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 SI unit of power?

Read the following text and answer briefly.

Context 1: For other uses, see Electricity (disambiguation) 
"Electric" redirects here  For other uses, see Electric (disambiguation) 
Lightning is one of the most dramatic effects of electricity 
Electricity is the set of physical phenomena associated with the presence and motion of matter that has a property of electric charge  In early days, electricity was considered as being not related to magnetism  Later on, many experimental results and the development of Maxwell's equations indicated that both electricity and magnetism are from a single phenomenon: electromagnetism  Various common phenomena are related to electricity, including lightning, static electricity, electric heating, electric discharges and many others 
The presence of an electric charge, which can be either positive or negative, produces an electric field  The movement of electric charges is an electric current and produces a magnetic field 
When a charge is placed in a location with a non-zero electric field, a force will act on it  The magnitude of this force is given by Coulomb's law  Thus, if that charge were to move, the electric field would be doing work on the electric charge  Thus we can speak of electric potential at a certain point in space, which is equal to the work done by an external agent in carrying a unit of positive charge from an arbitrarily chosen reference point to that point without any acceleration and is typically measured in volts 
electronics which deals with electrical circuits that involve active electrical components such as vacuum tubes, transistors, diodes and integrated circuits, and associated passive interconnection technologies 
Electrical phenomena have been studied since antiquity, though progress in theoretical understanding remained slow until the seventeenth and eighteenth centuries  Even then, practical applications for electricity were few, and it would not be until the late nineteenth century that electrical engineers were able to put it to industrial and residential use  The rapid expansion in electrical technology at this time transformed industry and society, becoming a driving force for the Second Industrial Revolution  Electricity's extraordinary versatility means it can be put to an almost limitless set of applications which include transport, heating, lighting, communications, and computation  Electrical power is now the backbone of modern industrial society 
Long before any knowledge of electricity existed, people were aware of shocks from electric fish  Ancient Egyptian texts dating from 2750 BCE referred to these fish as the "Thunderer of the Nile", and described them as the "protectors" of all other fish  Electric fish were again reported millennia later by ancient Greek, Roman and Arabic naturalists and physicians  Several ancient writers, such as Pliny the Elder and Scribonius Largus, attested to the numbing effect of electric shocks delivered by catfish and electric rays, and knew that such shocks could travel along conducting objects  Patients suffering from ailments such as gout or headache were directed to touch electric fish in the hope that the powerful jolt might cure them  Possibly the earliest and nearest approach to the discovery of the identity of lightning, and electricity from any other source, is to be attributed to the Arabs, who before the 15th century had the Arabic word for lightning ra‘ad (رعد) applied to the electric ray 
Ancient cultures around the Mediterranean knew that certain objects, such as rods of amber, could be rubbed with cat's fur to attract light objects like feathers  Thales of Miletus made a series of observations on static electricity around 600 BCE, from which he believed that friction rendered amber magnetic, in contrast to minerals such as magnetite, which needed no rubbing  Thales was incorrect in believing the attraction was due to a magnetic effect, but later science would prove a link between magnetism and electricity  According to a controversial theory, the Parthians may have had knowledge of electroplating, based on the 1936 discovery of the Baghdad Battery, which resembles a galvanic cell, though it is uncertain whether the artifact was electrical in nature 
Benjamin Franklin conducted extensive research on electricity in the 18th century, as documented by Joseph Priestley (1767) History and Present Status of Electricity, with whom Franklin carried on extended correspondence 
Electricity would remain little more than an intellectual curiosity for millennia until 1600, when the English scientist William Gilbert wrote De Magnete, in which he made a careful study of electricity and magnetism, distinguishing the lodestone effect from static electricity produced by rubbing amber  He coined the New Latin word electricus ("of amber" or "like amber", from ἤλεκτρον, elektron, the Greek word for "amber") to refer to the property of attracting small objects after being rubbed  This association gave rise to the English words "electric" and "electricity", which made their first appearance in print in Thomas Browne's Pseudodoxia Epidemica of 1646 
Further work was conducted in the 17th and early 18th centuries by Otto von Guericke, Robert Boyle, Stephen Gray and C  F  du Fay  Later in the 18th century, Benjamin Franklin conducted extensive research in electricity, selling his possessions to fund his work  In June 1752 he is reputed to have attached a metal key to the bottom of a dampened kite string and flown the kite in a storm-threatened sky  A succession of sparks jumping from the key to the back of his hand showed that lightning was indeed electrical in nature  He also explained the apparently paradoxical behavior of the Leyden jar as a device for storing large amounts of electrical charge in terms of electricity consisting of both positive and negative charges 
In 1791, Luigi Galvani published his discovery of bioelectromagnetics, demonstrating that electricity was the medium by which neurons passed signals to the muscles  Alessandro Volta's battery, or voltaic pile, of 1800, made from alternating layers of zinc and copper, provided scientists with a more reliable source of electrical energy than the electrostatic machines previously used  The recognition of electromagnetism, the unity of electric and magnetic phenomena, is due to Hans Christian Ørsted and André-Marie Ampère in 1819–1820  Michael Faraday invented the electric motor in 1821, and Georg Ohm mathematically analysed the electrical circuit in 1827  Electricity and magnetism (and light) were definitively linked by James Clerk Maxwell, in particular in his "On Physical Lines of Force" in 1861 and 1862 
While the early 19th century had seen rapid progress in electrical science, the late 19th century would see the greatest progress in electrical engineering  Through such people as Alexander Graham Bell, Ottó Bláthy, Thomas Edison, Galileo Ferraris, Oliver Heaviside, Ányos Jedlik, William Thomson, 1st Baron Kelvin, Charles Algernon Parsons, Werner von Siemens, Joseph Swan, Reginald Fessenden, Nikola Tesla and George Westinghouse, electricity turned from a scientific curiosity into an essential tool for modern life 
In 1887, Heinrich Hertz:843–44 discovered that electrodes illuminated with ultraviolet light create electric sparks more easily  In 1905, Albert Einstein published a paper that explained experimental data from the photoelectric effect as being the result of light energy being carried in discrete quantized packets, energising electrons  This discovery led to the quantum revolution  Einstein was awarded the Nobel Prize in Physics in 1921 for "his discovery of the law of the photoelectric effect"  The photoelectric effect is also employed in photocells such as can be found in solar panels and this is frequently used to make electricity commercially 
