A method for evaluating the overall complexity of multidimensional time series

Abstract

This paper discloses a method for evaluating the overall complexity of multidimensional time series. First, the original multidimensional time series is coarse-grained. Second, the coarse-grained multidimensional time series is compressed using sequence and symbolic state compression to obtain a one-dimensional global stateful sequence. Then, the multivariate complexity of the one-dimensional global stateful sequence is calculated using dimensionality complexity, entropy complexity, and symbolic complexity. Finally, the multivariate complexity of all time scales is calculated to obtain the multivariate multiscale complexity. This method can be run on multidimensional sequences with any number of variables; it is not affected by the order of the multidimensional sequence and only measures the complexity inherent in the multidimensional time series. It has a low computational complexity and is particularly suitable for multidimensional time series with large numbers of variables and high sampling rates.

Description

Technical Field

[0001] The present invention relates to the fields of multi-scale analysis methods, system overall complexity evaluation, and multi-dimensional data processing technology, and in particular to a multi-dimensional time series overall complexity evaluation method. Background Art

[0002] In most cases, the only way to understand a system's internal operating mechanisms is through measured time series signals of the system's macroscopic output. In reality, most complex systems and their output time series signals exhibit nonlinear, nonstationary, complex, chaotic, and highly random characteristics. Conventional linear analysis methods often overlook these inherent nonlinear characteristics of the time series. Nonlinear dynamic analysis methods, however, can effectively extract valuable information from time series and provide new technical tools for understanding complex systems and their output signals. Complexity is a nonlinear parameter widely used to characterize the complexity of a time series. The complexity of a system's output time series can often reflect the system's internal operating mechanisms. Currently, commonly used complexity analysis algorithms typically measure complexity from the perspectives of dimensional complexity, irregularity, and unpredictability.

[0003] Complex systems exhibit complex structures at different scales, known as fractal properties. The self-similarity of complex system structures can be measured through the study of small-scale structures. Nonlinear time series exhibit varying irregularities at different time scales. The complexity metrics obtained by traditional complexity analysis only represent the disorder of the time series at a specific time scale, and the measurement of disorder contradicts the assumptions of complexity theory. Therefore, traditional complexity analysis methods based on single-scale analysis often overlook the multi-scale complexity characteristics of a system and fail to accurately describe the true complexity of a complex system. The evaluation and characterization of system complexity requires a multi-scale, comprehensive analysis method to quantify the complexity of time series at different time scales and more intuitively and comprehensively describe the dynamic changes of the system.

[0004] With the continuous advancement of sensor technology, multivariate monitoring of complex systems is an inevitable trend. Complexity assessment methods based on one-dimensional time series only measure the local complexity of the system and are unable to assess the overall complexity of the system. Currently, most complexity assessment methods based on multidimensional time series typically perform spatial reconstruction based on multivariate embedding theory. Accurate spatial reconstruction is highly dependent on the spatial location information of the multivariate time series, which is computationally complex and unsuitable for multivariate time series with high dimensions. Therefore, a multivariate multiscale complexity algorithm is needed that can comprehensively assess the temporal and spatial dynamic complexity of multivariate time series output by the same system. Furthermore, this algorithm should be applicable to different types of complex systems, such as simulation, physiological, meteorological, and geological systems, and be suitable for multidimensional time series with high channel counts and high sampling rates. Physiological systems are a type of complex system that has attracted widespread attention because they can simultaneously output multidimensional time series. Traditional multiscale entropy algorithms and their improved algorithms often fail to fully assess the overall complexity of such multidimensional time series, and the results obtained often differ from the actual results due to the introduction of artificial interference. Ignoring the sequential effects of multiple variables and fully utilizing the structural information of multidimensional time series output by physiological systems has become a key focus in the field of multidimensional signal processing and analysis.

[0005] In summary, studying a method for evaluating the overall complexity of multidimensional time series has very important practical value. Summary of the Invention

[0006] Purpose of the invention: In view of the shortcomings of the above-mentioned existing multi-scale complexity analysis algorithms, the present invention provides a method for evaluating the overall complexity of multidimensional time series, which overcomes the shortcomings of the current multivariate and multi-scale complexity analysis algorithms that are unable to comprehensively evaluate the temporal and spatial dynamic complexity, and realizes the overall complexity evaluation of multidimensional time series output by multi-category complex systems.

