Estimating workers within Industry Sector using commuter data (combining probabilities?) I have some detailed data about the commuting patterns of workers.  If I know the following 3 things...


*

*The total number of workers commuting from one block to another.

*The total # and % of manufacturing-sector workers in the origin block.

*The total # and % of manufacturing-sector workers in the destination block.


... Is it possible to estimate the number of manufacturing-sector workers who commute from the origin block to the destination block?  Data structure is below.  I think it's a probability combination, but I'm too rusty to come up with an elegant solution.  Thanks a bunch for taking a look.  If successful, this analysis will be used to characterize the regional commuting patterns of specific industry clusters.


 A: To answer, you are going to have to make some kind of assumption.  You want to use the information on overall commuting patterns to estimate manufacturing commuting patterns.  One assumption you could make is that manufacturing workers are like other workers.  I'll use that assumption below.
Here is some notation for the problem:
\begin{array}{r r r}
\text{Symbol} & \text{Meaning} & \text{Comment}\\\hline
i & \text{Residence block} & i=1 \ldots N\\
j & \text{Work block}      & j=1 \ldots J\\
w_{i \rightarrow j} & \text{workers flowing $i$ to $j$} & \\
m_{i \rightarrow j} & \text{Mfg workers flowing $i$ to $j$} \\
M_i^r & \text{Total Mfg residence block $i$} & 
                        M_i^r=\sum_{j=1}^Jm_{i \rightarrow j} \\
M_j^w &\text{Total Mfg work block $j$}&
                        M_j^w=\sum_{i=1}^Nm_{i \rightarrow j} \\\hline
p_{i \rightarrow j}& \text{proportion workers $i$ flowing $j$} &
                     p_{i \rightarrow j}=w_{i \rightarrow j} / \sum_jw_{i \rightarrow j}\\
q_{i \rightarrow j}& \text{Mfg proportion workers $i$ flowing $j$} &
                     m_{i \rightarrow j}=q_{i \rightarrow j}M_i^r\\ \hline
\end{array}
The ultimate objective is to get $m_{i \rightarrow j}$ for each $i$ and $j$ using as data $M_i^r$ for each $i$ and $M_j^w$ for each $j$.  This is difficult because there are only $N+J$ pieces of information ($N$ from the $M_i^r$ and $J$ from the $M_j^w$) about the 
manufacturing workers but $NJ$ things to estimate ($m_{i \rightarrow j}$) for the manufacturing workers.  There are going to be an infinite number of ways to choose the  $m_{i \rightarrow j}$ (all $NJ$ of them) so that they satisfy the $N$ constraints $ M_i^r=\sum_{j=1}^Jm_{i \rightarrow j}$ and the $J$ constraints $M_j^w=\sum_{i=1}^Nm_{i \rightarrow j} $.  The idea is to narrow this infinity of solutions down to one by using the information in the $w_{i \rightarrow j}$, which you also observe.
To answer this problem "correctly," I would have to develop a statistical/behavioral model of worker choice of residence and work block, estimate it for overall workers, make some assumption about the similarity of preferences among overall and manufacturing workers, and then use the estimated model, along with the constraints, to estimate the $m_{i \rightarrow j}$.  After puttering with this for a while, I can't see a nice, simple way to do that.  I can see some hideously complicated ways, though.
So, I elect to solve the problem in an ad hoc way.  Let's just find the manufacturing worker flows most similar to the overall flows which respect the constraints.  So, first calculate the proportion of overall workers living in residence from $i$ who flow to work block $j$: $p_{i \rightarrow j}=w_{i \rightarrow j} / \sum_iw_{i \rightarrow j}$.  Now, we are going to solve a quadratic programming problem which finds the proportions of manufacturing workers living in residential block $i$ who flow to work block $j$, $q_{i \rightarrow j}$.  The problem finds them by minimizing the distance between these proportions and the similar proportions for overall workers, subject to $J$ of the constraints:
\begin{align}
min_{q_{i \rightarrow j} \; i=1 \ldots N \, j=1 \ldots J}&\sum_{i,j}(q_{i \rightarrow j}-p_{i \rightarrow j})^2\\
s.t.\;&M_j^w=\sum_{i=1}^Nq_{i \rightarrow j}M_i^w\;j=1 \ldots J
\end{align}
Finally, after finding the $q_{i \rightarrow j}$ from the above problem, we satisfy the other $N$ constraints automatically by the way we go from the $q_{i \rightarrow j}$ back to the $m_{i \rightarrow j}$.  That is $m_{i \rightarrow j}=q_{i \rightarrow j}M_i^r$.
This gives an answer.  The answer is intuitive.  Does it have desirable statistical properties?  To find out, you would have to specify a statistical model and etc as I described above.  Solving the problem above is pretty easy.  It is just a quadratic programming problem for which there are lots of canned, commercial solvers.
