CN108073695A - A kind of higher-dimension time-variable data method for visualizing of dimension reduction space visual perception enhancing - Google Patents

Abstract

The invention discloses a high-dimensional time-varying data visualization method for enhanced visual perception of dimensionality reduction space, comprising: reading high-dimensional time-varying data, using a multi-dimensional scaling algorithm to obtain two-dimensional space coordinates of the high-dimensional time-varying data, and using the formula (1) Solve the coordinates of high-dimensional time-varying data in two-dimensional space to obtain the target orthogonal matrix, , in formula (1), Q represents the target orthogonal matrix, Indicates the two-dimensional spatial coordinates of the i-th data of the high-dimensional time-varying data at the current moment, Represents the two-dimensional spatial coordinates of the i-th data of the high-dimensional time-varying data at the previous moment, N represents the number of high-dimensional time-varying data, D ^t (Q) represents the high-dimensional time-varying data after orthogonal matrix transformation The projection offset at the current moment relative to the previous moment; the projection offset of the high-dimensional time-varying data in the two-dimensional space is minimized by using the target orthogonal matrix, and the projection of the high-dimensional time-varying data at the current moment is obtained. The invention can significantly improve the visualization and analysis efficiency of high-dimensional time-varying data.

Description

A high-dimensional time-varying data visualization method with enhanced visual perception of dimensionality reduction space

technical field

The invention relates to a high-dimensional time-varying data visualization method in a dimensionality reduction space, and belongs to the technical fields of computer graphics and data visualization.

Background technique

High-dimensional data analysis usually considers the multidimensional attributes of the data at the same time, especially when there is a strong correlation between the dimensions of the data, which cannot be considered separately. It is possible to effectively display the multidimensional attributes of the data and use multidimensional attribute analysis to discover potential features in the data. Research hotspots in the field of visual analysis. Parallel coordinates (Mcdonnell K T, Mueller K. Illustrative Parallel Coordi-nates [J]. Computer Graphics Forum, 2008, 27(3): 1031-1038.) represent high-dimensional data using polylines spanning a set of parallel vertical axes, where each The parallel axes represent a dimension. Support users to extract the implicit information of the data along the direction of the axis arrangement, and then find the correlation between dimensions. However, unreasonable axis arrangement will cause serious data line confusion, which seriously reduces the readability of the visual system, and many scholars have conducted in-depth research on this. Scholars such as Graham use free curves instead of polylines to reduce occlusion in the same type of data and realize data differentiation in the same data set (Graham M, Kennedy J. Using Curves to Enhance Parallel Coordinate Visualizations[C]//International Conference on Information Visualization.IEEE Computer Society, 2003: 10). Peng and other scholars showed less confusion and enhanced data readability by rearranging parallel coordinates through density analysis (Peng W, Ward M O, Rundensteiner EA. Clutter Reduction in Multi-Dimensional Data Visualization Using Dimension Reordering[C]//IEEE Symposium on Information Visualization. IEEE Computer Society, 2004: 89-96.). Although parallel coordinates can display the distribution characteristics of multi-dimensional data, when the data set is large, the displayed data hierarchy is chaotic, and it is difficult to find the changing law of the data, which hinders people's understanding of information. Scatter plot matrix (Cleveland W C, Mcgill M E.Dynamic graphics for statis-tics[J].Journal of the Royal Statistical Society, 1990, 153(1)) Arrange the different dimensions of the data in a two-dimensional order in a certain order, and independently display the scatter points generated by any combination of two dimensions Graph, so as to support users to interpret the original data and analyze the correlation between any two dimensions. However, there is a lack of clustering display of data in the case of more than two dimensions. Although scholars such as Elmqvist propose to rotate the scatter plot in three-dimensional space to find the relationship between multiple dimensions, it is still difficult to deal with higher-dimensional data. There are limitations (Elmqvist N, Dragicevic P, Fekete JD. Rolling the Dice: Multidimensional Visual Exploration using ScatterplotMatrix Navigation [J]//. IEEE Transactions on Visualization & Computer Graphics, 2008, 14(6): 1141.).

Dimension reduction mapping technology can provide the representation of multi-dimensional data internal structure in low-dimensional space, and it is an effective multi-dimensional data visualization technology. It can help users effectively explore and understand abstract multi-dimensional information and its structure in low-dimensional space, enabling users to quickly and accurately discover the hidden features, relationships, and patterns in the data set in the process of knowledge discovery, information cognition, and information decision-making , trends and clustering information, etc. Principal component analysis is often used to reduce data to two or three dimensions to generate similarity layouts. However, this method discards other dimensions of the data, and it is difficult to maintain the high-dimensional distribution of the data in the original space. A classic method is locality preserving projection (locality preserving projection, LPP) (HeX, Niyogi P.Locality preserving projections[J]. Ad-vances in Neural Information Processing Systems, 2004, 16(1): 186-197), this The method linearly preserves the local high-dimensional distribution of the data. As a representative low-dimensional embedding algorithm, Isomap (Tenenbaum J B, Silva V D, Langford J C.A global geometric, for nonlinear dimensionality reduction[C]//2000:2319-23) can save data in the original Distributions in high-dimensional spaces. Compared with LPP, Isomap technology calculates the distance between all data pairs (which can be regarded as the difference of data pairs) to maintain the global high-dimensional distribution of data. Among them, the multidimensional scaling algorithm, as a commonly used algorithm in Isomap technology, effectively combines the advantages of parallel coordinates and scatterplot matrix, supports the use of low-dimensional space to approximate the expression of data distribution in multidimensional space, and shows the similarity of multidimensional data. sex. In order to verify and enhance the visual analysis ability of multidimensional scaling algorithms, domestic and foreign scholars have carried out in-depth research on multidimensional scaling algorithms. Scholars such as Keim designed a visualization method for large-scale database mining based on multidimensional scaling, and compared the display effects of parallel coordinates and scatter plot matrices, showing the efficiency of multidimensional scaling algorithms in high-dimensional data visualization (Keim D A , Kriegel H P. Visualization Techniques for Mining Large Databases: A Comparison [M]. IEEE Educational Activities Department, 1996). Vera and other scholars use interpolation method to smooth the two-dimensional mapping results of multidimensional scaling algorithm, and enhance the time series data analysis ability of multidimensional scaling algorithm (Vera J F, Angulo JM, Roldán J A.Stability analysis in nonstationary spatial covarianceestimation[J].Stochastic Environmental Research & Risk Assessment, 2016: 1-14).

