[arXiv:stat.ME:0910.2585]
Variable Selection and Updating In ModelBased Discriminant Analysis for High Dimensional Data with Food Authenticity Applications
by Murphy, Dean, and Raftery
Classifying or clustering (or semi supervised learning) spectra is a very challenging problem from collecting statisticalanalysisready data to reducing the dimensionality without sacrificing complex information in each spectrum. Not only how to estimate spiky (not differentiable) curves via statistically well defined procedures of estimating equations but also how to transform data that match the regularity conditions in statistics is challenging.
Another reason that astrophysics spectroscopic data classification and clustering is more difficult is that observed lines, and their intensities and FWHMs on top of continuum are related to atomic database and latent variables/hyper parameters (distance, rotation, absorption, column density, temperature, metalicity, types, system properties, etc). Frequently it becomes very challenging mixture problem to separate lines and to separate lines from continuum (boundary and identifiability issues). These complexity only appears in astronomy spectroscopic data because we only get indirect or uncontrolled data ruled by physics, as opposed to the the meat species spectra in the paper. These spectroscopic data outside astronomy are rather smooth, observed in controlled wavelength range, and no worries for correcting recession/radial velocity/red shift/extinction/lensing/etc.
Although the most relevant part to astronomers, i.e. spectroscopic data processing is not discussed in this paper, the most important part, statistical learning application to complex curves, spectral data, is well described. Some astronomers with appropriate data would like to try the variable selection strategy and to check out the classification methods in statistics. If it works out, it might save space for storing spectral data and time to collect high resolution spectra. Please, keep in mind that it is not necessary to use the same variable selection strategy. Astronomers can create better working versions for classification and clustering purpose, like Hardness Ratios, often used to reduce the dimensionality of spectral data since low total count spectra are not informative in the full energy (wavelength) range. Curse of dimensionality!.
]]>http://adsabs.harvard.edu/abs/2009MNRAS.395.1733W.
Title:Compressed sensing imaging techniques for radio interferometry
Authors: Wiaux, Y. et al.
Abstract: Radio interferometry probes astrophysical signals through incomplete and noisy Fourier measurements. The theory of compressed sensing demonstrates that such measurements may actually suffice for accurate reconstruction of sparse or compressible signals. We propose new generic imaging techniques based on convex optimization for global minimization problems defined in this context. The versatility of the framework notably allows introduction of specific prior information on the signals, which offers the possibility of significant improvements of reconstruction relative to the standard local matching pursuit algorithm CLEAN used in radio astronomy. We illustrate the potential of the approach by studying reconstruction performances on simulations of two different kinds of signals observed with very generic interferometric configurations. The first kind is an intensity field of compact astrophysical objects. The second kind is the imprint of cosmic strings in the temperature field of the cosmic microwave background radiation, of particular interest for cosmology.
As discussed, reconstructing images from noisy observations is typically considered as an illposed problem or an inverse problem. Owing to the personal lack of comprehension in image reconstruction of radio interferometry observation based on sample from Fourier space via inverse Fourier transform, I cannot judge how good this new adaption of compressed sensing for radio astronomical imagery is. I think, however, compressed sensing will take over many of traditional image reconstruction tools due to their shortage in forgiving sparsely represented large data/images .
Please, check my old post on compressed sensing for more references to the subject like the Rice university repository in addition to references from Wiaux et al. It’s a new exciting field with countless applications, already enjoying wide popularity from many scientific and engineering fields. My thought is that well developed compressed sensing algorithms might resolve bandwidth issues in satellite observations/communication by transmiting more images within fractional temporal intervals for improved image reconstruction.
]]>Having 13 Xray line and/or continuum ratios, a typical data display would be the 13 choose 2 combination of scatter plots as follows. Note that the upper left panels with three colors are drawn for the classification purpose (red: AL Lac, blue: IM Peg, green:Capella) while lower right ones are discolored for the clustering analysis purpose. These scatter plots are essential to exploratory data analysis but they do not convey information efficiently with these many scatter plots. In astronomical journals, thanks to astronomers’ a priori knowledge, a fewer pairs of important variables are selected and displayed to reduce the visualization complexity dramatically. Unfortunately, I cannot select physically important variables only.

