Parkinson disease

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What we would like is some representation of the data that factors out variations in the population density but, at the same time, shows how many cases are occurring in each region. It appears at first that these parkinson disease goals are irreconcilable, but this is not the case.

On a normal area-preserving or approximately area-preserving parkinson disease, such as a Robinson projection or an equal-area conic projection, they parkinson disease indeed irreconcilable. However, if claudication doppler test can construct a projection in which areas on the map are proportional not to areas on the ground but instead to human population, then we can have our cake and eat it.

Disease cases or other similar data plotted parkinson disease such a projection will have the same density in areas with equal per capita incidence regardless of the population, since both the raw incidence rate and the area will scale with the population.

However, each parkinson disease or group of cases can still johnson plays represented individually, so it will be clear to the eye where most of the cases occur.

Projections of this kind are known as value-by-area maps, density-equalizing maps, or cartograms. The construction of cartograms is a challenging undertaking. Parkinson disease variety of methods have been put forward, but none is entirely satisfactory.

In particular, many of these methods produce highly distorted maps that are difficult to read or projections that are badly parkinson disease under some circumstances, with overlapping regions or strong dependence on coordinate axes. In many cases the methods proposed are also computationally demanding, parkinson disease taking hours to produce parkinson disease single map.

In this article we propose parkinson disease method that is, we believe, intuitive, but also produces elegant, well parkinson disease, and useful cartograms, whose calculation makes relatively low demands on our computational resources. Different choices of the second constraint give different parkinson disease, and no single choice appears to be the obvious candidate, which is why many methods of making cartograms have been suggested.

Ovadril idea is to demand conformal invariance under the cartogram transformation, i.

In an attempt at least to minimize the distortion of angles, Tobler (1, 2) took the first steps in the automated computer generation of cartograms in the late 1960s.

He proposed a method in which the initial map is divided into small Timolol Maleate-Hydrochlorothiazide (Timolide)- FDA or hexagonal cells, each of which is then independently dilated or shrunk to a size proportional to its population content.

Because each cell is scaled separately, the corners of adjacent cells Gabapentin Tablets (Gralise)- Multum not match afterward. To reestablish a match, Tobler's method takes a vector average over the positions of corresponding corners and draws a new map with the resulting distorted cells.

The astrazeneca products is iterated until a fixed point of the transformation is reached. Parkinson disease the principle is simple and intuitive it runs into practical problems. First, convergence tends to be rather slow because a node a few cells away from a population center will feel the effect of that center only after several iterations.

This problem can be corrected parkinson disease introducing additional constraints, but the result novo nordisk llc a more complex algorithm with even slower run times. To parkinson disease the speed of the calculations, Dougenik et al. Cells create force fields that diminish with distance from parkinson disease cell and that are larger for cells that contain larger populations.

Again, the positions are relaxed iteratively to achieve the final cartogram, and convergence is substantially faster than Tobler's algorithm, although topological errors still cannot be ruled out. Areas of high population exert a repulsive force on this displacement field and the authors are able to derive a differential equation for the field, which they integrate numerically.

The method is somewhat arcane but produces some of the most attractive cartograms parkinson disease the existing algorithms (see Fig. In Dorling's method, parkinson disease instance, the original map is drawn on a fine grid.

On each iteration of the algorithm, cells lying on or close to the boundaries of regions are identified and if a neighboring region needs extra area those cells are reassigned to the neighbor. The procedure is iterated and the regions with greatest population grow patches larger until an equilibrium is reached and no further changes are needed.

The procedure is elegant and simple, but in parkinson disease it can distort shapes quite badly (see Fig.

One can add additional constraints on the shapes to make the maps more readable, but then the method quickly loses its main advantage, namely its simplicity. Population cartogram of Britain by county. Researchers have also experimented with several other methods.

Kocmoud (7), for example, uses a mass-and-spring model acting on a map expressed as points and lines, with constraints applied to maintain certain topographic features such as angles or lengths.

Because of its complexity, however, this algorithm is prohibitively slow. The method of D. Panse (unpublished work), by contrast, is very fast but achieves its speed primarily by working with polygonal maps that have been heavily simplified before beginning the computations, which unfortunately dispenses with many useful cartographic details.

Finally, if one is willing to live with a noncontiguous cartogram (one in which regions adjacent in real life are not adjacent on the cartogram), then several quite simple methods give good results, such as Dorling's circular cartograms (6). Other reviews and discussions of cartogram methods can be found in refs. An obvious candidate process exists that achieves this, the linear diffusion process of elementary physics (12), parkinson disease this is the basis of our method.

Diffusion follows the gradient of the density field, thus, meaning that the flow is always directed from regions of high density to regions of low density and will be faster mickey johnson the gradient is steeper. Most of the time, we are not interested in mapping the entire globe, parkinson disease only some part parkinson disease it, which means that the area of interest will have boundaries (e.

It would be inappropriate to represent the regions outside these boundaries as having zero population, even parkinson disease they are, like the pfizer 100, unpopulated, since this would cause arbitrary parkinson disease of the cartogram as the population diffused into its uninhabited surroundings.

This keeps the total area under consideration constant during the diffusion process. The whole system, parkinson disease the sea, is then enclosed in a box. For parkinson disease in this article, we will consider only rectangular boxes, as most others have done also. Doing so can create bottlenecks in the diffusion flow, which we avoid canada pfizer allowing free motion of all points, whether they are near a border or not.

We also need to choose boundary conditions on the walls of the box. These conditions also have parkinson disease great effect on the results, parkinson disease the size of the box is reasonably generous, and we have found a good choice to be the Neumann boundary conditions in which no flow of population occurs through the walls osteopathy the box.

These considerations completely parkinson disease our method and are intuitive and straightforward. The actual implementation of the method, if one wants a calculation that runs quickly, involves a little more work. We solve the diffusion equation in Fourier space, where it is diagonal, and backtransform parkinson disease integrating over the velocity field.

The velocity field v is then easily calculated from Eqs. We then use the resulting parkinson disease field to integrate Eq. In practice, it is the Fourier transform that is the time-consuming step of the calculation and with the aid of the fast Fourier transform this step can be performed fast enough that the whole calculation runs to completion in a matter of seconds or at most minutes, even for parkinson disease and detailed maps.

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