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Estimation Of The Box Cox Transformation Parameter And Application To Hydrologic Data 1

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Culminating PowerPoint presentation for my summer research with Dr. Richard McCuen.

Culminating PowerPoint presentation for my summer research with Dr. Richard McCuen.

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Estimation Of The Box Cox Transformation Parameter And Application To Hydrologic Data 1 Estimation Of The Box Cox Transformation Parameter And Application To Hydrologic Data 1 Presentation Transcript

  • Estimation of the Box-Cox Transformation Parameter and Application to Hydrologic Data Melanie Wong Friday, August 29, 2008
  • Introduction
    • Many statistical tests assume normality
    • Most hydrologic data are highly skewed
  • Detecting Normality
    • β1 (third moment) = skewness
    • β2 (fourth moment) = kurtosis
    • Characteristics of a normal distribution:
      • β1=0
      • β2=3
  • Moment Ratio Diagram
  • Transformation to Normality
    • Hydrologic data are not normal
    • Various transformations available
      • Logarithmic
      • Box-Cox
  • The Box-Cox Transformation
  • 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
  • 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
  • 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
  • Sampling Variation 10,000 simulations
  • Second Objective
      • To provide a method for estimating the Box-Cox transformation parameter for any set of data
  • The Importance of λ
    • Small changes in λ  Large changes in sampling variation
    • Need a more precise method to obtain optimum λ
  • 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
  • Simulation Results β1 values ↔ β2 values ↕
  • Variation of λ
  • 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
  • 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
  • Example 1: Monthly Rainfall Measurements
    • 36 monthly rainfall measurements (mm/month) from Lawrenceville, GA
    • β1: 0.56 β2: 2.91
  • Example 1: After Logarithmic Transformation
    • β1: -1.12 β2: 4.40
  • Example 1: After Box-Cox Transformation
    • λ: 0.55
    • β1: -0.09 β2: 2.82
  • Histogram of Rainfall Data
  • 90% Confidence Intervals on Rainfall Box-Cox Transformed: Log Transformed:
  • Example 2: Drainage Pipe Costs
    • The costs of 70 drainage systems
    • β1: 4.40 β2: 27.36
  • Example 2: After Logarithmic Transformation
    • β1: -0.167 β2: 4.04
  • Example 2: After Box-Cox Transformation
    • λ: 0.04
    • β1: 0.00379 β2: 3.96
  • Histogram of Pipe Cost Data
  • 90% Confidence Intervals on Pipe Cost Box-Cox Transformed: Log Transformed:
  • 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
    • QUESTIONS?