A high-dimensional data manifold clustering method based on multi-metric optimization

Abstract

The application belongs to the field of machine learning and data analysis, and relates to a high-dimensional data manifold clustering method based on multi-metric optimization, comprising: performing intrinsic dimension calculation on original high-dimensional isotope data after preprocessing; performing nonlinear dimension reduction on the preprocessed high-dimensional data to obtain low-dimensional manifold embedding representation; for key hyperparameters of the UMAP algorithm, a multi-metric optimization strategy and iterative calculation are adopted to determine the optimal value; the UMAP is re-run using the optimal parameters to obtain optimal low-dimensional embedding data, and a hierarchical agglomerative clustering algorithm is used in the low-dimensional space to divide the data into multiple cluster groups, and output clustering labels; the contribution degree of original high-dimensional features to the clustering labels is quantitatively calculated; a decision boundary is drawn in the low-dimensional embedding space, and an interpretable classification model is constructed in the original high-dimensional space. The beneficial effect is that the distance between data points in the low-dimensional space after UMAP dimension reduction has a practical meaning, and more truly reflects the distance and group relationship between samples.

Description

Technical Field

[0001] This invention relates to the fields of machine learning and data analysis technology, and in particular to a high-dimensional data manifold clustering method based on multi-metric optimization. Background Technology

[0002] Currently, in high-dimensional data analysis, mantle geochemical analysis (such as Sr-Nd-Pb-Hf isotopes) is inherently high-dimensional (5 dimensions or higher). In the early stages of technological development, visual inspection was typically carried out using traditional two-dimensional or three-dimensional isotope correlation maps (such as Sr-Nd maps). This method is highly subjective, and information loss due to projection may obscure true or imply false systematicity.

[0003] To overcome the subjectivity and information loss inherent in the aforementioned methods, classic mantle endmember models were developed, and linear statistical methods such as principal component analysis (PCA) and factor analysis (FA) were introduced. However, while these linear methods aim to find the direction of maximum data variance, they often fail to preserve or may ignore inherent low-order data trends or clusters in the data; in other words, they cannot capture the key local nonlinear structures in high-dimensional data.

[0004] In recent years, nonlinear dimensionality reduction algorithms, especially t-distributed random neighborhood embedding (t-SNE), have been introduced into high-dimensional data analysis and processing. The advantage of t-SNE is that it is a nonlinear technique and is good at preserving the local structure and pairwise similarity of data.

[0005] However, t-SNE, as one of the most advanced methods currently available, has three major shortcomings that urgently need to be addressed, including:

[0006] (1) Lack of global structure: The t-SNE algorithm prioritizes preserving the local structure of the data, but ignores the accuracy of the global topology. This results in the relative positions and distances between clusters in the t-SNE dimensionality reduction graph having no obvious practical meaning, which limits the understanding of the global relationships of the data.

[0007] (2) Subjectivity in parameter selection: The performance of t-SNE (and subsequent clustering) algorithms is highly dependent on the selection of key hyperparameters (such as perplexity or n_neighbors). In existing technologies, the selection of these parameters often relies on experience and lacks an objective, reproducible, and systematic optimization process, making it difficult to compare and verify the analysis results.

[0008] (3) Black Box Dilemma: t-SNE and clustering only provide the grouping results, but the model itself cannot explain the basis for the classification. It cannot quantitatively answer "which combinations of original isotopes (features) led to the separation of cluster A and cluster B?". This makes it difficult to translate the results of machine learning into an understanding of geochemical mechanisms.

[0009] Therefore, there is an urgent need in this field for a new technical solution to overcome the above-mentioned shortcomings of t-SNE. Summary of the Invention

[0010] In view of the above-mentioned shortcomings and deficiencies of the prior art, the present invention provides a high-dimensional data manifold clustering method based on multi-metric optimization, which solves the technical problems of ignoring global structure, subjective parameter selection, and black box results in existing high-dimensional data analysis.

[0011] To achieve the above objectives, the main technical solutions adopted by the present invention include:

[0012] In a first aspect, the present invention provides a high-dimensional data manifold clustering method based on multi-metric optimization, comprising the following steps:

[0013] Step 1: Calculate the intrinsic dimension after preprocessing the original high-dimensional isotope data;

[0014] Step 2: The Unified Manifold Approximation and Projection UMAP algorithm is used to perform nonlinear dimensionality reduction on the preprocessed high-dimensional data to obtain a low-dimensional manifold embedding representation;

[0015] Step 3: For the key hyperparameters of the UMAP algorithm, a multi-metric optimization strategy and iterative calculation are used to determine their optimal values;

[0016] Step 4: Rerun UMAP with optimal parameters to obtain the optimal low-dimensional embedding data, and use the hierarchical agglomerative clustering algorithm in this low-dimensional space to divide the data into multiple clusters and output cluster labels;

[0017] Step 5: Use the feature importance assessment method to quantitatively calculate the contribution of the original high-dimensional features to the clustering labels;

[0018] Step 6: Draw the decision boundary in the low-dimensional embedding space and build an interpretable classification model in the original high-dimensional space.

