Food Safety Monitoring Indicators and Early Warning Level Setting Based on Time Series Analysis

As we all know, food is necessary for people’s survival and development, so it is not difficult to conclude that the importance of food safety is equal to the importance of human health and life safety. Nowadays, the good development of national economy has begun to be gradually affected by food safety incidents, and people are increasingly concerned about food safety [1]. Foodborne diseases have become one of the six major threats to public health and safety faced by mankind at the beginning of the 21st century. Relevant studies have shown that in industrialized countries, the number of people suffering from foodborne diseases is increasing, and the increase can reach 30% per year. Due to the continuous enrichment of China’s material life, China has entered a period of food safety problems since 1998, and the number of important large-scale food poisoning or deaths has been on the rise [2]. Nowadays, people are striving to develop their economy and are pursuing a high quality of life, but along with this, more and more food safety related problems have come to light. Many unfamiliar terms such as Sudan red and melamine are known to more people. The occurrence of more and more food safety incidents in China and abroad has brought not only harm and fear to the people, but also the people have begun to pay attention to and learn how to recognize whether the food is safe or not compared to the previous superficial concept of food safety [3]. The food industry is arguably the industry with the highest gross national product, accounting for a large part of China’s economic progress and foreign trade. Then, in recent years, the reputation of the country’s food industry has been affected by one incident after another of food safety problems. The issue of the safety of the food supply has become increasingly serious and important due to the use of more sophisticated scientific and technological means in the production of food and the internationalization of the food supply. There are many measures that reflect our country’s clear attitude and strong determination to address food safety issues [4]. Food safety will be a long-term problem, and the management system of food safety is being improved under the adjustment of national policies. Food from the output to the consumption of this long-term complex process, some of the existing early warning system has been limited to its different objects and scope, the amount of information involved is insufficient, the comprehensive utilization rate is very low, mainly used only in the middle of the retail settlement, it is difficult to be used in the food supply chain throughout the beginning and end. Therefore, to set food safety monitoring indicators and early warning levels, the establishment of a combination of management, monitoring, risk assessment, prevention, control and regulation in one of the food safety early warning system to achieve early warning linkage will realize the effective management of a large number of data resources, but also the implementation of national policies, laws and regulations of the need [5-8].

This paper takes infant milk powder food safety as an example, conducts research on food safety monitoring indicators and early warning level setting, selects food safety monitoring indicators based on relevant literature, cases and expert interviews, and utilizes the improved hierarchical analysis method based on the three-scalar method to determine the weights of the indicators, so as to obtain the comprehensive index of food safety monitoring indicators. The ARIMA model and quadratic exponential smoothing model were constructed to predict the comprehensive index of food safety by analyzing the time series data.And compare the prediction effects of the two models. Finally, select the quadratic exponential smoothing method model, and set a reasonable warning level to ensure food safety monitoring and early warning.

2

Overview

The level of food safety monitoring and inspection is not only an important symbol of a citizen’s standard of living as well as the government’s ability to provide public safety services, but also the most important indicator in the international trade of food. Literature [9] highlights the seriousness of food safety problems and tries to establish a food safety early warning model through regional food safety standards and risk analysis to provide a theoretical basis for food safety research in China. Literature [10] verified that artificial intelligence, big data and internet of things can be used as food safety early warning and emerging risk identification tools and methods through practical cases, and they can improve the food system’s ability to resist food safety risks and provide technical support for the development of future food safety risk early warning systems. Literature [11] proposed a food safety risk early warning model based on the combination of entropy weight-based hierarchical analysis and self-coder-recursive neural network, and experimental analysis was carried out by taking the test data of a dairy brand in China as an example, and the results showed that the proposed model has practicability, and it can reduce the incidence of food safety events. Literature [12] takes the food safety testing data of chemical hazards as an example, and experimentally verifies that the data mining method is easier to obtain the warning information therein than the traditional statistical method, so as to facilitate the establishment of an effective early warning system to assist the regulatory authorities and related companies to ensure food safety and quality. Literature [13] proposes a food safety risk early warning and control method that combines the quality control analysis method and the improved hierarchical analysis method, and validates the effectiveness and reliability of the proposed method by taking the dairy product safety data in Guizhou Province, China, as an example, and researches the risk management technical support for the relevant quality inspection departments. Literature [14] designed an automatic food safety early warning system for dairy supply chain based on machine learning and verified the validity and feasibility of the proposed system by actual case study, and the study provides opportunities for food producers or inspectors to take timely measures to prevent food safety problems. Literature [15] proposed an improved clustering-radial basis function neural network based on hierarchical analysis and entropy weight method for food safety risk early warning and control, and empirically analyzed the meat products testing data in China as a case study, and the results showed that the proposed method can help the food safety regulatory authorities to control the food safety risks.

