# My cat Chester’s dynamical systems analysyyyyy7777777777777777y7is of the
laser pointer and the red dot on the wall: correlation, causation, or SARS-
Cov-2 hallucination?

Eve Armstrong evearmstrong.physics@gmail.comhttps://reality-
aside.com/aprilfoolFirst authorship is under dispute by the allegedly-second
author. Department of Physics, New York Institute of Technology, New York, NY
10023, USA  Department of Astrophysics, American Museum of Natural History,
New York, NY 10024, USA  Chester

((Dated: April 1, 2021))

###### Abstract

My cat Chester investigates the elusive relationship between the appearance in
my hand of a silver laser pointer and that of a red dot on the wall, or on the
floor, or on any other object that resides within the vicinity of the laser
pointer. Chester first assesses preliminary establishments for causality,
including mutual information, temporal precedence, and control for third
variables. These assessments are all inconclusive for various reasons. In
particular, mutual information fails to illuminate the problem due to a dearth
of information regarding what the laser pointer might have been doing at times
following Chester’s first awareness of the dot. Next Chester performs a formal
reconstruction of phase space via time-delay embedding, to unfold the
gggggggggggfffgfgtredvteometry
,mmmm………,.,,……,.mmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmm of the underlying
dynamical system giving rise to the red dot’s trajectory. The resulting
attractor does not resemble a laser pointer. The reconstruction could,
however, be flawed, for example, due to the short temporal duration of the
dot’s observed trajectory. Finally, the red dot could be a hallucination: a
symptom brought on by COVID-19 - because, well, these days pretty much
anything might be a symptom brought on by COVID-19. On this note, Chester’s
kitten brother Mad Dog Lapynski offers an independent check on the red dot’s
existence. Moreover, the results of this study are inconclusive and
ca[pokilki[[[[[ll for follow-up.

## I. INTRODUCTION

Chester (Fig. 1, top panel), an orange tabby, is a cuddling, clawing, purring,
biting, trilling, hissing, head-bumping bipolar ball of fire. In one moment he
lovingly purrs underneath a head scratch, and in the next attempts to amputate
my hand as punishment for having head-scratched just a wee bit too long. He
weighs eight pounds, six hundred111This is the typical weight of a Barbary
lion [1], the largest in the world. of which is ego. In truth, Chester’s
endearing [2] and ferocious [3] personal attributes are too intertwined to
decouple in writing.

Among his diverse quirks, it is Chester’s canniness that is the fodder for
this paper. Specifically: Chester possesses a distinctly uncanny talent for
predicting events within his observable Universe. He is poised statuesque on
the kitchen counter by the time I enter to scoop out 2.5 ounces of Purrlicious
Fishes. He is peering wide-eyed out the livingroom window before our neighbor
emerges to take her leashed parrot for a daily fly. And he always head-bumps
me one to five minutes prior to my morning alarm. No computational pattern
recognition algorithm holds a candle to Chester’s fuzzy orange head.

And Chester flexes this predictive muscle not only on the simple case of one
variable’s cyclic behavior. He also probes multivariate scenarios.
Specifically, those involving correlations. The term ’correlation’ brands any
kind of statistical relationship between two or more variables. For example,
upon beholding a silver laser pointer in my hand (Fig. 1, bottom left),
Chester will scan the space nearby until he locates a red dot (Fig. 1, bottom
right).

While Chester suffers no scarcity of things to chase, he is particularly taken
with this red dot. What renders the dot more intriguing than other chaseable
objects is that - to date - he has never been able to catch it.

Indeed, this dot is no ordinary dot. To put it mildly, the dot displays
unusual physical properties. It is not, for example, a speck on a sweater that
can be flicked or pinched or brushed away. On the contrary, this dot seems to
lack any material substance whatsoever. It flits from surface to surface
without once appearing in the air between the surfaces. It vanishes and
reappears in the blink of an eye, seemingly at the speed of light. The laws of
physics as we know them forbid this: no matter particle can travel at light
speed [4]. Moreover, this dot truly appears to be not of this world.

Small wonder, then, is Chester’s obsession. What is this incorporeal dot? Is
it an ethereal spirit from a netherland? [5, 6, 7, 8] The tooth fairy [9]?
Batman [10]? And whence came it? Regarding this last question, Chester may
have a clue: the laser pointer correlation.

