Journal of Mathematical Psychology 53 (2009) 106–118

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Journal of Mathematical Psychology

journal homepage: www.elsevier.com/locate/jmp

The dynamics of the stream of behaviorDavid BirchUniversity of Illinois at Urbana-Champaign, 3 Walnut Street, Exeter, NH 03833, USA

a r t i c l e i n f o

Article history:Received 25 August 2005Received in revised form20 August 2008Available online 12 February 2009

Keywords:Stream of behaviorDynamics of actionActivation timeMotivation theoryEthologyChoiceTransition times

a b s t r a c t

Motivational psychology and ethology conceive of behavior as a continuous stream of activities, aconceptualization that is taken up formally in the present paper. Modeling the stream of behavior,unlike modeling an individual activity selected out of the stream, requires processes of data generationthat govern the repeated overt appearances of activities through time. To this end the motivationtheory of the dynamics of action, with its self-contained data generating process, is employed as amathematical framework. The resulting theory,modeled as a continuous succession ofmutually exclusiveand exhaustive activities driven by the countdown of activation times, yields expectations for the relativefrequencies of transitions and the distributions of transition times in a stream segment. Results supportingthe theory were obtained in a series of critical tests using a unique data set provided by the artist MorganO’Hara who has made extensive recordings of her stream of behavior.

© 2009 Elsevier Inc. All rights reserved.

1. Introduction

The subject matter of this inquiry is behavior; specifically, thecontinuous stream of observable behavior. The aims of the inquiryare to set forth in mathematical terms certain essential featuresof a protocol recorded for a segment of the stream for a singleindividual and to formulate a theory of the motivational processesresponsible for those features. A requisite of the theory is thatit provide a data generating process intrinsic to the theory andappropriate to the genesis of the continuous stream of behavior.The research undertaken is of a type traditionally found in the

disciplines of ethology and motivational psychology and is basedon the systematic observation of behavior, either the detaileddescription of a single activity as it is going on or, as in this case,the organized appearances of several activities across a span oftime. The theory presented is built on the premise that livingorganisms are always active. Therefore, in the study of motivationthe question is not why a particular activity is occurring butwhy that activity rather than some other activity. The contextfor observing any activity is given by the continuous stream ofactivities into which it is embedded. It is this background streamof activities, defined by the relationships among the activities,that is under observation in the present research. The behavioralsituation to be analyzed is one in which the individual underobservation is unconstrained by the observer and is subject towhatever motivational influences the immediate circumstancesprovide, conditions that are standard in ethological studies. The

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0022-2496/$ – see front matter© 2009 Elsevier Inc. All rights reserved.doi:10.1016/j.jmp.2008.12.002

two essential elements of the research are, first, the new approachto identifying the processes producing the continuous stream ofbehavior, derived from existing motivation theory, and, second,the need to evaluate critically and rigorously that approach usingstream data of the form the theory has undertaken to explain.The present paper addresses issues of motivation and behavior,

as do the formulations of other researchers (e.g. McFarland (1971),McFarland and Bosser (1997) and Toates (1975)), but does so froma different perspective. Its emphasis is on the extended continuousstream of free behavior analyzed using conventional mathematicsrather than on episodes of behavior dealt with in control systemsterms (McFarland, 1974). Although their concepts and data andimmediate goals are different, the approaches all have motivationand behavior as their subject matter and are complementary.Under study is a stream of behavior segment defined as the

recording that results from imposing for a designated period oftime a coding system of K mutually exclusive and exhaustiveactivity categories on the continuous flux that is the behavioral lifeof a single individual. A stream segment both begins and ends withthe initiation of one of the K activities and a clock reading is notedas each intervening activity is initiated throughout the segmentduration.Stream segment protocols are rich in time and frequency

measures of the sort one expects to see in statistical summaries.Included are the total time spent in each activity over the courseof the interval of observation; the overall frequency of occurrenceof each activity; the number of transitions from every activityto every other activity; the relative frequencies of the transitionscontingent on the ongoing activity (choices among activities); thetime spent in an activity just prior to a transition to another activity

D. Birch / Journal of Mathematical Psychology 53 (2009) 106–118 107

(transition time); and a large number of higher order time andfrequency measures contingent on one or another aspect of thestream segment protocol.The theoretical analysis undertaken in the paper is directed

to, and is phrased in terms of a behavioral protocol recordedfor a single individual, but the formal structure of a streamprotocol (i.e. mutually exclusive and exhaustive events recordedcontinuously in time) can be found in other data collection settings.For example, subject matter could be as disparate as visual searchin which the sequence of eye movements and fixations is tracked;group discussion and decision making where who is talking aboutwhat and when is of interest; the succession of moods or emotionsthrough time; or, perhaps, the predominant area of investment ina financial portfolio; or, even, the nationality of the pope acrosscenturies. It may be that aspects of the theory to be presented willbe helpful in these and other areas of inquiry with the same formof protocol.It has been suggested (by a reviewer) that an area of application

of the proposed theory could be that of consumer behavior whereit would join the modeling carried out by Kahn, Kalwani, andMorrison (1986), McAlister (1982), and McAlister and Pessemier(1982) directed to consumer product purchases and brandchanging. Critically important to a successful application of thepresent theory to consumer behavior would be the identificationof the activities (or events) to be considered. They must bemutually exclusive and exhaustive and, taken together, constitutea continuous stream through time in order to conform to the datacollection paradigm required by the theory.

2. Critical features of a segment of the stream of behavior

Conceptually, the most important measure to be taken from astream of behavior protocol is the time that passes between aninstance of an activity’s termination and its next initiation. Termedthe activation time (Birch, 1984), this interval is a fundamentalfeature of a stream segment protocol in that it defines the timetaken to accomplish the countdown to the next instance of theactivity. The context used in 1984 to introduce activation timetheory was that of the scheduling of events in the workplace asanalyzed by Sheridan and Johannsen (1976) and Morgan (1981)and as covered by the listings of scheduling rules by Panwalkarand Iskander (1977). Activation time theory was presented as ascheduling rule not already under study and particularly as onein which events are interrupted rather than completed (Conway,Maxwell, & Miller, 1967). The theory put forth in the 1984 paperprovided for the analysis of the continuous stream of behavior byrelying directly on the countdown of fixed activation times for itsdata generating process; that theory can be viewed as a special caseof the activation time formulation of the present paper in whichanother layer of theory is developed.Activation times, however, are poorly suited to serve directly

as the building blocks for a theory of the generation of the streamof behavior protocol because of their theoretical complexity. Mostoften, a particular activation time spans a portion of the streamprotocol that contains transitions involving the other activities andthis means not only that there is a sequence of activities ongoingduring the activation time interval but that the sequence, and theactivation time itself, can be expected to be different from oneinstance to another. Further, it is plausible that the rate at whichan activation time is closing depends on the ongoing activity andthat the overall rate at which a particular protocol activation timeis closing is an average masking different rates for the differentintervening ongoing activities of the activation time interval.All the activation times found in a protocol are composed

of sequences of intervals identified by their associated ongoingactivities. In every case the final interval in an activation time

sequence defines the transition time that completes that particularactivation time countdown. This means that values for all thetransition times in a protocol are directly available, a findingthat provides the cornerstone for a theory of the generation ofcontinuous stream of behavior protocols.

3. Protocol generation via the dynamics of action

Consideration of the stream of behavior, up to this pointbased on the countdown of activation times, will be recast as anextension of the theory ofmotivation called the dynamics of action(Atkinson & Birch, 1970). For the sake of clarity of exposition it ishelpful to distinguish three vocabularies pertinent to the presentdiscussion. So far, because our discussion has been confined tothe stream segment protocol (i.e. data), we have needed onlythe vocabulary of the physical world (specification of activities,frequency counts, time values). In taking up the dynamics of actiona different vocabulary is utilized, one which is at the level ofthe motivation of behavior (action tendencies, instigating forces).Not needed in this particular inquiry but clearly required forother treatments of motivation framed by the dynamics of action(e.g. Vansteenkiste, Lens, De Witte, De Witte, and Deci (2004))is the vocabulary of conscious experience and mental activities(perceiving, calculating, recognizing, deciding).In the language of the dynamics of action, every activity that

appears in the stream of behavior is proposed to have associatedwith it a tendency to engage in that activity. Furthermore, theobservation that Activity A is ongoing is defined to mean thatthe tendency to engage in Activity A(TA) is the strongest in thehierarchy of tendencies. If, at some point in time later, the ongoingactivity changes from A to B, it must mean that something hasoccurred such that now TB > TA. Much of the theory of thedynamics of action spells out how motivational processes, forexample of instigation and consummation, bear on the changingstrengths of the tendencies to engage in activities.As regards the present inquiry into the critical features of

stream of behavior protocols, it is the linkage between activationtimes as found in the protocols and changing strengths oftendencies as found in the dynamics of action that is of immediaterelevance. That linkage is forged by noting that just as one can seean activation time counting down to zero in a protocol so one canconceive of an action tendency growing in strength to the pointwhere the associated activity is initiated. A connection between anactivation time and the strength of a tendency can be establishedsuch that the increasing strength of tendency for an activity acrosstime is mapped directly on to the decreasing remainder of thecountdown to zero for the activation time.Of most interest is the nature of the relationship between

activation times and tendency strengths during the final intervalof the activation time countdown, the interval which defines thetransition time as recorded in the stream segment protocol. Atthe beginning of the final interval the tendency strengths for allalternative activities will be positive or zero and less than that ofthe ongoing activity. The activation time for the destination activitywill have counted down to the transition time. The path taken bythe tendency strength for the destination activity from its initialvalue to the dominance value during the transition interval is whatcan be taken to the dynamics of action for theoretical explicationand elaboration.Every transition in a stream segment brings with it a new

ongoing activity, AO, and its dominant tendency, TO, with thefollowing four consequences, all attributable to the natureand character of the new ongoing activity and its associatedenvironment: (1) The strength of the new TO fixes the level of thecriterion tendency strength to be met in order to accomplish thenext transition and reflects the vigor with which the new ongoing

