### Affect-Relations: Temporal

Temporal affect-relations are represented by the temporal configuration of events in a temporal map.

Note:

- Temporal affect-relations can be used to describe processes, change with respect to time, and system dynamics. For example, the sequence of neurons firing across synaptic connections in the human nervous system is a temporal process (associated with human sensing, thinking, and acting). See below example of a teaching-studenting process.
- Temporal affect-relations contrast with representations for structural affect-relations.

An example of a temporal map is illustrated in the below table.

Temporal Map 1. Dynamic configuration of events for STUDENT ENGAGEMENT (*On-task, Off-task*) and TEACHER INSTRUCTION (*Direct, Non-Direct*) for TARGET STUDENT (*Mona*, others not shown).

CLOCK |
TARGET STUDENT |
TEACHER |
STUDENT ENGAGEMENT |

9:01 a.m. |
{ |
{ |
{ |

9:02 |
| |
| |
| |

9:03 |
| |
| |
{ |

9:04 |
| |
| |
| |

9:05 |
| |
| |
| |

9:06 |
| |
| |
{ |

9:07 |
| |
| |
{ |

9:08 |
| |
{ |
| |

9:09 |
| |
| |
| |

9:10 |
| |
| |
| |

9:11 |
| |
| |
{ |

9:12 |
| |
| |
| |

9:13 |
{ |
{ |
{ |

Each row, excluding column headings, in the temporal map indicates a 'joint temporal event' (JTE). For example, the JTE at 9:01 a.m. is: TARGET STUDENT is *Mona*, and TEACHER INSTRUCTION is *Direct*, and STUDENT ENGAGEMENT is *Off-task*. At 9:02 a.m. these 3 events continue, indicated by a vertical bar ( | ). At 9:03 a.m. STUDENT ENGAGEMENT changes ( { ) to *On-task*. Each cell in the temporal map (except in the first column and column headings) represents a 'singular temporal event' (STE). Every STE indicates the state of a CLASSIFICATION and its *value* at that point in time. An example of a STE is INSTRUCTION changes to *Non-direct* at 9:08 a.m. The classification is TEACHER INSTRUCTION, the state is 'changes to' (represented by { ), and the value is *Non-direct*. The *Null *value means that there is nothing relevant to a classification of events occuring at that time (and serves as a way of indicating the termination of a STE). A vertical bar ( | ) is a short-hand way of indicating that the previously coded temporal event is continuing to occur. For example, TEACHER INSTRUCTION changed to { *Non-direct* at 9:08 and | continued until 9:13 a.m., when it ended.

A chain of JTE's with their respective STE's in each row could also be represented by a di-graph, where the vertices (nodes) are temporal events, and the relation types are "is temporally followed by" and "occurs at the same time as". Each subsequent row in a temporal map implies the "is followed by" relation (e.g., JTE at 9:11 is followed by JTE at 9:12, etc.). Each row in the map implies the "occurs at the same time as" relation. By convention, a tabular representation of a temporal configuration is used, which is called a 'temporal map'. See further examples by Frick, Myers, Thompson and York (2006).

Analysis of Patterns in Time (APT) was invented as a method for mapping and analyzing temporal event patterns by Frick (1983, 1990). This was an attempt to provide measures of temporal trajectories in education systems in terms of probabilities of event occurrences. APT is a way to characterize and measure system dynamic properties.

While investigating the SIGGS theory model (Maccia & Maccia, 1966), Frick discovered that the measures of uncertainty in information theory were inadequate for predicting specific temporal patterns (Frick, 1983). SIGGS is grounded in *set* (S), *information* (I), *di-graph* (G) and *general systems* (GS) theories. SIGGS is a complex theory model with precise definitions of systems’ dynamic and structural properties such as toput, strongness, adaptibility, stress, wholeness, and so forth. SIGGS was used to develop a theory of education, consisting of 201 hypotheses.

In SIGGS, *information* is defined as a “characterization of occurrences” (Maccia & Maccia, 1966, p. 40), and in turn is further defined mathematically via set theory and probability theory (pp. 10-23, 40-53). Frick (1983) interpreted these occurrences as temporal events, characterized by classifications and categories used when observing empirical phenomena.

