Method for calculating correlation coefficient of aero-engine parameters based on task section

Abstract

The application discloses a kind of aviation engine parameter correlation coefficient calculation methods based on task section, belong to aviation engine load spectrum technical field, including: based on flight action division task section;Task section data is spliced to head and tail, and the flight performance parameter of task section is counted with engine performance parameter;The correlation coefficient of the flight performance parameter of task section and engine performance parameter is calculated, and the correlation of aviation engine flight performance parameter and engine performance parameter is obtained, the aviation engine parameter correlation coefficient calculation method based on task section is based on flight action task section, can be applied to aviation engine stress analysis, engine performance analysis, life research and subsequent test schedule preparation lays foundation, with higher engineering application value, and the correlation coefficient of the statistical calculation aviation engine airplane parameter and engine parameter has significant engineering to aviation engine load spectrum prediction and engine analysis.

Description

Technical Field

[0001] This invention belongs to the field of aero-engine load spectrum technology, specifically relating to a method for calculating the correlation coefficient of aero-engine parameters based on mission segment. Background Technology

[0002] Stress analysis and life studies of aero-engines rely on maneuvering flight load spectra. To determine the service life of an aero-engine, full-scale fatigue tests of structures or components must be conducted in the final design stage. To obtain reliable test results, full-scale fatigue tests should simulate real operating conditions as accurately as possible. However, due to the randomness of fatigue loads and the variability of real operating conditions, load spectrum simulation is necessary to apply random loading to test components. Single-parameter load spectra are very effective for stress analysis and life prediction of components subjected to simple loads, but they have significant limitations for components subjected to complex loads. In actual aero-engine operation, most components bear multi-component loads, and load parameters are strongly correlated with flight / engine performance parameters. However, the results calculated using the correlation coefficient method based on mathematical statistics, as shown in Table 1, are inconsistent with actual aero-engine usage, exhibiting low correlations in altitude Mach number and speed with normal overload coefficient. Therefore, a new method for calculating the correlation coefficient of aero-engine parameters based on mission phases needs to be developed to address these issues. Summary of the Invention

[0003] The purpose of this invention is to provide a method for calculating the correlation coefficient of aero-engine parameters based on the mission segment, so as to solve the problem that the low correlation between speed and normal overload coefficient does not conform to the actual use of aero-engines.

[0004] To achieve the above objectives, the present invention provides the following technical solution: a method for calculating the correlation coefficient of aero-engine parameters based on a mission segment, comprising:

[0005] Task segments are divided based on flight maneuvers;

[0006] The data from the first and last segments of the mission are spliced ​​together to statistically analyze the flight performance parameters and engine performance parameters of the mission segment.

[0007] The correlation coefficient between the flight performance parameters and engine performance parameters of the mission segment is calculated to obtain the correlation between the flight performance parameters and engine performance parameters of the aero-engine.

[0008] Preferably, the mission segment includes six types of typical flight maneuver mission segments; specifically, it includes: ascending flight maneuver mission segment, descending flight maneuver mission segment, level flight maneuver mission segment, circling flight maneuver mission segment, ascending and turning flight maneuver mission segment, and descending and turning flight maneuver mission segment.

[0009] Preferably, the method for classifying the flight maneuvers includes:

[0010] The flight mission profile is preprocessed, and the flight mission segment profile is divided into takeoff and landing mission segments. The intermediate mission segments are extracted for the identification of flight actions.

[0011] Complex maneuvers are categorized into five basic types based on the rate of change of heading angle, pitch angle, bank angle, and altitude: level flight, climb, glide, turn, and roll.

[0012] Based on the parameter ranges of the basic actions, MATLAB was used to identify the basic actions of the engine in the measured flight mission profile.

[0013] Preferably, the flight performance parameters and engine performance parameters include engine speed, normal overload coefficient, pitch angle, roll angle, heading angle, low-pressure turbine outlet temperature, high-pressure turbine inlet temperature, total temperature, altitude, and Mach number.

[0014] Preferably, the heading change rate range for level flight is (-1, 1) and the altitude change rate range is (-2, 2);

[0015] The rate of change of altitude for ascending categories is greater than 2;

[0016] The rate of change of altitude for descent-type gliding is less than 2.

