title: McCarthyism

text: In March 1950, McCarthy had initiated a series of investigations into potential infiltration of the Central Intelligence Agency (CIA) by communist agents and came up with a list of security risks that matched one previously compiled by the Agency itself. At the request of CIA director Allen Dulles, President Eisenhower demanded that McCarthy discontinue issuing subpoenas against the CIA. Documents made public in 2004 revealed that the CIA, under Dulles' orders, had broken into McCarthy's Senate office and fed disinformation to him in order to discredit him and stop his investigation from proceeding any further.


title: McCarthyism

text: The House Committee on Un-American Activities—commonly referred to as the HUAC—was the most prominent and active government committee involved in anti-communist investigations. Formed in 1938 and known as the Dies Committee, named for Rep. Martin Dies, who chaired it until 1944, HUAC investigated a variety of "activities", including those of German-American Nazis during World War II. The committee soon focused on Communism, beginning with an investigation into Communists in the Federal Theatre Project in 1938. A significant step for HUAC was its investigation of the charges of espionage brought against Alger Hiss in 1948. This investigation ultimately resulted in Hiss's trial and conviction for perjury, and convinced many of the usefulness of congressional committees for uncovering Communist subversion.


title: McCarthyism

text: HUAC achieved its greatest fame and notoriety with its investigation into the Hollywood film industry. In October 1947, the committee began to subpoena screenwriters, directors, and other movie-industry professionals to testify about their known or suspected membership in the Communist Party, association with its members, or support of its beliefs. At these testimonies, this question was asked: "Are you now or have you ever been a member of the Communist Party of the United States?" Among the first film industry witnesses subpoenaed by the committee were ten who decided not to cooperate. These men, who became known as the "Hollywood Ten", cited the First Amendment's guarantee of free speech and free assembly, which they believed legally protected them from being required to answer the committee's questions. This tactic failed, and the ten were sentenced to prison for contempt of Congress. Two of them were sentenced to six months, the rest to a year.


title: McCarthyism

text: In the future, witnesses (in the entertainment industries and otherwise) who were determined not to cooperate with the committee would claim their Fifth Amendment protection against self-incrimination. William Grooper and Rockwell Kent, the only two visual artists to be questioned by McCarthy, both took this approach, and emerged relatively unscathed by the experience. However, while this usually protected witnesses from a contempt-of-Congress citation, it was considered grounds for dismissal by many government and private-industry employers. The legal requirements for Fifth Amendment protection were such that a person could not testify about his own association with the Communist Party and then refuse to "name names" of colleagues with communist affiliations. Thus, many faced a choice between "crawl[ing] through the mud to be an informer," as actor Larry Parks put it, or becoming known as a "Fifth Amendment Communist"—an epithet often used by Senator McCarthy.


title: McCarthyism

text: In the Senate, the primary committee for investigating communists was the Senate Internal Security Subcommittee (SISS), formed in 1950 and charged with ensuring the enforcement of laws relating to "espionage, sabotage, and the protection of the internal security of the United States". The SISS was headed by Democrat Pat McCarran and gained a reputation for careful and extensive investigations. This committee spent a year investigating Owen Lattimore and other members of the Institute of Pacific Relations. As had been done numerous times before, the collection of scholars and diplomats associated with Lattimore (the so-called China Hands) were accused of "losing China", and while some evidence of pro-communist attitudes was found, nothing supported McCarran's accusation that Lattimore was "a conscious and articulate instrument of the Soviet conspiracy". Lattimore was charged with perjuring himself before the SISS in 1952. After many of the charges were rejected by a federal judge and one of the witnesses confessed to perjury, the case was dropped in 1955.


title: McCarthyism

text: McCarthy headed the Senate Permanent Subcommittee on Investigations in 1953 and 1954, and during that time, used it for a number of his communist-hunting investigations. McCarthy first examined allegations of communist influence in the Voice of America, and then turned to the overseas library program of the State Department. Card catalogs of these libraries were searched for works by authors McCarthy deemed inappropriate. McCarthy then recited the list of supposedly pro-communist authors before his subcommittee and the press. Yielding to the pressure, the State Department ordered its overseas librarians to remove from their shelves "material by any controversial persons, Communists, fellow travelers, etc." Some libraries actually burned the newly forbidden books. Though he did not block the State Department from carrying out this order, President Eisenhower publicly criticized the initiative as well, telling the graduating class of Dartmouth College President in 1953: “Don’t join the book burners! … Don’t be afraid to go to the library and read every book so long as that document does not offend our own ideas of decency—that should be the only censorship.” The president then settled for a compromise by retaining the ban on Communist books written by Communists, while also allowing the libraries to keep books on Communism written by anti-Communists.


title: McCarthyism

text: McCarthy's committee then began an investigation into the United States Army. This began at the Army Signal Corps laboratory at Fort Monmouth. McCarthy garnered some headlines with stories of a dangerous spy ring among the Army researchers, but ultimately nothing came of this investigation.


title: McCarthyism

text: McCarthy next turned his attention to the case of a U.S. Army dentist who had been promoted to the rank of major despite having refused to answer questions on an Army loyalty review form. McCarthy's handling of this investigation, including a series of insults directed at a brigadier general, led to the Army–McCarthy hearings, with the Army and McCarthy trading charges and counter-charges for 36 days before a nationwide television audience. While the official outcome of the hearings was inconclusive, this exposure of McCarthy to the American public resulted in a sharp decline in his popularity. In less than a year, McCarthy was censured by the Senate, and his position as a prominent force in anti-communism was essentially ended.


title: McCarthyism

text: On November 25, 1947, the day after the House of Representatives approved citations of contempt for the Hollywood Ten, Eric Johnston, president of the Motion Picture Association of America, issued a press release on behalf of the heads of the major studios that came to be referred to as the Waldorf Statement. This statement announced the firing of the Hollywood Ten and stated: "We will not knowingly employ a Communist or a member of any party or group which advocates the overthrow of the government of the United States..." This marked the beginning of the Hollywood blacklist. In spite of the fact that hundreds were denied employment, the studios, producers, and other employers did not publicly admit that a blacklist existed.


title: McCarthyism

text: At this time, private loyalty-review boards and anti-communist investigators began to appear to fill a growing demand among certain industries to certify that their employees were above reproach. Companies that were concerned about the sensitivity of their business, or which, like the entertainment industry, felt particularly vulnerable to public opinion, made use of these private services. For a fee, these teams investigated employees and questioned them about their politics and affiliations.


title: McCarthyism

text: At such hearings, the subject usually did not have a right to the presence of an attorney, and as with HUAC, the interviewee might be asked to defend himself against accusations without being allowed to cross-examine the accuser. These agencies kept cross-referenced lists of leftist organizations, publications, rallies, charities, and the like, as well as lists of individuals who were known or suspected communists. Books such as "Red Channels" and newsletters such as "Counterattack" and "Confidential Information" were published to keep track of communist and leftist organizations and individuals. Insofar as the various blacklists of McCarthyism were actual physical lists, they were created and maintained by these private organizations.


title: McCarthyism

text: Efforts to protect the United States from the perceived threat of communist subversion were particularly enabled by several federal laws. The Hatch Act of 1939 banned membership in subversive organizations, which was interpreted as being anti-labor legislation. The Hatch Act would allow for the reduction of influence of the Workers' Alliance, which was claimed to have been created by the Soviet Union based on a model of their unemployed councils. The Alien Registration Act or Smith Act of 1940 made the act of "knowingly or willfully advocate, abet, advise or teach the ... desirability or propriety of overthrowing the Government of the United States or of any State by force or violence, or for anyone to organize any association which teaches, advises or encourages such an overthrow, or for anyone to become a member of or to affiliate with any such association" a criminal offense.


title: McCarthyism

text: Hundreds of communists and others were prosecuted under this law between 1941 and 1957. Eleven leaders of the Communist Party were convicted under the Smith Act in 1949 in the Foley Square trial. Ten defendants were given sentences of five years and the eleventh was sentenced to three years. The defense attorneys were cited for contempt of court and given prison sentences. In 1951, 23 other leaders of the party were indicted, including Elizabeth Gurley Flynn, a founding member of the American Civil Liberties Union. Many were convicted on the basis of testimony that was later admitted to be false. By 1957, 140 leaders and members of the Communist Party had been charged under the law, of whom 93 were convicted.


title: McCarthyism

text: The McCarran Internal Security Act, which became law in 1950, has been described by scholar Ellen Schrecker as "the McCarthy era's only important piece of legislation" (the Smith Act technically antedated McCarthyism). However, the McCarran Act had no real effect beyond legal harassment. It required the registration of Communist organizations with the U.S. Attorney General and established the Subversive Activities Control Board to investigate possible communist-action and communist-front organizations so they could be required to register. Due to numerous hearings, delays, and appeals, the act was never enforced, even with regard to the Communist Party of the United States itself, and the major provisions of the act were found to be unconstitutional in 1965 and 1967. In 1952, the Immigration and Nationality, or McCarran–Walter, Act was passed. This law allowed the government to deport immigrants or naturalized citizens engaged in subversive activities and also to bar suspected subversives from entering the country.


title: McCarthyism

text: The Communist Control Act of 1954 was passed with overwhelming support in both houses of Congress after very little debate. Jointly drafted by Republican John Marshall Butler and Democrat Hubert Humphrey, the law was an extension of the Internal Security Act of 1950, and sought to outlaw the Communist Party by declaring that the party, as well as "Communist-Infiltrated Organizations" were "not entitled to any of the rights, privileges, and immunities attendant upon legal bodies." While the Communist Control Act had an odd mix of liberals and conservatives among its supporters, it never had any significant effect.


