prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: What is the relationship between the maximum velocity and the amplitude of the blob or depletion?

Read the following text and answer briefly.

Context 1: \section{Model equations} \label{sec:equations}

In drift-fluid models the continuity equation
\begin{align}
 \frac{\partial n}{\partial t} + \nabla\cdot\left( n \vec u_E  \right) &= 0 \label{eq:generala} 
\end{align}
describes the dynamics of the electron density $n$  Here
$\vec u_E := (\hat{\vec b} \times \nabla \phi)/B$ gives the electric drift
velocity in a magnetic field $\vec B := B \hat{\vec b}$ and an electric
potential $\phi$  We neglect contributions of the diamagnetic drift~\cite{Kube2016} 




Equation~\eqref{eq:generala} is closed by invoking quasineutrality, i e  the divergence of the ion polarization, 
the electron diamagnetic and the gravitational drift currents must vanish
\begin{align}
  \nabla\cdot\left( \frac{n}{\Omega} \left( \frac{\partial}{\partial t} 
  + \vec u_E \cdot\nabla  \right)\frac{\nabla_\perp \phi}{B}  + n\vec u_d - n\vec u_g\right) &=0
    
 
 
  \label{eq:generalb}
\end{align}
Here we denote 
$\nabla_\perp\phi/B := - \hat{\vec b} \times \vec u_E$, 
the electron diamagnetic drift
$\vec u_d := - T_e(\hat{\vec b} \times\nabla n ) /enB$
with the electron temperature $T_e$,
the ion gravitational drift velocity  
$\vec u_g := m_i \hat{\vec b} \times \vec g /B$
with ion mass $m_i$, and the ion gyro-frequency
$\Omega := eB/m_i$ 

Combining Eq ~\eqref{eq:generalb} with Eq ~\eqref{eq:generala} yields
\begin{align}
 \frac{\partial \rho}{\partial t} + \nabla\cdot\left( \rho\vec u_E \right) + \nabla \cdot\left( n(\vec u_\psi + \vec u_d + \vec u_g) \right) &= 0\label{eq:vorticity}
\end{align}
with the polarization charge density 
$\rho = \nabla\cdot( n\nabla_\perp \phi / \Omega B)$ 
and
$\vec u_\psi := \hat{\vec b}\times \nabla\psi /B$ 
with 
$\psi:= m_i\vec u_E^2 /2e$ 
We exploit this form of Eq ~\eqref{eq:generalb} in our numerical simulations 

Equations~\eqref{eq:generala} and \eqref{eq:generalb} respectively \eqref{eq:vorticity} have several invariants 
First, in Eq ~\eqref{eq:generala} the relative particle number 
$M(t) := \int \mathrm{dA}\, (n-n_0)$ is conserved over time
$\d M(t)/\d t = 0$  
Furthermore, we integrate 
$( T_e(1+\ln n) -T_e \ln B)\partial_t n$
as well as
$-e\phi \partial_t\rho - (m_i\vec u_E^2/2+gm_ix - T_e\ln B)\partial_t n$ 
over the domain to get, disregarding boundary contributions,
\begin{align}
  \frac{\d}{\d t}\left[T_eS(t) + H(t) \right] = 0, \label{eq:energya}\\ 
    \frac{\d}{\d t} \left[ E(t) - G(t) - H(t)\right] =  0,
    \label{eq:energyb}
\end{align}
where we define 
the entropy
$S(t):=\int \mathrm{dA}\, [n\ln(n/n_0) - (n-n_0)]$,  
the kinetic energy 
$E(t):=m_i \int \mathrm{dA}\, n\vec u_E^2/2$ 
and the potential energies
$G(t) := m_i g\int \mathrm{dA}\, x(n-n_0)$
and
$H(t) := T_e\int \mathrm{dA}\, (n-n_0) \ln (B^{-1})$ 
Note that $n\ln( n/n_0) - n + n_0 \approx (n-n_0)^2/2$ for $|(n-n_0)/n_0| \ll 1$ and $S(t)$ thus reduces to the 
local entropy form in Reference~\cite{Kube2016}  

We now set up a gravitational field $\vec g = g\hat x$ and a constant homogeneous background
magnetic field $\vec B = B_0 \hat z$ in a Cartesian coordinate system 
Then the divergences of the electric and gravitational drift velocities $\nabla\cdot\vec u_E$ and $\nabla\cdot\vec u_g$
and the diamagnetic current $\nabla\cdot(n\vec u_d)$ vanish, which makes the 
flow incompressible  Furthermore, the magnetic potential energy vanishes $H(t) = 0$ 

In a second system we model the inhomogeneous magnetic field present in tokamaks as
$\vec B := B_0 (1+ x/R_0)^{-1}\hat z$ and neglect the gravitational drift $\vec u_g = 0$ 
Then, the potential energy $G(t) = 0$  
Note that 
$H(t) = m_i \ensuremath{C_\mathrm{s}}^2/R_0\int\mathrm{dA}\, x(n-n_0) +\mathcal O(R_0^{-2}) $
reduces to $G(t)$ with the effective gravity $g_\text{eff}:= \ensuremath{C_\mathrm{s}}^2/R_0$ with $\ensuremath{C_\mathrm{s}}^2 := T_e/m_i$  
For the rest of this letter we treat $g$ and $g_\text{eff}$ as well as $G(t)$ and $H(t)$ on the same footing 
The magnetic field inhomogeneity thus entails compressible flows, which is 
the only difference to the model describing dynamics in a homogeneous magnetic field introduced above  
Since both $S(t)\geq 0$ and $E(t)\geq 0$ we further derive from Eq ~\eqref{eq:energya} and Eq ~\eqref{eq:energyb} that the kinetic energy
is bounded by $E(t) \leq T_eS(t) + E(t) = T_e S(0)$; a feature absent from the gravitational system with 
incompressible flows, where $S(t) = S(0)$  

We now show that the invariants Eqs ~\eqref{eq:energya} and \eqref{eq:energyb} present restrictions on the velocity and
acceleration of plasma blobs  
First, we define the blobs' center of mass (COM) via $X(t):= \int\mathrm{dA}\, x(n-n_0)/M$ and 
its COM velocity as $V(t):=\d X(t)/\d t$  
The latter is proportional to the total radial particle flux~\cite{Garcia_Bian_Fundamensky_POP_2006, Held2016a} 
We assume
that $n>n_0$ and $(n-n_0)^2/2 \leq [ n\ln (n/n_0) - (n-n_0)]n $ to show for both systems 
\begin{align}
  (MV)^2 &= \left( \int \mathrm{dA}\, n{\phi_y}/{B} \right)^2
  = \left( \int \mathrm{dA}\, (n-n_0){\phi_y}/{B} \right)^2\nonumber\\
 
&\leq 2 \left( \int \mathrm{dA}\, \left[n\ln (n/n_0) -(n-n_0)\right]^{1/2}\sqrt{n}{\phi_y}/{B}\right)^2\nonumber\\
 
  &\leq 4 S(0) E(t)/m_i 
 
  \label{eq:inequality}
\end{align}
Here we use the Cauchy-Schwartz inequality and 
$\phi_y:=\partial\phi/\partial y$  
Note that although we derive the inequality Eq ~\eqref{eq:inequality} only for amplitudes $\triangle n >0$  we assume that the results also hold for depletions  This is justified by our numerical results later in this letter  
If we initialize our density field with a seeded blob of radius $\ell$ and amplitude $\triangle n$ as 
\begin{align}
  n(\vec x, 0) &= n_0 + \triangle n \exp\left( -\frac{\vec x^2}{2\ell^2} \right), \label{eq:inita}
 
 
\end{align}
and  
$\phi(\vec x, 0 ) = 0$,
we immediately have $M := M(0) = 2\pi \ell^2 \triangle n$, $E(0) = G(0) = 0$ and 
$S(0) = 2\pi \ell^2 f(\triangle n)$, where $f(\triangle n)$ captures the amplitude dependence of 
the integral for $S(0)$  

The acceleration for both incompressible and compressible flows can be estimated
by assuming a linear acceleration $V=A_0t$ and $X=A_0t^2/2$~\cite{Held2016a} and using 
$E(t) = G(t) = m_igMX(t)$ in Eq ~\eqref{eq:inequality}
\begin{align}
  \frac{A_0}{g} =  \mathcal Q\frac{2S(0)}{M} \approx \frac{\mathcal Q}{2} \frac{\triangle n }{n_0+2\triangle n/9} 
  \label{eq:acceleration}
\end{align}
Here, we use the Pad\'e approximation of order $(1/1)$ of $2S(0)/M $
and define a model parameter $\mathcal Q$ with $0<\mathcal Q\leq1$ to be determined by numerical simulations 
Note that the Pad\'e approximation is a better approximation than a simple 
truncated Taylor expansion especially for large relative amplitudes of order unity 
Eq ~\eqref{eq:acceleration} predicts that $A_0/g\sim \triangle n/n_0$ for small 
amplitudes $|\triangle n/n_0| < 1$ and $A_0 \sim g $ for very large amplitudes $\triangle n /n_0 \gg 1$, 
which confirms the predictions in~\cite{Pecseli2016} and reproduces the limits discussed in~\cite{Angus2014} 

As pointed out earlier for compressible flows Eq ~\eqref{eq:inequality} can be further estimated
\begin{align}
  (MV)^2  \leq 4 T_eS(0)^2/m_i  
  \label{}
\end{align}
We therefore have a restriction on the maximum COM velocity for compressible flows, which is absent for incompressible flows
\begin{align}
  \frac{\max |V|}{\ensuremath{C_\mathrm{s}}} = {\mathcal Q}\frac{2S(0)}{M} \approx \frac{\mathcal Q}{2} \frac{|\triangle n| }{n_0+2/9 \triangle n } \approx \frac{\mathcal Q}{2} \frac{|\triangle n|}{n_0} 
  \label{eq:linear}
\end{align}
For $|\triangle n /n_0|< 1$ Eq ~\eqref{eq:linear} reduces to the linear scaling derived in~\cite{Kube2016}  
Finally, a scale analysis of Eq ~\eqref{eq:vorticity} shows that~\cite{Ott1978, Garcia2005, Held2016a}
\begin{align}
  \frac{\max |V|}{\ensuremath{C_\mathrm{s}}} = \mathcal R \left( \frac{\ell}{R_0}\frac{|\triangle n|}{n_0} \right)^{1/2} 
  \label{eq:sqrt}
\end{align}
This equation predicts a square root dependence of the center of mass velocity 
on amplitude and size  





We now propose a simple phenomenological model that captures the essential dynamics
of blobs and depletions in the previously stated systems  More specifically 
the model reproduces the acceleration Eq ~\eqref{eq:acceleration} with and without
Boussinesq approximation, the square root scaling for the COM velocity 
Eq ~\eqref{eq:sqrt} for incompressible flows as well as the relation between the 
square root scaling Eq ~\eqref{eq:sqrt} and the linear scaling 
Eq ~\eqref{eq:linear} for compressible flows  
The basic idea is that the COM of blobs behaves like 
the one of an infinitely long plasma column immersed in an ambient plasma  
The dynamics of this column reduces to the one of a two-dimensional ball 
This idea is similar to the analytical ``top hat'' density solution for
blob dynamics recently studied in~\cite{Pecseli2016} 
The ball is subject to buoyancy as well as linear and nonlinear friction
\begin{align}
  M_{\text{i}} \frac{d V}{d t} = (M_{\text{g}} - M_\text{p}) g - c_1 V  - \mathrm{sgn}(V ) \frac{1}{2}c_2 V^2 
  \label{eq:ball}
\end{align}
The gravity $g$ has a positive sign in the coordinate system; sgn$(f)$ is the sign function  
The first term on the right hand side is the buoyancy, where 
$M_{\text{g}} := \pi \ell^2 (n_0 + \mathcal Q \triangle n/2)$ 
is the gravitational mass of the ball with radius $\ell$ and 
$M_\mathrm{p} := n_0 \pi \ell^2 $ 
is the mass of the displaced ambient plasma 
Note that if $\triangle n<0$ the ball represents a depletion and the buoyancy term has a negative sign, i e  the depletion will rise  
We introduce an inertial mass 
$M_{\text{i}} := \pi\ell^2 (n_0 +2\triangle n/9)$ 
different from the gravitational mass $M_{\text{g}}$ in order to 
recover the initial acceleration in Eq ~\eqref{eq:acceleration}  
We interpret the parameters $\mathcal Q$ and $2/9$ as geometrical factors 
that capture the difference of the actual blob form from the idealized
``top hat'' solution  
Also note that the Boussinesq approximation appears in the model as a neglect of inertia, $M_{\text{i}} = \pi\ell^2n_0$ 