108 D. Birch / Journal of Mathematical Psychology 53 (2009) 106–118

activity is undertaken. (2) The hierarchy of subdominant tendencystrengths associated with the K − 1 alternative activities presentat the point of transition is expanded or contracted proportionallyin relation to the strength of the new TO. This means that theexisting hierarchy, bounded by the interval 0 to TO based on theprior ongoing activity, becomes bounded by the interval 0 to TObased on the new ongoing activity. (3) The new AO sets in placea configuration of basic instigating forces originating in the natureofAO and the physical, physiological, and experiential environmentbrought about byAO. (4) It is to be expected that an ongoing activitywill place constraints on the exposure of the individual to theinstigating forces for the alternative activities due to its nature.In the context of motivation theory generally, the strength of

TO is related to the vigor of the ongoing activity, to an individual’sinvolvement in the activity, and to the quality of the performanceassociated with the activity. Such concerns (not part of the presentinquiry) are taken up in two recent examples of theorizing inachievement motivation using the framework of the dynamicsof action wherein Sorrentino, Smithson, Hodson, Roney, andWalker (2003) present a detailed, mathematical analysis of thedeterminants of the instigating forces for certain achievementactivities and Tuerlinckx, De Boeck, and Lens (2002) address issuesrelated to the measurement of the achievement motive, one ofthe determinants of instigating forces, and so of the strength oftendencies, for achievement activities.

4. The instigation function

It is proposed in the present theory, as a mathematicalextension of the discussion of this topic in Atkinson and Birch(1970), that the configuration of basic instigating forces for thealternative activities, introduced with a new ongoing activity, issubject to exposure influences dictated by the nature of the newongoing activity as described by the instigation function. Mostimportant in this regard is that all the instigating forces for all thealternative activities are acted upon by the same time dependentfunction. Different ongoing activities are expected to have differentinstigation functions but any given function is applied uniformlyto all alternative activity instigating forces present. In this way anenvironmental scheduling effect on all the alternative activitiesthat is attributable exclusively to the ongoing activity is identified.The instigation function is an empirical construct such as is

commonly found inmany areas of science. It refers to the particularconditions present when the theory is being applied. The specificform of I(t) is not, at this point, theory based but is chosen becauseit can accommodate a variety of special case functions under oneumbrella.Intuitively, the prototypical instigation function, I(t), would

seem to be one in which exposure to instigation for alternativeactivities begins at or near zero, builds to a maximum over time,and then declines asymptotically to zero. Such a progressionof environmental scheduling of exposure to instigation for thealternative activities is captured in the proposed linear dualprocess function

I(t) = h[

(a+ bt)s−1

(v + wt)s+1

](1)

for constants h, a, b, v, w ≥ 0 and s > 1.In its most complete form, I(t) begins with an initial value of

h(as−1

vs+1

)at t = 0, grows to a maximum located at

t =s(vb− aw)− (vb+ aw)

2bw,

and then declines asymptotically to zero but special sets ofparameter values result in variations in the form taken by I(t)

over time: Most notably, I(t), with its initial value of h(as−1

vs+1

),

rises monotonically with positive acceleration if w = 0 anddeclinesmonotonicallywith negative acceleration to an asymptoteof zero if b = 0. The particular form taken by an instigationfunction depends on the nature of the newly ongoing activity andits associated environment.The first special case cited above, based onw = 0, encompasses

activities which, upon initiation, impose a prolonged period duringwhich all exposure to instigation for alternative activities is nearto zero followed by increasing instigation engagement. Sleepingis perhaps the best example of such an activity but any activitywhich places the individual in an environment protected fromdistractions to the activity in progress for a period of time wouldresult in the same kind of reduced instigation for alternativeactivities. The general instigation function with the parameterw = 0 takes on this form with the size of parameter v controllingthe duration of the reduced exposure. The second case, basedon b = 0, is that of newly ongoing activities which allowimmediate exposure to alternative activities’ instigation followedby a reduction in that exposure. Letting b = 0 in the generalinstigation function produces this form and a visit to the dentistis the sort of ongoing activity that sets such conditions: At thebeginning of the activity, while in the waiting room, exposure toinstigation for alternative activities is strongly present throughcellphones, magazines, conversation, etc. (perhaps even sufficientfor leaving the office), all of which are eliminated when one isactually seated in the dental chair and protected from outsideinfluences. Attending the theater is another activity in whichinitially, before the performance begins, there is a great dealof exposure to instigating forces for a variety of activities butsubsequently, when the lights dim, all such exposure is greatlyreduced.The special case of Eq. (1) in which parameters b and w both

equal zero is important theoretically and historically as a referencecondition in which I(t) = h

(as−1

vs+1

)= constant. These values

define conditions where a particular instigation function has asingle value across time and constitute what in the dynamicsof action is referred to as a constant environment, employed intheoretical discussions to highlight that variability in behavior canbe expected from the dynamic processes of the theory withoutpostulating variability as part of the theory. It is also the specialcondition imposed in the Birch (1984) paper.

5. Activation time theory

If the aim of deriving a data generating process for a streamsegment protocol based on activation time analysis is to beaccomplished, it is necessary that a rigorous connection betweenthe behavior theory of the dynamics of action and the datarecorded in a stream segment protocol be formed. In hand is Eq.(1), which provides the instigation function to be applied to theconstant basic instigating force, F ′, used to arrive at the strength oftendency for the associated alternative activity as specified in thedynamics of action:

T =∫F ′I(t)dt

which, when Eq. (1) is used to substitute for I(t), becomes

T =∫F ′h

[(a+ bt)s−1

(v + wt)s+1

]dt (2)

for constants F ′, h, a, b, v, w ≥ 0 and s > 1.Integrating Eq. (2) and determining the constant of integration

results in the theory’s expression for the strength of tendency for

D. Birch / Journal of Mathematical Psychology 53 (2009) 106–118 109

a non-ongoing activity across time as measured from the point atwhich the new ongoing activity is initiated. The result is

T = TI + F ′[

hs(vb− aw)

] [(a+ btv + wt

)s−

( av

)s]. (3)

Examination of Eq. (3) reveals that T growsmonotonically from theinitial value TI for t = 0 to approach

TI + F ′[

hs(vb− aw)

] [(bw

)s−

( av

)s]as t increases without limit.Theoretically, TI is very important. In general, T = TI for

t = 0 with the implication that the tendency to engage inan activity is maintained across changing ongoing activities andtheir accompanying environments. The initial strength of tendencyTI for an activity at the beginning of a new transition interval,what Atkinson and Birch (1970) call an inertial tendency, is theconsequence of prior exposure to instigating forces and sets thestage for further growth under the aegis of the new ongoingactivity.At the point of transition the strength of tendency for the new

AO assumes its value of TO with the tendencies for the alternativeactivities adjusting proportionally to the new 0 to TO intervalas stated earlier. The previously ongoing activity, the originatingactivity to the transition, has its tendency fall to zero strengthupon being replaced, based on the cessation consummatory lagprovision of the dynamics of action. The assumption of tendencyreduction to zero is stronger than necessary but allows a commonreference point for all tendencies and makes for simpler notation.

6. The bridge to transition times

The linkage between the dynamics of action and the observeddata of a stream segment protocol can now be established byderiving the theoretical value for a transition time (i.e. thetime taken for a transition to be accomplished measured fromthe initiation of the originating activity to the initiation of thedestination activity). With the newly ongoing activity designatedAO and its strength of tendency TO we place the remaining K − 1activities, A2 to AK , in ascending order according to their F ′ basicinstigating force values. Then, according to Eq. (3), for any of thesepotential destination activities, AD, a transition into the activityfrom the ongoing AO is projected to occur when

TD = TDI + F ′D

[h

s(vb− aw)

] [(a+ btDv + wtD

)s−

( av

)s]= TO. (4)

Although not so subscripted in Eq. (4), the parameter F ′D and thea, b, v, w, h and s of the instigation function all have constantvalues that are contingent on the particular AO currently ongoing.The difference between TO and TDI is the initial shortfall in TD

that must be surmounted in order for a transition to occur. Theshortfall for an activity is at its full value of TO immediately uponthat activity’s being supplanted in a transition (at which point thetendency for the activity is reduced to zero) following which the(TO − TD) shortfall proceeds toward zero in synchrony with thegrowth of TD and the countdownof the activation time. By denotingthe initial shortfall, or gap, in TD as GDI = (TO − TDI), Eq. (4) can berewritten as

GDI = F ′D

[h

s(vb− aw)

] [(a+ btDv + wtD

)s−

( av

)s]. (5)

The relationship expressed in Eq. (5) refers to the projectedtime, tD, it would take to bring GDI to zero and thus to accomplishthe transition given that a transition into AO has just taken place.

