# Error transfer in single-sphere neutron spectrometer-based solution problems

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David Zhang on 16 May 2023
Answered: Shaik on 16 May 2023
Hi! I am a graduate student. Since I am not a mathematics major, there are many things I don't know about this area, and thank you very much for any help and advice.
My project, in simple terms, is to solve an underdefinite equation that is itself linear. It's simple xR=b, where R is the corresponding matrix, simulated in Geant4, x is the amount of neutron injection we expect to get, and b is the count we expect from the detector.Since the detector has fewer spherical layer than the energy cluster we divide，That said, this is a very standard underdetermined problem.Unfortunately, however, there are various errors in the measurement process, whether it is in the counting, analog-to-digital conversion process, or reading process.This makes it difficult for us to solve this underdetermined equation.
How should I handle these errors and use them to measure the confidence level of the neutron spectrum I finally solved.
In addition, in addition to the least squares method, including methods such as annealing algorithms for selecting solutions later, I also want to use Bayesian methods and machine learning to solve this problem and compare their differences. Can these two methods be partially implemented in MATLAB?
Thanks for the help and advice, it was a great honor!

Shaik on 16 May 2023
Hi there! I'm glad to help you with your project. Solving underdetermined equations with measurement errors can indeed be a challenging task. I'll provide you with some guidance on handling errors, measuring confidence levels, and utilizing different solution methods including Bayesian and machine learning approaches.
1. Handling Errors:
• Error Characterization: Start by characterizing the errors in your measurement process. Understand the sources of errors and quantify their statistical properties (e.g., mean, standard deviation).
• Error Modeling: Incorporate the error model into your problem formulation. This could involve representing the errors as additional terms in the equation or treating them as noise in your measurements.
• Error Estimation: Estimate the errors using calibration data or statistical techniques to get an idea of their magnitude and impact.
1. Confidence Level Measurement:
• Uncertainty Analysis: Perform an uncertainty analysis to assess the confidence level of the solved neutron spectrum. This can involve techniques such as error propagation, Monte Carlo simulations, or statistical inference to quantify the uncertainty in the solution.
• Confidence Intervals: Use confidence intervals to provide a range of values within which the true neutron spectrum is likely to lie based on the uncertainty estimates.
1. Solution Methods:
• Least Squares Method: The least squares method is a common approach to solving underdetermined equations. MATLAB has built-in functions like lsqnonneg and lsqlin that can be used for this purpose.
• Bayesian Methods: MATLAB provides various tools for Bayesian inference, such as the Statistics and Machine Learning Toolbox. You can utilize Bayesian techniques like Markov chain Monte Carlo (MCMC) or variational inference to solve your underdetermined problem and obtain posterior distributions of the solution.
• Machine Learning: MATLAB is well-suited for implementing machine learning algorithms. You can explore regression techniques, such as ridge regression or lasso regression, which can handle underdetermined problems and incorporate regularization to mitigate overfitting.
Remember that the choice of solution method depends on your specific problem, available data, and computational resources. It's also beneficial to consult with your project advisor or domain experts to select the most appropriate approach.
MATLAB provides a rich set of functions, toolboxes, and libraries for various mathematical and statistical tasks. It supports implementing and comparing different solution methods, including those based on least squares, Bayesian inference, and machine learning.
If you need further assistance with specific implementation details or MATLAB usage, feel free to ask. Good luck with your project!

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