Health status assessment of unmanned aerial vehicle (UAV) attitude control system based on an improved multivariate state estimation method

2.2.1. An improved multivariate state estimation algorithm for the ACS of the FWUAV

First of all, the status observation needs to select the data to characterize the internal characteristics, which are generally reflected directly or indirectly based on the feedback signal data by the sensor of ACS. The status data generated by the normal operation of the system is taken as the historical health data set. By analyzing the relationships between the parameters in historical health data sets, the MSET^[24] applies the learned relationships between parameters to the observation of the test data, while using the learned patterns to assess the current status of the current test system data. Although there may be coupling and redundancy among the parameters, each quantity can reflect certain characteristics of the actual physical process. The results of the status estimation are obtained by comparing the patterns in the historical health data with the current test data and weighting the states of each parameter. The weights are determined by applying the learned patterns to the overlapping degree of the historical health data and the current test data of each parameter. To sum up, the MSET has certain learning ability and is a machine learning regression technology with supervision ability. In the field of machine learning, the historical health data is labeled as "normal state" and used as the training data, and the current test data is used as the observation data to be evaluated for obtaining final assessment results, which is the essential feature of supervision ability.

The basic principle of MSET is as follows: Supposing that the $$ n $$-dimensional measurement data of a part up to time $$ t_m $$ is obtained, and the data is expressed as a matrix form, in which the observation data of characteristic parameters at any time is a column vector, then the $$ n \times m $$ order observation matrix is obtained. For the ACS of the FWUAV, which is shown in Figure 2, where $$ n $$ is the number for the selected characteristic parameters of the ACS sensor ($$ n=19 $$), and $$ m $$ is the number of time points observed.

Figure 2. The observation matrix for FWUAV.

Taking $$ x_{ji} $$ to denote the observed value of the parameter $$ j $$ at the moment $$ t_i $$, then the observed value of each feature parameter at the moment $$ t_i $$ constitutes the observation vector $$ X({{t}_{i}})={{\left[ {{x}_{1i}}, {{x}_{2i}}, ..., {{x}_{ni}} \right]}^{\text{T}}} $$, where $$ n=19 $$. In addition to the observation data, it is set that all the historical data under the normal operation of the component for the ACS can be obtained, which is denoted as $$ T $$ and called the training data; i.e., $$ T $$ covers the dynamic range of the feature parameter values during the normal operation of the ACS, but it is not required to obtain all the values of the feature parameters under the normal state of the components for the ACS. A number of typical values of the feature parameters are selected from the training data $$ T $$, and the resulting matrix is called the memory matrix, denoted as $$ {\aleph} $$. If the training data that are not selected into the memory matrix, which are called the remaining training data, denoted as $$ L $$, then $$ T = {\aleph} \cup L $$. The linear weighting of the data in the memory matrix is used as an estimation of the state of the evaluation object, denoted as $$ X_{est} $$; i.e., $$ X_{est} = {\aleph} \cdot W $$, $$ W $$ is a weight vector, and the value is obtained by minimizing the estimation error vector $$ \epsilon_{X} $$.

Defining the state estimation error as $$ {{\varepsilon }_{X}}={{X}_{obs}}-{{X}_{est}} $$, where $$ X_{obs} $$ is the observed value, and the minimization of $$ \epsilon_{X} $$ implies the shortest length of the error vector, which can be written as

Clearly, (1) is equivalent to^[24]:

Because of $$ X_{est} = {\aleph} \cdot W $$, calculating the derivative of (2) with respect to $$ W $$ and setting it to zero to obtain the minimum value of the squared error, one has

Then, the weight vector $$ W $$ is obtained by solving (3); we have

The state estimation $$ X_{est} $$ can be obtained from the weight vector obtained as follows^[24]:

Then, the residual of the estimation can be described as

where $$ R_X $$ is the actual residual.

Applying the above estimation procedure to the remaining training data yields the following estimations:

Similarly, the residual of the remaining training data is obtained as

where $$ R_L $$ is the health residual.

