deletions | additions       diff --git a/Methodology Video Data.tex b/Methodology Video Data.tex  index 3365cde..5c2882e 100644  --- a/Methodology Video Data.tex  +++ b/Methodology Video Data.tex 
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\subsubsection{Derived Data: Velocity \& Acceleration}  Velocity and acceleration measures are derived through differentiation from position and velocity over time respectively. These are 2-dimensional vectors with a magnitude (speed and acceleration) and a heading.The heading of the velocity vector is typically used to determine the orientation of the vehicle.  It should be noted however that each successive derivation increases pixel precession error for that measure. A velocity measure requires twice as many pixels as a position measurement. Similarly, an acceleration measurement requires three times as many pixels as a position measurement. This type of error can be compensated for with moving average smoothing over a short window (e.g. (e.g.\  5 frames). At this time, acceleration measurements are still too noisy to be useful for instantaneous observations. Higher camera resolutions should solve this problem in future applications. \subsubsection{Size of Data}  \label{method-size_of_data}  Feature tracking provides a microscopic level of detail. Individual observations measured at a single site over the course of a normal day typically register in the tens of millions. The sample size (number) of individual tracking measurements (positions, velocities, etc.) per hour $n$ can be estimated with the following  equation \begin{equation} \label{eqn:data-size} n = fQd \end{equation}  where $f$ is the number of frames per second of the video, $Q$ is the average hourly flow-rate, and $d$ is the average dwell time of each vehicle in the scene (excluding full stops). Dwell time is affected by the size of the analysis area in the scene and the average speed. As such, the size of the analysis area needs to be carefully selected. Furthermore, over-representation of objects travelling below the average speed needs to be accounted for in all calculations. One option is to sample agregate  data per object with, for example, a simple mean, or alternatively to re-sample observations by position instead of time. For a series of equally spaced points in a grid, hex map, or along a spline the re-sampled value $m'_j$ at the point $p_j$ is the average \begin{equation} \label{eqn:resampling} m'_j = \frac{\sum_{i=1}^{n}{m_i} }{n} \end{equation}