text:
# Josiah Willard Gibbs

{{#invoke:Hatnote|hatnote}} Template:Featured article Template:Infobox scientist

Josiah Willard Gibbs (February 11, 1839 – April 28, 1903) was an American scientist who made important theoretical contributions to physics, chemistry, and mathematics. His work on the applications of thermodynamics was instrumental in transforming physical chemistry into a rigorous deductive science. Together with James Clerk Maxwell and Ludwig Boltzmann, he created statistical mechanics (a term that he coined), explaining the laws of thermodynamics as consequences of the statistical properties of large ensembles of particles. Gibbs also worked on the application of Maxwell's equations to problems in physical optics. As a mathematician, he invented modern vector calculus (independently of the British scientist Oliver Heaviside, who carried out similar work during the same period).

In 1863, Yale awarded Gibbs the first American doctorate in engineering. After a three-year sojourn in Europe, Gibbs spent the rest of his career at Yale, where he was professor of mathematical physics from 1871 until his death. Working in relative isolation, he became the earliest theoretical scientist in the United States to earn an international reputation and was praised by Albert Einstein as "the greatest mind in American history".[1] In 1901 Gibbs received what was then considered the highest honor awarded by the international scientific community, the Copley Medal of the Royal Society of London,[1] "for his contributions to mathematical physics".[2]

Commentators and biographers have remarked on the contrast between Gibbs's quiet, solitary life in turn of the century New England and the great international impact of his ideas. Though his work was almost entirely theoretical, the practical value of Gibbs's contributions became evident with the development of industrial chemistry during the first half of the 20th century. According to Robert A. Millikan, in pure science Gibbs "did for statistical mechanics and for thermodynamics what Laplace did for celestial mechanics and Maxwell did for electrodynamics, namely, made his field a well-nigh finished theoretical structure."[3]

## Biography

### Family background

Gibbs belonged to an old Yankee family that had produced distinguished American clergymen and academics since the 17th century. He was the fourth of five children and the only son of Josiah Willard Gibbs and his wife Mary Anna, née Van Cleve. On his father's side, he was descended from Samuel Willard, who served as acting President of Harvard College from 1701 to 1707. On his mother's side, one of his ancestors was the Rev. Jonathan Dickinson, the first president of the College of New Jersey (later Princeton University). Gibbs's given name, which he shared with his father and several other members of his extended family, derived from his ancestor Josiah Willard, who had been Secretary of the Province of Massachusetts Bay in the 18th century.[4]

The elder Gibbs was generally known to his family and colleagues as "Josiah", while the son was called "Willard".[5] Josiah Gibbs was a linguist and theologian who served as professor of sacred literature at Yale Divinity School from 1824 until his death in 1861. He is chiefly remembered today as the abolitionist who found an interpreter for the African passengers of the ship Amistad, allowing them to testify during the trial that followed their rebellion against being sold as slaves.[6]

### Early years

Willard Gibbs as a student, circa 1855

Willard Gibbs was educated at the Hopkins School and entered Yale College in 1854, aged 15. He graduated in 1858 near the top of his class, and was awarded prizes for excellence in mathematics and Latin.[7] He remained at Yale as a graduate student at the Sheffield Scientific School. At age 19, soon after his graduation from college, Gibbs was inducted into the Connecticut Academy of Arts and Sciences, a scholarly institution composed primarily of members of the Yale faculty.[8]

Relatively few documents from the period survive and it is impossible to reconstruct the details of Gibbs's early career with precision.[9] In the opinion of biographers, Gibbs's principal mentor and champion, both at Yale and in the Connecticut Academy, was probably the astronomer and mathematician Hubert Anson Newton, a leading authority on meteors, who remained Gibbs's lifelong friend and confidant.[8][9] After the death of his father in 1861, Gibbs inherited enough money to make him financially independent.[10]

Recurrent pulmonary trouble ailed the young Gibbs and his doctors were concerned that he might be susceptible to tuberculosis, which had killed his mother.[9] He also suffered from astigmatism, whose treatment was then still largely unfamiliar to oculists, so that Gibbs had to diagnose himself and grind his own lenses.[11][12] Though in later years he used glasses only for reading or other close work,[11] Gibbs's delicate health and imperfect eyesight probably explain why he did not volunteer to fight in the Civil War of 1861–65.[13] He was not conscripted and he remained at Yale for the duration of the war.[14]

Gibbs during his time as a tutor at Yale[15]

In 1863, Gibbs received the first Ph.D. degree in engineering granted in the US, for a thesis entitled "On the Form of the Teeth of Wheels in Spur Gearing", in which he used geometrical techniques to investigate the optimum design for gears.[16] This was also only the fifth Ph.D. granted in the US in any subject.[16] After graduation, Gibbs was appointed as tutor at the College for a term of three years. During the first two years he taught Latin and during the third "natural philosophy" (i.e., physics).[4] In 1866 he patented a design for a railway brake[17] and read a paper before the Connecticut Academy, entitled "The Proper Magnitude of the Units of Length", in which he proposed a scheme for rationalizing the system of units of measurement used in mechanics.[18]

After his term as tutor ended, Gibbs traveled to Europe with his sisters. They spent the winter of 1866–67 in Paris, where Gibbs attended lectures at the Sorbonne and the Collège de France, given by such distinguished mathematical scientists as Joseph Liouville and Michel Chasles.[19] Having undertaken a punishing regime of study, Gibbs caught a serious cold and a doctor, fearing tuberculosis, advised him to rest on the Riviera, where he and his sisters spent several months and where he made a full recovery.[20]

Moving to Berlin, Gibbs attended the lectures taught by mathematicians Karl Weierstrass and Leopold Kronecker, as well as by chemist Heinrich Gustav Magnus.[21] In August 1867, Gibbs's sister Julia was married in Berlin to Addison Van Name, who had been Gibbs's classmate at Yale. The newly married couple returned to New Haven, leaving Gibbs and his sister Anna in Germany.[22] In Heidelberg, Gibbs was exposed to the work of physicists Gustav Kirchhoff and Hermann von Helmholtz, and chemist Robert Bunsen. At the time, German academics were the leading authorities in the natural sciences, especially chemistry and thermodynamics.[23]

Gibbs returned to Yale in June 1869 and briefly taught French to engineering students.[24] It was probably also around this time that he worked on a new design for a steam-engine governor, his last significant investigation in mechanical engineering.[25][26] In 1871 he was appointed Professor of Mathematical Physics at Yale, the first such professorship in the United States. Gibbs, who had independent means and had yet to publish anything, was assigned to teach graduate students exclusively and was hired without salary.[27] Unsalaried teaching positions were common in German universities, on which the system of graduate scientific instruction at Yale was then being modeled.[28]

### Middle years

Maxwell's sketch of the lines of constant temperature and pressure, made in preparation for his construction of a solid model based on Gibbs's definition of a thermodynamic surface for water (see Maxwell's thermodynamic surface)

Gibbs published his first work in 1873, at the unusually advanced age of 34.[7] His papers on the geometric representation of thermodynamic quantities appeared in the Transactions of the Connecticut Academy. This journal had few readers capable of understanding Gibbs's work, but he shared reprints with correspondents in Europe and received an enthusiastic response from James Clerk Maxwell at Cambridge. Maxwell even made, with his own hands, a clay model illustrating Gibbs's construct. He then produced three plaster casts of his model and mailed one to Gibbs. That cast is on display at the Yale physics department.[29]

Maxwell included a chapter on Gibbs's work in the next edition of his Theory of Heat, published in 1875. He explained the usefulness of Gibbs's graphical methods in a lecture to the Chemical Society of London and even referred to it in the article on "Diagrams" that he wrote for the Encyclopædia Britannica.[30] Prospects of collaboration between him and Gibbs were cut short by Maxwell's early death in 1879, aged 48. The joke later circulated in New Haven that "only one man lived who could understand Gibbs's papers. That was Maxwell, and now he is dead."[31]

Gibbs then extended his thermodynamic analysis to multi-phase chemical systems (i.e., to systems composed of more than one kind of matter) and considered a variety of concrete applications. He described that research in a monograph titled "On the Equilibrium of Heterogeneous Substances", published by the Connecticut Academy in two parts that appeared respectively in 1875 and 1878. That work, which covers about three hundred pages and contains exactly seven hundred numbered mathematical equations,[32] begins with a quotation from Rudolf Clausius that expresses what would later be called the first and second laws of thermodynamics: "The energy of the world is constant. The entropy of the world tends towards a maximum."[33]

Gibbs's monograph rigorously and ingeniously applied his thermodynamic techniques to the interpretation of physico-chemical phenomena, explaining and relating what had previously been a mass of isolated facts and observations.[34] The work has been described as "the Principia of thermodynamics" and as a work of "practically unlimited scope".[32] Wilhelm Ostwald, who translated Gibbs's monograph into German, referred to Gibbs as the "founder of chemical energetics".[35] According to modern commentators,

It is universally recognised that its publication was an event of the first importance in the history of chemistry ... Nevertheless it was a number of years before its value was generally known, this delay was due largely to the fact that its mathematical form and rigorous deductive processes make it difficult reading for anyone, and especially so for students of experimental chemistry whom it most concerns.

— J. J. O'Connor and E. F. Robertson, 1997[7]

Gibbs continued to work without pay until 1880, when the new Johns Hopkins University in Baltimore, Maryland offered him a position paying $3,000 per year. In response, Yale offered him an annual salary of$2,000, which he was content to accept.[36]

### Later years

Yale's Sloane Physical Laboratory, as it stood between 1882 and 1931 at the current location of Jonathan Edwards College. Gibbs's office was on the second floor, to the right of the tower in the picture.[37]

From 1880 to 1884, Gibbs worked on developing the exterior algebra of Hermann Grassmann into a vector calculus well-suited to the needs of physicists. With this object in mind, Gibbs distinguished between the dot and cross products of two vectors and introduced the concept of dyadics. Similar work was carried out independently, and at around the same time, by the British mathematical physicist and engineer Oliver Heaviside. Gibbs sought to convince other physicists of the convenience of the vectorial approach over the quaternionic calculus of William Rowan Hamilton, which was then widely used by British scientists. This led him, in the early 1890s, to a controversy with Peter Guthrie Tait and others in the pages of Nature.[4]

Gibbs's lecture notes on vector calculus were privately printed in 1881 and 1884 for the use of his students, and were later adapted by Edwin Bidwell Wilson into a textbook, Vector Analysis, published in 1901.[4] That book helped to popularize the "del" notation that is widely used today in electrodynamics and fluid mechanics. In other mathematical work, he re-discovered the "Gibbs phenomenon" in the theory of Fourier series (which, unbeknownst to him and to later scholars, had been described fifty years before by an obscure English mathematician, Henry Wilbraham).[38]

The sine integral function, which gives the overshoot associated with the Gibbs phenomenon for the Fourier series of a step function on the real line

From 1882 to 1889, Gibbs wrote five papers on physical optics, in which he investigated birefringence and other optical phenomena and defended Maxwell's electromagnetic theory of light against the mechanical theories of Lord Kelvin and others.[4] In his work on optics just as much as in his work on thermodynamics, Gibbs deliberately avoided speculating about the microscopic structure of matter,[39] which proved a wise course in view of the revolutionary developments in quantum mechanics that began around the time of his death.[40]

