Fractal dimension of birds population sizes time series

Transcripción

Mathematical Biosciences 206 (2007) 155–171
www.elsevier.com/locate/mbs
Fractal dimension of birds population sizes time series
Alfonso Garmendia
a
a,*
,
Adela Salvador
b
Department of Agro-Forest Ecosystems, Higher Technical School of Rural Environments and Enology,
Polytechnic University of Valencia, Av. Blasco Ibáñez 21, 46010 Valencia, Spain
b
Higher Technical School of Civil Engineering, Madrid Polytechnic University, Spain
Received 31 March 2004; received in revised form 25 January 2005; accepted 3 March 2005
Available online 26 September 2005
Abstract
Information about fractal dimension is collected so that it can be applied to time series interpreting
Hurst coefficient. The population size of a species is modelled as a dynamic system. The Hurst coefficient
is calculated for these times series. A computer programme has been elaborated to compute the Hurst exponent of time series using the algorithms of range increment, second order moment increment and local second order moment increment. It has been applied to time series of birds populations.
2005 Elsevier Inc. All rights reserved.
Keywords: Fractal; Time series; Hurst coefficient; Time series of species populations
1. Introduction
Fractal geometry is a tool to quantitatively describe objects that are considered as extremely
complex and disorderly. The term fractal denotes something irregular, intricate, in which the
smallest parts are similar to the overall pattern.
The knowledge of self-similar fractals, the quantitative study of the singularities that naturally
appear on the iteration theory and the dynamic systems has been quickly developed through the
use of the computer, which produces a quick repetition of the process.
*
Corresponding author.
E-mail addresses: [email protected] (A. Garmendia), [email protected] (A. Salvador).
0025-5564/$ - see front matter 2005 Elsevier Inc. All rights reserved.
doi:10.1016/j.mbs.2005.03.014
156
A. Garmendia, A. Salvador / Mathematical Biosciences 206 (2007) 155–171
1.1. Fractal
Fractals are mathematic objects that fall within the Geometric Measure Theory. The exact and
definitive delimitation of it is yet to be established [4,5,29,26,13].
A self-similar fractal can be defined as the final product of an infinite iteration of a well-specified geometric process. It is the fixed point of a set of contractive applications [22]. It allows the
construction and control of extremely complex structures through very simple processes [25].
In literature, we can find different definitions for fractal: an object that has a fractionary
dimension, a fixed point of a set of contractive applications. . . The definition we have used
is a subset of Rn in which the topological dimension does not coincide with the Hausdorff
dimension.
1.2. Dimension
In 1919, Hausdorff introduced a key tool to measure those peculiar sets through the introduction of concepts we nowadays call Hausdorff dimension [29,26,57]. Besicovitch, during the 1920s,
continued these works and included the geometric measure theory.
Hausdorff dimension is a natural generalisation of the topological dimension. Both could are defined by the properties of their minimum open covering.
The open ball B(p, r) of radius r and centre p on a metric space is the set
Bðp; rÞ ¼ fx : distðx; pÞ < rg;
where dist(x, p) is the distance between point x and point p.
A set U in a metric space is known as an open set if, given a point p in U there is a distance r > 0
in the way that there is an open ball B(p, r) contained in U.
A family of open sets {Ua} is known as an open cover of a set X, if X is contained within the
union ¨{a}Ua of sets Ua.
The topological dimension of a space is inductively defined as follows:
The dimension of the empty set is 1, and the dimension of X is less or equal to n. This is the
same as saying that X has a series of open sets with borders less or equal to n 1.
Thus, it can be seen that a finite set has topological dimension of 0, a rectifiable curve has a
dimension of 1, and a differentiable surface has a dimension of 2.
The Hausdorff dimension (or Hausdorff–Besicovitch dimension) can be written as follows:
Let X be a subset of Rn covered by N(r) open covers of radius lesser or equal to r. When r tends
to 0, N(r) increments according to a power of approximately 1/r, let us say rD. Exponent D is
called the Hausdorff dimension or fractal dimension of the set X.
That is, for each r > 0, we calculate the lesser number of open covers N(r), of a radius less or
equal to r, which is necessary to cover the set X. We can see that the limit
log N ðrÞ
ð1:1Þ
D ¼ lim r!0
log r
exists. Value D is called the Hausdorff dimension for X. (Since log r ! 1, the sign is necessary
so that D is positive) [10,32,34,7,43,44,1,63].
