Smoothing methods in .NET Receive PDF 417 in .NET Smoothing methods

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Smoothing methods use visual studio .net pdf-417 2d barcode development todeploy barcode pdf417 for .net UPC-8 Case (1): x(i ) is bino .net framework pdf417 mial with parameters y(i ) and p and y(i ) is a constant. For example y(i ) is the population in area i and p is the probability that any randomly selected individual has a speci ed attribute so that x(i ) is the count of individuals with the speci ed attribute.

This might be the rate of some commonly occurring offence in a well-de ned at-risk population (e.g. household burglary).

Case (2): x(i ) is Poisson with parameter , w(i ) is Poisson with parameter and y(i ) = x(i ) + w(i ). The parameter is required to be much greater than (e.g.

/ 103 ). This arises when r(i ) is the incidence or mortality rate of a rare disease. x(i ) is the number of cases of the disease in area i whilst w(i ) is the number of healthy individuals in i.

Kafadar (1994) compares smoothing methods applied to directly standardized incidence ratios for areas using US county-level data on prostate cancer. In the following i denotes area, m denotes age cohort, (m) denotes the proportion of individuals in age group m from the standard population, d(i, m) denotes the number of cases in area i and age cohort m and n(i, m) denotes the number of person years at risk in area i and age cohort m. The age-adjusted rate for area i is given by: r (i).

m (m)[d(i,. m)/n(i, m)]. (7.4). The distance-weighted a PDF417 for .NET verage (see for example 7.2) of the age-adjusted rates gives the statistic: r (i).

j w(i,. j)r ( j)/. j w(i,. (7.5). Using distance-weighted averaging on the the numerator and denominator of (7.4) gives the statistic: r + (i). m (m) [d + (i, m)/n+ (i, m)]. (7.6). where d+ (i, m) = j w( j, i) d( j, m) and n+ (i, m) = j w( j, i) n( j, m). Assuming case (2) above and assuming that the bandwidth, D, is chosen so that age-specifc rates are constant within the selected window size then: var(r + (i)) < var(r (i)) < var(r (i)). The standard errors associated with rates computed from different sized base populations under the assumption of the Poisson model are not the same.

Standard errors are inversely proportional to the total population. Computing a map by smoothing across values with different standard errors is likely to be misleading and it is necessary to try to eliminate the effects of population size through a variance equalizing transformation of the original data values (see section 6.2.

1 and Cressie, 1991, p. 395). Kafadar (1996) considers a double.

ESDA: numerical methods weighting for distance and population size: w(i, j) [n( j)] [1.0 (di, j /D) ] 0. di, j < D otherwise (7.7). where n( j ) = m n( j, visual .net PDF417 m) is the total population in area j. She notes that with these weights the target area may not receive the largest weighting.

Kafadar s results indicate that (7.5) using simple disk averaging and (7.6) using simple disk averaging, (7.

3; = 3.0) and (7.7; = 1.

0; = 3.0) all indicate the presence of similar patterns in the data but differ in the intensity with which patches are displayed. Other choices for and may be made (Kafadar, 1999).

On the evidence of her analysis it seems that if the aim is to ensure patterns stand out then (7.6) with weights de ned by (7.7) performs well.

Non-linear smoothing: headbanging Linear smoothers like the moving mean tend to blur what might be meaningful sudden changes in a surface and smooth out real small-scale features. Where abrupt features are expected then non-linear smoothers are to be preferred (Kafadar, 1999). Median smoothing has been described above but we now turn to some other forms of non-linear smoothing.

Median-based head banging, is a non-linear smoother proposed by Tukey and Tukey (1981) and implemented by Hansen (1991) for spatial data. It can be used on data from a continuous surface or from a region partitioned into subareas. This smoother adapts the concept of a one-dimensional moving median smoother to the spatial situation where left and right are not well de ned.

This smoother has been advocated for geological data because it tends not to oversmooth zones of sharp transition, although it does tend to remove spikes. It also performed well in Kafadar s (1994) evaluations on non-gridded data, but the tendency to smooth out spikes is not necessarily a good property in the context of disease maps. Gelman et al.

(2000) have warned that it tends to produce high rates on the boundaries of study areas. The basic head banging algorithm is described, for example, in Kafadar (1999, p. 3173).

To smooth any data value z(i ) at location i, the head banging smoother starts by identifying J triples of datapoints near to i from amongst its N nearest neighbours. Each triple includes z(i ) plus a pair of z values taken from the set of nearest neighbours. Members of a triple are chosen so that the two locations from the set of nearest neighbours form with the location for i an angle within 45 of 180 .

If there are more than J triples that satisfy the criterion choose those closest to 180 . Now, the high screen is the centre of the highest values in the neighbourhood of i whilst the low screen is the centre of the lowest values. In particular, the high screen is the median across all the J (d).

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