time series

Time Series Analysis

Arrangement of statistical data in chronological order in accordance with occurrence of time is known as Time series.

• The objective of time series analysis is to find a pattern in the historical data and then extrapolate the pattern into the future.

Time Series Patterns

• Actual of the variable at time t = Mean value of the variable at time t + Random deviation from the mean value of the variable at time t.

i.e Actual = Pattern + error

Utilities:• Determine the type and nature of the

variations.• Compare the actual value and the expected

value.• Predict the behavior of the variable in the

future.• Compare the changes in the values of different

phenomenon at different times.

Components of a Time Series

• Trend

• Cyclical Variation

• Seasonal Variation

• Irregular Variation

• Trend we mean average tendency to increase or decrease over a long period of time.

• The oscillatory movements with period more than one year called cyclical fluctuations. Cyclical movements do not follow any regular pattern but move in a somewhat unpredictable pattern.

• Seasonal variation involves patterns of changes with in a year that operate in a regular and periodic manner.

• Irregular variations are unpredictable and are beyond the control of human hand.

Decomposition Models

• Additive ModelIn this model, it is assumed that the effect of various components can be estimated by adding the various components of a time-series. It is stated as:Y = T + C + S + I

Where Y is the time series value at time tT is the trend value C is the cyclical variation at time tS is the seasonal variation at time tI is random fluctuation at time t

.

• Multiplicative Model

The actual values of a time series, represented by Y can be found by multiplying four components at a particular time period. The effect of four components on the time series is interdependent. The multiplicative time series model is defined as:

Y = T × C × S × I

Smoothing MethodsThe objective of smoothing

methods is to smooth out the random variations due to irregular components of the time series.

1. Moving averages2. Weighted moving averages

Moving Averages

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n = length of time period

• The following table shows the production volume in ‘000 tonnes for a product. Use these data to compute a 3 year moving average for all available years. And also find the error. Year Production

1995 211996 221997 231998 251999 242000 222001 252002 262003 272004 26

Weighted Moving Averages

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• Vacuum cleaner sales for 12 months is given below. The owner of the supermarket decides to forecast sales by weighting the past three months as follows:

Weight applied Month3 Last month2 Two months ago1 Three months ago

Month 1 2 3 4 5 6 7 8 9 10 11 12

Sales 10 12 13 16 19 23 26 30 28 18 16 14

• A state government is studying the number of traffic fatalities in the state resulting from drunken driving for each of the last 12 months.

• Calculate 4 monthly moving average.

Month 1 2 3 4 5 6 7 8 9 10 11 12Accidents 280 300 280 280 270 240 230 230 220 200 210 200

• Trend Analysis:

It allows us to describe a historical pattern.

It permits us to project past patterns into the future.

It allows us to eliminate the trend component from the series.

• Measurement of Trend1. Principle of least squaresLinear trendQuadratic trendExponential trend

2. Method of Semi- averages

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• Plot the time series and comment on the appropriateness of a linear trend for the following data. If not what type of functional form do you believe would be most appropriate for the trend pattern of this time series?

Year 1 2 3 4 5 6 7 8 9 10

Sales 400 390 320 340 270 260 300 320 340 370

• The president of a small manufacturing firm is concerned about the continual increase in manufacturing costs over the past several years. The following figures provide a time series of the cost per unit for the firm’ s leading product over the eight years.

Does a linear trend appear to be present?

Year 1 2 3 4 5 6 7 8Cost/unit 20 24.5 28.2 27.5 26.6 30 31 36

• Below are given the figures of production in ‘000 quintals of a sugar factory.

• Fit a straight line trend • Plot these data and show the trend line• Estimate the production in 2004

Year 1995 1996 1997 1998 1999 2000 2001Production 80 90 92 83 94 99 92

Semi-Averages

• In this method, the whole data is divided into two parts with respect to time. In case of odd number of years the two parts are obtained by omitting the value corresponding to the middle year. We compute the A.M for each part and plot these two averages against the middle values of the respective periods by each part. The line obtained on joining these two points is the required trend line.

• The owner of a small company manufactures a product. Since he started the company, the number of units of the product he has sold is represented by the following time series:

• Find the trend line that describes the trend by using Semi-averages and forecast the demand for 2005

Year 1995 1996 1997 1998 1999 2000 2001Demand 100 120 95 105 108 102 112

• Fit a trend line to the following data by the method of semi-average and forecast the sales for the year 2002.Year Sales1993 1021994 1051995 1141996 1101997 1081998 1161999 112

• Measurement of Seasonal effects

Seasonal variation is defined as repetitive and predictable movement around the trend line in one year or less.

Seasonal effects on time series is essential for projection of past pattern into the future for short run predictions.

Seasonal effects are measured in terms of an index. A seasonal index is an average that indicates the percentage deviation of actual values of the time series from a base value which excludes the short term seasonal influences.

• Methods:

Method of simple average

Ratio-to –trend method

Ratio-to-moving average

Link relatives method

• Ratio-to-moving average method:

It provides an index for seasonality that describes the degree of seasonal variation.

• Procedure:1. Calculate the 4-quarter moving average for the

time series (for quarterly data). Calculate 12 month moving average for the data ( for monthly data).

2. Compute centered moving averages.

3. Calculate the % of the actual value to the moving average value for each quarter in the time series.Ratio-to-moving average=( Actual value/Moving average) *100

4. Arrange all the % of the actual to moving average values by quarter.

5. Calculate the modified mean by discarding the highest and lowest values for each quarter and average the remaining values.

6. Adjust the modified mean to 400 for quarterly data, 1200 for monthly data by multiplying each of the quarterly indices (monthly indices) by an adjusting component and is found by dividing the desired sum of the indices(400/1200) by the actual sum. This process brings to a total of 400 for quarterly data, 1200 for monthly data.

• The resort hotel wanted to establish the seasonal pattern of room demand by its clientele. Hotel management wants to improve customer service and is considering several plans to employ personnel during peak periods to achieve this goal. The following table contains the quarterly occupancy i.e the average number of guests during each quarter of the last 5 years.

Year I II III IV1991 1861 2203 2415 19081992 1921 2343 2514 19861993 1834 2154 2098 17991994 1837 2025 2304 19651995 2073 2414 2339 1967

No.of guests per quarter

Solution

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occupancy Actual

occupancy edseasonaliz De•

• Suppose the hotel management estimates from a de-seasonalized trend line that the de-seasonalized average occupancy for the 4 th quarter of the year 1996 as 2121. what will be the seasonalized average occupancy?

100

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• Method of Simple Averages:

1. Average the data by the years and months (quarters if quarterly are given).

2. Add the figures of each month (quarter) and obtain the average by dividing the monthly (quarterly) totals by the number of years.

3. Obtain an average of monthly (quarterly) averages by dividing the total of monthly(quarterly) averages by 12 (4).

4. Compute seasonal indices for different months (quarters) by expressing monthly averages (quarterly averages) as % of the grand average.

• The data below give the average quarterly prices of a commodity for four years. Calculate the seasonal indices.

Year I II III IV1980 40.3 44.8 46 481981 50.1 53.1 55.3 59.51982 47.2 50.1 52.1 55.21983 55.4 59 61.6 65.3

Quarter

Solution

• Measurement of Cyclic Variation:

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• A farmer’s marketing cooperative wants to measure the variations in its member’s wheat harvest over an 8 year period. The following table shows the volume harvested in each of the 8 years.

Year 1988 1989 1990 1991 1992 1993 1994 1995Actual Bushels '0000s) 7.5 7.8 8.2 8.2 8.4 8.5 8.7 9.1

Solution