A method and system for measuring similarity of spatiotemporal sequence data based on accumulation manifold

Abstract

This invention provides a method and system for measuring the similarity of spatiotemporal sequence data based on a product manifold. It acquires meteorological spatiotemporal sequence data, forming a fifth-order tensor containing time, meteorological elements, altitude, spatial latitudinal direction, and spatial longitudinal direction. The fifth-order tensor is then dimensionality-reduced to a fourth-order tensor containing time, meteorological features, spatial latitudinal direction, and spatial longitudinal direction. This fourth-order tensor is further dimensionality-reduced to obtain a second-order matrix containing spatial latitudinal and spatial longitudinal directions. The second-order matrix is ​​then expanded using a Chebyshev expansion to obtain a first-order vector containing Chebyshev coefficient arrays. All first-order vectors are combined to obtain a third-order temporal feature tensor. The similarity between two meteorological spatiotemporal sequence data points at different times is measured using geodesic distance on a Grassmann manifold. This invention combines two main approaches: product manifold and dimensionality reduction based on Chebyshev polynomial expansion. It has a wider applicability, better interpretability for specific application domains, and requires no training, making it easy to apply widely.

Description

Technical Field

[0001] This invention relates to the field of data processing technology, and in particular to a method and system for measuring the similarity of spatiotemporal sequence data based on the product manifold. Background Technology

[0002] Currently, many application areas (such as air quality forecasting, image processing, and computer vision) face a common problem: how to measure similarity based on spatiotemporal series data. Examples include measuring the similarity of air pollution processes, meteorological processes, gesture recognition, and action recognition. Typical spatiotemporal series data X∈R... T ×P×M×N It is a fourth-order tensor, with each order corresponding to time, feature channels, spatial rows, and spatial columns, respectively. Therefore, how to effectively and interpretably measure the similarity of such a high-order tensor has always been a significant challenge.

[0003] Furthermore, some existing literature tends to overlook the geometric structure of tensors and lacks consideration for multi-factor cases and the general case of multiple feature channels, resulting in limited effectiveness and applicability. Moreover, the interpretability of tensor decomposition-based dimensionality reduction methods for specific application areas (such as air quality prediction) is unclear. Meanwhile, currently widely used neural networks and deep learning also suffer from poor interpretability and the need for massive amounts of calibration samples in applications such as air quality prediction, image processing, and computer vision, thus limiting their in-depth application in these areas. Summary of the Invention

[0004] In view of the shortcomings of the prior art described above, the purpose of this invention is to provide a method and system for measuring the similarity of spatiotemporal sequence data based on the product manifold, so as to solve the problems in the existing schemes when measuring the similarity of spatiotemporal sequence data.

[0005] To achieve the above and other related objectives, this invention provides a method for measuring the similarity of spatiotemporal sequence data based on product manifolds, comprising the following steps:

[0006] Acquire meteorological spatiotemporal sequence data, and form a fifth-order tensor containing time, meteorological elements, altitude, spatial latitudinal direction, and spatial longitudinal direction based on the meteorological spatiotemporal sequence data;

[0007] The meteorological elements and altitude are combined into a meteorological feature, and the fifth-order tensor is reduced in dimensionality to be adjusted into a fourth-order tensor that includes time, meteorological features, spatial latitudinal and spatial longitudinal directions.

[0008] A single meteorological feature of a time slice is obtained from the fourth-order tensor, and the fourth-order tensor is reduced in dimensionality based on the obtained single meteorological feature to obtain a second-order matrix containing spatial latitudinal and spatial longitudinal directions.

[0009] Perform a Chebyshev expansion on the second-order matrix to obtain a first-order vector containing the Chebyshev coefficient array;

[0010] Spatial features are formed based on the Chebyshev coefficient array, and the first-order vectors of all meteorological features and corresponding time slice calculation results are combined to obtain a third-order time series feature tensor containing spatial features, meteorological features and time.

[0011] The third-order temporal feature tensor is represented as a Grassmann manifold, and the similarity of meteorological spatiotemporal sequence data at two different times is measured by the geodesic distance on the Grassmann manifold.

[0012] Optionally, before performing a Chebyshev expansion on the second-order matrix, the method further includes: obtaining a two-dimensional Chebyshev polynomial.

[0013] in, Let ψ be a one-dimensional Chebyshev polynomial in the x-direction. s (j) is the one-dimensional Chebyshev polynomial in the y direction; I0 represents the number of lattice points in the x direction, and J0 represents the number of lattice points in the y direction; k is the order of the one-dimensional Chebyshev polynomial in the x direction, and s is the order of the one-dimensional Chebyshev polynomial in the y direction.

[0014] Optionally, the recursive formula for the one-dimensional Chebyshev polynomial is:

[0015] Optionally, the process of performing a Chebyshev expansion on the second-order matrix to obtain a first-order vector containing the Chebyshev coefficient array includes:

[0016] Define a function Z(i,j) on a two-dimensional equidistant grid, where i = 1, 2, ..., I0; j = 1, 2, ..., J0;

[0017] By fitting the defined function Z(i,j) to a two-dimensional Chebyshev polynomial, the fitted function is obtained. have:

[0018]

[0019] Perform a Chebyshev expansion on a two-dimensional equidistant grid on the second-order matrix to obtain the Chebyshev coefficients A. ks ,have:

[0020]

[0021]

[0022]

[0023] Combine all the obtained Chebyshev coefficients into a one-dimensional array to obtain a first-order vector containing the Chebyshev coefficient array.

[0024] In the formula, K0 is the cutoff order of the Chebyshev polynomial in the x-direction, and 0 is the cutoff order of the Chebyshev polynomial in the y-direction; μ k Let v be the modulus of the Chebyshev polynomial in the x-direction. s Let be the modulus of the Chebyshev polynomial in the y-direction.

[0025] Alternatively, the process of measuring the similarity of two different time-space meteorological series data using geodesic distance on a Grassmanifold includes:

[0026] Acquire meteorological spatiotemporal series data at two different times;

[0027] A third-order temporal feature tensor X is obtained from the meteorological spatiotemporal sequence data at one time point, and a third-order temporal feature tensor Y is obtained from the meteorological spatiotemporal sequence data at another time point.

