[Martin Taylor 2017.08.14.14.14]
I have today sent the appended message to the three editors of Exp.
Brain. Res who are responsible for “Behavioural Sciences and
Neuropsychology”, the area in which I presume the Marken and Shaffer
paper on the power law of movement as a behavioural illusion was
handled. Since Rick has refused to provide the name of the actual
action editor, I can only hope that it was one of them.
Rick has seen a near-final draft of this letter, because I thought
that once the central mistake was made very obvious, he would prefer
to communicate with the journal himself, as though he had discovered
the error and wanted it to be corrected. Nevertheless, Rick still
insists that the paper is completely valid, so I have finally sent
the letter to the three editors. In the letter, I say: “Indeed, if
the paper is not retracted and is believed, it might inhibit further
research into the reasons the power law is so commonly observed.”
Rick said that this was, and remains, his intent and hope.
Since Rick earlier insisted that I make this public on CSGnet, I am
doing so, though I would have preferred this whole sorry incident to
have been kept private.
Martin
========letter follows=========
Sir or Madam,
I wish to bring to your attention a serious problem that I
believe negates the central thesis of the paper “The power law of
movement: an example of a behavioral illusion” (Marken and
Shaffer, Exp. Brain Res., 2017, DOI 10.1007/s00221-017-4939-y). I
do not know who was the the action Editor for the paper, so I am
sending this to the three editors listed on the journal Web site
under the heading “Behavioural Sciences and Neuropsychology”. If
this is incorrect, please would you forward this message to the
appropriate Editor.
···
The power law discussed in Marken and Shaffer's paper is a
long-known relation between R and V of the form V = kRx
that has been approximately found in a wide variety of experiments
under different conditions, with different organisms, and a wide
range of exponents “x”, usually between 1/4 and 1/3, but sometimes
as low as 1/6. That fact is interesting, and the reasons for it
remain worth investigating.Â
To explain this consistent velocity variation, Marken and Shaffer
create what they claim to be an equation for velocity as a
function of the radius of curvature, V=D1/3R1/3 .
Although this looks like an equation relating V and R, it is not,
because their D is proportional to V3 whatever the
radius of curvature. The equation therefore indicates that V is
functionally independent of R. Conceptually also, V must be
functionally independent of R, since velocity is a joint function
of distance and time, whereas the radius of curvature is defined
only by spatial variables.
I believe Marken and Shaffer's mathematical error invalidates
their paper as a contribution to the investigation of the power
law. Indeed, if the paper is not retracted and is believed, it
might inhibit further research into the reasons the power law is
so commonly observed.
What follows is a summary of the paper, followed by an analysis
of the critical equation.
I start with the critical section of the paper, quoted directly:
-----Quote-------
The equations used to calculate the
instantaneous velocity and curvature of a movement are
given in a paper by Gribble and Ostry (1996 ).
The instantaneous velocity of a two- dimensional curved
movement is calculated as follows:
V = (Ẋ 2 + Ẏ 2)1∕2 (2)
where Ẋ 2 and
Ẏ 2 are the instantaneous changes
in the position of the movement in the X and Y dimensions,
respectively. V is
a measure of tangential velocity at each instant during a
movement. The instantaneous curvature of a two-
dimensional curved movement is calculated as follows:
R=(Ẋ2+Ẏ2)3∕2∕|Ẋ⋅Ÿ−Ẍ⋅Ẏ| (3)
where Ẍ and
Ÿ are the instantaneous
accelerations of the movement in the X and Y dimensions,
respectively. R is
a measure of the radius of curvature at each instant
during a movement; the larger the value of R , the smaller the
curvature at that point in the movement.
An equation with tangential velocity, V ,
expressed as a function of radius of curvature, R ,
can be derived by appropriate substitution of Eq. (2 )
into Eq. (3 ). This is done by
recognizing that the numerator on the right side of Eq. (3 )
can be rewritten as (V2)3/2 or
V3 .
