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The history of telecommunication began with the use of smoke signals and drums in Africa, Asia, and the Americas. In the 1790s, the first fixed semaphore systems emerged in Europe. However, it was not until the 1830s that electrical telecommunication systems started to appear. This article details the history of telecommunication and the individuals who helped make telecommunication systems what they are today. The history of telecommunication is an important part of the larger history of communication.

Ancient systems and optical telegraphy

Early telecommunications included smoke signals and drums. Talking drums were used by natives in Africa, and smoke signals in North America and China. These systems were often used to do more than announce the presence of a military camp.

In Rabbinical Judaism a signal was given by means of kerchiefs or flags at intervals along the way back to the high priest to indicate the goat "for Azazel" had been pushed from the cliff.

Homing pigeons have occasionally been used throughout history by different cultures. Pigeon post had Persian roots, and was later used by the Romans to aid their military.

Greek hydraulic semaphore systems were used as early as the 4th century BC.  The hydraulic semaphores, which worked with water filled vessels and visual signals, functioned as optical telegraphs. However, they could only utilize a very limited range of pre-determined messages, and as with all such optical telegraphs could only be deployed during good visibility conditions.

During the Middle Ages, chains of beacons were commonly used on hilltops as a means of relaying a signal. Beacon chains suffered the drawback that they could only pass a single bit of information, so the meaning of the message such as "the enemy has been sighted" had to be agreed upon in advance. One notable instance of their use was during the Spanish Armada, when a beacon chain relayed a signal from Plymouth to London that signaled the arrival of the Spanish warships.

In 1774, the Swiss physicist Georges Lesage built an electrostatic telegraph consisting of a set of 24 conductive wires a few meters long connected to 24 elder balls suspended from a silk thread (each wire corresponds to a letter). The electrification of a wire by means of an electrostatic generator causes the corresponding elder ball to deflect and designate a letter to the operator located at the end of the line. The sequence of selected letters leads to the writing and transmission of a message.

French engineer Claude Chappe began working on visual telegraphy in 1790, using pairs of "clocks" whose hands pointed at different symbols. These did not prove quite viable at long distances, and Chappe revised his model to use two sets of jointed wooden beams. Operators moved the beams using cranks and wires. He built his first telegraph line between Lille and Paris, followed by a line from Strasbourg to Paris. In 1794, a Swedish engineer, Abraham Edelcrantz built a quite different system from Stockholm to Drottningholm. As opposed to Chappe's system which involved pulleys rotating beams of wood, Edelcrantz's system relied only upon shutters and was therefore faster.

However, semaphore as a communication system suffered from the need for skilled operators and expensive towers often at intervals of only ten to thirty kilometers (six to nineteen miles). As a result, the last commercial line was abandoned in 1880.

Electrical telegraph

Experiments on communication with electricity, initially unsuccessful, started in about 1726. Scientists including Laplace, Ampère, and Gauss were involved.

An early experiment in electrical telegraphy was an 'electrochemical' telegraph created by the German physician, anatomist and inventor Samuel Thomas von Sömmerring in 1809, based on an earlier, less robust design of 1804 by Spanish polymath and scientist Francisco Salva Campillo. Both their designs employed multiple wires (up to 35) in order to visually represent almost all Latin letters and numerals. Thus, messages could be conveyed electrically up to a few kilometers (in von Sömmerring's design), with each of the telegraph receiver's wires immersed in a separate glass tube of acid. An electric current was sequentially applied by the sender through the various wires representing each digit of a message; at the recipient's end the currents electrolysed the acid in the tubes in sequence, releasing streams of hydrogen bubbles next to each associated letter or numeral. The telegraph receiver's operator would visually observe the bubbles and could then record the transmitted message, albeit at a very low baud rate. The principal disadvantage to the system was its prohibitive cost, due to having to manufacture and string-up the multiple wire circuits it employed, as opposed to the single wire (with ground return) used by later telegraphs.

The first working telegraph was built by Francis Ronalds in 1816 and used static electricity.

Charles Wheatstone and William Fothergill Cooke patented a five-needle, six-wire system, which entered commercial use in 1838. It used the deflection of needles to represent messages and started operating over twenty-one kilometres (thirteen miles) of the Great Western Railway on 9 April 1839. Both Wheatstone and Cooke viewed their device as "an improvement to the [existing] electromagnetic telegraph" not as a new device.

On the other side of the Atlantic Ocean, Samuel Morse developed a version of the electrical telegraph which he demonstrated on 2 September 1837. Alfred Vail saw this demonstration and joined Morse to develop the register—a telegraph terminal that integrated a logging device for recording messages to paper tape. This was demonstrated successfully over three miles (five kilometres) on 6 January 1838 and eventually over forty miles (sixty-four kilometres) between Washington, D.C. and Baltimore on 24 May 1844. The patented invention proved lucrative and by 1851 telegraph lines in the United States spanned over 20,000 miles (32,000 kilometres). Morse's most important technical contribution to this telegraph was the simple and highly efficient Morse Code, co-developed with Vail, which was an important advance over Wheatstone's more complicated and expensive system, and required just two wires. The communications efficiency of the Morse Code preceded that of the Huffman code in digital communications by over 100 years, but Morse and Vail developed the code purely empirically, with shorter codes for more frequent letters.

The submarine cable across the English Channel, wire coated in gutta percha, was laid in 1851. Transatlantic cables installed in 1857 and 1858 only operated for a few days or weeks (carried messages of greeting back and forth between James Buchanan and Queen Victoria) before they failed. The project to lay a replacement line was delayed for five years by the American Civil War. The first successful transatlantic telegraph cable was completed on 27 July 1866, allowing continuous transatlantic telecommunication for the first time.

Telephone

The electric telephone was invented in the 1870s, based on earlier work with harmonic (multi-signal) telegraphs. The first commercial telephone services were set up in 1878 and 1879 on both sides of the Atlantic in the cities of New Haven, Connecticut in the US and London, England in the UK. Alexander Graham Bell held the master patent for the telephone that was needed for such services in both countries. All other patents for electric telephone devices and features flowed from this master patent. Credit for the invention of the electric telephone has been frequently disputed, and new controversies over the issue have arisen from time-to-time. As with other great inventions such as radio, television, the light bulb, and the digital computer, there were several inventors who did pioneering experimental work on voice transmission over a wire, who then improved on each other's ideas. However, the key innovators were Alexander Graham Bell and Gardiner Greene Hubbard, who created the first telephone company, the Bell Telephone Company in the United States, which later evolved into American Telephone & Telegraph (AT&T), at times the world's largest phone company.

Telephone technology grew quickly after the first commercial services emerged, with inter-city lines being built and telephone exchanges in every major city of the United States by the mid-1880s. The first transcontinental telephone call occurred on January 25, 1915.  Despite this, transatlantic voice communication remained impossible for customers until January 7, 1927 when a connection was established using radio. However no cable connection existed until TAT-1 was inaugurated on September 25, 1956 providing 36 telephone circuits.

In 1880, Bell and co-inventor Charles Sumner Tainter conducted the world's first wireless telephone call via modulated lightbeams projected by photophones. The scientific principles of their invention would not be utilized for several decades, when they were first deployed in military and fiber-optic communications.

The first transatlantic telephone cable (which incorporated hundreds of electronic amplifiers) was not operational until 1956, only six years before the first commercial telecommunications satellite, Telstar, was launched into space.

Radio and television

Over several years starting in 1894, the Italian inventor Guglielmo Marconi worked on adapting the newly discovered phenomenon of radio waves to telecommunication, building the first wireless telegraphy system using them.  In December 1901, he established wireless communication between St. John's, Newfoundland and Poldhu, Cornwall (England), earning him a Nobel Prize in Physics (which he shared with Karl Braun) in 1909. In 1900, Reginald Fessenden was able to wirelessly transmit a human voice.

Millimetre wave communication was first investigated by Bengali physicist Jagadish Chandra Bose during 18941896, when he reached an extremely high frequency of up to 60GHz in his experiments. He also introduced the use of semiconductor junctions to detect radio waves, when he patented the radio crystal detector in 1901.

In 1924, Japanese engineer Kenjiro Takayanagi began a research program on electronic television. In 1925, he demonstrated a CRT television with thermal electron emission. In 1926, he demonstrated a CRT television with 40-line resolution, the first working example of a fully electronic television receiver. In 1927, he increased the television resolution to 100 lines, which was unrivaled until 1931. In 1928, he was the first to transmit human faces in half-tones on television, influencing the later work of Vladimir K. Zworykin.

On March 25, 1925, Scottish inventor John Logie Baird publicly demonstrated the transmission of moving silhouette pictures at the London department store Selfridge's. Baird's system relied upon the fast-rotating Nipkow disk, and thus it became known as the mechanical television. In October 1925, Baird was successful in obtaining moving pictures with halftone shades, which were by most accounts the first true television pictures.  This led to a public demonstration of the improved device on 26 January 1926 again at Selfridges. His invention formed the basis of semi-experimental broadcasts done by the British Broadcasting Corporation beginning September 30, 1929.

For most of the twentieth century televisions used the cathode ray tube (CRT) invented by Karl Braun. Such a television was produced by Philo Farnsworth, who demonstrated crude silhouette images to his family in Idaho on September 7, 1927. Farnsworth's device would compete with the concurrent work of Kalman Tihanyi and Vladimir Zworykin.  Though the execution of the device was not yet what everyone hoped it could be, it earned Farnsworth a small production company. In 1934, he gave the first public demonstration of the television at Philadelphia's Franklin Institute and opened his own broadcasting station. Zworykin's camera, based on Tihanyi's Radioskop, which later would be known as the Iconoscope, had the backing of the influential Radio Corporation of America (RCA).  In the United States, court action between Farnsworth and RCA would resolve in Farnsworth's favour. John Logie Baird switched from mechanical television and became a pioneer of colour television using cathode-ray tubes.

After mid-century the spread of coaxial cable and microwave radio relay allowed television networks to spread across even large countries.

Semiconductor era
The modern period of telecommunication history from 1950 onwards is referred to as the semiconductor era, due to the wide adoption of semiconductor devices in telecommunication technology. The development of transistor technology and the semiconductor industry enabled significant advances in telecommunication technology, led to the price of telecommunications services declining significantly, and led to a transition away from state-owned narrowband circuit-switched networks to private broadband packet-switched networks. In turn, this led to a significant increase in the total number of telephone subscribers, reaching nearly 1billion users worldwide by the end of the 20th century.