[0028] The third-order temporal feature tensor X is expanded in order to obtain three corresponding expansion matrices; and the third-order temporal feature tensor Y is expanded in order to obtain three corresponding expansion matrices; wherein each expansion matrix corresponds to a Grassmann factor manifold;

[0029] Based on the Grassman factor manifold corresponding to each expansion matrix, calculate the correlation angle between the corresponding factor subspaces in the third-order temporal feature tensor X and the third-order temporal feature tensor Y;

[0030] The calculated correlation angles are sorted in ascending order to obtain the corresponding correlation angle vectors;

[0031] In the Grassmann product manifold, all relevant angular vectors are fused to calculate the chordal distance between the third-order temporal feature tensor X and the third-order temporal feature tensor Y, and the chordal distance is used as the similarity of meteorological spatiotemporal sequence data at two different times.

[0032] Optionally, the method further includes:

[0033] Expanding the third-order temporal feature tensor X by order, we obtain Xi. (1) ∈R T×(M×N) X (2) ∈R M×(T×N) and X ( 3 ) ∈R N×(T×M) These three expansion matrices;

[0034] Expanding the third-order temporal feature tensor Y by order, we obtain Y (1) ∈R T×(M×N) Y (2) ∈R M×(T×N) and Y (3) ∈R N ×(T×M) These three expansion matrices;

[0035] Based on the Grassman factor manifold corresponding to each expansion matrix, singular value decomposition (SVD) is used to calculate the correlation angle between the pairwise corresponding factor subspaces of the third-order temporal feature tensor X and the third-order temporal feature tensor Y, resulting in:

[0036]

[0037] In this array, the diagonal elements of the diagonal matrix Λ are arranged in descending order;

[0038] Based on the singular value λ of the j-th dimension j The correlation angle θ′ in the j-th dimension is obtained. j , where λ j =cosθ′ j In the formula, λ j ≥0, 0≤θ′ j ≤π / 2;

[0039] The correlation angle θ′ of the j-th dimension j Sort in ascending order to obtain the corresponding related angular vector θ i i = 1, 2, 3;

[0040] By fusing all relevant angular vectors in the Grassmann product manifold, the chordal distance between the third-order temporal feature tensor X and the third-order temporal feature tensor Y is calculated as: d(X,Y)=||(sinθ1,sinθ2,sinθ3)||2.

[0041] Optionally, the meteorological spatiotemporal series data includes: meteorological spatiotemporal series data for air quality prediction.

[0042] Optionally, the method further includes: using the Chebyshev coefficient to represent different wind directions, and / or using the Chebyshev coefficient to represent a linear change in temperature in a certain direction.

[0043] This invention also provides a spatiotemporal sequence data similarity measurement system based on product manifold, comprising:

[0044] The data acquisition module is used to acquire meteorological spatiotemporal sequence data;

[0045] The fifth-order tensor module is used to form a fifth-order tensor containing time, meteorological elements, altitude, spatial latitudinal direction, and spatial longitudinal direction based on the meteorological spatiotemporal sequence data.

[0046] The fourth-order tensor module is used to merge the meteorological elements and the altitude into meteorological features, and to reduce the dimensionality of the fifth-order tensor, adjusting the fifth-order tensor to include time, meteorological features, spatial latitudinal and spatial longitudinal directions.

[0047] The second-order matrix module is used to obtain a single meteorological feature of a time slice from the fourth-order tensor, and to reduce the dimension of the fourth-order tensor based on the obtained single meteorological feature to obtain a second-order matrix containing spatial latitudinal and spatial longitudinal directions.

[0048] The first-order vector module is used to perform a Chebyshev expansion on the second-order matrix to obtain a first-order vector containing an array of Chebyshev coefficients.

[0049] The third-order time series feature tensor module is used to form spatial features based on the Chebyshev coefficient array, and combine the first-order vectors of all meteorological features and corresponding time slice calculation results to obtain a third-order time series feature tensor containing spatial features, meteorological features and time.

[0050] The similarity measurement module is used to characterize the third-order temporal feature tensor as a Grassmann manifold and measure the similarity of meteorological spatiotemporal sequence data at two different times by using the geodesic distance on the Grassmann manifold.

[0051] Optionally, the meteorological spatiotemporal series data includes: meteorological spatiotemporal series data for air quality prediction.

[0052] As described above, the present invention provides a method and system for measuring the similarity of spatiotemporal sequence data based on product manifolds, which has the following beneficial effects:

[0053] This invention first acquires meteorological spatiotemporal sequence data, then forms a fifth-order tensor containing time, meteorological elements, altitude, spatial latitudinal direction, and spatial longitudinal direction based on the meteorological spatiotemporal sequence data; next, the meteorological elements and altitude are merged into meteorological features, and the fifth-order tensor is dimensionality-reduced, adjusting it into a fourth-order tensor containing time, meteorological features, spatial latitudinal direction, and spatial longitudinal direction; then, a single meteorological feature of a time slice is extracted from the fourth-order tensor, and the fourth-order tensor is dimensionality-reduced based on the acquired single meteorological feature to obtain a second-order matrix containing spatial latitudinal direction and spatial longitudinal direction; then, a Chebyshev expansion is performed on the second-order matrix to obtain a first-order vector containing Chebyshev coefficient arrays; then, spatial features are formed based on the Chebyshev coefficient arrays, and all meteorological features and the first-order vectors calculated from the corresponding time slices are combined to obtain a third-order temporal feature tensor containing spatial features, meteorological features, and time; finally, the third-order temporal feature tensor is represented as a Grassmann manifold, and the similarity of meteorological spatiotemporal sequence data at two different times is measured by the geodesic distance on the Grassmann manifold. Therefore, it can be seen that the present invention integrates two main schemes: manifold integration and dimensionality reduction based on Chebyshev polynomial expansion. It has a wider range of applications, better interpretability for specific application areas (such as air quality prediction), and requires no training, making it easy to apply widely. Attached Figure Description

[0054] Figure 1 A flowchart illustrating a spatiotemporal sequence data similarity measurement method based on product manifolds is provided in one embodiment.

