vstatmp_engl

Fig. 9.7. Deconvolution of a blurred pictur with the satellite method.

Measurement resolution and acceptance should stay approximately constant in the region in which the events migrate. When we start with a reasonably good approximation of the true distribution, this condition is usually satisfied. In exceptional cases it would be necessary to update the distribution of the satellites yik′ after each move, i.e. to simulate or correct them once again. It is more e cient, however, to perform the adaptation for all elements periodically after a certain number of migration steps.

The number K determines the maximal resolution after the deconvolution, it has

therefore a regularization e ect; e.g. for a measurement resolution σf and K = 16

√

the minimal sampling interval is σT = σf / K = σf /4.

If the true p.d.f. has several maxima, we may find several relative minima of the energy. In this case a new stochastic element has to be introduced in the minimization (see Sect. 5.2.7). In this case a move towards a position with smaller energy is not performed automatically, but only preferred statistically.

We have not yet explained how acceptance losses are taken into account. The simplest possibility is the following: If there are acceptance losses, we need Ki0 > K trials to generate the K satellites of the event yi. Consequently, we relate a weight wi = K0i/K to the element yi. After the deconvolution we then obtain a weighted sample.

A more detailed description of the satellite method is found in [53].

9.3.3 The Maximum Likelihood Method

In the rare cases where the transfer function t(x, x′) is known analytically or easily calculable otherwise, we can maximize the likelihood where the parameters are the locations of the true points. Neglecting acceptance losses, the p.d.f. for an observation x′, with the true values x1, . . . , xN as parameters is

240 9 Deconvolution

1.Likelihood fit of the true histogram with curvature sensitive or entropy regularization

2.Multiplication of the observed histogram vector with the inverted, regularized transfer matrix

3.Iterative deconvolution

4.Iterative binning-free deconvolution

5.The satellite method

6.The binning-free likelihood method

The first method is more transparent than the others. The user has the possibility to adapt the regularization function to his specific needs. With curvature regularization he may, for instance, choose a di erent regularization for di erent regions of the histogram, or for the di erent dimensions in a higher-dimensional histogram. He may also regularize with respect to an assumed shape of the resulting histogram. The statistical accuracy in di erent parts of the histogram can be taken into account. Regularization with the entropy approach is technically simpler but it is not suited for applications in particle physics, because it favors a globally uniform distribution while the local smearing urges for a local smoothing. It has, however, been successfully applied in astronomy and been further adjusted to specific problems there. An overview with critical remarks is given in [56].

The second method is independent from the shape of the distribution to be deconvoluted. It depends on the transfer matrix only. This has the advantage to be independent from subjective influences of the user. A disadvantage is that regions of the true histogram with high statistics are treated not di erently from those with only a few entries. A refined version which has successfully been applied in several experiments is presented in [8].

The third procedure is technically the simplest. It can be shown that it is very similar to the second method. It also suppresses small eigenvalues of the transfer matrix.

The binning-free, iterative method has the disadvantage that the user has to choose some parameters. It requires su ciently high statistics in all regions of the observation space. An advantage is that there are no approximations related to the binning. The deconvolution produces again single points in the observation space which can be subjected to selection criteria and collected into arbitrary histograms, while methods working with histograms have to decide on the corresponding parameters before the deconvolution is performed.

The satellite method has the same advantages. Important parameters must not be chosen, however. It is especially well suited for small samples and multidimensional distributions, where other methods have di culties. For large samples it is rather slow even on large computers.

The binning-free likelihood method requires an analytic transfer function. It is much faster than the satellite method, and is especially well suited for the deconvolution of narrow structures like point sources.

An qualitative comparison of the di erent methods does not show big di erences in the results. In the majority of problems the deconvolution of histograms with the fitting method and curvature regularization is the preferred solution.

Fig. 9.9. Result of a deconvolution with strong (top) and weak (bottom) regularization. The errors depend on the regularization strength.

As stated above, whenever the possibility exists to parametrize the true distribution, the deconvolution process should be avoided and replaced by a standard fit.

