CN111914402B - Dynamic topology estimation system and method based on signal characteristics and topology change priori - Google Patents
Dynamic topology estimation system and method based on signal characteristics and topology change priori Download PDFInfo
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Abstract
The invention discloses a dynamic topology estimation system and method based on signal characteristics and topology change priori, wherein the method considers that the actual topology is time-varying and hidden in data and cannot be directly measured, so that data-driven prior knowledge based on signal characteristics and topology change is taken into consideration for data processing. According to the method, the graph topology learning at different moments is obtained from the data driving perspective by utilizing the learning of historical data, so that the dynamic topology structure is corrected. From characteristics of the signal itself, such as smoothness of the signal on the structure, redundancy of the signal itself, etc., implicit structural features of the data, i.e. graph topology, can be estimated from the data. Unlike the topology learning of most of the current research fixed graphs, the method considers the prior knowledge of the topology change to dynamically correct the topology, and obtains more accurate dynamic topology estimation.
Description
[ field of technology ]
The invention belongs to the technical field of signal processing, and particularly relates to a dynamic topology estimation system and method based on signal characteristics and topology change priori.
[ background Art ]
Because of the vast amount of signal redundancy, massive raw data with structural features and the like widely existing in the internet of things nowadays, the traditional data acquisition and processing method is not applicable. Compared with the traditional data such as time series, voice, audio, radio, radar, biomedical signals or images and videos, the data in the Internet of things often has the characteristics of high data dimension, discretization, irregular spatial distribution and the like, and higher and more complex requirements are put forward on the analysis and the processing of the data.
With the increasing demand for signal and information processing in the irregular field, the graph gradually becomes an important tool for analyzing network data by virtue of the nature association attribute with the network structure, and can graphically represent application data, then solve the interesting problem by focusing on and analyzing the structure of the graph, or dynamically analyze the change of the network by researching the change of the graph. The network is mathematically abstracted through the topological structure of the graph, so that the structural relationship among signals can be represented, the sensor network, the traffic network, the social network and the like can be naturally represented, and the network-related data processing method is one of important tools for processing the network-related data. The topology of the graph on which it depends is discretized and the spatial distribution is irregular, unlike signals in conventional signal processing. Therefore, the structural characteristics of the data are fully utilized, and the massive redundant original data in the Internet of things are subjected to sparsification processing, so that the requirement on signal processing is reduced.
The topology of the graph signals has a great effect on the analysis of data, and specifically, the graph signal sampling and reconstruction are established on the graph topology, and can be specifically expressed as an adjacency matrix or a Laplace matrix. Therefore, how to obtain a suitable adjacency matrix is a problem to be solved in the internet of things.
Meanwhile, in the process of processing the data of the Internet of things, the situation of the actual Internet of things is considered, the graph topology is unknown and is not invariable, and the graph topology structure corresponding to the data changes with time. For example, the topology of the graph may also change due to damage to the sensors; meanwhile, the relevance among the nodes is difficult to define; furthermore, the effects of the external environment may also lead to changes in the topology of the graph, such as changes in the topology due to displacement of the tester during the measurement of sea level temperatures. Therefore, how to obtain a suitable adjacency matrix for a dynamic graph topology is a problem to be solved. Secondly, the topology learning is usually mainly performed by fixed topology; for the dynamic process, the change process is not considered, but the change is regarded as a mutation, and the specific topology change at the current moment is estimated through all the observation data. Researchers consider that in practice the situation of graph changes is time-varying, and that part of the historical data has a negative effect on the estimation of the dynamic topology, i.e. has an impact on the accuracy of the topology estimation.
[ invention ]
The invention aims to overcome the defects of the prior art, and discloses a dynamic topology estimation system and method based on signal characteristics and topology change priori, which are used for solving the problems that the topological structure of a graph signal in the prior art is mostly a fixed topology and cannot adapt to dynamic images.