Multidimensional data often has the characteristics of time-series changes, and the time-series evolution patterns of data are often difficult to explore. Therefore, it is of great significance to carry out visual analysis for multi-dimensional data with significant time series attributes. Animation (Bender S, Mcfarland D A. The art and science of dynamic network visualization [J]. Journal of SocialStructure, 2006, 7(2): 1206-1241) because it can effectively save the user's artistic conception map (mental map), Instead, it becomes an intuitive way to demonstrate the evolution of data over time. However, Archambault and other scholars have proved through experiments that the artistic conception map obtained in the animation may not be of great help to the visual cognition of a long time series. Not only that, the drastic changes of the projection points at multiple moments cannot support fast tracking of time nodes before and after. Change patterns (Archambault D, Purchase H, Pinaud B. Animation, Small Multiples, and the Effect of Mental MapPreservation in Dynamic Graphs [J]. IEEE Transactions on Visualization & Computer Graphics, 2011, 17(4): 539-552). Therefore, recent methods (Burch M, Fritz M, Beck F, et al. TimeSpiderTrees: A Novel Visual Metaphor for Dynamic Compound Graphs [C]//IEEE Symposium on Visual Languages and Human-Centric Computing. IEEE Computer Society, 2010: 168- 175. Liu S, Wu Y, Wei E, Liu M, and Liu Y. Storyflow: Tracking theevolution of stories[C]//IEEE Transactions on Visu-alization Computer Graphics, 19(12): 2436-45, 2013) multi-focus Use static graphs to display time series data. Time axis and small multiples are two options for encoding the time dimension in a static graph manner. Many scholars use the method of multi-small graphs for visual analysis. Scholars such as Hadlak proposed a visualization method based on multi-small graphs, which allows users to interactively select the focus area in multiple small graphs and establish a suitable layout for the selected data (Hadlak S , Schulz H J, Schumann H. In situ exploration of large dynamic networks [J]. IEEE Trans Vis ComputGraph, 2011, 17(12): 2334-2343). Scholars such as Walker introduced the advantages of spherical coordinates into parallel coordinates, which more intuitively demonstrated the timing characteristics of multidimensional data ([18]Walker J, Geng Z, Jones M, et al.Visualization of Large, Time-Dependent, Abstract Data with Integrated Spherical and Parallel Coordinates [C]//EuroVis-Short Papers. 2012: 43-47). However, in the face of long-term sequences, the multi-image method needs to display a large number of projection images, which seriously reduces the readability of visual effects, increases visual clutter, and limits the user's ability to track time series features.

The method based on the time axis uses time as an axis, and then compresses high-dimensional information into one-dimensional or two-dimensional space, which greatly enhances the readability. Therefore, a large number of research works using multi-dimensional scaling algorithms use this method to display multi-dimensional Time-varying data to enhance users' visual cognition in low-dimensional space. Scholars such as Dwyer proposed a technology similar to the space-time cube (space-time-cube), which regards time as the third dimension after the multi-dimensional scaling algorithm is projected into the two-dimensional space, thus showing the temporal changes of the data in the three-dimensional space ( Dwyer T. Gallagher D R. Visualizing Changes in Fund Manager Holdings in Two and a Half-Dimensions [J]. Information Visualization, 2004, 3(4): 227-244). Scholars such as Bernard designed a time-path-based method for displaying multi-dimensional time-varying data, connecting the same data entities at different times to form a time path to help users understand the temporal pattern changes of a single entity (Bernard J, Wilhelm N, Scherer M, et al. TimeSeriesPaths: Projection-Based Explorative Analysis of Multivariate Time Series Data[C]//Conference in Central Europe on Computer Graphics, Visualization and ComputerVision.2012). There are also some methods that use multidimensional scaling algorithms to project multidimensional data into two-dimensional space (Hu Y, WuS, Xia S, et al.Motion track: Visualizing variations of human motion data[C]//Visualization Symposium.IEEE, 2010 : 153-160; Ward M O, Guo Z. Visual Exploration of Time-Series Data with Shape Space Projections[J]. Computer Graphics Forum, 2011, 30(3): 701-710; Mao Y, Dillon J, Lebanon G. Sequential document visualization.[J].IEEE Transactions on Visualization&Computer Graphics, 2007, 13(6):1208-1215.), etc., focusing on the movement of single data in time and the paths they generate in time. While these techniques allow detection of cyclical patterns in data over time, they result in severe visual clutter that interferes with the user's visual tracking since data items are projected into arbitrary 2D coordinates. Scholars such as Jackle proposed a multidimensional scaling algorithm based on sliding windows (Jackle D, Fischer F, Schreck T, and Keim D.A.Temporal mds plots for analysis of multivariate data[C]//IEEE Transactions on Visualization Computer Graphics, 22(1): 141 , 2016), effectively compress multi-dimensional data into one dimension and visually analyze network security data in two-dimensional space with time as an axis. However, after mapping multidimensional data into one-dimensional space, the visual recognition of feature differences is suppressed, and after the use of sliding window technique, although the abruptness of multidimensional scaling algorithm is effectively smoothed, the outlier is suppressed. Show effectively.