With the development of the economy, all kinds of foodborne safety problems emerge. Therefore, it is urgent to strengthen food safety management and it is imperative to establish an early warning mechanism for food safety. Literature [16] established a food safety risk intelligent early warning model based on support vector machine, and experimentally verified the validity and accuracy of the validated early warning model, which can greatly reduce the possibility of food safety accidents. Literature [17] proposed an early warning model for rapid control of food safety risk based on anomaly scoring using unsupervised self-coder, and a batch of dairy products testing data in a province was used as an example for experimental analysis, and the results showed the reliability of the proposed model, and the study helped the market regulatory authorities to control the risk of food safety. Literature [18] synthesized a food safety risk early warning model based on long and short-term memory neural network and sum-product based hierarchical analysis method, and verified the scientific nature of the model through empirical analysis, which can realize accurate prediction and early warning of food safety risks. Literature [19] proposed an improved adaptive particle swarm optimization algorithm for optimizing the long and short-term memory neural network, and applied it to the construction of food safety risk analysis and early warning model, and after algorithm testing and model evaluation, the scientific and practicality of the proposed algorithm and model were verified, and it can help the relevant governmental departments to effectively warn of potential food safety risks. Literature [20] constructed a food safety network public opinion early warning evaluation model based on interval hierarchy analysis method (IAHP), and verified the effectiveness and feasibility of the constructed evaluation model through evaluation examples, which can improve the certainty of food safety network public opinion early warning and evaluation. Literature [21] developed a fresh food safety early warning and monitoring system based on k-mean clustering algorithm, and established a fresh product safety early warning indicator system through practical analysis to reduce the occurrence of fresh product safety events. Literature [22] first constructed a food security early warning indicator system using principal component analysis and hierarchical analysis, and then designed a Chinese food security risk assessment and early warning based on BP neural network model, aiming to break through the limitations of the existing research, and to provide signaling guidance and reference for early response to risks in Chinese food security.

3

Food safety monitoring and early warning mechanisms based on time series analysis

3.1

Limitations of traditional monitoring tools

Traditional means of food safety monitoring mainly depend on limited sample testing and laboratory analysis. However, sampling and testing can only cover a portion of food samples, making it difficult to fully capture potential risks in food. Laboratory analysis, on the other hand, takes a long time and cannot detect and respond to food safety problems in a timely manner. Therefore, the traditional methods of food safety monitoring cannot hardly meet the needs of modern food safety management, and more advanced and comprehensive monitoring technologies and methods need to be introduced to improve food safety.

3.2

Food safety monitoring and early warning based on time series analysis

In the context of food safety monitoring indicators and early warning levels, the construction of early warning models is a crucial link in achieving food safety monitoring and early warning. Early warning models aim to identify potential food safety risks based on historical data and real-time information, and issue early warning signals in advance. In this paper, the food safety early warning model is constructed using time series analysis.The steps for creating the early warning model are listed in order:

1) Determination of food safety monitoring indicators. A series of food safety monitoring indicators are determined by comprehensively considering the data of each link in the food supply chain.

