658
L. Lyons
to minimise a cost function C for the network. This is usually defined as C =
(z i − t i ) 2 , where z i is the trained network’s output for the ith event; t i is the target
output, usually chosen as 1 for signal and zero for background; and the summation
is over all testing events presented to the network. The problem with this is that
C is only loosely related to what we really want to optimise. For a search for
a new particle this could be the sensitivity of the experimental upper limit in the
absence of signal, while for a high statistics analysis measuring the properties (such
as mass or lifetime) of some well-established particle, we would be interested in
minimising the uncertainty (including systematic effects) on the result, without the
training procedure biassing the measurement.
As with all event separation methods. it is essential to check the performance
of a trained procedure by using a set of events that are independent of those
used for training. This is to ensure that the network does not use specific but
irrelevant features of the training events in its learning process, but can achieve
good performance on unseen data.
Some open questions are:
• How can we check that our multi-dimensional training samples for signal and
background are reliable descriptions of reality; and that they cover the region of
multi-dimensional space populated by the data?
• How should the ratio of the numbers of signal and background training events be
chosen, especially when there are several different sources of background?
• What is the best way of allowing for nuisance parameters in the models of the
signal and/or background?[25, 48]
• Are there useful and easy ways of optimising on what is really of interest?[49]
15.3.1 Understanding How Neural Networks Operate
It is useful to appreciate how neural networks operate in providing a good separation
of signal and background, as this can help in choosing a suitable architecture for the
network.
Figure 15.3a shows some hypothetical signal and background events in terms of
two measured variables x and y for each event. A network with two inputs (x and y),
a single hidden layer with 3 nodes, and a single output is used; it aims to give 1 for
signal and zero for background events (see Fig. 15.3b. This is achieved by training
the network with (x, y) values for known examples of signal and background; and
allowing the network to vary its internal parameters to minimise a suitably defined
cost function e.g. (z e − t e ) 2 , where the summation is over the training events, and
z e and t e are the network’s output and its target value (0 or 1) respectively.
training sample, because the network could then achieve a very small cost C simply by classifying
everything as background.
L. Lyons
to minimise a cost function C for the network. This is usually defined as C =
(z i − t i ) 2 , where z i is the trained network’s output for the ith event; t i is the target
output, usually chosen as 1 for signal and zero for background; and the summation
is over all testing events presented to the network. The problem with this is that
C is only loosely related to what we really want to optimise. For a search for
a new particle this could be the sensitivity of the experimental upper limit in the
absence of signal, while for a high statistics analysis measuring the properties (such
as mass or lifetime) of some well-established particle, we would be interested in
minimising the uncertainty (including systematic effects) on the result, without the
training procedure biassing the measurement.
As with all event separation methods. it is essential to check the performance
of a trained procedure by using a set of events that are independent of those
used for training. This is to ensure that the network does not use specific but
irrelevant features of the training events in its learning process, but can achieve
good performance on unseen data.
Some open questions are:
• How can we check that our multi-dimensional training samples for signal and
background are reliable descriptions of reality; and that they cover the region of
multi-dimensional space populated by the data?
• How should the ratio of the numbers of signal and background training events be
chosen, especially when there are several different sources of background?
• What is the best way of allowing for nuisance parameters in the models of the
signal and/or background?[25, 48]
• Are there useful and easy ways of optimising on what is really of interest?[49]
15.3.1 Understanding How Neural Networks Operate
It is useful to appreciate how neural networks operate in providing a good separation
of signal and background, as this can help in choosing a suitable architecture for the
network.
Figure 15.3a shows some hypothetical signal and background events in terms of
two measured variables x and y for each event. A network with two inputs (x and y),
a single hidden layer with 3 nodes, and a single output is used; it aims to give 1 for
signal and zero for background events (see Fig. 15.3b. This is achieved by training
the network with (x, y) values for known examples of signal and background; and
allowing the network to vary its internal parameters to minimise a suitably defined
cost function e.g. (z e − t e ) 2 , where the summation is over the training events, and
z e and t e are the network’s output and its target value (0 or 1) respectively.
training sample, because the network could then achieve a very small cost C simply by classifying
everything as background.