A multi-dimensional evaluation, attribution and anomaly detection method based on hierarchical Gaussian mixture model and Bayesian chain

Abstract

The application discloses a kind of multidimensional evaluation, attribution and abnormal detection method based on hierarchical Gaussian mixture model and bayesian chain, belong to multidimensional data analysis technical field.This method is with data-driven hierarchical clustering instead of artificial empowerment: first, global Gaussian mixture clustering is executed to obtain comprehensive score, then branch clustering is executed to four sub-dimension groups of institutional environment, macro economy, industrial base and risk management, and geometric mean is obtained to obtain branch score;After six-step Copula bayesian conditional probability chain correction, dimension attribution and missing penalty are realized by shapley value, and double conditional threshold is used to detect anti-consensus signal.In 169 objects, 30-dimensional standard data set, the within-group variance of this method is 0.0185 (about 10 times of equal weight method), ablation experiment confirms that bayesian chain is the core irreplaceable module, and Kendall τ Under 2 four dimension arrangements, it is 1.000.

Description

Technical Field

[0001] This invention belongs to the field of data mining and multidimensional assessment technology, specifically involving a multidimensional object comprehensive scoring method that integrates hierarchical clustering, probabilistic chain reasoning, and attribution analysis. It is applicable to data analysis scenarios requiring comprehensive ranking, attribution explanation, and anomaly pattern detection of a large number of objects across multiple dimensions. This method can be used in any data analysis scenario that requires comprehensive scoring, ranking, contribution attribution, and anomaly pattern detection of a batch of objects from multidimensional observation data. Typical scenarios include, but are not limited to: multidimensional assessment and ranking of the comprehensive development level of a country or region; cross-dimensional comprehensive scoring of corporate credit or investment value; selection and comparison of multiple indicators for projects or programs; and multidimensional risk profiling and abnormal behavior monitoring of customers or counterparties by financial institutions. The method does not limit the specific meaning of the data—as long as the data can be standardized into continuous numerical values, and the assessment objective is to provide a discriminative and interpretable comprehensive ranking across multiple dimensions, this method is applicable. Background Technology

[0002] In the field of multidimensional data evaluation, a common scenario is as follows: given a batch of evaluation objects—which could be enterprises, regions, projects, or any entities that need to be comprehensively compared—each object comes with dozens of observations across dimensions. The analyst's task is to rank them from "best" to "worst" and explain why. This requirement seems simple, but it is fraught with pitfalls at every step.

[0003] The first type is the weighted comprehensive scoring method, typically represented by ICRG (International Country Risk Guide, a multi-dimensional assessment system published monthly by the PRS Group since the 1980s). The approach is straightforward—mapping the raw values ​​of each dimension to scores, then summing them according to a set of weights provided by experts. The idea is direct and simple to implement, hence its widespread application. However, two drawbacks are unavoidable. First, the subjectivity of the weights: changing the set of experts or weights changes the ranking. Second, the conditional dependencies between dimensions are ignored: entities with good institutional conditions usually also have good macroeconomic performance, but the weighted summation treats these two factors as independent contributions, essentially counting the same set of information repeatedly. These two drawbacks combined result in poor discrimination—in typical datasets, the top 10 scores using the ICRG method have a range of only 0.043, barely showing any difference.

[0004] The second category is regression methods, exemplified by gravity models—which model bilateral flows as a function of the size and distance between the two parties. In time-series extrapolation, a certain log-log OLS gravity model achieved a goodness of fit of 0.45, with a correlation coefficient of 0.71 for predicting actual flows. Superficially, this seems impressive. The problem lies in the fact that this performance is almost entirely due to the contribution of the size term: a larger size of the evaluated object results in a larger predicted value and a higher ranking. Structural differences between dimensions—such as the conditional dependence of industrial base on institutional conditions—are glossed over in the regression framework by a linear covariance matrix. More importantly, regression can only answer "what is the ranking," not "why it's ranked here"—the economic meaning of the coefficients is not equivalent to attribution, especially in multidimensional scenarios with severe multicollinearity, where the sign and magnitude of individual coefficients are often unreliable.

[0005] The third type is the equal-weighted scorecard—which directly standardizes each dimension and then averages them equally. This method is the simplest, but its effectiveness is also the most limited. Tests showed that the equal-weighted method had a within-group variance of only 0.0018 on a standard multidimensional dataset, while the method in this invention reached 0.0956, approximately 53 times higher. The root of the difference lies in the fact that when there is a high correlation between dimensions, the equal-weighted method treats highly correlated dimensions as independent information sources, effectively counting the same signal repeatedly. Assuming ten dimensions and a Pearson correlation coefficient of 0.8 between any two, the equal-weighted method actually contributes information from only about 1.6 independent dimensions, yet ranks them according to the nominal precision of ten dimensions. The discriminative power is naturally compressed.

[0006] Looking back at these three methods, there are three common gaps. Gap one: None of them can deduce the conditional dependency structure between dimensions from the data itself—ICRG relies on experts, regression relies on linear assumptions, and equal weighting simply assumes independence. Gap two: None of them can explain "why this object is ranked in this position"—it's easy to assign a score, but difficult to explain how much each dimension contributes. Gap three: They all only focus on "who ranks high," completely ignoring the anti-consensus model of "who, although with a low total score, has an exceptionally outstanding local performance." In financial regulation and anomaly detection scenarios, this third type of information is often more valuable than the ranking itself. The motivation for this invention stems from the convergence of these three gaps—filling any one gap is not enough; a five-stage progressive framework is needed to address all three gaps simultaneously. Summary of the Invention

[0007] Weighted scoring methods have an inherent flaw: the weights are subjective. With the same set of data, changing the experts or the weights will alter the rankings. This invention bypasses weighting directly—it doesn't set weights, letting the data speak for itself. First, Gaussian mixture clustering is performed on all 30 dimensions of observation data, with a fixed number of seven categories—this seven isn't arbitrary; it's a value consistently pointed to by cross-validation of the Bayesian information criterion, silhouette coefficient, and cluster stability. After clustering, approximately 25,000 uniformly random particles are generated in the standardized space. The Mahalanobis distance from each particle to the seven cluster centers is calculated, and the chi-square critical value is used to determine the particle's affiliation. For each evaluated object, its category within the seven categories is determined by the proportion of particles falling within that category's ellipsoid. This yields the first-level comprehensive score—not a weighted sum, but rather "how probable is this object, in the multidimensional joint distribution, that belongs to the currently best-performing category."

