High accuracy detection technique for information data under the influence of multiple factors based on fractional partial differential equations

In engineering practice, several factors affect the various types of information during the collection process. For example, information data measurement errors are caused by equipment performance and the working environment. During the transmission of detection information, signal distortion caused by energy loss and signal interference causes unpredictable detection errors in collected data. Through the study of fractional calculus theory, it was found that it is suitable for studying nonlinear, noncausal, and nonstationary signals, and has the dual functions of improving detection information and enhancing signal strength. Therefore, under the influence of many factors, we applied the fractional difference algorithm to the field of information-data processing. A multisensor detection data fusion algorithm based on the fractional partial differential equation was adopted to establish online detection data. A multi-sensor detection data fusion algorithm based on a fractional partial differential equation was established, which effectively fuses the information data detection errors caused by various influencing factors and significantly improves the detection accuracy of information data. The effectiveness of this method was experimentally demonstrated by its application.


Introduction
Information detection technology is considered an important factor in ensuring the normal operation of the control system, * Author to whom any correspondence should be addressed.
Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. and the detection accuracy of multi-sensor information data is directly related to the working reliability of the information detection system. Online detection technology is a multisensor technology, which is the core technology of information detection systems, and the collection of real-time and accurate production information is directly related to the reliability of manufacturing systems. With the development of internet of things (IoT) technology, the use of IoT to collect multi-sensor detection information data in real time has become the development trend of information detection systems. However, in the online detection process of the information data of the detection object, the information is inevitably affected by various factors such as the measuring instrument performance, environmental interference, and information transmission distance, which directly affect the reliability of the information detection system. Therefore, research on the high-precision detection of information data in complex environments is of great significance for promoting the development of detection technology. The current research results can be divided into two categories: hardware and software.
The essence of the hardware method is to improve the detection accuracy of the information data using highperformance detection instruments. The main research results for the hardware methods are as follows: Hu [1] proposed a high-precision safety valve test architecture with three testing channels and effectively solved the problems of current safety valve testing. Huijun et al [2] developed a comprehensive sliding-separation test platform for RV reducers to realize high-precision and high-display test performance for various RV reducer parameters. Li et al [3,4] proposed a load differential radiation pulse on a transient electromagnetic highperformance radiation source for pulse-scanning detection to solve the problems of urban electromagnetic interference and insufficient harmonic components emitted by radiation sources. Ahmed et al [5][6][7] designed a hardware system based on radar and realized a real-time detection function for underground space-related information by enlarging the detection information. Jiaqi et al [8] proposed a one-stage remote sensing image object detection model: a multi-feature information complementary detector, which can improve the ability of the model to recognize long-distance dependent information and establish spatial-location relationships between features.
However, in engineering applications, we found that the hardware method had the following shortcomings.
(1) The detection accuracy of information data depends on the performance of the detection equipment. With improvements in detection accuracy, the cost of the detection system is higher. Therefore, they exhibit low-cost performance. (2) Their essence is to reduce the signal distortion caused by energy loss and signal interference in the information transmission process by improving signal strength. However, when collecting information, the measurement error of the data cannot be eliminated owing to differences in the equipment performance and working environment.
In recent years, most researchers have attempted to use software methods to achieve high-precision information-data detection to solve the shortage of hardware methods for information-data detection in complex environments. The essence of the software method is the information data-fusion algorithm. Up to now, there have been many research results Common mathematical algorithms are fuzzy set theory [9], fuzzy neural networks [10], probability model [11] and particle swarm optimization algorithm [12], et al and obtained a regrettable research review. For example, Huo et al [13] proposed an integral infinite log-ratio algorithm (IILRA) and an IILRA based on signal-to-noise ratio to improve the detection accuracy of the laser communication detection position in the atmosphere. Zhiyuan et al [14] proposed a normalizedvariance-detection method based on compression sensing measurements of received signals, and solved the problem of fast and accurate spectrum sensing technology under the condition of a low signal-to-noise ratio. Liu et al [15] proposed a target detection algorithm based on improved RetinaNet, which is suitable for transmission-line defect detection and improves the intelligent detection accuracy of UAV in power systems. Ru et al [16] proposed a lightweight ECA-YOLOX-Tiny model by embedding an efficient channel attention (ECA) module, which has a higher response rate for decision areas and special backgrounds, such as overlapping small target insulators, insulators obscured by tower poles, or insulators with high-similarity backgrounds. Liu Wenqiang et al [17] introduced a point cloud segmentation and recognition method based on three-dimensional convolutional neural networks to determine the different components of the catenary cantilever devices. Yin et al [18] proposed a complementary symmetric geometry-free method that made the detection of cycle slips more comprehensive and accurate. Lingfeng et al [19] established a junction temperature model based on a multiple linear stepwise regression algorithm, and used it to extract highprecision intersection online temperatures. However, through analysis of various current software methods, the following deficiencies were found in the detection of information in complex environments: (1) However, these methods do not improve the strength of the detection information, and cannot solve the problems of energy loss and signal interference during information transmission. Therefore, it is difficult to apply this method to engineering practice. (2) They did not analyze the cause of the information data detection error, the change rule of each influencing factor, or its influence on the detection value. Therefore, it is difficult to improve the detection accuracy of information data by reducing the detection error caused by various influencing factors.
Therefore, an ideal high-precision detection method for information under the joint action of multiple influencing factors has not been developed. To solve these problems, our team has been using the method of fractional calculus theory in data processing for many years [20][21][22][23][24][25][26] and found that fractional differential operators are suitable for studying nonlinear, non-causal, and non-stationary signals and have dual functions of improving detection information and enhancing signal strength. Therefore, by fusing the differences between the information and data, the information and data detection errors caused by various influencing factors can be eliminated. Improving the signal strength of the information can compensate for the energy loss of the signal during the transmission process and improve the anti-interference ability. Based on previous research, this study extends the fractional differential operator in one-dimensional space to the fractional partial differential field in multidimensional space to realize a high-precision detection function for information data under the joint action of multiple influencing factors.