[0007] Technical solution: The present invention provides a method for evaluating the overall complexity of a multidimensional time series, which specifically includes the following steps:

[0008] (1) Coarse-graining processing of multidimensional original time series;

[0009] (2) Using sequence method and symbol method to compress coarse-grained multidimensional time series to obtain one-dimensional global state series, including global state time series and the global state symbol sequence

[0010] (3) Calculate the multivariate complexity of one-dimensional global stateful sequences using dimensionality complexity, entropy complexity, and symbolic complexity;

[0011] (4) Calculate the multivariate complexity of all time scales to obtain the multivariate multiscale complexity.

[0012] Furthermore, the implementation process of step (1) is as follows:

[0013] The multidimensional original time series X is expressed as:

[0014]

[0015] Among them, M represents the number of variables in the multidimensional time series, N represents the data length of the multidimensional time series, and x k,u Represents the u-th data point in the k-th dimension of the multidimensional original time series;

[0016] Coarse-graining is performed on each dimension of data, and the k-th dimension v-th data point of the multidimensional coarse-grained time series with time scale s is Calculated by the following formula:

[0017]

[0018] Multidimensional coarse-grained time series Y s Expressed as:

[0019]

[0020] Furthermore, the implementation process of step (2) is as follows:

[0021] Remove the numerical offset of each variable to obtain the one-dimensional de-offset time series of the kth dimension and the multidimensional de-shifted time series B s :

[0022]

[0023] in, is the mean of the k-th dimension coarse-grained time series, is the standard deviation of the k-th dimension coarse-grained time series;

[0024] The specific calculation method of the sequence method is as follows:

[0025] Calculate the distance D between all variables in the system at the jth moment s,j , used to indicate the degree of chaos in the current system:

[0026]

[0027] in, Represents the distance between the two dimensions k1 and k2 at the jth moment;

[0028] The distance between two variables in the distance matrix is ​​divided into L intervals. The probability of the i-th interval at the j-th time is The degree of state at the jth moment Obtained by the following formula:

[0029]

[0030] Calculate the statefulness of all moments Get the global state time series

[0031] The specific calculation method of the symbolic method is as follows:

[0032] The state at each moment is defined as the system microstate at that moment; the system topology at all moments is extracted and input into the unsupervised clustering algorithm, and the number of target categories is set, and then it is reduced to the target number T through the clustering algorithm; the system category at any j-th moment is obtained At this time, the global state symbol sequence is a T-ary symbolic sequence.

[0033] Furthermore, the specific calculation method of the dimensional complexity in step (3) is as follows:

[0034] Based on the global state time series obtained in step (2) Delayed reconstruction of the new matrix

[0035]

[0036] Where t represents the fractal scale;

[0037] calculate The curve length L w (t):

[0038]

[0039] Calculate the total curve length L(t) for different t values ​​and take the logarithm of L(t) to get the multivariate dimensional complexity

[0040]

[0041] Here, β represents the power law exponent and C represents a constant.

[0042] Furthermore, the specific calculation method of the entropy complexity in step (3) includes approximate entropy, sample entropy and fuzzy entropy;

[0043] The specific calculation method of the approximate entropy complexity is as follows:

[0044] Based on global state time series The interval length is m, and the reconstructed sequence is obtained

[0045]

[0046] Calculate the bth subinterval and all subintervals The distance between

[0047]

[0048] The number of statistical distances less than or equal to the threshold r is calculated, and the similarity ratio is obtained according to the following formula

[0049]

[0050] For the entire global state sequence After counting the similar sequence ratios of all subintervals, the average similarity is obtained

[0051]

[0052] Changing the interval length to m+1, we get Multivariate entropy complexity for:

[0053]

[0054] The specific calculation method of the sample entropy complexity is as follows:

[0055] Based on global state time series The interval length is m, and the reconstructed sequence is obtained

[0056] Calculate the bth subinterval and other subintervals The distance between

[0057] The number of statistical distances less than or equal to the threshold r is calculated, and the similarity ratio is obtained according to the following formula

