How to do interpolation

Question

I have very little data for my analysis, and so I want to produce more data for analysis through interpolation.

My dataset contain 23 independent attributes and 1 dependent attribute.....how can this done interpolation?

Answer 1

Roughly speaking, to interpolate an array:

double[] data = LoadData();
double requestedIndex = /* set to the index you want - e.g. 1.25 to interpolate between values at data[1] and data[2] */;

int previousIndex = (int)requestedIndex; // in example, would be 1
int nextIndex = previousIndex + 1; // in example, would be 2

double factor = requestedIndex - (double)previousIndex; // in example, would be 0.25

// in example, this would give 75% of data[1] plus 25% of data[2]
double result = (data[previousIndex] * (1.0 - factor)) + (data[nextIndex] * factor);

This is really pseudo-code; it doesn't perform range-checking, assumes your data is in an object or array with an indexer, and so on.

Hope that helps to get you started - any questions please post a comment.

Answer 2

If the 23 independent variables are sampled in a hyper-grid (regularly spaced), then you can choose to partition into hyper-cubes and do linear interpolation of the dependent value from the vertex closest to the origin along the vectors defined from that vertex along the hyper-cube edges away from the origin. In general, for a given partitioning, you project the interpolation point onto each vector, which gives you a new 'coordinate' in that particular space, which can then be used to compute the new value by multiplying each coordinate by the difference of the dependent variable, summing the results, and adding to the dependent value at the local origin. For hyper-cubes, this projection is straightforward (you simply subtract the nearest vertex position closest to the origin.)

If your samples are not uniformly spaced, then the problem is much more challenging, as you would need to choose an appropriate partitioning if you wanted to perform linear interpolation. In principle, Delaunay triangulation generalizes to N dimensions, but it's not easy to do and the resulting geometric objects are a lot harder to understand and interpolate than a simple hyper-cube.

One thing you might consider is if your data set is naturally amenable to projection so that you can reduce the number of dimensions. For instance, if two of your independent variables dominate, you can collapse the problem to 2-dimensions, which is much easier to solve. Another thing you might consider is taking the sampling points and arranging them in a matrix. You can perform an SVD decomposition and look at the singular values. If there are a few dominant singular values, you can use this to perform a projection to the hyper-plane defined by those basis vectors and reduce the dimensions for your interpolation. Basically, if your data is spread in a particular set of dimensions, you can use those dominating dimensions to perform your interpolation, since you don't really have much information in the other dimensions anyway.

I agree with the other commentators, however, that your premise may be off. You generally don't want to interpolate to perform analysis, as you're just choosing to interpolate your data in different ways and the choice of interpolation biases the analysis. It only makes sense if you have a compelling reason to believe that a particular interpolation is physically consistent and you simply need additional points for a particular algorithm.

Answer 3

my main problem is of shortage of data, i hv to increase the size of my dataset, n attributes are categorical for example attribute A may be low, high, meduim, so interpolation is the right approach for it or not????

Answer 4

This is a mathematical problem but there is too little information in the question to properly answer. Depending on distribution of your real data you may try to find a function that it follows. You can also try to interpolate data using artificial neural network but that would be complex. The thing is that to find interpolations you need to analyze data you already have and that defeats the purpose. There is probably more to this problem but not explained. What is the nature of the data? Can you place it in n-dimensional space? What do you expect to get from analysis?

Answer 5

May I suggest Cubic Spline Interpolation http://www.coastrd.com/basic-cubic-spline-interpolation

unless you have a very specific need, this is easy to implement and calculates splines well.

Answer 6

Have a look at the regression methods presented in Elements of statistical learning; most of them may be tested in R. There are plenty of models that can be used: linear regression, local models and so on.