The development of metal–oxide–semiconductor (MOS) large-scale integration (LSI) technology, information theory and cellular networking led to the development of affordable mobile communications. There was a rapid growth of the telecommunications industry towards the end of the 20th century, primarily due to the introduction of digital signal processing in wireless communications, driven by the development of low-cost, very large-scale integration (VLSI) RF CMOS (radio-frequency complementary MOS) technology.

Videotelephony

The development of videotelephony involved the historical development of several technologies which enabled the use of live video in addition to voice telecommunications. The concept of videotelephony was first popularized in the late 1870s in both the United States and Europe, although the basic sciences to permit its very earliest trials would take nearly a half century to be discovered. This was first embodied in the device which came to be known as the video telephone, or videophone, and it evolved from intensive research and experimentation in several telecommunication fields, notably electrical telegraphy, telephony, radio, and television.

The development of the crucial video technology first started in the latter half of the 1920s in the United Kingdom and the United States, spurred notably by John Logie Baird and AT&T's Bell Labs. This occurred in part, at least by AT&T, to serve as an adjunct supplementing the use of the telephone. A number of organizations believed that videotelephony would be superior to plain voice communications. However video technology was to be deployed in analog television broadcasting long before it could become practical—or popular—for videophones.

Videotelephony developed in parallel with conventional voice telephone systems from the mid-to-late 20th century. Only in the late 20th century with the advent of  powerful video codecs and high-speed broadband did it become a practical technology for regular use. With the rapid improvements and popularity of the Internet, it became widespread through the use of videoconferencing and webcams, which frequently utilize Internet telephony, and in business, where telepresence technology has helped reduce the need to travel.

Practical digital videotelephony was only made possible with advances in video compression, due to the impractically high bandwidth requirements of uncompressed video. To achieve Video Graphics Array (VGA) quality video (480p resolution and 256 colors) with raw uncompressed video, it would require a bandwidth of over 92Mbps.

Satellite

The first U.S. satellite to relay communications was Project SCORE in 1958, which used a tape recorder to store and forward voice messages. It was used to send a Christmas greeting to the world from U.S. President Dwight D. Eisenhower. In 1960 NASA launched an Echo satellite; the  aluminized PET film balloon served as a passive reflector for radio communications.  Courier 1B, built by Philco, also launched in 1960, was the world's first active repeater satellite. Satellites these days are used for many applications such as GPS, television, internet and telephone.

Telstar was the first active, direct relay commercial communications satellite. Belonging to AT&T as part of a multi-national agreement between AT&T, Bell Telephone Laboratories, NASA, the British General Post Office, and the French National PTT (Post Office) to develop satellite communications, it was launched by NASA from Cape Canaveral on July 10, 1962, the first privately sponsored space launch. Relay 1 was launched on December 13, 1962, and became the first satellite to broadcast across the Pacific on November 22, 1963.

The first and historically most important application for communication satellites was in intercontinental long-distance telephony. The fixed Public Switched Telephone Network relays telephone calls from land line telephones to an earth station, where they are then transmitted a receiving satellite dish via a geostationary satellite in Earth orbit.  Improvements in submarine communications cables, through the use of fiber-optics, caused some decline in the use of satellites for fixed telephony in the late 20th century, but they still exclusively service remote islands such as Ascension Island, Saint Helena, Diego Garcia, and Easter Island, where no submarine cables are in service. There are also some continents and some regions of countries where landline telecommunications are rare to nonexistent, for example Antarctica, plus large regions of Australia, South America, Africa, Northern Canada, China, Russia and Greenland.

After commercial long-distance telephone service was established via communication satellites, a host of other commercial telecommunications were also adapted to similar satellites starting in 1979, including mobile satellite phones, satellite radio, satellite television and satellite Internet access. The earliest adaption for most such services occurred in the 1990s as the pricing for commercial satellite transponder channels continued to drop significantly.

Realization and demonstration, on October 29, 2001, of the first digital cinema transmission by satellite in Europe of a feature film by Bernard Pauchon, Alain Lorentz, Raymond Melwig and Philippe Binant.

Computer networks and the Internet

On September 11, 1940, George Stibitz was able to transmit problems using teletype to his Complex Number Calculator in New York City and receive the computed results back at Dartmouth College in New Hampshire. This configuration of a centralized computer or mainframe with remote dumb terminals remained popular throughout the 1950s. However it was not until the 1960s that researchers started to investigate packet switching a technology that would allow chunks of data to be sent to different computers  without first passing through a centralized mainframe. A four-node network emerged on December 5, 1969 between the University of California, Los Angeles, the Stanford Research Institute, the University of Utah and the University of California, Santa Barbara. This network would become ARPANET, which by 1981 would consist of 213 nodes. In June 1973, the first non-US node was added to the network belonging to Norway's NORSAR project. This was shortly followed by a node in London.

ARPANET's development centred on the Request for Comments process and on April 7, 1969, RFC 1 was published. This process is important because ARPANET would eventually merge with other networks to form the Internet and many of the protocols the Internet relies upon today were specified through this process. The first Transmission Control Protocol (TCP) specification,  (Specification of Internet Transmission Control Program), was written by Vinton Cerf, Yogen Dalal, and Carl Sunshine, and published in December 1974. It coined the term "Internet" as a shorthand for internetworking. In September 1981, RFC 791 introduced the Internet Protocol v4 (IPv4). This established the TCP/IP protocol, which much of the Internet relies upon today. The User Datagram Protocol (UDP), a more relaxed transport protocol that, unlike TCP, did not guarantee the orderly delivery of packets, was submitted on 28 August 1980 as RFC 768. An e-mail protocol, SMTP, was introduced in August 1982 by RFC 821 and http://1.0 a protocol that would make the hyperlinked Internet possible was introduced in May 1996 by RFC 1945.

However, not all important developments were made through the Request for Comments process. Two popular link protocols for local area networks (LANs) also appeared in the 1970s. A patent for the Token Ring protocol was filed by Olof Söderblom on October 29, 1974. And a paper on the Ethernet protocol was published by Robert Metcalfe and David Boggs in the July 1976 issue of Communications of the ACM. The Ethernet protocol had been inspired by the ALOHAnet protocol which had been developed by electrical engineering researchers at the University of Hawaii.

Internet access became widespread late in the century, using the old telephone and television networks.

Digital telephone technology

MOS technology was initially overlooked by Bell because they did not find it practical for analog telephone applications. MOS technology eventually became practical for telephone applications with the MOS mixed-signal integrated circuit, which combines analog and digital signal processing on a single chip, developed by former Bell engineer David A. Hodges with Paul R. Gray at UC Berkeley in the early 1970s. In 1974, Hodges and Gray worked with R.E. Suarez to develop MOS switched capacitor (SC) circuit technology, which they used to develop the digital-to-analog converter (DAC) chip, using MOSFETs and MOS capacitors for data conversion. This was followed by the analog-to-digital converter (ADC) chip, developed by Gray and J. McCreary in 1975.

MOS SC circuits led to the development of PCM codec-filter chips in the late 1970s. The silicon-gate CMOS (complementary MOS) PCM codec-filter chip, developed by Hodges and W.C. Black in 1980, has since been the industry standard for digital telephony. By the 1990s, telecommunication networks such as the public switched telephone network (PSTN) had been largely digitized with very-large-scale integration (VLSI) CMOS PCM codec-filters, widely used in electronic switching systems for telephone exchanges and data transmission applications.

Wireless revolution

The wireless revolution began in the 1990s, with the advent of digital wireless networks leading to a social revolution, and a paradigm shift from wired to wireless technology, including the proliferation of commercial wireless technologies such as cell phones, mobile telephony, pagers, wireless computer networks, cellular networks, the wireless Internet, and laptop and handheld computers with wireless connections. The wireless revolution has been driven by advances in radio frequency (RF) and microwave engineering, and the transition from analog to digital RF technology.

Advances in metal–oxide–semiconductor field-effect transistor (MOSFET, or MOS transistor) technology, the key component of the RF technology that enables digital wireless networks, has been central to this revolution. Hitachi developed the vertical power MOSFET in 1969, but it was not until Ragle perfected the concept in 1976 that the power MOSFET became practical. In 1977 Hitachi announce a planar type of DMOS that was practical for audio power output stages. RF CMOS (radio frequency CMOS) integrated circuit technology was later developed by Asad Abidi at UCLA in the late 1980s. By the 1990s, RF CMOS integrated circuits were widely adopted as RF circuits, while discrete MOSFET (power MOSFET and LDMOS) devices were widely adopted as RF power amplifiers, which led to the development and proliferation of digital wireless networks. Most of the essential elements of modern wireless networks are built from MOSFETs, including base station modules, routers, telecommunication circuits, and radio transceivers. MOSFET scaling has led to rapidly increasing wireless bandwidth, which has been doubling every 18 months (as noted by Edholm's law).

Timeline

Visual, auditory and ancillary methods (non-electrical)
 Prehistoric: Fires, beacons, smoke signals, communication drums, horns
 6th century BCE: Mail
 5th century BCE: Pigeon post
 4th century BCE: Hydraulic semaphores
 15th century CE: Maritime flag semaphores
 1672: First experimental acoustic (mechanical) telephone
 1790: Semaphore lines (optical telegraphs)
 1867: Signal lamps
 1877: Acoustic phonograph
 1900; optical
picture

Basic electrical signals
 1838: Electrical telegraph. See: Telegraph history
 1830s: Beginning of attempts to develop "wireless telegraphy", systems using some form of ground, water, air or other media for conduction to eliminate the need for conducting wires.
 1858: First trans-Atlantic telegraph cable
 1876: Telephone. See: Invention of the telephone, History of the telephone, Timeline of the telephone
 1880: Telephony via lightbeam photophones

Advanced electrical and electronic signals
 1896: First practical wireless telegraphy systems based on Radio. See: History of radio.
 1900: first television displayed only black and white images. Over the next decades, colour television were invented, showing images that were clearer and in full colour.
 1914: First North American transcontinental telephone calling
 1927: Television. See: History of television
 1927: First commercial radio-telephone service, U.K.–U.S.
 1930: First experimental videophones
 1934: First commercial radio-telephone service, U.S.–Japan
 1936: World's first public videophone network
 1946: Limited capacity Mobile Telephone Service for automobiles
 1947: First working transistor (see History of the transistor)
 1950: Semiconductor era begins
 1956: Transatlantic telephone cable
 1962: Commercial telecommunications satellite
 1964: Fiber optical telecommunications
 1965: First North American public videophone network
 1969: Computer networking
 1973: First modern-era mobile (cellular) phone
 1974: Internet (see History of Internet)
 1979: INMARSAT ship-to-shore satellite communications
 1981: First mobile (cellular) phone network
 1982: SMTP email
 1998: Mobile satellite hand-held phones
 2003: VoIP Internet Telephony

See also

 History of the Internet
 History of podcasting
 History of radio
 History of television
 History of the telephone
 History of videotelephony
 Information Age
 Information revolution
 Optical communication
 Outline of telecommunication

References

Sources
 Wenzlhuemer, Roland. Connecting the Nineteenth-Century World: The Telegraph and Globalization. Cambridge University Press, 2013.

