CN114694841A - Adverse event risk prediction method based on patient electronic health record - Google Patents
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Abstract
The invention discloses an adverse event risk prediction method based on an electronic health record of a patient, which comprises the following steps of: s1, preprocessing data; s2, performing K-means clustering sampling processing, and dividing data into 3 clusters to obtain 3 clustering centers; s3, dividing the 3 clustering centers into P*The maximum values in the three subsets are sorted from small to large and respectively used as an uncommon coding subset, a more common coding subset and a common coding subset, then the three subsets are respectively and correspondingly input into three basic classifiers of GRAM +, Dipole + and RNN + for pre-training, and then model fusion is carried out on the three basic classifiersAnd (6) mixing. According to the method, a clustering algorithm is used for sampling proper training samples for a basic learning device, a self-adaptive combination strategy is designed, and integration weights of different basic classifiers are generated in a self-adaptive mode according to the distance from the training samples to the center of a pre-training set, so that the model has stronger self-adaptability. In addition, through the sampling after clustering, the calculation amount can be obviously reduced when the basic embedding is trained.
Description
Technical Field
The invention relates to an adverse event risk prediction method based on an electronic health record of a patient.
Background
AIDS is a highly harmful infectious disease, is caused by infection of AIDS virus (HIV), and the main attack target of the AIDS is the most important CD4T lymphocyte in the human immune system, so that the human body loses the immune function, is easy to infect various diseases and has high fatality rate. After AIDS, if the patient is actively treated, a relatively good treatment effect can be obtained, but if adverse events such as serious complications occur, the treatment effect is affected. The method can predict adverse events such as complications which may occur in the future by combining conventional risk factors and AIDS patient specific factors, and can be used as powerful assistance for guiding the medical care of AIDS patients. Electronic Health Records (EHRs) for aids patients not only contain medical codes (including diagnosis, medication and program codes, such as 585.9 (chronic kidney disease), which refers to codes representing procedures such as intervention, treatment, etc.) for each visit of the patients, but also personalized data such as demographic data and vital signs of the patients, which can help doctors to make more reasonable decisions about their medical care by predicting possible adverse events in the future.
The chinese patent application CN109887606A, a diagnostic prediction method for bidirectional recurrent neural network based on attention, provides a prediction method for bidirectional recurrent neural network based on attention, which first embeds high-dimensional medical codes (i.e. clinical variables) into low-code layer space, and then inputs the coded representation into a bidirectional recurrent neural network based on attention to generate hidden state representation. The medical code for future visits is predicted by the softmax layer.
Edward Choi (E.Choi, M.T.Bahadori, L.Song, et al.UA-CRNN: GRAM: Graph-based assessment Model for Healthcare retrieval Learning [ C ] in: Proceedings of the 24th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, London,2018, pp.249-256) et al propose a Representation Learning method based on a Knowledge Graph Attention machine, mainly use the hierarchy information inherent to the medical ontology to learn an embedded Representation containing more informative medical codes, and then use a depth Learning method to predict. However, the above-mentioned technical method has the following problems: (1) the model has dependence on the training data volume, good prediction effect is achieved when the training data is sufficient, and the prediction performance is poor when the data volume is insufficient; (2) medical ontology knowledge contained in medical codes is ignored, and the prediction performance of medical codes with low occurrence frequency and rare cases is poor.
The representation learning method based on the knowledge graph needs larger calculation cost and training difficulty in order to learn the embedded representation of the medical code containing richer information. In addition, the above methods ignore individual differences between patients, which has an effect on the accuracy of the prediction.
Disclosure of Invention
The invention aims to overcome the defects of the prior art and provides an adverse event risk prediction method based on patient electronic health record, which is characterized in that a clustering algorithm is used for sampling proper training samples for a basic learner, a self-adaptive combination strategy is designed, and integration weights of different basic classifiers are generated in a self-adaptive manner according to the distance between the training samples and the center of a pre-training set, so that a model has stronger self-adaptability.
