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A Model of Extended, Semisystematic Visual Search

Brian J. Melloy, Sourav Das, Anand K. Gramopadhye, and Andrew T. Duchowski,

Clemson University, Clemson, South Carolina

Objective: A model of semisystematic search was sought that could account for both

memory retrieval and other performance-shaping factors. Background: Visual search

is an important aspect of many examination and monitoring tasks. As a result, visual

search performance has been the topic of many empirical investigations. These inves-

tigations have reported that individual search performance depends on participant

factors such as search behavior, which has motivated the development of models of

visual search that incorporate this behavior. Search behavior ranges from random to

strictly systematic; variation in behavior is commonly assumed to be caused by dif-

ferences in memory retrieval and search strategy. Methods: This model ultimately

took the form of a discrete-time nonstationary Markov process. Results: It yields both

performance and process measures that include accuracy, time to perception, task time,

and coverage while avoiding the statistical dif culties inherent to simulations. In par-

ticular, it was seen that as the search behavior becomes more systematic, expected

coverage and accuracy increase while expected task time decreases. Conclusion: In

addition to explaining these outcomes and their interrelationships from a theoretical

standpoint, the model can predict these outcomes in practice to a certain extent as it can

create an envelope de ned by best- and worst-case search performance. Application:

The model also has the capability of supporting assessment. That is, it can be used

to assess the effectiveness of an individual s search performance, and to provide pos-

sible explanations for this performance, through the use of one or more of the output

measures.

An extended search proceeds as a succession of

INTRODUCTION

focused gazes or xations in the person s effort

to perceive a target. The performance of such a

Human visual search is an important aspect of

search is measured by the accuracy achieved that

many civilian and military applications such as

is, the probability of discovering a target in a spe-

reconnaissance, tracking, information retrieval,

ci c length of time. Thus relating accuracy to time

aircraft inspection, medical image screening, in-

(or equivalently speed) is of central concern, espe-

dustrial inspection, and the monitoring of sonar,

cially in instances where the respective goals are

radar, and other displays. Even in instances where

in con ict (e.g., safety and productivity). More-

automation has replaced the human eye as the

over, search performance has been observed to

primary search instrument, the information is fre-

vary markedly, in part because of distinct search

quently still transferred to a human thorough a

behaviors and other individual differences (e.g.,

visual link. Thus, interest in the performance of

Wang, Lin, & Drury, 1997). Hence there is also a

humans in visual search tasks persists.

need to establish performance benchmarks that

In the context of this research, visual search is

represent the limits of search performance.

considered to be an extended examination of a

Search behavior, in particular, is commonly

eld with many elements (as opposed to those

assumed to be influenced by both memory re-

with a small number of visual elements requiring

trieval (e.g., Arani, Karwan, & Drury, 1984) and

few if any eye movements; e.g., Eriksen, 1990).

Address correspondence to Anand K. Gramopadhye, Clemson University, College of Engineering and Science, Advanced

Technology Systems Laboratory, Clemson, SC 29634; *******@*******.***. HUMAN FACTORS, Vol. 48, No. 3, Fall 2006,

SEMISYSTEMATIC SEARCH MODEL 541

search strategy (e.g., Williams, 1966). One aspect Wiener, 1975); attitude toward risk (e.g., Megaw

of search strategy is the degree of visual lobe over- & Richardson, 1979); individual differences in

lap. The visual lobe, or visual eld, is commonly search strategy (e.g.,Wang et al., 1997); specialized

defined as the area visible in a single fixation. search strategies such as left-to-right, line-by-line

Models have been developed that account for this patterns (e.g., Baveja et al., 1996); environmen-

overlap explicitly (Baveja, Drury, Karwan, & tal factors such as noise (which could have either

Malon,1996; Courtney & Guan,1996,1998; Sarac, a favorable or an unfavorable effect on perfor-

Batta, & Drury, 1997) and implicitly (e.g., Arani mance; e.g., Warner & Heimstra, 1972); and tem-

et al., 1984; Drury & Chi, 1995; Karwan, Moraw- poral factors such as arousal (e.g., Poulton, 1973).

ski, & Drury, 1995; Krendel & Wodinsky, 1960; Organizational factors such as training could also

Lin, 1991; Morawski, Drury, & Karwan, 1980, play a role, insofar as training would affect search

1992; Williams, 1966), both of which have been strategy, for example (e.g., Gramopadhye, Drury,

validated in practice (e.g., Baveja et al., 1996, and & Prabhu, 1997).

