CN114065925A - Brain function directed connection evaluation method based on three-dimensional adaptive partition algorithm - Google Patents
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Abstract
The invention discloses a brain function directed connection evaluation method based on a three-dimensional self-adaptive partition algorithm, which comprises the following steps of: step S1: discretizing a continuous time sequence; step S2: constructing a joint distribution space; step S3: dividing the whole distribution space into a plurality of sub-areas which are approximately uniformly distributed inside by using a three-dimensional self-adaptive partitioning algorithm; step S4: calculating a joint probability mass function; step S5: the transfer entropy is calculated. The method greatly reduces the problem of calculation result distortion caused by dimension disaster of the traditional discrete transfer entropy, and can more accurately and more robustly evaluate the directed connection of brain intervals, especially for neural signals with small sample size.
Description
Technical Field
The invention relates to the field of neural signal analysis, in particular to a brain function directed connection evaluation method based on a three-dimensional adaptive partition algorithm.
Background
Studying the functional connectivity of different brain regions is very important to our understanding of the mechanism of brain high cognitive function, because for normal brain high cognitive function, it is not a single brain region that works, but rather depends more on the cooperation between different brain regions. The transitive entropy provides a method for estimating directed functional connections, which is widely applied in the field of neuroscience because of the advantages that the method is independent of any specific data model and is sensitive to nonlinear interaction.
Because the continuous signal estimation transfer entropy has the problems of complex calculation, inherent error and the like, the signal is discretized firstly by a method of box separation or phase space reconstruction, and then the transfer entropy is calculated based on the occurrence frequency of all possible combinations. In order to obtain more information about the signal, a larger number of bin and phase space symbol classes are generally desired, but since the computation of the transfer entropy is based on a three-dimensional joint distribution, too many discrete classes can generate dimension disasters for a fixed-length actual signal, resulting in a frequency of many class combinations that is very small or even 0, eventually making the computation of the transfer entropy easily distorted, and the value of the transfer entropy distorted. Large sample sizes may solve this problem to some extent, but sample size problems may be a major obstacle to biomedical research. In the case where the sample size is small, the conventional calculation of the transfer entropy based on the occurrence frequency of all the class combinations is ineffective for the discrete variables of the plurality of classes.
Disclosure of Invention
The invention aims to provide a method for evaluating directed connection more accurately especially under the condition of small samples. According to the method, probability distribution is obtained in each sub-region which is approximately evenly distributed after being segmented by a three-dimensional adaptive partitioning algorithm, so that the transfer entropy is calculated, the problem of calculation result distortion caused by dimension disaster of the traditional discrete transfer entropy is greatly reduced, the discrete transfer entropy value is estimated more accurately and more reliably under the condition of less sample amount, and further directed function connection in the brain is analyzed correctly.
In order to realize the purpose, the following technical scheme is adopted:
a brain function directed connection assessment method based on a three-dimensional adaptive partition algorithm comprises the following steps:
in step S1, the continuous time series is discretized using a phase space reconstruction method. For recording in different brain areas respectivelyThe brain electrical signal xtAnd ytN, using the embedding dimension m and the lag τ to construct a vector [ x ═ 1,2t,xt+τ,...,xt+(m-1)τ]And [ yt,yt+τ,...,yt+(m-1)τ]Wherein t is 1,2, and n- (m-1) tau, then arranging m signal points in each vector in ascending order of amplitude to obtain
And
record the sequence number [ i ] of the sorted vector1,i2,...,im]And [ j ]1,j2,...,jm]And treating the different ordering patterns as different symbol classes instead of the previously constructed vectors, so that the original signal xtAnd ytIs discretized to have L ═ m! Discrete time series of different symbol classes X ═ X1,x2,...,xn-(m-1)τY ═ Y1,y2,...,yn-(m-1)τWherein each signal point in X and Y is a symbol, representing the sequential relationship of several adjacent signal points in the original time series.
And step S2, constructing a joint distribution space. Extracting X from discrete sequences X and Y by selecting prediction time uP、XFAnd YPWherein X isP＝{x1,x2,...,xn-(m-1)τ-uRepresents the past of X, XF＝{x1+u,x2+u,...,xn-(m-1)τRepresents the future of X, YP＝{y1,y2,...,yn-(m-1)τ-uRepresents the past of Y; then, according to the times of different classes of combinationP、XFAnd YPCombining frequency distributions to construct XP、XFAnd YPDue to a discrete variable XP、XFAnd YPAll have L categories, so the space has L in total3A possible combination of classes.
Step S3, using three-dimensional adaptive scoreThe region algorithm divides the entire distribution space into a number of sub-regions that are approximately uniformly distributed within. Taking into account a discrete variable XP、XFAnd YPThe constructed joint distribution space performs partitioning operation by the following steps:
(1) and (3) testing the uniformity of the distribution of the samples in the space by using a chi-square statistic, wherein the calculation formula of the chi-square statistic is as follows:
wherein
Representing a combination of categories
Number of samples of (i)
It is indicated that when the sample is evenly distributed,
the desired number of samples combined by category, i.e.