[0008] Single-layer clustering has a problem: it's coarse. A global clustering that combines all dimensions tells you that object A and object B are "in the same class," but why are they in the same class? Are they similar in terms of institutional dimensions, industrial dimensions, or everything? It's unclear. The global score erases structural differences between dimensions. The solution in this invention is to perform a second-layer branch clustering independently based on the results of the first-layer clustering, divided into four dimension groups: institutional conditions, macroeconomics, industrial base, and risk management. Branch clustering uses the seven global categories of the first layer as the grouping basis: objects within the same global category undergo another Gaussian mixture clustering within that sub-dimensional group. The score of each evaluated object in that sub-group is a 7:3 fusion of two sub-items: one is the base score weighted by the first-layer global score (maintaining global consistency), and the other is the particle affiliation score of the object in the branch clustering (capturing local structural differences). The geometric mean of the four sub-group scores is then aggregated to form the branch score. The beauty of geometric mean is that if any subgroup scores close to zero, the final score will be significantly lowered—it will not be "masked" by other high-scoring subgroups.

[0009] Existing methods also rely on a more subtle assumption: that dimensions are independent. The quality of institutional conditions is unrelated to macroeconomic performance; the strength of the industrial base is irrelevant to risk management. This assumption does not hold true in real-world data. Ignoring it leads to a systematic overestimation of objects with significant imbalances—their high scores in a few dimensions mask their low scores in most others through simple averaging. The solution proposed in this invention is to overlay a Bayesian conditional probability chain correction on top of the branch scores. Specifically, the six key evaluation steps are chained together in the logical order of "basic conditions → intermediate transmission → final performance." For each step, based on the conditional distribution of the preceding steps, the conditional expectation and conditional variance are derived using the Copula correlation coefficient matrix, and the conditional probability for that step is calculated. The chain probability is obtained by multiplying the six conditional probabilities together. After comparing this with the median chain probability of all objects, it is scaled using a damping factor (0.2) to become the final adjustment factor. The damping factor of 0.2 comes from Sobol global sensitivity analysis—this is the only sensitive parameter in the entire model (total effect index 0.867). Ablation experiments confirmed that after removing the Bayesian chain, the Kendall τ among the top 30 objects dropped from 1.00 to 0.70, and nine of the top ten objects experienced a significant change in ranking of more than five places. The information provided by the Bayesian chain is almost orthogonal to the clustering score—clustering captures "absolute levels," while chain inference discovers "relative equilibrium." A robust test across 24 dimensional permutations also confirmed this: the Kendall τ for any two permutations is 1.0000, and the method is completely insensitive to the order of steps.

[0010] Traditional evaluation methods simply assign a total score and call it a day. However, the total score doesn't answer the question "why." Decision-makers almost always ask, "My score is low, what's holding me back?" This invention uses Shapley values ​​to answer this. It treats the four dimension groups as participants in a cooperative game, iterating through all sixteen non-empty subset combinations—for each combination, all data from dimension groups outside that combination are replaced with the training set mean (i.e., a "masking" operation), and the score is recalculated. The Shapley value for each dimension group is the weighted average of its marginal contribution when it joins all possible alliances. The four Shapley values ​​are normalized and used as discount factors, ranging from 0.50 to 1.00. For objects with missing data, when less than 60% of a dimension group's available dimensions are available, the Shapley discount for that dimension group is activated, and the score is multiplied by the corresponding discount. This mechanism transforms the missing data penalty from "artificially defining a default value" to "deriving the actual marginal importance of that dimension group to the overall ranking from the data." At the attribution level, the Shapley value provides a dimensional-level decomposition of contribution for each object—this is not a "nominal interpretation" of linear regression coefficients, but a counterfactual attribution that has undergone a full combinatorial traversal, and is much more statistically rigorous.

[0011] From clustering to hierarchical decomposition, to Copula dependency chains, to Shapley attribution, to anti-consensus detection—these five stages are not a patchwork, but a progressively interlocking whole. The output of each stage is the input of the next, and the data structure of the previous stage determines what the next stage can do. Without the clustering results of the first layer, the second layer of branch clustering has no basis for grouping. Without branch scores, the Bayesian chain has no correctable baseline. Without robust scores after chain correction, Shapley attribution calculates marginal contributions on inaccurate scores, and the error will amplify along the line. Without Shapley's dimensional attribution, anti-consensus detection can only use a global threshold for hard cutting, resulting in a false positive rate as high as 22 / 23. The core innovation of this invention is not a list of five independent technical points, but rather the integration of them into an inseparable five-stage pipeline according to the data flow direction—removing any one stage will cause the pipeline to leak at the next bend.

[0012] Beneficial effects

[0013] This method is compared item by item with the three existing mainstream methods on a standard dataset covering 169 evaluation objects and 30 dimensions. The summary of each indicator is shown in Figure 8.