The second term is the linear friction term with coefficient $c_1(\ell)$, which
depends on the size of the ball 
If we disregard the nonlinear friction, $c_2=0$, Eq ~\eqref{eq:ball} directly yields a 
maximum velocity $c_1V^*=\pi \ell^2 n g \mathcal Q\triangle n/2$ 
From our previous considerations $\max V/\ensuremath{C_\mathrm{s}}=\mathcal Q \triangle n /2n_0$, we thus identify 
\begin{align}
  c_1 = \pi\ell^2 n_0 g/\ensuremath{C_\mathrm{s}}  
  \label{}
\end{align}
The linear friction coefficient thus depends on the gravity and the size of the
ball  

The last term in \eqref{eq:ball} is the nonlinear friction  The sign of the force depends on whether
the ball rises or falls in the ambient plasma  
If we disregard linear friction $c_1=0$, we have the maximum velocity 
$V^*= \sigma(\triangle n)\sqrt{\pi \ell^2|\triangle n| g\mathcal Q/c_2}$, 
which must equal 
$\max V= \sigma(\triangle n) \mathcal R \sqrt{g \ell |\triangle n/n_0|}$ 
and thus
\begin{align}
  c_2 = {\mathcal Q\pi n_0\ell }/{\mathcal R^2} 
  \label{}
\end{align}
Inserting $c_1$ and $c_2$ into Eq ~\eqref{eq:ball}
we can derive the maximum absolute velocity in the form 
\begin{align}
  \frac{\max |V|}{\ensuremath{C_\mathrm{s}}} = 
        \left(\frac{\mathcal R^2}{\mathcal Q}\right) \frac{\ell}{R_0} \left( 
        \left({1+\left( \frac{\mathcal Q}{\mathcal R} \right)^{2} \frac{|\triangle n|/n_0 }{\ell/R_0}}\right)^{1/2}-1 \right)
  \label{eq:vmax_theo}
\end{align}
and thus have a concise expression for $\max |V|$ that captures both the linear
scaling \eqref{eq:linear} as well as the square root scaling \eqref{eq:sqrt} 
With Eq ~\eqref{eq:acceleration} and Eq ~\eqref{eq:sqrt} respectively Eq ~\eqref{eq:vmax_theo} we 
finally arrive at an analytical expression for the time at which the maximum velocity is reached via 
$t_{\max V} \sim \max V/A_0$  Its inverse $\gamma:=t_{\max V}^{-1}$ gives the
global interchange growth rate, for which an empirical expression was
presented in Reference~\cite{Held2016a} 


We use the open source library FELTOR 
to simulate 
Eqs ~\eqref{eq:generala} and \eqref{eq:vorticity} with and without 
drift compression 
For numerical stabilty we added small diffusive terms on the right hand 
sides of the equations 
The discontinuous Galerkin methods employ three polynomial coefficients and a minimum of $N_x=N_y=768$ grid cells  The box size is $50\ell$ in order to mitigate 
influences of the finite box size on the blob dynamics  
Moreover, we used the invariants in Eqs  \eqref{eq:energya} and \eqref{eq:energyb} as consistency tests to verify the code and repeated simulations 
also in a gyrofluid model  
No differences to the results presented here were found  
Initial perturbations on the particle density field are given by Eq ~\eqref{eq:inita},
where the perturbation amplitude $\triangle n/n_0$ was chosen between $10^{-3}$ and $20$ for blobs and $-10^0$ and $ -10^{-3}$ for depletions  
Due to computational reasons we show results only for $\triangle n/n_0\leq 20$  


For compressible flows we consider two different cases $\ell/R_0 = 10^{-2}$ and
$\ell /R_0 = 10^{-3}$  
 For incompressible flows Eq ~\eqref{eq:generala} and \eqref{eq:vorticity}
 can be normalized such that the blob radius is absent from the equations~\cite{Ott1978, Kube2012}  
 The simulations of incompressible flows can thus be used for both sizes  
The numerical code as well as input parameters and output data can be found 
in the supplemental dataset to this contribution~\cite{Data2017} 

\begin{figure}[htb]
    \includegraphics[width=\columnwidth]{com_blobs}
    \caption{
      The maximum radial COM velocities of blobs for compressible and incompressible flows are shown  
      The continuous lines show Eq ~\eqref{eq:vmax_theo} while the 
      dashed line shows the square root scaling Eq ~\eqref{eq:sqrt} with 
      $\mathcal Q = 0 32$ and $\mathcal R=0 85$ 
    }
    \label{fig:com_blobs}
\end{figure}
In Fig ~\ref{fig:com_blobs} we plot the maximum COM velocity for blobs 
with and without drift compression 
For incompressible flows blobs follow the square root scaling almost 
perfectly  Only at very large amplitudes velocities are slightly below
the predicted values  
For small amplitudes we observe that the compressible blobs follow
a linear scaling  When the amplitudes increase there is a transition to the
square root scaling at around $\triangle n/n_0 \simeq 0 5$ for 
$\ell/R_0=10^{-2}$ and $\triangle n/n_0 \simeq 0 05$ for $\ell/R_0=10^{-3}$, which is consistent with Eq ~\eqref{eq:vmax_theo} and Reference~\cite{Kube2016}  
In the transition regions the simulated velocities are slightly larger than the predicted ones from Eq ~\eqref{eq:vmax_theo} 
Beyond these amplitudes
the velocities of compressible and incompressible blobs align  

\begin{figure}[htb]
    \includegraphics[width=\columnwidth]{com_holes}
    \caption{
      The maximum radial COM velocities of depletions for compressible and incompressible flows are shown  
      The continuous lines show Eq ~\eqref{eq:vmax_theo} while the 
      dashed line shows the square root scaling Eq ~\eqref{eq:sqrt} with 
      $\mathcal Q = 0 32$ and $\mathcal R=0 85$ 
      Note that small amplitudes are on the right and amplitudes close to unity are on the left side 
  }
    \label{fig:com_depletions}
\end{figure}
In Fig ~\ref{fig:com_depletions} we show the maximum radial COM velocity 
for depletions instead of blobs 
For relative amplitudes below $|\triangle n|/n_0 \simeq 0 5$ (right of unity in the plot) the velocities
coincide with the corresponding blob velocities in Fig ~\ref{fig:com_blobs}  
 For amplitudes larger than $|\triangle n|/n_0\simeq 0 5$ the 
velocities follow the square root scaling 
We observe that for plasma depletions beyond $90$ percent the velocities 
in both systems reach a constant value that is very well predicted by the
square root scaling  

\begin{figure}[htb]
    \includegraphics[width=\columnwidth]{acc_blobs}
    \caption{
      Average acceleration of blobs for compressible and incompressible flows are shown 
      The continuous line shows the acceleration in Eq ~\eqref{eq:acceleration} 
      with $\mathcal Q=0 32$
      while the dashed line is a linear reference line, which corresponds to the Boussinesq approximation  
  }
    \label{fig:acc_blobs}
\end{figure}
In Fig ~\ref{fig:acc_blobs} we show the average acceleration of blobs 
for compressible and incompressible flows computed
by dividing the maximum velocity $\max V$ by the time  
to reach this velocity $t_{\max V}$  
We compare the simulation results
to the theoretical predictions Eq ~\eqref{eq:acceleration} of our model with and without inertia  
The results of the compressible and incompressible systems coincide and fit very
well to our theoretical values  
For amplitudes larger than unity the acceleration deviates significantly from the prediction with Boussinesq approximation 

\begin{figure}[htb]
    \includegraphics[width=\columnwidth]{acc_holes}
    \caption{
      Average acceleration of depletions for compressible and incompressible flows are shown 
      The continuous line shows the acceleration in Eq ~\eqref{eq:acceleration} 
      with $\mathcal Q=0 32$
      while the dashed line is a linear reference line, which corresponds to the Boussinesq approximation  
    }
    \label{fig:acc_depletions}
\end{figure}
In Fig ~\ref{fig:acc_depletions} we show the simulated acceleration of depletions in the
compressible and the incompressible systems  We compare the simulation results
to the theoretical predictions Eq ~\eqref{eq:acceleration} of our model with and without inertia 
Deviations from our theoretical prediction Eq ~\eqref{eq:acceleration} are visible for amplitudes smaller than $\triangle n/n_0 \simeq -0 5$ (left of unity in the plot)  The relative deviations are small at around $20$ percent  
As in Fig ~\ref{fig:com_depletions} the acceleration reaches a constant values
for plasma depletions of more than $90$ percent 
Comparing Fig ~\ref{fig:acc_depletions} to Fig ~\ref{fig:acc_blobs} the asymmetry between blobs and depletions becomes 
apparent  While the acceleration of blobs is reduced for large 
amplitudes compared to a linear dependence the acceleration 
of depletions is increased  In the language of our simple buoyancy 
model the inertia of depletions is reduced but increased for blobs  



In conclusion  
  we discuss the dynamics of seeded blobs and depletions in a 
  compressible and an incompressible system 
  With only two fit parameters our theoretical results reproduce the 
  numerical COM velocities and accelerations over five orders of magnitude 
  We derive the amplitude dependence of the acceleration of blobs and depletions from 
  the conservation laws of our systems in Eq ~\eqref{eq:acceleration}  
  From the same inequality a linear regime is derived in the compressible system for 
  ratios of amplitudes to sizes smaller than a critical value 
   In this regime 
  the blob and depletion velocity depends linearly on the initial amplitude and 
  is independent of size  The regime is absent from the system with incompressible flows 
  Our theoretical results are verified by numerical simulations for all 
  amplitudes that are relevant in magnetic fusion devices 
  Finally, we suggest a new empirical blob model that captures the detailed dynamics of more complicated models  
  The Boussinesq approximation is clarified as the absence of inertia and a thus altered acceleration of blobs and depletions 
  The maximum blob velocity is not altered by the Boussinesq approximation 

The authors were supported with financial subvention from the Research Council of Norway under grant
240510/F20  M W  and M H  were supported by the Austrian Science Fund (FWF) Y398   The computational
results presented have been achieved in part using the Vienna Scientific Cluster (VSC)  Part of this work was performed on the Abel Cluster, owned by the University of Oslo and the Norwegian metacenter
for High Performance Computing (NOTUR), and operated by the Department for Research Computing at USIT,
the University of Oslo IT-department 
This work has been carried out within the framework of the EUROfusion Consortium and has received funding from the Euratom research and training programme 2014-2018 under grant agreement No 633053  The views and opinions expressed herein do not necessarily reflect those of the European Commission 
Context 2: Hugh Hilton Goodwin (December 21, 1900 – February 25, 1980) was a decorated officer in the United States Navy with the rank of Vice Admiral  A veteran of both World Wars, he commanded escort carrier  during the Mariana Islands campaign  Goodwin then served consecutively as Chief of Staff, Carrier Strike Group 6 and as Air Officer, Philippine Sea Frontier and participated in the Philippines campaign in the later part of the War 

Following the War, he remained in the Navy and rose to the flag rank and held several important commands including Vice Commander, Military Air Transport Service, Commander, Carrier Division Two and Commander, Naval Air Forces, Continental Air Defense Command 

Early life and career

Hugh H  Goodwin was born on December 21, 1900, in Monroe, Louisiana and attended Monroe High School there (now Neville High School)  Following the United States' entry into World War I in April 1917, Goodwin left the school without receiving the diploma in order to see some combat and enlisted the United States Navy on May 7, 1917  He completed basic training and was assigned to the battleship   Goodwin participated in the training of armed guard crews and engine room personnel as the Atlantic Fleet prepared to go to war and in November 1917, he sailed with the rest of Battleship Division 9, bound for Britain to reinforce the Grand Fleet in the North Sea 

Although he did not complete the last year of high school, Goodwin was able to earn an appointment to the United States Naval Academy at Annapolis, Maryland in June 1918  While at the academy, he earned a nickname "Huge" and among his classmates were several future admirals and generals including: Hyman G  Rickover, Milton E  Miles, Robert E  Blick Jr , Herbert S  Duckworth, Clayton C  Jerome, James P  Riseley, James A  Stuart, Frank Peak Akers, Sherman Clark, Raymond P  Coffman, Delbert S  Cornwell, Frederick J  Eckhoff, Ralph B  DeWitt, John Higgins, Vernon Huber, Albert K  Morehouse, Harold F  Pullen, Michael J  Malanaphy, William S  Parsons, Harold R  Stevens, John P  Whitney, Lyman G  Miller and George J  O'Shea 

Goodwin graduated with Bachelor of Science degree on June 3, 1922, and was commissioned Ensign in the United States Navy  He was subsequently assigned to the battleship  and took part in the voyage to Rio de Janeiro, Brazil, before he was ordered to the Naval Torpedo Station at Newport, Rhode Island for submarine instruction in June 1923  Goodwin completed the training several weeks later and was attached to the submarine   He then continued his further training aboard submarine  and following his promotion to Lieutenant (junior grade) on June 3, 1925, he qualified as submariner 

