Aerospace and Electronic Systems Magazine July 2018 - 11

Koleini, Roudbari, and Marefat
Table 2.

RMSE Associated with Predicted EGT Using Networks
Type α, β and γ
RMSE

Structures of Neural Network

0.1064

0.1055

0.1127

By using the Table 2, it can be considered that the neural
network type β has predicted the EGT parameter with RMSE of
0.1055 and thus has more accuracy than the two other networks
(i.e., α and γ). Therefore, the optimum network for predicting the
EGT will be the neural network type β.

Figure 8.

RMSE calculated based on predicted values of EGT using various
degrees of MPR.

ANALYSIS WITH MULTIPLE-POLYNOMIAL REGRESSION
(MPR) APPROACH
The regression models have a long history for applications such as
fitting data and mining relationships among features and variables.
As Hastie et al. [24] have mentioned, for cases with sparse data or
low signal-to-noise ratio, the regression models could lead to better
estimation results compared with some other nonlinear models with
more complexities. Also, from theoretical point of view, regression
models are supported by the results in statistical estimation theory
(the Gauss-Markov theorem) [29]. Details are given below.

Theorem.
If Y = Xβ, where Y is the output variable, X is the matrix of input
variables and β is the vector of parameters of the model (or the vec−1
T
T
ˆ
tor of regression coefficients), then the estimate β = X X X Y
is the Best Linear Unbiased Estimator among the linear class of
estimators for the vector of parameters β [29].
Application of regression models in gas turbine systems has
been reported in several researches [19]. In this study, MPR technique was used to estimate nonlinear relations between various parameters of the experimental gas turbine engine. For this purpose,
as mentioned in previous section, RPM was considered as input
variable of the system. EGT was also assumed as output variable.
In order to estimate nonlinear relation between input and output
parameters of engine, various degrees of MPR (from the first degree up to the sixth degree) were employed.
Just as before, (80%) of the experimental data set, which is
selected randomly, is used as training data to establish the MPR
models (i.e., MPR models with various degrees). Remaining data
were employed to evaluate MPR models' capability of estimating
the values of outputs associated with new sets of input. To avoid
over-fitting problems and to improve prediction capability of
various MPR models, K-fold cross-validation technique [24] was
employed. Various MPR model's degree of agreement between
predicted and measured values of EGT was studied by calculating
RMSE between predicted and measured values.

(

JULY 2018

)

Figure 9.

Predicted and measured values of EGT using the MPR technique.

Figure 8 compares values of RMSE generated by using various degrees of MPR to calculate EGT. As indicated in Figure 8,
MPR with degree of 3 has the lowest value of RMSE. However, by
increasing the degree of MPR, RMSE would enhance. Enhancement may occur due to over-fitting process. In other words, by increasing the degree of MPR, although resulted MPR model might
accurately approximate training data, but it would fail when it is
applied to the fraction of data set which is not used in training
procedures. Acceptable prediction capability and over-fitting occurrence prevention of MPR with degree of 3, makes it best of
them all to predict EGT.
Figure 9, for example, indicates EGT prediction performance
of MPR models with degree of 2 and 3. As previously mentioned,
this figure compares measured values of EGT with their associated predicted values, which have also been normalized by their
related characteristic quantities. To evaluate the degree of agreement between predicted and measured values of EGT, as displayed
in Figure 9, coefficient of determination (R2) was also calculated.

IEEE A&E SYSTEMS MAGAZINE

Table of Contents for the Digital Edition of Aerospace and Electronic Systems Magazine July 2018