[0017] The rate of change of course during turns is greater than 2.5;

[0018] The rate of change of the roll tilt angle is greater than 20.

[0019] Preferably, the calculation of the correlation coefficient between the flight parameters and engine parameters of the mission segment includes calculating the correlation coefficient between the two sets of sample data using formula (1);

[0020]

[0021] Where i represents the nth group, n represents the total number of samples, and x i Data values ​​in a set of samples, y is the mean of the first group of samples; i These are the data values ​​from the second set of samples. denoted as the mean of the second group of samples; r is the correlation coefficient between the two groups of sample data; when r > 0, it indicates a positive correlation between the two variables, and when r < 0, it indicates a negative correlation between the two variables; when |r| = 1, it indicates a perfect linear correlation between the two variables, that is, they satisfy the functional relationship.

[0022] Preferably, the complex motion recognition method includes: combining flight parameters based on basic motions into complex overall flight motions according to cumulative changes, range of changes, sequence, etc.; recognizing the time sequence of basic flight motions; determining the time sequence of complex motions based on the composition structure and sequence of basic motions; and refining the recognized time based on the cumulative changes of parameters during the complex motion process to obtain the complex motion.

[0023] The technical effects and advantages of this invention are as follows: This method for calculating the correlation coefficient of aero-engine parameters based on mission segments, by dividing the flight maneuver mission segments before calculating the correlation coefficient, can more accurately obtain the flight / engine correlation of the load spectrum. Based on the flight maneuver mission segments, it establishes the relationship between the aero-engine load spectrum and flight / engine parameters. Combined with the correlation coefficient calculation method, it calculates the correlation coefficient between aero-engine flight parameters and engine parameters. This method can be applied to aero-engine stress analysis, engine performance analysis, life study, and the compilation of subsequent test spectra, laying the foundation for aero-engines. It has high engineering application value, and the statistical calculation of the correlation coefficient between aero-engine aircraft parameters and engine parameters has significant engineering implications for aero-engine load spectrum prediction and engine analysis. Attached Figure Description

[0024] Figure 1 This is a flowchart illustrating an embodiment of the present invention;

[0025] Figure 2 This is a schematic diagram of the flight motion recognition of the present invention;

[0026] Figure 3 This is a schematic diagram illustrating the complex flight maneuver recognition method of the present invention. Detailed Implementation

[0027] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0028] To address the issue that most components of an aero-engine bear multiple loads during actual operation, and that load parameters are strongly correlated with flight / engine performance parameters, the correlation coefficient method based on mathematical statistics was used. The results shown in Table 1 are as follows: the correlation between altitude Mach number and speed and normal overload coefficient are low, which do not conform to the actual use of aero-engines.

[0029] Table 1. Results of correlation coefficient calculation based on mathematical statistics.

[0030]

[0031] This invention provides, for example Figure 1 The method for calculating the correlation coefficient of aero-engine parameters based on a mission segment, as shown, specifically includes the following steps:

[0032] S1. A certain type of aero-engine is divided into mission segments based on flight maneuvers. A complex action recognition method based on basic actions is adopted to divide it into six typical flight maneuver mission segments.

[0033] S2. Based on the division results of step S1, splice together the data of various typical mission segments and statistically analyze various flight performance parameters and engine performance parameters.

[0034] S3. Based on the statistical results of step S2, calculate the correlation coefficients of various flight performance parameters and engine performance parameters, and finally obtain the correlation between the flight performance parameters and engine performance parameters of the aero-engine.

[0035] In step S1, based on the task segmentation of flight maneuvers, a complex maneuver recognition method based on basic maneuvers is adopted to identify flight maneuvers on the measured load spectrum of a certain type of aero-engine. This mainly assesses the pilot's completion of various flight maneuvers by detecting the range of change of one or two flight parameter values. If the range of change does not exceed the parameter value requirements of the specified flight maneuver, the operation is considered qualified. However, in actual operation, the aircraft's flight maneuvers depend on multiple parameters. For example, climb maneuvers depend on parameters such as flight altitude, speed, pitch angle, roll angle, and pitch rate; cruise maneuvers depend on parameters such as flight altitude, speed, roll angle, and roll angle.

[0036] The specific steps of the task segment division standard based on flight actions mentioned in step S1 are as follows:

[0037] S11. After preprocessing the flight mission profile of a certain type of engine, the flight mission segment profile is divided into takeoff and landing mission segments, and the intermediate mission segments are extracted for the identification of flight actions.

