PQStat - Baza Wiedzy

Spis treści

Spatial smoothing
- Local smoothing

Spatial smoothing

The idea of spatial smoothing is obtaining a better (more stable and less noisy) value of the variable. The most common methods of building such a variable are based on borrowing the information from neighboring areas or on using a larger amount of information from the studied region (L.A. Waller 2004 ¹⁾, Luc Anslin 2006 ²⁾).

As a result, the values of the tested variable $X$ with elements $x_1,x_2,...,x_n$ will be transformed into a new, smoothed variable $smooth(X)$ with elements $smooth(x_1),smooth(x_2),...,smooth(x_n)$

Local smoothing

The researcher can control the analysis by selecting the distance/neighborhood matrix of the objects, setting the eigenpotential for the object to be smoothed, and indicating the smoothing method.

Spatial weighting matrix

Information about the neighborhood of objects and their mutual distances is defined in the spatial weights matrix. If a neighborhood matrix is used for smoothing - carrying only information information about neighbors (1) or not (0), then only the objects neighbouring with the tested one will have an influence on the obtained result, and the size of this influence will be the same for all neighbors. When a researcher wants to gradate the size of this influence, he should choose a matrix with arbitrary positive values. At the same time, it is important to remember that a larger value in the weight matrix gives a greater influence on the smoothing result. Therefore, in order for closer objects to have a greater influence on the obtained result than distant objects, they should have a higher weight in the matrix. Such an effect can be achieved by using, for example, an inverse Euclidean distance matrix inside a circle of radius $d$ . Then, closer objects will have a greater influence on the resulting score than distant ones, and the influence of objects outside the circle will be zero. For more extensive methods of constructing weight matrices, have a look at patial weight matrix and Similarity matrix.

Eigenpotential

The eigenpotential $p$ of the smoothed object determines the amount of influence of information about the test object on the smoothed value for that object.

Eigenpotential value

The eigenpotential value sets the size of the elements placed on the main diagonal of the weight matrix. By default, the eigenpotential value is set to 1. If it is set to zero ( $p=0$ ), the smoothed value of the tested object is calculated based only on the information contained in neighboring objects. On the other hand, increasing the value of the eigenpotential increases its share in the calculation of the smoothed value for that object.

Potential value correction

The setting of the eigenpotential value alone determines the influence of the tested object on the obtained result, but it does not define by how much this influence is to be greater/lesser than that of the neighbouring objects (elements off the main diagonal of the weight matrix). The dependence of the value on the main diagonal of the matrix both on the given value of the potential and on the values of other elements of the matrix allows to determine the size of the influence of the tested object in relation to neighbouring objects. Correction of the potential value is given by the formula:

$\begin{displaymath} w_{ii}=p\cdot\sum_{j=1,j\neq i}^n w_{ij} \end{displaymath}$

As a result, selecting the potential value correction option and setting the potential value to magnitude 3, for example, ensures that the effect of information about the test object on the smoothed value for that object will be three times that of its neighboring objects.

Methods

Locally weighted average

This transformation consists in calculating the arithmetic mean of the values of the variable $X$ for the object under study (according to the potential) and its neighboring objects (according to the given weight matrix). The observed value $x_i$ is transformed to a smoothed value $smooth(x_i)$ according to the formula:

$\begin{displaymath} smooth(x_i)=\frac{\sum_{j=1}^n w_{ij}x_j}{\sum_{j=1}^n w_{ij}} \end{displaymath}$

where:

$n$ –- number of spatial objects (number of points or polygons),

$x_j$ –- are the values of the variable for the objects being compared,

$w_{ij}$ –- elements of the spatial weight matrix.

Locally weighted median

This transformation consists in calculating the median of the values of the variable $X$ for the object under study (according to the potential) and its neighboring objects (according to a given matrix of weights). In order to determine it a neighborhood matrix is necessary, where weights are binary values. The value of one in the matrix means neighboring objects and zero means no neighboring.

Locally weighted average (corrected)

In the process of smoothing coefficients built on the basis of dividing two variables, determination of locally weighted average can be improved. The numerator and the denominator are smoothed and only then the quotient is created on the basis of these smoothed values. In this way, you can, for example, smooth the incidence rates determined in the course of epidemiological studies, where the numerator is the number of patients and the denominator is the size of the exposed population. In effect, objects with a larger population, will have a greater impact on the result of smoothing - therefore the denominator of the smoothed coefficient is called the adjustment variable.

The observed value of the $\frac{x_i}{y_i}$ coefficient is converted to a smoothed value $smooth\left(\frac{x_i}{y_i}\right)$ according to the formula:

$\begin{displaymath} smooth\left(\frac{x_i}{y_i}\right)=\frac{\sum_{j=1}^n w_{ij}x_j}{\sum_{j=1}^n w_{ij}y_j} \end{displaymath}$

where:

$n$ –- number of spatial objects (number of points or polygons),

$w_{ij}$ –- elements of the spatial weight matrix.

Empirical Local Bayes Smoothing (corrected)

The method of local Bayes smoothing was developed as one way to deal with the instability of the coefficients associated with small data counts and was described in detail by Waller (2004 ³⁾). Smoothing aims to improve the locally weighted mean (adjusted) so as to reduce its variance.

The observed value of the $\frac{x_i}{y_i}$ coefficient is converted to a smoothed value $smooth\left(\frac{x_i}{y_i}\right)$ according to the formula:

$\begin{displaymath} smooth\left(\frac{x_i}{y_i}\right)_{Bayes}=smooth\left(\frac{x_i}{y_i}\right)+C_i\left(\frac{x_i}{y_i}-smooth\left(\frac{x_i}{y_i}\right)\right) \end{displaymath}$

where:

$smooth\left(\frac{x_i}{y_i}\right)$ – locally weighted average (adjusted),

$C_i$ –- shrink factor,

$C_i=\frac{s^2-\frac{x_i/y_i}{\bar{y}_i}}{s^2-\frac{x_i/y_i}{\bar{y}_i}+\frac{x_i/y_i}{y_i}}$ jeśli $s^2-\frac{x_i/y_i}{\bar{y}_i}>0$

$s_i^2=\frac{\sum_{j=1}^ne_{ij}}{\sum_{j=1}^ny_jw_{ij}}$

$e_{ij}=y_i\left(\frac{x_i}{y_i}w_{ij}-smooth\left(\frac{x_i}{y_i}\right)\right)$

$\bar{y}_i=\frac{\sum_{i=1}^ny_i}{n}$ –- is the average population size,

$w_{ij}$ –- elements of the spatial weight matrix.

The shrinkage factor balances the local average $smooth(x_i/y_i)$ with the observed value of the coefficient $x_i/y_i$ . When the population size of the adjustment variable $y_i$ is small, then $C_i\to 0$ nd the estimated value is close to the locally weighted adjusted mean $smooth(x_i/y_i)$ . When the population size is large, then $C_i\to 1$ and the estimated value approaches the true value observed at that site $x_i/y_i$ .