hello everyone this is Hang Zhao
a phd student from stony brook
university advised by professor
Xiaojun Bi
this work is a collaboration between
michael wang xiaolei zhou and xiang
shi ren
steering law links the movement time with
id of a task
however it does not show how the
variance of movement time changes with
id
and how the distribution of movement
time changes with id
for example
the right figure shows that the
variance of the movement time increases
as the id increases
in this paper
we investigate how the variance of movement time
relates to id
how the distribution relates to id
and how to model the distribution of
movement time
our contributions are twofold
first we propose the quadratic variance
model which reveals that the variance of
movement time and id are quadratically
related
second we investigate how to model the
distribution of movement time based on
steering law and the quadratic variance
model
as shown in the figure the light blue
represents the histogram of the observed
data
blue curve represents the observed pdf
and the purple curve is the predicted
gamma distribution
gamma distribution well predicts the
distribution of movement time
we show how to obtain the quadratic
variance model
here we hypothesize this quadratic
valence relationship by assuming that
the movement of a pointer
in the steering task consists of
multiple sub-movements
and each sub-moon time take a unique time
which is a random variable t
with a fixed mean mu t and variance sigma
t square across all sub movements
take the constant tunnel as an example
the instantaneous speed form of steering
law shows that the instantaneous speed v
of the pointer is proportional to the
tunnel width w
under the aforementioned assumptions
the average travel distance of the
pointer in each sub movement is
calculated
where mu t is the mean duration of each
sub movement
w is the width of the tunnel and the b is
an empirically determined constant
then
the number of sub movements a user needs
to complete the task is calculated
where a is the length of the tunnel
therefore
the total movement time for completing
the task is n times t where t is a
random variable
the variance of movement time will be n
squared times sigma t squared where
sigma t squared is the variance of t
regardless of the distribution of t
plugging the number of sub movement
and the variance of t
we can get sigma square
notice this part is a constant
if it is assumed that the initial sub
movement takes some extra time
we can obtain the quadratic variance
model as follows
this equation is referred to as the
quadratic variance model
and it shows that the variance of
movement time is quadratically related
to id
for non-constant tunnels the explanation
is similar
besides the quadratic variance model we
also introduce five other candidates to
model the variance
they are constant
linear and quadratic form with different
complexity
levels model 4 is shown as the quadratic
variance model
model 1 and model 3 are introduced to
investigate whether quadratic term would
help to improve the fitting accuracy
model 2 5 and 6 are compared with the
model 4
to see whether a candidate with higher
or lower complexity will benefit modern
fitness
combining the variance models with
steering law we can predict the distribution
of movement time
we introduce six types of distributions
which are gaussian truncated gaussian
lognormal gamma extreme value and
exponential modified gaussian
distributions
among them
lognormal gamma extreme value and
exGaussian are positively skewed
distributions
where gaussian and truncated gaussian
are symmetric or near symmetric
distributions
take the gamma distribution with the
quadratic variance model as an example
we use steering law to predict the mean
and the quadratic variance model to
predict the variance of movement time
then we express the model parameters in
terms of the mean and variance
for other types of distribution models
we follow the same method
for exgaussian distribution it
naturally apply a quadratic relationship
between the variance and id
next we evaluate the variance model
using maximum likelihood estimation
and distribution model using bayesian
modeling of zhou and ren's
and our newly collected data sets
zhou and ren's steering data set includes
10 participants and nine conditions
our result shows that the quadratic
variance models has the highest or
second highest r squared value and the
lowest leave one (a, w) condition out cross
validation
root mean square error aic and waic and
it outperforms other variance models
the distribution modeling results
suggest that
the positive skewed distributions like
lognormal gamma extreme value and
exgaussian distributions
outperform gaussian and truncated
gaussian distributions
gamma distribution performs the best
among all the distribution models
the posterior preditive checks shows that
the gamma distribution in purple
resembles the observed pdf and performs
better than other distributions
similar to the evaluation of zhou and
ren's data set we carry out an
experiment on our new dataset
we recruited 12 participants with 12
conditions
our results show that the quadratic
variance model performs better than
other variance models
the distribution modeling result shows
that lognormal gamma extreme value and
exgaussian distribution outperform
gaussian and truncated gaussian
distributions
gamma distribution performs the best
the posterior predict checks also shows
that
the gamma distribution in purple looks
very similar to the observed pdf
and outperforms other distribution
models
in this research we propose the
quadratic variance model which reveals
that the variance of movement time and
id are quadratically related
also
our results suggest that the positively
skewed distributions such as lognormal
gamma extreme value and exgaussian
perform better than gaussian and
truncated gaussian distributions
gamma distribution performs slightly
better than other distributions in most
cases
overall our research advanced the
movement time prediction from a mean
estimate to the variance and
distribution estimates for steering
tasks