A mental illness detection method based on MF-TSK fuzzy system

Abstract

This invention discloses a method for detecting mental illnesses based on the MF-TSK fuzzy system, belonging to the field of EEG signal processing and intelligent detection technology. The method first segments multi-channel EEG signals, extracting multi-dimensional features in the time, frequency, and spatial domains. In the antecedent part of the fuzzy system, improved fuzzy C-means clustering (FCM) is used, combined with Mahalanobis distance, to cluster features and obtain stable and semantically clear fuzzy rule antecedent parameters. In the consequent part, a Gaussian mixture model (GMM) is used to probabilistically model the fuzzy rules, achieving the parsing of rule parameters. In the multi-dimensional feature fusion stage, a weighted optimization mechanism is constructed, integrating classification error constraints, output consistency constraints, and Shannon entropy weight constraints, to achieve adaptive fusion of multi-dimensional features. Through the above technical solution, high-precision and interpretable automatic detection of mental illnesses is achieved.

Description

Technical Field

[0001] This invention relates to the field of electroencephalogram (EEG) signal processing and intelligent detection technology, and in particular to a method for detecting mental illnesses based on the MF-TSK fuzzy system. Background Technology

[0002] Mental illness is a group of brain dysfunction disorders characterized primarily by abnormalities in cognition, emotion, volition, and behavior, severely impacting patients' social functioning and quality of life. Among these, mood disorders, a representative example of mental illness, have a high prevalence and disability rate, posing a significant challenge to global public health. According to the World Health Organization's website, there are hundreds of millions of people with mental illness worldwide, and the related disease burden continues to rise. It is projected that by 2030, mental disorders will become a major component of the global disease burden.

[0003] Currently, the clinical diagnosis of mental illness mainly relies on patient-report-based scale assessments and face-to-face interviews, such as structured assessment tools like the Patient Health Questionnaire and the Beck Scale. This diagnostic approach is not only time-consuming and costly, but also heavily depends on the clinician's subjective experience and the accuracy of the patient's description of their symptoms, making it susceptible to influences such as emotional state, cultural background, and cognitive biases. Meanwhile, the distribution of medical and patient resources in my country's mental health field has long been uneven, with a relative shortage of psychiatrists and significant regional disparities in medical resources and economic development. This results in some areas where patients with mental illness struggle to receive timely, objective, and accurate diagnoses, and may even miss the optimal intervention window due to misdiagnosis or missed diagnosis. Therefore, conducting research on mental illness detection based on objective physiological signals has significant social and clinical application value.

[0004] The combination of Multi-view Learning (MVL) and the Takagi-Sugeno-Kang (TSK) fuzzy system offers both multi-dimensional feature fusion and decision interpretability, making it widely used in mental illness research. MVL leverages the complementarity between different information sources for the same object, while the TSK fuzzy system excels in logical interpretability. However, traditional TSK fuzzy systems lack explicit semantic support for their parameters, resulting in a "grey box" characteristic that limits their reliability in clinical diagnosis. To address these issues, this invention proposes a mental illness detection method based on the MF-TSK fuzzy system for automated detection of mental illnesses. Summary of the Invention

[0005] To address the aforementioned technical problems, this invention provides a method for detecting mental illness based on the MF-TSK fuzzy system, comprising the following steps:

[0006] Step 1: Input multi-channel EEG data The EEG data from each channel is segmented with a time window of 6 seconds and a sliding step of 2 seconds.

[0007] Step 2: Extract multi-dimensional features from EEG data, including time domain, frequency domain, and spatial domain features, and construct a multi-dimensional feature set;

[0008] Step 3: Employ fuzzy C-means clustering (FCM) based on Minimum Minimum Similarity (MMS) and combine it with Mahalanobis distance to cluster the multidimensional feature set, thereby obtaining the cluster centers and widths of the TSK fuzzy system;

[0009] Step 4: Use a Gaussian Mixture Model (GMM) to probabilistically model the data distribution, obtaining the fuzzy rule set R, specifically the k-th fuzzy rule R. k Represented as:

[0010] (1)

[0011] in, and Corresponding to cluster centers c (d, k) and width σ (d, k) p k Let be the weight of the k-th rule;

[0012] Step 5: Design a Multi-Feature TSK Fuzzy System (MF-TSKs), defining the rule weight vector p and the multi-dimensional feature weight vector α. Integrating classification error constraints, output consistency constraints, and Shannon entropy weight constraints, the objective function of MF-TSKs is defined as:

[0013] (2)

[0014] Where CEC(•) is the classification error constraint, CR(•) is the output consistency constraint, H(•) is the Shannon entropy weight constraint, and λ1 and λ2 are hyperparameters used to control the strength of consistency and entropy constraints, respectively.

