Within 2 hours or less after an earthquake, responsible parties need to use earthquake emergency information for decision-making in highly uncertain situations. Since 2000, China has continuously developed various types of earthquake emergency information decision-making products, but only a few products can effectively aid in emergency decision-making. Therefore, it is necessary to establish a dynamic optimization information product and an evaluation model that considers the needs of the users.

After a moderate to strong earthquake ( 5.0 ≤ M ≤ 7.0 ) occurs in China, within an hour (also known as the decision-making time window), decision-makers in earthquake emergency response scenarios will employ their subjective judgments, emotional experiences, and personal preferences to choose suitable information products, transparently forming an estimation standard for the effectiveness of information (Cheng and Liu, 2010). This assumes that the utility value obtained through an expert scoring method in several typical earthquake disaster events is equivalent to the actual utility value.

The earthquake emergency information selected in this article was obtained from government-affiliated earthquake relief command centers and research institutions. The classification of metadata was based on more than 10 years of earthquake emergency information system services, and this study does not consider the technical investment and economic costs of information in the production and circulation processes.

The 34 selected earthquake disaster events have a spatial range of Sichuan and Yunnan Provinces, a time range of 2010–2021, and a magnitude range of 5.0–7.0. Products with the same name repeatedly appearing in multiple earthquake disasters and from multiple sources were recorded as one product. For products with different names but highly similar content, those with the most recent product were retained. Products produced 24 h after the earthquake were excluded. A total of 100 information products were obtained for statistical analysis.

Based on the theory of utility as a subdivision of information theory (Cheng et al., 1994), we propose that in earthquake emergency decision-making, an evaluation of the decision utility of information involves evaluating the degree to which information flows from the production department to the decision-making scenario and meets the needs of the user. This includes one batch of earthquake emergency information products (hereafter referred to as information products), one production department, one earthquake relief command organization and commander (hereinafter referred to as users), one decision-making scenario, and one set of evaluation methods. Therefore, the evaluation process of information decision utility designed in this study differs from existing evaluation methods by incorporating user needs and the perception of utility. Evaluation is a dynamic, cyclic process that comprises three modes (Figure 1). One is to construct a utility evaluation model, the focus of this article and the foundation of the evaluation process. The second is the “product demand” iterative mode, which is suitable for the emergency preparedness stage. Based on the model, the utility value of information products is quickly evaluated and tested to form a product utility list. The median principle method (Hu, 2011) is used to determine the utility threshold. Products with utility values not lower than the threshold are considered effective products, and the production department uses feedback information to update product information. This process is repeatedly optimized to obtain information products that approximate user preferences. The third is the parameter correction mode. After an earthquake event occurs, users will receive the latest effective information products by obtaining the decision utility value of the information product in the decision-making scenario, expanding the sample size, further correcting the utility parameters, updating the utility evaluation model, and feeding back the information to the production department.

First, information entropy calculation formulas and user demand preferences are separately established, after which user utility values are obtained through expert ratings. Then, data on the three variables is used to establish a regression model. The specific process is as follows.

1 Calculation of information entropy

The information content of a product is the leading indicator for evaluating its utility. The Shannon information entropy is used as a quantitative measure of information in mathematics (Fu, 2015). This study introduces the concept of information entropy to quantitatively measure the information content of products. To accurately measure the content of emergency information products, information is segmented into the smallest effective units, namely, metadata, based on content features.

The Shannon information entropy is calculated by multiplying the various possible amounts of information by their probabilities of occurrence, then summing the results to represent the expected value of the information in the entire data set (Equation 1). H x = − ∑ x ∈ N p x log ⁡ p x (1)

Here, x represents a random variable; p(x) represents the set of all possible outputs, the output probability function, and H measures the information content of the product. The previous text assumed that the probability of meta information appearing in information products is the same, and each type of meta information corresponds to different content features. Therefore, the meta information contained in each information product is positively correlated with the amount of information. Thus, the amount of information in each product can be determined only by the number of such information products, and then calculated using information entropy, i.e., the meta information in the product is exponentially calculated (Equation 2): H i = log 2 ⁡ n u m i (2) where n u m i is the number of types of meta information contained in the information product i.

