Optimization of continuous ranked probability score using PSO

Article history: Received December 10, 2014 Received in revised format: March 2, 2015 Accepted March 23, 2015 Available online April 1 2015 Weather forecast has been a major concern in various industries such as agriculture, aviation, maritime, tourism, transportation, etc. A good weather prediction may reduce natural disasters and unexpected events. This paper presents an empirical investigation to predict weather temperature using minimization of continuous ranked probability score (CRPS). The mean and standard deviation of normal density function are linear combination of the components of ensemble system. The resulted optimization model has been solved using particle swarm optimization (PSO) and the results are compared with Broyden–Fletcher–Goldfarb–Shanno (BFGS) method. The preliminary results indicate that the proposed PSO provides better results in terms of CRPS deviation criteria than the alternative BFGS method. Growing Science Ltd. All rights reserved. 5 © 201


Introduction
Weather forecast has been a major concern in various industries such as agriculture, aviation, maritime, tourism, transportation, etc.A good weather prediction may reduce natural disasters and unexpected events.Since 1950, with the advent of computers with high computing power, numerical weather prediction models have been presented.Numerical weather prediction models involve differential equations, which describe the physical laws and dynamics of atmosphere and they mainly depend on boundary conditions.Due to approximate solutions of numerical methods, results obtained from the implementation of numerical models approximate the real-world conditions and always contain some errors (Palmer, 1998;Zeng et al., 1993).Of course, the use of numerical methods is not the only cause of the error in the output but there are also some other sources of error.Among them the presence of errors in the initial/boundary values, the lack of data in some areas and chaotic nature of the dynamical system of the atmosphere are the most important.
According to Eckel et al. (2012) "Ambiguity is uncertainty in the prediction of forecast uncertainty, or in the forecast probability of a specific event, associated with random error in an ensemble forecast probability density function.In ensemble forecasting ambiguity arises from finite sampling and deficient simulation of the various sources of forecast uncertainty".
In a deterministic forecast, only one initial condition is considered as the input to the model regardless of the inherent uncertainty of the atmosphere.But in an ensemble prediction system the forecast is produced by drawing a finite sample from the probability distribution describing the uncertainty of the initial state of the atmosphere and the weather is forecasted by a probability distribution function.Since each ensemble member is a deterministic forecast, it involves systematic errors and needs to be post processed.There are some methods to remove bias and produce a probability distribution function such as rank histogram (Hamill, 2001), dressing method (Roulston & & Smith, 2003), Bayesian model averaging (BMA) (Raftery, et al., 2005), ensemble regression (Unger et al., 2009), variance inflation technique (Johnson & Bowler, 2009), ensemble model output statistics model (EMOS) (Gneiting et al., 2005) and ensemble kernel density model output statistics (EKDMOS) (Veenhuis, 2013).
In this research, an attempt is made to estimate a normal density function to forecast the surface temperature over Iran through minimization of continuous ranked probability score (CRPS) using particle swarm optimization (PSO) algorithm in the training period.

The proposed study
This paper presents an empirical investigation to predict weather temperature using continuous ranked probability score (CRPS) (Gneiting et al., 2005;Toth & Kalnay, 1997;Gneiting & Raftery, 2007).Let  1 , … ,  m be the components of ensemble system.In addition, let  2 be the variance of the components of ensemble system.Therefore, the linear combinations of ensemble system is defined as follows, where  +  1  1 + ⋯ +     and  +  2 are mean and variance of the ensemble system.Let  1 , … ,  m be the coefficients that present the performance of each ensemble member and c and d represent the coefficients of variance and are estimated in training period.The CRPS function is defined as follows, where ( − ) is a Heaviside function, which is defined as follows, Let F be a normal density distribution function with mean and standard deviation of μ and σ.Therefore we have, will be changed to mean absolute value.In fact, the average score is the mean absolute error, which is the generalized model and as it tends to zero the prediction will be more desirable.

Particle Swarm Optimization
Particle swarm optimization (PSO) is an evolutionary technique for optimizing functions designed based on social behavior of birds by Kennedy (2010).In this technique, a group of particles, as the variables of an optimization problem, are scattered in the search environment.More precisely, some particles have better positions than others do.Therefore, based on aggregative particles' behavior, other particles may try to raise their position to the prior particles' positions.The competence of X i The proposed study uses the following to adjust the speed and the position for each particle.

PSO Algorithm
Step 1.Initial values Generate X i in the interval [X min , X max ], i=1,…, I Step 2. Obtain Z(Xi) for i=1,…, I Step 3. Update pbest, if z(x i ) < z(p i ) then P i = X i , i=1,…, I Step 4. Update gbest, if z(p i ) < z(p i ) then P i = P i , i=1,…, I Step 5. Update velocity and position of each particle according to Eq. ( 5) and Eq. ( 6) If x id (t + 1) > X max then x id (t + 1) = X max and v id (t + 1) = 0 If x id (t + 1) < X min then x id (t + 1) = X min and v (t + 1) = 0 Step 6.If t=T Stop, otherwise t = t + 1 go to Step 2.
The proposed PSO method of this paper uses a vector for x(t) with the length of 12 where the first cell contains a, the second to tenth cells are devoted to b1 to bm and the last two items are devoted to c and d, respectively.Similarly, v(t) is also a vector of 12 cells, which contains the speed of each particle.X min and X max are as follows, In our survey, I=50, T=100, w = 0.5, cp = cg =2.Training program in this study is a rolling horizon of 60 days.In our survey, the ensemble system consists of 9 members, where each member contains weather temperature under various physical environments.For prediction, Weather Research and Forecasting (WRF) (Skamarock et al., 2005) model with initial Global forecast System (GFS) is used.
The data is gathered from September, 4 th , 2011 to February, 4 th , 2012 leading us to have 160 data.

The results
In order to measure the performance of the proposed study, we compare the results with Broyden-Fletcher-Goldfarb-Shanno (BFGS) method (Fletcher, 1970).MAE and RMSE have been calculated for both mean of raw ensemble and mean of bias corrected ensemble and compared with MAE and RMSE of the mean of the normal probability density function.

Conclusion
This paper has presented a survey to forecast weather temperature using continuous ranked probability score (CRPS).The mean and standard deviation of normal density function were linear combination of the components of ensemble system.The resulted optimization model has been solved using particle swarm optimization (PSO) and the results are compared with BFGS method.The preliminary results indicate that the proposed PSO has provided better results in terms of CRPS deviation criteria than the alternative BFGS method.
Ensemble members have been post processed by moving average method with a 15-day training period.Results show the minimum amounts obtained for the implementation of PSO on CRPS versus BFGS method are 1.19 and 1.21, respectively.Table1demonstrates the results of our survey when simple mean error is used for raw data and biascorrected data in the test period.As we can observe from the results of Table1, the mean of the normal probability density function has MAE 59% less and RMSE 51% less.