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manuscript/20210908_Adriaens_UWB_.tex

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@@ -69,9 +70,27 @@
 
 \subsection{Ultra-wide band data}
 \subsubsection{Data editing}
+
+
 \subsubsection{Data selection}	
 
 \subsection{Changepoint analysis}
+Changepoints are time instants or samples in which the statistical properties (i.e. distribution) of a (time) series change abruptly. In this study, we detected and combined the individual changepoints per cow per day in two time series of (x,y,z)-coordinate positioning data. Intuitively, one could argue to mainly rely on the position in the vertical (z) direction (height), as a cow that lies down is expected to remain in a lower and more stable position compared to when she is up. However, the z-position has been shown to be the most unreliable and noisy (range, variability,..) of all three coordinates, with its inaccuracy variable in time and space, and depending on e.g. the position in the barn, the behaviour and speed of the animals, the collar attachment and their interactions, etc. Similarly, relying on detection of a stable position in the (x,y)-direction (which is unmistakably true during lying bouts) is imprecise as well, as cow activity varies over the day, and oftentimes animals stand still for a longer period of time apart from their lying bouts, for example when grooming other animals, feeding, drinking or ruminating. These periods of “standing” inactivity might additionally depend on accessibility of cubicles and lying places, hierarchy, climate of the barn etc. In this study, we chose to work on a combination of two position derived time series. The first is the z-coordinate (height) of the animals, as this is the most straightforward one. The second time series is the ‘centre distance’ (CD), i.e. the position relative to the centre of the barn. The main advantage of using CD and not the raw (x,y)-position is that it summarizes position and movement of the animals in a single signal, less dependent on the actual direction of movement, and with a lower variability. Should a cow move in a perfect circle around the centre of the barn, however, CD remains constant (as is the case when a cow stands still or lies down). We assumed that this would be extremely rare, and when it would happen for a short period of time, this would not impair the analysis because movement as such causes the signal to be more variable, which also changes the statistical properties of the time series.
+The changepoint analysis relies on a parametric method that separates the time series in segments and computes either the level (average value, z-coordinate), or level and variability (mean and variance) of each segment from point \textit{m} to point \textit{n} as follows:
+
+\[\dfrac{1}{n-m+1}\sum_{r=m}^{n}x_{r}\]
+
+\[\dfrac{1}{n-m+1}\sum_{r=m}^{n}{(x_{r}-mean([x_{m}...x_{n}])^2}\]
+
+
+
+
+
+Based on the minimization of the sums of squares of the residual error (i.e. the maximum log likelihood), it finds the number of changepoints \textit{k} for which:
+
+
+
+Is smallest.
 
 \subsection{Statistical analysis}