[0019] As a further improvement to the method of this invention, the UMAP algorithm is based on Riemannian geometry and fuzzy topology theory, and simultaneously preserves the local fine structure and global topological relationships of the dataset during the dimensionality reduction process, specifically including:

[0020] Step 2.1: Use the Nearest-Neighbor-Descent algorithm to find the nearest neighbor for each high-dimensional data point, construct a weighted neighborhood graph to represent the manifold structure, and finally obtain a fuzzy neighborhood topology graph representing the high-dimensional manifold structure;

[0021] Step 2.2: Based on the fuzzy neighborhood topology graph, learn the manifold topology in its high-dimensional manifold structure, project it into the low-dimensional space, and obtain the low-dimensional representation of the manifold.

[0022] As a further improvement to the method of the present invention, obtaining the fuzzy neighborhood topology graph in step 2.1 includes the following methods:

[0023] Step 2.1.1: Use the Nearest-Neighbor-Descent algorithm to find the nearest neighbor for each high-dimensional data point and identify the nearest neighbor relationships of the high-dimensional data;

[0024] Step 2.1.2: Based on the established nearest neighbor relationships, construct a weighted neighborhood graph to represent the manifold structure, including:

[0025] Assuming that the high-dimensional data points are uniformly distributed on the manifold space, the hyperparameter local_connectivity ensures that each point is connected to at least one neighbor.

[0026] For each point, the association strength with its nearest neighbors decreases with increasing distance, forming a fuzzy neighborhood boundary;

[0027] To address the issue of inconsistent bidirectional edge weights between point pairs, a probabilistic union approach is used to merge edge weights.

[0028] Ultimately, a connected fuzzy neighborhood topology graph is obtained, representing the high-dimensional manifold structure.

[0029] As a further improvement to the method of the present invention, finding the low-dimensional representation of the manifold in step 2.2 includes:

[0030] In the low-dimensional embedding space, standard Euclidean distance is used, and the minimum allowable distance between embedding points is controlled by the hyperparameter min_dist.

[0031] The optimization of the low-dimensional manifold representation is achieved by minimizing the cross-entropy loss, and the loss function is shown in the following equation:

[0032]

[0033] In the formula, Represents the set of neighbors of a higher-dimensional point. Controlling the balance between the two parts, For high-dimensional space similarity, This represents low-dimensional spatial similarity.

[0034] Among them, high-dimensional space similarity Defined by the following formula:

[0035]

[0036] In the formula, For high-dimensional distance, The minimum radius threshold of the nearest neighbor point. Is it following Adjusted scale parameters, It is the number of nearest neighbors of a given point, n_neighbors.

[0037] Low-dimensional space similarity Defined by the following formula:

[0038]

[0039] In the formula, the parameters Depend on The fitting yields a distance between points in the low-dimensional space that is approximately [value missing]. At that time, the similarity decays to half;

[0040] Ultimately, the optimization problem is solved using the stochastic gradient descent algorithm to obtain the coordinates of each high-dimensional data point in a specified low-dimensional space, i.e., the low-dimensional representation.

[0041] As a further improvement to the method of the present invention, the multi-metric optimization strategy includes a comprehensive evaluation of the silhouette coefficient, the Calinski-Harabasz index, and the Davies-Bouldin index.

[0042] As a further improvement to the method of the present invention, step 3 employs a multi-metric optimization strategy and iterative calculation to determine its optimal value. Specifically, the optimal value of the key hyperparameter n_neighbors of the UMAP algorithm is determined through a multi-metric optimization strategy, including:

[0043] Set the fixed parameters of UMAP as follows: n_components=2, metric='minkowski', min_dist=0.1, and set the candidate range of n_neighbors to 5 to 55;

[0044] For each candidate n_neighbors value, perform UMAP dimensionality reduction and apply hierarchical agglomerative clustering algorithm to the generated two-dimensional result to calculate the silhouette coefficient, Calinski-Harabasz index and Davies-Bouldin index.

[0045] The optimal hyperparameter is selected as the n_neighbors value that maximizes the silhouette coefficient and the Calinski-Harabasz exponent while minimizing the Davies-Bouldin exponent.

[0046] As a further improvement to the method of the present invention, in step 4, the hierarchical agglomerative clustering algorithm is used to divide the data into multiple clusters, specifically the Ward algorithm is used to divide the data into multiple clusters.

[0047] The Ward algorithm tends to generate clusters of uniform size and compact structure, and its objective function is shown in the following equation:

[0048]

[0049] In the formula, and These are two clusters to be merged. and These are their sample sizes. and Each is its own centroid vector.

[0050] As a further improvement to the method of the present invention, the feature importance evaluation method in step 5 includes SHAP analysis and mutual information calculation;

[0051] The formula for calculating SHAP importance is:

[0052]

[0053] In the formula, The total number of samples, For the first In the nth sample The SHAP value of each feature.

