Table 1 Description of notation used in this paper and their interval, if applicable
From: Drivers of polar bear behavior and the possible effects of prey availability on foraging strategy
Variable | Interval | Description |
|---|---|---|
T | \(\{1,2,\ldots \}\) | Total number of time steps |
t | [1, T] | A time step |
\(X_t\) | – | Set of observations at time t |
\({{\textbf {X}}}\) | – | The set of all observations \((X_1,\ldots , X_T)\) |
l | \((0,\infty )\) | Step length between consecutive locations |
\(\phi\) | \((-\pi ,\pi ]\) | Turning angle (i.e., change in bearing) between consecutive steps |
r | \([0, \infty )\) | Magnitude of the stimulus |
\(\psi\) | \((-\pi ,\pi ]\) | Direction of a stimulus relative to the bearing of the previous time step |
\(\mu ^{(l)}\) | \((0,\infty )\) | Mean parameter of step length |
\(\sigma ^{(l)}\) | \((0,\infty )\) | Standard deviation parameter of step length |
\(\beta _1\) | \((-\infty , \infty )\) | Intercept coefficient for mean step length |
\(\beta _2\) | \((-\infty , \infty )\) | Slope coefficient for mean step length and r |
\(\mu ^{(\phi )}\) | \((-\pi ,\pi ]\) | Mean parameter of turning angle |
\(\kappa ^{(\phi )}\) | \((0,\infty )\) | Concentration parameter of turning angle |
\(\alpha _1\) | \((-\infty , \infty )\) | Bias coefficient in the same direction as the stimulus |
\(\alpha _2\) | \((-\infty , \infty )\) | Bias coefficient \(90^{\circ }\) left of the stimulus |
\(\vartheta\) | \((-\pi ,\pi ]\) | The direction of bias relative to stimulus |
\(M^*\) | [0, 1) | Scaled magnitude of attraction |
\(\hat{}\) | – | An estimated value, e.g., \({\hat{\mu }}^{(l)}\) is the estimated value of \(\mu ^{(l)}\) |
N | \(\{1,2,\ldots \}\) | Total number of behavioral states |
S | [1, N] | Behavioral state |
\(\varvec{S'}\) | [1, N] | Behavioral state with the highest within-cell proportion relative to across-cell proportion |
\(\gamma _{i,j}\) | [0, 1] | Transition probability from state i to state j |
\(\mathbf {\Gamma }\) | – | \(N \times N\) Transition probability matrix |
D | – | The drift state |
\(O^{(L)}, O^{(R)}\) | – | Olfactory search state with anemotaxis left of wind and anemotaxis right of wind, respectively |
ARS | – | Area-restricted search state |