redeem Model Terms — redeem

The help pages of rem and dem describe the model formulation and estimation details. This page documents all statistics available to be used in the model formulas, characterizing the intensities of event formation and dissolution.

In the redeem framework, models like DEM (fitted via dem) and REM (fitted via rem) are specified using R formulas. The right-hand side of these formulas defines the structural statistics and covariates, where each term must be specified separately as an explicit function call (e.g., ~ inertia() + reciprocity(window = 10)).

All terms support an optional transformation argument \(f\). The available transformations are:

"identity" (default): \(f(x) = x\)
"log": \(f(x) = \log(x + 1)\)
"recip": \(f(x) = 1/(x+1)\)
"bin": \(f(x) = I(x > 0)\)
"sig": sigmoid-like saturation, \(f(x) = x/(x + K)\)

Throughout, \(N_{i,j}(t)\) denotes the cumulative number of events from \(i\) to \(j\) up to (but not including) time \(t\); \(N_{i,j}^w(t)\) is the windowed analogue on \((t-w,\,t)\); \(d_i^{\mathrm{out}}(t) = |\{l: N_{i,l}(t)>0\}|\) is the historical out-degree of \(i\); and \(c_i^{\mathrm{out}}(t) = \sum_l N_{i,l}(t)\) is the total event count sent by \(i\). The superscript \(\mathrm{act}\) indicates that the quantity is computed on the currently active DEM network.

The implemented terms are grouped into five categories:

Baseline and Nuisance Terms: Intercept, time-varying baseline, and degree fixed effects.
Endogenous Dyadic Terms: Inertia, reciprocity, interaction duration, and participation shifts.
Triadic Closure and Shared Partners: Common partners and triangle statistics.
Degree and Centrality Statistics: Actor degree and event count statistics.
Exogenous Covariates: Dyadic and monadic covariate terms.

Arguments

K

Numeric; the evaluation point or scaling/saturation factor for the sufficient statistic (default is 1).

transformation

Character; specifies the transformation to apply to the statistic. One of:

"identity" (default): \(f(x) = x\)
"log": \(f(x) = \log(x+1)\)
"recip": \(f(x) = 1/(x+1)\)
"bin": \(f(x) = I(x>0)\)
"sig": sigmoid-like saturation, \(f(x) = x/(x+K)\)

event_stream

Optional matrix or data frame; an alternative event stream to use for calculating the statistic. If NULL (default), the modeled stream is used.

window

Numeric; time window for calculating the statistic (default Inf, i.e., use full history).

type

Character; the specific variation of the statistic or triangle type (e.g., "OSP", "ISP", "OTP", "ITP", "out_sender", "sum").

mode

Character; the participation shift mode (e.g., "ABBA", "ABBY").

data

For dyadic_cov, a numeric matrix of dimensions \(N \times N\), a scalar applied globally, or a named list of matrices for time-varying covariates. For monadic_cov, a numeric vector of length \(N\) or a named list of vectors for time-varying covariates.

fun

A function taking two arguments fun(v_i, v_j) to generate dyadic values.

change_points

Optional numeric vector; time points for time-varying covariates if data is a list.

changepoints

Numeric vector; time points where the baseline intensity is allowed to change.

labels

Character vector; optional labels for the resulting time intervals.

history

Character; "general" for cumulative history or "current" for currently active events.

count

Logical; if TRUE, uses count-based (weighted) versions of degree statistics (default FALSE).

...

Arguments passed to the underlying initialization function.

Multi-Stream Event Covariates

Most endogenous terms support covariates calculated from multiple event streams. By providing an event_stream argument to a term (e.g., inertia(event_stream = other_events)), users can model one event process while accounting for the history of another. The package automatically handles the splintering and union of these timelines.

Baseline and Nuisance Terms

Intercept(): Intercept: Constant log-intensity baseline. \(s_{i,j}(t) = 1\). Also available as intercept().
baseline(changepoints, labels): Baseline: Stepwise constant log-baseline with user-specified change points \(c_1 < c_2 < \ldots < c_K\). \(s_{i,j}(t) = \sum_{k=1}^{K} I(t \in [c_k, c_{k+1}))\). Coefficients are treated as nuisance parameters.
degree / degrees: Degree Fixed Effects: Node-specific sender and receiver baselines \(\alpha_i\) and \(\gamma_j\) estimated via Minorization-Maximization (MM). Contribution to linear predictor: \(\alpha_i + \gamma_j\). Treated as nuisance parameters.