Courtney & Guan, 1998, in the former case and At the core of the model is a function that char-

Drury & Chi, 1995, Krendel & Wodinsky, 1960, acterizes search behavior over time. This function

Morawski et al., 1980, and Williams, 1966, in the is not restricted to any particular form; as a result,

latter). The latter approach will be considered here. it may better serve to parallel actual performance.

The boundaries of search performance have The function can be estimated from accuracy (or

previously been established using models based other performance) data, which would be easier to

on diametric assumptions regarding search behav- obtain in practice than eye movement data. More-

ior. These two extreme cases are commonly re- over, the value of this function at a particular point

ferred to as systematic and random search. The in time corresponds to the systematic ef ciency

former is characterized by systematic xations of the searcher at that juncture, thus providing a

and the latter by random xations, as their names useful measure of individual performance. Final-

imply; these are analogous to sampling without ly, because a mathematical model is employed in

and with replacement, respectively. Naturally, lieu of a simulation, relationships between vari-

actual search behavior appears between these ex- ables are more transparent and certain statistical

tremes. problems inherent to simulations can be avoided.

Accordingly, Arani et al. (1984) developed a

variable-memory simulation model to represent MODEL DEFINITION

a search that is intended to be systematic but suf-

The process of searching a eld for targets is

fers from imperfect memory. (A mathematical

modeled as a series of xations. The search eld

model was derived as well, but it is tractable only

itself is assumed to be homogeneous; that is, there

under a very restrictive set of assumptions, thus

are no regions that are distinctive, visually or oth-

motivating the development of the simulation.)

erwise. (Textiles, glass, sheet metal, castings,

The model incorporates a standard two-parameter

roller bearings, and lap-splice joints of fuselage

decay/interference function memory model, which

structures are examples of homogeneous search

the authors stated could be estimated from eye

elds, provided that the targets are inconspicuous.)

movement data in practice. (In certain models

It is represented as a set of equal-sized cells, with

[Courtney & Guan,1996,1998] wherein lobe over-

the size of these cells corresponding to the area that

lap is modeled explicitly, the degree of overlap

can be encompassed in a single xation (com-

characterizes the extent of memory loss. Hence

monly referred to as a hard-shell visual lobe). Each

from a modeling standpoint, it is arguable that the

successive xation either deliberately glimpses a

converse would be true in cases in which memo-

cell not yet xated in a systematic manner or arbi-

ry is modeled explicitly.)

trarily glimpses a cell (which may or may not have

A mathematical model for semisystematic

been previously xated) in a random fashion.

search is proposed here that can account (primar-

In order for a particular target to be located, two

ily in an implicit manner) for memory retrieval

events must occur in succession: A cell containing

and errors therein, and for other factors that could

a target must be xated and the target subsequent-

potentially affect performance. The latter may in-

ly perceived. It is assumed that the targets are

clude participant factors such as motivation (e.g.,

542 Fall 2006 Human Factors

inconspicuous, which precludes the possibility of given that the cell that contains it has been x-

ated, j [0,1], j = 1, 2,, h.