It can be seen that the high-dimensional time-varying data visualization method can help domain experts quickly analyze and display data characteristics, enhance the visual perception of visualization, and improve the visualization and analysis efficiency of high-dimensional time-varying data to a certain extent. However, existing visualization methods such as parallel coordinates, scatterplot matrix, and dimensionality reduction mapping techniques still have certain limitations. It is difficult to display the time series characteristics of data, which is not conducive to user interactive analysis, and hinders the visualization efficiency of high-dimensional time-varying data. improvement.

Contents of the invention

The purpose of the present invention is to provide a high-dimensional time-varying data visualization method with enhanced spatial visual perception in reduced dimensionality.

To achieve the above object, the technical solution adopted in the present invention is:

The high-dimensional time-varying data visualization method for enhancing dimensionality reduction space visual perception of the present invention includes:

Read the high-dimensional time-varying data, use the multi-dimensional scaling algorithm to obtain the two-dimensional space coordinates of the high-dimensional time-varying data, use the formula (1) to solve the coordinates of the high-dimensional time-varying data in the two-dimensional space to obtain the target orthogonal matrix,

In formula (1), Q represents the target orthogonal matrix, Indicates the two-dimensional spatial coordinates of the i-th data of the high-dimensional time-varying data at the current moment, Represents the two-dimensional spatial coordinates of the i-th data of the high-dimensional time-varying data at the previous moment, N represents the number of high-dimensional time-varying data, D ^t (Q) represents the high-dimensional time-varying data after orthogonal matrix transformation The projection offset at the current moment relative to the previous moment;

The projection offset of high-dimensional time-varying data in two-dimensional space is minimized by using the target orthogonal matrix, and the projection of high-dimensional time-varying data at the current moment is obtained.

Further, the present invention uses the formula (1) to solve the coordinates of the two-dimensional space of the high-dimensional time-varying data in the two forms of rotation and flipping respectively, and obtains the corresponding rotation orthogonal matrix and flip orthogonal matrix, respectively calculates the rotated The projection offset of the high-dimensional time-varying data transformed by the orthogonal matrix and the flipped orthogonal matrix at the current moment relative to the previous moment, and the orthogonal matrix with the smaller projection offset is selected as the target orthogonal matrix.

Further, after minimizing the projection offset of the high-dimensional time-varying data in the two-dimensional space by using the target orthogonal matrix, the present invention divides the projection space of the high-dimensional time-varying data at the current moment into more than N subspaces, Different projection points are filled in different subspaces to obtain the projection of high-dimensional time-varying data at the current moment.

Further, the subspace of the present invention is a regular triangle, square or regular hexagon.

Further, in the present invention, when the subspace is a regular hexagon and there are more than two target projection points in the target subspace, each target projection point is filled one by one by the following method according to the traversal order of the target projection points in the target subspace into different regular hexagonal subspaces:

If the current target projection point is the first projection point, fill the current target projection point in the target subspace; otherwise, perform the following steps:

Step a. If the current target projection point falls at the center point of the current subspace, then execute step b, otherwise execute step c;

Step b. Search outward layer by layer from the current subspace until an unfilled outer regular hexagon is found; fill the current target projection point in the outer regular hexagon, and update the current subspace to the target subspace , and then judge whether the target projection point in the target subspace is filled, if not, then the next target projection point is the current target projection point and return to step a;

Step c. Find two regular hexagons with the smallest included angle from the adjacent regular hexagons in the current subspace, the regular hexagon with the smallest included angle means: in all adjacent regular hexagons in the current subspace, The angle between the line connecting the center point of the regular hexagon and the center point of the current subspace and the line connecting the current target projection point and the center point of the current subspace is the smallest;

Step d. Select the one whose center point and the distance from the current target projection point is smaller as the preferred regular hexagon from the two regular hexagons with the smallest included angle;

Step e. If the preferred regular hexagon is not filled, fill the current target projection point in the preferred regular hexagon, and update the current subspace to the target subspace, and then judge whether the target projection point in the target subspace is filled , if the filling is not completed, then the next target projection point is the current target projection point and return to step a; if the preferred regular hexagon has been filled, then perform step f;

Step f. From the two adjacent regular hexagons that have a common side with the preferred regular hexagon and the current subspace, select the one with the smaller distance between its center point and the current target projection point as the suboptimal regular hexagon; if the suboptimal If the regular hexagon is not filled, fill the current target projection point in the suboptimal regular hexagon, and update the current subspace to the target subspace, and then judge whether the target projection point in the target subspace is filled, if not After completion, the next target projection point is the current target projection point and return to step a; if the suboptimal regular hexagon has been filled, go to step g;

Step g. Update the current subspace to the preferred regular hexagon, then transfer the current target projection point from the target subspace to the current subspace, and make the current target projection point in the position of the current subspace before the transfer of the target The positions of the subspaces are the same, and then return to step c.

Compared with prior art, the beneficial effect of the present invention is:

(1) After using the multidimensional scaling algorithm to obtain the two-dimensional coordinates of the high-dimensional time series data in the dimensionality reduction space, the target orthogonal matrix is found by iterating the orthogonal transformation matrix to minimize the projection offset of adjacent time steps. Furthermore, the target orthogonal matrix is used to transform the coordinates of the high-dimensional time-varying data in the two-dimensional space, so as to minimize the projection offset of the high-dimensional time-varying data in the two-dimensional space, and reduce the offset of the projection points of adjacent time nodes. As a result, the present invention reduces the visual disorder of high-dimensional time-varying data in the dimensionality reduction space, quickly displays the time series features of the data, facilitates user interactive analysis, significantly improves the visualization and analysis efficiency of high-dimensional time-varying data, and can Effectively meet the user's application needs.