Theoretical guidance and analysis
At present, many experts and scholars' research on fractional calculus theory is limited to the fields of equation solving and information processing. However, how to apply it to the fusion of engineering information data and explore its application characteristics in data processing has not yet been found in the relevant research literature. Fractional differentials (FDs), also known as fractional derivatives, extend the differential order of integer-order differential equations to fractional order. FDs emerged in 1812. After hundreds of years of research, FDs do not have a unified definition, and many scholars have proposed their own definition methods and theoretical systems based on their own understanding and application fields. Currently, the most commonly used definitions are those by Grunwald-Letnikov (G-L), Caputo, and Riemann-Liouville.
Among the three definitions, because the G-L definition has the advantage of fast calculation speed, it is widely used in the engineering field; therefore, this study uses the G-L definition to study the application technology of fractional calculus theory in detection information data fusion.
If there is a real number In the above equation, 0 ⩽ n -1 < v < n and In equation (2),

Effect of FD on detection signal.
For years, many mathematicians and scientists have attempted to apply fractional calculus theory to signal processing and have achieved good application effects [13,14]. It is assumed that a detection signal S (t) exists, where S (t)∈ L 2 (R), and its Fourier transform is Let S v (t) be the v-order differential of S (t). According to the properties of the Fourier transform, we know that the v-order differential operator is equal to the multiplicative operator of the v-order differential multiplier function, d(ω) = (iω) v . Thus, the following equation was obtained: From the perspective of signal modulation, the physical meaning of the FD of a detection signal is equivalent to generalized amplitude and phase modulation. From the perspective of signal processing, the v-order fractional calculus operation of the detected signal is equivalent to establishing a linear time-invariant filtering system for the signal, and its filtering function is: The spectral characteristic curve of the FD operator can be obtained using equation (5), as shown in figure 1.
According to figure 1, FD operators have the following information processing characteristics: (1) During the processing of the detection signal, the FD operator enhances the medium-and high-frequency components of the signal. The lifting amplitude increased nonlinearly and rapidly with increasing frequency and FD order, thereby enhancing the signal strength. However, with an increase in the frequency, the gap between the signal enhancement values of the FD operators under different orders gradually narrows. (2) When ω > 1, with an increase in fractional order v and signal frequency ω, the variability between the signal enhancement coefficients of the fractional-order differential operators at different orders tends to decrease. With increasing frequency, the enhancement effect of differential operators of the same order on the signal intensity at different frequencies is essentially the same.