[0055] Figure 2 A schematic flowchart of a feature extraction process provided in one embodiment;

[0056] Figure 3 A schematic diagram of a geodesic is provided for one embodiment;

[0057] Figure 4 This is a schematic diagram of the hardware structure of a spatiotemporal sequence data similarity measurement system based on a product manifold, as provided in one embodiment. Detailed Implementation

[0058] The following specific examples illustrate the implementation of the present invention. Those skilled in the art can easily understand other advantages and effects of the present invention from the content disclosed in this specification. The present invention can also be implemented or applied through other different specific embodiments, and various details in this specification can also be modified or changed based on different viewpoints and applications without departing from the spirit of the present invention. It should be noted that, unless otherwise specified, the following embodiments and features described therein can be combined with each other.

[0059] It should be noted that the illustrations provided in this embodiment are only schematic representations of the basic concept of the present invention. Therefore, the drawings only show the components related to the present invention and are not drawn according to the actual number, shape and size of the components in the actual implementation. In the actual implementation, the form, quantity and proportion of each component can be arbitrarily changed, and the layout of the components may also be more complex.

[0060] Please see Figure 1 As shown, this embodiment provides a method for measuring the similarity of spatiotemporal sequence data based on product manifolds, including the following steps:

[0061] S100: Acquire meteorological spatiotemporal sequence data, and form a fifth-order tensor containing time, meteorological elements, altitude, spatial latitudinal direction, and spatial longitudinal direction based on the meteorological spatiotemporal sequence data. As an example, the meteorological spatiotemporal sequence data in this embodiment includes: meteorological spatiotemporal sequence data for air quality prediction; for example, it can be multi-feature "AQI (Air Quality Index) + meteorological" spatiotemporal sequence data.

[0062] S200, the meteorological elements and the altitude are merged into meteorological features, and the fifth-order tensor is reduced in dimensionality, and the fifth-order tensor is adjusted to include time, meteorological features, spatial latitudinal and spatial longitudinal directions;

[0063] S300, obtain a single meteorological feature of a time slice from the fourth-order tensor, and reduce the dimensionality of the fourth-order tensor based on the obtained single meteorological feature to obtain a second-order matrix containing spatial latitudinal and spatial longitudinal directions.

[0064] S400, Perform a Chebyshev expansion on the second-order matrix to obtain a first-order vector containing the Chebyshev coefficient array;

[0065] S500, spatial features are formed based on the Chebyshev coefficient array, and the first-order vectors of all meteorological features and corresponding time slice calculation results are combined to obtain a third-order time series feature tensor containing spatial features, meteorological features and time.

[0066] S600, the third-order temporal feature tensor is characterized as a Grassmann manifold, and the similarity of meteorological spatiotemporal sequence data at two different times is measured by the geodesic distance on the Grassmann manifold.

[0067] Specifically, before performing the Chebyshev expansion on the second-order matrix, step S400 further includes: obtaining the two-dimensional Chebyshev polynomial. i = 1, 2, ..., I0; j = 1, 2, ..., J0; where, Let ψ be a one-dimensional Chebyshev polynomial in the x-direction. s(j) is the one-dimensional Chebyshev polynomial in the y direction; I0 represents the number of lattice points in the x direction, and J0 represents the number of lattice points in the y direction; k is the order of the one-dimensional Chebyshev polynomial in the x direction, and s is the order of the one-dimensional Chebyshev polynomial in the y direction.

[0068] In this embodiment, with For example, a one-dimensional Chebyshev polynomial The values ​​for each order are:

[0069]

[0070] Therefore, in this embodiment, the recursive formula for the one-dimensional Chebyshev polynomial is:

[0071] According to the above description, step S400, which involves performing a Chebyshev expansion on the second-order matrix to obtain a first-order vector containing the Chebyshev coefficient array, includes:

[0072] Define a function Z(i,j) on a two-dimensional equidistant grid, where i = 1, 2, ..., I0; j = 1, 2, ..., J0; and I0 and J0 are positive integers.

[0073] By fitting the defined function Z(i,j) to a two-dimensional Chebyshev polynomial (similar to superposition function fitting using Taylor formulas and Fourier series), the fitted function is obtained. have: In the formula, K0 is the cutoff order of the Chebyshev polynomial in the x-direction, and S0 is the cutoff order of the Chebyshev polynomial in the y-direction.

[0074] Perform a Chebyshev expansion on a two-dimensional equidistant grid on the second-order matrix to obtain the Chebyshev coefficients A. ks ,have: in, In the formula, μ k Let v be the modulus of the Chebyshev polynomial in the x-direction. s Let be the modulus of the Chebyshev polynomial in the y-direction. In this embodiment, the fitting error can be measured by the sum of squared residuals, relative error, root mean square error, etc.

[0075] All the acquired Chebyshev coefficients are combined into a one-dimensional array to obtain a first-order vector containing the Chebyshev coefficient array. In this embodiment, the Chebyshev coefficients can be used to represent different wind directions, or they can be used to represent linear temperature changes in a certain direction.

[0076] According to the above description, in this embodiment, the process of measuring the similarity of meteorological spatiotemporal series data at two different times using geodesic distance on the Grassmann manifold includes:

[0077] Acquire meteorological spatiotemporal series data at two different times;

[0078] A third-order temporal feature tensor X is obtained from the meteorological spatiotemporal sequence data at one time point, and a third-order temporal feature tensor Y is obtained from the meteorological spatiotemporal sequence data at another time point.

[0079] The third-order temporal feature tensor X is expanded in order to obtain three corresponding expansion matrices; and the third-order temporal feature tensor Y is expanded in order to obtain three corresponding expansion matrices; wherein each expansion matrix corresponds to a Grassmann factor manifold. Specifically, the third-order temporal feature tensor X is expanded in order to obtain X (1) ∈R T×(M×N) X (2) ∈R M×(T×N) and X (3) ∈R N×(T×M) These three expansion matrices; by expanding the third-order temporal feature tensor Y in order, we obtain Y (1) ∈R T×(M×N) Y (2) ∈R M ×(T×N) and Y (3) ∈R N×(T×M) These three expansion matrices.