The purpose of the invention is realized by the following technical scheme: a method for adverse event risk prediction based on an electronic patient health record, comprising the steps of:
s1, preprocessing data: taking the data of each patient as a time-sequential diagnostic sequence in the electronic health record data; the diagnostic sequence was processed as follows:
s11 using C ═ C1,c2,...,cNDenotes the set of all diagnostic codes, ciRepresenting the ith diagnosis code, wherein i is more than or equal to 1 and less than or equal to N, and N represents the total number of the diagnosis codes; x ═ X1,x2,...,xT]Information representing a patient visit, wherein the tth visit information xt∈{0,1}N,{0,1}NRepresenting a vector of N elements, each element having a value of 0 or 1, i.e. xt={xt1,xt1,…,xti,…xtN}; if the diagnostic code c with serial number ii∈{c1,c2,...,cNX is present in the t-th visitti1, otherwise xti=0;
S12, using L ═ L1,l2,...,lT]Personalized data representing all visits of a patient, liA vector representation of the personalized data record representing the ith visit; the average value is obtained for each patient in T times of treatment, and the average value l of the same kind of data in different times of treatment is obtained*(ii) a Selecting missing values for numerical data, and selecting missing values to be filled by using an average value, and for non-numerical data, filling the missing values by using values with highest occurrence frequency in the patient data according to a mode principle in statistics;
s13, summing each diagnosis code in X to obtain the frequency of the unique diagnosis code in all the visit information of each patientNamely, it isTo all theSumming to obtain the frequency S of unique diagnosis codes in all data*Let P*=s*/S*Indicating that the frequency of occurrence of each diagnostic code in each patient data is allThe fraction in the partial data;
after the treatment is finished, the data of j th patient consists of three parts Xj、Lj、FjJ is more than or equal to 1 and less than or equal to M, and M represents the number of patients with collected data;mean value l representing the same data from different visits of the j-th patient*,Representing the proportion of the occurrence frequency of each diagnosis code in the jth patient data in all the data;
s2, performing K-means clustering sampling processing: with data for each patientPerforming K-means clustering as sample points, dividing data into 3 clusters to obtain 3 clustering centers theta1,θ2,θ3Then calculating F for each patient datajAnd F' of each cluster center at the same sampling rate for each cluster centerSelecting corresponding subdata sets from the data of all patients according to the sequence of the distances from small to large to obtain D' ═ D1'∪D2'∪D3', the generated plurality of subdata sets are used for training of a basic classifier;
s3, clustering 3 centers theta1,θ2,θ3According to P*The maximum values in the three subsets are sorted from small to large and respectively used as an uncommon code subset, a more common code subset and a common code subset, then the three subsets are respectively and correspondingly input into three basic classifiers of GRAM +, Dipole + and RNN + for pre-training, and then model fusion is carried out on the three basic classifiers.
Further, the GRAM + is based on GRAM, adds a global attention mechanism by using the personalized data of the patient as a guide, and is specifically designed as follows:
in the knowledge directed acyclic graph formed by the medical ontology, the leaf node is the element in the diagnosis code set in S11, and the ancestor node thereof represents the ontology represented by the leaf node derived from the leaf node; all nodes c are assigned a basic embedding vector e, representing the final representation of a leaf node as a basic embedded convex combination of itself and its ancestor nodes:
wherein g isiRepresents a medical code ciA (i) represents the code ciAnd ciIndex of ancestor node, αijIs the local attention weight, calculated by the Softmax function as follows:
f(ei,ej) Is a scalar value, representing eiAnd ejThe compatibility between two basic embeddings is obtained by a multilayer perceptron;
by concatenating the final representations g of all medical codes1,g2,...,gNTo obtain an embedded matrix G, and a future diagnosis vector vtExpressed as a vector xtMultiplied by the embedding matrix G and passed through a nonlinear tanh () function:
v1,v2,...,vT=tanh(G[x1,x2,...,xT])
the patient's personalized data L ═ L is then utilized1,l2,...,lT]To add a global attention weight betatObtaining a global representation u comprising patient-personalized data informationt:
ut=βtvt,t=1,2,...,T
βtBy the following Softmax function meterCalculating:
f(li,l*) Is a scalar value representing liAnd l*The compatibility between the sensors is obtained by a multilayer perceptron;
will u1,u2,...uTInputting the data into a GRU network to obtain a hidden state representation h1,h2,...,hTGenerating the first prediction information by the Softmax layerIs defined as:
h1,h2,...,hT=GRU(u1,u2,...,uT,θr)
upper labelRepresenting a transpose; the penalty calculation is back-propagated, the error between prediction and true is calculated, and back-propagation is learned and corrected until the GRAM + model converges.
