acmlicensed 978-1-4503-6708-0/20/04 $15.00

# An Intermittent Click Planning Model

Eunji Park and Byungjoo Lee∗

KAIST, Republic of Korea<EMAIL_ADDRESS>
(2020)

###### Abstract

Pointing is the task of tracking a target with a pointer and confirming the
target selection through a click action when the pointer is positioned within
the target. Little is known about the mechanism by which users plan and
execute the click action in the middle of the target tracking process. The
Intermittent Click Planning model proposed in this study describes the process
by which users plan and execute optimal click actions, from which the model
predicts the pointing error rates. In two studies in which users pointed to a
stationary target and a moving target, the model proved to accurately predict
the pointing error rates ($R^{2}=0.992$ and $0.985$, respectively). The model
has also successfully identified differences in cognitive characteristics
among first-person shooter game players.

###### doi:

https://doi.org/10.1145/3313831.3376725

###### keywords:

Click model; pointing; intermittent control; cue integration; temporal
pointing; internal clock.

††conference: CHI’20, April 25–30, 2020, Honolulu, HI, USA

<ccs2012> <concept>
<concept_id>10003120.10003121.10003122.10003332</concept_id>
<concept_desc>Human-centered computing User models</concept_desc>
<concept_significance>500</concept_significance> </concept> </ccs2012>

[500]Human-centered computing User models

##  1 Introduction

When pointing through an input device such as a mouse or trackpad, users
typically perform a click action to confirm the target selection. At that
time, for successful target selection, the mouse button must be pressed at the
appropriate moment when the pointer is located within the target. Here,
several questions arise: 1) when and how do users plan and execute a click
action during pointing? 2) what factors influence the accuracy and precision
of the click action? This study explores the user’s _click planning and
execution process_ , which has rarely been considered in previous pointing
studies in human-computer interaction (HCI).

Pointing is a task in which the tracking and the click action are combined
into one, and both actions can be blended in different ways in time and space.
For example, a careful user can plan and perform a click action after it is
certain that the pointer is positioned within the target. On the other hand,
users who are willing to take a risk can _anticipate_ when the pointer will be
positioned within the target and begin planning and executing the click action
before the pointer actually enters the target.

To the best of our knowledge, extant research has not explored the process of
planning and executing click in depth. For example, Fitts’ law [22, 39], a key
model for user’s pointing performance, yields an accurate prediction of the
overall pointing completion time. However, there is still room for providing
assumptions and explanations for the mechanism of the user clicking on the
target. Control theoretic models [42, 10, 4, 5], on the other hand,
successfully explain the fundamental principles of human motor planning and
execution, but only limited explanations have been given for click action so
far compared to tracking movement in pointing. ††*Corresponding author

###  1.1 Effect of Click Plan Quality on Pointing Performance

Since a click action is typically performed at very short distances and
duration (less than 130 ms) [58, 41], it may have a small impact on the
overall task completion time. However, since a pointing endpoint is ultimately
determined from a click action, the quality of the click plan can affect the
end point distribution or error rate ($ER$), another key performance metric of
pointing. The quality of a click here means how accurate and precise a click
action can be planned and executed in _time_ , as the click action itself
usually does not have to be spatially accurate or precise; usually, a finger
is already placed on the click button, and clicking requires a very short
travel distance. In other words, the Fitts’ index of difficulty (ID) of the
click action is close to zero.

Figure 1: This study present a computational model for predicting the pointing
error rates based on an understanding of the cognitive mechanism in which
click action is planned and executed by the user.

If users click in a situation where they can easily estimate _when_ their
pointer will be positioned within the target, the quality of the click plan
will be high, and the variance of the endpoint distribution and the pointing
error rate will be low. However, when users want to perform a quicker
pointing, they try to click in a situation where it is difficult to estimate
_when_ the pointer be located in the target, the quality of the click plan
will be low, and the end point variance and pointing error rate will have a
higher value. This study ultimately aims to present a model for predicting the
pointing error rates based on an understanding of the cognitive mechanism in
which click timing is estimated by the user.

###  1.2 Overview of Intermittent Click Planning Model

This study presents a cognitive model called the intermittent click planning
(ICP) model, which is a predictive model of pointing error rates that includes
explicit consideration of the click planning process.

The ICP model essentially regards the click planning process as a _timing
decision problem_ given to the user’s internal clock. The ICP model describes
the cognitive process in which the user’s internal clock estimates the optimal
timing of click actions while the pointer approaches the target. To this end,
the model considers six predictor variables (target position $\vec{p}_{t}$,
cursor position $\vec{p}_{c}$, target velocity $\vec{v}_{t}$, cursor velocity
$\vec{v}_{c}$, target size $W$, period of click repetition $P$) and four free
parameters ($c_{\sigma}$, $c_{\mu}$, $\nu$, $\delta$) that determine the
quality of the user’s click planning process by referring to previous studies
of sensorimotor synchronization [53, 47, 37] and human reaction [46, 35].
Predictor variables describe the situation where the user’s pointer is
tracking the target, such as the velocity of the pointer and the target, or
the target size (i.e., similar to $D$ and $W$ variables in Fitts’ law). Free
parameters of the model are constants representing the cognitive
characteristics of users, for example, the precision of the internal clock and
the drift rate of visual encoding (i.e., similar to $a$ and $b$ parameters in
Fitts’ law). Once these ten values have been determined, the ICP model can
predict the pointing error rates ($ER$).

The ICP model is based on two important assumptions about users. First, the
model assumes that the user is an _intermittent controller_. In studies of
human aimed movement, intermittent control refers to the process by which
humans discontinuously control the parameters that produce continuous motion
[10, 43, 17]. When analyzing the pointing trajectories, the user’s tracking
movement is comprised of several submovements, which manifests that
intermittent control is taking place. Due to such intermittency, the model
assumes that users build their click plan during the last submovement that was
executed just prior to the click. Otherwise, if click actions were planned
earlier than that, the plan would become obsolete because intermittent updates
of tracking movements later make the tracking situation different from the
situation where the click was planned. With this assumption, the model only
needs to consider the last submovement just prior to the click, not the whole
tracking trajectory when predicting the error rates.

Second, ICP model assumes that the user is a statistical encoder that makes
optimal use of the externally provided information that allows for estimation
of click timings. Based on _cue integration theory_ [18], the model shows how
users integrate click timings estimates from multiple sensory channels into a
single click timing estimate, through the maximum likelihood estimation (MLE)
process.

We conducted two user studies to verify the explanatory power of the model.
The first study deals with pointing at a stationary target, and the second
study concerns pointing at a moving target with constant velocity. In both
studies, our model proved to accurately predict the user’s pointing error
rates ($R^{2}=0.992$ and $0.985$, respectively). We also observed that the
user’s internal factors obtained by fitting the model to the data were _kept
at the same level regardless of whether the target was moving or not_. This
shows that our model successfully represents the actual cognitive process, not
simply overfitting the data with a number of free parameters.

Predicting error rates in pointing has been of great interest for many
reasons, from everyday interactions like touch [7, 8, 9] and typing [52, 56,
60, 21, 20, 25] to more dynamic interactions like games and music [38, 37,
35]. In general, error rate models has been used for evaluating the
performance of a given interface, designing an appropriate level of difficulty
for a game, or profiling different user populations.

To the best of our knowledge, this is the first model that describes the
mechanism by which users plan and execute a click action during pointing. Our
model does not conflict with existing pointing models, such as Fitts’ law or
control theoretical models; rather, it provides room for refinement of
existing models with a thorough consideration of the click process.

##  2 Related Work

###  2.1 Studies on Click Planning and Execution Process

A click is an input action in which a button is pressed on a keyboard or mouse
to carry out a computer operation. The click is the most basic task of HCI,
but it has been studied in the past as a secondary task required to confirm
the target selection in pointing. In this respect, a click is considered to be
performed sequentially based on minimal cognitive effort as long as the
pointer is located within the target. For example, early GOMS family models
measured total pointing time as a simple sum of pointer movement time and
click action duration [32, 12]. This is in line with the fact that in studies
of the Fitts’ law, a click action has been regarded as a non-information
component that simply adds to the trial completion time without affecting the
throughput of the pointing process [57].

