expect that (6.8) defines the derivative of an equivariant extension of
$W_{{\mathrm{AKV}}}(c)$, namely $W(c,\epsilon)$. This is in perfect agreement
with our formula (6.1).

To conclude, there is a reasonable analogy between our definition of the
equivariant superpotential and standard considerations based on open string
mirror symmetry. A key difference is that in both cases the relevant geometric
objects are non-compact. As a consequence of this, computations done in the
non-equivariant setting usually neglect semiclassical contributions from
constant maps, and the focus is entirely on the non-constant instanton
contributions in each phase separately. Instead in our setting we can keep
track of semiclassical contributions as well, since they get regularized by
equivariance. Thanks to this we obtain a function
$\mathcal{H}^{D}(\bm{t},c,\epsilon)$ that is globally defined, and whose
monodromies around different loops in the complexified $c$-plane encode all
disk superpotentials $W_{{\mathrm{AKV}}}$ in the different phases.

##  7 Summary and outlook

In this paper we introduced and studied a quantum uplift of the symplectic cut
construction in the setting of toric Calabi–Yau threefolds. Our approach is
based on the equivariant gauged linear sigma model, whose target is the
threefold after cutting. One the main characters in our story is the quantum
Lebesgue measure $\mathcal{H}^{D}(\bm{t},c,\epsilon)$ associated to the choice
of cut. The measure corresponds to the equivariant quantum volume of the
divisor that is invariant under the $U(1)$ action defining the cut.

The quantum measure encodes both closed and open topological string
invariants. Integrating $\mathcal{H}^{D}(\bm{t},c,\epsilon)$ over all values
of the transverse modulus $c$ recovers the equivariant quantum volume
$\mathcal{F}^{D}(\bm{t},\epsilon)$ of the ambient CY3

$\mathcal{F}^{D}(\bm{t},\epsilon)=\int_{-\infty}^{+\infty}\mathcal{H}^{D}(\bm{t},c,\epsilon)\,\mathop{}\\!\mathrm{d}c\,,$
(7.1)

which is further related to the closed Gromov–Witten potential. The monodromy
of $\mathcal{H}^{D}(\bm{t},c,\epsilon)$ along a circle in the complexified
$c$-plane defines instead an equivariant open string superpotential
$W(\bm{t},c,\epsilon)$

$W(\bm{t},c,\epsilon)=\frac{1}{2\pi\mathrm{i}}\int^{c+2\pi\mathrm{i}}_{c}\mathcal{H}^{D}(\bm{t},s,\epsilon)\mathop{}\\!\mathrm{d}s\,.$
(7.2)

The standard disk instanton potential for a toric Lagrangian associated to the
cut can be recovered from $W(\bm{t},c,\epsilon)$ by a suitable (phase
dependent) Laurent expansion in the equivariant parameters $\epsilon_{i}$.
Through equations (7.1)-(7.2), the quantum Lebesgue measure provides a direct
relation between closed and open string invariants.

Equivariant partition functions $\mathcal{F}^{D},\mathcal{H}^{D}$ are globally
defined across the entire Kähler moduli space (extended by open string moduli
for the latter). This is in contrast with conventional generating series of
closed and open Gromov–Witten invariants, which are typically defined in
specific regions of the moduli space, such as large radius. The apparent
tension between these results is resolved by observing that the equivariant
partition functions include semiclassical contributions due to constant maps
to non-compact manifolds, whose volumes are regularized by equivariance.

Neither the instanton contributions nor the semiclassical ones are
independently globally defined, since they exhibit jumps across different
phases of the geometry. However their sum is globally well-defined. This
perspective extends and unifies previous computations of GW invariants, in
very concrete ways. In particular, the partition function of the equivariant
GLSM produces all-instanton resummed expressions for
$\mathcal{F}^{D},\mathcal{H}^{D}$ that can be written down in closed form and
which can be analytically continued across different phases with no
difficulty. A prototypical example of this are the large-radius and the
conifold-point expressions (4.13)-(4.14) and (4.15) in terms of hypergeometric
functions.

The global nature of equivariant quantum volumes can be regarded as a
realization of the long-standing expectation that quantum cohomology rings are
invariants of birational equivalence classes [Mor94].

#### Further developments

Throughout this work we have come across several important questions that
deserve to be studied in greater detail.

_Normalization factors._ On a purely technical level, there are certain issues
with the normalization of the physical superpotentials obtained in the non-
equivariant limit. In this paper we have largely neglected these, since the
normalization can be unambiguously fixed by hand by matching the first open-
string instanton contribution with that of the toric brane in
$\mathbb{C}^{3}$. Nevertheless, it would certainly be satisfying to find a
coherent explanation for the overall factors.

_Mirror symmetry._ The global nature of quantum volumes
$\mathcal{F}^{D},\mathcal{H}^{D}$ hinges in a crucial way on equivariance. In
our story this is naturally introduced by a deformation of the GLSM, where
$\epsilon_{i}$ correspond to masses of chiral fields. A simple manipulation of
the GLSM Coulomb branch integral leads to an equivariant generalization of the
Hori–Vafa mirror geometry, as remarked around (2.44). On the one hand, this
would be key to place (6.8) on a more firm footing, by providing the
appropriate equivariant generalization of the right-hand side. On the other
hand, studying the equivariant deformation of Hori–Vafa mirror symmetry would
also open up the possibility to study non-compact $B$-branes as mirror non-
compact $A$-branes. Due to multivaluedness of the holomorphic top-form, we
expect calibration to produce non-compact mirror sLags very similar in nature
to the non-compact $A$-branes studied in the context of wrapped Fukaya
categories. For mirrors of toric threefolds these can be studied concretely by
a straightforward extension of the analysis of compact $A$-branes initiated in
[BLR23].

_Other phases, other cuts and framing._ In this work we focused on a few
examples, and analyzed the relation to open strings only in certain phases of
the $c$-plane. However a strength of our framework lies precisely in the
global global nature of $\mathcal{H}^{D}$, whose different monodromies tie
together equivariant superpotentials associated to different phases. Some of
the additional phases are most naturally accessible by changing the type of
cut under consideration. In addition to this, the open string superpotential
is known to change under so-called framing of the brane. Although we have
partially addressed some of these points in the examples considered in this
paper, it would be very interesting to extend our discussion further to
include all these global issues in a coherent and systematic fashion.

_Higher genus._ Although we have restricted attention to genus-zero
worldsheets, there are good reasons to expect an extension of our observations
to all-genera. From the viewpoint of GLSMs, the summation over genera incurs
in notorious technical complications related to the holomorphic anomaly
equations [Ber+94]. On the other hand, in recent years the study both closed
and open instanton partition functions has advanced significantly through a
variety of approaches, including the topological vertex [Aga+05], the TS/ST
correspondence [GHM16], the study of resurgent structures of both closed and
open strings [Mar08a, Mar08, PS10, Cou+16, GM22], and geometric
interpretations and extensions of the latter from in terms of Riemann-Hilbert
problems [Bri19, Bri20, CPT23, Ali+23, GHN22, AHT23, CLT20, GGH23]. In many of
these works, higher genus worldsheet instantons play an important role in the
analytic continuation across the moduli space. In our work we observed that
equivariance plays a similar role in the analytic continuation of genus zero
string partition functions. It would therefore be interesting to explore the
role of equivariance in the higher genus setting.

_Lift to M-theory._ It would be certainly desirable to gain a better grasp on
the physics behind quantum symplectic cuts in the context of string theory. A
promising direction seems to be through an uplift to M-theory [AKV02, AV01a]
where the symplectic cut of $X$ appears naturally. In presence of the D6 brane
on $L\times\mathbb{R}^{4}$, going to M-theory replaces $X$ with a $G_{2}$
manifold fibered by a circle over $X$ which shrinks at $L$. If we perform a
9/11 flip and compactify instead on a (choice of) $S^{1}$ from the $U(1)^{3}$
fibration of $X$ we obtain a new manifold whose topology resembles the
symplectic cut (2.25).

We plan to continue our discussion of these and other questions in upcoming
works [CLZ].

##  Appendix A Special function definitions and identities

In this appendix we collect technical formulas about special functions which
we use in the main text.