Gibbs coined the term statistical mechanics and introduced key concepts in the corresponding mathematical description of physical systems, including the notions of chemical potential (1876), statistical ensemble (1878), and phase space (1902).[41][42] Gibbs's derivation of the phenomenological laws of thermodynamics from the statistical properties of systems with many particles was presented in his highly-influential textbook Elementary Principles in Statistical Mechanics, published in 1902, a year before his death.[41]

Gibbs's retiring personality and intense focus on his work limited his accessibility to students. His principal protégé was Edwin Bidwell Wilson, who nonetheless explained that "except in the classroom I saw very little of Gibbs. He had a way, toward the end of the afternoon, of taking a stroll about the streets between his study in the old Sloane Laboratory and his home—a little exercise between work and dinner—and one might occasionally come across him at that time."[43] Gibbs did supervise the doctoral thesis on mathematical economics written by Irving Fisher in 1891.[44] After Gibbs's death, Fisher financed the publication of his Collected Works.[45] Another distinguished student was Lee De Forest, later a pioneer of radio technology.[46]

Gibbs died in New Haven on April 28, 1903, at the age of 64, victim of an acute intestinal obstruction.[43] A funeral was conducted two days later at his home on 121 High Street[47] and his body was buried in the nearby Grove Street Cemetery.[48] In May, Yale organized a memorial meeting at the Sloane Laboratory. The eminent British physicist J. J. Thomson was in attendance and delivered a brief address.[49]

### Personal life and character

Photograph taken around 1895. According to his student Lynde Wheeler, of the existing portraits this is the most faithful to Gibbs's kindly habitual expression.[50]

Gibbs never married, living all his life in his childhood home with his sister Julia and her husband Addison Van Name, who was the Yale librarian. Except for his customary summer vacations in the Adirondacks (at Keene Valley, New York) and later at the White Mountains (in Intervale, New Hampshire),[51] his sojourn in Europe in 1866–69 was almost the only time that Gibbs spent outside New Haven.[4] He joined Yale's College Church (a Congregational church) at the end of his freshman year[51][52] and remained a regular attendant for the rest of his life.[53] He generally voted for the Republican candidate in presidential elections, but he supported Grover Cleveland, a conservative Democrat.[54] Little else is known of his religious or political views, which he kept to himself.[53]

Gibbs did not produce a substantial personal correspondence and many of his letters were later lost or destroyed.[55] Beyond the technical writings concerning his research, he published only two other pieces: a brief obituary for Rudolf Clausius, one of the founders of the mathematical theory of thermodynamics, and a longer biographical memoir of his mentor at Yale, H. A. Newton.[56] In Edward Bidwell Wilson's view,

Gibbs was not an advertiser for personal renown nor a propagandist for science; he was a scholar, scion of an old scholarly family, living before the days when research had become search ... Gibbs was not a freak, he had no striking ways, he was a kindly dignified gentleman.

— E. B. Wilson, 1931[43]

According to Lynde Wheeler, who had been Gibbs's student at Yale, in his later years Gibbs

was always neatly dressed, usually wore a felt hat on the street, and never exhibited any of the physical mannerisms or eccentricities sometimes thought to be inseparable from genius ... His manner was cordial without being effusive and conveyed clearly the innate simplicity and sincerity of his nature.

— Lynde Wheeler, 1951[50]

He was a careful investor and financial manager, and at his death in 1903 his estate was valued at $100,000.[51] For many years he served as trustee, secretary, and treasurer of his alma mater, the Hopkins School.[57] US President Chester A. Arthur appointed him as one of the commissioners to the National Conference of Electricians, which convened in Philadelphia in September 1884, and Gibbs presided over one of its sessions.[51] A keen and skilled horseman,[58] Gibbs was seen habitually in New Haven driving his sister's carriage.[59] In an obituary published in the American Journal of Science, Gibbs's former student Henry A. Bumstead referred to Gibbs's personal character: Unassuming in manner, genial and kindly in his intercourse with his fellow-men, never showing impatience or irritation, devoid of personal ambition of the baser sort or of the slightest desire to exalt himself, he went far toward realizing the ideal of the unselfish, Christian gentleman. In the minds of those who knew him, the greatness of his intellectual achievements will never overshadow the beauty and dignity of his life. —H. A. Bumstead, 1903[4] ## Major scientific contributions ### Chemical thermodynamics Graphical representation of the free energy of a body, from the latter of the papers published by Gibbs in 1873. This shows a plane of constant volume, passing through the point A that represents the body's initial state. The curve MN is the section of the "surface of dissipated energy". AD and AE are, respectively, the energy (ε) and entropy (η) of the initial state. AB is the "available energy" (now called the Helmholtz free energy) and AC the "capacity for entropy" (i.e., the amount by which the entropy can be increased without changing the energy or volume). Gibbs's papers from the 1870s introduced the idea of expressing the internal energy U of a system in terms of the entropy S, in addition to the usual state-variables of volume V, pressure p, and temperature T.[41] He also introduced the concept of the chemical potential ${\displaystyle \mu }$ of a given chemical species, defined to be the rate of the increase in U associated with the increase in the number N of molecules of that species (at constant entropy and volume). Thus, it was Gibbs who first combined the first and second laws of thermodynamics[41] by expressing the infinitesimal change in the energy of a system in the form: where the sum in the last term is over the different chemical species. By taking the Legendre transform of this expression, he defined the concepts of enthalpy and "free energy", including what is now known as the "Gibbs free energy" (a thermodynamic potential which is especially useful to chemists since it determines whether a reaction will proceed spontaneously at a fixed temperature and pressure). In a similar way, he also obtained what later came to be known as the "Gibbs–Duhem equation".[34][41] The publication of the paper "On the Equilibrium of Heterogeneous Substances" (1874–78) is now regarded as a landmark in the development of physical chemistry.[7] In it, Gibbs developed a rigorous mathematical theory for various transport phenomena, including adsorption, electrochemistry, and the Marangoni effect in fluid mixtures.[34] He also formulated the phase rule for the number F of variables that may be independently controlled in an equilibrium mixture of C components existing in P phases. Awareness of this rule led to the widespread use of phase diagrams by chemists.[60] ### Statistical mechanics Together with James Clerk Maxwell and Ludwig Boltzmann, Gibbs founded "statistical mechanics", a term that he coined to identify the branch of theoretical physics that accounts for the observed thermodynamic properties of systems in terms of the statistics of large ensembles of particles. He introduced the concept of phase space and used it to define the microcanonical, canonical, and grand canonical ensembles, thus obtaining a more general formulation of the statistical properties of many-particle systems than Maxwell and Boltzmann had achieved before him.[41][42] According to Henri Poincaré, writing in 1904, even though Maxwell and Boltzmann had previously explained the irreversibility of macroscopic physical processes in probabilistic terms, "the one who has seen it most clearly, in a book too little read because it is a little difficult to read, is Gibbs, in his Elementary Principles of Statistical Mechanics."[61] Gibbs's analysis of irreversibility, and his formulation of Boltzmann's H-theorem and of the ergodic hypothesis, were major influences on the mathematical physics of the 20th century.[40][62] Gibbs was well aware that the application of the equipartition theorem to large systems of classical particles failed to explain the measurements of the specific heats of both solids and gases, and he argued that this was evidence of the danger of basing thermodynamics on "hypotheses about the constitution of matter".[42] Gibbs's own framework for statistical mechanics was so carefully constructed that it could be carried over almost intact after the discovery that the microscopic laws of nature obey quantum rules, rather than the classical laws known to Gibbs and to his contemporaries.[7] His resolution of the so-called "Gibbs paradox", about the entropy of the mixing of gases, is now often cited as a prefiguration of the indistinguishability of particles required by quantum physics.[63] ### Vector analysis Diagram showing the magnitude and direction of the cross product of two vectors, in the notation introduced by Gibbs British scientists, including Maxwell, had relied on Hamilton's quaternions in order to express the dynamics of physical quantities, like the electric and magnetic fields, having both a magnitude and a direction in three-dimensional space. Gibbs, however, noted that the product of quaternions always had to be separated into two parts: a one-dimensional (scalar) quantity and a three-dimensional vector, so that the use of quaternions introduced mathematical complications and redundancies that could be avoided in the interest of simplicity and to facilitate teaching. He therefore proposed defining distinct dot and cross products for pairs of vectors and introduced the now common notation for them. He was also largely responsible for the development of the vector calculus techniques still used today in electrodynamics and fluid mechanics. While he was working on vector analysis in the late 1870s, Gibbs discovered that his approach was similar to the one that Grassmann had taken in his "multiple algebra".[64] Gibbs then sought to publicize Grassmann's work, stressing that it was both more general and historically prior to Hamilton's quaternionic algebra. To establish Grassmann's priority, Gibbs convinced Grassmann's heirs to seek the publication in Germany of the essay on tides that Grassmann had submitted in 1840 to the faculty at the University of Berlin, in which he had first introduced the notion of what would later be called a vector space.[65] As Gibbs had advocated in the 1880s and 1890s, quaternions were eventually all but abandoned by physicists in favor of the vectorial approach developed by him and, independently, by Oliver Heaviside. Gibbs applied his vector methods to the determination of planetary and comet orbits. He also developed the concept of mutually reciprocal triads of vectors that later proved to be of importance in crystallography.[66] ### Physical optics A calcite crystal produces birefringence (or "double refraction") of light, a phenomenon which Gibbs explained using Maxwell's equations for electromagnetic phenomena. Though Gibbs's research on physical optics is less well known today than his other work, it made a significant contribution to classical electromagnetism by applying Maxwell's equations to the theory of optical processes such as birefringence, dispersion, and optical activity.[4][39] In that work, Gibbs showed that those processes could be accounted for by Maxwell's equations without any special assumptions about the microscopic structure of matter or about the nature of the medium in which electromagnetic waves were supposed to propagate (the so-called luminiferous ether). Gibbs also stressed that the absence of a longitudinal electromagnetic wave, which is needed to account for the observed properties of light, is automatically guaranteed by Maxwell's equations (by virtue of what is now called their "gauge invariance"), whereas in mechanical theories of light, such as Lord Kelvin's, it must be imposed as an ad hoc condition on the properties of the aether.[39] In his last paper on physical optics, Gibbs concluded that it may be said for the electrical theory [of light] that it is not obliged to invent hypotheses, but only to apply the laws furnished by the science of electricity, and that it is difficult to account for the coincidences between the electrical and optical properties of media unless we regard the motions of light as electrical. —J. W. Gibbs, 1889[4] Shortly afterwards, the electromagnetic nature of light was demonstrated by the experiments of Heinrich Hertz in Germany.[67] ## Scientific recognition Gibbs worked at a time when there was little tradition of rigorous theoretical science in the United States. His research was not easily understandable to his students or his colleagues and he made no effort to popularize his ideas or to simplify their exposition to make them more accessible.[7] His seminal work on thermodynamics was published mostly in the Transactions of the Connecticut Academy, a journal edited by his librarian brother-in-law, which was little read in the USA and even less so in Europe. When Gibbs submitted his long paper on the equilibrium of heterogeneous substances to the Academy, both Elias Loomis and H. A. Newton protested that they did not understand Gibbs's work at all, but they helped to raise the money needed to pay for the typesetting of the many mathematical symbols in the paper. Several Yale faculty members, as well as business and professional men in New Haven, contributed funds for that purpose.[68] Even though it had been immediately embraced by Maxwell, Gibbs's graphical formulation of the laws of thermodynamics only came into widespread use in the mid 20th century, with the work of László Tisza and Herbert Callen.[69] According to James Gerald Crowther, in his later years [Gibbs] was a tall, dignified gentleman, with a healthy stride and ruddy complexion, performing his share of household chores, approachable and kind (if unintelligible) to students. Gibbs was highly esteemed by his friends, but American science was too preoccupied with practical questions to make much use of his profound theoretical work during his lifetime. He lived out his quiet life at Yale, deeply admired by a few able students but making no immediate impress on American science commensurate with his genius. — J. G. Crowther, 1937[7] Burlington House, site of the Royal Society of London, in 1873 On the other hand, Gibbs did receive the major honors then possible for an academic scientist in the US. He was elected to the National Academy of Sciences in 1879 and received the 1880 Rumford Prize from the American Academy of Arts and Sciences for his work on chemical thermodynamics.[70] He was also awarded honorary doctorates by Princeton University and Williams College.[4] In Europe, Gibbs was inducted as honorary member of the London Mathematical Society in 1892 and as a foreign member of the Royal Society in 1897. He was elected as corresponding member of the Prussian and French Academies of Science and received honorary doctorates from the universities of Erlangen and Christiania (now Oslo).[4] The Royal Society further honored Gibbs in 1901 with the Copley Medal, then regarded as the highest international award in the natural sciences,[1] remarking that he had been "the first to apply the second law of thermodynamics to the exhaustive discussion of the relation between chemical, electrical and thermal energy and capacity for external work."[35] Gibbs, who remained in New Haven, was represented at the award ceremony by Commander Richardson Clover, the US naval attaché in London.[71] In his autobiography, mathematician Gian-Carlo Rota tells of casually browsing the mathematical stacks of Sterling Library and stumbling on a handwritten mailing list, attached to some of Gibbs's course notes, which listed over two hundred notable scientists of his day, including Poincaré, Boltzmann, David Hilbert, and Ernst Mach. From this, Rota concluded that Gibbs's work was better known among the scientific elite of his day than the published material suggests.[72] Lynde Wheeler reproduces that mailing list in an appendix to his biography of Gibbs.[73] That Gibbs succeeded in interesting his European correspondents in his work is demonstrated by the fact that his monograph "On the Equilibrium of Heterogeneous Substances" was translated into German (then the leading language for chemistry) by Wilhelm Ostwald in 1892 and into French by Henri Louis Le Châtelier in 1899.[74] ## Influence Gibbs's most immediate and obvious influence was on physical chemistry and statistical mechanics, two disciplines which he greatly helped to found. During Gibbs's lifetime, his phase rule was experimentally validated by Dutch chemist H. W. Bakhuis Roozeboom, who showed how to apply it in a variety of situations, thereby assuring it of widespread use.[75] In industrial chemistry, Gibbs's thermodynamics found many applications during the early 20th century, from electrochemistry to the development of the Haber process for the synthesis of ammonia.[76] When Dutch physicist J. D. van der Waals received the 1910 Nobel Prize "for his work on the equation of state for gases and liquids" he acknowledged the great influence of Gibbs's work on that subject.[77] Max Planck received the 1918 Nobel Prize for his work on quantum mechanics, particularly his 1900 paper on Planck's law for quantized black-body radiation. That work was based largely on the thermodynamics of Kirchhoff, Boltzmann, and Gibbs. Planck declared that Gibbs's name "not only in America but in the whole world will ever be reckoned among the most renowned theoretical physicists of all times."[78] Title page of Gibbs's Elementary Principles in Statistical Mechanics, one of the founding documents of that discipline, published in 1902 The first half of the 20th century saw the publication of two influential textbooks that soon came to be regarded as founding documents of chemical thermodynamics, both of which used and extended Gibbs's work in that field: these were Thermodynamics and the Free Energy of Chemical Processes (1923), by Gilbert N. Lewis and Merle Randall, and Modern Thermodynamics by the Methods of Willard Gibbs (1933), by Edward A. Guggenheim.[79] Under the influence of Lewis, William Giauque (who had originally wanted to be a chemical engineer) went on to become a professor of chemistry at Berkeley and won the 1949 Nobel Prize in Chemistry for his investigations into the properties of matter at temperatures close to absolute zero, studies guided by the third law of thermodynamics.[80] Gibbs's work on statistical ensembles, as presented in his 1902 textbook, has had a great impact in both theoretical physics and in pure mathematics.[40][62] According to mathematical physicist Arthur Wightman, It is one of the striking features of the work of Gibbs, noticed by every student of thermodynamics and statistical mechanics, that his formulations of physical concepts were so felicitously chosen that they have survived 100 years of turbulent development in theoretical physics and mathematics. — A. S. Wightman, 1990[40] Initially unaware of Gibbs's contributions in that field, Albert Einstein wrote three papers on statistical mechanics, published between 1902 and 1904. After reading Gibbs's textbook (which was translated into German by Ernst Zermelo in 1905), Einstein declared that Gibbs's treatment was superior to his own and explained that he would not have written those papers if he had known Gibbs's work.[81] Gibbs's early papers on the use of graphical methods in thermodynamics reflect a powerfully original understanding of what mathematicians would later call "convex analysis",[82] including ideas that, according to Barry Simon, "lay dormant for about seventy-five years".[83] Important mathematical concepts based on Gibbs's work on thermodynamics and statistical mechanics include the Gibbs lemma in game theory, the Gibbs inequality and Gibbs algorithm in information theory, as well as Gibbs sampling in computational statistics. The development of vector calculus was Gibbs's other great contribution to mathematics. The publication in 1901 of E. B. Wilson's textbook Vector Analysis, based on Gibbs's lectures at Yale, did much to propagate the use of vectorial methods and notation in both mathematics and theoretical physics, definitively displacing the quaternions that had until then been dominant in the scientific literature.[84] At Yale, Gibbs was also mentor to Lee De Forest, who went on to invent the triode amplifier and has been called the "father of radio".[85] De Forest credited Gibbs's influence for the realization "that the leaders in electrical development would be those who pursued the higher theory of waves and oscillations and the transmission by these means of intelligence and power."[46] Another student of Gibbs who played a significant role in the development of radio technology was Lynde Wheeler.[86] Gibbs also had an indirect influence on mathematical economics. He supervised the thesis of Irving Fisher, who received the first Ph.D. in economics from Yale in 1891. In that work, published in 1892 as Mathematical Investigations in the Theory of Value and Prices, Fisher drew a direct analogy between Gibbsian equilibrium in physical and chemical systems, and the general equilibrium of markets, and he used Gibbs's vectorial notation.[44][87] Gibbs's protegé Edwin Bidwell Wilson became, in turn, a mentor to leading American economist and Nobel Laureate Paul Samuelson.[88] In 1947, Samuelson published Foundations of Economic Analysis, based on his doctoral dissertation, in which he used as epigraph a remark attributed to Gibbs: "Mathematics is a language." Samuelson later explained that in his understanding of prices his "debts were not primarily to Pareto or Slutsky, but to the great thermodynamicist, Willard Gibbs of Yale."[89] Mathematician Norbert Wiener cited Gibbs's use of probability in the formulation of statistical mechanics as "the first great revolution of twentieth century physics" and as a major influence on his conception of cybernetics. Wiener explained in the preface to his book The Human Use of Human Beings that it was "devoted to the impact of the Gibbsian point of view on modern life, both through the substantive changes it has made to working science, and through the changes it has made indirectly in our attitude to life in general."[90] ## Commemoration Bronze memorial tablet, originally installed in 1912 at the Sloane Physics Laboratory, now at the entrance to the Josiah Willard Gibbs Laboratories, Yale University. When the German physical chemist Walther Nernst visited Yale in 1906 to give the Silliman lecture, he was surprised to find no tangible memorial for Gibbs. He therefore donated his$500 lecture fee to the university to help pay for a suitable monument. This was finally unveiled in 1912, in the form of a bronze bas-relief by sculptor Lee Lawrie, installed in the Sloane Physics Laboratory.[91] In 1910, the American Chemical Society established the Willard Gibbs Award for eminent work in pure or applied chemistry.[92] In 1923, the American Mathematical Society endowed the Josiah Willard Gibbs Lectureship, "to show the public some idea of the aspects of mathematics and its applications".[93]

