Target test model training method and device
A technology for testing models and targets, applied in the field of neural networks, can solve problems such as the influence of equipment temperature, achieve the effect of improving accuracy and reducing influence
Pending Publication Date: 2021-03-12
BEIJING AEROSPACE MEASUREMENT & CONTROL TECH
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AI-Extracted Technical Summary
Problems solved by technology
[0003] The purpose of the embodiment of the present application is to provide a training method ...
Method used
In this application, by carrying out iterative genetic operation to the individuality of initial test model, select optimal individuality, therefore can reduce the influence of temperature on pressure, improve the precision of pressure measurement, have solved high-density array type wind tunnel air pressure The problem of ...
Abstract
The invention provides a target test model training method and device, and belongs to the technical field of neural networks. The method comprises the following steps: inputting normalized numerical values of a sample temperature value and a sample pressure value into an initial test model, and obtaining a test pressure value output by the initial test model; determining a current fitness value ofan individual in a current group of the initial test model, the individual being a current weight value and a current threshold value of the initial test model, and an error value between the test pressure value and a preset pressure value under the current weight value and the current threshold value being inversely proportional to the current fitness value; if the current fitness value does notmeet the termination condition, performing iterative genetic operation on the individual of the initial test model until the current fitness value of the individual meets the termination condition toobtain a target weight and a target threshold; and taking the initial test model containing the target weight and the target threshold as a target test model. The influence of the temperature on thepressure can be reduced, and the precision of pressure measurement is improved.
Application Domain
Character and pattern recognitionArtificial life +3
Technology Topic
Computational physicsTarget weight +4
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Examples
- Experimental program(1)
Example Embodiment
[0059] In order to make the purposes, technical solutions and advantages of the embodiments of the present application clearer, the technical solutions in the embodiments of the present application will be described clearly and completely below with reference to the drawings in the embodiments of the present application. Obviously, the described embodiments It is a part of the embodiments of this application, but not all of the embodiments. Based on the embodiments in the present application, all other embodiments obtained by those of ordinary skill in the art without creative work shall fall within the protection scope of the present application.
[0060] In the following description, suffixes such as 'module', 'component' or 'unit' used to represent elements are used only to facilitate the description of the present application, and have no specific meaning per se. Therefore, "module" and "component" can be used interchangeably.
[0061] In order to solve the problems mentioned in the background art, according to an aspect of the embodiments of the present application, an embodiment of a training method of a target test model is provided.
[0062] Optionally, in the embodiment of the present application, the training method of the above-mentioned target test model can be applied to figure 1 In the hardware environment shown by the terminal 101 and the server 103. like figure 1 As shown, the server 103 is connected to the terminal 101 through the network, which can be used to provide services for the terminal or the client installed on the terminal, and the database 105 can be set on the server or independent of the server to provide data storage services for the server 103. The network includes but is not limited to: a wide area network, a metropolitan area network or a local area network, and the terminal 101 includes but is not limited to a PC, a mobile phone, a tablet computer, and the like.
[0063] A method for training a target test model in this embodiment of the present application may be executed by the server 103 , or may be executed jointly by the server 103 and the terminal 101 .
[0064] The embodiment of the present application provides a training method for a target test model, which can be applied to a server for constructing a target test model.
[0065] The following will describe in detail the training method of a target test model provided by the embodiment of the present application in conjunction with the specific implementation manner. figure 2 shown, the specific steps are as follows:
[0066] Step 201: Input the normalized values of the sample temperature value and the sample pressure value into the initial test model, and obtain the test pressure value output by the initial test model.
[0067] The server obtains the normalized values of the sample temperature value and the sample pressure value, and then inputs the normalized values into the initial test model to obtain the test pressure value output by the initial test model. The initial test model may be a BP (BackPropagation, a multilayer feedforward network trained by an error back propagation algorithm) neural network.
[0068] Exemplarily, the service uses a high and low temperature box to control temperature changes, sets temperature environments of 28°C, 35°C, 45°C, 52°C, and 61°C, and uses a standard air pressure source to provide the measurement device in these 5 different temperature environments. 0kPa, 20kPa, 40kPa, 60kPa, 80kPa standard pressure, a total of 25 sets of output results at different temperatures and different standard pressures can be obtained as the sample temperature value and the sample pressure value.
