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Metrics of trajectory deviation with respect to a reference trajectory

Source: R/resilience_metrics.R
resilience_metrics.Rd

Set of metrics to analyze the deviation of disturbed trajectories from an ecological dynamic regime (EDR) considering a representative trajectory or a predicted trajectory as the reference. These metrics include the resistance to the disturbance, amplitude, recovery, and net change. Altogether, these indices inform on the resilience of a system.

Usage

resistance(
  d,
  trajectories,
  states,
  disturbed_trajectories,
  disturbed_states,
  predisturbed_states = disturbed_states - 1
)

amplitude(
  d,
  trajectories,
  states,
  disturbed_trajectories,
  disturbed_states,
  predisturbed_states = disturbed_states - 1,
  reference,
  state_var = NULL,
  index = c("absolute", "relative"),
  method = "nearest_state"
)

recovery(
  d,
  trajectories,
  states,
  disturbed_trajectories,
  disturbed_states,
  reference,
  state_var = NULL,
  index = c("absolute", "relative"),
  method = "nearest_state"
)

net_change(
  d,
  trajectories,
  states,
  disturbed_trajectories,
  disturbed_states,
  predisturbed_states = disturbed_states - 1,
  reference,
  state_var = NULL,
  index = c("absolute", "relative"),
  method = "nearest_state"
)

Arguments

d

Either a symmetric matrix or an object of class dist containing the dissimilarities between each pair of states.

trajectories

Vector indicating the trajectory or site to which each state in d and state_var belongs.

states

Vector of integers indicating the order of the states in d and state_var for each trajectory.

disturbed_trajectories

Vector of the same class as trajectories indicating the identifier of the disturbed trajectories.

disturbed_states

Vector of integers included in states indicating the first state after the release of the disturbance for each value in disturbed_trajectories.

predisturbed_states

Vector of integers included in states indicating the last undisturbed state of each disturbed_trajectories. The previous states to disturbed_states are considered by default.

reference

Object of class RETRA or PETRA indicating the trajectory taken as the reference to compute amplitude, recovery, and net_change of the disturbed trajectories (see Details). PETRA objects must include the arguments (return_args = TRUE).

state_var

Object containing the state variables for each trajectory state (required if reference is of class PETRA).

index

Method to calculate amplitude, recovery, or net change ("absolute", "relative"; see Details).

method

Method to calculate the distance between the disturbed_states or predisturbed_states and the reference trajectory. One of "nearest_state", of "projection" (see Details).

Value

  • resistance() returns a data frame of two columns indicating the resistance value (Rt) for each disturbed_trajectory.

  • amplitude() returns a data frame of three columns indicating the amplitude value (A_abs; A_rel) for each disturbed_trajectory and reference. If index = c("absolute", "relative"), both values are included in a data frame of four columns.

  • recovery() returns a data frame of four columns indicating the recovery value (Rc_abs; Rc_rel) for each disturbed_trajectory, post-disturbance state (state) and reference. If index = c("absolute", "relative"), both values are included in a data frame of five columns.

  • net_change returns a data frame of four columns indicating the net change value (NC_abs; NC_rel) for each disturbed_trajectory, post-disturbance state (state), and reference. If index = c("absolute", "relative"), both values are included in a data frame of five columns.

Details

Resistance (resistance())

Resistance captures the immediate impact of the disturbance as a function of the changes in the state variables (Sánchez-Pinillos et al., 2019).

\( Rt = 1 - d_{pre,dist} \)

Amplitude (amplitude())

Amplitude indicates the direction in which the system is deviated during the disturbance in relation to the reference (Sánchez-Pinillos et al., 2024; Sánchez-Pinillos et al., 2026).

If reference is of class RETRA: Positive values indicate that the disturbance deviates the system towards the boundaries of the dynamic regime. Negative values indicate that the disturbance deviates the system towards the representative trajectory.

If reference is of class PETRA: Only values equal or greater than zero are possible. Positive values indicate that the system deviates from its predicted trajectory, whereas amplitude equal to zero indicates that the system remains on its predicted trajectory.

Two indices can be calculated:

If index = "absolute",

\( A = d_{dist,RT} - d_{pre,RT} \)

If index = "relative",

\( A = \frac{d_{dist,RT} - d_{pre,RT}}{d_{pre,dist}} \)

Recovery (recovery())

Recovery quantifies the ability of the system to evolve towards the reference following the relief of the disturbance (if positive) or move in the direction of the boundaries of the dynamic regime (if negative and reference is of class RETRA) or elsewhere (if negative and reference is of class PETRA) (Sánchez-Pinillos et al., 2024; Sánchez-Pinillos et al., 2026).

Two indices can be calculated:

If index = "absolute",

\( Rc = d_{dist,RT} - d_{post,RT} \)

If index = "relative",

\( Rc = \frac{d_{dist,RT} - d_{post,RT}}{d_{dist,post}} \)

Net change (net_change())

Net change quantifies the proximity of the system to the reference relative to the pre-disturbed state after the disturbance (Sánchez-Pinillos et al., 2024; Sánchez-Pinillos et al., 2026).

If reference is of class RETRA: Positive values indicate that the system eventually evolves towards the boundaries of the dynamic regime. Negative values indicate that the system eventually evolves towards the reference.