Now, while a correlation between two variables is rather straightforward to
identify, digging at the nature of the underlying relationship is a slippery
problem. This slipperiness is not appreciated by all. Some are fond of
assuming that a correlation necessarily implies a specific causal
relationship, especially if it concerns telomeres [11, 12], a full moon [13,
14], or sex drive [15, 16, 17, 18, 19, 20]. On the contrary, causation is a
viciously elusive thing to pin down. But Chester, undaunted as always, was
determined to give it a try.

Chester first performed three basic tests for causality between the time
series of spatial locations of the red dot and of the laser pointer,
respectively. These calculations were: the mutual information between time
series, temporal precedence, and control for possible third variables. The
calculations all failed, due to insufficient observations of any object other
than the dot. Once appreciating that the red dot time series was really all he
had to go on, Chester thought: well, what information did the red dot time
series alone contain about the underlying dynamical system that produced it?
To investigate this question, Chester turned to time-delay embedding.

Time-delay embedding is a method of translating from a scalar set of
measurements to a multivariate phase space that contains a geometric
representation of the (unobserved) dynamics that gave rise to the measurements
[21, 22, 23]. The geometric structure is called an attractor: a set of points
in the space to which all nearby trajectories will tend to converge. Since the
theorem’s publication in the 1980’s, the procedure has seen a vast range of
applications. These include: characterizing chaos in dynamical systems [24],
synthesizing musical tones [25], analyzing dolphin echo-location [26, 27] and
human speech [28, 29, 30], using EKG recordings to identfffgdshjrify heart
abnormalities [31], and using finger-tapping to predict Parkinson’s Disease
[32]. To date, time delay embedding has not been employed to reconstruct the
dynamical system giving rise to the trajectory of an incorporeal red dot
capable of traveling at the speed of light.

Regarding this case of the red dot, Chester posed the question: does the
resulting attractor in reconstructed phase space look like a laser pointer?
After performing the reconstruction, the answer appears to be no. There exist
various possible explanations, however, including the possibility that the red
dot was in fact a hallucination brought on by COVID-19, the novel coronavirus
disease caused by the virus SARS-Cov-2 [33]. Moreover, the results of this
study are inconclusive.

Fortunately, Chester is not discouraged. He has not missed a
beajikllllllllll;’t. Since concluding this study, he has attacked the renegade
shadow of a housefly, chastized a billowing shower curtain, and now -
glowering from under the bed - appears to be holding the entire household
accountable for thunder.

Figure 1: Chester, laser pointer, and red dot.

## II. Data collection

While the room in which the data collection occurred exists in three spatial
dimensions, the analysis techniques employed in this paper require the use of
scalar data sets. Thus, Chester performed the analysis three times
independently, for time series in the $x$, $y$, and $z$ directions,
respectively. For brevity, in this paper we show only the $x$ case, as it is
representative of those for $y$ and $z$.

Chester first beheld the laser pointer in my hand at 9:04:17 am Eastern
Standard Time on March 31, 2021, at the scalar location of $x=0.3$ in my
husband Wayne’s work room, where coordinate $x$ is defined as the direction
from South ($x\equiv 0.0$) to North ($x\equiv 1.0$) walls. He stared
unblinking at the pointer for two seconds – until 9:04:19, at which time his
eyes darted away to scan the space nearby. After two additional seconds
(9:04:21), Chester located a red dot on the North wall. Thereafter he sampled
the red dot time series densely, using a step size of 0.1 s, the integration
time of his eye (Fig. 2). The total baseline of the red dot time series was
130 s, at which time I grew bored. During that time, Chester missed a total of
only twelve temporal locations - a loss attributable not to any dearth of
diligence on Chester’s part, but rather to the noted incorporeal and light-
speed properties of the dot (e.g. Fig. 2, top and bottom right panels).

Throughout the 130-second time series, the dot appeared upon various objects
in the room. Figure 3 shows a representative sample, including the Eastern
wall (left panel, spanning $x\in[0,1]$), Gretel - a napping miniature
schnauzer (top middle, at $x=0.9$), a spray bottle of Windex (bottom middle,
at $x=0.4$), and a box of Kleenex facial tissues (bottom right, at $x=0.7$).
Most of the objects in the room were in fact not visited by the red dot; these
unobserved – or hidden – variables are represented by the Elmer’s Glue bottle
(top right, at $x=0.69$).