[0080] Based on the Grassmann factor manifold corresponding to each expansion matrix, the correlation angle between corresponding factor subspaces of the third-order temporal feature tensor X and the third-order temporal feature tensor Y is calculated. Specifically, based on the Grassmann factor manifold corresponding to each expansion matrix, singular value decomposition (SVD) is used to calculate the correlation angle between pairwise corresponding factor subspaces of the third-order temporal feature tensor X and the third-order temporal feature tensor Y, resulting in: In this array, the diagonal elements of the diagonal matrix Λ are arranged in descending order; based on the singular value λ of the j-th dimension... j The correlation angle θ′ in the j-th dimension is obtained. j , where λ j =cosθ′ j In the formula, λ j ≥0, 0≤θ′ j ≤π / 2.

[0081] The calculated correlation angles are sorted in ascending order to obtain the corresponding correlation angle vectors. Specifically, for the j-th dimension of the correlation angle θ′... j Sort in ascending order to obtain the corresponding related angular vector θ i , i = 1, 2, 3.

[0082] In the Grassmann product manifold, all relevant angle vectors are fused to calculate the chordal distance between the third-order temporal feature tensor X and the third-order temporal feature tensor Y, and this chordal distance is used as the similarity between meteorological spatiotemporal series data at two different times. Specifically, in the Grassmann product manifold, all relevant angle vectors are fused to calculate the chordal distance between the third-order temporal feature tensor X and the third-order temporal feature tensor Y, with: d(X,Y)=||(sinθ1,sinθ2,sinθ3)||2.

[0083] Therefore, this embodiment combines two main approaches: manifold integration and dimensionality reduction based on Chebyshev polynomial expansion. It has a wider range of applications, better interpretability for specific application areas (such as air quality prediction), and requires no training, making it easy to apply widely.

[0084] In another embodiment, the entire process of measuring the similarity of spatiotemporal sequence data can be roughly divided into the following two steps:

[0085] 1) Acquire data, then process the data, such as performing feature extraction or dimensionality reduction;

[0086] 2) Perform similarity measurement on the processed data.

[0087] Directly comparing similarity for five-dimensional gridded air pollution data is very difficult; therefore, a series of processing steps are needed to reduce the dimensionality of the data. The dimensionality reduction process in this embodiment is as follows: Figure 2 As shown, dimensionality reduction of the data facilitates the final similarity measurement of data from different time periods. For example, consider multi-feature spatiotemporal series data X∈R of "AQI + Weather". T×P×M×N It is a fourth-order tensor (time x feature x spatial latitudinal x spatial longitudinal). For a certain time t, then X t ∈R P×M×N It is a third-order tensor. For X t For each feature p, a matrix X can be obtained. tp ∈R M ×N Performing a "Chebyshev expansion on a two-dimensional equidistant grid" yields a Chebyshev coefficient vector V. p Therefore, X t Reduced to a matrix X′ t X then becomes a third-order tensor X′. After completing the above dimensionality reduction process, this embodiment can regard the third-order tensor X′ as corresponding to a Grassmann product manifold. In this way, the similarity between two third-order tensors X′ and Y′ can be directly measured by the "geodesic distance on the Grassmann product manifold".

[0088] In the feature extraction process of this embodiment, the elements and altitude information can be merged into meteorological features to reduce the dimension by one. Chebyshev expansion is then used to transform the grid data into feature coefficients, further reducing the dimension by one, resulting in a third-order temporal feature tensor, assumed to be Yt′∈R. P×V×T (Spatial features x meteorological features x time), Yt′ can be regarded as corresponding to a Grassmann manifold. In this way, the similarity of two third-order time series feature tensors Yt1′ and Yt2′ at different time periods can be directly measured by the "geodetic distance on the Grassmann manifold", that is, the feature data similarity comparison.

[0089] In terms of feature extraction, this embodiment selects the Chebyshev polynomial expansion. The Chebyshev polynomial has been used in meteorological work for a long time. As early as 1948, the Chebyshev polynomial was first used for weather forecasting. The polynomial expansion of the surface pressure field was used to analyze the results for similar forecasts. The application of the Chebyshev polynomial in meteorological work can be roughly divided into three aspects: (1) Weather and climate analysis: Many authors have expanded various meteorological fields and calculated the correlation between the Chebyshev coefficients of the fields and meteorological elements, and analyzed the relationship between circulation and weather. (2) Single-point forecast: Single-point forecasts mostly use regression equations, in which the forecast factor is the Chebyshev coefficient. (3) Meteorological element distribution forecast: One is to forecast point by point and then give the distribution on the plane; the other is to first forecast the Chebyshev coefficients and then use the coefficients to synthesize the plane distribution. This embodiment belongs to the first type. By expanding various meteorological field elements with the Chebyshev polynomial, the Chebyshev coefficients are calculated and used as the features of meteorological field elements at a certain time. The following is a brief introduction to the Chebyshev polynomial:

[0090] Chebyshev polynomials: Chebyshev polynomials are orthogonal polynomials of squared interpolation functions at discrete points. One-dimensional Chebyshev polynomials:

[0091] Where I0 represents the number of lattice points in the x-direction, and the subscript k is the order of the one-dimensional Chebyshev polynomial in the x-direction. For example, the values ​​of its various orders are:

[0092]

[0093] The recursive formula is:

[0094]

[0095] Two-dimensional Chebyshev polynomials:

[0096] in and ψ s(j) represents the one-dimensional Chebyshev polynomials in the x and y directions, respectively. Similarly, I0 and J0 represent the number of lattice points in the x and y directions, respectively, and the subscripts k and s represent the order of the one-dimensional Chebyshev polynomials in the x and y directions, respectively.

[0097] Chebyshev polynomial expansion: For atmospheric gridded data, we only need to discuss equidistant grids, i.e., Chebyshev expansion on two-dimensional equidistant grid points. Define a function Z(i,j), i=1,2,…,I0; j=1,2,…,J0 on the two-dimensional equidistant grid. Z(i,j) can be fitted using two-dimensional Chebyshev polynomials (similar to fitting superimposed functions such as Taylor series and Fourier series):

[0098]

[0099]

[0100]

[0101] Among them, A ks Here, K0 represents the Chebyshev coefficients, K0 is the cutoff order of the Chebyshev polynomial in the x-direction, and 0 is the cutoff order of the Chebyshev polynomial in the y-direction; μ k Let v be the modulus of the Chebyshev polynomial in the x-direction. s Let be the modulus of the Chebyshev polynomial in the y-direction. The fitting error can be measured by the sum of squared residuals, relative error, root mean square error, etc.