Further, the Dipole + utilizes patient-personalized data L ═ L1,l2,...,lT]As a guide, a bidirectional recurrent neural network and an attention mechanism are used simultaneously to predict the patient information; first, the visit information X is embedded into a representation vector v by a multi-layer perceptrontThen using the patient's personalized data L ═ L1,l2,...,lT]To add a global attention weight betatObtaining a global representation u comprising patient-personalized data informationt:
ut=βtvt,t=1,2,...,T
βtCalculated by the following Softmax function:
f(li,l*) Is a scalar value;
then vector utIs input to a bi-directional recurrent neural network and finally, the bi-directional outputs are concatenated to generate a potential vector for prediction using an attention mechanism based on the data sequence position.
Further, RNN + is based on RNN using patient personalized data L ═ L1,l2,...,lT]Guiding the patient visit information representation vector X to generate a global representation vector u containing patient personalized data informationtGlobal representation vector utThe algorithm of (c) is the same as the Dipole + method, and the vector u is represented globallytAn attention model based on the data sequence position is entered and then predictions are made using unidirectional GRUs.
Further, an adaptive weighted integration strategy is adopted in the model fusion stage, and for each sample Xi=[x1,x2,...,xT]Calculating it to each cluster centerDistance di=[δi1,δi2,δi3];
For each sample, the integrated weight w is generated using the following formulai:
The final integrated output result is expressed as:
The invention has the beneficial effects that: compared with the prior art, the technical scheme provided by the invention considers the difference between individuals of the patient, and utilizes the individualized data of the patient to guide the model to establish reasonable attention, thereby improving the accuracy of model prediction. In addition, the influence of different sample sizes on the model performance is considered, a proper training sample is sampled for the basic learning device through a clustering algorithm, a self-adaptive combination strategy is designed, and the integration weights of different basic classifiers are generated in a self-adaptive mode according to the distance between the training sample and the center of a pre-training set, so that the model has stronger self-adaptability. In addition, through the sampling after clustering, the calculation amount can be obviously reduced when the basic embedding is trained.
Drawings
FIG. 1 is a flow chart of an adverse event risk prediction method based on an electronic patient health record of the present invention;
FIG. 2 is a diagram illustrating the structure of the GRAM + classifier of the present invention;
FIG. 3 is a schematic diagram of the integration strategy of the present invention.
Detailed Description
The invention discloses an effective method for predicting adverse event risks of electronic health records of AIDS patients, which comprises the steps of firstly utilizing a clustering method to sample proper pre-training sets for different basic learners, integrating the prediction performances of different classifiers on codes with different frequencies, designing a self-adaptive combination strategy, and generating the integration weights of different basic classifiers in a self-adaptive manner according to the distance between a training sample and the center of the pre-training set so as to balance the difference of a single model on the prediction performances of medical codes with different frequency numbers. In addition, the model is added with an attention mechanism which takes personalized data as a guide to make up for individual differences, and the model accuracy is improved. The technical scheme of the invention is further explained by combining the attached drawings.
As shown in fig. 1, the method for predicting the risk of adverse events based on the electronic health record of the patient of the present invention comprises the following steps:
s1, preprocessing data: in the electronic health record data, the data of each patient is taken as a time sequence of diagnosis, and in each diagnosis, a plurality of diagnosis codes (including diagnosis, medication, program codes and the like) exist; the diagnostic sequence was processed as follows:
s11 using C ═ C1,c2,...,cNDenotes the set of all diagnostic codes, ciRepresenting the ith diagnosis code, wherein i is more than or equal to 1 and less than or equal to N, and N represents the total number of the diagnosis codes; x ═ X1,x2,...,xT]Representing the visit information of a patient, wherein the t-th visit information xt∈{0,1}NIs a binary vector, {0,1}NRepresenting a vector of N elements, each element having a value of 0 or 1, i.e. xt={xt1,xt1,…,xti,…xtN}; if the diagnostic code c with serial number ii∈{c1,c2,...,cNX is present in the t-th visitti1, otherwise xti=0;
S12, using L ═ L1,l2,...,lT]Personalized data representing all visits of a patient, liA vector representation of the personalized data record representing the ith visit; the average value of the T times of visits of each patient is obtained to obtain the average value of the same kind of data in different visitsValue l*(ii) a Selecting missing values for numerical data, and selecting missing values to be filled by using an average value, and for non-numerical data, filling the missing values by using values with highest occurrence frequency in the patient data according to a mode principle in statistics;
s13, summing each diagnosis code in X to obtain the frequency of the unique diagnosis code in all the clinic information of each patientNamely thatThen to allSumming to obtain the frequency S of unique diagnosis codes in all data*Let P*=s*/S*Representing the proportion of the frequency of occurrence of each diagnosis code in each patient data in all data;
after the treatment is finished, the data of j th patient consists of three parts Xj、Lj、FjJ is more than or equal to 1 and less than or equal to M, and M represents the number of patients with collected data; xjFor training classifiers, LjTo guide the attention mechanism of the classifier,is used for clustering and dividing the patients,mean value l representing the same data from different visits of the j-th patient*,Indicating the proportion of the frequency of occurrence of each diagnostic code in the jth patient data in the total data.