The requisite value of tD is found by solving Eq. (5) with the resultpresented as Eq. (6).

tD =v[(GDIF ′D

) ( sh

)(vb− aw)+

( av

)s] 1s− a

b− w[(GDIF ′D

) ( sh

)(vb− aw)+

( av

)s] 1s . (6)

Ideally, the theory of activation could be made to project atime to transition for each of the non-ongoing activities on eachtransition occasion. (See Birch (1968) and Timberlake and Birch(1967) for examples of experiments investigating single, isolatedchanges of activity carried out in the context of the dynamics ofaction.) This would be very important because it would designatethe destination activity for the next projected transition (the onewith the smallest projected transition duration) and the time itwould take to accomplish this transition. Such, however, is notpossible. The chief difficulty is that GDI will take on different valueson different instances of transition even as all the other parametervalues, while unknown, are taken to be constants tied to the natureof the ongoing activity.However, other, related critical features of a stream segment

protocol are observable and can be subjected to theoreticalanalysis. These are the aggregated measures of the relativefrequencies of transitions and the distributions of the transitiontimes forwhich it is possible to derive expectations from activationtime theory. As can be seen in Eq. (6), what is needed is thetheoretical specification of the distribution of the GDI basedon activation time theory analysis of the repeated instances oftransitions into a particular AO throughout the stream segmentinterval.It appears that the shape of the GDI density function over

repeated instances of transitions out of the same ongoing activityis best summarized as rectangular. This conclusion is drawnfrom the following considerations: Since GDI = (TO − TDI)and TO is a constant, the distribution of GDI will be givendirectly by the distribution of TDI . Since the appearances ofthe transitions during the repeated activation time countdownsof an activity are determined independently of the targetedactivity and derive from complex sequences of different ongoingactivities and their instigating functions and basic instigatingforce hierarchies, they should be spread evenly across the targetactivity’s activation interval. A rectangular distribution for theappearances of transitions across the activation time intervaltranslates into a rectangular distribution for TDI which, in turn,results in an expected rectangular distribution for GDI in the 0 toTO interval. In principle, the proposed rectangular distribution forTDI could be checked by a comprehensive program of calculation oftendency values directly from the theory but the endeavor seemshardly worthwhile. For the moment it seems more practical toadopt the equally likely hypothesis for G21, . . . ,GKI as the bestguess of what the theoretical distributions are; that is, what wouldbe found if a rigorous derivation from the activation time modelwere to be accomplished.

7. Relative frequencies of activity choices in a stream segment

7.1. Activation time theory derivations

Every transition recorded in a stream segment protocol is aninstance of behavioral choice signifying a preference for engagingin one activity over all the others at that particular moment intime. Activation time theory holds that a transition occurs whenthe strength of the tendency for an activity not ongoing catchesup to the strength of tendency for the ongoing activity. Thus,a transition from the ongoing activity to a specified destination

110 D. Birch / Journal of Mathematical Psychology 53 (2009) 106–118

activity signifies that TD = TO at that moment in time and thatthe theoretically projected transition time of tD as presented inEq. (6) has proved less than the transition durations projectedtheoretically for all the other K − 2 activities.Using the AO to A2 transition as a prototype, with AO denoting

the ongoing activity and activities A2 through AK ordered accordingto their basic instigating force values F ′2 ≤ F

′

3 ≤ · · · ≤ F′

K , wecan develop a theory of choice for the stream of behavior. Thenecessary and sufficient condition for a transition from AO to A2is that t2 ≤ t3, . . . , t2 ≤ tK . It is easily shown, using Eq. (6),that the condition t2 ≤ t3 reduces to the condition

G2IF ′2≤

G3IF ′3

and that the other conditions to be met for the transition AO to A2correspondingly reduce to G2IF ′2

≤G4IF ′4

, . . . ,G2IF ′2≤GKIF ′K.

The relative frequency of the AO to A2 transition, P(AO, A2), ina stream segment comprised of K activities can be determinedtheoretically by evaluating the multiple integral

P(AO, A2) =∫Rf (G2I , . . . ,GKI)d(GKI) · · · d(G2I)

over the proper region R. In the case at hand

f (G2I , . . . ,GKI) = f (G2I)f (G3I) . . . f (GKI) =(1TO

)K−1and, since the GDI ,D = 2 to K , are taken to have independentrectangular distributions over the interval 0 to TO, we have

P(AO, A2) =∫R

(1TO

)K−1d(GKI) . . . d(G2I). (7)

The region of integration, defined for A3 through AK relative toA2 since it is P(AO, A2) that is of interest, is given by

(F ′3F ′2

)G2I ≤

G3I ≤ TO for A3 through to(F ′KF ′2

)G2I ≤ GKI ≤ TO for AK and

0 ≤ G2I ≤(F ′2F ′K

)TO for A2.

The evaluation of Eq. (7) over G3I , . . . ,GKI has a result that canbe written

P(AO, A2) =∫f (G2I)d(G2I)

=

∫ (1TO

)K−1 ( 1T0 − G2I

) K∏k=2

[T0 −

(F ′kF ′2· G2I

)]d(G2I)

where K ≥ 2, and the interval of integration is 0 ≤ G2I ≤(F ′2F ′K

)TO.

This expression generalizes to the theoretical relative frequenciesfor all transitions out of AO into the K − 1 destination activities ADas follows:

P(AO, AD) =∫f (GDI)d(GDI)

=

∫ (1TO

)K−1 1(T0 − GDI)

×

K∏k=2

[T0 −

(F ′kF ′D· GDI

)]d(GDI) (8)

where K ≥ 2, 0 ≤ GDI ≤(F ′DF ′K

)TO, and F ′2 ≤ · · · ≤ F

′

K .The theoretical relative frequency of any destination activity,

P(AO, AD), is obtained by integrating Eq. (8) over the specifiedinterval. The resulting expression is of regular form but unwieldy.Examples of the specific equations for several low values of K ,however, suffice to reveal the nature of the regularities and providethe basis for extrapolating to any value of K . In particular, the

theoretical relative frequencies P(AO, A2) for K = 2, 3, 4, and 5are given by

K = 2: P(AO, A2) = 1

K = 3: P(AO, A2) =[1

(1)(2)

](F ′2F ′3

)K = 4: P(AO, A2) =

[1

(1)(2)

](F ′2F ′4

)−

[1

(2)(3)

](F ′2F ′4

)(F ′3F ′4

)K = 5: P(AO, A2) =

[1

(1)(2)

](F ′2F ′5

)−

[1

(2)(3)

] [(F ′2F ′5

)(F ′3F ′5

)+

(F ′2F ′5

)(F ′4F ′5

)]+

[1

(3)(4)

](F ′2F ′5

)(F ′3F ′5

)(F ′4F ′5

).

(9)

The corresponding equations for transitions AO to A3 throughAO to AK−1 for the various K can be found by simply interchangingF ′3 through F

′

K−1 with F′

2 in Eq. (9). The equations for the relativefrequencies of the AO to AK transitions, P(AO, AK ), are of a differentform because the interval of integration for GKI is 0 ≤ GKI ≤ TO andfor K = 2, 3, 4, and 5 appear as

K = 2: P(AO, A2) = 1

K = 3: P(AO, A3) = 1−(12

)(F ′2F ′3

)K = 4: P(AO, A4) = 1−

(12

)[(F ′2F ′4

)+

(F ′3F ′4

)]+

(13

)(F ′2F ′4

)(F ′3F ′4

)K = 5: P(AO, A5)

= 1−(12

)[(F ′2F ′5

)+

(F ′3F ′5

)+

(F ′4F ′5

)]+

(13

)[(F ′2F ′5

)(F ′3F ′5

)+

(F ′2F ′5

)(F ′4F ′5

)+

(F ′3F ′5

)(F ′4F ′5

)]−

(14

)(F ′2F ′5

)(F ′3F ′5

)(F ′4F ′5

).

(10)

The extrapolation to larger values of K is apparent and, as is tobe expected, the sum P(AO, A2)+· · ·+P(AO, AK ) = 1 for all K ≥ 2.There are now available, for any given originating activity, K−1

equations for the P(AO, A2) through P(AO, AK ) written in terms ofthe K − 2 unknowns(F ′2F ′K

), . . . ,

(F ′K−1F ′K

)which as a notational convenience are simplified to

R2,K =(F ′2F ′K

), . . . , RK−1,K =

(F ′K−1F ′K

).

The K − 2 expressions for the P(AO, A2), . . . , P(AO, AK−1) transi-tions out of a given originating activity can be used directly in con-junction with the observed relative frequencies of the transitionsto arrive at numerical values for the R2,K , . . . , RK−1,K . This deter-mination is made separately for each of the K originating activi-ties and therefore yields values for the basic instigating forces thatare relative to the largest basic instigating force value conditionalon the originating activity. Solving the equations simultaneouslyis formidable but methods of successive approximations are easilydeveloped.