He then served aboard submarine  off the coast of California, before he was ordered for the recruiting duty to San Francisco in September 1927  While in this capacity, Goodwin applied for naval aviation training which was ultimately approved and he was ordered to the Naval Air Station Pensacola, Florida in August 1928  Toward the end of the training, he was promoted to lieutenant on December 11, 1928, and upon the completion of the training in January 1929, he was designated Naval aviator 

Goodwin was subsequently attached to the Observation Squadron aboard the aircraft carrier  and participated in the Fleet exercises in the Caribbean  He was transferred to the Bureau of Aeronautics in Washington, D C  in August 1931 and served consecutively under the architect of naval aviation William A  Moffett and future Chief of Naval Operations Ernest J  King 

In June 1933, Goodwin was ordered to the Naval War College at Newport, Rhode Island, where he completed junior course in May of the following year  He subsequently joined the crew of aircraft carrier  and served under Captain Arthur B  Cook and took part in the Fleet exercises in the Caribbean and off the East Coast of the United States 

He was ordered back to the Naval Air Station Pensacola, Florida in June 1936 and was attached to the staff of the Base Commandant, then-Captain Charles A  Blakely  When Blakely was succeeded by William F  Halsey in June 1937, Goodwin remained in Halsey's staff and was promoted to Lieutenant Commander on December 1, 1937  He also completed correspondence course in International law at the Naval War College 

Goodwin was appointed Commanding officer of the Observation Squadron 1 in June 1938 and attached to the battleship  he took part in the patrolling of the Pacific and 
West Coast of the United States until September 1938, when he assumed command of the Observation Squadron 2 attached to the battleship  

When his old superior from Lexington, now Rear Admiral Arthur B  Cook, was appointed Commander Aircraft, Scouting Force in June 1939, he requested Goodwin as his Aide and Flag Secretary  He became Admiral Cook's protégé and after year and half of service in the Pacific, he continued as his Aide and Flag Secretary, when Cook was appointed Commander Aircraft, Atlantic Fleet in November 1940 

World War II

Following the United States' entry into World War II, Goodwin was promoted to the temporary rank of Commander on January 1, 1942, and assumed duty as advisor to the Argentine Navy  His promotion was made permanent two months later and he returned to the United States in early 1943 for duty as assistant director of Planning in the Bureau of Aeronautics under Rear admiral John S  McCain  While still in Argentina, Goodwin was promoted to the temporary rank of Captain on June 21, 1942 

By the end of December 1943, Goodwin was ordered to Astoria, Oregon, where he assumed command of newly commissioned escort carrier USS Gambier Bay  He was responsible for the initial training of the crew and was known as a strict disciplinarian, but the crew appreciated the skills he taught them that prepared them for combat  Goodwin insisted that everyone aboard has to do every job right every time and made us fight our ship at her best 

During the first half of 1944, Gambier Bay was tasked with ferrying aircraft for repairs and qualified carrier pilots from San Diego to Pearl Harbor, Hawaii, before departed on May 1, 1944, to join Rear admiral Harold B
Context 3: \section{Introduction}
In recent years, vehicular technology has attracted significant attention from the automotive and telecommunication industries, leading to the emergence of vehicle-to-everything (V2X) communications for improving road safety, traffic management services and driving comfort 
V2X supported by the sixth generation (6G) is envisioned to be a key enabler of future connected autonomous vehicles \cite{9779322}  Although its transformative benefits for leveraging intelligent transportation systems, V2X still face several technical issues mainly related to performance and security 

The integration of sensing and communication (ISAC) has emerged very recently as a revolutionary element of 6G that could potentially help enabling adaptive learning and intelligent decision-making in future V2X applications 
The combination of sensing and communication allows vehicles to perceive their surroundings better, predict manoeuvres from nearby users and make intelligent decisions, thus paving the way toward a safer transportation system \cite{9665433} 
Modernized vehicles are augmented with various types of sensors divided into exteroceptive to observe their surrounding environment and proprioceptive to observe their internal states 
The former like GPS, Lidar, and Cameras are conveyed to improve situational awareness, while latter sensors, such as steering, pedal, and wheel speed, convey to improve self-awareness  

While sensing the environment, vehicles can exchange messages that assist in improving situational- and self-awareness and in coordinating maneuvers with other vehicles 
Those messages like the basic safety (BSMs) and cooperative awareness messages (CAMs) are composed of transmitting vehicle's states such as position and velocity and other vehicles' states in the vicinity  Vehicles might use their sensors, such as cameras and Lidar, to detect road users (e g , pedestrians), which can be communicated with other road users via the V2X messages to improve the overall performance  However, V2X communication links carrying those messages are inherently vulnerable to malicious attacks due to the open and shared nature of the wireless spectrum among vehicles and other cellular users \cite{8336901}  For instance, a jammer in the vicinity might alter the information to be communicated to nearby vehicles/users or can intentionally disrupt communication between a platoon of vehicles making the legitimate signals unrecognizable for on-board units (OBUs) and/or road side units (RSUs) that endanger vehicular safety 
\cite{8553649} 

In addition, the integrity of GPS signals and the correct acquisition of navigation data to compute position, velocity and time information is critical in V2X applications for their safe operation  However, since civil GPS receivers rely on unencrypted satellite signals, spoofers can easily replicate them by deceiving the GPS receiver to compute falsified positions \cite{9226611} 
Also, the long distance between satellites and terrestrial GPS receivers leads to an extremely weak signal that can be easily drowned out by a spoofer  
Thus, GPS sensors' vulnerability to spoofing attacks poses a severe threat that might be causing vehicles to be out of control or even hijacked and endanger human life \cite{9881548} 
Therefore, GPS spoofing attacks and jamming interference needs to be controlled and detected in real-time to reach secured vehicular communications allowing vehicles to securely talk to each other and interact with the infrastructure (e g , roadside terminals, base stations) \cite{9860410} 

Existing methods for GPS spoofing detection include GPS signal analysis methods and GPS message encryption methods \cite{9845684}  However, the former requires the ground truth source during the detection process, which is not always possible to collect  In contrast, the latter involves support from a secured infrastructure and advanced computing resources on GPS receivers, which hinders their adoption in V2X applications  On the other hand, existing methods for jammer detection in vehicular networks are based on analysing the packet drop rate as in \cite{9484071}, making it difficult to detect an advanced jammer manipulating the legitimate signal instead of disrupting it 
In this work, we propose a method to jointly detect GPS spoofing and jamming attacks in the V2X network  A coupled generalized dynamic Bayesian network (C-GDBN) is employed to learn the interaction between RF signals received by the RSU from multiple vehicles and their corresponding trajectories  This integration of vehicles' positional information with vehicle-to-infrastructure (V2I) communications allows semantic learning while mapping RF signals with vehicles' trajectories and enables the RSU to jointly predict the RF signals it expects to receive from the vehicles from which it can anticipate the expected trajectories 

The main contributions of this paper can be summarized as follows: \textit{i)} A joint GPS spoofing and jamming detection method is proposed for the V2X scenario, which is based on learning a generative interactive model as the C-GDBN  Such a model encodes the cross-correlation between the RF signals transmitted by multiple vehicles and their trajectories, where their semantic meaning is coupled stochastically at a high abstraction level  \textit{ii)} A cognitive RSU equipped with the acquired C-GDBN can predict and estimate vehicle positions based on real-time RF signals  This allows RSU to evaluate whether both RF signals and vehicles' trajectories are evolving according to the dynamic rules encoded in the C-GDBN and, consequently, to identify the cause (i e , a jammer attacking the V2I or a spoofer attacking the satellite link) of the abnormal behaviour that occurred in the V2X environment  \textit{iii)} Extensive simulation results demonstrate that the proposed method accurately estimates the vehicles' trajectories from the predicted RF signals, effectively detect any abnormal behaviour and identify the type of abnormality occurring with high detection probabilities 
To our best knowledge, this is the first work that studies the joint detection of jamming and spoofing in V2X systems 

\section{System model and problem formulation}
The system model depicted in Fig ~\ref{fig_SystemModel}, includes a single cell vehicular network consisting of a road side unit (RSU) located at $\mathrm{p}_{R}=[{x}_{R},{y}_{R}]$, a road side jammer (RSJ) located at $\mathrm{p}_{J}=[{x}_{J},{y}_{J}]$, a road side spoofer (RSS) located at $\mathrm{p}_{s}=[{x}_{s},{y}_{s}]$ and $N$ vehicles moving along multi-lane road in an urban area  The time-varying positions of the $n$-th vehicle is given by $\mathrm{p}_{n,t}=[{x}_{n,t},{y}_{n,t}]$ where $n \in N$  Among the $K$ orthogonal subchannels available for the Vehicle-to-Infrastructure (V2I) communications, RSU assigns one V2I link to each vehicle  Each vehicle exchanges messages composed of the vehicle's state (i e , position and velocity) with RSU through the $k$-th V2I link by transmitting a signal $\textrm{x}_{t,k}$ carrying those messages at each time instant $t$ where $k \in K$  We consider a reactive RSJ that aims to attack the V2I link by injecting intentional interference to the communication link between vehicles and RSU to alter the transmitted signals by the vehicles  In contrast, the RSS purposes to mislead the vehicles by spoofing the GPS signal and so registering wrong GPS positions  RSU aims to detect both the spoofer on the satellite link and the jammer on multiple V2I links in order to take effective actions and protect the vehicular network  
The joint GPS spoofing and jamming detection problem can be formulated as the following ternary hypothesis test:
\begin{equation}
    \begin{cases}
        \mathcal{H}_{0}: \mathrm{z}_{t,k} = \mathrm{g}_{t,k}^{nR} \mathrm{x}_{t,k} + \mathrm{v}_{t,k}, \\
        \mathcal{H}_{1}: \mathrm{z}_{t,k} = \mathrm{g}_{t,k}^{nR} \mathrm{x}_{t,k} + \mathrm{g}_{t,k}^{JR} \mathrm{x}_{t,k}^{j} + \mathrm{v}_{t,k}, \\
        \mathcal{H}_{2}: \mathrm{z}_{t,k} = \mathrm{g}_{t,k}^{nR} \mathrm{x}_{t,k}^{*} + \mathrm{v}_{t,k},
    \end{cases}
\end{equation}
where $\mathcal{H}_{0}$, $\mathcal{H}_{1}$ and $\mathcal{H}_{2}$ denote three hypotheses corresponding to the absence of both jammer and spoofer, the presence of the jammer, and the presence of the spoofer, respectively  $\textrm{z}_{t,k}$ is the received signal at the RSU at $t$ over the $k$-th V2I link, $\textrm{g}_{t,k}^{nR}$ is the channel power gain from vehicle $n$ to the RSU formulated as: $\textrm{g}_{t,k}^{nR} = \alpha_{t,k}^{nR} \mathrm{h}_{t,k}^{nR}$, where $\alpha_{t,k}^{nR}$ is the large-scale fading including path-loss and shadowing modeled as \cite{8723178}: $\alpha_{t,k}^{nR}=G\beta d_{t,nR}^{-\gamma}$ 
\begin{figure}[t!]
    \centering
    \includegraphics[height=5 3cm]{Figures/SystemModel_V1 pdf}
    \caption{An illustration of the system model }
    \label{fig_SystemModel}
\end{figure}
$G$ is the pathloss constant, $\beta$ is a log normal shadow fading random variable, $d_{t,nR}=\sqrt{({x}_{n,t}-x_{R})^{2}+({y}_{n,t}-y_{R})^{2}}$ is the distance between the $n$-th vehicle and the RSU  $\gamma$ is the power decay exponent and
$\mathrm{h}_{t,k}$ is the small-scale fading component distributed according to $\mathcal{CN}(0,1)$  In addition, $\mathrm{x}_{t,k}$ is the desired signal transmitted by the $n$-th vehicle, and $\mathrm{v}_{t,k}$ is an additive white Gaussian noise with variance $\sigma_{n}^{2}$  $\mathrm{x}_{t,k}^{J}$ is the jamming signal, $\mathrm{x}_{t,k}^{*}$ is the spoofed signal (i e , the signal that carries the bits related to the wrong GPS positions), $\mathrm{g}_{t,k}^{JR} = \alpha_{t,k}^{JR} \mathrm{h}_{t,k}^{JR}$ is the channel power gain from RSJ to RSU where $\alpha_{t,k}^{JR}=G\beta d_{t,JR}^{-\gamma}$ such that $d_{t,JR}=\sqrt{({x}_{J}-x_{R})^{2}+({y}_{J}-y_{R})^{2}}$ 
We assume that the channel state information (CSI) of V2I links is known and can be estimated at the RSU as in \cite{8345717}  
The RSU is equipped with an RF antenna which can track the vehicles' trajectories after decoding the received RF signals  RSU aims to learn the interaction between the RF signals received from multiple vehicles and their corresponding trajectories 

