UNIT 5
Time Series Analysis

Learn about
Auto
regression and
Moving
average
Models
Understand
the business
scenarios
where Time
Series Analysis
is applicable
Learn about
ARIMA and
SARIMA
models for
forecasting
Get a solid
understanding
of Time Series
Analysis and
Forecasting
Learning Outcomes

Time Series Analysis
Time series is a series of data points in which each data point is associated with a
timestamp. A simple example is the price of a stock in the stock market at different
points of time on a given day. Another example is the amount of rainfall in a region at
different months of the year. R language uses many functions to create, manipulate and
plot the time series data. The data for the time series is stored in an R object called
time-series object. It is also a R data object like a vector or data frame.

Time Series Analysis
The time series object is created by using the ts() function.
The basic syntax for ts() function in time series analysis is −
timeseries.object.name <- ts(data, start, end, frequency)
Following is the description of the parameters used −
• data is a vector or matrix containing the values used in the time series.
• start specifies the start time for the first observation in time series.
• end specifies the end time for the last observation in time series.
•frequency specifies the number of observations per unit time.
Except the parameter "data" all other parameters are optional.

Time Series Analysis
Consider the annual rainfall details at a place starting from January 2012. We create an R time series object for a period of 12 months
and plot it.
# Get the data points in form of a R vector.
rainfall <- c(799,1174.8,865.1,1334.6,635.4,918.5,685.5,998.6,784.2,985,882.8,1071)
# Convert it to a time series object.
rainfall.timeseries <- ts(rainfall,start = c(2012,1),frequency = 12)
# Print the timeseries data.
print(rainfall.timeseries)
# Give the chart file a name.
png(file = "rainfall.png")
# Plot a graph of the time series.
plot(rainfall.timeseries)
# Save the file.
dev.off()

Time Series Analysis
Different Time Intervals
The value of the frequency parameter in the ts() function decides the time intervals at
which the data points are measured. A value of 12 indicates that the time series is for
12 months. Other values and its meaning is as below −
frequency = 12 pegs the data points for every month of a year.
frequency = 4 pegs the data points for every quarter of a year.
frequency = 6 pegs the data points for every 10 minutes of an hour.
frequency = 24*6 pegs the data points for every 10 minutes of a day.

Multiple time Series Analysis
We can plot multiple time series in one chart by combining both the series into a matrix.
# Get the data points in form of a R
vector.rainfall1<-
c(799,1174.8,865.1,1334.6,635.4,918.5,685.5,998.6,784.2,985,882.8,1071)
<-
rainfall2
c(655,1306.9,1323.4,1172.2,562.2,824,822.4,1265.5,799.6,1105.6,1106.7,1337.8)
# Convert them to a matrix.
combined.rainfall <- matrix(c(rainfall1,rainfall2),nrow = 12)
# Convert it to a time series object.
rainfall.timeseries <- ts(combined.rainfall,start = c(2012,1),frequency = 12)
# Print the timeseries data.print(rainfall.timeseries)
# Give the chart file a name.png(file = "rainfall_combined.png")
# Plot a graph of the time series.plot(rainfall.timeseries, main = "Multiple Time
Series")# Save the file.dev.off()

ARIMA
ARMA models are commonly used in time series modeling. In ARMA model, AR stands
for auto-regression and MA stands for moving average. If these words sound
intimidating to you, worry not – I’ll simplify these concepts in next few minutes for you!
Auto-Regressive Time Series Model
Let’s understanding AR models using the case below:
The current GDP of a country say x(t) is dependent on the last year’s GDP i.e. x(t – 1).
The hypothesis being that the total cost of production of products & services in a
country in a fiscal year (known as GDP) is dependent on the set up of manufacturing
plants / services in the previous year and the newly set up industries / plants / services
in the current year. But the primary component of the GDP is the former one.

Time Series Analysis
Hence, we can formally write the equation of GDP as:
x(t) = alpha * x(t – 1) + error (t)
This equation is known as AR(1) formulation. The numeral one (1) denotes that the next
instance is solely dependent on the previous instance. The alpha is a coefficient which
we seek so as to minimize the error function. Notice that x(t- 1) is indeed linked to x(t-
2) in the same fashion. Hence, any shock to x(t) will gradually fade off in future.

Time Series Analysis
For instance, let’s say x(t) is the number of juice bottles sold in a city on a particular
day. During winters, very few vendors purchased juice bottles. Suddenly, on a particular
day, the temperature rose and the demand of juice bottles soared to 1000. However,
after a few days, the climate became cold again. But, knowing that the people got used
to drinking juice during the hot days, there were 50% of the people still drinking juice
during the cold days. In following days, the proportion went down to 25% (50% of
50%) and then gradually to a small number after significant number of days. The
following graph explains the inertia property of AR series:

Moving Average Time Series Analysis
Let’s take another case to understand Moving average time series model.
A manufacturer produces a certain type of bag, which was readily available in the
market. Being a competitive market, the sale of the bag stood at zero for many days.
So, one day he did some experiment with the design and produced a different type of
bag. This type of bag was not available anywhere in the market. Thus, he was able to
sell the entire stock of 1000 bags (lets call this as x(t) ). The demand got so high that
the bag ran out of stock. As a result, some 100 odd customers couldn’t purchase this
bag. Lets call this gap as the error at that time point. With time, the bag had lost its woo
factor. But still few customers were left who went empty handed the previous day.
Following is a simple formulation to depict the scenario :

Moving Average Time Series Analysis
Difference between AR and MA models
The primary difference between an AR and MA model is based on the correlation
between time series objects at different time points. The correlation between x(t) and
x(t-n) for n > order of MA is always zero. This directly flows from the fact that
covariance between x(t) and x(t-n) is zero for MA models (something which we refer
from the example taken in the previous section). However, the correlation of x(t) and
x(t-n) gradually declines with n becoming larger in the AR model. This difference gets
exploited irrespective of having the AR model or MA model. The correlation plot can give
us the order of MA model.

Framework of ARIMA Modelling