However, the pointing time predicted by the GOMS models is longer than the
actual pointing time [27, 31]. This problem comes from the assumption that the
click action is performed sequentially after tracking movement. Therefore,
later GOMS models assumed the user’s anticipatory click action [31]. The
models assume that the click action can be planned in advance while tracking
the target and can also be performed in a chunk with the tracking movement
[12]. This is similar to the perspective of the ICP model in this study.
However, the existing GOMS models differ from the ICP model. They do not
describe the cognitive processes by which users plan and perform click
actions, and also assume that perfect users perform error-free clicks [12,
26].

###  2.2 Temporal Pointing Studies

Recently, several studies have modeled the user’s performance in a special
task where there is no physical movement other than click action (i.e., pure
clicking task). In that task, called _temporal pointing_ [37, 35, 38], the
user must acquire a target that is only selectable for a short time, such as a
blinking target or a target moving to a selection region, by activating a
click event. To successfully accomplish this task, the user must activate the
click at an accurate and precise timing by estimating _when_ the target will
be selectable, using his or her own internal clock.

In the temporal pointing process, two external cues are given that allow the
user to estimate the optimal click timing [35]: (1) a temporal repetition cue
and (2) a visually perceivable movement cue. The temporal pointing models
describe how the user encodes those external temporal cues to estimate when
the target will be selectable, and consequently predict the user’s click error
rates. Based on the click planning mechanism described by the temporal
pointing model, by adding additional assumptions about the user’s intermittent
control process, the ICP model in this study describes how users plan and
perform click actions during pointing.

###  2.3 Predicting Error Rates in Stationary Target Pointing

In the pointing task to a stationary target, the user’s pointing error rates
change with the time constraints given to the user. In general, the shorter
the time limit given to the user, the higher the error rates. This is called
the speed-accuracy trade-off [45] phenomenon. Based on empirical findings on
this phenomenon, there are various models that predict the pointing error
rates. The present study does not cover all of the models in detail, and we
refer interested readers to [54]. In this section, we introduce a state-of-
the-art model among them.

Wobbrock et al. [54, 55] derived a model that predicts the pointing error
rates of users in pointing when time pressure varies. Their model is derived
from Fitts’ law, and the error rate prediction can be done through a closed
form equation:

$\small ER=1-erf\Bigg{\\{}\frac{1}{D\sqrt{2}}\left[2.066\cdot
W(2^{\frac{MT_{e}-a}{b}}-1)\right]\Bigg{\\}}$ (1)

where $a$ and $b$ are free parameters inherited from Fitts’ law, $D$ is the
target distance, and $W$ is the width or size of the target. $1/b$ has been
called index of performance or throughput. $MT_{e}$ is the movement time
_actually_ taken in pointing and is an independent variable of the model. The
model describes the user’s error rate successfully for both one-dimensional
[54] and two-dimensional [55] pointing but essentially cannot hold in
situations where Fitts’ law is violated [16, 2, 15, 24, 51, 59]. The model
also does not account for the user’s error rate in pointing to a moving
target. Wobbrock’s model is used as the baseline for the first user study in
the present paper, which deals with pointing to a stationary target.

###  2.4 Predicting Error Rates in Moving Target Pointing

Pointing research has been conducted mainly on stationary targets. However,
pointing to moving targets is often done in applications such as games and
music. Moving target pointing has been studied in cognitive psychology, called
moving-target interception [6] or anticipation-coincidence [19] studies.
However, in the field of HCI, research on moving target pointing has started
only recently [28, 35, 37, 13, 14, 3].

Unlike a stationary target, in a pointing process on a moving target, the
positional relationship between the target and the pointer constantly changes
even if the pointer does not move. Therefore, it is difficult to model the
pointing situation with a few predictor variables such as the target distance
and target width. For example, four or more different pointing strategies
(pursuit, head-on, receding, and perpendicular) can be characterized [49].
This makes it difficult to build a model that predicts the error rates in a
moving target pointing task.

Recently, Huang and his colleagues [28, 29] have suggested a model called
Ternary-Gaussian that predicts endpoint distribution from the given target
velocity ($V$) and target width ($W$) condition. Based on the assumption that
the distance to the target does not affect the user’s pointing endpoint
distribution, the model can neglect the relative positional relationship
between the moving target and the pointer. The model predicts the pointing
endpoint distribution as a 2D Gaussian distribution:

$\small\begin{split}\sigma_{t}&=\sqrt{d_{t}+e_{t}V^{2}+f_{t}W^{2}+g_{t}V/W}\\\
\sigma_{n}&=\sqrt{d_{n}+e_{n}V^{2}+f_{n}W^{2}}\\\
\mu_{t}&=a_{t}+b_{t}V+c_{t}W\;\;\;\text{and}\;\;\;\mu_{n}=0\end{split}$ (2)

where $\mu_{t}$ and $\sigma_{t}$ are the mean and standard deviation of the
Gaussian distribution in the target velocity direction, respectively, and
$\mu_{n}$ and $\sigma_{n}$ are the mean and standard deviation of the Gaussian
distribution in the normal direction to the target velocity, respectively.
$a$, $b$, $c$, $d$, $e$, and $f$ are ten free parameters of the model that
must be determined from the user experiments. The pointing error rate can be
obtained by integrating the Gaussian distribution outside the target area. The
Ternary-Gaussian model is used as the baseline for the second user study in
this paper, which deals with pointing to a moving target.

##  3 Intermittent Click Planning Model

###  3.1 Task and Problem Formulation

The ICP model concerns the general 2D pointing task. The task involves a user
moving a pointer over a target and activating a click event. If a click event
occurs when the pointer is within the target, the trial is considered
successful; if it occurs outside the target, the trial is deemed failed. When
the user repeats the pointing, the ratio of failed trials to total number of
trials is the pointing error rate ($ER$).

The ICP model assumes that the target tracking movement and the click action
are each _separately_ planned and executed by the user. Among them, the ICP
model concentrates on the modeling of the click input movement. More
specifically, in the ICP model, it is assumed that the click action is planned
and executed in the middle of the target tracking movement. Thus, how the
target tracking movement of the pointer is being performed will affect the
quality of the planned click action. The situation where the pointer is
tracking the target can be characterized by the following variables:

  * •

Target velocity ($\vec{v}_{t}$): The two-dimensional vector representing the
velocity at which the target moves

  * •

Target position ($\vec{p}_{t}$): The two-dimensional vector representing the
location of the target

  * •

Target size ($W$): The width of the target (diameter if the target is
circular)

  * •

Pointer velocity ($\vec{v}_{p}$): The two-dimensional vector representing the
velocity at which the pointer moves

  * •

Pointer position ($\vec{p}_{p}$): The two-dimensional vector representing the
location of the pointer

  * •

Mean period of click repetition ($P$): When clicks are repeated this is the
average of the preceding inter-click time intervals (or average click period).

Note that the above variables can also explain the pointing situation for a
stationary target (i.e., $\vec{v}_{t}=0$).

###  3.2 Model Derivation

####  3.2.1 Two-level Process of Click Planning and Execution

The ICP model assumes that the planning and execution of click action is based
on a two-level process that contains a _central_ and a _peripheral_ level (see
Figure 2). The central process is responsible for estimating the optimal click
timing from a given target tracking situation. The peripheral process serves
to actually implement the click action at the timing estimated by the central
process.

The central process is performed by the user’s internal clock. In a given
tracking situation, the user’s internal clock is given several sensory signals
that allow for estimation of optimal click timing. The internal clock then
encodes the sensory signals to estimate the optimal click timing, and
generates a trigger pulse to activate the click action at the estimated
optimal click timing. This trigger pulse is sampled from a probabilistic
distribution and has a specific mean and variance because a given sensory
signal is not completely reliable.

The peripheral process is performed by the user’s motor system. When the
internal clock generates an activation pulse, the peripheral process executes
a predefined click action as soon as possible. The click action here refers to
the very short movement of the user’s finger pressing the button. However,
there is an inherent delay in the motor system. As a result, the user’s click
action is always slightly later than the internal clock’s intended timing.
This delay is also a stochastic value with a specific mean and standard
deviation. According to earlier models of sensorimotor synchronization [53,
50], the trigger pulse and the motor delay can be assumed to be probabilistic
independent.