(2) Furthermore, the target orthogonal matrix can be obtained more quickly by reducing the solution space through two forms of rotation and flipping.

(3) In order to obtain a better visual perception effect, the present invention divides the projection space of high-dimensional time-varying data at the current moment, comprehensively considers the factors of minimum angle and minimum distance, and projects different data items in different subspaces , thereby further avoiding visual confusion caused by two or more projection points falling on the same regular hexagon due to the increase of projection points.

Description of drawings

Fig. 1 is a comparison diagram of the present invention before and after minimizing the projection offset of high-dimensional time-varying data in two-dimensional space, wherein (a) is the projection of high-dimensional data using the classic multi-dimensional scaling algorithm at the current moment to two-dimensional space; (b) is the result of projecting high-dimensional data to two-dimensional space using classical multidimensional scaling algorithm at the next moment; (c) is using classical multidimensional scaling algorithm to project high-dimensional data to two-dimensional space at the current moment Projected to a two-dimensional space and then using the result map of the orthogonal transformation; (d) is the result map of the high-dimensional data projected to the two-dimensional space using the classical multidimensional scaling algorithm at the next moment and then using the orthogonal transformation result map.

Figure 2 is a visual effect diagram of directly stacking four projection points onto the same regular hexagon;

Fig. 3 is a visualization effect diagram of filling projection points using a scheme that comprehensively considers angles and distances in the present invention;

Fig. 4 is a flow chart of filling projection points using a scheme that comprehensively considers angles and distances in the present invention;

Fig. 5 is a comparison diagram of unoccluded rendering results, wherein (a) is a result image of a classic scatter plot, and (b) is an image of an unoccluded rendering result of the present invention.

Detailed ways

The method for visualizing high-dimensional time-varying data with enhanced visual perception in dimension-reduced space according to the present invention will be further described with specific embodiments below. The specific method is as follows:

Load the initial high-dimensional time-varying data, project the high-dimensional time-varying data into a two-dimensional space for display through the classic multi-dimensional scaling algorithm, and obtain the two-dimensional space coordinates of the high-dimensional time-varying data. The specific method is as follows:

First, the original high-dimensional time-varying data is normalized according to formula (2) to eliminate the influence of different dimensions.

In formula (2), x _i represents the value of the i-th data in dimension x; x _min and x _max respectively represent the maximum and minimum values of dimension x; Indicates the i-th data item in dimension x after normalization.

Then calculate the distance matrix of the normalized high-dimensional time-varying data in the original space. The present invention can adopt Euclidean distance array D, and calculation formula is as shown in formula (3):

In formula (3), respectively represent the i-th and j-th data items in the dimension x after normalization, and ||·|| represents the two-norm operation, Indicates the Euclidean distance between the i-th data item and the j-th data item in the dimension x after normalization, and n indicates the number of data items.

Then calculate the centered inner product matrix B represented by the low-dimensional space according to the distance matrix, as shown in formula (4):

In formula (4), Indicates the Euclidean distance between the i-th data item and the j-th data item, and n indicates the number of data items.

Finally, B is decomposed orthogonally, and the two largest eigenvalues and corresponding eigenvectors are selected to obtain the fitting composition in two-dimensional space The calculation formula is shown in formula (5):

In the formula (5), U, Λ, U' are respectively the transposition of the eigenvector array, the eigenvalue array, and the eigenvector array of the matrix B; Be the two largest absolute values of matrix B eigenvalues, U ₂ is the eigenvector corresponding to the eigenvalues; That is, the representation of high-dimensional time-varying data in a reduced-dimensional space (ie, a two-dimensional space).

In the multi-dimensional scaling algorithm, since only the invariance of the relative position between data is considered, the projection of the same data item at different times has strong arbitrariness. This difference is usually due to orthogonal transformation or normalization. It largely interferes with the user's visual tracking of the time-series features of interest, and brings serious visual disturbance to data analysts. Comparing Figure 1(a) and Figure 1(b), it can be seen that some data items are projected to the upper right corner at the previous moment, and suddenly mapped to the lower right corner at the next moment, compared with Figure 1(c) and Figure 1( d) There are serious visual disturbances, which to a certain extent restrict the improvement of the visualization efficiency of high-dimensional time-varying data.

For this reason, in order to improve the user's visual cognition of time-varying features, the present invention proposes to add an orthogonal transformation on the basis of the multi-dimensional scaling algorithm to correct the projection coordinates of the data, so as to realize the relative position of the data without changing the low-dimensional space. Based on this, the offset of multi-dimensional time-varying data in low-dimensional space is minimized, and the ability to explore the time-varying characteristics of data is enhanced.

Orthogonal transformation can correct the coordinates of projection points without changing the mutual positional relationship between data, thereby minimizing the offset distance of adjacent maps. For this reason, the present invention uses the iterative solution method shown in formula (1) to solve the coordinates of the high-dimensional time-varying data in the two-dimensional space to obtain the target orthogonal matrix to obtain the minimum projection offset.

In formula (1), Q represents the target orthogonal matrix, Indicates the two-dimensional spatial coordinates of the i-th data of the high-dimensional time-varying data at the current moment, Represents the two-dimensional spatial coordinates of the i-th data of the high-dimensional time-varying data at the previous moment, N represents the number of high-dimensional time-varying data, D ^t (Q) represents the high-dimensional time-varying data after orthogonal matrix transformation Projection offset at the current moment relative to the previous moment.