In order to achieve the purpose, the invention is realized by adopting the following technical scheme:
the dynamic topology estimation method based on the signal characteristics and the topology change priori is characterized by comprising the following steps of:
step 1, establishing an observation signal model;
step 2, based on the observation signal model, establishing a relation between the observation signal and the graph topology;
step 3, establishing a topology learning mathematical model aiming at the graph topology, and solving the mathematical model; the mathematical model is a punishment function and a punishment item for minimizing the observation signal at the t moment; the penalty function comprises maximum smoothness and minimum redundancy of the signal, and the penalty term comprises constraint topology scale and constraint topology change;
and 4, grouping the observation signal data according to time, substituting the grouped observation signal data into a mathematical model, and solving topology estimation in an iterative manner.
The invention further improves that:
preferably, in step 1, the observed signal model satisfies the following formula:
Y t ＝U′ t S′ t +ε (9)
wherein Y is t Represents the observed signal at time t, U' represents U t Is equivalent to the feature vector of (1) representing S t Equivalent plot signal frequency.
Preferably, in step 2, the relationship between the observed signal and the graph topology is:
in the method, in the process of the invention,p μ respectively represent the frequencies of the t-th time chartThe probability density of the value distribution and the probability density of the mean of the vertex values of the graph signals.
Preferably, when the observed signal average value is 0, formula (12) is expressed as:
wherein: parameter alpha 0 Is a regularization parameter, and the second term in equation (10) satisfies the following equation:
preferably, in step 3, the mathematical model is:
wherein f (L) t ,Y t ) Is a punishment function for the observation signal at the t moment; f (f) time (ΔW t ,L t ,Y t ) A penalty term representing a topology change; gamma represents the degree of time-varying speed of the graph topology, L t A Laplacian matrix representing the graph signal at time t; ΔW (delta W) t The amount of change in the adjacency matrix from time t-1 to time t is shown.
Preferably, f (L t ,Y t ) Expressed as:
preferably, f time (ΔW t ,L t ,Y t ) Expressed as:
preferably, the specific process of step 4 is as follows: grouping the observation signal data according to time, adopting an estimated value of a fixed topology as a data value at the initial moment, combining the divided 1 st group of observation signal data, and calculating to obtain an adjacent matrix of the first iteration through a mathematical model; and obtaining an adjacent matrix of the second iteration through the 2 nd group of observation signal data and the adjacent matrix of the first iteration, and analogizing to finally obtain the adjacent matrix which is the final dynamic topology.
A dynamic topology estimation system based on signal characteristics and topology change priors, comprising:
means for establishing an observation signal model;
means for establishing a relationship between the observed signal and the graph topology;
means for establishing a topology learning mathematical model and means for solving the mathematical model;
and the device is used for transmitting the observation signals to the mathematical model and carrying out iterative computation.
Compared with the prior art, the invention has the following beneficial effects:
the invention discloses a dynamic topology estimation system and method based on signal characteristics and topology change priori, wherein the method considers that the actual topology is time-varying and hidden in data and cannot be directly measured, so that data-driven prior knowledge based on signal characteristics and topology change is taken into consideration for data processing. According to the method, the graph topology learning at different moments is obtained from the data driving perspective by utilizing the learning of historical data, so that the dynamic topology structure is corrected. From characteristics of the signal itself, such as smoothness of the signal on the structure, redundancy of the signal itself, etc., implicit structural features of the data, i.e. graph topology, can be estimated from the data. Unlike the topology learning of most of the current research fixed graphs, the method considers the prior knowledge of the topology change to dynamically correct the topology, and obtains more accurate dynamic topology estimation. Secondly, the invention researches an approximation strategy for the time-varying topology, and approximates the time-varying topology gradually by adopting a dynamic topology estimation strategy for data grouping, so as to obtain more accurate topology.
[ description of the drawings ]
FIG. 1 is a flow chart of dynamic topology tracing of the present invention;
FIG. 2 is a graph comparing the dynamic topology of the present invention with the results of a direct topology estimation NMSE;
FIG. 3 is a graph comparing the relative error results of the dynamic topology and the direct topology estimation of the present invention;
FIG. 4 is a graph comparing the correlation results of the dynamic topology and the direct topology estimation of the present invention;
FIG. 5 is a graph comparing the results of the time-varying topology approximation and dynamic topology estimation NMSE of the present invention;
FIG. 6 is a graph comparing the relative error results of a time-varying topology approximation and a dynamic topology estimation of the present invention;
FIG. 7 is a comparison of correlation results of time-varying topology approximations and dynamic topology estimates of the present invention;
FIG. 8 is a historical data driven dynamic estimation;
fig. 9 is an incremental data driven time varying topology estimation.