Further reading
 Hilmes, Michele. Network Nations: A Transnational History of American and British Broadcasting (2011)
 John, Richard. Network Nation: Inventing American Telecommunications (Harvard U.P. 2010), emphasis on telephone
 Noll, Michael.  The Evolution of Media, 2007, Rowman & Littlefield
 Poe, Marshall T. A History of Communications: Media and Society From the Evolution of Speech to the Internet (Cambridge University Press; 2011) 352 pages; Documents how successive forms of communication are embraced and, in turn, foment change in social institutions.
 Wheen, Andrew. DOT-DASH TO DOT.COM: How Modern Telecommunications Evolved from the Telegraph to the Internet (Springer, 2011)
 Wu, Tim. The Master Switch: The Rise and Fall of Information Empires (2010)
 Lundy, Bert. Telegraph, Telephone and Wireless: How Telecom Changed the World (2008)

External links
 Katz, Randy H., "History of Communications Infrastructures" , Electrical Engineering and Computer Science Department (EECS) Department, University of California, Berkeley.
 International Telecommunication Union
 Aronsson's Telecom History Timeline
 From the Thurn & Taxis to the Phone Book of the World - 730 years of Telecom History
 Telecommunications History Group Virtual Museum
 Telecommunications History Germany
 Telecommunications History France

Telecommunications


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Home / News & Events / News / Closing the loop in British manufacturing: Sustainability & Circular Economy
Closing the loop in British manufacturing: Sustainability & Circular Economy
Adrian Curry is managing director at leading glass container manufacturer, Encirc, and has recently been appointed president of industry body, British Glass. Here, he discusses the importance of sustainable manufacturing in the UK, the benefits of a circular economy and what manufacturers can learn from the glass sector.
It's an uncomfortable truth to acknowledge, but industrial activity is widely seen as a fundamental obstacle in the fight for a sustainable future. In Britain, for example, the sector accounts for around a quarter of the UK's greenhouse gas emissions – a country which, worryingly, has recently been given a "final warning" by the European Commission for failing to address repeated breaches of legal air pollution limits.
With Brexit quickly becoming a reality, some doubt has been cast on our sector's commitment to current and upcoming international environmental regulations. In my mind, we cannot afford to let this doubt become justified. As UK manufacturers, we must all recognise the vital role we play when it comes to limiting our collective impact on the environment and become global champions of sustainable business.
"a truly circular economy could see a 70% reduction of carbon emissions"
To do this, we need to drive real and widespread behavioural change across our sector and develop more 'circular' modes of operation. The idea of creating a circular business model – a system where waste is no longer thrown away but instead put back into the supply chain – is not a new one. However, the benefits of turning this concept into a reality are fast becoming known. One global think-tank has estimated, that by implementing processes that lead to a truly circular economy, Europe could see a 70 per cent reduction of carbon emissions by 2030.
International bodies are, of course, rolling out policies that support increased resource efficiency. The EU is updating its waste and recycling legislation in the form of its new Circular Economy (CE) package, which will increase general recycling targets to 70 per cent by 2030. If this passes into EU law before the UK has formally left the Union, Britain is likely to adopt any agreed measures as its own. In addition, The British Standards Institute (BSI) is drawing up the world's first Circular Economy Standards – scalable guidelines that can be followed by businesses of different sizes around the world to help them boost sustainability.
A 2017 study by Aldersgate Group claims that increased levels of resource efficiency could boost the UK's economy by more than £80 billion by 2030. As President of British Glass and a representative of the glass sector, I have the responsibility to support the roll-out of this, helping manufacturers and consumers develop more circular ways of working. Fortuitously, the container glass industry has the potential to provide a perfect example of a scalable circular economy in action. As it is 100 per cent recyclable, glass can be melted down and remoulded an infinite number of times without losing any quality – something no other packaging material can offer.
To find out more about encirc's commitment to the environment, please visit http://www.encirc360.com/the-environment.php
Campaign launched to cut waste in workplaces
BT Chooses Northern Ireland for £28.6 Million Innovation Centre and Creation of Up to 50 New Graduate Jobs

redpajama_set_name: RedPajamaCommonCrawl


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text:
\section{Introduction}
\label{intro}
Graphical models \citep{Lauritzen96, Whittaker09} have become as a useful way of exploring and modeling the distribution. For instance, graphical models could be used to represent complex interactions among gene products resulted from biological processes. Such problems require us to infer an undirected graph from i.i.d. observations.

Let $X=(X_1,\ldots,X_p)^T$ be a random vector with some continuous distribution. An undirected graph $G$ has $p$ vertices, collected in a set $V$, one for each variable. We represent the edges as a set $E$ of unordered pairs: $(i,j)\in E$ if and only if there is an edge between $X_i$ and $X_j$. An edge between $X_i$ and $X_j$ is absent if $X_i$ and $X_j$ are independent, given the other variables.

The default model for graphical modeling of continuous data is the multivariate Gaussian. Let data $\mathcal{D}_n:=\{x_1,\ldots,x_n\}$ be the realizations of $n$ independent samples from a multivariate Gaussian distribution $\mathcal{N}_p(0,\Sigma)$, where $\Sigma$ is the covariance matrix. Then the log-likelihood of $\mathcal{D}_n$ is (up to a constant) given by
\begin{align}
l_n(\Theta):=\frac{n}{2}\log\det(\Theta)-\frac{n}{2}\text{tr}(\widehat{\Sigma}_n\Theta),
\end{align}
where $\Theta=\Sigma^{-1}$ is the precision matrix, i.e. inverse covariance matrix, and $\widehat{\Sigma}_n=\frac{1}{n}\sum_{i=1}^n x_i x_i^T$ is the empirical covariance matrix.

For Gaussian graphical models, it is well known that the edge between the $i$th and $j$th nodes is absent in the graph, meaning that the associated variables  are conditionally independent given the other variables, if and only if $\theta_{ij}=0$, where $\theta_{ij}$ is the $(i,j)$th element of $\Theta$. Therefore, the estimation and model selection problems in Gaussian graphical models are equivalent to estimation of the precision matrix and identification of its zero-pattern.

While it is one of the classical problems in multivariate statistics, with a renewed focus on high-dimensional data in recent years, a number of sparse estimators have been proposed to deal with the problem of precision matrix estimation. Among them, \citet{Yuan07} and \citet{Friedman08} impose an $\ell_1$ penalty on the entries of the precision matrix when maximizing the Gaussian log-likelihood, known as the \emph{graphical lasso}, encouraging some of the entries of the estimated precision matrix to be exactly zero. \citet{Meinshausen06} consider the neighborhood selection method via the lasso. \citet{Cai11} propose a constrained $\ell_1$ minimization approach for sparse inverse covariance matrix estimation. \citet{Yuan10} takes advantage of the connection between multivariate linear regression and entries of the inverse covariance matrix, developing an estimating procedure that can effectively exploit sparsity. Theoretical properties, including consistency in parameter estimation and sparsity structure recovery, are discussed in these and other papers \citep{Raskutti08, Rothman08}.

Given a collected family of estimators, linear or convex aggregation methods are another class of technique to address model selection problems and provide flexible ways to combine various models into a single estimator \citep{Rigollet11}. The idea of aggregating estimators was originally described in \citep{Nemirovski00}. The suggestion put forward by \citep{Nemirovski00} is to achieve two independent subsamples from the original sample by randomization: individual estimators are constructed from the first subsample while the second is used to perform aggregation on those individual estimators. This idea of two-step procedures carries over to models with i.i.d. observations where one can do sample splitting. Along with this method, one might aggregate estimators using the same observations for both estimation and aggregation. However, this would generally result in overfitting.

A primary motivation for aggregating estimators is that it can improve the estimation risk, as ``betting'' on multiple models can provide a type of insurance against a single model being poor \citep{Leung06}. Most of the recent work on estimator aggregation deals with regression learning problems. For example, \emph{Exponential Screening} (ES) for linear models provides a form of frequentist averaging over a large model class, which enjoys strong theoretical properties \citep{Rigollet11}. Also, an aggregation classifier is proposed in \citep{Lecue07} and an optimal rate of convex aggregation for the hinge risk is also obtained.

In this paper, we propose a new estimating procedure by considering a linear combination of a suitable set of individual estimators with different sparsity patterns. A sparsity pattern is defined as a binary vector with each element indicating whether the corresponding edge of the graph is absent or not. These individual estimators and the corresponding aggregating weights are determined to ensure a competitive rate of convergence for the risk of the aggregation estimator.

Our aggregation method is based on a sample-splitting procedure: the first subsample is set to construct individual estimators and the second subsample is then used to determine the weights and aggregate these estimators. To carry out the analysis of the aggregation step, it is enough to work conditionally on the first subsample so that the problem reduces to aggregation of deterministic estimators \citep{Rigollet12}. Namely, let $R(\cdot)$ denote a risk function, then given the deterministic estimator $\Theta_m$ with sparsity pattern $m$, one can construct an aggregation estimator $\widehat{\Theta}$ such that the excess risk
\begin{align}
r:=R(\widehat{\Theta})- \min_{m\in\mathcal{M}}R(\Theta_m),
\end{align}
is as small as possible, where $\mathcal{M}$ is a candidate set of sparsity patterns. Ideally, we wish to find an aggregation estimator whose risk is as close as possible (in a probabilistic sense) to the minimum risk of individual estimators. The risk function considered in the paper is the Kullback-Leibler divergence.