[0014] Performance comparison of different evaluation methods

[0015] Equal weighted scorecard 0.0018 0.550 0 Gravity model regression — -0.128 — ICRG Synthesis — 0.471* — This method (complete model) 0.0956 1.000 4

[0016] *ICRG is the reference baseline; ρ is the Spearman rank correlation coefficient.

[0017] (See attached figure 8 for details)

[0018] As shown in Figure 8, this method makes unique contributions in four dimensions: discriminative power, dimensional dependency modeling ability, attribution interpretability, and anomaly detection ability, which are unmatched by existing methods. In terms of discriminative power: the within-group variance of the equal-weighted method is only 0.0018, while the method of this invention reaches 0.0956, approximately 53 times higher; the range of the top 10 scores increases from 0.043 to 0.321, a difference of approximately 7.5 times. In terms of dimensional dependency modeling: ICRG relies on expert weighting, regression relies on linear assumptions, and the equal-weighted method assumes independence; none of these methods derive the conditional dependency structure between dimensions from the data itself—this invention solves this problem through Copula Bayesian chains and 24 permutations with full robustness (τ=1.000). In terms of attribution: ICRG and the equal-weighted method end with a total score, and the regression coefficients are distorted under multi-collinearity—this invention provides a dimensional-level contribution decomposition for each object through 16 full-combination counterfactual Shapley attributions. Anomaly detection: None of the three existing methods have anomaly detection mechanisms. This invention reduces the false positive rate from 22 / 23 to close to 0 through dual-condition anti-consensus detection.

[0019] Deep differences from existing methods

[0020] The root cause of the improved discriminative power is not parameter tuning, but the difference in modeling level. The equal-weighting method repeatedly counts the same information when dimensions are highly correlated—assuming ten dimensions and a Pearson correlation coefficient of 0.8 between any two, the equal-weighting method actually contributes only about 1.6 independent dimensions of information, yet ranks them according to the nominal precision of ten dimensions. Our method's Gaussian mixture clustering inherently possesses a covariance structure between dimensions. The cluster centers, in the joint distribution, seek "the direction in which a group of objects deviates from the mean," rather than a simple sum of deviations from each dimension. Hierarchical decomposition further amplifies discriminative power: global clustering coarsely divides 169 objects into seven major categories, while branch clustering further refines each category locally—this is equivalent to "first dividing into major categories, then finding details within those categories," similar to performing coarse-grained variance decomposition followed by fine-grained residual analysis. The combination of these two approaches naturally increases discriminative power.

[0021] The correlation coefficient with ICRG ranking is positive 0.471, which itself provides another important piece of information: the two methods are neither completely consistent nor completely orthogonal. About half of the ranking consensus comes from the "fundamentals" signal captured by both methods—objects with generally higher dimensionality will always rank higher. The other half of the divergence stems from their different uses of the same set of data: ICRG's weights are externally assigned by experts, while the branch scores of this method are endogenous to the data; ICRG assumes dimensionality independence, while this method captures conditional dependencies through Copula chains. This 47.1% correlation coefficient precisely illustrates that—changing the weighting method does not achieve the effect of this invention; the problem lies in the weighted summation paradigm itself. The correlation coefficient between the equal-weighting method and this invention is 0.550, which is higher because neither introduces external weights. However, the high correlation of the equal weight method is precisely its weakness—it and the "consensus" of this invention come from both being biased towards objects with a high overall level of dimensionality, but it cannot capture the "dark horse" that is locally biased but pulled up by chain reasoning and the "pseudo-excellence that appears balanced but is exposed as inflated by chain reasoning" that this method identifies. Attached Figure Description

[0022] Figure 1: Overall Method Architecture, Flowchart, and Technology Roadmap

[0023] Figure 2: Schematic diagram of confidence ellipsoid, Monte Carlo particle projection, and membership degree of the first layer Gaussian mixture clustering (N=25000, 95% confidence ellipsoid)

[0024] Figure 3: Data flow diagram of Bayesian conditional dependency chain, showing the conditional dependency flow and cumulative probability value of each node in the six-step chain structure.

[0025] Figure 4: Comparison of Dimensional Contribution Decomposition Examples for Shapley Attribution in Various Dimensions

[0026] Figure 5: Comparison of the scatter plot effect of dual-condition screening for anti-consensus signal detection

[0027] Figure 6: A schematic diagram of the topology of a Bayesian conditional dependency chain, showing a unidirectional acyclic directed chain consisting of six nodes. Each node represents a Bayesian conditional inference, and the directed edges represent the analytic dependencies between parameters.

[0028] Figure 7: Heatmap of conditional dependency coefficient matrix based on Gaussian Copula fitting, showing the pairwise conditional dependency coefficients between four dimensions: institutional conditions, macroeconomic conditions, industrial base, and external conditions. The color intensity represents the dependency strength (0.31-1.00), the diagonal self-dependency coefficient is 1.00, and the damping factor is 0.2.

[0029] Figure 8: Performance Comparison of Different Evaluation Methods

[0030] Figure 9: Summary of Sobol Global Sensitivity Analysis

[0031] Figure 10: Bayesian inverse high-frequency dimension statistics

[0032] Figure 11: Ablation Experiment Results Detailed Implementation

[0033] Example 1: Data Acquisition and Preprocessing

[0034] The implementation of this method is illustrated using a real multidimensional dataset covering 169 assessment objects and 30 observation dimensions as an example. The overall process is shown in Figure 1. The data comes from multiple public statistical databases, covering continuous observation indicators in multiple fields such as institutional quality, macroeconomics, industrial base, and risk management. Each record is a cross-sectional data point for an assessment object in a specific year. The dimensions are categorized into four sub-dimensional groups: institutional conditions (including several rule-of-law and governance-related indicators), macroeconomics (including several growth and stability indicators), industrial base (including several structural and efficiency indicators), and risk management (including several openness and external environment indicators). Each record in the dataset represents an observation of an assessment object in a specific year, totaling 2212 valid records, covering cross-sectional observations of 169 assessment objects over multiple years (unbalanced panel structure).

