Suppose I have vagueness in lifetime data and I assume the lifetimes to be fuzzy numbers(trapezoidal number say). Then how to determine the maximum likelihood estimates of parameter(of distribution and fuzzy number itself) of the fuzzy lifetimes.
Hi,
I suggest you see the links on the topic.
https://hal.science/hal-00654118/document
https://iopscience.iop.org/article/10.1088/1742-6596/2322/1/012022/pdf
https://computing4all.com/linear-regression-predictive-model/?gclid=Cj0KCQiAz9ieBhCIARIsACB0oGKolhv_-ut3P6SZLz9CqYZLc9CwqqoroJDXvcJxeqP82atW9kVw1kkaAr9wEALw_wcB
Article Fuzzy modeling, maximum likelihood estimation, and Kalman fi...
Best regards
Prof. Lafifi,
Thank you for your response. But what I want to estimate is the parameter of lifetime distribution as well as parameter of my trapezoidal numbers.
Hi,
Please take a look at rhe links.
Article Estimating the parameters of lifetime distributions under pr...
Article Parameter and reliability estimation of three-parameter life...
Article Estimating the parameters of Lomax distribution from impreci...
Article Parameters and reliability estimation for the weibull distri...
http://www.ukm.my/jsm/pdf_files/SM-PDF-45-11-2016/25%20Muhammad%20Shafiq.pdf
https://www.ajol.info/index.php/jfas/article/view/142578
Best regards
Prof. Lafifi,
I looked into this as well. I would like to share a paper of mine. In this paper we have taken three different membership functions and observed a change in bias. So what I believe is that there should exist some optimal values of parameters of the fuzzy number as well. How I can optimize them?
PFA
https://internet.bhu.ac.in/research_pub/jsr/Volumes/JSR_65_05_2021/35.pdf
Here is a general framework for estimating MLE with fuzzy data:
1. Define the Fuzzy Data Representation: Determine how the fuzzy data is represented. Fuzzy data can be represented using membership functions, fuzzy sets, or fuzzy numbers. The choice of representation depends on the nature of the data and the specific problem.
2. Specify the Likelihood Function: Define the likelihood function that represents the probability of observing the fuzzy data given the model parameters. This likelihood function should incorporate the uncertainty associated with the fuzzy data representation. The form of the likelihood function depends on the specific fuzzy data representation chosen.
3. Identify the Objective Function: The objective function to be maximized is the log-likelihood function, which is obtained by taking the logarithm of the likelihood function. This step is similar to traditional MLE.
4. Optimize the Objective Function: Apply optimization techniques to maximize the log-likelihood function and find the parameter values that maximize the likelihood of observing the fuzzy data. Common optimization methods include gradient-based methods (e.g., gradient descent) or numerical optimization algorithms (e.g., Newton-Raphson method). However, the choice of optimization method depends on the specific problem and the complexity of the likelihood function.
5. Assess Model Fit and Validity: Once the MLE is obtained, assess the model fit and validity by evaluating the goodness-of-fit measures or conducting model diagnostics. These steps help ensure that the estimated model parameters adequately capture the uncertainty in the fuzzy data and provide a good fit to the observed data.
Implementation details of MLE for fuzzy data can vary depending on the chosen fuzzy data representation and the nature of the problem. Additionally, specialized software or programming languages with fuzzy logic capabilities may be required to implement the estimation procedure.
good luck
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