a guided search (e.g., Wolfe, 1994). Thus, the like-

lihood of xating on a particular cell containing

a target is directly related to the number of xa- These parameters may also be considered in the

tions (which is directly proportional to the time en- context of performance shaping factors. For ex-

gaged in search), relative to the size of the search ample, a, bj, g, and h are task factors, whereas o

is a participant factor. The parameter j is affected

field, for any established search behavior. It is

further assumed that the targets are uniformly dis- both by task factors such as target conspicuity and

tributed over the search eld and that a cell may by participant factors such as visual acuity. Last-

ly, t is the functional parameter that characterizes

contain at most one target. (In cases where the

ratio of the number of cells to the number of tar- search behavior referred to in the previous sec-

gets is large, the probability that a single cell con- tion. Recall that its precise functional form will

tains more than one target is negligible; Morawski be de ned by memory retrieval and various other

et al., 1980.) Once a target is perceived, the search performance-shaping factors, such as individual

terminates. differences in search strategy. Thus the model can

However, it is not certain that a target will be implicitly incorporate the forenamed stationary

perceived, even though the cell containing the tar- and nonstationary (temporal) factors, as well as

get has been xated. This uncertainty is attribut- others, via this function.

able to factors such as the conspicuity of the target

and its distance from the center of xation. The MODEL FORMULATION

conditional probability that a particular target is

perceived, provided that the cell containing the tar- The fundamental modeling approach is based

get has been xated on, will be referred to as the on the concept of what will be referred to here as

perceptual sensitivity. (The value of the perceptu- a scan, the number of distinct cells xated within

al sensitivity is inversely related to the size of the a particular scan, the numbers and types of targets

hard-shell visual lobe.) There may be several such contained in the partition formed by these cells,

probabilities, as the values usually differ accord- and whether or not one of the targets in this parti-

ing to the type of target. However, the probabili- tion is perceived. The concepts of scan and distinct

ties do not vary with the location of a target, given cells will be clari ed before continuing. A scan is

that the search eld is homogeneous. Lastly, the essentially a measure of coverage that segments

conditions stated are consistent with those of both the search into blocks of n distinct fixations, in

Morawski et al. (1980) and Arani et al. (1984). which n corresponds to the number of cells in the

The model is intended to represent a semisys- search eld. The latter is established by comput-

tematic search that terminates upon detection of ing the ratio of the area of the entire eld to that

any target. Several parameters characterize the of the visual lobe:

search:

n = a/o . (1)

a: area of search eld, a, a > 0,

A xation is considered to be distinct if the newly

o: visual lobe; that is, area of an individual

cell, o, o > 0, xated cell has not already been glimpsed during

bj: number of type j targets in search eld, j = the current scan. Once n distinct xations have oc-

1, 2,, h, bj Z+, curred, the current scan is complete; the next xa-

g: search time limit (in seconds), g, g > tion demarcates a new scan. In other words, once

0, a new scan begins, the slate is wiped clean, so to

h: number of different target types in search speak, and all new xations are considered dis-

eld, h Z+, tinct until one of the cells in this new partition is

t: systematic search ef ciency that is, the re xated.

probability of a systematic xation at time t, Adopting this approach, the search process

t [0,1], t Z*, and will be modeled as a discrete-time nonstationary

j: perceptual sensitivity that is, the propor- Markov process (e.g., Ross, 2003). The states of

tion of time that a type j target is perceived, the process, Xt, will be represented by a 3-tuple

SEMISYSTEMATIC SEARCH MODEL 543

(k,l,m). The substance of these three indexes was or equivalently,

alluded to previously. Descriptions of the individ-

k n (b l ),

ual indexes and the relationships that exist be- (7)

tween them will now be presented by considering

the search as it progresses from the outset, through because (n k) and (b l ) represent the number

the completion of the rst scan, to the initiation of cells and targets that have not yet been xated,

of the second. respectively.

The initial scan commences as the search is ini- On each successive xation (with the excep-

tiated, with a series of xations. The rst index, k, tion of the xation that initiates a new scan), either

is a nonnegative integer, the value of which cor- one of the elements will be incremented by 1 if the

responds to the number of distinct cells that have newly xated cell is distinct and it contains a target,

been xated. It is therefore a measure of eld cov- or else, if not, the elements will remain unchanged.

erage, which of course cannot exceed the cumu- In the former case, this index will be expressed as

l + I j, in which I j designates the transpose of the

lative number of xations. The maximum value

of k corresponds to the maximum number of x- jth column of the identity matrix (or, i.e., its jth row,

ations, f, which is the quotient of in accordance with the convention of representing

all vectors in row form), indicating that the distinct

f = g/0.3, (2) cell xated contains a type j target. Once all of the

targets have been xated, lj = bj for j = 1, 2,, h,

in which 0.3 s is the duration of a single xation or equivalently, l = b.

