TimeSeries Analysis and Forecasting Part IV To read

  • Slides: 30
Download presentation
Time-Series Analysis and Forecasting – Part IV To read at home

Time-Series Analysis and Forecasting – Part IV To read at home

Method of moving average

Method of moving average

Method of moving average consists in replacement of initial levels of series by

Method of moving average consists in replacement of initial levels of series by the average values, which are calculated for the successively changing periods of time

Moving Averages § Example: Five-year moving average § First average: § Second average: §

Moving Averages § Example: Five-year moving average § First average: § Second average: § etc. Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 19 -4

(2 m+1)-Point Moving Average § A series of arithmetic means over time § Result

(2 m+1)-Point Moving Average § A series of arithmetic means over time § Result depends upon choice of m (the number of data values in each average) § Examples: § For a 5 year moving average, m = 2 § For a 7 year moving average, m = 3 § Etc. § Replace each xt with Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 19 -5

Example 16 19, 3 20, 5 22, 2 23 23, 6

Example 16 19, 3 20, 5 22, 2 23 23, 6

Example 17. Annual Data Year 1 2 3 4 5 6 7 8 9

Example 17. Annual Data Year 1 2 3 4 5 6 7 8 9 10 11 etc… Sales 23 40 25 27 32 48 33 37 37 50 40 etc… Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. … … Chap 19 -7

Calculating Moving Averages § Let m = 2 Year Sales Average Year 5 -Year

Calculating Moving Averages § Let m = 2 Year Sales Average Year 5 -Year Moving Average 1 23 3 29. 4 2 40 4 34. 4 3 25 5 33. 0 4 27 6 35. 4 5 32 7 37. 4 6 48 8 41. 0 7 33 9 39. 4 8 37 … … 9 37 10 50 11 40 etc… § Each moving average is for a consecutive block of (2 m+1) years Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 19 -8

Annual vs. Moving Average § The 5 -year moving average smoothes the data and

Annual vs. Moving Average § The 5 -year moving average smoothes the data and shows the underlying trend Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 19 -9

Centered Moving Averages (continued) § Let the time series have period s, where s

Centered Moving Averages (continued) § Let the time series have period s, where s is even number § i. e. , s = 4 for quarterly data and s = 12 for monthly data § To obtain a centered s-point moving average series Xt*: § Form the s-point moving averages § Form the centered s-point moving averages Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 19 -10

Centered Moving Averages § Used when an even number of values is used in

Centered Moving Averages § Used when an even number of values is used in the moving average § Average periods of 2. 5 or 3. 5 don’t match the original periods, so we average two consecutive moving averages to get centered moving averages Average Period 4 -Quarter Moving Average Centered Period Centered Moving Average 2. 5 28. 75 3 29. 88 3. 5 31. 00 4 32. 00 4. 5 33. 00 5 34. 00 5. 5 6 36. 25 6. 5 35. 00 etc… 37. 50 7 38. 13 7. 5 38. 75 8 39. 00 8. 5 39. 25 9 40. 13 9. 5 41. 00 Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 19 -11

Calculating the Ratio-to-Moving Average § Now estimate the seasonal impact § Divide the actual

Calculating the Ratio-to-Moving Average § Now estimate the seasonal impact § Divide the actual sales value by the centered moving average for that period Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 19 -12

Calculating a Seasonal Index Quarter Sales Centered Moving Average 1 2 3 4 5

Calculating a Seasonal Index Quarter Sales Centered Moving Average 1 2 3 4 5 6 7 8 9 10 11 … 23 40 25 27 32 48 33 37 37 50 40 … 29. 88 32. 00 34. 00 36. 25 38. 13 39. 00 40. 13 etc… … … Ratio-to. Moving Average 83. 7 84. 4 94. 1 132. 4 86. 5 94. 9 92. 2 etc… … … Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 19 -13

Calculating Seasonal Indexes (continued) Fall Quarter Sales Centered Moving Average 1 2 3 4

Calculating Seasonal Indexes (continued) Fall Quarter Sales Centered Moving Average 1 2 3 4 5 6 7 8 9 10 11 … 23 40 25 27 32 48 33 37 37 50 40 … 29. 88 32. 00 34. 00 36. 25 38. 13 39. 00 40. 13 etc… … … Ratio-to. Moving Average 83. 7 84. 4 94. 1 132. 4 86. 5 94. 9 92. 2 etc… … … Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. 1. Find the median of all of the same -season values 2. Adjust so that the average over all seasons is 100 Chap 19 -14

Interpreting Seasonal Indexes § Suppose we get these seasonal indexes: Seasonal Season Index §

Interpreting Seasonal Indexes § Suppose we get these seasonal indexes: Seasonal Season Index § Interpretation: Spring 0. 825 Spring sales average 82. 5% of the annual average sales Summer 1. 310 Summer sales are 31. 0% higher than the annual average sales Fall 0. 920 etc… Winter 0. 945 = 4. 000 -- four seasons, so must sum to 4 Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 19 -15