The original information classification was based on relevant research in the field of earthquakes (Su et al., 2003; Dong and Yang, 2014; Gong et al., 2018). Emergency information products were divided into eight categories: preparation information, earthquake situational information, basic information of the earthquake zone, assessment information, disaster situation, public opinion, emergency action, and intensity of investigation. Feature elements were extracted for each category of products to construct an earthquake emergency metadata table based on 210 categories (Table 1).

In accordance with the above, this study compiled the number of meta information products contained in 100 earthquake information product samples and measured the information content of the product samples using the information entropy H calculation method (Table 2).

2 Calculation of user demand preference

To accurately calculate customer demand preference P, it is necessary to collect user demand preference information and then calculate the corresponding probability based on a Bayesian formula.

Previous studies on the characteristics of earthquake emergency information have been conducted from multiple perspectives, and it is believed that earthquake emergency information products not only possess basic attributes such as nonindependence and sharing of general information but also unique attributes such as content complexity, timeliness, and polymorphic transmission (Wang et al., 2019). Based on the above analysis, in this article, we consider three elements: content, form, and time; we use Bayesian methods to construct a model for calculating user demand preferences.

In the model, the user’s demand for product content is taken as a prior probability, and the formal and temporal probabilities are considered as sample information. The posterior probability of the product is obtained after two rounds of Bayesian inference. The first round of computation is combined with formal probability sample information to produce a posterior probability that then serves as the prior probability for the new round of computation. The second round of computation is combined with the time probability sample information to generate the posterior probability under the product in the earthquake emergency response decision-making scenario.

We collected user preference information through a questionnaire. The valid sample data for this study were provided by 99 member units of the earthquake relief command headquarters in Sichuan and Yunnan provinces. Regarding the 100 information products mentioned earlier, we investigated whether the necessary information and preference for presentation forms such as documents, images, tables, and dynamic images, and preference for obtaining information in the time intervals of 0–2, 2–8, 8–12, and 12–72 h.

Applying the Bayesian formula to obtain the demand probability of products under formal and temporal conditions P I | F | T , constructing a user preference model, the specific calculation process is as follows (Equations 3, 4): P I i | F i k = P I i P F i k | I i P I i P F i k | I i + P I i ¯ P F i k | I i ¯ (3) P I i | F i   k | T i   j = P I i | F i   k P T i   j | I i | F i   k P I i | F i   k P T i   j | I i | F i   k + P I i | F i   k ¯ P T i   j I i | F i   k ¯ (4)

Here, i is the sample space of information products, i = 1, 2, 3, … , 100; k indicates various forms of products, including documents, images, tables, and dynamic images. For the demand time, including the 0–2, 2–8, 8–12, and 12–72 h periods; j is taken as the 0–2 h period in the scenario studied in this article. P I i is the probability of user demand for the product i; P I i ¯ is the probability of demand for products i that are unnecessary; P F i k | I i is the probability of demand for products presented in form; P F i k | I i ¯ is the probability of a demand for form k presentation that does not require a product i; P I i | F i k is the probability of demand when presenting the product i in form k; P I i | F i k ¯ is the probability of demand when the product i is not required to be presented in form k; P T i j | I i | F i k is the probability of demand for products in the required form within a time period, and P T i j | I i | F i k ¯ is the probability of a demand for information within a time period j for product i that does not require a formal form k.

The specific preference values of users for the product in earthquake emergency response decision-making scenarios are shown in Table 2.

3 Obtaining the decision utility value U

To obtain the decision utility values of the 100 earthquake emergency information products mentioned above, we adopted the indirect evaluation method (Yan, 2000). Twelve experts evaluated the comprehensive decision utility of the information products in the given historical earthquake disaster response decision-making scenarios based on their own experience. The experts gave the products a value of 0–10 points, and the average value was calculated. An expert product utility rating table was then formed after integrating the scenarios of 16 earthquake events in Sichuan and Yunnan. The formula for calculating utility value is as follows (Equation 5): U i = ∑ l = 1 M a l i M (5) where l is the set of scoring experts, l = 1, 2, 3, … , M = 12; i is an index of the sample space of information products, i = 1, 2, 3, … , N = 100. The decision utility value a l i is the average rating given by expert l to the product i in 16 earthquake events, and the decision utility value is the average rating given by all experts to the product. The specific values are listed in Table 2.