[0054] The formula for calculating mutual information is:

[0055]

[0056] In the formula, It is the first Characteristics of an isotope These are clustering labels. It is a joint probability distribution. and It is a marginal probability distribution.

[0057] In a second aspect, the present invention provides a computer-readable storage medium having a computer program stored thereon, wherein the program, when executed, implements the high-dimensional data manifold clustering method based on multi-metric optimization as described in any one of the first aspects above.

[0058] Thirdly, the present invention provides a storage device, including a storage medium and a processor, wherein the storage medium stores a computer program, and when the program is executed by the processor, it implements the high-dimensional data manifold clustering method based on multi-metric optimization described in any one of the first aspects above.

[0059] The beneficial effects of this invention are:

[0060] The UMAP algorithm, compared to t-SNE, can more accurately capture the global and topological structure of the dataset, while also excellently preserving the local features of the data.

[0061] In the low-dimensional space after UMAP dimensionality reduction, the distance between data points has practical meaning, more realistically reflecting the distance between samples and group relationships.

[0062] By introducing the silhouette coefficient, Calinski-Harabasz index, and Davies-Bouldin index, the key hyperparameter (n_neighbors) is optimized using multiple metrics, avoiding the arbitrariness of subjective parameter selection and making the optimization process more systematic and objective.

[0063] It can not only effectively reduce the dimensionality and cluster high-dimensional geochemical data, but also ensure the objectivity of the results through multi-metric optimization. By introducing SHAP and MI, it quantitatively explains the contribution of the original high-dimensional features (isotopes) to the two-dimensional structure and clustering results, turning the black box process of machine learning into a white box, and providing quantitative classification basis and model interpretation for subsequent professional field analysis (such as geochemistry). Attached Figure Description

[0064] Figure 1 This is a schematic diagram of the overall method flow provided in the embodiments of the present invention;

[0065] Figure 2 The comparison chart of the UMPA dimensionality reduction method and t-SNE dimensionality reduction results provided in the embodiments of the present invention is used to demonstrate that the present method is superior to t-SNE in preserving the global structure of the cluster.

[0066] Figure 3 For the cases where the number of clusters is 2 and 5, the following is an optimization piecewise line graph of the nearest neighbor value n_neighbor using the UMAP method provided in the embodiments of the present invention, wherein, Figure 3 Part (a) represents the optimization line graph of the results calculated using the silhouette coefficient as the statistical indicator when the number of clusters is 5. Figure 3 Part (b) in the figure represents the optimization line graph of the results calculated using the silhouette coefficient as the statistical indicator when the number of clusters is 2. Figure 3 Part (c) in the figure represents the optimization line graph of the results calculated using the Calinski-Harabasz index and the Davies-Bouldin index as statistical indicators when the number of clusters is 5. Figure 3 The (d) section in the figure represents the optimization line graph of the results calculated using the Calinski-Harabasz index and the Davies-Bouldin index as statistical indicators when the number of clusters is 2.

[0067] Figure 4 This is a visualization diagram of the HAC clustering results and decision boundary in the optimal UMAP dimensionality reduction space provided by an embodiment of the present invention, wherein... Figure 4 Part (a) in the figure represents the HAC clustering result with a clustering number of 2 in the UMAP dimensionality reduction space. Figure 4 Part (b) in the figure represents the HAC clustering result on the optimal UMAP dimensionality reduction space;

[0068] Figure 5 This is a graph showing the overall SHAP value and mutual information value of each isotopic feature pair for clustering labels provided in the embodiments of the present invention;

[0069] Figure 6 The contribution diagram of each isotopic feature to the SHAP value of different cluster categories provided in the embodiments of the present invention;

[0070] Figure 7 This is a graph showing the contribution of each isotopic feature to the SHAP values ​​of the two coordinate axes after dimensionality reduction of UMAP, as provided in the embodiments of the present invention. Detailed Implementation

[0071] To better explain and facilitate understanding of the present invention, the present invention will be described in detail below with reference to the accompanying drawings and specific embodiments.

[0072] To better understand the above technical solutions, exemplary embodiments of the present invention will be described in more detail below with reference to the accompanying drawings. Although exemplary embodiments of the present invention are shown in the drawings, it should be understood that the present invention can be implemented in various forms and should not be limited to the embodiments set forth herein. Rather, these embodiments are provided so that the present invention can be understood more clearly and thoroughly, and that the scope of the present invention can be fully conveyed to those skilled in the art.

[0073] This embodiment uses five isotopic data from plateau basalt. 206 Pb / 204 Pb, 207 Pb / 204Pb, 208 Pb / 204 Pb, 87 Sr / 86 Sr、 143 Nd / 144 The invention will be described in detail using Nd as an example for dimensionality reduction and clustering analysis. The application of the invention is not limited to this example.