title: McCarthyism

text: The act was successfully applied only twice. In 1954 it was used to prevent Communist Party members from appearing on the New Jersey state ballot, and in 1960, it was cited to deny the CPUSA recognition as an employer under New York state's unemployment compensation system. The "New York Post" called the act "a monstrosity", "a wretched repudiation of democratic principles," while "The Nation" accused Democratic liberals of a "neurotic, election-year anxiety to escape the charge of being 'soft on Communism' even at the expense of sacrificing constitutional rights."


title: McCarthyism

text: In addition to the federal laws and responding to the worries of the local opinion, several states enacted anti-communist statutes.


title: McCarthyism

text: By 1952, several states had enacted statutes against criminal anarchy, criminal syndicalism, and sedition; banned from public employment or even from receiving public aid, communists and "subversives"; asked for loyalty oaths from public servants, and severely restricted or even banned the Communist Party. In addition, six states had equivalents to the HUAC. The California Senate Factfinding Subcommittee on Un-American Activities and the Florida Legislative Investigation Committee were established by their respective legislatures.


title: McCarthyism

text: Some of these states had very severe, or even extreme, laws against communism. In 1950, Michigan enacted life imprisonment for subversive propaganda; the following year, Tennessee enacted the death penalty for advocating the violent overthrow of the government. The death penalty for membership in the Communist Party was discussed in Texas by Governor Allan Shivers, who described it as "worse than murder."


title: McCarthyism

text: Municipalities and counties also enacted anti-communist ordinances: Los Angeles banned any communist or "Muscovite model of police-state dictatorship" from owning arms and Birmingham, Alabama, and Jacksonville, Florida, banned any communist from being within the city's limits.


title: McCarthyism

text: McCarthyism was supported by a variety of groups, including the American Legion and various other anti-communist organizations. One core element of support was a variety of militantly anti-communist women's groups such as the American Public Relations Forum and the Minute Women of the U.S.A. These organized tens of thousands of housewives into study groups, letter-writing networks, and patriotic clubs that coordinated efforts to identify and eradicate what they saw as subversion.


title: McCarthyism

text: Although right-wing radicals were the bedrock of support for McCarthyism, they were not alone. A broad "coalition of the aggrieved" found McCarthyism attractive, or at least politically useful. Common themes uniting the coalition were opposition to internationalism, particularly the United Nations; opposition to social welfare provisions, particularly the various programs established by the New Deal; and opposition to efforts to reduce inequalities in the social structure of the United States.


title: McCarthyism

text: One focus of popular McCarthyism concerned the provision of public health services, particularly vaccination, mental health care services, and fluoridation, all of which were denounced by some to be communist plots to poison or brainwash the American people. Such viewpoints led to collisions between McCarthyite radicals and supporters of public-health programs, most notably in the case of the Alaska Mental Health Bill controversy of 1956.


title: McCarthyism

text: William F. Buckley Jr., the founder of the influential conservative political magazine "National Review", wrote a defense of McCarthy, "McCarthy and his Enemies", in which he asserted that "McCarthyism ... is a movement around which men of good will and stern morality can close ranks."


title: McCarthyism

text: In addition, as Richard Rovere points out, many ordinary Americans became convinced that there must be "no smoke without fire" and lent their support to McCarthyism. The Gallup poll found that at his peak in January 1954, 50% of the American public supported McCarthy, while 29% had an unfavorable opinion. His support fell to 34% in June 1954. Republicans tended to like what McCarthy was doing and Democrats did not, though McCarthy had significant support from traditional Democratic ethnic groups, especially Catholics, as well as many unskilled workers and small-business owners. (McCarthy himself was a Catholic.) He had very little support among union activists and Jews.


title: McCarthyism

text: Those who sought to justify McCarthyism did so largely through their characterization of communism, and American communists in particular. Proponents of McCarthyism claimed that the CPUSA was so completely under Moscow's control that any American communist was a puppet of the Soviet intelligence services. This view, if restricted to the Communist Party's leadership is supported by recent documentation from the archives of the KGB as well as post-war decodes of wartime Soviet radio traffic from the Venona project, showing that Moscow provided financial support to the CPUSA and had significant influence on CPUSA policies. J. Edgar Hoover commented in a 1950 speech, "Communist members, body and soul, are the property of the Party."


title: McCarthyism

text: According to historian Richard G. Powers, McCarthy added "bogus specificity" to "sweeping accusation[s]", gaining support among "countersubversive anticommunists" on one hand, who sought to find and punish perceived communists. On the other hand, "liberal anticommunists" believed that the Communist Party was "despicable and annoying" but ultimately politically irrelevant.


title: McCarthyism

text: President Harry Truman, who pursued the anti-Soviet Truman Doctrine, called McCarthy "the greatest asset the Kremlin has" by "torpedo[ing] the bipartisan foreign policy of the United States."


title: McCarthyism

text: Historian Landon R. Y. Storrs writes that the CPUSA's "secretiveness, its authoritarian internal structure, and the loyalty of its leaders to the Kremlin were fundamental flaws that help explain why and how it was demonized. On the other hand, most American Communists were idealists attracted by the party’s militance against various forms of social injustice." Furthermore, based on later declassified evidence, "The paradoxical lesson from several decades of scholarship is that the same organization that inspired democratic idealists in the pursuit of social justice also was secretive, authoritarian, and morally compromised by ties to the Stalin regime."


title: McCarthyism

text: In the mid 20th century, this attitude was not confined to arch-conservatives. In 1940, the American Civil Liberties Union ejected founding member Elizabeth Gurley Flynn, saying that her membership in the Communist Party was enough to disqualify her as a civil libertarian. In the government's prosecutions of Communist Party members under the Smith Act (see above), the prosecution case was based not on specific actions or statements by the defendants, but on the premise that a commitment to violent overthrow of the government was inherent in the doctrines of Marxism–Leninism. Passages of the CPUSA constitution that specifically rejected revolutionary violence were dismissed as deliberate deception.


title: McCarthyism

text: In addition, it was often claimed that the party didn't allow members to resign; thus someone who had been a member for a short time decades previously could be thought a current member. Many of the hearings and trials of McCarthyism featured testimony by former Communist Party members such as Elizabeth Bentley, Louis Budenz, and Whittaker Chambers, speaking as expert witnesses.


title: McCarthyism

text: Various historians and pundits have discussed alleged Soviet-directed infiltration of the U.S. government and the possible collaboration of high U.S. government officials.


title: McCarthyism

text: Estimating the number of victims of McCarthy is difficult. The number imprisoned is in the hundreds, and some ten or twelve thousand lost their jobs. In many cases, simply being subpoenaed by HUAC or one of the other committees was sufficient cause to be fired.


title: McCarthyism

text: For the vast majority, both the potential for them to do harm to the nation and the nature of their communist affiliation were tenuous. After the extremely damaging "Cambridge Five" spy scandal (Guy Burgess, Donald Maclean, Kim Philby, Anthony Blunt, et al.), suspected homosexuality was also a common cause for being targeted by McCarthyism. The hunt for "sexual perverts", who were presumed to be subversive by nature, resulted in over 5,000 federal workers being fired, and thousands were harassed and denied employment. Many have termed this aspect of McCarthyism the "lavender scare".


title: McCarthyism

text: Homosexuality was classified as a psychiatric disorder in the 1950s. However, in the context of the highly politicized Cold War environment, homosexuality became framed as a dangerous, contagious social disease that posed a potential threat to state security. As the family was believed to be the cornerstone of American strength and integrity, the description of homosexuals as "sexual perverts" meant that they were both unable to function within a family unit and presented the potential to poison the social body. This era also witnessed the establishment of widely spread FBI surveillance intended to identify homosexual government employees.


title: McCarthyism

text: The McCarthy hearings and according "sexual pervert" investigations can be seen to have been driven by a desire to identify individuals whose ability to function as loyal citizens had been compromised. McCarthy began his campaign by drawing upon the ways in which he embodied traditional American values to become the self-appointed vanguard of social morality.


title: McCarthyism

text: In the film industry, more than 300 actors, authors, and directors were denied work in the U.S. through the unofficial Hollywood blacklist. Blacklists were at work throughout the entertainment industry, in universities and schools at all levels, in the legal profession, and in many other fields. A port-security program initiated by the Coast Guard shortly after the start of the Korean War required a review of every maritime worker who loaded or worked aboard any American ship, regardless of cargo or destination. As with other loyalty-security reviews of McCarthyism, the identities of any accusers and even the nature of any accusations were typically kept secret from the accused. Nearly 3,000 seamen and longshoremen lost their jobs due to this program alone.


title: McCarthyism

text: Some of the notable people who were blacklisted or suffered some other persecution during McCarthyism include:


title: McCarthyism

text: In 1953, Robert K. Murray, a young professor of history at Pennsylvania State University who had served as an intelligence officer in World War II, was revising his dissertation on the Red Scare of 1919–20 for publication until Little, Brown and Company decided that "under the circumstances ... it wasn't wise for them to bring this book out." He learned that investigators were questioning his colleagues and relatives. The University of Minnesota press published his volume, "Red Scare: A Study in National Hysteria, 1919–1920", in 1955.


