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A confusion simulation at a level of one source per beam area. The input sources are shown as vertical lines, with the solid line representing the response when observed (convolved) with a Gaussian beam. Fig. 7.17. A confusion simulation at a level of one source per beam area. The input sources are shown as vertical lines, with the solid line representing the response when observed (convolved) with a Gaussian beam. The technique developed by Scheuer is known to astronomers as 'p(D)',...

N

So that the average of the data is distributed around i,, with variance a2 N. One of the exercises is to find the distribution of the variance, knowing the mean. This method is related to the classical technique of maximum likelihood. If the prior is 'diffuse', as in the example, then the posterior probability is proportional to the likelihood term f (data 3). Maximum likelihood picks out the mode of the posterior, the value of a which maximizes the likelihood. This amounts to characterizing...

The literature

Here we point to a few works which we have found useful, binning these into five types popular, the basic text, the rigorous text, the data analysis manual, and the books of specialist interest to astronomers. (1) The classic popular books have legendary titles (Sprent 1977) and (Rowntree 1981). They are all fun. A modern version with a twist in the title is (Utts 1996), which enter tains, serves as a statistics primer, and is almost a member of the next group. (2)...

Chisquare test

Pearson's (1900) paper in which chi-square was introduced is a foundation stone of modern statistical analysis1 a comprehensive and readable review (plus bibliography) is given by Cochran (1952). Consider observational data which can be binned, and a model hypothesis which predicts the population of each bin. The chi-square statistic describes the goodness-of-fit of the data to the model. If the numbers in each of k bins are Oi, and the values from the model are Ej,, then this statistic is The...

Onesample runs test of randomness

This delightfully simple test is contingent upon forming a binary (1-0) statistic from the sample data, e.g. heads-tails, or the sign of the residuals about the mean, or a best-fit line. It is to test H0 that the sample is random that successive observations are independent. Are there too many or too few Determine m, the number of or 1's n, the number of or 0's, N n + m and r, the number of runs. Look up the level of significance from the tabled probabilities (Table A2.8) for a one- or...

Detection and surveys

Watson, you are coming along wonderfully. You have really done very well indeed. It is true that you have missed everything of importance, but you have hit upon the method. . . (Sherlock Holmes in 'A Case of Identity', Sir Arthur Conan Doyle) 'Detection' is one of the commonest words in the practising astronomers' vocabulary. It is the preliminary to much else that happens in astronomy, whether it means locating a spectral line, a faint star or a gamma-ray burst. Indeed of its wide range of...

Highpass filters

A more difficult issue is presented in removing unwanted low-frequency components from observations. This is usually known as in the trade, and it is carried out to assess the continuum in spectra, for example. There is a long tradition of doing this by eye but least squares fits of polynomials, heavy smoothing and spline fitting are common ways of proceeding. The difficulty is inevitably Those parts of the scan with signal must be removed from consideration in order to place the continuum and...

Statistics on a spherical surface

Abstractly, the distribution of objects on the celestial sphere is simply the distribution of directions of a set of unit vectors. In this respect, other three-dimensional spaces may be of interest, like the Poincare sphere with unit vectors indicating the state of polarization of radiation. This is a thriving subfield of statistics and there is an excellent handbook (Fisher, Lewis & Embleton 1987). Much of the motivation comes from geophysical topics (orientation of palaeomagnetism, for...

Practical Statistics for Astronomers

Practical Statistics for Astronomers Astronomy, like any experimental subject, needs statistical methods to interpret data reliably. This practical handbook presents the most relevant statistical and probabilistic machinery for use in observational astronomy. Classical parametric and non-parametric methods are covered, but there is a strong emphasis on Bayesian solutions and the importance of probability in experimental inference. Chapters cover basic probability, correlation analysis,...