3.1 Principal component analysis
PCA is a dimensionality reduction technique that aims to transform original correlated
multi-dimensional data into a few uncorrelated principal components. Specifically,
PCA achieves this by decomposing the covariance matrix of the dataset to extract orthogonal
components, each capturing maximum variance in the data. By selecting components that
cumulatively explain most of the original information, PCA balances data fidelity
and dimensionality reduction, thereby minimizing computational complexity while preserving
critical features [27].
Correlation analysis quantifies the degree of linear dependence between two or more
variables, serving as a fundamental tool to identify interdependencies in multi-dimensional
datasets. When variables exhibit high collinearity, this analysis allows substituting
a representative variable for redundant ones, thereby reducing analytical complexity
while preserving data integrity. In SPSS, there are three types of correlation analysis
methods: Pearson, Spearman, and Kendall. The Pearson correlation coefficient has several
advantages over Spearman and Kendall: it accurately measures the strength and direction
of the linear relationship between continuous variables, serving as an important prerequisite
for regression analysis. It also provides higher test efficiency when the data meet
the assumptions of normal distribution and homoscedasticity. Its squared value can
intuitively explain the proportion of variance explained by the variables, and since
it is calculated based on the original data values, it fully utilizes the information
within the data. It is more suitable for analyzing the linear relationship between
continuous variables that follow a normal distribution. This paper adopts the Pearson
correlation analysis method [28].
The data processing algorithm in this study is designed to optimize the intrinsic
Y-parameters. Prior to optimization, the measured S-parameters undergo de-embedding
to derive the intrinsic Y-parameter matrix. The real and imaginary components of the
intrinsic Y-parameters ($Y_{11}$, $Y_{12}$, $Y_{21}$, $Y_{22}$) are then separated
into eight datasets, which are imported into SPSS for Z-score standardization to eliminate
dimensional unit effects. Subsequent PCA is performed, with real and imaginary parts
processed separately for each biasing condition. From Figs. 2 to 3, it can be seen that the extracted eigenvalue of the first principal component of
the real part or imaginary part is the largest among all principal component eigenvalues
and greater than 1. Similarly, the eigenvalues of the first principal component in
other cases follow the same pattern. The variance contribution rate and cumulative
contribution rate of the first principal component ($F_1$) are 97.335% for the real
part and 96.433% for the imaginary part. According to statistical criteria, when a
single principal component explains more than 80% of the total variance, it is considered
to effectively capture the primary information structure of the original data. Our
results show that $F_1$ accounts for over 96% of the total variance in both the real
and imaginary parts—far exceeding the commonly used threshold. In contrast, the second
principal component ($F_2$) contributes less than 4% in both cases, primarily reflecting
measurement noise or minor fluctuations. Therefore, retaining only $F_1$ not only
aligns with the PCA dimensionality reduction principle (Kaiser criterion: eigenvalue
> 1 and high explained variance) but also significantly reduces optimization complexity
while preserving model accuracy. Thus, this paper uses the first principal component
to optimize the intrinsic Y-parameters.
Fig. 2. Scree plot of real parts under bias conditions of V GS = 0 V and V DS = 12
V.
Fig. 3. Scree plot of imaginary parts under bias conditions of V GS = 0 V and V DS
= 12 V.
$F_1$ and the fitting coefficients $a_{ij}$, $b_{ij}$ can be derived from Eqs. (31) and (32).
After $F_1$ is known, Pearson correlation analysis is performed between $F_1$ and
each of the eight groups of intrinsic Y-parameter data. The analysis shows that each
group of data is significantly correlated with $F_1$, so the Pearson correlation coefficients
are the fitting coefficients $c_{ij}$ and $d_{ij}$. With the fitting coefficients
$a_{ij}$, $b_{ij}$, $c_{ij}$, and $d_{ij}$ all known, the real and imaginary parts
of $Y_{gs}$, $Y_{12}$, $Y_{gm}$, and $Y_{ds}$ derived from Eqs. (16)-(19) are shown in Eqs. (33)-(40):
The transformation relationships between $Y_{11}$, $Y_{12}$, $Y_{21}$, $Y_{22}$ and
$Y_{gs}$, $Y_{12}$, $Y_{gm}$, $Y_{ds}$ are delineated in Eqs. (41)-(44). Once the real and imaginary components of $Y_{gs}$, $Y_{12}$, $Y_{gm}$, and $Y_{ds}$
are obtained from Eqs. (33)-(40), substituting these values into Eqs. (41)-(44) yields the real and imaginary parts of the intrinsic Y-parameters $Y_{11}$, $Y_{12}$,
$Y_{21}$, $Y_{22}$ prior to optimization.
Subsequently, the eight expressions for the real and imaginary parts of $Y_{11}$,
$Y_{12}$, $Y_{21}$, and $Y_{22}$ are substituted into Eq. (31) to obtain the expression for $F_1$. Substituting the expression of $F_1$ into Eqs.
