M and M ++

Author: qgeissmann

math_sup/M&M_bullet_points.docx

@@ -1,4 +1,4 @@
-This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Arch Linux) (preloaded format=pdflatex 2018.3.29)  29 MAR 2018 17:52
+This is pdfTeX, Version 3.14159265-2.6-1.40.18 (TeX Live 2017/Arch Linux) (preloaded format=pdflatex 2018.3.29)  10 APR 2018 17:15
 entering extended mode
  restricted \write18 enabled.
  %&-line parsing enabled.
@@ -100,17 +100,17 @@ LaTeX Font Info:    Checking defaults for OMX/cmex/m/n on input line 3.
 LaTeX Font Info:    ... okay on input line 3.
 LaTeX Font Info:    Checking defaults for U/cmr/m/n on input line 3.
 LaTeX Font Info:    ... okay on input line 3.
-LaTeX Font Info:    Try loading font information for OMS+cmr on input line 21.
+LaTeX Font Info:    Try loading font information for OMS+cmr on input line 18.
  (/usr/share/texmf-dist/tex/latex/base/omscmr.fd
 File: omscmr.fd 2014/09/29 v2.5h Standard LaTeX font definitions
 )
 LaTeX Font Info:    Font shape `OMS/cmr/m/n' in size <10> not available
-(Font)              Font shape `OMS/cmsy/m/n' tried instead on input line 21.
+(Font)              Font shape `OMS/cmsy/m/n' tried instead on input line 18.
 
 [1
 
 {/var/lib/texmf/fonts/map/pdftex/updmap/pdftex.map}]
-Overfull \hbox (25.75497pt too wide) in paragraph at lines 76--77
+Overfull \hbox (25.75497pt too wide) in paragraph at lines 79--80
 []$\OML/cmm/m/it/10 T$ \OT1/cmr/m/n/10 is the set of all tested time in-ter-val
 s: $\OML/cmm/m/it/10 T \OT1/cmr/m/n/10 = \OMS/cmsy/m/n/10 f\OT1/cmr/m/n/10 [0\O
 ML/cmm/m/it/10 ; \OT1/cmr/m/n/10 0\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 25)\OML/cmm
@@ -121,9 +121,10 @@ T1/cmr/m/n/10 24)\OMS/cmsy/m/n/10 g\OML/cmm/m/it/10 h$\OT1/cmr/m/n/10 ,
  []
 
 
-Overfull \hbox (25.73744pt too wide) in paragraph at lines 79--80
+Overfull \hbox (35.73747pt too wide) in paragraph at lines 82--83
 []$\OML/cmm/m/it/10 X$ \OT1/cmr/m/n/10 is a the set of dis-crete be-haviours: $
-\OML/cmm/m/it/10 X \OT1/cmr/m/n/10 = []$. 
+\OML/cmm/m/it/10 X \OT1/cmr/m/n/10 = \OMS/cmsy/m/n/10 f\OML/cmm/m/it/10 quiesce
+nt; micro[]movement; walking\OMS/cmsy/m/n/10 g$\OT1/cmr/m/n/10 . 
  []
 
 [2] (./methods_equation.aux) ) 
@@ -134,7 +135,7 @@ Here is how much of TeX's memory you used:
  4469 multiletter control sequences out of 15000+600000
  5045 words of font info for 19 fonts, out of 8000000 for 9000
  1141 hyphenation exceptions out of 8191
- 27i,16n,21p,228b,201s stack positions out of 5000i,500n,10000p,200000b,80000s
+ 27i,16n,21p,228b,202s stack positions out of 5000i,500n,10000p,200000b,80000s
 </usr/share/texmf-dist/fonts/type1/public/amsfont
 s/cm/cmbx12.pfb></usr/share/texmf-dist/fonts/type1/public/amsfonts/cm/cmex10.pf
 b></usr/share/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi10.pfb></usr/share/
@@ -143,7 +144,7 @@ s/type1/public/amsfonts/cm/cmr10.pfb></usr/share/texmf-dist/fonts/type1/public/
 amsfonts/cm/cmr7.pfb></usr/share/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy
 10.pfb></usr/share/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy7.pfb></usr/sh
 are/texmf-dist/fonts/type1/public/amsfonts/cm/cmti10.pfb>
-Output written on methods_equation.pdf (2 pages, 100789 bytes).
+Output written on methods_equation.pdf (2 pages, 100929 bytes).
 PDF statistics:
  47 PDF objects out of 1000 (max. 8388607)
  33 compressed objects within 1 object stream

math_sup/mat_met_qg_eb.docx

@@ -3,27 +3,6 @@
 \begin{document}
 
 
-\section*{Behavioural state}
-
-\begin{align}
-B = 
-\begin{cases}
-quiescence,            & \text{if } V_{max} < T_V \forall i\\
-micro\text{-}movement, & \text{if } \sum^{i}{d(i, i-1)} < T_d\\
-walking,               & \text{otherwise}
-\end{cases}
-\end{align}
-
-
-
-Where,
-\begin{itemize}
-	\item $V_{max}$ is the maximal velocity,
-	\item $T_V$ the validated threshold under which immobility is scored,
-	\item $d$ is the Euclidean distance,
-	\item $T_d$ is a threshold of 15 mm on the cumulative distance above which walking is scored
-\end{itemize}
-
 
 \section*{Rebound calculation}
 Formally, the homoeostatic rebound $H_i$ of an individual $i$ was expressed as:
@@ -48,6 +27,30 @@ \section*{Rebound calculation}
 \end{align}
 
 
+\section*{Behavioural state}
+
+\begin{align}
+B = 
+\begin{cases}
+quiescence,            & \text{if } V_{max} < T_V \forall i\\
+micro\text{-}movement, & \text{if } \sum^{i}{|X_i - X_{i-1}|} < T_d\\
+walking,               & \text{otherwise}
+\end{cases}
+\end{align}
+
+
+
+Where,
+\begin{itemize}
+	\item $V_{max}$ is the maximal velocity,
+	\item $T_V$ the validated threshold under which immobility is scored,
+	\item $X$ is the position along the tube,
+	\item $T_d$ is a threshold of 15 mm on the distance moved above which $walking$ is scored.
+\end{itemize}
+
+The $T_d$ was defined empirically based on the observation of a bimodal distribution of the total distance moved in a minute.
+
+
 \section*{Relative position}
 
 \begin{align}
@@ -76,7 +79,7 @@ \section*{Hierarchical clustering}
 \item $T$ is the set of all tested time intervals: $T=\{[0, 0.25), [0.25,0.5), ..., [23.75, 24)\} h$,
 \item $BC_t$ is the Bhattacharyya coefficient at a time interval $t$,
 \item $p$ and $q$ are the observed distributions of behaviour for two different individuals, and
-\item $X$ is a the set of discrete behaviours: $X = {quiescent, micro\text{-}movement, walking}$.
+\item $X$ is a the set of discrete behaviours: $X = \{quiescent, micro\text{-}movement, walking\}$.
 \end{itemize}
 
 \end{document}