[ detailed description ] of the invention
The invention is described in further detail below with reference to the attached drawing figures:
in the description of the present invention, it should be noted that, directions or positional relationships indicated by terms such as "center", "upper", "lower", "left", "right", "vertical", "horizontal", "inner", "outer", etc., are based on directions or positional relationships shown in the drawings, are merely for convenience of description and simplification of description, and do not indicate or imply that the apparatus or element to be referred to must have a specific direction, be constructed and operated in the specific direction, and thus should not be construed as limiting the present invention; the terms "first," "second," and "third" are used for descriptive purposes only and are not to be construed as indicating or implying relative importance; furthermore, unless explicitly specified and limited otherwise, the terms "mounted," "connected," and "connected" are to be construed broadly, and may be either fixed or removable, for example; can be directly connected or indirectly connected through an intermediate medium, and can be communication between two elements. The specific meaning of the above terms in the present invention will be understood in specific cases by those of ordinary skill in the art.
To study the data on the nodes, it is assumed that the data is indexed by N nodes in the graph g= (V, W). The nodes thereof may characterize the data source, and the data on each node of V itself may be a time series, image, video, or other feature describing the node collected or generated by the agent. The edges of graph G represent the dependencies between data. And the edge weight value of G forms an adjacent matrix W, and the larger edge weight value indicates that the correlation of two correspondingly connected vertexes is stronger, so that the topological result of the graph signal can be intuitively represented.
For the adjacency matrix W, wherein the element W of the ith row and the jth column ij Defined as the weight of the edge connecting the ith node with the jth node. When there is an edge between the ith node and the jth node, w ij Is not 0; otherwise w ij Is 0. The degree matrix of the undirected weighted graph is defined as:
D＝diag(d 1 ,d 2 ,…,d N )＝diag(W·1) (1)
wherein the method comprises the steps ofDegree defined as the i-th node; vector 1 represents an N-dimensional matrix of all 1's.
The graph laplacian matrix, which is one of the graph topology representations, is defined as:
L＝D-W (2)
meanwhile, the laplace matrix and the adjacency matrix can be mutually converted, and the conversion relation thereof satisfies table 1 as follows.
Table 1 laplace matrix and adjacency matrix conversion contrast table
Meanwhile, considering the influence of dynamic topology change, for the Laplace matrix, the following is satisfied:
L t ＝L t-1 -ΔW t +diag(ΔW t ·1 N ) (3)
wherein L is t Represent the firstLaplacian matrix of graph signal at time t, deltaW t Vector 1 representing the change in the adjacency matrix from time t-1 to time t N Representing an N-dimensional matrix of all 1's.
For the determination of the topology estimation result, by comparing the topology matrix obtained by estimation with the actual topology matrix, the overall approximation degree of the matrix is mainly compared with the approximation degree of the specific numerical value of the matrix. The decision quantity is introduced as follows:
the (4) - (5) are used as decision criteria to measure the accuracy of topology estimation.
Based on the Laplace matrix and the adjacency matrix, the invention discloses a dynamic topology estimation system and a method based on signal characteristics and topology change priori, wherein the method specifically comprises the following steps:
step 1, establishing an observation signal model
For the undirected weighting graph G, an observed signal at time t is defined as:
Y t ＝[y t (1),y t (2),…,y t (N)] (7)
y t (i) Is the observed signal value at i vertices at time t; considering a Gaussian Markov random field model, the observed signal model satisfies the following conditions:
wherein U is t And S is t The eigenvector matrix of the laplacian matrix of the map signal at time t and the map signal frequency are represented, respectively. Parameter mu x Representing the mean value of the graph signal, the parameter epsilon represents the model building error and the influence of errors such as noise, and U' represents U t Is equivalent to the feature vector of (1) representing S t Equivalent plot signal frequency.