[0069] Step 202: Determine the current fitness value of the individuals in the current group of the initial test model.
[0070] The individual is the current weight and current threshold of the initial test model, and the error value between the test pressure value and the preset pressure value under the current weight and current threshold is inversely proportional to the current fitness value.
[0071] The initial test model includes an input layer, a hidden layer and an output layer, wherein the number of hidden layers is at least one. There are weights and thresholds between the input layer and the hidden layer, and there are weights and thresholds between the hidden layer and the output layer. The weights and thresholds between different model layers are different, and the weights and thresholds can also be changed. , different weights and thresholds correspond to different test pressure values.
[0072] The server determines the current weights and current thresholds of each model layer in the initial test model, and uses the current weights and current thresholds as an individual, and then determines its current fitness value for the individual. The initial test model is under the current weights and current thresholds The test pressure value is output, and the server determines the error value between the test pressure value and the preset pressure value. The smaller the error value is, the higher the current fitness value of the individual is.
[0073] Among them, the calculation formula of the error value is:
[0074] e=y n (j)-y(j), where e is the error value, y(j) is the test pressure value; y n (j) is the preset pressure value;
[0075] The formula for readjusting the current weights according to the error value is:
[0076] Among them, η is the learning efficiency, α is the situation factor, is the i+1th weight after the i-th weight is updated.
[0077] The formula for readjusting the current threshold based on the error value is:
[0078] Among them, η is the learning efficiency, α is the situation factor, is the i+1th threshold after the update of the ith threshold.
[0079] Step 203: If the current fitness value does not meet the termination condition, perform an iterative genetic operation on the individual of the initial test model until the current fitness value of the individual satisfies the termination condition, and obtain the target weight and target threshold.
[0080] The server judges whether the current fitness value satisfies the termination condition, and if the server judges that the current fitness value does not satisfy the termination condition, it indicates that under the current weight and the current threshold, the error value between the test pressure value and the preset pressure value is not less than the target error value, it is necessary to perform an iterative genetic operation on the individual of the initial test model to improve the fitness value of the individual until the current fitness value of the individual meets the termination condition, and the target weight and target threshold are obtained.
[0081] If the server determines that the current fitness value satisfies the termination condition, indicating that under the current weight and the current threshold, the error value between the test pressure value and the preset pressure value is less than the target error value, then the iterative inheritance of the individuals of the initial test model is stopped. operate.
[0082] Step 204: Use the initial test model containing the target weights and target thresholds as the target test model.
[0083] Under the target weight and target threshold, the current fitness value of the individual satisfies the termination condition, so the initial test model containing the target weight and target threshold is used as the target test model.
[0084] In this application, by performing iterative genetic operation on the individuals of the initial test model to select the optimal individual, the influence of temperature on the pressure can be reduced, the accuracy of pressure measurement can be improved, and the air pressure of the high-density array wind tunnel pressure measurement device can be solved. The problem of temperature drift during measurement.
[0085] As an optional implementation, as image 3 As shown, if the current fitness value does not meet the termination condition, the iterative genetic operation on the individuals of the initial test model includes:
[0086] Step 301: If the current fitness value does not meet the termination condition, select the candidate individuals in the current population through the fitness function. Wherein, the current population includes multiple individuals, and the individual to be selected is an individual whose current fitness value is higher than the preset fitness value.
[0087] If the server determines that the current fitness value does not meet the termination condition, it calculates the current fitness value of all individuals in the current population through the fitness function, and selects the individual whose current fitness value is higher than the preset fitness value as the candidate to be selected.
[0088] Wherein, determining the current fitness value of the individuals in the current group includes: determining the sum of the fitting residuals of all individuals and the number of individuals in the current group; and taking the ratio of the sum value to the number of individuals as the current fitness value.
[0089] The formula for calculating the fitness value is:
[0090] where f min is the current fitness value, is the fitting residual of the i-th individual, and N is the number of individuals.
[0091] Step 302: Perform genetic operations of crossover, mutation and replication on the individual to be selected to obtain the target individual.
[0092] The server selects two individuals from the individual to be selected for crossover operation, obtains the crossed individual, and then mutates the individual, which is beneficial to the algorithm to find the global optimal individual. In order to retain the optimal individual, the optimal individual can be replicated Inherited to the next generation to get the target individual. Illustratively, the roulette method can be used to replicate.