I am not a wellknowledged astronomer but believe in reducing dimensionality by the research objective. The goal is set from asking questions like “what do you want from this multivariate data set?” classification (classification rule/regression model that separates three stars, Capella, AL Lac, and IM Peg), clustering (are three stars naturally clustered into three groups? Or are there different number of clusters even if they are not well visible from above scatter plots?), hypothesis testing (are they same type of stars or different?), point estimation and its confidence interval (means and their error bars), and variable selection (or dimension reduction). So far no statistical question is well defined (it can be good thing for new discoveries). Prior to any confirmatory data analysis, we’d better find a way to display this multidimensional data efficiently. I thought parallel coordinates serve the purpose well but surprisingly, it was never discussed in astronomical literature, at least it didn’t appear in ADS.
Each 13 variable was either normalized (left) or standardized (right). The parallel coordinate plot looks both simpler and more informative. Capella observations occupy relatively separable space than the other stars. It is easy to distinguish that one Capella observation is an obvious outlier to the rest which is hardly seen from scatter plots. It is clear that discriminant analysis or classical support vector machine type classification methods cannot separate AL Lac and IM Pec. Clustering based on distance measures of dissimilarity also cannot be applied in order to see a natural grouping of these two stars whereas Capella observations form its own cluster. To my opinion, parallel coordinates provide more information about multidimensional data (dim>3) in a simpler way than scatter plots of multivariate data. It naturally shows highly correlated variables within the same star observations or across all target stars. This insight from visualization is a key to devising methods of variable selection or reducing dimensionality in the data set.
Personal opinion is that not having an efficient and informative visualization tool for visualizing complex high resolution spectra in many detailed metrics, smoothed bivariate (trivariate at most) information such as hardness ratios and quantiles are utilized in displaying Xray spectral data, instead. I’m not saying that the parallel coordinates are the ultimate answer to visualizing multivariate data but I’d like to emphasize that this method is more informative, intuitive and simple to understand the structure of relatively high dimensional data cloud.
Parallel coordinates has a long history. The earliest discussion I found was made in 1880ies. It became popular by Alfred Inselberg and gained recognition among statisticians by George Wegman (1990, Hyperdimensional Data Analysis Using Parallel Coordinates). Colorful images of the Sun, stars, galaxies, and their corona, interstellar gas, and jets are the eye catchers. I hope that data visualization tools gain equal spot lights since they summarize data and deliver lots of information. If images are well decorated cakes, then these tools from EDA are sophisticated and well baked cookies.
—————— [Added]
According to
[arxiv:0906.3979] The Golden Age of Statistical Graphics
Michael Friendly (2008)
Statistical Science, Vol. 23, No. 4, pp. 502535
it is 1885. Not knowing French – if I knew I’d like to read Gauss’ paper immediately prior to anything – I don’t know what the reference is about.
]]>The Drake eq. is used as an indicator that the chance of finding an organic society equivalent to the human society. As you guess, such chance is extremely low. What would be a chance that two obituaries of eminent statisticians who influenced many can appear in the same bulletin. Personal thought led that the obituary section of the bulletin is further extreme than the Drake equation.
If you are an astronomer who are interested in spectral analysis and looked for statistical or data analysis literature, you cannot miss I.J.Good’s bump hunting paper.
Density Estimation and BumpHunting by the Penalized Likelihood Method Exemplified by Scattering and Meteorite Data
by I.J.Good and R.A. Gaskins in JASA, Vol.75, No. 369, pp. 4256
The penalized likelihood approach for density estimation and bump hunting and its Bayesian interpretation has popularized statistical application to spectrum type natural science data.
Not by the popularity but by my personal interest in computational geometry and its statistical expansion, Worlsey’s publications became my reading list. Computational geometry pertains the goodness of nonparametric statistics for multivariate data which are not well explored compared to nonparametric methods for univariate data. His introductory paper about computational geometry like
Keith Worsley (1996)
The Geometry of Random images (zipped postscript), Chance, 9(1), pp.2740
can be informative and useful to some astronomers.
Speaking of the Drake equation, it was the first thing that gave me a notion of probability, it describes how one would simply formulate and compute the chance of finding life beyond the earth. The equation is a process of constructing a likelihood function. In fact, I didn’t think this equation to be a likelihood function at that time but its unique creativity carved my memory. The way this equation describes how to compute the chance of the existence of extraterrestrial intelligence is a good example of chain rule in modifying likelihood functions.
I have never met those scholars face to face but through their writings, their works shaped my way of thinking. This personal experience made me hard to believe obituaries of two respectful statisticians. It was like getting estimates of the chance of meeting ETs which I found very small when I played with the equation. Although their chances are extremely low, things can happen. Finding life outside of the earth and finding a sad news of two eminent scientists’ death are alike.