2) Establishment and optimization of the early warning model. With the help of machine learning algorithms, the early warning model of food safety is constructed. Supervised learning methods can be used to train the model using historical data, so that it learns the correlation between different indicators and potential risk patterns.The model can be optimized by adjusting algorithmic parameters and increasing training data to improve its ability to generalize to unknown risks.

3) Monitoring and analysis of real-time data flow. After creating the early warning model, a real-time data monitoring and analysis system should be established.This includes obtaining data in real time from various sources and analyzing it in real time through the early warning model.

4) Issuance of early warning signals. When the early warning model recognizes a potential food safety risk, the system should be able to automatically issue an early warning signal. The early warning signal can be in the form of alarm notification, report generation, or even automatic triggering of the corresponding risk response measures.

The above steps allow for the construction of an early warning model that draws information from multiple data sources, comprehensively assesses food safety risks, and detects and effectively warns potential problems in a timely manner. The construction of this model is a key link in the big data analysis system, providing a scientific and intelligent early warning mechanism for food safety monitoring.

4

Food safety early warning model based on time series analysis

The improved hierarchical analysis method and quadratic exponential smoothing method are combined in this paper to construct food safety detection and early warning models.

4.1

Improvement of Hierarchical Analysis based on Triple Scale Approach

The hierarchical analysis method (AHP) is a classical multi-attribute decision-making method, which can unify the measurement of tangible and intangible, quantitative and non-quantitative factors, and has a wide range of applications in many fields [23]. Constructing a judgment matrix is the first task of the AHP method, while the consistency test of the judgment matrix is the difficulty of the method. Therefore, this paper improves the hierarchical analysis method and adopts the three-scaled method to construct the comparison matrix, which greatly reduces the number of comparisons between indicators, simplifies the arithmetic process, and largely reduces the ambiguity in the judgment of the importance of indicators.

Using the three-scalar method to find out the comparison matrix and construct the indirect judgment matrix, directly applying mathematical methods to optimize the judgment matrix that meets the consistency requirements, and finally solving the eigenvector corresponding to the largest eigenvalue of the consistency matrix, we can get the weight value of each influential factor. The AHP method can be improved by taking the following specific steps:

1) Based on the principle of AHP, differentiate the degree of importance of mutual influencing factors at each level, analyze the relationship between the factors, establish the recursive hierarchical structure model of the system, and classify each indicator factor into the target layer, criterion layer and indicator layer.

2) According to the recursive hierarchical model, the importance of each factor at the same level is compared two by two using a three-scaled scale, and the corresponding comparison matrix is constructed, and the comparison matrix is set to be A, A = (a_ij)_n×n, which, by definition, has: (1) $A = [\begin{matrix} a_{11} & a_{12} & \dots & a_{1 n} \\ a_{21} & a_{22} & \dots & a_{2 n} \\ ⋮ & ⋮ & ⋮ & ⋮ \\ a_{n 1} & a_{n 2} & \dots & a_{n n} \end{matrix}]$

To realize the quantization of the elements of the comparison matrix, by the three-scalar method there: (2) $a_{i j} = {\begin{matrix} 0 & F a c t o r j i s m o r e i m p o r \tan t t h a n f a c t o r i \\ 0.5 & F a c t o r i i s e q u a l l y i m p o r \tan t a s f a c t o r j \\ 1 & F a c t o r i i s m o r e i m p o r \tan t t h a n f a c t o r j \end{matrix}$

3) Construct the indirect judgment matrices B, B = (b_ij)_n×n from the comparison matrices. Sum the comparison matrices A = (a_ij)_n×n by rows, denoted by: (3) ${\begin{array}{l} r_{i} = \sum_{j = 1}^{n} a_{i j} (i = 1, 2, 3, \dots, n) \\ r_{\max} = \max {r_{i}} \\ r_{\min} = \min {r_{i}} \end{array}$

Calculate the base point comparison scale b_m with the formula: (4) $b_{m} = \frac{r_{\max}}{r_{min}}$

The comparison matrix is transformed into an indirect matrix by mathematical transformation B, B = (b_ij)_n×n, Eq: (5) $b_{i j} = {\begin{array}{l} \frac{r_{i} - r_{j}}{r_{\max} - r_{\min}} (b_{m} - 1) + 1 & r_{i} \geq r_{j} \\ {[\frac{r_{j} - r_{i}}{r_{\max} - r_{\min}} (b_{m} - 1) + 1]}^{- 1} & r_{i} < r_{j} \end{array}$

Obviously, after the conversion of the indirect matrix B elements b_ij are satisfied b_ii = 1, b_ij = 1/b_ji.

