Archive for the ‘Uncertainty’ Category.

#### systematic errors

Ah ha~ Once I questioned, “what is systematic error?” (see [Q] systematic error.) Thanks to L. Lyons’ work discussed in [ArXiv] Particle Physics, I found this paper, titled Systematic Errors describing the concept and statistical inference related to systematic errors in the field of particle physics. It, gladly, shares lots of similarity with high energy astrophysics. Continue reading ‘systematic errors’ »

#### Likelihood Ratio Technique

I wonder what Fisher, Neyman, and Pearson would say if they see “Technique” after “Likelihood Ratio” instead of “Test.” A presenter’s saying “Likelihood Ratio Technique” for source identification, I couldn’t resist checking it out not to offend founding fathers of the likelihood principle in statistics since “Technique” sounded derogatory to be attached with “Likelihood” to my ears. I thank, above all, the speaker who kindly gave me the reference about this likelihood ratio technique. Continue reading ‘Likelihood Ratio Technique’ »

#### Lost in Translation: Measurement Error

You would think that something like “measurement error” is a well-defined concept, and everyone knows what it means. Not so. I have so far counted at least 3 different interpretations of what it means.

Suppose you have measurements X={Xi, i=1..N} of a quantity whose true value is, say, X0. One can then compute the mean and standard deviation of the measurements, E(X) and σX. One can also infer the value of a parameter θ(X), derive the posterior probability density p(θ|X), and obtain confidence intervals on it.

So here are the different interpretations:

1. Measurement error is σX, or the spread in the measurements. Astronomers tend to use the term in this manner.
2. Measurement error is X0-E(X), or the “error made when you make the measurement”, essentially what is left over beyond mere statistical variations. This is how statisticians seem to use it, essentially the bias term. To quote David van Dyk

For us it is just English. If your measurement is different from the real value. So this is not the Poisson variability of the source for effects or ARF, RMF, etc. It would disappear if you had a perfect measuring device (e.g., telescope).

3. Measurement error is the width of p(θ|X), i.e., the measurement error of the first type propagated through the analysis. Astronomers use this too to refer to measurement error.

Who am I to say which is right? But be aware of who you may be speaking with and be sure to clarify what you mean when you use the term!

#### Borel Cantelli Lemma for the Gaussian World

Almost two year long scrutinizing some publications by astronomers gave me enough impression that astronomers live in the Gaussian world. You are likely to object this statement by saying that astronomers know and use Poisson, binomial, Pareto (power laws), Weibull, exponential, Laplace (Cauchy), Gamma, and some other distributions.[1] This is true. I witness that these distributions are referred in many publications; however, when it comes to obtaining “BEST FIT estimates for the parameters of interest” and “their ERROR (BARS)”, suddenly everything goes back to the Gaussian world.[2]

Borel Cantelli Lemma (from Planet Math): because of mathematical symbols, a link was made but any probability books have the lemma with proofs and descriptions.

1. It is a bit disappointing fact that not many mention the t distribution, even though less than 30 observations are available.[]
2. To stay off this Gaussian world, some astronomers rely on Bayesian statistics and explicitly say that it is the only escape, which is sometimes true and sometimes not – I personally weigh more that Bayesians are not always more robust than frequentist methods as opposed to astronomers’ discussion about robust methods.[]

#### [SPS] Testing Completeness

There will be a special session at the 213th AAS meeting on meaning from surveys and population studies (SPS). Until then, it might be useful to pull out some interesting and relevant papers and questions/challenges as a preliminary to the meeting. I will not list astronomical catalogs and surveys only, which are literally countless these days but will bring out some if they change the way how science is performed with a description of the catalog (the best example would be SDSS, Sloan Digital Sky Survey, to my knowledge). Continue reading ‘[SPS] Testing Completeness’ »

#### It bothers me.

The full description is given http://cxc.harvard.edu/ciao3.4/ahelp/bayes.html about “bayes” under sherpa/ciao[1]. Some sentences kept bothering me and here’s my account for the reason given outside of quotes. Continue reading ‘It bothers me.’ »

1. Note that the current sherpa is beta under ciao 4.0 not under ciao 3.4 and a description about “bayes” from the most recent sherpa is not available yet, which means this post needs updates one new release is available[]

#### Redistribution

RMF. It is a wørd to strike terror even into the hearts of the intrepid. It refers to the spread in the measured energy of an incoming photon, and even astronomers often stumble over what it is and what it contains. It essentially sets down the measurement error for registering the energy of a photon in the given instrument.

