Mathematical Modeling of Human Performance and Cognition

 

 

[Features]

[Framework & Equations]

[Tutorials & Membership]

[Contributors & Users]

[Design Tools]

 

 

The goal of this document is to summarize and integrate all mathematical modeling work that quantifies the different aspects and components of human cognition and performance. It will also serve as a learning material platform for new users of mathematical modeling of human performance.

 

1. Unique Features of Mathematical Modeling of Human Performance and Cognition

Mathematical equations can predict, quantify and analyze human performance, workload, brain waves, and other indexes of human behavior in a rigorous way. Compared with computer simulation,

 

1) Mathematical models and equations of human behavior clearly quantify and extract the mechanisms of human behavior by clear quantifications of the relationships of variables including the relationship between the input and output of each equation. Users of these mathematical models will be much easier to understand and extract the relationships among variables than reading computer codes.

 

2) Mathematical models and equations of human behavior can be relatively easily to be edited, modified, improved, and integrated together to develop new mathematical equations.

 

3) Mathematical models and equations of human behavior and performance can be relatively easily be implemented in different programming languages and be imbedded in different intelligent systems to work together with system design.

 

4) Mathematical models and equations can lead to analytical solutions, which are more accurate than simulation results.

 

5) There are mathematical models and equations quantifying the entire human cognition system (See equations in the entire network) which is another unique feature of the mathematical modeling approach.

 

6) There are mathematical models and equations can be proved by mathematics derivation directly with no need to be verified by empirical data (See Equations in Wu, C., Berman, M., & Liu, Y., 2010).

 

 

2. Usage of Mathematical Models and Equations in this Summary Webpage

The equations summarized here can also serve as an index page and a guideline tool for modelers who can:

 

1) Use those mathematical models to quantify and predict new phenomena and tasks in human performance

 

2) Add and develop new equations and mathematical models to quantify new components of human cognition and performance to further grow with the framework of the Queuing Network-Model Human Processor (QN-MHP)

 

3) To be used and imbedded in different intelligent systems and tool design for human performance and behavior predictions

 

Tutorial and Free Membership:

 

1) How to build and verify models of human performance (General descriptions) (Wu, 2016)

 

2) How to build mathematical models (e.g., Page 9-10) (Wu & Liu, 2008a)

 

3) How to integrate and build new mathematical models in human performance modeling (Coming soon)

 

4) To become a member (user or contributor) of mathematical modeling group in human performance modeling, please email to changxu.wu@gmailcom (Please list your full name and institution/company name), we will send you recent updates, new modeling work, and new tutorials. All of them are free.

 

 

3. Human Machine System Design Tools based on the Equations on this Page [Link]

 

 

4. Mathematical Equations in the Queuing Network-Model Human Processors (QN-MHP) as the Framework

 

The General Structure of Queuing Network-Model Human Processor (QN-MHP)

 

(a.) Perceptual Subnetwork

 

(b.) Cognitive Subnetwork

 

(c.) Motor Subnetwork

1. Common visual processing

2. Visual recognition

3. Visual location

4. Visual recognition and location integration

5. Common auditory processing

6. Auditory recognition

7. Auditory location

8. Auditory recognition and location integration

 

A. Visuospatial sketchpad

B. Phonological loop

C. Central executive

D. Long-term procedural memory

E. Performance monitor

F. Complex cognitive function

G. Goal initiation

H. Long-term declarative & spatial memory

 

V. Sensorimotor integration

W. Motor program retrieval

X. Feedback

information collection

Y. Motor program assembling and error detecting

Z. Sending information to body parts

21-25 etc.: Body parts: eye, mouth, left hand, right hand, foot

 

Server Information Processing Time and Information Processing Capacities

Server Name

Processing Timea: Exponential Distribution (Mean, Min) (ms)

Capacity (Entitiesa)

 

Server Name

Processing Time: Exponential Distribution (Mean, Min) (ms)

Capacity (Entitiesb)

1

Exp (42, 25)

4

 

5

Exp (42, 25)

2

2

Exp (42, 25)

4

 

6

Exp (42, 25)

1

3

Exp (42, 25)

4

 

7

Exp (42, 25)

1

4

Exp (42, 25)

5

 