\section{Proposed method for joint detection of GPS spoofing and jamming}

\subsection{Environment Representation}
The RSU is receiving RF signals from each vehicle and tracking its trajectory (which we refer to as GPS signal) by decoding and demodulating the received RF signals  
The Generalized state-space model describing the $i$-th signal evolvement at multiple levels embodies the following equations: 
\begin{equation} \label{eq_discreteLevel}
    \mathrm{\Tilde{S}_{t}}^{(i)} = \mathrm{f}(\mathrm{\Tilde{S}_{t-1}}^{(i)}) + \mathrm{\tilde{w}}_{t},
\end{equation}
\begin{equation} \label{eq_continuousLevel}
    \mathrm{\Tilde{X}_{t}}^{(i)} = \mathrm{A} \mathrm{\Tilde{X}_{t-1}}^{(i)} + \mathrm{B} \mathrm{U}_{\mathrm{\Tilde{S}_{t}}^{(i)}} + \mathrm{\tilde{w}}_{t},
\end{equation}
\begin{equation} \label{eq_observationLevel}
    \mathrm{\Tilde{Z}_{t}}^{(i)} = \mathrm{H} \mathrm{\Tilde{X}_{t}}^{(i)} + \mathrm{\tilde{v}}_{t},
\end{equation}
where $i \in \{$RF, GPS$\}$ indicates the type of signal received by the RSU  The transition system model defined in \eqref{eq_discreteLevel} explains the evolution of the discrete random variables $\mathrm{\Tilde{S}_{t}}^{(i)}$ representing the clusters of the RF (or GPS) signal dynamics, $\mathrm{f}( )$ is a non linear function of its argument and the additive term $\mathrm{\tilde{w}}_{t}$ denotes the process noise  The dynamic model defined in \eqref{eq_continuousLevel} explains the RF signal dynamics evolution or the motion dynamics evolution of the $n$-th vehicle, where $\mathrm{\Tilde{X}_{t}}^{(i)}$ are hidden continuous variables generating sensory signals, $\mathrm{A} \in \mathbb{R}^{2d}$ and $\mathrm{B} \in \mathbb{R}^{2d}$ are the dynamic and control matrices, respectively, and $\mathrm{U}_{\mathrm{\Tilde{S}_{t}}^{(i)}}$ is the control vector representing the dynamic rules of how the signals evolve with time  The measurement model defined in \eqref{eq_observationLevel} describes dependence of the sensory signals $\mathrm{\Tilde{Z}_{t}}^{(i)}$ on the hidden states $\mathrm{\Tilde{X}_{t}}^{(i)}$ that is parametrized by the measurement matrix $\mathrm{B} \in \mathbb{R}^{2d}$ where $d$ stands for the data dimensionality and $\mathrm{\tilde{v}}_{t}$ is a random noise  

\subsection{Learning GDBN}
The hierarchical dynamic models defined in \eqref{eq_discreteLevel}, \eqref{eq_continuousLevel} and \eqref{eq_observationLevel} are structured in a Generalized Dynamic Bayesian Network (GDBN) \cite{9858012} as shown in Fig ~\ref{fig_GDBN_CGDBN}-(a) that provides a probabilistic graphical model expressing the conditional dependencies among random hidden variables and observable states  The generative process explaining how sensory signals have been generated can be factorized as:
\begin{equation} \label{eq_generative_process}
\begin{split}
    \mathrm{P}(\mathrm{\tilde{Z}}_{t}^{(i)}, \mathrm{\tilde{X}}_{t}^{(i)}, \mathrm{\tilde{S}}_{t}^{(i)}) = \mathrm{P}(\mathrm{\tilde{S}}_{0}^{(i)}) \mathrm{P}(\mathrm{\tilde{X}}_{0}^{(i)}) \\ \bigg[ \prod_{t=1}^{\mathrm{T}} \mathrm{P}(\mathrm{\tilde{Z}}_{t}^{(i)}|\mathrm{\tilde{X}}_{t}^{(i)}) \mathrm{P}(\mathrm{\tilde{X}}_{t}^{(i)}|\mathrm{\tilde{X}}_{t-1}^{(i)}, \mathrm{\tilde{S}}_{t}^{(i)}) \mathrm{P}(\mathrm{\tilde{S}}_{t}^{(i)}|\mathrm{\tilde{S}}_{t-1}^{(i)}) \bigg],
\end{split}
\end{equation}
where $\mathrm{P}(\mathrm{\tilde{S}}_{0}^{(i)})$ and $\mathrm{P}(\mathrm{\tilde{X}}_{0}^{(i)})$ are initial prior distributions, $\mathrm{P}(\mathrm{\tilde{Z}}_{t}^{(i)}|\mathrm{\tilde{X}}_{t}^{(i)})$ is the likelihood, $\mathrm{P}(\mathrm{\tilde{X}}_{t}^{(i)}|\mathrm{\tilde{X}}_{t-1}^{(i)}, \mathrm{\tilde{S}}_{t}^{(i)})$ and $\mathrm{P}(\mathrm{\tilde{S}}_{t}^{(i)}|\mathrm{\tilde{S}}_{t-1}^{(i)})$ are the transition densities describing the temporal and hierarchical dynamics of the generalized state-space model 
The generative process defined in \eqref{eq_generative_process} indicates the cause-effect relationships the model impose on the random variables $\mathrm{\tilde{S}}_{t}^{(i)}$, $\mathrm{\tilde{X}}_{t}^{(i)}$ and $\mathrm{\tilde{Z}}_{t}^{(i)}$ forming a chain of causality describing how one state contributes to the production of another state which is represented by the link $\mathrm{\tilde{S}}_{t}^{(i)} \rightarrow \mathrm{\tilde{X}}_{t}^{(i)} \rightarrow \mathrm{\tilde{Z}}_{t}^{(i)}$ 

The RSU starts perceiving the environment using a static assumption about the environmental states evolution by assuming that sensory signals are only subject to random noise  Hence, RSU predicts the RF signal (or vehciles trajectory) using the following simplified model:
$\mathrm{\tilde{X}}_{t}^{(i)} = \mathrm{A} \mathrm{\tilde{X}}_{t-1}^{(i)} + \mathrm{\tilde{w}}_{t}$, 
that differs from \eqref{eq_continuousLevel} in the control vector $\mathrm{U}_{\mathrm{\Tilde{S}_{t}}^{(i)}}$ which is supposed to be null, i e , $\mathrm{U}_{\mathrm{\Tilde{S}_{t}}^{(i)}} = 0$ as the dynamic rules explaining how the environmental states evolve with time are not discovered yet 
Those rules can be discovered by exploiting the generalized errors (GEs), i e , the difference between predictions and observations  The GEs projected into the measurement space are calculated as:
$\tilde{\varepsilon}_{\mathrm{\tilde{Z}}_{t}^{(i)}}^{} = \mathrm{\tilde{Z}}_{t}^{(i)} - \mathrm{H} \mathrm{\tilde{X}}_{t}^{(i)}$ 
Projecting $\tilde{\varepsilon}_{\mathrm{\tilde{Z}}_t}^{}$ back into the generalized state space can be done as follows:
\begin{equation}\label{GE_continuousLevel_initialModel}
    \tilde{\varepsilon}_{\mathrm{\tilde{X}}_t}^{(i)} = \mathrm{H}^{-1}\tilde{\varepsilon}_{\mathrm{\tilde{Z}}_{t}^{(i)}}^{}=\mathrm{H}^{-1}(\mathrm{\tilde{Z}}_{t}^{(i)}-\mathrm{H}\mathrm{\tilde{X}}_{t}^{(i)}) = \mathrm{H}^{-1}\mathrm{\tilde{Z}}_{t}^{(i)} - \mathrm{\tilde{X}}_{t}^{(i)} 
\end{equation}
The GEs defined in \eqref{GE_continuousLevel_initialModel} can be grouped into discrete clusters in an unsupervised manner by employing the Growing Neural Gas (GNG)  The latter produces a set of discrete variables (clusters) denoted by:
$\mathbf{\tilde{S}^{(i)}}=\{\mathrm{\tilde{S}}_{1}^{(i)},\mathrm{\tilde{S}}_{2}^{(i)},\dots,\mathrm{\tilde{S}}_{M_{i}}^{(i)}\}$,
where $M_{i}$ is the total number of clusters and each cluster $\mathrm{\tilde{S}}_{m}^{(i)} \in \mathbf{\tilde{S}^{(i)}}$ follows a Gaussian distribution composed of GEs with homogeneous properties, such that $\mathrm{\tilde{S}}_{m}^{(i)} \sim \mathcal{N}(\tilde{\mu}_{\mathrm{\tilde{S}}_{m}^{(i)}}=[\mu_{\tilde{S}_{m}^{(i)}}, \Dot{\mu}_{\tilde{S}_{m}^{(i)}}], \Sigma_{\mathrm{\tilde{S}}_{m}^{(i)}})$ 
\begin{figure}[t!]
    \begin{center}
        \begin{minipage}[b]{ 40\linewidth}
        \centering
           \includegraphics[width=2 5cm]{Figures/GDBN pdf}
        \\[-1 0mm]
        {\scriptsize (a)}
        \end{minipage}
        \begin{minipage}[b]{ 50\linewidth}
            \centering
            \includegraphics[width=5 0cm]{Figures/C_GDBN pdf}
           
            {\scriptsize (b)}
        \end{minipage}
        \caption{(a) The GDBN  (b) The coupled GDBN (C-GDBN) composed of two GDBNs representing the two signals received at the RSU where their discrete hidden variables are stochastically coupled }
        \label{fig_GDBN_CGDBN}
    \end{center}
\end{figure}
The dynamic transitions of the sensory signals among the available clusters can be captured in a time-varying transition matrix ($\Pi_{\tau}$) by estimating the time-varying transition probabilities $\pi_{ij}=\mathrm{P}(\mathrm{\tilde{S}}_{t}^{(i)}=i|\mathrm{\tilde{S}}_{t-1}^{(i)}=j, \tau)$ where $\tau$ is the time spent in $\mathrm{\tilde{S}}_{t-1}^{(i)}=j$ before transition to $\mathrm{\tilde{S}}_{t}^{(i)}=i$ 

\subsection{Learning Coupled GDBN (C-GDBN)}
The learning procedure described in the previous section can be executed for each signal type, i e , RF and GPS  After learning a separated GDBN model for each signal type, we analyse the interaction behaviour between RF signal and GPS signal received at the RSU by tracking the cluster firing among $\mathbf{\tilde{S}^{(1)}}$ and $\mathbf{\tilde{S}^{(2)}}$ during a certain experience  Such an interaction can be encoded in a Coupled GDBN (C-GDBN) as shown in Fig \ref{fig_GDBN_CGDBN}-(b) composed of the two GDBNs representing the two signals where their hidden variables at the discrete level are stochastically coupled (in $\mathrm{\tilde{C}}_{t}{=}[\mathrm{\tilde{S}}_{t}^{(1)},\mathrm{\tilde{S}}_{t}^{(2)}]$) as those variables are uncorrelated but have coupled means 
The interactive matrix $\Phi \in \mathbb{R}^{M_{1},M_{2}}$ encodes the firing cluster pattern allowing to predict the GPS signal from RF signal is defined as follows:
\begin{equation} \label{interactiveTM_fromRFtoGPS}
\Phi = 
        \begin{bmatrix} 
            \mathrm{P}(\mathrm{\Tilde{S}_{1}}^{(2)}|\mathrm{\Tilde{S}_{1}}^{(1)}) & \mathrm{P}(\mathrm{\Tilde{S}_{2}}^{(2)}|\mathrm{\Tilde{S}_{1}}^{(1)}) & \dots & \mathrm{P}(\mathrm{\Tilde{S}_{M_{2}}}^{(2)}|\mathrm{\Tilde{S}_{1}}^{(1)}) \\
            \mathrm{P}(\mathrm{\Tilde{S}_{1}}^{(2)}|\mathrm{\Tilde{S}_{2}}^{(1)}) & \mathrm{P}(\mathrm{\Tilde{S}_{2}}^{(2)}|\mathrm{\Tilde{S}_{2}}^{(1)}) & \dots & \mathrm{P}(\mathrm{\Tilde{S}_{M_{2}}}^{(2)}|\mathrm{\Tilde{S}_{2}}^{(1)}) \\
            \vdots & \vdots & \ddots & \vdots \\
            \mathrm{P}(\mathrm{\Tilde{S}_{1}}^{(2)}|\mathrm{\Tilde{S}_{M_{1}}}^{(1)}) & \mathrm{P}(\mathrm{\Tilde{S}_{2}}^{(2)}|\mathrm{\Tilde{S}_{M_{1}}}^{(1)}) & \dots & \mathrm{P}(\mathrm{\Tilde{S}_{M_{2}}}^{(2)}|\mathrm{\Tilde{S}_{M_{1}}}^{(1)}) 
        \end{bmatrix} 
\end{equation}