The first solid-state device was the "cat's-whisker detector" first used in the 1900s in radio receivers  A whisker-like wire is placed lightly in contact with a solid crystal (such as a germanium crystal) to detect a radio signal by the contact junction effect  In a solid-state component, the current is confined to solid elements and compounds engineered specifically to switch and amplify it  Current flow can be understood in two forms: as negatively charged electrons, and as positively charged electron deficiencies called holes  These charges and holes are understood in terms of quantum physics  The building material is most often a crystalline semiconductor 
The solid-state device came into its own with the invention of the transistor in 1947  Common solid-state devices include transistors, microprocessor chips, and RAM  A specialized type of RAM called flash RAM is used in USB flash drives and more recently, solid-state drives to replace mechanically rotating magnetic disc hard disk drives  Solid state devices became prevalent in the 1950s and the 1960s, during the transition from vacuum tubes to semiconductor diodes, transistors, integrated circuit (IC) and the light-emitting diode (LED) 
The presence of charge gives rise to an electrostatic force: charges exert a force on each other, an effect that was known, though not understood, in antiquity :457 A lightweight ball suspended from a string can be charged by touching it with a glass rod that has itself been charged by rubbing with a cloth  If a similar ball is charged by the same glass rod, it is found to repel the first: the charge acts to force the two balls apart  Two balls that are charged with a rubbed amber rod also repel each other  However, if one ball is charged by the glass rod, and the other by an amber rod, the two balls are found to attract each other  These phenomena were investigated in the late eighteenth century by Charles-Augustin de Coulomb, who deduced that charge manifests itself in two opposing forms  This discovery led to the well-known axiom: like-charged objects repel and opposite-charged objects attract 
The force acts on the charged particles themselves, hence charge has a tendency to spread itself as evenly as possible over a conducting surface  The magnitude of the electromagnetic force, whether attractive or repulsive, is given by Coulomb's law, which relates the force to the product of the charges and has an inverse-square relation to the distance between them :35 The electromagnetic force is very strong, second only in strength to the strong interaction, but unlike that force it operates over all distances  In comparison with the much weaker gravitational force, the electromagnetic force pushing two electrons apart is 1042 times that of the gravitational attraction pulling them together 
Study has shown that the origin of charge is from certain types of subatomic particles which have the property of electric charge  Electric charge gives rise to and interacts with the electromagnetic force, one of the four fundamental forces of nature  The most familiar carriers of electrical charge are the electron and proton  Experiment has shown charge to be a conserved quantity, that is, the net charge within an electrically isolated system will always remain constant regardless of any changes taking place within that system  Within the system, charge may be transferred between bodies, either by direct contact, or by passing along a conducting material, such as a wire :2–5 The informal term static electricity refers to the net presence (or 'imbalance') of charge on a body, usually caused when dissimilar materials are rubbed together, transferring charge from one to the other 
The charge on electrons and protons is opposite in sign, hence an amount of charge may be expressed as being either negative or positive  By convention, the charge carried by electrons is deemed negative, and that by protons positive, a custom that originated with the work of Benjamin Franklin  The amount of charge is usually given the symbol Q and expressed in coulombs; each electron carries the same charge of approximately −1 6022×10−19 coulomb  The proton has a charge that is equal and opposite, and thus +1 6022×10−19 coulomb  Charge is possessed not just by matter, but also by antimatter, each antiparticle bearing an equal and opposite charge to its corresponding particle 
The movement of electric charge is known as an electric current, the intensity of which is usually measured in amperes  Current can consist of any moving charged particles; most commonly these are electrons, but any charge in motion constitutes a current  Electric current can flow through some things, electrical conductors, but will not flow through an electrical insulator 
By historical convention, a positive current is defined as having the same direction of flow as any positive charge it contains, or to flow from the most positive part of a circuit to the most negative part  Current defined in this manner is called conventional current  The motion of negatively charged electrons around an electric circuit, one of the most familiar forms of current, is thus deemed positive in the opposite direction to that of the electrons  However, depending on the conditions, an electric current can consist of a flow of charged particles in either direction, or even in both directions at once  The positive-to-negative convention is widely used to simplify this situation 
The process by which electric current passes through a material is termed electrical conduction, and its nature varies with that of the charged particles and the material through which they are travelling  Examples of electric currents include metallic conduction, where electrons flow through a conductor such as metal, and electrolysis, where ions (charged atoms) flow through liquids, or through plasmas such as electrical sparks  While the particles themselves can move quite slowly, sometimes with an average drift velocity only fractions of a millimetre per second,:17 the electric field that drives them itself propagates at close to the speed of light, enabling electrical signals to pass rapidly along wires 
Current causes several observable effects, which historically were the means of recognising its presence  That water could be decomposed by the current from a voltaic pile was discovered by Nicholson and Carlisle in 1800, a process now known as electrolysis  Their work was greatly expanded upon by Michael Faraday in 1833  Current through a resistance causes localised heating, an effect James Prescott Joule studied mathematically in 1840 :23–24 One of the most important discoveries relating to current was made accidentally by Hans Christian Ørsted in 1820, when, while preparing a lecture, he witnessed the current in a wire disturbing the needle of a magnetic compass  He had discovered electromagnetism, a fundamental interaction between electricity and magnetics  The level of electromagnetic emissions generated by electric arcing is high enough to produce electromagnetic interference, which can be detrimental to the workings of adjacent equipment 
In engineering or household applications, current is often described as being either direct current (DC) or alternating current (AC)  These terms refer to how the current varies in time  Direct current, as produced by example from a battery and required by most electronic devices, is a unidirectional flow from the positive part of a circuit to the negative :11 If, as is most common, this flow is carried by electrons, they will be travelling in the opposite direction  Alternating current is any current that reverses direction repeatedly; almost always this takes the form of a sine wave :206–07 Alternating current thus pulses back and forth within a conductor without the charge moving any net distance over time  The time-averaged value of an alternating current is zero, but it delivers energy in first one direction, and then the reverse  Alternating current is affected by electrical properties that are not observed under steady state direct current, such as inductance and capacitance :223–25 These properties however can become important when circuitry is subjected to transients, such as when first energised 