[0015] The objective function is solved by a cross-optimization strategy. The regular weight vector p and the multidimensional feature weight vector α are iteratively updated by gradient descent to minimize the objective loss function.

[0016] Step 6: Generate the MF-TSKs rule set, specifically the k-th rule of the i-th dimension. Represented as:

[0017] (3);

[0018] Step 7: Achieve high-precision mental illness detection based on MF-TSKs rules.

[0019] Furthermore, the specific process of step three is as follows:

[0020] (1) Construct an undirected graph with samples (features) as nodes and similarity as edge weights. Use Prim's algorithm to generate a minimum spanning tree. The minimum spanning tree (MMS) between nodes is defined as follows:

[0021] (4)

[0022] Among them, P nm Represents node x n and x m The set of all paths between them, where p[h] represents the h-th node in path p, and a similarity matrix is ​​constructed based on the similarity between nodes;

[0023] (2) Initial cluster center selection: Randomly select a node as the first cluster center, and select the sample with the smallest similarity to the existing cluster center from the remaining nodes as the next cluster center. Repeat until k cluster centers are selected.

[0024] (3) After the initial cluster centers are selected, define the membership matrix U=[u nk ], where u nk Indicates sample x n The membership degree to the k-th cluster is used to characterize the distribution characteristics of samples within the cluster. The local covariance matrix S of each cluster is calculated. k :

[0025] (5)

[0026] Among them, v k The clustering process uses the Mahalanobis distance to alternately update the membership degree and the cluster center, with the membership degree update based on the cluster center.

[0027] (6)

[0028] The calculation process for Mahalanobis distance is as follows:

[0029] (7)

[0030] m is the fuzzy weighting coefficient, and we take m=2;

[0031] Cluster centers are updated using a weighted average:

[0032] (8)

[0033] The equations (5) to (8) are iterated alternately until the change in all cluster centers in two adjacent iterations is less than the preset threshold ε = 1 × 10. -5 ;

[0034] (4) Based on the membership matrix, determine the center and width of the antecedent parameters of each fuzzy rule:

[0035] (9)

[0036] (10).

[0037] Furthermore, the specific process of step four is as follows:

[0038] (1) Training samples (x) n , y n The joint probability density function of ) is characterized by GMM as:

[0039] (11)

[0040] Where, p k N represents the prior probability of the k-th Gaussian component. D+1 Represents a Gaussian distribution with dimension D+1. and Let be the input and output mean values ​​of the k-th Gaussian component, respectively. The covariance matrix of the k-th Gaussian component is:

[0041] (12)

[0042] Let covariance be the matrix between the inputs. This represents the covariance matrix between the input and output.

[0043] (2) In the joint probability density function of GMM, the joint distribution of the k-th Gaussian component can be expressed as:

[0044] (13)

[0045] The Gaussian components mentioned above, as a local region in the input-output space, have a mean value of , Covariance The central location and fuzzy range of the region were characterized respectively. This "local weighting" is highly consistent with the TSK fuzzy rule, that is, each Gaussian component can be regarded as the probabilistic form of a rule, and its probability density function corresponds to the membership function of the rule, with the prior probability p... k The weight corresponding to the k-th fuzzy rule is denoted as follows: And satisfy the following constraints:

[0046] (14);

[0047] (3) GMM under given input xn Under the given conditions, the expected output is:

[0048] (15)

[0049] in, Represents the covariance matrix The d-th diagonal element;

[0050] (4) Based on the above correspondence, generate the kth fuzzy rule R. k This leads to the fuzzy rule set R.