D. Birch / Journal of Mathematical Psychology 53 (2009) 106–118 111

7.2. Important property of certain stream segment protocols

The transitions in a stream segment protocol of K mutuallyexclusive and exhaustive activities can be summarized in a K × Ktransitionmatrix which has zeros down the diagonal. If the streamsegment, as recorded, begins and ends with the same activity,there will be, for each activity, the same number of initiationsas terminations of the activity; that is, the number of transitionsinto the activity will equal the number out of the activity sinceeach intervening activity is both entered and exited. The set of Kequations for the number of transitions out of each of the activitiesin such a balanced stream segment transition matrix refers to theequations for the rows of the matrix which have the form

nk,1 + nk,2 + · · · + nk,K = nk

where k successively takes on the values 1 through K and nk,k = 0or, alternatively,

pk,1nk + pk,2nk + · · · + pk,Knk = nk

where the pr,c are the relative frequencies of the transitionsconditional on the row activity and pk,k = 0.More importantly for present purposes, because of the special

circumstances of the balanced transition matrix by which the rowfrequencies equal the column frequencies, there exists a set ofequations for the number of transitions into the activities, writtenfor column k as

n1,k + n2,k + · · · + nK ,k = nk

with nk,k = 0, or as

p1,kn1 + p2,kn2 + · · · + pK ,knK = nk (11)

with k taking on the values 1 through K and pk,k = 0.If the pr,c are taken as known, the expressions of Eq. (11)

constitute a set ofK simultaneous equations inn1, . . . , nK , onlyK−1 of which are independent. Dividing through by nK gives Eq. (12),which is stated in terms of the frequencies relative to nK andthereby introduces the overall relative frequencies P(1), . . . , P(K)into the discussion,

p1,k

(n1nK

)+ p2,k

(n2nK

)+ · · · + pK ,k =

(nknK

). (12)

The first K − 1 expressions of Eq. (12) can be solvedsimultaneously for the K − 1 unknowns

(n1nK

), . . . ,

(nK−1nK

)with

these values directly convertible to their corresponding overallrelative frequency values P(1), . . . , P(K).That the overall relative frequencies of activities in a stream

segment protocol that begins and ends with the same activityare known, if the relative frequencies of the transitions out ofeach activity are known would seem to be a useful piece ofinformation about such stream segments. It means that for astream segment protocolwith a balanced transitionmatrix a singlesystem of relationships underlies both conditional and overallrelative frequencies such that knowledge of the first is sufficient forknowing the second. From the standpoint of activation time theoryit follows that the overall relative frequencies of activities, as wellas the conditional relative frequencies of activities, are derivablefrom the relative instigating force values of the theory via Eqs. (9)and (10).

7.3. Initial consideration of composite activities in the stream ofbehavior

The consequences of the conditions defining the balancedstreamsegment transitionmatrix go beyond the findings regarding

the dependence of the overall relative frequencies of activities ontheir conditional relative frequencies. They also encompass theresults of combining an original set of activities into a derivativeset of composite activities in which activities that are members ofa composite are treated as instances of the same activity in regardto transitions. It turns out that the relative frequency informationin the set of composite activities is contained in the conditionalrelative frequencies of the original set of activities.To illustrate this finding we take a given K × K balanced

transition matrix and construct a new (K − 1) × (K − 1) matrix,which is also balanced, by combining activities I and J into thecomposite activity (IJ). The only conditional relative frequenciesthat are different in the composite activity transition matrix fromthose in the original matrix are the transitions impacted by theI and J rows and columns: The conditional relative frequenciesof transitions into the composite activity (IJ) from the otheractivities are given by the sums of the original conditional relativefrequencies (i.e. pA,(IJ) = pA,I + pA,J , pB(IJ) = pB,I + pB,J , etc.).The conditional relative frequencies out of the composite activityinto the other activities are given by the sums of the separateoriginal conditioned relative frequencies into the activities minustheweighted sumof the original pI,J and pJ,I . For example, the valuefor p(I,J),A, the conditional relative frequency of the transitions fromthe composite activity (IJ) into original activity A is calculated from

p(I,J),A =nI,A + nJ,An(IJ)

where

n(IJ) = (nI − nI,J)+ (nJ − nJ,I)

giving

p(I,J),A =pI,AnI + pJ,AnJ

n(IJ)=

pI,A(nInK

)+ pJ,A

(nJnK

)n(IJ)nK

. (13)

Since(nInK

),(nJnK

), and

(n(IJ)nK

)are functions of the original con-

ditional relative frequencies exclusively, the new p(IJ),A, presentedas Eq. (13), is as well. Corresponding findings hold for all the otherconditional relative frequencies of transitions out of composite ac-tivity (IJ). With the conditional relative frequencies for the com-posite activity in hand, the overall relative frequencies of all theactivities can be calculated.The most significant general finding for balanced stream

segment transition matrices, both original and composite, is thatall the basic relative frequencies information in the matrices iscontained in the conditional relative frequencies of the originalmatrix. This constitutes a major conceptual efficiency in theconsideration of balanced stream segment transition matrices andhas potentially far reaching implications for activation time theory.From Eqs. (9) and (10) it follows that the conditional relativefrequencies for a given set of transitions can be written in terms ofthe instigating force values of activation time theory and, since therelative frequencies of composite activities can bewritten in termsof the conditional relative frequencies of the original activities,it follows that the relative frequencies of composite activitiescan be stated in terms of instigating force values as well. Theimplication is that, in principle, all activities can be conceptualizedas composite activities governed by instigating forces reducible tocombinations drawn from a primative set of elemental instigatingforces according to known rules.

112 D. Birch / Journal of Mathematical Psychology 53 (2009) 106–118

8. Transition time distributions in stream segment protocols

8.1. General case

Activation time theory provides equations that link directly theobserved relative frequencies of transitions in a stream segmentprotocol to the relative instigating force values for the activities.In the course of the derivation that culminates in Eqs. (9) and (10)the density f (GDI)d(GDI) for the initial shortfall present at a pointof transition for a destination activity, D, was defined as that givenin Eq. (8). Our present interest is with the distribution of the AO toAD transition times which is realized by the conditional density

f (GDI/AO, AD)d(GDI) =[

1P(AO, AD)

](1TO

)K−1×

1(T0 − GDI)

K∏k=2

[T0 −

(F ′kF ′D

)· GDI

]d(GDI) (14)

for 0 ≤ GDI ≤(F ′DF ′K

)TO.

To get from the density of the initial tendency shortfall, GDI ,for destination activity D to the density of the transition times,tD, aggregated over repeated instances of the AO to AD transitionin a stream segment, Eq. (6) is used. Eq. (6) makes possible thederivation of the relative frequencies of transition as presented inEqs. (9) and (10) and now, togetherwith Eq. (14), provides the basisfor the derivation of the conditional distributions of the transitiontimes.The path to the transition time conditional distributions is clear:

GDI and its derivative d(GDI) from Eq. (5) can be used to changevariables in Eq. (14) from GDI to tD. Although all is straightforwardin this process, the resulting equations along the way are lengthyand cumbersome, particularly as K becomes larger. The expansionof Eq. (14) produces a power function inGDI of (K−2)degreewhichis then elaborated by the substitutions for the GDI and d(GDI). Itis possible, however, even for K = 5, to discern regularities inthe f (tD/AO, AD) density that permit extrapolations to larger valuesof K .For K = 5 with the AO to A2 transition serving as prototype,

Eq. (14) expands to

f (G2I/AO, A2)d(G2I) =[

1P(AO, A2)

](1TO

)4 [TO −

(F ′3F ′2

)G2I

]×

[TO −

(F ′4F ′2

)G2I

] [TO −

(F ′5F ′2

)G2I

]d(G2I) (15)

for 0 ≤ G2I ≤(F ′2F ′5

)TO. A change of variable from the shortfall G2I

to the transition duration t2 is accomplished by using Eq. (6) andthe resulting derivative

d(G2I) = F ′2h⌊

(a+ bt2)s−1

(v + wt2)s+1

⌋d(t2).

Substitutions into Eq. (15) result in the conditional densityf (t2/AO, A2)d(t2). In order to make clear the regularities in thet2 density the variable Vc(t2) is introduced for purposes ofpresentation. Let

Vc(t2) =(a+ bt2)cs−1

(v + wt2)cs+1for c = 1, 2, . . . , K − 1

which permits f (t2/AO, A2)d(t2) to be written as

f (t2/AO, A2)d(t2) =[

hF2P(AO, A2)

]{V1(t2)−

[h(F3 + F4 + F5)s(vb− aw)

]

×

[V2(t2)−

( av

)sV1(t2)

]+

[h2(F3F4 + F3F5 + F4F5)

s2(vb− aw)2

]×

[V3(t2)− 2

( av

)sV2(t2)+

( av

)2sV1(t2)

]−

[h3F3F4F5

s3(vb− aw)3

]×

[V4(t2)− 3

( av

)sV3(t2)+ 3

( av

)2sV2(t2)

−

( av

)3sV1(t2)

]}d(t2) (16)

for 0 ≤ t2 ≤ U where

U =v

{[vss(vb−aw)

hF5+ as

] 1s− a

}vb− w

[vss(vb−aw)

hF5+ as

] 1s

.