[0074] Firstly, this embodiment provides a manifold clustering method based on multi-metric optimization, such as... Figure 1 As shown, it includes the following steps:

[0075] Step 1: After preprocessing the original high-dimensional isotope data, calculate the intrinsic dimension.

[0076] Specifically, firstly, the original five isotope data are processed using a mean-centered standardization method based on the range to eliminate the bias between the isotope systems caused by differences in numerical range and scale.

[0077] Its calculation formula is shown in equation (1):

[0078]

[0079] In the formula, These are the standardized isotope values. Represents the original isotope value, This is the mean of the isotope data series.

[0080] After processing, the intrinsic dimension ID of the high-dimensional data was calculated using the nearest neighbor distance (MLE) method to guide the necessity and effectiveness of dimensionality reduction. The calculated result ID≈3.66, which is much smaller than the original dimension of 5 (the original dimension is 5: the number of isotope data is 5, so the original dimension is 5), indicating that the data has a significant low-dimensional manifold structure, which is suitable for dimensionality reduction processing, and most of the structural information can be retained after dimensionality reduction.

[0081] The formula for calculating the intrinsic dimension is shown in equation (2):

[0082]

[0083] In the formula, It is the first in the dataset The sample point (the first) (isotope vectors of basalt samples) It is a point To the The distance to the nearest neighbor, It is a point Distance to the nearest neighbor.

[0084] Step 2: The Unified Manifold Approximation and Projection UMAP algorithm is used to perform nonlinear dimensionality reduction on the preprocessed high-dimensional data to obtain a low-dimensional manifold embedding representation.

[0085] The UMAP algorithm mentioned here is based on Riemannian geometry and fuzzy topology, and can simultaneously preserve the local fine structure and global topological relationships of the dataset during dimensionality reduction. Its core assumption is that the available data samples are uniformly distributed in the topological space Manifold, and that approximation and projection can be obtained from these finite data samples to a lower-dimensional space. Specifically, this includes the following methods:

[0086] Step 2.1: Use the Nearest-Neighbor-Descent algorithm to find the nearest neighbor for each high-dimensional data point, construct a weighted neighborhood graph to represent the manifold structure, and finally obtain a fuzzy neighborhood topology graph representing the high-dimensional manifold structure, including:

[0087] Step 2.1.1: Use the Nearest-Neighbor-Descent algorithm to find the nearest neighbor for each high-dimensional data point and identify the nearest neighbor relationships of the high-dimensional data;

[0088] First, UMPA uses the Nearest-Neighbor-Descent algorithm to find the nearest neighbors for each high-dimensional data point, thus clarifying the specific appearance of the high-dimensional data in the high-dimensional space. The neighborhood size is controlled by the UMPA hyperparameter `n_neighbors`, which specifies how many nearest neighbors to use; this parameter directly affects the algorithm's balance between local and global structure.

[0089] Smaller n_neighbors values ​​emphasize fine-grained capture of local structure; larger values ​​allow estimations to be based on a wider area, focusing more on preserving global structure.

[0090] Step 2.1.2: Based on the established nearest neighbor relationships, construct a weighted neighborhood graph to represent the manifold structure, including:

[0091] Assuming that the high-dimensional data points are uniformly distributed on the manifold space, the hyperparameter local_connectivity ensures that each point is connected to at least one neighbor.

[0092] The UMPA algorithm uses the hyperparameter local_connectivity (default value is 1) to ensure that each point is connected to at least one neighboring point, thus avoiding the creation of isolated points.

[0093] For each point, the association strength with its nearest neighbors decreases with increasing distance, forming a fuzzy neighborhood boundary.

[0094] To address the issue of inconsistent bidirectional edge weights between point pairs, a probabilistic union approach is used to merge them, resulting in the final edge weights. , where x and y are the weights of the two edges.

[0095] Ultimately, a connected fuzzy neighborhood topology graph is obtained, representing the high-dimensional manifold structure.

[0096] Step 2.2: Based on the fuzzy neighborhood topology graph, learn the manifold topology in its high-dimensional manifold structure, project it into a low-dimensional space, and obtain the low-dimensional representation of the manifold;

[0097] In the low-dimensional embedding space, standard Euclidean distance is used. The minimum allowable distance between embedding points is controlled by the hyperparameter min_dist (default value is 0.1) to control the minimum distribution of points, avoid the situation where many points overlap in the low-dimensional embedding, and ensure the clarity of visualization or downstream analysis.

[0098] After specifying the minimum distance, we begin searching for a low-dimensional manifold representation by minimizing the cross-entropy loss. Calculate according to formula (3):

[0099]

[0100] In the formula, Represents the set of neighbors of a higher-dimensional point. Controlling the balance between the two parts, For high-dimensional space similarity, This represents low-dimensional spatial similarity.

[0101] Among them, high-dimensional space similarity According to the definition in equation (4):

[0102]

[0103] In the formula, For high-dimensional distance, The minimum radius threshold of the nearest neighbor point. Is it following Adjusted scale parameters, It is n_neighbors (the number of nearest neighbors of a given point).

