Correlation Ellipse Research Articles

A biofilm reactor-based approach for rapid on-line determination of biodegradable organic pollutants

A new analytical approach utilizing a biofilm reactor (BFR) for rapid online determination of biochemical oxygen demand (BOD) was proposed and experimentally validated. The BFR was fabricated via a cultivation process using naturally occurring microbial seeds from locally collected wastewaters. The resultant BFR displays a remarkable rate of biodegradation towards a wide spectrum of organic substrates, capable of degrading over 20% of biodegradable organic substrates within 100s. More importantly, the BFR exhibits a superior indiscriminative biodegradation feature, enabling a precise prediction of BOD values of total biodegradable organics based on experimentally determined BOD values from partial degradation processes without a need for on-going calibration. The proposed approach was systematically validated using a range of individual organic substrates, their mixtures, synthetic samples and wastewaters. Highly significant linear correlations between the BFR and the standard BOD5 methods were obtained from diversified synthetic samples (r=0.988, p=0.000, n=45) and wastewaters (r=0.983, p=0.000, n=40). Near unity slope values of the principal axis of the correlation ellipse were obtained from all tested samples, suggesting both methods were essentially measuring the same BOD value. The reported method could be a useful online monitoring tool for determination of biodegradable organic pollutants.

Correlation Functions of Small-Scale Fluctuations of the Interplanetary Magnetic Field

The Interplanetary Magnetic Field shows complex spatial and temporal variations. Single spacecraft measurements reveal only a one-dimensional section of this rich four-dimensional phenomenon. Multi-point measurements of the four Cluster spacecraft provide a unique tool to study the spatiotemporal structure of the field. Using Cluster data we determined three-dimensional correlation functions of the fluctuations. By means of the correlation function one can describe and measure field variations. Our results can be used to verify theoretical predictions, to understand the formation and nature of solar wind turbulence. We found that the correlation length varies over almost six orders of magnitude. The IMF turbulence shows significant anisotropy with two distinct populations. In certain time intervals the ratio of the three axes of the correlation ellipse is 1/2.2/6 while in the remaining time we found extremely high correlation along one axis. We found favored directions in the orientation of the correlation ellipsoids.

Dynamics of wave packets in two-dimensional random systems with anisotropic disorder

A theoretical model is proposed to describe narrowband pulse dynamics in two-dimensional systems with arbitrary correlated disorder. In anisotropic systems with elongated cigarlike inhomogeneities, fast propagation is predicted in the direction across the structure where the wave is exponentially localized and tunneling of evanescent modes plays a dominant role in typical realizations. Along the structure, where the wave is channeled as in a waveguide, the motion of the wave energy is relatively slow. Numerical simulations performed for ultra-wide-band pulses show that even at the initial stage of wave evolution, the radiation diffuses predominantly in the direction along the major axis of the correlation ellipse. Spectral analysis of the results relates the long tail of the wave observed in the transverse direction to a number of frequency domain "lucky shots" associated with the long-living resonant modes localized inside the sample.

Localization of classical waves in weakly scattering two-dimensional media with anisotropic disorder

We study the localization of classical waves in weakly scattering two-dimensional systems with anisotropic disorder. The analysis is based on a perturbative path-integral technique combined with a spectral filtering that accounts for the first-order Bragg scattering only. It is shown that in the long-wavelength limit the radiation is always localized, and the localization length is independent of the direction of propagation, the latter in contrast to the predictions based on an anisotropic tight-binding model. For shorter wavelengths that are comparable to the correlation scales of the disorder, the transport properties of disordered media are essentially different in the directions along and across the correlation ellipse. There exists a frequency-dependent critical value of the anisotropy parameter, below which waves are localized at all angles of propagation. Above this critical value, the radiation is localized only within some angular sectors centered at the short axis of the correlation ellipse and is extended in other directions.