8

Exp (42, 25)

1

A

Exp (18, 6)

4

 

E

Exp (18, 6)

Infinitec

B

Exp (18, 6)

4

 

F

Exp (18, 6) per cycle

1

C

Exp (18, 6)

3

 

G

Exp (18, 6)

Infinitec

D

Exp (18, 6)e

Infinite

 

H

Exp (18, 6)e

Infinite

V

Exp (24, 10)

Infinitec

 

X

Exp (24, 10)

Infinitec

W

Exp (24, 10)

1c

 

21 (Eye Motor)

Saccade and Fixation Timed

1

Y

Exp (24, 10)

2

 

22 (Mouth)

As a function of number of syllables (Voice key closure time: 100 ms, Wu & Liu, 2008a)

1

Z

Exp (24, 10)

2

 

23 (Right Hand & Right Arm)

Arm and hand movement time, see Fitts's Law; Finger movement time, see (Wu & Liu, 2008b)

1 (If one movement per time)

25 (Right Foot)

Foot movement time, see (Zhang, Wu, & Wan, 2016; Zhao & Wu, 2013; Zhao, Wu, & Qiao, 2013)

1 (If one movement per time)

 

24 (Left Hand & Left Arm)

Arm and hand movement time, see Fitts's Law; Finger movement time, see (Wu & Liu, 2008b)

1 (If one movement per time)

26 (Left Foot)

Foot movement time, see (Zhang et al., 2016; Zhao & Wu, 2013; Zhao et al., 2013)

1 (If one movement per time)

 

27 (Head), 28 (Body), etc.

Head, body movement time etc.c

1 (If one movement per time)

a. Processing speed and capacities were set based on Model Human Processor (Card, et al., 1983), Wu et al (2008-2017), and Jacobson (1999).

b. Entity is defined as the smallest information processing unit in a given task. For example, in a typing task, one letter is an entity. In a speech warning responding task, each short word can be regarded as one entity. For long words in speech warning, each syllable can be represented as one entity.

c. Needs further modeling work and investigation.

d. See Model Human Processor (Card, et al., 1983).

e. Also depends on level of information retrieval (e.g., familiarity and number of time of retrieval).

 

Entire Network

Equation Set EN-1: Mental workload modeling measured by NASA-TLX: Equation (10-12) (Wu & Liu, 2007)

 

Variables

PD

Physical Demand

How much physical activity was required? Was the task easy or demanding, slack or strenuous?

TD

Temporal Demand

How much time pressure did you feel due to the pace at which the tasks or task elements occurred? Was the pace slow or rapid?

EF

Effort

How hard did you have to work (mentally and physically) to accomplish your level of performance?

PE

Performance

How successful were you in performing the task? How satisfied were you with your performance?

FR

Frustration

How irritated, stressed, and annoyed versus content, relaxed, and complacent did you feel during the task?

MD

Mental Demand

How much mental and perceptual activity was required? Was the task easy or demanding, simple or complex?

A

A factor of aging (A 1)

The value of A is directly proportional to age, set based on literature

Arrival rate

The arrival rate of the subnetwork i

Original processing speed

The original processing speed of server j for the young adults in QN-MHP

Number of servers

The total number of servers in the subnetwork m

T

Total time of a trial

The total task time of each trial

a

Constant

The constants in representing the direct proportional relation between the averaged utilizations and the subjective responses (a > 0), see the published work

b

Constant

Same above

 

Equation Set EN-2: Mental workload modeling measured by P300 amplitude and latency: Equation (10-11) (Wu, Liu, & Quinn-Walsh, 2008)

Variables

 

 

Amplitude of the ERP potential P300

Li

Latency of the P300

 

k

Constant

A constant in this relationship I = kN.

b

Constant

A constant in this inverse relationship

NE

Amount of NE

Modeling NE (norepinephrine) in Synaptic Transmission

Number

Number of information entities

Number

Number of information entities of other tasks concurrently processed in server j

Number

Number of processing cycles for each of those entities at server j

A random factor

Normally distributed random factor with mean being equal to zero

r

Distance

Distance from the electrical field point (the location where NE is released) to locations of the electrodes on the scalp

Processing times

Processing times of task i at the perceptual subnetwork, at Server A or B, and at Servers C and E, respectively