[0035] The preprocessing pipeline consists of three steps. Step 1, Missing Value Diagnosis—For each object and each sub-dimension group, if the available dimensions in that group are less than 60%, the group will trigger a Shapley missing value penalty (score multiplied by a discount factor) in subsequent evaluations. However, the data itself is not imputed to avoid introducing human bias. Step 2, Full-Dimensional Standardization—StandardScaler is used to standardize each dimension to zero mean and unit variance, eliminating dimensional differences. This fitting is performed on the training set, and the parameters are applied to all evaluation objects after being solidified to prevent data leakage. Step 3, Outlier Handling—Outliers are not deleted; instead, they are naturally weighted during clustering using Mahalanobis distance: outliers are naturally located away from all cluster centers in the standardized space, and their particle affiliation is automatically lower, eliminating the need for additional hard threshold truncation.

[0036] Example 2: Clustering and Particle Attribution Calculation

[0037] The first-level clustering—the global Gaussian mixture model—is performed in a joint space comprised of all 30 standardized dimensions. A full covariance matrix is ​​used (each cluster center possesses a complete inter-dimensional covariance structure), the number of categories is fixed at seven, and no regularization term is applied. The expectation-maximization algorithm is used for fitting, with a convergence threshold of 0.0001 (i.e., termination occurs when the difference in log-likelihood between two adjacent iterations is less than 0.0001). K-means++ is used for initialization to reduce the risk of local optima, and after multiple random initializations, the algorithm with the highest log-likelihood is selected.

[0038] The selection of the number of clusters underwent systematic cross-validation. For a candidate range of two to fifteen clusters, five indices were calculated: Bayesian Information Criterion (BIC), Aspect-Oriented Identification Criterion (AIC), silhouette coefficient, Gap statistic, and Jaccard bootstrap stability. BIC and AIC exhibited an elbow at cluster seven—the likelihood gain from adding new clusters after seven was offset by parameter penalties. The silhouette coefficient reached a local maximum of 0.153 at cluster seven. The Gap statistic exceeded the gap plus one standard error of cluster six at cluster seven, but failed to exceed the gap plus one standard error of cluster seven at cluster eight. Jaccard bootstrap stability (calculated by resampling 100 times, retaining 80% of the samples for each clustering) reached 0.601 at cluster seven, but plummeted to 0.72 at cluster eight. All five indices consistently pointed to cluster seven, therefore K=7 was adopted.

[0039] For each object to be evaluated, its standardized 30-dimensional vector is substituted into seven Gaussian probability density functions to calculate the log-likelihood value. The absolute value of the log-likelihood itself is not very meaningful—it is affected by the determinant of the covariance matrix and the number of dimensions. What is informative is its relative position in the full reference distribution. Therefore, the object likelihood value is mapped to the percentile of the likelihood values ​​of all 2212 records in the training set to obtain the belonging score. A more precise approach is Monte Carlo particle simulation (as shown in Figure 2): in the standardized space, the minimum value of each dimension of the training set is used as the lower bound and the maximum value as the upper bound, and each is extended outward by 0.5 standard deviations to generate 25,000 uniform random particles within the resulting hyperrectangle. The squared Mahalanobis distance from each particle to the seven cluster centers is calculated. If this value is less than the critical value of the chi-square distribution at the corresponding degree of freedom (30) and the 95% confidence level, then the particle is considered to "fall into" the confidence ellipsoid of the cluster center. The degree to which each object belongs to each category is equal to the number of particles falling within that category's ellipsoid divided by the total number of particles. The category with the highest degree of belonging is the cluster to which the object belongs. The first-level comprehensive score P1 is obtained by weighted averaging of the degree of belonging.

[0040] Example 3: Layered Decomposition and Branch Scoring

[0041] Branching projection is executed independently in four sub-dimensional groups, with Monte Carlo particle simulation at its core. Each sub-group is treated as a unit: in the normalized space, the minimum value of each dimension in the training set is used as the lower bound, and the maximum value as the upper bound, with each bound extended outward by 0.5 standard deviations as the boundary for uniform sampling. This "expansion by half a standard deviation" has a clear purpose—if only the extreme values ​​of the training set are used as boundaries, if any evaluation object has an extreme value exceeding the training set range in a certain dimension, the particle uniform sampling space will not cover it, leading to bias in the attribution calculation. After expanding by half a standard deviation, new objects within the normal fluctuation range can be covered.

[0042] 25,000 uniformly random particles were generated within the hyperrectangle. The choice of particle number underwent convergence testing: the entire pipeline was run independently with 1,000, 5,000, 10,000, 25,000, 50,000, and 100,000 particles respectively, and the ranking of the evaluation objects under each particle number was calculated and compared with the baseline ranking of 100,000 particles. The Kendall τ of 10,000 particles reached 0.994, and that of 25,000 particles reached 0.999—further increasing the number of particles has a marginal improvement in ranking approaching zero, so 25,000 was chosen as the default value. Sobol analysis also confirmed that the particle number is insensitive, with a total effect exponent of only 0.007.

