# Interpolating points observed with different time duration

I am doing spatial interpolation with a group of points.

Some points have complete observation of events:

• for a period of 24 hours we have observed N events in each point. These points have N events.

While others are incomplete:

• for a period of 6 hours we have observed M events in each point. This means that these points have >= M events, since we don't have 24h of observation we just know what is the value for the minimum amount of events.

Although the second group of points have incomplete observations, they are still important for the interpolation. I would like to use them in the spatial interpolation (kriging or co-kriging) with the condition that these are not complete observations.

I would like to interpolate new points and use both complete (24h) and incomplete (6h) points. How can I take into consideration, in a spatial interpolation, points that are incomplete?

How can this be done?

• Please clarify what exactly are you trying to interpolate: something like mean value of observations or values for particular hour or something else? – SS_Rebelious Nov 24 '13 at 11:33
• Can you assume that the rate at which events occur is constant over time? If so, there are some good methods available (such as a Poisson generalized linear model). If not, you are dealing with censored data and they will be much more difficult to analyze effectively (although it can be done, in principle, using maximum likelihood methods). – whuber Nov 25 '13 at 21:48
• @whuber I have no info on the rate, but lets assume it is not constant. Does it mean that there is no spatial interpolation method that allows me to use incomplete data? or the use of certain points were we just tell the method that for this points the minimum value observed is M. – Gago-Silva Nov 26 '13 at 7:58
• If the rate is not constant, you need a much more sophisticated and careful statistical analysis in order to use these data without risk of some unknown bias. (One additional issue of potential importance is to assess whether there is any correlation between the number of events and the occurrence of a six-hour value.) On the other hand, with a constant rate you have right-censored data. There exist statistical methods to make predictions with such data (such as methods of survival analysis) but their application to spatial prediction (interpolation) is not routine. – whuber Nov 26 '13 at 14:38