Composite beam reliability optimization method based on failure probability function

Abstract

The invention provides a composite beam reliability optimization method based on a failure probability function. The method comprises the following steps: 1) defining a random input variable, a design variable, a target function and a performance function; 2) calculating a failure sample and an extended failure probability based on an autonomous learning Kriging agent model and a Monte Carlo method according to the Bayesian theory; the method comprises the steps of (1) calculating a design parameter failure sample, (2) calculating a joint probability density implicit function on the basis of an adaptive K-nearest neighbor method and autonomous learning Kriging according to the calculated design parameter failure sample, and (3) calculating a failure probability implicit function on the basis of the Bayesian theorem.By means of the method, the problem that in the reliability optimization process, the calculation cost is difficult to bear due to reliability analysis is solved.

Description

Technical Field

[0001] This invention belongs to the field of composite beam reliability optimization, specifically relating to a composite beam reliability optimization method based on a failure probability function. Background Technology

[0002] In practical engineering, uncertainties are difficult to avoid. To address the impact of these uncertainties, the optimization design process must take them into account to design products that meet practical requirements. Reliability optimization design is a method to achieve the optimal design while ensuring a certain level of reliability. It balances economy and safety, and is an inevitable trend in optimization design development. Compared to traditional deterministic optimization design, it is more realistic and has gradually become a research hotspot in the optimization design of complex engineering systems, especially in safety-critical fields such as cranes and aviation, where it plays an irreplaceable role. The main method for solving reliability optimization problems is the double-loop method. In this method, the reliability optimization problem can be viewed as a nested loop problem, with the inner loop being a reliability analysis loop and the outer loop being a general constraint optimization loop. This means that each iteration in the optimization loop requires a reliability analysis of the iteration point, which significantly increases computational costs.

[0003] The failure probability function is a function of the failure probability changing with design variables. If the failure probability function can be obtained first, the internal reliability analysis in reliability optimization is transformed into a general constraint, and the reliability optimization problem is transformed into a general deterministic optimization problem. There is no need for a separate reliability analysis loop, thereby reducing the huge amount of computation caused by calculating reliability in the reliability optimization problem. Summary of the Invention

[0004] This invention provides a reliability optimization method for composite beams based on a failure probability function. This method solves the problem that the computational cost caused by reliability analysis is prohibitive during the reliability optimization process.

[0005] The technical solution of this invention is a reliability optimization method for composite beams based on a failure probability function, which is carried out according to the following steps: 1) Define random input variables, design variables, objective function, and functional function. A failure probability function is established using the difference between the allowable stress and the maximum normal stress experienced by the composite beam as the function of performance. , Where S is the allowable stress of the composite beam. The maximum normal stress on the composite beam. Random input variables are , in, A Let be the width of the beam.B L is the height of the beam, and L is the length of the beam. E w Let C be the Young's modulus of the beam, D be the width of the aluminum plate at the bottom of the beam, and E be the height of the aluminum plate at the bottom of the beam. a The Young's modulus of the aluminum plate at the bottom of the beam. P 1. P 2. P 3. P 4. P 5. P 6 represents six additional vertical forces along the beam. L 1. L 2. L 3. L 4. L 5. L 6 represents six different application points, where the six additional vertical forces follow a Gumbel distribution with a coefficient of variation of 0.2, while the other variables follow a normal distribution with a coefficient of variation of 0.01. Design variables are , Objective function: , Constraints: P f ( i ) < P f0 The objective function is to minimize the volume of the composite beam. P f ( i ) is the design variable i The probability of failure at that time, P f0 This represents the maximum allowable failure probability. 2) Based on the adaptive Kriging surrogate model The Monte Carlo method is used to calculate design variable failure samples and extended failure probabilities. Based on the random input variables determined in step 1), the prior distribution is set to a uniform distribution. Several design variable vectors, i.e., design points, are generated according to the joint probability density function of the design variables. For each design point, a corresponding input variable vector, i.e., random input points, is generated according to the conditional probability density function, and these vectors are combined into a random input point sample pool. Then, several random input points are randomly selected from the sample pool. The function function values ​​of these random input points are calculated based on the function function. A Kriging surrogate model that predicts the function function values ​​of the random input points is then constructed using these random input points and function function values. The proxy model is updated using the U function. The update stops when the minimum value of the U function is ≥2, and the updated proxy model is then used. Predict the function values ​​at all random input points to obtain the design variable vector sample set in the failure domain, i.e., failure samples and extended failure probabilities; 3) Based on adaptive AKNN method and autonomous learning Kriging surrogate model Calculate the implicit function of the conditional joint probability density. Normalize the failure samples in the failure domain obtained in step 2), and uniformly divide each design variable into several discrete points within its distribution range to form a design variable network dataset. After normalization, several design points are randomly selected. Then, based on the failure sample set, the probability density at these design points is calculated using the AKNN method. A surrogate model for predicting the probability density of design points is constructed using the extracted design points and their corresponding probability densities. And use the H function to update the proxy model. When the maximum value of the H function is ≤ 0.5, the update stops, and the implicit function of the conditional joint probability density is obtained; 4) Calculate the implicit function of failure probability Using the implicit function of the conditional joint probability density obtained in step 3) and the extended failure probability determined in step 2), the implicit function of failure probability is obtained based on Bayesian theory. 5) Reliability optimization of composite beams based on the established failure probability function. Based on the failure probability function, reliability optimization is transformed into a general optimization problem, and the objective function is solved using the general optimization method of the MATLAB platform.