Building housing the Josiah Willard Gibbs Laboratories, at Yale University's Science Hill

In 1945, Yale University created the J. Willard Gibbs Professorship in Theoretical Chemistry, held until 1973 by Lars Onsager. Onsager, who much like Gibbs focused on applying new mathematical ideas to problems in physical chemistry, won the 1968 Nobel Prize in chemistry.[94] In addition to establishing the Josiah Willard Gibbs Laboratories and the J. Willard Gibbs Assistant Professorship in Mathematics, Yale has hosted two symposia dedicated to Gibbs's life and work, one in 1989 and another on the centenary of his death, in 2003.[95] Rutgers University endowed a J. Willard Gibbs Professorship of Thermomechanics, held as of 2014 by Bernard Coleman.[96]

Gibbs was elected in 1950 to the Hall of Fame for Great Americans.[97] The oceanographic research ship USNS Josiah Willard Gibbs (T-AGOR-1) was in service with the United States Navy from 1958 to 1971.[98] The Gibbs crater, near the eastern limb of the Moon, was named in the scientist's honor in 1964.[99]

Edward Guggenheim introduced the symbol G for the Gibbs free energy in 1933, and this was used also by Dirk ter Haar in 1966.[100] This notation is now universal and is recommended by the IUPAC.[101] In 1960, William Giauque and others suggested the name "gibbs" (abbreviated gbs.) for the unit of entropy, calorie / Kelvin,[102] but this usage did not become common and the corresponding SI unit, Joule / Kelvin, carries no special name.

### In literature

In 1909, the American historian and novelist Henry Adams finished an essay entitled "The Rule of Phase Applied to History", in which he sought to apply Gibbs's phase rule and other thermodynamic concepts to a general theory of human history. William James, Henry Bumstead, and others criticized both Adams's tenuous grasp of the scientific concepts that he invoked, as well as the arbitrariness of his application of those concepts as metaphors for the evolution of human thought and society.[103] The essay remained unpublished until it appeared posthumously in 1919, in The Degradation of the Democratic Dogma, edited by Henry Adams's younger brother Brooks.[104]

File:Fortune June 1946.jpg
Cover of the June 1946 issue of Fortune, by artist Arthur Lidov, showing Gibbs's thermodynamic surface of water and his formula for the phase rule

In the 1930s, feminist poet Muriel Rukeyser became fascinated by Willard Gibbs and wrote a long poem about his life and work ("Gibbs", included in the collection A Turning Wind, published in 1939), as well as a book-length biography (Willard Gibbs, 1942).[105] According to Rukeyser:

Willard Gibbs is the type of the imagination at work in the world. His story is that of an opening up which has had its effect on our lives and our thinking; and, it seems to me, it is the emblem of the naked imagination—which is called abstract and impractical, but whose discoveries can be used by anyone who is interested, in whatever "field"—an imagination which for me, more than that of any other figure in American thought, any poet, or political, or religious figure, stands for imagination at its essential points.

