# Tangling-Untangling Cycle for Efficient Learning

Xin Li, 1
University at Albany
<EMAIL_ADDRESS>This work was partially supported by NSF IIS-2401748 and
BCS-2401398.

###### Abstract

The conventional wisdom of manifold learning is based on nonlinear
dimensionality reduction techniques such as IsoMAP and locally linear
embedding (LLE). We challenge this paradigm by exploiting the blessing of
dimensionality. Our intuition is simple: it is easier to untangle a low-
dimensional manifold in a higher-dimensional space due to its vastness, as
guaranteed by Whitney embedding theorem. A new insight brought by this work is
to introduce class labels as the context variables in the lifted higher-
dimensional space (so supervised learning becomes unsupervised learning). We
rigorously show that manifold untangling leads to linearly separable
classifiers in the lifted space. To correct the inevitable overfitting, we
consider the dual process of manifold untangling - tangling or aliasing- which
is important for generalization. Using context as the bonding element, we
construct a pair of manifold untangling and tangling operators, known as
tangling-untangling cycle (TUC). Untangling operator maps context-independent
representations (CIR) in low-dimensional space to context-dependent
representations (CDR) in high-dimensional space by inducing context as hidden
variables. The tangling operator maps CDR back to CIR by a simple integral
transformation for invariance and generalization. We also present the
hierarchical extensions of TUC based on the Cartesian product and the fractal
geometry. Despite the conceptual simplicity, TUC admits a biologically
plausible and energy-efficient implementation based on the time-locking
behavior of polychronization neural groups (PNG) and sleep-wake cycle (SWC).
The TUC-based theory applies to the computational modeling of various
cognitive functions by hippocampal-neocortical systems.

##  I Introduction

A key challenge in machine learning (ML) is balancing specification and
generalization (a.k.a. over-fitting and under-fitting). We want a model that
generalizes well (but not so simplistic that it underfits) and is well-
specified to capture the essential patterns in the training data (but not so
complex that it overfits). This subtle balance is often achieved through
statistical techniques like cross-validation, regularization, and choosing the
right model complexity. From a geometric perspective, the challenge is
intrinsically related to the manifold structure of the data acquired from the
physical world. To discover and exploit such manifold structure, nonlinearity
is commonly believed to be essential, as witnessed by kernel trick in support
vector machine [48] and nonlinear dimensionality reduction (e.g., IsoMAP [45]
and locally linear embedding [40]). However, nonlinearity is often at odds
with rigorous proof and the simplicity criterion in mathematical and
scientific research.

In this paper, we draw inspiration from the discovery of place cells [34] and
its related aliasing problem [29, 50]. Nature has discovered an elegant
solution to efficient learning through the evolution of mammalian brains, but
its secret has remained elusive. We shed some light on this greatest puzzle by
revisiting the role of specification and generalization from a manifold
untangling/tangling perspective [27]. Whitney’s embedding theorem in topology
[31] has shown that it is easier to untangle a manifold in a high dimensional
space. Such intuition inspires us to introduce context dependency as the clue
for manifold untangling. Using class labels C as contextual variables, we can
show an arbitrary manifold $P(\textbf{X})$ can be untangled (admitting linear
dichotomy [6]) in the lifted space $P(\textbf{X},\textbf{C})$. The opposite
direction (tangling operator) is a simple integral transform that
collapses/aliases all variables sharing the same context. By associating
tangling/untangling with context-independent representation (CIR) and context-
dependent representation (CDR), we rigorously show how to strike an optimal
tradeoff between specificity and generalization by a pair of tangling and
untangling operators, which we call tangling-untangling cycle (TUC).

Using TUC as the building block, we can construct more sophisticated learning
algorithms based on Cartesian products and fractal geometry [28]. It can be
shown that the Cartesian product of TUC is still a TUC, which supports the
multiscale extension from low-dimensional to high-dimensional space. To mirror
the nested structure of data acquired from the physical world, fractal-
inspired extension is constructed where the inter-scale context dependency is
induced by an index variable (a.k.a. positioning trick [26]). The efficiency
of the proposed learning algorithms is supported by their biological
implementations based on polychronization neural groups (PNG) [24]. The
implementation of TUC by sleep-wake cycle [13] finishes our journey at its
starting point - the discovery of place cells in the hippocampus [34]. The
presence and absence of external stimuli (i.e., default state vs. perturbation
state) offer an energy-efficient and biologically plausible implementation of
untangling and tangling operators in wake and sleep modes, respectively.
Overall, the main contributions of this paper can be summarized as follows:

  * •

A geometric insight about the blessing and curse of dimensionality leads to a
new understanding of specification and generalization in ML, which nicely fits
the emerging direct-fit-to-nature paradigm.

  * •

A construction of manifold tangling-untangling cycle (TUC) for efficiency
learning. Our construction represents a novel direct-fit-to-nature approach to
unsupervised learning. Tangling and untangling operators are implemented by
context dependency and integral transform, respectively.

  * •

A biological connection of TUC with sleep-wake cycle (SWC) and its related
cognitive functions. Untangling transforms CIR to CDR for specification in the
wake mode; and untangling maps CDR back to CIR for generalization in the sleep
mode.

  * •

An application of TUC theory to model sensorimotor and social interactions. We
use biological findings in various studies of cognitive functions as
experimental support for the new theoretical framework.

##  II Geometric Intuition

Figure 1: Kernel trick achieves linear separability by mapping input data to a
higher-dimensional feature space. $\phi:(x,y)\rightarrow(x,y,c)$, where
$c=0,1$ denote red and blue classes, is a simple lifting idea that can be
generalizable to a data manifold in arbitrary dimensions.

In graph theory, there is a well-known educational puzzle named “Three
Utilities Problem”. This puzzle can be formalized as a problem in topological
graph theory by asking whether the complete bipartite graph $K_{3,3}$, has a
graph embedding in the plane. The impossibility of the puzzle corresponds to
the fact that $K_{3,3}$ is not a planar graph. However, such a puzzle can be
easily solved if the planarity constraint is removed. This is a simple example
illustrating the blessing of dimensionality to the problem of manifold
untangling. In topology, Whitney Embedding Theorem [31] states that any smooth
manifold can be embedded into a Euclidean space of double its dimension. Based
on this classical result, we can prove the following lemma:

Lemma (Linear separability in high-dimensional space)

Given two tangled manifolds $A,B\in R^{n}$, one can always linearly separate
them in a high-dimensional space.

Proof. Using Whitney’s embedding theorem, we can also embed $A,B$ to a
Euclidean space in $\tilde{A},\tilde{B}\in R^{2n}$. Then it is trivial to
linearly separate $\tilde{A},\tilde{B}$ in $R^{4n}$.

In machine learning (ML), the kernel trick used by support vector machine [42]
represents a good example for illustrating the blessing of dimensionality. As
shown in Fig. 1, linearly non-separable data sets can become untangled
(linearly separable) after being mapped to a high dimensional feature space.
The construction or learning of features has been at the foundation of all
existing models, including both biological and artificial neural networks.
Recent direct-fit models suggest that we can solve the manifold untangling
problem without feature engineering [20]. This view is also consistent with
intelligence without representation [3] and supported by the discovery of
concept cells [38, 37]. As summarized in [9], identity-related feature
information of an object manifold can be simultaneously conveyed by the
gnostic units in the IT cortex along with contextual information (e.g.,
position, pose, and scale), implying a high-dimensional latent space.