 

Equation Set EN-3: Bold signal in fMRI modeling: Equation (27) (Wu & Liu, 2008a)

 

Variables

 

 

CB(t)

The integrated BOLD signal

Modeling of BOLD signal and its percentage of change: The integrated BOLD (blood oxygen level dependent) signal

s

Latency scale

 

M

Magnitude scale

 

k,a,b

Parameters

k, a, and b come from the equations of Cohen (1997) and Anderson et al. (2003), determined by the properties of the brain regions with certain fMRI measurement techniques

t

The duration of each trial

Modeling of BOLD Signal and Its Percentage of Change

The length of time being occupied at a server

In queuing networks can be quantified by Equation 28 (Gross & Harris, 1998):

 

Equation Set EN-4: Entity route selections and skill acquisitions based on reinforcement learning algorithms: Equation (9.7-9.8) (Wu, Berman, & Liu, 2010)

Variables

 

Processing speed of server i

The minimal of processing time of server i after intensive practice

The change of expected value of processing time of server i from the beginning to the end of the practice

Learning rate of server i

Number of entities processed by server i

 

Variables

 

 

Online Q value

is the online Q value if entity routes from server i to server j in t+1th transition

Maximum Q value

Maximum Q value routing from server j to next k server(s)

Processing speed

is the reward and is the processing speed of the server j if entity enters it at tth transition

Discount parameter

The discount parameter of routing to the next server()

Learning rate

The learning rate of Q online learning()

 

Equation Set EN-5: Information processing speed and its variability changes in learning process: Equation (6) (Wu & Liu, 2008b)

Variables

 

 

X

Summation of processing time of servers (Y)

Yi

Processing time of server i

k

Number of servers in the route

Arrival rates of entities/information

 

Equation Set EN-6: Modeling the expected utilization as the mental workload under the time stress. Equation (2) (Cao & Liu, 2015)

Variables

 

 

a,b

Parameter

Parameters a and b are the constants in representing the direct proportional relation between the averaged utilizations and the subjective responses (a > 0)

The average utilization of motor subnetwork

The score of PD reflects workload at the motor component, and therefore, it is in direct proportion to the average utilization of motor subnetwork

 

Equation Set EN-7: Modeling the response time of speech warnings. Equation (11, 12, 13) (Zhang, Wu & Wan, 2016)

Variables

Description

Tk

Notation of processing time of the stimulus at Server k (k =1-8, A,B,C,F,H, WZ)

T6(0)and T8(0)

The initial entity processing time in Server 6 and Server 8, respectively

UL

The perceived urgency as a function of warning loudness

US

The perceived urgency as a function of signal world choice

pi

Notation of probability of a warning stimulus traveling through a route i (i=I or II)

 

Perceptual Subnetwork

Visual perceptual subnetwork:

Server 1. Equation Set VP-1: Eye movement modeling in textual information perception: Equation (12) (Wu & Liu, 2008b)

Variables

Sources

E(FC)

The expected position of the first character in each chunk

Calculation of the Expected Position of the First Character i

E(FP)

The expected position of the fixation point

The half-range of each chunk under extensive practice condition

 

Server 1. Equation Set VP-2: Eye movement modeling in picture information perception: Equation (3) (Lim & Liu, 2009)

Variables

 

 

Importance index of function k

The relatively important function can be given a value 1, and a value 0 is given to the other. The importance index of function k can be calculated.

The importance value for function k

The importance value for function k obtained from each pair-wise comparisons, either 1 or 0.

 

Sever 3. Equation Set VP-3: Visual optical flow perception and speed perception: Equation (1) (Zhao & Wu, 2013)

Variables

 

Perceived speed

V

Actual speed

The current texture density

The texture density in the last driving scenario

The eye height in the last driving scenario

The current eye height

two constant parameters

 

Sever 4. Equation Set VP-4: Visual detection modeling with detection distance and image matrix: Equation (12) (Bi, Tsimhoni, & Liu, 2009)

Variables

 

 

RPOT

Square root of the number of pixels on a target

f

Focal length

S

Size of the area of a target object.