(e.g., Arani et al., 1984). (The number of xations The third index, m, is a binary variable that indi-

is often used herein to express the concept of time.) cates whether or not a xated target has been per-

Each successive xation will either increment k by ceived; that is,

1 if the newly xated cell is distinct or leave k un-

0 when target is not perceived

changed if it is not. Hence k is a nondecreasing

on tth xation, or

variable.

m= (8)

The second index, l, is a vector that has h ele- 1 when target is ( xated and)

ments, each of which corresponds to a different perceived on tth xation

target type. The jth element, lj, is a nonnegative

integer with a maximum value of bj, in which bj Clearly,

denotes the number of type j targets in the search

m l .

eld. (Herein, the convention will be to represent (9)

all vectors in row form.) These elements serve to

enumerate the various types of targets that are Recall that the search terminates whenever a tar-

contained in the partition formed by the cells that get is perceived.

have been xated. It follows therefore that Once all cells in the eld have been xated (at

least once), a scan is considered to be complete;

l k (3) hence at this stage (the end of the rst scan), k =

n. Moreover, because all of the targets must have

for k = 0, 1,, f, in which

been xated, it follows that l = b. Reaching this

h stage signi es that a target has not been perceived

l = (4) on the previous (t 1) xations. If a target is per-

j =1 ceived on the tth xation, then m = 1 and the search

is terminated. Otherwise, the next xation demar-

for lj = 0, 1,, bj. Similarly,

cates a new scan, whereupon l will be reinitial-

n k b l, (5) ized. After this xation, either l = 0, indicating that

the cell xated does not contain a target, or l = I j,

in which indicating that this cell contains a type j target.

The index k is not reinitialized, however, as it is a

b = (6) cumulative measure of coverage. Instead, a func-

tion, rk, is created,

j =1

544 Fall 2006 Human Factors

cur by means of either a random or a systematic

k(mod n) k = 0,1,, f, k n, 2n,

rk = (10) xation. Moreover, because the middle index re-

k = n, 2n,

n mains unchanged, this implies that the new cell

xated does not contain a target. As a result, the

so that its value corresponds to the number of dis- third index necessarily has a value of 0 because

tinct cells that have been xated during the cur- a target (that is not present) cannot be perceived.

Next, the transitions (k,l,0) (k + 1,l + I j,0) and

rent scan. Thus, in general, rk = n at the end of the

ith scan, i Z+, and (k,l,0) (k + 1,l + I j,1) differ from the previous

one in the respect that the distinct cell fixated

m l rk n (b l ), (11) contains a type j target, because the jth element

of the middle index of the destination 3-tuple has

because of Equations 3, 4, 6, 7, 9, and 10. (It will been incremented by one. In the former case the

be seen that this function also plays a central role target is not perceived, whereas in the latter case

in determining the transition probabilities.) Final- it is, as indicated by the respective values of the

ly, this description (and the speci cs) would apply third index of the destination 3-tuple.

Conversely, the transitions (k,l,0) (k,l,0)

to subsequent scans without loss of generality,

and (k,l,0) (k,l,1) signify instances in which a

other than k = n at the end of no scan other than

the rst. cell is refixated, because the first index is un-

Thus, the states of the Markov process may changed. Thus these particular transitions must be

now be represented as the result of a random xation. The second index

necessarily remains the same because the partition

Xt = (k,l,m), (12) (of distinct cells) has not been expanded to encom-

pass additional targets. In the former case, a target

for (k,l,m) t, in which t is the indexed set of is not perceived; this may be attributable either to

states (k,l,m) for all k, l, and m such that m l a failure to perceive a target when the cell that con-

rk n (b l ) and k t, for t = 0, 1,, f, because tains it is re xated or to simply re xating a cell that

of Equations 4, 6, 10, and 11. The transitions of does not contain a target. However, the latter tran-

the process form three distinct sets: sition re ects an instance in which the re xated

cell includes a target that is perceived.