\subsection{Joint Prediction and Perception}
RSU starts predicting the RF signals it expects to receive from each vehicle based on a Modified Markov Jump Particle Filter (M-MJPF) \cite{9858012} that combines Particle filter (PF) and Kalman filter (KF) to perform temporal and hierarchical predictions  Since the acquired C-GDBN allows predicting a certain signal's dynamic evolution based on another's evolution, it requires an interactive Bayesian filter capable of dealing with more complicated predictions  To this purpose, we propose to employ an Interactive M-MJPF (IM-MJPF) on the C-GDBN  The IM-MJPF consists of a PF that propagates a set of $L$ particles equally weighted, such that $\{\mathrm{\tilde{S}}_{t,l}^{(1)}, \mathrm{W}_{t,l}^{(1)}\}{\sim}\{\pi(\mathrm{\tilde{S}}_{t}^{(1)}), \frac{1}{L}\}$, where $\mathrm{\tilde{S}}_{t,l}^{(1)}$, $l \in L$ and $( ^{(1)})$ is the RF signal type  In addition, RSU relies on $\Phi$ defined in \eqref{interactiveTM_fromRFtoGPS} to predict $\mathrm{\tilde{S}}_{t}^{(2)}$ realizing the discrete cluster of vehicle's trajectory starting from the predicted RF signal according to: $\{\mathrm{\tilde{S}}_{t}^{(2)},\mathrm{W}_{t,l}^{(2)}\}{\sim} \{\Phi(\mathrm{\tilde{S}}_{t,l}^{(1)}){=}\mathrm{P}( |\mathrm{\tilde{S}}_{t,l}^{(1)}), \mathrm{W}_{t,l}^{(2)}\}$  For each predicted discrete variable $\mathrm{\tilde{S}}_{t,l}^{(i)}$, a multiple KF is employed to predict multiple continuous variables which guided by the predictions at the higher level as declared in \eqref{eq_continuousLevel} that can be represented probabilistically as $\mathrm{P}(\mathrm{\tilde{X}}_{t}^{(i)}|\mathrm{\tilde{X}}_{t-1}^{(i)}, \mathrm{\tilde{S}}_{t}^{(i)})$  The posterior probability that is used to evaluate expectations is given by:
\begin{multline} \label{piX}
    \pi(\mathrm{\tilde{X}}_{t}^{(i)})=\mathrm{P}(\mathrm{\tilde{X}}_{t}^{(i)},\mathrm{\tilde{S}}_{t}^{(i)}|\mathrm{\tilde{Z}}_{t-1}^{(i)})= \\ \int \mathrm{P}(\mathrm{\tilde{X}}_{t}^{(i)}|\mathrm{\tilde{X}}_{t-1}^{(i)}, \mathrm{\tilde{S}}_{t}^{(i)}) \lambda(\mathrm{\tilde{X}}_{t-1}^{(i)})d\mathrm{\tilde{X}}_{t-1}^{(i)},
\end{multline}
where $\lambda(\mathrm{\tilde{X}}_{t-1}^{(i)}){=}\mathrm{P}(\mathrm{\tilde{Z}}_{t-1}^{(i)}|\mathrm{\tilde{X}}_{t-1}^{(i)})$  
The posterior distribution can be updated (and so representing the updated belief) after having seen the new evidence $\mathrm{\tilde{Z}}_{t}^{(i)}$ by exploiting the diagnostic message $\lambda(\mathrm{\tilde{X}}_{t}^{(i)})$ in the following form: $\mathrm{P}(\mathrm{\tilde{X}}_{t}^{(i)}, \mathrm{\tilde{S}}_{t}^{(i)}|\mathrm{\tilde{Z}}_{t}^{(i)}) {=} \pi(\mathrm{\tilde{X}}_{t}^{(i)})\lambda(\mathrm{\tilde{X}}_{t}^{(i)})$  Likewise, belief in discrete hidden variables can be updated according to: $\mathrm{W}_{t,l}^{(i)}{=}\mathrm{W}_{t,l}^{(i)}\lambda (\mathrm{\tilde{S}}_{t}^{(i)})$ where:
$\lambda (\mathrm{\tilde{S}}_{t}^{(i)}) {=} \lambda (\mathrm{\Tilde{X}}_{t}^{(i)})\mathrm{P}(\mathrm{\Tilde{X}}_{t}^{(i)}|\mathrm{\tilde{S}}_{t}^{(i)}) {=} \mathrm{P}(\mathrm{\tilde{Z}}_{t}^{(i)}|\mathrm{\Tilde{X}}_{t}^{(i)})\mathrm{P}(\mathrm{\Tilde{X}}_{t}^{(i)}|\mathrm{\tilde{S}}_{t}^{(i)})$ 

\subsection{Joint GPS spoofing and jamming detection}
RSU can evaluate the current situation and identify if V2I is under attack, or the satellite link is under spoofing based on a multiple abnormality indicator produced by the IM-MJPF  The first indicator calculates the similarity between the predicted RF signal and the observed one, which is defined as:
\begin{equation}\label{eq_CLA1}
    \Upsilon_{\mathrm{\tilde{X}}_{t}^{(1)}} = -ln \bigg( \mathcal{BC} \big(\pi(\mathrm{\tilde{X}}_{t}^{(1)}),\lambda(\mathrm{\tilde{X}}_{t}^{(1)}) \big) \bigg),
\end{equation}
where $\mathcal{BC}( ){=}\int \sqrt{\pi(\mathrm{\tilde{X}}_{t}^{(1)}),\lambda(\mathrm{\tilde{X}}_{t}^{(1)}})d\mathrm{\tilde{X}}_{t}^{(1)}$ is the Bhattacharyya coefficient 
The second indicator calculates the similarity between the predicted GPS signal (from the RF signal) and the observed one after decoding the RF signal which is defined as:
\begin{equation}\label{eq_CLA2}
    \Upsilon_{\mathrm{\tilde{X}}_{t}^{(2)}} = -ln \bigg( \mathcal{BC} \big(\pi(\mathrm{\tilde{X}}_{t}^{(2)}),\lambda(\mathrm{\tilde{X}}_{t}^{(2)}) \big) \bigg),
\end{equation}
where $\mathcal{BC}( ){=}\int \sqrt{\pi(\mathrm{\tilde{X}}_{t}^{(2)}),\lambda(\mathrm{\tilde{X}}_{t}^{(2)}})d\mathrm{\tilde{X}}_{t}^{(2)}$ 
Different hypotheses can be identified by the RSU to understand the current situation whether there is: a jammer attacking the V2I link, or a spoofer attacking the link between the satellite and the vehicle or both jammer and spoofer are absent according to:
\begin{equation}
    \begin{cases}
        \mathcal{H}_{0}: \text{if} \ \ \Upsilon_{\mathrm{\tilde{X}}_{t}^{(1)}} < \xi_{1} \ \text{and} \ \Upsilon_{\mathrm{\tilde{X}}_{t}^{(2)}} < \xi_{2}, \\
        \mathcal{H}_{1}: \text{if} \ \ \Upsilon_{\mathrm{\tilde{X}}_{t}^{(1)}} \geq \xi_{1} \ \text{and} \ \Upsilon_{\mathrm{\tilde{X}}_{t}^{(2)}} \geq \xi_{2}, \\
        \mathcal{H}_{2}: \text{if} \ \ \Upsilon_{\mathrm{\tilde{X}}_{t}^{(1)}} < \xi_{1} \ \text{and} \ \Upsilon_{\mathrm{\tilde{X}}_{t}^{(2)}} \geq \xi_{2},
    \end{cases}
\end{equation}
where $\xi_{1} = \mathbb{E}[\Bar{\Upsilon}_{\mathrm{\tilde{X}}_{t}^{(1)}}] + 3\sqrt{\mathbb{V}[\Bar{\Upsilon}_{\mathrm{\tilde{X}}_{t}^{(1)}}]}$, and $\xi_{2} = \mathbb{E}[\Bar{\Upsilon}_{\mathrm{\tilde{X}}_{t}^{(2)}}] + 3\sqrt{\mathbb{V}[\Bar{\Upsilon}_{\mathrm{\tilde{X}}_{t}^{(2)}}]}$  In $\xi_{1}$ and $\xi_{2}$, $\Bar{\Upsilon}_{\mathrm{\tilde{X}}_{t}^{(1)}}$ and $\Bar{\Upsilon}_{\mathrm{\tilde{X}}_{t}^{(2)}}$ stand for the abnormality signals during training (i e , normal situation when jammer and spoofer are absent) 

\subsection{Evaluation metrics}
In order to evaluate the performance of the proposed method to jointly detect jammer and GPS spoofer, we adopt the jammer detection probability ($\mathrm{P}_{d}^{j}$) and the spoofer detection probability ($\mathrm{P}_{d}^{s}$), respectively, which are defined as:
\begin{equation}
    \mathrm{P}_{d}^{j} = \mathrm{Pr}(\Upsilon_{\mathrm{\tilde{X}}_{t}^{(1)}}\geq \xi_{1}, \Upsilon_{\mathrm{\tilde{X}}_{t}^{(2)}} \geq \xi_{2}|\mathcal{H}_{1}),
\end{equation}
\begin{equation}
    \mathrm{P}_{d}^{s} = \mathrm{Pr}(\Upsilon_{\mathrm{\tilde{X}}_{t}^{(1)}}< \xi_{1}, \Upsilon_{\mathrm{\tilde{X}}_{t}^{(2)}} \geq \xi_{2}|\mathcal{H}_{2}) 
\end{equation}
Also, we evaluate the accuracy of the proposed method in predicting and estimating the vehicles' trajectories and the expected RF signals by adopting the root mean square error (RMSE) defined as:
\begin{equation}
    RMSE = \sqrt{ \frac{1}{T} \sum_{t=1}^{T}\bigg( \mathrm{\tilde{Z}}_{t}^{(i)}-\mathrm{\tilde{X}}_{t}^{(i)} \bigg)^{2} },
\end{equation}
where $T$ is the total number of predictions 

\section{Simulation Results}
In this section, we evaluate the performance of the proposed method to jointly detect the jammer and the spoofer using extensive simulations  We consider $\mathrm{N}=2$ vehicles interacting inside the environment and exchanging their states (i e , position and velocity) with the RSU  The vehicles move along predefined trajectories performing various maneuvers which are picked from the \textit{Lankershim} dataset proposed by \cite{5206559}  The dataset depicts a four way intersection and includes about $19$ intersection maneuvers  RSU assigns one subchannel realizing the V2I link for each vehicle over which the vehicles' states are transmitted  The transmitted signal carrying the vehicle's state and the jamming signal are both QPSK modulated  
The simulation settings are: carrier frequency of $2$GHz, BW${=}1 4$MHz, cell radius of $500$m, RSU antenna height and gain is $25$m and $8$ dBi, receiver noise figure of $5$dB, vehicle antenna height and gain is $1 5$m and $3$dBi, vehicle speed is $40$Km/h, V2I transmit power is $23$dBm, jammer transmit power ranging from $20$dBm to $40$dBm, SNR of $20$dB, path loss model ($128 1{+}37 6log d$), Log-normal shadowing with $8$dB standard deviation and a fast fading channel following the Rayleigh distribution 
\begin{figure}[ht!]
    \begin{center}
        \begin{minipage}[b]{ 55\linewidth}
        \centering
            \includegraphics[width=5 0cm]{Results/ObservedTrajectories_reference}
        \\[-1 5mm]
        {\scriptsize (a)}
        \end{minipage}
        \begin{minipage}[b]{ 49\linewidth}
            \centering
            \includegraphics[width=4 9cm]{Results/ObservedRFsignal_Veh1_reference}
            \\[-1 5mm]
            {\scriptsize (b)}
        \end{minipage}
        \begin{minipage}[b]{ 49\linewidth}
            \centering
            \includegraphics[width=4 9cm]{Results/ObservedRFsignal_Veh2_reference}
            \\[-1 5mm]
            {\scriptsize (c)}
        \end{minipage}
        \caption{An example visualizing the received RF signals from the two vehicles and the corresponding trajectories: (a) Vehicles' trajectories, (b) received RF signal from vehicle 1, (c) received RF signal from vehicle 2 }
        \label{fig_receivedRFsignalandTrajectory}
    \end{center}
\end{figure}
\begin{figure}[t!]
    \begin{center}
        \begin{minipage}[b]{ 49\linewidth}
            \centering
                \includegraphics[width=4 5cm]{Results/clusters_trajectory_veh1}
            \\[-1 5mm]
            {\scriptsize (a)}
        \end{minipage}
        \begin{minipage}[b]{ 49\linewidth}
            \centering
                \includegraphics[width=4 5cm]{Results/clusters_trajectory_veh2}
            \\[-1 5mm]
            {\scriptsize (b)}
        \end{minipage}
        \begin{minipage}[b]{0 49\linewidth}
            \centering
            \includegraphics[width=4 5cm]{Results/clusters_RFsignal_veh1}
            \\[-1 5mm]
            {\scriptsize (c)}
        \end{minipage}
        \begin{minipage}[b]{0 49\linewidth}
            \centering
            \includegraphics[width=4 5cm]{Results/clusters_RFsignal_veh2}
            \\[-1 5mm]
            {\scriptsize (d)}
        \end{minipage}
       