Furthermore, the projection offset of the high-dimensional time-varying data in two-dimensional space is minimized by using the target orthogonal matrix, and the projection of the high-dimensional time-varying data at the current moment is obtained. As a result, the present invention reduces the visual disorder of high-dimensional time-varying data in the dimensionality reduction space, quickly displays the time series features of the data, facilitates user interactive analysis, significantly improves the visualization and analysis efficiency of high-dimensional time-varying data, and can Effectively meet the user's application needs.

As a preferred mode of the present invention, in order to obtain the target orthogonal matrix faster, the rotation and flip in the orthogonal transformation can be used to minimize the offset of the sequence map and reduce the solution space. That is, formula (1) is used to solve the coordinates of the two-dimensional space of the high-dimensional time-varying data through rotation and flipping respectively. Among them, the rotation transformation can adopt the matrix shown in formula (6):

In formula (6), Q _Rotation represents the orthogonal matrix of the rotation transformation, and θ represents the rotation angle in the rotation transformation, so the problem of minimizing the offset of the sequence mapping set is simplified to find the optimal rotation angle. Simultaneously flipping transformations can use a matrix as shown in equation (7):

In formula (7), Q _Overturn represents the orthogonal matrix of the flip transformation, and k represents the slope of the symmetry axis in the flip transformation.

The problem of minimizing offset by flipping transformations can be reduced to finding the optimal axis of symmetry. The present invention can obtain the most accurate orthogonal transformation scheme by traversing all possible angles and symmetry axes.

As a preferred embodiment, the present invention can use the average offset angle of data items in adjacent maps to reflect the rotation angle, thereby obtaining the optimal rotation angle, the calculation formula of which is shown in formula (8):

In formula (8), θ ^t represents the rotation angle at time t; respectively represent the projection coordinates of the i-th projection point at time t and time t-1; N is the number of projection points.

As a preferred embodiment of calculating the optimal symmetry axis, the present invention can use the approximate symmetry axis of the m data items with the largest offset on the adjacent map as the symmetry axis of the flip transformation, and its calculation formula is shown in formula (9) :

In formula (9), and are the projected points with the i-th largest offset on the map at time t and t-1, respectively. m is the number of data items with the largest offset defined by the user; k ^t indicates the slope of the symmetry axis at time t. The best axis of symmetry passing through the origin can be obtained by using the slope k ^t .

After the optimal rotation angle and axis of symmetry are obtained through formula (8) and formula (9), the appropriate orthogonal transformation is determined by comparing the offsets of rotation and flipping to achieve the best rotation or flipping transformation, and the minimum deviation is obtained. The dimensionality reduction mapping result of shifting enables users to get better visual cognition.

When projecting using the classic multi-dimensional scaling algorithm, the scatter diagram is selected for the result display. However, as the amount of data increases and the complex distribution of multi-dimensional data causes mutual occlusion of projection points, it is easy to cause visual ambiguity and interaction difficulties for users. And an appropriate projection point size is difficult to choose. Therefore, the present invention introduces space division technology to divide the scatter diagram projected by the multi-dimensional scaling algorithm into multiple subspaces, and each subspace can represent a projection point, thereby removing the mutual occlusion of the projection points. The specific process is as follows:

The present invention can divide the projection space by using the two simplest methods of dividing polygons: regular triangle and square. As a preferred embodiment of the present invention, using regular hexagons for space projection division can not only seamlessly divide the plane, but also further improve the aesthetic feeling of visualization. By using a one-to-one mapping method, each projected point can be filled into the corresponding regular hexagonal subspace. The present invention can divide the space by using as small a regular hexagon as possible so that there is at most one projection point on a regular hexagonal subspace. If there are two or more projection points falling on the same regular hexagon due to the increase of projection points, as a preferred embodiment of the present invention, the scheme shown in Figure 4 can be further adopted, taking into account the minimum included angle , The principle of the shortest distance fills other projection points into other regular hexagons, so as to ensure that a regular hexagon is filled with at most one projection point, effectively avoiding visual disorder. The following describes in detail how to divide the projection space and fill the projection points using the scheme shown in Figure 4:

When the subspace is a regular hexagon and there are more than two target projection points in the target subspace, according to the traversal order of the target projection points in the target subspace, each target projection point is filled to different regular hexagons one by one by the following method In the shape subspace:

If the current target projection point is the first projection point, fill the current target projection point in the target subspace; otherwise, perform the following steps:

Step a. If the current target projection point falls at the center point of the current subspace, then execute step b, otherwise execute step c;

Step b. Search outward layer by layer from the current subspace until an unfilled outer regular hexagon is found; fill the current target projection point in the outer regular hexagon, and update the current subspace to the target subspace , and then judge whether the target projection point in the target subspace is filled, if not, then the next target projection point is the current target projection point and return to step a;

Step c. Find two regular hexagons with the smallest included angle from the adjacent regular hexagons in the current subspace, the regular hexagon with the smallest included angle means: in all adjacent regular hexagons in the current subspace, The angle between the line connecting the center point of the regular hexagon and the center point of the current subspace and the line connecting the current target projection point and the center point of the current subspace is the smallest;

Step d. Select the one whose center point and the distance from the current target projection point is smaller as the preferred regular hexagon from the two regular hexagons with the smallest included angle;

Step e. If the preferred regular hexagon is not filled, fill the current target projection point in the preferred regular hexagon, and update the current subspace to the target subspace, and then judge whether the target projection point in the target subspace is filled , if the filling is not completed, then the next target projection point is the current target projection point and return to step a; if the preferred regular hexagon has been filled, then perform step f;

Step f. From the two adjacent regular hexagons that have a common side with the preferred regular hexagon and the current subspace, select the one with the smaller distance between its center point and the current target projection point as the suboptimal regular hexagon; if the suboptimal If the regular hexagon is not filled, fill the current target projection point in the suboptimal regular hexagon, and update the current subspace to the target subspace, and then judge whether the target projection point in the target subspace is filled, if not After completion, the next target projection point is the current target projection point and return to step a; if the suboptimal regular hexagon has been filled, go to step g;

Step g. Update the current subspace to the preferred regular hexagon, then transfer the current target projection point from the target subspace to the current subspace, and make the current target projection point in the position of the current subspace before the transfer of the target The positions of the subspaces are the same, and then return to step c.