(45) and (46) yields the expressions for the real and imaginary parts of the optimized intrinsic
Y-parameters. Finally, substituting the real and imaginary part expressions into Eq.
(47) gives the expressions for the optimized intrinsic Y-parameters $Y_{11}$, $Y_{12}$,
$Y_{21}$, and $Y_{22}$. These four expressions are the ones to be substituted into
the Osprey Optimization Algorithm (OOA) for subsequent calculations.
The fitting coefficients $a_{ij}$, $b_{ij}$, $c_{ij}$, and $d_{ij}$ under different
bias conditions are shown in Tables 3 and 4.
Table 3. Fitting coefficients aij, bij under three biasing conditions.
|
Coefficients
|
$V_{GS} = 0\text{ V}$, $V_{DS} = 12\text{ V}$
|
$V_{GS} = -1\text{ V}$, $V_{DS} = 12\text{ V}$
|
$V_{GS} = -2\text{ V}$, $V_{DS} = 28\text{ V}$
|
|
$a_{11}$
|
0.505294
|
0.499970
|
0.512651
|
|
$b_{11}$
|
0.504449
|
0.502817
|
0.557730
|
|
$a_{12}$
|
-0.504548
|
-0.501958
|
-0.511511
|
|
$b_{12}$
|
-0.487076
|
-0.481838
|
0.274238
|
|
$a_{21}$
|
0.483221
|
0.495960
|
0.462868
|
|
$b_{21}$
|
-0.504878
|
-0.504181
|
-0.550641
|
|
$a_{22}$
|
0.506656
|
0.502090
|
0.511186
|
|
$b_{22}$
|
0.503341
|
0.501539
|
0.557248
|
Table 4. Fitting coefficients cij, dij under three biasing conditions.
|
Coefficients
|
$V_{GS} = 0\text{ V}$, $V_{DS} = 12\text{ V}$
|
$V_{GS} = -1\text{ V}$, $V_{DS} = 12\text{ V}$
|
$V_{GS} = -2\text{ V}$, $V_{DS} = 28\text{ V}$
|
|
$c_{11}$
|
0.976998
|
0.975044
|
0.998166
|
|
$d_{11}$
|
0.998214
|
0.997014
|
0.991750
|
|
$c_{12}$
|
-0.976327
|
-0.978323
|
-0.985443
|
|
$d_{12}$
|
-0.929807
|
-0.902103
|
0.477317
|
|
$c_{21}$
|
0.961789
|
0.984532
|
0.876452
|
|
$d_{21}$
|
-0.990617
|
-0.987424
|
-0.979771
|
|
$c_{22}$
|
0.978096
|
0.977698
|
0.977358
|
|
$d_{22}$
|
0.996440
|
0.995358
|
0.989187
|
3.2 Osprey optimization algorithm
The Osprey Optimization Algorithm (OOA) is a meta-heuristic optimization framework
proposed by Mohammad Dehghani and Pavel Trojovský in October 2022 [29]. Inspired by the strategic foraging behavior of ospreys, OOA emulates the predatory
tactics of these cosmopolitan raptors, which specialize in catching live fish weighing
50 grams to 2 kilograms. Equipped with exceptional visual acuity, ospreys detect fish
positions in water while soaring 10 to 40 meters above the sea surface. Their predation
sequence unfolds in three stages: initial patrolling at a fixed altitude, rapid target
approach upon detection, and plunge-diving with talons extended to capture prey. Successful
hunts conclude with carrying prey to a perch for feeding. The series of actions from
an osprey capturing prey to carrying it to a suitable location for feeding are all
based on intelligent natural behaviors, and the Osprey Optimization Algorithm is also
created by drawing inspiration from these behaviors. In OOA, the research subject
is an osprey population model comprising multiple individuals, with its operational
process structured into three core phases [28]:
Step 1: Initialize the population size $N$ and the maximum number of iterations $T$,
then initialize the positions of the osprey population using Eqs. (48)-(50) and evaluate the objective function for each individual. Here, $X$ denotes the population
position matrix where $X_i$ is the $m$-dimensional position vector of the $i$-th osprey
($m$ being the total number of variable dimensions), $x_{i,j}$ represents the $j$-th
dimensional variable with lower and upper bounds $lb_j$ and $ub_j$, $r_{i,j}$ is a
random value in [0,1], $F$ is the objective function matrix storing values $F_i$ for
each osprey. The position corresponding to the optimal $F_i$ (minimum or maximum,
depending on the optimization goal) is designated as the current optimal solution,
which is iteratively updated in each iteration.