157
It can be seen that formula (1.1) is equivalent to the approximate exponent expression
NðrÞ const rD ;
ð1:2Þ
D
N(r) increments asymptotically with r .
Two values x and y are asymptotic when x tends to 0 if the limit
log y
ð1:3Þ
lim
x!0 log x
exists [41].
An observation that can be drawn is that usual sets have a Hausdorff dimension that coincides
with their topological dimension. In a segment, a square or a cube, N(r) increments by 1/r, 1/r2,
and 1/r3 respectively.
The decade of the 1970 is marked by Mandelbrots intuition, who was the first to notice some of
the application possibilities that this field had, and openly proposed them in a widely distributed
publication. [26]. Mandelbrot in 1982 [44] says that an object is fractal when there is a discrepancy
between these two dimensions. Thus, an object such as the curve of Kochs snowflake is an intermediate object between a line and a plane and its fractal dimension is log 4/log 3 [19,52,53]; the
Cantor set has as fractal dimension log 2/log 3; and Sierpinskys triangle, log 3/log 2 [20].
The Hausdorff dimension is difficult to work with, and therefore in self-similar object, it can be
calculated using the similarity dimension [36], since these fractals are the fixed point of a finite set
of contractive applications.
1.3. Fractals in nature
Finding natural fractal elements is easy, since the typical geometry of nature is fractal geometry.
We find them in oceanography [50], in the measuring of coasts, mountains [45], islands, coral reefs
[61].
Measuring of coasts: The classical example that appears in literature is the measuring of a coastline. It is observed that length L depends on the step size p, and that L is proportional to a power of p
LðpÞ ¼ kpd ; therefore lnðLÞ ¼ lnðkÞ þ d lnðpÞ.
Therefore, d is the slope of abscissa the logarithm of the step size and of ordinate the logarithm the
length of the coast measured [45].
Other examples are the spatial distribution of animals and plants, the steadiness of vegetation
patterns [47], the nature of fractures or fracture systems, the porosity of rocks, and the geometry
of reefs [9]. Morse et al. [49] studied the fractal character of several branches of trees and river
drainage networks. Lovejoy in 1982 [42], proved that the distribution of tropical clouds and rain
areas has a fractal structure.
The morphology of a flower corolla presents a certain corrugation [21]. The biologist C. Herrera from the Doñana Experimental Station, applied fractal geometry to his characterisation, and
classified some flowers through the measuring of the fractal dimension of the corolla perimeter,
proving that the fractal dimension was directly related to the absolute production of fruit, and
therefore pollination is favoured in flowers that are profusely dissected.
Fractal geometry has proved essential to understand chaotic population models [24], species
originations and extinctions in the fossil record [56], multi-resource competition models [33],
species–area relationships [40] and species diversity indices [8].
158
Fractal objects or behaviour also emerge in models not explicitly designed as such [51]. The
movement of animals, treated as a diffusion problem, was characterised as a fractal object, and
treated as a random walk [30].
2. Dimension of function graphs. Time series dimension
The problem of discerning the spectral (or fractal) properties of a model on the basis of a given
time series has been addressed by many authors in a number of different contexts
[16,14,13,28,65,23].
The graph of Brownian motion (Fig. 1), time continuous random process, is the prototype of
random fractals [15,31].
A continuous process {y(t)} is known as a random process or a Brownian process on continuous
time, if for a time step Dt, increments Dy(t) = y(t + Dt) y(t) are
(i) Normal distribution.
(ii) Zero average.
(iii) Proportional variance Dt.
Or the equivalent to (iii) (instead of (ii)):
Successive increments Dy(t) and Dy(t + Dt) are not correlated.
The axiom that characterizes random processes can be generalized with the characteristic of
a fractal process [43,44] introducing an additional parameter, Hurst exponent H, (0 < H < 1),
and replacing (iii) with
(iii 0 ) Variance proportional to Dt2H.
(Therefore, in the random process H = 1/2.)
(iv 0 ) In a fractal process, successive increments have a correlation q, independent of time t,
defined by
1
2H
ð2:1Þ
<q<1 .
2 ¼ 2 þ 2q
2
If {y(t)} is a fractal process with Hurst exponent H, then, "c > 0, the process
1
y c ¼ H yðctÞ
c
is another fractal process of similar statistical properties. It is renormalization.