Remark 1: In the estimation of the weight vector, $$ {{{\aleph}}^{\text{T}}}\cdot {\aleph} $$ is required to be invertible, which is generally difficult to be satisfied in practical applications. For example, when the number of states (columns of $$ {\aleph} $$) in the memory matrix $$ {\aleph} $$ is smaller than the number of state parameters (rows of $$ {\aleph} $$), $$ {\aleph} $$ is not full rank. To solve the matrix invertibility problem, it is common to use the MSET method with the nonlinear operator "$$ \otimes $$" instead of the vector product "$$ \cdot $$". In practice, the Euclidean distance operator^[25] and the Manhattan distance operator^[26] are often used as the nonlinear operators, which are defined respectively as

2.2.2. Dynamic Memory Matrix Construction Algorithm Based on Double-K Algorithm

The construction of the memory matrix is the key to the performance of MSET^[24]. The traditional method usually uses a fixed memory matrix, and the construction method is prone to cause information redundancy to reduce the overall computational speed on the one hand, and on the other hand, the estimation accuracy based on a fixed memory matrix may degrade over time because the operating conditions are constantly changing and may require frequent maintenance^[27]. Therefore, it is necessary to improve the flexibility of memory matrix construction.

Due to the similarity among many data points in the memory matrix, which are all derived from normal operation of the device, especially with a large sample size, the redundancy within the memory matrix can be significantly reduced by representing one data point as a representative for a cluster of highly similar data. A dynamic memory matrix construction method is designed based on the K-means and KNN algorithms. As two typical clustering methods, the K-means demonstrates the high computational efficiency and the strong adaptability, enabling the partitioning of the dataset into multiple categories based on their attributes^[28]. The core of the KNN algorithm is to infer the output based on the characteristics of several known samples that are nearest to the input^[29].

In this paper, the K-means algorithm is selected for constructing the initial dynamic memory matrix and the KNN algorithm is utilized for constructing the dynamic memory matrix based on different observations. To construct the initial memory matrix, the $$ m_1 $$ clustering centers in the normalized ACS training matrix are selected as the state vectors to construct the initial memory matrix, where $$ m_1 $$ is not more than 50% of the state vectors of the training matrix. This method is implemented using Euclidean distance and the improved K-means algorithm as given below:

Step 1: The same method of normalization is chosen for all matrices and vectors, which can be expressed as $$ {{z}_{ij}}=\frac{{{x}_{ij}}-{{\mu }_{i}}}{{{\sigma }_{i}}} $$ and $$ {{z}_{ij}}=\frac{{{x}_{ij}}-{{ min}_i}}{{ max}_{i}-{ min}_{i}} $$, where $$ x_{ij} $$ is the initial matrix vectors.

Step 2: The $$ m_1 $$ state vectors are randomly selected as clustering centers in the normalized ACS training matrix, $$ A = \left[ {{A_1}, {A_2}, \cdots , {A_{m_1}}} \right] $$.

Step 3: The distance $$ d\left( {{X_i}, {A_j}} \right) $$ between the remaining samples $$ X_i $$ to each cluster center $$ A_j $$ is calculated separately using Euclidean distance and classified into the class corresponding to the cluster center with the smallest distance.

Step 4: For each category $$ A_j $$, recalculating the cluster center (i.e., the mean value $$ {\mu _j} = {\left( {{\mu _{j, 1}}, {\mu _{j, 2}}, \cdots , {\mu _{j, p}}} \right)^T} $$ of each feature), finding the Euclidean distance $$ d\left( {{X_i}, {\mu _j}} \right) $$ of the respective state vectors to the cluster center and selecting the closest state vector to the mean value as the new cluster center $$ {{A'}_j} $$. Then, we have

where $$ k_j $$ denotes the number of state vectors in the $$ j $$-th cluster, and $$ {x_{i, l}^j} $$ indicates the $$ l $$-th characteristic parameter of the $$ i $$-th state vector in cluster $$ A_j $$, and $$ {l = 1, \cdots , p} $$.

Step 5: Repeating steps 3 and 4 until the sum of squares of the errors $$ I_{SSE} $$ in the cluster no longer changes, one has

Step 6: The initial memory matrix $$ \aleph $$ is constructed by taking the final $$ m_1 $$ cluster centers of state vectors.

The initial memory matrix constructed using this method not only has better flexibility, but also helps remove redundant vectors in the matrix by clustering, and it can reduce useless calculations in the traditional MSET^[27]. Moreover, to obtain the dynamic memory matrix, the steps for calculating the dynamic memory matrix of the observation process using the KNN algorithm are as follows:

Step 1: The same method of normalization is chosen for all matrices and vectors.