In Eq. (16) a refinement in notation andmeaning for instigatingforce has been made. As has already been discussed, a basicinstigating force, F ′D with the capability to strengthen the actiontendency for a non-ongoing, destination Activity D, is set in placeby the ongoing activity and its associated environment. However,it is recognized that the basic instigating force value is divided byTO, the maximum value attainable by TD, in the density for t2 andthat bywriting FD =

F ′DTOit ismade clear that the theory of activation

is stated in terms that are relative to TO, the strength of tendencyof the ongoing activity. Note that the change in notation to FD fromF ′D leaves Eqs. (9) and (10), the derived expressions for the relativefrequencies of transitions, unchanged in meaning.By presenting the t2 density in the form of Eq. (16) the

regularities of the density equations for increasing values of K areuncovered—just let F5 = 0 for the K = 4 density and F5 =0, F4 = 0 for the K = 3 density. Perhaps the only regularity inEq. (16) not immediately apparent as K increases relates to thepositive valued integer member of the coefficients for the Vc(t2)terms: The single instance of V1(t2) alone is based on (1); the twoinstances of V2(t2) and V1(t2) on (1 1); the three of V3(t2), V2(t2),and V1(t2) on (1 2 1); and the four of V4(t2), V3(t2), V2(t2), andV1(t2) on (1 3 3 1) all of which will be recognized as the sequenceof binomial coefficients. These patterns apply to K = 3 and K = 4as well as to K = 5 and would appear to allow extrapolation to allhigher values of K without difficulty. The density f (t2/AO, A2)d(t2)for K = 5 contains the complete set of regularities inherent in theprogression of increasing K and can be used as the basis for writingall the densities f (t3/AO, A3)d(t3), . . . , f (tK/AO, AK )d(tK ) for anyvalue of K . To do this one need only interchange F2 with F3, . . . , FKsuccessively in the density equation for the appropriate K .

8.2. Distribution of probability for transition times

Of more direct use than the density f (t2/AO, A2) given in Eq.(16) in bringing the theory to bear on stream segment protocolsis the distribution of probability F(u2/AO, A2) obtainable from theintegral of the density. The clock readings in every stream segmentprotocol will be recorded to the degree of fineness deemedappropriate for the activities being observed, whether seconds,minutes, hours or centuries, and integrating the density over theproper intervals coordinates the theory to the data. For example,if clock readings are recorded to the nearest hour, an entry of teno’clock is interpreted theoretically as encompassing the intervalnine thirty to ten thirty. The density is integrated accordinglyover that interval and the resulting probability assigned to the teno’clock value. In this way is the continuous density transformed

D. Birch / Journal of Mathematical Psychology 53 (2009) 106–118 113

into a discrete valued probability distribution for purposes ofanalysis.The cummulative distribution of probability F(u2/AO, A2),

originating in the density f (t2/AO, A2) is defined to be

F(u2/AO, A2) =∫ u2

Of (t2/AO, A2)d(t2)

and for f (t2/AO, A2) from Eq. (16)

F(u2/AO, A2) =[

F2P(AO, A2)

]{D1

[(a+ bu2v + wu2

)s−

( av

)s]−D2

[(a+ bu2v + wu2

)2s−

( av

)2s]

+D3

[(a+ bu2v + wu2

)3s−

( av

)3s]

−D4

[(a+ bu2v + wu2

)4s−

( av

)4s]}(17)

with distribution coefficients

D1 =[

hs(vb− aw)

]{1+

[h(F3 + F4 + F5)s(vb− aw)

] ( av

)s+

[h2(F3F4 + F3F5 + F4F5)

s2(vb− aw)2

] ( av

)2s+

[h3(F3F4F5)s3(vb− aw)3

] ( av

)3s},

D2 =[

h2s(vb− aw)

]{[h(F3 + F4 + F5)s(vb− aw)

]+ 2

[h2(F3F4 + F3F5 + F4F5)

s2(vb− aw)2

] ( av

)s+ 3

[h3(F3F4F5)s3(vb− aw)3

] ( av

)2s},

D3 =[

h3s(vb− aw)

]{[h2(F3F4 + F3F5 + F4F5)

s2(vb− aw)2

]+ 3

[h3(F3F4F5)s3(vb− aw)3

] ( av

)s},

and

D4 =[

h4s(vb− aw)

]{h3(F3F4F5)s3(vb− aw)3

}.

The coefficients D1,D2,D3, and D4, based on the instigationfunction and the configuration of instigating forces, tie together thecontinuous density f (t2/AO, A2) and the distribution of probabilityF(u2/AO, A2) to provide the means of analysis of the discrete timevalues recorded in a stream of behavior protocol.

8.3. Constant instigation exposure

An additional important note relates to the special cases arisingfrom the instigation function. The densities for many of thesespecial cases can be found by imposing the special case conditionsdirectly on the complete form of the density functions but cautionis in order. For example, one notable exception occurs for thetheoretically important special case of constant instigation acrossthe transition interval, I(t) = h

(as−1

vs+1

), obtained from Eq. (1) for

b = 0 andw = 0.The growth in tendency for a non-ongoing activity for this

‘‘constant environment’’ reference condition is linear,

T = TI + F ′h(as−1

vs+1

)t

= TI + F∗t with F∗ = F ′h(as−1

vs+1

);

for the prototype AO to A2 transition the transition time is t2 =G2IF∗2.

The derived conditional density of t2, given an AO to A2 transition,for K = 5 is

f (t2/AO, A2)d(t2) =[

F2P(AO, A2)

][1− (F3 + F4 + F5)t2

+ (F3F4 + F3F5 + F4F5)t22 − F3F4F5t32

]d(t2) (18)

for 0 ≤ t2 ≤ 1F5and F5 =

F∗5TO. For increasing K the power functions

for these reference condition densities continue to higher degreesand more extended coefficients.

9. Activation time theory analysis of a continuous stream ofbehavior segment

9.1. Morgan O’Hara stream of behavior protocols

Morgan O’Hara (O’Hara & Hewitt, 1983) has, for many yearsas part of her art, recorded her stream of behavior in terms of24 activity categories grouped by O’Hara into four compositecategories denoted Maintenance (M), Creation (C), Education (E),and Socialization (S) plus the category Night Sleep (N). Theindividual activities coded within Maintenance are identifiedby O’Hara as Earning Money (M1), Doing Errands (M2), DoingPaperwork (M3), Financial Accounting (M4), Commuting (M5),and Daytime Sleeping (M6). Within Creation she has Doing Art(C1), Explaining Her Art (C2), Written Meditation (C3), RecordingDreams (C4), Looking At Other Art (C5), and Doing Nothing (C6);within Education there are Studying (E1), Reading (E2), ListeningTo Music (E3), Working On Languages (E4), Exercising (E5), andSitting Meditation (E6); and Socialization is comprised of WithFriends (S1), With a Lover (S2), With Her Daughter (S3), PersonalCorrespondence (S4), and With Acquaintances (S5). Night Sleep isrecorded as an unanalyzed whole marking the interval betweenher retiring at night and arising the next morning.Not all of these individual activities, by their nature, are

mutually exclusive but O’Hara tended to record them as such. Forexample, in the present data set just over 8% of the entries refer tomore than one individual activity; each of these compounds, 375in number, was converted into an individual entry by designatingthe activity appearing first in the compound as the entry. Theindividual activity categories are exhaustive but do conformto O’Hara’s convention that no instance of an activity that isongoing for less than five minutes is recorded; in practice, theoverwhelming number of entries are to the nearest quarter hourclock reading.The present analyses are based on the O’Hara data for the five

composite activities for January, February, May, and June 1984during which time she was living in New York City; March, July,and August 1984 when she lived in San Francisco; and January,February, and March 1987 when she was in Europe. The data arearrived at by ignoring the subscripting of the individual activitydesignations and treating the stream segment as if the M, C, E,S, and N categories were recorded directly, including the clockreading for each instance of an activity transition.

9.2. Sufficiency of activation time theory for continuous stream ofbehavior data

Summaries of the observed frequencies of transitions amongthe five composite activities, their relative frequencies, and the

114 D. Birch / Journal of Mathematical Psychology 53 (2009) 106–118

Table 1Observed transition frequencies, relative frequencies, instigating force scale values and times spent in the activities forMorganO’Hara’s complete streamof behavior segment.