(k,l,0) (k + 1,l,0), (k + 1,l + I j,0), (k + 1,l + In contrast to the rst set of state transitions de-

I j,1), (k,l,0), (k,l,1) for k n,2n scribed previously, those included in the second

set, (k,b,0) (k + 1,0,0), (k,b,0) (k + 1,I j,0) and

(k,b,0) (k + 1,I j,1), occur only at the instant a

(k,b,0) (k + 1,0,0), (k + 1,I j,0), (k + 1,I j,1)

scan is completed. In this case, the second index

f 1

for k = n, 2n, and

of the origination 3-tuple must equal b because

the partition envelops the entire eld, and thus all

(k,l,1) (k,l,1). the targets, once a scan is completed. According-

ly, this index is reinitialized in the destination 3-

The rst set contains transitions that occur during tuples, because the commencement of a new scan

an ongoing search at any time other than when a creates a new partition. Similarly, the rst index

scan is completed, the second set includes those is incremented because the rst xation of a new

transitions that occur only at the time a scan is partition is necessarily distinct. The speci c real-

completed, and the transitions in the third set in- izations of the second and third indices (of the

dicate that a target has been perceived and the destination 3-tuples) of this set of transitions are

search terminated. interpreted in a manner identical to that of the

The conditions that de ne the transitions with- previous set. The last transition to be considered,

(k,l,1) (k,l,1), is characteristic of an absorbing

in these sets will now be described. To begin, the

transition (k,l,0) (k + 1,l,0) will be considered. state (in a Markov process). In the current context,

First observe that a distinct cell has been xated, absorption occurs when a target is perceived, be-

given that the rst index of the destination 3-tuple cause the search is terminated at that point.

has a value of (k + 1). Such a state change may oc- Finally, the likelihood of any particular state

SEMISYSTEMATIC SEARCH MODEL 545

change is governed by a set of transition proba- search task effort, irrespective of whether or not

bilities. These probabilities are obtained by con- a target is perceived, w = 1, 2 f; and

sidering the conjunction of several events. For Ct, coverage number of distinct cells xated

example, consider the transition (k,l,0) (k + by time t in the initial scan of a eld void of tar-

1,l,0) once again. Recall that this transition may gets, relative to the number of cells in the eld,

Ct (0,1];

occur via either a random or systematic xation.

Under the assumption of the former, four events

must occur: There will rst be a random xation; are output measures in the strictest sense. These

this xation will glimpse a distinct cell; the x- metrics are a function of the transition probabili-

ated cell will not include a target; and a target will ties; they are also a function of state probabilities.

Let q t(k,l,m) represent the probability that state

not be perceived. A systematic xation, of course,

alters the rst event but not the others. Because (k,l,m) is occupied at time t; that is,

random and systematic xations are exclusive, the

q t(k,l,m) = Pr[Xt = (k, l, m)],

respective probabilities of these events would be (15)

added. In this manner, the equation for the prob-

for (k,l,m) t, for all t. Now let q t(k,l,m) be the

ability of this particular transition, denoted by

pt(k,l,0),(k + 1,l,0), is obtained: (k,l,m)th element of the state probability vector qt.

Also let p t(k,l,0),(k + 1,l,0) be the (k,l,0)(k + 1,l,0)th

n rk element of the transition probability matrix Pt, let

pt(k,l,0),(k + 1,l,0) = (1 t)

p t(k,l,0),(k + 1, l + I,0) be the (k,l,0)(k + 1,l + I j,0)th ele-

n j

ment of Pt, and so on. Then

n rk (b l )

1 + t 1

n rk

(13) qt = qt 1 P t 1, (16)

n rk (b l )

1 =

n rk

for t = 1,2,, f, with

1 t

(n rk b + l ) t + .

n n rk q0 = [1, 0, 0 0], (17)

because q 0(0,0,0) = 1.