Step 2 (Stage 1): Exploration Phase (Osprey Position Recognition and Hunting). In
this phase, ospreys first identify the positions of prey on the sea surface and then
capture the fish. The second step of the OOA algorithm is established based on this
natural behavior, which also enhances the exploration ability of OOA to identify optimal
positions and escape local optima. For each osprey, the position of its prey fish
is the optimal position, and the set of fish positions for each osprey is listed in
Eq. (51):
where $FP_i$ represents the fish position set of the $i$-th osprey, and $X_{best}$
denotes the optimal position, which refers to the position corresponding to the optimal
objective function value. The osprey randomly selects a fish from the fish position
set to attack, and its new position is given by Eq. (52). However, since its position must not exceed the constraints, Eq. (53) is used to constrain its position.
where $x_{i,j}^{P1}$ represents the updated position of the $j$-th dimensional variable
for the $i$-th osprey in the first phase, $r_{i,j}$ is a random number between [0,1],
$SF_{i,j}$ denotes the position of the fish selected by the $i$-th osprey, and $I_{i,j}$
is a random number in [1,2]. If the objective function value of the osprey after position
update is better than the previous one, the previous position is replaced; otherwise,
the original position remains unchanged, as shown in Eq. (54).
where $X_i^{P1}$ represents the new position of the $i$-th osprey in the first phase,
and $F_i^{P1}$ denotes the objective function value of the $i$-th osprey in the first
phase.
Step 3 (Stage 2): Exploitation Phase (Carrying Fish to a Suitable Position). This
phase emulates ospreys transporting captured fish to a perch for feeding, thereby
enhancing OOA's exploitative capability to conduct local searches and converge toward
optimal regions. During this stage, ospreys probabilistically explore feasible positions,
with their updated coordinates defined by Eq. (55). To ensure positional validity, Eq. (56) enforces constraints that prevent variables from exceeding predefined bounds. This
mechanism mirrors the raptors' strategic selection of feeding sites, balancing local
refinement with solution feasibility to drive convergence while adhering to problem
constraints.
where $x_{i,j}^{P2}$ represents the new position of the $j$-th dimensional variable
for the $i$-th osprey in the second phase, $r$ is a random number between [0, 1],
and $t$ denotes the current iteration number. If the objective function value of the
osprey after position update is better than the previous one, the previous position
is replaced; otherwise, the original position remains unchanged, as shown in Eq. (57):
where $X_i^{P2}$ represents the new position of the $i$-th osprey in the second phase,
and $F_i^{P2}$ denotes the objective function value of the $i$-th osprey in the second
phase.
This study utilizes the Osprey Optimization Algorithm (OOA) to extract and optimize
intrinsic parameters. Before executing the algorithm, variables, constraints, and
the objective function must be input. Specifically, the variables consist of twelve
intrinsic parameters: $C_{gs}$, $C_{gd}$, $C_{ds}$, $R_i$, $R_{gd}$, $R_{ds}$, $G_m$,
$\tau$, $R_{gdf}$, $R_{gsf}$, $R_L$, and $L_{ds}$. The objective function is defined
as the error function [29], which quantifies the discrepancy between measured and simulated values to guide
the optimization process.
To ensure full reproducibility of the proposed PCA-assisted OOA calibration framework,
we provide complete implementation details as follows. The Osprey Optimization Algorithm
(OOA) is configured with a population size of 50 search agents and a maximum of 200
iterations. Parameter bounds are enforced via hard clipping: after each position update,
candidate solutions are projected back into the feasible range using element-wise
min and max operations with respect to predefined lower and upper bounds. The initial
population is generated by uniform random sampling within these bounds, and the MATLAB
random number generator seed is fixed to 12345 (via rng(12345)) to guarantee deterministic
results. All experiments were repeated over 10 independent runs with identical settings
but different random initializations; the relative standard deviation of the final
objective function value across these runs was consistently below 0.1%, confirming
high stability and robustness of the optimization process. Regarding PCA preprocessing,
the intrinsic admittance parameters (Y-parameters) are first decomposed into real
and imaginary parts and concatenated into a real-valued feature vector. Each feature
dimension is standardized to zero mean and unit variance prior to PCA. Only the first
principal component (PC1) is retained to guide the optimization, which accounts for
more than 96% of the total variance across all bias conditions. Crucially, the model
fitting error is evaluated on the full, uncompressed Y-parameter data, ensuring that
the performance metric remains unaffected by dimensionality reduction and that no
information loss biases the final assessment.
This study proposes two methodologies for intrinsic parameter extraction and optimization.
Method 1 (OOA) employs the Osprey Optimization Algorithm (OOA) for direct parameter
extraction and optimization, whereas Method 2 (PCA-OOA) integrates Principal Component
Analysis (PCA) and OOA in a two-stage framework: PCA first optimizes the intrinsic
Y-parameters, followed by OOA for subsequent parameter extraction and refinement.
To ensure comparability, initial conditions for both methods are uniformly set. The
workflow of the PCA-OOA algorithm for intrinsic parameter extraction is depicted in
Fig. 4, illustrating the sequential integration of PCA for pre-optimization and OOA for
fine-tuning, which highlights the synergistic effect of dimensionality reduction and
heuristic optimization. The extraction process using the PCA-OOA algorithm is shown
in Fig. 4.
Fig. 4. Flowchart of intrinsic parameter extraction for PCA-OOA algorithm.