]]>After meeting Prof. Herman Chernoff unexpectedly – I didn’t know he is Professor Emeritus at Harvard – the urge revived but I didn’t have data, still then. Alas, another absent mindedness: I don’t understand why I didn’t realize that I already have the data, XAtlas for trying Chernoff faces until today. Data and its full description is found from the XAtlas website (click). For Chernoff face, references suggested in Wiki:Chernoff face are good. I believe some folks are already familiar with Chernoff faces from a New York Times article last year, listed in Wiki (or a subset characterized by baseball lovers?).
Capella is a Xray bright star observed multiple times for Chandra calibration. I listed 16 ObsIDs in the figures below at each face, among 18+ Capella observations (Last time when I checked Chandra Data Archive, 18 Capella observations were available). These 16 are high resolution observations from which various metrics like interesting line ratios and line to continuum ratios can be extracted. I was told that optically it’s hard to find any evidence that Capella experienced catastrophic changes during the Chandra mission (about 10 years in orbit) but the story in Xray can’t be very different. In a dismally short time period (10 years for a star is a flash or less), Capella could have revealed short time scale high energy activities via Chandra. I just wanted to illustrate that Chernoff faces could help visualizing such changes or any peculiarities through interpretation friendly facial expressions (Studies have confirmed babies’ ability in facial expression recognitions). So, what do you think? Do faces look similar/different to you? Can you offer me astronomical reasons for why a certain face (ObsID) is different from the rest?
p.s. In order to draw these Chernoff faces, check descriptions of these R functions, faces() (yields the left figure) or faces2() (yields the right figure) by clicking on the function of your interest. There are other variations and other data analysis systems offer different fashioned tools for drawing Chernoff faces to explore multivariate data. Welcome any requests for plots in pdf. These jpeg files look too coarse on my screen.
p.p.s. Variables used for these faces are line ratios and line to continuum ratios, and the order of these input variables could change countenance but impressions from faces will not change (a face with distinctive shapes will look different than other faces even after the order of metrics/variables is scrambled or using different Chernoff face illustration tools). Mapping, say from an astronomical metric to the length of lips was not studied in this post.
p.p.p.s. Some data points are statistical outliers, not sure about how to explain strange numbers (unrealistic values for line ratios). I hope astronomers can help me to interpret those peculiar numbers in line/continuum ratios. My role is to show that statistics can motivate astronomers for new discoveries and to offer different graphics tools for enhancing visualization. I hope these faces motivate some astronomers to look into Capella in XAtlas (and beyond) in details with different spectacles, and find out the reasons for different facial expressions in Capella Xray observations. Particularly, ObsID 1199 is most questionable to me.
]]>When I was learning statistics, I never confronted such huge degrees of freedom. Well, given the facts that only a small amount of time is used for learning the chisquare goodnessoffit test, that the chisquare distribution is a subset of gamma distribution, and that statisticians do not handle a hundred of thousands (there are more low count spectra but I’ll discuss why I chose this big number later) of photons from Xray telescopes, almost surely no statistician would confront such huge degrees of freedom.
Degrees of freedom in spectral fitting are combined results of binning (or grouping into n classes) and the number of free parameters (p), i.e. np1. Those parameters of interest, targets to be optimized or to be sought for solutions are from physical source models, which are determined by law of physics. Nothing to be discussed from the statistical point of view about these source models except the model selection and assessment side, which seems to be almost unexplored area. On the other hand, I’d like to know more about binning and subsequent degrees of freedom.
A few binning schemes in spectral analysis that I often see are each bin having more than 25 counts (the same notion of 30 in statistics for CLT or the last number in a ttable) or counts in each bin satisfying a certain signal to noise ratio S/N level. For the latter, it is equivalent that sqrt(expected counts) is larger than the given S/N level since photon counts are Poisson distributed. There are more sophisticated adaptive binning strategies but I haven’t found mathematical, statistical, nor computational algorithmic justifications for those. They look empirical procedures to me that are discovered after many trials and errors on particular types of spectra (I often become suspicious if I can reproduce the same goodness of fit results with the same ObsIDs as reported in those publications). The point is that either simple or complex, at the end, if someone has a data file with large number of photons, n is generally larger than observations with sparse photons. This is the reason I happen to see inconceivable d.f.s to a statistician from some papers, like 4754.