If reference is of class PETRA: Only values equal or greater than zero are possible. Positive values indicate that the system eventually deviates from its predicted trajectory, whereas net change equal to zero indicates that the system reaches its predicted trajectory some time after the disturbance.

Two indices can be calculated:

If index = "absolute",

\( NC = d_{post,RT} - d_{pre,RT} \)

If index = "relative",

\( NC = \frac{d_{post,RT} - d_{pre,RT}}{d_{pre,post}} \)

In all cases:

  • \(d_{pre,RT}\) is the dissimilarity between the predisturbed_states and the reference.

  • \(d_{dist,RT}\) is the dissimilarity between the disturbed_states and the reference.

  • \(d_{post,RT}\) is the dissimilarity between the states after disturbed_states and the reference.

  • \(d_{pre,dist}\) is the dissimilarity contained in d between the predisturbed_states and the disturbed_states.

  • \(d_{dist,post}\) is the dissimilarity contained in d between the disturbed_states and the post-disturbed states.

  • \(d_{pre,post}\) is the dissimilarity contained in d between the predisturbed_states and the post-disturbed states.

\(d_{pre,RT}\), \(d_{dist,RT}\), and \(d_{post,RT}\) are calculated using the function state_to_trajectory() by two different methods:

  • If method = "nearest_state", \(d_{pre,RT}\), \(d_{dist,RT}\), and \(d_{post,RT}\) are calculated as the dissimilarity between the pre-disturbance, disturbed, or post-disturbance states and their nearest state in the reference.

  • If method = "projection", \(d_{pre,RT}\), \(d_{dist,RT}\), and \(d_{post,RT}\) are calculated as the dissimilarity between the pre-disturbance, disturbed, or post-disturbance states and their projection onto the reference or using the nearest state of the reference if the dissimilarity is smaller.

References

Sánchez-Pinillos, M., Leduc, A., Ameztegui, A., Kneeshaw, D., Lloret, F., & Coll, L. (2019). Resistance, resilience or change: Post-disturbance dynamics of boreal forests after insect outbreaks. Ecosystems 22, 1886-1901 https://doi.org/10.1007/s10021-019-00378-6

Sánchez-Pinillos, M., Dakos, V., & Kéfi, S. (2024). Ecological dynamic regimes: A key concept for assessing ecological resilience. Biological Conservation 289, 110409 https://doi.org/10.1016/j.biocon.2023.110409

Sánchez-Pinillos M., Fortin, M-J., Messier, C., Kneeshaw, D. 2026. Forecasting ecological trajectories from ecological dynamic regimes to improve resilience analysis. Methods in Ecology and Evolution. doi:10.1111/2041-210x.70372

See also

retra_edr() to identify representative trajectories in an ecological dynamic regime.

define_retra() to generate an object of classRETRA.

petra_edr() to predict ecological trajectories in an ecological dynamic regime.

MPD() for estimating the prediction accuracy of petra-edr() outputs.

state_to_trajectory() to calculate the position of a state with respect to a trajectory.

Author

Martina Sánchez-Pinillos

Examples

if (requireNamespace("vegan", quietly = TRUE)) {
  # Calculate the resistance of disturbed systems

  # Abundance matrix including disturbed and undisturbed trajectories
  abundance <- rbind(EDR_data$EDR3$abundance,
                     EDR_data$EDR3_disturbed$abundance, fill = TRUE)

  # State dissimilarities (Bray-Curtis) for disturbed and undisturbed trajectories
  d <- vegan::vegdist(abundance[, paste0("sp", 1:12)], method = "bray")

  # Resistance
  Rt <- resistance(d = d, trajectories = abundance$traj, states = abundance$state,
                   disturbed_trajectories = unique(abundance[!is.na(disturbed_states)]$traj),
                   disturbed_states = abundance[disturbed_states == 1]$state)

  # Calculate the amplitude, recovery, and net change of disturbed systems taking
  # a representative trajectory as the reference. Analogous metrics can be
  # calculated using a PETRA object as the reference.

  # Identify the representative trajectories of the EDR from undisturbed trajectories
  RT <- retra_edr(d = EDR_data$EDR3$state_dissim,
                  trajectories = EDR_data$EDR3$abundance$traj,
                  states = as.integer(EDR_data$EDR3$abundance$state),
                  minSegs = 5)

  # Amplitude
  A <- amplitude(d = d, trajectories = abundance$traj, states = abundance$state,
                 disturbed_trajectories = unique(abundance[!is.na(disturbed_states)]$traj),
                 disturbed_states = abundance[disturbed_states == 1]$state, reference = RT)

  # Recovery
  Rc <- recovery(d = d, trajectories = abundance$traj, states = abundance$state,
                 disturbed_trajectories = unique(abundance[!is.na(disturbed_states)]$traj),
                 disturbed_states = abundance[disturbed_states == 1]$state, reference = RT)

  # Net change
  NC <- net_change(d = d, trajectories = abundance$traj, states = abundance$state,
                   disturbed_trajectories = unique(abundance[!is.na(disturbed_states)]$traj),
                   disturbed_states = abundance[disturbed_states == 1]$state, reference = RT)

}