# Statistically back-calculating: Markov Chain?

I would like to calculate the value of bacteria on 4 surfaces $i=\{1..4\}$. A person touches some of those 4 surfaces at random and a count is made on their finger after each surface contact ($x_i$).

Someone lost the bacteria count ($x_i$) after each surface but I do know the total count (X) on a persons's finger after they've touched a number of surfaces. I also know which ones and in which order.

What I know:

1. Final bacteria count on a person's finger: $X$
2. Transfer efficiency from surface to finger: $PT_i=\displaystyle \frac{\text{Finger contact area}}{\text{Area of surface}_i}\frac{1}{\gamma_i}$ where $\gamma$ is a surface dependent constant.
3. The number of times the person touched a particular surface: $h_i$.

If I had surface counts $C_i$, the summation of bacteria is linear: ie $\begin{eqnarray} h_1C_1PT_1&=&x_1\\ h_2C_2PT_2&=&x_2\\ \vdots\quad &=& \quad \vdots\\ h_iC_iPT_i&=&x_i \end{eqnarray}$

such that summing over all surfaces $i$ the total count x is: $\displaystyle \sum_i h_iC_iPT_i=\sum_i x_i=X$.

Can I back calculate $C_i$, without $x_i$ even statistically or probabilistically?
Best regards.

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Did each subject touch all 4 surfaces, or did some touch some surfaces more than once? If the latter, simultaneous equations spring to mind. –  Michelle Jan 25 '12 at 19:12
Does the finger contact area in your expression for $PT_i$ vary with each touch? –  jbowman Jan 26 '12 at 1:04
@Michelle The subject(s) don't necessarily touch all four but any surface can be touched a number of times. –  HCAI Jan 26 '12 at 8:12
@jbowman It could do. –  HCAI Jan 26 '12 at 8:13
Do you have a measure of length of touch for each trial? –  Michelle Jan 26 '12 at 9:50

I've only dipped the top of my toe into operations research, so I welcome comments on this suggestion. Basically, you have what I think is an optimization problem (trying to get the most accurate estimates for each surface). I think you have a problem that can be solved through solving as a series of simultaneous equations using either linear or nonlinear programming. This falls squarely into operations research, and is outside my area of expertise.

I don't think you can use a Markov Chain process for this as you don't have transitional probabilities.

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Interesting. Could you advise as to what part of linear programming this might come under? I do actually have the transition probabilities but they're only a guess at best as I didn't observe the process. Would they help do you think? Thanks and best regards –  HCAI Jan 26 '12 at 19:01
On reflection I don't have the transition probabilities between surfaces. Only the probability of touching surface $i$ –  HCAI Jan 26 '12 at 19:19
Sorry, I can't be of more help with the LP/NLP as I have no experience or qualifications in that area. I'm hoping for an OR professional to chime in. –  Michelle Jan 26 '12 at 19:20
Thanks for your input Michelle, it's appreciated. –  HCAI Jan 28 '12 at 8:17