First, the chisquare goodness of fit test was designed for agricultural data (or biology considering Pearson’s eugenics) where the sample size is not a scale of scores of thousands. Please, note that bin in astronomy is called cell (class, interval, partition) in statistical papers and books showing applications of chisquare goodness fit tests.
I also like to point out that the chisquare goodness of fit test is different from the chisquare minimization even if they share the same equation. The former is for hypothesis testing and the latter is for optimization (best fit solution). Using the same data for optimization and testing introduces bias. That’s one of the reasons why with large number of data points, cross validation techniques are employed in statistics and machine learning^{[1]}. Since I consider binning as smoothing, the optimal number of bins and their size depends on data quality and source model property as is done in kernel density estimation or imminently various versions of chisquare tests or distance based nonparametric tests (KS test, for example).
Although published many decades ago, you might want to check this paper out to get a proper rule of thumb for the number of bins:
“On the choice of the number of class intervals in the application of the chi square test” (JSTOR link) by Mann and Wald in The Annals of Mathematical Statistics, Vol. 13, No. 3 (Sep., 1942), pp. 306317 where they showed that the number of classes is proportional to N^(2/5) (The underlying idea about the chisquare goodness of fit tests, detailed derivation, and exact equation about the number of classes is given in detail) and this is the reason why I chose a spectrum of 10^5 photons at the beginning. By ignoring other factors in the equation, 10^5 counts roughly yields 100 bins. About 4000 bins implies more than a billion photons, which seems a unthinkable number in Xray spectral analysis. Furthermore, many reports said Mann and Wald’s criterion results in too many bins and loss of powers. So, n is subject to be smaller than 100 for 10^5 photons.
The other issue with statistical analysis on Xray spectra is that although photons in each channel/bin can be treated as independent sample but the expected numbers of photons across bins are related via physical source model or so called link function borrowed from generalized linear model. However, well studied link functions in statistics do not match source models in high energy astrophysics. Typically, source models are not analytical. They are nonlinear, numerical, tabulated, or black box type that are incompatible with current link functions in generalized linear model that is a well developed, diverse, and robust subject in statistics for inference problems. Therefore, binning data and chisquare minimization seems to be an only strategy for statistical inference about parameters in source models so far (for some “specific” statistical or physical models, this is not true, which is not a topic of this discussion). Mann and Wald’s method for class size assumes equiprobable bins whereas channel or bin probabilities in astronomy would not satisfy the condition. The probability vector of multinomial distribution depends on binning, detector sensitivity, and source model instead of the equiprobable constraint from statistics. Well, it is hard to device an purely statistically optimal binning/grouping method for Xray spectral analysis.
Instead of individual group/bin dependent smoothing (S/N>3 grouping, for example), I, nevertheless, wish for developing binning/grouping schemes based on total sample size N particularly when N is large. I’m afraid that with the current chisquare test embedded in data analysis packages, the power of a chisquare statistic is so small and one will always have a good reduced chisquare value (astronomers’ simple model assessment tool: the measure of chisquare statistic divided by degrees of freedom and its expected value is one. If the reduced chisquare criterion is close to one, then the chosen source model and solution for parameters is considered to be best fit model and value). The fundamental idea of suitable number of bins is equivalent to optimal bandwidth problems in kernel density estimation, of which objective is accentuating the information via smoothing; therefore, methodology developed in the field of kernel density estimation may suggest how to bin/group the spectrum while preserving the most of information and increasing the efficiency. A modified strategy for binning and applying the chisquare test statistic for assessing model adequacy should be conceived instead of reporting thousands of degrees of freedom.
I think I must quit before getting too bored. Only I’d like to mention quite interesting papers that cited Mann and Wald (1942) and explored the chi square goodness of fit including Johnson’s A Bayesian chisquare test for GoodnessofFit (a link is made to the arxiv pdf file) which might provide more charm to astronomers who like to modify their chisquare methods in a Bayesian way. A chapter “On the Use and Misuse of ChiSquare” (link to google book excerpt) by KL Delucchi in A Handbook for Data Analysis in the Behavioral Sciences (1993) reads quite intriguing although the discussion is a reminder for behavior scientists.
Lastly, I’m very sure that astronomers explored properties of the chisquare statistic and chisquare type tests with their data sets. I admit that I didn’t make an expedition for such works since those are few needles in a mound of haystack. I’ll be very delighted to see an astronomers’ version of “use and misuse of chisquare,” a statistical account for whether the chisquare test with huge degrees of freedom is powerful enough, or any advice on that matter will be very much appreciated.