Ongoing activity Destination activityN C E S M Sum

N Freq x 52 14 31 198 295Rel freq x .176 .047 .105 .671 .999F-Scale x .226 .067 .142 .565 1.000Time spenta x 26,630 7355 13,920 96,440 144,345

C Freq 48 x 132 140 320 640Rel freq .075 x .206 .219 .500 1.000F-Scale .094 x .233 .245 .428 1.000Time Spent 2695 x 12,285 12,265 29,580 56,825

E Freq 70 142 x 163 284 659Rel freq .106 .215 x .247 .431 .999F-Scale .127 .234 x .261 .378 1.000Time Spent 3200 5445 x 8545 14,920 32,110

S Freq 50 127 150 x 369 696Rel freq .072 .182 .216 x .530 1.000F-Scale .092 .213 .245 x .449 .999Time Spent 8965 8245 12,210 x 32,145 61,565

M Freq 127 319 363 362 x 1171Rel freq .108 .272 .310 .309 x .999F-Scale .126 .274 .300 .300 x 1.000Time Spent 7550 37,855 41,250 43,590 x 130,245

a Times are in minutes; no individual activity duration shorter than 5 min was recorded.

times taken to accomplish the transitions for Morgan O’Hara’scomplete stream segment protocol can be found in Table 1.Also contained in the table are the instigating force scalevalues, denoted F-Scales, obtained from Eq. (9) and the relativefrequencies of Table 1 and presented as fractions so as to sumto unity. The observed relative frequencies of transitions can berecovered exactly from the instigating force scale values using Eqs.(9) and (10).Table 1 reveals clear differences in the relative number of

transitions into the five destination activities with the fewest,in general, into Night Sleep, the most into Maintenance, andmidnumbers into Creation, Education, and Socialization. There isin Table 1 remarkably little difference in the pattern of relativefrequencies across the destination activities for the differentongoing activities. Such a lack of interaction between the ongoingactivities and the destination activities points to the presence ofa single configuration of instigating forces that holds across thevarious ongoing activities. Also different in Table 1 are the totaltimes spent in the five activities, both overall and just procedingthe individual transitions.The theoretically expected transition time distributions to be

used in comparisons to the observed distributions were obtainedas follows: Eq. (16) gives the theoretical density for the AO, A2transition times written in terms of the instigation functionconstants h, a, b, v, w, and s, which are common to all transitionsout of AO and the instigating force parameters F2, F3, F4, and F5,which reduce to the single unknownparameter F5when advantageis taken of the results from the analysis of the activity choices.The conditional densities for t3, t4 and t5 can be obtained directlyfrom Eq. (16) by interchanging the subscripting for Activities 3, 4,and 5 successively with that for Activity 2. All the densities areconfined to the interval 0 < t < U , all have the same form inVc(t) = (a+bt)cs−1

(v+wt)cs+1for c = 1, 2, . . . , K−1 and the various densities

differ only in the constantsmaking up the coefficients for the Vc(t).This is important in that it allows the addition of the equationsterm by term over the different transitions to arrive at a singledensity for all transition times out of AO, weighted and pooled overthe destination activities. This density has the form common to thedensities for the individual transitions out of AO. Numerical valuesfor the parameters are achieved by fitting the pooled theoreticaldistribution to the pooled observed distribution.

9.3. Methods for determining instigation function parameter values

Converting the activation time theory densities for the individ-ual activities in order to test the theory’s representation of the ob-served discrete probability distributions requires two steps: Thedensities, Eq. (16), are integrated to arrive at the distributions ofprobability, Eq. (17), which are then evaluated over the intervalsappropriate to the observed discrete probability distributions. Inthe present case of the O’Hara data, the stream protocol recordingsof the time taken to accomplish transitions are most often madeto the nearest 15 min, except for a set of short transition timesrecorded as taking 5 min. To bring the theoretical densities intoline with the observed discrete distributions, the distributions ofprobability are evaluated for 5.0, 7.5, 22.5, 37.5, 52.5 min, etc. upto U , the upper limit for the densities, and the differences in prob-ability for 7.5− 5.0, 22.5− 7.5, 37.5− 22.5, 52.5− 37.5, etc. areassigned to the discrete distribution values 5, 15, 30, 45 min, etc.to conform to the written record of the protocol.The question raised at this point is whether the theory

of activation can accommodate the variety of transition timedistributions encountered in a stream of behavior protocol. Noattempt has been made to write a best fitting algorithm forspecifying a theoretical distribution; rather, what is sought is atheoretical depiction of the observed distributions of transitiontimes by assigning values to the parameters of the theory such thatthe most significant features of the distributions are represented.To this end a procedure was followed by which the observedtransition times were used to identify three points to be fit bythe distribution function: the upper limit, U , as boundry on thedistribution; the mode, to establish the height of the distribution;and a third point chosen to provide information regarding thesteepness of the slope in the neighborhood of the mode. The lattertwo points were then fit (each to within +.01 of its observedproportion), along with U , by selection of values for the h, a,b, v, w, and F constants of the theory. This procedure is a variationon what is known in the curve fitting literature as the method ofselected points (Lewis, 1960).Parameter values for the transition time distributions out of the

Creation, Education, Socialization andMaintenance activities werechosen by selecting an upper limit U for each distribution and thenfitting each of the observed proportions for distribution points 15and 30 min using the complete data weighted and pooled over the

D. Birch / Journal of Mathematical Psychology 53 (2009) 106–118 115

Table 2Activation theory parameter values calculated using the observed transition timedistribution out of each ongoing activity.

Ongoing activity Theoretical parametersh s a b v w F5 U

N 1 3.7 0 1 320 0 110 607.5C 1 1.1 1 0 130 1 263 787.5E 1 4.9 0 1 12 1 70 337.5S 1 5.0 6 1 21 1 86 547.5M 1 2.3 0 1 55 1 145 907.5

destination activities. In the case ofNight Sleep, adjustments to thisprocedure were introduced.The mode for the observed distribution of transition times

out of Night Sleep is not well defined, residing at 480 minwith a frequency of 26 out of a total frequency of 295 spreadover a 60–990 min range. In an attempt to arrive at a betterestimate of the location of the high point for the transition timedistribution than that given by the mode, the observed transitiontimes were grouped into successive intervals of 150 min width.The frequencies associated with these intervals in ascending orderare 4, 12, 90, 156, 28, 4 and 1; the first and fourth intervalswere selected for use in estimating the height of the distributionfunction and something of its slope.Table 2 presents the parameter values for each of the five

O’Hara activities as ongoing, solved for using the selected pointsprocedure. When these parameter values are inserted in theappropriate versions of Eq. (17), the theoretical proportions for thetransition time distributions are achieved.Of first concern is the degree of agreement between the

complete pooled observed transition time distributions andtheir theoretical counterparts. The percentage of the sums ofsquares of the observed frequencies that is accounted for by thecorresponding theoretical frequencies, % Acct, is the measure usedto evaluate the goodness of fit for each transition time distributionbased on the pooled transition times out of each activity. Fits of84.2, 93.5, 97.5, 95.1, and 91.9% for activities Night Sleep, Creation,Education, Socialization and Maintenance, respectively, establishthat these transition time data from the O’Hara stream of behaviorprotocols are fit sufficientlywell by determining suitable empiricalvalues for the parameters of the activation time theory.The corresponding % Acct values associated with the sums of

squares out of the five activities for the constant instigation versionof the theory, arrived at by fitting only one empirical constant, are−219.9% (due to the difficulty of fitting an observed function withincreasing frequencies by a theoretical function with decreasingfrequencies), 87.9, 93.4, 88.1 and 91.3%. These values, each ofwhich is below the value of its full theory counterpart, serve as atheoretically based reference for evaluating the % Acct results ofthe full theory.Also accommodated by the parameter values of Table 2 is the

variety of instigation functions associatedwith the differentO’Haraactivities. Included are instigation functions that aremonotonicallyincreasing (Night Sleep), monotonically decreasing (Creation),and nonmonotonically increasing–decreasing around a maximum(Education, Socialization and Maintenance).

9.4. Theoretically expected transition time distributions

The data generating process of activation time theory providesthe means for deriving a system of equations that covers boththe relative frequencies of the appearances of activities in thecontinuous stream of behavior and the distributions of thetransition times for those appearances. Of particular importance toevaluating the conceptual value of these equations is their successin predicting certain specific aspects of individual transition time

distributions, defined by their originating anddestination activitiesin combination.At this point in our analysis of the Morgan O’Hara data we

know only that the total transition time distributions out of eachof the five activities, arrived at by pooling frequencies over thefour destination activities in each case, are fit satisfactorily bythe equations of the full theory with over 90% of the observedfrequency variability accounted for by four of the activities andover 80% by the fifth activity and that these % Acct values aresuperior to those found for the constant instigation special caseof the theory. Testing one by one the goodness of fit of thetheoretically predicted individual transition time distributions,the constituents of the pooled distributions, to their observedcounterparts using the parameter values already arrived at in thefitting process, is the next step in the analysis. The percentageof the observed frequencies of transitions accounted for bya theoretical transition time distribution, % Acct, remains themeasure of fit as the analysis is extended to include the individualtransition time distributions.When Education is the originating activity, the observed

individual distributions are fit better than when any of the otherfour activities is ongoing as the pooled % Acct fit of 97.5% ismaintained at individual levels of 95.1, 97.3, 90.3 and 95.8% fortransition distributions out of E to N, C, S, and M respectively.Observed transition time distributions out of Maintenance are alsofit well by the theory with a pooled % Acct value of 91.9% andindividual values of 83.3, 90.7, 83.3 and 93.5% for distributionsout of M to N, C, S, and E. Fits with Creation and Socialization asongoing activities are less consistently good with one % Acct valueless than 80% appearing for each (70.6% for Creation and 42.4%for Socialization). When Night Sleep is the ongoing activity, % Acctdrops from its pooled value of 84.2% to 18.6, 47.6, 57.9 and 77.6%for the individual transition time distributions. All calculations aremade using the parameter values reported in Table 2 and all %Acct values for all the transitions can be found in the diagonals ofTable 3. Finally, 15 of the 20 individual % Acct values are higher forthe full theory than for the constant instigation special case in headto head comparisons.

