Fractals and multifractals in ecology and aquatic science

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Overall, Independently, fractality and multifractality were analyzed in oceanographic time series, including sea surface temperature and wind stress. At the end of Chapter 3, bivariate fractal analyses were performed to quantify how the fractal dynamics of sea surface temperature and wind stress were affecting the temporal fractal pattern of variability in smooth pink shrimp catches.

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Furthermore, LRD has been incorporated into climate variability models Fraedrich and Blender , Vyushin and Kushner and also used as a statistical tool to monitor the climate Kurnaz , Mudelsee In oceanographic research, LRD has not received much attention until the last decade when long-range memory fractionally differenced models and short-range memory autoregressive integrated moving average, ARIMA models were compared quantitatively as to their ability to explain oceanographic regime shifts and inter-decadal variability in the North Pacific Ocean Percival et al.

During the last decade, several oceanographic studies have detected and quantified LRD in time series of upwelling Denny et al. Usually, oceanographic studies focused on the estimation of long-range dependence parameters, but recently, LRD has also been used for the simulation of oceanographic responses to atmospheric forcing Legatt et al.

Long-range dependence in the dynamics of fish populations is frequently associated with changes in low-frequency oscillations of oceanographic variables. This means that changes in the temporal dynamics of fish catches, catch per-unit effort or spawning stock biomass can arise due to the emergence of oceanographic low-frequency components Rouyer et al. In these latter studies, oceanographic low-frequency oscillations were considered as the main source of long-range dependence LRD , but it is important to remark that from a statistical standpoint LRD can emerge without forcing by oceanographic conditions.

This transitional zone is strongly influenced by climate cycles superimposed one on another, whose interaction is not completely understood Patterson et al.

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Written in a step-by-step fashion, Fractals and Multifractals in Ecology and Aquatic Science provides scientists with a basic understanding of fractals and. Buy Fractals and Multifractals in Ecology and Aquatic Science on ✓ FREE SHIPPING on qualified orders.

The spring to fall period each year, which also corresponds to the smooth pink shrimp Pandalus jordani fishing season off the west coast of Vancouver Island Perry et al. Specific features of upwelling off the west coast of Vancouver Island are mainly driven by winds and the complex local topography Fang and Hsieh For the North-East Pacific ecosystem, significant shifts in climate indices representing sea surface temperature variability are associated with long-range memory process for which autocorrelation extends over multiple years Overland et al.

All of these can contribute to the emergence of long-range dependence in time series of fish populations. Under this framework, the first goal of this Chapter was to detect statistical long-range dependence LRD for sea surface temperature and wind stress from the west coast off 63 Vancouver Island. The second goal was the detection of long-range dependence in smooth pink shrimp total daily catches, and to verify if its scaling range -previously detected in Chapter 2- varies using a longer time series of catches.

The third objective was to detect if the fractal dynamics of Pandalus jordani catches is affected by the fractal dynamics of oceanographic time series. Fisheries and oceanographic time series are characterized by several processes acting on different time scales Rouyer et al. For that reason, a well-known method for the estimation of long-range dependence was selected: detrended fluctuation analysis DFA; Peng et al. In addition, DFA allows the estimation of fractal parameters in time series contaminated with data gaps.

Sea surface temperature records were obtained within an hour of local daytime high tide from the top 64 meter of the water column Cummins and Masson Figure 3. Leap days were omitted, and a total of missing observations approximately 3. For example, if no observation was registered for April 1, , the average value calculated considering every 1st of April from to was used to fill this missing observation.

We are aware that it is also possible to estimate long-range dependence parameters for time series with missing values without the need to fill missing 65 observations Chamoli et al. However, this implies the loss of temporal sequence between observations and for that reason, missing values were estimated to avoid mis-interpretations of results from bivariate fractal analyses. Vertical lines separate periods of ten years. The daily marine surface wind database, calculated from meteorological buoy number located Perry personal communication. This database corresponds to the wind stress i.

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Vertical grey lines separate periods of days, which correspond to two continuous fishing seasons. No fishing activity for a specific day of the fishing season means that a missing value for that day was obtained.

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In order to be consistent with the filling of missing observations applied to oceanographic time series, the same procedure was applied to the TDC time series; i. Definitions of variables used in this chapter are summarized in Table 3. Definitions of variables and parameters used in Chapter 3. The mean annual cycle for the smooth pink shrimp time series, which has a length of days formed by the stitching of continuous fishing seasons see Chapter 2 was removed, and the new time series defined as TDC anomaly Fig.

The filtering of trends of sea surface temperature and wind stress anomalies to isolate their stochastic variability and long-range dependence properties was also necessary Yuan et al. These latter lower and upper limits were selected because this range includes the sea surface temperature annual cycle or first harmonic days and its second harmonic of days Gower These periodicities were reconstructed and filtered out using a continuous wavelet transform CWT, Torrence and Compo according to: 73 3.

After filtering out periodic oscillations, they are called from here on as filtered sea surface temperature SSTF and filtered wind stress WSF time series. It is necessary to remark that when oceanographic time series were filtered with the objective to test their bivariate fractal relationship with the smooth pink shrimp time series, only daily observations matching fishing seasons April-October from to were used. Fractal analysis Long-range dependence LRD was detected and quantified using Detrended Fluctuation Analysis DFA methodology, used and recommended to estimate long-range dependence in complex geophysical time series Kantelhardt et al.

