MODELLING AND FORECASTING OF WEB TRAFFIC USING HOLT'S LINEAR, BATS AND TBATS MODELS

In the recent era, internet consumption has increased. Due to this heavy and regular use of internet web traffic is increased. Sometimes due to high web traffic server may also affect. In this study, cumulative data on the number of visitors to Wikipedia, Facebook, Energy, Android, and Apple, are analyzed in detail. Some descriptive statistics of visitor’s pages in Wikipedia are given such as mean, minimum, maximum, standard deviation, skewness 3888 BADR, MAKAROVSKIKH, MISHRA, ET AL. and kurtosis. The present study used different time series models like Holt’s Linear Trend, BATS and TBATS for different web pages. From the results, it was found that the Android page as well as apple page in Wikipedia holt’s Linear Trend model performance is better compared to other models. This kind of projection is helpful for web traffic to solve the server breakdown problem for larger users of the server. These Wikipedia pages have been chosen to Forecast the number of visitors to these pages through time series models BATS, TBATS, and Holt's Linear Trend Model in order to face future problems to mitigate over loading that may occur with the increase in the number of visitors to these pages and also to experiment the suitability of these models of Time series to Forecast the number of visitors and that to achieve the highest level of accuracy. MAPE for accuracy was used to compare model performance.


INTRODUCTION
Nowadays the web traffic consumption has been increased significantly and it became necessary to be forecasted in order to design the web server to balance and distribute the request loads [1]- [7]. Forecasting the load can allow building load balancing techniques to schedule client requests to meet the demands and to be dynamically scaled up and down for the server based on the forecasted visits [21]. Forecast of the upcoming traffic on multiple pages on a website is very challenging specifically if we are trying to forecast the traffic for Wikipedia pages [2] as it's becoming the primal source of knowledge for billions of users [3], who are constantly contributing and expanding the content. Wikipedia consented users to access the traffic by querying each article over a given window [7], but we used in this research the available dataset created by the Kaggle competition [6]. [22][23] compared times series models and used for forecasting purposes. In this research, we employed the time series forecasting techniques to estimate future request loads based on historical visits [4], traditional time series such as Autoregressive integrated moving average "ARIMA"[5], Seasonal ARIMA "SARIMA" [4] have been used extensively for forecasting and based on a recent research Abotaleb [9] found that 3889 MODELLING AND FORECASTING OF WEB TRAFFIC Holt's Linear Trend is giving accurate results In predicting the daily Infection cases for COVID-19 research in (Italy, China and USA) reaching to a conclusion that Holt's model is giving more accurate results than ARIMA models. So, we used Holt's Linear Model, and Trigonometric seasonality, Box-Cox transformation, ARMA errors, Trend and Seasonal components "TBATS" in which time series seasonality is not forced to be periodic, however, it's allowed to be dynamic, in which results proved to be more accurate.

MATERIAL AND METHODS
This study was performed on data obtained from a database by the Kaggle competition from01-07-2015 to 31-12-2016 [10]. We used cumulative data on the number of visitors to Wikipedia and four pages, namely Facebook, Energy, Android, and Apple, were selected. In this paper, we used time series models because these model a combination of Fourier with an exponential smoothing state-space model and a Box-Cox transformation, Both the BATS and TBATS models dealing with seasonality.