        \caption{GNG output after clustering the generalized errors obtained from different experiences: (a) clustered trajectory of vehicle 1, (b) clustered trajectory of vehicle 2, (c) clustered RF signal received from vehicle 1, (d) clustered RF signal received from vehicle 2 }
        \label{fig_GNG_of_receivedRFsignalandTrajectory}
    \end{center}
\end{figure}

The RSU aims to learn multiple interactive models (i e , C-GDBN models) encoding the cross relationship between the received RF signal from each vehicle and its corresponding trajectory  These models allow the RSU to predict the trajectory the vehicle will follow based on the received RF signal and evaluate whether the V2I is under jamming attacks or the satellite link is under spoofing  It is to note that the RSU is receiving only the RF signals from the two vehicles and obtaining their positions after decoding the RF signals  Thus, the RSU should be able to evaluate if the received RF signals are evolving according to the dynamic rules learned so far and if the vehicles are following the expected (right) trajectories to decide whether the V2I links are really under attack or whether the satellite link is under spoofing 

Fig ~\ref{fig_receivedRFsignalandTrajectory}-(a) illustrates an example of the interaction between the two vehicles performing a particular manoeuvre, and Fig ~\ref{fig_receivedRFsignalandTrajectory}-(b) shows the received RF signals by the RSU from the two vehicles  At the beginning of the learning process, RSU performs predictions according to the simplified model defined in \eqref{eq_continuousLevel} where $\mathrm{U}_{\mathrm{\Tilde{S}_{t}}^{(i)}} {=} 0$ 
After obtaining the generalized errors as pointed out in \eqref{GE_continuousLevel_initialModel}, RUS clusters those errors using GNG to learn two GDBN models encoding the dynamic rules of how the RF signal and the GPS signal evolve with time, respectively, as showed in Fig ~\ref{fig_GNG_of_receivedRFsignalandTrajectory} and Fig ~\ref{fig_graphicalRep_transitionMatrices}  RSU can couple the two GDBNs by learning the interactive transition matrix that is encoded in a C-GDBN as shown in Fig ~\ref{fig_interactiveMatrices} 
\begin{figure}[t!]
    \begin{center}
        \begin{minipage}[b]{ 49\linewidth}
            \centering
                \includegraphics[width=4 5cm]{Results/graphTransition_Trajectory_veh1}
            \\[-1 5mm]
            {\scriptsize (a)}
        \end{minipage}
        \begin{minipage}[b]{ 49\linewidth}
            \centering
                \includegraphics[width=4 5cm]{Results/graphTransition_Trajectory_veh2}
            \\[-1 5mm]
            {\scriptsize (b)}
        \end{minipage}
        \begin{minipage}[b]{0 49\linewidth}
            \centering
            \includegraphics[width=4 5cm]{Results/graphTransition_RFsignal_veh1}
            \\[-1 5mm]
            {\scriptsize (c)}
        \end{minipage}
        \begin{minipage}[b]{0 49\linewidth}
            \centering
            \includegraphics[width=4 5cm]{Results/graphTransition_RFsignal_veh2}
            \\[-1 5mm]
            {\scriptsize (d)}
        \end{minipage}
        \caption{Graphical representation of the transition matrices (TM): (a) TM related to the trajectory of vehicle 1, (b) TM related to the trajectory of vehicle 2, (c) TM related to the RF signal received from vehicle 1, (d) TM related to the RF signal received from vehicle 2 }
        \label{fig_graphicalRep_transitionMatrices}
    \end{center}
\end{figure}
\begin{figure}[t!]
    \begin{center}
        \begin{minipage}[b]{ 49\linewidth}
            \centering
                \includegraphics[width=3 8cm]{Results/interactiveMatrix_RFtoGPS_Neu5_veh1}
            \\[-1 0mm]
            {\scriptsize (a)}
        \end{minipage}
        \begin{minipage}[b]{0 49\linewidth}
            \centering
            \includegraphics[width=3 8cm]{Results/interactiveMatrix_RFtoGPS_Neu25_veh1}
            \\[-1 0mm]
            {\scriptsize (d)}
        \end{minipage}
       
        \caption{Interactive transition matrix defined in \eqref{interactiveTM_fromRFtoGPS} using different configurations: (a) $\mathrm{M_{1}}=5$, $\mathrm{M_{2}}=5$, (b) $\mathrm{M_{1}}=25$, $\mathrm{M_{2}}=25$ }
        \label{fig_interactiveMatrices}
    \end{center}
\end{figure}
\begin{figure}[t!]
    \begin{center}
        \begin{minipage}[b]{ 49\linewidth}
        \centering
            \includegraphics[width=4 9cm]{Results/RF_situation1_best_veh1}
        \\[-1 5mm]
        {\scriptsize (a)}
        \end{minipage}
        \begin{minipage}[b]{0 49\linewidth}
            \centering
            \includegraphics[width=4 9cm]{Results/RF_situation1_worst_veh1}
            \\[-1 5mm]
            {\scriptsize (b)}
        \end{minipage}
        \begin{minipage}[b]{0 49\linewidth}
            \centering
            \includegraphics[width=4 9cm]{Results/RF_situation1_best_veh2}
            \\[-1 5mm]
            {\scriptsize (c)}
        \end{minipage}
        \begin{minipage}[b]{0 49\linewidth}
            \centering
            \includegraphics[width=4 9cm]{Results/RF_situation1_worst_veh2}
            \\[-1 5mm]
            {\scriptsize (d)}
        \end{minipage}
        \caption{An example visualizing the predicted and observed RF signals transmitted by the 2 vehicles using different configurations  Predicted RF signal from: (a) vehicle 1 using $\mathrm{M_{1}}{=}5$, $\mathrm{M_{2}}{=}5$, (b) vehicle 1 using $\mathrm{M_{1}}{=}25$, $\mathrm{M_{2}}{=}25$, (c) vehicle 2 using $\mathrm{M_{1}}{=}5$, $\mathrm{M_{2}}{=}5$, (d) vehicle 2 using $\mathrm{M_{1}}{=}25$, $\mathrm{M_{2}}{=}25$ }
            \label{fig_situation1_PredictedRF}
    \end{center}
\end{figure}

\begin{figure}[t!]
    \begin{center}
        \begin{minipage}[b]{ 49\linewidth}
        \centering
            \includegraphics[width=4 8cm]{Results/GPSfromRF_situation1_best}
        \\[-1 0mm]
        {\scriptsize (a)}
        \end{minipage}
        \begin{minipage}[b]{0 49\linewidth}
            \centering
            \includegraphics[width=4 8cm]{Results/GPSfromRF_situation1_worst}
            \\[-1 0mm]
            {\scriptsize (b)}
        \end{minipage}
        %
        \caption{An example visualizing the predicted and observed trajectories of two vehicles interacting in the environment  (a) $\mathrm{M_{1}}{=}5$, $\mathrm{M_{2}}{=}5$, (b) $\mathrm{M_{1}}{=}25$, $\mathrm{M_{2}}{=}25$ }
            \label{fig_situation1_VehiclesTrajectories}
    \end{center}
\end{figure}

\begin{figure}[ht!]
    \begin{center}
        \begin{minipage}[b]{ 49\linewidth}
        \centering
            \includegraphics[width=4 8cm]{Results/rmse_on_trajectory}
        \\[-1 0mm]
        {\scriptsize (a)}
        \end{minipage}
        \begin{minipage}[b]{0 49\linewidth}
            \centering
            \includegraphics[width=4 8cm]{Results/rmse_on_RFSignal}
            \\[-1 0mm]
            {\scriptsize (b)}
        \end{minipage}
        \caption{The average RMSE after testing different experiences and examples of: (a) trajectories and (b) RF signals }
            \label{fig_rmse_onTraj_onSig}
    \end{center}
\end{figure}

Fig ~\ref{fig_situation1_PredictedRF} illustrates an example comparing between predicted RF signals and observed ones based on two different configurations in learning the interactive matrix (as shown in Fig ~\ref{fig_interactiveMatrices})  Also, Fig ~\ref{fig_situation1_VehiclesTrajectories} illustrates an example comparing between the predicted and observed trajectories of the two vehicles using the two interactive matrices depicted in Fig ~\ref{fig_interactiveMatrices}  From Fig ~\ref{fig_situation1_PredictedRF} and Fig ~\ref{fig_situation1_VehiclesTrajectories} we can see that using an interactive matrix with less clusters allows to perform better predictions compared to that with more clusters  This can be validated by observing Fig ~\ref{fig_rmse_onTraj_onSig} that illustrates the RMSE values versus different number of clusters related to the two models representing the dynamics of the received RF signals and the vehicles' trajectories  It can be seen that as the number of clusters increases the RMSE error increases, since adding more clusters decreases the firing probability that explains the possibility to be in one of the $M_{2}$ clusters of the second model conditioned in being in a certain cluster of the first model 

Fig ~\ref{fig_exNormal_Spoofed_JammedTrajectories} illustrates an example of vehicle's trajectory under normal situation (i e , jammer and spoofer are absent), under jamming attacks and under spoofing attacks  Also the figure shows the predicted trajectory which should follow the same dynamic rules learned during a normal situation  After that, we implemented the IM-MJPF on the learned C-GDBN to perform multiple predictions, i e , to predict the RF signal that the RSU is expecting to receive from a certain vehicle and the corresponding trajectory that the vehicle is supposed to follow  IM-MJPF through the comparison between multiple predictions and observations, produces multiple abnormality signals as defined in \eqref{eq_CLA1} and \eqref{eq_CLA2} which are used to detect the jammer and the spoofer 

Fig ~\ref{fig_abnormalitySignals_JammerSpoofer} illustrates the multiple abnormality signals related to the example shown in Fig ~\ref{fig_exNormal_Spoofed_JammedTrajectories}  We can observe that the abnormal signals related to both RF signal (Fig ~\ref{fig_abnormalitySignals_JammerSpoofer}-(a)) and trajectory (Fig ~\ref{fig_abnormalitySignals_JammerSpoofer}-(b)) are below the threshold under normal situations  This proves that RSU learned the correct dynamic rules of how RF signals and trajectories evolve when the jammer and spoofer are absent (i e , under normal situations)  Also, we can see that the RSU can notice a high deviation on both the RF signal and the corresponding trajectory due to a jamming interference from what it has learned so far by relying on the abnormality signals  In contrast, we can see that under spoofing attacks, RSU notice a deviation only on the trajectory and not on the RF signal since the spoofer has affected only the positions without manipulating the RF signal  In addition, it is obvious how the proposed method allows the RSU to identify the type of abnormality occurring and to explain the cause of the detected abnormality (i e , understanding if it was because of a jammer attacking the V2I link or a spoofer attacking the satellite link) 
\begin{figure}[t!]
    \centering
    \includegraphics[width=6 5cm]{Results/trajectories_underJamming_andSpoofing}
   
    \caption{Vehicle's trajectory under: normal situation, jamming and spoofing }
    \label{fig_exNormal_Spoofed_JammedTrajectories}
\end{figure}
\begin{figure}[t!]
    \begin{center}
        \begin{minipage}[b]{ 92\linewidth}
        \centering
            \includegraphics[height=2 6cm]{Results/abnSignal_onRF}
        \\[-1 5mm]
        {\scriptsize (a)}
        \end{minipage}
        \begin{minipage}[b]{ 92\linewidth}
            \centering
            \includegraphics[height=2 6cm]{Results/abnSignal_onGPS}
            \\[-1 5mm]
            {\scriptsize (b)}
        \end{minipage}
        %
        \caption{Abnormality Signals related to the example shown in Fig \ref{fig_exNormal_Spoofed_JammedTrajectories}: (a) abnormality indicators related to the RF signal, (b) abnormality indicators related to the trajectory }
            \label{fig_abnormalitySignals_JammerSpoofer}
    \end{center}
\end{figure}
\begin{figure}[t!]
    \centering
    \includegraphics[height=3 2cm]{Results/Detection_Probability_RFfromGPS_versusPj}
    \caption{Detection probability ($\mathrm{P_{d}}$) versus jammer's power ($\mathrm{P_{J}}$) using different number of clusters $\mathrm{M}_{2}$ }
    \label{fig_jammerDetectionProb}
\end{figure}
\begin{figure}[t!]
    \centering
    \includegraphics[height=3 2cm]{Results/spoofingDetectionProbability_falseAlarm_versusM2}
    \caption{Spoofing detection probability ($\mathrm{P}_{d}^{s}$) and spoofing false alarm ($\mathrm{P}_{f}^{s}$) versus the number of clusters $\mathrm{M}_{2}$ }
    \label{fig_spooferDetectionProb}
\end{figure}

Fig ~\ref{fig_jammerDetectionProb} shows the overall performance of the proposed method in detecting the jammer by testing many situations and examples and by considering different jamming powers which ranges from $20$dBm to $40$dBm  It can be seen that the proposed method is able to detect the jammer with high probabilities (near $1$) and by considering low and high jamming powers  Also, the figure compares the performance in detecting the jammer by varying the number of clusters ($M_{2}$) 
Fig ~\ref{fig_spooferDetectionProb} shows the overall performance of the proposed method in detecting the spoofer by testing different different examples of driving maneuvers  It can be seen that the RSU is able to detect the spoofer with high detection probability and null false alarm versus different number of clusters 

\section{Conclusion}
A joint detection method of GPS spoofing and jamming attacks is proposed  The method is based on learning a dynamic interactive model encoding the cross-correlation between the received RF signals from multiple vehicles and their corresponding trajectories  Simulation results show the high effectiveness of the proposed approach in jointly detecting the GPS spoofer and jammer attacks  
Subsequent work will extend the system model to consider more than two vehicles with different channel conditions and various modulation schemes to evaluate the effectiveness of the proposed method 

\bibliographystyle{IEEEtran}




ground_truth: The maximum velocity scales with the square root of the amplitude.