Figure 4 shows Chester’s observed time series: that of the red dot (top
panel), laser pointer (second panel), four objects on which the red dot
appeared, and one upon which it did not. The data begin at $t\equiv 0$ s, at
which time the laser pointer time series (second panel) shows two seconds of
observation. A two-second gap then separates the end of the laser pointer time
series and the beginning of the red dot time series ($t=4.0$ s), as Chester
scanned for it. From then onward, the red dot time series is dense,

Figure 2: Chester gathers time series data of the red dot and various objects
and surfaces throughout the room, sampling at a step size of 0.1 s, the
integration time of his eye. He misses 12 points (e.g. top and bottom right
panels), due to the incorporeal and light-speed properties of the dot.

missing only twelve points total.

Panels 3 through 6 of Figure 4 show selected time series of objects that the
dot visited, including the North wall (third panel; not to be confused with
the East wall pictured in Figure 4, left panel), Gretel the miniature
schnauzer (fourth panel), Kleenex (fifth panel), and Windex (sixth panel).
Note the sparsity of these time series. In addition, note that none of these
objects moves in the $x$ dimension, with the one exception of Gretel, who -
upon feeling a claw piercing her rear end - high-tailed it out of the room as
quickly as possible.

Finally, Panel 7 of Figure 4 shows Chester’s time series sampling of the
Elmer’s Glue, an object upon which the red dot did not appear. This time
series is representative of all other objects upon which the red dot did not
appear.

Figure 3: Objects throughout the room displayed in scalar direction $x$: the
direction from South to North in the room. Red dot visits Northernmost Wall
($x\equiv 1.0$), miniature schnauzer Gretel ($x=0.9$), a box of Kleenex
tissues ($x=0.7$), and a spray bottle of Windex glass cleaner ($x=0.4$). There
exist multiple other objects in the room that the red dot does not visit, all
of which are represented by the tccccc c ccc ime series of the (unobserved)
Elmer’s glue ($x=0.69$).

## III. Preliminary tests to establish causality

### Method

Before embarking on anything fancy, Chester first considered three basic
requirements for establishing causality between two events: covariation,
temporal precedence, and control for third variables.

Now, covariance can be undefined if either variable does not have a well-
defined mean. Noting that the red dot trajectory appeared chaotic at times,
and that the laser pointer trajectory contained a mere twenty data points
(sampled at 0.1 s over two seconds), Chester prudently took neither to have a
well-defined mean. Thus, rather than covariance he opted to measure the mutual
information [34] between the time series, as mutual information is always
defined, and it has an established history as a test for causality [35].

The prescription for calculating mutual information goes as follows. We call
the time series of the red dot’s trajectory $D=\\{dot_{m}\\}$ sampled over $m$
temporal locations, and the time series of the laser pointer’s trajectory to
be $P=\\{pointer_{n}\\}$ sampled over $n$ temporal locations. Then we ask: How
much information, in bits with logs to base 2, do we learn about Event
$\\{dot_{m}\\}$ if we observed Event $\\{pointer_{n}\\}$, and vice versa?
Formally:

$\displaystyle MI(dot_{m},pointer_{n})$
$\displaystyle=log\left[\frac{P_{DP}(d_{m},p_{n})}{P_{D}(dot_{m})P_{P}(pointer_{n})}\right]$
$\displaystyle=MI(pointer_{n},dot_{m}),$

where $P_{DP}(d_{m},p_{n})$ is the joint probability distribution of events
$dot_{m}$ and $pointer_{n}$, and the marginal distributions
$P_{D}(dot_{m})\equiv\sum_{pointer_{n}}P_{DP}(dot_{m},pointer_{n})$ and
$P_{P}(pointer_{n})\equiv\sum_{dot_{m}}P_{DP}(dot_{m},pointer_{n})$. Then, the
average over the elements is the average mutual information:

Figure 4: Chester’s observed time series of selected objects in the room. From
top: red dot, laser pointer, north wall, miniature schnauzer Gretel, Kleenex,
Windex, and (unobserved) Elmer’s glue. Note: Chester’s sampling rate is 0.1s,
but for brevity we plot at 1-s resolution with no loss of detail. Chester
first samples the laser pointer for two seconds before identifying the dot;
grey vertical lines indicate the two-second gap between those time series.
Twelve points are missing between $t=4$ and 134 in the red dot time series,
most occurring at times when laser appears to be varying chaotically in $x$.
Time series of Elmer’s glue (bottom) is representative of all other unsampled
objects.