The concept of the electric field was introduced by Michael Faraday  An electric field is created by a charged body in the space that surrounds it, and results in a force exerted on any other charges placed within the field  The electric field acts between two charges in a similar manner to the way that the gravitational field acts between two masses, and like it, extends towards infinity and shows an inverse square relationship with distance  However, there is an important difference  Gravity always acts in attraction, drawing two masses together, while the electric field can result in either attraction or repulsion  Since large bodies such as planets generally carry no net charge, the electric field at a distance is usually zero  Thus gravity is the dominant force at distance in the universe, despite being much weaker 
A hollow conducting body carries all its charge on its outer surface  The field is therefore zero at all places inside the body :88 This is the operating principal of the Faraday cage, a conducting metal shell which isolates its interior from outside electrical effects 
The principles of electrostatics are important when designing items of high-voltage equipment  There is a finite limit to the electric field strength that may be withstood by any medium  Beyond this point, electrical breakdown occurs and an electric arc causes flashover between the charged parts  Air, for example, tends to arc across small gaps at electric field strengths which exceed 30 kV per centimetre  Over larger gaps, its breakdown strength is weaker, perhaps 1 kV per centimetre  The most visible natural occurrence of this is lightning, caused when charge becomes separated in the clouds by rising columns of air, and raises the electric field in the air to greater than it can withstand  The voltage of a large lightning cloud may be as high as 100 MV and have discharge energies as great as 250 kWh 
A pair of AA cells  The + sign indicates the polarity of the potential difference between the battery terminals 
The concept of electric potential is closely linked to that of the electric field  A small charge placed within an electric field experiences a force, and to have brought that charge to that point against the force requires work  The electric potential at any point is defined as the energy required to bring a unit test charge from an infinite distance slowly to that point  It is usually measured in volts, and one volt is the potential for which one joule of work must be expended to bring a charge of one coulomb from infinity :494–98 This definition of potential, while formal, has little practical application, and a more useful concept is that of electric potential difference, and is the energy required to move a unit charge between two specified points  An electric field has the special property that it is conservative, which means that the path taken by the test charge is irrelevant: all paths between two specified points expend the same energy, and thus a unique value for potential difference may be stated :494–98 The volt is so strongly identified as the unit of choice for measurement and description of electric potential difference that the term voltage sees greater everyday usage 
For practical purposes, it is useful to define a common reference point to which potentials may be expressed and compared  While this could be at infinity, a much more useful reference is the Earth itself, which is assumed to be at the same potential everywhere  This reference point naturally takes the name earth or ground  Earth is assumed to be an infinite source of equal amounts of positive and negative charge, and is therefore electrically uncharged—and unchargeable 
Electric potential is a scalar quantity, that is, it has only magnitude and not direction  It may be viewed as analogous to height: just as a released object will fall through a difference in heights caused by a gravitational field, so a charge will 'fall' across the voltage caused by an electric field  As relief maps show contour lines marking points of equal height, a set of lines marking points of equal potential (known as equipotentials) may be drawn around an electrostatically charged object  The equipotentials cross all lines of force at right angles  They must also lie parallel to a conductor's surface, otherwise this would produce a force that will move the charge carriers to even the potential of the surface 
Ørsted's discovery in 1821 that a magnetic field existed around all sides of a wire carrying an electric current indicated that there was a direct relationship between electricity and magnetism  Moreover, the interaction seemed different from gravitational and electrostatic forces, the two forces of nature then known  The force on the compass needle did not direct it to or away from the current-carrying wire, but acted at right angles to it  Ørsted's slightly obscure words were that "the electric conflict acts in a revolving manner " The force also depended on the direction of the current, for if the flow was reversed, then the force did too 
Ørsted did not fully understand his discovery, but he observed the effect was reciprocal: a current exerts a force on a magnet, and a magnetic field exerts a force on a current  The phenomenon was further investigated by Ampère, who discovered that two parallel current-carrying wires exerted a force upon each other: two wires conducting currents in the same direction are attracted to each other, while wires containing currents in opposite directions are forced apart  The interaction is mediated by the magnetic field each current produces and forms the basis for the international definition of the ampere 
This relationship between magnetic fields and currents is extremely important, for it led to Michael Faraday's invention of the electric motor in 1821  Faraday's homopolar motor consisted of a permanent magnet sitting in a pool of mercury  A current was allowed through a wire suspended from a pivot above the magnet and dipped into the mercury  The magnet exerted a tangential force on the wire, making it circle around the magnet for as long as the current was maintained 
Experimentation by Faraday in 1831 revealed that a wire moving perpendicular to a magnetic field developed a potential difference between its ends  Further analysis of this process, known as electromagnetic induction, enabled him to state the principle, now known as Faraday's law of induction, that the potential difference induced in a closed circuit is proportional to the rate of change of magnetic flux through the loop  Exploitation of this discovery enabled him to invent the first electrical generator in 1831, in which he converted the mechanical energy of a rotating copper disc to electrical energy  Faraday's disc was inefficient and of no use as a practical generator, but it showed the possibility of generating electric power using magnetism, a possibility that would be taken up by those that followed on from his work 
Italian physicist Alessandro Volta showing his "battery" to French emperor Napoleon Bonaparte in the early 19th century 
The ability of chemical reactions to produce electricity, and conversely the ability of electricity to drive chemical reactions has a wide array of uses 
Electrochemistry has always been an important part of electricity  From the initial invention of the Voltaic pile, electrochemical cells have evolved into the many different types of batteries, electroplating and electrolysis cells  Aluminium is produced in vast quantities this way, and many portable devices are electrically powered using rechargeable cells 
A basic electric circuit  The voltage source V on the left drives a current I around the circuit, delivering electrical energy into the resistor R  From the resistor, the current returns to the source, completing the circuit 
An electric circuit is an interconnection of electric components such that electric charge is made to flow along a closed path (a circuit), usually to perform some useful task 
Electric power is the rate at which electric energy is transferred by an electric circuit  The SI unit of power is the watt, one joule per second 