##  III Context Dependency and Integral Transform for Untangling and Tangling

Our intuition is based on observing the duality between specification
(overfitting) and generalization (underfitting) in learning. An alternative
approach to over/under-fitting is direct-fit-to-nature [20], which greatly
inspired our approach in this work. Geometrically, we interpret
overfitting/underfitting as manifold untangling and tangling, as shown in Fig.
2. Assuming sensory raw inputs are data belonging to a low-dimensional
manifold embedded in a high-dimensional space, the objective of manifold
untangling is to overfit the data for specification. This objective can be
facilitated by embedding the manifold into a high-dimensional space, as
Whitney’s embedding theorem suggests. Therefore, manifold untangling maps
sensory raw input to a high-dimensional latent space where the fitting task
becomes trivial (e.g., by linear dichotomy [6]). The dual process of manifold
tangling is to underfit the data for generalization. A common example is
aliasing [29], which reduces the dimensionality by collapsing the variables
with a common state. We suggest an integral transform as a more general
strategy than aliasing for generalization. In this section, we will study the
pair of manifold untangling and tangling separately.

Figure 2: The tangling-untangling cycle (TUC) for solving
overfitting/underfitting problems in efficient ML. Note its biological
plausibility is consistent with sleep-wake cycle (SWC).

###  III-A Manifold Untangling: Dimensionality Expansion via Context
Dependency

The intuition behind context dependency can be easily explained by the toy
example of binary classification (e.g., cats vs. dogs). Suppose we collect
sensory raw data associated with two classes $\textbf{X}_{0},\textbf{X}_{1}\in
R^{n}$ corresponding to classes 0 and 1, respectively. The raw data in $R^{n}$
is tangled (e.g., blue vs. red dots as shown in Fig. 1) but they can be easily
untangled by lifting the dimensionality by one - i.e., $(\textbf{X}_{c},c)\in
R^{n+1}$ where $c=0,1$ (a linear classifier $c=1/2$ is sufficient). Such an
idea of dimensionality expansion can be better articulated as a strategy of
context dependency where the label is explicitly coupled with the training
data as a context variable. Note that the introduction of label-based context
dependency transforms supervised learning to unsupervised learning [2] by
direct-fit-to-nature [20]. We can generalize the strategy of context
dependency for high-dimensional labels/context as follows.

Proposition 1 (Specification). Context Dependency for Manifold Untangling.

Given sensory raw data $\textbf{X}\in R^{n}$ and its associated label
$\textbf{C}\in R^{k}$, we can untangle the tangled data distribution
$P(\textbf{X})$ by introducing context dependency as a joint PDF
$P(\textbf{X},\textbf{C})\in R^{n+k}$.

The proof of proposition 1 is a straightforward extension of the strategy used
in binary classification. By considering the discrete representation of label
or context C, linear separability is guaranteed in the expanded higher-
dimensional space.

To facilitate our discussion, we will call the tangled raw data X context-
independent representation (CIR) and the untangled data
$(\textbf{X},\textbf{C})$ context-dependent representation (CDR). A moment of
thought will reveal the overfitting trouble with CDR immediately. This is
indeed the intention of untangling the data manifold $P(\textbf{X})$ by
dimensionality expansion. It achieved the goal of classification by over-
fitting the data in the lifted dimensional space of CDR, which will have the
worst generalization performance (e.g., a random perturbation along the
decision boundary by observation noise will cause the incorrect classification
result). Such mishaps have a remedy of manifold tangling.

###  III-B Manifold Tangling for Generalization: Dimensionality Reduction via
Integral Transform

The dual operation of manifold tangling can be understood as the extension of
aliasing [29]. The basic idea behind aliasing is to provide useful invariants
by representing as equivalent states in which the same action/class is needed.
Aliasing is good for generalization but has the risk of over-generalization.
Such risk can be well managed by applying aliasing or its extended tangling
operator to the untangled data in the high-dimensional space. The trick is to
selectively remove or collapse hidden states where the context dependency
dictates the criterion of selectivity. Using contextual information C as the
guardrail, we are guaranteed a faithful generalization without the risk of
over-generalization.

Proposition 2 (Generalization). Integral Transform for Manifold Tangling.

Given untangled CDR $P(\textbf{X},\textbf{C})$, we can restore the tangled CIR
$P(\textbf{X})$ by an integral transform $P(\textbf{X})=\int
P(\textbf{X},\textbf{C})d\textbf{C}$.

The pairing of manifold untangling and tangling operators forms a complete
(TUC) cycle, as shown in Fig. 2. The connection of TUC with direct-fit models
[20] is appealing - the objective of learning is not human-interpretable rules
or world representations (i.e., “intelligence without representation” [3]).
Instead, evolution has invented a localized computation for embodied cognition
to interpolate over task-relevant manifolds (we will elaborate on this issue
later when studying sensorimotor interaction). Manifold untangling offers a
constructive solution to obtain over-parameterized models that are simple and
parsimonious. More importantly, the connection of TUC with the evolutionary
process is so powerful that we strongly believe it offers an important clue to
unlock hidden secrets in developmental and ecological psychology. In summary,
TUC forms the building block for our solution to efficient learning problems
in general. Before we study the application of this new theory into
neuroscience, we report several hierarchical extensions of TUC into chains and
fundamental group of torus [21] to enrich our weaponry of constructing over-
parameterized direct-fit models.

##  IV Hierarchical Extension and Connection with Fractal Geometry

First, we can spatially extend TUC with Cartesian products to form a
fundamental group of the circles [21]. Such a dimensional increase is similar
to the product PDF from the two marginals. Note that after manifold
untangling, factorized linear dichotomies are conceptually similar to naive
Bayes [39] (simple and parsimonious low-dimensional classifiers). Therefore,
the Cartesian product preserves the independence of factorized distributions
in the latent untangled space. In this manner, direct-fit models can be
arbitrarily scaled to a high-dimensional space without suffering from the
curse of dimensionality. Formally, we have the following result:

a) b)

Figure 3: Hierarchical extension of TUC: a) spatial extension by Cartesian
product of TUC, which forms the fundamental group of the circles; b) temporal
extension by alternating between untangling and tangling in time domain.

Proposition 3. (Cartesian Product of TUC forms the Fundamental Group of the
Circle).

Given sensory raw data $\textbf{X},\textbf{Y}\in R^{n}$ and their associated
labels $\textbf{C}_{X},\textbf{C}_{Y}\in R^{k}$, the Cartesian product of
their TUC is still a TUC. That is, $\textbf{Z}=\textbf{X}\oplus\textbf{Y}$ and
$\textbf{C}_{Z}=\textbf{C}_{X}\oplus\textbf{C}_{Y}$, then
$\textbf{Z}\leftrightarrows(\textbf{Z},\textbf{C}_{Z})$ is a TUC.

By recursively applying Cartesian products of TUC, we can construct a
topological structure with increasing dimensionalities. Fig. 3a illustrates a
toy example for such an extension of increasing dimensionalities (from a pair
of circles to a torus). Analogous to the extension of circles to the
fundamental group of circles, we can construct the fundamental group of TUC as
$\textbf{S}^{1}\times...\times\textbf{S}^{1}$.

An alternative hierarchical extension of TUC is based on the temporal depth
[35] where the process of untangling and tangling occurs in an alternating
manner. Temporal depth or hierarchy is formed based on allostasis [35], as
shown in Fig 3b. This construction will be useful for characterizing the
dynamics of TUC, such as error feedback and temporal stability, especially in
the presence of heterogeneous interactions (e.g., sensory interactions with
motor). The alternating between tangling and untangling operators is also
analogous to the alternating between sleep and wake modes in sleep-wake
cycles, as we will elaborate in the next section.

Proposition 4. (Allostatic concatenation of TUCs forms temporal hierarchy.

Temporally hierarchical extension of TUCs can support continual learning by
alternating between the processes of untangling and tangling in the time
domain.