[0058]

[0059] For the entire global state sequence After counting the similar sequence ratios of all subintervals, the average similarity is obtained

[0060]

[0061] Changing the interval length to m+1, we get Multivariate entropy complexity for:

[0062]

[0063] The specific calculation method of the fuzzy entropy complexity is as follows:

[0064] Based on global state time series The interval length is m, and the reconstructed sequence is obtained

[0065]

[0066] in, for The mean of

[0067] Calculate the bth subinterval and other subintervals The distance between Through the fuzzy function Define the similarity between two subintervals

[0068]

[0069] Where n is the fuzzy power; for the entire state sequence All subinterval statistical probabilities

[0070]

[0071] Changing the interval length to m+1, we get Multivariate entropy complexity for:

[0072]

[0073] Furthermore, the symbol complexity in step (3) includes permutation entropy complexity and symbol entropy complexity;

[0074] The specific calculation method of the permutation entropy complexity is as follows:

[0075] Based on global state symbol sequence The interval length is m, and the reconstructed sequence is obtained

[0076]

[0077] For each subinterval Sort the numbers in ascending order to get the arrangement pattern sequence There are E different permutation patterns, and the probability of the e-th permutation pattern is P e, multivariable symbolic complexity for:

[0078]

[0079] The specific calculation method of the symbol entropy complexity is as follows:

[0080] Based on global state symbol sequence The interval length is m, and the reconstructed sequence is obtained There are K possibilities for this m-ary sequence;

[0081] Statistical reconstruction sequence The probability P of the f-th m-ary sequence in f , multivariable symbolic complexity for:

[0082]

[0083] Beneficial effects: Compared with the prior art, the beneficial effects of the present invention are: 1. The present invention overcomes the shortcomings of the multivariate and multiscale complexity analysis algorithm that cannot comprehensively evaluate the temporal and spatial dynamic complexity, and realizes the overall complexity evaluation of the multidimensional time series output by the complex system; 2. The present invention is not affected by the order of the multidimensional time series, effectively measures the complexity of the multidimensional time series itself, and is applicable to multivariate time series with any number of variables; 3. The present invention can be applied to multidimensional time series with a large number of variables and a high sampling rate, with a small amount of calculation; 4. Compared with the traditional multiscale complexity analysis algorithm, the present invention can be applied to shorter time series; 5. The present invention has higher robustness and stability, and when used for multivariate output time series analysis, the individual differences of the results obtained are small. BRIEF DESCRIPTION OF THE DRAWINGS

[0084] Figure 1 is a flow chart of the present invention;

[0085] Figure 2 The results of multivariate and multiscale complexity analysis of 12-lead ECGs of healthy young people and healthy elderly people are shown. DETAILED DESCRIPTION

[0086] The present invention is further described in detail below with reference to the accompanying drawings:

[0087] This paper proposes a method for evaluating the overall complexity of multidimensional time series. Figure 1 As shown, the specific steps include:

[0088] Step 1: Coarse-graining multidimensional original time series.

[0089] The multidimensional original time series X is expressed as:

[0090]

[0091] Where M represents the number of variables in the multidimensional time series, N represents the data length of the multidimensional time series, and x k,u Represents the u-th data point in the k-th dimension of the multidimensional original time series.

[0092] Coarse-graining is performed on each dimension of data, and the k-th dimension v-th data point of the multidimensional coarse-grained time series with time scale s is Calculated by the following formula:

[0093]

[0094] Multidimensional coarse-grained time series Y s Expressed as:

[0095]

[0096] Step 2: Use the sequence method and symbolic method to compress the coarse-grained multidimensional time series to obtain a one-dimensional global state sequence GS s , including global stateful time series and the global state symbol sequence

[0097] Remove the numerical offset of each variable to obtain the one-dimensional de-offset time series of the kth dimension and the multidimensional de-shifted time series B s :

[0098]

[0099] in, is the mean of the k-th dimension coarse-grained time series, is the standard deviation of the k-th dimension coarse-grained time series.

[0100] Subsequently, the multidimensional coarse-grained time series after removing the offset can be compressed into a one-dimensional global state time series and a global state symbol sequence by using the sequence method and the symbol method respectively.