[0038] S12. After in-depth analysis of the characteristics of complex maneuver parameters in actual flight, combined with the experience of domain experts and the principle of ease of use, all complex maneuvers can be summarized into 5 basic maneuver categories by the rate of change of heading angle, pitch angle, bank angle and altitude: level flight, climb, glide, turn and roll, as shown in Table 2 below.

[0039] Table 2 Flight parameter range for basic maneuvers

[0040]

[0041] S13. Based on the parameter ranges of the above basic actions, MATLAB is used to perform basic action recognition on the measured flight mission profile of a certain type of turbofan engine. Figure 2 This is a single profile recognition result. Circles represent level flight maneuvers, triangles represent ascent maneuvers, rectangles represent descent maneuvers, trapezoids represent turns, and diamonds represent roll maneuvers. Due to the large number of profiles, only one flight mission profile recognition result is shown below. Figure 2 As shown;

[0042] Turbofan engines are primarily installed on aircraft with complex flight missions, such as fighter jets. These aircraft perform not only basic maneuvers but also complex combinations of these basic maneuvers, reflecting the real-world application of complex flight missions in fighter jets. This necessitates further refining complex flight maneuvers into simpler maneuvers. Based on the flight parameters of these basic maneuvers, complex overall flight maneuvers are then combined according to their cumulative changes, range of change, and sequence. The first step is to identify the time sequence of the basic flight maneuvers. The second step is to determine the time sequence of the complex maneuvers based on their composition and sequence. The third step is to refine the time sequence identified in the second step based on the cumulative changes in parameters during the complex maneuver, thus deriving the complex maneuver. The breakdown of complex maneuvers into basic maneuvers is shown in the table below.

[0043] Table 3. Breakdown of Complex Flight Maneuvers

[0044] hovering A constant-altitude turn greater than 90° but less than one full turn Turning Ascent Turn Ascend and turn until the cumulative change in track angle reaches 90° or greater. Ascent + Turn Descent Turn Descending and turning until the cumulative change in track angle reaches 90° or greater. Descent + Turn

[0045] Based on the above flight maneuver identification of an aircraft equipped with a certain type of turbofan engine, it can be found that the use of turbofan engines is relatively complex. Since the data in this invention all originate from the same type of engine and the same flight training mission, six flight maneuvers can be used to describe the mission profile, such as... Figure 3As shown, after identification using a MATLAB program, typical flight maneuver segments are defined. Segments with only circular dots are defined as level flight segments; segments with only triangular dots are defined as ascent segments; segments with only rectangular dots are defined as descent segments; segments with only trapezoidal dots are defined as circling segments; segments combining trapezoidal, rhombus, and triangular dots are defined as ascent and turn segments; and segments combining rectangular, rhombus, and trapezoidal dots are defined as descent and turn segments. A total of six typical mission segments are derived from all typical mission profiles, using the following method: Figure 3 As shown;

[0046] In this embodiment, the statistical analysis of various flight parameters and engine parameters for different flight maneuver segments, and the subsequent splicing of the beginning and end of each type of flight maneuver segment, includes:

[0047] The study analyzed 10 parameters that characterize engine performance and flight performance for six typical flight mission segments, including pitch angle, roll angle, yaw angle, altitude, Mach number, high-pressure rotor speed, low-pressure turbine after-temperature, inlet temperature, total temperature, and normal overload coefficient.

[0048] The 10 flight / engine parameters for six typical flight mission segments were spliced ​​together using a splicing program; among them, the full names of the flight / engine parameters are flight performance parameters and engine performance parameters.

[0049] The correlation coefficients for flight parameters and engine parameters of various spliced ​​flight mission segments were calculated, including:

[0050] The correlation coefficient is an indicator that measures the degree of correlation between multiple parameters. It is generally represented by the letter r and its value is generally between -1 and 1. The correlation coefficient is not an interval measure, but an ordinal data. When calculating the correlation coefficient, a large sample size is required to obtain a more reliable correlation coefficient value. Formula (1) is used to calculate the correlation coefficient of two sets of sample data. When calculating the correlation coefficient of two sets of samples, it is necessary to ensure that the sample size of the two sets of data is the same.