The corresponding p value is in the range of L3-1 degree of freedom calculated under a chi-squared distribution; if the p value is larger than the threshold th, the samples are considered to be uniformly distributed, and in this case, the next operation is not carried out; otherwise, the space is divided by the next step because the samples are not uniformly distributed;
(2) dividing the space into two regions (binary segmentation) at the best point that minimizes the sum of squared deviations; for this purpose, X is requiredP，XFAnd YPSearching the optimal segmentation points in three dimensions respectively; to find XPOptimal division point in dimension, XPClass variable of
i 1,2, L is ordered by their edge frequency in the space from small to large,
has an edge frequency of
Writing the sorted category variables
Wherein
Then, by minimizing
Can find XPUpper optimal segmentation point s:
wherein a is1And a2Is divided into two subspaces by X after s point divisionPAverage value of the edge frequency; likewise, calculate XFAnd YPThe optimal division point of (a); finally, D in three dimensions is compared, and the optimal division point in the dimension with the maximum D is selected for division;
(3) two subspaces obtained after partitioning are both a new distribution space; repeating steps (1) and (2) for the two new distribution spaces until all subspaces are uniformly distributed in the sense of a given threshold.
Step S4, a joint probability mass function is calculated. Through self-adaptive partitioning, the whole joint frequency distribution space is divided into a plurality of non-overlapping sub-areas; then dividing the sample amount in the sub-regions by the total sample amount, and calculating probability mass functions among the sub-regions; since the samples in each subregion are considered to be uniformly distributed, the probability of each subregion is divided by the number of class combinations in the region to yield a single numberThe probabilities of the individual class combinations ultimately constitute the overall probability mass function. For example, after adaptive partitioning, a certain sub-area A is partitioned by XP，XFAnd YPAre combined, provided that
Then the total probability of A is
Wherein N isTIs the total number of samples, since the samples in each sub-region are considered to be uniformly distributed, the probability of any one class combination in the sub-region is
Similar procedures are performed for all remaining sub-regions, constituting an overall probability mass function.
In step S5, the transfer entropy is calculated. The transfer entropy of Y → X is, according to the overall probability mass function:
in the same way, construct YP,YFAnd XPThen repeating step S3-4, resulting in a delivered entropy of X → Y of:
due to the adoption of the technical scheme, the invention has the following technical effects:
the whole distribution space is partitioned firstly and then the transfer entropy is calculated by adopting a self-adaptive partitioning algorithm, so that the problem of calculation result distortion caused by dimension disaster of the traditional discrete transfer entropy is greatly reduced.
The method can estimate the discrete transfer entropy more accurately and more robustly, especially the calculation of small samples. The invention can reliably estimate the transfer entropy value under the condition of less sample size, so the invention has great application prospect on the research of the directional function connection between brain regions in the field of neuroscience with limited sample size.
Drawings
FIG. 1 is a flow chart of the method of the present invention;
FIG. 2 is a fixed probability distribution model;
fig. 3-4 are graphs of the entropy of transfer calculated by the method of the present invention as a function of sample size.
Detailed Description
The technical scheme of the invention is further explained by combining the accompanying drawings as follows:
as shown in fig. 1, the present invention comprises the steps of:
step S1, continuous time series discretization.
For the EEG signals x recorded in different brain areas respectivelytAnd ytAnd t 1, 2.. times.n, which is discretized using a method of phase space reconstruction. In particular, vector [ x ] is constructed using embedding dimension m and lag τt,xt+τ,...,xt+(m-1)τ]And [ yt,yt+τ,...,yt+(m-1)τ]Wherein t is 1,2, and n- (m-1) tau, then arranging m signal points in each vector in ascending order of amplitude to obtain
And
record the sequence number [ i ] of the sorted vector1,i2,...,im]And [ j ]1,j2,...,jm]And treating the different ordering patterns as different symbol classes instead of the previously constructed vectors, so that the original signal xtAnd ytIs discretized to have L ═ m! Discrete time series of different symbol classes X ═ X1,x2,...,xn-(m-1)τY ═ Y1,y2,...,yn-(m-1)τWhere each signal point in X and Y is a symbol, representing several adjacent time points in the original time seriesThe order relationship of the near signal points.
And step S2, constructing a joint distribution space.
Extracting X from discrete sequences X and Y by selecting prediction time uP、XFAnd YPWherein X isP＝{x1,x2,...,xn-(m-1)τ-uRepresents the past of X, XF＝{x1+u,x2+u,...,xn-(m-1)τRepresents the future of X, YP＝{y1,y2,...,yn-(m-1)τ-uRepresents the past of Y; then, according to the times of different classes of combinationP、XFAnd YPCombining frequency distributions to construct XP、XFAnd YPDue to a discrete variable XP、XFAnd YPAll have L categories, so the space has L in total3A possible combination of classes.
