Gaussian distribution response functions

In radiation measurements, folding means to obtain the shape of the measured spectrum when the source and the detector response are known. Several examples of folding using a Gaussian distribution as the response function are presented next. 

The Bayesian time-domain approach presented in this chapter addresses this problem of parametric identification of linear dynamical models using a measured nonstationary response time history. This method has an explicit treatment on the nonstationarity of the response measurements and is based on an approximated probability density function (PDF) expansion of the response measurements. It allows for the direct calculation of the updated PDF of the model parameters. Therefore, the method provides not only the most probable values of the model parameters but also their associated uncertainty using one set of response data only. It is found that the updated PDF can be well approximated by an appropriately selected multi-variate Gaussian distribution centered at the most probable values of the parameters if the problem is... 

The disorder is Gaussian-distributed as in Eq. (8). The general response function for the force is... 

Fig. 4.6 The average difference between the transverse energy of the generated and the reconstructed jet as a function of the transverse energy of the generated jet. The energy response of TrackJets (triangles) is compared to the energy response of CaloJets (circles). The three levels of jet corrections are applied to the CaloJets. The errors correspond to the width of a Gaussian distribution fitted to the core of the distribution of the residuals...

A number of methods allow the estimation of probability densities, (a) A multivariate Gaussian distribution can be assumed the parameters are the class mean and the covariance matrix, (b) The p-dimensional probability density is estimated by the product of the probability densities of the p features, assuming they are independent, (c) The probability density at location x is estimated by a weighted sum of (Gaussian) kernel functions that have their centers at some prototype points of the class (neural network based on radial ba.sis functions, RBF ). (d) The probability density at location x is estimated from the neighboring objects (with known class memberships or known responses) by applying a voting scheme or by interpolation (KNN, Section 5.2). 

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