Fractals. Geostatistics. Time series analysis.

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Fractals. Geostatistics. Time series analysis.

• Edith Perrier, IRD France,
• Visiting scientist at UCT,
• Applied Maths Dept., room 417.1, Ext. 3205,
• E-mail edith_at_maths.uct.ac.za
• Web site where to download previous lectures
  lect.ppt (or lect.zip if links included)
  http//www.mth.uct.ac.za/Affiliations/BioMaths/Ind
  ex.html

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Time series and functions f(t)

• A time series plot of a sequence of measures x0,
  x1, x2, x3, ..consists of plotting the points
• (0, x0), (1, x1), (2, x2), (3, x3), ...
• fIRgtIR
• t-gtf(t), graph
• often a function of time

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Fractal curves as functions differentiable only
on a set of points of measure 0

• Pathogical curves Weierstrass described a
  function that was continuous but
  nondifferentiable -- no tangent could be
  described at any point. Cantor showed how a
  simple, repeated procedure could turn a line into
  a dust of scattered points, and Peano generated a
  convoluted curve that eventually touches every
  point on a plane. Whereas one can show that for
  ordinary function f 0,1-gtIR with a continuous
  derivation, the graph of f is a regular
  1-dimendional set, the Cantor graph has a non
  integer fractal dimension as well as the von
  Koch, Peano (Hilbert) curves.
• (http//plus.maths.org/) Hermite described these
  new functions as a "dreadful plague" and Poincaré
  wrote "Yesterday, if a new function was
  invented it was to serve some practical end
  today they are specially invented only to show up
  the arguments of our fathers, and they will never
  have any other use".
• http//www.math.ethz.ch/finance/talksSS2002.html.
  Talks in Financial and Insurance Mathematics,
  Summer Term 2002

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Weierstrass function
Exercise try different values for s
s1.5
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Solution
Run Mathematica
Web Example
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Weierstrass graph fractal dimension

• Hints from Falconer (1990) p.148
• Recent paper from Hunt (1996)
• The Weierstrass nowhere differentiable function,
  and functions constructed from similar infinite
  series, have been studied often as examples of
  functions whose graph is a fractal. Though there
  is a simple formula for the Hausdorff dimension
  of the graph which is widely accepted, it has not
  been rigorously proved to hold. We prove that if
  arbitrary phases are included in each term of the
  summation for the Weierstrass function, the
  Hausdorff dimension of the graph of the function
  has the conjectured value for almost every
  sequence of phases. The argument extends to a
  much wider class of Weierstrass-like functions.

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Temporal auto-correlations
We might expect a fractal graph with
( Proof using Box Counting
dimensions)
It is reasonable to expect a signal with a power
spectrum ( Proof using
Fourier transforms)
Falconer, 1990, p.158)
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Geostatistics

• Geostatistics is a collection of statistical
  methods which were traditionally used in
  geo-sciences. These methods describe spatial
  autocorrelation among sample data and use it in
  various types of spatial models. Geostatistical
  methods were recently adopted in ecology
  (landscape ecology) and appeared to be very
  useful in this new area.
• Regionalized random function Z(X) with
  assumptions of ergodicity (a single sample would
  reflect the statistical character of the random
  variable) thus stationarity (statistical
  homogeneity where on smaller subparts, we have
  the same RF with different realizations)
• First moment mean m(x)EZ(X),
• Second order moments variance varZ(X)E(Z(X)-m(
  x))2 and covariance C(x,xh)E(Z(X)-m(x))(Z(xh)
  -m(xh))
• Spatial autocorrelation can be analyzed using
  correlograms, covariance functions and variograms
  (semivariograms).

