deletions | additions       diff --git a/subsection_minimising_the_Jensen_Shannon__.tex b/subsection_minimising_the_Jensen_Shannon__.tex  index 8e21c10..e568ea7 100644  --- a/subsection_minimising_the_Jensen_Shannon__.tex  +++ b/subsection_minimising_the_Jensen_Shannon__.tex 
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\subsection{minimising the Jensen-Shannon distance between $dP$ $\phi(dP)$  and ($dP_+ $\phi(dP_+  + dP_-$) dP_-)$  and $\rho $\phi(\rho  c dU$ dU)$  and ($dP_+ $\phi(dP_+  - dP_-$)} dP_-)$}  The separation of the measured $dP$ and $dU$ can be done for any arbitrary value of $c$ usng using  the water hammer equations and the assumption that the forward and backward waves are additive. Thus for any value of $c$ (assuming that $\rho$ is a constant) we obtain distributions of $dP_+$ and $dP_-$. For noiseless measurements and the 'true' value of $c$, the distribution of the measured $dP$ should equal the sum of the distributions distribution  of $dP_+$ and $dP_-$ $(dP_+ + dP_-)$  and the distribution of $\rho c dU$ should equal the sum distribution of $(dP_+ - dP_-)$. For noisy data and an arbitrary value of $c$ we cannot expect the distribution to be equal, instead we look for the value  of $c$ for which  the distributions of $dP_+$ and $-dP_-$. are most similar.  That is, we minimise the sum of the two Jensen-Shannon distances \[  \Delta = \Delta_{JS}\big(dP;(dP_+ + dP_-)\big) + \Delta_{JS}\big(\rho cdU;(dP_+ - dP_-)\big)  \]  The Jensen-Shannon distance is zero only when the two distributions are identical and so we expect $\Delta > 0$ For noisy measurements. We argue, however, that the errors arising from the use of the incorrect wave speed $c$ will contribute to $\Delta$ so that it will be minimum when we use the true value of $c$.  This argument follows from looking at what happens as our estimate of $c$ deviates from its 'true' value. Consider first $c$ smaller than the true value. In this case, the calculated values of $dP_\pm$ will be influenced more by $dP$ and less by $dU$. In the limit $c = 0$, $dP_+ = dP_- = \frac{1}{2}dP$ and the separated pressure changes depend only on $dP$. As the separated pressure changes become more like $dP$, the Jensen-Shannon distance between $dP$ and the sum of the separated pressures and will decrease, being zero in the limit $c=0$. The distance between $\rho c dU$ and the difference of the separated pressures, however, will increase until it is equal to $\frac{1}{2}H(dU/\rho c)$ in the limit $c=0$.  The converse happens for $c$ larger than the true value. $dP_\pm$ is more influenced by $dU$ and less by $dP$. In the limit $c \rightarrow \infty$, $dP_+ = - dP_- = dU/\rho c$ and the separated pressure changes depend only on $dU$, being zero in the limit $c \rightarrow \infty$. The distance between the difference in the separated pressure changes and $dP$, however, will increase until it is equal to $\frac{1}{2}H(dP)$ in the limit $c \rightarrow \infty$.  In order to utilise this procedure we must estimate the probability density functions $\phi(dP)$, $\phi(\rho c dU)$, $\phi(dP_+)$ and $\phi(dP_-)$ where $dP_\pm$ are calculated for the assumed value of $c$ as described above. Given $\phi(dP_\pm)$, the probability density functions for their sum and difference are  \[  \phi(dP_+ + dP_-) = \frac{1}{2}(\phi(dP_+) + \phi(dP_-)) \phi(dP_+) \convolution \phi(dP_-)  \]  \[  \phi(dP_+ - dP_-) = \frac{1}{2}(\phi(dP_+) + \phi(-dP_-)) \phi(dP_+) \convolution \phi(-dP_-)  \]  The problem of estimating the probability density function (pdf) from a sample distribution has received considerable attention over the years. The simplest approach, generally called the maximum likelyhood estimator, is to estimate the pdf from the normalised histogram of the sampled distribution. For example, given the distribution of $dP$