title: McCarthyism

text: The nation was by no means united behind the policies and activities that have come to be associated with McCarthyism. The critics of various aspects of McCarthyism included many figures not generally noted for their liberalism. In his overridden veto of the McCarran Internal Security Act of 1950, President Truman wrote, "In a free country, we punish men for the crimes they commit, but never for the opinions they have." Truman also unsuccessfully vetoed the Taft–Hartley Act, which among other provisions denied trade unions National Labor Relations Board protection unless union leaders signed affidavits swearing they were not and had never been Communists. In 1953, after he left office, Truman criticized the current Eisenhower administration:


title: McCarthyism

text: On June 1, 1950, Senator Margaret Chase Smith, a Maine Republican, delivered a speech to the Senate she called a "Declaration of Conscience". In a clear attack upon McCarthyism, she called for an end to "character assassinations" and named "some of the basic principles of Americanism: The right to criticize; the right to hold unpopular beliefs; the right to protest; the right of independent thought". She said "freedom of speech is not what it used to be in America", and decried "cancerous tentacles of 'know nothing, suspect everything' attitudes". Six other Republican senators—Wayne Morse, Irving M. Ives, Charles W. Tobey, Edward John Thye, George Aiken, and Robert C. Hendrickson—joined Smith in condemning the tactics of McCarthyism.


title: McCarthyism

text: Elmer Davis, one of the most highly respected news reporters and commentators of the 1940s and 1950s, often spoke out against what he saw as the excesses of McCarthyism. On one occasion he warned that many local anti-communist movements constituted a "general attack not only on schools and colleges and libraries, on teachers and textbooks, but on all people who think and write ... in short, on the freedom of the mind".


title: McCarthyism

text: In 1952, the Supreme Court upheld a lower-court decision in "Adler v. Board of Education", thus approving a law that allowed state loyalty review boards to fire teachers deemed "subversive". In his dissenting opinion, Justice William O. Douglas wrote: "The present law proceeds on a principle repugnant to our society—guilt by association... What happens under this law is typical of what happens in a police state. Teachers are under constant surveillance; their pasts are combed for signs of disloyalty; their utterances are watched for clues to dangerous thoughts."


title: McCarthyism

text: One of the most influential opponents of McCarthyism was the famed CBS newscaster and analyst Edward R. Murrow. On October 20, 1953, Murrow's show "See It Now" aired an episode about the dismissal of Milo Radulovich, a former reserve Air Force lieutenant who was accused of associating with Communists. The show was strongly critical of the Air Force's methods, which included presenting evidence in a sealed envelope that Radulovich and his attorney were not allowed to open.


title: McCarthyism

text: On March 9, 1954, "See It Now" aired another episode on the issue of McCarthyism, this one attacking Joseph McCarthy himself. Titled "A Report on Senator Joseph R. McCarthy", it used footage of McCarthy speeches to portray him as dishonest, reckless, and abusive toward witnesses and prominent Americans. In his concluding comment, Murrow said:


title: McCarthyism

text: In April 1954, McCarthy was also under attack in the Army–McCarthy hearings. These hearings were televised live on the new American Broadcasting Company network, allowing the public to view first-hand McCarthy's interrogation of individuals and his controversial tactics. In one exchange, McCarthy reminded the attorney for the Army, Joseph Welch, that he had an employee in his law firm who had belonged to an organization that had been accused of Communist sympathies. In an exchange that reflected the increasingly negative public opinion of McCarthy, Welch rebuked the senator: "Have you no sense of decency, sir? At long last, have you left no sense of decency?"


title: McCarthyism

text: In the mid and late 1950s, the attitudes and institutions of McCarthyism slowly weakened. Changing public sentiments heavily contributed to the decline of McCarthyism. Its decline may also be charted through a series of court decisions.


title: McCarthyism

text: A key figure in the end of the blacklisting of McCarthyism was John Henry Faulk. Host of an afternoon comedy radio show, Faulk was a leftist active in his union, the American Federation of Television and Radio Artists. He was scrutinized by AWARE, Inc., one of the private firms that examined individuals for signs of communist "disloyalty". Marked by AWARE as unfit, he was fired by CBS Radio. Almost uniquely among the many victims of blacklisting, Faulk decided to sue AWARE in 1957 and finally won the case in 1962.


title: McCarthyism

text: With this court decision, the private blacklisters and those who used them were put on notice that they were legally liable for the professional and financial damage they caused. Although some informal blacklisting continued, the private "loyalty checking" agencies were soon a thing of the past. Even before the Faulk verdict, many in Hollywood had decided it was time to break the blacklist. In 1960, Dalton Trumbo, one of the best known members of the Hollywood Ten, was publicly credited with writing the films "Exodus" and "Spartacus".


title: McCarthyism

text: Much of the undoing of McCarthyism came at the hands of the U.S. Supreme Court under Chief Justice Earl Warren. As Richard Rovere wrote in his biography of Joseph McCarthy, "[T]he United States Supreme Court took judicial notice of the rents McCarthy was making in the fabric of liberty and thereupon wrote a series of decisions that have made the fabric stronger than before." Two Eisenhower appointees to the court—Earl Warren (who was made Chief Justice) and William J. Brennan, Jr.—proved to be more liberal than Eisenhower had anticipated.


title: McCarthyism

text: In 1956, the Warren Court heard the case of "Slochower v. Board of Education". Harry Slochower was a professor at Brooklyn College who had been fired by New York City for invoking the Fifth Amendment when McCarthy's committee questioned him about his past membership in the Communist Party. The court prohibited such actions, ruling "...we must condemn the practice of imputing a sinister meaning to the exercise of a person's constitutional right under the Fifth Amendment… The privilege against self-incrimination would be reduced to a hollow mockery if its exercise could be taken as equivalent either to a confession of guilt or a conclusive presumption of perjury." In addition, the 1956 "Cole v. Young" ruling also greatly weakened the ability to discriminate in the federal civilian workforce.


title: McCarthyism

text: Another key decision was in the 1957 case "Yates v. United States", in which the convictions of fourteen Communists were reversed. In Justice Black's opinion, he wrote of the original "Smith Act" trials: "The testimony of witnesses is comparatively insignificant. Guilt or innocence may turn on what Marx or Engels or someone else wrote or advocated as much as a hundred years or more ago… When the propriety of obnoxious or unfamiliar view about government is in reality made the crucial issue, …prejudice makes conviction inevitable except in the rarest circumstances."


title: McCarthyism

text: Also in 1957, the Supreme Court ruled on the case of "Watkins v. United States", curtailing the power of HUAC to punish uncooperative witnesses by finding them in contempt of Congress. Justice Warren wrote in the decision: "The mere summoning of a witness and compelling him to testify, against his will, about his beliefs, expressions or associations is a measure of governmental interference. And when those forced revelations concern matters that are unorthodox, unpopular, or even hateful to the general public, the reaction in the life of the witness may be disastrous."


title: McCarthyism

text: In its 1958 decision in "Kent v. Dulles", the Supreme Court halted the State Department from using the authority of its own regulations to refuse or revoke passports based on an applicant's communist beliefs or associations.


title: McCarthyism

text: The political divisions McCarthyism created in the United States continue to make themselves manifest, and the politics and history of anti-communism in the United States are still contentious. Portions of the massive security apparatus established during the McCarthy era still exist. Loyalty oaths are still required by the California Constitution for all officials and employees of the government of California (which is highly problematic for Quakers and Jehovah's Witnesses whose beliefs preclude them from pledging absolute loyalty to the state). At the federal level, a few portions of the McCarran Internal Security Act remain in effect. However, the act's detention provision was repealed in 1971. The McCarran Act's Communist registration requirement was declared unconstitutional in the 1965 Supreme Court ruling in "Albertson v. Subversive Activities Control Board" as well. The McCarran Act's Subversive Activities Control Board, which enforced the law's investigation requirement for persons alleged to be involved in "subversive activities," was officially abolished through Congressional legislation in 1972 as well.


title: McCarthyism

text: McCarthyism also attracts controversy purely as a historical issue. Through declassified documents from Soviet archives and Venona project decryptions of coded Soviet messages, the Soviet Union was found to have engaged in substantial espionage activities in the United States during the 1940s. The Communist Party USA also was substantially funded and its policies controlled by the Soviet Union, and accusations existed that CPUSA members were often recruited as spies.


title: McCarthyism

text: Liberal anti-communists like Edward Shils and Daniel Moynihan had a contempt for McCarthyism. Sociologist Edward Shils criticized an excessive policy of secrecy during the Cold War, that led to the misdirection of McCarthyism, which was addressed during the 1994–1997 Moynihan Commission. As Moynihan put it, "reaction to McCarthy took the form of a modish anti-anti-Communism that considered impolite any discussion of the very real threat Communism posed to Western values and security." After revelations of Soviet spy networks from the declassified Venona project, Moynihan wondered: "Might less secrecy have prevented the liberal overreaction to McCarthyism as well as McCarthyism itself?" He described the situation during the McCarthy era as "ignorant armies clashed by night". With McCarthy advocating an extremist view, the discussion of communist subversion was made into a civil rights issue instead of a counterintelligence one.


title: McCarthyism

text: In the view of some contemporary commentators, the revelations from Venona and other archives on espionage, stand as at least a partial vindication of McCarthyism. Some such as Goldberg feel that a genuinely dangerous subversive element was in the United States, and that this danger justified extreme measures. The opposing view holds that, recent revelations notwithstanding, by the time McCarthyism began in the late 1940s, the CPUSA was an ineffectual fringe group, and the damage done to U.S. interests by Soviet spies after World War II was minimal. Historian Ellen Schrecker states, ""in this country," McCarthyism did more damage to the constitution than the American Communist Party ever did."


title: McCarthyism

text: Historian John Earl Haynes, while acknowledging that inexcusable excesses occurred during McCarthyism, argues that some contemporary historians of McCarthyism underplay the un-democratic nature of the CPUSA. At the same time, Haynes, who studied the Venona decryptions extensively, argued that McCarthy's attempts to "make anti-communism a partisan weapon" actually "threatened [the post-War] anti-Communist consensus", thereby ultimately harming anti-Communist efforts more than helping them. Of the 159 people who were identified on lists used or referenced by McCarthy, evidence only substantially proved that nine of them had aided Soviet espionage efforts—while several hundred Soviet spies were actually known based on Venona and other evidence, most were never named by McCarthy.