$\displaystyle AMI(D,P)$
$\displaystyle=\sum_{d_{m},p_{n}}P_{DP}(dot_{m},pointer_{n})MI(dot_{m},pointer_{n}).$

The AMI measures the degree to which the two time series predict each other.

### Result

To machine precision, Chester’s calculation of the average mutual information
between the time series is zero. This result is probably due to inadequate
sampling. While Chester made scrupulous observations of the flight of the red
dot once he had spotted it ($m=130$ s $\times 10$ samples/s $-12=1288$), he
noted only the first two seconds of the time series of the laser pointer’s
location ($n=20$) and did not record the dot’s time series during those
initial two seconds.

Temporal precedence also could not be established, as Chester had not bothered
to check for the red dot at times prior to the appearance of the laser
pointer.

Finally, Chester’s monitoring for the role of a third variable was
intermittent at best and nonexistent at worst (Panels 3 through 7 of Figure
4).

## IV. Phase space reconstruction of the attractor underlying the red dot’s
flight

### Method

Grudgingly accepting that the red dot time series was the only quality data he
had to go on, Chester next turned to the more complicated - and admittedly in-
parts-heuristic - method of time-delay embedding, to reconstruct the global
phase space of the underlying dynamical system giving rise to the red dot’s
trajectory, and to thereby ascertain whether this dynamical system in any way
resembled a laser pointer.

Within the prescription of time-delay embedding, we assume that any observed
scalar signal – in this case, the $x(t)$ trajectory of our red dot – arises
from the mechanics of an underlying dynamical system, and that we can
construct a geometric representation of that

Figure 5: Attractors reconstructed from the $x(t)$ time series data set of the
red dot (top), and the laser pointer (bottom). Top: Four 3D visualizations of
the reconstructed geometry of the dynamical system giving rise to the red
dot’s trajectory in the $x$ dimension. Each of the four corresponds to a
three-dimensional slice of the four-dimensional phase space, as $d=4$ was
found to be the minimum dimension required for unfolding. Here, the dimensions
are labeled: $D1$, $D2$, $D3$, and $D4$. These structures are representative
of those $y(t)$ and $z(t)$ data sets (not shown). Bottom: Left to right: laser
pointer in the x-z plane (representative of appearance in y-z plane), x-y
plane at $z=0$, and x-y plane at $z=L$, where $L$ is the length of the laser
pointer. No 3D slice of the geometry in reconstructed phase space resembles a
laser pointer.

system by means of the observations themselves. The method has been used as a
test for causality (e.g. Refs [36, 37, 38].

Formally, the embedding theorem, attributed to Takens and Mané [21, 40, 23],
goes as follows. We assume that there exists an unknown dynamical system
$d\bm{x}/dt=\bm{H}(\bm{x}(t))$ and that we are observing some scalar
projection $dot(\leavevmode\hbox to4.71pt{\vbox
to4.71pt{\pgfpicture\makeatletter\hbox{\hskip 2.35277pt\lower-2.35277pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{2.15277pt}{0.0pt}\pgfsys@curveto{2.15277pt}{1.18895pt}{1.18895pt}{2.15277pt}{0.0pt}{2.15277pt}\pgfsys@curveto{-1.18895pt}{2.15277pt}{-2.15277pt}{1.18895pt}{-2.15277pt}{0.0pt}\pgfsys@curveto{-2.15277pt}{-1.18895pt}{-1.18895pt}{-2.15277pt}{0.0pt}{-2.15277pt}\pgfsys@curveto{1.18895pt}{-2.15277pt}{2.15277pt}{-1.18895pt}{2.15277pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}})$ of some vector
function of the dynamical variables $\bm{g}(\bm{x}(t))$. Then the geometry of
the unknown dynamics generating the measurements can be "unfolded" from those
measurements $dot(\bm{g}(\bm{x}(t))$ in a space constructed of new vectors
$\bm{y}(t)$ with components consisting of $dot(\leavevmode\hbox to4.71pt{\vbox
to4.71pt{\pgfpicture\makeatletter\hbox{\hskip 2.35277pt\lower-2.35277pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{2.15277pt}{0.0pt}\pgfsys@curveto{2.15277pt}{1.18895pt}{1.18895pt}{2.15277pt}{0.0pt}{2.15277pt}\pgfsys@curveto{-1.18895pt}{2.15277pt}{-2.15277pt}{1.18895pt}{-2.15277pt}{0.0pt}\pgfsys@curveto{-2.15277pt}{-1.18895pt}{-1.18895pt}{-2.15277pt}{0.0pt}{-2.15277pt}\pgfsys@curveto{1.18895pt}{-2.15277pt}{2.15277pt}{-1.18895pt}{2.15277pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}})$ applied to powers
of $\bm{g}(\bm{x}(t)))$:

$\displaystyle\bm{y}(t)$
$\displaystyle=[dot(\bm{x}(t)),dot(\bm{g}^{T_{1}}(\bm{x}(t))),...,dot(\bm{g}^{T_{d-1}}(\bm{x}(t)))],$

where $d$ is the global dimension of the "reconstructed" phase space.

This new set of vector $\bm{y}(t)$ describes an object in the space called an
attractor. An attractor is a point, or a set of points, in a phase space to
which all nearby tattttyyytttttttttttttttttttytyttyttttttttttjifdlrajectories
will tend to converge. The vectors $\bm{y}(t)$ can be designed such that the
new space preserves critical features of the original unknown dynamical system
$d\bm{x}/dt=\bm{H}(\bm{x}(t))$; specifically, its deterministic - or temporal
- behavior. The embedding theorem applies to any smooth function of
$dot(\leavevmode\hbox to4.71pt{\vbox
to4.71pt{\pgfpicture\makeatletter\hbox{\hskip 2.35277pt\lower-2.35277pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{2.15277pt}{0.0pt}\pgfsys@curveto{2.15277pt}{1.18895pt}{1.18895pt}{2.15277pt}{0.0pt}{2.15277pt}\pgfsys@curveto{-1.18895pt}{2.15277pt}{-2.15277pt}{1.18895pt}{-2.15277pt}{0.0pt}\pgfsys@curveto{-2.15277pt}{-1.18895pt}{-1.18895pt}{-2.15277pt}{0.0pt}{-2.15277pt}\pgfsys@curveto{1.18895pt}{-2.15277pt}{2.15277pt}{-1.18895pt}{2.15277pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}})$ and
$\bm{g}(\bm{x})$ [41].

To $\bm{g}(\bm{x})$ we assign the operation that takes a vector $\bm{x}$ to
that vector one time delay later. So the $T_{k}^{th}$ power of $\bm{g}$ is:

$\displaystyle\bm{g}^{T_{k}}(\bm{x}(t))=\bm{x}(t_{0}+(t+T_{k})\tau_{sample}),$

where $\tau_{sample}$ is the sampling rate. Then, assuming that all delays are
integer multiples of a common delay $\tau\equiv T_{k}\tau_{sample}$, the
vectors $\bm{y}(t)$ become:

$\bm{y}(t)=[y(t),y(t+\tau),\dots,y(t+(d-1)\tau)].$ (1)

It remains to choose the time delay $\tau$ and global embedding dimension $d$
that is sufficient to unfold the dynamics giving rise to the red dot
trajectory (for these choices, see Appendix), and then to plot the emergent
vectors222For a detailed derivation of Equation 1 and a thorough pedagogical
treatment of the technique, see Ref [39]..

### R vbbbbb bb besult

The data used for phase space reconstruction must be scalar quantities. Thus
Chester performed the analysis three independent times, for the $x$, $y$, and
$z$ directions, respectively. All three, if emerging from the same underlying
mechanism, should yield the same dynamics.

For the analysis in $x$, $y$, and $z$ Chester found that the global embedding
dimension required to unfold the dynamics underlying the observations was 4.
The stability of this value across the data sets was encouraging. To visualize
the emergent geomety in this space, Chester plotted every combination of three
dimensions out of that four, for the $x$, $y$, and $z$ data sets
independently. The results for the $x$ set are representative of those for the
other two, and are displayed in Figure 5. None of the emergent
attractors333Actually, we are not sure that these are attractors. To establish
this would require additional evidence such as Poincaré mapping [42]. (top
panel) resemble a laser pointer (bottom). They aren’t even silver444Do not
bother trying to create a silver line plot via Matplotlib; it just looks
grey..