4) The construction of the indirect matrix to meet the three conditions to have consistency, that is, b_ii = 1, b_ij = 1/b_ji, b_ij = b_ik×b_kj, the initial comparison of the first two conditions to meet the requirements, in order to make the judgment error is not large, to meet the requirements of the approximation, so that the judgment matrix is reasonable, need to be transformed into a consistency matrix of the indirect matrix C, C =(c_ij)_n×n, Eq: (6) $c_{i j} = e^{\frac{1}{n} \sum_{k = 1}^{n} \ln \frac{b_{i k}}{b_{j k}}} = e^{\frac{1}{n} \sum_{k = 1}^{n} [\ln (b_{i k}) - \ln (b_{j k})]}$

After mathematical optimization, for any i,j,s = 1,2,⋯,n, there is: (7) $\begin{matrix} c_{i j} = e^{\frac{1}{n} \sum_{k = 1}^{n} [\ln (b_{i k}) - \ln (b_{j k})]} = e^{\frac{1}{n} \sum_{k = 1}^{n} [\ln (b_{i k}) - \ln (b_{s k})]} \\ = e^{\frac{1}{n} \sum_{k = 1}^{n} [\ln (b_{s k}) - \ln (b_{j k})]} = b_{i s} \cdot b_{s j} \end{matrix}$

From Eqs. (6) to (7), we can get that the element c_ij in the optimized matrix C satisfies c_ij = c_ik×c_kj, c_ii = 1, c_ij = 1/c_ji, so the matrix C is a consistency matrix.

5) Calculate the relative weight of each comparison factor for its level. The square root method is used to calculate the eigenvector corresponding to the largest eigenvalue of the consistency matrix C and normalized. The eigenvector W is used as a vector of relative weights of the n influencing factor indicators of the level, which must satisfy: (8) ${\begin{array}{l} C W = λ_{\max} W \\ W = {[W_{1}, W_{2}, W_{3}, \dots, W_{n}]}^{T} \\ W_{i} = \frac{\sqrt[n]{\prod_{k = 1}^{n} c_{i k}}}{\sum_{k = 1}^{n} \sqrt[n]{\prod_{k = 1}^{n} c_{i k}}} \end{array}$

6) Calculate the synthetic weights of the factor indicators of each layer to the target layer of the system. The single-ordered relative weight of the n influencing factor indicators of the guideline layer to the target layer is W₁,W₂,W₃,⋯,W_n, and the single-ordered relative weight of the indicator layer factors to the guideline layer is w₁,w₂,⋯,w_i(i = 1,2,⋯,n). The synthetic weights of the factor indicators to the target layer of the system are ordered as follows: (9) $R_{j} = {[W_{1} w_{1}, W_{2} w_{2}, \dots, W_{n} w_{n}]}^{T}$

According to equation (9), the weights of the influencing factors can be derived, and then the ranking of the importance of the influencing factors of the indicator is carried out to determine the total ranking of the weights of each factor.