S2, performing K-means clustering sampling processing: with data for each patient F ═ l*,P*]Performing K-means clustering as sample points, and dividing data into 3 clusters(the number of the clusters is generally the same as that of the basic classifiers adopted later), data with similar personalized data and similar diagnostic code occurrence frequency in the diagnostic record are more likely to be gathered into the same cluster; given the number of clusters 3, the data of all patients were divided into D ═ D using the K-means algorithm in combination with D1∪D2∪D3To obtain 3 cluster centers theta1,θ2,θ3Then calculating F for each patient datajAnd F' of each cluster center at the same sampling rate for each cluster centerSelecting corresponding subdata sets from the data of all patients according to the sequence of the distances from small to large to obtain D' ═ D1'∪D2'∪D3' generating a plurality of subdata sets for training a basic classifier;
s3, clustering 3 centers theta1,θ2,θ3According to P*The maximum values in the three subsets are sorted from small to large and respectively used as an uncommon coding subset, a more common coding subset and a common coding subset, then the three subsets are respectively and correspondingly input into three basic classifiers of GRAM +, Dipole + and RNN + for pre-training to learn decision boundaries, and then model fusion is carried out on the three basic classifiers;
the GRAM + is based on GRAM, adds a global attention mechanism by using the personalized data of the patient as a guide, and is specifically designed as follows as shown in fig. 2:
in the knowledge directed acyclic graph formed by the medical ontology, leaf nodes and ancestor nodes are used for distinguishing, the medical ontology conforms to a tree structure when being named and coded, all the leaf nodes are elements in the diagnosis code set in S11, and the ancestor nodes represent ontologies represented by the leaf nodes and are derived from the leaf nodes; all nodes c are assigned a basic embedding vector e, representing the final representation of a leaf node as a basic embedded convex combination of itself and its ancestor nodes:
wherein g isiRepresents a medical code ci(i.e., leaf node) embedded representation, a (i) represents code ciAnd ciIndex of ancestor node, αijIs the local attention weight, calculated by the Softmax function as follows:
f(ei,ej) Is a scalar value, representing eiAnd ejCompatibility between two basic embeddings, derived by the multilayer perceptron (MLP); training basic embedding by using Glover, and learning coded representation by using a global co-occurrence matrix of all nodes c;
by concatenating the final representations g of all medical codes1,g2,...,gNTo obtain an embedded matrix G, and a future diagnosis vector vtExpressed as a vector xtMultiplied by the embedding matrix G and passed through a nonlinear hyperbolic tangent activation function tanh ():
v1,v2,...,vT=tanh(G[x1,x2,...,xT])
the patient's personalized data L ═ L is then utilized1,l2,...,lT]To add a global attention weight betatObtaining a global representation u comprising patient-personalized data informationt:
ut=βtvt,t=1,2,...,T
βtCalculated by the following Softmax function:
f(li,l*) Is a scalar value, representing liAnd l*The compatibility between the sensors is obtained by a multilayer perceptron;
u is to be1,u2,...uTInputting the data into a GRU network to obtain a hidden state representation h1,h2,...,hTGenerating the first prediction information by the Softmax layerIs defined as:
h1,h2,...,hT=GRU(u1,u2,...,uT,θr)
upper labelRepresenting a transposition; the loss calculation is back propagated, the error between prediction and reality is calculated, and back propagation is learned and corrected until the GRAM + model converges.