As a result, the quality of the user’s click actions depends on the following
factors in this model: (1) the quality of the trigger pulses generated by the
internal clock and (2) the quality of the motor delay. For the construction of
a simple and effective model, we assume that the motor process’s quality is
much higher (mean and variance close to zero) than that of the internal clock,
so that the effect of the motor delay on the click action can be ignored. This
is a plausible assumption because click action has a Fitts’ $ID$ that is
negligible.

Figure 2: The ICP model consists of two-level process: The _central process_
encodes a given sensory signal, producing activation pulses that trigger the
click action. The _peripheral process_ executes the click action as soon as
the internal clock generates an activation pulse.

####  3.2.2 Intermittent Click Planning

It is known that a person is an intermittent controller when performing target
tracking movements. Intermittent control refers to the process of
intermittently modifying the original motor plan while humans are performing
their movements. That is, the new motor plan intermittently replaces the
existing motor plan, and the human prepares the next motor plan while the
current motor plan is executed [43, 10, 11]. As a result of intermittent
control, the trajectory of a person’s aimed movement shows inherent
discontinuities and is comprised of several submovements. Because of such
intermittent updates of the motor plan, it is not correct to assume that
planning of a click action takes place during the entire target tracking
movement. Rather, we assume that the plan for the click action is made during
the execution of the last submovement just before the click. If the click
action is planned before that, it will be meaningless due to the subsequent
intermittent control process.

The start and end of a submovement can be found by tracking the local maximum
and minimum from the speed profile of the trajectory [36]. Assuming that the
time at which the last submovement started is $t_{sub}$, all model derivations
after this section deal only with the pointing situation since that point.

####  3.2.3 Implicit Aim Point in Time

In order for the click action to be attempted at the appropriate timing, the
user’s internal clock must first estimate the timing of two major events that
occur during the target tracking movement. The first event is the moment the
user’s pointer first contacts the target ($t=t_{enter}$). The second event is
the moment the user’s pointer exits the target ($t=t_{exit}$). The user must
activate the click action between these two events to make the target
acquisition successful.

Let $W_{t}$ be the time interval between two events. $W_{t}$ is the duration
from the moment the pointer first touches the target until it passes
completely through the target:

$W_{t}=W_{intersect}/||\vec{v}_{p}-\vec{v}_{t}||$ (3)

where $\vec{v}_{p}$ is the velocity vector of the pointer and $\vec{v}_{t}$ is
the velocity vector of the target. $W_{intersect}$ is the length of the line
where the movement trajectory of the pointer intersects with the target (see
Figure 3). If the pointer does not penetrate the target during the last
submovement, the value of $W_{intersect}$ is simply zero.

Figure 3: The ICP model describes the cognitive process in which a user
estimates the optimal click timing ($t_{click}$) in the last submovement of
the target tracking process.

The ICP model assumes that the user can encode the $t_{enter}$ and $t_{exit}$
from the sensory signals given in the pointing situation and can determine the
appropriate click timing ($t_{click}$) within $W_{t}$. The click timing is
expressed relative to $W_{t}$ as follows:

$t_{click}=c_{\mu}\cdot W_{t}$ (4)

In the above equation, time is defined with $t_{enter}$ as $0$; $c_{\mu}$ is
called the user’s _implicit aim point_. If $c_{\mu}$ is $0$,
$t_{click}=t_{enter}$; if $c_{\mu}$ is $1$, $t_{click}$ = $t_{exit}$.

####  3.2.4 Input Distribution in Time

Since negligible motor delay is assumed, a user’s click timing distribution is
determined solely from the distribution of click timings estimated by the
internal clock. In this case, all cases where $t_{click}$ is earlier than
$t_{enter}$ or later than $t_{exit}$ are failed inputs. In sensorimotor
synchronization studies [47], a person’s response timing distribution can
generally be regarded as Gaussian $\mathcal{N}(\mu,\sigma^{2})$, and its mean
($\mu$) and variance ($\sigma^{2}$) can be expressed as (see Figure 3 bottom):

$\mu=\text{E}[t_{click}]\;\;\;\text{and}\;\;\;\sigma^{2}=\text{Var}[t_{click}]$
(5)

####  3.2.5 Modeling Mean of Click Timing Estimates

The ICP model assumes a user’s _constant implicit aim point_ ($c_{\mu}$) to
account for the mean of the click timing distribution. In other words, the
mean of the user’s click timing distribution is presumed to be located at a
constant ratio of the length of $W_{t}$. For example, if the user’s implicit
aim point is 0.25, the mean value $\mu$ of the click distribution would be 25
ms for $W_{t}$ 100 ms, and 50 ms for $W_{t}$ 200 ms (from $t_{enter}=0$). The
constant implicit aim point of the user has been demonstrated experimentally
from previous temporal pointing studies [37, 35, 38]. Based on this
assumption, the mean ($\mu$) of the input distribution can be expressed as:

$\mu=\text{E}[t_{click}]=\text{E}[c_{\mu}W_{t}]=c_{\mu}W_{t}$ (6)

####  3.2.6 Modeling Variance of Click Timing Estimates

While approaching the target, users can estimate click timing from two
different sensory cues. The first cue is a _temporal structure cue_ ; if the
click has been performed at regular intervals or at a specific rhythm, the
user can also estimate the timing of the next click without additional
information [37]. The second is a _visually perceivable movement cue_ that is
encoded from the relative position and velocity between the pointer and the
target. Knowing the distance between the pointer and the target and the
relative velocity at which the pointer approaches the target, users can
estimate the timing when the click action should be performed [35, 48].

The click timings estimated from each cue have different standard deviations.
The standard deviation of the click timing estimated from the temporal
structure cue ($\sigma_{t}$) is known to increase in proportion to the period
of repetition ($P$). This is known as the scalar property of the internal
clock [47, 23]. For example, clapping every 5 seconds will have a higher
timing variability than clapping once every second:

$\sigma_{t}\propto P$ (7)

Next, the standard deviation of the click timing encoded from the visual
movement cue ($\sigma_{v}$) can be modeled as a function of the time interval
at which the user can observe the relative movement between the pointer and
the target (i.e., cue viewing time $t_{c}$):

$\sigma_{v}\propto 1/(\mathrm{e}^{\nu t_{c}}-1)+\delta$ (8)

This equation provides an intuitive understanding of the reliability of the
click timing encoded from the visual cue [35]. If cue viewing time is
sufficiently long ($t_{c}\rightarrow\infty$), $\sigma_{v}$ converges toward
$\delta$. $\delta$ represents the minimum standard deviation of the click
timing the user can encode from the visual cue. On the other hand, if cue
viewing time is very short ($t_{c}\rightarrow 0$) this is the same situation
as when no visual cue exists, where $\sigma_{v}$ diverges to infinity. $\nu$
represents the rate at which the user encodes click timing from the visual
cue. The higher this is, the more precise click timing can be planned for the
same $t_{c}$. $\nu$ is also called the drift rate in models of human reaction
[46].

Since we consider only the last submovement in the click model due to the
user’s intermittent control process, it can be assumed that the cue viewing
time ($t_{c}$) is the time from the start of the submovement ($t_{sub}$) to
the time of the click ($t_{click}$):

$t_{c}=t_{click}-t_{sub}$ (9)

Finally, by defining $c_{\sigma}$ as the parameter that determines the
proportionality of Equations 7 and 8, $\sigma_{t}$ and $\sigma_{v}$ can be
expressed as follows:

$\sigma_{t}=c_{\sigma}P\;\;\text{and}\;\;\sigma_{v}=c_{\sigma}(1/(\mathrm{e}^{\nu
t_{c}}-1)+\delta)$ (10)

####  3.2.7 Integration of Click Timing Estimates

Figure 4: The final click timing is encoded separately from each of the
temporal structure cue and the visual cue.

Cue integration theory [18, 35, 44] describes the process by which humans
integrate multiple estimates of a physical quantity into a single estimate.
According to the theory, humans encode the given information in a
statistically optimal way, which can be expressed by the maximum likelihood
estimation (MLE) process. For example, let’s say a person wants to estimate
the size of a ball. The person can estimate the size of the ball separately
through visual and haptic sensations. If the estimates from each sensory
channel have different variance, $\sigma_{1}^{2}$ and $\sigma_{2}^{2}$, then
the variance of the person’s final estimate of ball size is expressed by MLE
as:

$\sigma^{2}=\sigma^{2}_{1}\sigma^{2}_{2}/(\sigma^{2}_{1}+\sigma^{2}_{2})$ (11)

Through this process, the reliability of the integrated estimate is higher
than the reliability of individual estimates.