[0006] Specifically, as described in step 1) s max Calculate using the following equation. , in, A Let be the width of the beam. B The height of the beam, L Let be the length of the beam. E w For the Young's modulus of the beam, C The width of the aluminum plate at the bottom of the beam. D The height of the aluminum plate at the bottom of the beam. E a The Young's modulus of the aluminum plate at the bottom of the beam. P 1. P 2. P 3. P 4. P 5. P 6 represents six additional vertical forces along the beam. L 1. L 2. L 3. L 4.L 5. L 6 represents six different application points. Y max The distance between the centroid of the composite section and the bottom edge. W The equivalent moment of inertia of the composite section, s p The normal stress at different locations.

[0007] Step 2) specifically involves, The design variables are treated as random variables, and the failure probability function is expressed as follows: , in f Θ ( i The joint probability density function of the predefined design variables is given by the fact that all defined design variables are uniformly distributed. f Θ ( i ) is 1; F ( x , i ) is when x and i The space formed when both are random variables ( x , i The failure domain on ); P ( F ( x , i )) is the extended failure probability; f Θ ( i | F ( x , i )) is a design variable in the failure domain i The conditional probability density function; Choosing a uniform distribution as the prior probability density function, and based on a predefined joint probability density function... f Θ ( i )generate N One design point, m For the dimensions of the design points, , For each design point, based on the conditional probability density function of the random input points... f x ( x | i Generate a corresponding random input point. x k ( k =1,2, ‧‧‧,N ), and combine them into a full-space random input point sample pool. S , n Let the dimension of the random input points be . n≥m , ; The Kriging surrogate model, based on Gaussian process regression, models the spatial correlation between data points through a covariance function, thereby enabling the prediction of unknown points. The Kriging surrogate model decomposes the response value into two parts: a deterministic trend and a stochastic process. , f ( x ) are basis functions; β A vector of unknown coefficients; Y ( x () is the random component based on a Gaussian process with a mean of 0. The covariance function is: , Where σ 2 Process variance; R This is the correlation matrix; R ( x i, x j () is the Gaussian kernel for two sample points: , in, n The dimension of the input variables; i k For unknown related parameters; x i (k) , x j (k) They are respectively x i and x j The kth element; Based on failure surface and surrogate model The distance between them, using U The function and the variance of each predicted value obtained s g ( x To update the proxy model , , in It is the Kriging proxy model The predicted value at point X; sg ( x ) is the variance of each predicted value.