[0051]

[0052] Where i represents the nth group, n represents the total number of samples, and x i Data values ​​in a set of samples, y is the mean of the first group of samples; i These are the data values ​​from the second set of samples. is the mean of the second group of samples; r is the correlation coefficient of the two groups of sample data; when r > 0, it indicates that the two variables are positively correlated, when r < 0, it indicates that the two variables are negatively correlated; when |r| = 1, it indicates that the two variables are perfectly linearly correlated, that is, they satisfy the functional relationship. Table 3 shows the correspondence between the correlation coefficient values ​​and the degree of correlation.

[0053] Table 4. Correlation Relationship Table

[0054] r＝1 Perfectly positive correlation r＝-1 Perfectly negative correlation |r|≥0.8 Highly relevant 0.5≤|r|＜0.8 Moderately related 0.3≤|r|＜0.5 Low correlation |r|＜0.3 The correlation is extremely weak. r＝0 Wireless correlation

[0055] Select the speed spectrum representing the aerodynamic load of a certain type of turbofan aero-engine from the measured load spectrum; the normal overload spectrum representing the engine's maneuver load; the pitch angle, track angle, and roll angle representing the changes in the aircraft's flight maneuvers; the low-pressure turbine afterburner gas temperature and total temperature representing the engine's performance; and the altitude and Mach number representing flight parameters. Extract all these load parameters from all flight mission profiles, and use the correlation coefficient calculation method based on mathematical statistics to treat the load parameters as linearly related variables. Use formula (1) to calculate the correlation coefficient. The results of the six typical flight mission segments are shown in Tables 5, 6, 7, 8, 9, and 10.

[0056] Table 5. Correlation Coefficients for Climbing Task Sections

[0057] Pitch angle 1 -0.04618 -0.6236 0.26625 -0.45577 -0.46186 -0.43573 0.11277 -0.07879 0.28795 Roll angle -0.04618 1 0.12761 -0.07762 -0.08428 -0.05816 0.07294 -0.28797 -0.38847 -0.02509 rotational speed -0.6236 0.12761 1 -0.63385 0.42606 0.45524 0.88503 -0.64233 -0.43771 -0.12321 Heading angle 0.26625 -0.07762 -0.63385 1 -0.71485 -0.76471 -0.90014 0.71946 0.34034 0.14417 high -0.45577 -0.08428 0.42606 -0.71485 1 0.99498 0.79987 -0.26847 0.24364 -0.04881 Mach number -0.46186 -0.05816 0.45524 -0.76471 0.99498 1 0.83781 -0.32344 0.18877 -0.05092 Exhaust temperature -0.43573 0.07294 0.88503 -0.90014 0.79987 0.83781 1 -0.6591 -0.2461 -0.12657 Inlet temperature 0.11277 -0.28797 -0.64233 0.71946 -0.26847 -0.32344 -0.6591 1 0.80389 0.13805 Total temperature -0.07879 -0.38847 -0.43771 0.34034 0.24364 0.18877 -0.2461 0.80389 1 0.11909 Normal overload 0.28795 -0.02509 -0.12321 0.14417 -0.04881 -0.05092 -0.12657 0.13805 0.11909 1

[0058] Table 6 Correlation Coefficients for the Circling Mission Segment

[0059] Pitch angle 1 -0.11686 0.24229 -0.22371 -- 0.00544 0.18404 -0.06581 -0.07418 -0.11907 Roll angle -0.11686 1 0.37364 -0.07216 -- -0.19413 0.45135 -0.46545 -0.42315 0.49942 rotational speed 0.24229 0.37364 1 -0.05204 -- -0.26961 0.95723 -0.4601 -0.40907 0.32409 Heading angle -0.22371 -0.07216 -0.05204 1 -- 0.03759 -0.05854 0.21366 0.21822 0.0737 high -- -- -- -- -- -- -- -- -- -- Mach number 0.00544 -0.19413 -0.26961 0.03759 -- 1 -0.36733 0.87648 0.8695 0.05296 Exhaust temperature 0.18404 0.45135 0.95723 -0.05854 -- -0.36733 1 -0.53833 -0.49321 0.40677 Inlet temperature -0.06581 -0.46545 -0.4601 0.21366 -- 0.87648 -0.53833 1 0.98642 -0.02611 Total temperature -0.07418 -0.42315 -0.40907 0.21822 -- 0.8695 -0.49321 0.98642 1 0.06327 Normal overload -0.11907 0.49942 0.32409 0.0737 -- 0.05296 0.40677 -0.02611 0.06327 1

