REFLEXIONES ACERCA DE LAS FALACIAS MENCIONADAS POR MARTINEZ-CHAVANZ SOBRE LA MASA RELATIVA

Probabilistic neural network (PNN) was originally developed by Specht (1990) and is considered a hybrid technique that use a Bayesian classifier (Gelman et al. 1995) and a

Parzen-Cacoullos theory (Cacoullos 1966) on an NN platform to produce the probability distribution of each pattern or class. Both Bayesian classifier and Parzen- Cacoullos theory are considered statistical technique. PNN were successfully applied on some infrastructure modellings (Sinha and Pandey 2002; Kim et al. 2005a). Based on

in Section 3.3.4, the PNNDM in this study used the PNN to classify a pipe into one three possible conditions.

3.3.5.1 Bayesian Classifier

The Bayesian classifier (Wasserman 1993), as shown in Equation (3-26), was used to classify a pipe into one of the three possible pipe conditions. The purpose of Equation (3-26) is to minimize the expected risk in classification (Kim et al. 2005). From the

Bayesian theorem, the product of h_c and ( )f_c X is a posterior probability that allows the

updating of existing knowledge h_c with new information ( )f_c X . The existing

knowledge h_c could be obtained from a previous sample or expert opinion. The loss l_c

which is associated with misclassification can be interpreted with following examples. When a pipe in condition 1 is misclassified into condition 3, the loss is just an inspection cost. However, when a pipe in condition 3 is misclassified into condition 1, the loss can be substantially higher. This is because if no inspection or repair is done due to pipe is assumed in good condition, the pipe then fails and incurs repair cost and damage cost.

In this study, the loss l_c was assumed identical between the classes since no data for

calculating risk are available. This means that all pipes are treated equally. Furthermore, the prior probability h_c of occurrence in the class c was assumed to a follow uniform

distribution which means that a pipe is classified into condition c if its probability

distribution in that class has the highest value compared with those in other classes. This is because the effects of l_cand h_c were cancelled by the above assumptions.

( ) if _{c c c}( ) _{j j j}( ) [1,3]

D X =c l h f X ≥l h f X ∀ ∈j (3-26) where: X is a K-dimensional vector representing a pipe with K

contributing factors

D(X) is an image of X in a set of three classes C₁, C₂ and C₃ c

l is the loss associated with misclassifying a vector of the

class c into other classes c

h is the prior probability of occurrence in the class c

( )

c

3.3.5.2 Parzen-Cacoullos method for estimating PDF

As can be seen in the previous section, the challenge to the Bayesian classifier is the fact that the probability distribution function (PDF) ( )f_c X is not usually known.

Therefore, it is necessary to derive an estimate of ( )f_c X from the training data. This

can be done by using the Parzen-Cacoullos method (Kim et al. 2005a). The univariate

case of PDF was first developed by Parzen and then was extended to the multivariate case by Cacoullos. This method is given in Equation (3-27).

1 1 2 1 ( ) ... N i i K i f W Nσ σ σ = σ ⎛ − ⎞ = _{⎜ ⎟} ⎝ ⎠

∑

where: X is a K-dimensional vector representing a pipe with K

contributing factors

i

σ is standard deviation of a contributing factor i

N is the number of training data (i.e. number of pipes

with known condition and contributing factors) ( )

f X is the probability distribution function (PDF) of X W is (kernel) density function

This study assumed that all smoothing parameters are identical to a smoothing parameterσ and a bell-shaped Gaussian function is used for W. Equation (3-27) then

reduces to Equation (3-28). The meaning of the smoothing parameter σ in the case of the Gauss kernel, is that the Gaussian curve is sharply peaked with σ smaller than one, and tends to flatten out with increasing σ (Wasserman 1993).

2 / 2 2 1 1 || || ( ) exp (2 ) 2 N i K K i f N π σ = σ ⎛ − ⎞ = _{⎜ ⎟} ⎝ ⎠

∑

X X is the Euclidean distance from the vector X of the query pipe to a training vector Xi

Equation (3-28) was then used to compute the PDF ( )f_c X of a pipe represented by a

vector X of K-contributing factors in class c given a sample of training data

{

}

training vector of K-contributing factors. The Bayesian classifier was then applied to

classify or predict the pipe into one of the three possible conditions. 3.3.5.3 Topology of the PNNDM

An NN platform was used to implement the Bayesian classifier and the Parzen- Cacoullos method by Specht (1990), which was called PNN. The PNN is composed of many connected processing units or artificial neurons which are laid in four successive and fixed layers, namely input layer, pattern layer, summation layer and output layer as shown in Figure 3-11. For classifying a query pipe into one of the three conditions, the PNNDM computes the posterior probabilities of the query pipe to each of the condition classes. The condition class that the query pipe has the highest posterior probability is then assigned to the query pipe. The functions of four layers in the PNNDM are described below. Factor K Factor 3 Factor 2 Pipe condition Con trib u ting factors Factor 1

Input Layer Pattern Layer Summation Layer Output Layer

Arg-max

Figure 3-11: Topology of PNNDM A/ Input layer

In the input layer, the number of neurons is equal to the number of input factors. This layer does not perform any computations and simply distributes the query pipe with a vector of contributing factors to the neurons in the pattern layer.

B/ Pattern layer

Suppose that there are N training data in whichN₁,N₂andN₃ (N₁+N₂+N₃=N)

this layer is equal to N and these neurons are divided into three classes with

corresponding numbers of neuronsN₁,N₂andN₃as shown by three sets of neurons of

different shapes in Figure 3-11. The neurons with square shape are for training vectors of condition 1. The neurons with circular shape are for training vectors of condition 2. The neurons with elliptical shape are for training vectors of condition 3. This means that each neuron in a class was designed to hold a training vector of contributing factors. When the incoming signal of the query pipe is presented, these neurons compute the exponential part of the Equation (3-28) (i.e.exp || ₂ ||2

2 i σ ⎛ − ⎞ ⎜ ⎟ ⎝ ⎠

X X _{) and transfer the values to}

the summation layer C/ Summation Layer

The summation layer contains one neuron for each class. The number of neurons in the summation layer is equal to the number of classes. This layer computes the posterior probabilities of a query pipe being into one of the three possible conditions from Equation (3-28) on the incoming signals. The computed probabilities are then sent to the output layer.

D/ Output Layer

There is one neuron in this layer. The ‘Arg Max’ activation function of this neuron (Kim et al. 2005) simply compares the incoming probabilities and assigns the query pipe into the class that has the highest posterior probability.

3.3.5.4 Training PNNDM

Training of the PNNDM was actually to store the training data in the system by assigning their values into the neurons in the pattern layers. However, there is still one parameter, the smoothing parameterσof the Gaussian kernel, which needs to be estimated. This can be done by trial and error search so that the MSE of training data has the lowest value, or equivalently, the number of correct predictions on the training data has the highest possible value. A correct prediction is counted if the predicted pipe condition is same as the known pipe condition in the training data. The training of

PNNDM in this study was done by searching for the maximum number of correct predictions.