H. LIN ET AL.

140

Olkin (KMO) test and Bartlett’s test of sphericity to test the

data whether they are fit for FA. The values of KMO statistic

between 0.7 and 0.8 are good, values between 0.8 and 0.9 are

great and values above 0.9 are superb (Hutcheson & Sofroniou,

1999). For these samples, the value is 0.732, which falls into

the range of being good, so we are confident that the FA is

appropriate for these data.

The Bartlett’s test of sphericity measures the null hypothesis

that the original correlation matrix is an identity matrix. For a

satisfactory FA to proceed, some relationship between variables

are needed, in other words, we want this test to be significant as

a significa nt test tell s us the matrix is not an identity matrix. For

these data, Bartlett’s test is highly significant (p < .001). There-

fore, FA is appropriate (Field, 2005).

We adopt principal component analysis to extract factors and

varimax rotation method to progress factor rotation. Eigenval-

ues, the percent of variance attributable to each factor and the

cumulative variance of the first 14 factors are shown in Table 2.

From Table 2, we can see that the cumulative variance of the

first 14 factors has reached 91.129%. To ensure adequate in-

formation is maintained and the eigenvalue is greater than one,

ultimately there are 12 factors to replace the original 24 vari-

ables and 88.769% of total information is guaranteed.

According to rotated component score coefficient matrix,

every factor can be expressed by a linear combination of the

original variables. The main indices (loadings are more than 0.2)

explain the 12 factors are shown in Table 3. Further, if we use

each factor’s percentage of variance explained as weight and

sum up these 12 factors, the composite score of each sample

can be gained.

From the above analysis, only 12 factors can well reflect the

88.769% information of original 24 variables, thus greatly re-

ducing the dimension of evaluation and the correlation between

variables.

Back Propagation Neural Networks

Back propagation neural networks (BPNN) are multilayer

feed-forward neural networks (NN) based on back-propagation

algorithm. The nonlinear processing ability of BPNN can proc-

ess cognitive judgments in various complex environments ef-

fectively such as vague, incomplete and conflicting information.

It is the most widely used NN model.

BPNN is a supervised learning algorithm. It is necessarily a

multilayer perception (with input layer, hidden layers and out-

put layer). The network structure of BPNN is in Figure 1: input

vector is 1

, 2

,, m

and output vector is 1, 2,,

n. The learning process can be divided into two phases: (a)

the information flow goes through input layer, hidden layers,

output layer; (b) error back propagation network process, if the

NN model does not get expected output value in output layer,

the error signal propagates backward along the original path-

way layer by layer, and adjusts its weights and threshold value.

y y

y

Kolmogorov theorem of neural networks has proved that a full

learning three-layer BPNN can approximate any functions.

Therefore we choose a three-layer BPNN with only one hidden

layer. There is no theoretical guidance in selecting the number

of hidden layer nodes currently. Too many nodes will increase

the training time and weaken the networks’ generalization and

predictive ability, while too few nodes cannot reflect the corre-

lation between the follow-up value and previous value and the

model is insufficient. The number of nodes in hidden layer can

refer to the following formula: 1

mmna, whe1

m

is the number of hidden layer nodes, 1

m is the number of

input layer nodes, n is the number of output layer nodes, a

is a constant between one and ten (Rafael, 200

re

4).

Sample Classification and Neural

Networks Design

Choose 10 testing samples from 92 samples as a test set. In

test set each of the five states (not-at-all, a little bit, moderately,

quite-a-bit, extremely) has two samples respectively. The other

82 samples are as a training set.

To reflect the difference whether carrying out FA before

BPNN or not, we design two BPNN structures. In particular,

we name the former one as factor-analysis-back-propagation

(FABP) neural networks, while name the latter one as non-

factor-analysis-back propagation (NFABP) neural networks.

FABP: after FA, 12 factors are gained. Take the values

of 12 factors in training set as input, well then the num-

ber of input layer nodes is 12. Take the corresponding

composite scores of training samples as output and then

the number of output layer nodes is one. According to

the formula 1

mmna

and experiment results,

we decide the number of hidden layer nodes is eight. The

maximum training times are 1500.

NFABP: take the original data of training samples as in-

put, well then the number of input layer nodes is 24; take

the corresponding composite scores of training samples

Table 2.

Main variance explained.

Factors F1 F

2 F

3 F

4 F

5 F

6 F

7

Eigenvalues 3.025 2.874 2.226 2.100 2.067 1.994 1.917

Variance explained % 11.882 11.016 9.531 8.885 8.123 7.337 7.089

Accumulative variance explained% 11.882 22.898 32.429 41.314 49.437 56.774 63.863

Factors F8 F

9 F

10 F

11 F

12 F

13 F

14

Eigenvalues 1.882 1.843 1.756 1.087 1.052 0.876 0.853

Variance explained % 6.782 6.275 6.368 2.931 2.550 1.224 1.136

Accumulative variance explained % 70.645 76.920 83.288 86.219 88.769 89.993 91.129