Interpolation method: principal and computational algorithms

A Significant number of mathematical problems associated with the presence of unevenly distributed space of information. We are talking about information systems geographical focus, as they are able to measure the required value at certain points. To solve these problems, we often use one or another interpolation method.

Definition

Interpolation is a method of calculating intermediate values according to the available discrete set of values. The most common interpolation methods are: inverse distance weighted, trend surface and kriging.

Basic interpolation

So, take a closer look at the first method, its essence lies in the impact points closer to the estimate in comparison with further away. When using this method of interpolation involves the selection of some of the topography in a specific area around a specific point having the greatest influence on it. So select the maximum search radius or the number of points close to a certain point. Next, set the weight to the height at each specific point that is calculated depending on the distance from a given point. Only in this way can be achieved a greater contribution of the closest points to the predicted height when compared with the points removed to a greater distance from the target.

The Second method of interpolation is used when the researchers there is interest in the General trends of the surface. Similar to the first method for trend can be used points that are within a given plane. Here build the a lot of best approximation, based on mathematical equations (splines or polynomials). Mainly uses the method of least squares, based on the equations with nonlinear dependencies. The basis of this technique is the substitution of curves and other sequences of numeric type to simple. With the aim of building trend each value on the surface should be substituted into the equation. The result is a single value assigned to the prediction decision (period). All the other points the process continues.

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Another the above method of interpolation, kriging, involves the optimization of the interpolation procedure stemming from the statistical nature of the surface.

Using quadratic interpolation

There is another tool to identify specific points – method of quadratic interpolation, the essence of which is to replace a function in a certain interval quadratic parabola. Its extremum is calculated in an analytical way. After his approximate location (minimum or maximum) you need to set some period values, after which the search of finding a solution continue. Performing this procedure repeatedly, using the iterative procedure, the value of this equation to clarify to the result given in the problem statement accuracy.