A more subtle construction of hierarchical TUC is based on fractal geometry
[28]. We must extend the strategy context dependency across different scales
to support fractal TUC. This can be implemented by the positioning trick [26],
which was inspired by Hippocampal indexing theory [46]. The indexing variables
$I$ induce inter-scale context dependency - e.g., in structure-mapping theory
for analogy [16], higher-order relations are modeled based on the relation
between objects instead of attributes of objects (so-called systematicity
principle). The implementation of such principle requires the context to be
the object X instead of class label C, which requires a fractal design such as
the six-layered organization of cortical column. Such hierarchical
organization of structures into TUCs might explain the magical number of six
(e.g., six degrees of separation) or seven [30].

Figure 4: Illustration of sleep-wake cycle (SWC): untangling occurs during the
wake time (perturbation state) and tangling occurs during the sleep time
(default mode).

##  V Biological Connections with PNG and SWC

The biological implementation of the untangling operator is based on the
connection of the polychronization neural groups (PNG) theory in [24] with the
locally subspace untangling hypothesis in [9]. A direct consequence of direct-
fit-to-nature is over-parameterization [20] to increase the predictive power
without explicitly modeling the underlying generative structure of the world
(i.e., intelligence without representation [3]). PNG offers a plausible
computational model for supporting such mindless yet powerful generalizations.
The number of PNG [24] can exponentially increase as more neurons are
involved. Intuitively, one can argue that a high-level PNG with more neurons
will have more representation power than a low-level PNG, which makes it a
promising candidate for implementing the direct-fit-to-nature model [20].

We first extend Hebbian postulate [22] to make it work for PNG as follows - “A
low-level PNG that $drives$ a high-level PNG to fire tends to wire together”
[11]. Driving is hypothesized as a mechanism for generalizing cloning [50] to
achieve specificity (e.g., unidirectional place cells [4] and context-
dependent concept cells). The opposite direction from high-level to low-level
generalizes aliasing [29] to achieve invariance or generalization (e.g.,
omnidirectional place cells and context-independent concept cells [38, 37]).
Due to dimensionality expansion (the consequence of recursive untangling), the
support of CDR would increase exponentially. PNG-based implementation can
handle this combinatorial explosion because of its enormous capacity [24].

PNG-based implementation of untangling also aligns with the brain principle
advocated in [33] \- i.e., to learn efficient low-dimensional representations
for different cognitive tasks. Notably, all cognitions, including
consciousness, happen during the wake mode. The anti-thesis of cognition is to
study the phenomenon of unconsciousness (e.g., sleep mode) or perturbation
(e.g., cognitive impairment). The connection between sleep-wake cycle (SWC)
and cognitive impairment has been well documented in the literature [12]. The
new insight brought by this study is to recognize the connection of tangling
with the sleep mode, a process known as memory consolidation [49, 44]. We
hypothesize that the mapping from episodic memory to semantic memory can be
computationally modeled by the tangling operator (CDR$\rightarrow$ CIR), which
also admits a PNG-based implementation but without external input. If this
hypothesis holds, SWC [13] offers an energy-efficient implementation of both
hierarchical extensions of TUC in Fig. 3, as discovered by nature.

a) b)

Figure 5: Context dependency matters to both visual perception and motor
control. a) PNG-based untangling for visual perception; b) PNG-based
untangling for motor control.

The combination of PNG with SWC [13] allows us to study cognitive functions
from a novel TUC perspective. Using TUC as the vocabulary, we attempt to
construct a computational model for SWC as the first step toward a more
comprehensive neural syntax [4]. As application examples, we consider three
tasks here: spatial navigation, object recognition, and motor control.

Spatial Navigation. Navigation represents an easier learning problem that
evolution solved long before the vision problem. Our understanding about this
task has converged to a theory based on cognitive maps [47, 34, 50]. Cognitive
maps organize knowledge to support flexible and adaptive behavior. How do
mammalian brains encode 2D physical space information into 1D spike sequence?
This question is at the core of understanding the relationship between space
and time in its simplest form. The representation of spatial information
(Euclidean or non-Euclidean) plays an essential role in the building of
cognitive maps. A graph-based representation such as clone-structured causal
graphs (CSCG) [17] has been widely adopted for encoding the physical space
into action-related state space. During the construction of the CSCG model,
the aliasing problem is solved by the cloning trick, which can be interpreted
as a degenerated untangling operator.

Untangling during the wake mode (perturbation state) is demonstrated by place
cells with unidirectional place fields. The 1D travel sequence dictates the
spatio-temporal context, which helps resolve the ambiguity in space [4].
Tangling during the sleep mode (default state) is demonstrated by
omnidirectional place cells [34] that do not require a temporal context.
Memory consolidation [44, 49] implements the aliasing or tangling by binding
all PNG which shared the same context C together. The collapse of cloned
variables occurs at the hippocampus where the encoding of spatio-temporal
context (index to the local map or position on the global map) is handled by
CA3-CA1 collateral system (pp. 326 [4]). During the consolidation of semantic
memory (collection of CIR) from episodic memory (collection of CDR), the PNG-
based time-locking mechanism eliminates the spatio-temporal context to support
context-independent inference for generalization.

Object Recognition. The harder problem of object recognition in vision
requires the recursive use of untangling in multiple stages. Due to the local
receptive fields of the lateral geniculate nucleus (LGN), severe aliasing or
tangling is present. During the wake mode, the ventral stream of the visual
cortex has to take several steps
(V1$\rightarrow$V2$\rightarrow$V4$\rightarrow$IT) to untangle X into
$(\textbf{X},\textbf{C})$. During the sleep mode, memory consolidation plays
the aliasing trick (integral transform) to collapse $P(\textbf{X},\textbf{C})$
to $P(\textbf{X})$ (note this is the internal model of the external
environment). The formation of concept cells [38, 37] needs to take context C
into account. The invariance of object recognition is achieved by the
tangling/aliasing operator. This interpretation is consistent with the modern
view of inferior temporal (IT) neurons [9] \- i.e., the population of IT
neurons simultaneously convey explicit information about object identity and
its contextual information. Our SWC-based theoretical framework conclusively
dissolves the notorious binding problem [19] because untangling integrates all
information (color, shape, motion) into CDR. One can still call it binding but
there is no need for re-binding [8].

Motor control. The duality between visual perception and motor control was
observed in [36]. Based on this analogy, one can understand the untangling
problem in the motor cortex similarly to that in the sensory cortex [51], as
shown in Fig. 5b. The high-tangling state consists of options that are
context-independent; while the low-tangling state is the collection of actions
that are context-dependent. The importance of motor untangling to muscle-like
actions/commands has been demonstrated in [41]. Sleep mode plays an important
role in motor learning similarly to visual learning. During sleep,
particularly during rapid eye movement (REM) and deep stages of non-REM sleep,
the brain consolidates motor skills learned during the day. This process is
essential for motor learning, helping to improve skills and refine motor tasks
over time. Empirical evidence supporting this line of theoretical reasoning
can be found in [7].

##  VI Computational Modeling of Sensorimotor and Social Interactions

TUC-based local computation for direct-fit inspired us to propose a new
conceptual framework called “localized embodied cognition” with the locality
principle in mind from problem formulation to solution algorithms. We still
use the world “as its own model” [3], but the way local embodied cognition
never attempts to solve a global problem locally. An organism needs to couple
sensory with motor systems to achieve various goals but never stores knowledge
about the global environment. Instead, the bodily interactions with the
environment (situatedness) are persistently maintained by the neocortex (a
single book)’s interaction with the hippocampus (a large library). The
cognitive map in the hippocampus offers a situational context to interpret the
sensorimotor interaction occurring in the neocortex. In this section, we study
sensorimotor interaction and hippocampal-neocortical interaction using the
newly-constructed neural syntax.

Figure 6: Sensorimotor cycle (SMC) consists of sensory untangling-tangling
pair and motor untangling-tangling pair for predictive processing. Note that
sensory and motor context-dependent representations (CDR) are task-relevant
latent variables (the connection of chain); sensory and motor context-
independent representation (CIR) are the interface with the external
environment.

