title: The Cenotaph

text: The American historian Jay Winter described the Cenotaph as displaying a "striking minimalism". According to Winter, the Cenotaph "managed to transform a victory parade [...] into a time when millions could contemplate the [...] inexorable reality of death in war," and was "a work of genius because of its simplicity. It says so much because it says nothing at all. It is a form on which anyone could inscribe his or her own thoughts, reveries, sadness." He believed that, in designing an empty tomb, "the tomb of no one [...] became the tomb of all who had died in the war." He compared it favourably to another of Lutyens's major commemorative works, the Thiepval Memorial, built for the IWGC in France, and to Maya Lin's Vietnam Veterans Memorial in Washington, DC. Jenny Edkins, a British political scientist, also draws a parallel between the Cenotaph and the Vietnam Memorial and the unexpected public acclaim that both received immediately after their unveiling. Edkins believed that the apparent simplicity of, and lack of decoration on, the two memorials provided for a "collective act of mourning". Another architect, Andrew Crompton, of the University of Liverpool, re-evaluated the Cenotaph at the turn of the 21st century. He compared the diminishing tiers (when viewed from the ground up) to the hilt of a sheathed sword, its blade buried beneath the ground, which he felt resembled the mythical sword Excalibur.


title: The Cenotaph

text: The Cenotaph has been contrasted with the Royal Artillery Memorial by Charles Sargeant Jagger. Lutyens submitted a proposed design for that memorial, but the Royal Artillery rejected it on the grounds that it was too similar to the Cenotaph, and that they wanted a more realist monument, rather than Lutyens's abstract classicism. Whereas Lutyens placed the empty coffin high above the ground, distancing the observer from it, Jagger sculpted a dead soldier and placed it at eye level, confronting the observer with the reality of war. The Tomb of the Unknown Warrior in Westminster Abbey, inaugurated on the same day as the Cenotaph and another of London's most famous war memorials, has also been contrasted with the Cenotaph. Edkins observes that the Tomb was intended to "provide a grave for those who had none" and to become a focal point for the mourning of those buried overseas, but that the Cenotaph became much more popular as a site for both individual commemoration and public ceremonies.


title: The Cenotaph

text: The German-American historian George Mosse noted that most countries involved in the First World War eventually adopted the concept of burying an unidentified soldier, but in London the Cenotaph fulfilled the same purpose, despite the tomb in the abbey. Unlike elsewhere, it was the Cenotaph and not an unknown warrior that became the centre of national ceremonies, which Mosse considered was because the abbey was "too cluttered with memorials and tombs of famous Englishmen to provide the appropriate place for pilgrimages or celebrations" compared to the Cenotaph's location in the middle of a broad avenue. Ken Inglis, an Australian historian, and Gavin Stamp, a British architectural historian, both suggested that the Unknown Warrior was the Church of England's attempt to create a rival to the Cenotaph, which had no explicitly Christian symbolism, though another historian, David Lloyd, suggests that this was largely unsuccessful—the Church even petitioned for Armistice Day ceremonies to be held in Westminster Abbey rather than at the Cenotaph in 1923, but the proposal was rejected after it met with widespread public opposition. Instead, Lloyd noted that the two monuments became closely linked, and that "Together, the memorials reflect the complexity and ambiguity of the British response to the Great War."


title: The Cenotaph

text: No date was announced for the completion of the Cenotaph at first, but the British government was keen to have it in place for Remembrance Day (11 November). In September 1920, the announcement came that the Cenotaph would be unveiled on 11 November, the second anniversary of the armistice, and that the unveiling would be performed by King George V. For the occasion, the government designated the Cenotaph the official memorial to all British and empire dead lost during the First World War. It subsequently became the official memorial to British casualties from later conflicts.


title: The Cenotaph

text: Late into the planning, the government decided to exhume an unidentified serviceman—thenceforth to be known as the Unknown Warrior—from a grave in France, and inter him in Westminster Abbey. The last leg of the Unknown Warrior's journey to the abbey took place in coordination with events at the Cenotaph. The king was to unveil the Cenotaph, this time with Lutyens in attendance, along with other members of the royal family, the prime minister, and Randall Davidson, the Archbishop of Canterbury (the Church of England's most senior cleric). The Unknown Warrior was brought to Whitehall, his coffin resting on a gun carriage pulled by military horses, for the unveiling. The Cenotaph was shrouded in Union Flags until the king performed the unveiling at the stroke of 11 o'clock. This ceremonial act was followed by two minutes' silence, ending with the sounding of the "Last Post". The king placed a wreath of roses on the Unknown Warrior's coffin, and the cortege continued its journey—His Majesty, the other royals, Lloyd George, and the archbishop following the gun carriage to the abbey.


