Muhammad -

Is this perhaps a curve fitting problem where you would need to find a regression line/curve that minimizes the estimated sum of squared residuals, or perhaps the sum of absolute estimated residuals?  Perhaps, from a graphical analysis of estimated residuals, you may find weighted least squares to be more reasonable, and want to minimize the sum of squares of the estimated random factors of the estimated residuals?  Perhaps loess regression would be more reasonable for your situation? 

If you only have a univariate set of data, can you still plot them against time?  I assume that by "interpolation" you mean that there is some kind of sequence such that you are looking for a value between two that you already have. In order to have such a relationship, it sounds like you would have some other auxiliary data from which you could plot some kind of regression. 

Otherwise, it would seem that all you would have would be a missing number, presumably at random, for which you could only impute a mean.  But if you at least have enough other information to group the data into subsets, you could impute means by subset. 

I hope I did not too badly misunderstand your question.

Cheers - Jim 

PS -  You may want to check to see if "interpolation" is the word you meant to use.  You may also want to clarify what you meant by "reference data." 

You could remove one point which you already have in your dataset and try to interpolate its value from the other points. Than compare the interpolated value with the real one.

Then repeat the procedure by removing another point or more of them, depending on your dataset (point density and dislocation).

@James R Knaub  I have some missing data which i have marked as NaN. I interpolated this data set through 'linear, spline, cubic, near etc' interpolation techniques which are 1D interpolation techniques.

By interpolation i means based on known values find the values of missing or NaN values in my data sets.

About Reference data means original (without interpolated) data set through which cross validation i can checked which interpolation technique is best suited for my data set? This reference data set is missing from me and i have interpolated data how can i checked its cross validation and best interpolation technique for my data set?

@Tomislav Jogun please explain in detail your method of cross validations?

Let say we 1d data in hand and want to check its validation after interpolation. I understand you are encouraging me to plot uninterpreted data with interpolated data in order to check best suited interpolation?? Please tell me whether this procedure will be call cross validating or not?

Care must be taken when you use software if you are not certain how it works.  If you are using splines, for example, how can you look at a curve in one dimension? 

I hope the attached pictures will better explain my idea.

First, you have an original dataset with all points A, B, C, D, E, F, G marked as red circles. The blue line is interpolated from all values, what you have done.

Then, remove points C and E from the dataset. The missing points are called validation set and other points are training set.

You should make several interpolations with the training set - here, for example, spline A (green line) and spline B (yellow line). Then compare/validate the interpolated values from the training set with real values of C and E (validation set). As you can see, green line is closer to the real values of removed C and E points than the yellow.

The conclusion is that Spline A is better than Spline B because it has better predicted missing real values C and E.

Of course, you should try the procedure by removing other points.

I do not guarantee this is the real cross validation, but it's the closest thing you could do with your data.

@Tomislav Jogun your idea is quite right. Let i adopt this way. I try to make a plot of uninterpreted data with their corresponding interpolated data and checked which interpolation best fit in my data set.My data set consist on text file (lat,lon and air temperature) , NaN values appear in uninterpreted set randomly.Problem is that there are around 4000 text file if i start to make plots then it will consume too much time.

I have attached two text files. one is without interpolated containing NaN values and other is interpolated through linear interpolation. First column is latitude , second column is longitude and 3rd column is temperature which has interpolated

Here's your dataset's spatial distribution in QGIS software. As you can see, NaN values (purple) and existing values (green) aren't randomly distributed, which is quite a problem.

You should exclude some of green values. Why do you have 4000 files? Are they for the other parts of world or for the different moments in time?

If they're for the other parts of the world, you could append data into one file. For both cases, you could do it as a batch process, but it depends on your software.

@Tomislav Jogun  I have such kind of data set. Mostly of times it missing in sequence and some times it has randomly NaN values. I want to checked which interpolation technique will be best suited for this data set.  Tell me how can i handle interpolation of NaN in this case of data

In validation you must remove all NAN values (purple).

Then you have only true values (green). Divide true values in training and validation set, as I have described earlier. So, among green points choose some and remove their values. They become new NAN, but you actually know their true values. Make interpolation with rest of green points (training set), and compare obtained values with true removed values (validation set).

Then choose the interpolation method which has best predicted missing true values.

Make interpolation on the original whole dataset with the best method.

Dear Tomislav Jogun  ; i have bulk of data sets removing and manually checking will be time consume. I make to check scatter plot of two data sets i shared. I got straight line. But it is broken due to present of NaN values in one file. Please see my mind set. Training data set is my uninterpolated data set which contains NaN values and training data set is the data file which i have prepared after linear interpolation of training file.  

This sentence is confusing: "Training data set is my uninterpolated data set which contains NaN values and training data set is the data file which i have prepared after linear interpolation of training file." What is the validation set?

Which software do you use for interpolation?

Since you have spatial data in 2 dimensions, I recommend a GIS software (QGIS, GRASS, SAGA GIS...).