Endogenous Dyadic Terms

inertia(transformation, K, event_stream, window): Inertia: Cumulative count of past events from \(i\) to \(j\). \(s_{i,j}(t) = f(N_{i,j}(t))\); windowed: \(f(N_{i,j}^w(t))\).
reciprocity(transformation, K, event_stream, window): Reciprocity: Cumulative count of past events from \(j\) to \(i\). \(s_{i,j}(t) = f(N_{j,i}(t))\) (Directed only).
current_interaction(transformation, K, event_stream): Duration: Time elapsed since the currently active event \((i,j)\) started. \(s_{i,j}(t) = f(t - t_{\mathrm{start},i,j})\) (DEM only). Alias: duration().
Participation Shifts (for two consecutive events \((A \to B) \to (C \to D)\), each statistic is 1 if the specified pattern holds, 0 otherwise; REM only, Directed only):
- psABBA(event_stream): PS-ABBA: Receiver responds to sender. \(s_{C,D}(t) = I(C = B,\, D = A)\).
- psABBY(event_stream): PS-ABBY: Receiver initiates to a new target. \(s_{C,D}(t) = I(C = B,\, D \ne A)\).
- psABAY(event_stream): PS-ABAY: Sender initiates to a new target. \(s_{C,D}(t) = I(C = A,\, D \ne B)\).
- psABXA(event_stream): PS-ABXA: Outsider targets original sender. \(s_{C,D}(t) = I(C \ne A, C \ne B,\, D = A)\).
- psABXB(event_stream): PS-ABXB: Outsider targets original receiver. \(s_{C,D}(t) = I(C \ne A, C \ne B,\, D = B)\).
- psABXY(event_stream): PS-ABXY: Entirely new dyad. \(s_{C,D}(t) = I(C \ne A, C \ne B,\, D \ne A, D \ne B)\).
- ps(mode, event_stream): PS Shorthand: Dispatches to one of the six participation shift statistics above based on mode (one of "ABBA", "ABBY", "ABAY", "ABXA", "ABXB", "ABXY").

Triadic Closure and Shared Partners

general_common_partners( transformation, K, type, event_stream, window): Historical Common Partners: Number of nodes \(k\) sharing a historical directed path of the specified type with both \(i\) and \(j\). \(s_{i,j}(t) = f(|CP_{i,j}^{\mathrm{type}}(t)|)\).
- "OSP" (Outgoing Shared Partner): \(N_{i,k}(t)>0\) and \(N_{j,k}(t)>0\).
- "ISP" (Incoming Shared Partner): \(N_{k,i}(t)>0\) and \(N_{k,j}(t)>0\).
- "OTP" (Outgoing Two-Path): \(N_{i,k}(t)>0\) and \(N_{k,j}(t)>0\).
- "ITP" (Incoming Two-Path): \(N_{k,i}(t)>0\) and \(N_{j,k}(t)>0\).
Aliases: general_common_partner(), general_common_partner_OSP(), general_common_partner_ISP(), general_common_partner_OTP(), general_common_partner_ITP().
current_common_partners( transformation, K, type, event_stream): Active Common Partners: As general_common_partners but restricted to currently active edges. \(s_{i,j}(t) = f(|CP_{i,j}^{\mathrm{type,act}}(t)|)\) (DEM only). Aliases: current_common_partner(), current_common_partner_OSP(), current_common_partner_ISP(), current_common_partner_OTP(), current_common_partner_ITP().
general_triangle(transformation, K, type, event_stream, window): Historical Triangles: Number of closed triads around \((i,j)\) in the historical event network of the specified type. \(s_{i,j}(t) = f(|\Delta_{i,j}^{\mathrm{type}}(t)|)\) (Directed only).
current_triangle(transformation, K, type, event_stream): Active Triangles: As general_triangle but restricted to currently active edges. (DEM only, Directed only).
common_partner(history, type, ...): Common Partner Shorthand: Dispatches to general_common_partners() (history="general") or current_common_partners() (history="current").
triangle(history, type, ...): Triangle Shorthand: Dispatches to general_triangle() (history="general") or current_triangle() (history="current").