[0060] Table 7 Correlation Coefficients for Level Flight Mission Segment

[0061]

[0062]

[0063] Table 8. Correlation Coefficients of the Descent Mission Segment

[0064] Pitch angle 1 0.33923 0.76849 -0.40733 0.49934 -0.32048 0.44766 -0.73055 -0.74273 0.03894 Roll angle 0.33923 1 0.24149 -0.3327 0.28801 -0.1887 0.18573 -0.35834 -0.36232 -0.01796 rotational speed 0.76849 0.24149 1 -0.35127 0.93635 0.47111 0.97421 -0.82805 -0.77933 -0.08887 Heading angle -0.40733 -0.3327 -0.35127 1 -0.42144 0.09068 -0.31243 0.46812 0.47436 0.00546 high 0.49934 0.28801 0.93635 -0.42144 1 0.80114 0.95898 -0.86162 -0.81857 -0.0969 Mach number -0.32048 -0.1887 0.47111 0.09068 0.80114 1 0.59535 -0.04679 0.02729 -0.03527 Exhaust temperature 0.44766 0.18573 0.97421 -0.31243 0.95898 0.59535 1 -0.79576 -0.74125 -0.09958 Inlet temperature -0.73055 -0.35834 -0.82805 0.46812 -0.86162 -0.04679 -0.79576 1 0.98784 0.09257 Total temperature -0.74273 -0.36232 -0.77933 0.47436 -0.81857 0.02729 -0.74125 0.98784 1 0.08919 Normal overload 0.03894 -0.01796 -0.08887 0.00546 -0.0969 -0.03527 -0.09958 0.09257 0.08919 1

[0065] Table 9. Correlation Coefficients for Ascent and Turning Sections

[0066] Pitch angle 1 -0.2624 -0.11568 0.28171 -0.18932 -0.71855 -0.0731 0.66213 0.65881 0.28561 Roll angle -0.2624 1 0.06307 0.20712 0.50018 0.63185 -0.01664 -0.3618 -0.35164 -0.8999 rotational speed -0.11568 0.06307 1 -0.04037 0.04711 0.20963 0.97901 0.01035 0.0162 -0.01645 Heading angle 0.28171 0.20712 -0.04037 1 0.05442 -0.26053 -0.01295 0.49429 0.49365 -0.1188 high -0.18932 0.50018 0.04711 0.05442 1 0.67061 -0.04144 -0.2486 -0.24258 -0.36341 Mach number -0.71855 0.63185 0.20963 -0.26053 0.67061 1 0.10789 -0.8551 -0.8493 -0.57646 Exhaust temperature -0.0731 -0.01664 0.97901 -0.01295 -0.04144 0.10789 1 0.11213 0.11789 0.05476 Inlet temperature 0.66213 -0.3618 0.01035 0.49429 -0.2486 -0.8551 0.11213 1 0.99955 0.40273 Total temperature 0.65881 -0.35164 0.0162 0.49365 -0.24258 -0.8493 0.11789 0.99955 1 0.39472 Normal overload 0.28561 -0.8999 -0.01645 -0.1188 -0.36341 -0.57646 0.05476 0.40273 0.39472 1

[0067] Table 10 Correlation Coefficients for the Descending Turning Section

[0068]

[0069]

[0070] The data in the table shows that the correlation coefficient between flight performance parameters and engine performance parameters is significantly larger than the correlation coefficient between altitude and Mach number obtained in Table 1. For example, Tables 6 and 7 are the correlation coefficient tables for hovering and level flight. Since the altitude is constant, no calculation results are available. At the same time, the correlation coefficients of flight parameters such as pitch angle and roll angle under level flight maneuvers are relatively small with other parameters. Therefore, it can be concluded that it is reasonable and accurate to calculate the correlation coefficients of flight and engine parameters after classifying flight maneuvers, which is consistent with the actual use of aero engines.