[0101] (1) Sequential method:

[0102] Calculate the distance D between all variables in the system at the jth moment s,j , used to indicate the degree of chaos in the current system:

[0103]

[0104] in, Represents the distance between the two dimensions k1 and k2 at the jth moment.

[0105] The distance between two variables in the distance matrix is ​​divided into L intervals. The probability of the i-th interval at the j-th time is The degree of state at the jth moment Obtained by the following formula:

[0106]

[0107] Calculate the statefulness of all moments Get the global state time series

[0108] (2) Symbolic method:

[0109] The state at each moment is defined as the system microstate at that moment; the system topology at all moments is extracted and input into the unsupervised clustering algorithm, and the number of target categories is set, and then it is reduced to the target number T through the clustering algorithm; the system category at any j-th moment is obtained At this time, the global state symbol sequence is a T-ary symbolic sequence.

[0110] Step 3: Calculate the multivariate complexity C of the one-dimensional global state sequence using various computational methods such as dimensionality complexity, entropy complexity, and symbolic complexity. s .

[0111] (1) Dimensional complexity:

[0112] Based on global state time series Delayed reconstruction of the new matrix

[0113]

[0114] Here, t represents the fractal scale.

[0115] calculate The curve length L1(t):

[0116]

[0117] Calculate the total curve length L(t) for different t values ​​and take the logarithm of L(t) to get the multivariate dimensional complexity

[0118]

[0119] Here, β represents the power law exponent and C represents a constant.

[0120] (2) Entropy complexity:

[0121] Based on the one-dimensional global state time series extracted by the sequence method in step 2, the specific calculation methods of the entropy complexity in the multivariate complexity step of quantifying the one-dimensional global state sequence include approximate entropy, sample entropy, and fuzzy entropy.

[0122] (a) Approximate entropy complexity:

[0123] Based on global state time series The interval length is m, and the reconstructed sequence is obtained

[0124]

[0125] Calculate the bth subinterval and all subintervals The distance between

[0126]

[0127] The number of statistical distances less than or equal to the threshold r is calculated, and the similarity ratio is obtained according to the following formula

[0128]

[0129] For the entire global state sequence After counting the similar sequence ratios of all subintervals, the average similarity is obtained

[0130]

[0131] Changing the interval length to m+1, we get Multivariate entropy complexity for:

[0132]

[0133] (b) Sample entropy complexity:

[0134] Based on global state time series The interval length is m, and the reconstructed sequence is obtained Calculate the bth subinterval and other subintervals The distance between The number of statistical distances less than or equal to the threshold r is calculated, and the similarity ratio is obtained according to the following formula

[0135]

[0136] For the entire global state sequence After counting the similar sequence ratios of all subintervals, the average similarity is obtained

[0137]

[0138] Changing the interval length to m+1, we get Multivariate entropy complexity for:

[0139]

[0140] (c) Fuzzy entropy complexity:

[0141] Based on global state time series The interval length is m, and the reconstructed sequence is obtained

[0142]

[0143] in, for Calculate the mean of the b-th subinterval and other subintervals The distance between Through the fuzzy function Define the similarity between two subintervals

[0144]

[0145] Among them, n is the fuzzy power; for the entire state sequence All subinterval statistical probabilities

[0146]

[0147] Changing the interval length to m+1, we get Multivariate entropy complexity for:

[0148]

[0149] (3) Symbol complexity:

[0150] Based on the one-dimensional global stateful symbol sequence extracted in step 2, the specific calculation method of the symbol complexity in the multivariate complexity step of quantifying the one-dimensional global stateful symbol sequence includes permutation entropy and symbol entropy complexity.

[0151] (a) Permutation entropy complexity:

[0152] Based on global state symbol sequence The interval length is m, and the reconstructed sequence is obtained

[0153]

[0154] For each subinterval Sort the numbers in ascending order to get the arrangement pattern sequence

[0155] There are E different permutation patterns, and the probability of the e-th permutation pattern is P e , multivariable symbolic complexity for:

[0156]

[0157] (b) Symbolic entropy complexity.

