# Table 3 Formal definition of computed features

Preliminary symbols definitions
$$\mathbf{F}$$ is the set including the index of frames which contain one marker of the granule under study
$$\mathbf{x}$$, $$\mathbf{y}$$ and $$\mathbf{z}$$ are vectors containing the coordinates of the marker of the granule under study at each frame $$f \in \mathbf{F}$$
$${\mathbf{p}} _f = ({\mathbf{x}} _{f}, {\mathbf{y}} _{f}, {\mathbf{z}} _{f})$$, where $$f \in [0;|{\mathbf{F}} |-1]$$
Set of displacements: $${\mathbf{D}} = \{ ||{\mathbf{p}} _{f + 1} - {\mathbf{p}} _{f}||, f \in [0;|{\mathbf{F}} | - 1]\}$$
$$pdf(\cdot )$$: probability density function of $$\cdot$$
Set of angles: $${\mathbf{R}} = \left\{ \arccos \left( ({\mathbf{p}} _{f + 1} - {\mathbf{p}} _{f})\cdot ({\mathbf{p}} _{f + 2} - {\mathbf{p}} _{f + 1})\over {||{\mathbf{p}} _{f + 1} - {\mathbf{p}} _{f}||\cdot ||{\mathbf{p}} _{f + 2} - {\mathbf{p}} _{f + 1}||} \right) , f \in [0;|{\mathbf{F}} | - 2] \right\}$$
Set of velocity: $${\mathbf{V}} = \left\{ {\frac{{{\mathbf{P}}_{f} }}{{{\mathbf{F}}(f + {\mathbf{1}}) - {\mathbf{F}}(f)}},f \in [0;|{\mathbf{F}}| - 1]} \right\}$$
Set of acceleration: $${\mathbf{A}} = \left\{ {\frac{{{\mathbf{V}}_{{{\mathbf{F}}_{f} }} }}{{{\mathbf{F}}(f + {\mathbf{1}}) - {\mathbf{F}}(f)}},f \in [0;|{\mathbf{F}}| - 2]} \right\}$$
Features
Route-based  Hourly law-based
Total displacement $$(\Delta ) = \sum _{i = 1}^{|{\mathbf{D}} |} {\mathbf{D}} _i$$ Average velocity $$\left( \bar{{\mathbf{V }}} \right) = {{1}\over {|{\mathbf{V}} |}} \sum _{i = 1}^{|{\mathbf{V}} |} {\mathbf{V}} _i$$
Average angles $$\left( \bar{{\mathbf{R }}} \right) = {{1}\over {|{\mathbf{R}} |}} \sum _{i = 1}^{|{\mathbf{R}} |} {\mathbf{R}} _i$$ Velocity standard deviation $$= \sqrt{{{\sum _{i = 1}^{|{\mathbf{V}} |} ({\mathbf{V}} _i - \bar{{\mathbf{V} }})^2}\over {|{\mathbf{V}} |} - 1}}$$
Angles standard deviation $$= \sqrt{{{\sum _{i = 1}^{|{\mathbf{R}} |} ({\mathbf{R}} _i - \bar{{\mathbf{R} }})^2} \over {|{\mathbf{R}} |} - 1}}$$ Velocity skewness $$= {{{{1}\over {|{\mathbf{V}} |}}\sum _{i = 1}^{|{\mathbf{V}} |}({\mathbf{V}} _i - \bar{{\mathbf{V }}})^3}\over {\left( \sqrt{{{1}\over {|{\mathbf{V}} |}} \sum _{v = 1}^{|{\mathbf{V}} |}({\mathbf{V}} _i - \bar{{\mathbf{V} }})^2}\right) ^3}}$$
Tortuosity $$= {{\Delta }\over {|{\mathbf{p}} _{|{\mathbf{F}} |-1} - {\mathbf{p}} _1|}}$$ Velocity kurtosis $$= {{{{1}\over {|{\mathbf{V}} |}}\sum _{i = 1}^{|{\mathbf{V}} |}({\mathbf{V}} _i - \bar{{\mathbf{V} }})^4}\over {\left( {{1}\over {|{\mathbf{V}} |}}\sum _{i = 1}^{|{\mathbf{V}} |}({\mathbf{V}} _i - \bar{{\mathbf{V} }})^2\right) ^2}}$$
Energy $$= pdf({\mathbf{x}} )^2 + pdf({\mathbf{y}} )^2 + pdf({\mathbf{z}} )^2$$ Average acceleration $$\left( \bar{{\mathbf{A} }} \right) = {{1}\over {|{\mathbf{A}} |}} \sum _{i = 1}^{|{\mathbf{A}} |} {\mathbf{A}} _i$$
Entropy $$= -(pdf({\mathbf{x}} )\log _2{(pdf({\mathbf{x}} ))}~+$$ Acceleration standard deviation $$= \sqrt{{{\sum _{i = 1}^{|{\mathbf{A}} |} ({\mathbf{A}} _i - \bar{{\mathbf{A} }})^2}\over {|{\mathbf{A}} |} - 1}}$$
$$+~pdf({\mathbf{y}} )\log _2{(pdf({\mathbf{y}} ))}~+$$ Acceleration skewness $$= {{{{1}\over {|{\mathbf{A}} |}}\sum _{i = 1}^{|{\mathbf{A}} |}({\mathbf{A}} _i - \bar{{\mathbf{A} }})^3}\over {\left( \sqrt{{{1}\over {|{\mathbf{A}} |}}\sum _{i = 1}^{|{\mathbf{A}} |}({\mathbf{A}} _i - \bar{{\mathbf{A} }})^2}\right) ^3}}$$
$$+~pdf({\mathbf{z}} )\log _2{(pdf({\mathbf{z}} ))})$$ Acceleration kurtosis $$= {{{{1}\over {|{\mathbf{A}} |}}\sum _{i = 1}^{|{\mathbf{A}} |}({\mathbf{A}} _i - \bar{{\mathbf{A} }})^4}\over {\left( {{1}\over {|{\mathbf{A}} |}}\sum _{i = 1}^{|{\mathbf{A}} |}({\mathbf{A}} _i - \bar{{\mathbf{A} }})^2\right) ^2}}$$