The Dipole + utilizes patient-personalized data L ═ L1,l2,...,lT]As a guide, a bidirectional recurrent neural network and an attention mechanism are used simultaneously,to predict patient visit information; first, the visit information X is embedded into a representation vector v by a multi-layer perceptrontThen using the patient's personalized data L ═ L1,l2,...,lT]To add a global attention weight betatObtaining a global representation u comprising patient-personalized data informationt:
ut=βtvt,t=1,2,...,T
βtCalculated by the following Softmax function:
f(li,l*) Is a scalar value;
then vector utIs input to a bi-directional recurrent neural network and finally, the bi-directional outputs are concatenated to generate a potential vector for prediction using an attention mechanism based on the data sequence position.
RNN + is based on RNN using patient-specific data L ═ L1,l2,...,lT]Guiding a patient visit information representation vector X to generate a global representation vector u comprising patient personalized data informationtGlobal representation vector utThe algorithm of (c) is the same as the Dipole + method, and the vector u is represented globallytAn attention model based on the data sequence position is entered and then predictions are made using unidirectional GRUs.
Adopting a self-adaptive weighting integration strategy in a model fusion stage; as shown in FIG. 3, in the fusion phase, X is applied to each samplei=[x1,x2,...,xT]Calculating its distance d to each cluster centeri=[δi1,δi2,δi3](ii) a The distance measures the degree that the training sample data belongs to a certain cluster, the closer the center of the pre-training data subset is to the basic classifier of the training sample, the better adaptability to the sample is, and the indirect measurement measures that the classifier trained on the cluster predicts the new sample XiThe ability of the cell to perform. For each sample, the following formula was used to generateWeight w of the integrationi:
The final integrated output result is expressed as:
whereinThe output of the three basic classifiers is shown,is a prediction output which represents the probability of the occurrence of the medical code corresponding to the index in a future diagnosis, namely the risk of various adverse events which may occur in the future of the patient, thereby assisting the doctor to make more reasonable decisions on the medical care of the patient.
It will be appreciated by those of ordinary skill in the art that the embodiments described herein are intended to assist the reader in understanding the principles of the invention and are to be construed as being without limitation to such specifically recited embodiments and examples. Those skilled in the art can make various other specific changes and combinations based on the teachings of the present invention without departing from the spirit of the invention, and these changes and combinations are within the scope of the invention.
Claims (5)
1. A method for predicting risk of an adverse event based on an electronic health record of a patient, comprising the steps of:
s1, preprocessing data: taking the data of each patient as a time sequence of diagnosis in the electronic health record data; the diagnostic sequence was processed as follows:
s11 using C ═ C1,c2,...,cNDenotes all diagnostic codesSet, ciRepresenting the ith diagnosis code, wherein i is more than or equal to 1 and less than or equal to N, and N represents the total number of the diagnosis codes; x ═ X1,x2,...,xT]Representing the visit information of a patient, wherein the t-th visit information xt∈{0,1}N,{0,1}NRepresenting a vector of N elements, each element having a value of 0 or 1, i.e. xt={xt1,xt1,…,xti,…xtN}; if the diagnostic code c with serial number ii∈{c1,c2,...,cNIn the t-th visit, then xti1, otherwise xti=0;
S12, using L ═ L1,l2,...,lT]Personalized data representing all visits of a patient, liA vector representation of the personalized data record representing the ith visit; the average value is obtained for each patient in T times of treatment, and the average value l of the same kind of data in different times of treatment is obtained*(ii) a Selecting missing values for numerical data, and selecting missing values to be filled by using an average value, and for non-numerical data, filling the missing values by using values with highest occurrence frequency in the patient data according to a mode principle in statistics;
s13, summing each diagnosis code in X to obtain the frequency S of the unique diagnosis code in all the visit information of each patient*iI.e. byFor all s*iSumming to obtain the frequency S of unique diagnosis codes in all data*Let P*=s*/S*Representing the proportion of the occurrence frequency of each diagnosis code in each patient data in all data;
after the treatment is finished, the data of j th patient consists of three parts Xj、Lj、FjJ is more than or equal to 1 and less than or equal to M, and M represents the number of patients with collected data; fj=[l*j,P*j],l*jMean value l representing the same data from different visits of the j-th patient*,P*jIndicating the occurrence of each diagnostic code in the jth patient dataThe ratio of the frequency to the total data;
s2, performing K-means clustering sampling processing: with data F for each patientj=[l*j,P*j]Performing K-means clustering as sample points, dividing data into 3 clusters to obtain 3 clustering centers theta1,θ2,θ3Then calculating F for each patient datajAnd F' of each cluster center at the same sampling rate for each cluster centerSelecting corresponding subdata sets from the data of all patients according to the sequence of the distances from small to large to obtain D' ═ D1'∪D2'∪D3' generating a plurality of subdata sets for training a basic classifier;
s3, clustering 3 centers theta1,θ2,θ3According to P*The maximum values in the three subsets are sorted from small to large and respectively used as an rare coding subset, a more common coding subset and a common coding subset, then the three subsets are respectively and correspondingly input into three basic classifiers of GRAM +, Dipole + and RNN + for pre-training, and then the three basic classifiers are subjected to model fusion.