[0158] Based on global state symbol sequence The interval length is m, and the reconstructed sequence is obtained There are K possibilities for this m-ary sequence.

[0159] Statistical reconstruction sequence The probability P of the f-th m-ary sequence in f , multivariable symbolic complexity for:

[0160]

[0161] Step 4: Calculate the multivariate complexity of all time scales to obtain the multivariate multiscale complexity, that is, modify the time scale parameter of the coarse-graining process, repeat the above steps, obtain the multivariate complexity corresponding to different time scales, and obtain the multivariate multiscale complexity.

[0162] The following will further illustrate the technical solution of the present invention in conjunction with a specific implementation plan for 12-lead ECG time series analysis of healthy young people and healthy elderly people. The specific contents of the embodiment of the present invention are as follows:

[0163] The multivariate 12-lead ECG time series used in this embodiment has a dimension of 12, i.e., 12 variables, a sampling rate of 500 Hz, and each ECG time series segment is approximately 15 seconds long, meaning each time series segment has approximately 7,500 points. First, this embodiment preprocesses the two sets of ECG time series using conventional ECG time series preprocessing methods, including removing bad leads, removing power frequency interference with a notch filter, removing baseline drift and high-frequency noise with a bandpass filter, removing bad segments, and removing outliers. Subsequently, the coarse-grained multidimensional time series is obtained using the specific implementation steps of coarse-graining the original multidimensional time series in step 1. The coarse-grained multidimensional time series is compressed using the specific implementation steps of the sequence method in step 2 to obtain a one-dimensional global stateful sequence. The multivariate complexity of the one-dimensional global stateful sequence is calculated using the specific calculation method of sample entropy in step 3. Finally, the multivariate complexity of all time scales is calculated in step 4 to obtain the multivariate multiscale complexity, i.e., the above process is repeated with varying time scale factors. The results of multivariate and multiscale complexity analysis of 12-lead ECG in healthy young people and healthy elderly people are as follows: Figure 2 As shown in the figure, the results describe the changes in the overall complexity of the heart in the aging state relative to the normal state. At all time scales in the figure, the complexity of the healthy young group is higher than that of the healthy elderly group, indicating that the hearts of the healthy young group have higher overall complexity than those of the healthy elderly. Moreover, the complexity difference between the two groups increases with the increase of time scale until it tends to stabilize, and the difference between the two is small at small time scales, indicating that the multiscale correlation analysis method can comprehensively and truly characterize the complexity gap under different physiological states from a multi-scale perspective. In addition, the above multivariate and multi-scale complexity analysis results verify the conclusion that the aging state reduces the complexity of the system, indicating that this overall complexity assessment method can comprehensively consider the information contained in the multivariate signal to achieve a true and reliable overall complexity assessment.

[0164] The above results show that the multivariate and multiscale entropy analysis method can truly and reliably analyze the multidimensional time series (greater than 3 dimensions) output by the physiological system and achieve the purpose of effectively evaluating its overall complexity.

[0165] The above description only shows some preferred implementation schemes of the algorithm of the present invention. It should be pointed out that several algorithm improvements can be made without departing from the technical principles of the present invention, and these algorithm improvements should also be regarded as the scope of protection of the present invention.

Claims

1. A method for evaluating the overall complexity of a multidimensional time series, characterized by: The following steps are involved: (1) Coarse-graining the multidimensional original time series; the multidimensional original time series uses a multivariate 12-lead ECG time series, that is, there are 12 variables, the sampling rate is 500 Hz, and the number of points in each time series is 7500; preprocessing is completed on the two groups of ECG time series; (2) Using sequence method and symbol method to compress coarse-grained multidimensional time series to obtain one-dimensional global state series, including global state time series and the global state symbol sequence (3) Calculate the multivariate complexity of one-dimensional global stateful sequences using dimensionality complexity, entropy complexity, and symbolic complexity; (4) Calculate the multivariate complexity of all time scales to obtain multivariate multiscale complexity; The implementation process of step (2) is as follows: Remove the numerical offset of each variable to obtain the one-dimensional de-offset time series of the kth dimension and the multidimensional de-shifted time series B s : in, is the mean of the k-th dimension coarse-grained time series, is the standard deviation of the k-th dimension coarse-grained time series; The specific calculation method of the sequence method is as follows: Calculate the distance D between all variables in the system at the jth moment s,j , used to indicate the degree of chaos in the current system: in, Represents the distance between the two dimensions k1 and k2 at the jth moment; The distance between two variables in the distance matrix is ​​divided into L intervals. The probability of the i-th interval at the j-th time is The degree of state at the jth moment Obtained by the following formula: Calculate the statefulness of all moments Get the global state time series The specific calculation method of the symbolic method is as follows: The state at each moment is defined as the system microstate at that moment; the system topology at all moments is extracted and input into the unsupervised clustering algorithm, and the number of target categories is set, and then it is reduced to the target number T through the clustering algorithm; the system category at any j-th moment is obtained At this time, the global state symbol sequence is a T-ary symbolic sequence.