The other transition probabilities are derived in a

similar fashion. A complete set of transition prob- Now, the accuracyat time t is expressed in terms

abilities may be found in the Appendix. It is note- of the absorbing state probabilities as

worthy that models for both random and systematic

searches could be obtained by setting t = 0 and t = t

(18)

(k,l,1)

t = 1 in these equations, respectively, for all t. k l

(k,l,1) t

PERFORMANCE MEASURES

for t = 1, 2,, f, because absorption and target per-

There are several measures of interest, one of ception are synonymous in this context.

which, the mean systematic search ef ciency,, The mean has been selected to characterize the

can be determined directly from averaging the sys- remaining measures, as they are random variables.

tematic search ef ciency at each time epoch: The equation for the second measure, the expect-

ed time to perception, is

f 1

(14) f

V = V (v), (19)

t =0

v =1

The others,

t, accuracy (cumulative) probability of per- in which dV (v) denotes the mass function for the

ceiving a target by time t, t [0,1]; random variable. The equation for the mass func-

V, time to perception number of fixations tion

required to perceive a target, v = 1,2 f;

W, task time number of xations expended in dV (v) = Pr(V = v) = DV (v) DV (v 1), (20)

546 Fall 2006 Human Factors

E[W B = b]Pr(B = b),

for v = 1, 2,, f, can be readily found using the

W = E[E[W B]] = (25)

distribution function, DV(t), in which b

t in which E[W B = b] would be given by Equation

DV (t) = Pr(V t) = (21)

f 22. These performance measures will now be con-

sidered via a numerical example.

for t = 1, 2,, f, because the absorbing state prob-

abilities are cumulative over time. MODEL ILLUSTRATION

Next, recall that the search will terminate in one

of two ways: when either a target is perceived or

An example will be adapted from Arani et al.

time has lapsed as the consequence of an unsuc-

(1984) in which n = 50, f = 200 (i.e., g = 60 s), h =

cessful search. Hence, the expression for the ex-

2, Pr(B = [1, 1]) = Pr(B = [1, 2]) = Pr(B = [2, 1]) =

pected task time is a convex combination of the

Pr(B = [2, 2]) = 0.25, and = [0.8, 0.5]. The

mean time to perception and the time limit,weight-

corresponding state transition diagram is depicted

ed by the respective probabilities of a hit and a

in Figure 1. Now, for the current model, let the sys-

miss :

tematic search ef ciency be subject to exponential

decay; specifically, let t = xy z, x (0,1], y

W = f V + (1 f ) f. (22)

[0,1], z, z > 0, for t = 0, 1,, f 1. Although

x will be xed at 1 and z at 50 (that is, n) here, sev-

The nal measure is derived from a eld void

eral values of y will be considered in order to

of targets. Hence in this particular case l = b = 0,

demonstrate how different rates of decay affect

from which it follows that

the various performance measures. These values

are listed in Table 1, along with the corresponding

q t

average systematic ef ciencies. In addition, plots

(k,0,0) = 1 (23)

of the different systematic ef ciencies over time

k =1

are depicted in Figure 2. The rst value yields a

for t = 1, 2,, f. The resultant equation for the ex- random search (with the condition that 00 = 0).

Cases 2 through 4 produce searches that become

pected coverage at time t, then, is

random after 100, 150, and 200 xations, respec-

tively. Cases 5 through 9 generate searches that

1 have respective systematic search ef ciencies of

t n

(k, 0,0),

n 1%, 5%, 10%, 20%, and 40% after 60 s. Of course,

k =1

E [Ct] = the last value yields a search that is strictly sys-

(24)

n t

tematic.