4.2

Quadratic exponential smoothing

Exponential smoothing is a method often used in statistics to predict the trend of change based on time series observations for short and medium term trends, based on the principle of moving weighted average method, which assigns a larger weight to the nearest observations and a smaller weight to the observations that are farther away, as a way to predict the values of the next cycle [24]. The primary exponential smoothing method is able to predict values with insignificant trend changes well, but has a significant lag for values with significant trend changes. The quadratic exponential smoothing (DES) method finds out the development of the curve through the law of lag deviation, and corrects the lag deviation by using quadratic smoothing, so that it can predict the time series with more obvious changes [25]. Let the sequence of s time period in the delayed triggering process be 〈t₁,t₂t₃,⋯,t_s〉, and the load monitoring value of each time period be 〈y₁,y₂,y₃,⋯,y_s then the set of all observed data in the delayed triggering process is: (10) $Y_{t} = (〈 t_{1}, y_{1} 〉, 〈 t_{2}, y_{2} 〉, 〈 t_{3}, y_{3} 〉, \dots, 〈 t_{s}, y_{s} 〉)$

Primary smoothing is defined as: (11) $S_{t}^{(1)} = α y_{t} + (1 - α) S_{t - 1}^{(1)}$

Where: $S_{t}^{(1)}$ is the smoothed value for period t. α is the smoothing coefficient, which is a weight on the most recent observation. y_t is the actual observed value for period t. One time exponential smoothing is actually a moving average method with α as a weight. In the primary exponential smoothing method, the smoothed value of period t is used as the predicted value for the next period, which is ${\hat{y}}_{t + 1} = S_{t}^{(1)}$ .

The primary exponential smoothing can better predict the situation where the trend change is too obvious, for the situation where the trend change is obvious, the predicted value has a big lag, the secondary exponential smoothing method makes another exponential smoothing for the primary exponential smoothing, which is defined as: (12) $S_{t}^{(2)} = α S_{t}^{(1)} + (1 - α) S_{t - 1}^{(2)}$

Secondary exponential smoothing is to correct the lagged deviation generated by the primary exponential smoothing value, and use the law of lagged deviation to find out the development law of the curve, if the time series has a linear trend from a certain moment t, then the future time also has a linear trend, and therefore to establish a linear trend prediction model, the prediction model is: (13) ${\hat{y}}_{t + T} = a_{t} + b_{t} T$

Where: t is the current time. T is the number of periods separated by the current time period t to the future period T. ŷ_s+T is the predicted value for future period T. a_t is the intercept and b_t is the slope, i.e: (14) $a_{t} = 2 S_{t}^{(1)} - S_{t}^{(2)}$ (15) $b_{t} = \frac{\partial}{1 - \partial} (S_{t}^{(1)} - S_{t}^{(2)})$

5

Model application analysis

5.1

Determination of food safety monitoring indicators

5.1.1

Construction of the indicator system

This study starts from the theoretical perspective of “infant milk powder food safety” and refers to the existing literature and cases related to food safety monitoring to select human health hazards, the number and distribution of affected populations, potential economic hazards, the degree of attention paid by each country, the frequency of test failures, the degree of difficulty in detecting the food products (the availability of testing methods and accuracy), and the degree of difficulty in eradicating them as the standard factors for food safety assessment. The factors of food safety assessment criteria were selected, including the number and distribution of people affected, potential economic harm, the importance attached by countries, the frequency of detection failures, the ease of food detection (availability and accuracy of detection methods), and the ease of eradication.

Subsequently, a questionnaire survey was conducted with 30 experts from governmental regulatory departments who have senior research experience in food safety monitoring, and a preliminary food safety monitoring indicator system for infant milk powder was constructed, which included three major categories of risk factors, namely microbiological, physicochemical, and labeling items. The primary indicators are microbiological testing (A1), physical and chemical testing (A2) and labeling (A3), while the secondary indicators include non-pathogenic bacteria (B1), pathogenic bacteria (B2), vitamin-like substances (B3), essential and other ingredients (B4), carbohydrates and sugars (B5), vitamins (B6), minerals (B7), optional nutrients and additives (B8), toxic and hazardous substances (B9), and nutrients (B9). Harmful substances (B9), Nutrient fatty acids (B10), Labeling methods (B11), Instructions for use (B12), Packaging media (B13).

5.1.2

Determination of indicator weights

Combing the results of the Delphi method questionnaire of 30 experts and scholars, the comparison matrix of the first-level indicators can be obtained according to the three-scaled method as shown in Table 1.

Table 1.

The contrast matrix of the first level index

A	A1	A2	A3
A1	0.5	0	1
A2	1	0.5	1
A3	0	0	0.5

According to equation (3), summing by rows yields: r₁=1.5,r₂=2.5,r₃=0.5,r_max=2.5,r_min=0.5.

Then using equations (4) to (5), the judgment matrix is obtained as shown in Table 2.

Table 2.

The judgment matrix of the first level index

A	A1	A2	A3
A1	1	1/3	3
A2	3	1	5
A3	1/3	1/5	1

The elements of each row in the judgment matrix are then multiplied together and squared 3 times: $w_{1}^{*} = \sqrt[n]{\prod_{j = 1}^{n} c_{1 j}} = \sqrt[3]{1 \times 1 / 3 \times 3} = 1$ $w_{2}^{*} = \sqrt[n]{\prod_{j = 1}^{n} c_{2 j}} = \sqrt[3]{3 \times 1 \times 5} = 2.4662$ $w_{3}^{*} = \sqrt[n]{\prod_{j = 1}^{n} c_{3 j}} = \sqrt[3]{1 / 3 \times 1 / 5 \times 1} = 0.4055$

According to Equation (8), the weights of the three first-level food safety monitoring indicators were found to be: $w_{1} = \frac{w_{1}^{*}}{\sum_{i = 1}^{n} w_{i}^{*}} = \frac{1}{3.8717} = 0.2583$ $w_{2} = \frac{w_{2}^{*}}{\sum_{i = 1}^{n} w_{i}^{*}} = \frac{2.4662}{3.8717} = 0.6370$ $w_{3} = \frac{w_{3}^{*}}{\sum_{i = 1}^{n} w_{i}^{*}} = \frac{0.4055}{3.8717} = 0.1047$

The consistency test is performed on the judgment matrix of the first-level indicators, and the following are obtained: λ_max = 3.0385, CI = 0.0193, CR = 0.0332. Due to CR < 0.1, it indicates that the judgment matrix passes the consistency test.

Similarly, the relative weights corresponding to the 13 secondary indicators are obtained, and the weights of food safety testing indicators are finally obtained as shown in Table 3.

Table 3.

The weight of food safety detection indicators

Primary indicator	Weight	Secondary indicator	Weight	Absolute weight
A1	0.2583	B1	0.2264	0.0585
		B2	0.6845	0.1768
		B3	0.0891	0.0230
A2	0.6370	B4	0.0455	0.0290
		B5	0.0959	0.0611
		B6	0.4342	0.2766
		B7	0.0529	0.0337
		B8	0.2692	0.1715
		B9	0.0627	0.0399
		B10	0.0396	0.0252
A3	0.1047	B11	0.5339	0.0559
		B12	0.3849	0.0403
		B13	0.0812	0.0085

As shown in Table 3, B2 pathogenic bacteria, B6 vitamins, and B8 optional nutrients and additives are the main hidden dangers, and the absolute weights are 0.1768, 0.2766, and 0.1715, respectively, which are all greater than 0.1, which are also basically consistent with the food safety accidents that occurred. The reason for this is subjectively from certain non-self-disciplined food enterprises, objectively exposing the weakness of the risk management work of the food safety regulatory authorities, and urgently need to further put forward countermeasures in the logic of risk deployment and control, and synergistic governance.

The function model used for early warning determination is obtained by solving the weights of the food safety monitoring indicators: (16) $\begin{array}{l} Z = 0.0585 B_{1} + 0.1768 B_{2} + 0.0230 B_{3} + 0.0290 B_{4} + 0.0611 B_{5} \\ + 0.2766 B_{6} + 0.0337 B_{7} + 0.1715 B_{8} + 0.0399 B_{9} + 0.0252 B_{10} \\ + 0.0559 B_{11} + 0.0403 B_{12} + 0.0085 B_{13} \end{array}$

This is the weighted composite index of the improved hierarchical analysis method, which is solved by the measured values of the 13 secondary indicators of infant formula food safety. The full score for each indicator is set to 1, and the indicator food safety monitoring compliance score is set to 0.6.