title: The Cenotaph

text: The public response exceeded even that to the temporary Cenotaph in the aftermath of the armistice. Whitehall was closed to traffic for several days after the ceremony and wounded soldiers, other veterans, and members of the public began to file past the Cenotaph and lay flowers at its base. The volume of people wishing to lay tributes was such that there were delays of up to four hours to pass the Cenotaph; the procession continued through the night and into the next day. Within a week, the Cenotaph was deep in flowers and an estimated 1.25million people had visited it so far, while 500,000 had visited the Tomb of the Unknown Warrior. Lloyd George wrote to Lutyens, "The Cenotaph is the token of our mourning as a nation; the Grave of the Unknown Warrior is the token of our mourning as individuals."


title: The Cenotaph

text: Throughout the 1920s and 1930s, it was customary for men to doff their hats when passing the Cenotaph, even on a bus. The pilgrimage which began with the Cenotaph's unveiling continued on a smaller scale for the rest of the 1920s and into the 1930s. Pilgrimages continued until the outbreak of the Second World War. A particularly large crowd gathered on 11 November 1946, the year after that war ended, but attendance largely fell away thereafter. In the later 1920s, several proposals emerged for modifications to the Cenotaph, including the addition of life-size bronze statues at its corners, and installing a light inside the wreath at the top to emit a vertical beam, but all were rejected by the Office of Works on Lutyens's advice. The statues in particular would have added a literal element to the memorial which Greenberg (writing in 1989) believed would have been at odds with its "open symbolism and abstract character".


title: The Cenotaph

text: After the unveiling of the permanent memorial, members of the public continued to lay floral tributes as well as hand-written messages and personal memorials such as photographs, wreaths, and glass domes. The Office of Works struggled to decide what to do with the tributes and how to maintain an appropriate tone. It began preserving the messages so that they could be compiled into albums and given to the Imperial War Museum. By March 1921, officials had catalogued more than 30,000 items; the volume was such that they were forced to abandon their efforts at preservation. The office was keen to avoid being seen as a censor but also to preserve the character of the Cenotaph; officials thus removed some tributes which contained overtly political messages. A group of 5,000 unemployed men, on an anti-capitalist protest, paraded past the Cenotaph in 1921 and laid wreaths at its base; several with explicit political messages were removed. In 1933, Alfred Rosenberg, representing Nazi Germany, controversially laid a wreath at the Cenotaph. The accompanying card was removed overnight and the swastika on the wreath was scratched off. The following day, Captain James Sears, a First World War veteran and prospective Labour Party parliamentary candidate, removed the entire wreath and threw it in the river. He described his actions as "a deliberate protest against the desecration of our national war memorial" and against the views of the Nazi Party, which he believed were the same as those Britain had fought against. Sears was arrested, charged with malicious damage, and fined. Some newspaper columnists and letter writers sympathised with Sears's actions, though others felt that his actions themselves desecrated the Cenotaph by using it to make a political statement.


title: The Cenotaph

text: Following the Second World War, the Cenotaph was rededicated to include the British and empire dead from that war, and its dates in Roman numerals (MCMXXXIX and MCMXLV) were added to the inscription. King George VI unveiled the additions at a ceremony on 10 November 1946. No separate national memorial was built for the casualties of the second war; instead, remembrance services were expanded to commemorate the new dead, and veterans of that war and later conflicts joined an annual march-past.


title: The Cenotaph

text: Several political protests have taken place in the vicinity of the Cenotaph. In 2000, anti-capitalist protesters spray-painted slogans on it and on a statue of Winston Churchill. In a 2010 student protest, a man climbed the base and swung from one of the flags. In June 2020, the base was vandalised with spray paint during Black Lives Matter protests, and the following day a protester attempted to set fire to one of the Union Flags on the Cenotaph. The Cenotaph and several other monuments were covered up temporarily to prevent any further vandalism, though a group of far-right counter-protesters congregated around it a few days later. On 11 November 2020, Extinction Rebellion held an unauthorised protest at the Cenotaph that was condemned by politicians and the Royal British Legion.