Step 2: For the observation input $$ {{X}_{obs}}=[{{X}_{1}}\cdots {{X}_{m}}] $$, the Euclidean distance $$ d_i\left( {{X_{ob{s_i}}}, {Y_j}} \right) $$ between each of the state vectors $$ X_{obs_{i}} $$ and each of the state vector $$ Y_j $$ in the training set $$ T $$ is calculated separately, which can be written as

where $$ {x_{obs{_{i, k}}}} $$ is the $$ k $$-th characteristic parameter of the $$ i $$-th state vector in the observation matrix, and $$ y_{{j, k}} $$ is the $$ k $$-th characteristic parameter of the $$ j $$-th state vector in the training matrix.

Step 3: Upon arranging the distance vector $$ {D_i} = {\left( {{d_i}_{, 1}, {d_{i, 2}}, \cdots , {d_{i, m}}} \right)^T} $$, the $$ k $$ samples with the smallest distances are selected from the training set $$ T $$, thereby identifying the $$ k $$ nearest neighbors denoted as $$ {T_i} = \left[ {{Y_{i, 1}}}, {{Y_{i, 2}}}, \cdots, {{Y_{i, k}}} \right] $$.

Step 4: For the training matrix constructed to the observation vector $$ X_{obs_{i}} $$, the mean value for each characteristic parameter is taken as the corresponding training vector element, and finally, the corresponding training vector $$ {{\bar Y}_i} $$ will be obtained as follows:

Step 5: The matrix consisting of the training vectors corresponding to each observation vector is used as the memory matrix $$ \aleph $$.

According to the analysis above, the double-K algorithm is used to construct the dynamic memory matrix, which plays a pivotal role in minimizing information redundancy within the matrix. This construction process is paramount for optimizing the performance of the algorithm, as it systematically identifies and retains only the most relevant and non-repetitive data points from the input dataset. The resulting dynamic memory matrix serves as a refined representation of the dataset, which can facilitate more precise and efficient computation during the algorithm's execution. The specific construction process can be represented by Figure 3.

Figure 3. Dynamic memory matrix construction process.

2.2.3. MSET based on double-K dynamic memory matrix

Assuming that the object to be evaluated has $$ p $$ feature quantity, the implementation steps of the MSET based on the optimization of the double-K algorithm are as follows:

Step 1: Obtain the training data set and its statistical characteristics. The characteristic parameter data generated under normal working conditions is regarded as health status data, and the health status data of the ACS is taken as training data $$ T $$ to obtain the statistical characteristics of each characteristic quantity, including the mean value, the standard deviation, the maximum value and the minimum value. The training data $$ T $$ is a matrix containing the state quantity of each characteristic parameter at each moment, and the number of columns for the matrix $$ T $$ is the number of moments $$ K $$, the number of rows is the number of characteristic parameters $$ p $$; then, the dimension of the matrix $$ T $$ is $$ p \times K $$ as follows:

where $$ K>2p+2 $$.

In the step of obtaining the mean value, the standard deviation and the maximum and minimum values of the typical statistical characteristics for the assessment object under the health status from the historical health data, the health status does not need to contain all the historical health data, but the selection of health data sets needs to have certain typical significance. Then, the estimation of the mean and standard deviation for the $$ i $$-th characteristic quantity can be expressed as

Remark 2: In practice, the entire historical data of the ACS may not necessarily contain the full dynamic range, so the selection of the training data $$ T $$ needs to be addressed. The selection can consider part of the observation data and the proposed observations generated using these observations as the training data, and the data updating can take a moving window approach; i.e., after each time the observation data are obtained, the last columns of the observation matrix are used as the training data $$ T $$. This selection of $$ T $$ can help monitor the abnormal trend of the ACS.