[0093] Step 303: Form a sub-population with multiple target individuals, and use the sub-population as the current population.
[0094] The server forms a sub-population according to multiple target individuals, and takes the sub-population as the current population. The server continues to calculate the current fitness value of the individuals in the current population until the current fitness value satisfies the termination condition.
[0095] In this application, the server calculates the current fitness values of all individuals through the fitness function, and then determines the candidates whose current fitness values are higher than the preset fitness value, and performs the genetic operations of crossover, mutation and replication on the to-be-selected individuals, In this way, the individual with the best fitness value can be obtained by continuous iteration. Optimizing the BP neural network based on the genetic algorithm can effectively reduce the influence of temperature on the device measurement, improve the accuracy of the measurement device, and the algorithm is more robust.
[0096] As an optional implementation manner, after selecting the individual to be selected in the current population through the fitness function, the method further includes: performing an encoding operation on the individual to be selected to obtain the encoded individual to be selected; determining the current population of the initial test model Before the current fitness value of the individuals in the group, the method further includes: performing a decoding operation on the current group.
[0097] The server performs an encoding operation on the individual to be selected to obtain a compiled sub-population, and then decodes the sub-population to facilitate the calculation of the current fitness value of the current individual in the sub-population.
[0098] As an optional embodiment, after performing the iterative genetic operation on the individual of the initial test model, the method further includes: adding one to the original iteration number to obtain the current iteration number; if the current iteration number satisfies the termination condition, stopping Perform an iterative genetic operation on the individuals of the initial test model; take the weight and threshold of the current iteration number as the target weight and target threshold.
[0099] After the server forms a sub-population, it completes one iteration, and then adds one to the original number of iterations to obtain the current number of iterations. If the server determines that the current number of iterations satisfies the termination condition, it indicates that the number of iterations is sufficient, and the current weight and current threshold can be as the target weight and target threshold.
[0100] Figure 4 Flowchart for iterative genetic operations. like Figure 4As shown, determine the initial weights and initial thresholds of the BP neural network, encode the initial weights and initial thresholds to obtain initial individuals, build an initial population based on all initial individuals, decode the initial population, and perform fitness on all initial individuals value calculation, and judge whether the current fitness value satisfies the termination condition, if the server determines that the current fitness value satisfies the termination condition, then constructs the target test model based on the target weight and target threshold. If the server determines that the current fitness value does not meet the termination condition, it selects the candidate individuals whose fitness value is higher than the preset fitness value in the initial individuals, and performs the genetic operations of crossover, mutation and replication on the candidate individuals, and the obtained target individuals constitute The subpopulation is completed, which completes an iteration. The server continues to calculate the fitness value of the current individual in the subpopulation until the current fitness value or the number of iterations satisfies the termination condition. The termination condition includes that the current fitness value reaches the target fitness value or the current number of iterations reaches the target number of times.
[0101] As an optional implementation manner, before inputting the normalized values of the sample temperature value and the sample pressure value into the initial test model, the method further includes: acquiring multiple sample temperature values and multiple sample pressure values sent by the target sensor; Determine the maximum output calibration value, the minimum output calibration value and the input and output calibration value of the current sample of the target sensor; determine the first difference between the input and output calibration value and the minimum output calibration value, the second difference between the maximum output calibration value and the minimum output calibration value Difference; take the ratio of the first difference to the second difference as the normalized value of the input and output of the current sample.
[0102] During the pressure test experiment of the high-density array wind tunnel air pressure measurement device, the temperature and pressure are detected by multiple sensors, and the server obtains multiple sample temperature values and multiple sample pressure values sent by the target sensor. One of the sensors, the target sensor detects multiple sample temperature values and multiple sample pressure values. The server determines the maximum output calibration value and the minimum output calibration value sent by the target sensor, and the input output calibration value of one of the samples sent by the target sensor. The server determines the first difference between the input and output calibration values and the minimum output calibration value, and the second difference between the maximum output calibration value and the minimum output calibration value; and uses the ratio of the first difference to the second difference as the input and output of the current sample. Normalized values.
[0103] The calculation formula of the normalized value of the input and output is:
[0104]
[0105] in, Input and output normalized values for the mth sample neural network; X im is the input and output calibration value of the i-th sensor of the m-th sample; X imax , X imin Output the maximum and minimum calibration values for the ith sensor.