#### Neocortex as a Sensorimotor Machine.

One of the biggest mysteries in studying cortical columns is that they are
structures without a function [23]. Instead, several canonical microcircuits
such as Douglas-Martin model [10] and Haeusler-Maass model [18] have been
studied for predictive coding in the literature. These models have revealed a
remarkable correspondence between the microcircuitry of the cortical column
and the connectivity required by predictive coding. Cortical columns do not
have a function because they work like swiss-army knivies. By connecting with
different sensory and motor areas, they offer a universal processing algorithm
[32]. TUC-based modeling allows us to gain a deeper understanding of
sensorimotor interaction at the system level without the distraction from
lower-level implementations.

As shown in Fig. 6, the coupling of sensory and motor untangling-tangling
pairs naturally forms a bigger cycle for the next level along the hierarchy of
system neuroscience - modeling sensorimotor interaction by a pair of inverse
and forward models [25]. Since more is different [1], the sensory motor cycle
(SMC) is likely to be guided by a varying organizational principle, as
initially demonstrated by [43]. Based on the duality between sensory and motor
cortex [36], sensory CDR (after perceptual untangling) has to be linked with
motor CDR (after motor untangling). They are task-relevant latent variables in
the network dynamics of SMC [43]. We come up with the following conjecture but
leave computational modeling for the mechanism of inverse and forward models
in SMC and its relationship to SWC for a future study.

Conjecture (Context-dependent manifold untangling) The task of sensorimotor
learning can be facilitated by simultaneously untangling the neural manifold
of sensory and motor variables in the right positioning of latent space, where
the position is specified by the index from the hippocampus as in Hippocampal
indexing theory [46].

#### SMC-based Social Interaction

. Carrying the “more is different” principle one step further, we can envision
a bigger cycle connecting a pair of SMCs. When one person becomes another’s
environment, the communication between two SMCs forms another bigger cycle
known as non-verbal communication. The communication bandwidth of such non-
verbal communication is limited by the locality of our sensory organs. Even
though within each neocortex, the hierarchical organization maximizes the
information flow inward, social communication (flowing outward) has to wait
for evolution to improve its efficiency.

Figure 7: The cycle connecting speech production by motor cortex and language
comprehension by auditory cortex has a dual structure to SMC because social
communication between two people is connected by CIR (tangled states).

The fundamental role played by natural language is to facilitate communication
among people - from non-verbal to verbal. Fig. 7 shows a modification of SMC
to abstract the verbal communication between two people A and B, where motor
and auditory cortex are responsible for speech generation and language
comprehension, respectively. Note that unlike the original SMC in Fig. 6, the
interface between two TUCs is in the tangled state (CIR). This difference has
a fundamental impact on our internal (self-conscious) and external (social)
cognitive function. For internal communication between sensory and motor
cortex, the interface (inverse and forward in Fig. 6) connects two untangled
states and there is no bottleneck in terms of information flow. For external
(social) communication between two different person, the interface (inverse
and forward in Fig. 7) connects two tangled states and faces an information
flow bottleneck.

According to the simulation theory of mind-reading [15], mirror neurons can
help us understand and empathize with the emotions and intentions behind
others’ non-verbal cues. Modeling the new cycle in Fig. 7for understanding the
social nature of language might offer new insight to the neural exploitation
hypothesis [14]. According to this hypothesis, social cognition and language
are both connected with action through the support of mirror neurons. However,
the difference between non-verbal and verbal communication seems to be
determined by their organizational principles. Non-verbal communication such
as sign language does not admit hierarchical organization; while verbal
communication such as Chomsky’s hierarchy [5] in natural language offers an
evolutionary solution to maximize the information flow within the new cycle.

##  VII Conclusions and Future Works

In this paper, we have constructed a new TUC-based theory to solve the long-
standing learning problem. Exploiting the blessing of dimensionality by
inducing context dependency allows us to untangle any data manifold so linear
classifiers will be sufficient. Then the tangling operator based on integral
transform collapses the induced context variables for generalization. Such
mapping between tangled CIR and untangled CDR can be extended into
hierarchical representations by Cartesian product and fractal geometry. The
computational efficiency of TUC-based learning algorithms is supported by the
biological implementation by SWC. Sleep and wake modes in SWC elegantly
correspond to tangling and untangling operators due to the absence of sensory
inputs during the sleep mode. SWC allows us to construct a neural syntax for
an improved understanding of cognitive functions associated with sensorimotor
interaction and social interaction.

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# Platoon Fundamental Diagram estimation can be Markovian: evidence from
human- and self-driven vehicle trajectories

Michail A. Makridis
IVT, Civil Environmental and Geomatic Engineering
ETH Zurich, Switzerland
<EMAIL_ADDRESS>
&Anastasios Kouvelas
IVT, Civil Environmental and Geomatic Engineering
ETH Zurich, Switzerland
<EMAIL_ADDRESS>
&Jorge A. Laval
Civil and Environmental Engineering
Georgia Institute of Technology, US
<EMAIL_ADDRESS>

###### Abstract

We propose a simple and effective method to derive the Fundamental Diagram
(FD) from platoon vehicle trajectories. Average traffic state variables are
computed using Edie’s generalized definitions within time-dependent
trapezoidal space-time areas. To obtain a clear FD, we employ a bivariate data
aggregation technique to eliminate scatter. Our findings are as follows: (i)
The proposed method demonstrates a remarkably consistent relation between the
traffic variables and a clear triangular shape for autonomously-driven
vehicles. (ii) The FDs are invariant to several factors of heterogeneity such
as the platoon length, vehicle characteristics, road particularities, and data
acquisition accuracy. (iii) ACC-driven vehicle platoons with minimum headway
setting achieve much higher capacity, roughly 90% than those with a large
headway setting. (iv) Connectivity might increase capacity. (v) Human drivers
have a wider near-capacity operation area, showing different behaviors at high
speeds than low ones, and (vi) Safety concerns might arise due to high values
of backward wave speed for ACC-driven vehicles. Comparative analysis with the
state-of-the-art confirms the validity of our approach. The proposed method
stands out due to its simplicity and accuracy, which paves the way for
practical applications in real-time traffic flow monitoring and control within
modern intelligent transportation systems.

##  1 Introduction

Traffic investigations, initiated nearly 90 years ago, aimed to better
comprehend the intricate phenomena occurring in road transport systems. Among
early pioneers was Greenshields who in 1933 noticed a linear relation between
speed and traffic density and a parabolic relation between speed and traffic
flow using the assumption that traffic flow equals traffic density multiplied
by speed, also known as Fundamental Diagram (FD). Since then, numerous studies
in the literature exploit the concept of FD as a congestion modeling tool for
traffic estimation purposes at link or network levels Kouvelas et al. (2017);
Makridis and Kouvelas (2023); Du et al. (2023).

The study of vehicle platoon trajectories has been ongoing for over 50 years.
In 1966, Treiterer (1975) used aerial photographs from a helicopter, following
a specific vehicle and monitoring the traffic around it, as all vehicles
progress along the roadway. Subsequently, Treiterer and Myers (1974) observed
a generally retarded behavior displayed by a platoon of vehicles after
emerging from a kinematic disturbance as compared to the behavior of the same
vehicles approaching the disturbance, known today as traffic hysteresis
phenomenon. This discovery sparked a continued interest in platoon dynamics.

The exploration of platoon trajectories is appealing due to its ability to
offer insights into the behavior of different drivers under similar
conditions, which directly influences traffic state. The existence of diverse
driver behaviors on public roads has been widely recognized as a catalyst for
various traffic phenomena. Consequently, there is a strong interest in
accurately modeling and simulating this heterogeneity, as discussed in
Makridis et al. (2020a). Moreover, the integration of automation and
connectivity in vehicles further amplifies the impact of drivers’ behaviors on
road capacity and utilization Ioannou (1997); van Arem et al. (2006); Makridis
et al. (2020d). However, there is limited information in the existing
literature regarding how different driver modes may contribute to distinct
Fundamental Diagrams (FDs).