Fig. 1. The graph of Brownian motion.
ð2:2Þ
159
In order to introduce the concept of random fractal, let us consider FH as the family of all the
graphs of fractal processes with Hurst exponent H. The family FH is closed under renormalization (2.2) and all the elements of FH share the same statistical properties.
2.1. Powers (or power laws)
Relationships between the scale measurements of an invariant system take the form of power
laws.
Functions with self-similar graphs or similar with different scales must be of the form
y = f(x) = const xc for some exponent c. Since scale invariance requires
f ðaxÞ ¼ bf ðxÞ
ð2:3Þ
for a constant a and a related constant b dependent of a, scale invariance implies
f ðxÞ ¼ const xc ;
ð2:4Þ
where c = log b/log a.
The logarithmic transformation of a power is a linear function
log y ¼ logðconstÞ þ c log x.
ð2:5Þ
This permits the use of linear regressions to determine the fractal dimension of a process.
3. Fourier transform technique
This technique can be used to calculate the fractal exponent H for every continuous periodic
function of period 1. These functions can be written as a Fourier series through sine and cosine
functions [31]:
f ðxÞ ¼
1
X
ðan cos 2pnx þ bn sin 2pnxÞ.
ð3:1Þ
n¼0
The coefficients an and bn indicate the waves amplitudes and the Fourier coefficients set of a function is called spectrum. Therefore, any complex function can be written as a complex Fourier
series, using the exponential function
f ðzÞ ¼
þ1
X
cn e2pinz ;
ð3:2Þ
1
where cn is a complex number with real part an and imaginary part bn.
The discrete Fourier transform (DFT), requires only a finite sequence of data points to obtain
the Fourier coefficients.
A function Power spectrum is the succession of the amplitudes squares for the Fourier coefficients and it is explained as the energy associated to each frequency.
160
It is verified that:
(1) E(cn) = 0. The
mean is 0, both in the real and the imaginary parts.
coefficients
2
2
1
(2) Eðjcn j Þ ¼ n2 Eðjc1 j Þ. The coefficients variance satisfies a power law scaling. When n
increases, its module decreases.
(3) The phases are independent and uniformly distributed on the interval [0, 2p].
Let f be a fractal function on the interval [0, 1] with f(0) = f(1), and with fractal exponent H.
[31]. Let {cn; n > 0} be the spectrum. Then each cn is an independent sample from a complex valued normal distribution of expected value 0, and expected variance, the power spectrum, satisfies
the power law
Eðjcn j2 Þ ¼ const n12H .
ð3:3Þ
The phases are independent and distribute uniformly on the interval [0, 2p].
The fractal [29] is assembled from a series of sine wave components if different frequencies, the
amplitudes of which satisfy a power law relationship
Sðf Þ f ð2N þ12DÞ .
ð3:4Þ
4. Noise spectrum or colour
Noise spectrum or power spectrum in a time series can be split into a spectrum of basic waves
each associated with a characteristic wave frequency or colour [29]. A population reddened dynamic imply that power increases with decreasing frequency, what means that successive points in
the time series of population densities are more correlated than would be expected purely by
chance. In the case that successive entries in a time series are uncorrelated, it is referred to as
white noise, in which all frequencies occur with equal power. In the rare case of negative correlation between these terms, the data are said to be blue [59]. A spectrum in which this decrease
obeys a power law (y = axb) is called 1/f noise and is a signature of fractal behaviour [58].
Pink noise is a special class of red noise where the log(power) (power is the squared magnitude
of the Fourier spectrum) scales linearly with log(1/frequency) [59,54].
Frequency, f, is defined as the number of cycles per N points. To generate pink noise we set the
amplitude of the component sine waves as a function of the frequency
sffiffiffi rffiffiffiffi
1
i
.
ð4:1Þ
¼
Sðf Þ ¼
f
N
5. Second order moment techniques [31,61]
The third axiom offers a way to determine H on the postulate of a fractal model of experimental
data.
161
For a fractal process and any Dt, the corresponding Dy has an expectation value 0 and equals
EðDy 2 Þ ¼ cDt2H ;
ð5:1Þ
2
2
where the average of Dy is an unbiased estimator of E(Dy ).
Hence, Dx2 = cDt2H, where Dt is a time step and Dx2 is the second order moment of the corresponding spatial increment.
