Arboriculture & Urban Forestry 46(3): May 2020 Ɵq(B) = 1 − Ɵ1B −Ɵ2B2 − ... − ƟqBq and is the moving average operator of q order (MA[q]), which is a white noise process. If constant ϑ0 is different from zero, the integrated series provides a deterministic trend (i.e., the series presents an increasing or decreasing trend, which is independent of random disturbances)(Pin- dyck and Rubenfield 1991). For the verification of the stationarity of the model, visual analysis and decomposition of the time series were carried out. The Augmented Dickey-Fuller test (ADF)(Dickey and Fuller 1981) was applied to verify the stationarity of the series, along with the Phillips- Perron test (PP)(Phillips and Perron 1988) and the Kwiatkowski-Phillips-Schmidt-Shin test (KPSS) (Kwiatkowski et al. 1992). These tests verify whether series yt presents a unit root and, consequently, if sta- tionarity is confirmed. The identification of the model consists of deter- mining its order based on the “principle of parsi- mony.” This step is the most critical for the use of the model and determines the types of generator model series: yt AR MA ARMA ARIMA ...SARIMA What is the order of the model, i.e., what are the values of (p) (q) (p, q) (p, d, q) (p, d, q) × (P, D, Q)s To assist in this identification step, time domain analysis was utilized (Box and Pierce 1970), which is the fundamental approach for the analysis of time series. After identification and selection of the appropriate model, process parameters (AR) and (MA) were esti- mated. Parameter estimates were obtained from the Gaussian distribution for the maximum likelihood method, for all possible combinations, to fulfill the conditions of invertibility and uniqueness for the parameters. Standardized residues, residues of autocorrelation function (ACF), residues of partial autocorrelation function (PACF), and portmanteau’s test (Dickey and Fuller 1981) were analyzed to verify whether the model proposed was white noise: Qk = n Ʃ k l and ck c2 k n = number of observations; k = number of lags; = autocorrelation of residuals. The model is accepted if Q ≤ χ2 (l, k − n), where χ2 165 is the chi- square, l is the significance level (with a 95% confi- dence interval), k the lag order, and n the number of parameters. Another way to verify the model utilizes the Akaike’s Information Criterion (AIC): AIC = −2ln(L) + 2(p + q) L = maximum likelihood; p and q = model param- eters to obtain the minimum AIC value (Akaike 1977). After the iterative process of identifying, esti- mating, and checking the model, if the model can provide an estimate of the series that satisfactorily adjusts to real data, this model can be used to forecast variable values. Forecasting processes, based on time series models, are procedures that aim to extend (to future values) the model described and adjusted to the present and past values of the variable. Therefore, forecasts enable determination of the expected value of a future observation. The mean square error (MSE) was calculated for the forecasts obtained, enabling comparison between forecasted and observed values for the adjusted series and further selection of the model with lowest MSE (Soares et al. 2010). MSE =Ʃ(yt − ye n t )2 RESULTS AND DISCUSSION Figure 1 shows the evolution and behavior of the urban pruning waste mass series for João Pessoa (MPA-JP) and the logarithmic series (Ln [MPA-JP]) from January 2008 to December 2013, expressed in 1,000 tons. The napierian logarithm was necessary to stabilize the variance while preserving the properties of the series data. MPA-JP presented an estimated average of 2,151 tons, a median of 2,115 tons, a min- imum value of 1,256 tons, and a maximum value of 2,772 tons. Analysis of Figure 1 reveals an increasing trend for the MPA-JP series over time, which provides indi- cations of nonstationary character. It can also be observed that the mass of UPW increased in the period between April and July in comparison with the other months of the year. This occurred due to the rainy season in the city, increasing the occurrences of broken branches and fallen trees. It is also worth not- ing that when the mass of UPW generated annually is analyzed, the demand of the population for the ©2020 International Society of Arboriculture
May 2020
| Title Name |
Pages |
Delete |
Url |
| Empty |
Search Text Block
Page #page_num
#doc_title
Hi $receivername|$receiveremail,
$sendername|$senderemail wrote these comments for you:
$message
$sendername|$senderemail would like for you to view the following digital edition.
Please click on the page below to be directed to the digital edition:
$thumbnail$pagenum
$link$pagenum
Your form submission was a success. You will be contacted by Washington Gas with follow-up information regarding your request.
This process might take longer please wait
