Sequential variational class

[jgomezdans]

Strategy:

1. Start at the first file 1.1. Retrieve the observation information for the first file. 1.2. Use the "inverse GPs" to provide initial estimates of the state (where observations exist). Maybe use downsample to resample to state grid size if the observations are coarse with respect to the state grid. 1.3 Go to next file
2. At this stage, what we have is a first pass estimate of the state for each image location (so gaps in space & time). We use the fast smoother code to infer a complete time series of data.
3. Once the complete (and gap filled) initial state grid is calculated, estimate the smoother parameters for 3.1 Each image (spatial smoothing, variance of the state w.r.t neighbouring pixels over all pixels) 3.2 Over time (temporal "uncertainty inflation", where we calculate the variance of the difference between state at time t and t+1 for each parameter)
4. Start at the first image 4.1 Use the smoothed estimates of the state as your starting point for the minimisation 4.2 Set up observation constraint function for the observation 4.2 Save state & posterior stuff to disk 4.3 If there's another observation on the same time step: 4.3.1 Use the posterior mean & hessian as the prior mean and inverse covariance matrix (no inflation of the hessian) 4.3.2 Set up observation constraint function for the observation 4.3.3 Use the prior mean as the starting point for the minimisation

5. Advance time by one unit 5.1 Inflate hessian $H$ by using the "inflation matrix" $F$. $I$ is the identity matrix, and $C{posterior}$ we don't have access to as it's large. According to some paper, we can calculate only with the Hessian as this (remember that $F$ is purely diagonal, and so sparse) $$ (C{posterior}+ F)^{-1} = H - F[I + HF]H $$ 5.2 Is there an image at this time step? 5.2.1 YES->Use the previous step state mean and the newly inflated covariance as the prior, re-use the state mean as starting point, minimise, save results

   5.2.2 NO -> Save the state as it is, and save the updated inflated covariance

[maxchernet]

Hi Jose:-) Thanks for this so detailed "issue"! On the stage2 when you talking about fast smoother code. Do you mean classes TemporalSmoother and SpatialSmoother? On the stage 5.2 you said that if we don't have an image for this time step, step will be saved as it is. Does it mean that we will have discrete "steps" in the resulted time series?