Based on the cue integration theory, we can also model the variance of the
final click timing that the user estimates during the target tracking process.
As described earlier, click timing can be estimated separately from two
different sensory cues, and the variance of those estimates can be expressed
as $\sigma_{t}^{2}$ and $\sigma_{v}^{2}$ in Equation 10. When those two
estimates of click timings are integrated into a single click timing by MLE,
the variance of the final click timing estimate ($\sigma^{2}$) can be
expressed as:

$\sigma^{2}=\frac{\sigma^{2}_{t}\sigma^{2}_{v}}{(\sigma^{2}_{t}+\sigma^{2}_{v})}=\frac{c_{\sigma}^{2}\cdot
P^{2}}{(1+(P/(1/(\mathrm{e}^{\nu t_{c}}-1)+\delta))^{2})}$ (12)

####  3.2.8 End Point Distribution and Pointing Error Rate

Finally, the user’s click timing distribution is a 1D Gaussian distribution
$\mathcal{N}(\mu,\sigma^{2})$ with mean $\mu$ (Equation 6) and variance
$\sigma^{2}$ (Equation 12) on the time axis. In this distribution, cases where
the click input timing $t_{click}$ is between 0 and $W_{t}$ are successful in
acquiring the target, and cases where the click input timing $t_{click}$ is
smaller than 0 or greater than $W_{t}$ are failing in the target acquisition.
So by subtracting the integral of the input distribution from 0 to $W_{t}$
from 1.0, we can get the expected pointing error rate ($ER$):

$\begin{split}ER&=1-\int_{0}^{W_{t}-\mu}\mathcal{N}(t)dt-\int_{0}^{\mu}\mathcal{N}(t)dt\\\
&=1-\frac{1}{2}\left[erf(\frac{W_{t}-\mu}{\sigma\sqrt{2}})+erf(\frac{\mu}{\sigma\sqrt{2}})\right]\end{split}$
(13)
$=1-\frac{1}{2}\left[erf(\frac{(1-c_{\mu})W_{t}}{\sqrt{{\frac{2c_{\sigma}^{2}\cdot
P^{2}}{1+(\frac{P}{\frac{1}{\mathrm{e}^{\nu
t_{c}}-1}+\delta})^{2}}}}})+erf(\frac{c_{\mu}W_{t}}{\sqrt{\frac{2c_{\sigma}^{2}\cdot
P^{2}}{1+(\frac{P}{\frac{1}{\mathrm{e}^{\nu t_{c}}-1}+\delta})^{2}}}})\right]$

Table 1: Comparison between baseline models and ICP model |  No. free parameters |  Moving Target |  Stationary Target |  Physical meaning of free parameters
---|---|---|---|---
Baseline 1 [54] |  2 |  No |  Yes |  Yes
Baseline 2 [29] |  10 |  Yes |  No |  No
ICP model |  4 |  Yes |  Yes |  Yes

###  3.3 Summary of Free Parameters in the ICP Model

The ICP model has _four_ free parameters ($c_{\sigma}$, $c_{\mu}$, $\nu$,
$\delta$) that must be calibrated through user experiments, such as the slope
and $y$-intercept in Fitts’ law. The larger the number of free parameters, the
better the model can describe the data, but at the same time, overfitting
problems can arise. The baseline model of stationary target pointing [54] has
two free parameters, and the baseline model of moving target pointing [29] has
ten free parameters. With just four parameters, the ICP model can effectively
predict the pointing error rate for both moving and stationary targets (see
Table 1). Also, each parameter of the ICP model has a clear cognitive meaning.
The meaning of each free parameter of the ICP model is:

  * •

$c_{\sigma}$ represents the encoding precision of the user’s internal clock.
The higher the $c_{\sigma}$, the lower the performance of estimating the click
timing.

  * •

$c_{\mu}$ represents the user’s implicit aim point. If $c_{\mu}$ is 0 (or
1.0), it means that the user estimated on average the click timing the moment
the pointer first contacted (or first exited) the target. A $c_{\mu}$ of 0.5
gives the lowest error rate, but users typically have a $c_{\mu}$ value lower
than 0.5 [37, 35, 38], which is called the negative mean asynchrony (NMA)
phenomenon [50, 47, 33].

  * •

$\nu$ is the rate at which the user encodes sensory information to estimate
click timing from the _visual cue_. The higher this value, the more reliable
the user can perform with a shorter cue viewing time ($t_{c}$).

  * •

$\delta$ is the minimum standard deviation of the click timing estimated from
the visual cue when the user is given enough cue viewing time ($t_{c}$).

##  4 Implementation of Prediction Software

This section describes how to predict the pointing error rate on a real system
using the ICP model. We need to implement a software that runs in three steps:
(1) _real-time trajectory logging_ , (2) _submovement segmentation_ , and (3)
_predicting pointing error rate_.

###  4.1 Step 1: Real-Time Trajectory Logging

The software first logs the trajectory of the target and the pointer in real
time until a click event occurs. If the sampling rate of the input device
being used is $f$ (unit: Hz), more specifically the software collects the
following data from the moment of the previous click until the next click is
observed:

$\centering\left[x_{p}^{i},\,y_{p}^{i},\,x_{t}^{i},\,y_{t}^{i},\,t^{i}\right],\,\text{for
$i$ = $1$ to $N$}\@add@centering$

($x_{p}^{i},y_{p}^{i}$) and ($x_{t}^{i},y_{t}^{i}$) represent the coordinates
of the $i$-th sampled pointer and the target, respectively. $t^{0}$ is the
moment the preceding click occurred and $t^{N}$ is the moment when the current
click event occurred. When a total of $N$ points are sampled, $t^{i}$ is the
time stamp of the $i$-th sampling (i.e., $t^{i}=1/f\cdot i$).

###  4.2 Step 2: Submovement Segmentation

When a click occurs, the speed profile of the pointer is obtained by
numerically differentiating the logged trajectory of the pointer. The
resulting speed profile is often contaminated with sensor noise from the input
device. Since this reduces the performance of submovement segmentation, noise
in the speed profile must be removed through a low-pass filter. Various types
of filters can be applied, but simple forms such as Gaussian kernel filters
(e.g., $\sigma=3$) can work well enough.

The system then identifies the local minima and maxima in the smoothed speed
profile and each neighboring minimum-maximum-minimum triplet is considered to
be a possible submovement (see Figure 5). We use Persistence1D [34] as an
algorithm to find local extrema in the speed profile. This algorithm returns
all pairs of minima and maxima that exceed the pre-defined persistence value
(e.g., 0.2 for a computer mouse). To prevent jitter of click motion from being
missegmented into a submovement, only triplets with a duration of at least 50
ms are considered as submovements.

Figure 5: The actual result of the submovement segmentation

###  4.3 Step 3: Predicting Pointing Error Rate

Among the submovements segmented in the previous step, the submovement started
just before the click event is analyzed to predict the error rate (i.e., the
last submovement). If the last submovement of the pointer started at the
$s$-th sample, the timestamp at that moment is $t^{s}$. Because the user could
observe the movement of the pointer during the last submovement, the cue
viewing time ($t_{c}$) can be calculated from Equation 9:

$t_{c}=t^{N}-t^{s}$ (14)

The software then computes the average velocity vector of the pointer
($\vec{v}_{p}$) and target ($\vec{v}_{t}$) during the submovement:

$\begin{split}&\vec{v}_{p}=\left[(x_{p}^{N}-x_{p}^{s}),\,(y_{p}^{N}-y_{p}^{s})\right]/t_{p}\\\
&\vec{v}_{t}=\left[(x_{t}^{N}-x_{t}^{s}),\,(y_{t}^{N}-y_{t}^{s})\right]/t_{p}\end{split}$

The relative velocity vector $\vec{v}$ between the target and the pointer can
be simply obtained as follows:

$\vec{v}=\vec{v}_{p}-\vec{v}_{t}$ (15)

From the point ($x_{p}^{s},y_{p}^{s}$) where the last submovement starts, the
pointer approaches the target with a relative velocity of $\vec{v}$. The
software then calculates $W_{t}$, which is the time it takes the pointer to
pass through the target. $W_{t}$ can be obtained by dividing $W_{intersect}$
by the magnitude of the relative velocity $\lVert\vec{v}\rVert$, where
$W_{intersect}$ is the length of the line segment defined by the intersection
of the extended straight line $\vec{v}$ and the target (Figure 3). Next, $P$
is calculated, which is the period in which the click is repeated. This can be
calculated as the average of the time intervals between all preceding clicks
[37, 35].