This study prioritizes parametric simplicity and high interpretability. Similar to the empirical calibration approach employed by Haroon et al. (2017) for model frictional losses and turbulent dissipation in pipelines, our polynomial model enables scenario-driven utility estimation and optimization without requiring the construction of a fully mechanistic framework.

Polynomial-fitting model was adopted to capture the statistical relationships between decision utility and influencing factors. It is a commonly used method for building models of the relationship between independent and dependent variables. According to the number of independent variables, the fitting process can be divided into univariate and multivariate fitting. This process can be further classified as linear and nonlinear fitting (Wang and Ruan, 2004; Huang and Liu, 2021). Here, we construct a univariate polynomial model by analyzing the correlation between product decision utility and user preference, and then add information to conduct a bivariate polynomial model analysis, aiming to obtain a more accurate model that simulates the statistical relationships between decision utility (U), user preference (P), and information (H). Therefore, in this study, the following general formulas for univariate and bivariate polynomials are as follows (Equations 6, 7): f x = ∑ i = 1 n a i x i , i = 1 , 2 , 3 , . . . n (6) f x = ∑ i = 1 n a i x i y j , i 、 j = 1 , 2 , 3 , . . . n , i + j ≤ n (7) where i is the order of the polynomial. When the order of a univariate polynomial is 1, the function is linear.

To quantitatively evaluate the effectiveness of the model, this study used three statistical factors: the correlation coefficient (R2), the root mean square error (RMSE), and bias to evaluate the relationship between actual utility and assessed utility. The formulas are as follows (Equations 8–10): R 2 = ∑ u i − u i ¯ U i − U i ¯ ∑ u i − u i ¯ 2 ∑ U i − U i ¯ 2 (8) R M S E = ∑ i = 1 N u i − U i ¯ 2 N (9) b i a s = ∑ i = 1 N u i − U i N (10) where i = 1, 2, 3,. ., N is the quantity of products; u _i is the simulated utility value estimated using the model, and U _i is the decision utility value of the product. The closer R ² is to 1, and the closer the absolute values of RMSE and bias are to 0, the better the evaluation performance of the model.

In the model, the dependent variables are the user’s preference for the product (P) and the information content of the product (H), and the independent variable is the decision utility of the product (U). The model was evaluated using the software MATLAB, and the descriptive statistics of each variable are shown in Table 3.

The maximum, minimum, and average values of P are 1, 0.7344, and 0.0270, respectively; The maximum, minimum, and average values of H are 5.2095, 2.3414, and 1, respectively; The maximum, minimum, and average values of U are 9.7230, 1.7500, and 6.1027, respectively.

1 The U-P Model

To find the relationship between U and P, we use scatter plot data for polynomial regression fitting and establish a U-P model. The scatter plot (Figure 2a) of product decision utility U and user preference P shows that there was a significant positive correlation between product decision utility and user preference. Therefore, in this study, we attempted to establish a model using a univariate polynomial fitting method, and selected the first-order polynomial (linear) model and the second-order polynomial (nonlinear) model with relatively good performance.

Further analysis shows that the residual distribution of the second-order polynomial model was between −2.44 and 1.92, with 45% of the residuals distributed between −0.5 and 0.5 (Figures 2b,c), which can better reflect the relationship between product utility and user preferences. Therefore, the U − P model relationship can be represented by the following equation (Equation 11): U = c 1 * P + c 2 * P 2 + c 3 (11)

In the formula, c1 and c2 are constant coefficients, and c3 is a constant.

When fitting with a polynomial, the model and statistical factor are as follows (Equation 12): U = 6.509 * P + 1.322 , R 2 = 0.7999 , R M S E = 0.8984 , bias = 0.0002 (12)

Meanwhile, when fitting with a second-order polynomial, the model and statistical factors are as follows (Equation 13): U = 4.525 * P 2 + 1.266 * P + 2 . 387 , R 2 = 0.8205 , R M S E = 0.8509 , bias = 0.0003 (13)

Comparing two models, the second model has a smaller RMSE, it was concluded that expressing the U − P model using a second-order polynomial was more appropriate.