The beta function has the following integral representations

$B(a,b)=\frac{\operatorname{\Gamma}(a)\operatorname{\Gamma}(b)}{\operatorname{\Gamma}(a+b)}=\int^{1}_{0}y^{a-1}(1-y)^{b-1}\mathop{}\\!\mathrm{d}y=\int^{\infty}_{0}\frac{x^{a-1}}{(1+x)^{a+b}}\mathop{}\\!\mathrm{d}x$
(A.1)

where we have used the following change of variables $x=y/(1-y)$ under the
integral. The Appell $F_{1}$ function has the integral representation

$F_{1}(a;b_{1},b_{2};c;z_{1},z_{2})=\frac{\operatorname{\Gamma}(c)}{\operatorname{\Gamma}(a)\operatorname{\Gamma}(c-a)}\int_{0}^{1}\frac{w^{a-1}(1-w)^{c-a-1}}{(1-wz_{1})^{b_{1}}(1-wz_{2})^{b_{2}}}\,\mathop{}\\!\mathrm{d}w$
(A.2)

for $\Re(a)>0\land\Re(c-a)>0$.

We use the integral representation of the ${}_{2}F_{1}$ hypergeometric
function

$B(b,c-b)\,{}_{2}F_{1}(a,b;c;z)=\int_{0}^{1}x^{b-1}(1-x)^{c-b-1}(1-zx)^{-a}~{}\mathop{}\\!\mathrm{d}x$
(A.3)

with $\Re(c)>\Re(b)>0$ and $\Re(z)\leq 1$.

Below we list some relevant identities for hypergeometric functions. First we
have

${}_{2}F_{1}(a,b;c;z)=\frac{\operatorname{\Gamma}(c)\operatorname{\Gamma}(a+b-c)}{\operatorname{\Gamma}(a)\operatorname{\Gamma}(b)}\,{}_{2}F_{1}(c-a,c-b;c-a-b+1;1-z)\,(1-z)^{c-a-b}\\\
+\frac{\operatorname{\Gamma}(c)\operatorname{\Gamma}(c-a-b)}{\operatorname{\Gamma}(c-a)\operatorname{\Gamma}(c-b)}\,{}_{2}F_{1}(a,b;a+b-c+1;1-z)$
(A.4)

where $c-a-b$ is not an integer. Next we recall

${}_{2}F_{1}\left(a,b;c;1-z\right)=z^{-a}{}_{2}F_{1}\left(a,c-b;c;1-\frac{1}{z}\right)~{},~{}~{}~{}~{}z\notin(0,\infty)$
(A.5)

and

${}_{2}F_{1}(a,b;c;z)=\frac{\operatorname{\Gamma}(c)\operatorname{\Gamma}(b-a)}{(-z)^{a}\operatorname{\Gamma}(b)\operatorname{\Gamma}(c-a)}\,{}_{2}F_{1}\left(a,a-c+1;a-b+1;\frac{1}{z}\right)\\\
+\frac{\operatorname{\Gamma}(c)\operatorname{\Gamma}(a-b)}{(-z)^{b}\operatorname{\Gamma}(a)\operatorname{\Gamma}(c-b)}\,{}_{2}F_{1}\left(b,b-c+1;b-a+1;\frac{1}{z}\right)$
(A.6)

Finally, we have a different integral representation for the function
${}_{2}F_{1}$ as

$B(a,c+b-a)\,{}_{2}F_{1}(c,a;c+b;1-z)=\int_{0}^{\infty}\frac{x^{a-1}}{(1+x)^{b}(1+zx)^{c}}~{}\mathop{}\\!\mathrm{d}x$
(A.7)

with $\Re(b+c)>\Re(a)>0$ and $\Re(z)\geq 0$.

##  Appendix B Disk partition function for conifold

Using the conventions from Section 4 we define the disk function for $t>0$ as
the contour integral

$\mathcal{F}^{D}(t,\epsilon)=\oint_{(1),(2)}\frac{\mathop{}\\!\mathrm{d}\phi}{2\pi\mathrm{i}}\,\mathrm{e}^{t\phi}\operatorname{\Gamma}(\epsilon_{1}+\phi)\operatorname{\Gamma}(\epsilon_{2}+\phi)\operatorname{\Gamma}(\epsilon_{3}-\phi)\operatorname{\Gamma}(\epsilon_{4}-\phi)$
(B.1)

where the contour includes two towers of poles $-\epsilon_{1}-n$ and
$-\epsilon_{2}-n$ for $n=0,1,2,\dots$. After performing the integrals the
answer can be rewritten as follows

$\displaystyle\mathcal{F}^{D}(t,\epsilon)=$
$\displaystyle\mathrm{e}^{-\epsilon_{1}t}\operatorname{\Gamma}(\epsilon_{2}-\epsilon_{1})\operatorname{\Gamma}(\epsilon_{1}+\epsilon_{3})\operatorname{\Gamma}(\epsilon_{1}+\epsilon_{4})\,{}_{2}F_{1}\left(\epsilon_{1}+\epsilon_{3},\epsilon_{1}+\epsilon_{4};\epsilon_{1}-\epsilon_{2}+1;\mathrm{e}^{-t}\right)$
(B.2)
$\displaystyle+\mathrm{e}^{-\epsilon_{2}t}\operatorname{\Gamma}(\epsilon_{1}-\epsilon_{2})\operatorname{\Gamma}(\epsilon_{2}+\epsilon_{3})\operatorname{\Gamma}(\epsilon_{2}+\epsilon_{4})\,{}_{2}F_{1}\left(\epsilon_{2}+\epsilon_{3},\epsilon_{2}+\epsilon_{4};\epsilon_{2}-\epsilon_{1}+1;\mathrm{e}^{-t}\right)\,.$

The disk partition function satifies the following equivariant PF equation

$\Big{(}\operatorname{\mathcal{D}}_{1}\operatorname{\mathcal{D}}_{2}-\mathrm{e}^{-t}\operatorname{\mathcal{D}}_{3}\operatorname{\mathcal{D}}_{4}\Big{)}\mathcal{F}^{D}(t,\epsilon)\equiv\Big{(}(\epsilon_{1}+\partial_{t})(\epsilon_{2}+\partial_{t})-\mathrm{e}^{-t}(\epsilon_{3}-\partial_{t})(\epsilon_{4}-\partial_{t})\Big{)}\mathcal{F}^{D}(t,\epsilon)=0$
(B.3)

and semiclassically it goes to the equivariant volume (with some insertions of
$\operatorname{\Gamma}$-class). It is natural to write it in the following way

$\mathcal{F}^{D}(t,\epsilon)={\rm{s.c.}}+t\operatorname{Li}_{2}(\mathrm{e}^{-t})-2\operatorname{Li}_{3}(\mathrm{e}^{-t})+O(\epsilon)~{},$
(B.4)

where the semiclassical terms contain singularities in $\epsilon$’s and
instanton terms (the poles $n=1,2,\dots$) are regular (for more details see
[CPZ22]).

Using the functional definition (B.2) and the identity (A.4) we can rewrite
the disk function as

$\mathcal{F}^{D}(t,\epsilon)=\frac{\operatorname{\Gamma}(\epsilon_{3}+\epsilon_{2})\operatorname{\Gamma}(\epsilon_{4}+\epsilon_{1})\operatorname{\Gamma}(\epsilon_{1}+\epsilon_{3})\operatorname{\Gamma}(\epsilon_{2}+\epsilon_{4})}{\operatorname{\Gamma}(\epsilon_{1}+\epsilon_{2}+\epsilon_{3}+\epsilon_{4})}\times\\\
\times\mathrm{e}^{-\epsilon_{1}t}\,{}_{2}F_{1}(\epsilon_{1}+\epsilon_{4},\epsilon_{1}+\epsilon_{3};\epsilon_{1}+\epsilon_{2}+\epsilon_{3}+\epsilon_{4};1-\mathrm{e}^{-t})\,,$
(B.5)

or, equivalently, using (A.5) we can write

$\mathcal{F}^{D}(t,\epsilon)=\frac{\operatorname{\Gamma}(\epsilon_{1}+\epsilon_{3})\operatorname{\Gamma}(\epsilon_{2}+\epsilon_{3})\operatorname{\Gamma}(\epsilon_{1}+\epsilon_{4})\operatorname{\Gamma}(\epsilon_{2}+\epsilon_{4})}{\operatorname{\Gamma}(\epsilon_{1}+\epsilon_{2}+\epsilon_{3}+\epsilon_{4})}\times\\\
\times\mathrm{e}^{\epsilon_{3}t}\,{}_{2}F_{1}(\epsilon_{1}+\epsilon_{3},\epsilon_{2}+\epsilon_{3};\epsilon_{1}+\epsilon_{2}+\epsilon_{3}+\epsilon_{4};1-\mathrm{e}^{t})~{}.$
(B.6)

