CN112149893A - Design flood value prediction method and device - Google Patents

Abstract

The invention provides a design flood value prediction method and device, and relates to the technical field of hydrological analysis. The method includes: obtaining an information sequence of annual maximum flood peak flow in a target watershed; and obtaining a Pearson type III distribution according to the annual maximum flood peak flow information sequence. According to the likelihood function of the Pearson type III distribution, obtain a penalized likelihood function; according to the penalized likelihood function, predict the design flood value of the target watershed; output the predicted design flood value . The present invention can reduce the error of predicting the design flood value.

Description

A design flood value prediction method and device

technical field

The present invention relates to the technical field of hydrological analysis, and in particular, to a method and device for predicting design flood values.

Background technique

Since the 20th century, with the change of climate, the global temperature has shown a significant upward trend, and the frequency of extreme hydrometeorological events has also increased. Human activities have caused significant changes in the basin environment. For example, the impervious area has increased significantly with urban expansion, accelerating the process of runoff and runoff; changes in the range of grasslands, forests, and cultivated land have changed runoff capacity in the basin; the construction of a large number of water conservancy projects has affected the process of runoff and runoff. The process of confluence of watersheds.

The interannual variation trend of watershed floods brings new challenges to the design of water conservancy projects. For a long time, the water conservancy project needs to weigh its economy and safety in the planning and design stage. Overestimating the design flood value will increase unnecessary engineering investment, while underestimating the design value will make the project face higher risks. Therefore, the predicted flood value will increase. Often determines the scale of a project. The traditional design method assumes that the flood sequence obeys the assumption of consistency and independent and identical distribution, and fails to take into account the fact that most extreme hydrometeorological events present a significant change trend. Due to the long planning life of water conservancy projects, the conclusion obtained by traditional statistical inference methods It may not be possible to apply in future environments, while non-uniform distribution models with changing parameters have been gradually applied in the last two decades. The parameters of the non-uniform distribution model can be expressed as a time term or a function of other covariates. Under the condition that the future trend of covariates can be clarified, the application of the non-uniform distribution model can obtain better fitting results.

At present, many different distribution models have been widely used in hydrological frequency analysis, among which the Pearson type III distribution is widely used. Under the assumption of consistency, the parameter estimators of the distribution can be obtained based on the method of moments and the maximum likelihood method, but the method of moments cannot be applied under the assumption of non-uniformity. The maximum likelihood method estimates the non-uniform Pearson type III distribution also get unreasonable solutions. When the maximum likelihood method is used to estimate the non-uniform Pearson type III distribution, it is necessary to obtain the calculation results based on the numerical method. However, in the optimization process based on the numerical method, the optimal solution is often obtained because the optimization process is disturbed and the wrong optimal solution is obtained. . It can be seen that the error in predicting the design flood value is too large at present.

SUMMARY OF THE INVENTION

Embodiments of the present invention provide a design flood value prediction method and device to solve the problem of design flood value prediction.

In a first aspect, an embodiment of the present invention provides a design flood value prediction method, including:

Obtain the annual maximum flood peak flow information sequence of the target watershed;

Obtain the likelihood function of the Pearson III distribution according to the maximum flood peak flow information sequence in the year;

Obtain a penalty likelihood function according to the likelihood function of the Pearson III distribution;

predicting the design flood value of the target watershed according to the penalized likelihood function;

The predicted design flood value is output.

In a second aspect, an embodiment of the present invention provides a design flood value prediction device, including:

The first acquisition module is used to acquire the annual maximum flood peak flow information sequence of the target watershed;

The second acquisition module is used to obtain the likelihood function of Pearson III distribution according to the maximum flood peak flow information sequence of the year;

The 3rd acquisition module is used to obtain the penalty likelihood function according to the likelihood function of the Pearson III distribution;

The fourth acquisition module is used to predict the design flood value of the target watershed according to the penalty likelihood function;

An output module for outputting the predicted design flood value.

In the embodiment of the present invention, the annual maximum flood peak flow information sequence of the target watershed is obtained; according to the annual maximum flood peak flow information sequence, the likelihood function of the Pearson III distribution is obtained; according to the likelihood of the Pearson III distribution function to obtain a penalized likelihood function; according to the penalized likelihood function, predict the design flood value of the target watershed and output the predicted design flood value. In this way, since the penalized likelihood function obtained by the likelihood function of the Pearson type III distribution is used to predict the design flood value, the error of predicting the design flood value can be reduced.