title: The Cenotaph

text: In 2013, just before the centenary of the First World War, English Heritage carried out £60,000 of restoration work on the Cenotaph. Contractors cleaned the stonework using steam and a poultice to remove dirt and algae and counter the effects of weathering and pollution. Somewhat controversially, the German president, Frank-Walter Steinmeier, was invited to lay a wreath at the Cenotaph on Remembrance Sunday 2018 to mark the centenary of the armistice, the first time a German representative had been present at the commemorations.


title: The Cenotaph

text: The Cenotaph is the focal point for the National Service of Remembrance held annually on Remembrance Sunday, the closest Sunday to 11 November. In the Cenotaph's early years, the service was informal and crowds gathered round the memorial to pay their respects and lay tributes, but the ceremony gradually became more formal, and has changed little since the 1930s. Whitehall is closed to vehicle traffic and a two-minute silence is observed at 11:00am. After the silence, the crowd sings traditional hymns, accompanied by military musicians. The monarch and the prime minister (or their representatives) then lay wreaths at the Cenotaph, followed by other members of the royal family, politicians, and Commonwealth high commissioners. Afterwards, serving military personnel, veterans' associations, and other organisations march past and lay their own wreaths. Until the Second World War, the service was held on 11 November. It was moved to a Sunday to avoid interrupting wartime production, and has been held on a Sunday ever since. According to Paul Fussell, an American literary historian specialising in the cultural effects of the world wars, "to sense the British obsession with the Great War, all that is necessary is to stand at the Cenotaph [...] on any Remembrance Sunday and listen to the two minutes of silence," which he describes as "appropriately shocking in the context of the customary hum of traffic."


title: The Cenotaph

text: Other annual ceremonies are also held at the Cenotaph, such as commemorations by individual regiments or veterans' associations, or on anniversaries such as Anzac Day (25 April). In 2000, relatives of soldiers executed for desertion or cowardice during the First World War joined the Remembrance Sunday parade for the first time, and in 2014 a representative of the Irish government laid a wreath on Remembrance Sunday for the first time, in memory of Irishmen who fought in the British armed forces in the First World War.


title: The Cenotaph

text: The BBC began broadcasting special radio programming for Armistice Day in 1923, and began broadcasting the events at the Cenotaph live from 1928. Radio broadcasting enabled the silence to be observed simultaneously across the country, and allowed millions of listeners to feel part of the ceremony. The BBC began broadcasting television pictures of the ceremony from 1937. The broadcast has run almost continually since its inception, interrupted only for the Second World War, making it one of the longest-running annual broadcasts in the world.


title: The Cenotaph

text: The Cenotaph was designated a grade I listed building on 5 February 1970. Listing provides legal protection from unauthorised demolition or modification. Grade I is the highest possible grade, reserved for buildings of "exceptional" historical or architectural interest and applied to 2.5 per cent of listings. The Cenotaph is in the care of English Heritage, which manages historic buildings for the nation. To mark the centenary of the First World War, Historic England conducted research into war memorials with the aim of listing 2,500. As part of the project, they identified 44 freestanding war memorials in England designed by Lutyens, which they declared to be a national collection. All 44 are listed buildings and had their list entries enhanced with new research; five (including Southampton) were upgraded to grade I on Remembrance Sunday 2014, joining the Cenotaph and the Arch of Remembrance in Leicester.


title: The Cenotaph

text: The renowned architectural historian Nikolaus Pevsner described the Cenotaph as "the chief national war memorial". Gavin Stamp, a British architectural historian and the author of Lutyens's entry in the "Oxford Dictionary of National Biography", wrote that Lutyens's work commemorating the British war dead (the Cenotaph, his work with the IWGC and his memorial commissions elsewhere) was responsible for Lutyens's elevation to the status of a national figure. A few days after the unveiling, Lloyd George wrote to Lutyens: "the Cenotaph, by its very simplicity, fittingly expresses the memory in which the people hold all those who so bravely fought and died" in the war. In 1921, Lutyens was awarded the Royal Institute of British Architects' highest award, the Royal Gold Medal, for his body of work. Presenting the medal, the institute's president, John Simpson, described the Cenotaph as "the most remarkable of all [Lutyens's] creations [...] austere yet gracious, technically perfect, it is the very expression of repressed emotion, of massive simplicity of purpose, of the qualities which mark those whom it commemorates and those who raised it."


title: The Cenotaph

text: According to Jane Brown, in her biography of the architect, Lutyens was faced with a "constant stream" of war memorial commissions from the unveiling of the temporary Cenotaph until at least 1924. He went on to design more than 130 war memorials and cemeteries, many influenced by his work on the Cenotaph. His Southampton Cenotaph was unveiled in 1920, while the permanent monument on Whitehall was still under construction. His later cenotaphs include Rochdale, Manchester, and the Midland Railway War Memorial in Derby. Lutyens also used the design for monuments in several of his cemeteries in Belgium and France for the IWGC, most famously at Étaples.