title: McCarthyism

text: A number of observers have compared the oppression of liberals and leftists during the McCarthy period to 2000s-era actions against suspected terrorists, most of them Muslims. In "The Age of Anxiety: McCarthyism to Terrorism", author Haynes Johnson compares the "abuses suffered by aliens thrown into high-security U.S. prisons in the wake of 9/11" to the excesses of the McCarthy era. Similarly, David D. Cole has written that the Patriot Act "in effect resurrects the philosophy of McCarthyism, simply substituting 'terrorist' for 'communist'."


title: McCarthyism

text: From the opposite pole, conservative writer Ann Coulter devotes much of her book "" to drawing parallels between past opposition to McCarthy and McCarthyism and the policies and beliefs of modern-day liberals, arguing that the former hindered the anti-communist cause and the latter hinder the War on Terrorism. Other authors who have drawn on a comparison between current anti-terrorism policies and McCarthyism include Geoffrey R. Stone, Ted Morgan, and Jonah Goldberg.


title: McCarthyism

text: Since the time of McCarthy, the word "McCarthyism" has entered American speech as a general term for a variety of practices: aggressively questioning a person's patriotism, making poorly supported accusations, using accusations of disloyalty to pressure a person to adhere to conformist politics or to discredit an opponent, subverting civil and political rights in the name of national security, and the use of demagoguery are all often referred to as "McCarthyism".


title: McCarthyism

text: The 1951 novel "The Troubled Air" by Irwin Shaw tells the story of the director of a (fictional) radio show, broadcast live at the time, who is given a deadline to investigate his cast for alleged links to communism. The novel recounts the devastating effects on all concerned.


title: McCarthyism

text: The 1952 Arthur Miller play "The Crucible" used the Salem witch trials as a metaphor for McCarthyism, suggesting that the process of McCarthyism-style persecution can occur at any time or place. The play focused on the fact that once accused, a person had little chance of exoneration, given the irrational and circular reasoning of both the courts and the public. Miller later wrote: "The more I read into the Salem panic, the more it touched off corresponding images of common experiences in the fifties."


title: McCarthyism

text: The 1976 film "The Front" starring Woody Allen dealt with the McCarthy-era Hollywood blacklist. The film was made by those blacklisted: producer and director Martin Ritt; writer Walter Bernstein; and actors Zero Mostel, Herschel Bernardi, Michael Murphy, John Randolph, Lloyd Gough, and Joshua Shelley.


title: McCarthyism

text: "Guilty by Suspicion" is a 1991 American drama film about the Hollywood blacklist, McCarthyism, and the activities of the HUAC. Written and directed by Irwin Winkler, it starred Robert De Niro, Annette Bening, and George Wendt.


title: McCarthyism

text: The 2005 film "Good Night, and Good Luck" by George Clooney starred David Strathairn as broadcast journalist Edward R. Murrow and contained archival footage of McCarthy.


title: Principal component analysis

text: Principal component analysis (PCA) is a popular technique for analyzing large datasets containing a high number of dimensions/features per observation, increasing the interpretability of data while preserving the maximum amount of information, and enabling the visualization of multidimensional data. Formally, PCA is a statistical technique for reducing the dimensionality of a dataset. This is accomplished by linearly transforming the data into a new coordinate system where (most of) the variation in the data can be described with fewer dimensions than the initial data. Many studies use the first two principal components in order to plot the data in two dimensions and to visually identify clusters of closely related data points. Principal component analysis has applications in many fields such as population genetics, microbiome studies, and atmospheric science.


title: Principal component analysis

text: The principal components of a collection of points in a real coordinate space are a sequence of formula_1 unit vectors, where the formula_2-th vector is the direction of a line that best fits the data while being orthogonal to the first formula_3 vectors. Here, a best-fitting line is defined as one that minimizes the average squared perpendicular distance from the points to the line. These directions constitute an orthonormal basis in which different individual dimensions of the data are linearly uncorrelated. Principal component analysis (PCA) is the process of computing the principal components and using them to perform a change of basis on the data, sometimes using only the first few principal components and ignoring the rest.


title: Principal component analysis

text: In data analysis, the first principal component of a set of formula_1 variables, presumed to be jointly normally distributed, is the derived variable formed as a linear combination of the original variables that explains the most variance. The second principal component explains the most variance in what is left once the effect of the first component is removed, and we may proceed through formula_1 iterations until all the variance is explained. PCA is most commonly used when many of the variables are highly correlated with each other and it is desirable to reduce their number to an independent set.


title: Principal component analysis

text: PCA is used in exploratory data analysis and for making predictive models. It is commonly used for dimensionality reduction by projecting each data point onto only the first few principal components to obtain lower-dimensional data while preserving as much of the data's variation as possible. The first principal component can equivalently be defined as a direction that maximizes the variance of the projected data. The formula_2-th principal component can be taken as a direction orthogonal to the first formula_3 principal components that maximizes the variance of the projected data.


title: Principal component analysis

text: For either objective, it can be shown that the principal components are eigenvectors of the data's covariance matrix. Thus, the principal components are often computed by eigendecomposition of the data covariance matrix or singular value decomposition of the data matrix. PCA is the simplest of the true eigenvector-based multivariate analyses and is closely related to factor analysis. Factor analysis typically incorporates more domain specific assumptions about the underlying structure and solves eigenvectors of a slightly different matrix. PCA is also related to canonical correlation analysis (CCA). CCA defines coordinate systems that optimally describe the cross-covariance between two datasets while PCA defines a new orthogonal coordinate system that optimally describes variance in a single dataset. Robust and L1-norm-based variants of standard PCA have also been proposed.


title: Principal component analysis

text: PCA was invented in 1901 by Karl Pearson, as an analogue of the principal axis theorem in mechanics; it was later independently developed and named by Harold Hotelling in the 1930s. Depending on the field of application, it is also named the discrete Karhunen–Loève transform (KLT) in signal processing, the Hotelling transform in multivariate quality control, proper orthogonal decomposition (POD) in mechanical engineering, singular value decomposition (SVD) of X (invented in the last quarter of the 20th century), eigenvalue decomposition (EVD) of XX in linear algebra, factor analysis (for a discussion of the differences between PCA and factor analysis see Ch. 7 of Jolliffe's "Principal Component Analysis"), Eckart–Young theorem (Harman, 1960), or empirical orthogonal functions (EOF) in meteorological science, empirical eigenfunction decomposition (Sirovich, 1987), empirical component analysis (Lorenz, 1956), quasiharmonic modes (Brooks et al., 1988), spectral decomposition in noise and vibration, and empirical modal analysis in structural dynamics.


title: Principal component analysis

text: PCA can be thought of as fitting a "p"-dimensional ellipsoid to the data, where each axis of the ellipsoid represents a principal component. If some axis of the ellipsoid is small, then the variance along that axis is also small.


title: Principal component analysis

text: To find the axes of the ellipsoid, we must first center the values of each variable in the dataset on 0 by subtracting the mean of the variable's observed values from each of those values. These transformed values are used instead of the original observed values for each of the variables. Then, we compute the covariance matrix of the data and calculate the eigenvalues and corresponding eigenvectors of this covariance matrix. Then we must normalize each of the orthogonal eigenvectors to turn them into unit vectors. Once this is done, each of the mutually-orthogonal unit eigenvectors can be interpreted as an axis of the ellipsoid fitted to the data. This choice of basis will transform the covariance matrix into a diagonalized form, in which the diagonal elements represent the variance of each axis. The proportion of the variance that each eigenvector represents can be calculated by dividing the eigenvalue corresponding to that eigenvector by the sum of all eigenvalues.


title: Principal component analysis

text: PCA is defined as an orthogonal linear transformation that transforms the data to a new coordinate system such that the greatest variance by some scalar projection of the data comes to lie on the first coordinate (called the first principal component), the second greatest variance on the second coordinate, and so on.


title: Principal component analysis

text: Consider an formula_8 data matrix, X, with column-wise zero empirical mean (the sample mean of each column has been shifted to zero), where each of the "n" rows represents a different repetition of the experiment, and each of the "p" columns gives a particular kind of feature (say, the results from a particular sensor).


title: Principal component analysis

text: Mathematically, the transformation is defined by a set of size formula_9 of "p"-dimensional vectors of weights or coefficients formula_10 that map each row vector formula_11 of X to a new vector of principal component "scores" formula_12, given by


title: Principal component analysis

text: in such a way that the individual variables formula_14 of t considered over the data set successively inherit the maximum possible variance from X, with each coefficient vector w constrained to be a unit vector (where formula_9 is usually selected to be strictly less than formula_1 to reduce dimensionality).


title: Principal component analysis

text: The quantity to be maximised can be recognised as a Rayleigh quotient. A standard result for a positive semidefinite matrix such as XX is that the quotient's maximum possible value is the largest eigenvalue of the matrix, which occurs when w is the corresponding eigenvector.


title: Principal component analysis

text: With w found, the first principal component of a data vector x can then be given as a score "t" = x ⋅ w in the transformed co-ordinates, or as the corresponding vector in the original variables, {x ⋅ w} w.


title: Principal component analysis

text: It turns out that this gives the remaining eigenvectors of XX, with the maximum values for the quantity in brackets given by their corresponding eigenvalues. Thus the weight vectors are eigenvectors of XX.