[0043] For each category in the clustering results of each subgroup, the squared Mahalanobis distance from each particle to the category center is calculated. The advantage of Mahalanobis distance is that it naturally incorporates the inter-dimensional correlation structure into the calculation—two particles that are equidistant in the original Euclidean space will have a shorter Mahalanobis distance if one of them is located along the principal axis of the covariance matrix (high-density direction), and vice versa. After this natural weighting, particles falling into the core region of the cluster receive a higher belonging weight. The score of each evaluated object in the subgroup is a fusion of two sub-items: the cluster belonging score (proportion within the particle) is weighted at 0.7, and the first-level global score is weighted at 0.3. This 7:3 weighting is not subjectively set—Sobol global sensitivity analysis scans the weights as continuous parameters across the entire range of zero to one, and the results show that the total effect exponent of the weight parameters is only 0.042. As long as neither side is made completely zero, the deviation of the final ranking Kendall τ under any weight ratio from the benchmark (7:3) does not exceed 0.051. 7:3 is just a convention.

[0044] The scores of the four subgroups are taken as a geometric mean and aggregated into a branch score P2. The use of the geometric mean instead of the arithmetic mean is carefully considered: any subgroup score close to zero will be carried over to the final score through multiplication and will not be "masked" by other high-scoring subgroups. This property is crucial in multidimensional evaluation scenarios—an object that is extremely weak in institutional conditions but performs reasonably well in other dimensions might be "saved" under the arithmetic mean, but will have nowhere to hide under the geometric mean. Based on the principle of conservative evaluation, the geometric mean is more in line with the practical logic that "the weakest link determines the overall capacity of the system."

[0045] The number of clusters in each subgroup is determined by two constraints: it cannot exceed the subgroup's dimension (i.e., three dimensions) to prevent overfitting in low-dimensional space; and it cannot exceed one-tenth of the number of training samples in the subgroup to ensure statistical support. For subgroups with very few samples, the number of clusters degenerates to one—at this point, the particle ellipsoid determination degenerates into multivariate normal outlier detection: a Mahalanobis distance exceeding the chi-square critical value indicates an outlier, and the membership score decays with distance. The confidence level of the confidence ellipsoid is uniformly set to 95%. If it is too small (e.g., 68%), the ellipsoid is too tight, and the membership score is overly sensitive to sampling noise; if it is too large (e.g., 99.7%), the ellipsoid is too loose, encompassing almost all particles, and the score loses its discriminative power. Sobol analysis confirms that this parameter is insensitive (total effect exponent is only 0.038).

[0046] Example 4: Conditional Dependency Chain Correction

[0047] The topology of the conditional dependency chain is shown in Figure 6, which is a unidirectional acyclic directed link: Node 1 (policy approval environment) → Node 2 (financing availability) → Node 3 (site selection and land acquisition conditions) → Node 4 (compliance and environmental protection requirements) → Node 5 (construction capability) → Node 6 (operation and management level). Each directed edge represents the conditional constraint of the previous node's score on the expected value of the next node, and the edge weight is analytically derived from the conditional distribution of the Copula correlation coefficient matrix (as shown in Figure 7). This structure reflects the real-time logic of investment project implementation: policy approval is the entry point, financing depends on policy permission, site selection and land acquisition are constrained by the financing scale, compliance standards are affected by local regulations of the site selection, construction capability is constrained by compliance requirements, and operation and management are ultimately affected by the cumulative impact of all the aforementioned links. The specific approach is as follows. The six steps are arranged in the logical order of the investment process: ① policy approval → ② financing → ③ site selection and land acquisition → ④ compliance and environmental protection → ⑤ construction and construction → ⑥ operation and management, forming a conditional dependency chain. Each step is associated with a set of dimensional factors. The percentile of the target object on these factors is taken as the mean, and linearly mapped to the interval between 0.50 and 0.95 to obtain the marginal probability P_step_i for that step. Four sets of ViCo GaussianCopula correlation coefficient matrices (a total of 16 Copula pairs) are imported. For each step, using the score vector of completed steps as a condition, the current conditional expectation of that step is calculated using the multivariate normal conditional distribution formula, and then weighted and fused with the marginal probability to obtain the conditional correction probability for that step: P_cond_i = (1-λ)P_step_i + λ·E[P | score of previous steps], where λ is the conditional weight, increasing with the cumulative number of conditions. The chain probability is obtained by multiplying the six conditional probabilities: P_chain = ∏ P_cond_i.

[0048] To ensure the positive semidefiniteness of the correlation coefficient matrix in high-dimensional scenarios (e.g., 30-dimensional), the Ledoit-Wolf shrinkage estimation is used to regularize the sample covariance matrix, and the shrinkage coefficient is determined by minimizing the Frobenius loss. Likelihood maximization employs the BFGS quasi-Newton algorithm, with convergence criteria being a gradient norm less than 10 or a log-likelihood change of less than 10 over two iterations.

[0049] The mapping relationships between each step and the dimensional factors (as shown in Figure 3) are as follows: Step ① (Policy Approval) is associated with 6 institutional dimensions—political stability, corruption index, democracy index, WGI corruption control, WGI rule of law index, and WGI discourse power; Step ② (Financing) is associated with 7 macro-financial dimensions—GDP growth rate, GDP per capita, FDI inflow ratio, exchange rate, external debt / GNI, government debt / GDP, and foreign exchange reserves; Step ③ (Site Selection and Land Acquisition) is associated with 4 scale and infrastructure dimensions—total GDP, labor force, electricity coverage, and total population; Step ④ (Compliance and Environmental Protection) is associated with 5 institutional and development dimensions—corruption index, tax / GDP, human development index, WGI government efficiency, and WGI regulatory quality; Step ⑤ (Construction) is associated with 3 industry dimensions—industry / GDP ratio, fixed broadband coverage, and electricity coverage; Step ⑥ (Operation and Management) is associated with 4 trade openness dimensions—GDP per capita, trade / GDP ratio, exports / GDP, and imports / GDP. The various dimensions can be reasonably reused across different steps (e.g., GDP per capital appears in both financing and operational management steps, and CPI score appears in both policy approval and compliance and environmental protection steps), reflecting the different meanings of the same dimension at different investment stages.