5.1. Determination of H
EðDyðDtÞ2 Þ ¼ cDt2H ) ln EðDyðDtÞ2 Þ ¼ ln c þ 2H ln Dt;
2
2H
EðDyð2DtÞ Þ ¼ cð2DtÞ
2
) ln EðDyð2DtÞ Þ ¼ ln c þ 2H ln 2Dt.
Subtracting these two equations
ln E½yðt þ 2DtÞ yðtÞ2 ln E½yðt þ DtÞ yðtÞ2
¼ 2H ðln 2Dt ln DtÞ ¼ 2H ðln 2Dt=DtÞ ¼ 2H ln 2.
Therefore
2
2
H ¼ 1=ð2 ln 2Þfln E½yðt þ 2DtÞ yðtÞ ln E½yðt þ DtÞ yðtÞÞ g.
ð5:2Þ
As the Hurst exponent of a fractal process is independent of the time step used, local calculations
of the form (5.2) can be made to test whether a process is fractal.
Precaution: In the case of long time series, the increment Dy is very small compared to the second order moment, and the terms second order moment and variance are interchangeable. It
is not so for short series, in which expectation and second order moment must be tested
independently.
6. Local second order moment techniques
Hurst exponent can also be determined from the correlation coefficient between successive
increments, using formula (2.1). Under the assumption that the Dy has an expected value of 0,
we can determine q through the definition
Eð½yðt þ 2DtÞ yðt þ DtÞ½yðt þ DtÞ yðtÞÞ
q ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi .
Eð½yðt þ 2DtÞ yðt þ DtÞ2 ÞEð½yðt þ DtÞ yðtÞ2 Þ
The expectation of [y(t + 2Dt) y(t + Dt)] Æ [y(t + Dt) y(t)],
[y(t + Dt) y(t)]2 are unbiased estimators for the expectations.
Local exponent H is obtained from formula (2.1)
22H ¼ 2 þ 2q;
which is :
H ¼ lnð2 þ 2qÞ= ln 4.
ð6:1Þ
[y(t + 2Dt) y(t + Dt)]2,
ð6:2Þ
The axiom for a fractal process can be tested by repeating this local computation for different
values of Dt.
162
In general, formulae (6.1) and (6.2) compute the local fractal exponent even when the expectation value is not zero and q is not the correlation coefficient.
7. Range increment
Range is the difference between the maximum and the minimum values of y(t).
R(Dt) denotes the average range of the process {y(t)} on all intervals of deviation Dt.
In the equation
1
y c ¼ H yðctÞ
c
processes {y(t)} and {yc(t)} must have the same expected range. This implies that the range of the
process {yc(t)} in an interval of duration Dt is 1/cH times the range of the process {y(t)} in an
interval of duration Dt/c. Replacing Dt/c by Dt we obtain
RðDtÞ ¼ cDtH ;
ð7:1Þ
where Dt is the time step and R(Dt) is the average range value in the interval of time of duration
Dt.
The formula (7.1), for the case of a fractal process in discrete time, is only maintained in long
enough intervals of time. In short times, the range cannot be adequately determined, which implies that R(Dt) will increment quicker than DtH. For this reason, in Brownian motion in discrete
times, the result is H = 0.63 instead of H = 0.5.
8. Fractal dimension and Hurst coefficient on northern europe passerine species
Natural objects are not ideal fractals, but their properties are often sufficiently similar across a
wide range of feasible scales that the tools of fractal geometry can be used, providing novel insights where Euclidean tools were found to be insufficient for describing such objects [18,24,3].
The fractal model, like the straight line in linear regression, may be seen as a simplifying frame
that help us understand certain features of reality, without necessarily having to be strictly true
itself [29,31].
In many ecological cases there is not a stationary state, necessary to understand their dynamics
by Euclidean tools. In these cases variance is more informative that mean [6] and stochastic processes defined in term of fractal geometry (1/f noise models) can thus describe much ecological
variability [27].