The preferred scheme of the present invention, which comprehensively considers the included angle and the distance, will be described in detail below with specific examples.

As a comparison, FIG. 2 shows a visual effect diagram of directly stacking the four projection points A, B, C, and D into the same regular hexagon (that is, they all fall within the regular hexagon 1).

As shown in Figure 3(a), the four projection points A, B, C, and D fall in the target subspace in sequence (that is, they all fall in the regular hexagon 1). The projection space of high-dimensional time-varying data at the current moment is divided into no less than four regular hexagonal subspaces. In Fig. 3 (a), regular hexagons 2, 3, 4, 5, 6, 7 are adjacent regular hexagons having a shared side with the current subspace (regular hexagon 1), and they are relative to the current subspace (regular hexagon 1). In terms of polygon 1), it is the outer regular hexagon of the first level; with respect to the current subspace (regular hexagon 1), the regular hexagon 8-19 is the outer regular hexagon of the second level.

When adopting the preferred solution shown in Fig. 4 of the present invention, according to the traversal order of the target projection points, the first target projection point A, the second target projection point B, the third target projection point C, the fourth target projection point The target projection point D is filled. The specific method is as follows:

(1) First, fill the first target projection point A. As shown in Fig. 3(a), since A is the first target projection point, the target projection point A is filled in the current subspace. At this time, the current subspace is the target subspace (regular hexagon 1).

(2) Next, fill in the second target projection point B. Since the current target projection point B does not fall at the center point of the current subspace, the following steps are performed;

Step a. It is judged that the current target projection point B does not fall at the center point O ₁ of the regular hexagon 1 in the current subspace, so step c is executed.

Step c. Connect the current target projection point B with the center point O ₁ of the regular hexagon 1 in the current subspace to obtain the connection line O ₁ B. Connect the center points of all adjacent regular hexagons 2-7 in the current subspace to the center point O ₁ of the current subspace respectively, and compare the angles between these connecting lines and the connecting line O ₁ B. Among them, the angle between α and β (α=β) is the smallest. As shown in Figure 3(a), α is the angle between the line connecting the center point O ₁ of the regular hexagon 1 and the center point O ₂ of the regular hexagon 2 (the line O ₁ O ₂ ) and the line O ₁ B , β is the angle between the line connecting the center point O ₁ of the regular hexagon 1 and the center point O ₅ of the regular hexagon 5 (the line O ₁ O ₅ ) and the line O ₁ B. Since the included angles α and β are the smallest, the regular hexagon 2 and the regular hexagon 5 are both so-called "regular hexagons with the smallest included angle". Execute step d.

Step d. Since both the regular hexagon 2 and the regular hexagon 5 are regular hexagons with the smallest included angle, it is necessary to further select the distance between the center point of the regular hexagon 2 and the regular hexagon 5 and the current target projection point B The one with the smaller distance is the preferred regular hexagon. As shown in Figure 3(a), since the distance between the current target projection point B and the center point _O2 is smaller than the distance between B and the center point _O5 , according to the principle of the shortest distance, the regular hexagon 2 is selected as The preferred regular hexagon to be filled by the current target projection point B.

Step e. Since the preferred regular hexagon (regular hexagon 2) is not filled, the current target projection point B is filled in the preferred regular hexagon (regular hexagon 2), and the current subspace is updated to the target subspace ( Namely regular hexagon 1). [See Figure 3(b)].

(3) Next, fill the third projection point C.

Step a. After judging, the current target projection point C does not fall at the center point of the current subspace, so the following steps are performed:

Wherein, step c to step d can refer to the above-mentioned filling process of the target projection point B, according to the principles of the smallest included angle and the shortest distance, the regular hexagon 2 is an alternative preferred regular hexagon for filling the target projection point C. Next, step e is performed. details as follows:

Step e. Since the preferred regular hexagon (regular hexagon 2) has already been filled by the target projection point B, the target projection point C can no longer be filled in the regular hexagon 2, thus continuing to execute step f.

Step f. Select one of the two adjacent regular hexagons (regular hexagon 3 and regular hexagon 4) that have a common side with the preferred regular hexagon (regular hexagon 2) and the current subspace (regular hexagon 1) The one with the smaller distance between the center point and the current target projection point is the suboptimal regular hexagon. Since the length of the line O 3 C connecting the center point O ₃ of the regular hexagon ₃ and the target projection point C is shorter than the length of the line O 4 C connecting the center point O ₄ of the regular hexagon ₄ and the projection point C, the regular hexagon Shape 3 is a suboptimal regular hexagon.

Since the suboptimal regular hexagon (regular hexagon 3) is not filled, the current target projection point C is filled in the suboptimal regular hexagon (regular hexagon 3), and the current subspace is updated to the target subspace (regular hexagon 3) Polygon 1). [See Figure 3(c)].

(4) Filling the fourth projection point D.