2. A method for evaluating the overall complexity of a multidimensional time series according to claim 1, characterized in that: The implementation process of step (1) is as follows: The multidimensional original time series X is represented as: Where M represents the number of variables in the multidimensional time series, N represents the data length of the multidimensional time series, and x k,u Represents the u-th data point in the k-th dimension of the multidimensional original time series; Coarse-graining is performed on each dimension of data, and the k-th dimension v-th data point of the multidimensional coarse-grained time series with time scale s is Calculated by the following formula: Multidimensional coarse-grained time series Y s Expressed as:

3. A method for evaluating the overall complexity of a multidimensional time series according to claim 1, characterized in that: The specific calculation method of the dimensional complexity in step (3) is as follows: Based on the global state time series obtained in step (2) Delayed reconstruction of the new matrix Where t represents the fractal scale; calculate The curve length L w (t): Calculate the total curve length L(t) for different t values ​​and take the logarithm of L(t) to get the multivariate dimensional complexity Here, β represents the power law exponent and C represents a constant.

4. A method for evaluating the overall complexity of a multidimensional time series according to claim 1, characterized in that: The specific calculation methods of the entropy complexity in step (3) include approximate entropy, sample entropy and fuzzy entropy; The specific calculation method of the approximate entropy complexity is as follows: Based on global state time series The interval length is m, and the reconstructed sequence is obtained Calculate the bth subinterval and all subintervals The distance between The number of statistical distances less than or equal to the threshold r is calculated, and the similarity ratio is obtained according to the following formula For the entire global state sequence After counting the similar sequence ratios of all subintervals, the average similarity is obtained Changing the interval length to m+1, we get Multivariate entropy complexity for: The specific calculation method of the sample entropy complexity is as follows: Based on global state time series The interval length is m, and the reconstructed sequence is obtained Calculate the bth subinterval and other subintervals The distance between The number of statistical distances less than or equal to the threshold r is calculated, and the similarity ratio is obtained according to the following formula For the entire global state sequence After counting the similar sequence ratios of all subintervals, the average similarity is obtained Changing the interval length to m+1, we get Multivariate entropy complexity for: The specific calculation method of the fuzzy entropy complexity is as follows: Based on global state time series The interval length is m, and the reconstructed sequence is obtained in, for The mean of Calculate the bth subinterval and other subintervals The distance between Through the fuzzy function Define the similarity between two subintervals Where n is the fuzzy power; for the entire state sequence All subinterval statistical probabilities Changing the interval length to m+1, we get Multivariate entropy complexity for:

5. A method for evaluating the overall complexity of a multidimensional time series according to claim 1, characterized in that: The symbol complexity in step (3) includes permutation entropy complexity and symbol entropy complexity; The specific calculation method of the permutation entropy complexity is as follows: Based on global state symbol sequence The interval length is m, and the reconstructed sequence is obtained For each subinterval Sort the numbers in ascending order to get the arrangement pattern sequence There are E different permutation patterns, and the probability of the e-th permutation pattern is P e , multivariable symbolic complexity for: The specific calculation method of the symbol entropy complexity is as follows: Based on global state symbol sequence The interval length is m, and the reconstructed sequence is obtained There are F possibilities for this m-ary sequence; Statistical reconstruction sequence The probability P of the f-th m-ary sequence in f , multivariable symbolic complexity for:

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