[0008] Step 3) specifically refers to The range of values ​​for each design variable [ i j (L) , i j (U) ] ( j =1,2, ‧‧‧, m The interior is evenly divided into n One point, combined into a high-dimensional network dataset. Ɒ ,in N 1= n 1 m , , Points are obtained based on the distribution of these adjacent failure samples. i The probability density, , in K It is the number of nearby failed samples; N S It is the total number of samples in the failed sample set; V K ( i ) is included around the design point K The volume of the most recently failed sample, i.e., the volume of the hypersphere. , in R It is the radius of the hypersphere; that is, the distance from the design point to the first... K The Euclidean distance between the nearest failed sample; m It is the dimension of the design variable vector; C (‧) is the Gamma function. Adaptive K The values ​​are as follows: , N S , m Same as above; m 1 represents the number of design variables that fall within their respective distribution ranges, both upper and lower bounds. The expression for the H function is: , In the formula, F It is the cumulative distribution function; ϕ It is the probability density function; m ( x)and s ( x These are the Kriging proxy models. The predicted values ​​and the variance of the predicted values, when all design points H When the learning function value is less than or equal to 0.5, the surrogate model is considered to be... Once the required accuracy is achieved, stop adding points. max( H ( x ))≤0.5.

[0009] Step 3) Establishing the proxy model The implicit function representing the conditional joint probability density is the joint probability density function of the predefined design variables, since the prior distribution is chosen as a uniform distribution. f Θ ( i The value is 1.

[0010] The composite beam reliability optimization method of this invention is based on a failure probability function calculation method combining a self-learning Kriging surrogate model and the Monte Carlo method (AK-MCS) with the adaptive K-nearest neighbor method and the self-learning Kriging surrogate model method (AKNN-AK). This method transforms reliability optimization into a deterministic optimization problem, reducing the number of function calls, significantly reducing computational costs in reliability analysis, and greatly improving the efficiency of reliability optimization design. This invention is applicable to the calculation of failure probability functions for composite beams, and can also be applied to the calculation of failure probability functions in other engineering cases. Attached Figure Description

[0011] Figure 1 This is a schematic diagram of the stress on a composite beam.

[0012] Figure 2 This is a flowchart of the calculation of failure samples and extended failure probability in this invention.

[0013] Figure 3 This is a flowchart of the implicit function for calculating the conditional joint probability density of the present invention.

[0014] Figure 4 It is assumed that only A , B The failure probability function graph for the design variables.

[0015] Figure 5 The failure probability of the case shown. P f Curves varying with design variables Figure 1 .

[0016] Figure 6 The failure probability of the case shown. Pf Curves varying with design variables Figure 2 .

[0017] Figure 7 The failure probability of the case shown. P f Curves varying with design variables Figure 3 .

[0018] Figure 8 The failure probability of the case shown. P f Curves varying with design variables Figure 4 .

[0019] Figure 9 The failure probability of the case shown. P f Curves varying with design variables Figure 5 . Detailed Implementation

[0020] The following embodiments are merely preferred technical solutions of the present invention and are not intended to limit the present invention in any way. For those skilled in the art, the present invention can have various modifications and variations. 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.

[0021] The following examples are combined Figure 1~Figure 5 A reliability optimization method for composite beams based on a failure probability function is presented, including the following steps: 1) Define random input variables, design variables, objective function, and functional function. A failure probability function is established using the difference between the allowable stress and the maximum normal stress experienced by the composite beam as the function of performance. The function of the composite beam is, , In this example S Set to 0.0195 GPa. s max The following equation can be used to calculate it. , Random input variables: x = [ A , B , C , D , L , E w , E a , L 1, L 2,L 3, L 4, L 5, L 6, P 1, P 2, P 3, P 4, P 5, P 6], in, A Let be the width of the beam. B The height of the beam, L Let be the length of the beam. E w For the Young's modulus of the beam, C The width of the aluminum plate at the bottom of the beam. D The height of the aluminum plate at the bottom of the beam. E a The Young's modulus of the aluminum plate at the bottom of the beam. P 1. P 2. P 3. P 4. P 5. P 6 represents six additional vertical forces along the beam. L 1. L 2. L 3. L 4. L 5. L 6 represents six different application points. Y max The distance between the centroid of the composite section and the bottom edge. W The equivalent moment of inertia of the composite section, s p The normal stresses at different locations are represented by six additional perpendicular forces, which follow a Gumbel distribution with a coefficient of variation of 0.2. All other variables follow a normal distribution with a coefficient of variation of 0.01. The variable information is shown in Table 1; ; Design variables: i = [ i 1, i 2, i 3, i 4, i 5] = [ A , B , C , D , L ], The upper and lower bounds of the design variable are respectively i U =[105,200,85,20,1420] and i L =[95,190,75,19,1400]; Objective function: , Constraints: P f ( i ) < P f0， in, P f ( i ) is the design variable i The probability of failure at that time, P f0 The maximum allowed failure probability is 0.15.

