The previous article continues, the book is connected last time... The last time, on the basis of improving the global regression, GWR finally came out, and finally the space analysis field finally has its own dedicated regression algorithm. If spatial statistics are different from the two characteristics of classical statistics: spatial correlation and spatial heterogeneity, Moran index can be used to quantify spatial correlation, then geographically weighted regression can be used to quantify spatial heterogeneity. Sex.
In the improvement of the global regression problem, local regression can be said to be the simplest method. GWR continues to apply the idea of ​​local regression, but in the local window mode, it follows the so-called "first law of geography" and returns. At the time, the spatial relationship is used as a weight to be added to the operation. The following is an example to explain the basic idea of ​​GWR.
First look at global regression and local regression:
In the local regression, set a window, and then perform regression calculation in each part according to the set window size. In fact, it seems to be a reduced version of the global regression.
Looking at geo-weighted regression:
Geographical weighting, like other regression analysis, first delineates a study area. Of course, this area can also contain the entire area of ​​the entire study data (in this way, you can use spatial relationships (such as k-near) to perform local geography. Weighted calculations... The next most important thing is to use the different spatial positions of each element to calculate the attenuation function. This is a continuous function. With this attenuation function, when you put the spatial position of each element (usually After the coordinate information (x, y)) and the value of the feature are brought into this function, you can get a weight value, which can be brought into the regression equation.
So you can see that the most important thing is the distance attenuation function. Just because there is a decay function that gives different weights, this method will be called “geographically weighted regression analysisâ€. The theoretical basis of this decay function is exactly what Tobler puts the so-called "Tobler's First Law or Tobler's First Law of Geography": the closer the data is, the greater the impact on the results than the distant data. In mathematics, it is turned into weight.
Using these formulas, all sample points can be calculated point by point. When each sample point is calculated, other samples participating in the calculation will be given different weights according to different spatial relationships with the sample points. The relevant regression coefficients for each of the different samples can be derived. Finally, by interpreting these coefficients, the entire analysis process of the entire geographically weighted regression analysis is completed.
I have been emphasizing this attenuation function, so consider if there is no attenuation? Without attenuation, it is found that all weights are the same (weights are all 1, 1 multiplied by any number, which is equal to itself)... Then the equation becomes a global regression equation. This deviated from the first law of geography, and immediately returned to the classical statistical theory.
Now let's see how this attenuation function is calculated.
The following formula is posted first. Students with math phobia should skip it:
Among them, W (ui, vi) is the space weight matrix, this concept please go back and look at the vernacular space statistics seventeen ... but in view of everyone difficult to turn back, I posted the previous content directly here:
Weight matrix, let's see if this spatial weight matrix is ​​a trick:
The thing on the left, called the undirected graph, is the so-called distance matrix. Because we have said before, in spatial analysis, the concept of spatial relations needs to be conceptualized, so it is also commonly called the spatial weight matrix.
Of course, this weight matrix is ​​simple and straightforward, so the direct use of the elements in the matrix with the shortest distance, such as the distance between B and C, can be directly queried through the matrix to WBC = 2.
After having the weight matrix, bring it into the matrix and get the following equation:
In practical applications, the common spatial weighting functions are mainly the following:
1, Gaussian function:
2, double square function (bi-square)
This method uses the fitted values ​​to perform the calculation, where
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