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We have a sequence $\lbrace b \rbrace$ which is defined in a relation $b_i = b_{i-1} * 0.9 + t_i * 0.1$. If we have observations of the full sequence of $\lbrace b \rbrace$, it would be quite simple to calculate the values of $\lbrace t \rbrace$. However, the case is that we only have around $50\%$ to $80\%$ observations in $\lbrace b \rbrace$ and we still want to calculate the values of $\lbrace t \rbrace$ as accurately as possible. Are there any well-known mathematical methods to solve this problem? Thanks!

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What do you mean by "now"? What did we have before? –  Yuval Filmus Mar 1 '12 at 21:15
    
@YuvalFilmus It's more like a misconception...I should edit it... –  derekhh Mar 1 '12 at 21:16
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1 Answer

up vote 3 down vote accepted

The simplest would be to decide that if you know $b_i$ and $b_j$ for $i<j$ but none of the intermediate indices, then $t_{i+1}$ through $t_j$ will be estimated as the same value. It is then simple to telescope the equations and find that common value given $b_i$ and $b_j$.

Whether this is a reasonable model depends on how your data were produced, of course.

In some applications, it may be better to require that $t_i$ through $t_j$ is an arithmetic (or even geometric) sequence. Assuming that you already know $t_i$, you can again telescope the definitions of $b_{i+1}$ to $b_j$ and find how much of a per-step increase in $t$ (or multiplier on $t$) you need for $b_j$ to come out right.

Beware that the latter method carries a risk that if there are long runs where you know only isolated values of $b_i$, the entire sequence of $t$ estimates will be sensitive to the first known $b$, because you're using already-estimated $t_i$s to produce new ones. Thus, you could end up with wildly oscillating $t$s even for $b$s that don't vary much. In contrast, the "equal $t$s" approach don't depend on anything but the two neighboring $b$s and is therefore more stable.

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