A geomorhologic approach to estimate landslide hazard and risk in urban and rural areas

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Time Series

• Many attributes, including
• Frequency-size probability distribution of
  values. Ordering of values does not matter.
• Pattern of the data series ordering
  matterscorrelations (persistence), long-range
  vs. short range

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Long-range correlations

• Many ways to measure the strength of (long-range)
  correlations of a time series, including
• Fourier Power-Spectral Analysis
• Detrended Fluctuation Analysis
• Autocorrelation
• Semivariogram Analysis
• Hurst Rescaled-Range (R/S) Analysis
• Wavelet Spectral Analysis
• Interevent occurrence times

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Fourier Power Spectral Analysis
S f b S power spectral density (Amplit
ude of Fourier coefficients)2 f frequency b
constant b 5 2D
b 0 White noise. Uncorrelated b 1 Pink
(1/f ) noise. Medium strength long-range
persistence. b 2 Brownian motion. Strong
long-range persistence, e.g. topography.
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Power-Spectral Analysis
Data Series
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Wavelet analysis of annual tree-ring widths for
bristle cone pines (pinus longaeva) on Campito
Mtn, California ITRDB USA 37N,118W Graybill
LaMarche See also papers by Harlan Ferguson.
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Wavelet variance analysis of annual tree-ring
data for Campito Mtn, California (2111 BC to 1969
AD). Larger b indicates stronger long-range
persistence .
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Measures of long-range persistence

• Fourier Power-Spectral Analysis
• bPS bPS52D
• Detrended Fluctuation Analysis
• aDFA bDFA 2aDFA1
• Semivariogram Analysis
• HaHausdorff Exponent bSemivariogram 2Ha1 0
  lt Ha lt 1
• Hurst Rescaled-Range (R/S) Analysis
• HuHurst Exponent bHurst 2Hu1 0 lt Hu lt 1
• Wavelet Spectral Analysis bWavelet
• Autocorrelation

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Create Fractional Noises

• Why? As benchmarks with given properties to
  examine the many techniques.
• Fourier Filtering Fourier Filter a white noise
  so that is has a prescribed strength of
  long-range persistence
• Others Discrete Wavelets, random additions,
  Surrogate data techniques, stochastic
  differential equations, Mandelbrot cartoon
  Brownian motions, etc.

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Poster Performance Tests for Techniques that
Measure Long-Range Persistence
Poster Detrended Fluctuation Analysis
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S f b (S power spectral density f
frequency b constant )
Fractional Gaussian Noises created by Fourier
Filtering
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Fractional log-normal Noises, cv mean/s
1.0 created from the fGNs from Fourier Filtering
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Discussion

• Methods for synthetic time series construction
  (heavy tails?)
• Techniques for quantifying (long-range)
  correlations in time series
• Fitting data
• Statistical tests (fitting data, hypothesis
  tests, error bars, etc.)