title: The Cenotaph

text: Examples of artworks featuring the Cenotaph include "Immortal Shrine" (1928) by Will Longstaff (held at the Australian War Memorial) and "The Cenotaph (Morning of the Peace Procession)" (1919) by Sir William Nicholson. The latter work by Nicholson sold at auction at Christie's in London in 2018 for £62,500. The Cenotaph also featured on the reverse of the 1928 Armistice Day memorial medal by Charles Doman. Examples of the Cenotaph featuring in artworks commemorating national events include the ceremonial paintings commissioned in 1920 by the government and the king from Frank Owen Salisbury to mark the unveiling of the Cenotaph, titled "The Passing of the Unknown Warrior, 11 November 1920". A study for the work hangs in Buckingham Palace; the main work is in the Ministry of Defence Main Building off Whitehall.


title: The Cenotaph

text: A 1936 novel by Irene Rathbone with an anti-war theme, "They Call it Peace", concluded with a scene set at the Cenotaph in which two women complete pilgrimages to the monument, one to honour the dead and one feeling that the deaths were in vain. The cultural response to the Cenotaph also includes poetry such as "The Cenotaph" (1919) by Charlotte Mew, "The Cenotaph in Whitehall" (1920) by Max Plowman, "The Cenotaph" (1922) by Ursula Roberts, "London Stone" (1923) by Rudyard Kipling, "At the Cenotaph" (1933) by Siegfried Sassoon, and "At the Cenotaph" (1935) by Hugh MacDiarmid. Laurence Binyon's "For the Fallen" (1914) is closely associated with the Cenotaph, having been recited at its unveiling, and commonly features in remembrance services, particularly the fourth stanza, which concludes: According to the literary historian Alex Moffett, the poems about the Cenotaph convey the different narratives of the First World War and the way in which it should be remembered, in much the same way that the monument itself is open to interpretation. The poetry also expresses the conflict between sombre commemoration of the dead and celebration of victory, "a tension that many have read within the Cenotaph itself".


title: The Cenotaph

text: According to one study of British war memorials, the Cenotaph's "deceptively simple design and deliberately non-sectarian message ensured that its form would be adopted widely, with local variations". From its unveiling, the Cenotaph proved highly influential on other war memorials in Britain. The art historian Alan Borg wrote that the Cenotaph was the "one memorial that proved to be more influential than any other". Several towns and cities erected war memorials based to some extent on Lutyens's design for Whitehall, though the term "cenotaph" came to be applied to almost any war memorial that was not itself a tomb. Lutyens designed several other cenotaphs in England and one in Wales, while replicas, of varying quality and accuracy, were built across Britain, along with many other monuments inspired to some extent by Lutyens's design. Examples include Leeds War Memorial and Glasgow Cenotaph.


title: The Cenotaph

text: Replicas were also built in other countries of the British Empire, usually by local architects with input from Lutyens. The government of Bermuda opted for a two-thirds scale replica, unveiled in 1925, having paid Lutyens a fee for his plans and for advice on the site. A wooden replica was erected in London, Ontario, until a permanent version, a three-quarters scale replica of Whitehall's, could be erected in 1934. Hong Kong's Cenotaph, unveiled in 1928, was built by a local architectural practice with input from Lutyens. The cenotaph at the Auckland War Memorial Museum in New Zealand is a copy of Whitehall's, though Lutyens was not involved in its inception. Purchasing the designs from Lutyens was deemed too expensive so a local architect, Keith Draffin, sketched it from cinema newsreels. At least four other copies exist in New Zealand. Several temporary replicas were built as placeholders until permanent memorials could be built, including one in Toronto, Canada, replaced with the Old City Hall Cenotaph, and one in Melbourne, Australia, which stood until 1937, three years after the completion of the Shrine of Remembrance.


title: The Cenotaph

text: Borg observed that there was no agreed standard for war memorials, with wide variations in design, though Lutyens's Cenotaph and Sir Reginald Blomfield's Cross of Sacrifice came closest. Such was the impact of the Cenotaph that even Blomfield, a great rival of Lutyens, drew on it for his Royal Air Force Memorial a short distance away on the bank of the River Thames. According to King, the Cenotaph's popularity with the public and its widespread use and adaptation by other artists, including professional rivals, showed the extent to which it became common property rather than a concept exclusive to Lutyens. The Imperial War Museum's War Memorials Register identified at least 55 replica or similar cenotaphs in Britain alone.


