Turnaround Predictions for India and Maharashtra using Optimization Tools

Turnaround Predictions for India and Maharashtra using Optimization Tools

Dr Himanshu Shekhar

With previous posts getting superb response, I am exploring mathematical tools of optimization further for arriving at mathematically correct numbers keeping my emotions and sentiments aside for the prediction of turnaround.

Introduction: The exponential variation of cumulative confirmed cases for India has been projected many times and it is always accepted by readers. However, when prediction about turnaround is made, there have been split opinions. Two schools of thoughts are seen – one expecting or rather wishing a quick turnaround and other school extending the prevailing situation to August, September and so on. The current activity is taken to apply optimization technique of minimizing square of error and obtain the optimized turnaround prediction. For test study, confirmed cases from India and Maharashtra is considered and analyzed. Data till 08.06.2020 is taken.

Approach: The URL of post in the blog for prior estimates for Maharashtra and India is given below.

https://himanshushekharscience.blogspot.com/2020/06/cumulative-normal-distribution-and.html

First analysis is carried out for Maharashtra, where it was predicted earlier that turnaround is expected on 08.06.2020. The earlier prediction is reproduced below. It is having a normal distribution curve fitted to daily confirmed cases, with peak value of 3000 cases and standard deviation of 1200.

The error function or the square of difference between observed and calculated values are added and indicated as Chi-square value. Minimization of Chi-square is taken as objective function with variables mean, standard deviation and amplitude. The optimized result is reproduced below, which indicates that turnaround in Maharashtra is already achieved on 04.06.2020, with much lower standard deviation and peak daily confirmed cases.

Similar optimization exercise is repeated for India. In earlier prediction, it was assumed that peak daily confirmed cases can be 11000 and turnaround is possible by 21.06.2020. Although, India was on the rising limb of the normal distribution curve, but an optimistic estimate is made for the same and presented in one of the previous posts. The same is reproduced below.

However, this is just an optimistic estimate and the reality from mathematical point of view is obtained by optimization of square of error of daily confirmed cases, as objective function. The variables for optimization is again, mean, standard deviation and amplitude. The optimization gives peak daily confirmed cases of 21648 with turnaround on 28 July 2020. The curve is given below.

Conclusion: Although mathematically, Chi-square minimization can be taken as objective function with mean, standard deviation and amplitude as optimization variables, but the results may be mathematically correct, only. It seems scary for India, as peak daily confirmed cases of 21648 and turnaround of 28.07.2020 is too far. However, the same optimization gives turnaround in Maharashtra on 04.06.2020 with peak cases of 2710. Mathematical correctness and reality is at stake for such exercises.

Dr Himanshu Shekhar