In different fields of study, cumulative growth models over time play an important role, and many researchers have contributed to the knowledge of the developed models [1-2]. Among the most common nonlinear models are: Gompertz, Weibull, negative exponential, Richard's model, logistic, monomolecular, Brody, Mitcherlich, von Betalanffy, S-shaped, among others [2-3]. There are about 77 known equations referring to sigmoidal growth models, which are used in epidemics, bioassays, agriculture, engineering, tree diameter, forest height distribution, etc. [1,2,4,5]. Among the most widely used models are those of Gompertz, Richard and Weibull [2,3]. These models have been implemented in COVID-19, a new disease; where it was shown for COVID-19 cases in Iraq, that ROR modeling gave better results than nonlinear models, so for meteorology we will use this linear ROR model [6-7].

In this study we developed a linear ROR model of eight temporal variables, these were: maximum and minimum extreme temperatures, daily precipitation, maximum and minimum humidity, daily mean, wind strength and cloudiness, which we will compare with the persistence model, which is a good model obtained for meteorological prediction, and we will show how it is possible to predict it on a given day well in advance which shows that chaos theory can be defeated if this way of prediction is used [7-8]. 

In the case of this research we have data from the Yabu weather station (located in the city of Santa Clara, Villa Clara province, Cuba) for that day over the years, and it is with this data that we demonstrate that long-term weather prediction for this particular day (December 10) is also possible with an acceptable margin of error, which shows that chaos can be overcome if this form of prediction is used.

The objective of the research was to demonstrate that the long-term meteorological prediction for December 10 is possible by means of the methodology of the Regressive Objective Regression (ROR) with an acceptable margin of error for eight meteorological variables in the Yabu station, Santa Clara, Villa Clara, Cuba, thus overcoming the Chaos theory.

For this research, data from the Yabu weather station (located in the city of Santa Clara, Villa Clara province, Cuba) were used (Figure 1), for this day (December 10) over the years, in order to demonstrate that long-term weather forecasting for this particular day is also possible with an acceptable margin of error.

The data used were eight meteorological variables (extreme maximum and minimum temperatures, daily precipitation, maximum and minimum humidity, average daily humidity, wind strength and cloudiness). The forecast was made using the methodology of Regressive Objective Regression (ROR) that has been implemented in different variables, such as viruses circulating in the province of Villa Clara, the population dynamics of culicidae, the entomofauna of culicidae and copepods [9-11], and particularly in COVID-19 in Cuba [6,12]. A long-term forecast of up to 15 years was made, from December 10, 2020 to December 10, 2035.

In the linear ROR methodology, first the dichotomous variables DS, DI and NoC must be created, where: NoC: Number of base cases (its coefficient in the model represents the trend of the series). DS = 1, if NoC is odd; DI = 0, if NoC is even, and vice versa. DS represents a saw tooth function and DI this same function, but in inverted form, so that the variable to be modeled is trapped between these parameters and a large amount of variance is explained, a detailed explanation of which can be found in Fimia et al. [10].

General Description of Climate

According to Köppen's classification (modified), in most of Cuba the predominant climate is of the warm tropical type, with a rainy season in the summer. In general, it is quite accepted to express that Cuba's climate is tropical, seasonally humid, with maritime influence and semi-continental features, where Villa Clara province also responds to these characteristics.

Figure 1: Geographical Distribution of Meteorological Stations

Due to its geographical position, the city of Santa Clara is located at a latitude very close to the Tropic of Cancer, which conditions the reception of high values of solar radiation throughout the year, determining the warm character of its climate. In addition, it is located in the limit between the tropical and extratropical circulation zones, receiving the influence of both in a seasonal way. In the season from approximately November to April, the meteorological and climatic variations become more noticeable, with abrupt changes in the daily weather, associated with the passage of frontal systems, the anticyclonic influence of continental origin and the extratropical low pressure centers. From May to October, on the other hand, there are few variations in the weather, with the more or less marked influence of the North Atlantic anticyclone. The most important changes are related to the presence of disturbances in the tropical circulation (easterly waves and tropical cyclones).

The analysis and processing of the data was carried out with the help of the IBM SPSS statistical package, version 19.