Determination of values of SIGGS properties of *feedin*, *feedout*, *feedthrough* and *feedback* requires measures of temporal patterns. More specifically *feedin* is defined as transmission of information (occurrences of elements) from *toput* at *time 1* to *input* at *time* *2. *For example, the distribution of students who *apply* to various degree programs at a university in the spring are part of *toput*, and those students who are subsequently *admitted and attend* in the fall then become part of the *input* distribution of students in those degree programs in that particular education system. Similarly, *feedthrough* is defined in SIGGS as *feedin *followed later by *feedout*. For example, students matriculate (*feedin*), and later they graduate, drop out, or flunk out (*feedout*—*fromput* followed by *output*); this entire set of trajectories constitutes that system’s student *feedthrough*.

In set theory, a *relation* is the Cartesian Product of two or more sets of elements. Such a relation consists of a set of ordered pairs of elements, or more generally, *tuples*. Each *n*-tuple characterizes a pattern—that is, a conjoining of elements. For example, a *4*-tuple characterizes the *feedthrough* of a particular student from *toput* at *time* *1*, to *input* at *time* *2*, to *fromput* at *time* *3*, to *output* at *time* *4*. One student might apply to a university music program (*toput*), be admitted as a music major (*input*), later change her major, completing a bachelor’s degree in computer science (*fromput*), then get a good-paying job as a software engineer after graduation (*output*). Another *4*-tuple is characterized by a different student who applies for a computer science major, but instead gets admitted to a general studies program, later leaves the university with no degree, and then is employed in a low-paying job.

When occurrences of students moving through the university are mapped into categories of classifications which represent *4*-tuples, a *joint probability distribution* can be formed (from the Cartesian Product of *toput*, *input*, *fromput*, and *output* classifications which determine student *feedthrough* for the university). However, the *T* and *B* measures from information theory (Maccia & Maccia, 1966; Coombs, Dawes, & Tversky, 1970) do not provide specific predictions of temporal patterns (or trajectories); rather *T* and *B* coefficients are *measures of* *overall* *uncertainty* in the joint probability distributions of temporal occurrences. This is analogous to how an *F*-test in ANOVA indicates overall statistical significance, but does not tell us which contrasts are significant when there are more than two group means being compared.

Moreover, Frick (1983) subsequently proved mathematically that marginals (e.g., *toput*, *input*, *fromput*, *output*) of joint probability distributions cannot dependably predict cell values, that is, probabilities of conjoint occurrences of temporal events (e.g., *feedin*, *feedout*, *feedthrough*, *feedback*). He concludes:

There is no unique solution to this set of equations [18 – 21, from the calculus of probability theory], since the determinate of the matrix of coefficients is zero.… The mathematical conclusion is that there is no way to uniquely determine the joint probability distribution given only the marginal probability distributions, except in a few special cases where the marginal probabilities are zeros and ones, or all equal. (p. 79)

Hence, the need for alternative methods was justified theoretically. APT was invented as such an alternative approach, which has been further developed into MAPSAT in recent years.

#### Historical Note

Frick (1983) originally referred to this method of measuring system dynamics as Nonmetric Temporal Path Analysis (NTPA). The name was later changed to Analysis of Patterns in Time (Frick, 1990), to prevent confusion with path analysis which used statistical regression methods.

When Frick began collaborating with Thompson (circa 1995-2015), this methodology was expanded to include Analysis of Patterns in Configuration (APC), based on ATIS Graph Theory (see also structural affect-relations).

For several years, these two methodologies were referred to as APT&C, but the nomenclature was changed in 2007 to: Map and Analyze Patterns and Structures Across Time (MAPSAT), which includes APT, APC and potentially future methodologies to be investigated, such as topological analyses of system structures.

For an overview of MAPSAT, see Frick's presentation in 2007: Predicting Patterns in Education: Linking Theory to Practice (requires Adobe Flash Player plugin).