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: Which air unit did Goodwin command during the initial landings of Marines on Saipan?

Read the following text and answer briefly.

Context 2:  Sallada's Carrier Support Group 2, staging in the Marshalls for the invasion of the Marianas 

The air unit, VC-10 Squadron, under Goodwin's command gave close air support to the initial landings of Marines on Saipan on June 15, 1944, destroying enemy gun emplacements, troops, tanks, and trucks  On the 17th, her combat air patrol (CAP) shot down or turned back all but a handful of 47 enemy planes headed for her task group and her gunners shot down two of the three planes that did break through to attack her 

Goodwin's carrier continued in providing of close ground support operations at Tinian during the end of July 1944, then turned her attention to Guam, where she gave identical aid to invading troops until mid-August that year  For his service during the Mariana Islands campaign, Goodwin was decorated with Bronze Star Medal with Combat "V" 

He was succeeded by Captain Walter V  R  Vieweg on August 18, 1944, and appointed Chief of Staff, Carrier Division Six under Rear admiral Arthur W  Radford  The Gambier Bay was sunk in the Battle off Samar on October 25, 1944, during the Battle of Leyte Gulf after helping turn back a much larger attacking Japanese surface force 

Goodwin served with Carrier Division Six during the Bonin Islands raids, the naval operations at Palau and took part in the Battle of Leyte Gulf and operations supporting Leyte landings in late 1944  He was later appointed Air Officer of the Philippine Sea Frontier under Rear admiral James L  Kauffman and remained with that command until the end of hostilities  For his service in the later part of World War II, Goodwin was decorated with Legion of Merit with Combat "V"  He was also entitled to wear two Navy Presidential Unit Citations and Navy Unit Commendation 

Postwar service

Following the surrender of Japan, Goodwin assumed command of Light aircraft carrier  on August 24, 1945  The ship was tasked with air missions over Japan became mercy flights over Allied prisoner-of-war camps, dropping food and medicine until the men could be rescued  She was also present at Tokyo Bay for the Japanese surrender on September 2, 1945 

Goodwin returned with San Jacinto to the United States in mid-September 1945 and he was detached in January 1946  He subsequently served in the office of the Chief of Naval Operations until May that year, when he entered the instruction at National War College  Goodwin graduated in June 1947 and served on Secretary's committee for Research on Reorganization  Upon promotion to Rear admiral on April 1, 1949, Goodwin was appointed Chief of Staff and Aide to Commander-in-Chief, Atlantic Fleet under Admiral William H  P  Blandy 

Revolt of the Admirals

In April 1949, the budget's cuts and proposed reorganization of the United States Armed Forces by the Secretary of Defense Louis A  Johnson launched the wave of discontent between senior commanders in the United States Navy  Johnson proposed the merging of the Marine Corps into the Army, and reduce the Navy to a convoy-escort force 

Goodwin's superior officer, Admiral Blandy was call to testify before the House Committee on Armed Services and his harsh statements for the defense of the Navy, costed him his career  Goodwin shared his views and openly criticized Secretary Johnson for having power concentrated in a single civilian executive, who is an appointee of the Government and not an elected representative of the people  He also criticized aspects of defense unification which permitted the Joint Chiefs of Staff to vote on arms policies of individual services, and thus "rob" the branches of autonomy 

The outbreak of the Korean War in summer 1950 proved the proposal of Secretary Johnson as incorrect and he resigned in September that year  Also Secretary of the Navy, Francis P  Matthews resigned one month earlier 

Later service

Due to the Revolts of the admirals, Blandy was forced to retire in February 1950 and Goodwin was ordered to Newport, Rhode Island for temporary duty as Chief of Staff and Aide to the President of the Naval War College under Vice admiral Donald B  Beary in April 1950  Goodwin was detached from that assignment two months and appointed member of the General Board of the Navy  He was shortly thereafter appointed acting Navy Chief of Public Information, as the substitute for Rear Admiral Russell S  Berkey, who was relieved of illness, but returned to the General Board of the Navy in July that year  Goodwin served in that capacity until February 1951, when he relieved his Academy class, Rear admiral John P  Whitney as Vice Commander, Military Air Transport Service (MATS) 

While in this capacity, Goodwin served under Lieutenant general Laurence S  Kuter and was co-responsible for the logistical support of United Nations troops fighting in Korea  The MATS operated from the United States to Japan and Goodwin served in this capacity until August 1953, when he was appointed Commander Carrier Division Two  While in this assignment, he took part in the Operation Mariner, Joint Anglo-American exercise which encountered very heavy seas over a two-week period in fall 1953 

Goodwin was ordered to the Philippines in May 1954 and assumed duty as Commander, U S  Naval Forces in the Philippines with headquarters at Naval Station Sangley Point near Cavite  He held that command in the period of tensions between Taiwan and China and publicly declared shortly after his arrival, that any attack on Taiwan by the Chinese Communists on the mainland would result in US participation in the conflict  The naval fighter planes under his command also provided escort for passing commercial planes  Goodwin worked together with retired Admiral Raymond A  Spruance, then-Ambassador to the Philippines, and accompanied him during the visits to Singapore, Bangkok and Saigon in January 1955 

On December 18, 1955, Goodwin's classmate Rear admiral Albert K  Morehouse, then serving as Commander, Naval Air Forces, Continental Air Defense Command (CONAD), died of heart attack and Goodwin was ordered to CONAD headquarters in Colorado Springs, Colorado to assume Morehouse's position  While in this capacity, he was subordinated to Army General Earle E  Partridge and was responsible for the Naval and Marine Forces allocated to the command designated for the defense of the Continental United States 

Retirement

Goodwin retired on June 1, 1957, after 40 years of active service and was advanced to the rank of Vice admiral on the retired list for having been specially commended in combat  A week later, he was invited back to his Monroe High School (now Neville High School) and handed a diploma showing that he had been graduated with the class of 1918  He then settled in Monterey, California where he taught American history at Stevenson school and was a member of the Naval Order of the United States 

Vice admiral Hugh H  Goodwin died at his home on February 25, 1980, aged 79  He was survived by his wife, Eleanor with whom he had two children, a daughter Sidney and a son Hugh Jr , who graduated from the Naval Academy in June 1948, but died one year later, when the Hellcat fighter he was piloting collided with another over the Gulf of Mexico during training 

Decorations

Here is the ribbon bar of Vice admiral Hugh H  Goodwin:

References

1900 births
1980 deaths
People from Monroe, Louisiana
Military personnel from Louisiana
United States Naval Academy alumni
Naval War College alumni
United States Naval Aviators
United States Navy personnel of World War I
United States Navy World War II admirals
United States Navy vice admirals
United States submarine commanders
Recipients of the Legion of Merit



ground_truth: VC-10 Squadron.

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: What is the name of the generative interactive model used in the method?

Read the following text and answer briefly.