This form of the disk partition function is naturally related to the chamber
$t<0$ and can also be obtained via the following contour integral

$\displaystyle\mathcal{F}^{D}(t,\epsilon)=$
$\displaystyle-\oint_{(3)(4)}\frac{\mathop{}\\!\mathrm{d}\phi}{2\pi\mathrm{i}}\,\mathrm{e}^{t\phi}\operatorname{\Gamma}(\epsilon_{1}+\phi)\operatorname{\Gamma}(\epsilon_{2}+\phi)\operatorname{\Gamma}(\epsilon_{3}-\phi)\operatorname{\Gamma}(\epsilon_{4}-\phi)$
(B.7) $\displaystyle=$
$\displaystyle\mathrm{e}^{\epsilon_{3}t}\operatorname{\Gamma}(\epsilon_{1}+\epsilon_{3})\operatorname{\Gamma}(\epsilon_{2}+\epsilon_{3})\operatorname{\Gamma}(\epsilon_{4}-\epsilon_{3})\,{}_{2}F_{1}\left(\epsilon_{1}+\epsilon_{3},\epsilon_{2}+\epsilon_{3};\epsilon_{3}-\epsilon_{4}+1;\mathrm{e}^{t}\right)$
$\displaystyle+\mathrm{e}^{\epsilon_{4}t}\operatorname{\Gamma}(\epsilon_{1}+\epsilon_{4})\operatorname{\Gamma}(\epsilon_{2}+\epsilon_{4})\operatorname{\Gamma}(\epsilon_{3}-\epsilon_{4})\,{}_{2}F_{1}\left(\epsilon_{1}+\epsilon_{4},\epsilon_{2}+\epsilon_{4};\epsilon_{4}-\epsilon_{3}+1;\mathrm{e}^{t}\right)$

where one can use (A.4) to related the above expression to (B.6). The extra
minus in front of the contour integral (B.7) appears due to the fact that
closing the contour along the imaginary axis either to the left or to the
right would carry opposite orientation.

We remark that all these different formulas can obtained from the following
integral expression

$\mathcal{F}^{D}(t,\epsilon)=\int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\frac{\mathop{}\\!\mathrm{d}\phi}{2\pi\mathrm{i}}\int_{0}^{\infty}\mathop{}\\!\mathrm{d}y_{1}\mathop{}\\!\mathrm{d}y_{2}\mathop{}\\!\mathrm{d}y_{3}\mathop{}\\!\mathrm{d}y_{4}\,\mathrm{e}^{t\phi-
y_{1}-y_{2}-y_{3}-y_{4}}y_{1}^{\epsilon_{1}+\phi-1}y_{2}^{\epsilon_{2}+\phi-1}y_{3}^{\epsilon_{3}-\phi-1}y_{4}^{\epsilon_{4}-\phi-1}$
(B.8)

which is the representation (2.44) written for the conifold.

##  Appendix C Disk partition function for local $\mathbb{P}^{2}$

Using the conventions from Section 5 we define the disk function for local
$\mathbb{P}^{2}$ for $t>0$ by the contour integral

$\mathcal{F}^{D}(t,\epsilon)=\oint_{(1),(2),(3)}\frac{\mathop{}\\!\mathrm{d}\phi}{2\pi\mathrm{i}}\,\mathrm{e}^{t\phi}\operatorname{\Gamma}(\epsilon_{1}+\phi)\operatorname{\Gamma}(\epsilon_{2}+\phi)\operatorname{\Gamma}(\epsilon_{3}+\phi)\operatorname{\Gamma}(\epsilon_{4}-3\phi)~{},$
(C.1)

where we take into account three towers of the poles: $(-\epsilon_{1}-n)$,
$(-\epsilon_{2}-n)$ and $(-\epsilon_{3}-n)$ for $n=0,1,\dots$. The disk
partition function $\mathcal{F}^{D}(t,\epsilon)$ satisfies the following
equivariant PF equation

$\Big{(}\operatorname{\mathcal{D}}_{1}\operatorname{\mathcal{D}}_{2}\operatorname{\mathcal{D}}_{3}-\mathrm{e}^{-t}(2+\operatorname{\mathcal{D}}_{4})(1+\operatorname{\mathcal{D}}_{4})\operatorname{\mathcal{D}}_{4}\Big{)}\mathcal{F}^{D}(t,\epsilon)=0~{}.$
(C.2)

The expansion in equivariant parameters $\epsilon$ and the relation of
$\mathcal{F}^{D}(t,\epsilon)$ to closed GW invariant is discussed in [CPZ22].
Alternatively, we can write the disk partition function as

$\displaystyle\mathcal{F}^{D}$
$\displaystyle=\int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\frac{\mathop{}\\!\mathrm{d}\phi}{2\pi\mathrm{i}}\int_{0}^{\infty}\mathop{}\\!\mathrm{d}y_{1}\mathop{}\\!\mathrm{d}y_{2}\mathop{}\\!\mathrm{d}y_{3}\mathop{}\\!\mathrm{d}y_{4}~{}\mathrm{e}^{\phi
t-\sum
y_{i}}y_{1}^{\epsilon_{1}+\phi-1}y_{2}^{\epsilon_{2}+\phi-1}y_{3}^{\epsilon_{3}+\phi-1}y_{4}^{\epsilon_{4}-3\phi-1}$
(C.3)
$\displaystyle=\int_{0}^{\infty}\mathop{}\\!\mathrm{d}y_{1}\mathop{}\\!\mathrm{d}y_{2}\mathop{}\\!\mathrm{d}y_{3}\mathop{}\\!\mathrm{d}y_{4}~{}\mathrm{e}^{-\sum
y_{i}}\prod_{i}y_{i}^{\epsilon_{i}-1}\delta\left(\log\left(\frac{\mathrm{e}^{-t}y_{4}^{3}}{y_{1}y_{2}y_{3}}\right)\right)$
$\displaystyle=\mathrm{e}^{-\epsilon_{1}t}\int_{0}^{\infty}\mathop{}\\!\mathrm{d}y_{2}\mathop{}\\!\mathrm{d}y_{3}\mathop{}\\!\mathrm{d}y_{4}~{}y_{2}^{\epsilon_{2}-\epsilon_{1}-1}y_{3}^{\epsilon_{3}-\epsilon_{1}-1}y_{4}^{\epsilon_{4}+3\epsilon_{1}-1}\exp\left(-\frac{\mathrm{e}^{-t}y_{4}^{3}}{y_{2}y_{3}}-y_{2}-y_{3}-y_{4}\right)$
$\displaystyle=\mathrm{e}^{-\epsilon_{1}t}\operatorname{\Gamma}(\epsilon_{2}-\epsilon_{1})\operatorname{\Gamma}(\epsilon_{3}-\epsilon_{1})\operatorname{\Gamma}(3\epsilon_{1}+\epsilon_{4})\times$
$\displaystyle\quad\times\,{}_{3}F_{2}\left(\epsilon_{1}+\frac{\epsilon_{4}}{3},\epsilon_{1}+\frac{\epsilon_{4}}{3}+\frac{1}{3},\epsilon_{1}+\frac{\epsilon_{4}}{3}+\frac{2}{3};\epsilon_{1}-\epsilon_{2}+1,\epsilon_{1}-\epsilon_{3}+1;-27\mathrm{e}^{-t}\right)$
$\displaystyle+\mathrm{e}^{-\epsilon_{2}t}\operatorname{\Gamma}(\epsilon_{1}-\epsilon_{2})\operatorname{\Gamma}(\epsilon_{3}-\epsilon_{2})\operatorname{\Gamma}(3\epsilon_{2}+\epsilon_{4})\times$
$\displaystyle\quad\times\,{}_{3}F_{2}\left(\epsilon_{2}+\frac{\epsilon_{4}}{3},\epsilon_{2}+\frac{\epsilon_{4}}{3}+\frac{1}{3},\epsilon_{2}+\frac{\epsilon_{4}}{3}+\frac{2}{3};\epsilon_{2}-\epsilon_{1}+1,\epsilon_{2}-\epsilon_{3}+1;-27\mathrm{e}^{-t}\right)$
$\displaystyle+\mathrm{e}^{-\epsilon_{3}t}\operatorname{\Gamma}(\epsilon_{1}-\epsilon_{3})\operatorname{\Gamma}(\epsilon_{2}-\epsilon_{3})\operatorname{\Gamma}(3\epsilon_{3}+\epsilon_{4})\times$
$\displaystyle\quad\times\,{}_{3}F_{2}\left(\epsilon_{3}+\frac{\epsilon_{4}}{3},\epsilon_{3}+\frac{\epsilon_{4}}{3}+\frac{1}{3},\epsilon_{3}+\frac{\epsilon_{4}}{3}+\frac{2}{3};\epsilon_{3}-\epsilon_{1}+1,\epsilon_{3}-\epsilon_{2}+1;-27\mathrm{e}^{-t}\right)~{},$

where we use the ${}_{3}F_{2}$ hypergeometric function. This explicit
expression matches with the residue computation (C.1) after resummation.