Analytical smoothing of time series

Analytical smoothing of time series

Levels of time series are considered as the function of time

Levels of time series are considered as the function of time

The procedure of smoothing has 3 steps Ø choice of the form of function;

The procedure of smoothing has 3 steps Ø choice of the form of function; Ø determination of parameters of the function; Ø receiving the smoothed values of the levels of series on the basis of the function

Let’s consider this method on the example of linear trend equation where a &

Let’s consider this method on the example of linear trend equation where a & b – parameters; t – time

The best way to use linear trend in cases, when the preliminary investigation shows,

The best way to use linear trend in cases, when the preliminary investigation shows, that levels of series change with approximately the same speed, i. e. when chain absolute increases are approximately equal

Parameters a & b are determined by the least square method LSM

Parameters a & b are determined by the least square method LSM

The usage of LSM gives the following system of equations for determining the parameters:

The usage of LSM gives the following system of equations for determining the parameters:

The given system of equations can be significantly simplified, if we enumerate the time

The given system of equations can be significantly simplified, if we enumerate the time in the way, that

If the series contains odd number of levels, then the central level of series

If the series contains odd number of levels, then the central level of series is enumerated as 0. Levels to the side of decrease of time are enumerated by -1; 2; -3. . . , to the side of increase of time – by 1; 2; 3. . .

If the series contains even number of levels, then the closest levels to the

If the series contains even number of levels, then the closest levels to the center are enumerated by -1 and 1, then numeration is the same as with odd number of levels but only with the step 2: . . . -5, -3, -1, +3, +5. . .

Example 17

Example 17

End of Part IV § To be continued… Statistics for Business and Economics, 6

End of Part IV § To be continued… Statistics for Business and Economics, 6 e © 2007 Pearson Education, Inc. Chap 19 -30

Forecasting Replenishment Forecasting Replenishment Forecasting Forecasting Replenishment ForecastingForecasting Replenishment Forecasting Replenishment Forecasting Forecasting Replenishment Forecasting
TimeSeries Analysis and Forecasting Part V To readTimeSeries Analysis and Forecasting Part V To read
forecasting longterm forecasting middleterm forecasting shortterm forecasting 2forecasting longterm forecasting middleterm forecasting shortterm forecasting 2
Business and Economic Forecasting Lecture 5 TimeSeries ForecastingBusiness and Economic Forecasting Lecture 5 TimeSeries Forecasting
Forecasting Forecasting Problems and Methods New product forecastingForecasting Forecasting Problems and Methods New product forecasting
Introduction Definition Forecasting Technology Technological Forecasting Process ForecastingIntroduction Definition Forecasting Technology Technological Forecasting Process Forecasting
Forecasting Forecasting A Basis of Forecasting In businessForecasting Forecasting A Basis of Forecasting In business
Forecasting Forecasting A Basis of Forecasting In businessForecasting Forecasting A Basis of Forecasting In business
DEMAND FORECASTING DEMAND FORECASTING Demand forecasting means estimationDEMAND FORECASTING DEMAND FORECASTING Demand forecasting means estimation
Forecasting Models FORECASTING Forecasting is attempting to predictForecasting Models FORECASTING Forecasting is attempting to predict
Forecasting Qualitative Analysis Quantitative Analysis Predictions or ForecastingForecasting Qualitative Analysis Quantitative Analysis Predictions or Forecasting
FORECASTING Qualitative Analysis Quantitative Analysis Predictions or ForecastingFORECASTING Qualitative Analysis Quantitative Analysis Predictions or Forecasting
g MultiModel Fusion for Robust TimeSeries Forecasting Weizhongg MultiModel Fusion for Robust TimeSeries Forecasting Weizhong
Chapter 16 TimeSeries Forecasting Yandell Econ 216 ChapChapter 16 TimeSeries Forecasting Yandell Econ 216 Chap
Budget and Forecasting Part 2 In Year ForecastingBudget and Forecasting Part 2 In Year Forecasting
Financial Forecasting Forecasting and Pro Forma Analysis lFinancial Forecasting Forecasting and Pro Forma Analysis l
Chapter 13 Weather Forecasting and Analysis Weather forecastingChapter 13 Weather Forecasting and Analysis Weather forecasting
Forecasting Approaches to Forecasting A Judgmental Analysis SubjectiveForecasting Approaches to Forecasting A Judgmental Analysis Subjective
CSC 2515 Lecture 10 Part 2 Making timeseriesCSC 2515 Lecture 10 Part 2 Making timeseries
TimeSeries Analysis J C Clint Sprott Department ofTimeSeries Analysis J C Clint Sprott Department of
Applications of Neural Networks in TimeSeries Analysis AdamApplications of Neural Networks in TimeSeries Analysis Adam
Using rasterbased timeseries analysis to define temporal streamflowUsing rasterbased timeseries analysis to define temporal streamflow
13 Introduction to TimeSeries Analysis What is in13 Introduction to TimeSeries Analysis What is in
Time Series Analysis TimeSeries Numerical data obtained atTime Series Analysis TimeSeries Numerical data obtained at