9.5. Matched and mismatched predictions

Seeking an additional context to assist in interpreting thedegree of support for the theory to be adduced from thesenumbers, we introduce the idea of matched and mismatchedpredictions. Matched pairs compare predicted transition time dis-tributions to their theoretically designated observed distributioncounterparts; mismatched pairs compare predicted distributionsto observed distributions not designated by the theory asmatched.If the theory is on track, it is to be expected that predictions formatched pairs will be more successful than those for mismatchedpairs because the distinctions regarding the coefficients of Eqs. (16)and (17) beingmade in the theory are borne out empirically. On theother hand, if matched pairs predictions are not more successfulthan mismatched pairs when tested against data, certain distinc-tionsmade in the theorywould seem to be irrelevant ormisguided.At issue in the present instance of the Morgan O’Hara data

is whether what appear to be only minor differences in thevalues of certain coefficients for different theoretical transitiontime distributions are reflected in the data as called for by thetheory. Presented as a prototype for transition time distributions,the density of Eq. (16), written for the transition (AO, A2), serves togenerate the densities for transitions (AO, A3), (AO, A4) and (AO, A5)simply by interchanging successively A3 with A2, A4 with A2 andA5 with A2. The resulting set of four densities all have the sameform, dictated by the parameter values of their common instigationfunction, and differ only in the combinations of the values of

116 D. Birch / Journal of Mathematical Psychology 53 (2009) 106–118

Table 3Percentage of observed transition time distribution frequencies accounted for bymatched and mismatched predictions from activation time theory for ongoingactivities: Night Sleep, Creation, Education, Socialization and Maintenance.

Theory transition Observed transition

N, E 18.6a 46.3 53.3 71.8N, S 19.2 47.6 55.4 73.3N, C 20.1 49.3 57.9 75.0N, M 28.5 65.2 68.0 77.6Out of N 84.2C, N 85.1 77.3 64.1 83.9C, E 81.7 83.5 70.1 88.1C, S 81.3 84.0 70.6 88.5C, M 71.0 93.7 79.3 93.7Out of C 93.5E, N 95.1 97.8 85.0 91.0E, C 93.6 97.3 89.2 93.2E, S 93.0 97.1 90.3 93.8E, M 88.7 94.0 95.7 95.8Out of E 97.5S, N 42.4 89.8 74.6 86.0S, C 44.8 91.5 78.9 88.8S, E 45.5 91.9 80.1 89.6S, M 56.8 91.8 89.1 95.0Out of S 95.1M, N 83.3 86.2 75.3 91.8M, C 79.2 90.7 81.9 93.2M, S 78.1 91.7 83.3 93.5M, E 78.1 91.7 83.4 93.5Out of M 91.9a Matched prediction entries are designated by underlining.

the instigating forces F2, F3, F4 and F5 as they enter into thecoefficients. These are distinctionsmade in the theory, the questionis whether they are of sufficient import that they distinguishbetween the matched and mismatched pairs’ predictions madein regard to the O’Hara data. The answer would seem to be thatthey do.Table 3 is comprised of five subtables, one each for N, C, E, S,

and M. Each subtable contains the complete set of matched andmismatched pairs for the associated ongoing activity. The rowsidentify the theoretical transitions of a pair and the columns theobserved so that the diagonal contains the matched pairs and theoff diagonal the mismatched. Entries in a subtable give the % Acctand allow comparisons within a column across the rows so thatdifferent theoretical distributions can be brought up against thesame observed distribution. All calculations use the parametervalues to be found in Table 2.Matched pairs appear to do somewhat better than mismatched

pairs in predicting the observed transition time distributions. Thehighest % Acct value within a column is found for the matchedpairs in 8 of the observed transition distributions with 12 for themismatched pairswhich, on a per pair basis, gives an 8:4 advantageto the matched pairs. When the comparison is extended to includethe top two ranks,which is the sameas askingwhichhas a greater %Acct value thematched pair or themedian of the threemismatchedpairs, the answer is the matched pair in 14 of the 20 instancesfor a 70% to 30% superiority. In a similar vein, of the total of60 mismatched pairs 2/3 have % Acct values smaller than theircorresponding match pair values.These results support the activation time theory in what is a

series of subtle distinctions based on the individual coefficients tobe applied to a common density form. At the same time it needsto be noted that, overall, the matched pairs have only a 78.6% to77.2% advantage over mismatched pairs when the 20 instances ofmatched pairs and the 20 medians for the mismatched pairs areeach averaged. It is the consistency of thematched pairs advantage,not its absolute magnitude, that gives credence to the theory’sparticular account of the process responsible for the individualtransition time distributions.

10. Hazard function considerations

The progression of conditional likelihoods, observed and the-oretical, that constitutes a hazard function offers the opportunityfor a different test of the suitability of activation time theory forstream of behavior data; it also reveals important descriptive fea-tures of a stream of behavior not immediately apparent in the pro-tocols. For example, given that Morgan O’Hara has been engagedin Night Sleep for 2 h, what is the likelihood that she will rise inthe next hour, and, failing that, successively in each of the hoursfollowing? Are these likelihoods rising or falling? The same or dif-ferent for the other activities? These are hazard function questionsto be answered using the observed data and theoretical equationsalready in hand.Results of the hazard function analysis are easily summarized:

The observed hazard function, calculated from 60-min intervals(to make it easier to discern the overall direction taken by thelikelihoods) of the pooled transition time distributions, increasesfor ongoing activity Night Sleep and decreases for ongoingactivities Creation, Education, Socialization, and Maintenance. Thetheoretical hazard functions agree with the observed in each case.These results stand in contrast to those obtained for the con-

stant environment reference case of the instigation functionwhichproduces theoretical hazard functions that increase monotonicallyfor all five of the activities, results that are clearly not in agreementwith the observed functions for Creation, Education, Socialization,andMaintenance. Thus, the constant instigation hypothesis fails tobe applicable in general.

11. A critical feature of stream of behavior data

11.1. Theoretical

The fundamental premise of the dynamics of action and ofactivation time theory is that, isomorphic with the countdownof activation times observable in a stream of behavior protocol,action tendencies grow in strength as a consequence of exposureto instigating forces and persist unless reduced by consummatoryforces. At a point of transition in a streamof behavior a new activitywith its dominant tendency becomes ongoing and all the otheractivities, each potentially the next ongoing activity, have actiontendencies that continue to grow in strength from their transitionpoint levels. In the analyses undertaken thus far the fundamentalpremise has served implicitly in the derivations of quantitativeexpectations about the relative frequencies of transitions and thetimes taken to complete those transitions—expectations that farewell when put to test against data.Still sought, however, is a test whose outcome bears directly

on the fundamental premise of the theory, one where the premiseitself is more directly at risk. Such a test is now proposed,introduced by the following conjecture: It is suggested that, for aspecified ongoing activity and any two alternative activities, theactivity less preferred as a destination will have proportionallymore short transition times than themore favored alternative. Thisis, perhaps, not intuitively obvious in that the correlation betweenthe relative popularity of an alternative activity and the relativepromptnesswithwhich it is undertaken is proposed to be negative.The basic idea behind the conjecture – that the shapes of

the conditional transition time distributions are systematicallyrelated to transition preferences – can be looked at in detailin the context of the transition time densities derived from thetheory. Using Eq. (16) for K = 5, it is straightforward to showfor F2 < F3 < F4 < F5 that f (t2/AO, A2) > f (t3/AO, A3)when t2 = t3 = 0, a finding that is easily extended to allfive activities. Thus, the intercepts for the conditional densitiesf (t2/AO, A2), . . . , f (t5/AO, A5) proceed from greater to lesser even

D. Birch / Journal of Mathematical Psychology 53 (2009) 106–118 117

Table 4Splits of frequencies of support and not support for the activation time theorycalculated from the six comparisons of pairs of transitions classified as to ongoingactivity and protocol source.

Protocol source

Ongoing Complete NYC SF EU Sumactivity +− +− +− +− +−

N 2–4 3–3 2–4 3–3 8–10C 5–1 4–2 6–0 4–2 14–4E 5–1 5–1 6–0 4–2 15–3S 2–4 3–3 2–4 6–0 11–7M 4–2 5–1 5–1 5–0a 15–2Sum 18–12 20–10 21–9 22–7 63–26a TransitionsM, S andM, C recorded the same relative frequencies and so provideno basis for making a prediction.

as P(AO, A2) < P(AO, A3) < P(AO, A4) < P(AO, A5). Furthermore,since f (t2/AO, A2), . . . , f (t5/AO, A5) all have total areas equalto unity, are unimodal and span the common interval 0 ≤t2, t3, t4, t5 ≤ U , it follows that each density crosses paths withevery other density within the 0 to U interval. The result of thesecrossings is that the densities differ systematically as just noted.