Fortunately, a growing number of experimental datasets that include human-
driven and automated vehicles emerge, as in Makridis et al. (2021); Gunter et
al. (2021). Such datasets provide valuable opportunities to gain fresh
insights into the topic. While there is extended research on the microscopic
properties of partially or fully automated vehicles such as reaction time
Brunner et al. (2022); Makridis et al. (2020c), string instability Li et al.
(2021); Gunter et al. (2021), safety Lu et al. (2021); Mattas et al. (2020);
Mullakkal-Babu et al. (2020), and energy efficiency Apostolakis et al. (2023);
He et al. (2020); Ma et al. (2021), there are very few works that explicitly
relate their microscopic behavior with the traffic state. Although there are
some recent efforts, see Chen et al. (2017); Shi and Li (2021), achieving
generalization is not a straightforward task due to challenges from various
factors such as road geometry, platoon length, diverse vehicle specifications,
or data acquisition frequency. This paper takes advantage of the operational
properties of commercial ACC (and CACC) systems to aim for a constant headway
policy. It proposes a simple and robust numerical method to describe
macroscopic traffic flow properties of ACC- and CACC-driven platoons of
vehicles.

By analyzing observations from either fully or partially automated vehicles
and applying an aggregation technique, we establish a clear relationship
between the three traffic flow variables: speed, density, and flow. This
results in clear FD for homogeneous drivers in platoons, using only
inexpensive raw trajectory observations. Due to the absence of ground-truth
information, the validity of the method in terms of traffic state inference is
assessed using one of the currently most reliable methods for traffic state
inference in the literature Laval (2011); Maiti et al. (2023), and the
resemblance is remarkably high.

Our method proves to be robust against heterogeneity from means of
observations, including variations in noise levels, platoon synthesis, platoon
length, and traffic conditions. Additionally, it eliminates the need for
estimating the backward wave speed and minimizes computational complexity. The
results reveal a triangular shape for the FD of partially automated vehicles
and provide concrete insights into the expected capacity and speed-dependent
dynamic behavior of human drivers compared to partially automated vehicles. At
the same time, safety concerns arise for ACC-driven vehicles with minimum
distance settings.

In the remainder of this paper, we present in Section 2 the proposed
methodology, while Section 3 discusses the results and finally, conclusions
are summarized in Section 4.

##  2 Methodology

First, we adopt the well-known Edie’s generalized definitions for raw movement
data aggregation to derive fundamental traffic variables, namely, flow,
density, and space-mean speed, see Edie (1965); Cassidy and Coifman (1997).

It is important to note that in this procedure, a rectangular area needs to be
defined, and the flow, density, and speed are calculated based on the number
of vehicle-kilometers traveled and vehicle-time traveled within this region.
Based on the most acknowledged measurement method in the literature, see Laval
(2011), the shape of this region should be a sided rectangle including the
platoon vehicles with side slopes representing the expected backward wave
speed. This approach is crucial to avoid blending different traffic states and
to obtain a reliable approximation of the prevailing traffic state within the
given space-time area. However, implementation-wise there are numerical
difficulties for discrete observations. Additionally, defining a global
representation of the fundamental diagram (FD) for a driver mode remains an
unresolved matter, particularly when analyzing large trajectory datasets due
to the significant variability observed.

In this study, we first attempt to estimate the traffic state at a given time
based on two consecutive observations. The data resolution (usually 10$Hz$)
does not affect the final results. Through this process, the estimated traffic
states are derived, which are equal in number to the total number of
observations minus one.

Here, we present an example that illustrates how to determine the effective
platoon length, which establishes the boundaries of our designated area.
Figure 1(a) depicts a platoon of vehicles with their antennas used for data
acquisition. The observed platoon length represents the distance between the
first and last antenna, excluding a portion of the first and last vehicle.
Assuming equal vehicle lengths and identical antenna positions on each
vehicle, this corresponds to the distance between the back bumper of the
leader and the back bumper of the last follower. To account for assumptions’
variability and measurement noise, we incorporate a small buffer space
(approximately 3 meters). This adjusted length is referred to as the effective
platoon length, which varies dynamically as the vehicles move. Figure 1(b)
provides a schematic representation of the area used for the computation of
speed, density, and flow. For the mathematical representation, we refer the
reader to A.

(a) Illustration of the observed and effective platoon length.

(b) Illustration of moving scalene trapezoid space-time area.

Figure 1: Schematic representation of an observed platoon and a dynamic area
for instantaneous traffic state estimation.

The computation of the instantaneous traffic states described above is carried
out for all available datasets, classified according to the driver mode
(human, ACC with fixed distance setting, and CACC). This generates a large
number of potential states and a significant amount of scatter. We aim to
verify if all the observations that correspond to the same driver mode, e.g.,
ACC with the minimum setting, correspond to the same fundamental diagram. To
derive a single fundamental diagram (FD) relationship, we employ a simple and
efficient data aggregation technique across all available data.

In the flow-density representation, we divide the observed density range into
bins of length $\delta_{k}$. Within each density interval, we calculate the
average of all observed flow values, yielding a single estimated flow value
per density bin. An exemplary illustration is given in Figure 2. As expected,
the accuracy of the fundamental traffic state improves with more observations.
The results show that the inferred fundamental traffic states per driver mode
are independent of the bin length, $\delta_{k}$. A similar aggregation process
is applied to the observed speed range with aggregation of observed flow
values within each speed interval. The aggregation is applied to the density
and speed ranges since we expect a concave flow function over each of these
quantities. The aggregation process is formally described in A.

Figure 2: Schematic representation of the aggregation method. Blue dots
correspond to instantaneous traffic state estimates for the same driver mode.
Each dot corresponds to an observation without considering the experiment,
platoon size, powertrain, etc. All estimates have instantaneous density
withing the bin range, i.e., $k\in(n\delta_{k},(n+1)\delta_{k})$. The inferred
fundamental traffic state for the corresponding bin is derived by averaging
all observed instantaneous densities. Therefore each density bin corresponds
to one flow value.

##  3 Results

The method we propose is implemented using data gathered from nine independent
campaigns and across five driver modes for platooning. These clusters consist
of ACC-driven vehicles with minimum, median, and maximum settings, cooperative
ACC-driven vehicles, and human drivers. Although the number of observations
for ACC-driven vehicles with minimum and maximum settings is significantly
higher than for the other clusters, we present results incorporating all
available data. Further information regarding the data used can be found in B
and the relevant literature.

Figure 3 illustrates the flow-density plot of the FD for two methods, the
observed driver modes and a $\delta_{k}=\delta_{v}=0.3$. Please note that
despite the absence of traffic state ground-truth, we compare the proposed
method with Laval (2011) as a benchmark technique and we verify the realistic
traffic state inference. The inferred FDs with the proposed method (Figure
3(a)) and the existing state-of-the-art in the literature (Figure 3(b)) are
very similar. The results for the other two bi-level plots, namely Speed-
Density and Flow-Speed are provided in C.

The black lines in Figure 3(a) correspond to the triangular fundamental
diagram (TFD) that was calibrated on the data from ACC-driven vehicles with a
minimum and maximum setting (two driver modes with available observations).
Calibration was performed with minimization of the sum of normalized root mean
square errors of the flow and speed. Further information on the calibration
process can be found in D.

(a) Proposed

(b) Maiti et al. (2023)

Figure 3: Flow-Density plot with the proposed method and the measurement
method proposed by Laval (2011) after the application of the Shear
transformation described in Maiti et al. (2023) and the aggregation process
described in this paper.