Description of drawings

Fig. 1 is a flow chart of a design flood value prediction method provided by an embodiment of the present invention;

Fig. 2 is the change graph of the penalty function of different coefficient values;

FIG. 3 is a structure of a designed flood value prediction device provided by an embodiment of the present invention.

Detailed ways

The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are part of the embodiments of the present application, rather than 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.

The terms "first", "second" and the like in the description and claims of the present application are used to distinguish similar objects, and are not used to describe a specific order or sequence. It is to be understood that the data so used are interchangeable under appropriate circumstances so that the embodiments of the present application can be practiced in sequences other than those illustrated or described herein, and "first", "second" distinguishes Usually it is a class, and the number of objects is not limited. For example, the first object may be one or multiple.

Please refer to FIG. 1. FIG. 1 is a flowchart of a design flood value prediction method provided by an embodiment of the present invention. As shown in FIG. 1, the following steps are included:

Step 101: Obtain the information sequence of the annual maximum flood peak flow of the target watershed.

Step 102: Obtain the likelihood function of the Pearson III distribution according to the maximum peak flow information sequence of the year.

The above-mentioned Pearson type III distribution is a probability distribution curve of asymmetric distribution with one end being finite and one end being infinite. Under the assumption of consistency, the parameter estimates of the distribution can be obtained based on the method of moments, but the method of moments cannot be used in non-uniform conditions. Therefore, under non-uniform conditions, Pearson type III distribution is widely used in the field of hydrological analysis technology.

Step 103: Obtain a penalty likelihood function according to the likelihood function of the Pearson III distribution.

Step 104: Predict the design flood value of the target watershed according to the penalized likelihood function.

Step 105: Output the predicted design flood value.

The predicted design flood value output from the above steps can be used as important reference data for water conservancy project construction, for example, according to the predicted design flood value, it is checked whether the design of the flood control building meets the design requirements.

In the embodiment of the present invention, through the above steps, it is possible to obtain the penalized likelihood function to predict the design flood value according to the likelihood function of the Pearson type III distribution, so that the error of predicting the design flood value can be reduced.

As an optional implementation, obtain the annual maximum flood peak flow information sequence of the target watershed, including:

In the actual measured flood sequence of the target watershed, the annual maximum value of the flood flow is selected to constitute the annual maximum flood peak flow information sequence.

Runoff is formed by atmospheric precipitation and enters rivers, lakes or oceans through different paths in the watershed. It is also customary to represent the amount of water passing through a section of a river in a certain period of time, that is, runoff. Therefore, runoff refers to the amount of water passing through a certain section of the target watershed per unit time, and runoff is one of the most important hydrological elements on land. Daily runoff is the unit time of runoff in days, that is, the total amount of water passing through a certain section of the target watershed every day. The main factors affecting runoff are climatic factors and underlying surface factors. Climatic factors include rainfall intensity and evaporation, and underlying surface factors include geometric factors, natural geographical location factors and human activity factors. It can be seen that due to the existence of different influencing factors, the daily total water volume passing through a certain section of the target watershed is different.

The actual measured flood information of the target watershed is aggregated to form a flood dataset for each year. In the flood data set of each year, the maximum value of the flood flow in each year is selected to form the information sequence of the maximum annual flood peak flow.

In this embodiment, since the maximum value of the flood flow in each year is selected as the information sequence of the annual maximum flood peak flow, the accuracy of the obtained annual maximum flood peak flow information can be improved.

Optionally, according to the maximum flood peak flow information sequence of the year, obtain the likelihood function of Pearson III distribution, including:

According to the maximum flood peak flow information sequence of the year, the probability density function of Pearson III distribution is obtained by the following calculation:

Among them, f(x _t ) is the probability density function, t is the covariate, α(t) is the shape parameter, β(t) is the scale parameter, r(t) is the location parameter, and x _t is the annual maximum peak flow information sequence ;

According to the probability density function, the likelihood function of the Pearson III distribution is obtained by the following calculation:

where L(α(t), β(t), r(t)) is the likelihood function, f(x _t ) is the probability density function, t is the covariate, α(t) is the shape parameter, β(t) ) is the scale parameter, r(t) is the location parameter, and x _t is the annual maximum flood peak flow information sequence.