— Muriel Rukeyser, 1949[106]

In 1946, Fortune magazine illustrated a cover story on "Fundamental Science" with a representation of the thermodynamic surface that Maxwell had built based on Gibbs's proposal. Rukeyser had called this surface a "statue of water"[107] and the magazine saw in it "the abstract creation of a great American scientist that lends itself to the symbolism of contemporary art forms."[108] The artwork by Arthur Lidov also included Gibbs's mathematical expression of the phase rule for heterogeneous mixtures, as well as a radar screen, an oscilloscope waveform, Newton's apple, and a small rendition of a three-dimensional phase diagram.[108]

Gibbs's nephew, Ralph Gibbs Van Name, a professor of physical chemistry at Yale, was unhappy with Rukeyser's biography, in part because of her lack of scientific training. Van Name had withheld the family papers from her and, after her book was published in 1942 to positive literary but mixed scientific reviews, he tried to encourage Gibbs's former students to produce a more technically oriented biography.[109] Rukeyser's approach to Gibbs was also sharply criticized by Gibbs's former student and protégé Edwin Wilson.[110] With Van Name's and Wilson's encouragement, physicist Lynde Wheeler published a new biography of Gibbs in 1951.[111][112]

Both Gibbs and Rukeyser's biography of him figure prominently in the poetry collection True North (1997) by Stephanie Strickland.[113] In fiction, Gibbs appears as the mentor to character Kit Traverse in Thomas Pynchon's novel Against the Day (2006). That novel also prominently discusses the birefringence of Iceland spar, an optical phenomenon that Gibbs investigated.[114]

### Gibbs stamp (2005)

In 2005, the United States Postal Service issued the American Scientists commemorative postage stamp series designed by artist Victor Stabin, depicting Gibbs, John von Neumann, Barbara McClintock, and Richard Feynman. The first day of issue ceremony for the series was held on May 4 at Yale University's Luce Hall and was attended by John Marburger, scientific advisor to the President of the United States, Rick Levin, president of Yale, and family members of the scientists honored, including physician John W. Gibbs, a distant cousin of Willard Gibbs.[115]

Kenneth R. Jolls, a professor of chemical engineering at Iowa State University and an expert on graphical methods in thermodynamics, consulted on the design of the stamp honoring Gibbs.[116][117][118] The stamp identifies Gibbs as a "thermodynamicist" and features a diagram from the 4th edition of Maxwell's Theory of Heat, published in 1875, which illustrates Gibbs's thermodynamic surface for water.[117][118] Microprinting on the collar of Gibbs's portrait depicts his original mathematical equation for the change in the energy of a substance in terms of its entropy and the other state variables.[119]

## References

1. Template:Cite web
2. Template:Cite web
3. {{#invoke:Citation/CS1|citation |CitationClass=journal }}
5. Cropper 2001, p. 121
6. Template:Cite web
7. Template:Cite web
8. Rukeyser 1988, p. 104
9. Wheeler 1998, pp. 23–24
10. Rukeyser 1998, pp. 120, 142
11. Wheeler 1998, pp. 30–31
12. Rukeyser 1988, p. 143
13. Wheeler 1998, p. 30
14. Rukeyser 1998, p. 134
15. Wheeler 1998, p. 44
16. Wheeler 1998, p. 32
17. US Patent No. 53,971, "Car Brake", Apr. 17, 1866. See The Early Work of Willard Gibbs in Applied Mechanics, (New York: Henry Schuman, 1947), pp. 51–62.
18. Wheeler 1998, appendix II
19. Wheeler 1998, p. 40
20. Wheeler 1998, p. 41
21. Wheeler 1998, p. 42
22. Rukeyser 1988, p. 151
23. Rukeyser 1988, pp. 158–161
24. {{#invoke:citation/CS1|citation |CitationClass=book }}
25. Template:Cite jstor
26. Wheeler 1998, pp. 54–55
27. Rukeyser 1988, pp. 181–182
28. Wheeler 1998, pp. 57–59
29. Template:Cite web
30. Rukeyser 1988, p. 201
31. Rukeyser 1988, p. 251
32. Cropper 2001, p. 109
33. Quoted in Rukeyser 1988, p. 233
34. Wheeler 1998, ch. V
35. Template:Cite EB1911
36. Wheeler 1998, p. 91
37. Wheeler 1998, p. 86
38. Template:Cite doi
39. Wheeler 1998, ch. VIII
40. {{#invoke:citation/CS1|citation |CitationClass=book }}
41. Klein 2008
42. Wheeler 1998, ch. X
43. Wilson 1931
44. {{#invoke:Citation/CS1|citation |CitationClass=journal }}
45. {{#invoke:citation/CS1|citation |CitationClass=book }}
46. {{#invoke:Citation/CS1|citation |CitationClass=journal }}
47. Wheeler 1998, p. 197
48. Template:Cite web
49. Wheeler 1998, pp. 197–199
50. Wheeler 1998, pp. 179–180
51. Seeger 1974, pp. 15–16
52. {{#invoke:citation/CS1|citation |CitationClass=book }}
53. Wheeler, 1998, p. 16
54. {{#invoke:citation/CS1|citation |CitationClass=book }}
55. Rukeyser 1988, pp. 254, 345, 430
56. Wheeler 1998, p. 95. See also the Collected Works, vol. II
57. Wheeler, 1998, p. 144
58. Rukeyser 1988, p. 191
59. Rukeyser 1988, p. 224
60. Template:Cite web
61. {{#invoke:citation/CS1|citation |CitationClass=book }}
62. {{#invoke:citation/CS1|citation |CitationClass=book }}
63. See, e.g., {{#invoke:citation/CS1|citation |CitationClass=book }}
64. Letter by Gibbs to Victor Schlegel, quoted in Wheeler 1998, pp. 107–109
65. Wheeler 1998, pp. 113–116
66. {{#invoke:citation/CS1|citation |CitationClass=book }}
67. {{#invoke:citation/CS1|citation |CitationClass=book }}
68. Rukeyser 1998, pp. 225–226
69. Wightman 1979, pp. xiii, lxxx
70. {{#invoke:citation/CS1|citation |CitationClass=book }}
71. Rukeyser 1998, p. 345
72. {{#invoke:citation/CS1|citation |CitationClass=book }}
73. Wheeler 1998, appendix IV
74. Wheeler 1998, pp. 102–104
75. {{#invoke:citation/CS1|citation |CitationClass=book }}
76. Template:Cite doi
77. Template:Cite web
78. {{#invoke:citation/CS1|citation |CitationClass=book }}
79. {{#invoke:citation/CS1|citation |CitationClass=book }}
80. Template:Cite web
81. {{#invoke:Citation/CS1|citation |CitationClass=journal }}
82. Wightman 1979, pp. x–xxxiv
83. {{#invoke:citation/CS1|citation |CitationClass=book }}
84. {{#invoke:citation/CS1|citation |CitationClass=book }}
85. Seeger 1974, p. 18
86. Template:Cite doi
87. {{#invoke:Citation/CS1|citation |CitationClass=journal }}
88. {{#invoke:citation/CS1|citation |CitationClass=book }}
89. {{#invoke:citation/CS1|citation |CitationClass=book }}
90. Seeger 1974, p. 21
91. Template:Cite web
92. Template:Cite web
93. Template:Cite doi
94. {{#invoke:Citation/CS1|citation |CitationClass=journal }}
95. Template:Cite web
96. Template:Cite web
97. Template:Cite web
98. Template:Cite web
99. Seeger 1974, p. 96
100. Template:Cite doi
101. Template:Cite doi
102. Template:Cite jstor
103. {{#invoke:citation/CS1|citation |CitationClass=book }}
104. {{#invoke:citation/CS1|citation |CitationClass=book }}
105. Template:Cite doi
106. Rukeyser 1988, p. 203
107. {{#invoke:Citation/CS1|citation |CitationClass=journal }}
108. Template:Cite web
109. {{#invoke:Citation/CS1|citation |CitationClass=journal }}
110. Wheeler 1998, pp. ix–xiii
111. Template:Cite jstor
112. {{#invoke:citation/CS1|citation |CitationClass=book }}
113. {{#invoke:citation/CS1|citation |CitationClass=book }}
114. Template:Cite news
115. Template:Cite news
116. Template:Cite news
117. {{#invoke:Citation/CS1|citation |CitationClass=journal }}
118. {{#invoke:Citation/CS1|citation |CitationClass=journal }}

## Bibliography

### Primary

Gibbs's other papers are included in both:

• The Scientific Papers of J. Willard Gibbs, in two volumes, eds. H. A. Bumstead and R. G. Van Name, (Woodbridge, CT: Ox Bow Press, 1993 [1906]). ISBN 0-918024-77-3, ISBN 1-881987-06-X. For scans of the 1906 printing, see vol. I and vol. II.
• The Collected Works of J. Willard Gibbs, in two volumes, eds. W. R. Longley and R. G. Van Name, (New Haven: Yale University Press, 1957 [1928]). For scans of the 1928 printing, see vol. I and vol. II.

### Secondary

• H. A. Bumstead, "Josiah Willard Gibbs", American Journal of Science (ser. 4) 16, 187–202 (1903) Template:Hide in printTemplate:Only in print. Reprinted with some additions in both The Scientific Papers, vol. I, pp. xiii–xxviiii (1906) and The Collected Works of J. Willard Gibbs, vol. I, pp. xiii–xxviiii (1928). Also available here.
• D. G. Caldi and G. D. Mostow (eds.), Proceedings of the Gibbs Symposium, Yale University, May 15–17, 1989, (American Mathematical Society and American Institute of Physics, 1990).
• W. H. Cropper, "The Greatest Simplicity: Willard Gibbs", in Great Physicists, (Oxford: Oxford University Press, 2001), pp. 106–123. ISBN 0-19-517324-4
• M. J. Crowe, A History of Vector Analysis: The Evolution of the Idea of a Vectorial System, (New York: Dover, 1994 [1967]). ISBN 0-486-67910-1
• J. G. Crowther, Famous American Men of Science, (Freeport, NY: Books for Libraries Press, 1969 [1937]). ISBN 0-8369-0040-5
• F. G. Donnan and A. E. Hass (eds.), A Commentary on the Scientific Writings of J. Willard Gibbs, in two volumes, (New York: Arno, 1980 [1936]). ISBN 0-405-12544-5. Only vol I. is currently available online.
• P. Duhem, Josiah-Willard Gibbs à propos de la publication de ses Mémoires scientifiques, (Paris: A. Herman, 1908).
• C. S. Hastings, "Josiah Willard Gibbs", Biographical Memoirs of the National Academy of Sciences, 6, 373–393 (1909).
• M. J. Klein, "Gibbs, Josiah Willard", in Complete Dictionary of Scientific Biography, vol. 5, (Detroit: Charles Scriber's Sons, 2008), pp. 386–393.
• M. Rukeyser, Willard Gibbs: American Genius, (Woodbridge, CT: Ox Bow Press, 1988 [1942]). ISBN 0-918024-57-9
• R. J. Seeger, J. Willard Gibbs, American mathematical physicist par excellence, (Oxford and New York: Pergamon Press, 1974). ISBN 0-08-018013-2
• L. P. Wheeler, Josiah Willard Gibbs, The History of a Great Mind, (Woodbridge, CT: Ox Bow Press, 1998 [1951]). ISBN 1-881987-11-6
• A. S. Wightman, "Convexity and the notion of equilibrium state in thermodynamics and statistical mechanics". Published as an introduction to R. B. Israel, Convexity in the Theory of Lattice Gases, (Princeton, NJ: Princeton University Press, 1979), pp. ix–lxxxv. ISBN 0-691-08209-X
• E. B. Wilson, "Reminiscences of Gibbs by a student and colleague", Bulletin of the American Mathematical Society, 37, 401–416 (1931).


text:
# Considerations When Using Lat/Long Cords on Classifier

I'm interested in using Latitude and Longitude points as features to build a classifier model, but would like to better understand if I need to be taking any precautions when using Lat/Long cords in this way.