In the first stage of the DFA methodology, the cumulative sum, also called the nonstationary profile function Y i , was constructed according to: 3. Then, Y i obtained using equation 3. In other words, the fluctuation of the time series is determined as the variance upon the estimated local trend. In the final stage, the variances of all segments were averaged and the square root was taken to obtain the mean fluctuation function F s as: 3.

In this final stage, we average F s calculated over several segments of different size s and analyze the relationship between the mean fluctuation function and the segment size. F s 75 obtained using equation 3. Local trends can be estimated by least squares fitting of polynomials of different orders. In this research DFA-1 was used, which means that linear polynomials were removed from time series defined in equation 3. In other words, a large fluctuation observed in the time series in relation to its mean value is likely to be followed by a large fluctuation and vice versa.

These deviations are intrinsic to the methodology because the scaling behaviour is only approached asymptotically; i.

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When time scales larger than this threshold were used, the number of segments used in the averaging of F s obtained with equation 3. Multifractal analysis The DFA methodology allows the estimation of a single scaling exponent to detect fractal monofractal dynamics of a time series. It is also possible to generalize the DFA analysis to analyze time series under a multifractal approach. After the detrending procedure it is possible to obtain the qth order fluctuation function as: 77 3. Then the relationship between Fq s and the time scale s was analyzed.

If the time series under investigation is long-range correlated, Fq s will increase for larger values of s in an analogous way to equation 3. For stationary time series, h 2 corresponds to the Hurst exponent and hence h q is designated as the generalized Hurst exponent Kantelhardt et al. For fractal monofractal time series the h q exponent is independent of q. For these types of time series the scaling behaviour of the variances is equal for all time scales and q values.

Soil aggregates as mass fractals

Considering that the type of information used in the elaboration of self-similar-based indicators is available for many other fisheries, and most indicators usually fail to detect ecosystem changes when annual data are used to predict such changes Perreti and Munch , fine scale catch data for other fisheries should be used to test the conditions, performance and generality of proposed early-warning indicators. Tell us if something is incorrect. Moreover, the KG estimator shows the best tracking performance among twelve H estimators when applied to processes characterized by time-varying Hurst estimators Sheng et al. On a logarithmic scale, scaling dynamics will manifest as an approximately linear curve when plotting E w versus w. The residual time series, free of these unwanted oscillations, can be observed in Figure 4. Local slopes fluctuating around zero can be associated with indicators following a linear trend Takalo , Maraun et al. Using Dispersional Analysis Caccia et al.

For multifractal time series a single scaling exponent like h 2 is not sufficient to completely characterize their dynamics since many subsets of the time series have different scaling behaviours. Instead of calculating the fluctuation function of the detrended variance using equation 3. Then, the qth order detrended covariance for non-zero q values was calculated as: 3.

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Again, in an analogous way to equation 3. It is clear that when the same time series is used in equation 3. The generalized Hurst coefficient defined in equation 3. This is clearly shown by the continuous wavelet global spectrum Torrence and Compo , where the annual cycle is marked as a continuous red band centered on a scale equal to days Fig. Sea surface temperature wavelet power spectrum for period. The white dotted line indicates the cone of influence COI , below which power spectrum coefficients should be interpreted with caution.

A frequency band in dark red associated with the annual cycle is clearly visible. The reconstructed SST anomaly time series is shown in Figure 3. The filtered sea surface temperature time series SSTF , used for the estimation of long-range dependence Fig.

Fractals and Multifractals in Ecology and Aquatic Science

Maximum likelihood estimation of H is possible only when the analyzed time series behaves as fGn or summed fGn. Due to the specificity of this method, it cannot be used for a wide class of fractal time series. Accordingly it was not selected in the methodology section as the principal method for the estimation of LRD.


In consequence, it was used here as an auxiliary methodology when fGn time series were obtained, enabling the estimation of Hurst coefficients that are independent of selected scaling ranges. The intrinsic assumption that SSTF can be represented by a single LRD parameter that does not vary over time -which is valid only for long and exactly simulated LRD time series- cannot be sustained as expected Fig. However, relatively constant Hurst coefficients above a long-term mean were obtained for prolonged periods: , and visualized as runs with similar values in Fig.

After and until the behaviour of H changed markedly from one year to the next; and again marked low H values were obtained for years dominated by anomalous oceanographic conditions: and Fig. Hurst coefficients estimated for filtered sea surface temperature SSTF time series during the period. Lines were fitted to scaling ranges to estimate the Hurst coefficient. A scaling break was detected for sea surface temperature time series around days 28; Fig.

The Hurst coefficient obtained for the scaling range between 16 and 80 days estimated with Detrended Fluctuation Analysis method was equal to 0. Maximum likelihood estimates of H were not calculated for the filtered wind stress time series WSF because it showed deviations from a Gaussian distribution Fig. Two scaling ranges were detected Fig.

For smooth pink shrimp total daily catch anomalies TDCa two clear scaling ranges were 86 detected. The first extended between 16 and days Fig.

This result confirms 87 that long-range dependence is present in smooth pink shrimp total daily catches and it does not depend on the phase or period of the fishery. The h q calculated for smooth pink shrimp total daily catch anomalies TDCa showed similar variability to that estimated for the oceanographic time series. As a consequence, TDCa is also classified as mono fractal , and the estimation of multifractal exponents was not necessary Fig. The estimated bivariate Hurst exponent equation 3.

This means that scaling ranges were well defined, and consequently the generalized Hurst exponent was accurately estimated Fig.