Step 2: Obtain the current test data set of the ACS. The data set of the ACS is selected from the test data of the current state characteristics to form the $$ p \times m $$-dimensional observation matrix $$ {{X}_{obs}}={{({{x}_{ij}})}_{p\times m}} $$, where $$ p $$ is also the number of characteristic parameters, $$ m $$ denotes the number of time points in the current time period, and the matrix element $$ x_ij $$ stands for the $$ x_{ij} $$ observation of the $$ j $$-th observed value, and the observation matrix $$ X_{obs} $$ is as follows:

Step 3: Standardization of the historical health data and the test data. Since the characteristic parameters are all FWUAV system parameters, but are of different types and involve diverse aspects of physical concepts, and, thus, have varying scales and levels of quantity. Thus, the training data $$ T $$ and the observation matrix $$ X_{obs} $$ need to be standardized first if the historical health data and test data are to be used to obtain the assessment results. The standardization method can adopt the z-score or min-max standardization formula, and the results of $$ x_{ij} $$ standardizing the observation matrix are $$ {{z}_{ij}}=\frac{{{x}_{ij}}-{{\mu }_{i}}}{{{\sigma }_{i}}} $$ and $$ {{z}_{ij}}=\frac{{{x}_{ij}}-{{ min}_i}}{{ max}_{i}-{ min}_{i}} $$, respectively. The obtained normalized observation data is denoted as $$ {{Z}_{obs}}={{({{Z}_{1}}\cdots , {{Z}_{m}})}^{\text{T}}} $$. Similarly, the normalized training data set is $$ T' $$, and the same normalization method needs to be used for the training data and the test data; otherwise, it may cause the problem of different data scales.

Step 4: The observation residual estimation method based on MSET is as follows: The dynamic memory matrix $$ {\aleph} $$ is constructed according to the double-K algorithm, where $$ {\aleph} $$ is a $$ p\times M $$ matrix; due to the dynamic memory matrix constructed by the double-K algorithm, the number of columns $$ m $$ of the matrix may change during the operation, but it must be guaranteed that $$ M>2p+2 $$ during the operation. The data from the data matrix $$ T' $$ that is removed from the memory matrix $$ {\aleph} $$ is called residual data and is denoted by the capital letter $$ L $$, i.e., $$ T_{p\times K}^{'}={{{\aleph}}_{p\times M}}\cup {{L}_{p\times N}} $$, where $$ M+N=K $$.

Remark 3: The construction of the memory matrix is key to the performance of the algorithm. In order to avoid the overfitting problem, the memory matrix should not have too much data.

In addition, the observation residuals $$ R_x $$ and $$ Z_{est} $$ are calculated. The observation residual estimation $$ R_X $$ is obtained from the observation matrix $$ Z_{obs} $$ and the memory matrix $$ {\aleph} $$ with:

where $$ \alpha >0 $$ is the constant regularization factor to be selected.

2.2.4. Component anomaly state estimation based on MEST & SPRT

The sequential probability ratio test (SPRT)^[30] is a method of measuring the degree of deviation for an observation from the normal state by comparing the probability of the observation occurring in the normal state and the abnormal state with the ratio of the probabilities. The basic principle is as follows: Assuming that $$ {{H}_{0}}: {\theta ={{\theta }_{0}}, }\; {{H}_{1}}:\theta =\lambda {{\theta }_{0}}\overset{\wedge}{=}{{\theta }_{1}}(\lambda >1) $$, denoting $$ {{Y}_{n}}=({{Y}_{1}}, ..., {{Y}_{n}}) $$ as the sample data, and taking the likelihood function as $$ L({{Y}_{n}};{{H}_{i}})=\prod\limits_{i=1}^{n}{f({{y}_{i}}\left| {{H}_{i}} \right.)} $$, where $$ f({{y}_{i}}\left| {{H}_{i}} \right.) $$ is the density function of the distribution for $$ {{Y}_{i}} $$. Defining the likelihood ratio as $$ \Lambda ({{Y}_{n}})=\frac{L({{Y}_{n}};{{H}_{1}})}{L({{Y}_{n}};{{H}_{0}})} $$, the determination of $$ H_0 $$ and $$ H_1 $$ is obtained by comparing $$ \Lambda ({{Y}_{n}}) $$ with an upper limit value $$ {\cal B} $$ and a lower limit value $$ {\cal A} $$. The decision-making rules are as follows: (ⅰ) If $$ \Lambda ({{Y}_{n}})\ge {\cal B} $$, then $$ H_1 $$ is accepted; (ⅱ) If $$ \Lambda ({{Y}_{n}})\le {\cal A} $$, then $$ H_1 $$ is accepted; (ⅲ) If $$ {\cal A}<\Lambda ({{Y}_{n}})<{\cal B} $$, then make no decision, continue with the observation and repeat the above steps. Moreover, the upper limit value $$ {\cal B} $$ and lower limit value $$ {\cal A} $$ are defined as $$ {\cal A} \approx \frac{\beta }{1-\alpha } $$ and $$ {\cal B}\approx \frac{1-\beta }{\alpha } $$, respectively, for a given false alarm rate $$ \alpha $$ and missed alarm rate $$ \beta $$.