The rest of the paper is organized as follows. In Section~\ref{method}, we describe the graphical aggregation in detail. Theoretical properties are provided in Section~\ref{theory}. Numerical experiments are presented in Sections~\ref{num}. Finally, we conclude in Section~\ref{conclude}. Detailed proofs are collected in Appendix section.

\section{Graphical Exponential Screening Estimator}
\label{method}

\subsection{Notations and preliminaries}
Let $|\cdot|_1$ denote the $\ell_1$ vector norm, $\|\cdot\|$ denote the spectral matrix norm, and $\|\cdot\|_F$ denote the Frobenius matrix norm. For any symmetric positive-definite matrix $A$ we use the notation $A\succ 0$. For any two real numbers $a$ and $b$, we use the notation $a \vee b:=\max(a,b)$. Let $\mathbf{1}(\cdot)$ denote the indicator function.

We call a \emph{sparsity pattern} any binary vector $m\in\mathcal{M}$, where $\mathcal{M}$ is a candidate set of sparsity patterns and $\mathcal{M}\subset\{0, 1\}^{p(p-1)/2}$. The $k$th coordinate of $m$ can be interpreted as an indicator of presence ($m_k = 1$) or absence ($m_k = 0$) of the edge $(i_k,j_k)\in E$ such that $(i_k,j_k)$ is the $k$th element of the ordered list $\mathcal{S}_p$, where $\mathcal{S}_p=\{(i,j):1\leq i<j \leq p\}$.

We partition the sample $\mathcal{D}_n$ into two independent subsamples, $\mathcal{D}_{n_1}^{(1)}$ and $\mathcal{D}_{n_2}^{(2)}$, of size $n_1$ and $n_2$, respectively, where $n_1+n_2=n$.

\subsection{Graphical aggregation method}
In the aggregation procedure, $\mathcal{D}_{n_1}^{(1)}$ is utilized to construct individual estimators and $\mathcal{D}_{n_2}^{(2)}$ is then used to aggregate these estimators. Given each sparsity pattern $m\in\mathcal{M}$, we first define the individual estimator of the precision matrix.

\begin{definition} For each $m\in\mathcal{M}$, let $E_m$ be the edge set of the graph with sparsity pattern $m$. Let $\widehat{\Theta}_m^{(1)}$ be the constrained maximum likelihood estimator, defined by
\begin{align}
\widehat{\Theta}_m^{(1)} = \underset{\Theta\in\mathcal{C}_m}{\operatorname{argmax}} \;\{\log\det(\Theta)-\emph{\text{tr}}(\widehat\Sigma_{n_1}^{(1)}\Theta)\},
\end{align}
where
\begin{align}
\mathcal{C}_m&=\{\Theta\succ0:\theta_{ij}=0\emph{\text{ for any }}(i,j)\not\in E_m\emph{\text{ and }}i\not=j\},
\end{align}
and $\widehat\Sigma_{n_1}^{(1)}$ denotes the empirical covariance matrix using the first subsample $\mathcal{D}_{n_1}^{(1)}$.
\end{definition}

Notice that each individual estimator maximizes the log-likelihood under the constraint that some pre-defined subset of the parameters are zero. If $n_1\geq q_m$ where $q_m$ is the maximal clique size of a minimal chordal cover of the graph with edge set ${E}_m$, estimator $\widehat{\Theta}_m^{(1)}$ exists and is unique \cite{Uhler12}.
The following relationships hold regarding $\widehat{\Theta}_m^{(1)}$ and its inverse $(\widehat{\Theta}_m^{(1)})^{-1}$:
\begin{align}
[\widehat{\Theta}_m^{(1)}]_{ij} &= 0,\;\forall(i,j)\not\in{E}_m,\text{ and} \\
[(\widehat{\Theta}_m^{(1)})^{-1}]_{ij} &= [\widehat\Sigma_{n_1}^{(1)}]_{ij},\;\forall(i,j)\in{E}_m\cup\{(i,i),i=1,\ldots,p\}.
\end{align}
Indeed we can drive the above properties via the Lagrange form, where we add Lagrange constants for all missing edges of ${E}_m$
\begin{align}
\label{cmle}
l_{n_1,m}(\Theta):=\log\det(\Theta)-\text{tr}(\widehat\Sigma_{n_1}^{(1)}\Theta)-\sum_{(i,j)\not\in{E}_m}\gamma_{ij}\theta_{ij},
\end{align}
where $\gamma_{ij}$ is a Lagrange constant. The gradient equation for maximizing (\ref{cmle}) can be written as
\begin{align}
\Theta^{-1}-\widehat\Sigma_{n_1}^{(1)}-\Gamma=0,
\end{align}
where $\Gamma$ is a matrix of Lagrange parameters with nonzero values for all pairs with edges absent. It is an equality-constrained convex optimization problem, and a number of methods have been proposed for solving it, for example, in Hastie et~al.~\cite{Hastie09}.

Now we define the aggregation estimator, which linearly combines a set of individual estimators $\widehat{\Theta}_m^{(1)}$ for $m\in\mathcal{M}$.

\begin{definition}
\label{maindef}
Let $\widehat{\Theta}_{\emph{\text{gES}}}^{(1,2)}$ be the graphical Exponential Screening (gES) estimator that linearly combines a set of individual estimators $\widehat{\Theta}_m^{(1)}$, defined by
\begin{align}
\widehat{\Theta}_{\emph{\text{gES}}}^{(1,2)}=\sum_{m\in\mathcal{M}}v_m^{(1,2)}\widehat{\Theta}_m^{(1)},
\end{align}
where the superscripts denote which subsamples are used for constructions, and the weights
\begin{align}
&v_m^{(1,2)}: =
\frac{\exp\{(n_2/2)(\emph{\text{logdet}}(\widehat{\Theta}_m^{(1)})-\emph{\text{tr}}(\widehat{\Theta}_m^{(1)}\widehat{S}_{n_2}^{(2)}))\}\pi_m}{\sum_{m'\in\mathcal{M}}\exp\{(n_2/2)(\emph{\text{logdet}}(\widehat{\Theta}_{m'}^{(1)})-\emph{\text{tr}}(\widehat{\Theta}_{m'}^{(1)}\widehat{S}_{n_2}^{(2)}))\}\pi_{m'}}.
\end{align}
Here, $S_{n_2}^{(2)}$ is an estimator of $\Sigma$, and $\pi_m$ is a (prior) probability distribution on the set of sparsity pattern $\mathcal{M}$, defined by
\begin{align}
\pi_m:= \frac{1}{H}\left(\frac{|m|_1}{e p(p-1)}\right)^{|m|_1},
\end{align}
where $H$ is a normalization factor to ensure that $\pi_m$ add up to one.
\end{definition}

Observe that the gES estimator is a linear combination of the set of individual estimators $\widehat{\Theta}_m^{(1)}$ with weights $v_m^{(1,2)}$. It is indeed a convex combination since the weights add up to one.

A natural choice of $S_{n_2}^{(2)}$ would be the empirical covariance matrix $\widehat{\Sigma}_{n_2}^{(2)}$. In this scenario, ignoring the prior distribution $\pi_m$, the individual weight is proportional to the likelihood of estimator $\widehat{\Theta}_m^{(1)}$ evaluated on the second subsample. Thus, the higher the predictive ability, the more weighting will be put on the corresponding individual estimator. However, as is shown in the next section, this would lead to a deterioration of convergence rate in high-dimensional settings. Instead, we can use the hard thresholding estimator proposed in \citet{Bickel08}. To be more specific, the thresholding estimation of $\widehat{\Sigma}_{n_2}^{(2)}$ thresholded at $\gamma$ is defined by
\begin{align}
S_{n_2}^{(2)}=T_\gamma(\widehat\Sigma_{n_2}^{(2)}):=\{\hat\sigma_{ij}^{(2)}\cdot\mathbf{1}(|\hat\sigma_{ij}^{(2)}|\geq \gamma)\}_{1\leq i,j\leq p},
\end{align}
where $\hat\sigma_{ij}^{(2)}$ is the $(i,j)$th element of $\widehat{\Sigma}_{n_2}^{(2)}$.
In practice, we can apply the following procedure for the problem of threshold selection \cite{Bickel08}: we split the second subsample randomly into two pieces of size $n_2(1-1/\log n_2)$ and $n_2/\log n_2$, respectively, and repeat this $B$ times. Let $\widehat\Sigma_{1,v}^{(2)}$ and $\widehat\Sigma_{2,v}^{(2)}$ be the empirical covariance matrices based on the two pieces, from the $v$th split. Then the thresholding parameter $\gamma$ is determined by
\begin{align}
\gamma=\underset{\gamma'}{\text{argmin}}\left\{\frac{1}{B}\sum_{v=1}^B\|T_{\gamma'}(\widehat\Sigma_{1,v}^{(2)})-\widehat\Sigma_{2,v}^{(2)}\|_F^2\right\}.
\end{align}

In Definition~\ref{maindef}, we also incorporate a deterministic factor $\pi_m$ into the weighted averaging to account for (prior) model complexity, in a manner that facilitates desirable risk properties \cite{Leung06, Rigollet11}. Here, low-complexity models are favored. Alternatively, if the set of sparsity patterns $\mathcal{M}=\{0, 1\}^{p(p-1)/2}$, the following deterministic factor specifies a uniform distribution on the cardinality of the subset
and a conditionally uniform distribution on the subsets of that size:
\begin{align}
\pi_m^{\text{U}}=\left[\left(p(p-1)/2+1\right){p(p-1)/2 \choose |m|_1}\right]^{-1}.
\end{align}
In addition, a simple way is to choose a flat prior, where we set $\pi_m^\text{F}=\pi_{m'}^\text{F}$ for any $m,m'\in\mathcal{M}$.

We show in the next section that, under some technical conditions, the risk of the gES estimator is bounded by the risk of the best individual estimator plus a low price for aggregating.