[0050] The chain probability itself is not the final adjustment factor. The ratio = P_chain / P_median is calculated using the median chain probability of all objects as the baseline. If the ratio is greater than one, it means that the object performs better than "average level" in chain reasoning and should be adjusted upward; if it is less than one, it should be adjusted downward. However, directly multiplying by the ratio is too aggressive - the probability of an extremely unbalanced object may be close to zero in one or two steps, causing the ratio to be close to zero and the score to be "reset". A damping factor D (taken as 0.2) is introduced: adjustment = 1 + D × (ratio 1). The damping factor reduces the influence of the ratio to twenty percent of the original range, ensuring that the score of an extremely unbalanced object will not be over-penalized due to one or two abnormal steps. (3)

[0051] The damping factor D is the only parameter in the entire method that requires adjustment. The Sobol global sensitivity analysis scanned D between 0.05 and 0.50, achieving a total effect index of 0.867, while the confidence level (0.038), fusion weight (0.042), correlation coefficient (0.018), missing number penalty (0.029), and indicator factor (0.006) were all insignificant. This, in turn, validates a key judgment in the method design: the Bayesian chain is not an added-on module, but rather the most information-dense node in the entire pipeline. The Sobol analysis is summarized in Figure 9.

[0052] Summary of Sobol Global Sensitivity Analysis

[0053] Damping factor D 0.867 Unique sensitive parameter Fusion weights (7:3) 0.042 Not significant Confidence ellipsoid level (95%) 0.038 Not significant Missing number penalty 0.029 Not significant Copula correlation coefficient 0.018 Not significant Wind vane factor 0.006 Not significant

[0054] (See attached figure for detailed Figure 9)

[0055] Data from the ablation experiment: After removing the Bayesian chain (i.e., setting all chain adjustment factors to one), the Kendall τ among the top 30 objects dropped from 1.00 to 0.70, and the rankings of nine of the top ten objects changed significantly (more than five places). This result shows that the ranking information provided by the Bayesian chain is almost orthogonal to the clustering score—clustering captures "absolute levels," while chain inference discovers "relative equilibrium." There is no overlap between the two.

[0056] Robustness test of dimensional order: There are twenty-four possible permutations for the four sub-dimensional groups. The complete pipeline is run for each of the twenty-four permutations, calculating the ranking for each. Kendall's τ is 1.0000 for any two permutations—completely consistent. The method is completely insensitive to the order of steps. This property stems from the commutativity of the Copula conditional distribution: the joint probability density is decomposed into a product of conditional probabilities regardless of the decomposition order. With finite samples and discretization truncation errors, the small numerical differences caused by different orders are further smoothed out by a damping factor.

[0057] It should be noted that the 24-permutation result in τ = 1.000000 is not an experimental coincidence, but rather an inherent mathematical property of the method. This is because the marginal probability P_step_i of each step in the six-step chain depends only on its respective associated dimensional subset, with no overlap. Regardless of the permutation order, the six marginal probabilities are completely consistent, and the product chain probability ∏P_step_i naturally remains constant. The Copula conditional expectation term is added as an additional adjustment term to the marginal probabilities in this method, and its value also depends only on the distribution of the associated dimensional subsets at each step, regardless of the step order. Therefore, the final ranking of any permutation is completely consistent—this is a natural consequence of the commutativity of the chain product structure, not a random experimental phenomenon.

[0058] Example 5: Shapley Attribution and Result Output

[0059] The Shapley value attribution (as shown in Figure 4) serves two functions: in the evaluation phase, it applies a discount penalty to objects with missing data; in the output phase, it provides dimensional interpretability for each object. The calculation process is as follows: The 30 objects with the highest data completeness are selected as the attribution baseline set. For the four sub-dimensional groups, all sixteen non-empty subsets (including the empty set, a total of sixteen "coalition" combinations) are traversed. For each combination, all data from the dimensional groups outside that combination are replaced with the mean of that group in the training set (i.e., a "masking" operation), and the first-level clustering and score calculation process is rerun. The Shapley value for each dimensional group is equal to the weighted average of the score increments when that dimensional group is added to all coalitions that do not contain it, with the weight being the reciprocal of the number of combinations of that coalition size. The four Shapley values ​​are normalized and mapped to discount factors, ranging from 0.50 to 1.00—0.50 for the worst-case dimensional group contribution and 1.00 for the best. The discount range is set to 0.50 instead of 0 to prevent the propagation effect of scores dropping to zero.

[0060] Final composite score: P_final = P_fused × chain_adjustment × Πpenalty_i, where P_fused = P1 × 0.70 + P2 × 0.30. The range of P_fused is controlled between 0.02 and 0.95, the range of chain_adjustment is approximately between 0.85 and 1.15, and the range of penalty_i is between 0.50 and 1.00. After multiplying the three, the final score is distributed between 0.008 and 1.09, with a median of approximately 0.25 and an upper quartile of approximately 0.42. (2)

[0061] The output consists of four parts: ① A P_final ranking table of all evaluated objects (including ranking, score, and cluster category); ② Shapley attribution decomposition of each object in the four sub-dimensions (including contribution value and contribution percentage of each dimension); ③ Detailed score of each item in the six steps of the Bayesian chain (marginal probability, conditional probability, and cumulative chain probability of each step); ④ A list of objects marked as anti-consensus signals and details of specific triggering conditions.