Step a. After judging, the current target projection point D does not fall at the center point of the current subspace, so the following steps are performed:

Similarly, steps b to f can refer to the above filling process of the target projection point C. According to the principles of the smallest included angle and the shortest distance, the regular hexagon 2 is an alternative preferred regular hexagon for filling the target projection point D. However, since the preferred regular hexagon (regular hexagon 2 ) is already filled with the target projection point B, the target projection point D cannot be filled in the regular hexagon 2 any more. Since the suboptimal regular hexagon (regular hexagon 3 ) has already been filled by the target projection point C, the target projection point D cannot be filled in the regular hexagon 3 . Thus, proceed to step g.

Step g. Update the current subspace to the preferred regular hexagon (ie regular hexagon 2), and use the regular hexagon 2 as the new current subspace. Then transfer the current target projection point D from the target subspace (regular hexagon 1) to the current subspace (regular hexagon 2), and make the current target projection point D in the position of the current subspace (regular hexagon 2) The position in the target subspace (regular hexagon 1) before the transfer is the same, and then continue to execute step h according to the method of step c.

Step h. As shown in Figure 3(c), the regular hexagons 1, 3, 4, 8, 9, and 10 are the adjacent regular hexagons of the current subspace (regular hexagon 2). The center points of all adjacent regular hexagons 1, 3, 4, 8, 9, 10 of the current subspace (regular hexagon 2) are respectively connected with the center point _O2 of the current subspace (regular hexagon 2), and compared The size of the angle between these connecting lines and the connecting line O ₂ D. Among them, the angle between γ and δ is the smallest (γ=δ). As shown in Figure 3(c), the angle between the line O 9 O 2 connecting _the center point O ₉ of _the regular hexagon 9 and the center point O ₂ of the current subspace (regular hexagon 2) and the line O ₂ D is γ, the angle between the line O 2 O 1 connecting _the center point O ₂ of the current subspace (regular hexagon 2 ) and the center point O ₁ of the regular hexagon ₁ , and the line O ₂ D is δ, and γ=δ. Therefore, the two regular hexagons with the smallest included angle are regular hexagon 1 and regular hexagon 9.

Step i. According to the method of step d, since the distance between the current target projection point D and the center point _O9 is smaller than the distance between D and the center point _O1 , therefore, according to the principle of the shortest distance, select the regular hexagon 9 as The preferred regular hexagon to be filled by the current target projection point D.

Step j. Since the regular hexagon 9 has not been filled, fill the current target projection point D in the regular hexagon 9 . After the projection is completed, update the current subspace to the target subspace (ie regular hexagon 1).

If there are more target projection points in the current regular hexagon 1, return to step a, select the corresponding regular hexagon and fill the target projection points according to the above method.

As a preferred embodiment of the present invention, according to the method shown in Figure 4, each target projection point is filled into each regular hexagon one by one, which can remove the visual problems caused by mutual occlusion of projection points after high-dimensional data is degraded to low-dimensional space. Ambiguity, users can obtain unobstructed high-dimensional data projection display in low-dimensional space, which enhances users' visual cognition of multi-dimensional data.

Fig. 1 shows the high-dimensional time-varying data displayed using scatter diagrams using the classical multidimensional scaling algorithm and the result image comparison obtained by the method of the present invention, wherein Fig. 1 (a) and Fig. 1 (b) use the formula ( 3), the drawing results obtained by the classic multidimensional scaling algorithm shown in formula (4) and formula (5), because only the invariance of the relative position between data is considered, the projection of the same data item at different times has a strong Arbitrariness creates a great obstacle to the user's visual perception, and it is difficult to visually track the timing pattern of the data; Figure 1(c) and Figure 1(d) are shown by formula (8) and formula (9) The dimensionality reduction space enhances the visual perception of high-dimensional time-varying data to visualize the rendering result image, calculates the orthogonal transformation matrix, minimizes the projection point offset of adjacent time steps, and enhances the user's vision of the time-series features of high-dimensional time-varying data Perception, comparing Fig. 1(b) and Fig. 1(d), it can be found that by applying the algorithm of the present invention, the projection points of adjacent time steps of high-dimensional time-varying data can be clearly tracked.

Fig. 5 shows the result image comparison obtained by using the modified multi-dimensional scaling algorithm and formula (1) to display the high-dimensional data using a scatter diagram and dividing the space by hexagons. Among them, Fig. 5(a) is the drawing result obtained by using the ordinary scatter diagram. Since the size of the projected points in the scatter diagram is difficult to define, there are often serious occlusion phenomena. Figure 5(b) is the rendering result image obtained by using a regular hexagon to divide the space. When multiple points are projected on the same regular hexagon, the mutual occlusion of the projection points is removed by using the above-mentioned optimal scheme that comprehensively considers the included angle and distance. .

It can be seen that the result image obtained by using the method of the present invention does not have the occlusion phenomenon in the scatter diagram, and interactive analysis can be conveniently performed, which improves the visualization and analysis efficiency of high-dimensional time-varying data.

Compared with the traditional high-dimensional time-varying data visualization process, the biggest advantage of the present invention is to obtain the expression of high-dimensional time-varying data in low-dimensional space, and make the high-dimensional time-varying data The projection offset in the two-dimensional space is minimized to help users effectively visually perceive and track the time patterns of interest, and enhance the user's visual perception of high-dimensional time-varying data in the reduced-dimensional space. In addition, in order to avoid mutual occlusion between projection points, regular hexagons are further introduced to divide the projection space of high-dimensional time-varying data, so as to enhance users' visual cognition and interaction with the characteristics of high-dimensional data in the dimensionality reduction space, which can Quickly display the implicit timing patterns in the data, improving the visualization and analysis efficiency of high-dimensional time-varying data. Utilizing the method of the present invention can enhance the user's visual perception of time-varying features of high-dimensional data in low-dimensional space, and calculate an orthogonal matrix to minimize the projection offset of high-dimensional time-varying data in two-dimensional space, reducing multi-dimensional The visual disorder caused by the scaling algorithm when analyzing time-varying data, and the projection space is divided by hexagons, the mutual occlusion between projection points is removed, and the visualization and analysis efficiency of high-dimensional time-varying data is improved.

