Post on 18-Nov-2014
description
Allama Iqbal Open University
Subject Title:Subject Title:StatisticsStatistics
Presented By:Presented By:Adeel AhmadAdeel Ahmad
Roll No:Roll No:AD-514968AD-514968
Topic
• Measures of trend and seasonal variation
Area of discussion
• Time Series• Components of time series• SECULAR TREND OR TREND• Reasons for studying trends• Seasonal Variations• Reasons for studying seasonal variations
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.• A series of values over a period of time.• In time series, Time act as an independent
variable to estimate dependent variable. • Example: Monthly sales of a company for
the last year.
Time Series
• 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.
Causes in variations in time series data
• Social Customs, Festivals• Seasons• The four phase of business:
1. Prosperity, decline ,depression ,recovery• Natural Climate:
1. Earthquake ,food etc• Political Movement/Changes, War etc
Importance of Time series Analysis
• A very popular tool for business forecasting
• Basis for understanding past behavior• Can forecast future activities/Planning for
future operation
Components of time series
• SECULAR TREND OR TREND• SEASONAL VARIATION/FLUCTUATIONS• CYCLICAL VARIATION/FLUCTUATIONS• IRREGULAR VARIATION AND MOVEMENT
Variation in time Series
Type of Variation
Long term Short term
SECULAR CYCLICAL SEASONAL IRREGULAR
SECULAR TREND OR TREND
• SECULAR TREND (or) TREND Secular trend is the smooth, regular and long term movement of series showing a continuous growth or decline over a long period of time.
• The general tendency of the data to grow or decline over a long period of time.
• 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.
Reasons for studying trends
• The study of secular trends allows us to describe a historical pattern
• Studying secular trends permits us to project past patterns or trends in to the future.
• 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/n• b= ΣXY/ΣX2 • n: number of observations
Fit a straight line trends by the method of least squares to the following data:
Year 2003 2004 2005 2006 2007 2008
Production
Tones
24 25 29 26 22 24
Year Production
X
X-x
Xy x² Yc Fluctuation
1991 24 -2.5 -60 6.25 25.858 1.8575
1992 25 -1.5 -37.5 2.25 25.514 0.514
1993 29 -0.5 -14.5 0.25 25.172 -3.828
1994 26 0.5 13 0.25 24.828 -1.172
1995 22 1.5 33 2.25 24.486 2.486
1996 24 2.5 60 6.25 24.142 0.142
Total 150 0 -6 17.5
X = 1993.5 n= 6 a= Y = 25 b= (ΣXY/ ΣX2) = -0.343
Y=25 - 0.343X
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.
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.
• Two methods: • Method of Averages. • Ratio-to-Moving-Average Method.
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
The Quarterly sales for a graphics software company are given below. Determine the seasonal components.
Year 2003 2004 2005 2006 2007 2008 Total Average
Q1 500 450 350 550 550 750 3150 525
Q2 350 350 200 350 400 500 2150 358.33
Q3 250 200 150 250 350 400 1600 266.67
Q4 400 300 400 550 600 650 2900 483.33
G=(525+358.33+266.67+483.33) 4 = 408.3325
Seasonal IndexQUARTER SEASONAL INDEX
Q1 128.575
Q2 87.454
Q3 65.307
Q4 118.367
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.