[0104] Low-dimensional space similarity According to the definition in equation (5):

[0105]

[0106] In the formula, the parameters Depend on The fitting yields a distance between points in the low-dimensional space that is approximately [value missing]. At that time, the similarity decays to half.

[0107] Therefore, n_neighbors directly determines the size of the neighbor set, thus affecting the attraction radius; min_dist determines the minimum distance between low-dimensional points, affecting the density of high-dimensional data clusters.

[0108] This optimization problem is usually solved using the stochastic gradient descent algorithm, which ultimately yields the coordinates of each data point in a specified low-dimensional space, i.e., low-dimensional embedded data.

[0109] In this embodiment, the output of this step is a low-dimensional embedded coordinate array, with the number of rows equal to the number of samples and the number of columns equal to the set low-dimensional dimension (usually 2 or 3), which can be directly used for visualization or subsequent analysis.

[0110] Step 3: For the key hyperparameters of the UMAP algorithm, a multi-metric optimization strategy and iterative calculation are adopted to determine their optimal values.

[0111] Specifically, during the dimensionality reduction process, a multi-metric optimization strategy is introduced for the key hyperparameter of the UMAP algorithm (n_neighbors, i.e., the size of the local neighborhood).

[0112] The multi-metric optimization strategy refers to evaluating multiple clustering evaluation indices comprehensively, rather than relying on a single indicator. These clustering evaluation indices include the Silhouette Coefficient, the Calinski-Harabasz index, and the Davies-Bouldin index.

[0113] Among them, the optimal values ​​of key hyperparameters of the UMAP algorithm are determined through a multi-metric optimization strategy, including the following methods:

[0114] Set the fixed parameters of UMAP as follows: n_components=2 (to reduce high-dimensional data to two dimensions), metric='minkowski', min_dist=0.1. Set the candidate range of the key hyperparameter n_neighbors to 5 to 55.

[0115] For each candidate n_neighbors value, parameter optimization is performed, UMAP dimensionality reduction is executed, and hierarchical agglomerative clustering algorithm is applied to the generated two-dimensional results to calculate the silhouette coefficient, Calinski-Harabasz index, and Davies-Bouldin index.

[0116] The optimal hyperparameters are selected as those that maximize the silhouette coefficient and the Calinski-Harabasz exponent while minimizing the Davies-Bouldin exponent. These include:

[0117] For each n_neighbors value, UMAP dimensionality reduction is performed, and hierarchical agglomerative clustering algorithm and multi-metric optimization strategy are applied to the generated two-dimensional results. The n_neighbors value that maximizes the silhouette coefficient and Calinski-Harabasz exponent while minimizing the Davies-Bouldin exponent is selected as the optimal hyperparameter.

[0118] The silhouette score comprehensively measures both the cohesion and separation of a cluster. For each sample point... Its profile coefficient is defined according to equation (6):

[0119]

[0120] In the formula, For sample points The average distance to all other points within its cluster indicates the density of the cluster. For sample points The average distance to all points in the nearest other clusters represents the degree of inter-cluster separation.

[0121] The silhouette coefficients of the entire dataset (i.e., the dataset consisting of five isotopic data of plateau basalt to be processed in this embodiment) are for all sample points. The average value, whose range is . A value closer to 1 indicates a better clustering result (close clustering within clusters and separation between clusters), a value closer to 0 indicates overlap between clusters, and a negative value indicates that some samples may have been incorrectly classified.

[0122] The Calinski-Harabasz index (variance ratio criterion) evaluates clustering performance by measuring the ratio of inter-cluster dispersion to intra-cluster dispersion. It is defined according to equation (7):

[0123]

[0124] In the formula, This represents the sum of squares of the centroids of each cluster relative to the global centroid, reflecting the differences between clusters. This represents the sum of squares of each sample within a cluster relative to the cluster centroid, reflecting intra-cluster differences; For the number of clusters, The total number of samples. The larger the index value, the greater the differences between clusters and the closer the clusters are, thus resulting in better clustering performance.

[0125] The Davies-Bouldin index measures the similarity between clusters. For cluster i, its internal dispersion is defined according to equation (8):

[0126]

[0127] In the formula, For clusters The center of mass, This represents the number of samples within a cluster. The cluster spacing is defined as... That is, the Euclidean distance between the centroids of the clusters. For cluster pairs According to the definition in equation (9):

[0128]

[0129] The worst-case separation degree for each cluster is taken as... The final DB index is the average of these values ​​across all clusters. A smaller DB index indicates greater inter-cluster separation compared to intra-cluster dispersion, resulting in better clustering performance.

[0130] like Figure 3 As shown in the figure, the dimensionality reduction method provided in this embodiment of the invention is compared with the dimensionality reduction results of t-SNE. This is used to demonstrate that the present method is superior to t-SNE in maintaining the global structure of the cluster. The red dashed line indicates the position where the comprehensive evaluation index is optimal.

