Using Transformations to Predict and Smooth Time Series

Time series has a leading position in statistical Analysis. Nowadays, many economic and industrial operations have been built based on time series. These operations include predicting the product demand variation, the future product prices oscillation, the stock storing control etc. This paper presents a study to show the effect of transformation and smoothing on the performance of the time series. The research results have shown a significant improvement in time-series operation can be noticed when the principles of transformation and smoothing are applied on time series. Introduction Time series topic is one of the essential topics nowadays because it is starting to be applied to different types of science widely. Mathematical-statistical possesses in analyzing the time series have begun to provide important functions of estimation. Furthermore, other involved significant points have made important decisions, and they used to simulate some mathematics and statistics samples for the given problem. Time series use parameters to predict the future and make the right resolution for stable series. Stationary of series is very important in Analysis because it provides a stable mathematical model of series. Smoothing models may be classified into two categories. The first is which contains constant parametric and the second is of variable parameters, median and variance. The constant parametric model has main ways to solve because its parameters are stationary for each domain of data, i.e. its work on making series stationery by making differences and mathematics relating such as logarithms transformations and square roots etc. Making transformations is very important in time series analysis because it makes data prepared to analyze and more accurate to get the estimation function. In this research, the transformation will be used to make a function stationary then getting smoothing function. From another view, methods of smooth ratios may deal with variables and states that, difficult to deal with it because parametric estimations smooth in a matter that easy to solve. There are many ways to use these methods in time series analyzing, which working on smoothing data before applying statistic samples on it and deal with the varying in parameters. So, this paper searches for sample preferences and if smoothing methods treated it, then compared it with the ARIMA model. Theatrical introduction The time series is a set of observations for a specific phenomenon during a time period. Mathematically, it is a sequence of random variables defined over a probability space of multiple parameters and indicated by the index t, which is an item of T. Time series usually written as {x(t); t ∈ T} and it consists of two variables. One of them varies with time, and the others are dependent (y=f(x)). 1. Changes which affect the improvement of time series Any time series is affected by multiple factors natural, economic, seasonal etc., and that effects on the general structure of time series even in long or short ranges. Figure 1 shows the seasonal variables which are iterative changes in year seasons. While figure 2 highlights the cyclic variables which are influencing the time series vibrations and repeated every regular period, it may occur every 1 to 10 years. In random variable time series, there Using Transformations to Predict and Smooth Time Series 648 are some changes that occur suddenly in data which are difficult to be predicted. Such types of data are important in spite of their difficulty, but it may appear as small waves in time series data. Figure 1, the seasonal time series Figure 2, the cyclic variable time series 2. Stationary time series Stationary and non-stationary of data are essential and important in time series analysis. Graphing time series for a period [t, t+h] may sometimes be matched with graphing the series in another interval [s, s+h] and that refers to time homogenous in series behaviour, which called stationary. This time series is further divided into two parts: 2.1. Strictly stationary The series { x(t)} considered as strict if the probability distribution xt1 , xt2 ... ... . xtn is the same for xt1+k , xt2+k , ... . xtn+k for all selected time intervals t1, t2, ... . , tn and for any constant. The time series said to be strict when the three previous conditions are applicable. 2.2. Weak stationary It means the common variable probability distribution (xt1 , xt2 ... ... xtn) to some extent may be changed with time if the mean and variance were constant, and if covariance (xt1 , xt+k) is a function to slowing down the Interval K and independent on time, t. 3. Nonstationary time series Practically, most time series is non-stationary, and it might be impossible to prove using diagrams or statistic tests. For example, economic parametric usually considered a non-stationary time series because of its generality. For such reason, it has to be converted into stationary one to facilitate its modelling process. This series includes two cases: 3.1 Difference Modifications In 1976, Box and Jenkins completely described the samples and introduced guides to understanding non-stationary series and converting it. If {xt1} expressed a non-stationary time series. Then it can be converted using the following equation: Yt = ∆ xt Where (∆= 1 − B) and B is called the indicator of back difference. If Bxt = xt−1 and B xt = xt−2, then the process of time decreasing will be suitable when the process is described by the difference. For instance, if the time series was non-stationary, then the no-stationarity can be processed so that the time series becomes stationary by using the first difference as explained in the following equation:


Introduction
Time series topic is one of the essential topics nowadays because it is starting to be applied to different types of science widely. Mathematical-statistical possesses in analyzing the time series have begun to provide important functions of estimation. Furthermore, other involved significant points have made important decisions, and they used to simulate some mathematics and statistics samples for the given problem. Time series use parameters to predict the future and make the right resolution for stable series.
Stationary of series is very important in Analysis because it provides a stable mathematical model of series.
Smoothing models may be classified into two categories. The first is which contains constant parametric and the second is of variable parameters, median and variance. The constant parametric model has main ways to solve because its parameters are stationary for each domain of data, i.e. its work on making series stationery by making differences and mathematics relating such as logarithms transformations and square roots etc.
Making transformations is very important in time series analysis because it makes data prepared to analyze and more accurate to get the estimation function. In this research, the transformation will be used to make a function stationary then getting smoothing function. From another view, methods of smooth ratios may deal with variables and states that, difficult to deal with it because parametric estimations smooth in a matter that easy to solve.
There are many ways to use these methods in time series analyzing, which working on smoothing data before applying statistic samples on it and deal with the varying in parameters. So, this paper searches for sample preferences and if smoothing methods treated it, then compared it with the ARIMA model.