Fractional partial differential
From the analysis of the amplitude-frequency characteristics of FD operators, we know that in the signal enhancement stage, when the influencing factors reach a certain value, the FD operator can effectively fuse the differences between the detection data caused by the influencing factor. Therefore, fractional calculus is used to describe many phenomena in engineering and science [15,16]. FD equations have been used extensively over the last two decades because of their varied applications in many spheres of physical and biological sciences [27,28], but they can only be applied to processing one-dimensional signals. In the process of data collection, the detection information is affected by many factors, such as testing equipment performance and equipment working environment. Therefore, fractional partial differential equations must be applied to expand the FD operators in a multidimensional space.

Fractional partial differential equation.
To simplify the calculation workload, we assumed that the information data value is affected by two factors. According to the G-L definition of the fractional partial differential, we extend the FD operators to a two-dimensional space. Assuming that the information acquisition system collects any given twodimensional energy signal S (x, y) and S (x, y)∈L 2 (R), we can obtain the fractional partial differential equation S v (x, y) of the signal as: Refer to equation (1), can be obtained the following equation . (7) 2.2.2. Applications of the fractional partial differential equations. Compared to an integer-order partial differential equation, an outstanding advantage of the fractional partial differential equation is that it can better simulate the physical and dynamic system processes of nature. Compared with FD equations, a fractional partial differential equation can take the unknown quantity in the equation as the influencing factor to realize the FD treatment of multiple influence factors in the equation. Because fractional partial differential equations have the above advantages, in recent years, research on the characteristics and applications of fractional partial differential equations has become the focus of many experts and scholars. Some research results have been widely used in the research of technical problems in the fields of temperature field distribution, image processing, mechanical analysis, and detection technology [27][28][29]. For example, Bao et al [30] applied fractional partial differentials to achieve the research goal of thickness design of high-temperature protective clothing under actual limited conditions. Zhou et al [31] applied them to demonstrate their advantages in image denoising and reducing the step effect as well as in denoising and superresolution reconstruction. Tianlong [32] applied fractional partial differential to fluid mechanics.

Fusion algorithm model based on fractional partial differential equations
The IOT is used to collect a group of production information S i (i = 1, 2,.. n) of the same type owing to the influence of the test instrument performance, working environment, and information transmission distance, which leads to significant differences between the data collected by the information systems. Therefore, relevant information and data fusion models must be established to effectively remove the detection errors caused by the aforementioned factors and to improve the detection accuracy of the information. Suppose we detect a spatial signal J (t), and the parameters x and y are the influencing factors of the measured value of signal J (t). Therefore, the measured value of signal Because the influences of the influencing factors x and y on the detection value are independent of each other, the calculation method for the function of the influencing factors x and y is interdependent. Referring to equation (1), the FD equation of function J (x, y) in each space can be obtained as: (1) When the influence factor is x In the above equation, 0 ⩽ n -1 < v < n and In equation (9), when h 1 → 0, n 1 → ∞, let h 1 = [b 1 − a 1 ]/n 1 , and n 1 = [b 1 − a 1 ]/h 1 .
(2) When the influence factor is y In the above equation, 0 ⩽ n -1 < v < n and In equation (11) Based on equations (6) and (7), the model of the equipment monitoring information data fusion algorithm based on the fractional partial differential equation under IoT is as follows: In the above equation