Electricity generation is often done with electric generators, but can also be supplied by chemical sources such as electric batteries or by other means from a wide variety of sources of energy  Electric power is generally supplied to businesses and homes by the electric power industry  Electricity is usually sold by the kilowatt hour (3 6 MJ) which is the product of power in kilowatts multiplied by running time in hours  Electric utilities measure power using electricity meters, which keep a running total of the electric energy delivered to a customer  Unlike fossil fuels, electricity is a low entropy form of energy and can be converted into motion or many other forms of energy with high efficiency 
Electronics deals with electrical circuits that involve active electrical components such as vacuum tubes, transistors, diodes, optoelectronics, sensors and integrated circuits, and associated passive interconnection technologies  The nonlinear behaviour of active components and their ability to control electron flows makes amplification of weak signals possible and electronics is widely used in information processing, telecommunications, and signal processing  The ability of electronic devices to act as switches makes digital information processing possible  Interconnection technologies such as circuit boards, electronics packaging technology, and other varied forms of communication infrastructure complete circuit functionality and transform the mixed components into a regular working system 
Today, most electronic devices use semiconductor components to perform electron control  The study of semiconductor devices and related technology is considered a branch of solid state physics, whereas the design and construction of electronic circuits to solve practical problems come under electronics engineering 
Thus, the work of many researchers enabled the use of electronics to convert signals into high frequency oscillating currents, and via suitably shaped conductors, electricity permits the transmission and reception of these signals via radio waves over very long distances 
Early 20th-century alternator made in Budapest, Hungary, in the power generating hall of a hydroelectric station (photograph by Prokudin-Gorsky, 1905–1915) 
In the 6th century BC, the Greek philosopher Thales of Miletus experimented with amber rods and these experiments were the first studies into the production of electrical energy  While this method, now known as the triboelectric effect, can lift light objects and generate sparks, it is extremely inefficient  It was not until the invention of the voltaic pile in the eighteenth century that a viable source of electricity became available  The voltaic pile, and its modern descendant, the electrical battery, store energy chemically and make it available on demand in the form of electrical energy  The battery is a versatile and very common power source which is ideally suited to many applications, but its energy storage is finite, and once discharged it must be disposed of or recharged  For large electrical demands electrical energy must be generated and transmitted continuously over conductive transmission lines 
Electrical power is usually generated by electro-mechanical generators driven by steam produced from fossil fuel combustion, or the heat released from nuclear reactions; or from other sources such as kinetic energy extracted from wind or flowing water  The modern steam turbine invented by Sir Charles Parsons in 1884 today generates about 80 percent of the electric power in the world using a variety of heat sources  Such generators bear no resemblance to Faraday's homopolar disc generator of 1831, but they still rely on his electromagnetic principle that a conductor linking a changing magnetic field induces a potential difference across its ends  The invention in the late nineteenth century of the transformer meant that electrical power could be transmitted more efficiently at a higher voltage but lower current  Efficient electrical transmission meant in turn that electricity could be generated at centralised power stations, where it benefited from economies of scale, and then be despatched relatively long distances to where it was needed 
Since electrical energy cannot easily be stored in quantities large enough to meet demands on a national scale, at all times exactly as much must be produced as is required  This requires electricity utilities to make careful predictions of their electrical loads, and maintain constant co-ordination with their power stations  A certain amount of generation must always be held in reserve to cushion an electrical grid against inevitable disturbances and losses 
Electricity is a very convenient way to transfer energy, and it has been adapted to a huge, and growing, number of uses  The invention of a practical incandescent light bulb in the 1870s led to lighting becoming one of the first publicly available applications of electrical power  Although electrification brought with it its own dangers, replacing the naked flames of gas lighting greatly reduced fire hazards within homes and factories  Public utilities were set up in many cities targeting the burgeoning market for electrical lighting  In the late 20th century and in modern times, the trend has started to flow in the direction of deregulation in the electrical power sector 
The resistive Joule heating effect employed in filament light bulbs also sees more direct use in electric heating  While this is versatile and controllable, it can be seen as wasteful, since most electrical generation has already required the production of heat at a power station  A number of countries, such as Denmark, have issued legislation restricting or banning the use of resistive electric heating in new buildings  Electricity is however still a highly practical energy source for heating and refrigeration, with air conditioning/heat pumps representing a growing sector for electricity demand for heating and cooling, the effects of which electricity utilities are increasingly obliged to accommodate 
Electricity is used within telecommunications, and indeed the electrical telegraph, demonstrated commercially in 1837 by Cooke and Wheatstone, was one of its earliest applications  With the construction of first intercontinental, and then transatlantic, telegraph systems in the 1860s, electricity had enabled communications in minutes across the globe  Optical fibre and satellite communication have taken a share of the market for communications systems, but electricity can be expected to remain an essential part of the process 
The effects of electromagnetism are most visibly employed in the electric motor, which provides a clean and efficient means of motive power  A stationary motor such as a winch is easily provided with a supply of power, but a motor that moves with its application, such as an electric vehicle, is obliged to either carry along a power source such as a battery, or to collect current from a sliding contact such as a pantograph  Electrically powered vehicles are used in public transportation, such as electric buses and trains, and an increasing number of battery-powered electric cars in private ownership 
Electronic devices make use of the transistor, perhaps one of the most important inventions of the twentieth century, and a fundamental building block of all modern circuitry  A modern integrated circuit may contain several billion miniaturised transistors in a region only a few centimetres square 
A voltage applied to a human body causes an electric current through the tissues, and although the relationship is non-linear, the greater the voltage, the greater the current  The threshold for perception varies with the supply frequency and with the path of the current, but is about 0 1 mA to 1 mA for mains-frequency electricity, though a current as low as a microamp can be detected as an electrovibration effect under certain conditions  If the current is sufficiently high, it will cause muscle contraction, fibrillation of the heart, and tissue burns  The lack of any visible sign that a conductor is electrified makes electricity a particular hazard  The pain caused by an electric shock can be intense, leading electricity at times to be employed as a method of torture  Death caused by an electric shock is referred to as electrocution  Electrocution is still the means of judicial execution in some jurisdictions, though its use has become rarer in recent times 