[0106] As an optional implementation manner, after taking the ratio of the first difference value and the second difference value as the normalized value of the input and output of the current sample, the method further includes: determining the normalized value and the input layer in the original test model and the first weight, the first threshold and the first transfer function between the hidden layer, and determine the first output of the hidden layer according to the normalized value, the first weight, the first threshold and the first transfer function value; determine the second weight, the second threshold and the second transfer function before the hidden layer and the output layer in the original test model, and according to the first output value, the second weight, the second threshold and the second transfer function, Determine the second output value of the output layer; build an initial test model according to the normalized value and the second output value.
[0107] Each model layer of the BP neural network model may include at least one node, for example, the input layer has 2 nodes, the hidden layer has 5 nodes, and the output layer has 1 node. There is a transfer function between the input layer and the hidden layer, and between the hidden layer and the output layer, for example, the transfer function is a tansig function. Figure 5 Schematic diagram of testing the model for the target.
[0108] The formula for calculating the first output value of the hidden layer is:
[0109]
[0110] Among them, Z K is the first output value, f 1 ( ) is the first transfer function between the input layer and the hidden layer, w ki , θ k are the weights and thresholds between the input layer and the hidden layer, respectively, x i is the normalized value of input and output, n and q are the number of nodes in the input layer and hidden layer respectively, k is the kth node of the hidden layer,
[0111] The calculation formula of the second output value of the output layer is:
[0112]
[0113] where y j is the second output value, f 2 ( ) is the second transfer function between the hidden layer and the output layer, w jk , θ j are the weights and thresholds between the hidden layer and the output layer, respectively, m is the number of nodes in the output layer, j is the jth node of the output layer,
[0114] As an optional implementation manner, after using the initial test model containing the target weight and the target threshold as the target test model, the method further includes: inputting the normalized values of the target temperature value and the target pressure value into the target test model, And obtain the first pressure value output by the target test model; denormalize the first pressure value to obtain the final pressure value after temperature compensation.
[0115] The server embeds the target test model into the software of the host computer of the high-density array wind tunnel air pressure measurement device, and then inputs the normalized values of the target temperature value and target pressure value output by the high-density array wind tunnel air pressure measurement device into the target test model, the target test model outputs the first pressure value after calculation, and the server denormalizes the first pressure value to obtain the final pressure value after temperature compensation.
[0116] The present application uses the software compensation method to perform temperature compensation on the high-density array wind tunnel air pressure measurement device, which can reduce the complexity and cost of the device hardware.
[0117] Figure 6-1 It is a schematic diagram of the processing time of the BP neural network. Figure 6-2 Schematic diagram of the processing time for optimizing a BP neural network for a genetic algorithm. It can be seen from the figure that the genetic algorithm optimized BP neural network has faster convergence speed and higher data fusion accuracy.
[0118] Figure 7-1 It is a schematic diagram of the output pressure value of the calibrated pressure of 40kPa. Figure 7-2 It is a schematic diagram of the output pressure value of the calibrated pressure of 80kPa. It can be seen from the figure that under two pressures, the present application can effectively suppress the influence of temperature on the measuring device.
[0119] Figure 8-1 It is a schematic diagram of the mean absolute error of the pressure value under each channel. Figure 8-2 It is a schematic diagram of the maximum absolute error of the pressure value under each channel. It can be seen from the figure that the output accuracy of the measurement device has been improved after temperature compensation.
[0120] Based on the same technical concept, the embodiment of the present application also provides a training device for a target test model, such as Figure 9 As shown, the device includes:
[0121] An input-output module 901, configured to input the normalized values of the sample temperature value and the sample pressure value into the initial test model, and obtain the test pressure value output by the initial test model;
[0122] The first determination module 902 is used to determine the current fitness value of the individual in the current group of the initial test model, wherein the individual is the current weight and the current threshold of the initial test model, and the test pressure is under the current weight and the current threshold The error value between the value and the preset pressure value is inversely proportional to the current fitness value;
[0123] The iteration module 903 is configured to perform an iterative genetic operation on the individual of the initial test model if the current fitness value does not meet the termination condition, until the current fitness value of the individual satisfies the termination condition, and obtain the target weight and the target threshold;
[0124] The first is as a module 904, which is used to use the initial test model containing the target weights and target thresholds as the target test model.