From the obtained $P$, $t_{c}$, and $W_{t}$, the software can finally predict
the pointing error rate through Equation 13. If free parameters are already
available, the error rate is calculated immediately. If free parameters are
not obtained in advance, the values can be obtained through a general
optimization process that minimizes the difference between the error rate
predicted by the ICP model and the error rate of users observed in actual
experiments. In the following sections, we demonstrate those processes in
greater detail by conducting actual user studies and verify the error rate
prediction performance of the model.

##  5 Study 1: Pointing on a Stationary Target

Based on the implemented system, two user studies were conducted (Figure 6).
In all studies, the data analysis was performed using the implementation
described in the previous section. In Study 1, participants performed a two-
dimensional pointing task on a stationary target. Users were given a time
limit, which resulted in a wide range of error rates. We used Wobbrock’s error
rate model [54] as the baseline for performance comparison.

Figure 6: Task screen of Study 1 (pointing on a stationary target) and Study 2
(pointing on a moving target)

###  5.1 Method

####  5.1.1 Participants

Twelve paid participants from a local university (7 males, 5 females) were
recruited. The average age of the participants was 24.42 years
($\sigma$=3.26). All of the participants were right-handed. Their average
mouse usage time per day was 5.63 hours ($\sigma$=3.59). The participants
played games 6.86 hours ($\sigma$=2.61) per a week with a computer mouse. All
participants had normal or corrected vision.

####  5.1.2 Design

The experiment followed a 2$\times$3$\times$6 within-subject design with three
independent variables: _Target Width_ , _Target Distance_ , and _Time Limit_.
The levels were the following:

  * •

Target Width: 4.8 and 8.4 mm

  * •

Target Distance: 48, 144, and 240 mm

  * •

Time Limit: 300, 400, 500, 600, 700, and 800 ms

Twenty angle of approaches were tested for each _Time Limit_ -_Target Width_
-_Target Distance_ condition. A _Time Limit_ condition did not change to the
next condition until all corresponding width-distance conditions had been
completed (240 pointing trials per each _Time Limit_ condition). Within a
_Time Limit_ condition, the _Target Width_ and _Target Distance_ are given in
random order. The _Time Limit_ conditions are given in a random order. Each
_Time Limit_ condition was repeated twice. The angle of approach was given to
the participant in a clockwise sequence of 360 degrees divided into 20 steps.
In the end, 17,280 (=$2\times 3\times 6\times 2\times 20\times 12$) input
events from 12 participants were logged.

####  5.1.3 Task

Participants were given two circular targets. After clicking on the blue
target, clicking on the red target ended the trial (Figure 6). If the
participant did not click on the red target within the given _Time Limit_
after clicking on the blue target, the red target disappeared. Even if the red
target disappeared, participants had to click to go to the next trial. If a
participant clicked inside the red target (or the disappeared red target), the
trial was considered successful. Participants were asked to perform pointing
as quickly and accurately as possible. They were also asked to complete each
trial within the _Time Limit_.

####  5.1.4 Apparatus

The application was implemented on a 3 GHz desktop computer (Mac mini,
10.13.1). A 27-inch LED monitor (LG 27UD69P) was used, and the resolution of
the task screen was 2,560$\times$1,440 pixels. The pointing device was a two-
button wired optical mouse (Samsung SNJ-B138) with a resolution of 1,000 DPI
and a the polling rate of 125 Hz. The pointer was a standard arrowhead
pointing to the upper left. The mouse gain function maintained the default
setting of the OS. The refresh rate of the application was maintained at 60
Hz.

####  5.1.5 Procedure

Participants sat on a regular office chair and the monitor was installed at
the participant’s eye level. Before the experiment, the experimenter briefly
introduced the task to the participants. Subsequently, the participants filled
out a pre-questionnaire. Participants also signed a consent form. A practice
session was given until participants were accustomed to the task. The
experiment for each individual took about an hour and each participant was
rewarded with $10.

###  5.2 Results

####  5.2.1 Data Validation

For all trials, the movement time that the participants actually performed the
task was about 124% of the given _Time Limit_. However, as the last
submovement already started at 72% (SD=51%) of the _Time Limit_ , we know that
the participants did not intentionally wait for the target to disappear.

####  5.2.2 Descriptive Statistics

The overall average error rate for all participants’ trials was 37%. This is
about two times higher than the error rate in other studies [54, 55]. The
duration of the last submovement, or the cue viewing time $t_{c}$, was
measured to be 296 ms (SD=119 ms) on average, which is similar to the known
values in previous studies [30, 36]. $W_{t}$ was measured to be 244 ms on
average (SD=285). In 2,613 trials (15.1 %), the pointer moved in a direction
that could not intersect the target (i.e., $W_{intersect}$=0). In that case
$W_{t}$ was considered zero. Except for those cases, the average of the
measured $W_{t}$ was 288 ms (SD=289). The mean of the input repetition period
$P$ was measured as 636 ms (SD=207). The average magnitude of the relative
velocity between the target and the pointer was 64.98 mm/s (SD=103).

####  5.2.3 Overall Model Fit to the Baseline Model

As a baseline, Wobbrock’s model [54, 55] was fitted to 36 data points using
Equation 1 (2 _Target Widths_ $\times$ 3 _Target Distances_ $\times$ 6 _Time
Limits_). In the equation, $MT_{e}$ is calculated as the mean time actually
spent in a pointing trial at each condition, not the given _Time Limit_ value.
We used the Global Optimization Toolbox of MATLAB for the fitting.

As reported in the previous studies [54, 55], the model successfully explains
the error rate of the user (Adjusted $R^{2}$ = 0.954, see Figure 7). The free
parameters of the model were $a$=130 ms and $b$=157 ms/bit, respectively.
Because these values are based on Fitts’ law, throughput can be calculated as
the reciprocal of $b$. As a result, we obtained 6.37 bit/s similar to the
previously measured value for the mouse [40].

Table 2: The fitting results from the two user studies in this paper  | $c_{\mu}$ | $c_{\sigma}$ | $\nu$ | $\delta$ | $R^{2}$
---|---|---|---|---|---
Study 1 | 0.129 | 0.0873 | 14.532 | 0.461 | 0.992
Study 2 | 0.241 | 0.093 | 25.33 | 0.337 | 0.985

####  5.2.4 Overall Model Fit to ICP Model

In our model, the variable that determines the error rate of the user is
$W_{t}/D_{t}$ (see Equation 13). Therefore, the total data is arranged in
order of $W_{t}/D_{t}$, and then the error rate is obtained through the
sequential binning from the left. This allows us to get 36 final data points
as we fit the baseline model. The following values from a previous study [35]
were used as the free parameters for the initial sorting: $c_{\mu}=$0.25,
$c_{\sigma}=$0.08, $\nu=$20.2, and $\delta=$0.366.

As a result, our model fitted with the observed error rate with a high
coefficient of determination (adjusted $R^{2}=0.992$, see Figure 7). The free
parameters obtained as a result of fitting are summarized in the Table 2.

Figure 7: Results from Study 1: Both the baseline model and the model proposed
in this study explained the observed error rate well.

We also conducted two-fold cross validation with random sampling for each
model. The mean absolute error (MAE) values obtained were 4.86% and 2.56% for
the baseline model and our model, respectively.

###  5.3 Discussion

In the task of pointing to a stationary target, the baseline model and the ICP
model predicted the error rate of the participants with high accuracy.
However, our model predicted error rates more accurately than the baseline
model. This supports our hypothesis that the pointing error occurs from the
user’s click timing estimation during the last submovement.

##  6 study 2: Pointing on a Moving Target

In Study 2, participants pointed to a circular moving target. Regardless of
the target movement, we analyzed the last submovement with the same algorithm
as in Study 1 and fitted it to the model. We used Huang’s model [29] as a
baseline.

To verify that the free parameters obtained from the fittings successfully
reflect the cognitive characteristics of the users, we recruited participants
into two groups: gamers and non-gamers. Participants in the gamer group were
recruited as experts in the first-person shooter (FPS) games due the
similarities of the tasks performed in such games and in the study.