Electricity is not a human invention, and may be observed in several forms in nature, a prominent manifestation of which is lightning  Many interactions familiar at the macroscopic level, such as touch, friction or chemical bonding, are due to interactions between electric fields on the atomic scale  The Earth's magnetic field is thought to arise from a natural dynamo of circulating currents in the planet's core  Certain crystals, such as quartz, or even sugar, generate a potential difference across their faces when subjected to external pressure  This phenomenon is known as piezoelectricity, from the Greek piezein (πιέζειν), meaning to press, and was discovered in 1880 by Pierre and Jacques Curie  The effect is reciprocal, and when a piezoelectric material is subjected to an electric field, a small change in physical dimensions takes place 
§Bioelectrogenesis in microbial life is a prominent phenomenon in soils and sediment ecology resulting from anaerobic respiration  The microbial fuel cell mimics this ubiquitous natural phenomenon 
Some organisms, such as sharks, are able to detect and respond to changes in electric fields, an ability known as electroreception, while others, termed electrogenic, are able to generate voltages themselves to serve as a predatory or defensive weapon  The order Gymnotiformes, of which the best known example is the electric eel, detect or stun their prey via high voltages generated from modified muscle cells called electrocytes  All animals transmit information along their cell membranes with voltage pulses called action potentials, whose functions include communication by the nervous system between neurons and muscles  An electric shock stimulates this system, and causes muscles to contract  Action potentials are also responsible for coordinating activities in certain plants 
In the 19th and early 20th century, electricity was not part of the everyday life of many people, even in the industrialised Western world  The popular culture of the time accordingly often depicted it as a mysterious, quasi-magical force that can slay the living, revive the dead or otherwise bend the laws of nature  This attitude began with the 1771 experiments of Luigi Galvani in which the legs of dead frogs were shown to twitch on application of animal electricity  "Revitalization" or resuscitation of apparently dead or drowned persons was reported in the medical literature shortly after Galvani's work  These results were known to Mary Shelley when she authored Frankenstein (1819), although she does not name the method of revitalization of the monster  The revitalization of monsters with electricity later became a stock theme in horror films 
As the public familiarity with electricity as the lifeblood of the Second Industrial Revolution grew, its wielders were more often cast in a positive light, such as the workers who "finger death at their gloves' end as they piece and repiece the living wires" in Rudyard Kipling's 1907 poem Sons of Martha  Electrically powered vehicles of every sort featured large in adventure stories such as those of Jules Verne and the Tom Swift books  The masters of electricity, whether fictional or real—including scientists such as Thomas Edison, Charles Steinmetz or Nikola Tesla—were popularly conceived of as having wizard-like powers 
With electricity ceasing to be a novelty and becoming a necessity of everyday life in the later half of the 20th century, it required particular attention by popular culture only when it stops flowing, an event that usually signals disaster  The people who keep it flowing, such as the nameless hero of Jimmy Webb’s song "Wichita Lineman" (1968), are still often cast as heroic, wizard-like figures 
Ampère's circuital law, connects the direction of an electric current and its associated magnetic currents 
^ Diogenes Laertius  R D  Hicks (ed )  "Lives of Eminent Philosophers, Book 1 Chapter 1 "  Perseus Digital Library  Tufts University  Retrieved 5 February 2017  Aristotle and Hippias affirm that, arguing from the magnet and from amber, he attributed a soul or life even to inanimate objects 
^ Aristotle  Daniel C  Stevenson (ed )  "De Animus (On the Soul) Book 1 Part 2 (B4 verso)"  The Internet Classics Archive  Translated by J A  Smith  Retrieved 5 February 2017  Thales, too, to judge from what is recorded about him, seems to have held soul to be a motive force, since he said that the magnet has a soul in it because it moves the iron 
^ a b c Guarnieri, M  (2014)  "Electricity in the age of Enlightenment"  IEEE Industrial Electronics Magazine  8 (3): 60–63  doi:10 1109/MIE 2014 2335431 
^ Srodes, James (2002), Franklin: The Essential Founding Father, Regnery Publishing, pp  92–94, ISBN 0-89526-163-4 It is uncertain if Franklin personally carried out this experiment, but it is popularly attributed to him 
^ a b Guarnieri, M  (2014)  "The Big Jump from the Legs of a Frog"  IEEE Industrial Electronics Magazine  8 (4): 59–61, 69  doi:10 1109/MIE 2014 2361237 
^ Hertz, Heinrich (1887)  "Ueber den Einfluss des ultravioletten Lichtes auf die electrische Entladung"  Annalen der Physik  267 (8): S  983–1000  Bibcode:1887AnP   267  983H  doi:10 1002/andp 18872670827 
^ "The Nobel Prize in Physics 1921"  Nobel Foundation  Retrieved 2013-03-16 
^ John Sydney Blakemore, Solid state physics, pp  1–3, Cambridge University Press, 1985 ISBN 0-521-31391-0 
^ Richard C  Jaeger, Travis N  Blalock, Microelectronic circuit design, pp  46–47, McGraw-Hill Professional, 2003 ISBN 0-07-250503-6 
^ "The repulsive force between two small spheres charged with the same type of electricity is inversely proportional to the square of the distance between the centres of the two spheres " Charles-Augustin de Coulomb, Histoire de l'Academie Royal des Sciences, Paris 1785 
^ Sewell, Tyson (1902), The Elements of Electrical Engineering, Lockwood, p  18   The Q originally stood for 'quantity of electricity', the term 'electricity' now more commonly expressed as 'charge' 
^ a b Berkson, William (1974), Fields of Force: The Development of a World View from Faraday to Einstein, Routledge, p  370, ISBN 0-7100-7626-6 Accounts differ as to whether this was before, during, or after a lecture 
^ "Lab Note #105 EMI Reduction – Unsuppressed vs  Suppressed"  Arc Suppression Technologies  April 2011  Retrieved March 7, 2012 
^ Almost all electric fields vary in space  An exception is the electric field surrounding a planar conductor of infinite extent, the field of which is uniform 
^ Paul J  Nahin (9 October 2002)  Oliver Heaviside: The Life, Work, and Times of an Electrical Genius of the Victorian Age  JHU Press  ISBN 978-0-8018-6909-9 
^ "The Bumpy Road to Energy Deregulation"  EnPowered  2016-03-28 
^ a b c d e f g h Van Riper, op cit , p  71 
Look up electricity in Wiktionary, the free dictionary 
Basic Concepts of Electricity chapter from Lessons In Electric Circuits Vol 1 DC book and series 
Context 2: Brooksley Elizabeth Born (born August 27, 1940) is an American attorney and former public official who, from August 26, 1996, to June 1, 1999, was chair of the Commodity Futures Trading Commission (CFTC), the federal agency which oversees the U S  futures and commodity options markets  During her tenure on the CFTC, Born lobbied Congress and the President to give the CFTC oversight of off-exchange markets for derivatives, in addition to its role with respect to exchange-traded derivatives, but her warnings were ignored or dismissed, and her calls for reform resisted by other regulators <ref name="nytimes">Goodman, Peter S  The Reckoning - Taking Hard New Look at a Greenspan Legacy, The New York Times, October 9, 2008 </ref> Born resigned as chairperson on June 1, 1999, shortly after Congress passed legislation prohibiting her agency from regulating derivatives 