Fusion process based on fractional partial differentials
To obtain the information data of the detected object in real time, an information detection system often uses IoT to collect all types of information data. Based on the idea of integration under a distributed system, we classified the collected data, obtained the required information data first, and then analyzed the main factors leading to the differences between the data. We fused the differential information data using the detection data fusion model based on the fractional partial differential equation to effectively reduce the difference in detection data between the sensors. In the multisensor detection data fusion algorithm model based on fractional partial differential equations, the premise of data fusion is to obtain the function S (x,y) for a data detected value S and influencing factors x and y. A fusion model with different detection data was developed using fractional partial differential calculations. The fusion process for the detection data is as follows: (1) IoT is applied to obtain all types of detection information in real time and to select the required experimental data by analyzing the detection data. (2) By analyzing the characteristics of the information data, we determined the main factors, x and y that affect the measured value of the data. (3) Using the fitting method, the functional relationship S (x, y) between the detected value of information data S and its influencing factors x and y. (4) Apply the fractional partial differential equation to fuse information data S and obtain fused information data S v . (5) We analyzed the fused information data and evaluated the application of fractional calculus theory in information data fusion. (6) End.

Problem description
To compare the performance of the data fusion algorithm, we selected experimental data from literature [33]. The measured temperature data of the thermostat were collected by eight smart temperature sensors, and the data were measured every hour eight times. IoT was applied to the information collection system. The measured values of each sensor are listed in table 1. The average value of the measured values is taken as the real value, that is 53.07 • C. From the data shown in table 1, it can be seen that the data 46.52, 47.35, 48.43, 56.05, 57.66, and 58.34, respectively, exceed the international standard of the common deviation; therefore, they should be removed before fusion. From table 1, we can see that the total measurement error of the eight sensors at different time points was T = 1.58, the standard deviation of the detected data between the sensors was S x = 0.251, and the standard deviation of the measurement data at different times was S y = 0.333.

Analysis and processing of test data
Using the data shown in table 1, we take the measured average value of each sensor as its measured value F i and draw the distribution curve of the measured data of each sensor, as shown in figure 2. The average of the detected values of the eight sensors at different time points was the measured true value F j , and a data distribution curve was drawn, as shown in figure 3. It was found that their measured values presented an irregular discrete distribution above and below the mean value of 53.07 • C.
According to the experimental data given in table 1 and the data distribution curves shown in figures 2 and 3, the measured values have the following characteristics.
(1) In the measurement of thermostat temperature, owing to the difference in the performance of the detection instruments, there was a large difference between the same detection data, indicating that the performance of the detection equipment and instruments was an important factor affecting the temperature measurement value. (2) When measuring at different times, owing to the differences in the detection environment, there is a large difference between the measured values of the same information at different time points. Therefore, the detection environment is an important factor that affects the measured values.