By definition, population levels of a species persistent in time are maintained within limits and
therefore the fluctuation range of the population (supposed as random) has an asymptotic value
[37]. The variance of population time series increases with observation time, apparently without
limit [38,55]. Traditional models of density-dependent growth imply the existence of a basin of
attraction, which confines the fluctuation of population abundance to a well-defined range of
values about equilibrium [37]. Thus, for tightly regulated populations, the variance
should converge to a clear limit in long enough series, but inside these fluctuations there is
163
density-independent stochastic growth, the prime example of which is a random walk, for which
the variance grows linearly with time [2]. It is therefore a challenge to predict the variation range
of a population in the long term, using relatively short time series. The 1/f noise process has been
used in simulation models of extinction rates [28] and in laboratory experiments for testing
population-dynamic hypotheses [11].
Environmental noise is one of the components affecting the population abundances. This means
that changes in extrinsic abiotic factors after the removal of diurnal, lunar and seasonal cycles
alter the environment surrounding the population. These changes can be for example variations
in the temperature, fires and floods. Environmental forcing has been argued to be one of the main
reasons behind the redness of the animal populations [60,2]. Steele [60] examined the records of
both terrestrial and marine physical systems and observed that marine environment shows red
dynamics and the terrestrial one white up to several decades and after that red spectrum. This
gives evident propositions for what kind of autocorrelated noise is appropriate for these two physical systems.
Hurst coefficient, measured with the method of range increment, measures the increment of
population fluctuations when the time interval increments Dt. This means that a population with
a greater Hurst exponent has larger fluctuation range increments. Disregarding the value of constant c of formula (7.1), for the same population sizes, the greater values of Hurst coefficient could
be related to a greater danger of extinction [61].
On a time series with a fractal structure, it could be expected that Hurst coefficients, measured
by the second order moment method and the range method and the local second order moment
method, be similar. Hastings and Sugihara [31] advise us that for short time series, Hurst coefficient calculated by the range increment method offers values that are greater than the real ones.
However, there is also a prevailing tendency, across a wide variety of species, for temporal variability to increase with the length of the census [46,17,2,55,62]. It has usually been associated with
spectral reddening, (a tendency for low or high abundances to be followed by more of the same)
of population dynamics. Dynamics can become reddened in several ways: Redness can be inherited from variation in the environment [60], it may arise through certain types of stochastic density
dependence [48,2], or it may be generated through long-range spatial interactions [64]. One would
expect a population whose numbers fluctuate more over time to have a greater risk of extinction
[37,38,55,39], although this need not always be so [28].
The fractal exponent (Hurst coefficient), i.e. the range increment of population variations, can
be due to characteristics intrinsic to the species, or to environmental characteristics. If the range
increment rate for the variations of a population depends more on environmental (local) characteristics, the Hurst coefficients of the different populations will not be related. In the case they
were, we would be dealing with a characteristic intrinsic to the species (at least in the studied
areas).
There are three hypotheses to be tested:
(1) Check the fractal structure of the time series of passerine populations. In order to carry out
this check, we use the three measurements of the fractal exponent by the three aforementioned methods (range increment, second moment and local second moment) and their
comparison.
(2) Check if Hurst coefficient is maintained for the same species in different places.
164
(3) Check if the range increment of the fluctuations of a population can be related to the danger
of extinction or some other population parameter.
The data come from real census and therefore are short time series. This means that fractal
dimension estimates may be inaccurate due to wandering intercepts [12], but they are still interesting for comparisons (see [29]).
9. Materials and methods
To measure Hurst coefficients of different passerine species, the data of the Bird Census News
[35] regarding three countries of Northern Europe (Finland (f), Sweden (sw) and Denmark (d)),
which form a latitudinal gradient, has been used. In Sweden the time series of both register methods used have been utilized: counts at count points (swp) and parcels registered through the mapping method (swm).
The different measurements of Hurst coefficients have been carried out by updating a computer
programme. This programme measures Hurst coefficient through range and second order moment
increment and through the method of local second order moment.
9.1. Programme
The programmes proposed by Hastings and Sugihara [31] to calculate fractal exponents of time
series using Hurst exponent with the techniques of range increment, second order moment increment, and local second order increment have been elaborated. In these programmes, in Pascal language, (Turbo Pascal 5.0 de Borland), they have been unified in one, and the correct functioning
has been checked with the data used by [31]. It was modified to adapt to the new data that needed
to be evaluated. Easy access menus were introduced, graphic proceedings were incorporated, and
the screen display and the printing results were improved.
The program used to measure Hurst coefficient by the three mentioned methods is freely available on: http://www.bi.upv.es/~algarsal/hurst/hurst.zip.