title: John Reynolds (actor)

text: John Paul Reynolds (born August 5, 1991) is an American actor and writer. He is known for his starring role in the series "Search Party" (2016–2022) and his recurring role on "Stranger Things" (2016–present). Reynolds also starred in the miniseries "Four Weddings and a Funeral" in 2019. Born in Madison, Wisconsin, he began his career at the Annoyance Theatre in Chicago.


title: John Reynolds (actor)

text: Reynolds began his career in Chicago at the age of 18. He graduated from the IO Training Center and the Second City Conservatory in Chicago, and performed regularly at IO Chicago, The Annoyance, Chemically Imbalanced Comedy, The Playground, and Upstairs Gallery. He was also a part of the Kill All Comedy collective along with comedians Conner O'Malley and Gary Richardson.


title: John Reynolds (actor)

text: Reynolds has said he based his "Search Party" character Drew Gardner on his experience growing up in the Midwest: "I grew up in Madison, Wisconsin, and I based him as a guy I would know in the Midwest who sees his brothers grow up a certain way, and wants to maintain that status quo with his family, and doesn’t want to be in any spotlight."


title: Parcelforce

text: Parcelforce Worldwide is a courier and logistics service in the United Kingdom. Parcelforce Worldwide is a trading name of the Royal Mail and is organised within the UK Parcels, International and Letters division of the group. The company delivers to destinations worldwide, using an international partner network. Parcelforce Worldwide is a provider of express delivery services covering 99.6% of the world's population, with over 6500 employees.


title: Parcelforce

text: Its European delivery partner, General Logistics Systems (GLS), is also a subsidiary of Royal Mail and delivers more than one million parcels a day across 34 countries in Europe. Parcelforce Worldwide is a direct competitor of other worldwide delivery brands, such as DHL, DX Group, FedEx and UPS.


title: Parcelforce

text: Parcelforce Limited operates a "hub and spoke" collection and delivery system with two hubs based at Coventry, adjacent to the airport. One hub is for parcels for the United Kingdom, and the other for international parcels. The hub of the United Kingdom, one of the country's largest buildings, is a highly automated tracking and sorting centre covering and can handle up to 58,500 parcels an hour.


title: Parcelforce

text: The Parcel Post service of Royal Mail was started in 1883, though parcel services operated by the railway companies, later Red Star Parcels and British Road Services, were also heavily used for many decades. Royal Mail was separated into three divisions in 1986 and in August 1990, Royal Mail Parcels was rebranded as Parcelforce.


title: Parcelforce

text: In July 1992, the Secretary of State for Trade and Industry, Michael Heseltine, proposed a privatisation of the Parcelforce business. He argued that the company provided most of its services to businesses rather than individuals, and that the competitive market it operated in meant there was no need for it to remain a publicly owned company.


title: Parcelforce

text: In March 2002, the universal parcels service was transferred to Royal Mail leaving Parcelforce to concentrate on time guaranteed, next day and two day express deliveries.


title: Parcelforce

text: In January 2007, Parcelforce Worldwide became the first express carrier in the United Kingdom to give its customers the option to offset the carbon emissions associated with the delivery of their parcels. The company allowed customers to donate 5p for a delivery for the United Kingdom, or 10p for an international delivery to The Woodland Trust. Parcelforce would then match the donation.


title: Parcelforce

text: In October 2012, Royal Mail announced an investment of £75 million in Parcelforce Worldwide planned over four years, which will involve the opening of new processing centres in Chorley, Lancashire and depots in Basingstoke, Hampshire and Bodmin, Cornwall.


title: Parcelforce

text: Parcelforce Worldwide operates its collection and delivery services via a network of 54 “local” depots. These have been strategically located around the United Kingdom from Inverness in the north of Scotland to Plymouth in the southwest of England. There are two depots in Wales, one central depot in Northern Ireland (serving Belfast) and two smaller satellite depots.


title: Parcelforce

text: The largest posters also have the option of a direct feed collection. In other words, rather than a local depot collecting parcels for sorting, a trailer is left on site, filled up during the course of the working day and then taken directly to Coventry for processing.


title: United States Consumer Price Index

text: The United States Consumer Price Index (CPI) is a set of consumer price indices calculated by the U.S. Bureau of Labor Statistics (BLS). To be precise, the BLS routinely computes many different CPIs that are used for different purposes. Each is a time series measure of the price of consumer goods and services. The BLS publishes the CPI monthly.