The calibrated inferred FDs exhibit remarkable consistency with the TFD shape.
However, depending on the aggregation strength (defined by $\delta$), some TFD
parameters might lose their significance during calibration when there is
insufficient available data. For instance, the observed speeds for the maximum
ACC setting only range up to 120 $[km/h]$, while for the minimum ACC setting,
they reach 140 $[km/h]$, see Figure 5(a) and 5(b). As a result, the
calibration of the free-flow part for the former might not be accurate.
Similarly, it is currently not meaningful to provide calibrated TFD for the
other driver modes, namely, medium ACC setting and CACC, due to scarcity of
data.

Another interesting observation relates to the calibrated backward wave speed
values for the ACC-driven vehicles that are given in Table 2. ACC-driven
vehicles with maximum setting have realistic wave speed of around 15-17$km/h$.
However, the same variable for ACC-driven vehicles with minimum setting has
been found to be over 30$km/h$. This reveals safety concerns for such
platoons. An in-depth investigation is out of the present paper’s scope but
will be part of future research.

While the available data for human drivers may seem sufficient, it appears
that the TFD shape only poorly describes their behavior. The respective FD has
a more smooth shape with a very wide near-capacity area. The behavioral
analysis of the FD for human drivers reveals commonalities and differences
between human behavior and control behavior which eventually is also reflected
in traffic flow. By referring to Figure 5(d) (provided in C), it can be
observed that human drivers behave similarly to ACC-driven vehicles with a
maximum setting at higher speeds (above 50 $[km/h]$). However at lower speeds
(below 30 $[km/h]$), reflecting congested conditions, human drivers take
greater safety risks by maintaining much shorter distances from vehicles
ahead. Consequently, their behavior aligns more closely with ACC-driven
vehicles with a minimum setting. In terms of traffic implications, such
behavior leads to better road utilization compared to partially automated
vehicles under over-saturated conditions nearing jam density. Unfortunately,
there is no available data to observe the corresponding behavior of CACC-
driven vehicles but it would be interesting to explore this in future
research. Finally, the above analysis conceptually aligns with empirical
evidence, showing human drivers being more relaxed at higher speeds, allowing
for larger time gaps, while being more agile at lower speeds to minimize their
travel time, see Xu et al. (2021).

Consistent with existing literature, the findings presented in this study
suggest that connectivity in ACC systems has the potential to improve traffic
flow. Although the empirical data available is limited, it is evident that the
observed behavior of CACC-driven vehicles differs significantly from both ACC-
driven and human-driven vehicles. While we cannot derive precise values for
capacity, critical density, and jam density due to the scarcity of data, it is
evident that the potential for improvement is substantial. The observed flow
reaches around 4000 $[veh/h]$ (possibly higher overall) as can be seen in
Figure 3. Furthermore, Figure 5(d) in Annex C shows that they manage to
maintain the same platoon speed as the other modes (ACC and human drivers)
under very high density, thus very close inter-vehicle spacing. This, of
course, can be attributed to the effect of connectivity that provides more
information to the vehicles, so that, they can keep a close distance from
their leaders without compromising safety and stability. It should be noted,
however, that further empirical data is necessary to validate the safety and
robustness of CACC systems in the face of communication interference threats.

It is important to acknowledge that the nominal capacity values mentioned
earlier assume ideal uniform driving conditions, which are not reflective of
real-world environments. However, the relevant FD characteristics provide
valuable insights into the variations in efficiency among different driver
modes of transportation and the potential each mode holds. Such information is
essential for the design and implementation of optimal traffic management and
control strategies.

###  3.1 Analysis of different platoon states

In the literature, the estimation of stationary conditions in a platoon
dataset is necessary for the inference of traffic variables as mentioned in
Laval (2011). However, this is not a requirement for the proposed method.
Nevertheless, during platoon acceleration, deceleration, and stable (cruising)
states, different behaviors arise and therefore it is interesting to be able
to observe if and how such states affect a unified FD representation.
Consequently, we post-process all available data and categorize the platoon
trajectories into three platoon states, namely acceleration, deceleration, and
stable state events. Categorization is performed by inferring partial
trajectories where the average platoon speed is increasing, decreasing, or
remaining constant. The idea is then to aggregate traffic variables per
platoon state so that we analyze their impact on the unified FD
representation. For more in-depth information about the processing details, we
refer the reader to Annex E.

The results with the proposed method per platoon state for ACC-driven vehicles
with minimum setting are illustrated in Figure 4. The traffic states
corresponding to stable platoon state lie between those of accelerating and
decelerating states, at (visually) equal distances. Interestingly, by looking
only at stable platoon states, we would not have been able to infer the
complete FD.

(a) Flow-Density plot - proposed Figure 4: Flow-Density plot for different
platoon states. The plots refer to observed vehicle platoons driven with the
ACC minimum setting.

###  3.2 Sensitivity of $\delta$ parameter in calibrated FD

The results presented so far refer to $\delta_{k}$ and $\delta_{v}$ parameters
equal to 0.3 $[veh/km]$ and 0.3 $[km/h]$ values for data aggregation. Here we
test the consistency of the calibrated TFD parameters through a sensitivity
analysis for different $\delta$ values ranging from 0.3 to 3.5 density/speed
units. The results validate completely the consistency of the inferred FD in
all cases and they are summarized in Table 2 in F. It is interesting to
observe that the calibrated TFD parameters for the minimum and maximum
$\delta$s are really close, even if higher $\delta$ leads to very few points
for calibration.

##  4 Conclusion

We study different experimental observations with platoons of vehicles driven
by five driver modes, i.e., Adaptive Cruise Control (ACC) with minimum, medium
and maximum setting, Cooperative-ACC (CACC) and human drivers. We propose a
numerically simple and effective method to derive the Fundamental Diagram (FD)
per driver mode after the application of a bivariate aggregation technique.
The proposed method demonstrates a remarkably consistent relation between the
traffic variables and a clear triangular shape for autonomously-driven
vehicles. Comparison with a state-of-the-art method from the literature
confirms the validity of the inferred states. Two important benefits are the
absence of a need to assume a backward wave speed and to estimate stationary
conditions.

The driver mode with minimum ACC setting points to much higher capacity values
(roughly 90%) than ACC-driven vehicles with maximum setting or human drivers.
However, this also leads to much higher backward wave speed, raising safety
concerns. Preliminary results with limited data show that connectivity can
significantly improve capacity. The FD shape for automated vehicles is
triangular, while human drivers exhibit a wider near-capacity area.

##  Appendix A Methodology

Edie’s generalized definition is given for reference below:

$\displaystyle k$ $\displaystyle=\sum_{i=1}^{N}t_{i}\Big{/}\mid A\mid$ (1a)
$\displaystyle q$ $\displaystyle=\sum_{i=1}^{N}x_{i}\Big{/}\mid A\mid$ (1b)
$\displaystyle v$
$\displaystyle=q/k=\sum_{i=1}^{N}x_{i}\Big{/}\sum_{i=1}^{N}t_{i}$ (1c)

where $k$, $q$ and $v$ stand for density, flow, and speed in the region: $A$,
$\mid A\mid$ is the area of $A$ and $t_{i},x_{i}$ is the travel time and
distance traveled respectively, of vehicle $i$ inside $\mid A\mid$.

The effective platoon length $l_{p}(t)$, is shown in Figure 1(b) and it is
dynamic over time as the vehicles move. We define $\mid A\mid$ as a time-
variant scalene trapezoid area described but two consecutive platoon steps.
Such a simplification is by design particularly useful and inexpensive for the
analysis of discrete trajectory observations.

