# An alternate method of evaluating Lagrange multipliers of MEP

**Part of the following topical collections:**

## Abstract

We present an alternate method for evaluating Lagrange multipliers of probability distributions usually used in maximum entropy principle. The Lagrange multipliers are evaluated using least square method. Both the methods are used to calculate Lagrange multipliers in fitting wind speed data. Kolmogorov–Smirnov test, Chi squared goodness of fit test, and root mean square error indicate the usefulness of this alternate method.

## Keywords

Maximum entropy principle Maximum entropy method Wind data Wind speed Hansweert Least square method Least square fitting## 1 Introduction

Maximum entropy principle has a variety of applications in modeling of stochastic data. These applications involve many branches of natural and social sciences. Pressé et al. [1] reviewed the origins and use of Maximum Entropy Principle in Statistical Physics and beyond in fields such as Biology. Karlin et al. [2] used MEP for constructing equilibria in lattice kinetic equations. Chen and Dai [3] investigated MEP of uncertainty distributions for uncertain variables. Županović et al. [4] took an alternative path to derive Kirchhoff’s loop rule where they employed MEP to show that in network branches currents are distributed to achieve maximum entropy. Alves et al. [5] found a connection between the maximum entropy principle and the Higgs boson mass without introducing any extra assumptions into the Standard Model. Hanel etal. [6] generalized the notion of entropy to complex and non-Ergodic systems and showed that the MEP is a consistent method for such systems. Roux and Weare investigated the consistency of the MEP and restrained-ensemble simulations. They showed that the conditions of the restrained-ensemble simulations were compatible with the maximum entropy principle [7]. Aldana-Bobadilla and Kuri-Morales proposed a method of clustering based upon the MEP and showed their method was more effective than supervised ones [8]. Pontzen and Governato used MEP to derive the phase-space distribution for a halo of dark matter. They used physically motivated constraints and found a good match with three simulations of dark matter [9]. Rahmati, Pourghasemi and Melesse applied random forest and maximum entropy models to map groundwater potential in the Mehran region of Iran and found MEP provided more successful predictions [10]. Ruggeri used non-linear MEP to model a polyatomic gas under changing pressure conditions. He found the framework to be in agreement with the phenomenological Extended Thermodynamics theory [11].

## 2 MEP and new method

- 1.
Define the range (x

_{min}, x_{max}) and step size dx. - 2.
Use some standard functions for \( \phi_{n} \left( x \right). \)

- 3.
Start iterative procedure with some \( \varvec{\lambda}_{\varvec{o}} . \)

- 4.
Calculate following two integrals.

$$ G_{n} \left(\varvec{\lambda}\right) = \smallint \phi_{n} \left( x \right)exp\left[ { - \mathop \sum \limits_{n = 0}^{N} \lambda_{n} \phi_{n} \left( x \right)} \right]dx = \mu_{n} , n = 0, \ldots ,N $$(4)$$ g_{nk} = g_{kn} = \frac{{\partial G_{n} \left(\varvec{\lambda}\right)}}{{\partial \lambda_{k} }} = - \smallint x^{k} x^{n} exp\left[ { - \mathop \sum \limits_{m = 0}^{N} \lambda_{m} x^{m} } \right]dx = - G_{n + k} \left(\varvec{\lambda}\right) $$(5) - 5.
Solve the equation \( \varvec{G\delta } = \varvec{v} \), where \( v = \left[ {\mu_{o} - G_{o} \left( {\lambda^{o} } \right), \mu_{1} - G_{1} \left( {\lambda^{o} } \right), \ldots \ldots , \mu_{N} - G_{N} \left( {\lambda^{o} } \right)} \right]^{t} \) to find \( \varvec{\delta} \)

- 6.
Calculate \( \varvec{\lambda}=\varvec{\lambda}^{0} +\varvec{\delta} \) The process is repeated with this new value as \( \varvec{\lambda}^{0} \) ntil \( \varvec{\delta} \) becomes negligible.

In this procedure, the first \( \varvec{\lambda}_{\varvec{o}} \) is basically an initial guess for the vector \( \varvec{\lambda} \). Then the new values of \( \varvec{\lambda} \) are taken as the new \( \varvec{\lambda}_{\varvec{o}} \cdot\varvec{\delta} \) are uncertainties in \( \varvec{\lambda}_{\varvec{o}} \).

## 3 Least square method

## 4 Wind speed data

Many different statistical distributions have been used to model wind speed data. The most widely used distribution is Weibull distribution [12, 13]. MEP more accurately models wind speed data [14, 15, 16, 17]. Djafari developed a matlab program to model wind power density using maximum entropy distributions [18]. Wind speed data for this study has been downloaded from the site [19]. This site contains wind speed data of various cities of The Netherlands and it has been made available for free for educational and research purposes. As a sample we randomly picked the city, Hansweert and used hourly wind data for the year 2000. It has a longitude of 3.998 and latitude of 51.446. Hansweert is a village in southwest Netherlands with overall pleasant weather. The average temperature in winter is 2 °C, while in summer it is 18 °C. In Hansweert winters are windier.

## 5 Goodness of fit tests

Three different goodness of fit tests have been used to determine how well the datasets fit on the classical distribution of maximum entropy principle using both methods.

## 6 Kolmogorov–Smirnov test

To check the validity of the hypothesis that the given data follows a specific distribution, a test of hypothesis with given level of significance is conducted. If \( KS < KS_{\alpha } \) then the two distributions are similar. At 99% level of significance, \( KS_{99} = \frac{1.63}{\sqrt n } \). For n = 6 and \( \alpha = 99\% \), the critical value is 0.618. The best feature of KS test is that it does not depend on sample size.