BATS and TBATS Models
TBATS is an improvement modification of BATS that allows multiple seasonal incorrect cycles.
Equation (2) represents the seasonal pattern Equations (3), (4) and (5) are global trends and local trends where 1 , … , denote that seasonal period, and denote that the level and trend of components of the time series at time t, ( ) denote that seasonal component at time t, represents to ARMA(p, q) component and is white noise process.
The smoothing parameters are given by , , for i=1…T and ∅ is the dampening parameter, which gives more control over trendextrapolation when the trend component is damped [12]. For seasonal data the following equations representing Trigonometric exponential smoothing models , , where 1 ( ) and 2 ( ) are the smoothing parameters.
( ) = 2 / . This is an extended, modified single source of error version of single seasonal multiple sources of error representation suggested by [13]and is equivalent to index seasonal approaches when = /2 for evenvalues of and when = ( − 1)/2 for odd values of . But most seasonal terms will require much smaller values of , thus reducing the number of parameters to be estimated.
In the single seasonal multiple sources of error setting [14] an alternative, but equivalent formulation of representation (2) is preferred [15] which can be obtained hyper-parameterizing the single seasonal multiple sources of error version of (2) using 3891 MODELLING AND FORECASTING OF WEB TRAFFIC Equations (16) and (17) are seasonal patterns modeled by the Fourier model.

Holt's Linear Trend Method
The exponentially weighted moving average is also the averages of smoothing random variability with the following properties: (1) that is very important that older data have a declining weight; (2) it is very simple to calculate; and (3) the most important for data set is that minimal data is needed. Holt, C.E. 1957 had given equation.

Forecast Equation
Level Equation Trend Equation where represent an estimate of the level of series at time t, denotes an estimate of the trend (slope) of the series at time t, is the smoothing parameter for level, 0 ≤ ≤ 1 and * is the smoothing parameter for the trend, 0 ≤ * ≤ 1 that's with simple exponential smoothing. shows that is a weighted average of observation and the one-step-ahead training forecast for time ̂ and the one-step-ahead training forecast for time t. here given by −1 + −1 . The trend equation shows that is a weighted average of the estimated trend at time t based on − −1 and −1 the previous estimate of the trend. The smoothing parameters and * , and the initial values 0 and 0 are estimated by minimising the SSE for the one-step training errors.

Overfit problem
If the model is overfitting on training data when the model performs well on the training data but does not perform well on the evaluation data so when we comparing training data with testing data the accuracy of testing data is better than training data

RESULT AND DISCUSSION
From table 1, we find that: from 01-07-2015 to 31-12-2016, the visitors to Facebook page in Wikipedia have increased during the period from (65) to (28328). Average daily visitors to the Facebook page in Wikipedia are (13447.63). Kurtosis value is (1.6) indicates the data follows a platykurtic distribution which shows a tail that's thinner than a normal distribution which means the number of outliers will not be large. Followed by a positive value of skewness (0.12) which 3893 MODELLING AND FORECASTING OF WEB TRAFFIC is between -0.5 and 0.5, the distribution is approximately symmetric.
The Visitors of Energy page in Wikipedia have increased from (16) to (5014) during the same period, with average daily Visitors of Energy page in Wikipedia about (2284). Kurtosis value is (1.6) indicates the data follows a platykurtic distribution which shows a tail that's thinner than a normal distribution which means the number of outliers will not be large. With the positive value of skewness (0.16) which is between -0.5 and 0.5, the distribution is approximately symmetric.
The visitors of Android page in Wikipedia have increased from (8) 7) indicates the data follows a platykurtic distribution which shows a tail that's thinner than a normal distribution which means the number of outliers will not be large.
With the positive value of skewness (0.04) which is between -0.5 and 0.5, the distribution is approximately symmetric.
The parameter for the level smoothing is denoted by Alpha, and the parameter for the trend smoothing is denoted by Beta, α and β are constrained to 0-1 with higher values giving faster learning and lower values providing slower learning. From Table 2   In Table 5 In These results show that the best forecasting for BATS linear model is Wikipedia's Facebook page. In Table 8, MAPEs are also presented for the TBATS model for Facebook page in Wikipedia, Energy page in Wikipedia, Android page in Wikipedia, Apple page in Wikipedia. In Table 8, similar results were seen in Tables 6 and 7, and it was seen that all three methods gave consistent results.

HOLT'S LINEAR TREND AND TBATS MODELS
• Holt's Linear Trend Model were fitted using R software.
• BATS and TBATS Model were fitted using R software.