5.2

Experimental setup

5.2.1

Data sources

The data used for the experiment were obtained from the food safety sampling information of infant milk powder issued by the Market Supervision Administration of Province L from 2021 to 2022, totaling 24 issues. The data used in the validation were obtained from the 1st to 5th issues of the food safety sampling information of infant formula issued by the Market Supervision Administration of Province L in 2023. All the data used were transformed into a comprehensive index of infant milk powder food safety according to Equation (19).

5.2.2

Time-Series Characteristics of Food Safety Sampling and Inspection Pass Rates

The information on food safety supervision and sampling of infant milk powder in Province L from 2021 to 2022 was organized, and the time series of sampling qualified rate was obtained as shown in Figure 1. Observing Figure 1, it can be seen that the sampling qualified rate generally shows horizontal downward fluctuation.

5.3

ARIMA model setup and testing

5.3.1

Stability tests

From Figure 1, the overall slope of the series is downward, indicating that the ADF test is performed, where the difference order is 1. The t-statistic of the ADF test for this series is -6.367, and the critical values of 1%, 5%, and 10% are -3.629, -2.946, and -2.658, respectively. And the p-value is 0.001<0.01, so it is inferred that the series is smooth at this point.

5.3.2

Travel tests

A swim test was conducted on this time series and the results showed that P=0.008<0.05, indicating that the development of this time series is not random and has a pattern that can be further analyzed.

5.3.3

Model Identification and Fitting

According to the principle that the lower the value of AIC and BIC of the information criterion, the optimal model is known as ARMA(2,1,1). Combined with the characteristics of the timing diagram, analyzing and comparing the model optimization when different parameters, the parameters of the ARMA(2,1,1) model are obtained as shown in Table 4.

Table 4.

ARMA (2,1,1) model parameter

Term	Symbol	Coefficient	Standard error	z	P	95% CI
Constant term	c	-0.001	0.002	-3.659	0.001	-0.003~-0.001
AR parameter	α₁	0.284	0.108	2.637	0.013	-0.003~-0.001
AR parameter	α₂	0.051	0.119	0.414	0.697	-0.003~-0.001
MA parameter	β₁	-1.037	0.028	-32.523	0.001	-0.003~-0.001

According to the parameter results in Table 4, the ARMA(2,1,1) model formula is obtained as: (17) $y^{(t)} = - 0.001 + 0.284 y^{(t - 1)} + 0.051 y^{(t - 2)} - 1.037 ε^{(t - 1)}$

In Eq. (20): y^(t) denotes the current value in the differential operation, t denotes the number of timings, and ε denotes the error.

5.3.4

Residual white noise test

The white noise test was performed using the Q statistic and the results of the white noise test are shown in Table 5. As shown in Table 5, the P-values of Q6, Q12, and Q18 are 0.974, 0.625, and 0.832, respectively, which are all greater than 0.1, indicating that the residuals of the model are white noise at the significance level of 0.1 and meet the requirements.

Table 5.

White noise test results

Term	Symbol	Coefficient	Standard error	z	P	95% CI
Constant term	c	-0.001	0.002	-3.659	0.001	-0.003~-0.001
AR parameter	α₁	0.284	0.108	2.637	0.013	-0.003~-0.001
AR parameter	α₂	0.051	0.119	0.414	0.697	-0.003~-0.001
MA parameter	β₁	-1.037	0.028	-32.523	0.001	-0.003~-0.001

5.4

Quadratic exponential smoothing method model construction and forecast comparison

5.4.1

Exponential smoothing method modeling results

A total of 24 periods of food safety sampling data of infant milk powder in province L from 2021 to 2022 are taken as the time series for modeling, which is noted as Y=(Y₁,Y₂,…Y_t),t=24. In order to select the value of the better smoothing coefficient α, this paper obtains the optimal smoothing coefficient α=0.75 through the planning and solving of EXCEL, and at this time, the MSE=0.017. The value of a_t and b_t, which are obtained through the calculation of the year 2022, is selected as the constant, and at this time, the quadratic index of the mathematical model is: (18) $Y_{t + T} = Y_{24 + T} = 0.8255 + 0.0637 T$