Moving random surfaces and correlation analysis

Two models of moving random surfaces with known velocity and known correlation function are described. The random surfaces are formed from an infinite set of plane waves with decorrelation arising from random temporal and spatial changes. For model 1 the random changes are independent of the wave frequency. For model 2 they increase with frequency. It is shown that if there are random spatial changes in the direction of motion, then speeds derived by the full correlation method will be less than the true speed of the surface. If temporal series based on the models are subjected to a high‐pass filter before correlation analysis, the correlation speeds increase in value. Correlation functions from one of these models (model 2) fail the requirements for full correlation analysis because the half widths of the cross‐correlation functions are greater than the half width of the autocorrelation function. However, from a survey of experimental ionospheric drift measurements it is argued that radio interference patterns appear to have the form of this model. For this model, (1) correlation speeds increase with increasing antenna spacing, (2) the correlation ellipse tends to line up with that antenna pair with the greatest spacing, (3) the straight line of the straight‐line method will not be straight, and as stated above, (4) the speeds will be low. These features have been found experimentally and are contrary to the assumptions of the correlation method. It is shown that an alternate analysis, the “similar fades average velocity method,” gives a better estimate of the surface speed than does the correlation method.

Judgments of salary bias and test bias from statistical evidence

Academics examined correlation scatter plots and made judgments of group bias. Half of the subjects judged bias in salaries from graphs that showed salaries plotted against merit with a separate correlation ellipse for each of two groups. The others judged test bias from graphs that showed job performance vs test scores. Relative positions of the centroids for the two groups and the within-group correlation were systematically varied. Judgments of bias were not consistent with either “forward” or “reverse” regression definitions of bias. Judged bias is not a monotonic function of the group difference in salary between persons of equal merit nor is it a monotonic function of the group difference in merit between persons of equal salary. Instead, judgments in both tasks were consistent with a difference of differences model; for example, when the group difference in standard deviation units is equal on both salary and merit, judges say the situation is unbiased. The results are consistent with a one-mediator model of group equity, which assumes that both measures are imperfectly and equally correlated with the mediator.

The physical significance of the correlation ellipse in ionospheric drift analysis

It is argued that the conventional interpretation of the correlation ellipse in ionospheric drift analysis is correct, and that this ellipse provides a useful description of the spatial properties of the amplitude pattern over the ground.

The physical significance of the correlation ellipse in ionospheric drift analysis

It is shown that the correlation ellipse bears no relationship to the “average” shape of irregularities in the amplitude pattern formed at the ground from the interference of radio waves reflected or scattered from the ionosphere. The concept that the correlation ellipse should be related to the average shape arose because it was considered that the cross‐correlation coefficient for zero delay between two fading records (with spatial separation r) was related to the average normalized amplitude of the irregularities at a displacement r from the center of the average irregularity. As shown in this paper, the cross‐correlation coefficient for zero delay depends only on the mean fading period and the mean and variance of the time delays between corresponding fades. It retains its usual significance only if r is in the direction of drift.

Some tests of methods of analysis of ionospheric drift records using an array of 89 aerials

Results from an array of 89 aerials are used to test the validity of the usual methods of measuring ionospheric drifts, in which only three aerials are used. It is shown that the magnitude of the ‘true’ velocity determined by ‘full correlation analysis’ increases as the size of the aerial triangle increases, and tends to a limit which is the correct value. The ‘apparent’ velocity, which is equal to the velocity obtained by the ‘method of similar fades’, shows no variation with triangle size, but gives consistently high values. The ‘correlation ellipse’ shows a tendency to be aligned along the hypotenuse when small right-angled triangles are used. An explanation of these results is given in terms of instrumental effects in the recording and digitizing process. It is suggested that there is an optimum aerial separation for spaced receiver experiments, and that this is considerably larger than has usually been employed in the past. For E-region measurements, an equilateral triangle of side about 300 m is recommended.

The analysis of ionospheric drift data in the closely spaced receiver method

Data are presented which show that in many cases the orientation of the correlation ellipse shows a degree of dependence on the relative disposition of the three receiving aerials. It is suggested that this phenomenon is to be expected when the extremal lines of the diffraction pattern are curved rather than straight.

Notes on correlation methods for evaluating ionospheric winds from radio fading records

A correlation method requiring six values of the correlation coefficient is developed. Expressions for the drift velocity, fading velocity, and characteristic velocity are obtained from a correlation theory extended to include an elliptical contour in the horizontal plane. The physical significance of the derived velocities is considered. Preliminary data indicate that the correlation ellipse exhibits a preferred orientation and that fading associated with random changes is as important as fading associated with a drifting pattern.