Step S3, the entire distribution space is divided into a number of sub-regions with approximately uniform distribution inside using a three-dimensional adaptive partitioning algorithm.
Taking into account a discrete variable XP、XFAnd YPThe constructed joint distribution space performs partitioning operation by the following steps:
(1) and (3) testing the uniformity of the distribution of the samples in the space by using a chi-square statistic, wherein the calculation formula of the chi-square statistic is as follows:
wherein
Representing a combination of categories
Number of samples of (i)
It is indicated that when the sample is evenly distributed,
the desired number of samples combined by category, i.e.
The corresponding p value is in the range of L3-1 degree of freedom calculated under a chi-squared distribution; if the p value is larger than the threshold th, the samples are considered to be uniformly distributed, and in this case, the next operation is not carried out; otherwise, the space is divided by the next step because the samples are not uniformly distributed;
(2) dividing the space into two regions (binary segmentation) at the best point that minimizes the sum of squared deviations; for this purpose, X is requiredP，XFAnd YPSearching the optimal segmentation points in three dimensions respectively; to find XPOptimal division point in dimension, XPClass variable of
i 1,2, L is ordered by their edge frequency in the space from small to large,
has an edge frequency of
Writing the sorted category variables
Wherein
Then, by minimizing
Can find XPUpper optimal segmentation point s:
wherein a is1And a2Is divided into two subspaces by X after s point divisionPAverage value of the edge frequency; likewise, calculate XFAnd YPThe optimal division point of (a); finally, D in three dimensions is compared, and the optimal division point in the dimension with the maximum D is selected for division;
(3) two subspaces obtained after partitioning are both a new distribution space; repeating steps (1) and (2) for the two new distribution spaces until all subspaces are uniformly distributed in the sense of a given threshold.
Step S4, a joint probability mass function is calculated.
Through self-adaptive partitioning, the whole joint frequency distribution space is divided into a plurality of non-overlapping sub-areas; then dividing the sample amount in the sub-regions by the total sample amount, and calculating probability mass functions among the sub-regions; since the samples in each sub-region are considered to be uniformly distributed, the probability of each sub-region is divided by the number of class combinations in the region to obtain the probability of a single class combination, which ultimately constitutes the overall probability mass function. For example, after adaptive partitioning, a certain sub-area A is partitioned by XP，XFAnd YPAre combined, provided that
Then the total probability of A is
Wherein N isTIs the total number of samples, since the samples in each sub-region are considered to be uniformly distributed, the probability of any one class combination in the sub-region is
Similar procedures are performed for all remaining sub-regions, constituting an overall probability mass function.
In step S5, the transfer entropy is calculated.
The transfer entropy of Y → X is, according to the overall probability mass function:
in the same way, construct YP,YFAnd XPThen repeating step S3-4, resulting in a delivered entropy of X → Y of:
by XP，XFAnd YPGenerates two discrete time series X and Y with directional information transfer, wherein each series has 8 categories, and XFRatio XPLagging by 5 sample points. However, as shown in FIG. 2, there are two superclasses for sequences X and Y, respectively, namely X ∈ { α ∈ [ alpha ]1＝(x1,x2,x3,x4),α2＝(x5,x6,x7,x8)}，Y∈{β1＝(y1,y2,y3,y4),β2＝(y5,y6,y7,y8)}，XP、XFAnd YPIs given by the 2 x 2 joint probability distribution of the super class combinations, and in each combination all samples are evenly distributed. According to the model, there is a directed information transfer between Y and X, and the true transfer entropy should be TEY→X＝0.5174，TEX→Y＝0。
The entropy of the discrete sequences X and Y is calculated by the method of the invention, and the result is shown in FIGS. 3-4. The result shows that when the sample size is small, the traditional discrete transfer entropy calculation method generates larger deviation with the true value, and slowly converges to the true value along with the increase of the sample size; compared with the prior art, the method can obtain more reliable transfer entropy estimated values under the condition of less sample size, and under the condition of different threshold values (th is 0.1,0.5 and 0.9), the method can converge faster than the traditional method and is closer to the true value under the given sample size. This shows that the invention is more accurate and robust in estimating discrete transfer entropy, especially in computing small samples. The invention can reliably estimate the transfer entropy value under the condition of less sample size, so the invention has great application prospect on the research of the directional function connection between brain regions in the field of neuroscience with limited sample size.