Early life and education
Born graduated from Abraham Lincoln High School (San Francisco, California) at the age of 16  She then attended Stanford University, where she majored in English and was graduated with the class of 1961   She initially wanted to become a doctor, but a guidance counsellor at Stanford advised her against medicine, so she majored in English literature instead 

She then attended Stanford Law School, one of only seven women in her class   She was the first female student ever to be named president of the Stanford Law Review  She received the "Outstanding Senior" award and graduated as valedictorian of the class of 1964 

Legal career
Immediately after law school Born was selected as a law clerk to judge Henry Edgerton of the U S  Court of Appeals for the District of Columbia Circuit  It was during this time that she met her first husband, Jacob C  Landau, who was a journalist covering the Federal courts at the time  Following her clerkship, she became an associate at the Washington, D C -based international law firm of Arnold & Porter  Born was attracted to Arnold & Porter because it was one of the few major law firms to have a woman partner at that time, Carolyn Agger, who was the head of the tax practice  Born took a two-year leave of absence from Arnold & Porter to accompany her first husband to Boston, where he had received a fellowship  During that time she worked as a research assistant to law professor Alan Dershowitz 

Born's early career at Arnold & Porter focused on international trade law, in which she represented a number of Swiss industries and the government of Switzerland  She developed a practice representing clients in numerous complex litigation and arbitration cases involving financial market transactions  Among her high-profile cases was the matter of the Hunt Brothers attempt to corner the market in silver in the 1970s  She made partner at Arnold & Porter, after moving to a three-day schedule to help raise her second child, and eventually rose to be the head of the firm's derivatives practice 