D

Distance of the image forming

 

Auditory perceptual subnetwork:

Sever 6. Equation Set AP-1: Modeling the effect of loudness on speech warning perception: Equation (1, 11) (Zhang, Wu, & Wan, 2016)

Variables

 

 

The perceived urgency

Modeling the relationship between loudness and perceived urgency

,

Constants

The relationship between intensity and perceived urgency was quantified: = 1.33, = 0.64,

Random factors

distributed random factors following distribution [0, 0.7]

L

Loudness level

 

Variables

 

The effect of loudness on reaction time

The initial entity processing time in Server 6

The effect of loudness on perceived urgency

 

Sever 8. Equation Set AP-2: Modeling the effect of signal word on speech warning perception: Equation (12) (Zhang, Wu, & Wan, 2016)

Variables

 

The effect of signal word choice on reaction time

The entity processing time in Server 8

The urgency level expressed by the initial words

The number of words in the ith speech warning

 

 

Cognitive Subnetwork

Server B:

Equation Set C-1: Modeling of optimal chunking of textual information: Equation (22) (Wu & Liu, 2008b)

Variables

Description

Z

Objective function of task completion time

Overall duration of processing each chunk at servers after server B

N

Total number of entities processed

x

Chunk size

Rate of retrieval failure at server B

R

Average duration to correct an error caused by a wrongly processed entity or character

 

Equation Set C-2: Modeling of memory decay of speech information: Equation (7) ( Zhang, Wu, & Wan, 2016)

Variables

Description

The probability of memory decay

Lead time of a speech warning

 

Equation Set C-3: Modeling the probability of route choice in reinforcement learning of the speech warnings: Equation (4-5) ( Zhang, Wu, & Wan, 2016)

Variables

The route choice error rate

The error rate when a speech warning travels via route i

The probability of a speech warning entity processed via route i

 

Variables

The error rate of route choice

L

Loudness level in dB

Perceived urgency level with different signal word choice

,

Parameters to quantify the power law of perceived urgency and loudness

,

Parameters to quantify the power law of perceived annoyance and loudness

pI, pII

Probabilities of choosing route I (the shorter route) and route II (the longer route)

 

Server C:

Equation Set C-4: Inhibiting incompatible responses modeling: Equation (4-6) (Wu & Liu, 2008a)

 

Variables

 

T2,C-comp and T2,F-comp,

Processing times of Server C and F in the compatible conditions

T2,C-incomp and T2,F-incomp

Processing times of Server C and F in the incompatible conditions

SOA (stimulus onset asynchrony)

The delay between the presentation of the stimuli of T1 and T2

Tk

Processing time at server k (k=AP, VP, A, B, C, F, W, Y, Z, X)

 

Equation Set C-5: Dual task interference modeling: Equation (8-9) (Lin & Wu, 2012)

Variables

Description

Sources

 

DLi

Delay time

 

Ti,C

The entity processing time needed at Server C

 

PTi-1

Time lapse for the previous key to be pressed

 

Iv

Inter stimulus interval

 

Iv + TAP+ TB

Time lapse for the entity of the on-going stimulus to leave Server B

 

PTi-1-(Iv +TAP+TB)

The least duration that the current stimulus needs to wait at Server C

 

TC

Cycle time at Server C

 

 

Server E:

Equation Set C-6: Background noise in motor control: Equation (15) (Lin & Wu, 2012)

Variables

 

 

The extent of SDN added with muscle activation level u;

Modelling baseline errors in numerical typing

,

Experimental constants

Modelling baseline errors in numerical typing

u

Muscle activation level

Modelling baseline errors in numerical typing

c

The extent of temporal noise

c is the extent of TN which accumulates as movement time increases

I

Interference index

I was an interference index accounting for the relative extent of the dual-task interference in background noise (CN).