2 The U-P-H Model

Based on the U-P model, with the addition of information content H, due to the scattered distribution of information content and product decision utility U not being concentrated, the correlation between the two is not as strong (Figure 3a). Therefore, the information content H was taken as the joint action term of user preferences P, forming the following model relationship (Equation 14): U = c 1 * P + c 2 * P 2 + c 3 * P * H + c 4 (14)

In the formula, c ₁, c ₂, and c ₃ are constant coefficients, and c ₄ is a constant.

When constructing the utility evaluation model with added information, the model and statistical factor are as follows (Equation 15): U = 0.4723 * P + 2.903 * P 2 + 0.9144 * P * H + 2.307 (15)

The coefficient of determination of the model was 0.8955, the root mean square error RMSE was 0.6492, and the bias was 0.0033, indicating that it exhibited better performance than the U-P model. The evaluation of the U-P-H model indicated that the simulated utility and decision utility of the evaluation were evenly distributed around a straight line with a slope of 1 (Figures 3b,c), with residuals ranging from −1.7868 to 1.6434, and more than 60% of the residuals were distributed between −0.5 and 0.5. The evaluation model results showed no significant overestimation or underestimation, indicating that the model was effective.

3 The Applicability of Polynomial Regression

The objective of this study is to support product prioritization by estimating decision utility. Compared to models like logistic regression that require binarization and incorporate scenario-specific thresholds during training, our method preserves the full resolution of utility scores, enhancing adaptability. Due to the limited sample size and low-dimensional feature space, complex models such as Random Forests or Support Vector Machines carry a higher risk of overfitting and tend to obscure the contributions and monotonic relationships of individual factors, thereby reducing interpretability in time-critical decision-making. In contrast, polynomial regression provides an explicit and interpretable mapping. Empirical results show a clear curvature in the U–P–H relationship, with decision utility (U) increasing as information (H) rises, and the effect becoming more sensitive with higher preference (P), reflecting a monotonic trend rather than absolute magnitude.

To verify that the utility prediction model does not suffer from overfitting and possesses robust generalization capability, a 5-fold cross-validation (CV) procedure was employed. Compared with conventional train–test splitting, k-fold cross-validation effectively reduces data selection bias and makes more efficient use of limited datasets. In this study, the entire dataset was randomly partitioned into ten equally sized subsets. In each fold, one subset was used as the validation set while the remaining nine served as the training set. Importantly, the fixed-structure polynomial regression model was directly applied without re-fitting or parameter tuning; the coefficients used were those previously derived from the empirical fitting process. Three key performance metrics—coefficient of determination (R²), root mean square error (RMSE), and bias—were calculated in each fold to evaluate both the fitting accuracy and generalization ability. The results demonstrate consistent performance across folds, confirming that the proposed model maintains stability and avoids overfitting.

The detailed evaluation results are illustrated in Figure 4, which consists of three subplots:

Figure 4a presents the R² values of both training and validation sets across the five folds. It is observed that the model maintains a consistently high degree of explanatory power, with all training folds yielding R² values between 0.88 and 0.91, and validation folds remaining above 0.88 in most cases. Although slight fluctuations are visible, particularly in fold two and fold four for the validation set, the overall trend suggests that the model does not rely excessively on any specific data subset and performs robustly across different data partitions.

Figure 4b shows the RMSE for each fold, reflecting the prediction error magnitude. The training RMSE ranges mostly from 0.62 to 0.66 while the validation RMSE shows a similar but slightly wider distribution, from approximately 0.42–0.73. Notably, folds one and two demonstrate relatively higher validation RMSE, which might be attributed to local data variability. However, no fold displays extreme deviation, and most errors stay within a reasonable range, suggesting adequate model precision on unseen samples.

Figure 4c analyzes the bias—i.e., the mean signed error—between predicted and actual utility values. The training bias fluctuates narrowly around zero (mostly within ±0.05), while the validation bias exhibits more pronounced deviations in certain folds, such as fold 2 (approx. −0.10) and fold 5 (approx. +0.08). These variations reflect some degree of asymmetry in prediction, yet the overall magnitude remains small, indicating no systematic overestimation or underestimation across folds.

Together, the cross-validation results demonstrate that the model maintains high generalization capability with minimal overfitting. The use of K-fold cross-validation provides an objective and rigorous measure of model robustness. These findings further confirm that the model’s performance is not confined to specific sample partitions.