Born was among the first female attorneys to systematically address inequities regarding how the laws treated women  Born and another female lawyer, Marna Tucker, taught what is considered to have been the first "Women and the Law" course at Catholic University’s Columbus School of Law  The class exclusively concerned prejudicial treatment of women under the laws of the United States, past and present   Born and Tucker were surprised to discover that there was no textbook on the issue at the time   Born is also one of the co-founders of the National Women's Law Center  Born also helped rewrite the American Bar Association rules to make it possible for more women and minorities to sit on federal bench 

During her long legal career, and into her retirement, Born did much pro bono and other types of volunteer work  She was active in the American Bar Association, the largest professional organization of lawyers in the United States   Initially Born was named a member of the governing council of the ABA's Individual Rights Section, eventually becoming chairperson   Born and Tucker founded the ABA Women's Caucus, the first organization of female lawyers in the ABA   She held several other senior positions in the ABA, including being named the first woman member of the ABA's Standing Committee on the Federal Judiciary   As a member of the Judiciary Committee, Born provided testimony and opinion on persons nominated for federal judgeships  In 1980 she was named chair of the committee   As chair of the committee, Born was invited to address the U S  Congress regarding the nomination of Judge Sandra Day O'Connor to the U S  Supreme Court 

In 1993, Born's name was floated as a possible candidate for Attorney General of the United States, but Janet Reno was nominated 

In July 2009, Nancy Pelosi appointed Brooksley Born as a commissioner to the Financial Crisis Inquiry Commission (FCIC) 

Born and the OTC derivatives market
Born was appointed to the CFTC on April 15, 1994, by President Bill Clinton  Due to litigation against Bankers Trust Company by Procter and Gamble and other corporate clients, Born and her team at the CFTC sought comments on the regulation of over-the-counter derivatives, a first step in the process of writing CFTC regulations to supplement the existing regulations of the Federal Reserve System,  the Options Clearing Corporation, and the National Association of Insurance Commissioners  Born was particularly concerned about swaps, financial instruments that are traded over the counter between banks, insurance companies or other funds or companies, and thus have no transparency except to the two counterparties and the counterparties' regulators, if any   CFTC regulation was strenuously opposed by Federal Reserve chairman Alan Greenspan, and by Treasury Secretaries Robert Rubin and Lawrence Summers  On May 7, 1998, former SEC Chairman Arthur Levitt joined Rubin and Greenspan in objecting to the issuance of the CFTC's concept release
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



ground_truth: Watt, one joule per second.

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 award did Brooksley Born receive in 2009?

Read the following text and answer briefly.

Context 2:  Their response dismissed Born's analysis and focused on the hypothetical possibility that CFTC regulation of swaps and other OTC derivative instruments could create a "legal uncertainty" regarding such financial instruments,  hypothetically reducing the value of the instruments  They argued that the imposition of regulatory costs would "stifle financial innovation" and encourage financial capital to transfer its  transactions offshore  The disagreement between Born and the Executive Office's top economic policy advisors has been described not only as a classic Washington turf war, but also a war of ideologies,  insofar as it is possible to argue that Born's actions were consistent with Keynesian and neoclassical economics while Greenspan, Rubin, Levitt, and Summers consistently espoused neoliberal, and neoconservative policies 