Claims (5)

1. A high-dimensional time-varying data visualization method for enhanced spatial visual perception, characterized in that it comprises: Read the high-dimensional time-varying data, use the multi-dimensional scaling algorithm to obtain the two-dimensional space coordinates of the high-dimensional time-varying data, use the formula (1) to solve the coordinates of the high-dimensional time-varying data in the two-dimensional space to obtain the target orthogonal matrix, <mrow><munder><mrow><mi>arg</mi><mi>min</mi></mrow><mi>Q</mi></munder><msup><mi>D</mi><mi>t</mi></msup><mrow><mo>(</mo><mi>Q</mi><mo>)</mo></mrow><mo>=</mo><munderover><mo>&amp;Sigma;</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></munderover><mrow><mo>(</mo><mo>|</mo><mo>|</mo><msubsup><mi>Qv</mi><mi>i</mi><mi>t</mi></msubsup><mo>-</mo><msubsup><mi>v</mi><mi>i</mi><mrow><mi>t</mi><mo>-</mo><mn>1</mn></mrow></msubsup><mo>|</mo><mo>|</mo><mo>)</mo></mrow><mo>-</mo><mo>-</mo><mo>-</mo><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow> In formula (1), Q represents the target orthogonal matrix, Indicates the two-dimensional spatial coordinates of the i-th data of the high-dimensional time-varying data at the current moment, Represents the two-dimensional spatial coordinates of the i-th data of the high-dimensional time-varying data at the previous moment, N represents the number of high-dimensional time-varying data, D ^t (Q) represents the high-dimensional time-varying data after orthogonal matrix transformation The projection offset at the current moment relative to the previous moment; The projection offset of high-dimensional time-varying data in two-dimensional space is minimized by using the target orthogonal matrix, and the projection of high-dimensional time-varying data at the current moment is obtained. 2. The high-dimensional time-varying data visualization method according to claim 1, characterized in that: use formula (1) to solve the coordinates of the two-dimensional space of the high-dimensional time-varying data by rotating and flipping respectively, Corresponding to the obtained rotation orthogonal matrix and flipped orthogonal matrix, respectively calculate the projection offset of the high-dimensional time-varying data transformed by the rotated orthogonal matrix and flipped orthogonal matrix at the current moment relative to the previous moment, and select one of them with a relatively The orthomatrix of the small projected offset serves as the target orthomatrix. 3. The high-dimensional time-varying data visualization method according to claim 1 or 2, characterized in that: after using the target orthogonal matrix to minimize the projection offset of the high-dimensional time-varying data in two-dimensional space, the high-dimensional The projection space of time-varying data at the current moment is divided into more than N subspaces, and different projection points are filled in different subspaces to obtain the projection of high-dimensional time-varying data at the current moment. 4. The high-dimensional time-varying data visualization method according to claim 3, wherein the subspace is a regular triangle, a square or a regular hexagon. 5. The high-dimensional time-varying data visualization method according to claim 4, wherein when the subspace is a regular hexagon and there are more than two target projection points in the target subspace, according to the target in the target subspace The traversal order of the projection points fills each target projection point into different regular hexagonal subspaces one by one by the following method: If the current target projection point is the first projection point, fill the current target projection point in the target subspace; otherwise, perform the following steps: Step a. If the current target projection point falls at the center point of the current subspace, then execute step b, otherwise execute step c; Step b. Search outward layer by layer from the current subspace until an unfilled outer regular hexagon is found; fill the current target projection point in the outer regular hexagon, and update the current subspace to the target subspace , and then judge whether the target projection point in the target subspace is filled, if not, then the next target projection point is the current target projection point and return to step a; Step c. Find two regular hexagons with the smallest included angle from the adjacent regular hexagons in the current subspace, the regular hexagon with the smallest included angle means: in all adjacent regular hexagons in the current subspace, The angle between the line connecting the center point of the regular hexagon and the center point of the current subspace and the line connecting the current target projection point and the center point of the current subspace is the smallest; Step d. Select the one whose center point and the distance from the current target projection point is smaller as the preferred regular hexagon from the two regular hexagons with the smallest included angle; Step e. If the preferred regular hexagon is not filled, fill the current target projection point in the preferred regular hexagon, and update the current subspace to the target subspace, and then judge whether the target projection point in the target subspace is filled , if the filling is not completed, then the next target projection point is the current target projection point and return to step a; if the preferred regular hexagon has been filled, then perform step f; Step f. From the two adjacent regular hexagons that have a common side with the preferred regular hexagon and the current subspace, select the one with the smaller distance between its center point and the current target projection point as the suboptimal regular hexagon; if the suboptimal If the regular hexagon is not filled, fill the current target projection point in the suboptimal regular hexagon, and update the current subspace to the target subspace, and then judge whether the target projection point in the target subspace is filled, if not After completion, the next target projection point is the current target projection point and return to step a; if the suboptimal regular hexagon has been filled, go to step g; Step g. Update the current subspace to the preferred regular hexagon, then transfer the current target projection point from the target subspace to the current subspace, and make the current target projection point in the position of the current subspace before the transfer of the target The positions of the subspaces are the same, and then return to step c.