11.2. Test of the critical feature

As a test of the relationship between transition preferences andtransition times called for by activation time theory, comparisonsusing the six possible pairs of transitions within each of thefive ongoing activities were carried out, first for the completeprotocol and then for New York City, San Francisco, and Europeseparately. The crossover point for each of the six pairs of transitiontime distributions was located using the theoretical proportionsfor the 15-min intervals. Then the corresponding two sets ofobserved frequencies were summed to the crossover point, and adetermination made as to whether the relative frequency for theless favored alternative was greater than that for the more favoredto that point, as predicted from the theory.Table 4 summarizes the results of the comparisons for the

complete protocol and for the separate New York City, SanFrancisco, and Europe protocols in terms of the numbers of thesix outcomes that support and do not support the theory. For eachongoing activity there are four transitions out of that activity andso six pairs of transitions to be compared. Results for the completeprotocol show an overall 18–12 or 60% predominance of successfulover not successful comparisons with some apparent differencesattributable to the ongoing activities.When comparisons aremadefor New York City, San Francisco, and Europe separately, anapproximately 2:1 success rate holds from location to location forthe values summed over the five ongoing activities.In an attempt to devise a measure that reflects the nearness of

an observed support-nonsupport split to the 6–0 split predictedby the theory, an index of closeness is proposed in which thefraction of possible splits that are closer to 6–0 than the observedsplit is calculated. Thus, for our present K = 5 example the totalnumber of possible splits is 64 distributed 6–0 (1), 5–1 (6), 4–2(15), 3–3 (20), 2–4 (15), 1–5 (6) and 0–6 (1) calculated from thecombinations 6Cx for x = 0 through 6. The corresponding numbersand fractions of the 64 splits closer to 6–0 than the specified splitare 6–0 (0) (0), 5–1 (1) (.02), 4–2 (7) (.11), 3–3 (22) (.34), 2–4 (42)(.66), 1–5 (57) (.89), and 0–6 (63) (.98).Table 5 uses the index to measure the closeness of each + —

split of Table 4 to the predicted 6–0 split. The findings for thecomplete protocol,which also describewell those for theNewYorkCity, San Francisco, and Europe protocols separately, indicate thatwhen Creation, Education, or Maintenance is the ongoing activitythe fractions of splits closer to 6–0 than the observed split arequite small giving strong support to the theory. Ongoing activities

Table 5Closenessa of the agreement between the observed support-not support splits andthe 6–0 splits predicted by theory classified as to ongoing activity and protocolsource.

Protocol source

Ongoing activity Complete NYC SF EU Aver

N .66 .34 .66 .34 .45C .02 .11 0 .11 .07E .02 .02 0 .11 .04S .66 .34 .66 0 .33M .11 .02 .02 0 .01Aver .29 .17 .27 .11 .18a The closeness index is calculated as the fraction of possible splits that are closerto the predicted 6–0 split than is the observed split.

Socialization and Night Sleep suffer by contrast but, at least for theaverages (.33 and .45) calculated over the three protocol locations,enter on the positive side of the ledger. The overall index ofcloseness of .18 means that only roughly 20% of all possible splitsbetween successful and not successful comparisons are closer tothe theory’s 6–0 split than are the splits observed.When the same procedures are applied to the reference

condition of constant exposure to instigation using the completeprotocol, the result is less good than for the general case. Slippageoccurs for the Night Sleep and Creation splits making the overalltally 17–13 as compared to the earlier 18–12; the overall indexof closeness goes from .29 to .31. The reference condition drawson activation time theory and should show a negative correlationbetween the popularity of a transition and the presence ofshort transition times, which it does. The constant instigationfunctions of the reference condition, however, appear to be lesseffective than the instigation functions fit using the full theoryin locating the various density crossing points. The generalizationthat proportionally more short duration transition times should befound for less preferred transitions does seem to hold for MorganO’Hara’s stream of behavior protocols; activation time theoryattributes this to the influence of persisting action tendencies inthe countdown of activation times.

12. Discussion

The new data generating process introduced by activation timetheory provides the foundation for a successful mathematicalmodel of the data from the continuous streamof behavior recordedin the Morgan O’Hara protocol. The modeling of the countdown ofactivation times, which are directly observable in stream segmentprotocols, is accomplished by utilizing parts of motivation theoryas they appear in the dynamics of action, most notably theconcepts and functions of persisting tendency and instigatingforce. Activation times are critical features of stream of behaviorprotocols and their identification with action tendencies permitsthem to be studied in the wider context of motivation theory.Direct support for activation time theory is found in its handling

of various aspects of the transition time distributions tallied fromthe O’Hara protocols, most notably the fits of the full theoryto the data using the parameter values of Table 2. Over 90% ofthe observed frequency sums of squares is accounted for by thefull theory for the transition time distributions out of Creation,Education, Socialization, and Maintenance and over 80% for outof Night Sleep. In each of the five cases just cited the fit of thefull theory is superior to that of the constant instigation referenceversion of the theory where only one parameter is used to fit thedata. Similar results favorable to the theory hold for the individualtransition time distribution fits as well.Other, especially compelling direct evidence supporting the

theory comes from empirical tests of three sets of derivationsfrom the theory. The results related to matched and mismatched

118 D. Birch / Journal of Mathematical Psychology 53 (2009) 106–118

predictions regarding the observed transition time distributions,hazard function expectations, and the prediction of shortertransition times for lesser preferred destination activities alldepend on the specific properties of activation time theory and allsupport the theory.It is worthwhile for discussion purposes to separate the

contributions of activation and instigation to the generation ofa stream segment protocol. Activation refers to the interfacebetween overt, observable activities and the theoretical tendenciesto engage in those activities, instigation to the processes thatincrease the strengths of action tendencies. There is somethingof a division of labor related to the two that is illustrated by thetheoretically derived hypothesis that a less preferred activity ina stream segment will be chosen more promptly than a morepreferred activity, measured in terms of transition times. Thatsuch a prediction is made at all is attributable to processes ofactivation which generate a family of transition time densitiesthat intersect systematically as time passes; precisely wherethe intersections occur is governed by the particulars of theenvironmental scheduling of exposure to instigating forces asembodied in the instigation function.Theoretically, instigating forces summarize in mathematical

terms the motivational impact of the internal and externalenvironments of an individual on tendencies to action andtherefore on action itself. Stimuli, in the broad sense of theterm, are the agents of application for instigating forces and theinstigation function is a proposal regarding the environmentalscheduling of exposure to instigating forces immediately followingupon a transition. Selected out for special attention is theimportant influence of the ongoing activity on the extent towhich instigating forces are engaged at any particular time. Eachongoing activity provides the setting for the imposition of aconfiguration of instigating forces and the instigation functiondescribes the nature of the exposure to the forces through time forthat particular condition. The theory does not specify a particularinstigation function, an empirical construct, for an ongoing activitybut does provide for the discovery of suitable instigation functions,accomplished by fitting a theoretical transition time density toa set of observed transition time values. In the case of MorganO’Hara’s stream segment protocol the five composite activitiesgive rise to five different instigation functions, all special casesof Eq. (1). Each of the functions serves well in accounting forits associated observed transition time distributions, includingsecondary characteristics of shape and location.In the paper, activation time theory has been directed narrowly

to the continuous stream of observable behavior as it is takenup in studies of ethology and motivation in order to establish itsvalidity when applied to a particular domain. As was suggestedin the Introduction, however, the theory may have other areasof application because many events, not just the activities ofindividuals, occur as a continuous stream. If so, the stage wouldbe set for the generalization of the present theory.Activation time theory and its concrete realization in the

observable stream of overt behavior have relationships to otherresearch in two distinct directions. On the one hand, the theoryhas communality with research in a broad range of disciplinesbecause its data generating process of counting down activationtimes matches the nature of the observations made in thosedisciplines. As a result, the probability theory developed for thestream of behavior using activation time theory is applicable tothe data of these other disciplines. The other set of relationshipsis with research specifically directed to the study of motivationand behavior where activation time theory joins with all theother research directed to this topic. The theory applies to datacollected as a stream of activities and adds to the scope andfocus of research on motivation and behavior generally. Concepts

of tendency, instigating force, and the environmental schedulingof exposure to instigating forces, as embodied in the instigationfunction, are central to this particular realization of the generalactivation time theory. The first direction relationships define thebreadth of applicability of the theory and the second the focus onthe details of data analysis available from the theory.These initial steps in the analysis of stream of behavior type

protocols appear promising especially in light of the scope andcoherence provided the theory by the availability of streamsegment balanced transition matrices. Critical features of streamsegment data such as the relative frequencies of transitionsand the form of transition time distributions have been givena theoretical foundation based on the countdown of activationtimes. Many other measures are yet to be derived from the theory,including activation time itself, measures which would providemore complete summaries of stream of behavior data and allowfurther tests of the theory.

Acknowledgments

I thank my collaborator Morgan O’Hara for making availableto me examples of her art in the form of protocols of her streamof behavior and for enlightening me about her artistic endeavorsresulting in this unique data set. I am also most grateful to Prof. Dr.Willy Lens for making it possible for me to spend time workingon activation time theory as a visitor at the Katholieke UniversiteitLeuven and to the Research Board of the University of Illinois,Linda S. Wilson, Secretary, for research support that enabled thetranscription of the O’Hara protocols.

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