[0102] Physical interpretation of Chebyshev coefficients: In practical problems, it is extremely important to make appropriate interpretations of the obtained Chebyshev coefficients of each order for specific expansion objects Z(i,j), such as the correlation of corresponding physical meanings or the correlation of natural phenomena. This is an important aspect of applying Chebyshev polynomials to practical problems.

[0103] For two-dimensional Chebyshev coefficients, the 0th-order Chebyshev coefficient represents the overall average, the 1st-order coefficient represents linear increase or decrease, and the 2nd-order coefficient represents a parabolic weight variation. In conjunction with physical phenomena, such as coefficient A... 01 A 10 It can represent different wind directions, or it can represent a linear change in temperature in a certain direction. Therefore, this coefficient can be regarded as a characteristic coefficient of the corresponding meteorological field.

[0104] Chebyshev coefficients of order higher than 2 represent the weights of surface variations and higher-order changes, resulting in a greater number of combinations and more complex interpretations, currently difficult to correlate with physical phenomena. In analyzing meteorological issues, generally only the results of fitting to order 2 and below, along with their corresponding coefficients, are needed. This paper uses a two-dimensional second-order Chebyshev polynomial for expansion, which yields A... 00 A01 A 02 A 10 A 11 A 12 A 20 A 21 A 22 There are a total of 9 coefficients.

[0105] Therefore, in the feature extraction process of this embodiment, the grid data of a certain meteorological element mxn at a certain moment can be extracted and transformed into 9 coefficients, that is, a vector of length 9.

[0106] Feature data similarity comparison: In the similarity comparison, this embodiment uses the geodesic distance between two third-order tensors as the similarity between the two tensors. This geodesic distance is the chord distance of the Grassman manifold.

[0107] First, we need to introduce the mathematical concepts related to Grassmann manifolds:

[0108] (1) Manifold

[0109] Since its introduction in Science in 2000, manifold learning has become a hot topic in machine learning and data mining. Its basic assumption is that existing data is sampled from a high-dimensional space, and therefore possesses a low-dimensional manifold structure within that high-dimensional space. A manifold is a geometric object (something that people can imagine and observe). Simply put, since people can't readily perceive what data looks like in its current form, they can imagine it existing in a high-dimensional space where it has a shape. A good example is constellations. How do we describe a sky full of stars? We can imagine them having shapes in a higher-dimensional universe, thus creating constellations like Vega and Orion.

[0110] A manifold is a space that locally possesses properties of Euclidean space; it is a generalization of the concepts of curves and surfaces in Euclidean space. The English name "manifold" means "many + fold," that is, the superposition of many surface patches.

[0111] For example, the Earth is a two-dimensional manifold (which can be described using two coordinates—longitude and latitude), but locally it can be considered as Euclidean space, and approximate distance calculations can be performed using Euclidean distance.

[0112] (2) Grassmannian manifold

[0113] The Grassmann manifold is named after the mathematician Hermann Grassmann, who first studied it. A Grassmann manifold is all linear subspaces of a given dimension of a vector space V. For example, the Grassmann manifold G(1, d) is the space of lines passing through the origin in the d-dimensional vector space V, while G(2, d) is the space of planes passing through the origin in the d-dimensional vector space V. In three-dimensional Euclidean space, a plane is completely defined by one of its perpendiculars (and vice versa); thus, G(1, 3) is isomorphic to G(2, 3). It can be simply considered as a space composed of vectors of many dimensions.

[0114] (3) Geodesic lines

[0115] What's the shortest distance between two points? In two dimensions, it's a straight line (line segment). But in three dimensions? The shortest distance between two points on Earth isn't a straight line; it's the straight line drawn when the Earth is unfolded into a two-dimensional plane. That line becomes a curve on a three-dimensional Earth. This curve represents the shortest distance between two points; in this example, we call it a geodesic. In simpler terms, the geodesic is the shortest distance between two points. For example... Figure 3 As shown, the shortest distance from A to C is the line segment after b is unfolded, but it appears as a curve when viewed from the perspective of a three-dimensional sphere.

[0116] The geodesic distance on the Grassmann manifold G(p,d), which is the length of the geodesic line, can be viewed as the minimum rotation amplitude when rotating from one linear subspace to another. This minimum rotation amplitude can be calculated using the correlation angle θ (also known as the principal angle or canonical angle) between the two linear subspaces Span(X1) and Span(X2). In this embodiment, the chordal distance is used as the geodesic distance: d(X1,X2)=||sinθ||2,θ=[θ1,θ2,…,θ p ] T Where θ i The calculation uses singular value decomposition.

[0117] Similarity calculation steps:

[0118] Input: X, Y ∈ R T×M×N ;

[0119] Output: The distance d between X and Y.

[0120] ① Expand by order; if it is a 3rd order tensor, it will be expanded into three 2nd order matrices:

[0121] X (1) ∈R T×(M×N) Y (1) ∈R T×(M×N) ;

[0122] X (2) ∈R M×(T×N) Y (2) ∈R M×(T×N) ;

[0123] X (3) ∈R N×(T×M) Y (3) ∈R N×(T×M) ;

[0124] ② For the matrix obtained in the previous step, use Singular Value Decomposition (SVD) to calculate the correlation angles between corresponding subspaces pairwise:

[0125]

[0126] The diagonal elements of the diagonal matrix Λ are arranged in descending order, where λ j =cosθ′ j (greater than or equal to 0, 0 ≤ θ′) j ≤π / 2) represents the singular value of the j-th dimension, thus yielding the correlation angle θ′ of the j-th dimension. j These correlated angles form a correlated angle vector θ arranged in ascending order. i i = 1, 2, 3.

[0127] ③ Calculate the distance, which is the chord distance: d(X,Y)=||(sinθ1,sinθ2,sinθ3)||2.