In the probability density function of the above Pearson III distribution, t is a covariate, α(t) shape parameter, β(t) scale parameter and r(t) position parameter are all expressed as functions of covariate t. Covariates are generally expressed as time. According to the change of time, the shape parameters, scale parameters and position parameters will also be in a state of change. Since the information sequence of the annual maximum flood peak flow belongs to the annual maximum value sequence, the Pearson III distribution will show a positive bias, that is, the corresponding β(t)>0, and x _t -r(t)≥0.

A set of parameter values where the likelihood function of the above-mentioned Pearson type III distribution obtains a maximum value is the maximum likelihood estimation result of the parameters of the Pearson type III distribution.

In this embodiment, because the likelihood function of the Pearson type III distribution can determine the function parameter estimation result under non-uniform conditions, the obtained parameter estimation result is more in line with the actual situation, thereby improving the reliability of the design flood value prediction result.

Optionally, obtaining a penalty likelihood function according to the likelihood function of the Pearson III distribution, including:

The penalty function is obtained by the following calculation;

Among them, π(x _t -r(t)) is the penalty function, t is the covariate, r(t) is the location parameter, x _t is the information sequence of the annual maximum flood peak flow, and a, c, and B are respectively in the penalty function. coefficient;

According to the likelihood function of the Pearson III distribution and the penalty function, the penalty likelihood function is obtained by the following calculation:

Among them, PL(α(t), β(t), r(t)) is the penalized likelihood function, t is the covariate, α(t) is the shape parameter, β(t) is the scale parameter, and r(t) is the position parameter, f(x _t ) is the probability density function, and π(x _t -r(t)) is the penalty function.

In the above penalty function, the coefficients a and c in the penalty function reflect the penalty intensity of the penalty function when (x _t -r(t))≈0, B is the action threshold of the penalty function, when x _t -r(t) When ≤B, the penalty function will take effect and limit the value of the position parameter in the distribution.

The above penalty function is combined and multiplied by the likelihood function of the Pearson III distribution to obtain the above penalty likelihood function.

The value range of the above penalty function π(x _t -r(t)) is between 0 and 1. When x _t -r(t)>B, there is π(x _t -r(t))=1 , the penalty likelihood function is exactly the same as the likelihood function of the Pearson III distribution above. When x _t -r(t)≤B, the penalty function π(x _t -r(t)) is between 0 and 1, thereby reducing the value of the penalty likelihood function.

Please refer to Figure 2. Figure 2 is a graph of the change of the penalty function with different coefficient values. As shown in Figure 2, Figure 2 reflects that when the coefficients a and c in the penalty function take different values, the change of the penalty function in x _t -r(t ) ∈ [0,0.05] on the interval. It can be seen that the larger the values of a and c are, the greater the penalty function is for (x _t -r(t))≈0, that is, the more likely it is that (x _t -r(t))≈0 Small. Generally speaking, the values of a and c are recommended as a>50, and c is an odd number greater than or equal to 1.

The principle of using the maximum likelihood method to obtain the optimal parameter estimation is to take the parameter with the maximum value from the likelihood function of the Pearson III distribution as the optimal parameter estimation result. Therefore, the introduction of a penalty function into the likelihood function can make x The probability of _t - r(t) falling between the interval [0,B] decreases. In order to avoid the situation of (x _t -r(t))≈0, the value of the action threshold B of the penalty function should not be too large, usually B∈[0.03,0.1].

Therefore, the coefficients in the penalty function can take a=100, B=0.05, and c=3. In this case, when x _t -r(t)∈[0,0.03], there is π(x _t -r(t ))≈0, at this time the penalized likelihood function PL(α(t),β(t),r(t))≈0, so as to avoid the occurrence of (x _t -r (t)) ≈ 0 case.

In this embodiment, since the penalized likelihood function is obtained according to the likelihood function of the Pearson type III distribution, the accuracy of the optimal parameter can be improved.