\subsection{Approximation algorithm}

To implement the estimating procedure, note that exact computation of the aggregation estimator might require the calculation of as many as $2^{p(p-1)/2}$ individual estimators. In many cases this number could be extremely large, and we must make a numerical approximation. Observing that the aggregation estimator is actually the expectation of a random variable that has a probability mass proportional to $v_m^{(1,2)}$ on individual estimator $\widehat{\Theta}_m^{(1)}$ for $m\in\mathcal{M}$, then the Metropolis-Hastings algorithm can be exploited to provide such an approximation. The detail algorithm is shown in Algorithm~\ref{alg}.

\begin{algorithm}
\label{alg}
\vskip 0.1in
If the set of sparsity patterns $\mathcal{M}=\{0, 1\}^{p(p-1)/2}$, then the gES estimator can be approximated by running a Metropolis-Hastings algorithm on a $p(p-1)/2$-dimensional hypercube:
\vspace{-\topsep}
\begin{itemize}
\item[(S1)] Initialize $m_t=\{0\}^{\frac{p(p-1)}{2}}, t=0$;
\item[(S2)] For each $t\geq 0$, generate $m_t'$ with the uniform distribution on the neighbours of $m_t$;
\item[(S3)] Generate an $[0,1]$-uniformly distributed number $r$;
\item[(S4)] Put $m_{t+1} \leftarrow m_t'$, if $r < \min\{1,v_{m_t'}^{(1,2)}/v_{m_t}^{(1,2)}\}$; otherwise, $m_{t+1} \leftarrow m_t;$
\item[(S5)] Compute $\widehat\Theta_{m_{t+1}}^{(1)}$. Stop if $t>T_0+T$;
otherwise, update $t \leftarrow t+1$ and go to step (S2).
\end{itemize}
\vspace{-\topsep}
Then we can approximate $\widehat\Theta_{\emph{\text{gES}}}^{(1,2)}$ by
\begin{align}
\widehat{\widehat{\Theta}_{\emph{\text{gES}}}^{(1,2)}}=\frac{1}{T}\sum_{t=T_0+1}^{T_0+T}\widehat\Theta_{m_t}^{(1)},
\end{align}
where $T_0$ and $T$ are positive integers.
\vskip 0.1in
\end{algorithm}
Here, the neighbours of $m_t$ consists of all the sparsity patterns with a Manhattan distance of one to $m_t$. The following proposition shows that the resulting Markov chain ensures the ergodicity.
\begin{proposition}
The Markov chain $\{m_t\}_{0\leq t\leq T_0+T}$ generated by Algorithm~\ref{alg} satisfies
\begin{align}
\lim_{T\to \infty}\frac{1}{T}\sum_{t=T_0+1}^{T_0+T}\widehat\Theta_{m_t}^{(1)}=\sum_{m\in\mathcal{M}}v_m^{(1,2)}\widehat{\Theta}_m^{(1)},\;\text{almost surely}.
\end{align}
\end{proposition}

The proof is straightforward as the Markov chain is clearly $v^{(1,2)}$-irreducible.

The Metropolis-Hastings algorithm incorporates a trade-off between prediction and sparsity to decide whether to add or discard an edge. Observe that the gES estimator would always estimate a precision matrix in which all the elements are nonzero, since all possible individual estimators are linearly mixed. However, the Metropolis-Hastings algorithm would lead to a sparse precision matrix estimation as in the regression case \cite{Rigollet11}.

\section{Theoretical Properties}
\label{theory}

In this section, we show that under some technical conditions, the risk of the gES estimator is bounded by the risk of the best individual estimator in the dictionary plus a low aggregating price.

Following the notations in \citet{Bickel08}, we define the uniformity class of covariance matrices invariant under permutations by
\begin{align}
\mathcal{U}(q,\delta,M)=\Bigl\{\Sigma\in\mathbb{R}^{p\times p}:\;\Sigma\succ0,\;\sigma_{ii}\leq M,\;
\sum_{j=1}^p|\sigma_{ij}|^q\leq \delta,\;\text{for all }i\Bigr\},
\end{align}
for $0\leq q<1$, where $M$ and $\delta$ are constants.

For any estimator $\widehat\Theta\succ0$, we define the risk function
\begin{align}
R(\widehat\Theta)=\text{tr}(\widehat\Theta\Sigma)-\text{logdet}(\widehat\Theta).
\end{align}
Note that $\Sigma$ is the true covariance matrix. Consider the Kullback-Leibler (KL) divergence
\begin{align}
\text{KL}(\widehat\Theta)=R(\widehat\Theta)-R(\Theta)\geq0.
\end{align}

For each individual estimator corresponding to $m\in\mathcal{M}$, we assume $\widehat{\Theta}_m^{(1)}$ exists and is unique. The following proposition relates the KL risk of the aggregation estimator $\widehat{\Theta}_{{\text{gES}}}^{(1,2)}$ to the KL risks of individual estimators $\widehat{\Theta}_{m}^{(1)}$.

\begin{proposition}
\label{prop1}
The gES estimator $\widehat{\Theta}_{\emph{\text{gES}}}^{(1,2)}$ in Definition~\ref{maindef} satisfies the following inequality
\begin{align}
\emph{\text{KL}}(\widehat{\Theta}_{\emph{\text{gES}}}^{(1,2)})\leq &
\min_{m\in\mathcal{M}}\Bigl \{\emph{\text{KL}}(\widehat{\Theta}_{m}^{(1)})+\frac{2}{n_2}\log\frac{1}{\pi_{m}}+
\emph{\text{tr}}((\widehat{\Theta}_{m}^{(1)}-\widehat{\Theta}_{\emph{\text{gES}}}^{(1,2)})(\widehat{S}_{n_2}^{(2)}-\Sigma))\Bigr\} \\
\leq&\min_{m\in\mathcal{M}}\Bigl \{\emph{\text{KL}}(\widehat{\Theta}_{m}^{(1)})+\frac{2}{n_2}\log\frac{1}{\pi_{m}}+
\emph{\text{tr}}(\widehat{\Theta}_{m}^{(1)}(\widehat{S}_{n_2}^{(2)}-\Sigma))
+\|\widehat{S}_{n_2}^{(2)}-\Sigma\|\cdot\emph{\text{tr}}(\widehat{\Theta}_{\emph{\text{gES}}}^{(1,2)}).
\end{align}
\end{proposition}

It is shown in \citet{Vershynin12} that under some conditions
\begin{align}
\|\widehat{\Sigma}_{n_2}^{(2)}-\Sigma\|=O_P\left(\sqrt{\frac{p}{n_2}}\right),
\end{align}
thus if we use the empirical covariance matrix as $\widehat{S}_{n_2}^{(2)}$, Proposition~\ref{prop1} implies that this would lead to a deterioration of convergence rate in high dimensions. Then we choose to use the thresholded covariance matrix estimation in \citet{Bickel08}.

We consider the following assumption for further analysis of the remainder term.
\begin{assumption}
\label{asm}
The set of sparsity patterns $\mathcal{M}$ satisfies the following condition
\begin{align}
\max_{m\in\mathcal{M}}{\emph{\text{tr}}}(\widehat{\Theta}_m^{(1)})=O_P(p).
\end{align}
\end{assumption}

This assumption ensures a fast convergence rate of the aggregation estimator. It is shown in Dahl et~al.~\cite{Dahl05} that the inverse of $\widehat{\Theta}_m^{(1)}$ is a solution to the following problem that maximizes the determinant of a symmetric positive definite matrix $Z$:
\begin{equation}
\begin{aligned}
& \underset{Z\succ 0}{\text{max}}
& & \text{logdet}\;Z, \\
& \text{  s.t.}
& & Z_{ij} = [\widehat{\Sigma}_{n_1}^{(1)}]_{ij},\;\forall(i,j)\in{E}_m\cup\{(i,i),i=1,\ldots,p\}.
\end{aligned}
\end{equation}
In general, note that for any two graphs $G$ and $G'$ with sparsity patterns $m$ and $m'$, respectively, if $G$ is a subgraph of $G'$, then the trace of $\widehat{\Theta}_{m'}^{(1)}$ would always be larger than that of $\widehat{\Theta}_{m}^{(1)}$. Thus the most dense graph in the dictionary would always be able to achieve the maximum trace among all individual estimators. This assumption claims that the diagonal entries of the true precision matrix $\Theta$ are well estimated by all individual estimators in the dictionary $\mathcal{M}$.

Let ${m}^*\in\mathcal{M}$ be the sparsity pattern that attains the minimum KL risk in the dictionary $\mathcal{M}$:
\begin{align}
{m}^*=\underset{m\in\mathcal{M}}{\text{argmin }}{\text{KL}}(\widehat{\Theta}_{m}^{(1)}).
\end{align}

The following theorem shows the oracle inequality that the aggregation estimator satisfies.

\begin{theorem}
\label{main}
Suppose Assumption~\ref{asm} hold. Uniformly on $\mathcal{U}(q,\delta, M)$, for sufficiently large $K$, if the thresholding parameter $\lambda=K\sqrt{\frac{\log p}{n_2}}$, and $\frac{\log p}{n_2}=o(1)$, then the gES estimator $\widehat{\Theta}_{\emph{\text{gES}}}^{(1,2)}$ satisfies
\begin{align}
\emph{\text{KL}}(\widehat{\Theta}_{\emph{\text{gES}}}^{(1,2)})-\min_{m\in\mathcal{M}}\emph{\text{KL}}(\widehat{\Theta}_{m}^{(1)})
=O_P\left(\frac{{s}^*\log p}{n_2}+p\left(\frac{\log p}{n_2}\right)^{(1-q)/2}\right),
\end{align}
where ${s}^*=|{m}^*|_1$ is the number of nonzero off-diagonal elements of $\widehat{\Theta}_{{m}^*}$.
\end{theorem}

This theorem yields a rate of convergence of the excess risk. In particular, if the set of sparsity patterns $\mathcal{M}$ also includes the true sparsity pattern $m^0$. Let $s^0$ be the number of nonzero off-diagonal elements of $\Theta$. It is shown in Zhou et~al.~\cite{Zhou11} that under some technical conditions
\begin{align}
{\text{KL}}(\widehat{\Theta}_{m^0}^{(1)})=O_P\left(\frac{(s^0+p)\log p}{n_2}\right).
\end{align}
Combine it with Theorem~\ref{main} and assume that $s^0=O(p)$, we obtain
\begin{align}
{\text{KL}}(\widehat{\Theta}_{{\text{gES}}}^{(1,2)})=O_P\left(p\left(\frac{\log p}{n_2}\right)^{(1-q)/2}\right),
\end{align}
for $0\leq q<1$.