Context 4: S  393, 406–07 (1932) (Brandeis, J , dissenting)  The Court has long since adopted his argument,52× 52  See, e g , Smith v  Allwright, 321 U S  649, 665 (1944)  as well as its corollary — that stare decisis commands “special force in the area of statutory interpretation” where “Congress remains free to alter what [the Court has] done ”53× 53  Patterson v  McLean Credit Union, 491 U S  164, 172–73 (1989)  For normative evaluations of heightened stare decisis for statutory precedents, see generally Einer Elhauge, Statutory Default Rules: How to Interpret Unclear Legislation 211–23 (2008); and William N  Eskridge, Jr , Overruling Statutory Precedents, 76 Geo  L J  1361, 1364–1409 (1988)  Justice Brandeis’s logic demands that dormant commerce clause cases, where Congress is free to act, be granted the weightier stare decisis 54× 54  Scholars have noted the curious fact that Justice Brandeis included many dormant commerce clause cases as examples of overruled constitutional precedents  See, e g , Earl M  Maltz, Commentary, Some Thoughts on the Death of Stare Decisis in Constitutional Law, 1980 Wis  L  Rev  467, 468–469, 469 n 11  One explanation for this is that Justice Brandeis sought the authority of Chief Justice Taney’s dictum that the Court’s “opinion upon the construction of the Constitution is always open to discussion” — which referred to the dormant commerce clause  See Burnet, 285 U S  at 408 n 3 (Brandeis, J , dissenting) (quoting The Passenger Cases, 48 U S  (7 How ) 283, 470 (1849) (Taney, C J , dissenting))  In Chief Justice Taney’s time, it was thought that Congress could not override the Court’s dormant commerce clause decisions, see Cooley v  Bd  of Wardens, 53 U S  (12 How ) 299, 321 (1852), so the context of Chief Justice Taney’s dictum does not conflict with Justice Brandeis’s theory of stare decisis  The Court applied this reasoning in Quill, as Chief Justice Roberts underscored 55× 55  Wayfair, 138 S  Ct  at 2102 (Roberts, C J , dissenting) 
Yet the Wayfair majority refused to consider Congress’s authority to legislate as a relevant factor for stare decisis 56× 56  Even Justice Kennedy’s earlier opinion in Direct Marketing contemplated judicially overruling Quill, conspicuously neglecting a possible legislative solution  See supra p  278  The Court even insisted that to do so “is inconsistent with the Court’s proper role,” since Quill embodied “a false constitutional premise of th[e] Court’s own creation ”57× 57  Wayfair, 138 S  Ct  at 2096 (emphasis added)  This refusal breaks from the practical Brandeisian wisdom that has guided the Court’s treatment of precedent for the better part of a century  The point is not that stare decisis should have ultimately propped up Bellas Hess yet again, as Wayfair’s dissenting Justices maintained  After all, a realistic approach that is alert to each branch’s institutional capacities might have led to the conclusion that Congress was actually ill-equipped to overrule Quill  In this vein, the Court could have sensibly pointed out that Congress is unlikely to stick its neck out with a tax hike (or a look-alike) from which only the states would benefit 58× 58  For two practical arguments to this effect, see Brian Galle, Essay, Kill Quill, Keep the Dormant Commerce Clause: History’s Lessons on Congressional Control of State Taxation, 70 Stan  L  Rev  Online 158, 160–62 (2018), https://review law stanford edu/wp-content/uploads/sites/3/2018/03/70-Stan -L -Rev -Online-158-Galle pdf [https://perma cc/22YP-P4V5]; Edward A  Zelinsky, The Political Process Argument for Overruling Quill, 82 Brook  L  Rev  1177, 1191–92 (2017)  Indeed, South Dakota advanced such practical arguments in its brief 59× 59  See Petitioner’s Brief at 54, Wayfair, 138 S  Ct  2080 (No  17-494) (“Congress has little incentive to act here because it would be (or appear to be) authorizing new or greater tax collections from its constituents, while receiving none of the revenue in return ”)  More generally, the Court might have discussed the limits of the states’ influence in the federal system as a reason not to wait for congressional intervention, a topic it has debated on other occasions 60× 60  See Richard H  Pildes, Institutional Formalism and Realism in Constitutional and Public Law, 2013 Sup  Ct  Rev  1, 30–32; see also Galle, supra note 58, at 159 (“Congress is not a trustworthy guardian of state fiscal power, making continuing judicial involvement a more appealing prospect ”)  Or it could have argued that new facts on the ground — namely, the blast of e-commerce that hit like a comet after Quill — overpowered stare decisis of any force, special or plain 61× 61  Two recent studies of stare decisis highlighted the physical presence rule as exemplifying a precedent that may reasonably be overruled due to changed facts  See Bryan A  Garner et al , The Law of Judicial Precedent 364–65 (2016); Randy J  Kozel, Settled Versus Right: A Theory of Precedent 112–13 (2017)  It should be noted that the authors of The Law of Judicial Precedent classify the physical presence rule as a constitutional precedent for stare decisis purposes, thus anticipating the Court’s misstep in Wayfair  Garner et al , supra, at 354–65  Because even statutory precedents may sometimes be overruled,62× 62  See Patterson v  McLean Credit Union, 491 U S  164, 173–74 (1989) (discussing justifications for overruling statutory precedents)  Contra Lawrence C  Marshall, “Let Congress Do It”: The Case for an Absolute Rule of Statutory Stare Decisis, 88 Mich  L  Rev  177 (1989)  the Court could have killed Quill without first planting its constitutional kiss of death 63× 63  Cf  Thomas R  Lee, Stare Decisis in Historical Perspective: From the Founding Era to the Rehnquist Court, 52 Vand  L  Rev  647, 704 (1999) (“Justice Brandeis’       memorable prose has since become a mandatory part of the burial rite for any constitutional precedent ”) 
The Court resisted such arguments  Instead, Wayfair reasoned that Congress’s total ability to correct an erroneous decision counts for nothing when the Court gets the Constitution wrong  That such a theory sprouts from a case like Wayfair, which repudiated a “formalistic distinction,”64× 64  Wayfair, 138 S  Ct  at 2092  is ironic  Wayfair’s stare decisis analysis resorts to the formalism of making constitutional a “magic” word65× 65  See Transcript of Oral Argument, supra note 21, at 12  rather than asking whether Congress can step in 
Moreover, the Court’s new thinking on stare decisis threatens other constitutional default rules  Wayfair now stands for the proposition that a “constitutional default rule” — a term the Court apparently lifted from South Dakota’s reply brief on the merits66× 66  Reply Brief at 22, Wayfair, 138 S  Ct  2080 (No  17-494) (“Congress is polarized, which makes it critical       to get the constitutional default rule right ”)  — gets only weakened stare decisis  To appreciate why this holding matters, it is worth exploring the concept and scope of constitutional default rules  Contract theory describes default rules as legal rules that the parties may “contract around ”67× 67  See, e g , Ian Ayres & Robert Gertner, Filling Gaps in Incomplete Contracts: An Economic Theory of Default Rules, 99 Yale L J  87, 87 (1989)  Although “constitutional default rule” could be read broadly to include a variety of actors and contracting mechanisms,68× 68  See John Ferejohn & Barry Friedman, Toward a Political Theory of Constitutional Default Rules, 33 Fla  St  U  L  Rev  825, 826 (2006) (“When we speak of default rules in constitutional law, we typically are talking about specifications of ways the government can act (or modify its behavior) to get around a constitutional prohibition ”)  the Court’s use of the term for purposes of stare decisis may be narrowly defined as judicial precedents of constitutional law that “are ultimately subject to congressional control ”69× 69  Gillian E  Metzger, Congress, Article IV, and Interstate Relations, 120 Harv  L  Rev  1468, 1525 (2007) (describing judicially enforceable “constitutional default rules imposing obligations on the states in the name of union [that] are ultimately subject to congressional control”)  The dormant commerce clause is a paradigmatic constitutional default rule because what the Court does today Congress may undo tomorrow  Justice Scalia declared this fact “[t]he clearest sign that the negative Commerce Clause is a judicial fraud,” for “[h]ow could congressional consent lift a constitutional prohibition?”70× 70  Comptroller of the Treasury of Md  v  Wynne, 135 S  Ct  1787, 1808 (2015) (Scalia, J , dissenting)  But that’s what a constitutional default rule is  The Court has allowed Congress to overturn its dormant commerce clause cases since 1891 71× 71  See In re Rahrer, 140 U S  545, 560–62 (1891) 
Dormant commerce clause cases are not the only constitutional default rules  Professor Laurence Tribe’s treatise identifies two others 72× 72  1 Laurence H  Tribe, American Constitutional Law § 6-35 (3d ed  2000)  And in a groundbreaking article, Professor Henry Monaghan revealed “a substructure of substantive, procedural, and remedial rules” forming “a constitutional common law subject to amendment, modification, or even reversal by Congress ”73× 73  Henry P  Monaghan, The Supreme Court, 1974 Term — Foreword: Constitutional Common Law, 89 Harv  L  Rev  1, 2–3 (1975); see also Susan R  Klein, Identifying and (Re)Formulating Prophylactic Rules, Safe Harbors, and Incidental Rights in Constitutional Criminal Procedure, 99 Mich  L  Rev  1030 (2001) (further developing Monaghan’s theory in criminal procedure context)  What follows is a list of six lines of cases beyond the dormant commerce clause that may be fairly described as constitutional default rules  The first two are drawn from Tribe’s treatise while the next four are found in Monaghan’s article:
(1) State Taxation of Federal Instrumentalities: States may not tax instrumentalities of the federal government74× 74  McCulloch v  Maryland, 17 U S  (4 Wheat ) 316, 436 (1819)  — unless Congress consents 75× 75  See, e g , Helvering v  Gerhardt, 304 U S  405, 411 n 1 (1938) (“Congress may curtail an immunity which might otherwise be implied or enlarge it beyond the point where, Congress being silent, the Court would set its limits ” (citations omitted))  One court has described such judicial decisions as setting a “constitutional default rule ” United States v  Delaware, 958 F 2d 555, 560 n 9 (3d Cir  1992) (“[W]e must decide the constitutional default rule for this type of tax, fully aware that Congress could decide at any time to reverse our decision statutorily ”)  (2) Article I, Section 10 Cases: Article I, Section 10 provides that certain prohibitions on the states may be waived by Congress 76× 76  See U S  Const  art  I, § 10, cls  2–3  The Court has taken note of this when considering whether to overrule, for instance, an Import-Export Clause precedent 77× 77  See Hooven & Allison Co  v  Evatt, 324 U S  652, 668 (1945) (“In view of the fact that the Constitution gives Congress authority to consent to state taxation of imports and hence to lay down its own test for determining when the immunity ends, we see no convincing practical reason for abandoning the test which has been applied for more than a century        ”), overruled on other grounds by Limbach v  Hooven & Allison Co , 466 U S  353 (1984)  In Michelin Tire Corp  v  Wages, 423 U S  276 (1976), the Court left open the question whether “Congress may authorize, under the Import-Export Clause, an exaction that it could not directly impose under the Tax Clause ” Id  at 301 n 13  Metzger, however, argues that the Import-Export Clause is free of other clauses’ limits on congressional power  See Metzger, supra note 69, at 1500 & n 120  (3) Bivens Cases: In Bivens v  Six Unknown Named Agents of Federal Bureau of Narcotics,78× 78  403 U S  388 (1971)  the Court held that a violation of the Fourth Amendment gives rise to a right to sue for damages 79× 79  Id  at 397  But the Court has also held that “[s]uch a cause of action may be defeated       when       Congress has provided an alternative remedy which it explicitly declared to be a substitute for recovery directly under the Constitution and viewed as equally effective ”80× 80  Carlson v  Green, 446 U S  14, 18–19 (1980)  (4) Miranda Cases: The Miranda Court famously “encourage[d]” Congress and the states to explore alternative “procedures which are at least as effective in apprising accused persons of their right of silence and in assuring a continuous opportunity to exercise it ”81× 81  Miranda v  Arizona, 384 U S  436, 467 (1966)  In Dickerson v  United States, 530 U S  428 (2000), the Court struck down a congressional attempt to effectively abolish Miranda, holding that “Miranda announced a constitutional rule that Congress may not supersede legislatively ” Id  at 444  But Dickerson also stood by Miranda’s “invitation for legislative action” to replace Miranda with an adequate substitute  Id  at 440; see also Michael C  Dorf & Barry Friedman, Shared Constitutional Interpretation, 2000 Sup  Ct  Rev  61 (discussing legislative alternatives to Miranda)  (5) The Police Lineup Case: In United States v  Wade,82× 82  388 U S  218 (1967)  the Court created an exclusionary rule for evidence obtained from a police lineup in violation of the Sixth Amendment right to counsel but acknowledged that it could be replaced by “[l]egislative or other regulations       which eliminate the risks of abuse ”83× 83  Id  at 239  (6) The Exclusionary Rule Cases: Mapp v  Ohio made the Fourth Amendment “exclusionary rule” binding on the states,84× 84  367 U S  643, 655 (1961)  yet Congress is thought to have the power to replace it 85× 85  See Bivens v  Six Unknown Named Agents of Fed  Bureau of Narcotics, 403 U S  388, 422–24 (1971) (Burger, C J , dissenting) (inviting Congress to replace the Fourth Amendment exclusionary rule); Harold J  Krent, How to Move Beyond the Exclusionary Rule: Structuring Judicial Response to Legislative Reform Efforts, 26 Pepp  L  Rev  855, 864–71 (1999) 
All of the above are arguably constitutional default rules set by the Court that remain, to one degree or another, open to congressional revision  The list could be longer or shorter, depending on which default rules the Court will view as constitutional86× 86  A shorter list could be produced by whittling away at the constitutional status of the cases identified by Monaghan  While the Court has held that Miranda is a constitutional decision, Dickerson, 530 U S  at 444, some of the other cases may be viewed as nonconstitutional  See, e g , Collins v  Virginia, 138 S  Ct  1663, 1675–80 (2018) (Thomas, J , concurring) (arguing that Mapp is “nonconstitutional,” id  at 1678 n 5); Richard H  Fallon, Jr  et al , Hart and Wechsler’s The Federal Courts and the Federal System 775–77 (7th ed  2015) (discussing whether Bivens is constitutionally required)  Conversely, a longer list might include any constitutional right that can be waived by a party  See, e g , Daniel A  Farber, Another View of the Quagmire: Unconstitutional Conditions and Contract Theory, 33 Fla  St  U  L  Rev  913, 918 (2006) (describing the Eleventh Amendment as “just a contractual default rule that the states are free to barter away”)  Such a list might also include various constitutionally inspired judicial presumptions  See, e g , Jack Goldsmith & John F  Manning, The President’s Completion Power, 115 Yale L J  2280, 2299 (2006) (describing the Chevron presumption of delegated interpretive power to administrative agencies as “a constitutionally inspired default rule”); Nicholas Quinn Rosenkranz, Federal Rules of Statutory Interpretation, 115 Harv  L  Rev  2085, 2097–98 (2002) (describing clear statement rules as “constitutional default rules” reversible by Congress)  Many other decisions could likely be characterized as constitutional default rules; the list above is only an initial stab  and on how it will answer open questions about congressional authority over certain constitutional provisions 87× 87  See, e g , Thomas v  Wash  Gas Light Co , 448 U S  261, 272 n 18 (1980) (plurality opinion) (leaving unresolved whether Congress may limit constitutional full faith and credit obligations); White v  Mass  Council of Constr  Emp’rs, Inc , 460 U S  204, 215 n 1 (1983) (Blackmun, J , concurring in part and dissenting in part) (leaving unresolved “whether Congress may authorize       what otherwise would be a violation” of the Privileges and Immunities Clause); 1 Tribe, supra note 72, § 6-35, at 1243–44 (arguing that Congress cannot override judicial constructions of the Privileges and Immunities Clause); Metzger, supra note 69, at 1486–89 (arguing the opposite)  But the takeaway is clear: weaker stare decisis for constitutional default rules  Pre-Wayfair, one would have thought that stare decisis applies with special force to such precedents, given congressional power to set them straight  Not anymore  Why? Because it is improper to “ask Congress to address a false constitutional premise of th[e] Court’s own creation ”88× 88  Wayfair, 138 S  Ct  at 2096  The Latin for Wayfair’s doctrine is not stare decisis, which should reflect a realistic, working relationship between the legislative and judicial branches  It is mea culpa 
In its zeal to update the Constitution for “the Cyber Age,”89× 89  Id  at 2097  the Court deleted Congress from stare decisis doctrine in constitutional cases  The Court had better options  It could have left Quill on Congress’s doorstep, as the dissent argued  Or it could have justified overruling Quill notwithstanding the special force of stare decisis  Instead, the Court reasoned that it doesn’t matter whether Congress is willing and able to do the job: a constitutional mess calls for a judicial clean-up crew  For constitutional default rules — a category of decisions embracing the dormant commerce clause and sweeping far beyond — Wayfair’s new theory of stare decisis makes the Court’s precedents less sticky and Congress less relevant 



ground_truth: The generative interactive model used in the method is called the Coupled Generalized Dynamic Bayesian Network (C-GDBN).