Optionally, predicting the design flood value of the target watershed according to the penalty likelihood function, including:

According to the penalty likelihood function, obtain the penalty likelihood function estimation parameter corresponding to the maximum value of the penalty likelihood function;

According to the estimated parameters of the penalized likelihood function, the design flood value is calculated as follows:

Among them, Q _p is the design flood value, gaminv is the inverse function of the cumulative distribution function of the gamma distribution, p is the transcendence probability, t is the covariate, β(t) is the estimated scale parameter of the penalized likelihood function, and r(t) is the penalized likelihood function. Then the function estimates the position parameter.

Based on the principle of maximum likelihood, when the penalized likelihood function takes a maximum value, the corresponding parameter is the estimated parameter of the penalized likelihood function. Further, the parameter is estimated according to the penalized likelihood function to obtain the maximum value in the next year under the transcendence probability p. The predicted design flood value corresponding to the flood peak flow information sequence. The maximum likelihood principle refers to selecting the parameter value that maximizes the function within the possible value range of the parameter as the estimated value of the parameter.

The information evaluation criterion can be used to verify the pros and cons of the distribution model. The commonly used ones are the Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC). When there are multiple groups of distribution models, the AIC/BIC value is the smallest. The set of models is the optimal model:

AIC=-2LLF+2numPar

BIC=-2LLF+numPar×log(n)

In the formula, LLF is the maximum log-likelihood estimation function value. For example, in the penalized likelihood function, LLF is expressed as the logarithm of the maximum value of the penalized likelihood function, and n is the length of the annual maximum flood peak flow information sequence. , numPar is the number of parameters to be estimated in the distribution. For the Pearson III distribution under the consistency condition, when the scale and shape parameters of the non-consistent Pearson III distribution are fixed values, and the position parameter is expressed as a linear function of time t , that is, r(t)=at+b, then numPar=4. For flood sequences with significant change trends, the Pearson III distribution under consistent conditions and the non-consistent Pearson III distribution based on penalized likelihood function can be used to fit the information sequence of the annual maximum flood peak flow. Based on AIC /BIC test results or other model diagnostic methods to verify the theoretical rationality and feasibility of the penalized likelihood function.

In this embodiment, since the parameter corresponding to the maximum value of the penalized likelihood function is used as the estimated parameter of the penalized likelihood function, the accuracy of the flood prediction value can be improved.

Please refer to Fig. 3. Fig. 3 is a structural diagram of a design flood value prediction device provided by an embodiment of the present invention. As shown in Fig. 3, the design flood value prediction device includes:

The first obtaining module 301 is used to obtain the information sequence of the annual maximum flood peak flow of the target watershed;

The second obtaining module 302 is configured to obtain the likelihood function of the Pearson type III distribution according to the maximum flood peak flow information sequence of the year;

The third acquisition module 303 is used to obtain a penalty likelihood function according to the likelihood function of the Pearson type III distribution;

The fourth acquisition module 304 is used to predict the design flood value of the target watershed according to the penalty likelihood function;

The output module 305 is used for outputting the predicted design flood value.

Optionally, the first acquisition module is configured to select the annual maximum value of the flood flow from the actually measured flood sequence of the target watershed to form the annual maximum flood peak flow information sequence.

Optionally, the second acquisition module is used to obtain the probability density function of Pearson type III distribution according to the maximum flood peak flow information sequence of the year, by the following calculation:

Among them, f(x _t ) is the probability density function, t is the covariate, α(t) is the shape parameter, β(t) is the scale parameter, r(t) is the location parameter, and x _t is the annual maximum peak flow information sequence ;

According to the probability density function, the likelihood function of the Pearson III distribution is obtained by the following calculation:

where L(α(t), β(t), r(t)) is the likelihood function, f(x _t ) is the probability density function, t is the covariate, α(t) is the shape parameter, β(t) ) is the scale parameter, r(t) is the location parameter, and x _t is the annual maximum flood peak flow information sequence.

