ForecastIT 2. Linear Regression & Model Statistics
- 1. Linear Regression and Model StatisticsLesson #2Linear Regression MethodCopyright 2010 DeepThought, Inc.1
- 2. Linear Regression and Model StatisticsMethod IntroductionOne of the simpler methods to use for forecasting
- 3. Estimates a line through the data
- 4. Uses the estimated line equation to forecast future values.
- 6. Y = a + b × tCopyright 2010 DeepThought, Inc.2
- 7. Linear Regression and Model StatisticsModel CharacteristicsMethod characteristics
- 8. Fits a line to the data
- 9. Estimating a line which minimizes the errors between actual data points and model estimates
- 10. When to use method
- 11. Estimate trend
- 13. When not to use
- 14. Estimate anything beyond a simple linear relationship Copyright 2010 DeepThought, Inc.3
- 15. Linear Regression and Model StatisticsForecasting StepsSet an objectiveBuild modelEvaluate modelUse modelCopyright 2010 DeepThought, Inc.4
- 16. Linear Regression and Model StatisticsObjective SettingSimpler is better
- 17. Linear regression allows to test whether a line fitted to the data works as a model. Objectives should take that principal under consideration
- 18. Example objectives for M2 Money Stock (see next slide):
- 19. Test if M2 has a linear trend over time
- 20. If M2 exhibits a statistically significant trend, review and interpret results
- 21. If model looks good, create a forecast based off modelCopyright 2010 DeepThought, Inc.5
- 22. Linear Regression and Model StatisticsExample: M2 Money StockCopyright 2010 DeepThought, Inc.6
- 23. Linear Regression and Model StatisticsMethod SelectionObserve time series qualities: trend, seasonality, cyclicality, and randomness
- 24. Adjust time frame, units, periods to forecast as needed
- 25. Determine if linear regression is a possible candidate based on method characteristics
- 26. Determine if transforming the units will enable use of model
- 27. Eight different unit transformation techniquesCopyright 2010 DeepThought, Inc.7
- 28. Linear Regression and Model StatisticsBuild ModelSoftware finds us the best fit line to the data; minimizing the sum of squared errorsCopyright 2010 DeepThought, Inc.8
- 29. Linear Regression and Model StatisticsEvaluate ModelDescriptive Statistics
- 30. Mean
- 31. Variance & Standard Deviation
- 32. Accuracy / Error
- 33. SSE
- 34. RMSE
- 35. MAPE
- 36. R2; Adjusted R2
- 38. F-Test
- 39. P-Value F-TestCopyright 2010 DeepThought, Inc.9
- 40. Linear Regression and Model StatisticsDescriptive StatisticsMeanThe average value of the data set ×http://images.google.com/imgres?imgurl=http://www.cs.princeton.edu/introcs/11gaussian/images/stddev.png&imgrefurl=http://www.cs.princeton.edu/introcs/11gaussian/&usg=__7JZMBeSrlQKPfVL2YCVuV8HVXkY=&h=206&w=570&sz=18&hl=en&start=54&um=1&tbnid=5jb7PXr6kgP08M:&tbnh=48&tbnw=134&prev=/images%3Fq%3Dstandard%2Brandom%2Bdistribution%26ndsp%3D18%26hl%3Den%26client%3Dfirefox-a%26rls%3Dorg.mozilla:en-US:official%26hs%3DXpO%26sa%3DN%26start%3D36%26um%3D1Copyright 2010 DeepThought, Inc.10
- 41. Linear Regression and Model StatisticsVariance & Standard DeviationThe sum of squared deviations of the data from the mean
- 42. Estimates the variation the data exhibits from the mean
- 43. Standard deviation is the squared root of the variance
- 44. Used to measure the distribution of the variable away from the mean, most observations of the variable will be within ± 3 standard deviationsCopyright 2010 DeepThought, Inc.11
- 45. Linear Regression and Model StatisticsM2 ExampleMean
- 46. 4214.38
- 47. Variance
- 48. 3346475.10
- 50. 1829.34 Copyright 2010 DeepThought, Inc.12
- 51. Linear Regression and Model StatisticsAccuracy/ErrorSSESum of Square Errors (SSE)
- 52. Sums the errors between the actual values and model values
- 53. Measures the total error of the model
- 54. M2 Example:
- 55. SSE: 316778645.89 Copyright 2010 DeepThought, Inc.13
- 56. Linear Regression and Model StatisticsRMSEThe square root of the sum of square error divided by the number of observations
- 57. An averaged out total of errors based upon the number of observations
- 58. Simple way to compare models based on error
- 59. M2 Example:
- 60. RMSE: 456.82 Copyright 2010 DeepThought, Inc.14
- 61. Linear Regression and Model StatisticsMAPEThe average percentage error of the model
- 62. Describes the average percentage of variation exhibited between actual and forecasted values
- 63. M2 Example:
- 64. MAPE: 10.09% Copyright 2010 DeepThought, Inc.15
- 65. Linear Regression and Model StatisticsR-Squared & Adjusted R-SquaredA proportion between unexplained and explained errors
- 66. Measures the percentage of variation captured by the model
- 67. Adjusted R2incorporated the number of variables used and sample size to adjust the R2 valueCopyright 2010 DeepThought, Inc.16
- 68. Linear Regression and Model StatisticsM2 ExampleR2
- 69. 93.76%
- 70. Adjusted R2
- 71. 93.76% Copyright 2010 DeepThought, Inc.17
- 72. Linear Regression and Model StatisticsStatistical SignificanceF-TestA proportion between explained and unexplained errors of model
- 73. Used to test if model build is statistically significant from being equal to zero
- 74. The larger the F-test the betterCopyright 2010 DeepThought, Inc.18
- 75. Linear Regression and Model Statistics F-Test P-ValueThe F-Test P-Value representsthe percentage of significance of the F-test (blue area on graph) The higher the value of the F-test the lower the shaded blue area is. As the blue area decreases, confidence about our model being statistically significant increases
- 76. 1 – p-value = Significance Level of the Model (%)
- 77. Significance level of the model (%) represents the amount of confidence we have that our model is different from a model with no impact, or zero impactCopyright 2010 DeepThought, Inc.19
- 78. Linear Regression and Model StatisticsM2 ExampleF-Test
- 79. 22778.98
- 80. F-Test P-Value
- 81. 0.00 Copyright 2010 DeepThought, Inc.20
- 82. Linear Regression and Model StatisticsCompare Multiple ModelsSkip this step until have knowledge of multiple methods
- 83. Will use accuracy/error statistics to compare multiple models to find best modelsCopyright 2010 DeepThought, Inc.21
- 84. Linear Regression and Model StatisticsUse ModelUnderstand limitations of model
- 85. Only measures a trend
- 86. A long term average
- 88. Does M2 has a linear trend
- 89. If trend exists, what is its magnitude
- 90. If model statistically significant, forecastCopyright 2010 DeepThought, Inc.22
- 91. Linear Regression and Model StatisticsM2 ExampleM2 = 1145.31 + 4.04 × Time
- 92. Next Period is 1519
- 93. Forecast for that period is:
- 94. Y = 1145.31 + 4.04 × 1519Y = 7283.446866 Copyright 2010 DeepThought, Inc.23