\section{Experimental Results}
\label{num}

In this section, we provide empirical evidence to illustrate the usefulness of the proposed gES estimator and compare it with other state-of-the-art methods in parameter estimation and graph recovery using simulated and real datasets.

\subsection{Numerical simulations}
We generate synthetic datasets with sample size $n=200$ or $400$, and number of nodes $p=50,100$ or $200$. We use the following three models for simulating graphs and precision matrices. Figure~\ref{fig:simulation} displays a typical run of the generated graphs of the precision matrices when $p=100$.

\begin{itemize}
\item[(a)] ``AR'': The off-diagonal $(i,j)$th element of the adjacency matrix is set to be 1 if $|i-j|= 1$ and 0 otherwise;
\item[(b)] ``Hub'': The vertices are evenly partitioned into $p/10$ disjoint groups. Each group is associated with a center vertex $i$ in that group and the off-diagonal $(i,j)$th element of the adjacency matrix is set to 1 if $j$ also belongs to the same group as $i$ and 0 otherwise;
\item[(c)] ``Random": The off-diagonal $(i,j)$th element of the adjacency matrix is randomly set to be 1 with certain probability  and 0 otherwise.
\end{itemize}

\begin{figure}[!h]
\begin{center}
\begin{tabular}{ccc}
\includegraphics[width=.25\textwidth]{figs/ar} & $\quad$
\includegraphics[width=.25\textwidth]{figs/hub} & $\quad$
\includegraphics[width=.25\textwidth]{figs/random} \\ \\
(a) AR & $\quad$ (b) Hub & $\quad$ (c) Random
\end{tabular}
\end{center}
\caption{{An illustration of the three graph patterns with $p=100$ nodes.}}
\label{fig:simulation}
\end{figure}

We compare the proposed gES estimator with the graphical lasso (glasso) \cite{Friedman08, Yuan07} and constrained $\ell_1$ minimization estimator (CLIME) \citep{Cai11} with the tuning parameters determined by 10-fold cross-validation. We also consider the method of multiple testing of hypotheses about vanishing partial correlation coefficients \cite{Drton04, Drton07}, where the family-wise error rates for incorrect edge inclusion are controlled by Bonferroni correction at level $\alpha=0.05$. We refer this method as ``pcorTest'' in this paper. For the gES estimator, the full dataset is random partitioned into two subsamples with equal size, while for other methods, the graphs are estimated using the full dataset in order to make the comparison fair.

Notice that when $p$ is large, the Metropolis-Hastings algorithm for the gES estimator might take a large number of iterations to generate a good random tour over the hypercube with as many as $p(p-1)/2$ dimensions, and thus could become computationally unsuitable. In this case, we need to approximate the model space and select a candidate set of edges before the execution of the Metropolis-Hastings algorithm. We identify a candidate set of edges by applying some pre-screening method to the first subsample $\mathcal{D}_{n_1}^{(1)}$. Let the ordered list $\mathcal{Q}_p\subset\mathcal{S}_p$ be the set of selected edges. Then our aggregation process is constructed on this subset of edges. Then the set of sparsity patterns is given by
\begin{align}
\mathcal{M}=\prod_{k=1}^{p(p-1)/2}\mathcal{C}_k,
\end{align}
where
\begin{align}
\mathcal{C}_k: = \begin{cases} \{0,1\}\quad &\text{if }(i_k,j_k)\in\mathcal{Q}_p\text{, where }(i_k,j_k)\text{ is the }k\text{th}
\text{ element of }\mathcal{S}_p\\ \{0\}\quad &\text{otherwise} \end{cases}.
\end{align}
In practice where $p$ is large, the Metropolis-Hastings algorithm introduced in Algorithm~\ref{alg} will be applied to this reduced set of sparsity patterns instead of the set of all possible edges $\{0, 1\}^{p(p-1)/2}$. Many pre-screening methods could be considered here, for instance, the glasso. Note that we prefer to choose a small regularization parameter to prevent any true edge being ruled out in this stage. An alternative algorithm could be based on the $\ell_l$ regularization paths from the glasso method. Let $\mathcal{B}$ be a set of regularization parameters. For each $\beta\in\mathcal{B}$, let $m_\beta$ be the sparsity pattern resulted from glasso with tuning parameter $\beta$. Then the aggregation method is constructed on the set of sparsity patterns $\mathcal{M}=\{m_\beta:\beta\in\mathcal{B}\}$. However, in practical applications, the Metropolis-Hastings algorithm based on the $\ell_l$ regularization paths always concentrations on a single model after convergence, and generally does not perform well.

For any estimator $\widehat\Theta\succ0$, we use the following criteria for the comparisons.
\vspace{-\topsep}
\begin{itemize}
\item[(1)] Squared Frobenius norm error:
\begin{align}
\text{Frobenius}=\|\widehat\Theta-\Theta\|_F^2;
\end{align}
\item[(2)] Kullback-Leibler loss:
\begin{align}
\text{KL}&=-\log\text{det}(\widehat{\Theta})+\text{trace}(\widehat{\Theta}\Sigma)-(-\log\text{det}(\Theta)+p);
\end{align}
\item[(3)] Precision: the number of correctly estimated edges divided by the total number of edges in the estimated graph. Let $G=(V, E)$ be a $p$-dimensional graph and let $\widehat{G}=(\widehat{V},\widehat{E})$ be an estimated graph. We define
\begin{align}
\text{Precision}=\frac{|\widehat{E}\cap E|}{|\widehat{E}|};
\end{align}
\item[(4)] Recall: the number of correctly estimated edges divided by the total number of edges in the true graph, defined by
\begin{align}
\text{Recall}=\frac{|\widehat{E}\cap E|}{|E|};
\end{align}
\item[(5)] $F_1$-score: a weighted average of the precision and recall, defined by
\begin{align}
F_1\text{-score}=2\cdot\frac{\text{Precision}\cdot\text{Recall}}{\text{Precision}+\text{Recall}},
\end{align}
where an $F_1$-score reaches its best value at 1 and worst score at 0.
\end{itemize}