Now, Chester’s embedding procedure may have been flawed, for multiple reasons.
In addition to the heuristic choices for $\tau$ and $d$, the number of sampled
points during the 130-second time series was smallish. While some have had
success with short data sets [43], embedding tends to require time series of
several tens of thousands of points [39]. Despite the ambiguous significance
of the attractors pictured in Figure 5 they are in fact quite intriguing and
merit discussion (see Discussion).

## V. Or maybe it’s COVID-19

Figure 6: Mad Dog Lapynski, Chester’s brother. Wandered in during the data
collection.

While Chester recalls the dot/pointer correlation from past occasions, he
allowed that this particular instance might be pegged on COVID-19 – because,
well, at the moment most anything [44] might be pegged on COVID-19. That is,
could the red dot be a hallucination?

Fortunately, as Chester pondered this possibility, his younger brother Mad Dog
Lapynski (Figure 6) wandered onto the scene. Mad Dog Lapynski also noted the
red dot (Figure 7), inadvertently establishing an independent verification of
its existence555Unless the brothers are displaying identical symptoms..

Mad Dog Lapynski opted not to perform his own analysis of the data. He only
stayed long enough to comment snidely that Chester’s study neglected to
consider topological causality [45], and to smugly decline our offer of co-
authorship.

Figure 7: Mad Dog Lapynski also sees the red dot, an independent check that
renders the COVID-19 hypothesis unlikely.

## VI. DISCUSSION

While the strategies employed in this paper have yielded inconclusive results,
they by no means represent an exhaustive examination of tests for causality.
Other lines of attack exist in the literature, for example, topological
causality [45], as noted by Mad Dog Lapynski. Nevertheless, this study is
noteworthy as the first dynamical systems approach to probing the confounding
nature of incorporeal light-speed red dots.

In addition, while the significance of the attractors in Chester’s
reconstructed phase space is unclear, the results do carry intriguing
implications that might not be readily obvious to the reader. Allow us to
elucidate.

FTake, for example, the structure that emerges in the $D_{1}$-$D_{3}$-$D_{4}$
slice of the four-dimensional phase space (Figure 5, middle left). Thisf
particular viewing angfffle reveals the startfffffffffffling property
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fffscinating and merits further investigation.

## VII. ACKNOWLEDGMENTS

E.A. thanks her attorney overseeing the action filed by her co-author
concerning the appropriate assignment of first authorship of this manuscript.
(Her case is that she did all of the write-up; C. argues that he did all of
the work and some of the write-up.) E.A. also thanks BAND-AID ® regarding the
multiple injuries incurred by her forearms throughout the litigation thus far.
C., as usual, thanks no one.

## Appendix

Chester’s choices for the values of time delay $\tau$ and global embedding
dimension $d$ for phase space reconstruction were rather heuristic, as they
have been in all the literature cited in this paper.

To choose $\tau$ there exists no single formal prescription. The delay should
be sufficiently long to render the coordinates essentially independent, so
that they form an orthonormal basis that spans the phase space. Too short a
delay will not permit sufficient new information to have been generated in the
interim, while too long a delay will erase evidence of causality, assuming
that there was any to begin with. To strike a balance between these extremes,
Chester plotted the average mutual information between coordinates
$AMI(x(t)\mid x(t+\tau))$ as a function of $\tau$, and chose the value of
$\tau$ that yielded the first minimum of this distribution [46, 47]. This
strategy has been used successfully for time-delay embedding of the chaotic
Hénon map [24], chaotic Lorenz-63 system [24], and an audio tape taken inside
an aviary of ornery cowbirds [48]. (Note: here we are considering the mutual
information between successive points in the red-dot time series, and not
between the dot time series and laser pointer time series, as was the case in
Preliminary tests.)

Next, the choice for $d$ performs the unfolding. There exists no single
preferred methodology for this calculation either. Chester followed the
prescription of minimizing the percentage of “false nearest neighbors” [49] in
a time series. On a trajectory, points may appear sequential for one of two
reasons: 1) the dynamics truly renders them sequential, or 2) errors in the
sequence were acquired via the projection from the $\bm{x}(t)$ space to the
scalar $h(\bm{g}(\bm{x}(t)))$ space – and so these sequential points are
"false neighbors". If a sufficient number of components $d$ is used to
construct the vectors $\bm{y}(t)$ that define our new phase space, then the
tangles can be unfolded - i.e. the imposter neighbors eliminated.

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