Step 2, establishing the relation between the observation signal and the topology
For the maximum posterior probability of the graph frequency (equivalent graph frequency), the calculation result is as follows:
wherein the probability density p e 、p St′ Respectively representing model establishment errors and frequency value distribution probability density of the equivalent graph at the t moment. Due to the independence of the observed signals, i.e. the mean value mu of the plot signals is considered x Is independent, i.e. satisfiesThat is, the expression (11) is expressed as:
p μ the probability density of the graph frequency value distribution at the t moment and the probability density of the average value of the graph signal vertex values respectively, namely the maximum posterior probability of the equivalent graph frequency is related to the probability distribution of the model building error, the probability density of the graph signal average value and the probability of the graph frequency, wherein the probability distribution of the model building error is expressed as log p e (Y t -U′ t S′ t ) The probability density of the mean of the graph signal is expressed as log p μ (μ x ) And the probability of the graph frequency is expressed as +.>
For probability of graph frequency, when considering fixed graph topology representation, i.e. considering that the graph frequencies of the graph signals at different moments are independently and equidistributed, taking a Gaussian Markov random field model as an example, the graph frequencies satisfyWherein the function is + Representing the pseudo-inverse, the parameter Λ represents the eigenvector of the laplace matrix L. At this time, for (12), under the condition of the signal 0 mean value, the following is satisfied:
the second term in the above formula (11) satisfies the following formula:
S t T ΛS t ＝((U t T Y t ) T Λ(U t T Y t ))＝(Y t T U t ΛU t T Y t )＝Y t T L t Y t (14)
through the above steps, the relation between the observation signal and the topology is initially established, and then the initially established relation is dynamically corrected.
Step 3, establishing and solving a mathematical model of specific topology learning
The impact of topology estimation comes mainly from two aspects of time t. In combination with practical consideration, in the dynamic topology model, the graph frequencies of graph signals at different moments are related and cannot be intuitively represented, and a general data-driven topology learning model is built, so that the following conditions are satisfied:
further analysis of the above, whichWherein f (·) is a penalty function for the current signal, which is a penalty function of the above formula (13), f time (. Cndot.) represents a penalty term for topology changes, the parameter γ represents how fast the graph topology changes over time, and for topology changes γ=0, i.e. fast changes, it appears to solve each topology separately; when γ= infinity, i.e. approximately unchanged, it appears as an "average" result for the same graph. Now utilize f time (. About.) correcting the topology change, and estimating the topology change in addition to the current topology structure to obtain the dynamic topology correction result.
Step 3.1, modeling taking into account the current signal f (·).
First, in GMRF, for topology learning of a fixed topology, from the above discussion, a quadratic form Y is introduced according to probability distribution of graph frequency as shown in expression (14) T LY, this quadratic form also represents the smoothness of the signal. Second, consider the redundancy of the signal itself. Since the observed signal is typically superfluous, it is necessary to enforce constraints on the observed signal sparsity, such as the TV-GL algorithm. The function TV-GL satisfiesWherein l ij And y ij Representing the elements of matrix L and matrix Y, respectively. This is also the case for the observed signal 1 And (5) a norm constraint, which represents the self sparseness degree of the observed signal. Furthermore, to avoid meaningless solutions, the F-norm constraint topology scale is introduced while avoiding occurrence of meaningless solutions like the solution of all 0 solutions. The pair function f (·) can be expressed as follows:
step 3.2 penalty term f for dynamic topology changes time (. Cndot.) a model is built through a priori knowledge of the topology dynamics.
Considering the specific model, firstly considering the model conditions such as thermal diffusion, taking the temperature as vertex data, conductingThe heat tends to act as a weighted edge, resulting in a temperature near the heat source that tends to stabilize due to thermal diffusion, resulting in a change in topology that tends to smooth. In this class of models, by exploiting Tikhonov smoothness conditionsRepresenting a dynamic topology change. Further, a case where the topology change is sparse is considered. Considering a specific scene, for a sensor network, the position of each sensor moves to change the relative relation between the sensor and other nodes, so that topology change is caused, and a typical sparse signal model is utilized 0 And convex relaxation is DeltaW|| 1 Or using the kernel norm ΔW * And (5) realizing model establishment.
In summary, for dynamic topology estimation model satisfaction (taking sparse change as an example), the latter part of equation (15) can be expressed as:
wherein the parameter alpha 1 ,β 1 ,δ 1 Gamma is a regularization parameter. The first three constraint conditions are constraint on the characteristics of the undirected graph Laplacian matrix, and the last constraint condition is the change relation between the previous moment topology and the next moment topology when dynamically changing.