[0128] In summary, this invention provides a method for measuring the similarity of spatiotemporal sequence data based on integral manifolds. First, meteorological spatiotemporal sequence data is acquired. Then, a fifth-order tensor containing time, meteorological elements, altitude, spatial latitudinal direction, and spatial longitudinal direction is formed based on the meteorological spatiotemporal sequence data. Next, the meteorological elements and altitude are merged into meteorological features, and the fifth-order tensor is reduced in dimensionality to a fourth-order tensor containing time, meteorological features, spatial latitudinal direction, and spatial longitudinal direction. Finally, a single meteorological feature for a time slice is extracted from the fourth-order tensor, and the fourth-order tensor is then used to measure the similarity of the fourth-order tensor based on the extracted single meteorological feature. The method involves dimensionality reduction to obtain a second-order matrix containing spatial latitudinal and spatial meridional axes. This second-order matrix is ​​then expanded using Chebyshev expansion to obtain a first-order vector containing Chebyshev coefficient arrays. Spatial features are then formed based on the Chebyshev coefficient arrays, and all meteorological features are combined with the first-order vectors calculated from the corresponding time slices to obtain a third-order temporal feature tensor containing spatial, meteorological, and temporal features. Finally, the third-order temporal feature tensor is represented as a Grassmann manifold, and the similarity between two different time-series meteorological spatiotemporal data is measured using the geodesic distance on the Grassmann manifold. Therefore, this method combines two main approaches: dimensionality reduction based on the product manifold and Chebyshev polynomial expansion. It has a wider range of applications, better interpretability for specific application areas (such as air quality forecasting), and requires no training, making it easy to apply widely.

[0129] like Figure 4 As shown, a spatiotemporal sequence data similarity measurement system based on product manifolds includes:

[0130] The data acquisition module M10 is used to acquire meteorological spatiotemporal series data. As an example, the meteorological spatiotemporal series data in this embodiment includes: meteorological spatiotemporal series data for air quality prediction; for example, it can be multi-feature "AQI (Air Quality Index) + meteorological" spatiotemporal series data.

[0131] The fifth-order tensor module M20 is used to form a fifth-order tensor containing time, meteorological elements, altitude, spatial latitudinal direction, and spatial longitudinal direction based on the meteorological spatiotemporal sequence data.

[0132] The fourth-order tensor module M30 is used to merge the meteorological elements and the altitude into meteorological features, and to reduce the dimensionality of the fifth-order tensor, adjusting the fifth-order tensor into a fourth-order tensor that includes time, meteorological features, spatial latitudinal and spatial longitudinal directions.

[0133] The second-order matrix module M40 is used to obtain a single meteorological feature of a time slice from the fourth-order tensor, and to reduce the dimension of the fourth-order tensor based on the obtained single meteorological feature to obtain a second-order matrix containing spatial latitudinal and spatial longitudinal directions.

[0134] The first-order vector module M50 is used to perform a Chebyshev expansion on the second-order matrix to obtain a first-order vector containing a Chebyshev coefficient array.

[0135] The third-order time series feature tensor module M60 is used to form spatial features based on the Chebyshev coefficient array, and combine the first-order vectors of all meteorological features and corresponding time slice calculation results to obtain a third-order time series feature tensor containing spatial features, meteorological features and time.

[0136] The similarity measurement module M70 is used to characterize the third-order temporal feature tensor as a Grassmann manifold and measure the similarity of two different time-space meteorological series data by the geodesic distance on the Grassmann manifold.

[0137] Specifically, before performing the Chebyshev expansion on the second-order matrix, the second-order matrix module M40 further includes: obtaining the two-dimensional Chebyshev polynomial. in, Let ψ be a one-dimensional Chebyshev polynomial in the x-direction. s (j) is the one-dimensional Chebyshev polynomial in the y direction; I0 represents the number of lattice points in the x direction, and J0 represents the number of lattice points in the y direction; k is the order of the one-dimensional Chebyshev polynomial in the x direction, and s is the order of the one-dimensional Chebyshev polynomial in the y direction.

[0138] In this embodiment, with For example, a one-dimensional Chebyshev polynomial The values ​​for each order are:

[0139]

[0140] Therefore, in this embodiment, the recursive formula for the one-dimensional Chebyshev polynomial is:

[0141] According to the above description, the process by which the second-order matrix module M40 performs a Chebyshev expansion on the second-order matrix to obtain a first-order vector containing a Chebyshev coefficient array includes:

[0142] Define a function Z(i,j) on a two-dimensional equidistant grid, where i = 1, 2, ..., I0; j = 1, 2, ..., J0; and I0 and J0 are positive integers.

[0143] By fitting the defined function Z(i,j) to a two-dimensional Chebyshev polynomial (similar to superposition function fitting using Taylor formulas and Fourier series), the fitted function is obtained. have: In the formula, K0 is the cutoff order of the Chebyshev polynomial in the x-direction, and 0 is the cutoff order of the Chebyshev polynomial in the y-direction.

[0144] Perform a Chebyshev expansion on a two-dimensional equidistant grid on the second-order matrix to obtain the Chebyshev coefficients A. ks ,have: in, In the formula, k Let v be the modulus of the Chebyshev polynomial in the x-direction. s Let be the modulus of the Chebyshev polynomial in the y-direction. In this embodiment, the fitting error can be measured by the sum of squared residuals, relative error, root mean square error, etc.

[0145] All the acquired Chebyshev coefficients are combined into a one-dimensional array to obtain a first-order vector containing the Chebyshev coefficient array. In this embodiment, the Chebyshev coefficients can be used to represent different wind directions, or they can be used to represent linear temperature changes in a certain direction.

[0146] According to the above description, in this embodiment, the process of measuring the similarity of meteorological spatiotemporal series data at two different times using geodesic distance on the Grassmann manifold includes:

[0147] Acquire meteorological spatiotemporal series data at two different times;

[0148] A third-order temporal feature tensor X is obtained from the meteorological spatiotemporal sequence data at one time point, and a third-order temporal feature tensor Y is obtained from the meteorological spatiotemporal sequence data at another time point.

[0149] The third-order temporal feature tensor X is expanded in order to obtain three corresponding expansion matrices; and the third-order temporal feature tensor Y is expanded in order to obtain three corresponding expansion matrices; wherein each expansion matrix corresponds to a Grassmann factor manifold. Specifically, the third-order temporal feature tensor X is expanded in order to obtain X (1) ∈R T×(M×N) X (2) ∈R M×(T×N) and X (3) ∈R N×(T×M) These three expansion matrices; by expanding the third-order temporal feature tensor Y in order, we obtain Y (1) ∈R T×(M×N) Y (2) ∈R M ×(T×N) and Y (3) ∈R N×(T×M) These three expansion matrices.