[0131] Figure 3 In the case of cluster numbers of 2 and 5 provided in the embodiments of the present invention, the optimization line graph of the nearest neighbor value n_neighbor using the UMAP method is used. The normalized silhouette coefficient is used to quantify the comprehensive parameter of the combined calculation result of the Calinski-Harabasz index and the Davies-Bouldin index. The normalized silhouette coefficient is used to quantify the combined calculation result of the Calinski-Harabasz index and the Davies-Bouldin index.

[0132] in, Figure 3 Part (a) represents the optimization line graph of the results calculated using the silhouette coefficient as the statistical indicator when the number of clusters is 5. The larger the value, the better the value. Figure 3 Part (c) in the figure represents the optimization line graph of the results calculated using the Calinski-Harabasz index and the Davies-Bouldin index as statistical indicators when the number of clusters is 5. The smaller the value, the better the value. In summary, when the number of clusters is 5, n_neighbor is the best value.

[0133] Figure 3 Part (b) in the figure represents the optimization line graph of the results calculated using the silhouette coefficient as the statistical indicator when the number of clusters is 2. The larger the value, the better the value. Figure 3 The (d) section represents the optimization line graph of the results calculated using the Calinski-Harabasz index and the Davies-Bouldin index as statistical indicators when the number of clusters is 2. The smaller the value, the better the value. Overall, when the number of clusters is 2, n_neighbor is best at 17.

[0134] Step 4: Rerun UMAP with optimal parameters to obtain the optimal low-dimensional embedding data, and use the hierarchical agglomerative clustering algorithm in this low-dimensional space to divide the data into multiple clusters and output cluster labels.

[0135] The hierarchical agglomerative clustering algorithm is used to divide the data into multiple clusters, specifically the Ward algorithm.

[0136] The Ward algorithm tends to generate clusters of uniform size and compact structure. Its objective function (merging variance increments) is calculated as shown in equation (10):

[0137]

[0138] In the formula, and These are two clusters to be merged. and These are their sample sizes. and Each is its own centroid vector.

[0139] Step 5: Use the feature importance assessment method to quantitatively calculate the contribution of the original high-dimensional features to the clustering labels.

[0140] To ensure the clustering results have good interpretability, a dual-validation feature importance assessment method was adopted, which combines SHAP analysis with mutual information calculation to quantitatively reveal the contribution of each feature (i.e., different isotope ratios) in the original high-dimensional data to the clustering labels. This process is also known as white-boxing of the clustering results, thus turning the black-box process of machine learning into a white-box process.

[0141] The calculation of SHAP importance is based on game theory principles and can quantify the marginal contribution of each feature to the model's prediction results. The calculation method is as follows:

[0142] First, a random forest classifier is trained using the original isotopic data as features and the labels generated by the clustering algorithm as targets. Then, the SHAP value of each feature in each sample is calculated using Python's shap library. The global importance of a feature is determined by the absolute average of its SHAP values ​​across all samples, as shown in equation (11).

[0143]

[0144] In the formula, The total number of samples, For the first In the nth sample The SHAP value of each feature.

[0145] The mutual information (MI) is used to measure the amount of information shared between a feature and the target category label, and it can effectively capture the non-linear dependency between them. In this step, the mutual information value between each isotope feature and the cluster label is directly calculated using Python's scikit-learn library, and the calculation formula is shown in equation (12):

[0146]

[0147] In the formula, It is the first Characteristics of an isotope These are clustering labels. It is a joint probability distribution. and It is a marginal probability distribution.

[0148] By combining SHAP and mutual information, two methods that evaluate feature importance from different perspectives, this invention achieves a robust and quantitative interpretation of clustering results, significantly enhancing the credibility and practicality of the model in professional domain analysis.

[0149] A schematic diagram illustrating the results of feature importance interpretation is shown below. Figures 5 to 7 As shown:

[0150] Overall importance of features, such as Figure 5 As shown, the SHAP value and mutual information value are compared. The results show that the most significant influence on the clustering label is... 87 Sr / 86 Sr (SHAP and MI values ​​are both >0.3); followed by ² 06 Pb / ² 04 Pb (all > 0.25); then ¹ 4 ³Nd / ¹ 44 Nd (all > 0.2); while² 07 Pb / ² 04 Pb and² 08 Pb / ² 04 The effects of Pb were all relatively small (all <0.1).

[0151] Feature classification importance such as Figure 6 As shown, the effects of isotopes on different categories are analyzed. For the HIMU type, the greatest influence is ². 06 Pb / ² 04 Pb; for the EM type, the most significant impact is ¹ 4 ³Nd / ¹ 44 Nd.

[0152] The impact of features on the importance of low-dimensional embedded data, such as Figure 7As shown, the influence of isotopes on low-dimensional coordinate axis variables is analyzed. The results show that for UMAP-1 (horizontal axis), the greatest influence is... 87 Sr / 86 Sr, followed by ² 06 Pb / ² 04 Pb; for UMAP-2 (vertical axis), the one with the greatest impact is¹ 4 ³Nd / ¹ 44 Nd, followed by ² 06 Pb / ² 04 Pb.