title: Principal component analysis

text: The "k"-th principal component of a data vector x can therefore be given as a score "t" = x ⋅ w in the transformed coordinates, or as the corresponding vector in the space of the original variables, {x ⋅ w} w, where w is the "k"th eigenvector of XX.


title: Principal component analysis

text: where W is a "p"-by-"p" matrix of weights whose columns are the eigenvectors of XX. The transpose of W is sometimes called the whitening or sphering transformation. Columns of W multiplied by the square root of corresponding eigenvalues, that is, eigenvectors scaled up by the variances, are called "loadings" in PCA or in Factor analysis.


title: Principal component analysis

text: XX itself can be recognized as proportional to the empirical sample covariance matrix of the dataset X.


title: Principal component analysis

text: The sample covariance "Q" between two of the different principal components over the dataset is given by:


title: Principal component analysis

text: where the eigenvalue property of w has been used to move from line 2 to line 3. However eigenvectors w and w corresponding to eigenvalues of a symmetric matrix are orthogonal (if the eigenvalues are different), or can be orthogonalised (if the vectors happen to share an equal repeated value). The product in the final line is therefore zero; there is no sample covariance between different principal components over the dataset.


title: Principal component analysis

text: Another way to characterise the principal components transformation is therefore as the transformation to coordinates which diagonalise the empirical sample covariance matrix.


title: Principal component analysis

text: where Λ is the diagonal matrix of eigenvalues "λ" of XX. "λ" is equal to the sum of the squares over the dataset associated with each component "k", that is, "λ" = Σ "t" = Σ (x ⋅ w).


title: Principal component analysis

text: The transformation T = X W maps a data vector x from an original space of "p" variables to a new space of "p" variables which are uncorrelated over the dataset. However, not all the principal components need to be kept. Keeping only the first "L" principal components, produced by using only the first "L" eigenvectors, gives the truncated transformation


title: Principal component analysis

text: where the matrix T now has "n" rows but only "L" columns. In other words, PCA learns a linear transformation formula_27 where the columns of matrix formula_28 form an orthogonal basis for the "L" features (the components of representation "t") that are decorrelated. By construction, of all the transformed data matrices with only "L" columns, this score matrix maximises the variance in the original data that has been preserved, while minimising the total squared reconstruction error formula_29 or formula_30.


title: Principal component analysis

text: Such dimensionality reduction can be a very useful step for visualising and processing high-dimensional datasets, while still retaining as much of the variance in the dataset as possible. For example, selecting "L" = 2 and keeping only the first two principal components finds the two-dimensional plane through the high-dimensional dataset in which the data is most spread out, so if the data contains clusters these too may be most spread out, and therefore most visible to be plotted out in a two-dimensional diagram; whereas if two directions through the data (or two of the original variables) are chosen at random, the clusters may be much less spread apart from each other, and may in fact be much more likely to substantially overlay each other, making them indistinguishable.


title: Principal component analysis

text: Similarly, in regression analysis, the larger the number of explanatory variables allowed, the greater is the chance of overfitting the model, producing conclusions that fail to generalise to other datasets. One approach, especially when there are strong correlations between different possible explanatory variables, is to reduce them to a few principal components and then run the regression against them, a method called principal component regression.


title: Principal component analysis

text: Dimensionality reduction may also be appropriate when the variables in a dataset are noisy. If each column of the dataset contains independent identically distributed Gaussian noise, then the columns of T will also contain similarly identically distributed Gaussian noise (such a distribution is invariant under the effects of the matrix W, which can be thought of as a high-dimensional rotation of the co-ordinate axes). However, with more of the total variance concentrated in the first few principal components compared to the same noise variance, the proportionate effect of the noise is less—the first few components achieve a higher signal-to-noise ratio. PCA thus can have the effect of concentrating much of the signal into the first few principal components, which can usefully be captured by dimensionality reduction; while the later principal components may be dominated by noise, and so disposed of without great loss. If the dataset is not too large, the significance of the principal components can be tested using parametric bootstrap, as an aid in determining how many principal components to retain.


title: Principal component analysis

text: The principal components transformation can also be associated with another matrix factorization, the singular value decomposition (SVD) of X,


title: Principal component analysis

text: Here Σ is an "n"-by-"p" rectangular diagonal matrix of positive numbers "σ", called the singular values of X; U is an "n"-by-"n" matrix, the columns of which are orthogonal unit vectors of length "n" called the left singular vectors of X; and W is a "p"-by-"p" whose columns are orthogonal unit vectors of length "p" and called the right singular vectors of X.


title: Principal component analysis

text: where formula_33 is the square diagonal matrix with the singular values of X and the excess zeros chopped off that satisfies formula_34. Comparison with the eigenvector factorization of XX establishes that the right singular vectors W of X are equivalent to the eigenvectors of XX, while the singular values "σ" of formula_35 are equal to the square-root of the eigenvalues "λ" of XX.


title: Principal component analysis

text: so each column of T is given by one of the left singular vectors of X multiplied by the corresponding singular value. This form is also the polar decomposition of T.


title: Principal component analysis

text: Efficient algorithms exist to calculate the SVD of X without having to form the matrix XX, so computing the SVD is now the standard way to calculate a principal components analysis from a data matrix, unless only a handful of components are required.


title: Principal component analysis

text: As with the eigen-decomposition, a truncated score matrix T can be obtained by considering only the first L largest singular values and their singular vectors:


title: Principal component analysis

text: The truncation of a matrix M or T using a truncated singular value decomposition in this way produces a truncated matrix that is the nearest possible matrix of rank "L" to the original matrix, in the sense of the difference between the two having the smallest possible Frobenius norm, a result known as the Eckart–Young theorem [1936].


title: Principal component analysis

text: The singular values (in Σ) are the square roots of the eigenvalues of the matrix XX. Each eigenvalue is proportional to the portion of the "variance" (more correctly of the sum of the squared distances of the points from their multidimensional mean) that is associated with each eigenvector. The sum of all the eigenvalues is equal to the sum of the squared distances of the points from their multidimensional mean. PCA essentially rotates the set of points around their mean in order to align with the principal components. This moves as much of the variance as possible (using an orthogonal transformation) into the first few dimensions. The values in the remaining dimensions, therefore, tend to be small and may be dropped with minimal loss of information (see below). PCA is often used in this manner for dimensionality reduction. PCA has the distinction of being the optimal orthogonal transformation for keeping the subspace that has largest "variance" (as defined above). This advantage, however, comes at the price of greater computational requirements if compared, for example, and when applicable, to the discrete cosine transform, and in particular to the DCT-II which is simply known as the "DCT". Nonlinear dimensionality reduction techniques tend to be more computationally demanding than PCA.


title: Principal component analysis

text: PCA is sensitive to the scaling of the variables. If we have just two variables and they have the same sample variance and are completely correlated, then the PCA will entail a rotation by 45° and the "weights" (they are the cosines of rotation) for the two variables with respect to the principal component will be equal. But if we multiply all values of the first variable by 100, then the first principal component will be almost the same as that variable, with a small contribution from the other variable, whereas the second component will be almost aligned with the second original variable. This means that whenever the different variables have different units (like temperature and mass), PCA is a somewhat arbitrary method of analysis. (Different results would be obtained if one used Fahrenheit rather than Celsius for example.) Pearson's original paper was entitled "On Lines and Planes of Closest Fit to Systems of Points in Space" – "in space" implies physical Euclidean space where such concerns do not arise. One way of making the PCA less arbitrary is to use variables scaled so as to have unit variance, by standardizing the data and hence use the autocorrelation matrix instead of the autocovariance matrix as a basis for PCA. However, this compresses (or expands) the fluctuations in all dimensions of the signal space to unit variance.


title: Principal component analysis

text: Mean subtraction (a.k.a. "mean centering") is necessary for performing classical PCA to ensure that the first principal component describes the direction of maximum variance. If mean subtraction is not performed, the first principal component might instead correspond more or less to the mean of the data. A mean of zero is needed for finding a basis that minimizes the mean square error of the approximation of the data.


title: Principal component analysis

text: Mean-centering is unnecessary if performing a principal components analysis on a correlation matrix, as the data are already centered after calculating correlations. Correlations are derived from the cross-product of two standard scores (Z-scores) or statistical moments (hence the name: "Pearson Product-Moment Correlation"). Also see the article by Kromrey & Foster-Johnson (1998) on ""Mean-centering in Moderated Regression: Much Ado About Nothing"". Since covariances are correlations of normalized variables (Z- or standard-scores) a PCA based on the correlation matrix of X is equal to a PCA based on the covariance matrix of Z, the standardized version of X.


title: Principal component analysis

text: PCA is a popular primary technique in pattern recognition. It is not, however, optimized for class separability. However, it has been used to quantify the distance between two or more classes by calculating center of mass for each class in principal component space and reporting Euclidean distance between center of mass of two or more classes. The linear discriminant analysis is an alternative which is optimized for class separability.


title: Principal component analysis

text: The statistical implication of this property is that the last few PCs are not simply unstructured left-overs after removing the important PCs. Because these last PCs have variances as small as possible they are useful in their own right. They can help to detect unsuspected near-constant linear relationships between the elements of , and they may also be useful in regression, in selecting a subset of variables from , and in outlier detection.


title: Principal component analysis

text: Then, perhaps the main statistical implication of the result is that not only can we decompose the combined variances of all the elements of into decreasing contributions due to each PC, but we can also decompose the whole covariance matrix into contributions formula_59 from each PC. Although not strictly decreasing, the elements of formula_59 will tend to become smaller as formula_61 increases, as formula_59 is nonincreasing for increasing formula_61, whereas the elements of formula_64 tend to stay about the same size because of the normalization constraints: formula_65.