For example I'd like to predict a certain type of event/actor (actor1) mentioned in data below. I ran them through a number of models like KNN, NB, CART, Random Forest, etc...and got decent results, but I'm not sure if I need to take any precautions (e.g. scaling, transformations, or something else) when feeding spatial coordinates into a classifier.

Any thoughts, feedback or suggestions on how to best handle this are welcome.

• One thing we have found often works with latitude and longitude is to convert it into distance and direction ($0-2\pi$) from various relevant landmarks, in this case probably larger cities in Nigeria. Thus, if events happen near Lagos, or alternatively Kano or Ibaden (for example), it doesn't require two splits of latitude and longitude to get the city, just one split on "Lagos/not Lagos" and, farther down the tree, a split on distance. – jbowman Jan 4 at 4:22

format: plain


text:
Here's the question you clicked on:

## orb10925 Group Title What is the solution of -4x + 1 ≥ 7? 2 years ago 2 years ago

• This Question is Closed
1. pfenn1 Group Title

Subtract 1 from both sides, then divide both sides by -4 to solve for x. Remember If both sides of an inequality are multiplied by the same negative number, then:$> \text{ becomes } <$$< \text{ becomes }$etc.

2. tishara429 Group Title

x=-1.75

3. pfenn1 Group Title

It can't be x=-1.75. It has to be an inequality.


text:
Problems – We Solve Problems
Filter Problems
Showing 1 to 5 of 5 entries

#### Chessboard colouring , Examples and counterexamples. Constructive proofs , Pigeonhole principle (finite number of poits, lines etc.)

What is the minimum number of 1×1 squares that need to be drawn in order to get an image of a 25×25 square divided into 625 smaller 1×1 squares?

My Problem Set reset
No Problems selected


text:
# Mathematics Colloquia and Seminars

A number field is monogenic (over $\mathbb{Q}$) if the ring of integers admits a power integral basis, i.e., a $\mathbb{Z}$-basis of the form $\{1, \alpha, \alpha^2, \ldots, \alpha^{n−1}\}$. The first portion of the talk will be spent revisiting some classical examples of monogeneity and non-monogeneity. We will pay particular attention to obstructions to monogeneity and relations to other arithmetic questions. With some of the classical context for monogeneity and power integral bases in hand, we will investigate the monogeneity of division fields of elliptic curves. This will culminate in two results: one describing non-monogeneity “horizontally” and the other non-monogeneity “vertically.” We will finish with some of the difficulties of generalizing to abelian varieties of dimension greater than one, highlighting a partial generalization.


text:
Home

# Electron-Muon Correlations in Proton+Proton and Deuteron+Gold Collisions at PHENIX

## Tatia Engelmore

Title:
Electron-Muon Correlations in Proton+Proton and Deuteron+Gold Collisions at PHENIX
Author(s):
Engelmore, Tatia
Cole, Brian A.
Date:
Type:
Dissertations
Department:
Physics
Permanent URL:
Notes:
Ph.D., Columbia University.
Abstract:
This dissertation presents the first measurement of electron-muon azimuthal correlations at the PHENIX experiment at RHIC in 200 GeV proton-proton and deuteron-gold collisions. Electron-muon pairs result from the semileptonic decay of D mesons, which come from correlated charm pairs. The pairs are measured at forward rapidity, with η < 0.5 for the electron and 1.4 < η < 2.1 for the muon. Electron-muon pairs exhibit a characteristic peak at Δφ = π in the azimuthal distribution due to momentum conservation in the c\bar{c} decay, and this enables clear identification. The shape of the azimuthal pair distribution in p+p collisions helps us determine which hard scattering processes contribute to charm production, and it allows us to test NLO QCD predictions. The p+p result also serves as a baseline measurement for understanding heavy ion collisions. Pairs were also measured in d+Au collisions at forward rapidity in the deuteron-going direction, which is a kinematic region at which we expect suppression effects to be evident. The pair yield in d+Au was found to be suppressed relative to that in p+p. Also the peak in Δφ almost disappears in d+Au, indicating either a change in charm production mechanisms or interactions with the nuclear matter.
Subject(s):
Nuclear physics
Item views:
431


text:
A-weighting

A graph of the A-, B-, C- and D-weightings across the frequency range 10 Hz – 20 kHz
Video illustrating A-weighting by analyzing a sine sweep (contains audio)

A-weighting is the most commonly used of a family of curves defined in the International standard IEC 61672:2003 and various national standards relating to the measurement of sound pressure level. A-weighting is applied to instrument-measured sound levels in an effort to account for the relative loudness perceived by the human ear, as the ear is less sensitive to low audio frequencies. It is employed by arithmetically adding a table of values, listed by octave or third-octave bands, to the measured sound pressure levels in dB. The resulting octave band measurements are usually added (logarithmic method) to provide a single A-weighted value describing the sound; the units are written as dB(A). Other weighting sets of values – B, C, D and now Z – are discussed below.

The curves were originally defined for use at different average sound levels, but A-weighting, though originally intended only for the measurement of low-level sounds (around 40 phon), is now commonly used for the measurement of environmental noise and industrial noise, as well as when assessing potential hearing damage and other noise health effects at all sound levels; indeed, the use of A-frequency-weighting is now mandated for all these measurements, because decades of field experience have shown a very good correlation with occupational deafness in the frequency range of human speech. It is also used when measuring low-level noise in audio equipment, especially in the United States.[not verified in body] In Britain, Europe and many other parts of the world, broadcasters and audio engineers[who?] more often use the ITU-R 468 noise weighting, which was developed in the 1960s based on research by the BBC and other organizations. This research showed that our ears respond differently to random noise, and the equal-loudness curves on which the A, B and C weightings were based are really only valid for pure single tones.[not verified in body]

History

A-weighting began with work by Fletcher and Munson which resulted in their publication, in 1933, of a set of equal-loudness contours. Three years later these curves were used in the first American standard for sound level meters.[1] This ANSI standard, later revised as ANSI S1.4-1981, incorporated B-weighting as well as the A-weighting curve, recognising the unsuitability of the latter for anything other than low-level measurements. But B-weighting has since fallen into disuse. Later work, first by Zwicker and then by Schomer, attempted to overcome the difficulty posed by different levels, and work by the BBC resulted in the CCIR-468 weighting, currently maintained as ITU-R 468 noise weighting, which gives more representative readings on noise as opposed to pure tones.[citation needed]

Deficiencies

A-weighting is valid to represent the sensitivity of the human ear as a function of the frequency of pure tones, but only for relatively quiet levels of sound. In effect, the A-weighting is based on the 40-phon Fletcher–Munson curves which represented an early determination of the equal-loudness contour for human hearing. However, because decades of field experience have shown a very good correlation between the A scale and occupational deafness in the frequency range of human speech, this scale is employed in many jurisdictions to evaluate the risks of occupational deafness and other auditory problems related to signals or speech intelligibility in noisy environments.

Because of perceived discrepancies between early and more recent determinations, the International Organization for Standardization (ISO) recently revised its standard curves as defined in ISO 226, in response to the recommendations of a study coordinated by the Research Institute of Electrical Communication, Tohoku University, Japan. The study produced new curves by combining the results of several studies, by researchers in Japan, Germany, Denmark, UK, and USA. (Japan was the greatest contributor with about 40% of the data.) This has resulted in the recent acceptance of a new set of curves standardized as ISO 226:2003. The report comments on the surprisingly large differences, and the fact that the original Fletcher–Munson contours are in better agreement with recent results than the Robinson-Dadson, which appear to differ by as much as 10–15 dB especially in the low-frequency region, for reasons that are not explained. Fortuitously, the 40-phon Fletcher–Munson curve is particularly close to the modern ISO 226:2003 standard.[2]

Nevertheless, A-weighting would be a better match to the loudness curve if it fell much more steeply above 10 kHz, and it is likely that this compromise came about because steep filters were difficult to construct in the early days of electronics.[citation needed] Nowadays, no such limitation need exist, as demonstrated by the ITU-R 468 curve. If A-weighting is used without further band-limiting it is possible to obtain different readings on different instruments when ultrasonic, or near ultrasonic noise is present. Accurate measurements therefore require a 20 kHz low-pass filter to be combined with the A-weighting curve in modern instruments. This is defined in IEC 61012 as AU weighting and while very desirable, is rarely fitted to commercial sound level meters.

B-, C-, D- and Z-weightings

A-frequency-weighting is mandated by the international standard IEC 61672 to be fitted to all sound level meters and are approximations to the equal loudness contours given in ISO 226.[3] The old B- and D-frequency-weightings have fallen into disuse, but many sound level meters provide for C frequency-weighting and its fitting is mandated — at least for testing purposes — to precision (Class one) sound level meters. D-frequency-weighting was specifically designed for use when measuring high level aircraft noise in accordance with the IEC 537 measurement standard. The large peak in the D-weighting curve is not a feature of the equal-loudness contours, but reflects the fact that humans hear random noise differently from pure tones, an effect that is particularly pronounced around 6 kHz. This is because individual neurons from different regions of the cochlea in the inner ear respond to narrow bands of frequencies, but the higher frequency neurons integrate a wider band and hence signal a louder sound when presented with noise containing many frequencies than for a single pure tone of the same pressure level.[citation needed] Following changes to the ISO standard, D-frequency-weighting should now only be used for non-bypass engines and as these are not fitted to commercial aircraft — but only to military ones — A-frequency-weighting is now mandated for light civilian aircraft measurements, while a more accurate loudness-corrected weighting EPNdB is required for certification of large transport aircraft[4]

Z- or ZERO frequency-weighting was introduced in the International Standard IEC 61672 in 2003 and was intended to replace the "Flat" or "Linear" frequency weighting often fitted by manufacturers. This change was needed as each sound level meter manufacturer could choose their own low and high frequency cut-offs (–3 dB) points, resulting in different readings, especially when peak sound level was being measured. It is a flat frequency response between 10 Hz and 20 kHz ±1.5dB.[5] As well, the C-frequency-weighting, with –3 dB points at 31.5 Hz and 8 kHz did not have a sufficient bandpass to allow the sensibly correct measurement of true peak noise (Lpk).

B- and D-frequency-weightings are no longer described in the body of the standard IEC 61672 : 2003, but their frequency responses can be found in the older IEC 60651, although that has been formally withdrawn by the International Electro-technical Commission in favour of IEC 61672 : 2003. The frequency weighting tolerances in IEC 61672 have been tightened over those in the earlier standards IEC 179 and IEC 60651 and thus instruments complying with the earlier specifications should no longer be used for legally required measurements.

Environmental and other noise measurements

Label related to a portable air compressor

A-weighted decibels are abbreviated dB(A) or dBA. When acoustic (calibrated microphone) measurements are being referred to, then the units used will be dB SPL referenced to 20 micropascals = 0 dB SPL. dBrn adjusted is not a synonym for dBA, but for dBa (in telecommunications dBa denotes "decibels adjusted" i.e. weighted absolute noise power, which has nothing to do with A-weighting).

The A-weighting curve has been widely adopted for environmental noise measurement, and is standard in many sound level meters. The A-weighting system is used in any measurement of environmental noise (examples of which include roadway noise, rail noise, aircraft noise). A-weighting is also in common use for assessing potential hearing damage caused by loud noise.

A-weighted SPL measurements of noise level are increasingly found on sales literature for domestic appliances such as refrigerators, freezers and computer fans. In Europe, the A-weighted noise level is used for instance for normalizing the noise of tires on cars.

The A-weighting is also used for noise dose measurements at work. A noise level of more than 85 dB(A) each day increases the risk factor for hearing damage.