kq t(k, 0,0) + q t(k, 0,0), t>n

Figure 3 reveals that coverage is directly relat-

k =1 k = n +1

ed to the degree of systematic ef ciency, as expect-

ed. Moreover, this gure con rms that in the case

for t = 1, 2,, f. of strictly systematic search, the eld coverage is

equal to 1 (or 100%) when the number of xations

Heretofore, it has been assumed that the num-

corresponds to the eld size. (This also coincides

bers of the various types of targets present (in the

with the point at which the difference in coverage

eld) are known with certainty. Indeed, this would

yielded by the extreme behaviors reaches a max-

be the case in a synthetic task environment in the

imum.) The expected eld coverage of the other

context of training, for example. However, if the

cases will approach but never achieve a value

numbers are not known (with certainty), then b

of 1 (for any nite number of xations), because

would not be xed but instead would represent a

complete coverage is not certain when any random

realization of a random vector, B. Nonetheless, all

behavior is exhibited. Next, observe the striking

of the output measures could still be found by using

similarities between the expected coverage curves

a theorem of conditional expectation. For exam-

and the corresponding accuracy curves, shown in

ple, the equation for the expected value of the task

Figure 4. In particular, observe how closely the

time would become

SEMISYSTEMATIC SEARCH MODEL 547

Figure 1. State transition diagram.

curves of Cases 2 and 3 parallel that of the ran- atively less systematic exhibit perception times

dom search, and that Cases 6 through 9 converge that are initially smaller, and later larger, than

at 75 xations. Thus these gures suggest a direct their counterparts. The reason for this is that

link between expected coverage and accuracy. when the less systematic searches are successful,

The expected perception times for the various it is more likely that they will be successful early

cases, however, tend to diverge at first as the on, as illustrated by the extreme cases in Figure

number of fixations increases, as displayed in 6a. This initial advantage is negated as the time

Figure 5. Moreover, although the curves of the horizon is extended, however, because protract-

different cases maintain their respective positions ed searches are more likely to be the by-product

with respect to accuracy and coverage, the per- of less efficient behavior, as demonstrated in

ception time curves do not. The cases that are rel- Figure 6b. Moreover, because it is less likely that

548 Fall 2006 Human Factors

process. The time-dependent, semisystematic

TABLE 1: Selected Values of y With Correspond-

ing Average Systematic Search Ef ciencies search behavior is expressed by an embedded

function. Given its generic nature, this function

Case y

is capable of generating not only time-dependent

decreases in ef ciency but increments as well, if

1 .0 .0

2 .028747 .073 appropriate. The function can be estimated from

3 .076524 .100 performance data such as accuracy, or process

4 .171375 .144

data such as coverage, although the latter are usu-

5 .318314 .219

ally more dif cult to obtain in practice.

6 .473083 .320

The present model requires no assumptions

7 .562446 .393

8 .668760 .499 beyond those applied by Morawski et al. (1980)

9 .795242 .656 to their models for random and strictly systemat-

10 1.0 1.0

ic search, despite its capacity to reproduce these

behaviors, as well as those characteristic of semi-

systematic search. (These particular models are

these searches will be successful, the expected underscored because they are extensions of ear-

task time curves (which represent a weighted lier models of random [e.g., Krendel & Wodinsky,

combination of the perception and unsuccessful 1960] and strictly systematic search [e.g., Wil-

termination times) re ect the fact that less ef - liams, 1966].) The same is also true with regard to

cient searches are consistently more time con- the assumptions imposed in the variable-memory

suming on average, as shown in Figure 7. simulation model developed by Arani et al. (1984).

Nevertheless, the mathematical model proposed

CONCLUSION

here embodies both memory-related factors and

An extended semisystematic search was mod- other determinants, and it is not subject to the sta-

eled with a discrete-time nonstationary Markov tistical dif culties intrinsic to simulation methods.

random systematic

1 1

0.9 0.9

0.8 0.8

0.7 0.7

0.6 0.6

t 0.5 0.5

0.4 0.4

0.3 0.3

0.2 0.2

0.1 0.1

0 0

0-25-50-75-100 125-***-***-***

number of fixations ( t )

Figure 2. Systematic search ef ciency versus number of xations.

