wolter-et-all.md

wolter-et-all.md

Wolter, M., Assenmacher, J., Hentschel, B., Shirski, M., & Kuhlen, T. (2009). A Time Model for Time-Varying Visualization. Computer Graphics Forum, 28(6), 1561–1571.

Motivation

In 2009 existing frameworks for dealing with complex, real-world temporal data treated time as a simple, continuous, linear dimension—essentially a fourth spatial coordinate to be mapped or animated sequentially.

Wolter et al were motivated by visualization needs around:

Asynchronous and Event-Based Data: Many datasets are driven by discrete, irregularly spaced events rather than smooth, continuous sampling.

Cyclical or Periodic Phenomena: Domains like environmental science or business analytics require representations of seasonal or daily cycles, where linear time mapping obscures crucial periodic relationships.

Uncertainty and Indeterminacy: In fields such as medical planning or climate simulation, temporal data often includes uncertainty, estimated durations, or constrained possibilities, which cannot be expressed using fixed time points.

The architecture of the time in the paper

1. Temporal Primitives and Granularity

The model formalizes the distinction between fundamental temporal units: Time Points (instantaneous events) versus Time Intervals (durations).

For visual computing, points might be marked by specific glyphs, while intervals require representations of duration, overlap, and relationship. A requirement of their work is that the model must handle data at varying granularities, supporting the user's ability to seamlessly transition from viewing aggregate trends to second-by-second measurements.

2. Temporal Topology and Structure

Their topology concerns itself with:

Linear Time: The standard continuous or discrete chronological sequence (historical records).

Cyclic Time: Time structured by recurring patterns (e.g., hours in a day, months in a year), essential for visualizations like spirals.2

Branching Time: Complex topologies necessary for representing simulations, multi-scenario planning, or decision pathways, where time itself is not a single, linear axis but a set of potential histories.

3. Temporal Indeterminacy

The Wolter et al model formalizes temporal indeterminacy and uncertainty. Specifically it seeks to handle constraints and flexibility around intervals (e.g., an operation must happen between T1 and T2, but its precise start is uncertain).

This seems to imply that the model defined necessary metadata fields to capture:

Fuzzy Boundaries: Probabilistic start or end times for events.

Interval Constraints: Relationships between intervals (e.g., Event A must precede Event B by a duration of $\Delta t$).

This requirement to model complex relationships moves the data structure from simple time series tuples to a framework capable of handling Allen's Interval Algebra or similar constraint-based temporal representations. The capacity to formalize uncertainty allows visualization systems designed around the Wolter model is intended to support highly complex decision-making and predictive analysis.

The Model

Definitions

Time frame

a frame of reference, rather than a frame of measurement

Mutual Conversion Scheme

some scheme that can convert from one time frame to another one

User Time

The real time a user lives in and perceives. user time is continuous and linear but not ordinal. It does not allow cycles.

$$ U \subset \mathcal{R}^{+} $$

Visualization Time

A normalized time frame describing the complete time dependent process. It may be cyclic.

$$ V = [0,1] \subset \mathcal{R} $$

User Window

The visualization time has a start point $u_0$ and end point $u_1$ of the visualization interval $[0,1]$ in user time, where

$$ u_0, u_1 \in U. $$

Time Mapping to the User Frame

Let time instant $u \in [u_0, u_1]$. A mapping $\hat{u}\colon U \to V$ is defined as

$$ \hat{u}(u) = \frac{u-u_0}{u_1-u_0} $$

Time instants $u_0, $u_1$ change with every repetition of looping as cyclic times are unrolled to become strictly non-cyclic for the user frame.

Simulation Time Frame

The change of time in a simulated process. Simulation time is continuous and linear but not ordinal.

Simulation time may be cyclic, because some processes repeat.

$$ S = [s_{start}, s_{end}] \subset \mathbb{R} $$

All data derived from a simulation are defined with respect to this time frame. The Time Frame must have a metric to enable mutual conversion.

Discrete Time Step

A single time instant in a simulation. It is valid only in a single simulation time.

Discrete Time Data

A set of discrete data $D$ contains discrete pairs of a value and a time step. The pairs may be non-uniformly distributed, and sparse. The position of a pair is ordinal.

Time Instant

Time instants belong to the Simulation Time Frame, and have no duration.

Time index Frame

A time index frame $I$ is a set of ordinals into discrete data $D$. The individual elements of $I$ are chosen arbitrarily, $i \subseteq \textit{count}(D)$ and are typically specific elements of interest.

Surjective mapping of time index frame to simulation frame

Time indices admit a mapping to simulation time $\hat{s} \colon S \to I$ where $\hat{s}{start} = i_0$ and $\hat{s}{end} = i_{n-1}$. This is a total mapping and surjective as all indices correspond to a simulation time, and monotonically increasing as both time frames proceed forward in time.

Reversed surjective mapping

The reversed surjective mapping $\hat{s}^{-1}$ (not an inverse) assigns an interval of simulation time in which the a time index is valid. The codomain of $\hat{s}^{-1}$ is a set of time intervals forming a partition on the complete simulation interval $[s_{start}, s_{end}]$. This function is bijective and the partition covers the interval.

The selection of interval partitions may use any likely strategy, such as nearest neighbour mapping.

Domain Mapping

A parametric domain, such as crank angle, may map to simulation time.

No Branching

Time is linear without branching. Branching time is handled as variations, and treated as independent.

Continuous Data

Continuous data is defined as a function of time $d = f(t)$ for some time-dependent function $f$ and a time instant $t$.

Display of Simultaneous Timelines

For each distinct simulation $k$, the simulation time $S^{(k)}$ is associated with an interval of visualization time $[{v_0}^{(k)}, {v_1}^{(k)}] \subseteq V$ with a mapping function $\hat{v}^{(k)} \colon V \to S$ with $\hat{v}^{(k)}({v_0}^{(k)}) = {s_{start}}^{(k)}$ and $\hat{v}^{(k)}({v_1}^{(k)}) = {s_{end}}^{(k)}$.

This allows several simulations to be displayed in tracks or sequentially on a timeline.

Modeling Operations

Embedding

The index mapping allows a new partition to be generated by creating a new Time Index Frame $I$ where indices are repeated in order to create a new reversed surjective mapping after suitable remapping of the indices to a particular interval.

Movement

A given instant may be moved on the timeline via the application of a matrix. An unmoved instant implicitly is moved by the Identity matrix.

Events

Time instants belong to the simulation time frame. Time instants have no duration, therefore to be observable, an event is modeled by devising an observable interval to which the event may be assigned.

Interaction Operations

Control

The flow of time is controlled by a user time mapping $\hat{u}$. If $\hat{u}$ evaluates to zero time is stopped. By changing the user window length $u_1 - u_0$, the process is displayed slower or faster. Backwards display of simulation data is accomplished by mapping to proceed from visualization time 1 to 0.

Selection

For the selection of a specific time instant, two bounding time frames are used. By using normalized visualtion time $v \in [0,1]$ fuzzy queries may be consulted, such as "in the first quarter of the process."

Range

A subinterval of a process may be restricted such that $[{sub}{start}, {sub}{end} \subseteq [0, 1]$. The user time mapping $\hat{u}$ must be adapted as the codomain changes in order that a process is displayed in its original speed.

Subset

A sparse reduction of a time interval frame yields a coarse view into simulation time.

Applications

A number of applications follow, illustrating how the Model and Operations may be used to visualize various sorts of time-varying information on a timeline.