Overview
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explicitly makes the assumption that things are close to one another are more alike than those that are farther apart
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to predict a value for any unmeasured location
- IDW uses the measured values surrounding the prediction location
measured values closest to the prediction location have more influence on the predicted value that those farther away
- it is assumed each measured point has a local influence that diminishes with distance
Definition
the expected result is a discrete assignment of the unknown function \(u\) in a study region:
\(u(x): x \rightarrow \mathbb{R}, x \in D \subset \mathbb{R}^n\)
where \(D\) is the study region
The set of \(N\) known data points can be described as a list of tuples
\([(x_1,u_1),(x_2,u_2),...,(x_N,u_N)]\)
the function is to be "smooth" (continuous and once differentiable), to be exact \((u(x_i)=u_i)\) and to meet the user's intuitive expectations about the phenomenom under investigation.
Power function
Weights are proportional to the inverse of the distance
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raised to the power value \(p\)
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as a result, as the distance increases, the weights decrease rapidly
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the rate at which the weights decrease is dependent on the value of \(p\)
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if \(p = 0\) , there is no decrease with distance
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and because each weight \(\gamma_i\) is the same
- the prediction will be the mean of all data values in the search neighborhood
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and because each weight \(\gamma_i\) is the same
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if \(p = 0\) , there is no decrease with distance
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the rate at which the weights decrease is dependent on the value of \(p\)
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as \(p\) increases, the weights for distant points decrease rapidly
- if the \(p\) value is very high, only the immediate surrounding points will influence the prediction
when \(p=2\) the method is known as the inverse distance squared weighted interpolation
Search Neighborhood
Things close to one another are more alike than those that are farther away
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as the locations get far away, the measure values will have little relationship to the value of the prediction location
exclude more distant points that have little influence for speedup
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it is common practice to limit the number of measure values by specifying a search neighborhood
- restricting how far and where to look for the measured values to be used in the prediction
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it is common practice to limit the number of measure values by specifying a search neighborhood
when to use IDW
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a surface calculated using IDW depends on selection of power value \(p\) and the search neighborhood strategy
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the output surface is sensitive to clustering and the presence of outliers
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IDW assumes that the phenomenom being modeled is driven by local variation
- which can be modeled by defining an adequate search neighborhood
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IDW assumes that the phenomenom being modeled is driven by local variation
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since it does not produce prediction standard errors
- justifying its use is problematic