The State of the Art in Intrusion Prevention and Detection

264 The State of the Art in Intrusion Prevention and Detection

by the averaged (over the set of observations) square of differences of actual values ŷ

j

(X, w), j = 1,

2,…, M of the outputs calculated for particular observations X with the weights w (X = (x

1

x

2

,…, x

N

)

being the vector of all inputs of the neuron/neural network while w is the vector of all weights, and

their desired values y

j

. Thus, the cost C(.) for a single training sample (X, Y) can be expressed as

Cy

Xy

jj

j

M

() (

ˆ

(,

))ww=−

=

∑

2

1

(11.7)

and in a case of a larger training set (X

(i)

, Y

(i)

), i = 1, 2,…,I it will be averaged over all samples:

C

I

yX y

j

i

j

i

j

M

i

I

()

ˆ

(,)

() ()

ww=−

()

==

∑∑

1

2

11

.