title: Principal component analysis

text: As noted above, the results of PCA depend on the scaling of the variables. This can be cured by scaling each feature by its standard deviation, so that one ends up with dimensionless features with unital variance.


title: Principal component analysis

text: The applicability of PCA as described above is limited by certain (tacit) assumptions made in its derivation. In particular, PCA can capture linear correlations between the features but fails when this assumption is violated (see Figure 6a in the reference). In some cases, coordinate transformations can restore the linearity assumption and PCA can then be applied (see kernel PCA).


title: Principal component analysis

text: Another limitation is the mean-removal process before constructing the covariance matrix for PCA. In fields such as astronomy, all the signals are non-negative, and the mean-removal process will force the mean of some astrophysical exposures to be zero, which consequently creates unphysical negative fluxes, and forward modeling has to be performed to recover the true magnitude of the signals. As an alternative method, non-negative matrix factorization focusing only on the non-negative elements in the matrices, which is well-suited for astrophysical observations. See more at Relation between PCA and Non-negative Matrix Factorization.


title: Principal component analysis

text: PCA is at a disadvantage if the data has not been standardized before applying the algorithm to it. PCA transforms original data into data that is relevant to the principal components of that data, which means that the new data variables cannot be interpreted in the same ways that the originals were. They are linear interpretations of the original variables. Also, if PCA is not performed properly, there is a high likelihood of information loss.


title: Principal component analysis

text: PCA relies on a linear model. If a dataset has a pattern hidden inside it that is nonlinear, then PCA can actually steer the analysis in the complete opposite direction of progress. Researchers at Kansas State University discovered that the sampling error in their experiments impacted the bias of PCA results. "If the number of subjects or blocks is smaller than 30, and/or the researcher is interested in PC's beyond the first, it may be better to first correct for the serial correlation, before PCA is conducted". The researchers at Kansas State also found that PCA could be "seriously biased if the autocorrelation structure of the data is not correctly handled".


title: Principal component analysis

text: Dimensionality reduction results in a loss of information, in general. PCA-based dimensionality reduction tends to minimize that information loss, under certain signal and noise models.


title: Principal component analysis

text: that is, that the data vector formula_67 is the sum of the desired information-bearing signal formula_68 and a noise signal formula_69 one can show that PCA can be optimal for dimensionality reduction, from an information-theoretic point-of-view.


title: Principal component analysis

text: In particular, Linsker showed that if formula_68 is Gaussian and formula_69 is Gaussian noise with a covariance matrix proportional to the identity matrix, the PCA maximizes the mutual information formula_72 between the desired information formula_68 and the dimensionality-reduced output formula_74.


title: Principal component analysis

text: If the noise is still Gaussian and has a covariance matrix proportional to the identity matrix (that is, the components of the vector formula_69 are iid), but the information-bearing signal formula_68 is non-Gaussian (which is a common scenario), PCA at least minimizes an upper bound on the "information loss", which is defined as


title: Principal component analysis

text: The optimality of PCA is also preserved if the noise formula_69 is iid and at least more Gaussian (in terms of the Kullback–Leibler divergence) than the information-bearing signal formula_68. In general, even if the above signal model holds, PCA loses its information-theoretic optimality as soon as the noise formula_69 becomes dependent.


title: Principal component analysis

text: The following is a detailed description of PCA using the covariance method (see also here) as opposed to the correlation method.


title: Principal component analysis

text: The goal is to transform a given data set X of dimension "p" to an alternative data set Y of smaller dimension "L". Equivalently, we are seeking to find the matrix Y, where Y is the Karhunen–Loève transform (KLT) of matrix X:


title: Principal component analysis

text: Suppose you have data comprising a set of observations of "p" variables, and you want to reduce the data so that each observation can be described with only "L" variables, "L" < "p". Suppose further, that the data are arranged as a set of "n" data vectors formula_82 with each formula_83 representing a single grouped observation of the "p" variables.


title: Principal component analysis

text: Mean subtraction is an integral part of the solution towards finding a principal component basis that minimizes the mean square error of approximating the data. Hence we proceed by centering the data as follows:


title: Principal component analysis

text: In some applications, each variable (column of B) may also be scaled to have a variance equal to 1 (see Z-score). This step affects the calculated principal components, but makes them independent of the units used to measure the different variables.


title: Principal component analysis

text: That is, the first column of formula_99 is the projection of the data points onto the first principal component, the second column is the projection onto the second principal component, etc.


title: Principal component analysis

text: Let X be a "d"-dimensional random vector expressed as column vector. Without loss of generality, assume X has zero mean.


title: Principal component analysis

text: We want to find formula_100 a orthonormal transformation matrix P so that PX has a diagonal covariance matrix (that is, PX is a random vector with all its distinct components pairwise uncorrelated).


title: Principal component analysis

text: This is very constructive, as cov(X) is guaranteed to be a non-negative definite matrix and thus is guaranteed to be diagonalisable by some unitary matrix.


title: Principal component analysis

text: In practical implementations, especially with high dimensional data (large ), the naive covariance method is rarely used because it is not efficient due to high computational and memory costs of explicitly determining the covariance matrix. The covariance-free approach avoids the operations of explicitly calculating and storing the covariance matrix , instead utilizing one of matrix-free methods, for example, based on the function evaluating the product at the cost of operations.


title: Principal component analysis

text: One way to compute the first principal component efficiently is shown in the following pseudo-code, for a data matrix with zero mean, without ever computing its covariance matrix.


title: Principal component analysis

text: This power iteration algorithm simply calculates the vector , normalizes, and places the result back in . The eigenvalue is approximated by , which is the Rayleigh quotient on the unit vector for the covariance matrix . If the largest singular value is well separated from the next largest one, the vector gets close to the first principal component of within the number of iterations , which is small relative to , at the total cost . The power iteration convergence can be accelerated without noticeably sacrificing the small cost per iteration using more advanced matrix-free methods, such as the Lanczos algorithm or the Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) method.


title: Principal component analysis

text: Subsequent principal components can be computed one-by-one via deflation or simultaneously as a block. In the former approach, imprecisions in already computed approximate principal components additively affect the accuracy of the subsequently computed principal components, thus increasing the error with every new computation. The latter approach in the block power method replaces single-vectors and with block-vectors, matrices and . Every column of approximates one of the leading principal components, while all columns are iterated simultaneously. The main calculation is evaluation of the product . Implemented, for example, in LOBPCG, efficient blocking eliminates the accumulation of the errors, allows using high-level BLAS matrix-matrix product functions, and typically leads to faster convergence, compared to the single-vector one-by-one technique.


title: Principal component analysis

text: "Non-linear iterative partial least squares (NIPALS)" is a variant the classical power iteration with matrix deflation by subtraction implemented for computing the first few components in a principal component or partial least squares analysis. For very-high-dimensional datasets, such as those generated in the *omics sciences (for example, genomics, metabolomics) it is usually only necessary to compute the first few PCs. The non-linear iterative partial least squares (NIPALS) algorithm updates iterative approximations to the leading scores and loadings t and r by the power iteration multiplying on every iteration by X on the left and on the right, that is, calculation of the covariance matrix is avoided, just as in the matrix-free implementation of the power iterations to , based on the function evaluating the product .


title: Principal component analysis

text: The matrix deflation by subtraction is performed by subtracting the outer product, tr from X leaving the deflated residual matrix used to calculate the subsequent leading PCs.


title: Principal component analysis

text: For large data matrices, or matrices that have a high degree of column collinearity, NIPALS suffers from loss of orthogonality of PCs due to machine precision round-off errors accumulated in each iteration and matrix deflation by subtraction. A Gram–Schmidt re-orthogonalization algorithm is applied to both the scores and the loadings at each iteration step to eliminate this loss of orthogonality. NIPALS reliance on single-vector multiplications cannot take advantage of high-level BLAS and results in slow convergence for clustered leading singular values—both these deficiencies are resolved in more sophisticated matrix-free block solvers, such as the Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) method.


title: Principal component analysis

text: In an "online" or "streaming" situation with data arriving piece by piece rather than being stored in a single batch, it is useful to make an estimate of the PCA projection that can be updated sequentially. This can be done efficiently, but requires different algorithms.


title: Principal component analysis

text: In PCA, it is common that we want to introduce qualitative variables as supplementary elements. For example, many quantitative variables have been measured on plants. For these plants, some qualitative variables are available as, for example, the species to which the plant belongs. These data were subjected to PCA for quantitative variables. When analyzing the results, it is natural to connect the principal components to the qualitative variable "species".


title: Principal component analysis

text: These results are what is called "introducing a qualitative variable as supplementary element". This procedure is detailed in and Husson, Lê & Pagès 2009 and Pagès 2013.


title: Principal component analysis

text: Few software offer this option in an "automatic" way. This is the case of SPAD that historically, following the work of Ludovic Lebart, was the first to propose this option, and the R package FactoMineR.


title: Principal component analysis

text: The earliest application of factor analysis was in locating and measuring components of human intelligence. it was believed that intelligence had various uncorrelated components such as spatial intelligence, verbal intelligence, induction, deduction etc and that scores on these could be adduced by factor analysis from results on various tests, to give a single index known as the Intelligence Quotient (IQ). The pioneering statistical psychologist Spearman actually developed factor analysis in 1904 for his two-factor theory of intelligence, adding a formal technique to the science of psychometrics. In 1924 Thurstone looked for 56 factors of intelligence, developing the notion of Mental Age. Standard IQ tests today are based on this early work.