SUBJECT(bayes) CONTEXT(sherpa)
SYNOPSIS
A Bayesian maximum likelihood function.
Maximum likelihood function is common for both Bayesian and frequentist methods. I don’t know get the point why “Bayesian” is particularly added with “maximum likelihood function.”
DESCRIPTION
(snip)
We can relate this likelihood to the Bayesian posterior density for S(i) and B(i)
using Bayes’ Theorem:p[S(i),B(i)  N(i,S)] = p[S(i)B(i)] * p[B(i)] * p[N(i,S)  S(i),B(i)] / p[D] .
The factor p[S(i)B(i)] is the Bayesian prior probability for the source model
amplitude, which is assumed to be constant, and p[D] is an ignorable normalization
constant. The prior probability p[B(i)] is treated differently; we can specify it
using the posterior probability for B(i) offsource:p[B(i)] = [ A (A B(i))^N(i,B) / N(i,B)! ] * exp[A B(i)] ,
where A is an “area” factor that rescales the number of predicted background
counts B(i) to the offsource region.IMPORTANT: this formula is derived assuming that the background is constant as a
function of spatial area, time, etc. If the background is not constant, the Bayes
function should not be used.
Why not? If I rephrase it, what it said is that B(i) is a constant. Then why one bothers to write p[B(i)], a probability density of a constant? The statement sounds self contradictory to me. I guess B(i) is a constant parameter. It would be suitable to write that Background is homogeneous and the Background is describable with homogeneous Poisson process if the above pdf is a correct model for Background. Also, a slight notation change is required. Assuming the Poisson process, we could estimate the background rate (constant parameter) and its density p[B(i)], and this estimate is a constant as stated for p[S(i)B(i)], a prior probability for the constant source model amplitude.
I think the reason for “Bayes should not used” is that the current sherpa is not capable of executing hierarchical modeling. Nevertheless, I believe one can script the MCMC methodologies with SLang/Python to be aggregated with existing sherpa tools to incorporate a possible space dependent density, p[B(i,x,y)]. I was told that currently a constant background regardless of locations and background subtraction is commonly practiced.
To take into account all possible values of B(i), we integrate, or marginalize,
the posterior density p[S(i),B(i)  N(i,S)] over all allowed values of B(i):p[S(i)  N(i,S)] = (integral)_0^(infinity) p[S(i),B(i)  N(i,S)] dB(i) .
For the constant background case, this integral may be done analytically. We do
not show the final result here; see Loredo. The function log p[S(i)N(i,S)] is
minimized to find the bestfit value of S(i). The magnitude of this function
depends upon the number of bins included in the fit and the values of the data
themselves. Hence one cannot analytically assign a `goodnessoffit’ measure to a
given value of this function. Such a measure can, in principle, be computed by
performing Monte Carlo simulations. One would repeatedly sample new datasets from
the bestfit model, and fit them, and note where the observed function minimum
lies within the derived distribution of minima. (The ability to perform Monte
Carlo simulations is a feature that will be included in a future version of
Sherpa.)Note on Background Subtraction
Bayesian computation means one way or the other that one is able to get posterior distributions in the presence of various parameters regardless of their kinds: source or background. I wonder why there’s a discrimination such that source parameter has uncertainty whereas the background is constant and is subtracted (yet marginalization is emulated by subtracting different background counts with corresponding weights). It fell awkward to me. Background counts as well as source counts are Poisson random. I would like to know what justifies constant background while one uses probabilistic approaches via Bayesian methods. I would like to know why the mixture model approach – a mixture of source model and background model with marginalization over background by treating B(i) as a nuisance parameter – has not been tried. By casting eye sights broadly on Bayesian modeling methods and basics of probability, more robustly estimating the source model and their parameters is tractable without subtracting background prior to fitting a source model.
The background should not be subtracted from the data when this function is used
The background only needs to be specified, as in this example:
(snip)EXAMPLES
EXAMPLE 1
Specify the fitting statistic and then confirm it has been set. The method is then
changed from “LevenbergMarquardt” (the default), since this statistic does not
work with that algorithm.sherpa> STATISTIC BAYES
sherpa> SHOW STATISTIC
Statistic: Bayes
sherpa> METHOD POWELL
(snip)
I would like to know why it’s not working with LevenbergMarquardt (LM) but working with Powell. Any references that explain why LM does not work with Bayes?