The input data of SPRT-based anomaly detection are health residuals $$ R_L $$ and actual residuals $$ R_x $$, and the basic idea of the test is to examine whether they come from the same object, if so, the assessment object is in a normal state; otherwise, it is in an abnormal state. The realization process of SPRT is given below for the two cases of normal overall and non-parametric. It should be noted that the following test method is only for one-dimensional scalar; for the detection of multi-dimensional observation vectors, one way is to introduce weight vectors to turn multi-dimensional vectors into one-dimensional vectors, and the other way is to test each feature parameter separately. In the traditional MSET&SPRT method, it is assumed that the residuals follow a normal distribution. The null hypothesis describes the health status of the assessment object, i.e., the mean value is 0 and the standard deviation is $$ \sigma $$. And the four following alternative assumptions describe the degradation state of the ACS: (ⅰ) the mean value shifts to +$$ \mathcal{M} $$, the standard deviation is $$ \sigma $$; (ⅱ) the mean value shifts to -$$ \mathcal{M} $$, the standard deviation is $$ \sigma $$; (ⅲ) the mean value is 0, and the standard deviation is $$ \mathcal{V} \sigma $$; (ⅳ) the mean value is 0, and the standard deviation is $$ \sigma / \mathcal{V} $$, where $$ \mathcal{M} $$ and $$ \mathcal{V} $$ are disturbance parameters determined before testing, which need to be analyzed according to the specific system and the behavior. In addition, the mean value of the healthy residuals $$ R_L $$ is first obtained and the null hypothesis is acquired by subtracting the mean value from the healthy residuals $$ R_L $$ before implementing the SPRT; the observation residuals of the observed data are used as inputs, and the mean value of the healthy residuals needed to be subtracted before performing the SPRT test. The SPRT implementation algorithm is as follows:

Step 1: The observation residual $$ R_X $$ is obtained.

Step 2: The health residual $$ R_L $$ is calculated. The estimation matrix of the remaining training data $$ L_{est} $$ and the health residual $$ R_L $$ are obtained from the remaining data $$ L $$ and the memory matrix $$ {\aleph} $$, the expressions for $$ L_{est} $$ and $$ R_L $$ are

Step 3: Calculate the degree of anomaly $$ \Theta $$ from the historical health status based on the SPRT. Calculating SPRT as an indicator of an abnormal state, and the $$ p\times m $$-dimensional observation residual matrix and $$ p\times N $$-dimensional health observation residual matrix are written in the following vector form:

Supposing that each column vector of the observation residual matrix $$ R_X $$ and the health residual matrix $$ R_L $$ have the multivariate normal distribution, we have

The probability ratio of the $$ R_X $$ to the $$ R_L $$ is taken as the anomaly degree as follows:

where $$ f(\cdot) $$ is the $$ p $$-dimensional multivariate normal distribution function, representing the probability density of the $$ p $$-dimensional vector $$ \xi $$, and its expression can be written as

where $$ \mu $$ and $$ \sum{_{\xi }} $$ are the mean value and the covariance of the distribution for $$ \xi $$ subject to the multivariate normal distribution, respectively.

Step 4: Calculate the component abnormal state index $$ H $$. The logistic function of the degree of deviation $$ \Theta $$ from the normal state of the component is used as the component abnormal state indicator, i.e., $$ H=\frac{2}{1+{{e}^{\Theta }}} $$.

2.3.1. Establishment of a set of indicators for assessing the ACS health status

The health status assessment indexes are both the direct parameters for assessing the health status of components and the basic parameters for assessing the overall health status of the ACS. The ACS enables the FWUAV to follow the attitude desired signal, and the dynamic performance, the steady-state performance, and the cumulative effect of time involved in the tracking process, as well as the key measurement parameters of each component for the control system, collectively constitute the set of health assessment indexes for the ACS of the FWUAV. Therefore, the index parameters are chosen in the health status assessment index system as the steady-state tracking error, the overshooting amount, the regulation time and the time error integral related to the attitude angle. Next, the above indicators are explained as follows:

(ⅰ) The steady-state error is an important dynamic measure of the accuracy for a system and is usually expected to be zero or not to exceed a certain range. It refers to the deviation from the commanded attitude signal and the actual attitude after the FWUAV has returned to a steady state after perturbations.