To check if these time series have a fractal structure, despite the fact that the results obtained
through the different methods to estimate Hurst coefficient are completely different, we have used
linear regressions between the Hurst coefficient obtained by the second order moment increment
method and the range increment method.
A comparison between both Hurst coefficients obtained from the four sets of data has
been carried out though a Spearman correlation analysis and Wilcoxons non-parametric
analysis.
9.2. Estimate confidence limits for fractal exponents [31]
With
P the data set (x2 i, yi) is constructed the regression line y = a + bx, that minimize the squares
sum: (yi a bxi) obtaining the distribution functions of a, b and r (correlation coefficient).
Let B denote the random variable corresponding to the slope b of a regression line through n
points, then the transformed random variable
1=2
ðB b0 Þ ðn 2Þr2x
T ¼
h
i1=2
ð1 q2 Þr2y
has a Students t-distribution with n 2 freedom degrees. Solving for B yields
h
i1=2
ð1 q2 Þr2y
B ¼ b0 þ T 1=2
ðn 2Þr2x
165
ð9:1Þ
ð9:2Þ
a distribution of mean 0 and variance
ð1 q2 Þr2y
varðBÞ ¼
.
ðn 4Þr2x
ð9:3Þ
Confidence limits are readily obtained from tables of the Students t-distribution. For n larger
than 25–30 fitness between the normal approximation and the t-distribution is enough and implies
that T is approximately normal with mean 0 and variance 1. Therefore, B is approximately normal
with mean b0 and variance
varðbÞ ¼
ð1 r2 Þr2y
.
ðn 4Þr2x
ð9:4Þ
In this case, confidence limits for b0 are readily obtained using tables for the normal distribution:
pffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffiffiffi
b 1:96 varðbÞ 6 b0 6 b þ 1:96 varðbÞ
ð9:5Þ
B statistics can be understood using the residual variance (variance of non explained errors or the
differences between predicted and observed values)
r2 ¼ ð1 q2 Þr2y
ð9:6Þ
to rewrite variance of B as
varðBÞ ¼
ð1 q2 Þr2y
r2
¼
.
ðn 4Þr2x
ðn 4Þr2x
ð9:7Þ
This result can be used to calculate the confidence intervals for the fractal exponent. In the program the standard deviation of the regression line (sdb) is measured as
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð1 r2 Þ ðsumysq k ybar2 Þ
sdb ¼
ð9:8Þ
ðk 4Þ ðsumxsq k xbar2 Þ
being r the correlation coefficient and k the time step used to measure the fractal exponent. Then
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Pk
2 !
u
u
P
ln
y
i
k
2
i¼1
uð1 r2 Þ i¼1 ðln y i Þ k k
u
u
ð9:9Þ
sdb ¼ u
Pk
2 !
u
ln xi
t ðk 4Þ Pk ðln x Þ2 k i¼1
i
i¼1
k
166
when H is calculated by the second order moment technique (mom2), for the confidence interval it
has to be multiplied by a correction factor, expfactor = 0.5. For H calculated by the range increment, expfactor = 1.
For k 6 4, sdb cannot be calculated. For 4 < k 6 27, the value is on the tables of the Students
t-distribution, with k 2 freedom degrees. For k > 27, normal table is used and the confidence
interval is
H 1:96 sdb expfactor 6 H 6 H þ 1:96 sdb expfactor
ð9:10Þ
10. Results
The Hurts exponent has been measured by three different methods: Range Increment, second
moment growth, and local second moment growth of 20 * 4 real temporal series of passerine populations, two of Sweden (different census methods), one of Denmark and one of Finland (see
Table 1).
When analysing the data of the different measurements, it can be said that the time series do not
seem to have a fractal structure due to the great differences existing between Hurst coefficients
measured through the different methods. The correlation coefficients of the regression lines of
the range increment method are very high, whereas for the method of second order moment increments are not acceptable in many cases.
However, the regression analysis of the Hurst coefficients calculated by the methods of range
increment and second order moment is highly significant (p < 0,0001, R2 = 71%),
H moment ¼ 60:6 þ 1:4H range.
This implies that the great differences observed are due to the shortness of the time series used in
this work. In these regressions, it could be clearly observed that the coefficient measured through
the method of range increment is larger than the coefficient measured through the method of the
second order moment, which was predictable when using time series that were too short (see [31]
for similar results).