I do look forward your comments and references, particularly reasons for Bayesian maximum likelihood function and Bugs with LM. Also, I look forward to see off the norm approaches such as modeling fully in Bayesian ways (like van Dyk et al. 2001, yet I see its application rarely) or marginalizing Background without subtraction but simultaneously fitting the source model. There are plenty of rooms to be improved in source model fitting under contamination and distortion of xray photon incidents through space, telescope, and signal transmission.
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.
The Redistribution Matrix File (RMF) is, at its most basic, a description of how the detector responds to incoming photons. It describes the transformation from the photons that are impinging on the detector to the counts that are recorded by the instrument electronics. Ideally, one would want there to be a onetoone mapping between the photon’s incoming energy and the recorded energy, but in the real world detectors are not ideal. The process of measuring the energy introduces a measurement error, which is encoded as the probability that incoming photons at energy E are read out in detector channels i. Thus, for each energy E, there results an array of probabilities p(iE) such that the observed counts in channel i,
$$c_id_E \sim {\rm Poisson}(p(iE) \cdot d_E) \,,$$
where d_{E} is the expected counts at energy E, and is the product of the source flux at the telescope and the effective area of the telescope+detector combination. Equivalently, the expected counts in channel i,
$${\rm E}(c_id_E) = p(iE) \cdot d_E \,.$$
The full format of how the arrays p(iE) are stored in files is described in a HEASARC memo, CAL/GEN/92002a. Briefly, it is a FITS file with two tables, of which only the first one really matters. This first table (“SPECRESP MATRIX”) contains the energy grid boundaries {E_{j}; j=1..N_{E}} where each entry j corresponds to one set of p(iE_{j}). The arrays themselves are stored in compressed form, as the smallest possible array that excludes all the zeros. An ideal detector, where $$p(iE_j) \equiv \delta_{ij}$$ would be compressed to a matrix of size N_{E} × 1. The FITS extension also contains additional arrays to help uncompress the matrix, such as the index of the first nonzero element and the number of nonzero elements for each p(iE_{j}).
The second extension (“EBOUNDS”) contains an energy grid {e_{i}; i=1..N_{channels}} that maps to the channels i. This grid is fake! Do not use it for anything except display purposes or for convenient shorthand! What it is is a mapping of the average detector gain to the true energy, such that it lists the most likely energy of the photons registered in that bin. This grid allows astronomers to specify filters to the spectrum in convenient units that are semiinvariant across instruments (such as [Å] or [keV]) rather than detector channel numbers, which are unique to each instrument. But keep in mind, this is a convenient fiction, and should never be taken seriously. It is useful when the width of p(iE) spans only a few channels, and completely useless for lowerresolution detectors.
]]>Studying that Smile (subscription needed)
A tutorial on multispectral imaging of paintings using the Mona Lisa as a case study
by Ribes, Pillay, Schmitt, and Lahanier
IEEE Sig. Proc. Mag. Jul. 2008, pp.1426
Conclusions: In this article, we have presented a tutorial description of the multispectral acquisition of images from a signal processing point of view.
I wonder if there’s literature in astronomy matching this tutorial from which we may expand and improve current astronomical photometry processes by adopting strategies developed by more populated signal/image processing engineers and statisticians. (Considering good textbooks on statistical signal processing, and many fundamental algorithms born thanks to them, I must include statisticians. Although not discussed in this tutorial, Hidden Markov Model (HMM) is often used in signal processing but from ADS, with such keywords, no astronomical publication is aware of HMM – please, confirm my finding that HMM is not used among astronomers because my search scheme is likely imperfect.)
]]>– In the XSPEC packages, the parametric bootstrap is command FAKEIT, which makes Monte Carlo simulation of specified spectral model.
– XSPEC does not provide a nonparametric bootstrap capability.
Parametric Bootstrap: $$X_1^*,…,X_n^* \sim F(\cdot;\theta_n)$$
Both $$\sqrt{n} \sup_x F_n(x)F(x;\theta_n)$$ and $$\sqrt{n} \sup_x F_n^*(x)F(x;\theta_n^*)$$ have the same limiting distribution.^{[1]}
Nonparametric Bootstrap:$$X_1^*,…,X_n^* \sim F_n.$$
A bias correction $$B_n(x)=F_n(x)F(x;\theta_n)$$ is needed.
$$\sqrt{n} \sup_x F_n(x)F(x;\theta_n)$$ and $$\sqrt{n} \sup_x F_n^*(x)F(x;\theta_n^*)B_n(x)$$ have the same limiting distribution.^{[2]}