[0062] Example 6: Detection of Anti-Consensus Signals

[0063] Anti-consensus signal detection—an optional but highly informative add-on module. Its motivation stems from a common data analysis dilemma: traditional comprehensive rankings invariably prioritize those with balanced performance across all aspects—which is certainly reasonable, but analysts are often truly interested in those "undervalued by the overall ranking but excelling in specific areas" anomalies. In financial regulation, if an institution with high overall risk only shows exceptional performance in a single dimension (such as industry concentration)—this "uneven strength" is itself a signal worth exploring, but the comprehensive ranking will suppress it, and analysts may completely miss it.

[0064] The detection is achieved through a dual-condition threshold, as shown in Figure 5. Condition 1 (Low Consensus Zone Threshold): The overall score P1 of the first layer must be lower than the left-tail threshold determined based on extreme value theory. This is achieved by fitting a generalized Pareto distribution to the left tail (lowest 15% interval) of the full distribution of P1, and using the fifth percentile as the threshold. In a typical dataset, the measured value is approximately 0.10. This means that only the "hardest" objects, those with the worst overall scores in the bottom 10%, will be detected—objects with already good overall scores, even if significantly increased by chain correction, are not considered "anti-consensus," but rather "should be high to begin with." Condition 2 (Significant Correction Threshold): The positive difference between the final score P_final after chain correction and P1 must exceed the median of this difference for all positively corrected objects. In a standard dataset, the measured median is approximately 0.062. Condition 2 eliminates false candidates that are "low-scoring but the chain correction doesn't lift them up much"—they are indeed low in the global distribution, but the Bayesian chain does not consider them to be underestimated. The chain correction is very small, indicating that all dimensions are indeed relatively weak, not "local flashes".

[0065] The two conditions are logically ANDed. The old single-condition scheme (with only a lenient threshold of P1 below 0.35) labeled twenty-three objects on the standard dataset, of which twenty-two were confirmed as false positives after manual verification of each dimension: their scores were balanced across dimensions, and their low global ranking was due to their overall weak absolute level, not constituting the "low consensus, high local potential" pattern. After the dual-condition hardening, only one object was labeled—this object had the third-lowest P1 globally (approximately 0.08), but after chain correction, P_final jumped to the global median level (approximately 0.30), and manual verification confirmed that it performed well in multiple indicators in the industry foundation sub-dimensional group. The false positive rate was compressed from 22 / 23 to close to 0.

[0066] As an optional extension, Bayesian inverse analysis provides another observation window. From the posterior distribution of the global Gaussian mixture model, Monte Carlo sampling was performed according to the mixture weights of each of the seven classes, generating approximately 10,000 virtual objects. Each virtual object underwent a full evaluation pipeline, and approximately 590 "high-scoring virtual objects" were selected based on their P_final ranking within the top 5.9% (corresponding to approximately the top ten percentiles in the real data). The frequency distribution of these high-scoring objects in the highest percentile of each of the four sub-dimensional groups was statistically analyzed, and the results are summarized in Figure 10.

[0067] Bayesian inverse high-frequency dimension statistics

[0068] Institutional conditions Approximately 23% Approximately 14% -9pp Macroeconomics Approximately 27% Approximately 12% -15pp Industrial base Approximately 37% Approximately 66% +29pp external conditions Approximately 13% Approximately 8% -5pp

[0069] (See attached figure for detailed Figure 10)

[0070] The frequency of the industrial foundation dimension jumped from about 37% to about 66%, an increase of nearly 30 percentage points. A common characteristic of the high-scoring subjects is their outstanding performance in the industrial foundation dimension—this finding was not pre-planned, but emerged from the data, providing a quantitative basis for subsequent subject profiling and dimension weight optimization.

[0071] Example 7 Verification Experiment

[0072] The ablation experiments are summarized in Figure 11. For the standard dataset (169 objects, 30 dimensions), individual modules were removed sequentially and the rankings were recalculated, then compared with the rankings of the complete method using Kendall τ.

[0073] Ablation test results

[0074] None (complete model) — 1.000 — Shapley attribution GMM+Layered+Copula Chain 1.000 0 Anti-consensus detection GMM + Layered + Copula Chain + Shapley 1.000 0 Layered disassembly GMM+Copula chain 0.921 0.079 Bayesian chain correction GMM + Layering + Shapley + Anti-Consensus 0.700 0.300 GMM clustering Layered + Copula Chain 0.563 0.437

[0075] (See attached figure for detailed Figure 11)

[0076] Conclusion: The Bayesian chain is the only irreplaceable core component. Removing it caused a sharp 30% drop in ranking consistency, with 90% of the top ten objects experiencing significant ranking changes. While the removal of branch scores was not as drastic as the chain component, the within-group variance plummeted from 0.0956 to 0.0035 (a decrease of approximately 27 times), confirming that hierarchical decomposition is the primary source of discriminative power. The Shapley attribution discount only affected about 10% of objects with missing data, having a limited impact on the overall ranking; its value lies not in ranking but in attribution explanation. Anti-consensus detection does not affect ranking and is an independent signal discovery module. In the temporal extrapolation validation, using 2212 records from 2022 and earlier years as the training set, after calibrating all model parameters, the target ranking of the same batch of objects was calculated on observation data from the following year (2023), which was not used in training. The results show that the Kendall τ for the ranking during training and the ranking during extrapolation is +0.87, which is a significant improvement compared to the same-period repeatability τ=0.31 of the equal-weight method, indicating that the model has robust extrapolation capabilities in the time dimension.

[0077] In terms of comprehensive comparison: The comparison results of ICRG weighted scoring, equal-weighted scoring cards, and the method of this invention were run on the same dataset. These results are fully presented in Figure 8 of the "Beneficial Effects" section above and will not be repeated here. The correlation coefficient with ICRG ranking is positive 0.471, and the correlation coefficient with the equal-weighted method ranking is 0.550. A detailed comparison of the within-group discrimination, top 10 range, and anti-consensus detection capability of the three methods is also shown in Figure 8.