title: Principal component analysis

text: In 1949, Shevky and Williams introduced the theory of factorial ecology, which dominated studies of residential differentiation from the 1950s to the 1970s. Neighbourhoods in a city were recognizable or could be distinguished from one another by various characteristics which could be reduced to three by factor analysis. These were known as 'social rank' (an index of occupational status), 'familism' or family size, and 'ethnicity'; Cluster analysis could then be applied to divide the city into clusters or precincts according to values of the three key factor variables. An extensive literature developed around factorial ecology in urban geography, but the approach went out of fashion after 1980 as being methodologically primitive and having little place in postmodern geographical paradigms.


title: Principal component analysis

text: One of the problems with factor analysis has always been finding convincing names for the various artificial factors. In 2000, Flood revived the factorial ecology approach to show that principal components analysis actually gave meaningful answers directly, without resorting to factor rotation. The principal components were actually dual variables or shadow prices of 'forces' pushing people together or apart in cities. The first component was 'accessibility', the classic trade-off between demand for travel and demand for space, around which classical urban economics is based. The next two components were 'disadvantage', which keeps people of similar status in separate neighbourhoods (mediated by planning), and ethnicity, where people of similar ethnic backgrounds try to co-locate.


title: Principal component analysis

text: About the same time, the Australian Bureau of Statistics defined distinct indexes of advantage and disadvantage taking the first principal component of sets of key variables that were thought to be important. These SEIFA indexes are regularly published for various jurisdictions, and are used frequently in spatial analysis.


title: Principal component analysis

text: PCA has been the only formal method available for the development of indexes, which are otherwise a hit-or-miss "ad hoc" undertaking.


title: Principal component analysis

text: The City Development Index was developed by PCA from about 200 indicators of city outcomes in a 1996 survey of 254 global cities. The first principal component was subject to iterative regression, adding the original variables singly until about 90% of its variation was accounted for. The index ultimately used about 15 indicators but was a good predictor of many more variables. Its comparative value agreed very well with a subjective assessment of the condition of each city. The coefficients on items of infrastructure were roughly proportional to the average costs of providing the underlying services, suggesting the Index was actually a measure of effective physical and social investment in the city.


title: Principal component analysis

text: The country-level Human Development Index (HDI) from UNDP, which has been published since 1990 and is very extensively used in development studies, has very similar coefficients on similar indicators, strongly suggesting it was originally constructed using PCA.


title: Principal component analysis

text: In 1978 Cavalli-Sforza and others pioneered the use of principal components analysis (PCA) to summarise data on variation in human gene frequencies across regions. The components showed distinctive patterns, including gradients and sinusoidal waves. They interpreted these patterns as resulting from specific ancient migration events.


title: Principal component analysis

text: Since then, PCA has been ubiquitous in population genetics, with thousands of papers using PCA as a display mechanism. Genetics varies largely according to proximity, so the first two principal components actually show spatial distribution and may be used to map the relative geographical location of different population groups, thereby showing individuals who have wandered from their original locations.


title: Principal component analysis

text: PCA in genetics has been technically controversial, in that the technique has been performed on discrete non-normal variables and often on binary allele markers. The lack of any measures of standard error in PCA are also an impediment to more consistent usage. In August 2022, the molecular biologist Eran Elhaik published a theoretical paper in Scientific Reports analyzing 12 PCA applications. He concluded that it was easy to manipulate the method, which, in his view, generated results that were 'erroneous, contradictory, and absurd.' Specifically, he argued, the results achieved in population genetics were characterized by cherry-picking and circular reasoning.


title: Principal component analysis

text: Market research has been an extensive user of PCA. It is used to develop customer satisfaction or customer loyalty scores for products, and with clustering, to develop market segments that may be targeted with advertising campaigns, in much the same way as factorial ecology will locate geographical areas with similar characteristics.


title: Principal component analysis

text: PCA rapidly transforms large amounts of data into smaller, easier-to-digest variables that can be more rapidly and readily analyzed. In any consumer questionnaire, there are series of questions designed to elicit consumer attitudes, and principal components seek out latent variables underlying these attitudes. For example, the Oxford Internet Survey in 2013 asked 2000 people about their attitudes and beliefs, and from these analysts extracted four principal component dimensions, which they identified as 'escape', 'social networking', 'efficiency', and 'problem creating'.


title: Principal component analysis

text: Another example from Joe Flood in 2008 extracted an attitudinal index toward housing from 28 attitude questions in a national survey of 2697 households in Australia. The first principal component represented a general attitude toward property and home ownership. The index, or the attitude questions it embodied, could be fed into a General Linear Model of tenure choice. The strongest determinant of private renting by far was the attitude index, rather than income, marital status or household type.


title: Principal component analysis

text: In quantitative finance, principal component analysis can be directly applied to the risk management of interest rate derivative portfolios. Trading multiple swap instruments which are usually a function of 30–500 other market quotable swap instruments is sought to be reduced to usually 3 or 4 principal components, representing the path of interest rates on a macro basis. Converting risks to be represented as those to factor loadings (or multipliers) provides assessments and understanding beyond that available to simply collectively viewing risks to individual 30–500 buckets.


title: Principal component analysis

text: PCA has also been applied to equity portfolios in a similar fashion, both to portfolio risk and to risk return. One application is to reduce portfolio risk, where allocation strategies are applied to the "principal portfolios" instead of the underlying stocks. A second is to enhance portfolio return, using the principal components to select stocks with upside potential.


title: Principal component analysis

text: A variant of principal components analysis is used in neuroscience to identify the specific properties of a stimulus that increases a neuron's probability of generating an action potential. This technique is known as spike-triggered covariance analysis. In a typical application an experimenter presents a white noise process as a stimulus (usually either as a sensory input to a test subject, or as a current injected directly into the neuron) and records a train of action potentials, or spikes, produced by the neuron as a result. Presumably, certain features of the stimulus make the neuron more likely to spike. In order to extract these features, the experimenter calculates the covariance matrix of the "spike-triggered ensemble", the set of all stimuli (defined and discretized over a finite time window, typically on the order of 100 ms) that immediately preceded a spike. The eigenvectors of the difference between the spike-triggered covariance matrix and the covariance matrix of the "prior stimulus ensemble" (the set of all stimuli, defined over the same length time window) then indicate the directions in the space of stimuli along which the variance of the spike-triggered ensemble differed the most from that of the prior stimulus ensemble. Specifically, the eigenvectors with the largest positive eigenvalues correspond to the directions along which the variance of the spike-triggered ensemble showed the largest positive change compared to the varince of the prior. Since these were the directions in which varying the stimulus led to a spike, they are often good approximations of the sought after relevant stimulus features.


title: Principal component analysis

text: In neuroscience, PCA is also used to discern the identity of a neuron from the shape of its action potential. Spike sorting is an important procedure because extracellular recording techniques often pick up signals from more than one neuron. In spike sorting, one first uses PCA to reduce the dimensionality of the space of action potential waveforms, and then performs clustering analysis to associate specific action potentials with individual neurons.


title: Principal component analysis

text: PCA as a dimension reduction technique is particularly suited to detect coordinated activities of large neuronal ensembles. It has been used in determining collective variables, that is, order parameters, during phase transitions in the brain.


title: Principal component analysis

text: and is conceptually similar to PCA, but scales the data (which should be non-negative) so that rows and columns are treated equivalently. It is traditionally applied to contingency tables.


title: Principal component analysis

text: Because CA is a descriptive technique, it can be applied to tables for which the chi-squared statistic is appropriate or not.


title: Principal component analysis

text: Several variants of CA are available including detrended correspondence analysis and canonical correspondence analysis. One special extension is multiple correspondence analysis, which may be seen as the counterpart of principal component analysis for categorical data.


title: Principal component analysis

text: Principal component analysis creates variables that are linear combinations of the original variables. The new variables have the property that the variables are all orthogonal. The PCA transformation can be helpful as a pre-processing step before clustering. PCA is a variance-focused approach seeking to reproduce the total variable variance, in which components reflect both common and unique variance of the variable. PCA is generally preferred for purposes of data reduction (that is, translating variable space into optimal factor space) but not when the goal is to detect the latent construct or factors.


title: Principal component analysis

text: Factor analysis is similar to principal component analysis, in that factor analysis also involves linear combinations of variables. Different from PCA, factor analysis is a correlation-focused approach seeking to reproduce the inter-correlations among variables, in which the factors "represent the common variance of variables, excluding unique variance". In terms of the correlation matrix, this corresponds with focusing on explaining the off-diagonal terms (that is, shared co-variance), while PCA focuses on explaining the terms that sit on the diagonal. However, as a side result, when trying to reproduce the on-diagonal terms, PCA also tends to fit relatively well the off-diagonal correlations. Results given by PCA and factor analysis are very similar in most situations, but this is not always the case, and there are some problems where the results are significantly different. Factor analysis is generally used when the research purpose is detecting data structure (that is, latent constructs or factors) or causal modeling. If the factor model is incorrectly formulated or the assumptions are not met, then factor analysis will give erroneous results.


title: Principal component analysis

text: It has been asserted that the relaxed solution of -means clustering, specified by the cluster indicators, is given by the principal components, and the PCA subspace spanned by the principal directions is identical to the cluster centroid subspace. However, that PCA is a useful relaxation of -means clustering was not a new result, and it is straightforward to uncover counterexamples to the statement that the cluster centroid subspace is spanned by the principal directions.


title: Principal component analysis

text: <nowiki> </nowiki>Non-negative matrix factorization (NMF) is a dimension reduction method where only non-negative elements in the matrices are used, which is therefore a promising method in astronomy, in the sense that astrophysical signals are non-negative. The PCA components are orthogonal to each other, while the NMF components are all non-negative and therefore constructs a non-orthogonal basis.


