.. _full_sample_unfolding_paragraph: Full Sample Unfolding ##################### The burnsample unfolding has been presented at the ICRC and the proceedings is available here: `Unfolding the Atmospheric Muon Flux with IceCube: Investigating Stopping Muons and High-Energy Prompt Contributions `_. In the following, the determination of the regularization strength and two robustness tests are presented on MC. The effective area and the corresponding systematic uncertainties are described in :ref:`Unfolding/Effective Area `. MC --- Regularization ============== At first, the regularization parameter tau is determined by finding the minimum of the global correlation .. math:: \rho = \sum_{i>j} V_{ij}\,, where V is the covariance matrix of the unfolded distribution, with i and j being the indices of the unfolding bins. This does not include the under- and overflow bin, nor the systematic bins. For the burnsample MC, the global correlation is presented in :numref:`full_sample_global_correlation_mc`. Here, 280 tau values from 1e-6 to 1 are tested. It is checked whether the minimization of the unfolding using minuit is successful, otherwise the point is discarded. Due to some jumps in the distribution, instead of the exact minimum, a rolling average with a window size of 8 is used to determine the optimal tau value. This results in a value of tau=0.0022 on MC. However, this needs to be determined for all changes in the unfolding, thus whenever the binning changes, the cosmic-ray models changes, the spectral index of the cosmic-ray flux changes, or the unfolding is applied to experimental data. .. _full_sample_global_correlation_mc: .. figure:: images/plots/unfolding/new/full_sample/global_correlations_single_tau_mc.png :width: 600px : Global correlation as a function of the regularization parameter tau on MC. Robustness Tests ================ Afterwards, two robustness tests are performed. At first, the impact of a shift in the spectral index of the cosmic-ray flux on the unfolded muon flux is investigated. The unfolding algorithm is trained on the nominal spectrum with H3a, and then the spectrum is varied by ±0.1. This re-weighting is achieved by multiplying the weight :math:`w` of each event by the primary energy :math:`E_{\text{P}}` to the power of the shift in the spectral index :math:`gamma_{\text{s}}`, via .. math:: w_{\text{s}} = w \times E_{\text{P}}^{\gamma_{\text{s}}}\,, resulting to the shifted weights :math:`w_{\text{s}}`. The results are shown in :numref:`full_sample_delta_gamma_shift`. Within the uncertainties, the results are compatible. Thus, the burnsample unfolding is robust against small changes in the spectral index of the cosmic-ray flux. .. _full_sample_delta_gamma_shift: .. figure:: images/plots/unfolding/new/full_sample/test_delta_gamma/unfolding_flux_systematics_gamma_shift_comparison.png :width: 600px : The impact of a shift in the spectral index of the cosmic-ray flux on the unfolded muon flux on MC is investigated. The unfolding algorithm is trained on the nominal spectrum with H3a, and then the spectrum is varied by ±0.1. This re-weighting is achieved by multiplying the weight of each event by the the primary energy to the power of the shift in the spectral index. The ratio to the true H3a MC distribution is shown. The uncertainties come from the inverse of the Hesse matrix (not systematic scaling of the effective area is included here). Within the uncertainties, the results are compatible. Thus, the unfolding is robust against changes in the spectral index of the cosmic-ray flux. Secondly, the impact of the primary cosmic-ray model on the unfolded flux is studied. The unfolding algorithm is trained on the four different primary models, and then the same pseudo test dataset sampled from an H3a distribution is unfolded. The results are shown in :numref:`full_sample_primary_model_shift`. Within the uncertainties, the results are compatible. Thus, the unfolding is robust against changes in the primary cosmic-ray model. .. _full_sample_primary_model_shift: .. figure:: images/plots/unfolding/new/full_sample/test_primary_model/unfolding_flux_systematics_weight_col_shift_primary_models_comparison_zoom.png :width: 600px : Study of the impact of the primary cosmic-ray model on the unfolded flux on MC. The unfolding algorithm is trained on the four different primary models, and then the same pseudo test dataset sampled from an H3a distribution is unfolded. The ratio to the true H3a MC distribution is shown. The uncertainties come from the inverse of the Hesse matrix (not systematic scaling of the effective area is included here). Within the uncertainties, the results are compatible. Thus, the unfolding is robust against changes in the primary cosmic-ray model. Sensitivity Tests ================= Here, a test is performed to make sure the unfolding is sensitive to the prompt component. For this, two pseudo test datasets are created. One set is sampled from the total H3a distribution including the prompt and conventional component, and one set is sampled from a distribution that only includes the conventional component. This is done via the extended history option in CORSIKA. Hence, for the set without prompt muons, all muons that do not have a pion or kaon as a parent particle are removed. When we then unfold both sets with the same response matrix that includes the prompt component as in our nominal analysis, we expect that the unfolded set without the prompt component follows the conventional distribution, while the unfolded set with the prompt component should be above that, following the total distribution including prompt and conventional. For the comparisons MCEq with H3a and SIBYLL 2.3c is utilized. In :numref:`full_sample_sensitivity_no_prompt`, the result for the set without prompt muons is shown. The unfolded distribution follows the conventional distribution, and no significant prompt contribution is observed, as expected. .. _full_sample_sensitivity_no_prompt: .. figure:: images/plots/unfolding/new/full_sample/prompt_sensitivity/unfolding_flux_mceq_02-1_H3a_SIBYLL23c_A_eff_unc_no_prompt.png :width: 600px : Unfolded muon flux for a pseudo test dataset sampled from a distribution that **only includes the conventional component**. The unfolding is performed with the same response matrix that includes the prompt component as in our nominal analysis. The unfolded distribution follows the conventional distribution, and no significant prompt contribution is observed, as expected. Uncertainties are included from minuit and the effective area. In :numref:`full_sample_sensitivity_with_prompt`, the result for the set with prompt muons is shown. The unfolded distribution is significantly above the conventional distribution, following the total distribution including prompt and conventional, as expected. .. _full_sample_sensitivity_with_prompt: .. figure:: images/plots/unfolding/new/full_sample/prompt_sensitivity/unfolding_flux_mceq_02-1_H3a_SIBYLL23c_A_eff_unc.png :width: 600px : Unfolded muon flux for a pseudo test dataset sampled from the total H3a distribution **including the prompt and conventional component**. The unfolding is performed with the same response matrix that includes the prompt component as in our nominal analysis. The unfolded distribution is significantly above the conventional distribution, following the total distribution including prompt and conventional, as expected. Uncertainties are included from minuit and the effective area. Data ---- Unblinded results will be presented here.