Estimation Of The Box Cox Transformation Parameter And Application To Hydrologic Data 1
- 1. Estimation of the Box-Cox Transformation Parameter and Application to Hydrologic Data Melanie Wong Friday, August 29, 2008
- 2. Introduction Many statistical tests assume normality Most hydrologic data are highly skewed
- 3. Detecting Normality β1 (third moment) = skewness β2 (fourth moment) = kurtosis Characteristics of a normal distribution: β1=0 β2=3
- 5. Transformation to Normality Hydrologic data are not normal Various transformations available Logarithmic Box-Cox
- 7. Transformation Procedure Decide if sample data is normal Obtain value of λ Transform the data Apply confidence intervals, statistical tests, or tolerance limits Perform inverse transformation
- 8. Determining the Optimal λ “ Snap-to-the-grid” method Box and Cox (1967): “... fix one, or possibly a small number, of λ's and go ahead with the detailed estimation...” Distribution-based method λ=-1.0 reciprocal transform λ=-0.5 reciprocal square root transform λ=0 natural log transform λ=+0.5 square root transform λ=1.0 no transformation needed
- 9. Research Goal and Objectives Goal: To develop a better understanding of the Box-Cox transformation so that it can be applied with greater confidence Objectives: To characterize the sampling variation To provide a method for estimating the Box-Cox transformation parameter for any set of data
- 10. Sampling Variation 10,000 simulations
- 11. Second Objective To provide a method for estimating the Box-Cox transformation parameter for any set of data
- 12. The Importance of λ Small changes in λ Large changes in sampling variation Need a more precise method to obtain optimum λ
- 13. Optimizing λ Procedure: Monte Carlo simulation to identify sampling distributions of β1 and β2 values for different λ values Distribution types: Gamma, exponential, uniform Population sizes: 1000, 500, 200, 100, 50, 10
- 14. Simulation Results β1 values ↔ β2 values ↕
- 15. Variation of λ
- 16. Confidence Intervals Procedure: Logarithmic 1) Make Logarithmic transformation of x 2) Use normal theory for confidence intervals on y 3) Inverse transform the confidence intervals to values of x
- 17. Confidence Interval Procedure: Box-Cox 1) Make BCT of x to y using optimum λ 2) Use normal theory for confidence intervals on y 3) Inverse transform the confidence intervals to values of x
- 18. Example 1: Monthly Rainfall Measurements 36 monthly rainfall measurements (mm/month) from Lawrenceville, GA β1: 0.56 β2: 2.91
- 19. Example 1: After Logarithmic Transformation β1: -1.12 β2: 4.40
- 20. Example 1: After Box-Cox Transformation λ: 0.55 β1: -0.09 β2: 2.82
- 21. Histogram of Rainfall Data
- 22. 90% Confidence Intervals on Rainfall Box-Cox Transformed: Log Transformed:
- 23. Example 2: Drainage Pipe Costs The costs of 70 drainage systems β1: 4.40 β2: 27.36
- 24. Example 2: After Logarithmic Transformation β1: -0.167 β2: 4.04
- 25. Example 2: After Box-Cox Transformation λ: 0.04 β1: 0.00379 β2: 3.96
- 26. Histogram of Pipe Cost Data
- 27. 90% Confidence Intervals on Pipe Cost Box-Cox Transformed: Log Transformed:
- 28. Conclusions Box-Cox transform is more suitable than logarithmic transform for: Normalizing data Determining confidence intervals, tolerance limits, outliers, and other tests Sampling distributions of λ were determined Optimum values of λ vs. β1 and β2 were developed
- 29. QUESTIONS?