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Ajit, Mouli, Knickerbocker, and Wood

# Machine learning framework for end-to-end implementation of Incident
duration prediction

Smrithi Ajit
<EMAIL_ADDRESS>
Varsha R Mouli, Research Engineer, Institute of Transportation
<EMAIL_ADDRESS>
Skylar Knickerbocker, Research Engineer, Institute of Transportation
<EMAIL_ADDRESS>
Jonathan S. Wood, Ph.D, Assistant Professor, Dept.of Civil,Construction and
Environmental Engineering, Iowa State University
<EMAIL_ADDRESS>
##  1 Abstract

Traffic congestion caused by non-recurring incidents such as vehicle crashes
and debris is a key issue for Traffic Management Centers (TMCs). Clearing
incidents in a timely manner is essential for improving safety and reducing
delays and emissions for the traveling public. However, TMCs and other
responders face a challenge in predicting the duration of incidents (until the
roadway is clear), making decisions of what resources to deploy difficult. To
address this problem, this research developed an analytical framework and end-
to-end machine-learning solution for predicting incident duration based on
information available as soon as an incident report is received. Quality
predictions of incident duration can help TMCs and other responders take a
proactive approach in deploying responder services such as tow trucks,
maintenance crews or activating alternative routes. The predictions use a
combination of classification and regression machine learning modules. The
performance of the developed solution has been evaluated based on the Mean
Absolute Error (MAE), or deviation from the actual incident duration as well
as Area Under the Curve (AUC) and Mean Absolute Percentage Error (MAPE). The
results showed that the framework significantly improved incident duration
prediction compared to methods from previous research.

Keywords: non-recurring, crash, predicts, incident duration, machine learning,
MAE

##  2 Introduction and Background

According to the Traffic Incident Management Handbook, an incident is defined
as a non-recurring event resulting in either a reduction in roadway capacity
or an abnormal increase in demand [1]. These incidents include, but are not
limited to, vehicle crashes, disabled vehicles, debris, and spilled cargo.
Incidents not only result in traveler delay but also increase the likelihood
of secondary crashes and other secondary effects due to increased
opportunities for secondary events to occur [2]. Secondary events can lead to
increased demand for police, fire, and emergency services, reduced air
quality, and other environmental impacts.

Total incident duration is comprised of incident notification time, response
time, and clearance time, as illustrated in Figure 1. As shown, the total
incident duration is the total time from the start of the incident until the
reported time of clearance for the event [3] and includes incident
notification, response, and clearance times. The incident notification time is
from the start of the incident until the time it is reported. The response
time is from the report time until the response unit’s arrival. The clearance
time is the time taken to clear the incident after emergency responders have
arrived on the scene.

While the incident duration time is not controlled by the responding agencies,
dispatching the correct personnel and equipment can reduce the total incident
duration by minimizing the sum total of response and clearance times. This
requires planning, preparedness, and coordination between responders.
Information on what resources should be dispatched can be improved using
predictive models of the total incident duration. For instance, having
accurate predictions of the total incident duration can assist Traffic
Management Center (TMC) operators in selecting the appropriate actions from
potential options such as the following: (1) diverting traffic to an alternate
route, (2) providing a warning of a potential delay to travelers planning to
take a congested route and (3) ensuring helper services, such as safety
service patrol or maintenance crews, arrive at the incident spot on time. For
example, if an incident will be cleared within a half hour, it may not be
reasonable to detour traffic on a route that increases the travel time over
that amount of time. Helper services may also not be requested if the incident
is cleared before the time it takes the service patrol to arrive at the
incident location.

Figure 1: Illustration of traffic incident duration

As indicated above, incident duration prediction is critically important to
the TMC for timely mitigation of traffic congestion, to not only forewarn
people of the crashes on a particular route in advance but reduce the
likelihood of secondary crashes. Past studies have focused on understanding
and identifying factors related to incident type, roadway data, time of the
day, weather conditions, speed, traffic volume, blocked lane, location,
environment, weather, road characteristics, temporal and spatial factors, and
so on, either through associative mining or through prediction based
descriptive models that are either statistical or machine learning based
models.

Early work in the area of incident duration prediction used linear regression
models and related statistical tests such as Analysis of Variance (ANOVA).
Many of these models were limited by the number of data points [4],[5] [6].
Coefficient of determination, Root Mean Squared Error (RMSE), Mean Absolute
Error (MAE), and Mean Absolute Percentage Error (MAPE) were the performance
metrics commonly used to report the efficiency of the ML models. The lowest
MAPE that was reported with these models include that by [7], who reported
34.1% for a total of 2,512 accidents using a cluster-based log-normal
distribution model and a MAPE of 37% reported by Khattak et al. for a total of
59,804 accidents.

Other methods that have been used in the published literature include fuzzy
logic, Artificial Neural Networks (ANN), Bayesian methods, Survival or Hazard
models, tree-based machine learning methods, and text mining. Among fuzzy
logic-based approaches, the lowest MAPE was 36% with an average error of 0.3
minutes reported by the authors in [8] and [9], respectively. However, these
studies were also limited by fewer data points. Methods using ANN have also
been attempted on small and large datasets, with Pereira et.al[10] achieving a
median error of 9.9 minutes for a dataset of size 10,139 and Lopes et al. [11]
achieving within 10 to 20 minutes error for a dataset of 10,762 incidents.

Bayesian models are useful in applying prior knowledge of associations between
variables in developing predictive models. Several authors have developed the
incident duration prediction problem into a classification problem and have
used Bayesian approaches for predictions achieving accuracy ranging between
74% to 80%. Authors of [12] used the Bayesian ANN model for a large dataset
achieving a low MAPE of 0.18 to 0.29. Other Bayesian approaches reported
include Bayesian propagation neural network, and Bayesian Support Vector
Regression [13].

Survival models or Hazard based models are used to predict the length of the
duration between the occurrence of the event, and its clearance while also
being capable of predicting when the incident duration will end given it has
proceeded for time t. This ability of the models helps prediction of incident
duration at different stages of the incident as presented in [14] using an AFT
hazard-based model. Fully parametric Hazard based models using log-normal and
log-logistic distributions were used in [15] and reported a MAPE value of 47%.
A logistic Accelerated Failure Time model (AFT) has been presented in [16]
that reported a MAPE of 43.7%, and another Weibull distribution parametric
model using gamma heterogeneity was used in [17] to account for unobserved
heterogeneity. In real-time, as time progresses, the traffic conditions
change, and the new set of factors influence the incident duration, due to
which time sequential models are useful. Incorporating additional features as
the incident progresses can be very helpful in traffic decision-making.
Inverse Gaussian frailty AFT model, Multilevel mixed-effect AFT model, and
FMAM models were used to predict the incident duration with sequential TIM
information in [18]. Multiple studies also identified that a single method
could not suit all the incident duration ranges and came up with hybrid
approaches that combine two or more methods. In the study [19], Hazard Based
Duration Models (HBDMs) were used as leaves of the M5P Tree, and in the work
of [20], fuzzy entropy was used to select the features, and ANN was used to
predict the duration based on these features. A competing mixture model was
presented in [21] to analyze the influence of the clearance method and other
covariates on traffic incidents.