Table 1 shows the model parameters, where the coefficients of the explained variance R are high, while the model errors were small. Temperatures depend on lags in 15 years the minimum, and 13 years the maximum, precipitation in eight years, minimum humidity in ten years, wind force (FF) in 13 years and cloudiness in 15 years.

Table 1: Parameters of the Models of the Different Time Variables in Yabu, Cuba

The analysis of variance of the model was highly significant, with Fisher's F of 1350.6 significant at 100 % (Table 2).

Table 2: Results of the Analysis of Variance

ANOVA a, b, a. Dependent variable: Tmax, b. Linear regression through the origin, c. Predictors: Lag13Tmax, DS, DI, d. This total sum of squares is not corrected for the constant because the constant is zero for the regression through the origin.

In table 3, it can be seen that all variables are significant, but the temperature trend was not significant, and therefore is not shown. The linear model depends on the data returned in 13 years (Lag13 Tmax).

Table 3: Results of the Linear Model for the Variables Included in the Study

Coefficient ^{a, b,} a. Dependent variable: Tmax, b. Linear regression through the origin

Figure 2: Maximum Temperature Forecast According to ROR Linear Model for the Yabu, Cuba Meteorological Station

In figure 2, the modeling results show that the predicted values coincide with the actual values of the variable, where the errors are very small, close to zero. Since no trend is observed in the extreme temperatures, we can state that on December 10 throughout history there is no cooling or warming trend, which does not mean that for any other specific day in the daily data series there may be a significant trend.

In the 13 year forecast, it was obtained that December 10 will present days with high and low maximum temperatures. This is the first time that an ROR model is applied to the long-term weather forecast in Cuba 13 years in advance for a specific day, as it had already been done for climatic variables such as sand dumping on Varadero beach [13], also for electricity consumption in Villa Clara province [14]; in addition, other works also model the number of people in the long term with cerebrovascular accidents [7,12] as well as the modeling of atmospheric pressure and its impact on mosquito population density well in advance [10,11]. Another long-term model was carried out in Sancti Spiritus, Cuba, where a forecast of daily variables is made one year in advance using ROR regression [14], as well as in the prediction of earthquakes [1516,], and of different meteorological disturbances, also of epidemic outbreaks in different geographical latitudes, where global warming of the planet has a high share of responsibility, and therefore, modeling is an essential tool, and even human cloning [16-23].

Table 4 shows the results of the errors taking an independent sample of 11 cases, from 2010 to 2021 to see how the model behaves in an independent sample of 25 % of the cases. As can be seen, the errors are small for all variables, as are the standard deviations of the variables.

Table 4: Descriptive Statistics

The forecast results for the remaining meteorological variables in Yabu, Cuba are shown below (Figures 3-9).

Figure 3: Extreme Minimum Temperature

Figure 4: Daily Precipitation

Figure 5: Daily Wind Force

Figure 6: Maximum Daily Relative Humidity

 Figure 7: Minimum Relative Humidity

Figure 8: Mean Relative Humidity

Figure 9: Daily Cloudiness on December 10 in Yabu, Cuba

Finally, the rate of improvement of one model over another was calculated as follows, which is nothing else than the modeling ABILITY. Here a slight modification was made to the Wilks (1987) formula, adding the mean instead of the mean square error (MSE) according to the original formula, obtaining that the model established is the persistence model and the model to be tested is the ROR, this was done in the independent sample of 11 cases (Table 5).

Table 5: Results of Mean Error in an Independent Sample of 11 Cases

The improvement ranged from 14.5 % for minimum temperature to 60.2 % for maximum temperature, so we can state that in this case a linear ROR model outperforms a persistence model for most of the variables studied, except for humidities where persistence outperformed the ROR model by a narrow margin.

We can affirm in this case that the linear ROR model outperformed the persistence model in most of the variables studied, except in the case of humidity, where persistence outperformed the ROR model by a narrow margin, thus overcoming the chaos in weather forecasting, all of which is very positive, since it is the first time that a ROR model has been applied to weather forecasting processes for a specific day 8, 10, 13 and 15 years in advance.

Acknowledgement

We would like to thank the Villa Clara Cuba Provincial Meteorological Center for providing the computing resources without which this work would not have been possible.

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