Time series models have been the basis for the study of a behavior or metrics over a period of time. In decisions that involve a factor of uncertainty about the future, time series models have been found to be one of the most effective methods of forecasting. We often encounter different time series models in sales forecasting, weather forecasting, inventory studies, and so on. In the field of information science, Jeong and Kim (2010) reviewed a selected annotated bibliography of core books in order to conduct a time series analysis.

Often the first step in a time series analysis is to plot the data and observe any patterns that might occur over time by using a plot. This helps us to determine whether there appears to be a long-term upward or downward movement or whether the series seems to vary around a horizontal line over time.

The following scenario will show how the plot illustrates the direction of data. In this example, we follow the number of books borrowed from the library from 1982 to 2014 as found in the public library log in New York City. Table 14.1.

	Year	Numbers of books borrowed from the library
1	1982	1558
2	1984	1564
3	1994	1407
4	1995	1309
5	1986	1424
6	1987	1321
7	1988	1543
8	1989	1321
9	1990	1488
10	1991	1562
11	1992	1618
12	1993	1686
13	1994	1840
14	1995	1865
15	1996	1636
16	1997	1652
17	1998	1453
18	1999	1698
19	2000	1523
20	2001	1437
21	2002	1387
22	2003	1312
23	2004	1212
24	2005	1201
25	2006	1009
26	2007	1001
27	2008	999
28	2009	1764
29	2010	1675
30	2011	1218
31	2012	1212
32	2013	1200

You will import the file to R by the following command:
>library_borrowing <-read.csv(“C:/Table14.1″, header=T, dec=”,”, sep=”;”)
Note that paths use forward slashes “/” instead of backslashes
>plot(library_borrowing[, 5], type=”1″, lowd=2, col=”red”, xlab=Years”, ylab=”Number of books”, main=”Number of books borrowed from the library” xl)

The result:

 

 

 

 

 

As we can see, this illustration provides an indication and direction of the decline of book borrowing from the library. This trend started in 2000 and stopped in 2009.

The goal of the time series can be classified in five steps:
1. Descriptive: identifying patterns in correlated data—trends and seasonal variations
2. Explanation: understanding and modeling the data
3. Forecasting: prediction of short-term trends from previous patterns
4. Intervention analysis: discovering if a single event changes the time series
5. Quality control: deviations of a specified size indicate a problem

Another aspect of time series is smoothing, a common technique. Smoothing always involves some form of local averaging of data such that the nonsystematic components (variations away from main trend) of individual observations cancel each other out. The most common technique is moving average smoothing, which replaces each element of the series by either the simple or weighted average of n surrounding elements, where n is the width of the smoothing “window.” All of these methods will filter out the noise and convert the data into a smooth curve that is relatively unbiased by outliers.

Among the methods for fitting a straight line to a series of data, Least Square Method is the one used most frequently. The equation of a straight line is Y = a + bx where x is the time period, say year, and Y is the value of the item measured against time, a is the Y intercept, and b is the coefficient of x indicating slope of the trend line. In order to find a and b, the following two equations are calculated:

ΣY = ax + b Σx
ΣXY = a Σx + bΣx2

The code in R:
Based on the library’s example above, we selected the first five numbers:
>time <- c(1986, 1987, 1988, 1989, 1999)
>number <- c(270, 285, 295, 315, 330)
>res=lm(time~number)
>res=lm(time~number)
>res
Coefficients:
(Intercept) 1933.293
Number 0.189

In order to calculate the smoothing of Least Square data, we often use a linear operation. This process converts a single time series {yt} into another time series {xt} by a linear operation. The formula for smoothing using Moving Average:

 

 

 

 

Next, Chapter 15, Visualization Display
Previous, Chapter 13, Analysis of Variances and Chi-Square Test