[0022] 2) Based on the adaptive Kriging surrogate model The Monte Carlo method (AK-MCS) is used to calculate the design variable failure samples and extended failure probabilities. Based on the input variables determined in step 1), and using Bayesian theory, design variables. i Treated as a random variable, the failure probability function can be expressed as: , Assuming the prior distribution is uniform, then i = [ i 1, i 2, i 3, i 4, i 5] = [ A , B , C , D , L Each design variable is generated according to a uniform distribution. N= 10 5 Design points ( i 1, i 2, ‧‧‧, i N ) T , , For each design point, according to the conditional probability density function f x ( x | i Generate a corresponding random input point. x k ( k =1,2, ‧‧‧, N ), and combine them into a random input sample pool S,n =19, , Randomly drawn from the sample pool b =50 sample variables X b = ( x 1, x 2, ‧‧‧, x b ), and calculate the function value G for these 50 random input points. b ={g( x 1), g( x 2), ‧‧‧,g( x b )},Will X b and G b Kriging surrogate model for constructing the function value of predicting random input points The initial training sample points are then used to construct the initial proxy model. For the sample pool S All random input points use a surrogate model Perform function value prediction, obtain predicted values ​​and standard deviations, and calculate... U The function value will U The random input point with the smallest function value x min and its actual functional value g ( x min Add to the training set and update the proxy model. The update will stop when the minimum value of the U function is greater than or equal to 2. , in, It is a proxy model exist x Predicted value at point; s g ( x ) is the variance of each predicted value. Use the trained agent model Predict the function values ​​for all random input points to obtain the failure sample set and extended failure probability in the failure domain.

[0023] 3) Based on adaptive AKNN method and autonomous learning Kriging surrogate model (AKNN-AK) computes the implicit function of the joint probability density. Normalize the failure sample set obtained in step 2), and evenly divide the distribution range of each design variable into...n 1 point ( n 1=10), forming a high-dimensional network sample set of design variables. Ɒ ,in N 1=10 5 , , Will Ɒ Design point ( k 1 = 1, 2, ... N 1) After normalization, 100 design points are randomly selected, and the probability density of the failure sample set at these design points is calculated using the AKNN method. , in K It is the number of nearby failed samples; N S It is the total number of samples in the failed sample set; V K ( i ) is included around the design point K The volume of the most recently failed sample (i.e., the hypersphere volume). , in R It is the radius of the hypersphere; that is, the distance from the design point to the first... K The Euclidean distance between the nearest failed sample; m It is the dimension of the design variable vector; C (‧) is the Gamma function;

[0024] Adaptive K The values ​​are as follows: , N S , m Same as above; m 1 represents the number of design variables that fall within the upper and lower bounds of their distribution range. A Kriging surrogate model is constructed using the extracted design points (before normalization) and their corresponding probability densities. ,use H The function updates the proxy model. ,when H The proxy model will stop updating when the maximum value of the function is less than or equal to 0.5. This yields the implicit function of the conditional joint probability density.

[0025] 4) Calculate the implicit function of failure probability. Using the implicit function of the conditional joint probability density obtained in step 3) and the extended failure probability determined in step 2), the implicit function of the failure probability is obtained based on Bayesian theory. , in f Θ ( i ) is the joint probability density function of predefined design variables. In this embodiment, the design variables are independent of each other, assuming there are n design variables; because all samples have been normalized when calculating the probability density, f Θ ( i )=1, , F ( x , i ) is when x and i The space formed when both are random variables ( x , i The failure domain on ); P ( F ( x , i The failure probability of the extended failure domain is ()), which can be estimated using the Monte Carlo method. It is calculated by dividing the total number of failure samples in the failure domain by the total number of parameter samples generated by a uniform distribution. N ; f Θ ( i | F ( x , i )) is the conditional joint probability density function of the failure domain, and is the implicit probability density function obtained in step 3).

[0026] This embodiment calculates the average relative error. E MRE The average accuracy of the proposed method in calculating the multi-point failure probability is verified, as shown in the following expression: , P f (i j ) and They represent different design points. i j The failure probability estimated by the dual-cycle method (DLMCS method) and the invented method is the expected value of the failure probability calculated by the dual-cycle method. M For calculation E MRE The number of design points at time Indicate different design points i j The average value of the failure probability is calculated below. Nd This indicates the number of times the function is called. Tables 2 and 3 show the failure probability when a certain design variable changes while all other design variables are at their upper and lower bounds. P f The calculation results vary with the design variable.