Claims

1. A multi-dimensional evaluation, attribution, and anomaly detection method based on a hierarchical Gaussian mixture model and Bayesian chains, characterized in that, Includes the following steps: The first layer of Gaussian mixture clustering is performed on the multi-dimensional raw data to divide the evaluation objects into several reference groups. The Monte Carlo particle sampling method is used to calculate the belonging degree of each evaluation object in the reference groups to obtain the first layer comprehensive score. The multi-dimensional data is divided into several dimension groups according to domain attributes. Based on the results of the first layer of Gaussian mixture clustering, the second layer of Gaussian mixture clustering is performed on each dimension group. The particle belonging degree score and the first layer comprehensive score are fused to obtain the value of each dimension component. The geometric mean of the values ​​of each dimension component is taken as the branch score. The conditional probability of each step is recursively corrected for the dimension scores of each step using the Copula correlation coefficient matrix. The conditional probability of each step is derived from the correction score of the previous step through a multivariate normal conditional distribution. After smoothing with a damping coefficient, the chain correction factor is obtained and multiplied by the branch score to obtain the corrected evaluation value. The Shapley value attribution is used to perform missing penalty and contribution decomposition on each dimension group. The object with low score but high correction is marked with an anti-consensus signal by a two-condition threshold.

2. The method according to claim 1, characterized in that, The Monte Carlo particle sampling method includes: generating uniform random particles in a standardized space; calculating the Mahalanobis distance from each particle to the center of the group for each group obtained by the first layer of Gaussian mixture clustering; if the square of the Mahalanobis distance is less than the chi-square distribution critical value of the corresponding degree of freedom and the preset confidence level, then the particle is determined to fall into the confidence ellipsoid of the group; the degree of belonging of each evaluation object to each group is equal to the number of particles falling into the confidence ellipsoid of the group divided by the total number of particles.

3. The method according to claim 2, characterized in that, In the Monte Carlo particle sampling method, the number of particles is determined by a convergence test, preferably 2500; the preset confidence level is 95%; and the fusion weight of the first layer comprehensive score and the particle affiliation score is 7:

3.

4. The method according to claim 1, characterized in that, In the recursive correction of the conditional probability, the conditional expectation at each step is derived from the Copula correlation coefficient matrix using the conditional distribution formula of the multivariate normal distribution. The Copula correlation coefficient matrix is ​​a Gaussian Copula based on the maximum likelihood estimation of the training dataset, and its multivariate normal conditional distribution formula is μ_cond = μ2 + Σ_{21}·Σ_{11}^{-1}·(z1 − μ1), where Σ 11 Σ 21 These are the covariance block matrices of the conditional variable and the conditionally affected variable, respectively; the damping coefficient ranges from 0.05 to 0.50, preferably 0.

2.

5. The method according to claim 1, characterized in that, The number of categories in the first layer of Gaussian mixture clustering is determined by cross-validation, preferably 7 categories; each category of the first layer clustering result corresponds to a combination pattern of dimensional features.

6. The method according to claim 1, characterized in that, The Shapley value attribution step includes: treating each dimension group as a participant in a cooperative game, traversing all subset combinations of dimension groups, recalculating the first-level comprehensive score for each combination, obtaining the marginal contribution of each dimension group when joining each combination, and obtaining the attribution weight of each dimension group to the final comprehensive score after weighted averaging according to the alliance size.

7. The method according to claim 1, characterized in that, The anti-consensus signal detection step includes: when the object to be evaluated simultaneously meets (a) the first-layer comprehensive score is lower than the left tail threshold determined based on the extreme value theory, and (b) the positive difference between the final comprehensive score and the first-layer comprehensive score is greater than the median of the global positive difference, the object is marked as an anti-consensus signal.

8. The method according to claim 4, characterized in that, The damping coefficient was determined through Sobol global sensitivity analysis: within the damping coefficient range of [0.05, 0.50], the total effect index S_T of the damping coefficient was calculated using Kendall τ ranked in the Top 30 as the output variable, and the damping coefficient value corresponding to the peak value of S_T was taken as the optimal value; the current calibration result is 0.2, and the total effect index S_T is 0.867, which is the only significantly sensitive parameter among the 6 parameters.

9. The method according to claim 1, characterized in that, The fusion weight of the clustering affiliation score within each dimension group and the global score of the first layer is 7:

3. The final comprehensive score P_fused is obtained by fusing the comprehensive score P1 and the branch score P2 of the first layer according to an empirical ratio.

10. A multi-dimensional data comprehensive evaluation system based on the method of any one of claims 1 to 9, characterized in that, include: The data preprocessing module is used to standardize multi-dimensional raw data; The clustering scoring module is used to perform the first-level global Gaussian mixture clustering, Monte Carlo particle affiliation calculation, and the second-level branch clustering and geometric mean fusion based on the first-level clustering results; the chain correction module is used to perform multi-step conditional probability recursive correction through the Copula correlation coefficient matrix; the attribution module is used to calculate the attribution weight and missing discount of each dimension group by traversing the entire combination through the Shapley value. The detection module is used to filter anti-consensus signal objects through a dual-condition threshold to achieve anomaly pattern detection. Specifically, the output of the data preprocessing module is connected to the input of the clustering scoring module; the output of the clustering scoring module is simultaneously connected to the inputs of the conditional dependency chain correction module and the Shapley attribution module; the output of the conditional dependency chain correction module is connected to another input of the Shapley attribution module to provide a corrected score after Copula conditional expectation adjustment; and the output of the Shapley attribution module is the final comprehensive score and dimensional attribution decomposition result.

Images