Theatrical introduction
The time series is a set of observations for a specific phenomenon during a time period. Mathematically, it is a sequence of random variables defined over a probability space of multiple parameters and indicated by the index t, which is an item of T. Time series usually written as { ( ); ∈ } and it consists of two variables. One of them varies with time, and the others are dependent (y=f(x)).

Changes which affect the improvement of time series
Any time series is affected by multiple factors natural, economic, seasonal etc., and that effects on the general structure of time series even in long or short ranges. Figure 1 shows the seasonal variables which are iterative changes in year seasons. While figure 2 highlights the cyclic variables which are influencing the time series vibrations and repeated every regular period, it may occur every 1 to 10 years. In random variable time series, there are some changes that occur suddenly in data which are difficult to be predicted. Such types of data are important in spite of their difficulty, but it may appear as small waves in time series data. Figure 1, the seasonal time series Figure 2, the cyclic variable time series

Stationary time series
Stationary and non-stationary of data are essential and important in time series analysis. Graphing time series for a period [t, t+h] may sometimes be matched with graphing the series in another interval [s, s+h] and that refers to time homogenous in series behaviour, which called stationary. This time series is further divided into two parts: 2.1. Strictly stationary The series { ( )} considered as strict if the probability distribution 1 , 2 … … . is the same for 1+ , 2+ , … . + for all selected time intervals 1 , 2 , … . , and for any constant. The time series said to be strict when the three previous conditions are applicable.

Weak stationary
It means the common variable probability distribution ( 1 , 2 … … ) to some extent may be changed with time if the mean and variance were constant, and if covariance ( 1 , + ) is a function to slowing down the Interval K and independent on time, t.

Non-stationary time series
Practically, most time series is non-stationary, and it might be impossible to prove using diagrams or statistic tests. For example, economic parametric usually considered a non-stationary time series because of its generality. For such reason, it has to be converted into stationary one to facilitate its modelling process. This series includes two cases:

Difference Modifications
In 1976, Box and Jenkins completely described the samples and introduced guides to understanding non-stationary series and converting it. If { 1 } expressed a non-stationary time series. Then it can be converted using the following equation: = ∆ Where (∆= 1 − ) and B is called the indicator of back difference. If = −1 and 2 = −2 , then the process of time decreasing will be suitable when the process is described by the difference. For instance, if the time series was non-stationary, then the no-stationarity can be processed so that the time series becomes stationary by using the first difference as explained in the following equation: And by applying the process of time reflecting on the previous equation, it becomes: It can be noticed that the first difference expressed by (1-B) and comparing with the second difference, the last equation becomes: The purpose of taking the first and second differences is to satisfy the stationary of time series.

The case of non-stationary variance
It is one of the most important problems which avoids obtaining the accurate sample. But using converting as (logarithms, square roots etc.) for time series data may fix the problem. = + 1 −1 + 2 −2 + − + Where: : represents the random error or noise. It is usually distributed (0, 2 ) : a constant and −1 < < 1 ; 1 , 2 , 3 represents parameters of the autoregressive sample. The autocorrelation function gradually exponentially decreases while , while the partial autocorrelation has been lost after the period (P). For an instant, when P=1, the above equation becomes: 2) Autoregressive-Moving average models ((ARIMA) Slutzky and Wold participated in improving sample to three-dimension in the estimation and named it by "auto regressive -mixed moving average and autoregressive used when data are stationary. Where: The function of partial autocorrelation generally decreased, as an example, ARIMA becomes = 1 −1 − 1 −1 + +

3) Autoregressive integrated moving average models
Box and Jenkins (1976) described the overall samples and affected guides to understanding and dealing with data stationery. A sample has been affected by them capable of dealing with non-stationary series and converting it into stationary one by using differences odd degrees (d=1,2). It can be written by ARIMA (p,d,q) as: Exponential smoothing series ‫االسية‬ Pegels (1969) classified smoothing methods, which is one of the most important methods to estimate the time series and contained different ways to deal with all series types. It is a method which smoothing the non-seasonal series and gives a pervious = + (1 − ) 1 Single exponential smoothing method Provides by C.C. Holt (1958) which used non-seasonal time series and then Browns (1963) worked on using it for most time series types. Harrison (1965) provides guides to apply the method as follow: 1. Let − are -----------and weak to be available, so at − an approach value has to be used. Such as Ft will be an alternative data +1 = + ( − ) Data obtained from the previous equation are more stationary because it depends on a weight which is a fractional value. The equation can be simplified as: Smoothing +1 depending on a specific ratio which is (1/N) for all real data found, and previous data weight depending on (1 − 1 ).

N: is a positive number
(1/N): is a value between zero and one.

Paper Methodology
The paper methodology is based on applying the rainfall data on the theoretical equation presented and discussed above. The first process is to draw the observations on Y-axis and time at X-axis.

Matching sample of ARIMA
To find the best match sample of time series for rainfall, two criteria have been used. They are the Aiki and MSE criteria.

Results and discussion
Based on the paper methodology, the following results were obtained 1 Finding the best rainfall sample Using ARIMA (p,d,q) and by using AIC(K) and MSE criteria, it has been found that (from the table (1)) sample ARIMA A (5,0,2) was the best because it has the minimum value.