[0051] Furthermore, the specific process of defining the objective function in step five is as follows:

[0052] (1) Define the rule weight vector ,in This represents the weight vector of all rules in the i-th feature class, and also introduces the feature weight vector. and satisfy constraints To ensure that multiple features can effectively participate in inference during training, a classification error constraint based on parameters p and α is defined:

[0053] (16)

[0054] For the corresponding label vector as input, Activate the intensity vector for the rule;

[0055] (2) To enhance the consistency of output results among multiple features, a regularization term based on output variance is defined:

[0056] (17);

[0057] (3) To prevent some features from being ignored due to low information content, and to avoid excessive concentration of weights, a weight constraint based on Shannon entropy is defined:

[0058] (18);

[0059] (4) By combining the classification error constraint, output consistency constraint, and Shannon entropy weight constraint, the objective function of MF-TSKs is obtained.

[0060] Compared with the prior art, the beneficial effects of the present invention using the above technical solution are as follows:

[0061] (1) This invention solves the problem that a single EEG feature is difficult to fully characterize the differences between disease patients and normal controls (NC). It constructs a multidimensional EEG feature representation from three dimensions: time domain, frequency domain and spatial domain, so as to realize the information complementarity and fusion of multidimensional EEG features and provide a more comprehensive information representation for the automatic detection of mental illness.

[0062] (2) This invention addresses the difficulty in balancing performance and interpretability in existing classifiers by proposing a multi-dimensional feature TSK fuzzy system (MF-TSKs). In the antecedent, the system uses MMS-FCM combined with Mahalanobis distance for fuzzy partitioning to improve clustering stability. In the consequent, a Gaussian mixture model (GMM) is introduced to model the parameters, enhancing the clarity and interpretability of the rule semantics, and imposing weight constraints on the rules. The interpretable fuzzy rule structure achieves a balance between high performance and interpretability, thereby enhancing the model's physiological interpretability and credibility. It also addresses the problem of insufficient correlation and complementarity fusion mechanisms among multi-dimensional features. Through multi-dimensional rule fusion and weight adaptive mechanisms, information complementarity and collaborative optimization among features of different dimensions are achieved, thereby improving the system's ability to discriminate mental illnesses and its interpretability. Attached Figure Description

[0063] Figure 1 This is a flowchart of the method of the present invention;

[0064] Figure 2 This invention provides a one-dimensional interpretable TSK fuzzy system for embodiments of the present invention.

[0065] Figure 3 This is the multidimensional feature learning process in an embodiment of the present invention;

[0066] Figure 4 This is the fuzzy rule generation process for MF-TSKs in an embodiment of the present invention;

[0067] Figure 5 This is an embodiment of the mental illness detection process based on the MF-TSKs fuzzy system of the present invention;

[0068] Figure 6 This invention provides a comparison of the MF-TSKs fuzzy system with other methods for the study of mental illness outcomes. Detailed Implementation

[0069] The specific embodiments of the present invention will be further described in detail below with reference to the accompanying drawings and examples. The following examples use depression (DP) detection as a specific application scenario to illustrate the implementation process and technical effects of the mental illness detection method proposed in this invention, but are not intended to limit the scope of the invention.

[0070] Please see Figure 1As shown, the present invention provides a technical solution: The present invention provides a method for detecting mental illness based on the MF-TSKs fuzzy system, using DP detection as a typical embodiment for illustration. The steps of the method are as follows:

[0071] Step 1: Input DP patient and NC multichannel EEG data The EEG data from each channel is segmented with a time window of 6 seconds and a sliding step of 2 seconds.

[0072] Step 2: Extract multi-dimensional features from EEG data, including time domain, frequency domain, and spatial domain features, and construct a multi-dimensional feature set;

[0073] Step 3: Use Minimum Minimum Similarity (MMS) fuzzy C-means clustering (FCM) combined with Mahalanobis distance to cluster the multidimensional feature set, obtaining the cluster centers and widths of the TSK fuzzy system. The specific process is as follows:

[0074] (1) Construct an undirected graph with samples (features) as nodes and similarity as edge weights. Use Prim's algorithm to generate a minimum spanning tree. The minimum spanning tree (MMS) between nodes is defined as follows:

[0075] (4)

[0076] Among them, P nm Represents node x n and x m The set of all paths between them, where p[h] represents the h-th node in path p, and a similarity matrix is ​​constructed based on the similarity between nodes;

[0077] (2) Initial cluster center selection: Randomly select a node as the first cluster center, and select the sample with the smallest similarity to the existing cluster center from the remaining nodes as the next cluster center. Repeat until k cluster centers are selected.