Degree and Centrality Statistics

general_degree_out_sender( transformation, K, event_stream, window): Sender Out-Degree: Historical out-degree of sender \(i\). \(s_{i,j}(t) = f(d_i^{\mathrm{out}}(t))\) (Directed only).
general_degree_out_receiver( transformation, K, event_stream, window): Receiver Out-Degree: Historical out-degree of receiver \(j\). \(s_{i,j}(t) = f(d_j^{\mathrm{out}}(t))\) (Directed only).
general_degree_in_sender( transformation, K, event_stream, window): Sender In-Degree: Historical in-degree of sender \(i\). \(s_{i,j}(t) = f(d_i^{\mathrm{in}}(t))\) (Directed only).
general_degree_in_receiver( transformation, K, event_stream, window): Receiver In-Degree: Historical in-degree of receiver \(j\). \(s_{i,j}(t) = f(d_j^{\mathrm{in}}(t))\) (Directed only).
general_degree_sum(transformation, K, event_stream, window): Degree Sum: Sum of historical degrees of both endpoints. \(s_{i,j}(t) = f(d_i(t) + d_j(t))\) (Undirected only).
general_degree_absdiff( transformation, K, event_stream, window): Degree Absolute Difference: Absolute difference in historical degrees. \(s_{i,j}(t) = f(|d_i(t) - d_j(t)|)\) (Undirected only).
general_count_out_sender( transformation, K, event_stream, window): Sender Out-Count: Total events sent by sender \(i\). \(s_{i,j}(t) = f(c_i^{\mathrm{out}}(t))\) (Directed only).
general_count_out_receiver( transformation, K, event_stream, window): Receiver Out-Count: Total events sent by receiver \(j\). \(s_{i,j}(t) = f(c_j^{\mathrm{out}}(t))\) (Directed only).
general_count_in_sender( transformation, K, event_stream, window): Sender In-Count: Total events received by sender \(i\). \(s_{i,j}(t) = f(c_i^{\mathrm{in}}(t))\) (Directed only).
general_count_in_receiver( transformation, K, event_stream, window): Receiver In-Count: Total events received by receiver \(j\). \(s_{i,j}(t) = f(c_j^{\mathrm{in}}(t))\) (Directed only).
general_count_sum(transformation, K, event_stream, window): Count Sum: Sum of total event counts of both endpoints. \(s_{i,j}(t) = f(c_i(t) + c_j(t))\) (Undirected only).
general_count_absdiff( transformation, K, event_stream, window): Count Absolute Difference: Absolute difference in total event counts. \(s_{i,j}(t) = f(|c_i(t) - c_j(t)|)\) (Undirected only).
current_degree_out_sender(transformation, K, event_stream): Active Sender Out-Degree: Out-degree of \(i\) in the active DEM network. \(s_{i,j}(t) = f(d_i^{\mathrm{out,act}}(t))\) (DEM only, Directed only).
current_degree_out_receiver( transformation, K, event_stream): Active Receiver Out-Degree: Out-degree of \(j\) in active DEM network. \(s_{i,j}(t) = f(d_j^{\mathrm{out,act}}(t))\) (DEM only, Directed only).
current_degree_in_sender(transformation, K, event_stream): Active Sender In-Degree: In-degree of \(i\) in the active DEM network. \(s_{i,j}(t) = f(d_i^{\mathrm{in,act}}(t))\) (DEM only, Directed only).
current_degree_in_receiver( transformation, K, event_stream): Active Receiver In-Degree: In-degree of \(j\) in active DEM network. \(s_{i,j}(t) = f(d_j^{\mathrm{in,act}}(t))\) (DEM only, Directed only).
current_degree_sum(transformation, K, event_stream): Active Degree Sum: Sum of active degrees of both endpoints. \(s_{i,j}(t) = f(d_i^{\mathrm{act}}(t) + d_j^{\mathrm{act}}(t))\) (DEM only, Undirected only).
current_degree_absdiff(transformation, K, event_stream): Active Degree Absolute Difference: Absolute difference in active degrees. \(s_{i,j}(t) = f(|d_i^{\mathrm{act}}(t) - d_j^{\mathrm{act}}(t)|)\) (DEM only, Undirected only).
current_count_out_sender(transformation, K, event_stream): Active Sender Out-Count: Total active events sent by \(i\). \(s_{i,j}(t) = f(c_i^{\mathrm{out,act}}(t))\) (DEM only, Directed only).
current_count_out_receiver( transformation, K, event_stream): Active Receiver Out-Count: Total active events sent by \(j\). \(s_{i,j}(t) = f(c_j^{\mathrm{out,act}}(t))\) (DEM only, Directed only).
current_count_in_sender(transformation, K, event_stream): Active Sender In-Count: Total active events received by \(i\). \(s_{i,j}(t) = f(c_i^{\mathrm{in,act}}(t))\) (DEM only, Directed only).
current_count_in_receiver( transformation, K, event_stream): Active Receiver In-Count: Total active events received by \(j\). \(s_{i,j}(t) = f(c_j^{\mathrm{in,act}}(t))\) (DEM only, Directed only).
current_count_sum(transformation, K, event_stream): Active Count Sum: Sum of active event counts of both endpoints. \(s_{i,j}(t) = f(c_i^{\mathrm{act}}(t) + c_j^{\mathrm{act}}(t))\) (DEM only, Undirected only).
current_count_absdiff(transformation, K, event_stream): Active Count Absolute Difference: Absolute difference in active event counts. \(s_{i,j}(t) = f(|c_i^{\mathrm{act}}(t) - c_j^{\mathrm{act}}(t)|)\) (DEM only, Undirected only).
degree( history, type, count, transformation, K, event_stream, window): Degree Shorthand: Dispatches to the appropriate degree statistic based on history ("general" or "current") and type ("out_sender", "out_receiver", "in_sender", "in_receiver", "sum", "absdiff"). Set count = TRUE for weighted (count-based) variants. Alias: degrees().
count(history, type, transformation, K, event_stream, window): Count Shorthand: Equivalent to degree(..., count = TRUE).

Exogenous Covariates

dyadic_cov(data, change_points): Dyadic Covariate: Time-constant or time-varying external dyadic covariate matrix \(X\). \(s_{i,j}(t) = X_{i,j}(t)\).
monadic_cov(data, fun, change_points): Monadic Covariate: External monadic covariate vector \(x\) converted to a dyadic matrix via user-supplied function \(g\). \(s_{i,j}(t) = g(x_i(t),\, x_j(t))\).