BA 9101STATISTICS FOR MANAGEMENT : BA 9101STATISTICS FOR MANAGEMENT TIME SERIES
TIME SERIES : TIME SERIES A time series is a set of observations taken at specified times usually at equal intervals.
It is a set of data depending upon time.
Example: Monthly sales of a company for the last year.
TIME SERIES (contd.) : TIME SERIES (contd.) The Time Series given as a variable Y, is usually given as a function of time t.
Yt denotes the value of Y at time t.
According to additive model, the time series can be expressed as:
Y=T+S+C+I , where
Y= Value of original time series.
T= Trend value.
S= Seasonal variation.
C= Cyclical variation.
I= Irregular variation.
VARIATIONS IN TIME SERIES : VARIATIONS IN TIME SERIES TYPES OF VARIATIONS
LONG TERM SHORT TERM
SECULAR CYCLICAL SEASONAL IRREGULAR
(TREND)
SECULAR TREND (or) TREND : SECULAR TREND (or) TREND Secular trend is the smooth, regular and long term movement of series showing a continuous growth stagnation or decline over a long period of time.
Graphically, it exhibits general direction and shape of time series.
Examples:
Upward trend in economic growth due to increasing population, price, etc.
Downward trends in a time series is relating to death and birth rates, etc.
CYCLICAL VARIATIONS : CYCLICAL VARIATIONS One complete period which is more than a year is called as a cycle.
Example: A business cycle.
Over time, there are years when the business cycle hits a peak above the trend line. At other times, business activity is likely to slump, hitting a low point below the trend line.
Cyclical movements do not follow any regular patterns but move in an unpredictable way.
SEASONAL VARIATIONS : SEASONAL VARIATIONS Involves patterns of change within a year that tend to be repeated from year to year.
Short term periodic movement.
Example:
Increase in the number of flu and viral fever cases in winter every year.
RANDOM (OR) IRREGULAR VARIATIONS : RANDOM (OR) IRREGULAR VARIATIONS They are purely random and unpredictable.
They do not reveal any pattern of the repetitive tendency and be considered as residual variations.
They are caused by unforeseen events like floods, wars, earthquakes, etc.
Example: Prices of LPG, Petrol and Diesel.
TREND ANALYSIS : TREND ANALYSIS Represents the long term direction of the series.
Reasons for studying trends:
To describe a historical pattern.
To project past patterns and trends into future.
Sometimes to eliminate the trend component for accuracy.
Two methods:
Freehand (or) graphic method.
Least square method.
FITTING THE TREND BY LEAST SQUARE METHOD : FITTING THE TREND BY LEAST SQUARE METHOD Principle: The sum of squares of the deviations of the actual and computed values is least for the line of the fit.
To fit a straight line,
Y=a+bx, where
a= ΣY b= ΣXY
n ΣX2
n: number of observations
Slide 11: Fit a straight line trend by the method of least squares to the following data:
Slide 13: X = 1993.5
n= 6 Y=25 - 0.343X
a= Y = 25
b= (ΣXY/ ΣX2) = -0.343
Slide 15: 2. Obtain the straight line trend equation and tabulate against each year after estimation of the trend and short term fluctuations.
CYCLICAL VARIATIONS : CYCLICAL VARIATIONS The component of a time series that tends to oscillate above and below the secular trend line for periods longer than 1 year.
2 Methods:
Residual Method (% of trend).
Relative Cyclical Residual Method.
Residual Method: % of trend = (Y/Ŷ)*100
Relative Cyclical Residual: {(Y-Ŷ) / Ŷ} * 100
Y: Actual value Ŷ: Estimated / Expected value
Slide 18: 3. The details of grains received by Farmer’s Co-operative over 8 years has been given in the table below. Find the percent of trend and the relative cyclical residual for the data.
SEASONAL VARIATION : SEASONAL VARIATION For detecting seasonal variations, time intervals must be measured in small units, such as days, weeks, months or quarters.
Reasons for studying seasonal variations:
To describe a historical pattern.
To project past patterns into future.
For deseasonalizing the time series.
Two methods:
Method of Averages.
Ratio-to-Moving-Average Method.
SEASONAL VARIATIONS : SEASONAL VARIATIONS METHOD OF AVERAGES
Steps:
Find the quarterly averages for the 4 quarters of the given years: X1,X2,X3,X4.
Calculate the Grand Average, G.
G= X1 + X2 + X3 + X4
4
Find the seasonal indices for each quarter.
Seasonal Index for ith quarter = Xi * 100
G
Slide 22: 4. The Quarterly sales for a graphics software company are given below. Determine the seasonal components.
Slide 23: G=(525+358.33+266.67+483.33)
4
= 408.3325
SEASONAL VARIATIONS : SEASONAL VARIATIONS Ratio-to-Moving-Average Method
Steps:
Calculate the 4-quarter moving total.
Compute the 4-quarter moving average.
Center the 4-quarter moving average.
Calculate the percentage of actual value to the moving average value.
Calculate the modified mean.
Adjust the modified mean.
IRREGULAR VARIATIONS : IRREGULAR VARIATIONS These occur over short intervals and follow a random pattern.
Because of their unpredictability, we do not attempt to explain them mathematically.
However, we can often isolate its causes.
It is what is “left over” after we eliminate trend cyclical and seasonal variations from a time series.
A problem containing all the 4 components of variations ! : A problem containing all the 4 components of variations ! The procedure for describing this time series will consist of these 3 stages:
Deseasonalizing the time series.
Developing the trend line.
Finding the cyclical variation around the trend line