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Outline

Outline. Objectives Test-statistic Criterion Change Rate Calculation Published Statistical Methods New Statistical Methods Comparison of All Methods Erosion Hazard Maps Conclusion. Objectives. Introduce the Akaike Information Criterion (AICc) as an objective test statistic

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Outline

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  1. Outline • Objectives • Test-statistic Criterion • Change Rate Calculation • Published Statistical Methods • New Statistical Methods • Comparison of All Methods • Erosion Hazard Maps • Conclusion

  2. Objectives • Introduce the Akaike Information Criterion (AICc) as an objective test statistic • Introduce new statistical methods to reduce over-fitting of data and improve prediction of hazard zone

  3. Corrected Akaike Information Criterion (AICc) AICc = Nlog[RSS/N] + 2KN/(N-K-1) RSS= Residual Sum of Squares N= # of data points K= # of parameters +1 1st term 2nd term As K , 1st term and 2nd term Tradeoff between under-fitting and over-fitting

  4. Corrected Akaike Information Criterion (AICc) AICc = Nlog[RSS/N] + 2KN/(N-K-1) • The model with the lowest AICc fits the data the best • We use AICc two ways: • to determine the number of parameters in the new statistical methods • to distinguish which statistical method is best • To compare statistical methods, the change rate calculation should be the same

  5. Change Rate Calculation

  6. Published Statistical Methods Single-Transect Method Rates are calculated at each transect A beach with 30 transects will have 30 rates This method assumes each transect is independent of each other

  7. Published Statistical Methods Single-Transect Method AICc = 1041.093 K= 3 * # transects If adjacent transects have similar parameters, data is over-fitted Large data scatter due to short term changes can mask the long term signal

  8. Published Statistical Methods T-Binning Method Bins are created by grouping transects that have indistinguishable rates The Student’s T-test statistic is used to identify bins Transects grouped at different window sizes are compared to the rest of the beach by computing a t-test R=binned erosion rate T-test compares R1 to R2

  9. Published Statistical Methods T-Binning Method Transects # 26-36 overlap within the two bins. We call this the transition zone

  10. Published Statistical Methods T-Binning Method The transition zone is identified as its own bin

  11. Published Statistical Methods T-Binning Method AICc = 797.699 K= #transects + 2*#bins Rates are discontinuous at bin borders, which might not be the case within a beach system This technique requires significant user input and is not practical for long stretches of beach

  12. New Statistical Methods A-Binning Method Unlike T-binning, A-binning uses the AICc values to identify bins For each bin size, all possible bin models are calculated. The best bin configuration is the one with the lowest AICc. For bin size=1, all transects within the beach system are binned together In the example above, there are 30 transects. For a 2-bin model, there are 29 possible bin configurations

  13. New Statistical Methods A-Binning Method As the # of bins increases, the AICc values first decline, then begin to rise The rise In AICc is due to over-fitting of the data Once AICc starts to rise, we can stop examining models

  14. New Statistical Methods A-Binning Method The bin size with the lowest AICc score is considered the best model

  15. New Statistical Methods A-Binning Method AICc = 774.048 K= #transects + 2*#bins AICc value will be lower than the AICc for T-binning A-binning removes user input, which makes it more objective and easier to use than T-binning

  16. New Statistical Methods PXT PXT uses a polynomial in space and time to simultaneously model all transects The beach is expanded as a linear sum of basis functions called Legendre Polynomials, sampled at each transect The coefficients of the Legendre Polynomials are used for prediction Scaled transect location

  17. New Statistical Methods PX – A Special Case of PXT PX is a special case of PXT The rates are time-independent (the fits are linear at each transect) AICc is used to identify the optimal polynomial fit

  18. New Statistical Methods PX – A Special Case of PXT The rates are modeled with respect to transect location For this example, the best-fit polynomial has 7 coefficients

  19. New Statistical Methods PX – A Special Case of PXT AICc = 791.231 K= #transects + #coefficients + 1 Unlike binning, the rates are continuous throughout the beach system

  20. New Statistical Methods PXT In PXT, the rate is modeled with respect to transect location and time First, the coefficients of PX are calculated to identify the time-independent terms Next, the time-dependent coefficents are calculated, using AICc to determine the best fit In the example above, the time-independent coefficients (PX) = 7 and the time-dependent coefficient = 1. The rate is changing with time.

  21. New Statistical Methods PXT AICc = 789.990 K= #transects + #coefficients + 1 Because the parameters are reduced with this method, we can test whether change rates are changing with time (acceleration)

  22. New Statistical Methods Eigenbeaches Eigenbeaches Method is similar to PXT Eigenbeaches generate their own polynomial from the data that carries information about the unknown beach physics Eigenvectors are the principal components of the beach data Scaled transect location

  23. New Statistical Methods Eigenbeaches The eigenvectors are used the same way as Legendre Polynomials (PXT) for prediction First, the time-independent coefficients are identified by the AICc Next, the time-dependent coefficients are calculated, using AICc to determine the best fit In the example above, the time-independent coefficients = 1 and the time-dependent coefficients = 0. The rate is not changing with time.

  24. New Statistical Methods Eigenbeaches AICc = 757.720 K= #transects + #coefficients + 1 Eigenbeaches Method is a parsimonious model for the data

  25. Comparison of All Methods Rates of all Methods All methods, except for T-binning, closely resemble each other

  26. Comparison of All Methods Hazard Line of all Methods The baseline is the average of all shorelines used in the analysis

  27. Erosion Hazard Maps Single-Transect Method

  28. Erosion Hazard Maps T-Binning

  29. Erosion Hazard Maps A-Binning

  30. Erosion Hazard Maps PX

  31. Erosion Hazard Maps PXT

  32. Erosion Hazard Maps Eigenbeaches

  33. Conclusions • Single-Transect Method has the most parameters and is more prone to over-fitting • An objective criterion, such as AICc, identifies which statistical method is best for a particular beach • Utilizing AICc to find the best fit in A-binning, PXT, and Eigenbeaches reduces the effects of noise • The reduction in the number of parameters in PXT and Eigenbeach allows the investigation of rates changing with time

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