Tree-based models have also received significant attention. Classification
Tree methods have reported MAPE values ranging between 42.7% to 65% [22, 23,
24]. A more recent focus has been on ensemble techniques like RF, Gradient
Boost method, Extreme Gradient Boost Method, and Ada Boost owing to the
superiority of performance with datasets with a large number of categorical
variables[25, 26, 27]. In [26], RF models developed for long and short-
duration datasets reported an MAE of 36.652 minutes and 14.97 minutes,
respectively.

Besides the approaches discussed above, text-based analysis has also been
used. In [28], the authors illustrated an improvement of more than 35% over
non-textual models, and a knowledge-based method was used in [29] to estimate
incident clearance duration on Maryland I-95. In [30], a spatio-temporal
feature learning model called TITAN was proposed that considers hidden spatio-
temporal associations by considering connectivity between road segments in
addition to identifying high-level features. In [31], a copula-based tri-
variate framework was used to generate a stochastic dependence relationship
between the various variables capable of predicting incident duration.

In the context of ML models, since no prior assumptions are made about the
distribution of the data, the ML models will be able to identify underlying
relationships only if data challenges like skewness and the heterogenous
nature of the data are adjusted for prior to training the ML models. The
heterogenous nature of the data is handled through the use of feature
selection methods that help restrict the variables to a few most significant
ones. In order to group similar data points, a general approach is to use
supervised or unsupervised clustering methods. Supervised clustering of data
points followed by application of regression models can be seen in [32] while
an unsupervised approach can be seen in [33]. A supervised approach that makes
use of blending of results from multiple ML models for the prediction of
incident duration has not been applied in the published literature.

##  3 Objectives

The overall objective of this study is to develop a total incident duration
prediction framework to support TMC operations by utilizing machine learning
in a two-step process. The first step uses supervised classification to
predict which of three time ranges the total incident duration falls within.
The ranges include within a half hour (class 1), between half an hour and 2
hours (class 2), and more than 2 hours (class 3) based on [1]. This can be
used to assist the operator in identifying incidents that need immediate
attention and what type of resources may be required. This step also serves to
improve the prediction in the regression module through a prior grouping of
similar incidents. The second step utilizes a regression model to produce a
more accurate prediction of the incident duration in minutes. Three years of
historical incident data are used to train, test, validate, and finalize the
machine-learning components used in the framework. Two different feature sets
have been considered as part of the analysis. The first one is utilized in the
initial prediction and involves basic time and location information while the
second feature set is used in the second step as more details are available
about the event which includes additional features such as road conditions and
environmental factors. To show that the analysis framework works well even
with minimal features, each model framework has been evaluated using both
feature sets. An ensemble technique called blending has been utilized to
improve the performance of regular ML models used in the framework. The
performance of the developed supervised ML framework has also been validated
against an alternative framework, an unsupervised ML framework, and a basic
regression model that allows censorship, known as a Tobit model.

A few applications of the developed framework have been provided in the
Discussion section. Additionally, to understand if the classification step
improves predictions, the Mean Absolute Error (MAE) results from the developed
framework have been compared with MAE results obtained from a framework that
predicts without classifying the data into ranges.

##  4 Data Description and Pre-processing

The Advanced Traffic Management System (ATMS) events data is the primary
source of information used in this paper. It is maintained by the Iowa
Department of Transportation Traffic Management Center (TMC) and contains
detailed records of traffic incidents that occurred in the state. The data is
collected by TMC operators who are actively managing incidents on the roads
and includes information such as the number of lanes closed, start and end
times, the presence of emergency responders, and the severity of the incident.
This information is useful for creating after-action reports, which allow the
Iowa DOT to review and analyze their response to incidents in order to
identify areas for improvement. The ATMS data are made available on a daily
basis and includes all incidents that have been closed from the previous day
[34]. The data used in this study was collected from 2017 to 2019.

The Advanced Traffic Management System (ATMS) data used in this study includes
both recurring and non-recurring events, such as work zones, collisions,
debris, stopped vehicles, and special events. However, the study focuses
specifically on collisions involving 1, 2, or 3 vehicles and debris, as these
types of incidents can have a significant impact on traffic flow and are
within the control of TMC operators. The study also limits the analysis to
incidents that occurred on rural and municipal interstates within the state of
Iowa and excludes days with severe weather events as these can significantly
impact incident duration. To identify severe weather days, the study used the
National Oceanic and Atmospheric Administration’s (NOAA) storm database and
excluded incidents that occurred in affected counties. After removing
incomplete records, the study included a total of 6,832 incidents that
occurred over a three-year period.

The data used in the study was imbalanced, with more incidents in the short
and medium-duration classes compared to the long-duration class. To address
this issue, the study used a method called Synthetic Minority Oversampling
Technique (SMOTE), which is a way to balance the data by duplicating samples
from the minority class (in this case, the long-duration class) without adding
any new information. This helps to ensure that the model is not biased towards
the majority class. To address the correlation between variables in the data,
the study removed features that had a high correlation value (greater than
0.4) and used the pycaret tool to further address this issue. This helps to
ensure that the model is not influenced by correlated features, which can
impact the accuracy of the model.

For the analysis, the dependent variable will be the duration of each
incident, calculated as the time between when the incident started and when
the roadway was cleared. The dataset includes several variables, which are
listed in Table 1, along with their data types and any transformations that
were applied. Columns with text data, such as information about responders,
were converted to categorical variables. Binary variables were used to encode
information about the presence of specific responders (e.g., police,
firefighters, etc.) at the incident, with a value of 1 indicating presence and
0 indicating absence. This responder information is described in more detail
in Table 5, which also includes statistics about the duration of incidents for
each type of responder.

Categorical variables that were not ordered were one-hot encoded. Missing
values for numerical variables were imputed with the mean value while missing
values for categorical variables were imputed with a constant based on the
mode. For the variables vehicles, trucks, injuries, fatalities, and AADT, the
variables were converted to categorical variables as described in Table 1. The
statistics for the Average Annual Daily Traffic (AADT) information in the
dataset are provided in Table 5. It can be seen from these statistics that the
majority of incidents have a duration in short to medium range.

The basic statistics related to the dataset have been provided in Table 5.
These descriptive statistics show the wide variance in roadway clearance times
with values ranging from 1 to 1,358 minutes, along with a positive skew of
5.637 minutes. This is addressed by removing the skew in the data using a box
cox transformation of the data, which reduces the skew of the dataset to
-0.035.

Table 1: Explanation of variables in the dataset, their datatypes and
transformations

Feature Group |  Variable name |  Derived |  Datatype |  Categories
---|---|---|---|---
Lane Information |  Number of lanes(lanes) |  N |  Integer |  NA
|  OnlyShouldersClosed |  Y |  Binary |  1-responded, 0-did not respond
Location information |  Direction |  N |  Binary |  1: North, 2: South, 3: West, 4: East,
|  County region |  Y |  Factor |  North East, North West, Central, South East, South West
|  City number |  N |  Factor |  Numerical binning
Incident |  Event Type |  Y |  Factor |  Crash(1vehicle, 2 vehicle and 3 vehicle crash), Debris
Vehicles involved |  Vehicles |  N |  Factor |  0,1,2 and more than 3
|  Trucks |  N |  Factor |  0,1,2 and more than 3
Severity |  Injuries |  N |  Factor |  1-injured, 0-not injured
|  Fatalities |  N |  Factor |  1-1 or more fatality, 0- no fatality
Responder Information |  Responder Type: Police, TOW, DOT, DPS |  Y |  Factor |  1-responded, 0-did not respond
Detection |  Detection method |  N |  Factor |  Police, Highway Helper, Automated, DOT, Cameras, Others
Temporal information |  Time Of Day(TOD) |  Y |  Factor |  Morning: 7am to 9am, Early Afternoon:10am to 12pm, Afternoon: 1pm to 3pm, Evening rush:4 pm to 6pm, Evening: 7pm to 9pm, Night: 10pm to 6 am
|  Day Of Week(DOW) |  Y |  Factor |  Integer encoding 0 to 6 represent Monday to Sunday
|  Season |  Y |  Integer |  1=Winter (December to February), 2=Spring (March to May), 3=Summer (June to August), 4=Autumn (September to November)
|  Year |  Y |  Integer |  2017, 2018, 2019
Traffic information |  Hourly traffic volume |  Y |  Integer |  NA
|  AADT |  Y |  Factor |  1:<8,000, 2:8,000-12,000, 3:12,000-24,000, 4:24,000-48,000, 5:>48,000
Road characteristics |  Surface width |  N |  Float |  NA
|  Surface type |  Y |  Factor |  Grade and drained earth, gravel or stone, bituminous over gravel or stone, etc
|  Terrain |  Y |  Factor |  flat, rolly, hilly