Optionally, the third obtaining module is used to obtain the penalty function through the following calculation;

Among them, π(x _t -r(t)) is the penalty function, t is the covariate, r(t) is the location parameter, x _t is the information sequence of the annual maximum flood peak flow, and a, c, and B are respectively in the penalty function. coefficient;

According to the likelihood function of the Pearson III distribution and the penalty function, the penalty likelihood function is obtained by the following calculation:

Among them, PL(α(t), β(t), r(t)) is the penalized likelihood function, t is the covariate, α(t) is the shape parameter, β(t) is the scale parameter, and r(t) is the position parameter, f(x _t ) is the probability density function, and π(x _t -r(t)) is the penalty function.

Optionally, the fourth acquisition module is used to obtain the penalty likelihood function estimation parameter corresponding to the penalty likelihood function maximum value according to the penalty likelihood function;

According to the estimated parameters of the penalized likelihood function, the design flood value is calculated as follows:

Among them, Q _p is the design flood value, gaminv is the inverse function of the gamma cumulative distribution function, p is the transcendence probability, t is the covariate, β(t) is the estimated scale parameter of the penalized likelihood function, and r(t) is the penalized likelihood The function estimates positional parameters.

The designed flood value prediction device provided in the embodiment of the present invention can implement each process in the method embodiment of FIG. 1 , and in order to avoid repetition, details are not repeated here.

It should be noted that, the device for predicting the design flood value in the embodiment of the present invention may be a device, and may also be a component, an integrated circuit, or a chip in an electronic device.

It should be noted that, herein, the terms "comprising", "comprising" or any other variation thereof are intended to encompass non-exclusive inclusion, such that a process, method, article or device comprising a series of elements includes not only those elements, It also includes other elements not expressly listed or 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. Furthermore, it should be noted that the scope of the methods and apparatus in the embodiments of the present application is not limited to performing the functions in the order shown or discussed, but may also include performing the functions in a substantially simultaneous manner or in the reverse order depending on the functions involved. To perform functions, for example, the described methods may be performed in an order different from that described, and various steps may also be added, omitted, or combined. Additionally, features described with reference to some examples may be combined in other examples.

From the description of the above embodiments, those skilled in the art can clearly understand that the method of the above embodiment can be implemented by means of software plus a necessary general hardware platform, and of course can also be implemented by hardware, but in many cases the former is better implementation. Based on this understanding, the technical solutions 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, and the computer software products are stored in a storage medium (such as ROM/RAM, magnetic disk, CD-ROM), including several instructions to make a terminal (which may be a mobile phone, a computer, a server, an air conditioner, or a network device, etc.) execute the methods described in the various embodiments of this application.

The embodiments of the present application have been described above in conjunction with the accompanying drawings, but the present application is not limited to the above-mentioned specific embodiments, which are merely illustrative rather than restrictive. Under the inspiration of this application, without departing from the scope of protection of the purpose of this application and the claims, many forms can be made, which all fall within the protection of this application.

Claims (10)

1. a design flood value prediction method, is characterized in that, comprises: Obtain the annual maximum flood peak flow information sequence of the target watershed; Obtain the likelihood function of the Pearson III distribution according to the maximum flood peak flow information sequence in the year; Obtain a penalty likelihood function according to the likelihood function of the Pearson III distribution; predicting the design flood value of the target watershed according to the penalized likelihood function; The predicted design flood value is output. 2. The design flood value prediction method as claimed in claim 1, wherein the acquisition of the annual maximum flood peak flow information sequence of the target watershed comprises: In the actually measured flood sequence of the target watershed, the annual maximum value of the flood flow is selected to constitute the information sequence of the annual maximum flood peak flow. 3. the design flood value prediction method as claimed in claim 1, is characterized in that, described according to described annual maximum flood peak flow information sequence, obtains the likelihood function of Pearson type III distribution, comprises: According to the maximum flood peak flow information sequence of the year, the probability density function of the Pearson III distribution is obtained by the following calculation:

Among them, f(x _t ) is the probability density function, t is the covariate, α(t) is the shape parameter, β(t) is the scale parameter, r(t) is the location parameter, and x _t is the annual maximum peak flow information sequence ; According to the probability density function, the likelihood function of the Pearson III distribution is obtained by the following calculation:

where L(α(t), β(t), r(t)) is the likelihood function, f(x _t ) is the probability density function, t is the covariate, α(t) is the shape parameter, β(t) ) is the scale parameter, r(t) is the location parameter, and x _t is the annual maximum flood peak flow information sequence. 4. The method for predicting design flood value as claimed in claim 1, characterized in that, according to the likelihood function of the Pearson type III distribution, obtaining a penalty likelihood function, comprising: The penalty function is obtained by the following calculation;