\begin{table*}[!ht]
\caption{{Quantitative comparison of different methods on simulated data; Averaged quantities with their standard errors (in parentheses) are reported.}}
\vspace{0.2cm}
\label{tab:simresult}
\centering
\resizebox{9.22cm}{!}{
\begin{tabular}{llrrccc}\toprule
& & \multicolumn{5}{c}{$(n,p)=(200,50)$} \\
\cmidrule(l){3-7}
Model & Estimator & Frobenius\; & KL$\qquad$ & Precision & Recall & $F_1$-score \\
\midrule
AR& gES & 6.06 (0.97) & 1.61 (0.29) & 0.73 (0.05) & 0.97 (0.02) & 0.83 (0.03) \\
& pcorTest & 5.91 (1.48) & 1.73 (0.49) & 0.99 (0.01) & 0.89 (0.04) & 0.94 (0.02) \\
& glasso & 6.39 (0.82) & 1.51 (0.14) & 0.21 (0.03) & 1.00 (0.00) & 0.35 (0.04) \\
& CLIME & 5.46 (0.76) & 1.77 (0.25) & 0.15 (0.02) & 1.00 (0.00) & 0.26 (0.03) \\
\midrule
Hub& gES & 6.29 (1.57) & 1.44 (0.31) & 0.65 (0.06) & 0.97 (0.02) & 0.78 (0.04) \\
& pcorTest & 26.56 (3.77) & 6.02 (0.67) & 0.98 (0.02) & 0.53 (0.06) & 0.69 (0.05) \\
& glasso & 16.43 (1.84) & 1.53 (0.13) & 0.18 (0.02) & 1.00 (0.00) & 0.31 (0.03) \\
& CLIME & 7.34 (0.99) & 3.03 (0.43) & 0.14 (0.01) & 1.00 (0.00) & 0.24 (0.02) \\
\midrule
Random& gES & 5.94 (1.72) & 1.72 (0.46) & 0.73 (0.05) & 0.91 (0.06) & 0.81 (0.04) \\
& pcorTest & 8.55 (3.63) & 2.45 (0.93) & 0.99 (0.02) & 0.72 (0.13) & 0.83 (0.09) \\
& glasso & 7.09 (1.18) & 1.31 (0.16) & 0.20 (0.03) & 1.00 (0.01) & 0.33 (0.04) \\
& CLIME & 4.59 (0.91) & 1.68 (0.32) & 0.13 (0.02) & 1.00 (0.00) & 0.23 (0.03) \\
\midrule
\midrule
& & \multicolumn{5}{c}{$(n,p)=(200,100)$} \\
\cmidrule(l){3-7}
Model & Estimator & Frobenius\; & KL$\qquad$ & Precision & Recall & $F_1$-score \\
\midrule
AR& gES & 13.94 (2.58) & 3.10 (0.55) & 0.84 (0.03) & 0.98 (0.02) & 0.90 (0.02)\\
& pcorTest & 17.48 (2.70) & 4.55 (0.72) & 0.91 (0.03) & 0.88 (0.03) & 0.89 (0.02) \\
& glasso & 20.89 (1.77) & 3.71 (0.21) & 0.17 (0.02) & 1.00 (0.00) & 0.29 (0.02) \\
& CLIME & 13.90 (1.47) & 4.79 (0.50) & 0.13 (0.01) & 1.00 (0.00) & 0.23 (0.02) \\
\midrule
Hub& gES & 14.27 (2.77) &  3.29 (0.58) & 0.77 (0.03) & 0.97 (0.02) & 0.86 (0.02) \\
& pcorTest & 72.07 (5.98) & 14.92 (0.93) & 0.81 (0.08) & 0.42 (0.04) & 0.55 (0.04) \\
& glasso & 38.56 (1.67) & 3.61 (0.17) & 0.15 (0.01) & 1.00 (0.00) & 0.27 (0.02) \\
& CLIME & 17.27 (2.14) & 8.48 (0.89) & 0.12 (0.01) & 1.00 (0.00) & 0.22 (0.02) \\
\midrule
Random& gES & 12.89 (3.05) & 3.94 (0.82) & 0.85 (0.04) & 0.88 (0.05) & 0.86 (0.04) \\
& pcorTest & 26.05 (7.44) & 7.15 (1.70) & 0.82 (0.06) & 0.58 (0.10) & 0.68 (0.08) \\
& glasso & 16.58 (2.24) & 3.04 (0.22) & 0.18 (0.02) & 1.00 (0.00) & 0.30 (0.04) \\
& CLIME & 10.39 (1.12) & 4.27 (0.50) & 0.11 (0.01) & 1.00 (0.00) & 0.20 (0.02) \\
\midrule
\midrule
& & \multicolumn{5}{c}{$(n,p)=(400,100)$} \\
\cmidrule(l){3-7}
Model & Estimator & Frobenius\; & KL$\qquad$ & Precision & Recall & $F_1$-score \\
\midrule
AR& gES & 5.54 (1.12) & 1.14 (0.18) & 0.82 (0.03) & 1.00 (0.00) & 0.90 (0.02) \\
& pcorTest & 2.46 (0.30) & 0.51 (0.06) & 0.99 (0.01) & 1.00 (0.00) & 1.00 (0.00) \\
& glasso & 12.17 (0.95) & 1.93 (0.10) & 0.17 (0.01) & 1.00 (0.00) & 0.29 (0.02) \\
& CLIME & 7.55 (0.58) & 2.08 (0.21) & 0.14 (0.01) & 1.00 (0.00) & 0.25 (0.02) \\
\midrule
Hub& gES & 5.27 (1.12) & 1.16 (0.18) & 0.82 (0.03) & 0.99 (0.01) & 0.90 (0.02) \\
& pcorTest & 11.20 (2.66) & 2.96 (0.69) & 0.99 (0.01) & 0.90 (0.03) & 0.94 (0.02) \\
& glasso & 23.76 (1.76) & 1.91 (0.09) & 0.15 (0.02) & 1.00 (0.00) & 0.26 (0.03) \\
& CLIME & 8.31 (0.78) & 3.20 (0.27) & 0.13 (0.00) & 1.00 (0.00) & 0.23 (0.01) \\
\midrule
Random& gES & 5.46 (0.85) & 1.98 (0.33) & 0.84 (0.03) & 0.94 (0.04) & 0.88 (0.03) \\
& pcorTest & 6.53 (1.76) & 2.42 (0.65) & 0.99 (0.01) & 0.82 (0.06) & 0.90 (0.04) \\
& glasso & 6.48 (0.75) & 1.69 (0.15) & 0.19 (0.02) & 1.00 (0.00) & 0.32 (0.03) \\
& CLIME & 5.03 (0.54) & 1.95 (0.17) & 0.14 (0.02) & 1.00 (0.00) & 0.24 (0.02) \\
\midrule
\midrule
& & \multicolumn{5}{c}{$(n,p)=(400,200)$} \\
\cmidrule(l){3-7}
Model & Estimator & Frobenius\; & KL$\qquad$ & Precision & Recall & $F_1$-score \\
\midrule
AR& gES & 14.27 (2.42) & 2.99 (0.53) & 0.89 (0.02) & 0.99 (0.01) & 0.94 (0.01) \\
& pcorTest & 6.02 (1.13) & 1.36 (0.28) & 0.92 (0.02) & 1.00 (0.00) & 0.96 (0.01) \\
& glasso & 29.93 (2.37) & 4.62 (0.19) & 0.14 (0.02) & 1.00 (0.00) & 0.25 (0.03) \\
& CLIME & 16.33 (1.53) & 5.66 (0.61) & 0.10 (0.03) & 1.00 (0.00) & 0.18 (0.05) \\
\midrule
Hub& gES & 13.49 (2.63) & 2.89 (0.54) & 0.83 (0.04) & 0.99 (0.01) & 0.90 (0.03) \\
& pcorTest & 44.47 (4.72) & 11.25 (1.11) & 0.89 (0.03) & 0.79 (0.02) & 0.84 (0.02) \\
& glasso & 54.98 (1.84) & 4.45 (0.16) & 0.13 (0.00) & 1.00 (0.00) & 0.23 (0.01) \\
& CLIME & 19.65 (1.68) & 8.17 (0.50) & 0.14 (0.00) & 1.00 (0.00) & 0.25 (0.01) \\
\midrule
Random& gES & 10.43 (1.24) & 3.88 (0.51) & 0.95 (0.02) & 0.90 (0.04) & 0.92 (0.02) \\
& pcorTest & 13.21 (1.81) & 4.95 (0.74) & 0.87 (0.04) & 0.77 (0.06) & 0.81 (0.04) \\
& glasso & 11.67 (1.18) & 3.24 (0.20) & 0.16 (0.01) & 1.00 (0.00) & 0.28 (0.02) \\
& CLIME & 9.77 (0.97) & 3.56 (0.27) & 0.12 (0.01) & 1.00 (0.00) & 0.21 (0.02) \\
\bottomrule
\end{tabular}}
\end{table*}

Table~\ref{tab:simresult} shows the simulation results of the quantitative comparison of different methods, where we repeat the experiments 50 times and report the averaged values with their standard errors. We can see that the gES and CLIME estimators perform better than glasso and pcorTest in term of the squared Frobenius norm errors, while gES and glasso are better than CLIME and pcorTest regarding the Kullback-Leibler loss. For the comparison of graph structure recovery, the gES and pcorTest estimators outperform other methods.

Figure~\ref{fig:mcmc} displays the evolution of the Metropolis-Hastings algorithm, showing evidence that the algorithm converges after 4000 iterations. Figure~\ref{fig:sensitivity} shows a typical realization of the gES method, varying the regularization parameter $\lambda$ in the pre-screening glasso, where we can see that the results for graphical aggregation are not quite sensitive to the pre-screening method.

\begin{figure}[!h]
\begin{center}
\vspace{-0.2cm}
\begin{tabular}{cc}
\includegraphics[width=.7\textwidth]{figs/mcmc} \\
\end{tabular}
\end{center}
\vspace{-0.4cm}
\caption{Number of selected edges by the Metropolis-Hastings algorithm as a function of iterations, given a typical run of the gES estimator for Hub graph on simulated data with $(n,p)=(400,100)$.}
\label{fig:mcmc}
\end{figure}

\begin{figure}[!h]
\vskip 0.2in
\begin{center}
\begin{tabular}{cc}
\includegraphics[width=.45\textwidth]{figs/kl} & $\quad$
\includegraphics[width=.45\textwidth]{figs/precision}  \\
\end{tabular}
\end{center}
\caption{{Typical realization of the gES estimator for Hub graph on simulated data with $(n,p)=(400,100)$, varying the regularization parameter $\lambda$ in the pre-screening glasso; Number of selected candidate edges (marked as $E$) are also reported.}}
\label{fig:sensitivity}
\end{figure}

The computational complexity of glasso \cite{Friedman08} which uses a row-by-row block coordinate algorithm is roughly $O(p^3)$. For the gES estimator, the complexity is roughly $O(L p^3)$, where $L=T+T_0$ is the number of iterations in the Metropolis-Hastings algorithm. Note that $L$ can be dramatically reduced if we keep track of and store the individual estimators to avoid duplicated computation of precision matrix with the same sparsity pattern.

\subsection{Analysis of microarray data}

In this study, we consider a real-world dataset based on Affymetrix GeneChip microarrays for the plant \emph{Arabidopsis thaliana} \cite{Wille04}. The sample size is $n = 118$. A nonparanormal transformation is estimated and the expression levels for each chip are replaced by their respective normal scores, subject to a Winsorized truncation \cite{Liu09}. A subset of $p=40$ genes from the isoprenoid pathway are chosen, and we study the associations among them using the proposed gES estimator and the glasso method with tuning parameter determined by cross-validation.

The results show that glasso selects 378 edges and gES selects 111 edges.  Among those selected edges, 102 edges are identified by both methods. Figure~\ref{fig:image} shows grids of rectangles with gray scale corresponding to the absolute values in the estimated precision matrix for each method.

\begin{figure}[!h]
\begin{center}
\begin{tabular}{cc}
\includegraphics[width=.45\textwidth]{figs/realglimage1} & $\quad$
\includegraphics[width=.45\textwidth]{figs/realgesimage1} \\
(a) glasso & $\quad$ (b) gES \\
\end{tabular}
\end{center}
\caption{Grids of rectangles with gray scale corresponding to the absolute values in the estimated precision matrix for each method.}
\label{fig:image}
\end{figure}

We also analyze a dataset on microarrays for the gene expression levels \cite{Nayak09}. Of the 4238 genes in immortalized B cells for $n=295$ normal individuals, we select $p=318$ genes that are associated with the phenotypes in genome-wide association studies. We study the estimated graphs obtained by the glasso and gES estimators. The expression levels for each gene are pre-processed by log-transformation and standardization.

The results indicate that glasso selects 12514 edges and the gES estimator selects 1631 edges. Among those, 1542 edges are identified by both methods. Figure~\ref{fig:realgraph2} provides the estimated graphs.

\begin{figure}[!h]
\begin{center}
\begin{tabular}{cc}
\includegraphics[width=.45\textwidth]{figs/realgl2} & $\quad$
\includegraphics[width=.45\textwidth]{figs/realges2} \\ \\
(a) glasso & $\quad$ (b) gES \\
\end{tabular}
\end{center}
\caption{Estimated graphs for microarray data example for immortalized human B cells.}
\label{fig:realgraph2}
\end{figure}

\section{Conclusion}
\label{conclude}

In this paper, we propose a new aggregation method for estimating the precision matrix in Gaussian graphical models, by considering a convex combination of a suitable set of individual estimators with different underlying sparsity patterns. We investigate the risk of this aggregation estimator and show by an oracle that it is comparable to the risk of the best estimator based on a single graph. Experimental results validate the usefulness of our method in practice.