## 7 Chi square goodness of fit test

_{i}) and corresponding expected frequency (E

_{i}). The Chi square test statistic is given by

## 8 Root mean square error (RMSE)

## 9 Results and discussion

Goodness of fit statistics of MEP fit and least square fit for the month of January

Cities | Fits | Chi square | Kolmogorov–Smirnov | RMSE |
---|---|---|---|---|

Hansweert | MEP fit | 0.058554 | 0.013204 | 0.029344 |

Least square fit | 3.05E − 16 | 6.70E − 08 | 6.86E − 08 | |

Karachi | MEP fit | 2.48E − 2 | 2.66E − 3 | 3.62E − 4 |

Least square fit | 5.63E − 4 | 6.63E − 4 | 1.263E − 5 |

Month-wise test statistics for goodness of fit for Least Square fit

Month | N | Chi square | Kolmogorov–Smirnov test | RMSE |
---|---|---|---|---|

January | 5 | 3.05E − 16 | 6.70E − 08 | 6.86E − 08 |

February | 5 | 3.05E − 16 | 6.70E − 08 | 6.86E − 08 |

March | 5 | 7.88E − 18 | 1.48E − 08 | 1.26E − 08 |

April | 5 | 1.97E − 17 | 1.85E − 08 | 1.33E − 08 |

May | 5 | 1.52E − 17 | 3.11E − 08 | 1.49E − 08 |

June | 5 | 1.42E − 16 | 3.92E − 08 | 4.43E − 08 |

July | 5 | 3.35E − 17 | 3.23E − 08 | 2.54E − 08 |

August | 5 | 1.69E − 17 | 2.04E − 08 | 1.70E − 08 |

September | 5 | 8.65E − 18 | 1.86E − 08 | 1.17E − 08 |

October | 5 | 4.62E − 16 | 1.35E − 07 | 1.02E − 07 |

November | 6 | 1.44E − 11 | 3.28E − 05 | 1.64E − 05 |

December | 5 | 8.57E − 18 | 3.63E − 08 | 1.70E − 08 |

Values of Lagrange coefficients obtained by Least Square Method

\( \varvec{\lambda}_{0} \) | \( \varvec{\lambda}_{1} \) | \( \varvec{\lambda}_{2} \) | \( \varvec{\lambda}_{3} \) | \( \varvec{\lambda}_{4} \) | \( \varvec{\lambda}_{5} \) | \( \varvec{\lambda}_{6} \) | \( \varvec{\lambda}_{7} \) | |
---|---|---|---|---|---|---|---|---|

Jan | 0.2586 | − 3.7199 | 1.1559 | − 0.1965 | 1.7840E − 02 | − 7.9752E − 04 | 1.3813E − 05 | |

Feb | − 0.0962 | − 2.6092 | 0.7384 | − 0.1229 | 1.0853E − 02 | − 4.6586E − 04 | 7.7139E − 06 | |

Mar | − 2.2530 | − 1.4277 | 0.2821 | − 0.0390 | 3.7522E − 03 | − 1.9167E − 04 | 3.8228E − 06 | |

Apr | − 1.4752 | − 1.4661 | 0.0975 | 0.0120 | − 1.5274E − 03 | 4.6887E − 05 | ||

May | − 8.5501 | 8.9182 | − 4.8420 | 1.0956 | − 1.2571E − 01 | 7.7100E − 03 | − 2.4044E − 04 | 2.9919E − 06 |

Jun | 0.6504 | − 4.0858 | 1.2287 | − 0.2112 | 2.1028E − 02 | − 1.0907E − 03 | 2.2989E + 00 | |

Jul | − 1.2563 | − 2.3945 | 0.5998 | − 0.0938 | 9.9086E − 03 | − 5.8817E − 04 | 1.4429E − 05 | |

Aug | 0.1333 | − 5.0964 | 1.9391 | − 0.3807 | 3.9318E − 02 | − 1.9633E − 03 | 3.7214E − 05 | |

Sep | − 0.5050 | − 2.7926 | 0.6578 | − 0.0698 | 1.2434E − 03 | 3.0950E − 04 | − 1.5337E − 05 | |

Oct | 2.4246 | − 3.9655 | 0.9971 | − 0.1661 | 1.7848E − 02 | − 1.1256E − 03 | 3.7408E − 05 | − 5.0212E − 07 |

Nov | 38.0074 | − 31.1175 | 9.8836 | − 1.6705 | 1.5398E − 01 | − 7.2824E − 03 | 1.3832E − 04 | |

Dec | 3.1023 | − 3.7479 | 0.7224 | − 0.0822 | 5.5612E − 03 | − 1.9647E − 04 | 2.7869E − 06 |

## 10 Conclusion

- (i)
MEP should be used if statistical information of the distribution is needed, and LSM is preferable when fitting of dataset is required.

- (ii)
The results show the evaluation of Lagrange multipliers through LSM is better than that obtained by MEP (see Table 1).

A Python program has been developed that fits various degree polynomials as classical probability distributions and displays the degree for which the errors are least. It also displays the calculated values of Lagrange multipliers. For error calculation Chi square, Kolmogorov–Smirnov and Root Mean Square Error test statistics were used. The alternative method consists of using least square method to find the Lagrange multipliers of maximum entropy principle and has proven to provide a better fit of the dataset, as evidenced by lower values of tests of significance statistics.

## Notes

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

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