[0125] Optionally, the iteration module 903 includes:
[0126] The selection unit is used to select the candidate individuals in the current population through the fitness function if the current fitness value does not meet the termination condition, wherein the current population includes multiple individuals, and the candidate individuals are those whose current fitness value is higher than the expected value. Set the fitness value of the individual;
[0127] The operation unit is used to perform genetic operations of crossover, mutation and replication on the selected individuals to obtain the target individual;
[0128] The forming unit is used to form a sub-population with multiple target individuals, and use the sub-population as the current population.
[0129] Optionally, the device further includes:
[0130] The encoding module is used to perform encoding operation on the individual to be selected to obtain the encoded individual to be selected;
[0131] The decoding module is used for decoding the current group.
[0132] Optionally, the device further includes:
[0133] an acquisition module for acquiring multiple sample temperature values and multiple sample pressure values sent by the target sensor;
[0134] The second determination module is used to determine the maximum output calibration value, the minimum output calibration value and the input and output calibration value of the current sample of the target sensor;
[0135] a third determination module, configured to determine the first difference between the input and output calibration values and the minimum output calibration value, and the second difference between the maximum output calibration value and the minimum output calibration value;
[0136] The second is as a module, which is used for taking the ratio of the first difference value and the second difference value as the normalized value of the input and output of the current sample.
[0137] Optionally, the device further includes:
[0138] The fourth determination module is used for determining the normalized value and the first weight, the first threshold and the first transfer function between the input layer and the hidden layer in the original test model, and according to the normalized value, the first weight value, the first threshold and the first transfer function to determine the first output value of the hidden layer;
[0139] The fifth determination module is used for determining the second weight, the second threshold and the second transfer function before the hidden layer and the output layer in the original test model, and according to the first output value, the second weight, the second threshold and the The second transfer function determines the second output value of the output layer;
[0140] A building block for building an initial test model based on the normalized value and the second output value.
[0141] Optionally, the device further includes:
[0142] Add a module to add one to the original number of iterations to get the current number of iterations;
[0143] The stop module is used to stop the iterative genetic operation on the individuals of the initial test model if the current number of iterations satisfies the termination condition;
[0144] The third is as a module, which is used to use the weight and threshold of the current number of iterations as the target weight and target threshold.
[0145] Optionally, the first determining module 902 includes:
[0146] A determination unit for determining the sum of the fitting residuals of all individuals and the number of individuals in the current group;
[0147] As a unit, it is used to take the ratio of the sum value to the number of individuals as the current fitness value.
[0148] Optionally, the device further includes:
[0149] The input module is used to input the normalized values of the target temperature value and the target pressure value into the target test model, and obtain the first pressure value output by the target test model;
[0150] The denormalization module is used to denormalize the first pressure value to obtain the final pressure value after temperature compensation.
[0151] According to another aspect of the embodiments of the present application, the present application provides an electronic device, as shown in FIG. 6 , including a memory 1003 , a processor 1001 , a communication interface 1002 and a communication bus 1004 . The computer program running on 1001, the memory 1003, the processor 1001 communicate with the communication bus 1004 through the communication interface 1002, and the processor 1001 implements the steps of the above method when executing the computer program.
[0152] The memory and the processor in the above electronic device communicate through a communication bus and a communication interface. The communication bus may be a Peripheral Component Interconnect (PCI for short) bus or an Extended Industry Standard Architecture (Extended Industry Standard Architecture, EISA for short) bus or the like. The communication bus can be divided into an address bus, a data bus, a control bus, and the like.
[0153] The memory may include random access memory (Random Access Memory, RAM for short), and may also include non-volatile memory (non-volatile memory), such as at least one disk memory. Optionally, the memory may also be at least one storage device located away from the aforementioned processor.
[0154] The above-mentioned processor may be a general-purpose processor, including a central processing unit (Central Processing Unit, referred to as CPU), a network processor (Network Processor, referred to as NP), etc.; may also be a digital signal processor (Digital Signal Processing, referred to as DSP) , Application Specific Integrated Circuit (ASIC for short), Field-Programmable Gate Array (FPGA for short) or other programmable logic devices, discrete gate or transistor logic devices, and discrete hardware components.
[0155] According to yet another aspect of the embodiments of the present application, there is also provided a computer-readable medium having non-volatile program code executable by a processor.