\section*{Acknowledgements}
This paper is based on part of my Ph.D dissertation. I want to thank my advisor, Dr. John Lafferty, for his help and guidance on this project.

redpajama_set_name: RedPajamaArXiv


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By Jeffrey St. Clair, CounterPunch, May 19, 2009
Things go missing. It's to be expected. Even at the Pentagon. Last October, the Pentagon's inspector general reported that the military's accountants had misplaced a destroyer, several tanks and armored personnel carriers, hundreds of machine guns, rounds of ammo, grenade launchers and some surface-to-air missiles. In all, nearly $8 billion in weapons were AWOL.
Those anomalies are bad enough. But what's truly chilling is the fact that the Pentagon has lost track of the mother of all weapons, a hydrogen bomb. The thermonuclear weapon, designed to incinerate Moscow, has been sitting somewhere off the coast of Savannah, Georgia for the past 40 years. The Air Force has gone to greater lengths to conceal the mishap than to locate the bomb and secure it.
On the night of February 5, 1958 a B-47 Stratojet bomber carrying a hydrogen bomb on a night training flight off the Georgia coast collided with an F-86 Saberjet fighter at 36,000 feet. The collision destroyed the fighter and severely damaged a wing of the bomber, leaving one of its engines partially dislodged. The bomber's pilot, Maj. Howard Richardson, was instructed to jettison the H-bomb before attempting a landing. Richardson dropped the bomb into the shallow waters of Warsaw Sound, near the mouth of the Savannah River, a few miles from the city of Tybee Island, where he believed the bomb would be swiftly recovered.
The Pentagon recorded the incident in a top secret memo to the chairman of the Atomic Energy Commission. The memo has been partially declassified: "A B-47 aircraft with a [word redacted] nuclear weapon aboard was damaged in a collision with an F-86 aircraft near Sylvania, Georgia, on February 5, 1958. The B-47 aircraft attempted three times unsuccessfully to land with the weapon. The weapon was then jettisoned visually over water off the mouth of the Savannah River. No detonation was observed."
Soon search and rescue teams were sent to the site. Warsaw Sound was mysteriously cordoned off by Air Force troops. For six weeks, the Air Force looked for the bomb without success. Underwater divers scoured the depths, troops tromped through nearby salt marshes, and a blimp hovered over the area attempting to spot a hole or crater in the beach or swamp. Then just a month later, the search was abruptly halted. The Air Force sent its forces to Florence, South Carolina, where another H-bomb had been accidentally dropped by a B-47. The bomb's 200 pounds of TNT exploded on impact, sending radioactive debris across the landscape. The explosion caused extensive property damage and several injuries on the ground. Fortunately, the nuke itself didn't detonate.
The search teams never returned to Tybee Island, and the affair of the missing H-bomb was discreetly covered up. The end of the search was noted in a partially declassified memo from the Pentagon to the AEC, in which the Air Force politely requested a new H-bomb to replace the one it had lost. "The search for this weapon was discontinued on 4-16-58 and the weapon is considered irretrievably lost. It is requested that one [phrase redacted] weapon be made available for release to the DOD as a replacement."
There was a big problem, of course, and the Pentagon knew it. In the first three months of 1958 alone, the Air Force had four major accidents involving H-bombs. (Since 1945, the United States has lost 11 nuclear weapons.) The Tybee Island bomb remained a threat, as the AEC acknowledged in a June 10, 1958 classified memo to Congress: "There exists the possibility of accidental discovery of the unrecovered weapon through dredging or construction in the probable impact area. … The Department of Defense has been requested to monitor all dredging and construction activities."
But the wizards of Armageddon saw it less as a security, safety or ecological problem, than a potential public relations disaster that could turn an already paranoid population against their ambitious nuclear project. The Pentagon and the AEC tried to squelch media interest in the issue by a doling out a morsel of candor and a lot of misdirection. In a joint statement to the press, the Defense Department and the AEC admitted that radioactivity could be "scattered" by the detonation of the high explosives in the H-bombs. But the letter downplayed possibility of that ever happening: "The likelihood that a particular accident would involve a nuclear weapon is extremely limited."
In fact, that scenario had already occurred and would occur again.
That's where the matter stood for more than 42 years until a deep sea salvage company, run by former Air Force personnel and a CIA agent, disclosed the existence of the bomb and offered to locate it for a million dollars. Along with recently declassified documents, the disclosure prompted fear and outrage among coastal residents and calls for a congressional investigation into the incident itself and why the Pentagon had stopped looking for the missing bomb. "We're horrified because some of that information has been covered up for years," said Rep. Jack Kingston, a Georgia Republican.
The cover-up continues. The Air Force, however, has told local residents and the congressional delegation that there was nothing to worry about.
"We've looked into this particular issue from all angles and we're very comfortable," said Major Gen. Franklin J. "Judd" Blaisdell, deputy chief of staff for air and space operations at Air Force headquarters in Washington. "Our biggest concern is that of localized heavy metal contamination."……………………………………………………………………
In short, there aren't any easy answers. The problem is exacerbated by the Pentagon's failure to conduct a comprehensive analysis of the situation and reluctance to fully disclose what it knows. "I believe the plutonium capsule is in the bomb, but that a nuclear detonation is improbable because the neutron generators used back then were polonium-beryllium, which has a very short half-life," said Don Moniak, a nuclear weapons expert with the Blue Ridge Environmental Defense League in Aiken, South Carolina. "Without neutrons, weapons grade plutonium won't blow. However, there could be a fission or criticality event if the plutonium was somehow put in an incorrect configuration. There could be a major inferno if the high explosives went off and the lithium deuteride reacted as expected. Or there could just be an explosion that scattered uranium and plutonium all over hell."
This essay is featured in the forthcoming book, Loose Nukes published by Count Zero Press.
Jeffrey St. Clair is the author of Been Brown So Long It Looked Like Green to Me: the Politics of Nature and Grand Theft Pentagon. His newest book, Born Under a Bad Sky, is just out from AK Press / CounterPunch books. He can be reached at: sitka@comcast.net.
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Q: Compilation error: Generic array creation I would like to create an array of ArrayList<String>. I tried the following:
static ArrayList<String>[] displayBlocks = new ArrayList<String>[3];

However, I'm getting a compile time error:

generic array creation

I have added import java.util.*;. How can I get it to compile?

A: This construct is indeed not allowed in Java. You could use the varargs hack:
static List<String>[] displayBlocks = createArray(3);

with
public static <E> E[] createArray(int length, E... elements) {
    return Arrays.copyOf(elements, length);
}

Needless to say, a List<List<String>> is better, unless you're extremely tight on memory, but then I'd wonder why you don't use a String[][].

A: A side from the other answer one can also create the list like this, which might feel a little bit more familiar
static ArrayList<String>[] displayBlocks = (ArrayList<String>[]) new ArrayList[3];

A: if you want an array of arraylist:
import java.util.ArrayList;
import java.util.List;

public class Foo{

    List [] arrayOfLists = new ArrayList[10];

}

Here is a related post. you cant create a generic array of arraylist.
You can do this:
import java.util.ArrayList;
import java.util.List;

public class Foo{

    ArrayList<ArrayList<String>> ll = new ArrayList<ArrayList<String>>();

}

A: From the Java Language Specification section on array creation expressions:

It is a compile-time error if the ClassOrInterfaceType does not denote a reifiable type (§4.7). . . . The rules above imply that the element type in an array creation expression cannot be a parameterized type, other than an unbounded wildcard.

Like others have said, use an ArrayList of ArrayLists, or else use a cast (where you might want to also suppress unchecked conversion warnings).

A: What you are doing wrong :
You have not created an object.  Remember that in java
1) All objects have constructors.
2) Constructors are methods.
3) To create an object, you must call the constructor , thus, you are calling a method !
Your desire is ambiguous . But it will be good to read both of these answers :
1) You could make an Array of array lists : This is a data structure with 10 array lists.  it is essentially a 2D array where the amount of columns per row is variable, but there will only be 10 rows.
This would be an odd data structure to create.  It might represent, for example, a domain model where you have exactly 10 people and each of those people had a variable number of dvds in their dvd collection. so you would have 10 rows, each one with an array list in it.
static ArrayList[] displayBlocks = new ArrayList[10];

*

*not that there is no () here, just a [] * This is because we havent populated our array, rather, we just declare that there is an array with 10 slots, and each slot will have type "ArrayList". To populate, you would have to iterate and add displayBlock[0]=new ArrayList(), for 0-9.

2) Probably, you just wanted a simple array list , which is a linear collection of items.
To do this you would simply declare :
Collection a = new ArrayList();
//or
List a = new ArrayList();
Now ... you should ask yourself why you can declare the variable as either a Collection or a List, if you want to really java's collections work.

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'It's like someone has died': Kerry is heartbroken without George Kay
Every break-up is hard, but it appears that Kerry Katona is really struggling with her latest split-up.
The former Atomic Kitten announced that she had ended her one-year marriage to George Kay last month after it was reported that he had assaulted her.
The singer spoke to OK! Magazine and admitted that the break-up was difficult but it was "the right thing to do."
"George was with me 24/7. It's like somebody has died. He was my best friend," said the mother-of-five.
"There is no going back with George and me. That is what hurts so much, as I know he is a good person."
Ms Katona also admitted that she will never get married again after the breakdown of her most recent marriage but will always miss having George by her side.
"I think the hardest thing is walking away from somebody you love so much, but it's the right thing to do."
george kay


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http://www.nationsreportcard.gov/reading_math_2015/?utm_sou…
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Forums » Professional Video Editing & Finishing Forums » Avid Media Composer - Mac » 2018 Macbook Pros STILL not whitelisted?
Latest post Sun, Aug 19 2018 6:12 PM by benrichardson. 5 replies.
2018 Macbook Pros STILL not whitelisted?
Hi, I recently got a Macbook Pro 2018 13 inch and learned that since their release, they haven't been whitelisted by Avid.
A rep said they were 'hardware testing' the machines on July 26th, which is more than 2 weeks ago. They still haven't whitelisted the new Macbook Pros. People like me have deadlines to reach and without Media Composer working that obviously is impossible. Can we please come to a resolution for this?
Re: 2018 Macbook Pros STILL not whitelisted?
The testing is on its final stage, soon we will be posting the updates.
Also, people seem to be able to use 2018.2 with their new 2018 MBP's, so that might be a workaround for now?
2018.8 was released today. However, while it is compatible with 10.13.6, we have not officially aproved the 2018 Macbooks and iMac's.