A comparison of forecast models to predict weather parameters

Transcription

1 A comparison of forecast models to predict weather parameters GUIDO GUIZZI 1, CLAUDIO SILVESTRI 1, ELPIDIO ROMANO 2, ROBERTO REVETRIA 3 1 Dipartimento di Ingegneria Chimica, dei Materiali e della Produzione Industriale 1 University of Naples Federico II P.le Tecchio 80, 80125, Naples ITALY 2 Industrial Plants, Logistics and Transportation International Telematic University UniNettuno Corso Vittorio Emanuele II, 39, Roma ITALY 3 Dipartimento di Ingegneria Meccanica, Energetica, Gestionale e dei Trasporti University of Genoa Via dell Opera Pia 15, 16145, Genoa ITALY [email protected], [email protected], [email protected], [email protected] Abstract: - Weather forecasting is a really important matter for who is involved in the electrical energy market. The target of this study is to give concrete results about weather forecasting methods to people who needs an accurate estimate of weather parameters to forecast to sell electrical energy in a day-ahead market contest. Approximately one month of meteorological data are analyzed to forecast temperature, pressure and humidity of a month. models and exponential smoothing models have been compared to comprehend which method is more adapted to model the temperature behaviour in Caserta, Italy. Key-Words: - models, exponential smoothing, time series, R, forecasting, weather parameters 1 Introduction The accuracy of meteorological forecasts is having major economic impact in recent years during which the energy companies need to predict the amount of energy that may be sold. Although during this decades the sensibility about a world with more clean energies is increased, in Italy over the 75% of the energy is produced in fossil fuel based stations. The amount of energy produced by turbines of these stations is often function of meteorological parameters, which may be predicted using statistical methods. The main parameters that determine the significant quantity of produced energy are three: temperature, pressure and humidity. While the behaviour of climate changes is irregular and chaotic, if we focus the attention on the forecast of these three parameters it's possible to use linear methods to reach great results. The crucial point for many energy companies is the choosing of the method that assures a better performance in terms of accuracy. In this study, three methods to analyze and forecast the time series of temperature, pressure and humidity are compared. The first method is the model, approached to the seasonally adjusted data. The second method is the additive seasonal model. The third method is a more general exponential smoothing technique, the model described in Hyndman et al. (2008). The available dataset includes 4 years of observations that was obtained from an energy station built in Caserta, Italy. This station continuously collects meteorological data every 15 minutes. This high frequency ensures a great performance of the seasonal exponential smoothing as shown in the on-going study. 2 State of art ISBN:

2 In literature, there are many methods used to analyze and forecast time series. A brief illustration of the mentioned models is useful to understand the results of the forecasts. 2.1 model The autoregressive integrated moving average () model is a generalization of an autoregressive moving average (ARMA) model. The difference between these two models is the use of an integration term that needs to differentiate the time series, letting them become stationary. The stationary condition is necessary to operate with an model. The parameters of an model are three (p,d,q) and represent respectively the autoregressive parameter, the integration parameter and the moving average Differencing Differencing is an excellent way of transforming a non-stationary series to a stationary one. This is obtained by subtracting the observation in the current period from the previous one. If this transformation is done only once to a series, the data has been first differenced. yy tt = yy tt yy tt 1 (1) This process essentially eliminates the trend if your series is growing at a fairly constant rate. If it is growing at an increasing rate, you can apply the same procedure and difference the data again, obtaining a second differenced data. yy = yy tt yy tt 1 (2) ARMA model Once differenced the time series it's possible to describe the obtained time series with an ARMA model: yy tt = φφ 1 yy tt φφ pp yy tt pp + θθ 1 ee tt θθ qq ee tt qq + ee tt (3) where yy tt is the differenced series. The terms on the right side of the expression include both lagged values of yy tt and lagged errors. The choosing of appropriate parameters p, d and q can be difficult. The best way to choose them is to use the Box- Jenkins technique (1971) analyzing the autocorrelation function (ACF) and the partial autocorrelation function (PACF) of the differenced time series. 2.2 Exponential Smoothing Exponential smoothing is a forecasting method used when the single components of the time series (trend and seasonal factors) may be changing over time. More recent observations are weighted more heavily than remote observations. This unequally consideration of the observations is possible by using smoothing constants. There are many studies about exponential smoothing, from the simple exponential smoothing (SES) to the Holt's trend corrected exponential smoothing and lasting the method performed by Holt (1957) and (1960) model The additive method is used for time series that exhibit a linear trend and a fixed seasonal pattern, with constant variation. The time series may be described by this model: yy tt = (ββ 0 + ββ 1 tt) + SSSS tt + εε tt (4) where ββ 0 is the point estimate of the mean in time period t-1, ββ 1 is the growth rate in time period t-1, SSSS tt the seasonal factor and εε tt the error term in time t. To implement the additive method, we let l TT 1 denote the estimate of the level (the mean) in time TT 1, and bb TT 1 the growth rate in the same time. If we use SSSS TT LL as the most recent estimate of the seasonal factor corresponding to the time period T, we may describe iteratively the level of the time series at the time T: l TT = αα(yy TT ssss TT LL ) + (1 αα)(l TT 1 + bb TT 1 ) (5) where α is a smoothing constant between 0 and 1, (yy TT ssss TT LL ) is the deseasonalized observation in time period T, and (l TT 1 + bb TT 1 ) is the estimate of the level of time series in time period T. The estimate of the growth rate in time period T uses the smoothing constant γ and is bb TT = γγ(l TT l TT 1 ) + (1 γγ)bb TT 1 (6) ISBN:

3 The new estimate for the seasonal factor SSSS TT in time period T uses the smoothing constant δ and is ssss TT = δδ(yy TT l TT ) + (1 δδ)ssss TT LL (7) where (yy TT l TT ) is an estimate of the newly observed seasonal variation model Exponential smoothing models may be different depending on the presence of the smoothing parameters α, γ and δ, already mentioned, but mainly on the type of the components of a time series: error, trend and seasonality (). The seasonality, if it's present in time series, may be differenced in two types: additive (A) and multiplicative (M). Whereas the trend, if it's present, may be differenced in: additive (A), additive damped (A d ), multiplicative (M), and multiplicative damped (M d ). The combination of the types of these two components, seasonality and trend, are illustrated in Table 1. Table 1 Exponential smoothing methods Trend Component Seasonal Component N A M N (N,N) (N,A) (N,M) A (A,N) (A,A) (A,M) A d (A d,n) (A d,a) (A d,m) M (M,N) (M,A) (M,M) M d (M d,n) (M d,a) (M d,m) The most recent studies of Hyndman et al. (2002), Taylor (2003), Hyndman et al. (2008), about exponential smoothing developed these fifteen methods. The presence of an additive or multiplicative error component (A,M) causes an increment of the possible methods, bring them from fifteen to thirty. Then it's possible to describe all the types of the exponential smoothing models by using three letters. For example the additive error model (A,A d,n) represent the damped Holt's method. This method uses the maximum likelihood function to estimate the starting parameters and then it may estimate iteratively all the parameters to forecast future values of time series. In addition with the method we may have linear and non-linear models, and then time series that exhibit non-linear characteristics can be fitted well by these models. The framework provides an automatic way of selecting the best exponential smoothing model, including Holt's model, method (additive and multiplicative), damped trend method of Gardner and McKenzie. 3 Analysis and results The software used to verify the performance of the mentioned models is R-studio. The available dataset includes temperature, pressure and humidity values Fig. 1 Temperature values of a random monthly dataset ISBN:

4 from 19 January 2011 to 12 February 2015, thousands of observations (N=142,656). The target of this study is to determinate the amount of energy that may be sold in the italian day-ahead market (in italian MGP), so we focus the attention on the forecast parameters (the temperature in particular) of next day. The temperature exhibits a daily seasonal behaviour as shown in Fig.1. examining a random monthly dataset. This seasonal time series is not stationary, then it's necessary to make a decomposition and a deseasonalization of the time series before the application of the models. Moreover we have to underline the existent, but not much relevant, of a correlation factor between the three parameters (Table 2). 3.1 Dataset size Because of the great amount of available datas and the cyclical daily pattern of the time series, it's important to decide the best dataset size on which models reach better performances. The behavoiur of the time series during a single day is surely more comparable with the pattern of more recent previous days than more remote datas. If we have the real observations of a day k, it's possible to see what's the best dataset size (using datas until the day k-1) for a model to reach best forecast values for the day k. We may use the same dataset size to forecast the values of the day k+1 (in which we haven't real observations yet), and then day-to-day. Table 2 Correlation of parameters Correlation Temperature Pressure Humidity Temperature Pressure Humidity Decomposition of time series The decomposition of time series is useful to transform time series into deseasonalized time series by subtracting the seasonal component from the original time series. The software R uses a function, stl( ), that provides to decompose time series, as shown in Fig. 2. Fig. 2 Trend, seasonal and remainder components using STL function ISBN:

5 Fig. 3 Results of the comparison between models in 2 July Forecast models in R It's necessary to transform vectors of parameters temperature, pressure and humidity in time series by the function ts( ), specifying the frequency of the time series (96 in this case). The forecast package needs to use the functions of the forecast models. The function used to fit datas are mainly two: Holt( ) and stlm( ). The first function needs to use the method, and it's possible to use the additive method or the multiplicative Winter method by setting an input parameter of the function. The second function, instead, needs to use both models and, by setting the right input parameters. the function stlm( ) works with stl objects: it takes a time series, applies the STL decomposition, models the seasonally adjusted data using the specified model (in this case or ). For models it's possible to improve the fitting wellness by setting in input regressors of the time series (pressure and humidity), whereas for models it's possible to choose which model to use by setting the three letters ("Z" denotes an automatic selection of the component type by the function). The forecast( ) function needs to forecast parameters, setting the model used to fit the dataset and the number of forecast values (in this case 96), and then reseasonalizes the results by adding back the last period of the estimated seasonal component. To complete the analysis we used a last R function, stlf( ). This function combines the functions stlm( ) and forecast( ), but it's not possible to choose the model: the function selects automatically the model that is considered the best choice. ISBN:

6 3.3 Results of check Determining the performances of the models is the most significant stage in forecasting. Although there are many performance measures that evaluate forecast models, the mean absolute error () and the mean absolute percentage error () are the most common and revealing ones in this case, and these are computed below: MMMMMM = 1 NN yy NN ii=1 ii ŷ ii (8) MMMMMMMM = 1 NN NN yy ii ŷ ii ii=1 (9) yy ii To compare the models we choosed a random month (July 2014) to forecast day by day, with a 5-possible dataset size (7,15,30,45,60 previous days). The plot illustrated in Fig. 3 is related to a random day of the selected month, 2 July Performance parameters Table 4 Forecasting error parameters of temperature in July 2014 Min value Mean value Max value As shown in the plot, the and stlf( ) functions reach great forecast results, with a really small. The models selected by the four functions are reported in the Table 3. A summary of the monthly forecast in July for all the models used may be seen in the Table 4 on top. Table 3 Temperature forecast model in July Model used Model STL + (3,1,1) model STL + (A,N,N) STL + (A,A d,n) Pressure and humidity forecast The same comparison with same methods was done to forecast pressure and humidity values in the same month (July 2014). The time series plots are illustrated in Fig. 4 (humidity) and in Fig. 5 (pressure). ISBN:

7 Fig. 4 Humidity values of a random monthly dataset Fig. 5 Pressure values of a random monthly dataset ISBN:

8 Performance parameters Table 5 Forecasting error parameters of humidity in July 2014 Min value Mean value Max value Performance parameters Table 6 Forecasting error parameters of pressure in July 2014 Min value Mean value Max value Even if the pressure behaviour seems to be less predictable than the humidity behavoiur it's important to underscore that the scale between these time series is different. In fact the humidity daily fluctuations and the resulting variance are higher in the humidity time series. This is why the humidity performance parameters are worse than the pressure ones, as it's possible to see in Table 5 and Table 6. 4 Conclusions In this study, meteorological data of Caserta was analyzed using different models which belong to two forecasting classes: and exponential smoothing. The additive model is a model that may be represented by an (A,A,M) model, so it's normal that has worse performances than more general function. The real comparison is between the model and the model. The linear exponential smoothing models have an counterpart, but the non-linear exponential smoothing models don't have an counterpart. The reason why exponential smoothing models (using function) have better performances is that a non-stationary model to fit the values is required and the behaviour of the time series has a non-linear tendency. models work better if the model needed has to be stationary. Moreover, although many models haven't got an exponential smoothing counterpart, they can't be non-linear models. Another proof of better performances of exponential models to forecast temperature is given by the results of the method choice: in all days of July 2014 the function stlf( ) selected always the model as the best temperature forecast method. There is also a remark that we have to do: function doesn't allow inclusion of regressors data, differently to function. In this case the correlation between meteorological parameters is low, especially between temperature and pressure. But there can be parameters with a higher correlation and, consequently, function may reach better performances. References [1] Gardner Jr, E. S. (1985). Exponential smoothing: The state of the art. Journal of Forecasting 4(1), ISBN:

9 [2] Gardner Jr, E. S. (2006). Exponential smoothing: The state of the art Part II.International Journal of Forecasting 22(4), [3] Hyndman, R. J., A. B. Koehler, J. K. Ord and R. D. Snyder (2008). Forecasting with exponential smoothing: the state space approach. Berlin: Springer-Verlag. [4] Box, G. E. P., G. M. Jenkins and G. C. Reinsel (2008). Time series analysis: forecasting and control. 4th. Hoboken, NJ: John Wiley & Sons. [5] Brockwell, P. J. and R. A. Davis (2002). Introduction to time series and forecasting. 2nd ed. New York: Springer. [6] Chatfield, C. (2000). Time-series forecasting. Boca Raton: Chapman & Hall/CRC. [7] Pena, D., G.C. Tiao and R.S. Tsay, eds. (2001). A course in time series analysis. New York: John Wiley & Sons. [8] Shumway, R. H. and D. S. Stoffer (2011). Time series analysis and its applications: with R examples. 3rd ed. New York: Springer. [9] Cleveland, R. B., W. S. Cleveland, J. E. McRae and I. J. Terpenning (1990). STL : A seasonal-trend decomposition procedure based on loess. Journal of Official Statistics 6(1), [10] Gomez, V. and A. Maravall (2001). Seasonal adjustment and signal extraction in economic time series. In: A course in time series analysis. Ed. by D. Pena, G.C. Tiao and R.S. Tsay. New York: John Wiley & Sons. Chap. 8, pp [11] Ladiray, D. and B. Quenneville (2001). Seasonal adjustment with the X- 11 method. Lecture notes in statistics. Springer-Verlag. [12] Miller, D. M. and D. Williams (2003). Shrinkage estimators of time series seasonal factors and their effect on forecasting accuracy. International Journal of Forecasting, 19(4), [13] Theodosiou, M. (2011). Forecasting monthly and quarterly time series using STL decomposition. International Journal of Forecasting, 27(4), ISBN: