percival, W. J.; Brown, M. L)

Lastly, I want to point out that estimating the covariance matrix and its inversion can be very challenging in various problems, which has lead people to develop numerous algorithms, strategies, and applications. These mathematical or computer scientific challenges and prescriptions are not presented in this post. Please, be aware that estimating the (inverse) covariance matrix is not simple as they are presented with real data.

• [astro-ph:0806.1140] N.Bonhomme, H.M.Courtois, R.B.Tully
      Derivation of Distances with the Tully-Fisher Relation: The Antlia Cluster
  (Tully Fisher relation is well known and one of many occasions statistics could help. On the contrary, astronomical biases as well as measurement errors hinder from the collaboration).
• [astro-ph:0806.1222] S. Dye
      Star formation histories from multi-band photometry: A new approach (Bayesian evidence)
• [astro-ph:0806.1232] M. Cara and M. Lister
      Avoiding spurious breaks in binned luminosity functions
  (I think that binning is not always necessary and overdosed, while there are alternatives.)
• [astro-ph:0806.1326] J.C. Ramirez Velez, A. Lopez Ariste and M. Semel
      Strength distribution of solar magnetic fields in photospheric quiet Sun regions (PCA was utilized)
• [astro-ph:0806.1487] M.D.Schneider et al.
      Simulations and cosmological inference: A statistical model for power spectra means and covariances
  (They used R and its package Latin hypercube samples, lhs.)
• [astro-ph:0806.1558] Ivan L. Andronov et al.
      Idling Magnetic White Dwarf in the Synchronizing Polar BY Cam. The Noah-2 Project (PCA is applied)
• [astro-ph:0806.1880] R. G. Arendt et al.
      Comparison of 3.6 – 8.0 Micron Spitzer/IRAC Galactic Center Survey Point Sources with Chandra X-Ray Point Sources in the Central 40×40 Parsecs (K-S test)

I’m glad to see this week presented a paper that I had dreamed of many years ago in addition to other interesting papers. Nowadays, I’m more and more realizing that astronomical machine learning is not simple as what we see from machine learning and statistical computation literature, which typically adopted data sets from the data repository whose characteristics are well known over the many years (for example, the famous iris data; there are toy data sets and mock catalogs, no shortage of data sets of public characteristics). As the long list of authors indicates, machine learning on astronomical massive data sets are never meant to be a little girl’s dream. With a bit of my sentiment, I offer the list of this week:

• [astro-ph:0804.4068] S. Pires et al.
  FASTLens (FAst STatistics for weak Lensing) : Fast method for Weak Lensing Statistics and map making
• [astro-ph:0804.4142] M.Kowalski et al.
  Improved Cosmological Constraints from New, Old and Combined Supernova Datasets
• [astro-ph:0804.4219] M. Bazarghan and R. Gupta
  Automated Classification of Sloan Digital Sky Survey (SDSS) Stellar Spectra using Artificial Neural Networks
• [gr-qc:0804.4144]E. L. Robinson, J. D. Romano, A. Vecchio
  Search for a stochastic gravitational-wave signal in the second round of the Mock LISA Data challenges
• [astro-ph:0804.4483]C. Lintott et al.
  Galaxy Zoo : Morphologies derived from visual inspection of galaxies from the Sloan Digital Sky Survey
• [astro-ph:0804.4692] M. J. Martinez Gonzalez et al.
  PCA detection and denoising of Zeeman signatures in stellar polarised spectra
• [astro-ph:0805.0101] J. Ireland et al.
  Multiresolution analysis of active region magnetic structure and its correlation with the Mt. Wilson classification and flaring activity

A relevant post related machine learning on galaxy morphology from the slog is found at svm and galaxy morphological classification

< Added: 3rd week May 2008>[astro-ph:0805.2612] S. P. Bamford et al.
Galaxy Zoo: the independence of morphology and colour