9.5 Error Estimation for the Deconvoluted Distribution

The fitting methods produce error estimates automatically, for the other methods the uncertainties can be obtained by the usual error propagation. But we run into another unavoidable di culty connected to the regularization: The size of the errors depends on the strength of the regularization which on the other hand is unrelated to the statistical accuracy of the data. This is illustrated in Fig. 9.9 where the deconvolution of a structure function with di erent regularization strengths is shown. A strongly regularized distribution may exhibit smaller errors than the distribution before the convolution. This is unsatisfactory, as we loose information by the smearing. We should present errors which do not depend on data manipulations.

As described above, the errors of neighboring bins of a histogram are negatively correlated. The goodness-of-fit changes only slightly if we, for instance, enhance a bin content and accordingly reduce both the two neighboring bin contents or vice versa. The regularization has the e ect to minimize the di erence while keeping the sum of entries nearly unchanged, as can be seen in the example of Fig. 9.9. The e ect of the regularization is sketched in Fig. 9.10. Even a soft regularization will reduce the area of the error ellipse considerably.

Besides the pure Poisson fluctuations the graphical presentation should also show the measurement resolution. We represent it by a horizontal bar for each point. Fig. 9.11 shows a deconvolution result for two regularizations of di erent strength. The curve represents the true distribution. Contrary to the representation of Fig. 9.9, the vertical error bars are now independent of the regularization strength. The horizontal bars indicate the resolution. With the strong regularization the valley is somewhat filled up due to the suppression of curvature and the points are following the curve better than expected from the error bars. The experienced scientist is able to judge also for the weak regularization that the curve is compatible with the data.

In multi-dimensional and in binning-free applications a graphical representation of the resolution is di cult but the tranfer function has to be documented in some way.

246 10 Hypothesis Tests

Usually a test is associated with a decision: accept or reject. We will not always attempt a decision but confine ourselves to fix the parameters which form the bases for a possible decision.

As mentioned, we will primarily deal with a part of test theory which is especially important in natural sciences and also in many other empirical research areas, namely that only one hypothesis, we call it the null hypothesis H0, is tested while the admitted alternative is so vague or general that it cannot be parameterized. The alternative hypothesis H1 is simply “H0 is false”. The question is whether the sample at hand is in agreement with H0 or whether it deviates significantly from it. The corresponding tests are called goodness-of-fit (GOF) tests.

Strongly related to GOF tests are two-sample tests which check whether two samples belong to the same population.

At the end of this chapter we will treat another case in which we have a partially specified alternative and which plays an important role in physics. There the goal is to investigate whether a small signal is significant or explainable by a fluctuation of a background distribution corresponding to H0. We call this procedure signal test.

10.2 Some Definitions

Before addressing GOF tests, we introduce some notations.

10.2.1 Single and Composite Hypotheses

We distinguish between simple and composite hypotheses. The former fix the population uniquely. Thus H0: “The sample is drawn from a normal distribution with mean zero and variance one, i.e. N(0, 1).” is a simple hypothesis. If the alternative is also simple, e.g. H1 : “N(5, 1)”, then we have the task to decide between two simple hypotheses which we have already treated in Chap. 6, Sect. 6.3. In this simple case there exists an optimum test, the likelihood ratio test.

Composite hypotheses are characterized by free parameters, like H0: “The sample is drawn from a normal distribution.” The user will adjust mean and variance of the normal distribution and test whether the adjusted Gaussian is compatible with the data.

The hypothesis that we want to test is always H0, the null hypothesis, and the alternative H1 is in most cases the hypothesis that H0 does not apply. H1 then represents an infinite number of specified hypotheses.

10.2.2 Test Statistic, Critical Region and Significance Level

After we have fixed the null hypothesis and the admitted alternative H1, we must choose a test statistic t(x), which is a function of the sample values x ≡ {x1, . . . , xN }, possibly in such a way that the di erence between the distribution f(t|H0) and distributions belonging to H1 are as large as possible. To simplify the notation, we consider one-dimensional distributions. The generalization to multi-dimensional observations is trivial.