Using Wilcoxons analysis and Spearmans correlation analysis to compare the four time series,
only a certain positive correlation (p < 0.1) is observed between the values for Finland and
Denmark. Surprisingly, no correlation is observed between the two time series of Sweden.
To see the relationship between Hurst coefficient and the different population parameters, the
coefficient measured from the range increment has been used, since it is the measure with less variation. No relations are observed with any population parameter (body size, colonisation capability, food type or phylogeny).
Fractional Brownian motion can be divided into three quite distinct categories: H < 1/2, H =
1/2 and H > 1/2. The case H = 1/2 is the ordinary Brownian motion, which has independent
increments, i.e. y(t + 2Dt) y(t + Dt) and y(t + Dt) y(t) being independent in the sense of probability theory; their correlation is 0 [53].
For H > 1/2 there is a positive correlation between these increments, i.e. if the graph of y(t) increases for same t, then it tends to continue to increase for t 0 > t (Fig. 2).
For H < 1/2 the opposite is true. There is a negative correlation between the increments (Fig. 3).
Finland
Anthus trivialis
Motacilla alba alba
Prunella modularis
Erithacus rubecula
Phoenicurus phoenicurus
Turdus philomelos
Turdus iliacus
Sylvia curruca
Sylvia borin
Phylloscopus sibilatrix
Phylloscopus collibita
Phylloscopus trochilus
Regulus regulus
Muscicapa striata
Ficedula hypoleuca
Parus montanus
Parus major
Garrulus glandarius
Fringilla coelebs
Carduelis spinus
Emberiza citrinella
Sweden mapping
method
Sweden point counts
Denmark
Second moment
growth
Range
increment
Second
moment
growth
Range
increment
Second
moment
growth
Range
increment
Second
moment
growth
Range
increment
0.40 ± 0.09
0.41 ± 0.06
0.28 ± 0.03
0.11 ± 0.14
0.17 ± 0.07
0.33 ± 0.07
0.44 ± 0.09
0.13 ± 0.17
0.11 ± 0.08
0.08 ± 0.13
0.49 ± 0.02
0.23 ± 0.07
0.44 ± 0.11
0.14 ± 0.19
0.20 ± 0.08
0.34 ± 0.13
0.02 ± 0.13
0.30 ± 0.13
0.45 ± 0.05
0.01 ± 0.13
0.13 ± 0.10
0.83 ± 0.05
0.71 ± 0.01
0.68 ± 0.04
0.58 ± 0.06
0.38 ± 0.03
0.66 ± 0.05
0.81 ± 0.05
0.47 ± 0.05
0.54 ± 0.03
0.47 ± 0.08
0.79 ± 0.05
0.73 ± 0.05
0.72 ± 0.02
0.39 ± 0.04
0.59 ± 0.03
0.63 ± 0.04
0.45 ± 0.07
0.63 ± 0.02
0.75 ± 0.07
0.48 ± 0.02
0.61 ± 0.04
0.24 ± 0.09
0.63 ± 0.03
0.36 ± 0.09
0.46 ± 0.10
0.22 ± 0.07
0.46 ± 0.10
0.31 ± 0.03
0.62 ± 0.04
0.81 ± 0.04
0.75 ± 0.05
0.71 ± 0.05
0.66 ± 0.04
0.75 ± 0.04
0.60 ± 0.02
0.90 ± 0.03
0.93 ± 0.02
0.73 ± 0.02
0.66 ± 0.02
0.70 ± 0.05
0.73 ± 0.02
0.57 ± 0.06
0.61 ± 0.02
0.71 ± 0.04
0.64 ± 0.02
0.80 ± 0.02
0.60 ± 0.03
0.53 ± 0.01
0.63 ± 0.04
0.76 ± 0.08
0.49 ± 0.04
0.54 ± 0.00
0.66 ± 0.01
0.72 ± 0.02
0.81 ± 0.02
0.50 ± 0.04
0.34 ± 0.06
0.30 ± 0.11
0.22 ± 0.13
0.11 ± 0.19
0.24 ± 0.06
0.22 ± 0.14
0.39 ± 0.07
0.59 ± 0.06
0.31 ± 0.07
0.13 ± 0.07
0.03 ± 0.11
0.43 ± 0.19
0.07 ± 0.21
0.18 ± 0.03
0.42 ± 0.04
0.29 ± 0.07
0.30 ± 0.10
0.07 ± 0.07
0.13 ± 0.12
0.34 ± 0.03
0.15 ± 0.05
0.66 ± 0.03
0.66 ± 0.05
0.42 ± 0.05
0.68 ± 0.05
0.68 ± 0.02
0.50 ± 0.03
0.13 ± 0.10
0.51 ± 0.04
0.60 ± 0.02
0.37 ± 0.05
0.38 ± 0.03
0.85 ± 0.06
0.62 ± 0.02
0.72 ± 0.03
0.90 ± 0.06
0.62 ± 0.04
0.71 ± 0.02
0.68 ± 0.01
0.67 ± 0.02
0.57 ± 0.03
0.79 ± 0.03
0.54 ± 0.03
0.54 ± 0.02
0.46 ± 0.04
0.47 ± 0.07
0.62 ± 0.03
0.72 ± 0.01
0.57 ± 0.02
0.58 ± 0.06
0.71 ± 0.04
0.59 ± 0.04
0.68 ± 0.02
0.63 ± 0.03