title: United States Consumer Price Index

text: The BLS started the statistic in 1919. It currently computes thousands of consumer price indices, beginning with monthly average prices for each of 8,018 category-area combinations (211 categories of consumption items in 38 urban geographical areas). They also track how much of each of these category area combinations is in the "market basket" consumed by different groups of people. Different published consumer price indices differ in the weights, including the target consumer group and how frequently the weights are updated. The weights for many indices are modified only in January of even-numbered years and are held constant for the next two years. However, weights for the chained CPI (C-CPI-U) are updated each month, so they more accurately track short term shifts in consumption patterns.


title: United States Consumer Price Index

text: The urban wage earner and clerical worker population consists of consumer units consisting of clerical workers, sales workers, craft workers, operative, service workers, or laborers. (Excluded from this population are professional, managerial, and technical workers; the self-employed; short-term workers; the unemployed; and retirees and others not in the labor force.) More than one half of the consumer unit's income has to be earned from the above occupations, and at least one of the members must be employed for 37 weeks or more in an eligible occupation. The consumer price index for urban wage earners and clerical workers (CPI-W) is a continuation of the historical index that was introduced after World War I for use in wage negotiation. As new uses were developed for the CPI, the need for a broader and more representative index became apparent. The Social Security Administration uses the CPI-W as the basis for its periodic COLA (cost-of-living adjustment).


title: United States Consumer Price Index

text: The all-urban consumer population consists of all urban households in metropolitan statistical areas (MSAs) and in urban places of 2,500 inhabitants or more. Non-farm consumers living in rural areas within MSAs are included, but the index excludes rural consumers and the military and institutional population. The consumer price index for all urban consumers (CPI-U) was introduced in 1978. In 2020 it represented the buying habits of approximately 93 percent of the non-institutional population of the United States, compared with 29 percent represented in the CPI-W. The methodology for producing the index is the same for both populations.


title: United States Consumer Price Index

text: The core CPI index excludes goods with high price volatility, such as food and energy. This measure of core inflation systematically excludes food and energy prices because, historically, they have been highly volatile and non-systemic. More specifically, food and energy prices are widely thought to be subject to large changes that often fail to persist and do not represent relative price changes. In many instances, large movements in food and energy prices arise because of supply disruptions such as drought or OPEC-led cutbacks in production. This was introduced in the early 1970s when food and especially oil prices were quite volatile, and the Fed wanted an index that was less subject to short term shocks. However, on January 25, 2012, the Fed announced they would stop using the core CPI and rely instead on the personal consumption expenditures price index.


title: United States Consumer Price Index

text: This index applies to the same target population as the CPI-U, but the weights are updated each month. This allows the weights to evolve more gracefully with people's consumption patterns; for CPI-U, the weights are changed only in January of even-numbered years and are held constant for the next two years.


title: United States Consumer Price Index

text: Since at least 1982, the BLS has also computed a consumer price index for the elderly to account for the fact that the consumption patterns of seniors are different from those of younger people. For the BLS, "elderly" means that the reference person or a spouse is at least 62 years of age; approximately 24 percent of all consumer units meet this definition. Individuals in this group consume roughly double the amount of medical care as all consumers in CPI-U or employees in CPI-W.


title: United States Consumer Price Index

text: In January of each year, Social Security recipients receive a cost of living adjustment (COLA) "to ensure that the purchasing power of Social Security and Supplemental Security Income (SSI) benefits is not eroded by inflation. It is based on the percentage increase in the consumer price index for urban wage earners and clerical workers (CPI-W)".


title: United States Consumer Price Index

text: However, from December 1982 through December 2011, the all-items CPI-E rose at an annual average rate of 3.1 percent, compared with increases of 2.9 percent for both the CPI-U and CPI-W. This suggests that the elderly have been losing purchasing power at the rate of roughly 0.2 (=3.1–2.9) percentage points per year.


title: United States Consumer Price Index

text: In 2003 Hobijn and Lagakos estimated that the social security trust fund would run out of money in 40 years using CPI-W and in 35 years using CPI-E.


title: United States Consumer Price Index

text: The Consumer Price Index was initiated during World War I, when rapid increases in prices, particularly in shipbuilding centers, made an index essential for calculating cost-of-living adjustments in wages. To provide appropriate weighting patterns for the index, it reflected the relative importance of goods and services purchased in 92 different industrial centers in 1917–1919. Periodic collection of prices was started, and in 1919 the Bureau of Labor Statistics began publication of separate indexes for 32 cities. Regular publication of a national index, the U.S. city average began in 1921, and indexes were estimated back to 1913 using records of food prices.