[0150] Based on the Grassmann factor manifold corresponding to each expansion matrix, the correlation angle between corresponding factor subspaces of the third-order temporal feature tensor X and the third-order temporal feature tensor Y is calculated. Specifically, based on the Grassmann factor manifold corresponding to each expansion matrix, singular value decomposition (SVD) is used to calculate the correlation angle between pairwise corresponding factor subspaces of the third-order temporal feature tensor X and the third-order temporal feature tensor Y, resulting in: In this array, the diagonal elements of the diagonal matrix Λ are arranged in descending order; based on the singular value λ of the j-th dimension... j The correlation angle θ′ in the j-th dimension is obtained. j , where λ j =cosθ′ j In the formula, λ j ≥0, 0≤θ′ j ≤π / 2.

[0151] The calculated correlation angles are sorted in ascending order to obtain the corresponding correlation angle vectors. Specifically, for the j-th dimension of the correlation angle θ′... j Sort in ascending order to obtain the corresponding related angular vector θ i , i = 1, 2, 3.

[0152] In the Grassmann product manifold, all relevant angle vectors are fused to calculate the chordal distance between the third-order temporal feature tensor X and the third-order temporal feature tensor Y, and this chordal distance is used as the similarity between meteorological spatiotemporal series data at two different times. Specifically, in the Grassmann product manifold, all relevant angle vectors are fused to calculate the chordal distance between the third-order temporal feature tensor X and the third-order temporal feature tensor Y, with: d(X,Y)=||(sinθ1,sinθ2,sinθ3)||2.

[0153] Therefore, this embodiment combines two main approaches: manifold integration and dimensionality reduction based on Chebyshev polynomial expansion. It has a wider range of applications, better interpretability for specific application areas (such as air quality prediction), and requires no training, making it easy to apply widely.

[0154] In summary, this invention provides a spatiotemporal sequence data similarity measurement system based on integral manifolds. First, meteorological spatiotemporal sequence data is acquired. Then, a fifth-order tensor containing time, meteorological elements, altitude, spatial latitudinal direction, and spatial longitudinal direction is formed based on the meteorological spatiotemporal sequence data. Next, the meteorological elements and altitude are merged into meteorological features, and the fifth-order tensor is dimensionality reduced, adjusting it to a fourth-order tensor containing time, meteorological features, spatial latitudinal direction, and spatial longitudinal direction. Finally, a single meteorological feature for a time slice is extracted from the fourth-order tensor, and the fourth-order tensor is then used to measure the similarity of the fourth-order tensor data based on the acquired single meteorological feature. The system performs dimensionality reduction to obtain a second-order matrix containing spatial latitudinal and spatial meridional axes. Then, it performs a Chebyshev expansion on the second-order matrix to obtain a first-order vector containing Chebyshev coefficient arrays. Spatial features are then formed based on the Chebyshev coefficient arrays, and all meteorological features are combined with the first-order vectors calculated from the corresponding time slices to obtain a third-order temporal feature tensor containing spatial, meteorological, and temporal features. Finally, the third-order temporal feature tensor is represented as a Grassmann manifold, and the similarity between two different time-series meteorological spatiotemporal data is measured using the geodesic distance on the Grassmann manifold. Therefore, this system integrates two main approaches: dimensionality reduction based on the product manifold and Chebyshev polynomial expansion. This results in a wider range of applications, better interpretability for specific fields (such as air quality forecasting), and the system requires no training, facilitating widespread application.

[0155] Therefore, this invention effectively overcomes the various shortcomings of the prior art and has high industrial applicability. The above embodiments are merely illustrative of the principles and effects of this invention and are not intended to limit the invention. Any person skilled in the art can modify or change the above embodiments without departing from the spirit and scope of this invention. Therefore, all equivalent modifications or changes made by those skilled in the art without departing from the spirit and technical concept disclosed in this invention should still be covered by the claims of this invention.

[0156] It should be understood that although terms such as first, second, third, etc., may be used in the embodiments of the present invention to describe the preset range, these preset ranges should not be limited to these terms. These terms are only used to distinguish the preset ranges from one another. For example, without departing from the scope of the embodiments of the present invention, the first preset range may also be referred to as the second preset range, and similarly, the second preset range may also be referred to as the first preset range.

Claims

1. A method for measuring the similarity of spatiotemporal sequence data based on product manifolds, characterized in that, Includes the following steps: Acquire meteorological spatiotemporal sequence data, and form a fifth-order tensor containing time, meteorological elements, altitude, spatial latitudinal direction, and spatial longitudinal direction based on the meteorological spatiotemporal sequence data; The meteorological elements and altitude are combined into a meteorological feature, and the fifth-order tensor is reduced in dimensionality to be adjusted into a fourth-order tensor that includes time, meteorological features, spatial latitudinal and spatial longitudinal directions. A single meteorological feature of a time slice is obtained from the fourth-order tensor, and the fourth-order tensor is reduced in dimensionality based on the obtained single meteorological feature to obtain a second-order matrix containing spatial latitudinal and spatial longitudinal directions. Perform a Chebyshev expansion on the second-order matrix to obtain a first-order vector containing the Chebyshev coefficient array; Spatial features are formed based on the Chebyshev coefficient array, and the first-order vectors of all meteorological features and corresponding time slice calculation results are combined to obtain a third-order time series feature tensor containing spatial features, meteorological features and time. The third-order temporal feature tensor is represented as a Grassmann manifold, and the similarity of meteorological spatiotemporal series data at two different times is measured by the geodesic distance on the Grassmann manifold. This includes: acquiring meteorological spatiotemporal series data at two different times, and obtaining the third-order temporal feature tensor based on the meteorological spatiotemporal series data at one of the times. And obtain the third-order temporal feature tensor based on meteorological spatiotemporal sequence data at another time. ; for the third-order temporal feature tensor Expanding by order yields three corresponding expansion matrices; and the third-order temporal feature tensor... Expanding by order yields three corresponding expansion matrices; each expansion matrix corresponds to a Grassmann factor manifold; the third-order temporal feature tensor is calculated based on the Grassmann factor manifold corresponding to each expansion matrix. and the third-order temporal feature tensor The correlation angles between corresponding factor subspaces are calculated; the calculated correlation angles are sorted in ascending order to obtain the corresponding correlation angle vectors; all correlation angle vectors are fused in the Grassmann product manifold to calculate the third-order temporal feature tensor. and the third-order temporal feature tensor The chord distance is used as the similarity between meteorological spatiotemporal sequence data at two different times.

