Exercise 7: Maximum likelihood fits

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1 Kirchhoff-Institut, Physikalisches Institut and MPI für Kernphysik Winter semester KIP CIP Pool.40 Exercises for Statistical Methods in Particle Physics Dr. Oleg Brandt Dr. Markward Britsch Exercise 7: Maximum likelihood fits November 24, 204 To be handed in by by 4:00h, Sunday, November 30 Please send your solutions by to until Sunday, November 30, 4:00h, so that we can discuss them on the following Monday. Make sure that you use SMIPP:Exercise07 as subject line. It is important that you test all program code, macros, etc. before you hand them in. Please also comment the code reasonably well. Make sure that your solution code compiles, either by a command like > root -l exercise7_.c+ (mind the +) or using a Makefile attached to your . If plots are requested, please include print statements to produce pdf files in your code. Preface There are three main applications for the maximum likelihood (ML) fit. Firstly, an unbinned fit of a pdf to measured values can be done, as in Exercise. Secondly, in Exercise 3 we will fit a general function (i.e., not a pdf) to (x, y)-data points. Finally, one can fit a pdf to the shape of a histogram, which we will do on the next exercise sheet. We will also use different tools to do the fits. In Exercise we will construct our own negative log likelihood (NLL) function as a TF and use a build-in method to find the minimum. In Exercise 3 we will use the Minuit package via root s TMinuit implementation as we have done so in the hands-on exercises. An unbinned maximum likelihood fit of a pdf: a neutrinoless double beta decay signal Many experiments nowadays are looking for the so called neutrinoless double beta decay, where a nucleus decays by emitting two electrons, but no neutinos. As this process violates lepton number (L) conservation without violating the baryon number (B) conversation, the observation of such a decay would imply physics beyond the standard

2 model, as in the standard model B L is conserved. In addition, it would also allow the determination of the absolute neutrino masses. One type of experiments use germanium (e.g., the GERDA experiment, see recent results in Ref. []) for this looking for decays 76 Ge 76 Se e e, where the two electrons are detected and their energy is measured. The corresponding normal double beta decay (where in addition two neutrinos are emitted) has already been observed. Since the two neutrinos are not detected, the energy spectrum of this decay is very wide. While the energy spectrum of the neutrinoless double beta decay has a sharp peak, as all the libarated energy is contained in the electrons (see Fig. ). From the mass difference of the two nuclei, this expected liberated kinetic energy, also called Q-value, i.e., the total kinetic energy of the two electrons, can be determined very precisely to Q = ± kev. Assume that a future experiment finds a surprisingly large signal of 20 counts in the region around the expected Q-value with negligible background. In Double the following we want Beta to make a fitdecay to the data to obtain Detection an estimator for the peak value in the data, to compare it to the expected Q-value, to see if this is really the long-sought neutrinoless double beta decay. Spectrum of summed electron energies. 2νββ produces Figure a : continuous The energy distribution spectrum for the double as the betaν decay. energy The left is part not comes detected. from 0νββ results in a peak at the Q-value of 76 Ge at 2039 kev. Ge has several advantages: the one where two neutrinos are emitted. The right part illustrates the hypothetical neutrinoless double beta decay (also shown in the insert). As this decay has a very long life time, the width is completely negligible and thus the width of the signal is determined by the detector resolution. Typical resolutions of calorimetric detectors have a gaussian core, though often with corrections to the tails. Ge detectors Here we have are the typical used correction as both of a long the tail decay to the low source energy side and duethe to energy detection losses in the apparatus. To model this we use a special simplified version of the so called device. crystal ball pdf. It has a gaussian part above a certain energy value and a power law behavior below this value, as shown in Fig. 2. The energy resolution of this detector has been measured to be σ = 3.3 kev, the tail is measured to start at α =.5 standard deviations below the peak and this tail drops off as a power of n = More precisely, the form of our pdf that we use is given Ge has a very high energy resolution: The higher the energy resolution, the higher the signal to background ratio. 2 anuel Walter The GERDA Experiment for the Search of Neutrinoless Double Beta Decay Patras Workshop July 203 Waldthausen

3 A crystal ball pdf pdf value Power law pdf Gaussian pdf E [kev] Figure 2: A crystal ball function with the two behaviors above and below the energy value indicated by the blue line. by ( exp (x µ)2 f(x; µ) = N ( A x µ σ 2σ 2 ) if x µ σ ) n if x µ σ > α α with σ = 3.3 kev, α =.5, n = 2.25, A = 08396, N = 0.05 kev. The parameters A and N are chosen as to guarantee that the total pdf is normalized, continuous and differentiable. The measured kinetic energies of these 20 events are given in to you kev by the definitions const int doublebeta_numdatapoints = 20; double doublebeta_data[ doublebeta_numdatapoints ] = { , , , 2035, , , , , , , , , , , , , , , , }; in File doublebeta_data.h. Create a TF object that represents the negative log likelihood (NLL) function and minimize it using TF::GetMinimumX(). The statistical error can be read off the plot of the NLL function as in Fig. 3. The error boundaries are where the NLL function value is 2 its value at the minimum. Determine this error from the plot. Provide the plot including arrows (TArrow) or lines (TLine) where your error boundaries are. Compare 3