Step 5.3 solving the dynamic topology estimation model
Specifically solving the equation (17), by the optimization model of the equation (17), the signal Y and the topology representation L in the model t Not co-convex, so consider an optimization model that implements (17) with alternating minimisation. Representing (17) as two optimization models:
but in a specific implementation, only the specific topological representation L is of interest t Moreover, without losing generality, the observation signal is assumed to have no error, all errors only exist in model establishment, and the error in the formula (16) is unavoidable, so that only verification is carried out at the later stage, and only the formula (17) is considered in actual calculation. In the result of the above alternate minimization, there is eventually necessarily a signal value Y approaching the observed signal X t Therefore, all observation signals are directly utilized for modeling, and the model meets the following conditions:
wherein Y is t The observed signal value at time t is shown. For models as in (20), convex problem solving is achieved directly through the cvx toolbox.
And (5) completing the establishment and the solving of the model.
Step 4, establishing an approaching time-varying topology strategy
Since the topology is time-varying, when the topology changes significantly, some historical data is no longer available, which can adversely affect the topology estimation. As shown in fig. 1, in the solving process, by using the optimized model formula (13), if the final topology estimation value is directly calculated from the known topology, the solving is inaccurate; therefore, for time-varying topology, all data are divided according to time, incremental data are utilized to gradually carry out dynamic topology estimation, specifically, an estimated value obtained by fixed topology estimation is adopted as an initial adjacent matrix W at the initial moment 0 Substituting the 1 st group of observation signal data into the model established in the step 3 and solving to obtain a 1 st adjacency matrix W by combining the divided 1 st group of observation signal data 1 The method comprises the steps of carrying out a first treatment on the surface of the Then the 1 st adjacent matrix and the 2 nd group of observation signal data are combined and substituted into the model established in the step 3 to be solved, and the 2 nd adjacent matrix W is obtained 2 Repeating the above process until the nth adjacent matrix W is obtained n I.e. to achieve an optimization of the time-varying topology.
Represented in this stepIs a stepwise dynamic topology tracking process of n steps. N-th step result W obtained by observation data estimation of each step n ' topology estimate W at the moment directly from all historical data estimates n "compared to the actual topology W, which should be closer to the current moment n 。
The main idea of the step is to correct the time-varying dynamic topology by combining the dual driving of the historical data and the updated data in a step-by-step process, thereby achieving the purpose of better estimation than that obtained by all the historical data. In this step, the estimated value obtained by adopting the fixed topology estimation for the initial time is combined as W 0 . The historical data packet serves as incremental data according to time, and the incremental data is utilized to combine with the dynamic topology optimization model to estimate.
Examples
In order to verify the estimation performance of the dynamic topology estimation strategy and the time-varying topology approximation strategy provided by the invention, firstly, a test is carried out in a synthetic data set. First, synthetic data is generated. Map signals of 50 points and 100 points in scale are established, respectively. The signal meets the GMRF condition for the magnitude of the change in the graph topology between intervals 0, 1.
First is the effect of the dynamic topology estimation strategy. In the experiment, 50 point diagram simulation is carried out, and the difference between fixed topology estimation and dynamic topology estimation is compared through different observed signal values. The results are shown in FIGS. 2-4.
By comparing the solid and dashed lines in fig. 2-4, it can be concluded that the results of the dynamic topology estimation are significantly better than the results of the direct estimation, especially when the number of observed datasets is low. This illustrates the feasibility and superiority of dynamic topology estimation.
And secondly, the effect of a time-varying topology approximation strategy. In the experiment, 100 point diagrams are simulated, small sample observation conditions are fixed, 90 groups of observation data are selected, and the observation data are divided into 3 groups of incremental data according to time. The estimated effects of the two are compared by different scales of topology changes, as shown in fig. 5-7.
First, since the topology is assumed to be sparsely changed, the estimation result is reduced as the number of sides to be changed increases. By comparing the curves in the graph, the result verifies the dynamic topological effect and accords with the expectation. By comparing the step-wise tracking results with the direct estimation results, the tracking results are better, both in correlation, NMSE and relative error. The results indicate that dynamic topology tracking has a better advantage in approximating time-varying topologies.
