Hi Aniket,

   I always used bilinear interpolation for my work and found good result. But again I guess this would depend also on which variables and distribution that you are working on. A sensible way to deal with this is to carry out a testing with your datasets and figured out which methods work best. 

Hi Aniket,

I'm not sure I understand what you mean exactly by "observed" and "modeled" data - I'm guessing you mean something like "raw" vs. "numbers after some calculation has been made".

As I understand it, however, neither of the above two types of data are what really drive this question; rather it's a matter of the scale of measurement used in the data (nominal, ordinal, invterval, ratio), and whether or not the phenomena reflected in the data is continuous or discrete.

Nearest neighbor interpolation is generally a bad choice for a spatially continuous variable (e.g., elevation), and only really makes sense for nominal data (e.g., classified, where the original ratio data the classification is driven by [if there was any] isn't available).

Have a look at these discussions:

http://www.quantdec.com/SYSEN597/GTKAV/section9/map_algebra.htm

and

http://webapps.fundp.ac.be/geotp/SIG/interpolating.pdf

Cheers,

P

Hi Aniket,

Just this about 1), 2), and 3).

1) Interpolation.

In the mathematical field of numerical analysis, interpolation is a method of constructing new data points within the range of a discrete set of known data points. In engineering and natural sciences, one often has a number of data points, obtained by sampling or experimentation, which represent the values of a function for a limited number of values of the independent variable. It is often required to interpolate (i.e. estimate) the value of that function for an intermediate value of the independent variable. This may be achieved by curve fitting, regression analysis or when in more dimensions than two, by

In statistics, resampling is any of a variety of methods for doing one of the following:

Estimating the precision of sample statistics (medians, variances, percentiles) by using subsets of available data (jackknifing) or drawing randomly with replacement from a set of data points (bootstrapping)

Exchanging labels on data points when performing significance tests (permutation tests, also called exact tests, randomization tests, or re-randomization tests)

Validating models by using random subsets (bootstrapping, cross validation)

Common resampling techniques include bootstrapping, jackknifing and permutation tests.

2) Resampling

In statistics, resampling is any of a variety of methods for doing one of the following:

- Estimating the precision of sample statistics (medians, variances, percentiles) by

  using subsets of available data (jackknifing) or drawing randomly with replacement

  from a set of data points (bootstrapping).

- Exchanging labels on data points when performing significance tests (permutation

  tests, also called exact tests, randomization tests, or re-randomization tests)

 - Validating models by using random subsets (bootstrapping, cross validation).

- Giving new values to data points in Remote Sensing or GIS is performed when a dataset of raster or vector data is reprojected  in ea new projection georefereced coordinate system. In that case resampling is required to attribute new values to the reprojected pixel values.

Hence, do not mix up  mix up interpolation with resampling. They are two totally different statistical techniques used in a totally different context and with totally different objectives. So you will first have to answer the question whether you want to resample or interpolate to solve your problem!

3) Spatial interpolation

By the way  do not forget that spatial interpolation is probably what you are looking for. In numerical analysis, multivariate interpolation or spatial interpolation is interpolation on functions of more than one variable. The function to be interpolated is known at given points (xi, yi, zi) and the interpolation problem consist of yielding values at arbitrary points (x, y, z).

Code can be found here:

- Example C++ code for several 1D, 2D and 3D spline interpolations (including Catmull-Rom splines).

- Multi-dimensional Hermite Interpolation and Approximation from Prof. Chandrajit Bajaja, Purdue University

And last but not  least a very good spatial interpolation routine can be found at: http://surgeweb.sweb.cz/

The program SurGe is intended for the approximation (interpolation) of points  XYZ  into nodes of a square grid. It contains a method named ABOS (Approximation Based On Smoothing). ABOS is a method for approximating (interpolating) z-coordinates of irregularly spaced points (usually measured in the terrain) by using a continuous function with two independent variables. The interpolation function is defined by a matrix of real numbers P, which represents the surface at nodes of a square grid. The domain of the interpolation function is defined by the outer lines of the grid:

The value of the surface at any point (x0,y0) within the grid can be evaluated from the equation of the bilinear polynomial f(x,y) = axy + bx + cy + d, which is defined by corner points of the grid square containing the point (x0,y0). This algorithm covers ground which many geospatial scientists want to cover.

So Aniket, I figger you still have some thinking to do. But good luck, lots of perseverance and may the Forces of Nature be with you.

Frank

In the case of data with a clear tendency to be grouped into sub-areas, nearest-neighbor renders best performance than bi-linear (i. e. when modeling land surface, with the nearest neighbor the small hills and promontories tend to be better reproduced). If you are looking for some kind of distribution such like that, then you might want to use natural neighbor. On the other hand, if you have a coarser data grid and you want to see how data might be distributed over your entire limits, the krigging or bi-linear is much better, although some expected features in local areas might be lost.

 Thanks you all

I guess I was unable to put my question forward properly, so let me explain a little:

In some applications, like uncertainty quantification of satellite observed data like soil moisture, its advised not to use bi-linear interpolation for remapping the observed data as it imposes weights to the data points or averages the observed data points which can change the distribution of error in the new remapped data, so it is advisable to use nearest neighbor.

But I was trying to remap a reanalysis product, and as the reanalysis product is LIKE a modeled data product where the output is from a continuous function. So I was a bit confused whether it would be a good idea to use bi-linear interpolation rather than nearest neighbor. And from the above discussion I think bi-linear should be the preferred remapping technique for remapping a modeled data.

@Frank: For explaining the difference between resampling and spatial interpolation. 

To what end are you doing the remapping? Would-bi-linear interpolation add another layer of smoothing? How different would the results of the two techniques be? How much of a difference can your research goals tolerate?

Hello Jack, 

Apparently there is not much difference between the results of the two methods when applied to the modeled data.

I was just being cautious, as i am dealing with uncertainty in the observed data sets. but if you are using modeled data set, then it doesn't matter that much.

Nearest neighbor interpolation is an ad-hoc, empirical method, i.e. it is not based on any theory or any theoretical assumptions. It is just based on the assumption that values at two locations close together are more "similar" than values for two locations far apart. It is highly sensitive to the spatial pattern of the data locations. It has two somewhat dubious advantages; since it not based on any theory you don't have to check whether any assumptions are satisfied, secondly you don't have to do any computations, essentially you can do it with a map of the data locations and the interpolation locations and a scheme to find the "nearest neighbor" for any interpolation location. Note that "simialr" is not a precisely determined term.

Inverse distance weighting is a somewhat more sophisticated version of nearest neighbor in that it uses the data values at several locations to determine the interpolated value, i.e. a weighted average. Again IDW is not based on any theory and can be highly sensitive to the spatial pattern of the data locations

For a simple explanation of bi-linear interpolation see

https://en.wikipedia.org/wiki/Bilinear_interpolation

As you will see it is not based on any theory but it  is based on the assumption that the data locations are the vertices of a regular grid

None of these provide any measure of accuracy or errors

You need to look at either the Radial Basis function literature or the geostatistics literature

1994, Myers,D.E., Statistical Methods for Interpolation of Spatial Data Inter. J. Applied Science and Computations 1, 283- 318

1991, Myers,D.E.Interpolation and Estimation with Spatially Located Data Chemometrics and Intelligent Laboratory Systems 11, 209-228

1994, Myers,D.E., Spatial Interpolation: An Overview Geoderma 62, 17-28

A critical issue you need to consider, is your data "point data" or does it represent a spatial average, e.g. over a pixel or does it represent a count over a subregion (quadrat or pixel)? Most non-geostatistical methods do not account for the shape/geographical size of the sub-region.