[0078] (3) After the initial cluster centers are selected, define the membership matrix U=[u nk ], where u nk Indicates sample x n The membership degree to the k-th cluster is used to characterize the distribution characteristics of samples within the cluster. The local covariance matrix S of each cluster is calculated. k :

[0079] (5)

[0080] Among them, v k The clustering process uses the Mahalanobis distance to alternately update the membership degree and the cluster center, with the membership degree update based on the cluster center.

[0081] (6)

[0082] The calculation process for Mahalanobis distance is as follows:

[0083] (7)

[0084] m is the fuzzy weighting coefficient, and we take m=2;

[0085] Cluster centers are updated using a weighted average:

[0086] (8)

[0087] The equations (4) to (8) are iterated alternately until the change in all cluster centers in two adjacent iterations is less than the preset threshold ε = 1 × 10. -5 ;

[0088] (4) Based on the membership matrix, determine the center and width of the antecedent parameters of each fuzzy rule:

[0089] (9)

[0090] (10);

[0091] Step 4: Use a Gaussian Mixture Model (GMM) to probabilistically model the data distribution, and the k-th fuzzy rule R k The generation process is as follows:

[0092] (1) Training samples (x) n , y n The joint probability density function of ) is characterized by GMM as:

[0093] (11)

[0094] Where, p k N represents the prior probability of the k-th Gaussian component. D+1 Represents a Gaussian distribution with dimension D+1. and Let be the input and output mean values ​​of the k-th Gaussian component, respectively. The covariance matrix of the k-th Gaussian component is:

[0095] (12)

[0096] Let covariance be the matrix between the inputs. This represents the covariance matrix between the input and output.

[0097] (2) In the joint probability density function of GMM, the joint distribution of the k-th Gaussian component can be expressed as:

[0098] (13)

[0099] The Gaussian components mentioned above, as a local region in the input-output space, have a mean value of , Covariance The central location and fuzzy range of the region were characterized respectively. This "local weighting" is highly consistent with the TSK fuzzy rule, that is, each Gaussian component can be regarded as the probabilistic form of a rule, and its probability density function corresponds to the membership function of the rule, with the prior probability p... k The weight corresponding to the k-th fuzzy rule is denoted as follows: And satisfy the following constraints:

[0100] (14);

[0101] (3) GMM under given input x n Under the given conditions, the expected output is:

[0102] (15)

[0103] in, Represents the covariance matrix The d-th diagonal element;

[0104] (4) Based on the above correspondence (e.g.) Figure 2 As shown), generate the kth fuzzy rule R. k :

[0105] (1)

[0106] in, and Corresponding to cluster centers c (d, k) and width σ (d, k) ,

[0107] Furthermore, based on R k Obtain the fuzzy rule set R;

[0108] Step 5: Design a multi-dimensional feature-based fuzzy system (MF-TSKs), such as... Figure 3 As shown, defining the rule weight vector p and the multidimensional feature weight vector α, and considering the classification error constraint, output consistency constraint, and Shannon entropy weight constraint, the objective function of MF-TSKs is defined as:

[0109] (2)

[0110] Where CEC(•) is the classification error constraint, CR(•) is the output consistency constraint, H(•) is the Shannon entropy weight constraint, and λ1 and λ2 are hyperparameters used to control the strength of consistency and entropy constraints, respectively.

[0111] A cross-optimization strategy is employed to solve the objective function. The regular weight vector p and the multi-dimensional feature weight vector α are iteratively updated using gradient descent to minimize the objective loss function. The specific process for defining the objective function is as follows:

[0112] (1) Define the rule weight vector ,in This represents the weight vector of all rules in the i-th feature class, and also introduces the feature weight vector. and satisfy constraints To ensure that multiple features can effectively participate in inference during training, a classification error constraint based on parameters p and α is defined:

[0113] (16)

[0114] For the corresponding label vector as input, Activate the intensity vector for the rule;

[0115] (2) To enhance the consistency of output results among multiple features, a regularization term based on output variance is defined:

[0116] (17);

[0117] (3) To prevent some features from being ignored due to low information content and to avoid excessive concentration of weights, a weight constraint based on Shannon entropy is defined:

[0118] (18);

[0119] (4) Combining the classification error constraint, output consistency constraint, and Shannon entropy weight constraint, the objective function of MF-TSKs is obtained;