0.36 ± 0.06
0.45 ± 0.05
0.39 ± 0.08
0.05 ± 0.23
0.56 ± 0.07
0.19 ± 0.08
0.12 ± 0.06
0.01 ± 0.06
0.29 ± 0.10
0.21 ± 0.11
0.28 ± 0.04
0.14 ± 0.05
0.38 ± 0.08
0.40 ± 0.02
0.24 ± 0.06
0.21 ± 0.09
0.41 ± 0.02
0.28 ± 0.11
0.56 ± 0.02
Table 1
Hurst coefficients calculated from the passerine populations time series
167
168
H2ndM = 0.40; HRange = 0.83
H2ndM = 0.49; HRange = 0.79
Anthus Trivialis Finland
140
120
100
80
60
40
20
0
Phylloscopus collibita Finland
150
100
50
0
1
1 2 3 4 5 6 7 8 9 10 11 12 13
2
3
4
5
6
7
8
9 10 11 12 13
Fig. 2. Population time series of different passerine species with Hurst coefficient H > 1/2.
H2ndM = 0.14; HRange = 0.39
H2ndM = 0.01; HRange = 0.48
200
Muscicapa striata Finland
150
150
100
100
50
50
Carduelis spinus Finland
0
0
1 2 3 4 5 6 7 8 9 10 11 12 13
1 2 3 4 5 6 7 8 9 10 11 12 13
Fig. 3. Population temporal series of different passerine species with Hurst coefficient H < 1/2.
Theoretically H can be related with the extinction risk [28,61]. If this is true, the species with
higher extinction risk would be the species with higher H on the three countries: A. trivialis,
M. alba, P. modularis, and on the other side, S. curruca, S. borin, M. striata and P. major are
the species with more stable populations.
There is not a latitudinal trend for all the species, although some of them are more stable on
higher latitudes (P. phoenicurus, P. sibilatrix) or on lower latitudes (E. citrinella).
11. Conclusions
The time series of the passerine that we have worked with are too short to permit the confirmation of their fractal structure (if they have it). It can be observed by the linear regressions that
the H estimations from different methods are well related in all the cases. The coefficient measured
by the range increment method is always bigger than the coefficient measured through the method
of the second order moment increase, which was predictable when using time series that were too
short (see [31] for similar results).
These differences in the H estimations are linearly related. Although the real H value cannot
be obtained with these time series (too short), the relation between the values obtained by different methods suggest that still is possible to compare the values from different countries and
species.
169
On most cases, for each specie, the Hurst coefficient maintain a high or a low value for all the
countries, suggesting that it is an intrinsic variable of the species, independent from environmental
factors. But for other species the differences are very conspicuous suggesting precisely the opposite: it is a variable so highly sensitive that it offers different values on the same populations, with
the same species but with different registering methods.
Moreover, this coefficient does not correlate with any other population parameter studied.
However, this does not imply that no further research must be done in this area. In fact, we
encourage all those in possession of time series to carry out an analysis of their fractal dimension,
behaviour and structure, using the programme elaborated for this work.
Acknowledgment
We would like to thank Dr. J.L. Telleria for his directives, and the Foreign Language Coordination Office at the Polytechnic University of Valencia for their help in translating this paper.
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