[0156] Optionally, for specific examples in this embodiment, reference may be made to the examples described in the foregoing embodiments, and details are not described herein again in this embodiment.
[0157] When the embodiments of the present application are specifically implemented, reference may be made to the above-mentioned embodiments, which have corresponding technical effects.
[0158] It will be appreciated that the embodiments described herein may be implemented in hardware, software, firmware, middleware, microcode, or a combination thereof. For hardware implementation, the processing unit may be implemented in one or more Application Specific Integrated Circuits (ASIC), Digital Signal Processing (DSP), Digital Signal Processing Device (DSP Device, DSPD), programmable logic Devices (Programmable Logic Device, PLD), Field-Programmable Gate Array (Field-Programmable Gate Array, FPGA), general purpose processors, controllers, microcontrollers, microprocessors, other electronic units for performing the functions described in this application or a combination thereof.
[0159] For a software implementation, the techniques described herein may be implemented by means of units that perform the functions described herein. Software codes may be stored in memory and executed by a processor. The memory can be implemented in the processor or external to the processor.
[0160] Those of ordinary skill in the art can realize that the units and algorithm steps of each example described in conjunction with the embodiments disclosed herein can be implemented in electronic hardware, or a combination of computer software and electronic hardware. Whether these functions are performed in hardware or software depends on the specific application and design constraints of the technical solution. Skilled artisans may implement the described functionality using different methods for each particular application, but such implementations should not be considered beyond the scope of this application.
[0161] Those skilled in the art can clearly understand that, for the convenience and brevity of description, the specific working process of the above-described systems, devices and units may refer to the corresponding processes in the foregoing method embodiments, which will not be repeated here.
[0162] In the embodiments provided in this application, it should be understood that the disclosed apparatus and method may be implemented in other manners. For example, the apparatus embodiments described above are only illustrative. For example, the division of the modules is only a logical function division. In actual implementation, there may be other division methods. For example, multiple modules or components may be combined or Can be integrated into another system, or some features can be ignored, or not implemented. On the other hand, the shown or discussed mutual coupling or direct coupling or communication connection may be through some interfaces, indirect coupling or communication connection of devices or units, and may be in electrical, mechanical or other forms.
[0163] The units described as separate components may or may not be physically separated, and components displayed as units may or may not be physical units, that is, may be located in one place, or may be distributed to multiple network units. Some or all of the units may be selected according to actual needs to achieve the purpose of the solution in this embodiment.
[0164] In addition, each functional unit in each embodiment of the present application may be integrated into one processing unit, or each unit may exist physically alone, or two or more units may be integrated into one unit.
[0165] The functions, if implemented in the form of software functional units and sold or used as independent products, may be stored in a computer-readable storage medium. Based on this understanding, the technical solutions of the embodiments of the present application can be embodied in the form of software products in essence, or the parts that make contributions to the prior art or the parts of the technical solutions, and the computer software products are stored in a storage medium , including several instructions for causing a computer device (which may be a personal computer, a server, or a network device, etc.) to execute all or part of the steps of the methods described in the various embodiments of the present application. The aforementioned storage medium includes: a U disk, a removable hard disk, a ROM, a RAM, a magnetic disk, or an optical disk and other mediums that can store program codes. It should be noted that, in this document, relational terms such as "first" and "second" etc. are only used to distinguish one entity or operation from another entity or operation, and do not necessarily require or imply these There is no such actual relationship or sequence between entities or operations. Moreover, the terms "comprising", "comprising" or any other variation thereof are intended to encompass a non-exclusive inclusion such that a process, method, article or device that includes a list of elements includes not only those elements, but also includes not explicitly listed or other elements inherent to such a process, method, article or apparatus. Without further limitation, an element qualified by the phrase "comprising a..." does not preclude the presence of additional identical elements in a process, method, article or apparatus that includes the element.
[0166] The above descriptions are only specific embodiments of the present application, so that those skilled in the art can understand or implement the present application. Various modifications to these embodiments will be readily apparent to those skilled in the art, and the generic principles defined herein may be implemented in other embodiments without departing from the spirit or scope of the present application. Therefore, this application is not intended to be limited to the embodiments shown herein, but is to be accorded the widest scope consistent with the principles and novel features claimed herein.
PUM


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