title: United States Consumer Price Index

text: Because people's buying habits had changed substantially, a new study was made covering expenditures in the years 1934–1936, which provided the basis for a comprehensively revised index introduced in 1940. During World War II, when many commodities were scarce and goods were rationed, the index weights were adjusted temporarily to reflect these shortages. In 1951, the BLS again made interim adjustments, based on surveys of consumer expenditures in seven cities between 1947 and 1949, to reflect the most important effects of immediate postwar changes in buying patterns. The index was again revised in 1953 and 1964.


title: United States Consumer Price Index

text: In 1978, the index was revised to reflect the spending patterns based upon the surveys of consumer expenditures conducted in 1972–1974. A new and expanded 85-area sample was selected based on the 1970 Census of Population. The Point-of-Purchase Survey (POPS) was also introduced. POPS eliminated reliance on outdated secondary sources for screening samples of establishments or outlets where prices are collected. A second, more broadly based CPI for All Urban Consumers, the CPI-U was also introduced. The CPI-U took into account the buying patterns of professional and salaried workers, part-time workers, the self-employed, the unemployed, and retired people, in addition to wage earners and clerical workers.


title: United States Consumer Price Index

text: In January 1983, housing prices were replaced with "owners' equivalent of rent" because rents are more stable. Because house prices rose and fell more than rents during the housing bubble and crash, housing's effects on inflation and deflation are not reflected in the CPI.


title: United States Consumer Price Index

text: In 1995, the Senate Finance Committee appointed a commission to study CPI's ability to estimate inflation. The CPI commission found in their study that the index overestimated the cost of living by a value between 0.8 and 1.6 percentage points.


title: United States Consumer Price Index

text: If CPI overestimates inflation, then claims that real wages have fallen over time could be unfounded. An overestimation of only a few tenths of a percentage point per annum compounds dramatically over time. In the 1970s and 80s the federal government began indexing several transfers and taxes including social security (see below "Uses of the CPI"). The overestimation of CPI would imply that the increases in these taxes and transfers have been greater than necessary, meaning the government and taxpayers have overpaid for them.


title: United States Consumer Price Index

text: The Commission concluded that more than half of the overestimation was due to slow adjustments in the index to new products or changes in product quality. At that time, the weights for indices like CPI-U and CPI-W were updated only once per decade; today, they are updated in January of even-numbered years. However, even with this more frequent updates, the CPI-U and CPI-W might still be excessively slow in responding to new technologies. For example, by 1996 there were over 47 million cellular phone users in the United States, but the weights for the CPI did not account for this new product until 1998. This new product lowered costs of communication when away from the home. The commission recommended that the BLS update weights more frequently to prevent upward bias in the index from a failure to properly account for the benefits of new products.


title: United States Consumer Price Index

text: Additional upward biases were said to come from several other sources. Fixed weights do not accommodate consumer substitutions among commodities, such as buying more chicken when the price of beef increases. Because the CPI assumes that people continue to buy beef, it would increase even if people are buying chicken instead. However, this is by design: the CPI measures the change in expenses required for people to maintain the same standard of living. The Commission also found that 99% of all data were collected during the week, although an increasing amount of purchases happen during the weekend. Additional bias was said to stem from changes in retailing that were unaccounted for in the CPI.


title: United States Consumer Price Index

text: Some critics believe however, that because of changes to the way that the CPI is calculated, and because energy and food price changes were excluded from the Federal Reserve's calculation of "core inflation", that inflation is being dramatically underestimated. The second argument is unrelated to the CPI, except insofar as the calculation of CPI is modified in response to a perceived overstatement of inflation.


title: United States Consumer Price Index

text: The Federal Reserve's policy of ignoring food and energy prices when making interest rate decisions is often confused with the measurement of the CPI by the Bureau of Labor Statistics. The BLS publishes both a headline CPI which "counts" food and energy prices, and also a CPI for "all items less food and energy", or "core" CPI. None of the prominent legislated uses of the CPI excludes food and energy. However, with regard to calculating inflation, the Federal Reserve no longer uses the CPI, preferring to use core PCE instead.