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Spatial autocorrelations

• Correlogram is estimated using equation
• where indexes -h and h refer to sample points
  located at the tail and head of vector h z-h and
  zh are organism counts in samples separated by
  lag vector h summation is performed over all
  pairs of samples separated by vector h Nh is the
  number of pairs of samples separated by vector h
  M-h and Mh are mean values for samples located
  at the tail and head of vector h s-h and sh are
  standard deviations of samples located at the
  tail and head of vector h.
• Other measures of spatial dependence are
  covariance function
• and variogram (semivariogram)
• The correlogram, covariance function, and
  variogram are all related. If the population mean
  and variance are constant over the sampling area
  (there is no trend) then
• where C(0) is the covariance at zero lag
  variance squared standard deviation.
• Fromhttp//www.gypsymoth.ento.vt.edu/sharov/PopEc
  ol/lec3/geostat.html Alexei A. Sharov, Department
  of Entomology at Virginia Tech.

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Variograms
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Examples of one-dimensional spatial distributions
with different nugget effects and corresponding
variograms
http//www.gypsymoth.ento.vt.edu/sharov/PopEcol/l
ec3/geostat.html
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Geostatistics, estimations and kriging

• Geostatistics changes the entire methodology of
  sampling. Traditional sampling methods don't work
  with autocorrelated data and therefore, the main
  purpose of sampling plans is to avoid spatial
  correlations. In geostatistics there is no need
  in avoiding autocorrelations and sampling becomes
  less restrictive. Also, geostatistics changes the
  emphasis from estimation of averages to mapping
  of spatially-distributed populations.
• The value z0 at unsampled location 0 is estimated
  as a weighted average of sample values zi at
  locations i around it Weights depend on the
  degree of correlations among sample points and
  estimated point.

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Variograms and fractals

• Search for theoretical links
• The Fractal/Facies Concept An Alternative to
  Using Variograms for Generating Subsurface
  Heterogeneity from Fred J. Molz, III,
  Environmental Engineering Science, Clemson
  University, fredi_at_clemson.edu, James W. Castle,
  Geological Sciences, Clemson University, and
  Silong Lu, Geotrans, Inc,. Past applications of
  stochastic theory in geology have been based
  mainly on treating heterogeneous property
  distributions as stationary, correlated, random
  processes, with short correlation lengths. A
  convenient way to characterize such distributions
  is through computation of a variogram, which
  should reach a sill as one approaches a
  measurement separation equal to the correlation
  length. This approach has had limited success,
  because often a sill is not reached or is poorly
  defined. An explanation for such behavior is
  given by modern fractal-based theories that
  conceptualize natural heterogeneity as
  non-stationary stochastic processes with
  stationary increments, the mathematical basis for
  stochastic fractals.
• Multifractality and spatial statistics Cheng QM
  COMPUTERS GEOSCIENCES 25 (9) 949-961 NOV 1999.
  . In the present paper, a theoretical
  investigation is developed to illustrate (1) the
  characteristics of multifractality as measured by
  the parameter tau "(q) (2) relationships between
  multifractality and spatial statistics including
  semivariogram and autocorrelation in
  geostatistics, indexes used in lacunarity
  analysis and correlation coefficients. It can be
  shown that these statistics primarily are related
  to multifractality as determined by tau "(1).
  This is an important result because not only does
  it provide the link between multifractals and
  spatial statistics but it also shows that
  statistics based on second-order moments are
  restrictive in that they only characterize a
  multifractal measure around the mean value. In
  applications where extreme values need to be
  taken into account, the entire multifractal
  spectrum should be used rather than local
  properties of the spectrum around the mean only
  alternatively, statistics defined on the basis of
  higher-order moments can be employed for analysis
  of extreme values.
• Even softwares
• Management

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Static patterns and dynamical processes?
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Lecture series

• Lecture 1. March. 3rd. Introduction to fractal
  geometry . Measures and power laws.
• Lecture 2. March 5th. Definitions of non-integer
  dimensions and mathematical formalisms
• Lecture 3. March 10th. Iteration of functions and
  fractal patterns
• Lecture 4. March 12th. Extensions Self-similar
  and self-affine sets . Multifractals.
• Lecture 5. March 17th. Fractals / Geostatistics
  / Time series analysis
• Lecture 6. March 19th. Dynamical processes
  Fractal and random walks
• Lecture 7. March 24th. Dynamical processes
  Fractal and Percolation
• Lecture 8. March 26th. Dynamical processes
  Fractal and Chaos