2. The method as claimed in claim 1, wherein the GRAM + is a global attention mechanism added by using personalized data of patients as guidance based on GRAM, and is specifically designed as follows:
in the knowledge directed acyclic graph formed by the medical ontology, the leaf node is the element in the diagnosis code set in S11, and the ancestor node thereof represents the ontology represented by the leaf node derived from the leaf node; all nodes c are assigned a basic embedding vector e, representing the final representation of a leaf node as a basic embedded convex combination of itself and its ancestor nodes:
wherein g isiRepresents a medical code ciA (i) represents the code ciAnd ciIndex of ancestor node, αijIs the local attention weight, calculated by the Softmax function as follows:
f(ei,ej) Is a scalar value, representing eiAnd ejThe compatibility between two basic embeddings is obtained by a multilayer perceptron;
by concatenating the final representations g of all medical codes1,g2,...,gNTo obtain an embedded matrix G, and a diagnosis vector vtExpressed as a vector xtMultiplied by the embedding matrix G and passed through a nonlinear hyperbolic tangent activation function tanh ():
v1,v2,...,vT=tanh(G[x1,x2,...,xT])
the patient's personalized data L ═ L is then utilized1,l2,...,lT]To add a global attention weight betatObtaining a global representation u comprising patient-personalized data informationt:
ut=βtvt,t=1,2,...,T
βtCalculated by the following Softmax function:
f(li,l*) Is a scalar value representing liAnd l*The compatibility between the sensors is obtained by a multilayer perceptron;
will u1,u2,...uTInputting the data into a GRU network to obtain a hidden state representation h1,h2,...,hTGenerating the first prediction information by the Softmax layerIs defined as:
h1,h2,...,hT=GRU(u1,u2,...,uT,θr)
3. The method as claimed in claim 1, wherein the Dipole + utilizes the patient's personality to predict the risk of adverse eventsChange data L ═ L1,l2,...,lT]As a guide, a bidirectional recurrent neural network and an attention mechanism are used simultaneously to predict the patient information; first, the information X of the doctor is embedded into a representation vector v by a multi-layer perceptrontThen using the patient's personalized data L ═ L1,l2,...,lT]To add a global attention weight betatObtaining a global representation u comprising patient-personalized data informationt:
ut=βtvt,t=1,2,...,T
βtCalculated by the following Softmax function:
f(li,l*) Is a scalar value;
then vector utIs input to a bi-directional recurrent neural network and finally, the bi-directional outputs are concatenated to generate a potential vector for prediction using an attention mechanism based on the data sequence position.
4. The method as claimed in claim 3, wherein RNN + is based on RNN using patient-personalized data L ═ L1,l2,...,lT]Guiding a patient visit information representation vector X to generate a global representation vector u comprising patient personalized data informationtGlobal representation vector utThe algorithm of (c) is the same as the Dipole + method, and the vector u is represented globallytAn attention model based on the data sequence position is entered and then a prediction is made using unidirectional GRUs.
5. The method of claim 1, wherein an adaptive weighted integration strategy is applied in the model fusion stage for each sample Xi=[x1,x2,...,xT]Calculating its distance d to each cluster centeri=[δi1,δi2,δi3];
For each sample, the integrated weight w is generated using the following formulai:
The final integrated output result is expressed as:
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