Wing Sheung Chan

76 Event selection and classification Table 4.5.: Numbers of simulated events used in the NN training. Process eτ 1P eτ 3P µτ 1P µτ 3P Z → `τ 31643 11927 31643 11927 Z → τ τ 3515217 1379724 3344264 1373453 W +jets 100204 39213 100289 40273 Z → `` 36751 not used 13953 not used The training samples are randomly (but reproducibly) split into two sets of equal size. For each classifier, a pair of NNs are constructed and trained using the different sets of events. For each NN, the set of events that is not used in the training is considered the “evaluation set”. The performance of the NN is estimated using the evaluation set. This ensures that the NNs are evaluated using unseen samples to avoid bias and allows cross validation of the two NNs for each classifier. The evaluation set is further split into two halves, one of which is used to optimise the hyperparameters for the NN architecture and training, while the other half is used to make other optimisation such as adding or dropping input variables. 4.2.2. Input variables The input to the NNs is a mix of low-level and high-level kinematic variables. The low-level variables are the four-momenta of the reconstructed ` , τ had - vis and E mis s T † in each event. In order to remove known spatial symmetries for optimal training, the low- level variables are first transformed before they are fed into the NNs. The transformation consists of the following steps in the listed order: 1. The ` – τ had - vis – E miss T system is boosted in a direction in the transverse plane such that the total transverse momentum of the system becomes zero. 2. The system is rotated about the z -axis such that E miss T is aligned with the x -axis. 3. If the τ had - vis momentum has a negative z -component, the entire system is rotated about the (new) x -axis by π . To preserve Lorentz invariance, no boosting in the z -direction or rotation about the x - or y -axis is involved, given that the true momentum of undetected particles in the z -direction is unknown to us. By removing known spatial symmetries, we spare the NNs the extra effort to “learn” the equivalence between events that are boosted and/or rotated with respect to each other but are otherwise the same. This essentially increases the effective training sample size. After the transformation, only six independent non-vanishing components are † The four-momentum of E miss T is defined with a zero mass and a zero z -component (i.e. η = 0 ).