Noise exposure for visitors of venues with loud music is usually also expressed in dB(A), although the presence of high levels of low frequency noise does not justify this.

Although the A-weighting curve, in widespread use for noise measurement, is said to have been based on the 40-phon Fletcher-Munson curve, research in the 1960s demonstrated that determinations of equal-loudness made using pure tones are not directly relevant to our perception of noise.[6] This is because the cochlea in our inner ear analyses sounds in terms of spectral content, each 'hair-cell' responding to a narrow band of frequencies known as a critical band.[citation needed] The high-frequency bands are wider in absolute terms than the low frequency bands, and therefore 'collect' proportionately more power from a noise source.[citation needed] However, when more than one critical band is stimulated, the outputs of the various bands are summed by the brain to produce an impression of loudness. For these reasons equal-loudness curves derived using noise bands show an upwards tilt above 1 kHz and a downward tilt below 1 kHz when compared to the curves derived using pure tones.

This enhanced sensitivity to noise in the region of 6 kHz became particularly apparent in the late 1960s with the introduction of compact cassette recorders and Dolby-B noise reduction. A-weighted noise measurements were found to give misleading results because they did not give sufficient prominence to the 6 kHz region where the noise reduction was having greatest effect, and did not sufficiently attenuate noise around 10 kHz and above (a particular example is with the 19 kHz pilot tone on FM radio systems which, though usually inaudible is not sufficiently attenuated by A-weighting, so that sometimes one piece of equipment would even measure worse than another and yet sound better, because of differing spectral content.

ITU-R 468 noise weighting was therefore developed to more accurately reflect the subjective loudness of all types of noise, as opposed to tones. This curve, which came out of work done by the BBC Research Department, and was standardised by the CCIR and later adopted by many other standards bodies (IEC, BSI) and, as of 2006, is maintained by the ITU. It became widely used in Europe, especially in broadcasting, and was adopted by Dolby Laboratories who realised its superior validity for their purposes when measuring noise on film soundtracks and compact cassette systems. Its advantages over A-weighting are less accepted in the US, where the use of A-weighting still predominates.[citation needed] It is used by broadcasters in Britain, Europe, and former countries of the British Empire such as Australia and South Africa.

Function realisation of some common weightings

The standard[7] defines weightings (${\displaystyle A(f),C(f)}$ ) in dB units by tables with tolerance limits (to allow a variety of implementations). Additionally, the standard describes weighting functions ${\displaystyle R_{X}(f)}$ [7] to calculate the weightings. The weighting function ${\displaystyle R_{X}(f)}$  is applied to the amplitude spectrum (not the intensity spectrum) of the unweighted sound level. The offsets ensure the normalisation to 0 dB at 1000 Hz. Appropriate weighting functions are:[8]

A

{\displaystyle {\begin{aligned}R_{A}(f)&={12194^{2}f^{4} \over \left(f^{2}+20.6^{2}\right)\ {\sqrt {\left(f^{2}+107.7^{2}\right)\left(f^{2}+737.9^{2}\right)}}\ \left(f^{2}+12194^{2}\right)}\ ,\\[3pt]A(f)&=20\log _{10}\left(R_{A}(f)\right)-20\log _{10}\left(R_{A}(1000)\right)\\&\approx 20\log _{10}\left(R_{A}(f)\right)+2.00\end{aligned}}} [7]

B

{\displaystyle {\begin{aligned}R_{B}(f)&={12194^{2}f^{3} \over \left(f^{2}+20.6^{2}\right)\ {\sqrt {\left(f^{2}+158.5^{2}\right)}}\ \left(f^{2}+12194^{2}\right)}\ ,\\[3pt]B(f)&=20\log _{10}\left(R_{B}(f)\right)-20\log _{10}\left(R_{B}(1000)\right)\\&\approx 20\log _{10}\left(R_{B}(f)\right)+0.17\end{aligned}}}

C

{\displaystyle {\begin{aligned}R_{C}(f)&={12194^{2}f^{2} \over \left(f^{2}+20.6^{2}\right)\ \left(f^{2}+12194^{2}\right)}\ ,\\[3pt]C(f)&=20\log _{10}\left(R_{C}(f)\right)-20\log _{10}\left(R_{C}(1000)\right)\\[3pt]&\approx 20\log _{10}\left(R_{C}(f)\right)+0.06\end{aligned}}} [7]

D

{\displaystyle {\begin{aligned}h(f)&={\frac {\left(1037918.48-f^{2}\right)^{2}+1080768.16\,f^{2}}{\left(9837328-f^{2}\right)^{2}+11723776\,f^{2}}}\\[3pt]R_{D}(f)&={\frac {f}{6.8966888496476\cdot 10^{-5}}}{\sqrt {\frac {h(f)}{\left(f^{2}+79919.29\right)\left(f^{2}+1345600\right)}}}\\D(f)&=20\log _{10}\left(R_{D}(f)\right).\end{aligned}}} [9]

Transfer function equivalent

The gain curves can be realised[10] by the following s-domain transfer functions. They are not defined in this way though, being defined by tables of values with tolerances in the standards documents, thus allowing different realisations:[citation needed]

A

${\displaystyle H_{A}(s)\approx {k_{A}\cdot s^{4} \over (s+129.4)^{2}\quad (s+676.7)\quad (s+4636)\quad (s+76655)^{2}}}$
kA ≈ 7.39705×109

B

${\displaystyle H_{B}(s)\approx {k_{B}\cdot s^{3} \over (s+129.4)^{2}\quad (s+995.9)\quad (s+76655)^{2}}}$
kB ≈ 5.99185×109

C

${\displaystyle H_{C}(s)\approx {k_{C}\cdot s^{2} \over (s+129.4)^{2}\quad (s+76655)^{2}}}$
kC ≈ 5.91797×109

D

${\displaystyle H_{D}(s)\approx {k_{D}\cdot s\cdot \left(s^{2}+6532s+4.0975\times 10^{7}\right) \over (s+1776.3)\quad (s+7288.5)\quad \left(s^{2}+21514s+3.8836\times 10^{8}\right)}}$
kD ≈ 91104.32

The k values are constants which are used to normalize the function to a gain of 1 (0 dB). The values listed above normalize the functions to 0 dB at 1 kHz, as they are typically used. (This normalization is shown in the image.)

References

1. ^ Richard L. St. Pierre, Jr. and Daniel J. Maguire (July 2004), The Impact of A-weighting Sound Pressure Level Measurements during the Evaluation of Noise Exposure (PDF), retrieved 2011-09-13
2. ^ Precise and Full-range Determination of Two-dimensional Equal Loudness Contours (PDF), archived from the original (PDF) on 2007-09-27
3. ^ Rimell, Andrew; Mansfield, Neil; Paddan, Gurmail (2015). "Design of digital filters for frequency weightings (A and C) required for risk assessments of workers exposed to noise". Industrial Health. 53 (53): 21–7. doi:10.2486/indhealth.2013-0003. PMC 4331191. PMID 25224333. S2CID 13997453.
4. ^ http://www.icao.int/Meetings/EnvironmentalWorkshops/Documents/NoiseCertificationWorkshop-2004/BIP_2_2_jb.pdf
5. ^ Lauer, Amanda; El‐Sharkawy, AbdEl‐Monem M.; Kraitchman, Dara; Edelstein, William (2012). "MRI Acoustic Noise Can Harm Experimental and Companion Animals". Journal of Magnetic Resonance Imaging. 36 (3): 743–7. doi:10.1002/jmri.23653. PMID 22488793.
6. ^ Bauer, B.; Torick, E. (1966). "Researches in loudness measurement". IEEE Transactions on Audio and Electroacoustics. 14 (3): 141–151. doi:10.1109/TAU.1966.1161864.
7. ^ a b c d IEC 61672-1:2013 Electroacoustics - Sound level meters - Part 1: Specifications. IEC. 2013.
8. ^ "Frequency weighting equations". Cross Spectrum. 2004. Archived from the original on 2011-03-30.[unreliable source?]
9. ^ RONALD M. AARTS (March 1992). "A Comparison of Some Loudness Measures for Loudspeaker Listening Tests" (PDF). Audio Engineering Society. Archived from the original (PDF) on 2016-03-04. Retrieved 2013-02-28. Cite journal requires |journal= (help)
10. ^ Noise Measurement Briefing, Product Technology Partners Ltd., archived from the original on 2008-06-30
• Audio Engineer's Reference Book, 2nd Ed 1999, edited Michael Talbot Smith, Focal Press
• An Introduction to the Psychology of Hearing 5th ed, Brian C. J. Moore, Elsevier Press


text:
## The role of law in the process of development: with special reference to the transfer of technology to underdeveloped countries

Publication year:
1968
In:
In: Science and technology in developing countries. - Cambridge : Cambridge Univ. Pr., 1968. - P. 493-519
Call number:
500 HEIA 82169
Record appears in:


text:
Frequency-Severity Modeling in consideration of COVID-19 induced effects

This post is supposed to give a brief introduction in Frequency-Severity models. These models are very popular for determine the optimal price for an insurance. We will take a look at the general idea of frequency-severity modeling and for illustration explore a simple example by using rudimentary assumptions. There is a vast amount of literature that deals with how to calculate the model putting different assumptions. For example by assuming that frequency and severity is not independent [1]. To model covariates this article uses Generalized Linear Models (GLM) [2] and as an extension Generalized Linear Mixed Models (GLMM), which could be used to model corona effects, solved using Maximum likelihood estimation. Other methods use Bayesian methods [3], with the advantage that asymptotic results can be derived even if the observed data is sparse, or tree methods that allows a non linear relationship [4, 5].

1. Frequency-Severity Modeling

For illustration purposes we lay out a simple univariate model (we are going to model only a single risk), extending the model is straightforward. We want to model the Loss Cost of a given entity. Let be the number of claims over the time interval , , the amount of each claim (loss). Our aim is model the aggregated loss , since is unbounded, for asymptotics, it is meaningful to consider the average loss per claim . By construction we are facing two random variables, is continues while is discrete, the idea of Frequency-Severity Modeling is to model the joined distribution as product of frequency and conditional severity distribution

Instead one can also model directly, often done using a Tweedie distribution, however by separating frequency and severity additional insights are expected.

1.1 Example

A popular approach is to model as a compound Poisson process. Assume that is a counting of a Poisson process with intensity , while where is a probability distribution with density independent of . Then and a popular choice here is to assume a gamma distribution which leads to for and if . Statistical measures of interest are for example expected loss and variance given by

(1)

1.1.1 Estimation

Assume we observed claims , then the maximum likelihood estimators are given by , , while an optimal has to fulfill where is Digamma function . In practice this equation is solved numerically. This leads to estimators for expectation and variance.

2. Introducing Covariates

The previous analysis was carried out without covariates . In the insurance context, there is usually more information available than just the amount and number of claims. Covariates are for example the age of a customer or the car type when looking at motor insurance contracts. An important aspect when modeling insurance data is how long a contract had existed in a certain state of covariates. Thus additionally we introduce an artificial covariates labeled as “exposure” used to calibrate the size of a potential outcome variable. Based on these covariates we are interested in the conditional expectation

.

2.1 Modeling covariates using GLM

A common way to model covariates is to use a GLM. Let the outcome be a random variable with a particular distribution of the exponential family, e.g. the density can be written as where is a scaling parameter also known as dispersion parameter. We assume that there exists some link function such that for outcome with independent covariates , . Besides, a nice feature is that the variance function has the form

(2)

meaning that the variance is a function of the mean.

