Lag distance is the average of the distance between pair of samples at a given (average) direction among your samples...so yes, you can. Try different lag distances to improve your experimental variograms. good luck!

Dear Jhalendra, of course you can use lags higher than the sampling distance. The reason why you observe an R2 better if you increase the lag is that you decrease the points in the plot and the model will fit better. My advice is do not exagerate with the increase of lags. From 5 to 6 you don't loose much of information and probably you can see better the autocorrelation range between samples.

cheers

Hey Jhalendra. In addition, you should take a look at the number of pairs your get when defining the lag distances, as they relate to how reliable the variogram is for a particular lag distance. If you want to do kriging, in general you want to specify lag distances that give your a smooth empirical variogram, and sufficient number of pairs of points for each lag. This is specially important for smaller lag distances, as they are frequently the most important ones in guiding in the choice of the model's functional form in variogram analysis.

Good luck.

Hi Jhalendra,

I think you are not choosing appropriate model for fitting, which one are you using? Spherical or Exponential? Please recheck your model fitting by evaluating different model based on sum of squared error. Because it's never possible that you increase uncertainty in variogram modelling by decreasing minimum lag. Since 1970s it has been recommended to increase the number of spatial points (minimum 100) with the objective of decreasing lag distance in variogram modelling at the same time for increasing the number of paired comparison. In the gstat package of R, variogram fitting is quite trivial and you can easily control the lag distance with the 'width' parameter.

Hope that helps.

There are two components of distance in the variogram: distance threshold and distance interval. There is no rule of the thumb to decide them, though the default R takes the 10% of the range of distance across all sampled points. You need to begin to explore, and choose the one that fits well with your theory. For example, in the air pollution work that I do, I find about 6 mile as the great threshold for PM10 (particulate < 10 micron), because beyond this distance there is no autocorrelation. I found this based on the trail and error, and also the variogram became flat at this distance, this also fitted well with my theory, given the coarse size of the particle they settle with gravity faster than the gaseous pollutant.

Also, with regard to the interval (i.e. lag separation) you want to make sure that neither you have too dense data points or too sparse data points within a given lag, certainly more than 3 points within each lag.

Yes you can. Howevr, watch the descriptive statistics and sample pairs ib your bins. No less than 32 pairs is allowed. This could guarantee the robustness of your variogram model. Increasing lag is possible, but your sill and nugget will differ from what you expect. In this case judge about the physical nature of your experimental field and results. Since you definitely expect a level of certainty of your estimations. As long as you get better rss and r2 (both together) this will guarantee a definitely very acceptable reduced variances. Don't worry go ahead.

All of the preceding comments and suggestions are useful but to summarize, (1) it is the behavior of the sample variogram at short distances that is the most important but you can't use a lag distance that is shorter than the minimal distance between two data locations, i.e. there are no pairs of locations with a shorter lag distance. (2) The total number of pairs is fixed by the total number of data locations so the only choice is how to apportion the pairs into distance classes. While there is no theoretical reason to require all the distance classes to have the same width, it would be much more difficult to program if non equal widths were allowed. As you decrease the width of the distance class the number of pairs per distance class will decrease and the number of plotted points will increase. Conversely if the distance class widths are increased the number of pairs per distance class will decrease hence you want to look for a medium choice. Fewer pairs per distance class can make the plot of the sample variogram look more jagged, conversely more pairs per distance class means you will have less information about the shape of the variogram for short distances and less information about the shape over all. (3) Ideally you want a large number of data locations but generally this increases the cost of collecting the data and often one has data that has already been collected with no option to collect more. There is no fixed minimal number to ensure a good sample variogram that will work in every application. (4) Don't over look the possibility of a geometric anistropy, you need to compute and examine directional sample variograms but again the total number of pairs of data locations is fixed and then is split up among the different directional sample variograms. Sometimes the directional sample variograms simply are not very good and you end up using an isotropic model for lack of better information. Also as a rule of thumb don't rely on lag distances that exceed half the diagonal distance across the region of interest because at about that distance the number of pairs begins to decrease sometimes rather fast, i..e the reliability of the plotted points decreases (because the number of pairs decreases)

Bottom line: (a) try different widths for the distance classes to see how much effect there is, (b) consider using a weighted least squares fitting but note that there is no absolute best choice for the weighting so try different possibilities, (c) there are at least six different diagnostics you can get from cross-validation so examine those and see how they change as you change the model type and/or the variogram model parameters.

Footnote: "semivariogram" is outdated terminology since the geostatistics conference in Avignon in 1988

Thank you very much for your response with regard to the question I posted here. Your answers are very helpful!

The sample variogram is not unique for a given data set because the user must make decisions about (1) the lag spacing , (2) the maximum number of lags, to say nothing of whether directional sample variograms are computed to check for possible anisotropy.. If you choose a lag spacing that is smaller than the shortest distance between data locations then there will be lags where there aren't any pairs and hence no computed value for the sample variogram. If you use lag spacings that are much greater then you will increase the number of pairs per distance class but with fewer plotted points. Not also that if you use a lag distance that is greater than about half the maximum distance between data locations then the number of pairs per plotted point begins to decrease, i.e. the reliability of the estimated values of the variogram will decrease. For a given data set the total number of pairs is fixed hence you can only decide how to apportion them for the different distance classes. While you will find various numbers quoted as being the minimum number of pairs for a sample variogram point, there is no number that will always work. Remember while it is common to refer to "fitting a sample variogram", the variogram is a function but the sample variogram is not. It is simply a collection of estimates of variogram values for specific lag distances. The validity of the R2 value is based on assumptions that are not satisifed and while it is probably helpful it is not a real statistic for variogram fitting. You will also want to look at the cross validation statistics, there are at least six and they may not tell you the same thing.