[0027] , .

[0028] As can be seen from Tables 2 and 3, the average failure probability calculated by the method of this invention is very close to that calculated by the DLMCS method. The lower bound average relative error is about 0.016, and the upper bound average relative error is about 0.04. Furthermore, the method of this invention calls the function 323 times, which is much lower than the 10 times of the DLMCS method. 10 Second-rate.

[0029] 5) Based on the established failure probability function, the reliability of the composite beam is optimized. Objective function: min f ( i ) = ( i 1× i 2+ i 3× i 4)× i 5. Constraints: P f ( i ) < P f0 The objective function is to minimize the volume of the composite beam. P f ( i ) is the design variable i The probability of failure at that time, P f0 The maximum allowable failure probability is 0.15.

[0030] Table 4 Performance values ​​of the original design method and the invented method .

Claims

1. A reliability optimization method for composite beams based on a failure probability function, comprising the following steps: 1) Define random input variables, design variables, objective function, and functional function. A failure probability function is established using the difference between the allowable stress and the maximum normal stress experienced by the composite beam as the function of performance. ， in, S is the allowable stress of the composite beam. The maximum normal stress on the composite beam. Random input variables are ， in, A Let be the width of the beam. B L is the height of the beam, and L is the length of the beam. E w Let C be the Young's modulus of the beam, D be the width of the aluminum plate at the bottom of the beam, and E be the height of the aluminum plate at the bottom of the beam. a The Young's modulus of the aluminum plate at the bottom of the beam. P 1. P 2. P 3. P 4. P 5. P 6 represents six additional vertical forces along the beam. L 1. L 2. L 3. L 4. L 5. L 6 represents six different application points, where the six additional vertical forces follow a Gumbel distribution with a coefficient of variation of 0.2, while the other variables follow a normal distribution with a coefficient of variation of 0.

01. Design variables are , Objective function: , Constraints: P f ( θ ) < P f0 The objective function is to minimize the volume of the composite beam. P f ( θ ) is the design variable θ The probability of failure at that time, P f0 This represents the maximum allowable failure probability. 2) Based on the adaptive Kriging surrogate model The Monte Carlo method is used to calculate design variable failure samples and extended failure probabilities. Based on the random input variables determined in step 1), the prior distribution is set to a uniform distribution. Several design variable vectors, i.e., design points, are generated according to the joint probability density function of the design variables. For each design point, a corresponding input variable vector, i.e., random input points, is generated according to the conditional probability density function, and these vectors are combined into a random input point sample pool. Then, several random input points are randomly selected from the sample pool. The function function values ​​of these random input points are calculated based on the function function. A Kriging surrogate model that predicts the function function values ​​of the random input points is then constructed using these random input points and function function values. The proxy model is updated using the U function. The update stops when the minimum value of the U function is ≥2, and the updated proxy model is then used. Predict the function values ​​at all random input points to obtain the design variable vector sample set in the failure domain, i.e., failure samples and extended failure probabilities; 3) Based on adaptive AKNN method and autonomous learning Kriging surrogate model Calculate the implicit function of the conditional joint probability density. Normalize the failure samples in the failure domain obtained in step 2), and uniformly divide each design variable into several discrete points within its distribution range to form a design variable network dataset. After normalization, several design points are randomly selected. Then, based on the failure sample set, the probability density at these design points is calculated using the AKNN method. A surrogate model for predicting the probability density of design points is constructed using the extracted design points and their corresponding probability densities. And use the H function to update the proxy model. When the maximum value of the H function is ≤ 0.5, the update stops, and the implicit function of the conditional joint probability density is obtained; 4) Calculate the implicit function of failure probability Using the implicit function of the conditional joint probability density obtained in step 3) and the extended failure probability determined in step 2), the implicit function of failure probability is obtained based on Bayesian theory. 5) Reliability optimization of composite beams based on the established failure probability function. Based on the failure probability function, reliability optimization is transformed into a general optimization problem, and the objective function is solved using the general optimization method of the MATLAB platform.