[0071] The technical effects and advantages of this invention are as follows: This method for calculating the correlation coefficient of aero-engine parameters based on the mission segment establishes the relationship between the aero-engine load spectrum and flight / engine parameters based on the flight maneuver mission segment. Combined with the correlation coefficient calculation method, it calculates the correlation coefficient between the aero-engine flight parameters and engine parameters. This method can be applied to aero-engine stress analysis, engine performance analysis, life study, and lays the foundation for the compilation of subsequent test spectra. It has high engineering application value, and the statistical calculation of the correlation coefficient between aero-engine aircraft parameters and engine parameters has significant engineering implications for aero-engine load spectrum prediction and engine analysis.

[0072] Although preferred embodiments of this application have been described, those skilled in the art, upon learning the basic inventive concept, can make other changes and modifications to these embodiments. Therefore, the appended claims are intended to be interpreted as including the preferred embodiments as well as all changes and modifications falling within the scope of this application.

[0073] Obviously, those skilled in the art can make various modifications and variations to this application without departing from the spirit and scope of this application. Therefore, if such modifications and variations fall within the scope of the claims of this application and their equivalents, this application also intends to include such modifications and variations.

[0074] Finally, it should be noted that the above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art can still modify the technical solutions described in the foregoing embodiments or make equivalent substitutions for some of the technical features. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A method for calculating the correlation coefficient of aero-engine parameters based on a mission segment, characterized in that: include: Task segments are divided based on flight maneuvers; The data from the first and last segments of the mission are spliced ​​together to statistically analyze the flight performance parameters and engine performance parameters of the mission segment. The correlation coefficient between the flight performance parameters and engine performance parameters of the mission segment is calculated to obtain the correlation between the flight performance parameters and engine performance parameters of the aero-engine. The mission segment includes six types of typical flight maneuver mission segments; specifically, it includes: ascent flight maneuver mission segment, descent flight maneuver mission segment, level flight maneuver mission segment, circling flight maneuver mission segment, ascent and turn flight maneuver mission segment, and descent and turn flight maneuver mission segment. The criteria for classifying flight maneuvers include: The flight mission profile is preprocessed, and the flight mission segment profile is divided into takeoff and landing mission segments. The intermediate mission segments are extracted for the identification of flight actions. Based on the rate of change of heading angle, pitch angle, bank angle and altitude, complex maneuvers are divided into 5 basic maneuvers: level flight, climb, glide, turn and roll. Based on the parameter ranges of the basic actions, MATLAB was used to identify the basic actions of the engine in the measured flight mission profile. The heading change rate range for level flight is (-1° / s, 1° / s) and the altitude change rate range is (-2m / s, 2m / s); The rate of change of altitude for ascending categories is greater than 2 m / s; The rate of change of altitude for gliding types is less than 2 m / s; The rate of change of heading during turns is greater than 2.5° / s; Roll-type tilt angle change rate is greater than 20° / s; The complex motion recognition method includes: based on the flight parameters of basic motions, combining them into complex overall flight motions according to the cumulative changes, range of changes, and sequence; recognizing the time sequence of basic flight motions; determining the time sequence of complex motions according to the composition structure and sequence of basic motions; and refining the recognized time based on the cumulative changes of parameters during the complex motion process to obtain the complex motion.

2. The method for calculating the correlation coefficient of aero-engine parameters based on a mission segment according to claim 1, characterized in that: The flight performance parameters and engine performance parameters include engine speed, normal overload coefficient, pitch angle, roll angle, heading angle, low-pressure turbine outlet temperature, high-pressure turbine inlet temperature, total temperature, altitude, and Mach number.

3. The method for calculating the correlation coefficient of aero-engine parameters based on a mission segment according to claim 1, characterized in that: The calculation of the correlation coefficient between the flight performance parameters and engine performance parameters of the mission segment includes: The correlation coefficient between the two sets of sample data is calculated using formula (1); Where i represents the nth group, n represents the total number of samples, and x i Data values ​​in a set of samples, y is the mean of the first group of samples; i These are the data values ​​from the second set of samples. denoted as the mean of the second group of samples; r is the correlation coefficient between the two groups of sample data; when r > 0, it indicates a positive correlation between the two variables, and when r < 0, it indicates a negative correlation between the two variables; when |r| = 1, it indicates a perfect linear correlation between the two variables, that is, they satisfy the functional relationship.

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