[0153] Step 6: Draw the decision boundary in the low-dimensional embedding space and build an interpretable classification model in the original high-dimensional space;

[0154] like Figure 4 As shown, to visually demonstrate the separation of different clusters in the dimensionality-reduced space, a linear SVM is used to plot the decision boundary in the UMAP two-dimensional space. Simultaneously, a multinomial logistic regression model is trained in the high-dimensional original isotope space to obtain a system of linear equations that can quantitatively calculate the scores of samples belonging to each category, providing a direct mathematical classification tool for subsequent research.

[0155] Specifically, in the two-dimensional space obtained by UMAP dimensionality reduction, a support vector machine (SVM) with a linear kernel function is used to generate a linear decision boundary, so as to intuitively understand the relative position of the clusters in the dimensionality reduction space.

[0156] Meanwhile, in order to directly apply the clustering results to the classification of the original high-dimensional data and obtain clear discrimination criteria, an interpretable classification model is constructed in the high-dimensional original isotopic two-dimensional space. This is achieved through the following two methods:

[0157] 1) Rule extraction: The decision tree algorithm is used to determine which variables and thresholds determine the cluster affiliation. Multinomial logistic regression is used, and the regularization parameters are optimized through 10-fold cross-validation to construct the decision boundary in the high-dimensional space. The linear discriminant equation set shown in equation (13) is obtained, which provides a quantitative classification basis for subsequent geochemical analysis.

[0158] For a given sample (A vector containing multiple isotope ratios), which belongs to the category Score Calculated by the following linear equation (13):

[0159]

[0160] In the formula, It is an augmented isotopic eigenvector. It is the first Isotope ratios It corresponds to the category The weight coefficient vector.

[0161] 2) Quantitative Classifier Construction: A multinomial logistic regression model is employed to obtain a linear discriminant equation system that can be directly used for calculation and classification. The model automatically optimizes its regularization parameters through 10-fold cross-validation to ensure its generalization ability and stability.

[0162] The high-dimensional data manifold clustering method based on multi-metric optimization provided in this embodiment can not only effectively achieve dimensionality reduction and clustering of high-dimensional geochemical data, but also ensure the objectivity of the results through multi-metric optimization, and provide clear quantitative interpretation through methods such as SHAP and MI, transforming the black-box model into a white-box model.

[0163] In a second aspect, embodiments of the present invention provide a computer-readable storage medium having a computer program stored thereon, wherein the program, when executed, implements the high-dimensional data manifold clustering method based on multi-metric optimization described in any of the first aspects above.

[0164] Thirdly, embodiments of the present invention provide a storage device, including a storage medium and a processor, wherein the storage medium stores a computer program, and when the program is executed by the processor, it implements the high-dimensional data manifold clustering method based on multi-metric optimization described in any of the first aspects above.

[0165] Those skilled in the art will understand that embodiments of the present invention can be provided as methods, systems, or computer program products. Therefore, the present invention can take the form of a completely hardware embodiment, a completely software embodiment, or an embodiment combining software and hardware aspects. Furthermore, the present invention can take the form of a computer program product embodied on one or more computer-usable storage media (including, but not limited to, disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code.

[0166] Obviously, those skilled in the art can make various modifications and variations to this invention without departing from its spirit and scope. Therefore, if these modifications and variations fall within the scope of the claims of this invention and their equivalents, then this invention should also include these modifications and variations.

[0167] Although embodiments of the present invention have been shown and described above, it is understood that the above embodiments are exemplary and should not be construed as limiting the present invention. Those skilled in the art can make modifications, alterations, substitutions and variations to the above embodiments within the scope of the present invention.

Claims

1. A high-dimensional data manifold clustering method based on multi-metric optimization, characterized in that, Includes the following steps: Step 1: Calculate the intrinsic dimension after preprocessing the original high-dimensional isotope data; Step 2: The Unified Manifold Approximation and Projection UMAP algorithm is used to perform nonlinear dimensionality reduction on the preprocessed high-dimensional data to obtain a low-dimensional manifold embedding representation; Step 3: For the key hyperparameters of the UMAP algorithm, a multi-metric optimization strategy and iterative calculation are used to determine their optimal values; Step 3 specifically employs a multi-metric optimization strategy, including the silhouette coefficient, Calinski-Harabasz index, and Davies-Bouldin index, along with iterative calculations, to determine their optimal values, including: Set the fixed parameters of UMAP as follows: n_components=2, metric='minkowski', min_dist=0.1, and set the candidate range of n_neighbors to 5 to 55; For each candidate n_neighbors value, parameter optimization is performed, UMAP dimensionality reduction is executed, and hierarchical agglomerative clustering algorithm is applied to the generated two-dimensional results to calculate the silhouette coefficient, Calinski-Harabasz index, and Davies-Bouldin index. The optimal hyperparameter is selected as the n_neighbors value that maximizes the silhouette coefficient and the Calinski-Harabasz exponent while minimizing the Davies-Bouldin exponent. Step 4: Rerun UMAP with optimal parameters to obtain the optimal low-dimensional embedding data, and use the hierarchical agglomerative clustering algorithm in the low-dimensional space corresponding to the optimal low-dimensional embedding data to divide the data into multiple clusters and output cluster labels. Step 5: Use the feature importance assessment method to quantitatively calculate the contribution of the original high-dimensional features to the clustering labels; Step 6: Draw the decision boundary in the low-dimensional embedding space and build an interpretable classification model in the original high-dimensional space.

