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Geospatial Data Analysis: A Review of Theory and Methods

3) Geographically Weighted Regression: Spatial nonstationarity is a condition in which a simple ”global” model cannot explain the relationships between some sets of variables. The model must weight over the space to reflect the structure within the data. Geographically weighted regression( GWR) is a technique which attempts to capture this variation by calibrating a multiple regression model which allows different relationships to exist at different points in space. This technique is loosely based on kernel regression. GWR is a simple technique that extends the traditional regression framework by allowing local variations in rates of change so that the coefficients in the model rather than being global estimates are specific to a location i. The regression equation is then given as:

where $a _i._k$ is the value of the kth parameter at location i. For this model, it seems intuitively appealing to base estimates of aik on observations close to i. In GWR weighting an observation in accordance with its proximity to i would allow an estimation of $a _i._k$ to be made that meets the criterion of ”closeness of calibration points”. In kernel regression, y is modelled as a nonlinear function of x by weighted regression, with weights for the ith observation depending on the proxmity of x and $x _i$ for each i with the estimator being,

The typical output from the above equation will be set of parameter estimates that can be mapped in geographic space to represent the nonstationarity or parameter ”drift”. The choice of weighting scheme is based on the proximity of i to the sampling locations around i. One of the schemes is to specify $w _i._j$ as a continuous function of $d _i._j$ , the distance between i and j. One of the weighting based on distance can be defined as

so that if i is a point in space at which data are observed, the weighting of that data will be unity and the weighting of other data will decrease according to a Gaussian curve as the distance between i and j increases. One another weighting scheme having the computationally desirable property of excluding all data pints greater than some distance from i and also the analytically desirable property of continuity as in bisquare function defined by

This excludes points outside radius d but tapers the weighting of points inside the radius, so that $w _i._j$ is a continuous and once differentiable function for all points less than d units from i.

In the weighting schemes so far, the weighting function once calibrated is assumed to be constant throughout the study area. However there can be places where this is not a reasonable assumption. For example, in economic applications, pricing structures may be dependent on local markets, but the notion of locality may vary regionally i.e, the geographical extent of a London market may be broader than that for Newcastle.
In these cases, a more reasonable approach to GWR might be to have a spatially variable weighting function so that $\beta i$ is computed instead of $\beta$ . Although this is computationally complex, the results will be informative, not only of the nature of relationships between attributes but also the nature of how locations interact with each other.

4) Fractal models: A fractal cab be described as an entity that possesses self-similarity on all scales and non-integer dimension. It is important that a fractal needs to only exhibit similar(but not exactly the same) type of structure at all scales, Moreover according to Madelbrot [24]: A fractal set is one for which the dimension strictly exceeds the topological dimension. This means that while a line feature has a dimension of 1 in classical geometry, it must have a dimension larger than 1 if it is to have fractal properties.

The rough description of a fractal object is the exponent in the expression of the form as shown in the below equation [25] [26]:

in which r is the radius, N is the number quantifying the object x under consideration at the radius r, a is a constant, and D is the fractal dimension.

The self-similarity as stated by Mandelbrot is that each part is a reduced-size copy of the whole, i.e, the spread of any component of the system at a given point is proportional to its distance from that center. By calculating the quantity of any given component of the system as a function of the distance from the center, it should be possible to verify its fractal properties and extract its fractal dimension [27].

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