Table 2: Responder Information

Responder |  % of total |  Incident duration |  Incident duration
---|---|---|---
(Presence) |  incidents |  Mean (min) |  Median (min)
DOT-No |  91% |  39.3 |  26
DOT-Yes |  9% |  88.7 |  61
EMS-No |  88% |  38.9 |  23
EMS-Yes |  12% |  76.2 |  57
HH-No |  51% |  40.0 |  22
HH-Yes |  49% |  47.7 |  33
Police-No |  67% |  33.6 |  17
Police-Yes |  33% |  65.5 |  47
TOW-No |  77% |  76.3 |  19
TOW-Yes |  23% |  23.7 |  51

Table 3: Feature Set Description

Feature group |  Description
---|---
Basic Features (FS1) |  Basic temporal and spatial information,vehicles,trucks,injuries
Full Features(FS2) |  AADT, hourly volume, road and environment factors, Responder information

Table 4: Statistical Information

Metric |  Value(minutes)
---|---
Standard deviation |  53.7
Mean |  45.2
Median |  31
Minimum |  1
25th percentile |  10
75th percentile |  59
Maximum |  542

Table 5: AADT versus Incident Duration

AADT |  Avg. Total |  Median Total |  No. of Records
---|---|---|---
16,000 |  47.8 |  28 |  153
27,700 |  28.8 |  4 |  131
45,300 |  54.6 |  43 |  151
57,000 |  28.4 |  14 |  266
100,500 |  30.4 |  20 |  194

Figure 2: Overall workflow of developed framework

##  5 Methodology

The framework for estimating event duration shown in Figure 2 can be separated
into two core components, including the classification and regression modules.
The goal of the first part of the framework in the classification module is to
classify an event as being of short, medium, or long duration based on its
total incident duration. Events with a duration of less than 30 minutes are
classified as short duration, those between 30 minutes and 2 hours are
classified as medium duration, and those lasting more than 2 hours are
classified as long duration.

The initial classification is based on only limited information available
about an event, including its location, date, and time of day, since not all
details about the event are available when the incident occurs. This minimal
information is used to assign the event to one of the three duration
categories.

After the initial classification, the data is then processed through a linear
referencing system (LRS) module, which removes the spatial dependencies
(latitude and longitude coordinates) of the data by adding two additional
features: a Route ID and a Measure. The Route ID represents the route on which
the event occurred, while the Measure provides the linear distance of the
event’s location from a reference point. These parameters are used to extract
additional features, such as Average Annual Daily Traffic (AADT), terrain,
surface width, and surface type, from the Roadway Asset Management System
(RAMS).

Figure 3: Illustration of the concept of blending

With the initial classification and additional attributes from the asset
management system, the event then moves to the regression module. The
regression module runs a regression model that is fine-tuned to predict the
exact duration of the event based on the subset to which the data point
belongs. Each model is trained on the training data, tested on holdout or test
data, and then retrained on both the train and holdout data before saving the
model.

To develop a model with the best efficiency, the solution employs a process
called blending, which combines the strengths of multiple models to create a
single model with hybrid features.

In the classification module, it was found that the Random Forest model, the
Extra Tree Classifier model, and the Light GBM model achieved the lowest mean
absolute error (MAE). The predictions of these three models are combined by a
meta-learner to give a blended final prediction, as shown in Figure 3.
Similarly, for the regression module, the predictions from the combinations of
Random Forest model in combination with Catboost model are fed into meta-
learner for short-duration prediction. Predictions from Random Forest model
and Huber model are fed into meta-learner for medium-duration prediction. For
long-duration event prediction, XG Boost model alone is used since it
performed better than blended models. A Tobit model is used as a reference to
evaluate the MAE value and determine how the methodology is improving the
results.

###  5.1 Model Evaluation Metrics

One commonly used performance metric for evaluating classification models is
the Area Under the Curve (AUC) of the Receiver Operating Characteristics (ROC)
plot. The ROC curve plots the true positive rate against the false positive
rate at different classification thresholds, and the AUC is calculated by
finding the area under the ROC curve. A higher AUC value indicates that the
model is better at distinguishing between the classes. In the problem at hand,
accurate classification of an event into the correct duration category is
crucial because it ensures that the appropriate regression model, which has
been specifically trained for that category, is applied.

$TPR=\frac{TP}{FP+TN}$

$FPR=\frac{FP}{FP+TN}$

$Precision=\frac{TP}{TP+FP}$

$Recall=\frac{TP}{TP+FN}$

Additionally, precision, accuracy, and recall are useful parameters for
evaluating model performance. While accuracy indicates the performance of the
model in predicting the correct class or category, precision indicates the
ratio of what the model predicted correctly to what it predicted overall, and
recall indicates what the model predicted correctly to what the true
classifications are. Along with high AUC values, the best possible value for
recall is also desired because the model needs to be sensitive toward the
higher severity incidents that are associated with longer durations. In the
case of regression models, commonly used evaluation metrics include Mean
Absolute Error (MAE) and Root Mean Squared Error (RMSE).

$MAE=(\frac{1}{n})\sum_{i=1}^{n}\left|t_{pi}-t_{oi}\right|$

###  5.2 Model description

The best-performing machine learning model for the classification and
regression modules was selected based on the highest AUC value for the
classification module and the lowest mean absolute error (MAE) for the
regression module. Tree-based models generally performed better. To further
improve model performance, an ensemble technique called blending was applied.
The following section provides a general overview of the top-performing models
used in the classification and regression modules.

####  5.2.1 Random Forest classifier and regressor

Random Forest is an ensemble method that uses bootstrap aggregation, also
known as bagging, to train multiple decision trees on different samples of the
data with replacement. The final output is obtained by combining the
predictions of these decision trees. Two common hyperparameters used to tune
and improve the performance of a Random Forest model are max_depth and
n_estimators. Max_depth determines the maximum depth of each tree, while
n_estimators specifies the number of trees in the forest. Adjusting these
hyperparameters can help optimize the model’s performance.

####  5.2.2 CatBoost classifier and regressor

The CatBoost algorithm is designed to perform well on categorical data and has
several advantages. It can produce high-quality results without the need for
extensive parameter tuning, it provides fast and improved predictions by
reducing overfitting, and it can handle datasets with less data. This model is
typically effective on heterogenous data, which is characterized by low
quality, high variability of data types, ambiguous data, and data with high
redundancy. In our case, the incident duration dataset can be considered
heterogenous, and it has been found that the CatBoost model performs well on
this type of data.

####  5.2.3 Light Gradient Boost classifier and regressor

Light GBM is a gradient boosting machine learning framework based on tree-
based algorithms. It grows the tree leaf-wise, which is different from other
tree-based algorithms that grow level-wise. Light GBM has the advantages of
high speed and the ability to handle large datasets with lower memory usage
and it focuses on the accuracy of the results.

####  5.2.4 Blending

Blending is an ensemble technique that uses a linear regression or logistic
regression model as a meta model to blend or combine the predictions of
multiple machine learning models to estimate more accurate predictions. The
concept of blending has been demonstrated in Figure 3 where the base ML models
RF, ET and Light GBM are individually trained on the training data and make
individual predictions on the holdout or test dataset. The meta model, which
is usually a logistic regression model, is trained on these predictions to
make a single final prediction.

####  5.2.5 Huber regressor

Traditional regression models can produce misleading results if the underlying
assumptions are not met. Robust regression methods are designed to minimize
the impact of these violations. One commonly used method for robust regression
is the Huber loss, which is less sensitive to outliers than the squared error
loss. This makes it a useful tool for minimizing the effect of outliers on the
regression analysis. The Huber loss function can be defined as below:

$L_{\delta}(y,f(x))=\begin{cases}\frac{1}{2}{a^{2}},&\text{for
|y-f(x)|}\leq\delta\\\
\delta(|y-f(x)|-\frac{1}{2}\delta),&\text{otherwise}\end{cases}$

Figure 4: Comparison of AUC (basic feature set(left), all features(right)
(validation dataset)

####  5.2.6 Tobit model

Tobit model is a class of regression models developed in 1958 with the
intention of mitigating the problem truncated data. The Tobit model can also
be considered a special case of a censored linear regression model that can be
represented by the equation below:

$\begin{cases}y_{i}^{*}\quad if\quad y_{L}<y_{i}^{*}<y_{U},\\\ y_{L}\quad
if\quad y_{i}^{*}<y_{L},\\\ y_{U}\quad if\quad y_{i}^{*}>=y_{U}\\\
\end{cases}$

Here, $y_{i}^{*}$ is a latent variable that is not always observable and,
$y_{L}$ and $y_{U}$ are the lower and upper limits to which the model is
censored.