Thankfully, its usage is robustly built into analysis software such as Sherpa or XSPEC and most people don’t have to deal with the nitty gritty on a daily basis. But given the profusion of statistical software being written for astronomers, it is perhaps useful to go over what it means. Continue reading ‘Redistribution’ »

#### GSL – GNU Scientific Library

I’ve talked about IMSL on my pyIMSL post, which is a commercial scientific library. There is a GNU version of IMSL, GSL. Finding GSL is the courtesy of Jiangang, who was the author of the poster that I most liked from the 212th AAS, (see My first AAS. V. measurement error and EM and his comment.) Continue reading ‘GSL – GNU Scientific Library’ »

#### [tutorial] multispectral imaging, a case study

Without signal processing courses, the following equation should be awfully familiar to astronomers of photometry and handling data:
$$c_k=\int_\Lambda l(\lambda) r(\lambda) f_k(\lambda) \alpha(\lambda) d\lambda +n_k$$
Terms are in order, camera response (c_k), light source (l), spectral radiance by l (r), filter (f), sensitivity (α), and noise (n_k), where Λ indicates the range of the spectrum in which the camera is sensitive.
Or simplified to $$c_k=\int_\Lambda \phi_k (\lambda) r(\lambda) d\lambda +n_k$$
where φ denotes the combined illuminant and the spectral sensitivity of the k-th channel, which goes by augmented spectral sensitivity. Well, we can skip spectral radiance r, though. Unfortunately, the sensitivity α has multiple layers, not a simple closed function of λ in astronomical photometry.
Or $$c_k=\Theta r +n$$
Inverting Θ and finding a reconstruction operator such that r=inv(Θ)c_k leads spectral reconstruction although Θ is, in general, not a square matrix. Otherwise, approach from indirect reconstruction. Continue reading ‘[tutorial] multispectral imaging, a case study’ »

#### [Q] Objectivity and Frequentist Statistics

Is there an objective method to combine measurements of the same quantity obtained with different instruments?

Suppose you have a set of N1 measurements obtained with one detector, and another set of N2 measurements obtained with a second detector. And let’s say you wanted something as simple as an estimate of the mean of the quantity (say the intensity) being measured. Let us further stipulate that the measurement errors of each of the points is similar in magnitude and neither instrument displays any odd behavior. How does one combine the two datasets without appealing to subjective biases about the reliability or otherwise of the two instruments? Continue reading ‘[Q] Objectivity and Frequentist Statistics’ »

#### Why Gaussianity?

Physicists believe that the Gaussian law has been proved in mathematics while mathematicians think that it was experimentally established in physics — Henri Poincare

#### An anecdote on entrophy

My greatest concern was what to call it. I thought of calling it “information”, but the word was overly used, so I decided to call it “uncertainty”. When I discussed it with John von Neumann, he had a better idea. Von Neumann told me, “You should call it entropy, for two reasons. In the first place your uncertainty function has been used in statistical mechanics under that name, so it already has a name. In the second place, and more important, nobody knows what entropy really is, so in a debate you will always have the advantage.”

#### loess and lowess and locfit, oh my

Diab Jerius follows up on LOESS techniques with a very nice summary update and finds LOCFIT to be very useful, but there are still questions about how it deals with measurement errors and combining observations from different experiments:

#### On the history and use of some standard statistical models

What if R. A. Fisher was hired by the Royal Observatory in spite that his interest was biology and agriculture, or W. S. Gosset[1] instead of brewery? An article by E.L. Lehmann made me think this what if. If so, astronomers could have handled errors better than now. Continue reading ‘On the history and use of some standard statistical models’ »

1. Gosset’s pen name was Student, from which the name, Student-t in t-distribution or t-test was spawned.[]

#### my first AAS. VI. Normalization

One realization of mine during the meeting was related to a cultural difference; therefore, there is no relation to any presentations during the 212th AAS in this post. Please, correct me if you find wrong statements. I cannot cover all perspectives from both disciplines but I think there are two distinct fashions in practicing normalization. Continue reading ‘my first AAS. VI. Normalization’ »