In 1998, a trillion-dollar hedge fund called Long Term Capital Management (LTCM) was near collapse   Using mathematical models to calculate debt risk, LTCM used derivatives to leverage $5 billion into more than $1 trillion, doing business with fifteen of Wall Street's largest financial institutions   The derivative transactions were not regulated, nor were investors able to evaluate LTCM's exposures   Born stated, "I thought that LTCM was exactly what I had been worried about"   In the last weekend of September 1998, the President's working group was told that the entire American economy hung in the balance   After intervention by the Federal Reserve, the crisis was averted   In congressional hearings into the crisis, Greenspan acknowledged that language had been introduced into an agriculture bill that would prevent CFTC from regulating the derivatives which were at the center of the crisis that threatened the US economy   U S  Representative Maurice Hinchey (D-NY) asked "How many more failures do you think we'd have to have before some regulation in this area might be appropriate?"  In response, Greenspan brushed aside the substance of Born's warnings with the simple assertion that "the degree of supervision of regulation of the over-the-counter derivatives market is quite adequate to maintain a degree of stability in the system"  Born's warning was that there wasn't any regulation of them   Born's chief of staff, Michael Greenberger summed up Greenspan's position this way: "Greenspan didn't believe that fraud was something that needed to be enforced, and he assumed she probably did  And of course, she did "  Under heavy pressure from the financial lobby, legislation prohibiting regulation of derivatives by Born's agency was passed by the Congress   Born resigned on June 1, 1999 

The derivatives market continued to grow yearly throughout both terms of George W  Bush's administration  On September 15, 2008, the bankruptcy of Lehman Brothers forced a broad recognition of a financial crisis in both the US and world capital markets   As Lehman Brothers' failure temporarily reduced financial capital's confidence, a number of newspaper articles and television programs suggested that the failure's possible causes included the conflict between the CFTC and the other regulators Faiola, Anthony, Nakashima, Ellen and Drew, Jill  The Crash: Risk and Regulation - What Went Wrong, The Washington Post, October 15, 2008 

Born declined to publicly comment on the unfolding 2008 crisis until March 2009, when she said: "The market grew so enormously, with so little oversight and regulation, that it made the financial crisis much deeper and more pervasive than it otherwise would have been " She also lamented the influence of Wall Street lobbyists on the process and the refusal of regulators to discuss even modest reforms 

An October 2009 Frontline documentary titled "The Warning"  described Born's thwarted efforts to regulate and bring transparency to the derivatives market, and the continuing opposition thereto  The program concluded with an excerpted interview with Born sounding another warning: "I think we will have continuing danger from these markets and that we will have repeats of the financial crisis -- may differ in details but there will be significant financial downturns and disasters attributed to this regulatory gap, over and over, until we learn from experience "

In 2009 Born, along with Sheila Bair of the FDIC, was awarded the John F  Kennedy Profiles in Courage Award in recognition of the "political courage she demonstrated in sounding early warnings about conditions that contributed" to the 2007-08 financial crisis   According to Caroline Kennedy, "Brooksley Born recognized that the financial security of all Americans was being put at risk by the greed, negligence and opposition of  powerful and well connected interests     The catastrophic financial events of recent months have  proved them [Born and Sheila Bair] right "  One member of the President's working group had a change of heart about Brooksley Born   SEC Chairman Arthur Levitt stated "I've come to know her as one of the most capable, dedicated, intelligent and committed public servants that I have ever come to know", adding that "I could have done much better  I could have made a difference" in response to her warnings 

In 2010, a documentary film Inside Job further alleged that derivatives regulation was ineffective from the Clinton administration on  Along with fellow whistleblower, former IMF Chief Economist Raghuram Rajan, who was also scorned by the economic establishment, Brooksley Born was cited as one of the authorities arguing that financial derivatives increase economic risk 

 Personal life 
Born is married to Alexander E  Bennett (also retired from Arnold & Porter)   She has five adult children - two from a previous marriage to Jacob Landau and three stepchildren  Notably, Born was named a partner at Arnold & Porter while working part-time so she could raise her two young children   When both of her children were school-age, Born returned to practice full-time 

References

External links
Attorney profile at Arnold & Porter
Brooksley Born (2009 Winner) of the Profiles in Courage Award, with acceptance speech transcript and NECN video

Profile at MarketsWiki
Speeches and statements
"Testimony Of Brooksley Born Chairperson of the CFTC Concerning The Over-The-Counter Derivatives Market", before the House Committee On Banking And Financial Services, July 24, 1998 
"The Lessons of Long Term Capital Management L P ", Remarks of Brooksley Born, Chairperson of the CFTC, Chicago-Kent-IIT Commodities Law Institute, Chicago, Illinois, October 15, 1998 
 Interview: Brooksley Born for "PBS Frontline: The Warning", PBS, (streaming VIDEO 1 hour), October 20, 2009 
Articles
Manuel Roig-Franzia  "Credit Crisis Cassandra:Brooksley Born's Unheeded Warning Is a Rueful Echo 10 Years On", The Washington Post, May 26, 2009
 Taibbi, Matt  "The Great American Bubble Machine", Rolling Stone'', July 9–23, 2009

1940 births
American women lawyers
Arnold & Porter people
Clinton administration personnel
Columbus School of Law faculty
Commodity Futures Trading Commission personnel
Heads of United States federal agencies
Lawyers from San Francisco
Living people
Stanford Law School alumni
21st-century American women
Stanford University alumni
Context 3: , 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



ground_truth: In 2009, Brooksley Born received the John F. Kennedy Profiles in Courage Award.

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