##  6 Results

During the training phase of the analysis, several models, including Random
Forest (RF), CatBoost Regressor, Gradient Boost, Ada Boost, and XGBoost, were
trained and tested on the short, medium, and long duration datasets, as well
as other base models such as SVM, Decision Tree, and KNN. The models were then
used to make predictions on the test dataset and were ranked in descending
order of AUC values for classification models and ascending order of MAE
values for regression models. The classification model(s) with the highest AUC
values and the regression model(s) with the lowest MAE values were identified
as the best models. Additionally, blending was performed on the top 2 (BM2),
top 3 (BM3), top 4 (BM4), and top 5 (BM5) best machine learning models to
create four new models. These models were also evaluated in the same way and
compared against the individual models to evaluate the improvement resulting
from blending.

This remainder of this section discusses the training, testing and validation
results provided in Tables 6, 7 and 8.

The results were obtained through the process of deciding the ML components of
the developed framework under the availability of both the feature sets
explained in Table 5. The basic feature set includes information collected
from the site of the incident by the reporting policeman. This includes
information about the time and location of crash, type of crash, vehicle
information, involvement of trucks in the incident and the number of injured
occupants and so on. This information can be used to make the first
prediction. In the next phase, more information relating to the incident is
obtained like the AADT, hourly volume, road and environmental factors related
to the location of the incident from RAMS, and whether a responder arrived or
not at the incident site. This additional information along with the basic
features (full features) helps make a more refined prediction of the time
required to clear the incident.

###  6.1 Classification Module Results

In the classification module, a blended model obtained from combining Random
Forest model with an Extra Tree model and Light GBM yields the best AUC of
81.9% for the basic features and an AUC of 86% when the full features are
available as shown in Table 6. The corresponding AUC graphs can be found in
Figure 4. These models are trained, saved and used to predict on the
validation dataset. The results of the prediction are shown in Figure 5 with
S, M and L indicating short, medium and long duration events. It is evident
from comparing the results in Table 6 that the classification accuracy of
short duration incidents improved by 7.14% and medium duration events improved
by 0.97% when additional features are available beyond the basic features.

Figure 5: Confusion matrices (basic feature set(left), all features(right) (validation dataset) with Short (S), Medium (M), and Long (L) classifications. Observed classifications are on the x-axis and predicted classifications are on the y-axis Table 6: Results from classification module (based on FS1 and FS2) Data |  Features |  Best Model |  AUC |  Precision |  Accuracy |  Recall
---|---|---|---|---|---|---
Train |  Basic |  RF+ET+LGBM |  0.81 |  0.683 |  0.684 |  0.6026
Test |  Basic |  RF+ET+LGBM |  0.819 |  0.702 |  0.7 |  0.611
Train |  All |  RF+ET+LGBM |  0.87 |  0.76 |  0.753 |  0.662
Test |  All |  RF+ET+LGBM |  0.86 |  0.739 |  0.737 |  0.661

###  6.2 Regression Module Results

Table 7: Results from regression module (based on basic features) Duration |  Model |  MAE(test) |  MAE(valid)
---|---|---|---
Short |  RF |  5.66 |  24.58
Medium |  RF |  14.07 |  42.27
Long |  RF+XGBoost |  41.9 |  129.1
Table 8: Results from regression module (based on all features) Duration |  Model |  MAE(test) |  MAE(valid)
---|---|---|---
Short |  RF+Catboost |  5.76 |  22.11
Medium |  RF+Huber |  15.73 |  47.3
Long |  XGBoost |  33.27 |  96.09

UnsupSup_MCTobit_MC$0$$50$$100$$150$MAEShortMediumLong

UnsupSup_MCTobit_MC$0$$50$$100$$150$MAEShortMediumLong

Figure 6: Comparison of MAE values (basic feature set(left), all
features(right) (test dataset)

UnsupSup_MCTobit_MC$50$$100$$150$MAEShortMediumLong

UnsupSup_MCTobit_MC$0$$50$$100$$150$MAEShortMediumLong

Figure 7: Comparison of MAE values (basic feature set(left), all
features(right) (validation dataset)

From Table 7, it was found that for short and medium duration incidents, the
RF model had the best performance, while a blend model consisting of the RF
model and XGBoost performed the best in the case of long duration incidents.
The best MAEs reported on the test dataset were 5.66, 14.07, and 41.9 minutes,
respectively. When all the variables were considered, the MAE values improved
to 5.76, 15.73, and 33.27 minutes, respectively. When the performance of the
models on the validation dataset was evaluated, it was found that the MAE for
the validation dataset was significantly higher than the MAE of the test
dataset. This deviation from the expected values could be due to the large
variance in the data. When additional features were available, an improvement
in MAE values was observed for long duration incidents, as shown in Table 8.
In addition to blending, hyperparameter tuning of the best model, as reported
by the pycaret package, was also performed using the Optuna library. However,
hypertuning did not help improve the performance of the model in any of the
cases considered.

ShortMediumLong$0$$50$$100$$150$Incident DurationMAE (Test)
(min)Without_classWith_class (a) MAE (test) for developed framework

ShortMediumLong$0$$50$$100$$150$Incident DurationMAE
(Validation)(min)Without_classWith_class (b) MAE (validation) for developed
framework

ShortMediumLong$0$$50$$100$$150$Incident DurationMAE
(Test)(min)Without_classWith_class (c) MAE (test) for Tobit model

ShortMediumLong$50$$100$$150$Incident DurationMAE
(Validation)(min)Without_classWith_class (d) MAE (validation) for Tobit model

Figure 8: Comparison of MAE values corresponding to test and validation
datasets for framework model and Tobit model with and without prior
classification when basic feature set available

ShortMediumLong$0$$50$$100$$150$Incident DurationMAE
(Test)(min)Without_classWith_class (a) MAE (test) for developed framework

ShortMediumLong$0$$50$$100$$150$Incident DurationMAE
(Validation)(min)Without_classWith_class (b) MAE (validation) for developed
framework

ShortMediumLong$0$$50$$100$$150$Incident DurationMAE
(Test)(min)Without_classWith_class (c) MAE (test) for Tobit model

ShortMediumLong$50$$100$$150$Incident DurationMAE
(Validation)(min)Without_classWith_class (d) MAE (validation) for Tobit model

Figure 9: Comparison of MAE values corresponding to test and validation
datasets for framework model and Tobit model with and without prior
classification when full feature set is available

###  6.3 Evaluation of Classification Module in Developed Framework

To understand the usefulness of the classification module in the developed
framework, a Random Forest model was used to predict incident duration without
performing any prior classification (indicated as Without_class). The mean
absolute error (MAE) values obtained from the test, and validation datasets
were compared in Figure 8 and Figure 9. It was observed that for the basic
feature set, including the classification step prior to regression helped to
reduce the MAE of the model for short, medium, and long-duration incidents by
56%, 39.76%, and 57.131%, respectively, as evident in the test data results
shown in Figure 8(a). When all the features were considered, prior
classification helped to reduce MAE by 45.2, 23.1, and 63.2% for the
respective duration categories, as shown in Figure 9(a). However, when
considering the validation dataset, the overall MAE was found to improve only
for the long-duration dataset when all the features were considered together,
as shown in Figures 8(b) and 9(b).

The same analysis was applied to the Tobit model, and it was found that the
pre-classification model improved the MAE values for both the basic and all
feature sets in both the test and validation datasets, as shown in Figures
8(c), 8(d), 9(c), 9(d) with the exception of two cases.