(ⅱ) The overshoot is an indicator describing the maximum degree of the deviation for the controlled variable from the steady state, and it is a dynamic indicator reacting to the stability of the transient process. The overshoot is one of the important dynamic indicators of the attitude maneuvering, and the overshooting too much may result in the attitude angle being too large and exceeding the flight envelope, which will affect the flight safety.

(ⅲ) The regulation time mainly measures whether the system can quickly return to a stable state when it is subjected to frequent perturbations, reflecting the rapidity of the system's response. After the system encounters perturbations, a shorter regulation time implies a stronger attitude maneuvering performance, while a longer regulation time implies a poor attitude maneuvering performance.

(ⅳ) The time integral index is the integral of the controlled quantity deviation from the static value along the time axis in the transient process, and the absolute value error $$ e $$ integral is chosen here, and the integral model is $$ f(e, t)=\left| e \right|, J=\int_{0}^{\infty }{\left| e \right|}dt $$, which is increased by the increase of the error amplitude and the growth of time. The larger the integral value, the control effect will be worse, and the control effectiveness will gradually become weaker with time.

Once the assessment hierarchy of the system has been determined, the data for the indicators to be assessed can be normalized and the weights assigned to the indicators can be determined.

2.3.2. Determination of the weights for the health assessment indicators based on the analytic hierarchy process

Health assessment indicators are juxtaposed, but the importance of each indicator to the ACS of the FWUAV is different, so the establishment of the health assessment indicator methodology actually needs to analyze the importance of each indicator parameter in the indicator set. The determination of indicator weights can enable the health assessment algorithm to effectively capture the important aspects of FWUAV system health. The determination of the weights for the health assessment indicators in this paper will be based on the analytic hierarchy process (AHP) method^[31]. The steps of the AHP methodology applied to health status assessment are as follows:

Step 1: Establish a hierarchical model based on the functional structure. A hierarchical structure model is constructed based on the functional components of the assessment system. Taking the ACS of the FWUAV as the top level, and the systematically expanding downward layer by layer according to the functional structure. Then, the complete system hierarchy is obtained and the appropriate assessment indicators are defined for each level.

Step 2: Normalize the indicators based on utility functions. A suitable utility function is selected based on the physical meaning of each indicator to normalize the indicator data. The choice of utility functions should comprehensively consider the impact of the indicator on the system. Additionally, the threshold values for the indicator attributes should be determined in accordance with practical considerations.

Step 3: Construct all judgment matrices for each level. The process of constructing judgment matrices involves applying pairwise comparison methods. The constructed judgment matrices are positive reciprocal matrices. The definition of a judgment matrix $$ B $$ is as follows:

where $$ n=19 $$ represents the number of indicators; $$ {{C}_{ij}}=f({{x}_{i}}, {{x}_{j}}) $$ denotes the scale of importance for comparing indicators $$ x_i $$ and $$ x_j $$, when $$ i=j $$, $$ C_{ij}=1 $$; and when $$ i\neq j $$, $$ C_{ij}=1/C_{ji} $$. The selection of $$ f({{x}_{i}}, {{x}_{j}}) $$ is determined by the importance level between indicators $$ x_i $$ and $$ x_j $$.

Step 4: Consistency Check. After constructing judgment matrix $$ B $$, a consistency check is required to ensure the logical validity of the weight selection. The consistency index ($$ CI $$) is calculated based on the maximum eigenvalue \(\lambda_{\max} \) of the judgment matrix $$ B $$ using \(CI = \frac{\lambda_{\max} - n}{n - 1} \). Then, the consistency ratio ($$ CR $$) is calculated using \(CR = \frac{CI}{RI} \), where $$ RI $$ is the random index corresponding to the order of the matrix. If $$ CR < 0.1 $$, the judgment matrix is considered consistent^[32]. If it does not meet this criterion, the matrix needs to be updated to achieve consistency.

Step 5: Perform the assessment based on the weights. The weight vector is calculated for a single layer using the following formula, known as the eigenvector method:

where $$ \omega_i $$ is the weight parameter and the $$ a_{ij} $$ is element of the eigenvector.