For estimation we rely on Maximum likelihood, in general the log likelihood for the -th covariate given observations given by

(3)

using the identities , .

1.2.1 Estimate the example using GLM

Recap our previous example, by using a compound Poisson process it was assumed that is a counting of a Poisson process while follows a distribution. It is easy to verify, that both distributions are member of the exponential family since the distribution of the Poisson process is given by , , while for the Gamma distribution we have to use , , . Finally, based on (1) and (2),

We assume that we observe types of policyholder. Since in this example, the observed claims per policyholder is independent from the corresponding claim amount the two GLM are estimated separately. Further we assume that there exists covariates . We use a log-link for either GLM, such that with the link function is given by and thus where and .

In order to find the maximum likelihood estimate for , we use the log likelihood to calculate

(4)

with a limiting distribution .
Accordingly the maximum likelihood estimate for using the log likelihood is given by

with a limiting distribution .

2.2 Incorporate Random Effects

Mixed effect models are used in cases where the response variables arise from different distributions. Therefore they are particularly useful provided that a repeated measurement is made on the same or on clusters of related statistical units. A nice intuition whether to model a random or a fixed effect can be found here. In a nutshell, model a factor as a random effects if:

• If the factor (group) is expected to have expression not yet observed (not entire population of groups is observed)
• If for a particular reason we are not interested in the coefficient of the factor rather than the anticipated structure that may be imposed on our observations.

In the insurance context we can use random effects to model a setup where several observations belong to the same few policyholder. In that case the unique identifier for each policyholder is used as covariates for random effects. Another important application where we may use random effect to create better models is due to the corona crisis. Here we observe polices which where issued under corona restrictions installed by the government while others are not. It is very likely that, for example because people do no longer drive to work, this has an effect on the occurred claims. A possible way out is to add the information of government policies as additional covariates to be modeled using random effects.
Thus, as an extension of the GLM in this section we will now include both fixed and random effects, leading to a Generalized Mixed Effect Model (GLMM). Similar to the GLM case we assume that each expression of follows a particular distribution of the exponential family such that conditioned on the random effect

(5)

where the covariates are separated for fixed effects and random effects , distributed with , such that , . For random effects we require that . Similar to (3) the likelihood is given by

Therefore fitting GLMMs via maximum likelihood is very similar to solve the GLM, but involves an additional integration over the random effects. This is usually extremely computationally intensive, especially when more covariates are modeled using random effects.

2.2.1 Extension of the Example

For illustration we will now extend our frequency example using a Poisson process and allow further for a single independent normal distributed random effect. Thus

(6)

Using the link , the Likelihood is comparable to (4) and is given by

3. Coding example in R

As an illustrative example we will use the well know Swedish Insurance Dataset. These data were compiled by the Swedish Committee on the Analysis of Risk Premium in Motor Insurance, summarized in Hallin and Ingenbleek (1983) and Andrews and Herzberg (1985). The data are cross-sectional, describing third party automobile insurance claims for the year 1977. In this example we will model the number of claims (the frequency) using a GLM as described above based on the covariates Kilometers, Zone, Bonus, Make and with a GLMM using “Zones” as random effects.

library('lme4')
## Loading required package: Matrix
library('doParallel')
## Loading required package: foreach
## Loading required package: iterators
## Loading required package: parallel
library('foreach')

# Fit the Model using the entire Dataset to check for significant factors
glm =glm(Claims ~ Kilometres+factor(Zone)+
summary(glm)
##
## Call:
## glm(formula = Claims ~ Kilometres + factor(Zone) + factor(Bonus) +
##     factor(Make), family = poisson(link = log), data = SwedishMotorInsurance,
##     offset = log(Insured))
##
## Deviance Residuals:
##     Min       1Q   Median       3Q      Max
## -7.9809  -0.8745  -0.1800   0.5774   6.5738
##
## Coefficients:
##                 Estimate Std. Error  z value Pr(>|z|)
## (Intercept)    -1.915417   0.014012 -136.697  < 2e-16 ***
## Kilometres      0.139678   0.002596   53.805  < 2e-16 ***
## factor(Zone)2  -0.238441   0.009495  -25.111  < 2e-16 ***
## factor(Zone)3  -0.387248   0.009669  -40.049  < 2e-16 ***
## factor(Zone)4  -0.582763   0.008653  -67.345  < 2e-16 ***
## factor(Zone)5  -0.326750   0.014529  -22.489  < 2e-16 ***
## factor(Zone)6  -0.526907   0.011876  -44.366  < 2e-16 ***
## factor(Zone)7  -0.731561   0.040698  -17.976  < 2e-16 ***
## factor(Bonus)2 -0.476283   0.012090  -39.395  < 2e-16 ***
## factor(Bonus)3 -0.689975   0.013502  -51.102  < 2e-16 ***
## factor(Bonus)4 -0.824070   0.014577  -56.534  < 2e-16 ***
## factor(Bonus)5 -0.923048   0.013960  -66.121  < 2e-16 ***
## factor(Bonus)6 -0.992035   0.011622  -85.359  < 2e-16 ***
## factor(Bonus)7 -1.326230   0.008673 -152.910  < 2e-16 ***
## factor(Make)2   0.073345   0.021238    3.454 0.000553 ***
## factor(Make)3  -0.251061   0.025091  -10.006  < 2e-16 ***
## factor(Make)4  -0.671032   0.024120  -27.821  < 2e-16 ***
## factor(Make)5   0.153801   0.020234    7.601 2.94e-14 ***
## factor(Make)6  -0.338687   0.017372  -19.496  < 2e-16 ***
## factor(Make)7  -0.056468   0.023343   -2.419 0.015561 *
## factor(Make)8  -0.052694   0.031581   -1.669 0.095216 .
## factor(Make)9  -0.073278   0.009939   -7.373 1.67e-13 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for poisson family taken to be 1)
##
##     Null deviance: 34070.6  on 2181  degrees of freedom
## Residual deviance:  3087.6  on 2160  degrees of freedom
## AIC: 10769
##
## Number of Fisher Scoring iterations: 4
glmm <- glmer(Claims ~ Kilometres+factor(Bonus)
+factor(Make)+ offset(log(Insured)) + (1|Zone), family=poisson(link=log), data=SwedishMotorInsurance)
summary(glmm)
## Generalized linear mixed model fit by maximum likelihood (Laplace
##   Approximation) [glmerMod]
##  Family: poisson  ( log )
## Formula:
## Claims ~ Kilometres + factor(Bonus) + factor(Make) + offset(log(Insured)) +
##     (1 | Zone)
##    Data: SwedishMotorInsurance
##
##      AIC      BIC   logLik deviance df.resid
##  10810.2  10906.9  -5388.1  10776.2     2165
##
## Scaled residuals:
##     Min      1Q  Median      3Q     Max
## -7.7295 -0.7598 -0.1686  0.5935  9.9509
##
## Random effects:
##  Groups Name        Variance Std.Dev.
##  Zone   (Intercept) 0.04945  0.2224
## Number of obs: 2182, groups:  Zone, 7
##
## Fixed effects:
##                 Estimate Std. Error  z value Pr(>|z|)
## (Intercept)    -2.313183   0.085212  -27.146  < 2e-16 ***
## Kilometres      0.139678   0.002596   53.806  < 2e-16 ***
## factor(Bonus)2 -0.476295   0.012090  -39.397  < 2e-16 ***
## factor(Bonus)3 -0.689984   0.013502  -51.103  < 2e-16 ***
## factor(Bonus)4 -0.824074   0.014576  -56.535  < 2e-16 ***
## factor(Bonus)5 -0.923050   0.013960  -66.122  < 2e-16 ***
## factor(Bonus)6 -0.992032   0.011622  -85.360  < 2e-16 ***
## factor(Bonus)7 -1.326260   0.008673 -152.916  < 2e-16 ***
## factor(Make)2   0.073348   0.021237    3.454 0.000553 ***
## factor(Make)3  -0.251008   0.025090  -10.004  < 2e-16 ***
## factor(Make)4  -0.671048   0.024119  -27.822  < 2e-16 ***
## factor(Make)5   0.153803   0.020234    7.601 2.93e-14 ***
## factor(Make)6  -0.338714   0.017372  -19.498  < 2e-16 ***
## factor(Make)7  -0.056462   0.023342   -2.419 0.015569 *
## factor(Make)8  -0.052737   0.031581   -1.670 0.094938 .
## factor(Make)9  -0.073280   0.009939   -7.373 1.67e-13 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Correlation matrix not shown by default, as p = 16 > 12.
## Use print(x, correlation=TRUE)  or
##     vcov(x)        if you need it
ranef(glmm)
## Zone
##   (Intercept)
## 1  0.39743524
## 2  0.15919148
## 3  0.01052576
## 4 -0.18486041
## 5  0.07079332
## 6 -0.12886749
## 7 -0.32328705
##
## with conditional variances for "Zone"
# Create a helper function for model evaluation
model_eval = function(train, test) {
glm =glm(Claims ~ Kilometres+factor(Zone)+

glmm <- glmer(Claims ~ Kilometres+factor(Bonus)
+factor(Make)+ offset(log(Insured)) + (1|Zone), family=poisson(link=log), data=train)

N=dim(test)[1]
mse_glm = 1/N*sum( (predict( glm  ,test, type = "response") -test["Claims"] )^2 )
mse_glmm = 1/N*sum( (predict( glmm ,test, type = "response") -test["Claims"] )^2 )
return( c(mse_glm, mse_glmm) )
}
# Compare the fit
model_eval(SwedishMotorInsurance,SwedishMotorInsurance)
## [1] 223.3410 223.3951
# Eval the model under different conditions

## Resample with test and train
cl <- makeCluster(15) #number of cores
registerDoParallel(cl)

runs <- foreach(1:100, .combine='cbind') %dopar% {
library('lme4')
N= dim(SwedishMotorInsurance)[1]
sample=sample(1:N, floor(0.8*N), replace = FALSE, prob = NULL)
train=SwedishMotorInsurance[sample,]
test=SwedishMotorInsurance[-sample,]
model_eval(train, test)
}
boxplot(t(runs), names= c("glm", "glmm") )

In this example the Mixed Model does not improve the prediction quality of the model. This is not surprising since “Zones” is distributed equally and the test dataset resemble the training data very well. The take away message here is that in this example we are not worse off using a glmm instead of a glm.

We will now push the luck a bit in favor n the direction of the GLMM by modifying the sample strategy to evaluate the model. An important application of the GLMM if not the entire population of a factor is observed,

Since we still want to compare both models and a GLM is not able to perform a prediction using an unknown factor not used for training we will give the glm only very few observations for a certain group instead. To evaluate the model, then only observations from that group are used.

trails = 500
n_train = c( rep(2,trails), rep(5,trails),  rep(20,trails),  rep(30,trails) )

runs_z6 <- foreach(n_train=n_train, .combine='cbind' ) %dopar% {
library('lme4')
train_o6=SwedishMotorInsurance[SwedishMotorInsurance["Zone"]!=6,]
data_m6=SwedishMotorInsurance[SwedishMotorInsurance["Zone"]==6,]
sample=sample(1:dim(data_m6)[1], n_train, replace = FALSE, prob = NULL)
train=rbind(train_o6, data_m6[sample,])
test=data_m6[-sample,]
c(n_train/dim(data_m6)[1], model_eval(train, test))
}

data=t(runs_z6)
boxplot( c(data[,2],data[,3]) ~ paste( rep(round(data[,1],2) ,2), c( rep("GLM", length(n_train)), rep("GLMM", length(n_train))) ) , outline=FALSE,ylab="MSE", xlab="Model",cex.axis = 0.6)
rect(0.5,-3,2.5,3300,col="grey90",lty=0)
rect(4.5,-3,6.5,3300,col="grey90",lty=0)
boxplot( c(data[,2],data[,3]) ~ paste( rep(round(data[,1],2) ,2), c( rep("GLM", length(n_train)), rep("GLMM", length(n_train))) ) , outline=FALSE,ylab="MSE", xlab="Model",add=TRUE,cex.axis = 0.6)

The figure shows how the two models perform. For training only a fraction, ranging from 0.01 to 0.1, of observations from one particular group where available. We can conclude, that the GLMM performs better if only a fraction of observation for a certain group are present for training. If more data becomes available the predictive power of GLM and GLMM coincide.