###  6.1 Method

####  6.1.1 Participants

We recruited a total of 16 participants divided into two groups: (1) gamers (8
males) and (2) non-gamers (1 male, 7 females). All the participants were
right-handed. The average age of participants in the gamer group was 24.4
years (SD=3.81) and in the non-gamer group was 25.63 years (SD=4.53).
Participants of the gamer group played FPS games an average of 15 hours per
week, and their average mouse usage time per day was 7.13 hours (SD=2.23).
Meanwhile, non-gamer group participants did not play FPS game at all and they
use a mouse for an average of 4.25 hours (SD=3.99) per week.

####  6.1.2 Participant Recruiting Criteria

We recruited gamers based on the following criteria: (1) a player of the game
PlayerUnknown’s Battlegrounds who was within the top 5% rating, or (2) a
player of the game Overwatch who owned a higher level than master (the top 2
to 3%) and who mainly focused on characters who need excellent aiming ability
(such as Hanzo or Mccree). Meanwhile, non-gamers were recruited with people
who had never played FPS games before.

####  6.1.3 Design

The experiment followed a within-subject design with two independent
variables: _Target Speed_ and _Target Width_. These two factors were randomly
determined for each trial in the following ranges:

  * •

Target Speed: from 0 mm/s to 510 mm/s

  * •

Target Width: from 9.6 mm to 24 mm

In order to satisfy ecological validity, we reproduced the speed range of the
target in commercial games such as Fruit Ninja (107 mm/s) and Piano Tiles (314
mm/s). The location where the target was generated and the direction the
target moved were randomly determined for each trial. Participants performed a
total of 9 _Block_ s of trials and each _Block_ consisted of 200 trials. As a
result, 28,800 input events from 16 participants were logged (=$16\times
9\times 200$).

####  6.1.4 Task

Participants were instructed to click on a blue circular target moving
straight at a constant speed (see Figure 6). If the target collided with a
wall (edge of the screen), the target bounced at the same angle as the
incident angle. The trial was considered successful only when the participant
clicked inside the target. Regardless of success, if a click event occurred,
the current target disappeared and a new target was created with randomized
size and speed. Participants were asked to perform pointing as quickly and
accurately as possible.

####  6.1.5 Apparatus

The application was implemented on the same 3 GHz desktop computer as in Study
1 (Mac mini, 10.13.1 High Sierra). A 24-inch LED-backlit LCD gaming monitor
(ASUS ROG SWIFT PH248Q) was used and the resolution of the task screen was
1920$\times$1080 pixels. The pointing device was a wired optical mouse
(Logitec G502) with a resolution of 1000 DPI and a polling rate of 125 Hz.
Mouse acceleration was disabled and the tracking speed of the mouse was 4/10
(the default setting of the Mac OS). The refresh rate of the application used
in the experiment was maintained at 60 Hz.

####  6.1.6 Procedure

Participants sat on a regular chair. The display was installed at the
participant’s eye level. Participants signed a consent form before the
experiment. After the participants completed the pre-questionnaire, the
experimenter briefly demonstrated the task. 50 trials were given to
participants as a practice session before starting the main study. A one-
minute break was provided at the end of each _Block_. It took about an hour
per person to finish the study. The amount of compensation for participation
was the same as in Study 1.

###  6.2 Results

Figure 8: Results from Study 2

####  6.2.1 Descriptive Statistics

The overall average error rate for all participants’ trials was 37%. This
value was almost the same as in Study 1. The duration of the last submovement,
or $t_{c}$, was measured to be 275 ms (SD=238 ms) on average, which is similar
to the submovement duration reported in previous studies [30, 36]. $W_{t}$ was
measured to be 107 ms on average (SD=148 ms). In 4,658 trials (16.2 %), the
pointer moved in a direction that could not intersect the target (i.e.,
$W_{intersect}$=0). In that case, $W_{t}$ was considered zero. The average
interval from click to click was 905 ms (SD=460 ms). The average magnitude of
the relative velocity between the target and the pointer
($\lVert\vec{v}\rVert$) was 144 mm/s (SD=74 mm/s).

####  6.2.2 Removing Learning Effect

Pointing to a moving target is a challenging task, so there can be a
significant learning effect. In fact, the effect of _Block_ on error rate was
significant (F(8,120)=5.183, p<0.001). From the Helmert contrast of the
_Block_ effect, we confirmed that the learning effect becomes insignificant
from the third _Block_ (p=0.376) The following results were obtained by
analyzing the remaining seven _Block_ s after excluding the first two.

####  6.2.3 Overall Model Fit to the Baseline Model

We used Huang’s model as the baseline model for the Study 2 [29]. In Huang’s
model, the difficulty of a task can be determined according to $\mu_{t}$,
$\sigma_{t}$, and $\sigma_{n}$ (see Related work section). Therefore, we
arranged the data by the average of these three values. Then, we binned in
order from the smallest value to the larger one and obtained 36 fit points to
plot. In this study as well, we used the Global Optimization Toolbox of MATLAB
for the fitting. Huang’s model does not provide explicit error rate prediction
for circular targets. Therefore, we calculated the error rate based on Monte-
Carlo simulation from the distribution predicted by the model and the average
target size of the bin. The data was fitted to the baseline model with
adjusted $R^{2}$=0.558 and the MAE value obtained from two-fold cross
validation was 6.33%. The free parameters of the model were $a_{t}$=0.13,
$b_{t}$=1.0-4, $c_{t}$=-0.19, $d_{t}$=3.6-10, $e_{t}$=0, $f_{t}$=0.041,
$g_{t}$=0, $d_{n}$=0.003, $e_{n}$=0, and $f_{n}$=0.

####  6.2.4 Overall Model Fit to the ICP Model

In the same manner as in Study 1, the data of all trials were binned by
$W_{t}/D_{t}$, and finally, 36 averaged data points were obtained. As a
result, our model fitted the empirical error rate with a high coefficient of
determination (adjusted $R^{2}=0.985$, see Figure 8). The free parameters
obtained as a result of fitting are summarized in the Table 2. The free
parameters obtained were similar to those in the previous study [35] and in
Study 1. We performed two-fold cross validation with random sampling. The MAE
was 3.04% for our model.

####  6.2.5 Comparing Gamers and Non-gamers

By fitting our model to individual data, we obtained four free parameters
($c_{\sigma},c_{\mu},\nu,\delta$) for each participant. Considering the
reduced number of individual data points, the error rates were obtained from
the 18 bins of $W_{t}/D_{t}$. As a result, the error rate for each participant
was fitted to the model with a high coefficient of determination
($R^{2}$=0.938 to 0.990, M=0.973, SD=0.016). An independent-samples t-test was
conducted to compare free parameters and error rates between gamers and non-
gamers. There was a significant difference in the error rates for gamer
(M=27.4%, SD=6.9%) and non-gamer (M=44.1%, SD=14.1%) groups; $t$(14)=3.021,
$p$ = 0.009. There was no significant difference in the period of input
repetition ($P$) between gamers (M=871 ms, SD=186 ms) and non-gamers (M=886
ms, SD=64 ms); $t$(8.641)=-0.218, $p$ = 0.833. There was also a significant
difference in the $c_{\sigma}$ for gamer (M=0.058, SD=0.029) and non-gamer
(M=0.11, SD=0.031) groups; $t$(14)=-3.14, $p$ = 0.007. No significant
difference was found for the other three parameters $c_{\mu}$ ($t$(14)=0.36,
$p$=0.73), $\nu$ ($t$(14)=-0.22, $p$=0.83), and $\delta$ ($t$(14)=1.48,
$p$=0.16). The results are summarized in Figure 9.

Figure 9: Comparison of four free parameters between the gamer group and the
non-gamer group

###  6.3 Discussion

Unlike Study 1, in Study 2, there was a significant difference in explanatory
power between the ICP model and the baseline model. The results of the cross
validation also showed that the ICP model predicted the error rate better than
the baseline model. The baseline model, however, predicted the error rate in
moving target pointing with high accuracy in the original paper [29]
($R^{2}$=0.94). The difference between their results and ours seems to be due
to differences in the speed range of the targets used in the experiment. Their
experiment used targets with a maximum velocity of approximately 200 mm/s when
the pixels were roughly converted to millimeters. However, our experiments
used a target speed of up to 500 mm/s, which is 2.5 times faster than the
target speed they used.