2. The spatiotemporal sequence data similarity measurement method based on product manifold according to claim 1, characterized in that, Before performing the Chebyshev expansion on the second-order matrix, the method further includes: obtaining the two-dimensional Chebyshev polynomial. ; in, Let be a one-dimensional Chebyshev polynomial in the x-direction. Let be a one-dimensional Chebyshev polynomial in the y-direction; This represents the number of grid points in the x-direction. This represents the number of grid points in the y-direction; Let be the order of the one-dimensional Chebyshev polynomial in the x-direction, and s be the order of the one-dimensional Chebyshev polynomial in the y-direction.

3. The spatiotemporal sequence data similarity measurement method based on product manifold according to claim 2, characterized in that, The recursive formula for the one-dimensional Chebyshev polynomial is: .

4. The spatiotemporal sequence data similarity measurement method based on product manifold according to claim 2 or 3, characterized in that, The process of performing a Chebyshev expansion on the second-order matrix to obtain a first-order vector containing the Chebyshev coefficient array includes: Define a function on a two-dimensional equidistant grid. ; Using two-dimensional Chebyshev polynomials to define the function Perform fitting to obtain the fitting function. ,have: Perform a Chebyshev expansion on a two-dimensional equidistant grid on the second-order matrix to obtain the Chebyshev coefficients. ,have: Combine all the obtained Chebyshev coefficients into a one-dimensional array to obtain a first-order vector containing the Chebyshev coefficient array. In the formula, Let be the cutoff order of the Chebyshev polynomial in the x-direction. Let be the cutoff order of the Chebyshev polynomial in the y-direction; Let x be the modulus of the Chebyshev polynomial in the x-direction. Let be the modulus of the Chebyshev polynomial in the y-direction.

5. The spatiotemporal sequence data similarity measurement method based on product manifold according to claim 4, characterized in that, The method further includes: For the third-order temporal feature tensor Expanding by order, we obtain , and These three expansion matrices; For the third-order temporal feature tensor Expanding by order, we obtain , and These three expansion matrices; Based on the Grassman factor manifold corresponding to each expansion matrix, the third-order temporal feature tensor is calculated using singular value decomposition (SVD). and the third-order temporal feature tensor In the equation, the correlation angles between pairwise corresponding factors and their subspaces are: Among them, diagonal array The diagonal elements are arranged in descending order; According to the j Singular values ​​of dimension , obtained the j Related angles of dimensions ,in, In the formula, ; For the j Related angles of dimensions Arrange in ascending order to obtain the corresponding related angle vectors. ; The third-order temporal feature tensor is calculated by fusing all relevant angle vectors in the Grassmann product manifold. and the third-order temporal feature tensor The chord distance is: .

6. The spatiotemporal sequence data similarity measurement method based on product manifold according to claim 1, characterized in that, The meteorological spatiotemporal series data includes: meteorological spatiotemporal series data used for air quality forecasting.

7. The spatiotemporal sequence data similarity measurement method based on product manifold according to claim 1, characterized in that, The method further includes: using the Chebyshev coefficient to represent different wind directions, and / or using the Chebyshev coefficient to represent a linear change in temperature in a certain direction.

8. A spatiotemporal sequence data similarity measurement system based on product manifold, characterized in that, Including: The data acquisition module is used to acquire meteorological spatiotemporal sequence data; The fifth-order tensor module is used to form a fifth-order tensor containing time, meteorological elements, altitude, spatial latitudinal direction, and spatial longitudinal direction based on the meteorological spatiotemporal sequence data. The fourth-order tensor module is used to merge the meteorological elements and the altitude into meteorological features, and to reduce the dimensionality of the fifth-order tensor, adjusting the fifth-order tensor to include time, meteorological features, spatial latitudinal and spatial longitudinal directions. The second-order matrix module is used to obtain a single meteorological feature of a time slice from the fourth-order tensor, and to reduce the dimension of the fourth-order tensor based on the obtained single meteorological feature to obtain a second-order matrix containing spatial latitudinal and spatial longitudinal directions. The first-order vector module is used to perform a Chebyshev expansion on the second-order matrix to obtain a first-order vector containing an array of Chebyshev coefficients. The third-order time series feature tensor module is used to form spatial features based on the Chebyshev coefficient array, and combine the first-order vectors of all meteorological features and corresponding time slice calculation results to obtain a third-order time series feature tensor containing spatial features, meteorological features and time. The similarity measurement module is used to represent the third-order temporal feature tensor as a Grassmann manifold and measure the similarity of meteorological spatiotemporal series data at two different times by using the geodesic distance on the Grassmann manifold. This includes: acquiring meteorological spatiotemporal series data at two different times; and obtaining the third-order temporal feature tensor based on the meteorological spatiotemporal series data at one of the times. And obtain the third-order temporal feature tensor based on meteorological spatiotemporal sequence data at another time. ; for the third-order temporal feature tensor Expanding by order yields three corresponding expansion matrices; and the third-order temporal feature tensor... Expanding by order yields three corresponding expansion matrices; each expansion matrix corresponds to a Grassmann factor manifold; the third-order temporal feature tensor is calculated based on the Grassmann factor manifold corresponding to each expansion matrix. and the third-order temporal feature tensor The correlation angles between corresponding factor subspaces are calculated; the calculated correlation angles are sorted in ascending order to obtain the corresponding correlation angle vectors; all correlation angle vectors are fused in the Grassmann product manifold to calculate the third-order temporal feature tensor. and the third-order temporal feature tensor The chord distance is used as the similarity between meteorological spatiotemporal sequence data at two different times.

9. The spatiotemporal sequence data similarity measurement system based on product manifold according to claim 8, characterized in that, The meteorological spatiotemporal series data includes: meteorological spatiotemporal series data used for air quality forecasting.

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