2. The composite beam reliability optimization method based on failure probability function according to claim 1, characterized in that, The steps described in step 1) σ max Calculate using the following equation. ， in, A Let be the width of the beam. B The height of the beam, L Let be the length of the beam. E w For the Young's modulus of the beam, C The width of the aluminum plate at the bottom of the beam. D The height of the aluminum plate at the bottom of the beam. E a The Young's modulus of the aluminum plate at the bottom of the beam. P 1. P 2. P 3. P 4. P 5. P 6 represents six additional vertical forces along the beam. L 1. L 2. L 3. L 4. L 5. L 6 represents six different application points. Y max The distance between the centroid of the composite section and the bottom edge. W The equivalent moment of inertia of the composite section, σ p The normal stress at different locations.

3. The composite beam reliability optimization method based on failure probability function according to claim 1, characterized in that, Step 2) specifically involves, The design variables are treated as random variables, and the failure probability function is expressed as follows: ， in f Θ ( θ The joint probability density function of the predefined design variables is given by the fact that all defined design variables are uniformly distributed. f Θ ( θ ) is 1; F ( x , θ ) is when x and θ The space formed when both are random variables ( x , θ The failure domain on ); P ( F ( x , θ )) is the extended failure probability; f Θ ( θ | F ( x , θ )) is a design variable in the failure domain θ The conditional probability density function; Choosing a uniform distribution as the prior probability density function, and based on a predefined joint probability density function... f Θ ( θ )generate N One design point, m For the dimensions of the design points, ， For each design point, based on the conditional probability density function of the random input points... f x ( x | θ Generate a corresponding random input point. x k ( k =1,2, ‧‧‧, N ), and combine them into a full-space random input point sample pool. S , n Let the dimension of the random input points be . n≥ m , ； The Kriging surrogate model, based on Gaussian process regression, models the spatial correlation between data points through a covariance function, thereby enabling the prediction of unknown points. The Kriging surrogate model decomposes the response value into two parts: a deterministic trend and a stochastic process. ， f ( x ) are basis functions; β A vector of unknown coefficients; Y ( x () is the random component based on a Gaussian process with a mean of 0. The covariance function is: ， Where σ 2 Process variance; R This is the correlation matrix; R ( x i, x j () is the Gaussian kernel for two sample points: ， in, n The dimension of the input variables; θ k For unknown related parameters; x i (k) , x j (k) They are respectively x i and x j The kth element; Based on failure surface and surrogate model The distance between them, using U The function and the variance of each predicted value obtained σ g ( x To update the proxy model , ， in It is the Kriging proxy model The predicted value at point X; σ g ( x ) is the variance of each predicted value.

4. The composite beam reliability optimization method based on failure probability function according to claim 1, characterized in that, Step 3) specifically refers to The range of values ​​for each design variable [ θ j (L) , θ j (U) ] ( j =1,2, ‧‧‧, m The interior is evenly divided into n One point, combined into a high-dimensional network dataset. Ɒ ,in N 1= n 1 m , ， Points are obtained based on the distribution of these adjacent failure samples. θ The probability density, ， in K It is the number of nearby failed samples; N S It is the total number of samples in the failed sample set; V K ( θ ) is included around the design point K The volume of the most recently failed sample, i.e., the volume of the hypersphere. ， in R It is the radius of the hypersphere; that is, the distance from the design point to the first... K The Euclidean distance between the nearest failed sample; m It is the dimension of the design variable vector; Γ (‧) is the Gamma function. Adaptive K The values ​​are as follows: ， N S , m Same as above; m 1 represents the number of design variables that fall within their respective distribution ranges, both upper and lower bounds. The expression for the H function is: ， In the formula, Φ It is the cumulative distribution function; ϕ It is the probability density function; μ ( x )and σ ( x These are the Kriging proxy models. The predicted values ​​and the variance of the predicted values, when all design points H When the learning function value is less than or equal to 0.5, the surrogate model is considered to be... Once the required accuracy is achieved, stop adding points. max( H ( x ))≤0.

5.

5. The composite beam reliability optimization method based on failure probability function according to claim 1, characterized in that, Step 3) Establishing the proxy model The implicit function representing the conditional joint probability density is the joint probability density function of the predefined design variables, since the prior distribution is chosen as a uniform distribution. f Θ ( θ The value is 1.

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