Again, by example verification. Sea level temperature data is selected, 30 groups of observation data are selected under the condition of small samples, the observation data are divided into 3 groups of incremental data, the time interval of all the data is larger, and the dynamic estimation driven by historical data is compared with the time-varying topology estimation result driven by the incremental data, as shown in fig. 8 and 9.
In the two graphs, square points with different gray depths represent different values, and the two topology estimation result graphs are compared to find that the weights of most edges are similar, i.e. the weight edge results of most of the two graphs are the same. However, there is still a significant difference which can be seen as an effect of a continuous topology change. In combination with the results of the synthetic data, most of the differences are found to be points of constant variation, which is consistent with expectations, and the dynamic topology tracking method is an effective modification of the topology estimation. Comparing the results of the two graphs, it can be considered that the topology can be estimated effectively, and fig. 9 is closer to the topology estimation when viewed.
The foregoing description of the preferred embodiments of the invention is not intended to be limiting, but rather is intended to cover all modifications, equivalents, alternatives, and improvements that fall within the spirit and scope of the invention.
Claims (3)
1. The dynamic topology estimation method based on the signal characteristics and the topology change priori is characterized by comprising the following steps of:
step 1, establishing an observation signal model;
in step 1, the observed signal model satisfies the following formula:
Y t ＝U′ t S t ′+ε(9)
wherein Y is t Represents the observed signal at time t, U' represents U t Etc. of (2)Valence feature vector, representing S t Equivalent plot signal frequencies;
step 2, based on the observation signal model, establishing a relation between the observation signal and the graph topology;
the relationship between the observed signal and the graph topology is:
wherein p is St 、p μ Respectively representing the probability density of the frequency value distribution of the chart at the t moment and the probability density of the average value of the vertex values of the chart signals;
when the observed signal mean is 0, formula (12) is expressed as:
wherein: parameter alpha 0 Is a regularization parameter;
step 3, establishing a topology learning mathematical model aiming at the graph topology, and solving the mathematical model; the mathematical model is a punishment function and a punishment item for minimizing the observation signal at the t moment; the penalty function comprises maximum smoothness and minimum redundancy of the signal, and the penalty term comprises constraint topology scale and constraint topology change;
the mathematical model is as follows:
wherein f (L) t ,Y t ) Is a punishment function for the observation signal at the t moment; f (f) time (ΔW t ,L t ,Y t ) A penalty term representing a topology change; gamma represents the degree of time-varying speed of the graph topology, L t A Laplacian matrix representing the graph signal at time t; ΔW (delta W) t Representing the variation of the adjacency matrix from the t-1 time to the t time;
f(L t ,Y t ) Expressed as:
f time (ΔW t ,L t ,Y t ) Expressed as:
and 4, grouping the observation signal data according to time, substituting the grouped observation signal data into a mathematical model, and solving topology estimation in an iterative manner.
2. The method for estimating dynamic topology based on signal characteristics and topology change priori according to claim 1, wherein the specific process of step 4 is as follows: grouping the observation signal data according to time, adopting an estimated value of a fixed topology as a data value at the initial moment, combining the divided 1 st group of observation signal data, and calculating to obtain an adjacent matrix of the first iteration through a mathematical model; and obtaining an adjacent matrix of the second iteration through the 2 nd group of observation signal data and the adjacent matrix of the first iteration, and analogizing to finally obtain the adjacent matrix which is the final dynamic topology.
3. A dynamic topology estimation system based on signal characteristics and topology change priors for implementing the method of claim 1, comprising:
means for establishing an observation signal model;
means for establishing a relationship between the observed signal and the graph topology;
means for establishing a topology learning mathematical model and means for solving the mathematical model;
and the device is used for transmitting the observation signals to the mathematical model and carrying out iterative computation.
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| 冯振宇 ; 沈浚 ; 汪东耀 ; 刘英 ; 温桂平 ; .基于稀疏自适应学习的台区用户拓扑结构校验.电测与仪表.2020,(07),全文. * |
| 刘远 ; 王天真 ; 汤天浩 ; 李继方 ; 陈嘉琦 ; .基于PCA-SVM模型的多电平逆变***故障诊断.电力***保护与控制.2013,(03),全文. * |
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