Let us assume that we have platoon observations with $N$ vehicles and a
constant data acquisition frequency equal to $f=\frac{1}{\delta t}$. Figure
1(b) provides a schematic representation of the area used for the computation
of speed, density, and flow. Considering that the platoon $l_{p}(t)$ at two
consecutive time instances, we derive $\mid A\mid$ numerically as follows:

$\begin{gathered}\mid A(t)\mid=\frac{l_{p}(t)+l_{p}(t+\delta t)}{2}\delta
t\end{gathered}$ (2)

Speed, density, and flow variables for a platoon of $N$ vehicles and for a
fixed observation interval $\delta t$ are computed accordingly:

$\displaystyle k(t)$ $\displaystyle=(N\delta t)\Big{/}\mid
A(t)\mid=2N\Big{/}(l_{p}(t)+l_{p}(t+\delta t))$ (3a) $\displaystyle q(t)$
$\displaystyle=\sum_{i=1}^{N}\delta x_{i}(t)\Big{/}\mid A(t)\mid$ (3b)
$\displaystyle v(t)$ $\displaystyle=q(t)/k(t)=\sum_{i=1}^{N}\delta
x_{i}(t)\Big{/}(N\delta t)$ (3c)

where $\delta x_{i}(t)=x_{i}(t+\delta t)-x_{i}(t)$, derived by the individual
vehicle positions at times $t$ and $t+\delta t$.

In the remaining part of this Appendix, we describe formally the aggregation
process applied on the available observation per driver mode. Let us define a
set of $M+1$ discrete equally spaced values as follows:

$\begin{gathered}B=\\{0*\delta,1*\delta,\ldots M*\delta\\}\end{gathered}$ (4)

where $\delta$ is the discrete quantity interval and $B$ is the corresponding
set of discrete quantity values. In our case, we create two sets, one for
density discretization and another for speed discretization, with the
corresponding $\delta_{k}$ and $\delta_{v}$ intervals, i.e.
$B_{k}=\\{0*\delta_{k},1*\delta_{k},\ldots M*\delta_{k}\\}$ and
$B_{v}=\\{0*\delta_{v},1*\delta_{v},\ldots M*\delta_{v}\\}$.

In order to compute the aggregated values for $v$ \- $k$ and $q$ \- $k$ plots
based on observed density, speed and flow variables, i.e. $k(t)$, $v(t)$,
$q(t)$, for each $\delta_{k}$ within $B_{k}$, we average all available
measurements across all datasets to derive $M+1$ points that describe the
central data tendency.

$\\{k,v,q\\}_{i}=\begin{cases}\frac{1}{T_{i}}\sum_{t=1}^{T_{i}}k(t),&\mbox{where
}i*\delta_{k}\leq k(t)\leq(i+1)*\delta_{k},\forall i\in\\{0,\ldots M_{k}\\}\\\
\frac{1}{T_{i}}\sum_{t=1}^{T_{i}}v(t),&\mbox{where }i*\delta_{k}\leq
k(t)\leq(i+1)*\delta_{k},\forall i\in\\{0,\ldots M_{k}\\}\\\
\frac{1}{T_{i}}\sum_{t=1}^{T_{i}}q(t),&\mbox{where }i*\delta_{k}\leq
k(t)\leq(i+1)*\delta_{k},\forall i\in\\{0,\ldots M_{k}\\}\\\ \end{cases}$ (5)

where $T_{i}$ refers to the total number of $k(t)$, $v(t)$, $q(t)$
measurements when $k(t)$ lies within $(i*\delta_{k},(i+1)*\delta_{k}]$.

Finally, for the computation of aggregated values for $v$ \- $q$, since it is
expected a two-valued function across the flow, the aggregation is performed
on the speed in a similar way as Eq. 5 with $\delta_{v}$ interval according to
the following equation:

$\\{v,q\\}_{i}=\begin{cases}\frac{1}{T_{i}}\sum_{t=1}^{T_{i}}v(t),&\mbox{where
}i*\delta_{v}\leq v(t)\leq(i+1)*\delta_{v},\forall i\in\\{0,\ldots M_{v}\\}\\\
\frac{1}{T_{i}}\sum_{t=1}^{T_{i}}q(t),&\mbox{where }i*\delta_{v}\leq
v(t)\leq(i+1)*\delta_{v},\forall i\in\\{0,\ldots M_{v}\\}\\\ \end{cases}$ (6)

where $T_{i}$ refers to the total number of available $k(t)$, $v(t)$, $q(t)$
measurements when $v(t)$ lies within $(i*\delta_{v},(i+1)*\delta_{v}]$. Please
note that $M_{k}=M_{v}$ in the case that $\delta_{k}=\delta_{v}$. For
simplicity, in the results of this work, we use the same $\delta$ values on
the density and flow axes.

##  Appendix B Data

The proposed methodology is demonstrated through empirical observations from
car-following campaigns with platoons of vehicles. Eleven campaigns from five
major datasets are considered in this work. The observations refer to human-
driven, ACC-driven and CACC-driven vehicles. The total distance traveled per
platoon in all experiments is 844.8 $[km]$ for ACC-driven platoons, 38.2
$[km]$ for human-driven platoons and 29.3 $[km]$ for CACC-driven platoons. A
brief introduction of each campaign follows while all datasets are summarized
in TABLE 1.

Table 1: Summary of experimental campaigns used in this work. | Driver | ACC settings | Number of vehicles | Homogenous
---|---|---|---|---
AstaZero |  ACC:172 $[km]$/HV:22.9 $[km]$ | min | 5 | No
ZalaZone (Handling) |  ACC:49.5 $[km]$/HV:15.3 $[km]$ | min/max | 4-10 | No
ZalaZone (Dynamic) |  ACC:67.3 $[km]$ | min/med/max | 4-11 | No
Cherasco |  ACC:6.5 $[km]$ | min | 3 | No
Casale |  ACC:142.5 $[km]$ | min | 2 | Yes
JRC |  ACC:10.9 $[km]$ | min/med | 2/3 | Yes/No
CARMA |  CACC:29.3 $[km]$ | - | 5 | Yes
VU1 |  ACC:390 $[km]$ | min/max | 2 | Yes
VU2 |  ACC:6.1 $[km]$ | min | 7 | Yes

####  B.0.1 OpenACC dataset

The OpenACC dataset was initiated by the open-access policy of the Joint
Research Centre of the European Commission and involves a set of car-following
experiments with human-driven and ACC-driven vehicles on public roads, test
tracks, and the JRC private urban network. The dataset is constantly populated
and this work includes campaigns that are not mentioned in the original
publication. In this work, we use car-following observations from AstaZero,
ZalaZone, Cherasco, Casale, and JRC campaigns. More details about the OpenACC
can be found in Makridis et al. (2021). Currently, available campaigns can be
downloaded from the Makridis et al. (2020b) website.

####  B.0.2 CARMA dataset

The United States Department of Transportation has developed a proof-of-
concept platooning system that, additionally to the familiar ACC, includes V2V
and V2I communication. This CACC implementation is based on the Cooperative
Automation Research and Mobility Applications (CARMA) platform. As described
in Tiernan et al. (2017), the first version of CARMA dataset was acquired in
2016 at the Aberdeen Test Center in Maryland. In this work, we use the second
subsequent campaign, CARMA2, a platooning experiment presented in Tiernan et
al. (2019) in 2018. The experiment included five vehicles of the same model
and was conducted on the 7.2 $[km]$ two-lane road of the proving ground. The
features of the test track are assumed to be similar to the typical US
highway. Spacing measurements were obtained with PinPoint GPS, while for speed
measurements the production wheel speed was used. More details on the dataset
can be found in the above-mentioned reports.

####  B.0.3 Vanderbilt University dataset

The Vanderbilt University dataset includes two campaigns with car-following
data of ACC-driven vehicles. During the first campaign (VU1) four speed
profiles are recorded to observe the behavior of each vehicle in the two-
vehicle tests. For all tests, the vehicles begin on the track and start at a
low speed with a full-size sedan as the lead vehicle, and the test vehicle as
the follower vehicle. To validate the emergent traffic flow behavior of ACC
vehicles at the aggregate (system) level, a second campaign (VU2) was
conducted where seven identical ACC vehicles follow a lead vehicle that
creates a velocity slow down event. Position and speed data for each vehicle
were collected using high-accuracy uBlox EVK-M8T GPS receivers. For more
information we refer the reader to the corresponding publication Gunter et al.
(2021).