Fitting of functional relationship between detected value and influence factor
According to the common sense of surveying, the standard deviation is the best performance index for measuring the accuracy of information data; therefore, in this case, we used the performance of the test equipment as the influencing factor   x and the working environment of the equipment as the influencing factor y to effectively deal with the test value of the thermostat temperature, and used the standard deviations S x and S y as the influencing factors to establish the functional equation between them and the measured value S i . The least-squares method has the advantage of not requiring a priori information about the data during data processing, and can obtain ideal data fusion accuracy [34]. Therefore, it is widely used to fit measurement data using polynomial functions. Therefore, this study used the least-squares method to fit the functional equations of S (x) and S (y) as follows: From the data in table 1, it can be seen that each experimental datum is equal to S (x i ) or S (y j ). Thus, S (x i ) = S (y j ), and we obtain the following equation: S(x, y) = (S(x) + S(y)) /2 = ( a 0 + a 1 x + a 2 x 2 + · · · + a n x n + b 0 + b 1 y + b 2 y 2 + · · · + b m y m ) /2 . (1) Functional relationship between influence factor x and detection value S According to equation (14), the function S (x) is obtained by calculating the values of coefficients a i and n. According to the basic definition of the least squares method, the order of the polynomial in the S (x) equation should be less than the number of samples. Therefore, n < 8. According to the experimental data listed in table 1, using the polyfit function in MATLAB, we can conclude that when the order of polynomial n = 1, the error between the fitting values of each sensor's detection data is 1.0779, which is the smallest. In this case, the functional relationship between the detection data S and the influence factor x is: (2) Functional relationship between influencing factor y and detection value S According to the above theory, function S (y) is obtained by calculating the values of the coefficients b i and m used in equation (14). According to the basic definition of the least-squares method, the order of polynomial S (y) shown in equation (14) should be less than the number of samples; thus, m < 8. According to the experimental data listed in table 1 and the polyfit function in MATLAB, we can conclude that when the order of the polynomial m = 1, the error between the fitting values of the detection data of each sensor is 0.3667, which is the smallest. In this case, the functional relationship between the detection data S and influence factor y is S (y) = 0.667y + 52.873. (1) Select fractional-order v According to the amplitude and frequency characteristics of the FD operator, when the differential order is 0 < v < 1, the variability between the enhancement coefficients of the different FD operators tends to decrease as the signal frequency increases. To save space, this case considers the fractional order v = 0.5, as the middle value of the range [0, 1], and studies the application effect of the fractional partial differential equation in the data fusion processing of bearing oil temperature sampling under the IoT.
(2) Select the value of the discrete step h From equation (13) (9) and related parameters (v, h 1 , h 2 , m, n) were substituted into the mathematical model of mobile device detection data fusion processing based on the fractional partial differential equation shown in equations (6) and (7). The following equation is obtained: In the above equation, .
(19) According to the above equations, we can calculate the measured temperature data of thermostat S (x i , y j ) of each temperature sensor using a 0.5-order partial differential equation. The results of the 0.5-order partial differential equation fusion under influence factor x are listed in table 2, and the results of the 0.5-order partial differential equation fusion under influence factor y are listed in table 3.
By combining equations (16)- (19), we obtained the values of the 0.5-order partial differential of the measured value for each sensor at different time points, as shown in table 4.

Fusion processing results of detection data by 0.5-order
partial differential equation. Comparing the mean value F of table 1 with the mean value F 0.5 of table 2, we can conclude that after the 0.5-order partial differential fusion, the numerical magnification is K = 1.166, which can improve the antiinterference ability of the signal and the reliability of information transmission. To compare the effect of data fusion with those of the other algorithms, we divided the data in table 4 by K and calculated the final data processing results, as shown in table 5. From table 5, we can see that, after fusion using the 0.5-order partial differential equation, the fusion accuracy improved effectively. Analysis from measurement error, the total absolute error of the methods used in this paper was 0.175, which is significantly lower than the total absolute error  of 1.58 obtained by the average value algorithm, the total absolute error of 0.395 obtained by the algorithm described in [33], and the total absolute error of 0.335 obtained by the algorithm described in [35]. After processing using the 0.5-order partial differential algorithm, the standard deviation between the measured values of each sensor and the information data at different time points was approximately one-tenth that before processing. According to the data shown in table 4, a comparison chart of the standard deviation between the measured values of each sensor and different time points before and after processing by the 0.5 order partial differential algorithm was drawn, as shown in figures 4 and 5. It can be observed that the information data were randomly distributed near the average value. Experiments show that the 0.5-order partial differential algorithm has a strong fusion ability for experimental data.

Conclusion
In engineering practice, the detection information cannot avoid the common influence of multiple irregular change factors, resulting in unpredictable measurement errors. In this study, we found that FD operators are suitable for studying nonlinear, noncausal, and nonstationary signals. Therefore, the FD algorithm was applied to the processing of complex detection information. According to the number and type of influencing factors, the FD operator is extended to the fractional partial differential field, which realizes effective fusion processing of information data under the joint action of multiple influencing factors. By enhancing the strength of the detection signal, the anti-interference ability of the detection information in the transmission process is improved, and the information and data distortion caused by energy loss are compensated. In engineering applications, we can adjust the value of the parameter fractional order v and step h in the algorithm to satisfy the information data detection target required by the system. The differences between the data were combined to improve the measurement accuracy of detection information. The effectiveness of this method is demonstrated using an example.

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).