The residual in estimation is the difference between the measurement and the previous estimate. My question is when I plot this quantity, what is ,if any, information I can extract or infer from this quantity? No beneficial information I can see from the figure. Any suggestions?
If your estimate is based on a linear time invariant model then you could use the following measures for the quality of your estimate: cross-correlation between the estimate and residual, and the autocorrelation of the residual. Namely the cross-correlation should be close to zero at all time, otherwise there would still be some residual dynamics that could be modeled. The autocorrelation should be close to zero at all time except at zero, where it should be close to one. If this is the case then this would mean that the residual would be white noise, which can't be predicted/modeled.
Your residual tells you if your model ,is a good fit to the data. It should meet the orthogonality criteria as described in @fiboranic’s answer if your model is linear and noise is Gaussian.
In Kalman Filter Tracking, the Interacting Multiple Model Technique (IMM) the residuals are the innovations sequence and they are tested as the filters run.
In Statistics, most packages like SAS will test an ANOVA set of residuals for Normality (Gaussian. ality) and independence for goodness of fit. The same is true for Linear Regression.
Even in some nonlinear problems, like an EKF, the innovations are nearly as diagnostic as for the KF.