Huang’s model predicts that the faster the target speed ($V$ in Equation 2),
the greater the variance of the pointing end point. However, according to the
ICP model, even if the target moves fast, the pointing error rate can be
lowered if the user moves well to reduce the relative speed between the
pointer and the target. In fact, the coefficients of the Huang model, which
are multiplied by the target velocity ($e_{t}$, $e_{n}$, $g_{t}$), were mostly
optimized to be zero at the fitting, indicating that the variance of the
endpoint distribution did not increase as the model predicted when the target
velocity was increased.

The parameters of the ICP model obtained from Study 1 were very similar to the
values obtained from Study 2 (see Table 2). This supports the assumption of
the ICP model that the planning and execution of click actions is performed
with the same cognitive-motor process regardless of whether the target is
moving or not.

The model also succeeded in uncovering differences in cognitive
characteristics in the click process between gamers and non-gamers. Compared
to non-gamers, gamers had higher $c_{\sigma}$ value and lower $\delta$ value.
As shown in Equation 10, the reliability of the click timing obtained from the
visual cue ($\sigma_{v}$) can be thought of as the product of $c_{\sigma}$ and
$\delta$. However, the reliability of the click timing obtained from the
temporal cue ($\sigma_{t}$) is determined from $c_{\sigma}$ alone. As shown in
Figure 9, multiplying the $c_{\sigma}$ and $\delta$ values yields similar
values for gamers and non-gamers. Therefore, we can interpret that gamers have
similar ability to estimate click timing from visual cues to that of non-
gamers, but have better ability to encode the rhythm of clicks with an
internal clock.

##  7 Conclusion

The model proposed in this study accurately predicted users’ pointing error
rate with a simple algorithm regardless of the target motion ($R^{2}$= 0.985
to 0.992 and MAE=2.56% to 3.04%). In particular, the four free parameters
obtained from the data fitting remained similar for different pointing
situations (see Table 2). Based on this robust explanatory power, the model
revealed significant cognitive differences between gamers and non-gamers.

Nonetheless, this study has some limitations. First, it is difficult to apply
our model in a situation where the trajectory of the cursor is difficult to
track. Secondly, further validation is needed as to whether this model is
generally applicable for more complex patterns of target motion. Third, our
model did not explain how a user’s internal clock encodes the temporal
structure cue when the input is randomly repeated. Fourth, the ICP model was
currently only tested for young users in their 20s. For users from other age
groups, further research will be needed to determine whether the ICP model
predicts error rates well and extracts meaningful parameters. Fifth, the ICP
model only predicts the error rate of the click action and does not predict
how the user’s tracking movement will be performed before the click action. We
envision that this limitation can be overcome by integrating the ICP model
with existing control theoretical [42, 10, 11] pointing models that can
simulate the user’s target tracking movements.

##  8 Acknowledgements

This research was funded by the National Research Foundation of Korea
(2017R1C1B2002101) and the Korea Creative Content Agency (R2019020010).