5.4.2

Comparative results of model predictions

The ARMA(2,1,1) model and the quadratic exponential smoothing (DES) model were used to compare the prediction efficacy of the composite index of food safety of infant formula sampled in Province L from the 1st to the 5th period of 2023, and the results of the model prediction comparison are shown in Figure 2.

As can be seen from Figure 2, the relative and absolute errors of the quadratic exponential smoothing model prediction range from [0.02%,0.99%] and [0.0004,0.0125], respectively, which are smaller than that of the ARMA(2,1,1) model, which indicates that the quadratic exponential smoothing method is more effective in prediction.

Thus, this paper chooses the resulting quadratic exponential smoothing model for monitoring and early warning of infant formula food safety composite index, with the early warning level set at 0.7, i.e., the early warning is issued immediately when the food safety composite index is close to substandard.

6

Conclusion

This paper uses infant milk powder food safety as an example to demonstrate how to set food safety monitoring indicators and an early warning level.

1) In the setting of food safety monitoring indicators, this paper constructs a food safety monitoring indicator system for infant milk powder, including three primary indicators of microbiological testing, physical and chemical testing and labeling, as well as 13 secondary indicators of non-pathogenic bacteria, pathogenic bacteria, vitamin-like substances, essential ingredients and other ingredients, carbohydrates and sugars, vitamins, minerals, optional nutrients and additives, toxic and hazardous substances, nutrient Fatty acids, labeling methods, instructions for use, and packaging media were 13 secondary indicators, and the food safety composite index function for indicator assignment was determined using hierarchical analysis based on the three-scaling method.

2) In terms of food safety early warning level setting, this paper first conducts ADF test on the time series, and the P value is 0.001<0.01, indicating that the series is smooth at this time. And the P-value of the swim test is 0.008<0.05, which indicates that the time series is regular and can be further analyzed. According to the information criterion AIC and BIC value the lower the better principle, determine the optimal ARMA model for ARMA (2,1,1). By analyzing and comparing the model optimization at different parameters, this paper derives the formula of ARMA(2,1,1) model and conducts the white noise test on the model, and the P-values of Q6, Q12, and Q18 are all greater than 0.1, which indicates that the residuals of the model are white noise under the significance level of 0.1, which meets the requirements. Then, this paper obtains the optimal smoothing coefficient of the quadratic exponential smoothing model through the planning solution of EXCEL as α = 0.75, at which time MSE=0.017, so as to construct the food safety warning model based on the quadratic exponential smoothing method. Finally, by comparing the prediction effect of the model, it can be seen that the relative and absolute errors of the prediction of the quadratic exponential smoothing model are smaller than that of the ARMA(2,1,1) model, and thus this paper chooses the constructed quadratic exponential smoothing model for the monitoring and early warning of food safety, and sets the warning level to 0.7.

Lingua:: Inglese

Frequenza di pubblicazione:: 1 volte all'anno
Argomenti della rivista:: Scienze biologiche, Scienze della vita, altro, Matematica, Matematica applicata, Matematica generale, Fisica, Fisica, altro

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Food Safety Monitoring Indicators and Early Warning Level Setting Based on Time Series Analysis

Xian Liu

Chun Bian

Yanfeng Qu

Pubblicato online: 17 mar 2025

Ricevuto: 16 nov 2024

Accettato: 14 feb 2025

DOI: https://doi.org/10.2478/amns-2025-0297

Parole chiaveTri-scalar method, Hierarchical analysis method, ARIMA model, Quadratic exponential smoothing method, Food safety

© 2025 Xian Liu et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Parole chiave
Tri-scalar method, Hierarchical analysis method, ARIMA model, Quadratic exponential smoothing method, Food safety