title: Principal component analysis

text: In PCA, the contribution of each component is ranked based on the magnitude of its corresponding eigenvalue, which is equivalent to the fractional residual variance (FRV) in analyzing empirical data. For NMF, its components are ranked based only on the empirical FRV curves. The residual fractional eigenvalue plots, that is, formula_106 as a function of component number formula_61 given a total of formula_108 components, for PCA has a flat plateau, where no data is captured to remove the quasi-static noise, then the curves dropped quickly as an indication of over-fitting and captures random noise. The FRV curves for NMF is decreasing continuously when the NMF components are constructed sequentially, indicating the continuous capturing of quasi-static noise; then converge to higher levels than PCA, indicating the less over-fitting property of NMF.


title: Principal component analysis

text: It is often difficult to interpret the principal components when the data include many variables of various origins, or when some variables are qualitative. This leads the PCA user to a delicate elimination of several variables. If observations or variables have an excessive impact on the direction of the axes, they should be removed and then projected as supplementary elements. In addition, it is necessary to avoid interpreting the proximities between the points close to the center of the factorial plane.


title: Principal component analysis

text: The iconography of correlations, on the contrary, which is not a projection on a system of axes, does not have these drawbacks. We can therefore keep all the variables.


title: Principal component analysis

text: The principle of the diagram is to underline the "remarkable" correlations of the correlation matrix, by a solid line (positive correlation) or dotted line (negative correlation).


title: Principal component analysis

text: A strong correlation is not "remarkable" if it is not direct, but caused by the effect of a third variable. Conversely, weak correlations can be "remarkable". For example, if a variable Y depends on several independent variables, the correlations of Y with each of them are weak and yet "remarkable".


title: Principal component analysis

text: A particular disadvantage of PCA is that the principal components are usually linear combinations of all input variables. Sparse PCA overcomes this disadvantage by finding linear combinations that contain just a few input variables. It extends the classic method of principal component analysis (PCA) for the reduction of dimensionality of data by adding sparsity constraint on the input variables.


title: Principal component analysis

text: The methodological and theoretical developments of Sparse PCA as well as its applications in scientific studies were recently reviewed in a survey paper.


title: Principal component analysis

text: Most of the modern methods for nonlinear dimensionality reduction find their theoretical and algorithmic roots in PCA or K-means. Pearson's original idea was to take a straight line (or plane) which will be "the best fit" to a set of data points. Trevor Hastie expanded on this concept by proposing Principal curves as the natural extension for the geometric interpretation of PCA, which explicitly constructs a manifold for data approximation followed by projecting the points onto it, as is illustrated by Fig.


title: Principal component analysis

text: See also the elastic map algorithm and principal geodesic analysis. Another popular generalization is kernel PCA, which corresponds to PCA performed in a reproducing kernel Hilbert space associated with a positive definite kernel.


title: Principal component analysis

text: In multilinear subspace learning, PCA is generalized to multilinear PCA (MPCA) that extracts features directly from tensor representations. MPCA is solved by performing PCA in each mode of the tensor iteratively. MPCA has been applied to face recognition, gait recognition, etc. MPCA is further extended to uncorrelated MPCA, non-negative MPCA and robust MPCA.


title: Principal component analysis

text: "N"-way principal component analysis may be performed with models such as Tucker decomposition, PARAFAC, multiple factor analysis, co-inertia analysis, STATIS, and DISTATIS.


title: Principal component analysis

text: While PCA finds the mathematically optimal method (as in minimizing the squared error), it is still sensitive to outliers in the data that produce large errors, something that the method tries to avoid in the first place. It is therefore common practice to remove outliers before computing PCA. However, in some contexts, outliers can be difficult to identify. For example, in data mining algorithms like correlation clustering, the assignment of points to clusters and outliers is not known beforehand.


title: Principal component analysis

text: A recently proposed generalization of PCA based on a weighted PCA increases robustness by assigning different weights to data objects based on their estimated relevancy.


title: Principal component analysis

text: Robust principal component analysis (RPCA) via decomposition in low-rank and sparse matrices is a modification of PCA that works well with respect to grossly corrupted observations.


title: Principal component analysis

text: Independent component analysis (ICA) is directed to similar problems as principal component analysis, but finds additively separable components rather than successive approximations.


title: Principal component analysis

text: Given a matrix formula_109, it tries to decompose it into two matrices such that formula_110. A key difference from techniques such as PCA and ICA is that some of the entries of formula_111 are constrained to be 0. Here formula_101 is termed the regulatory layer. While in general such a decomposition can have multiple solutions, they prove that if the following conditions are satisfied :


title: Principal component analysis

text: Discriminant analysis of principal components (DAPC) is a multivariate method used to identify and describe clusters of genetically related individuals. Genetic variation is partitioned into two components: variation between groups and within groups, and it maximizes the former. Linear discriminants are linear combinations of alleles which best separate the clusters. Alleles that most contribute to this discrimination are therefore those that are the most markedly different across groups. The contributions of alleles to the groupings identified by DAPC can allow identifying regions of the genome driving the genetic divergence among groups


title: Principal component analysis

text: In DAPC, data is first transformed using a principal components analysis (PCA) and subsequently clusters are identified using discriminant analysis (DA).


title: Sodium

text: Sodium is a chemical element with the symbol Na (from Latin "natrium") and atomic number 11. It is a soft, silvery-white, highly reactive metal. Sodium is an alkali metal, being in group 1 of the periodic table. Its only stable isotope is Na. The free metal does not occur in nature, and must be prepared from compounds. Sodium is the sixth most abundant element in the Earth's crust and exists in numerous minerals such as feldspars, sodalite, and halite (NaCl). Many salts of sodium are highly water-soluble: sodium ions have been leached by the action of water from the Earth's minerals over eons, and thus sodium and chlorine are the most common dissolved elements by weight in the oceans.


title: Sodium

text: Sodium was first isolated by Humphry Davy in 1807 by the electrolysis of sodium hydroxide. Among many other useful sodium compounds, sodium hydroxide (lye) is used in soap manufacture, and sodium chloride (edible salt) is a de-icing agent and a nutrient for animals including humans.


title: Sodium

text: Sodium is an essential element for all animals and some plants. Sodium ions are the major cation in the extracellular fluid (ECF) and as such are the major contributor to the ECF osmotic pressure and ECF compartment volume. Loss of water from the ECF compartment increases the sodium concentration, a condition called hypernatremia. Isotonic loss of water and sodium from the ECF compartment decreases the size of that compartment in a condition called ECF hypovolemia.


title: Sodium

text: By means of the sodium-potassium pump, living human cells pump three sodium ions out of the cell in exchange for two potassium ions pumped in; comparing ion concentrations across the cell membrane, inside to outside, potassium measures about 40:1, and sodium, about 1:10. In nerve cells, the electrical charge across the cell membrane enables transmission of the nerve impulse—an action potential—when the charge is dissipated; sodium plays a key role in that activity.


title: Sodium

text: Sodium at standard temperature and pressure is a soft silvery metal that combines with oxygen in the air and forms grayish white sodium oxide unless immersed in oil or inert gas, which are the conditions it is usually stored in. Sodium metal can be easily cut with a knife and is a good conductor of electricity and heat because it has only one electron in its valence shell, resulting in weak metallic bonding and free electrons, which carry energy. Due to having low atomic mass and large atomic radius, sodium is third-least dense of all elemental metals and is one of only three metals that can float on water, the other two being lithium and potassium.


title: Sodium

text: The melting (98 °C) and boiling (883 °C) points of sodium are lower than those of lithium but higher than those of the heavier alkali metals potassium, rubidium, and caesium, following periodic trends down the group. These properties change dramatically at elevated pressures: at 1.5 Mbar, the color changes from silvery metallic to black; at 1.9 Mbar the material becomes transparent with a red color; and at 3 Mbar, sodium is a clear and transparent solid. All of these high-pressure allotropes are insulators and electrides.


title: Sodium

text: In a flame test, sodium and its compounds glow yellow because the excited 3s electrons of sodium emit a photon when they fall from 3p to 3s; the wavelength of this photon corresponds to the D line at about 589.3 nm. Spin-orbit interactions involving the electron in the 3p orbital split the D line into two, at 589.0 and 589.6 nm; hyperfine structures involving both orbitals cause many more lines.


title: Sodium

text: Twenty isotopes of sodium are known, but only Na is stable. Na is created in the carbon-burning process in stars by fusing two carbon atoms together; this requires temperatures above 600 megakelvins and a star of at least three solar masses. Two radioactive, cosmogenic isotopes are the byproduct of cosmic ray spallation: Na has a half-life of 2.6 years and Na, a half-life of 15 hours; all other isotopes have a half-life of less than one minute.


title: Sodium

text: Two nuclear isomers have been discovered, the longer-lived one being Na with a half-life of around 20.2 milliseconds. Acute neutron radiation, as from a nuclear criticality accident, converts some of the stable Na in human blood to Na; the neutron radiation dosage of a victim can be calculated by measuring the concentration of Na relative to Na.


title: Sodium

text: Sodium atoms have 11 electrons, one more than the stable configuration of the noble gas neon. The first and second ionization energies are 495.8 kJ/mol and 4562 kJ/mol, respectively. As a result, sodium usually forms ionic compounds involving the Na cation.


title: Sodium

text: Metallic sodium is generally less reactive than potassium and more reactive than lithium. Sodium metal is highly reducing, with the standard reduction potential for the Na/Na couple being −2.71 volts, though potassium and lithium have even more negative potentials.


title: Sodium

text: The thermal, fluidic, chemical, and nuclear properties of molten sodium metal have caused it to be one of the main coolants of choice for the fast breeder reactor. Such nuclear reactors are seen as a crucial step for the production of clean energy.