4 your results to the expectation. Plot the data into a histogram and plot the best fit function, as well as the true Q-value function with it. - ln L ln L = / E [kev] Figure 3: The minimum of the NLL gives the fit result (black arrow), while the x-values, where the NLL function is 2 above the minimal y-value, give the error range (blue arrows). Mind that the errors can be different to the left than to the right side. Hints: When implementing your fit function as a TF, you will need to define it via a C-function with parameters like that: Double_t specialcb(double_t *x, Double_t *par); where x[0] will be the x-value given to the function and par is an array with the parameters. Then define the TF like TF fitfunc("fitfunc", specialcb, 2004, 2054, 2); The 2 here gives the number of parameters. One parameter for µ which we want to determine, and a second one to scale the pdf to fit the histogram entries, i.e., to re-normalize this function to have the same integral as the histogram. For this do not forget that the size of the bins changes this integral. Check that this re-normalization works by looking at plots! You do not want to fit the re-normalization. Fix it to the correct value in the fit by calling TF::FixParameter(). Although it is performance wise stupid, for ease of coding, you can implement the NLL using the logarithm of the fit function that you have defined. 4

5 2 Determining the bias of a maximum likelihood fit: a neutrinoless double beta decay signal To determine the bias, make 000 toy MC experiments drawing 20 times from your fit function using your fit result from above as µ. Do the fit for each toy data set. You can get the bias as the mean of the results of the fits to your toy MC data minus µ from your fit to data. Is your bias significant taking into account the error on the mean (TH::GetMeanError())? If yes, take the bias into account and quote a reasonable systematic error. Compare to the expectation and to the results form Exercise. Also compare your determined statistical error with the RMS form the toy MC results. Do they match? Hints: It is probably a good idea to define a global pointer pointing to the data. It will point first to the measured data, then to the current toy MC data set and can be used in your main code as well as in the function that implements the NLL. 3 A maximum likelihood fit of a function to data points Here we re-use the code already used in the hands-on 7.2 this Monday, starting with the same setup. Suppose we have 20 independent measurements r i ± σ i with the same σ, but different mean values µ i. Now consider there is a linear dependence of µ i on some variable x of the mean values, i.e., r i = µ i = + x with standard deviation σ = 0.. The points will have equal distances in x, x = 0.. An Example is shown in Fig. 4 +x distribution of fitted parameter Figure 4: 20 data points fluctuation of with individual point a linear dependence. hist_flucpoint 4 Entries 20 Mean RMS In the following we 3want to investigate the behavior of the maximum likelihood 2.5 (ML) method under different circumstances by using toy MC data that follows the 2 above. First we will use.5 gaussian error distributions and a linear model with two fit 0.6 parameters (r = a 0 + a x). Mind that finding the global minimum of a two dimensional function is already far less -0.5 trivial then -0.2 finding the 0. global minimum of a one 0.2dimensional function, that is why we are using the software Minuit here. Have a short look at the a0 distribution hist_a0 corresponding root documentation: Entries 0 Mean 0 RMS ln L fit (- ln L ) - (- ln L ) true fit a distribution hist_a Entries 0 Mean 0 RMS

6 Later we will have a look at what happens with non-gaussian errors. Another option is to use gaussian errors, but a non-linear model (r = (a + a 4 0 x)( a 0+a ) ) in the fit, designed in a way that again a 0 =, a = is the correct answer. Eventually we will try non-gaussian errors with the non-linear model. To do this, we need to construct a NLL function and then minimize this function. We have done this already in the hands-on exercise. Now do the following. a) Change the existing program so it does a large number of toy MC experiments and plot: for one case the points with errors, the true linear dependence and the fitted dependence (already done in the hands-on exercise) fluctuations of the individual points (already done in the hands-on exercise) distribution of the fit parameter a 0 distribution of the fit parameter a distribution of a 0 vs. a ln L fit mean ( ln L true ) ( ln L fit ) in a 0 a plane The numbers are already obtained from Minuit in the code, you only have to fill the pre-defined histograms! b) Now we will change the production of our measurements to non-gaussian errors. Use a function nongaussfunc in neg_lnl.cc that implements the function f(y) = exp ( ) y4 σ (where y will be y = ri µ 4 i later on). To get the correct RMS = σ = 0., we need to set σ Γ( 4 = σ ) Γ( 3 4 ) (where Γ is the gamma function, for which you can use TMath::Gamma()). Do the same plots as before. c) Now do the same with gaussian errors, but fitting the non-linear function from above (fcn_nonlin in neg_lnl.cc). d) And now non-linear, non-gaussian. For all these observe and compare. Do the distributions look reasonable? Is there evidence for a bias? References [] Phys. Rev. Lett. (203)