 

Server F

Equation Set C-7: Choice reaction modeling in multiple tasks: Equation (B16) (Wu & Liu, 2008a)

Variables

 

 

E(RT2)

Expected reaction time

 

SOA

Stimulus-onset asynchrony

The time difference between the onset of the two stimuli from two tasks

Ti

Processing time of servers see (Wu & Liu, 2008a)

TFst

 

Equation Set C-8: Modeling the effects of response complexity (using a single finger or multiple fingers at the same time): Equation (1-13) (Lin & Wu, 2012)

Variables

 

 

Response time to i th stimulus with a finger strategy under an urgency condition

Finger strategy

 

i

Response order

 

T

Processing time

 

 

Equation Set C-9: Complex decision making with value matrix: Equation (5) (Zhao & Wu, 2013)

Variables

 

P(t)

Speed choice at time t

V(t)

Momentary valence

M(n)

Human subjective attribute matrix

W(t)

Attention weight matrix

S

Feedback matrix

 

Equation Set C-10: Perceived risk modeling: Equation (5) (Zhuang & Wu, 2013)

Variables

 

PRv

Human perceived risk increases with higher risk from vehicles

PRl

Human perceived risk increases with higher risk from local-defined risk

ag

A coefficient adjusting effect of group size of human

Ngroup

Group size of human

 

Equation Set C-11: Decision making in lateral control: Equation (1, 2) (Bi, Gan, Shang, & Liu, 2012)

Variables

 

Increment of steering angle

kp, kd

The coefficients of proportional derivative controller

a'y

The first derivative of acceleration

E

Error between the desired lateral position gained with the predefined desired path and predictive lateral position computed with the internal vehicle dynamics model

v

Current lateral velocity

tp

Preview time

 

Equation Set C-12: Modeling hazard evaluation accuracy: Equation (8, 16, 21) ( Zhang, Wu, & Wan, 2016)

Variables

 

 

The effect of hazard evaluation accuracy on error rate

 

Perceived value of hazard

 

Actual value of hazard

 

Estimated distance

 

Threshold of perceived distance

 

Actual distance between the current position of warning receiving vehicle

v(t)

Instant speed

The instant speed (v) and acceleration (at) at time t is modeled in [23] as follows:

Global optic flow rate of the textured ground surface

φ is the global optic flow rate of the textured ground surface, a proportion of speed as long as eye height is constant

k

Parameter

The parameter k is quantified by the annual mileage divided by a maximum value of annual mileage in general

Perceived time-to-collision

The perceived time-to-collision (TTCp) will be affected by the existence of the lead vehicle. TTC is the actual time to collision that the vehicle will be able to avoid a collision without exceeding the assumed maximum deceleration

LV

Lead vehicle status

LV is a dichotomous variable of the lead vehicle in order to model the effect of the lead vehicle on TTCp (0 = without lead vehicle; 1 = with lead vehicle)

Lead time of speech warning

 

Server G:

Equation Set C-13: Urgency and Motivation Modeling: Equation (12) (Lin & Wu, 2012)

Variables

 

 

Response time to i th stimulus with a finger strategy under an urgency condition

RT

Reaction time

 

DL

Delay time caused by dual-task interference

 

MT

Movement time

 

Key-closure Time

 

Finger strategy

notation of finger strategy. =0 Single finger typing; =1 Multi-finger typing

Urgency

notation of urgency. =1non-urgent condition; =0urgent condition

i

Response order

notation of response order. i=1first response in 9-digit number, and so on.

 

Motor Subnetwork

Server W:

Equation Set M-1: Motor program retrieval modeling in the learning process: Equation (2) (Wu & Liu, 2008b)

Variables

 

 

Processing time in each server

Reduction of Server Processing Time.

Expected minimal processing time (Ti) at server i after intensive practice

Feyen (2002)

Change in the expected processing time from the beginning

to the end of practice

Reduction of Server Processing Time.

Learning rate of server i

Heathcote et al. (2000)

Number of entities processed by server i

Reduction of Server Processing Time.

 

Server X:

Equation Set M-2: Error correction modeling in close-loop motor control: Equation (24-32) (Lin & Wu, 2012)

Variables

 

 

The uncorrected portion of endpoint variability

Endpoint variability in different conditions

MT

Movement time

Modelling response time of numerical typing: the general equation of response time

DL

Delay time caused by dual-task interference

Modelling response time of numerical typing: the general equation of response time

Err%

Estimations of error rates

Estimations of error rates (Err%) in jth typing conditions

P(), P()

Parameters

The probability of errors in X-direction and Y-direction during jth experimental condition

 

Hand Servers (Server 23, 24):

Equation Set M-3: Hand and finger movement time and errors in QWERTY keyboard typing: Equation (19)