Range of values in your original dataset is from 285.75 to 305.188. By definition, "interpolation is a method of constructing new data points within the range of a discrete set of known data points." How did you manage to get values exceeding the range of the original dataset? For example, you have the value of 328.976, which is impossible.

Also, linear interpolation is proven as the worst.

You must do a spatial interpolation (bilinear or bicubic spline, Kriging etc.).

Regarding your scatterplot, it just reveals that you have correlated data because it's from the same source. It doesn't mean that you should use linear interpolation.

@Tomislav Jogun  , I have used matlab in order to computed linear,nearest,cubic,spline,next, previous interpolations on my data sets. Can i use programmings in GIS softwares? To me training data set is the orginal data set(containing NaN values) and testing data sets which has interpolated through these techniques i mentioned above. 

I have attached interpolationed files of the same data set?

I hope you now understand my problem. Tell me how can i checked which interpolated technique like, near, previous, next, linear, cubic, spline will be best for my data set.

All attached files are created from the original(containing NaN values) file. I want to select which will be best interpolation for me? But no idea what can i do?

Matlab is a good program, but you must treat your data as 2D, not 1D.

There are functions for Gridded Data Interpolation: http://www.mathworks.com/help/matlab/ref/interp2.html

I have made a simple visual analysis of your interpolations and it seems to me that method "next" yielded the best results.

The problem is that you have too much NAN values and that most of the techniques alter true values in the same way. It's hard to make any comparisons.

I have tried to help by my idea of manual check, but it would be time consuming. The only advice I can give that you should treat your data as 2D. I have used simple assesment methods, like visual analysis or analysis of minimum and maximum values. For example, spline has returned values from -161 to 482, which is impossible.

Whish you a good luck,

Tomislav

What is meant by 2D interpolation? As you can see my data consist on latitude , longitude and temperature value. I often confuse with term " 2D interpolation" . NaN values appear only in  temperature column(3rd in each text file). And i want to interpolate only NaN value but with out changing its latitude and longitude. Please tell me in complete explainations what is meant by 2D interpolation?

Whether i have to interpolate over latitude and longitude also with NaN values?

over this

http://www.mathworks.com/help/matlab/ref/interp2.html

Vq = interp2(X,Y,V,Xq,Yq) exited for 2 D interpolation. Please tell me also with example what will X,Y,V,Xq,Yq in arugements of this function(keeping view of my data set?

Usman

Your data have locations defined in 2D coordinate system (longitude = X, latitude = Y). I have already sent you an image of your data visualized as regular grid in QGIS.

You don't change or interpolate the locations (first two rows in your .txt files). The X and Y are the first arguments.

Every location has some true value or NAN value of temperature (third row). True values are the third argument (V).

You interpolate only NAN values on Xq and Yq locations by taking into account true values in 2 dimensions (X, Y). That's why it's called a 2D interpolation. If you'd have values only along a single line, that would be a 1D interpolation.

The program should return only interpolated values instead of NAN values, non altering their locations or true values.

I haven't worked in Matlab, but you have an extensive documentation.

Thanks..

I think Xq,Yq will be coordinates of NaN values which will be interpolated from V point in above formula. I confirm from documentation of matlab

Whenever trying to solve a problem it is tempting to ask "which is the best method" as though there is a universal notion of "best" but there isn't. So you must first decide your criterion for "best", e.g once choice would be the least amount of computation but for most instances I don't think that is a good choice.

Most algorithms for spatial interpolation depend in some way on spatial distances (e.g. between data locations, between data locations and interpolation locations), in Euclidean space the formula for distance is simple but not so if you are using latitude and longitude. In particular do not use latitude and longitude as though they were Euclidean coordinates this will give very wrong answers. If you are using software such as ArcGIS it includes algorithms to convert latitude and longitude to UPM coordinates but one must still be careful since these are only valid in limited regions. One minute of latitude is one nautical mile at any longitude but one minute of longitude varies substantially depending on the latitude.

Matlab is not really a program, it is more like a programming language although it does have some builtin functions and you need to know and understand how they work. If you use Matlab you are likely re-inventing the wheel and using software that has not been tested. A much better choice would be the gstat package in R which is well documented (the source code is readily available), has good documentation and tutorials. However it is designed for data wherein the data locations have Euclidean coordinates

The term "2D interpolation" is probably not a good one although you will find it many places in the literature. the "2D" refers to the space of data locations. Unless you are considering only data in a limited region with nearly constant elevation you should think in terms of 3D since there is a clear physical relationship between temperature and elevation.

Depending on what you mean by "linear interpolation"  Jogun is right, it is a simple but poor method.

I suggest you begin by looking at the meteorology literature, you will find a lot of papers on interpolating temperature data and you will learn a lot about your problem. Forget abou "training" and "validation" sets, they will tell you almost nothing about how well the interpolation method works. You are having to make a number of unspecified very strong assumptions that are almost certainly wrong, i.e. that the phenomenon that generates the two sets is exactly the same and under the same conditions.