###  6.4 Comparison of our framework with unsupervised approach and Tobit
model

For the unsupervised approach, K-means clustering model was applied instead of
supervised classification. The optimal number of clusters for both the basic
feature set and the feature set involving all the features was found to be 4.
The optimal number of clusters was chosen based on the elbow fit method and
the Silhouette Factor. The Silhouette coefficient obtained for both the cases
have been presented in Table 9. A higher value closer to 1 obtained when more
features are available gives better distinction between clusters.

Table 9: Comparison of Silhouette factors and optimal number of clusters Feature Set |  Number of clusters |  Silhouette factor
---|---|---
Basic |  4 |  0.07
All |  4 |  0.81

The MAE values obtained from the supervised and unsupervised approaches for
the test and validation datasets have been plotted in Figures 6 and 7.
"Unsup", "Sup_MC", and "Tobit_MC" stand for the MAE in the unsupervised
approach, the supervised approach with misclassification error, and the Tobit
model with misclassification error, respectively. In all three cases for the
test dataset, the MAE values obtained for our framework are lower than those
for the unsupervised and Tobit approaches. As the number of available features
increases, the classification error decreases, resulting in a lower MAE value.
This effect is more pronounced in the case of long duration incidents.
However, for the validation dataset, our framework performs slightly worse
than the unsupervised approach and Tobit model. This is more noticeable in the
case of medium duration incidents (30 minutes to 2 hours).

##  7 Discussion

In this section, the results of the classification and regression models that
were developed and tested in the previous section are discussed. The
motivation for choosing certain models or combinations of models over others
is also explained.

A variety of models were tested for the classification module and ranked based
on the highest AUC for both the basic and full feature sets. Ultimately, the
model that combines a random forest, an extra tree, and a light gradient
boosting machine had the highest AUC and was selected for the classification
module. For the regression module, models were selected based on the lowest
MAE, which varied based on the initiation duration classification from the
classification module and the basic or full feature set. When only basic
features are available, Table 7 shows the selected models with the lowest MAE
for the short, medium, and long classifications. When the full feature set is
available, the best regression models for the short, medium, and long
durations are shown in Table 8. The combination of the classification and
regression modules combine the best models for estimating the incident
duration based on the pre-trained dataset.

For validating the results against other models, the performance of the
developed supervised framework was compared to two additional frameworks that
included an unsupervised clustering with regression and a Tobit model. The
mean absolute error (MAE) was used as the evaluation metric for comparison.
The results of the test data showed that the supervised framework performs
better than the unsupervised approach and the Tobit model. With the validation
dataset, the performance of the supervised framework did not match the
improved performance seen on the test dataset. However, the Tobit model with
the pre-classification module showed improved efficiency on both the test and
validation datasets. The Tobit model’s performance was similar to that of the
supervised framework only when the pre-classification module was present.

It was also observed that for short and medium-duration data, all three
approaches produced similar MAE values. However, the pre-classification module
improved the results of long-duration events more than the other two
categories.

Overall, the Tobit model’s results were similar to those of the supervised
network’s regression module, and it produced improved results on both the test
and validation datasets. Therefore, there may be value in considering a simple
Tobit model after the classification module.

The performance of the models in the classification and regression modules was
evaluated using the area under the curve (AUC) and mean absolute error (MAE),
respectively. The highest AUC achieved was 86% when additional variables, such
as road characteristics and responder information, were included. In terms of
MAE, the lowest values achieved were 5.66 minutes for short-duration events,
14.07 minutes for medium-duration events, and an average of 41.9 minutes for
long-duration events on the test dataset when using only basic features. When
additional features were available, the MAE increased to 5.76 and 15.73 for
short and medium incidents, respectively, but decreased to 33.27 minutes for
long-duration incidents.

To compare the results to those obtained in previous research and to
understand if the findings were consistent with those reported in the
literature, the mean absolute percentage error (MAPE) values were also
calculated for each of the models. The MAPE values achieved with the basic
dataset were 99.16% for short duration, 24.45% for medium duration, and 18.34%
for long duration. The inclusion of additional variables, such as road
characteristics and highway helper information, resulted in improved MAPE
values of 96.88% and 16.21% for short and long incidents, respectively.

In a study involving the dynamic prediction of incident duration using an
adaptive feature set, the MAPE values reported for short-duration events were
100.9% for incidents lasting 5-15 minutes and between 75% and 96% for
incidents lasting 16-35 minutes [13]. For medium-duration events lasting
36-200 minutes, a MAPE between 20% and 50% was reported. In comparison, the
methods used in this paper achieved MAPE values in the range of 16% to 26% for
events of a wide range of durations lasting from 30 minutes up to a day when
all variables were considered, including variables that are likely not
available at the initial stages of responding to incidents (included to ensure
comparability with previous research). The MAPE, MAE, and mean absolute
relative error (MARE) results reported by other researchers working in this
area have been provided in Table 10 for comparison.

Table 10: Comparison of our results with existing studies

Reference |  Results
---|---
Khattak et al. [35] |  MAPE 5-15min:329%, >120min:80%
Valenti et al. [36] |  MAPE ANN:44%,SVR:36%
Perera et al. [28] |  MAPE between 100% and 40%
Qing et al. [37] |  KNN : 59.2%,CART: 57.1% Quantile Reg :49.1%
Tang et al. [33] |  MAPE : 34.8%
Hamad et al. [38] |  MAE : 36.52 min
Park et al. [12] |  MAE : 0.18 to 0.29
Ma et al. [39] |  MARE : 16.44%(<15 min) , 33.13%(>15 min)
Banishree et al. [39] |  MAPE : 100.9%(5-15 min) , 75-96%(16-35 min),20-50%(>200 min),61% o 27.58%(overall)
Our Performance (basic features), MAPE |  Short:99.16%(< 30min), Medium:24.45%(30-120 min), Long:18.34%(>120 min)
Our Performance (all features), MAPE |  Short:96.88% ,Medium:26.4% ,Long:16.21%
Our Performance (basic features), MAE |  Short:5.66 min , Medium: 14.07 min, Long: 41.9 min
Our Performance (all features), MAE |  Short: 5.76 min , Medium: 15.73 min , Long: 33.27 min

##  8 Conclusions and Limitations

In this research, an ML-based classification and regression framework was
proposed for incident duration prediction. The developed framework is capable
of generating real-time predictions of incident duration based on the
information provided to the response agency in the initial incident call. It
additionally integrates other factors including road type, surface type, AADT,
hourly volume information, etc., that are not available directly from the
incident report enhancing the accuracy of prediction of the incident duration.
The predicted incident duration can be used for developing a simple priority-
based ranking system that helps traffic operators know which incidents to
prioritize and what resources will likely be required. The incident duration
information can also be combined with other information, such as the Level of
Service (LOS) information, to generate additional actionable insights.

The framework utilized in this research involved two modules: a classification
followed by regression; the classification to provide a quick estimation of
the incident duration using a supervised approach and the regression to
provide a more accurate prediction as shown in Figure 2.

The success of the incident duration prediction has a strong dependence on the
performance of the machine learning models. Among the various ML models
trained, tested, and validated, the Random Forest model exhibited high
efficiency and consistent performance across all the case scenarios tested.
The ensembling technique of blending models exhibiting the lowest MAE values
further improved the predictions.

Overall, the results indicate that the prediction of incident duration is more
accurate when there is a pre-classification module that helps determine the
class of incident duration. Though an unsupervised classification of events is
an option to consider, it is not performing consistently across the three
classes. The authors feel that the predictions benefit from the presence of
the supervised classification module. In the regression module, however, the
Tobit model performs almost on par with other single and blended regression
models, as discussed earlier.

One of the challenges faced in developing and implementing the analytical
framework was the skewness of the incident duration data. This was addressed
by implementing a box-cox transformation on the target variable. The range of
values of incident duration was large: therefore, applying a single model for
the whole dataset was not effective. This challenge was addressed by including
a classification module to assign incidents to 3 ranges - short (<30 minutes),
medium (30 minutes - 2 hours), and long(>2 hours). Categorical factors were
one hot encoded, and textual data (e.g., responder information) was extracted
and converted to categorical variables for predictive model development.

A limitation of this research is the use of data for a single state (Iowa).
While the results are validated for Iowa, other states may have different
trainings, and TIM resources and plans - leading to the results obtained in
this research potentially not being applicable to other locations. Thus,
future research should apply the framework and methods used in this paper to
additional datasets from other geographical areas. Additionally, future
research could evaluate the temporal transferability and stability of
prediction models developed using this framework and methods.

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