[0120] Step Six: Generate corresponding fuzzy rules from the multidimensional features, some of which are as follows: Figure 4 As shown, a total of seven fuzzy rules are generated: 2 rules in the time domain, 3 rules in the frequency domain, and 2 rules in the spatial domain. Specifically, the k-th rule in the i-th dimension... Represented as:

[0121] (3)

[0122] This leads to the generation of MF-TSKs rule sets;

[0123] Step Seven: Based on the MF-TSKs rule set, the reasoning process in the testing (mental illness detection) phase is as follows: Figure 5 As shown, DP patients are labeled as vectors [1 0]. T The NC marker vector is [0 1]. T After using the spatiotemporal three-dimensional features of DP patients as input to MF-TSKs, the resulting decision output label vector is y=[0.61 0.38]. T The result is output from each dimension y. (i) The values ​​(i=1,2,3) are weighted and fused together. Then, according to the decision rule, the largest element in y is set to 1, and the remaining elements are set to 0, resulting in the final output y=[1 0]. T This corresponds to the DP patient label. The reasoning process achieves complete interpretability from input features, fuzzy rule layer, multi-dimensional fusion to the global decision layer. In other words, it achieves high-precision DP detection based on MF-TSKs rules.

[0124] To verify the performance of MF-TSKs in mental illness detection provided by this invention, the performance of MF-TSKs in mental illness detection was compared with that of three single-dimensional detection methods (SVM, KNN, and Random Forest) and three multi-dimensional detection methods (MV-L2-SVM, MV-CNN, and MV-TSK-FS). The results are as follows: Figure 6 As shown in the figure, in the single-dimensional experiments, SVM, KNN, and RF showed similar detection accuracy, AUC, and F1-score in the temporal and spatial domains, but their performance in the frequency domain decreased significantly. This result indicates that single-dimensional methods are highly dependent on feature type. In contrast, MF-TSKs achieved accuracies of 91.04%, 75%, and 92.33% in the temporal, frequency, and spatial domains, respectively, fully leveraging dimensional information and demonstrating superior DP detection performance under the same conditions. In the multi-dimensional experiments, MF-TSKs showed significant performance advantages in multi-dimensional scenarios, with its multi-dimensional learning performance outperforming MV-L2-SVM, MV-TSK-FS, and MV-CNN, achieving a maximum DP detection accuracy of 98.60%.

[0125] It should be noted that the above embodiments are merely preferred implementations of the technical solution of the present invention and are not intended to limit the scope of protection of the present invention. Those skilled in the art should understand that, without departing from the core design concept and scope of the claims, adaptive adjustments or equivalent substitutions can be made to the specific implementation of the technical solution based on actual application scenarios, including but not limited to structural optimization, parameter adjustment, and functional module reorganization. All equivalent modifications or inventive extensions implemented based on the technical solution of the present invention fall within the scope of protection of the patent rights of this invention. The scope of protection of the present invention should be determined by the complete content of the claims; the embodiments described in the specification and drawings are only used to explain the legal connotation and technical features of the claims.

Claims

1. A method for detecting mental illness based on the MF-TSK fuzzy system, characterized in that, The specific steps are as follows: Step 1: Input multi-channel EEG data The EEG data from each channel is segmented with a time window of 6 seconds and a sliding step of 2 seconds. Step 2: Extract multi-dimensional features from EEG data in the time domain, frequency domain, and spatial domain, and construct a multi-dimensional feature set; Step 3: Employ fuzzy C-means clustering (FCM) based on minimizing maximum similarity (MMS) and combine it with Mahalanobis distance to cluster the multidimensional feature set, thereby obtaining the cluster centers and widths of the TSK fuzzy system; Step 4: Use a Gaussian Mixture Model (GMM) to probabilistically model the data distribution, obtaining the fuzzy rule set R, specifically the k-th fuzzy rule R. k Represented as: (1) in, and Corresponding to cluster centers c (d, k) and width σ (d, k) p k Let be the weight of the k-th rule; Step 5: Design a Multi-Feature TSK Fuzzy System (MF-TSKs), defining the rule weight vector p and the multi-dimensional feature weight vector α. Integrating classification error constraints, output consistency constraints, and Shannon entropy weight constraints, the objective function of MF-TSKs is defined as: (2) Where CEC(•) is the classification error constraint, CR(•) is the output consistency constraint, H(•) is the Shannon entropy weight constraint, and λ1 and λ2 are hyperparameters used to control the strength of consistency and entropy constraints, respectively. The objective function is solved by a cross-optimization strategy. The regular weight vector p and the multidimensional feature weight vector α are iteratively updated by gradient descent to minimize the objective loss function. Step 6: Generate the MF-TSKs rule set, specifically the k-th rule of the i-th dimension. Represented as: (3)； Step 7: Achieve high-precision mental illness detection based on MF-TSKs rules.

