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Table 1 Metrics for measuring dyad joint movement

From: Metrics for describing dyadic movement: a review

Metric Range Parameters fixed ad hoc and Assumptions
\(Prox = K_{\delta }^{+}/T\) [0, 1] i) δ: distance threshold, ii) K : kernel
\(Cs = \frac { D_{chance} - \left (\sum _{t=1}^{T} d_{t}^{A,B} \right)/T}{D_{chance} + \left (\sum _{t=1}^{T} d_{t}^{A,B} \right)/T} \) ]−1,1] Dchance definition
\(HAI = \frac {K_{\delta }^{+}}{K_{\delta }^{+} + (n_{A0} + n_{0B})/2}\) [0,1] i) Reference area, ii) δ: distance threshold
\(L_{ixn}T = \text {logistic}\left (\ln {\left (\frac {n_{AB}/p_{AB} + n_{00}/p_{00}}{n_{A0}/p_{A0} + n_{0B}/p_{0B} }\right)}\right)\) [0, 1] Reference area;
\(jPPA = \frac {S\left \{ \bigcup \limits _{t=1}^{T-1} \left (E_{\phi ^{A}}\left (X_{t}^{A},X_{t+1}^{A}\right) \cap E_{\phi ^{B}}\left (X_{t}^{B},X_{t+1}^{B}\right) \right) \right \}}{S\left \{ \bigcup \limits _{t=1}^{T-1} \left (E_{\phi ^{A}}\left (X_{t}^{A},X_{t+1}^{A}\right) \cup E_{\phi ^{B}}\left (X_{t}^{B},X_{t+1}^{B}\right) \right) \right \}}\) [0, 1] i) Every zone within ellipse has same odd of being transited, ii) ϕ: maximum velocity
\(CSEM = \frac {\max \left \{m; N_{m} >0\right \} }{T-1}\) [0, 1] distance threshold
\(r_{V} = \frac {\sum _{t=1}^{T}\left (V^{A}_{t} - \bar {V}^{A}\right)\left (V^{B}_{t} - \bar {V}^{B}\right)}{\sqrt {\sum _{t=1}^{T}\left (V^{A}_{t} - \bar {V}^{A}\right)^{2}}\sqrt {\sum _{t=1}^{T}\left (V^{B}_{t} - \bar {V}^{B}\right)^{2}}}\) [-1, 1]  
\(DI_{d} = \left (\sum _{t=1}^{T-1} \left [1 - \left (\frac {\mid d_{t,t+1}^{A}-d_{t,t+1}^{B}\mid }{d_{t,t+1}^{A}+d_{t,t+1}^{B}}\right)^{\beta }\right ]\right)/(T-1)\) [0, 1] β: scaling parameter
\(DI_{\theta } = \left (\sum _{t=1}^{T-1} \cos \left (\theta _{t,t+1}^{A} - \theta _{t,t+1}^{B}\right)\right)/(T-1)\) [-1, 1]  
\(DI = \frac {\sum _{t=1}^{T-1} \cos (\theta _{t,t+1}^{A} - \theta _{t,t+1}^{B})\left [1 - \left (\frac {\mid d_{t,t+1}^{A}-d_{t,t+1}^{B}\mid }{d_{t,t+1}^{A}+d_{t,t+1}^{B}}\right)^{\beta }\right ]}{T-1}\) [-1, 1]  
   β: scaling parameter
  1. Note: The formulas assume simultaneous fixes. \(K_{\delta }^{+} = \sum _{t=1}^{T} K_{\delta }\left (X^{A}_{t},X^{B}_{t}\right)\); T is the number of (paired) fixes in the dyad; δ is a distance-related parameter. K is a kernel function. A, B: the two individuals in the dyad; T: number of fixes in the dyad; Dchance is the chance-expected distance between A and B; nAB: number of observed fixes where A and B are simultaneously in the reference area (when a subscript is 0, it represents the absence of the corresponding individual from the reference area); pAB: probability of finding A and B simultaneously in the reference area (same interpretation as for n when a subscript is 0); \(E_{\phi ^{A}}\left (X_{t}^{A},X_{t+1}^{A}\right)\) is the ellipse formed with positions Xt and Xt+1, and maximum velocity ϕ from individual A (analogous for B); S represents the surface of the spatial object between braces; VA (and VB, resp.) represents the analysed motion variable of A (and B); \(\bar {V}^{A}\) (and \(\bar {V}^{B}\)) represent their average; β is a scale parameter; θ, the absolute angle; Nm is the number of m-similar consecutive segments within the series of analysed steps