## References

  * [1]
  * [2] Jos J Adam, Robin Mol, Jay Pratt, and Martin H Fischer. 2006.  Moving farther but faster: an exception to Fitts’s law.  Psychological science 17, 9 (2006), 794–798.
  * [3] Abir Al Hajri, Sidney Fels, Gregor Miller, and Michael Ilich. 2011.  Moving target selection in 2D graphical user interfaces. In IFIP Conference on Human-Computer Interaction. Springer, 141–161.
  * [4] Stanislav Aranovskiy, Rosane Ushirobira, Denis Efimov, and Géry Casiez. 2016.  Modeling pointing tasks in mouse-based human-computer interactions. In 2016 IEEE 55th Conference on Decision and Control (CDC). IEEE, 6595–6600.
  * [5] Stanislav Aranovskiy, Rosane Ushirobira, Denis Efimov, and Géry Casiez. 2019.  A switched dynamic model for pointing tasks with a computer mouse.  Asian Journal of Control (2019).
  * [6] James J Belisle. 1963.  Accuracy, reliability, and refractoriness in a coincidence-anticipation task.  Research Quarterly. American Association for Health, Physical Education and Recreation 34, 3 (1963), 271–281.
  * [7] Xiaojun Bi, Yang Li, and Shumin Zhai. 2013.  FFitts law: modeling finger touch with fitts’ law. In Proceedings of the SIGCHI Conference on Human Factors in Computing Systems. ACM, 1363–1372.
  * [8] Xiaojun Bi and Shumin Zhai. 2013.  Bayesian touch: a statistical criterion of target selection with finger touch. In Proceedings of the 26th annual ACM symposium on User interface software and technology. ACM, 51–60.
  * [9] Xiaojun Bi and Shumin Zhai. 2016.  Predicting finger-touch accuracy based on the dual Gaussian distribution model. In Proceedings of the 29th Annual Symposium on User Interface Software and Technology. ACM, 313–319.
  * [10] Robin T Bye and Peter D Neilson. 2008.  The BUMP model of response planning: Variable horizon predictive control accounts for the speed–accuracy tradeoffs and velocity profiles of aimed movement.  Human movement science 27, 5 (2008), 771–798.
  * [11] Robin T Bye and Peter D Neilson. 2010.  The BUMP model of response planning: intermittent predictive control accounts for 10 Hz physiological tremor.  Human movement science 29, 5 (2010), 713–736.
  * [12] Stuart K Card, Thomas P Moran, and Allen Newell. 1980.  The keystroke-level model for user performance time with interactive systems.  Commun. ACM 23, 7 (1980), 396–410.
  * [13] Juan Sebastián Casallas. 2015.  Prediction of user action in moving-target selection tasks.  Ph.D. Dissertation. Iowa State University.
  * [14] Juan Sebastián Casallas, James H Oliver, Jonathan W Kelly, Frédéric Merienne, and Samir Garbaya. 2013.  Towards a model for predicting intention in 3D moving-target selection tasks. In International Conference on Engineering Psychology and Cognitive Ergonomics. Springer, 13–22.
  * [15] Olivier Chapuis and Pierre Dragicevic. 2008.  Small targets: why are they so difficult to acquire.  Laboratoire de Recherche en Informatique, Tech. Rep (2008).
  * [16] Olivier Chapuis and Pierre Dragicevic. 2011.  Effects of motor scale, visual scale, and quantization on small target acquisition difficulty.  ACM Transactions on Computer-Human Interaction (TOCHI) 18, 3 (2011), 13.
  * [17] Kenneth JW Craik. 1947.  THEORY OF THE HUMAN OPERATOR IN CONTROL SYSTEMS 1: I. THE OPERATOR AS AN ENGINEERING SYSTEM.  British Journal of Psychology. General Section 38, 2 (1947), 56–61.
  * [18] Marc O Ernst and Martin S Banks. 2002.  Humans integrate visual and haptic information in a statistically optimal fashion.  Nature 415, 6870 (2002), 429.
  * [19] Brett R Fajen and William H Warren. 2007.  Behavioral dynamics of intercepting a moving target.  Experimental Brain Research 180, 2 (2007), 303–319.
  * [20] Leah Findlater and Jacob Wobbrock. 2012.  Personalized input: improving ten-finger touchscreen typing through automatic adaptation. In Proceedings of the SIGCHI Conference on Human Factors in Computing Systems. ACM, 815–824.
  * [21] Leah Findlater, Jacob O Wobbrock, and Daniel Wigdor. 2011.  Typing on flat glass: examining ten-finger expert typing patterns on touch surfaces. In Proceedings of the SIGCHI conference on Human factors in computing systems. ACM, 2453–2462.
  * [22] Paul M Fitts. 1954.  The information capacity of the human motor system in controlling the amplitude of movement.  Journal of experimental psychology 47, 6 (1954), 381.
  * [23] John Gibbon, Russell M Church, and Warren H Meck. 1984.  Scalar timing in memory.  Annals of the New York Academy of sciences 423, 1 (1984), 52–77.
  * [24] Cheryl M Glazebrook, Dovin Kiernan, Timothy N Welsh, and Luc Tremblay. 2015.  How one breaks Fitts’s Law and gets away with it: Moving further and faster involves more efficient online control.  Human movement science 39 (2015), 163–176.
  * [25] Asela Gunawardana, Tim Paek, and Christopher Meek. 2010.  Usability guided key-target resizing for soft keyboards. In Proceedings of the 15th international conference on Intelligent user interfaces. ACM, 111–118.
  * [26] Paul Holleis, Friederike Otto, Heinrich Hussmann, and Albrecht Schmidt. 2007.  Keystroke-level model for advanced mobile phone interaction. In Proceedings of the SIGCHI Conference on Human Factors in Computing Systems. ACM, 1505–1514.
  * [27] Anthony J Hornof and David E Kieras. 1999.  Cognitive modeling demonstrates how people use anticipated location knowledge of menu items. In Proceedings of the SIGCHI conference on Human Factors in Computing Systems. ACM, 410–417.
  * [28] Jin Huang, Feng Tian, Xiangmin Fan, Xiaolong (Luke) Zhang, and Shumin Zhai. 2018.  Understanding the Uncertainty in 1D Unidirectional Moving Target Selection. In Proceedings of the 2018 CHI Conference on Human Factors in Computing Systems. ACM, to be determined.
  * [29] Jin Huang, Feng Tian, Nianlong Li, Zhang, and Xiangmin Fan. 2019.  Modeling the Uncertainty in 2D Moving Target Selection. In The 32st Annual ACM Symposium on User Interface Software and Technology. ACM, to be determined.
  * [30] Richard J Jagacinski, Daniel W Repperger, Martin S Moran, Sharon L Ward, and Betty Glass. 1980.  Fitts’ law and the microstructure of rapid discrete movements.  Journal of Experimental Psychology: Human Perception and Performance 6, 2 (1980), 309.
  * [31] Bonnie E John, Konstantine Prevas, Dario D Salvucci, and Ken Koedinger. 2004.  Predictive human performance modeling made easy. In Proceedings of the SIGCHI conference on Human factors in computing systems. ACM, 455–462.
  * [32] David Kieras and others. 2001.  Using the keystroke-level model to estimate execution times.  University of Michigan 555 (2001).
  * [33] Sunjun Kim, Byungjoo Lee, and Antti Oulasvirta. 2018.  Impact Activation Improves Rapid Button Pressing. In Proceedings of the 2018 CHI Conference on Human Factors in Computing Systems. ACM, to appear.
  * [34] Y Kozlov and T Weinkauf. 2015.  Persistence1D: Extracting and filtering minima and maxima of 1d functions.  h ttp://people. mpi-inf. mpg. de/ weinkauf/notes/persistence1d. html, accessed (2015), 11–01.
  * [35] Byungjoo Lee, Sunjun Kim, and Antti Oulasvirta. 2018.  Moving Target Selection: A Cue Integration Model. In Proceedings of the 2018 CHI Conference on Human Factors in Computing Systems. ACM, to appear.
  * [36] Byungjoo Lee, Mathieu Nancel, and Antti Oulasvirta. 2016.  AutoGain: Adapting Gain Functions by Optimizing Submovement Efficiency.  arXiv preprint arXiv:1611.08154 (2016).
  * [37] Byungjoo Lee and Antti Oulasvirta. 2016.  Modelling error rates in temporal pointing. In Proceedings of the 2016 CHI Conference on Human Factors in Computing Systems. ACM, 1857–1868.
  * [38] Injung Lee, Sunjun Kim, and Byungjoo Lee. 2019.  Geometrically Compensating Effect of End-to-End Latency in Moving-Target Selection Games. In Proceedings of the 2019 CHI Conference on Human Factors in Computing Systems. ACM, 560.
  * [39] I Scott MacKenzie. 1992.  Fitts’ law as a research and design tool in human-computer interaction.  Human-computer interaction 7, 1 (1992), 91–139.
  * [40] I Scott MacKenzie and Poika Isokoski. 2008.  Fitts’ throughput and the speed-accuracy tradeoff. In Proceedings of the SIGCHI Conference on Human Factors in Computing Systems. ACM, 1633–1636.
  * [41] I Scott MacKenzie and Shawn X Zhang. 1999.  The design and evaluation of a high-performance soft keyboard. In Proceedings of the SIGCHI conference on Human Factors in Computing Systems. ACM, 25–31.
  * [42] Jörg Müller, Antti Oulasvirta, and Roderick Murray-Smith. 2017.  Control theoretic models of pointing.  ACM Transactions on Computer-Human Interaction (TOCHI) 24, 4 (2017), 27.
  * [43] PD Neilson, MD Neilson, and NJ O’dwyer. 1988.  Internal models and intermittency: A theoretical account of human tracking behavior.  Biological Cybernetics 58, 2 (1988), 101–112.
  * [44] Antti Oulasvirta, Sunjun Kim, and Byungjoo Lee. 2018.  Neuromechanics of a Button Press. In Proceedings of the 2018 CHI Conference on Human Factors in Computing Systems. ACM, to appear.
  * [45] Réjean Plamondon and Adel M Alimi. 1997.  Speed/accuracy trade-offs in target-directed movements.  Behavioral and brain sciences 20, 2 (1997), 279–303.
  * [46] Roger Ratcliff. 1978.  A theory of memory retrieval.  Psychological review 85, 2 (1978), 59.
  * [47] Bruno H Repp. 2005.  Sensorimotor synchronization: a review of the tapping literature.  Psychonomic bulletin & review 12, 6 (2005), 969–992.
  * [48] James R Tresilian. 1991.  Empirical and theoretical issues in the perception of time to contact.  Journal of Experimental Psychology: Human Perception and Performance 17, 3 (1991), 865.
  * [49] James R Tresilian. 2005.  Hitting a moving target: perception and action in the timing of rapid interceptions.  Perception & Psychophysics 67, 1 (2005), 129–149.
  * [50] Dirk Vorberg and Alan Wing. 1996.  Modeling variability and dependence in timing.  In Handbook of perception and action. Vol. 2. Elsevier, 181–262.
  * [51] Stephen A Wallace and Karl M Newell. 1983.  Visual control of discrete aiming movements.  The Quarterly Journal of Experimental Psychology 35, 2 (1983), 311–321.
  * [52] Daryl Weir, Simon Rogers, Roderick Murray-Smith, and Markus Löchtefeld. 2012.  A user-specific machine learning approach for improving touch accuracy on mobile devices. In Proceedings of the 25th annual ACM symposium on User interface software and technology. ACM, 465–476.
  * [53] Alan M Wing and Alfred B Kristofferson. 1973.  Response delays and the timing of discrete motor responses.  Perception & Psychophysics 14, 1 (1973), 5–12.
  * [54] Jacob O Wobbrock, Edward Cutrell, Susumu Harada, and I Scott MacKenzie. 2008.  An error model for pointing based on Fitts’ law. In Proceedings of the SIGCHI conference on human factors in computing systems. ACM, 1613–1622.
  * [55] Jacob O Wobbrock, Alex Jansen, and Kristen Shinohara. 2011.  Modeling and predicting pointing errors in two dimensions. In Proceedings of the SIGCHI Conference on Human Factors in Computing Systems. ACM, 1653–1656.
  * [56] Xin Yi, Chun Yu, Mingrui Zhang, Sida Gao, Ke Sun, and Yuanchun Shi. 2015.  Atk: Enabling ten-finger freehand typing in air based on 3d hand tracking data. In Proceedings of the 28th Annual ACM Symposium on User Interface Software & Technology. ACM, 539–548.
  * [57] Shumin Zhai. 2004.  Characterizing computer input with Fitts’ law parameters—the information and non-information aspects of pointing.  International Journal of Human-Computer Studies 61, 6 (2004), 791–809.
  * [58] Shumin Zhai, Michael Hunter, and Barton A Smith. 2002.  Performance optimization of virtual keyboards.  Human–Computer Interaction 17, 2-3 (2002), 229–269.
  * [59] Shumin Zhai, Jing Kong, and Xiangshi Ren. 2004.  Speed–accuracy tradeoff in Fitts’ law tasks—on the equivalency of actual and nominal pointing precision.  International journal of human-computer studies 61, 6 (2004), 823–856.
  * [60] Suwen Zhu, Tianyao Luo, Xiaojun Bi, and Shumin Zhai. 2018.  Typing on an invisible keyboard. In Proceedings of the 2018 CHI Conference on Human Factors in Computing Systems. ACM, 439.