SEMISYSTEMATIC SEARCH MODEL 549

random systematic

0.9

0.8

0.7

0.6

accuracy

0.5

0.4

0.3

0.2

0.1

0-25-50-75-100 125-***-***-***

number of fixations ( t )

Figure 3. Accuracy versus number of xations.

random systematic

0.9

0.8

0.7

expected coverage

0.6

0.5

0.4

0.3

0.2

0.1

0-25-50-75-100 125-***-***-***

number of fixations ( t )

Figure 4. Expected empty eld coverage versus number of xations.

550 Fall 2006 Human Factors

random systematic

expected time to perception

0-25-50-75-100 125-***-***-***

number of fixations ( f )

Figure 5. Expected time to perception versus maximum number of xations.

Moreover, this model generates both perfor- In addition to explaining these outcomes and

mance and process measures, whereas the vari- their interrelationships from a theoretical stand-

able-memory model yields only accuracy. point, the model can predict these outcomes in

Although the other models yield mean and medi- practice to a certain extent as it can create an

an times to perception, in addition to accuracy, envelope de ned by best- and worst-case search

these values represent approximations that are performances. The practical value of the model

apparently based on an infinite time horizon. for predicting intermediate performance is

Consequently, a single value is produced, irrespec- arguable, however, because doing so would

tive of the time limit. require the estimation of the search efficiency

Specifically, the measures that the present parameter by means of either a pilot study or past

model is able to produce are accuracy, eld cov- data from similar tasks. Nevertheless, it is note-

erage, time to perception, and task time. In par- worthy that the model also has the capability of

ticular, it was seen that as the search behavior supporting assessment. That is, it can be used to

becomes more systematic, expected coverage assess the effectiveness of an individual s search

and accuracy increase and expected task time performance, and to provide possible explana-

decreases. These outcomes are consistent with tions for this performance, through the use of one

empirical studies (e.g., Megaw & Richardson, or more of the output measures. In this manner,

1979; Schoonard & Gould, 1973; Wang et al., the model can serve both initially to screen can-

1997) and hence support the validity of the didates for visual search tasks and, subsequently,

model. It was also observed that whereas increas- to identify interventions for those who routinely

ingly systematic behavior ultimately yields perform these tasks. Finally, although the appli-

smaller expected perception times, the reverse is cation of this model is currently confined to

true initially, which represents a nding that is homogeneous search elds, it potentially could be

neither con rmed nor contradicted by the litera- adapted to incorporate heterogeneous regions.

ture (to the best of our knowledge). Such a model would not only have intrinsic value

SEMISYSTEMATIC SEARCH MODEL 551

(a) random systematic

0.07

0.06

probability of target perception

0.05

0.04

0.03

0.02

0.01

1 3 5 7 *-**-**-**-**-** 21 23 25

number of fixations ( t )

(b) random systematic

0.045

0.04

probability of target perception

0.035

0.03

0.025

0.02

0.015

0.01

0.005

1 5 *-**-**-**-**-** **-**-**-**-**-** 57 61 65 69 73

number of fixations ( t )

Figure 6. Probability of target perception versus number of xations for random and strictly systematic searches

over abbreviated (a, above) and expanded (b, below) horizons.

552 Fall 2006 Human Factors

random systematic

expected task time

0-25-50-75-100 125-***-***-***

number of fixations ( f )

Figure 7. Expected task time versus maximum number of xations.

p t(k,l,0),(k+1,l+I j,0) = (1 j)(bj lj)

but would also represent the next step in the pro-

spective development of a model for extended (27)

1 t

+ t, j,

guided search.

n n rk

APPENDIX

p t(k,l,0),(k+1,l+I j,1) = j (bj lj)

Recall that the transitions of the process consti- (28)

1 t

+ n rk, j,

tute three distinct sets:

(k,l,0) (k + 1,l,0), (k + 1,l + I j,0), (k + 1,l +

I j,1), (k,l,0), (k,l,1), h

1 t

p t(k,l,0),(k,l,0) = rk

(k,b,0) (k + 1,0,0), (k + 1,I j,0), (k + 1,I j,1),, (29)

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