Finally, it should be noted that: the above-mentioned embodiments are only used for illustrating the technical solution of the present invention, and not for limiting the same; although the present invention has been described in detail with reference to the foregoing embodiments, it will be understood by those of ordinary skill in the art that: the technical solutions described in the foregoing embodiments may still be modified, or some or all of the technical features may be equivalently replaced; and the modifications or the substitutions do not make the essence of the corresponding technical solutions depart from the scope of the technical solutions of the embodiments of the present invention.
Claims (6)
1. A brain function directed connection assessment method based on a three-dimensional adaptive partition algorithm is characterized by comprising the following steps: which comprises the following steps:
step S1, giving electroencephalogram signals recorded by two different brain areas, discretizing the electroencephalogram signals by a phase space reconstruction method to obtain discrete sequences X and Y with L categories;
step S2, extracting X from the discrete sequences X and Y respectivelyP、XFAnd YPAnd constructing XP、XFAnd YPA joint distribution space of (a);
step S3, dividing the whole united distribution space into a plurality of sub-areas with approximately uniform distribution inside by using a three-dimensional self-adaptive partition algorithm;
step S4, calculating probability mass functions among all the sub-regions, and averaging the probability of each sub-region in each region to obtain an overall probability mass function;
in step S5, the transfer entropy of Y → X is calculated from the overall probability mass function.
2. The brain function oriented connection assessment method based on the three-dimensional adaptive partition algorithm according to claim 1The method is characterized in that: in step S1, electroencephalogram signals x recorded in different brain regions are recordedtAnd ytN, discretizing the two time series using a phase space reconstruction method with an embedding dimension m and a lag τ, resulting in discrete time series X and Y with L different classes, where L ═ m! X ═ X1,x2,...,xn-(m-1)τ}，Y＝{y1,y2,...,yn-(m-1)τ}。
3. The brain function directed connection assessment method based on the three-dimensional adaptive partition algorithm according to claim 1, characterized in that: in step S2, the prediction time u is selected to extract X from the discrete sequences X and YP、XFAnd YPWherein X isP＝{x1,x2,...,xn-(m-1)τ-uRepresents the past of X, XF＝{x1+u,x2+u,...,xn-(m-1)τRepresents the future of X, YP＝{y1,y2,...,yn-(m-1)τ-uRepresents the past of Y; then, according to the times of different classes of combinationP、XFAnd YPCombining frequency distributions to construct XP、XFAnd YPDue to a discrete variable XP、XFAnd YPAll have L categories, so the space has L in total3A possible combination of classes.
4. The brain function directed connection assessment method based on the three-dimensional adaptive partition algorithm according to claim 1, characterized in that: in step S3, consider the discrete variable XP、XFAnd YPThe constructed joint distribution space performs partitioning operation by the following steps:
(1) and (3) testing the uniformity of the distribution of the samples in the space by using a chi-square statistic, wherein the calculation formula of the chi-square statistic is as follows:
wherein
Representing a combination of categories
Number of samples of (i)
It is indicated that when the sample is evenly distributed,
the desired number of samples combined by category, i.e.
The corresponding p value is in the range of L3-1 degree of freedom calculated under a chi-squared distribution; if the p value is larger than the threshold th, the samples are considered to be uniformly distributed, and in this case, the next operation is not carried out; otherwise, the space is divided by the next step because the samples are not uniformly distributed;
(2) dividing the space into two regions (binary segmentation) at the best point that minimizes the sum of squared deviations; for this purpose, X is requiredP，XFAnd YPSearching the optimal segmentation points in three dimensions respectively; to find XPOptimal division point in dimension, XPClass variable of
The edges are sorted from small to large according to the frequency of the edges in the space,
has an edge frequency of
Writing the sorted category variables
Wherein
Then, by minimizing
Can find XPUpper optimal segmentation point s:
wherein a is1And a2Is divided into two subspaces by X after s point divisionPAverage value of the edge frequency; likewise, calculate XFAnd YPThe optimal division point of (a); finally, D in three dimensions is compared, and the optimal division point in the dimension with the maximum D is selected for division;
(3) two subspaces obtained after partitioning are both a new distribution space; repeating steps (1) and (2) for the two new distribution spaces until all subspaces are uniformly distributed in the sense of a given threshold.
5. The brain function directed connection assessment method based on the three-dimensional adaptive partition algorithm according to claim 1, characterized in that: in step S4, the entire joint frequency distribution space is divided into several non-overlapping sub-regions by adaptive partitioning; then dividing the sample amount in the sub-regions by the total sample amount, and calculating probability mass functions among the sub-regions; since the samples in each sub-region are considered to be uniformly distributed, the probability of each sub-region is divided by the number of class combinations in the region to obtain the probability of a single class combination, which ultimately constitutes the overall probability mass function.
6. The brain function directed connection assessment method based on the three-dimensional adaptive partition algorithm according to claim 1, characterized in that:
in step S5, the transfer entropy of Y → X is, based on the overall probability mass function:
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