What does the locality condition mean in interpolation?
The locality condition means that each data point influences the interpolated surface only up to a certain distance. Equivalently, the value at a given point will be similar to the values at nearby points, and distant points do not continue to influence one another. This condition is fulfilled by almost all interpolation methods and is the basis for local interpolation techniques.
Conditions
- Presented as a basic condition fulfilled by almost all interpolation methods.
- Applies to local interpolation methods.
Reasoning, step by step
- Identify the need for additional conditions to resolve the non-uniqueness of interpolation.
- Apply the locality condition: restrict the influence of each data point to a limited distance.
- Ensure that the value at any target point is primarily determined by nearby data points.
- Use this condition to develop local interpolation methods that compute values using only a small subset of the total data points.
Example
The slide lists the locality condition as “each point influences the surface only up to certain distance; or value at a given point will be similar to values at nearby points,” which the lecturer explains as distant points no longer influencing one another.
Common misconceptions
- Assuming that every data point in a dataset equally influences the interpolated value at any arbitrary location.
- Believing that locality is an optional condition that rarely applies to standard interpolation methods.
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