2. The high-dimensional data manifold clustering method based on multi-metric optimization according to claim 1, characterized in that, The UMAP algorithm, based on Riemannian geometry and fuzzy topology, preserves both the fine local structure and global topological relationships of the dataset during dimensionality reduction. Specifically, it includes: Step 2.1: Use the Nearest-Neighbor-Descent algorithm to find the nearest neighbor for each high-dimensional data point, construct a weighted neighborhood graph to represent the manifold structure, and finally obtain a fuzzy neighborhood topology graph representing the high-dimensional manifold structure; Step 2.2: Based on the fuzzy neighborhood topology graph, learn the manifold topology in its high-dimensional manifold structure, project it into the low-dimensional space, and obtain the low-dimensional representation of the manifold.

3. The high-dimensional data manifold clustering method based on multi-metric optimization according to claim 2, characterized in that, The method for obtaining the fuzzy neighborhood topology graph in step 2.1 includes the following: Step 2.1.1: Use the Nearest-Neighbor-Descent algorithm to find the nearest neighbor for each high-dimensional data point and identify the nearest neighbor relationships of the high-dimensional data; Step 2.1.2: Based on the established nearest neighbor relationships, construct a weighted neighborhood graph to represent the manifold structure, including: Assuming that the high-dimensional data points are uniformly distributed on the manifold space, the hyperparameter local_connectivity ensures that each point is connected to at least one neighbor. For each point, the association strength with its nearest neighbors decreases with increasing distance, forming a fuzzy neighborhood boundary; To address the issue of inconsistent bidirectional edge weights between point pairs, a probabilistic union approach is used to merge edge weights. Ultimately, a connected fuzzy neighborhood topology graph is obtained, representing the high-dimensional manifold structure.

4. The high-dimensional data manifold clustering method based on multi-metric optimization according to claim 3, characterized in that, Finding the low-dimensional representation of the manifold in step 2.2 includes: In the low-dimensional embedding space, standard Euclidean distance is used, and the minimum allowable distance between embedding points is controlled by the hyperparameter min_dist. The optimization of the low-dimensional manifold representation is achieved by minimizing the cross-entropy loss, and the loss function is shown in the following equation: In the formula, Represents the set of neighbors of a higher-dimensional point. Controlling the balance between the two parts, For high-dimensional space similarity, Similarity in low-dimensional space; Among them, high-dimensional space similarity Defined by the following formula: In the formula, For high-dimensional distance, The minimum radius threshold of the nearest neighbor point. Is it following Adjusted scale parameters, It is the number of nearest neighbors of a given point, n_neighbors; Low-dimensional space similarity Defined by the following formula: In the formula, the parameters Depend on The fitting yields a result that ensures the distance between points in the low-dimensional space is constant. At that time, the similarity decays to half; Finally, the optimization problem of the loss function is solved by the stochastic gradient descent algorithm to obtain the coordinates of each high-dimensional data point in a specified low-dimensional space, i.e., the low-dimensional representation.

5. The high-dimensional data manifold clustering method based on multi-metric optimization according to claim 1, characterized in that, In step 4, the hierarchical agglomerative clustering algorithm is used to divide the data into multiple clusters, specifically the Ward algorithm is used to divide the data into multiple clusters. The Ward algorithm tends to generate clusters of uniform size and compact structure, and its objective function is shown in the following equation: In the formula, and These are two clusters to be merged. and These are their sample sizes. and Each is its own centroid vector.

6. The high-dimensional data manifold clustering method based on multi-metric optimization according to claim 1, characterized in that, The feature importance assessment methods in step 5 include SHAP analysis and mutual information calculation; The formula for calculating SHAP importance is: In the formula, The total number of samples, For the first In the nth sample The SHAP value of each feature; The formula for calculating mutual information is: In the formula, It is the first Characteristics of an isotope These are clustering labels. It is a joint probability distribution. and It is a marginal probability distribution.

7. A computer-readable storage medium having a computer program stored thereon, characterized in that, When the program is executed, it implements a high-dimensional data manifold clustering method based on multi-metric optimization as described in any one of claims 1 to 6.

8. A storage device, comprising a storage medium and a processor, characterized in that, The storage medium stores a computer program that, when executed by a processor, implements a high-dimensional data manifold clustering method based on multi-metric optimization as described in any one of claims 1 to 6.

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