References

[1] J. Garrido, C. Genest, and J. Schulz, “Generalized linear models for dependent frequency and severity of insurance claims,” Insurance: mathematics and economics, vol. 70, pp. 205-215, 2016.
[Bibtex]
@article{Garrido2016,
title = {Generalized linear models for dependent frequency and severity of insurance claims},
journal = {Insurance: Mathematics and Economics},
volume = {70},
pages = {205-215},
year = {2016},
issn = {0167-6687},
doi = {https://doi.org/10.1016/j.insmatheco.2016.06.006},
url = {https://www.sciencedirect.com/science/article/pii/S0167668715303358},
author = {J. Garrido and C. Genest and J. Schulz},
keywords = {Aggregate claims model, Claim frequency, Claim severity, Dependence, Exponential dispersion models, Generalized linear model, Loss cost},
abstract = {Traditionally, claim counts and amounts are assumed to be independent in non-life insurance. This paper explores how this often unwarranted assumption can be relaxed in a simple way while incorporating rating factors into the model. The approach consists of fitting generalized linear models to the marginal frequency and the conditional severity components of the total claim cost; dependence between them is induced by treating the number of claims as a covariate in the model for the average claim size. In addition to being easy to implement, this modeling strategy has the advantage that when Poisson counts are assumed together with a log-link for the conditional severity model, the resulting pure premium is the product of a marginal mean frequency, a modified marginal mean severity, and an easily interpreted correction term that reflects the dependence. The approach is illustrated through simulations and applied to a Canadian automobile insurance dataset.}
}
[2] S. Haberman and A. E. Renshaw, “Generalized linear models and actuarial science,” Journal of the royal statistical society: series d (the statistician), vol. 45, iss. 4, pp. 407-436, 1996.
[Bibtex]
@article{Haberman1996,
author = {Haberman, Steven and Renshaw, Arthur E.},
title = {Generalized Linear Models and Actuarial Science},
journal = {Journal of the Royal Statistical Society: Series D (The Statistician)},
volume = {45},
number = {4},
pages = {407-436},
keywords = {Generalized linear models, Life-insurance, non-life-insurance models},
doi = {https://doi.org/10.2307/2988543},
abstract = {SUMMARY The authors review the applications of generalized linear models to actuarial problems. This rich class of statistical model has been successfully applied in recent years to a wide range of problems, involving mortality, multiple-state models, lapses, premium rating and reserving. Selective examples of these applications are presented.},
year = {1996}
}
[3] C. Gao, Q. Li, and Z. Guo, “Automobile insurance pricing with bayesian general linear model,” in Innovative computing and information, Berlin, Heidelberg, 2011, p. 359–365.
[Bibtex]
@InProceedings{Gao2011,
author="Gao, Cheng
and Li, Qi
and Guo, Zirui",
editor="Dai, Minli",
title="Automobile Insurance Pricing with Bayesian General Linear Model",
booktitle="Innovative Computing and Information",
year="2011",
publisher="Springer Berlin Heidelberg",
pages="359--365",
abstract="This paper applies Bayesian Model to the automobile insurance pricing in the purpose of solving the problem that the Generalized Linear Model (GLM) applied in actuarial pricing cannot reveal prior information and has too much confidence in the information from data. Under the assumption of Squared Error Loss, the estimation of the parameter in the model is the mean of the posterior distribution which was calculated by Markov Chain Monte Carlo Methods (MCMC). Finally, take the auto-insurance of WASA Insurance Company in Switzerland as an example. Run MCMC in WINBUGS software to get the estimation of the parameters, and design the comparative auto insurance tariff for this company. The comparison between the pricing utilizing Bayesian Model and that according to GLM reveals that owing to the function of prior information, the two automobile tariffs differ distinctively. Moreover, the prior information has been elegantly reflected in the Bayesian Model.",
isbn="978-3-642-23993-9"
}
[4] L. Guelman, “Gradient boosting trees for auto insurance loss cost modeling and prediction,” Expert systems with applications, vol. 39, iss. 3, pp. 3659-3667, 2012.
[Bibtex]
@article{Guelman2012,
title = {Gradient boosting trees for auto insurance loss cost modeling and prediction},
journal = {Expert Systems with Applications},
volume = {39},
number = {3},
pages = {3659-3667},
year = {2012},
issn = {0957-4174},
doi = {https://doi.org/10.1016/j.eswa.2011.09.058},
url = {https://www.sciencedirect.com/science/article/pii/S0957417411013674},
author = {Leo Guelman},
keywords = {Statistical learning, Gradient boosting trees, Insurance pricing},
abstract = {Gradient Boosting (GB) is an iterative algorithm that combines simple parameterized functions with “poor” performance (high prediction error) to produce a highly accurate prediction rule. In contrast to other statistical learning methods usually providing comparable accuracy (e.g., neural networks and support vector machines), GB gives interpretable results, while requiring little data preprocessing and tuning of the parameters. The method is highly robust to less than clean data and can be applied to classification or regression problems from a variety of response distributions (Gaussian, Bernoulli, Poisson, and Laplace). Complex interactions are modeled simply, missing values in the predictors are managed almost without loss of information, and feature selection is performed as an integral part of the procedure. These properties make GB a good candidate for insurance loss cost modeling. However, to the best of our knowledge, the application of this method to insurance pricing has not been fully documented to date. This paper presents the theory of GB and its application to the problem of predicting auto “at-fault” accident loss cost using data from a major Canadian insurer. The predictive accuracy of the model is compared against the conventional Generalized Linear Model (GLM) approach.}
}
[5] R. Henckaerts, M. Côté, K. Antonio, and R. Verbelen, “Boosting insights in insurance tariff plans with tree-based machine learning methods,” North american actuarial journal, pp. 1-31, 2020.
[Bibtex]
@article{Henckaerts2020,
author = { Roel Henckaerts and Marie-Pier Côté and Katrien Antonio and Roel Verbelen },
title = {Boosting Insights in Insurance Tariff Plans with Tree-Based Machine Learning Methods},
journal = {North American Actuarial Journal},
volume = {0},
number = {0},
pages = {1-31},
year = {2020},
publisher = {Routledge},
doi = {10.1080/10920277.2020.1745656},
URL = {
https://doi.org/10.1080/10920277.2020.1745656
},
eprint = {
https://doi.org/10.1080/10920277.2020.1745656
}
}

This site uses Akismet to reduce spam. Learn how your comment data is processed.


text:
# Why is the sequence divergent?

Well, the sequence is:

$a_n = n! - n^n$

I can't seem to figure it out. $n!$ goes to infinity, $n^n$ goes to infinity, I know the the result should be negative infinity, but I can't really find a way to explain it.

Just to note: I am embarrassed to even ask that question.

-

Hint: $\dfrac{n^n}{n!} \ge n$.

Added: Let $n \ge 2$. The number $n!$ is the product of the $n-1$ terms $n, n-1, n-2, \dots, 2$, all of which are $\le n$. Thus $n! \le n^{n-1}$, and therefore $\dfrac{n^n}{n!} \ge n$.

Now $$a_n=n!\left(1-\frac{n^n}{n!}\right) \le 1-n.$$ But $\displaystyle \lim_{n\to \infty}(1-n) =-\infty$. Since $a_n\le 1-n$, it follows that $\displaystyle \lim_{n\to \infty}(n!-n^n) =-\infty$.

Comment: In situations of this type, some people prefer to say that the limit does not exist. That assertion is less informative, since it tells us much less about the behaviour of $a_n$ as $n$ gets very large.

-
Just to be sure, you mean the comparison test, right? Does that apply to sequences? –  PawelMysior Nov 28 '11 at 14:48
A version of comparison does apply to sequences, but it is best not to think that way. We have $n!-n^n=n!(1-\frac{n^n}{n!})\le n!(1-n)$ (the last is by the Hint). But $\displaystyle \lim_{n\to\infty}n!(1-n)=-\infty$ (or doesn't exist, it depends what conventions are used in your course). So depending on those conventions, you can say your original limit is $-\infty$, or that it doesn't exist. –  André Nicolas Nov 28 '11 at 15:45
Sorry to be bugging You again, but I can't really get the grip on your theory. I get that the smaller sequence is divergent to negative infinity, but why is the bigger one, I think I'm missing some basic knowledge here... could You please point me in the right direction? –  PawelMysior Nov 28 '11 at 17:26
No problem! I will put more detail in the main post. –  André Nicolas Nov 28 '11 at 18:07

HINT: write $a_n = n^n(n!/n^n - 1)$ and try to prove that the sequence $b_n = n!/n^n$ converges to zero.

-


text:
http://iet.metastore.ingenta.com
1887

## Field trials of the VHF data exchange system (VDES) satellite downlink component

• Author(s):
• DOI:

$16.00 (plus tax if applicable) ##### Buy Knowledge Pack 10 chapters for$120.00
(plus taxes if applicable)

IET members benefit from discounts to all IET publications and free access to E&T Magazine. If you are an IET member, log in to your account and the discounts will automatically be applied.

Recommend Title Publication to library

You must fill out fields marked with: *

Librarian details
Name:*
Email:*
Name:*
Email:*
Department:*
Why are you recommending this title?
Select reason:

Advances in Communications Satellite Systems Proceedings of The 36th International Communications Satellite Systems Conference (ICSSC-2018) — Recommend this title to your library

## Thank you

This paper focuses on analysing and characterising the communication channel of the VDES satellite downlink component by carrying out a measurement campaign of the VDE-SAT signal. The VHF downlink signal was transmitted from the Norwegian satellite NORSAT-2 operated by Space Norway. Three simultaneous carriers, specified in ITU-R M.2092-0, were transmitted in the test waveform: carrier 1 with BPSK/CDMA-8 modulation, carrier 2 with π/4 QPSK modulation and carrier 3 with 8PSK modulation. In addition, the satellite also transmits a CW carrier for test purposes. During this work a COTS-based receiver station has been designed, and installed on the roof of a house next to the sea. An embedded software application was implemented to automatically capture the test waveform every time the satellite was over the horizon. All the captured data was processed offline and the received waveform was thoroughly analysed, in order to assess the quality of the received signal and model the communications channel. The most relevant results derived from this work are presented in the paper.

Preview this chapter:

Field trials of the VHF data exchange system (VDES) satellite downlink component, Page 1 of 2

| /docserver/preview/fulltext/books/te/pbte086e/PBTE086E_ch26-1.gif /docserver/preview/fulltext/books/te/pbte086e/PBTE086E_ch26-2.gif

### Related content

content/books/10.1049/pbte086e_ch26
pub_keyword,iet_inspecKeyword,pub_concept
6
6
This is a required field