Among them, π(x _t -r(t)) is the penalty function, t is the covariate, r(t) is the location parameter, x _t is the information sequence of the annual maximum flood peak flow, and a, c, and B are respectively in the penalty function. coefficient; According to the likelihood function of the Pearson III distribution and the penalty function, the penalty likelihood function is obtained by the following calculation:

Among them, PL(α(t), β(t), r(t)) is the penalized likelihood function, t is the covariate, α(t) is the shape parameter, β(t) is the scale parameter, and r(t) is the position parameter, f(x _t ) is the probability density function, and π(x _t -r(t)) is the penalty function. 5. The method for predicting design flood value according to claim 1, wherein, predicting the design flood value of the target watershed according to the penalty likelihood function, comprising: According to the penalty likelihood function, obtain the penalty likelihood function estimation parameter corresponding to the maximum value of the penalty likelihood function; According to the estimated parameters of the penalized likelihood function, the design flood value is calculated as follows:

Among them, Q _p is the design flood value, gaminv is the inverse function of the cumulative distribution function of the gamma distribution, p is the transcendence probability, t is the covariate, β(t) is the estimated scale parameter of the penalized likelihood function, and r(t) is the penalized likelihood function. Then the function estimates the position parameter. 6. A design flood value prediction device, characterized in that, comprising: The first acquisition module is used to acquire the annual maximum flood peak flow information sequence of the target watershed; The second acquisition module is used to acquire the likelihood function of the Pearson III distribution according to the maximum flood peak flow information sequence of the year; The third obtaining module is used to obtain the penalty likelihood function according to the likelihood function of the Pearson III distribution; The fourth acquisition module is used to predict the design flood value of the target watershed according to the penalty likelihood function; An output module for outputting the predicted design flood value. 7. The designed flood value prediction device according to claim 6, wherein the first acquisition module is used to select the annual maximum value of flood flow in the actual measured flood sequence of the target watershed to form the annual maximum value. Sequence of peak flow information. 8. The design flood value prediction device according to claim 6, wherein the second acquisition module is configured to obtain the probability density function of Pearson type III distribution by the following calculation according to the annual maximum flood peak flow information sequence :

Among them, f(x _t ) is the probability density function, t is the covariate, α(t) is the shape parameter, β(t) is the scale parameter, r(t) is the location parameter, and x _t is the annual maximum peak flow information sequence ; According to the probability density function, the likelihood function of the Pearson III distribution is obtained by the following calculation:

where L(α(t), β(t), r(t)) is the likelihood function, f(x _t ) is the probability density function, t is the covariate, α(t) is the shape parameter, β(t) ) is the scale parameter, r(t) is the location parameter, and x _t is the annual maximum flood peak flow information sequence. 9. The design flood value prediction device as claimed in claim 6, wherein the third acquisition module is used to obtain a penalty function by calculating as follows;

Among them, π(x _t -r(t)) is the penalty function, t is the covariate, r(t) is the location parameter, x _t is the information sequence of the annual maximum flood peak flow, and a, c, and B are respectively in the penalty function. coefficient; According to the likelihood function of the Pearson III distribution and the penalty function, the penalty likelihood function is obtained by the following calculation:

Among them, PL(α(t), β(t), r(t)) is the penalized likelihood function, t is the covariate, α(t) is the shape parameter, β(t) is the scale parameter, and r(t) is the position parameter, f(x _t ) is the probability density function, and π(x _t -r(t)) is the penalty function. 10. The design flood value prediction device according to claim 6, wherein the fourth obtaining module is used to obtain the penalty likelihood function estimation corresponding to the maximum value of the penalty likelihood function according to the penalty likelihood function. parameter; According to the estimated parameters of the penalized likelihood function, the design flood value is calculated as follows:

Among them, Q _p is the design flood value, gaminv is the inverse function of the gamma cumulative distribution function, p is the transcendence probability, t is the covariate, β(t) is the estimated scale parameter of the penalized likelihood function, and r(t) is the penalized likelihood The function estimates positional parameters.

Images