2. The method for detecting mental illness based on the MF-TSK fuzzy system according to claim 1, characterized in that, The specific process of step three is as follows: (1) Construct an undirected graph with samples (features) as nodes and similarity as edge weights. Use Prim's algorithm to generate a minimum spanning tree. The minimum spanning tree (MMS) between nodes is defined as follows: (4) Among them, P nm Represents node x n and x m The set of all paths between them, where p[h] represents the h-th node in path p, and a similarity matrix is ​​constructed based on the similarity between nodes; (2) Initial cluster center selection: Randomly select a node as the first cluster center, and select the sample with the smallest similarity to the existing cluster center from the remaining nodes as the next cluster center. Repeat until k cluster centers are selected. (3) After the initial cluster centers are selected, define the membership matrix U=[u nk ], where u nk Indicates sample x n The membership degree to the k-th cluster is used to characterize the distribution characteristics of samples within the cluster. The local covariance matrix S of each cluster is calculated. k : (5) Among them, v k The clustering process uses the Mahalanobis distance to alternately update the membership degree and the cluster center, with the membership degree update based on the cluster center. (6) The calculation process for Mahalanobis distance is as follows: (7) m is the fuzzy weighting coefficient, and we take m=2; Cluster centers are updated using a weighted average: (8) The equations (5) to (8) are iterated alternately until the change in all cluster centers in two adjacent iterations is less than the preset threshold ε = 1 × 10. -5 ; (4) Based on the membership matrix, determine the center and width of the antecedent parameters of each fuzzy rule: (9) (10)。 3. The method for detecting mental illness based on the MF-TSK fuzzy system according to claim 1, characterized in that, The specific process for step four is as follows: (1) Training samples (x) n , y n The joint probability density function of ) is characterized by GMM as: (11) Where, p k N represents the prior probability of the k-th Gaussian component. D+1 Represents a D+1 dimensional Gaussian distribution. and Let be the input and output mean values ​​of the k-th Gaussian component, respectively. The covariance matrix of the k-th Gaussian component is: (12) Let covariance be the matrix between the inputs. This represents the covariance matrix between the input and output. (2) In the joint probability density function of GMM, the joint distribution of the k-th Gaussian component can be expressed as: (13) The Gaussian components mentioned above, as a local region in the input-output space, have a mean value of , Covariance The central location and fuzzy range of the region were characterized respectively. This "local weighting" is highly consistent with the TSK fuzzy rule, that is, each Gaussian component can be regarded as the probability form of a rule, and its probability density function corresponds to the membership function of the rule, with the prior probability p k The weight corresponding to the k-th fuzzy rule is denoted as follows: And satisfy the following constraints: (14)； (3) GMM under given input x n Under the given conditions, the expected output is: (15) in, Represents the covariance matrix The d-th diagonal element; (4) Based on the above correspondence, generate the kth fuzzy rule R. k This leads to the fuzzy rule set R.

4. The method for detecting mental illness based on the MF-TSK fuzzy system according to claim 1, characterized in that, Step five involves the following steps: (1) Define the rule weight vector ,in This represents the weight vector of all rules in the i-th feature class, and also introduces the feature weight vector. and satisfy constraints To ensure that multiple features can effectively participate in inference during training, a classification error constraint based on parameters p and α is defined: (16) For the corresponding label vector as input, Activate the intensity vector for the rule; (2) To enhance the consistency of output results among multiple features, a regularization term based on output variance is defined: (17)； (3) To prevent some features from being ignored due to low information content, and to avoid excessive concentration of weights, a weight constraint based on Shannon entropy is defined: (18)； (4) By combining the classification error constraint, output consistency constraint, and Shannon entropy weight constraint, the objective function of MF-TSKs is obtained.

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