Therefore, Eq. (3 )
can be rewritten as follows:
R=V3∕|Ẋ⋅Ÿ−Ẍ⋅Ẏ|. (4)
Letting the denominator of Eq. 4 be called D and solving for V, Eq. 4 can be
written as follows:
V = D1∕3 ⋅
R1∕ 3  Â
(5)
where D = |Ẋ ⋅ Ÿ − Ẍ ⋅ Ẏ |,
------End Quote--------
Marken and Shaffer then claim that equation (5), V = D1/3R1/3 ,
together with an equivalent equation (6) relating
curvature C to tangential angular velocity A, shows why V
has a 1/3 power law relationship with R. They write:Â “* Equations
5 and 6 show that measures of velocity—V and Aâ—are
mathematically related to measures of curvature—R aand C,
respectively—as power functions with exponents thatt are
exactly equal to the values that researchers have found
when analyzing actual movement data* .”. [omitting
reference to the crucial “D”].Â
Subsequently, however, they re-include D, but only as a kind
of adjunct variable, saying: “* However, Eqs. 5 and 6 show
that the measures of velocity are a function not only of
curvature but also of the cross-product variable, D. The
existence of this “extra� cross-product variable in the
mathematical relationship between measures of curvature
and velocity will prove to be important in explaining the
results obtained in power law studies of movement*.”
Marken and Shaffer use equation (5) to argue that every
experimental observation relating R to V should be
corrected to give the “true” 1/3 power relationship
between R and V by incorporating the "’* extra’ cross
product variable* " D, as though D were just a minor
correction factor independent of V.
In order for their analysis to work as claimed, D must be
constant or nearly so, and at least must be independent of
V, whereas in fact, for any radius R, D actually is
proportional to V3.
-------discussion-------
Marken and Shaffer's "extra" cross-product variable D is
central to their argument. I now examine how D is
constructed.
In Cartesian space, any curve can be represented
parametrically in terms of x and y as a function of the
distance “s” along the curve from an arbitrary zero point.
The x-y coordinates of the point at the along-curve
distance s are functions of s, namely x(s) and y(s). Those
x-y values as functions of s define the trace of the curve
in space.
Any parameter of which s is a doubly differentiable
function can serve in place of s. For example, if an
object moves along the curve with some velocity profile as
a function of time t, its x and y positions would be
functions of time: x = x(s(t)), y = y(s(t)). The velocity
of the object along the curve would be V = ds/dt, by
definition.
Marken and Shaffer’s D is defined as D = |Ẋ â‹… Ÿ − Ẍ â‹… Ẏ |,
Written in the more usual Leibniz notation,
D = |(dx/dt)(d2y/dt2)-(dy/dt)(d2x/dt2 )|Â
Using the chain rule for differentiation to separate out
the contributions to D of s and V, this can be written
D = |(dx/ds)(d2y/ds2)(ds/dt)3
- (dy/ds)(d2x/ds)2(ds/dt)3|
  = |V3((dx/ds)(d2x/ds2 )
- (dy/ds)(d2x/ds2))|
  = V3 times a purely spatial property of the
curve, a function of s alone, independent of VÂ Â
If we call the “spatial property” f(s), then
D = V3 f(s), so Marken and Shaffer’s equation
(3), which determines the radius of curvature R. becomes
R = V3/V3f(s) = 1/f(s), demonstrating that R is
functionally independent of V.
Their equation (5) becomes
V = Vf(s)1/3R1/3
The appearance of V on both sides of their equation (5) thus
rewritten further illustrates the opposite of Marken and
Shaffer’s claim that their equation 5 (and the analogous
equation 6 for the relation between curvature C and
tangential angular velocity A) shows that "* measures
of velocity—V and A—are mathematically relateated to measures
of curvature—R and C, respectively—as power fr functions with
exponents that are exactly equal to the values that
researchers have found when analyzing actual movement data* ".
What their equations (3) and (5) show instead is that, as
one logically should expect, measures of velocity have no
necessary relation at all to the radius of curvature.
Even without mathematical demonstration, it is apparent that
measures of spatial curvature are not functions of time,
whereas measures of velocity depend on time. Curvature and
velocity cannot be functionally related, however closely
they are found to be related in experiments and observations
of natural movements.
Since the central thesis of the paper depends on Marken and
Shaffer’s misinterpretation of their equation (5) as showing
a necessary relationship between V and R, I believe the
paper should be retracted entirely.
------------
Martin Taylor, Ph.D., P. Eng.
Senior Experimental Psychologist (retired)
Defence Research and Development Canada - Toronto.