Abstract-Mean shift, a simple iterative procedure that shifts each data point to the average of data points in its neighborhood, is generalized and analyzed in this paper. This generalization makes some k-means like clustering algorithms its special cases. It is shown that mean shift is a mode-seeking process on a surface constructed with a "shadow" kernel. For Gaussian kernels, mean shift is a gradient mapping. Convergence is studied for mean shift iterations. Cluster analysis is treated as a deterministic problem of finding a fixed point of mean shift that characterizes the data. Applications in clustering and Hough transform are demonstrated. Mean shift is also considered as an evolutionary strategy that performs multistart global optimization.

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Taylors' frozen turbulence hypothesis suggests that all turbulent eddies are advected by the mean streamwise velocity, without changes in their properties. This hypothesis has been widely invoked to compute Reynolds averaging using temporal turbulence data measured at a single point in space. However, in the atmospheric surface layer, the exact relationship between convection velocity and wave number k has not been fully revealed since previous observations were limited by either their spatial resolution or by the sampling length. Using Distributed Temperature Sensing (DTS), acquiring turbulent temperature fluctuations at high temporal and spatial frequencies, we computed convection velocities across wave numbers using a phase spectrum method. We found that convection velocity decreases as k−1/3 at the higher wave numbers of the inertial subrange instead of being independent of wave number as suggested by Taylor's hypothesis. We further corroborated this result using large eddy simulations. Applying Taylor's hypothesis thus systematically underestimates turbulent spectrum in the inertial subrange. A correction is proposed for point‐based eddy‐covariance measurements, which can improve surface energy budget closure and estimates of CO2 fluxes.

The convergence and ordering of Kohonen's batch-mode self-organizing map with Heskes and Kappen's (1993) winner selection are proved. Selim and Ismail's (1984) objective function for k-means clustering is generalized in the convergence proof of the self-organizing map. It is shown that when the neighborhood relation is doubly decreasing, order in the map is preserved. An unordered map becomes ordered when a degenerate state of ordering is entered, where the number of distinct winners is one or two. One strategy to enter this state is to run the algorithm with a broad neighborhood relation.

We present the numerical solutions of the equations for turbulence developed in papers I and II. (1) The model predicts the Kolmogorov law and Ko=5/3, in accord with recent data; (2) in the inertial-conductive regime, the model predicts the Corrsin spectrum for the temperature variance and the Batchelor constant Ba=σt Ko, where σt=0.72 is the turbulent Prandtl number; (3) the predicted energy spectrum in the dissipation region is in agreement with recent laboratory measurements; (4) in the inertial-convective region, the temperature variance spectrum is closer to the spectrum (−11/3) obtained by LES when the velocity field is rapidly stirred at all scales than(−17/3), which holds when the velocity field is frozen in time and has a Gaussian statistics; (5) for freely decaying turbulence, the power law spectra for energy and temperature variance, as well as the velocity and temperature integral scales, agree with the most recent LES data; (6) after a few evolutionary times, the skewness S reaches S=0.5, in accord with a variety of data; (7) for shear-driven flows, the Reynolds stress spectrum E12(k) has an inertial regime with a power −7/3, in accord with recent data; (8) for two shear-driven flows, plane strain and axisymmetric contraction, turbulent kinetic energy, Reynolds stress tensor, and dissipation rate εij versus time compare very well with DNS data; (9) the slow and rapid parts of the pressure–strain correlation tensor compare with DNS data better than with the three most widely used phenomenological models. The rapid parts are also in excellent agreement with the DNS data; (10) for homogeneous shear, turbulent kinetic energy and Reynolds stress tensor versus time match quite closely LES data. We recall that the model does not contain any free parameters.