There are couple of aggregation techniques, when developing composite indica- tors. However, one of two major aggregation techniques is often used:
5. Data and Methods of Development
- Linear Aggregation (LA): It is a summative aggregation method usually used with equal weighting criteria. LA can be formulated using the following formula: SCIc= Q X q=1 wix0ic (5.10)
- Geometric Mean (GM): It is a multiplicative aggregation technique and can be performed using the GM equation, which is stable and highly recom- mended for aggregating SCI, and such technique is widely studied in fuzzy set theory (Zimmermann and Zysno,1983). The GM can be expressed using the following equation:
SCIc= I Y i=1 xwi i,c (5.11)
where SCI is the aggregated composite score for a certain country c, x is the value of each variable or indicator i for country c raised to the power w, which is the weight assigned for each variable or indicator i.
5.7
Computational Intelligence Techniques
Computational Intelligence (CI) methods are becoming popular for their precision modelling, clustering, predictions and trend analysis. Some methods could be used as an alternative to the statistical techniques. The proceeding sections presents a review of the methods used in this study.
5.7.1
Fuzzy c-Means Algorithm
Fuzzy c-Means (FCM) is a clustering algorithm introduced by Dunn (1973) ini- tially and improved by Bezdek et al. (1984). FCM stems from the famous K- means algorithm, but it differs in that the data point has partial membership in a cluster, with grades between 0 and 1. Therefore, FCM allows one piece of data to be a member of two or more clusters according to its degree of member- ship, which is determined based on the distance (usually the Euclidean) between
5. Data and Methods of Development
a data point and the cluster centre. At each iteration, an objective function is minimized to find the best location for the clusters and its values are returned in objective function.
Fuzzy clusters can be characterised by class membership function matrix, and cluster centres are determined first at the learning stage, and then the classifi- cation is made by the comparison of euclidean distance between the incoming features and each cluster centre.
For a data set represented as X = {x1, x2, . . . , xj. . . , xn} ⊂ Rs into c clusters,
where 1 < c < n; the fuzzy clusters can be characterized by a c × n membership function matrix U , whose entries satisfy the following conditions:
c X i=1 ui,j = 1, j = 1, 2, . . . , n (5.12) 0 < n X j=1 ui,j < n, i = 1, 2, . . . , c (5.13)
where ui,j is the grade of membership for xj data entry in the ith cluster. Cluster
centres are determined initially at the learning stage. Then, the classification is made by comparison of distance between the data points and cluster centres. Clusters are obtained by the minimisation of the following cost function via an iterative scheme. J (U, V ) = n X j=1 c X i=1 (ui,j)2kxj − vik (5.14)
where V = {v1, v2, . . . , vi, . . . vc} are c vectors of cluster centres with vi represent-
ing the centre for ith cluster.
To calculate the centre of each cluster, the following iterative algorithm is used.
1. Estimate the class membership U . 2. Calculate vectors of cluster centres
5. Data and Methods of Development
V = {v1, v2, . . . , vi, . . . vc} using the following expression:
vi = Pn j=1(ui,j)2xj Pn j=1(ui,j)2 i = 1, 2, . . . , c (5.15)
3. Update the class membership matrix U with:
ui,j = 1 Pc r=1 kx j−vik kxj−vrk 2 i = 1, . . . , c; j = 1, . . . , n (5.16)
4. If control error (defined as the difference between two consecutive iterations of the membership matrix U ) is less that a pre-specific value, then the process can stop. Otherwise process will repeat again from step 2.
After a number of iterations, cluster centres will satisfy the minimisation of the cost function J to a local minimum.
5.7.2
Vector Quantization
Vector Quantization (VQ) is a classic technique from signal processing and usually used for data compression, to recode data into more reduced forms. One such technique is which maps groups of input symbols, called vectors, onto a small set of vectors, called the “codebook”. Each vector in the codebook is a codeword (Zheng et al., 1997). VQ is an approximator, similar to that of rounding-off to the nearest integer (Linde et al., 1980).
To illustrate the concept let us assume that there is a training sequence consist- ing of n source vectors T = x1, x2, ..., xn. The training sequence can be obtained
from some large database. n is assumed to be sufficiently large so that all the statistical properties of the source are captured by the training sequence. It is assumed that the source vectors are k-dimensional, e.g.,
Xn = {xn1, xn2, . . . , xnk}, n = 1, 2, ..., N
let n be the number of code-vectors and let C = {C1, C2, ..., Cn}, for the codebook.
Each code-vector is k-dimensional, e.g., cn = (cn1, cn2, ..., cnk), n = 1, 2, ..., N .
5. Data and Methods of Development
Figure 5.3: Illustration of the LBG LVQ clustering process.
Let Sm be the encoding region associated with code-vector cm and let P =
{S1, S2, ..., Sm} denote the partition of the space. If the source vector xm is
in the encoding region Sm, then its approximation represented by Q(xn) is Cm:
Q(xn) = cm, if xn∈ Sm.
(Gray, 1984); (Wu and Guan, 1994).
LBG VQ Design Algorithm
The main purpose of utilising the Linde Buzo and Gray Vector Quantization (LBG VQ) algorithm is because of its ability to learn to detect and produce the centroid, the codebooks and codevectors between given scores. The LBG VQ algorithm is an iterative method which alternatively solves the above two opti- mality criteria (Linde et al., 1980). The algorithm requires an initial codebook. This initial codebook is obtained by the splitting method. In this method, an initial codevector is set as the average of the entire training sequence. This code- vector is then split into two as the initial codebook. The final two codevectors are splitted into four and the process is repeated until the desired number of codevectors is obtained (Ramamurthi and Gersho, 1986). Figure 5.3 illustrates the iteration and splitting process for some randomly generated data points. The LBG VQ algorithm is summarized in Appendix A.
5. Data and Methods of Development