##  Appendix C Fundamental Diagrams

The flow-density plots are given in Figure 3(a) and 3(b). Here we provide
additionally the Speed-Density and Flow-Speed plots for both methods in Figure
5(d). Calibrated FDs fit very nicely on the data with a minimum and maximum
ACC setting. Unfortunately, there are limited data on CACC-driven vehicles but
figures show great potential for increased capacity and critical density for
this technology. It can safely be assumed that this result is due to the
additional information from vehicle connectivity As mentioned in Section 3,
the FD for human drivers seems to be between the FD of ACC-driven vehicles
with minimum setting and those with a maximum setting. Attempting to interpret
this finding intuitively, we can assume that humans drive keeping larger time
spaces at high speeds, while they keep shorter spacings at low speeds, i.e.,
the congested branch of the FD. Finally, as expected ACC-driven vehicles with
a medium setting generate FD points between those with a maximum and a minimum
setting.

(a)  $v$ \- $k$ plot - proposed

(b)  $v$ \- $q$ plot - proposed

(c) $v$ \- $k$ plot - Maiti et al. (2023)

(d) $v$ \- $q$ plot - Maiti et al. (2023)

Figure 5: Bi-level plots with the proposed method after aggregation

.

##  Appendix D Calibration of the Triangular Fundamental Diagram

For calibration purposes, we use the triangular fundamental diagram (TFD)
proposed by Daganzo (1995), which is probably the most widespread. The TFD has
been chosen due to its simplicity in the implementation, describing the
density-flow plane with a bi-linear relation. More specifically, the flow-
density function is adequately described by the following:

$Q(k)=\begin{cases}v_{f}k,&\mbox{if }k\leq k_{cr}\\\ w(k_{jam}-k),&\mbox{if
}k\geq k_{cr}\\\ \end{cases}$ (7)

where $v_{f}$ is the free-flow speed, $w$ is the wave speed, $k_{cr}$ is the
critical density and $k_{jam}$ is the jam density.

Based on trigonometry and assuming having critical density $k_{cr}$, the first
and second leg of Equation 7 are equal. Therefore, we can derive the following
equation for the wave speed and reduce the number of variables by one:

$\begin{gathered}w=\frac{v_{f}k_{cr}}{k_{jam}-k_{cr}}\end{gathered}$ (8)

From Equations 7 and 8 we have:

$Q(k)=\begin{cases}v_{f}k,&\mbox{if }k\leq k_{cr}\\\
\frac{v_{f}k_{cr}}{k_{jam}-k_{cr}}k,&\mbox{if }k\geq k_{cr}\\\ \end{cases}$
(9)

Consequently we perform the following simple optimization operation on the
$v$-$k$ (see Figure 3) and the parameter vector $x=[v_{f},k_{cr},k_{jam}]$,
which contains the parameters of the TFD:

$x=\operatorname*{arg\,min}_{x:\text{lb}\leq x\leq\text{ub}}\mathcal{Q_{x}}$
(10)

where lb and ub are the lower and upper bounds for the three variables of the
TFD. $\mathcal{Q_{x}}$ is the objective function, based on the root mean
square errors between observations and the TFD:

$\mathcal{Q_{x}}=\frac{\sqrt{\frac{1}{(M+1)}\sum_{m=0}^{M}(q_{m}-Q(k_{m}))}}{\frac{1}{(M+1)}\sum_{m=0}^{M}(q_{m})}+\frac{\sqrt{\frac{1}{(M+1)}\sum_{m=0}^{M}(v_{m}-Q(k_{m})/k_{m})}}{\frac{1}{(M+1)}\sum_{m=0}^{M}(v_{m})}$
(11)

##  Appendix E Analysis of acceleration, deceleration and stable platoon
states

Figure 6(a) shows the segmented speed profiles from a five-vehicle platoon
from the AstaZero campaign. The segmentation was performed on the basis of the
average platoon speed (platoon profile). We obtain acceleration and
deceleration parts using a local min/max algorithm on the platoon profile
111https://www.csc.kth.se/ weinkauf/notes/persistence1d.html. Furthermore, we
manually label the stable parts, that is, partial speed profiles where the
vehicles follow each other in platoon formation with only slight speed
variations.

(a) Speed profile segmentation based on platoon average speed in acceleration,
deceleration and following state events.

(b) The derived $v$ \- $k$ diagram for a partial platoon trajectory from the
AstaZero campaign.

Figure 6: Acceleration, deceleration and stable platoon states.

Figure 6(b) shows the speed-density ($v$-$k$) plot as they are computed by Eq.
3a and Eq. 3c respectively, for the platoon measurements shown in Figure 6(a).
The colors correspond to the three different states mentioned above. Noticing
the green points that correspond to equilibrium conditions, it is already
visible a quasi-linear relation between platoon production and platoon
density, (similar to the speed-spacing relation in the analysis of
trajectories at link level). The speed-density and flow-speed plots are shown
for completion in Figure 7.

(a) $v$ \- $k$ plot - proposed

(b) $q$ \- $v$ plot - proposed

Figure 7: Bi-level plots for different platoon states. The plots refer to
observed vehicle platoons driven with the ACC minimum setting.

##  Appendix F Sensitivity analysis on the aggregation process

The calibration results for different $\delta$ values are summarized in Table
2. It is interesting to observe that the calibrated TFD parameters for the
minimum and maximum $\delta$s remain consistent, even for higher $\delta$
values, that normally lead to very few points for calibration. A visual
illustration of the $q$ \- $k$ plot for different $\delta$ values is shown in
Figure 8.

(a) $q$ \- $k$ plot with $\delta_{k,v}=0.6$.

(b) $q$ \- $k$ plot with $\delta_{k,v}=1.0$.

(c) $q$ \- $k$ plot with $\delta_{k,v}=1.5$.

(d) $q$ \- $k$ plot with $\delta_{k,v}=2$.

(e) $v$ \- $k$ plot with $\delta_{q,v}=3$.

(f) $q$ \- $k$ plot with $\delta_{k,v}=3.5$.

Figure 8: Sensitivity analysis for the consistency of the calibrated $q$ \- $k$ plots of the FD for 6 additional $\delta_{k,v}$ values. Table 2: Calibrated TFD parameters that remain consistent for different $\delta_{k,v}$ values. $\delta_{k,v}$ | setting | $v_{f}[km/h]$ | $k_{cr}[veh/km]$ | $k_{jam}[veh/km]$ | $w[km/h]$
---|---|---|---|---|---
0.3 | min | 126.0 | 21.3 | 104.4 | 32.4
0.3 | max | 110.1 | 12.9 | 101.0 | 16.2
0.6 | min | 125.8 | 21.4 | 104.2 | 32.5
0.6 | max | 110.6 | 12.9 | 101.3 | 16.2
1.0 | min | 126.2 | 21.3 | 103.9 | 32.6
1.0 | max | 106.8 | 13.4 | 100.2 | 16.4
1.5 | min | 126.2 | 21.3 | 103.8 | 32.6
1.5 | max | 108.7 | 13.2 | 100.3 | 16.4
2 | min | 125.4 | 21.6 | 103.0 | 33.3
2 | max | 104.0 | 13.7 | 99.2 | 16.6
3 | min | 126.2 | 21.4 | 103.0 | 33.1
3 | max | 112.9 | 12.8 | 98.5 | 16.9
3.5 | min | 126.2 | 21.4 | 103.0 | 33.1
3.5 | max | 106.0 | 13.6 | 98.9 | 16.8

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