title: United States Consumer Price Index

text: Some critics believe that changes in CPI calculation due to the Boskin Commission have led to dramatic cuts in inflation estimates. They believe that using pre-Boskin methods, which they also think are still used by most other countries, the current U.S. inflation is estimated to be around 7% per year. The BLS maintains that these beliefs are based on misunderstandings of the CPI. For example, the BLS has stated that changes made due to the introduction of the geometric mean formula to account for product substitution (one of the Boskin recommended changes) have lowered the measured rate of inflation by less than 0.3% per year, and the methods now used are commonly employed in the CPIs of developed nations.


title: United States Consumer Price Index

text: In the first stage, elementary indices are created to show the price levels of very similar goods in the same area. For instance, there is an elementary index for "sports equipment in Seattle". As of June 2007, there are 8,018 of these elementary indices [8,018 = 211 * 38, where 211 is the number of categories ("item strata") and 38 is the number of geographical areas considered]. All but a few of the elementary indices are based on geometric mean formulas.


title: United States Consumer Price Index

text: In the second stage, the elementary indices are combined to create a number of aggregate indices, including the CPI. (The CPI is an aggregate of all 8,018 basic indices. BLS also computes other aggregates computed uses smaller subsets of the basic indices. For instance, there is an all-items index for Boston, and an all-areas index for electricity.)


title: United States Consumer Price Index

text: The weight (or quantities, to use the above terminology) of an item in the CPI is derived from the expenditure on that item as estimated by the Consumer Expenditure Survey. This survey provides data on the average expenditure on selected items, such as white bread, gasoline and so on, that were purchased by the index population during the survey period. In a fixed-weight index such as CPI-U, the implicit quantity of any item used in calculating the index remains the same from month to month.


title: United States Consumer Price Index

text: A related concept is the relative importance of an item. The relative importance shows the share of total expenditure that would occur if quantities consumed were unaffected by changes in relative prices and actually remained constant. Although the implicit quantity weights remain fixed, the relative importance changes over time, reflecting average price changes. Items registering a greater than average price increase (or smaller decrease) become relatively more important.


title: United States Consumer Price Index

text: This two-stage method is relatively new. Before 1999, CPI used only Laspeyres indices, measures of the price changes in a fixed market basket of consumption goods and services of constant quantity and quality bought on average by urban consumers, either for all urban consumers (CPI-U) or for urban wage earners and clerical workers (CPI-W). It is argued that Laspeyres index systematically overstates inflation because it does not take into account changes in the quantities consumed that may occur as a response to price changes. The Laspeyres formula works under the assumption that consumers always buy the same amount of each good in the market basket, no matter what the price. The geometric mean price index formula used to calculate many of the elementary indices, in contrast, assumes that consumers will always spend the same amount of money on a good and shift the quantity they buy of that good based on the price. Critics argue that consumers standard of living has declined if price increases force them from preferred to less preferred goods. This logic suggests that the geometric mean price formula understates inflation.


title: Amy Carmichael

text: Amy Beatrice Carmichael (16 December 1867 – 18 January 1951) was an Irish Christian missionary in India who opened an orphanage and founded a mission in Dohnavur. She served in India for 55 years without furlough and wrote 35 books about the missionary work there.


title: Amy Carmichael

text: Amy Beatrice Carmichael was born in the small village of Millisle, County Down, Ireland, in 1867, as the oldest of seven siblings. Her parents were David Carmichael, a miller, and his wife Catherine, both devout Christians. Amy attended Harrogate Ladies College for four years in her youth.


title: Amy Carmichael

text: Amy's father moved the family to Belfast when she was 16 years old, but he died two years later. In Belfast, the Carmichaels founded the Welcome Evangelical Church. In the mid-1880s, Carmichael started a Sunday-morning class for the ‘Shawlies’ (mill girls who wore shawls instead of hats) in the church hall of Rosemary Street Presbyterian. This mission grew quickly to include several hundred attendees. At this time Amy saw an advertisement in "The Christian" for an iron hall that could be erected for £500 and would seat 500 people. Two donations, £500 from Miss Kate Mitchell and one plot of land from a mill owner, led to the erection of the first "Welcome Hall", on the corner of Cambrai Street and Heather Street in 1887.


title: Amy Carmichael

text: Amy continued at the Welcome until she received a call to work among the mill girls of Manchester in 1889, from which she moved on to overseas missionary work, despite suffering from neuralgia, a disease that made her whole body weak and achy and often put her in bed for weeks on end. At the Keswick Convention of 1887, she heard Hudson Taylor, founder of the China Inland Mission (CIM), speak about missionary life; soon afterwards, she became convinced of her calling to missionary work. She applied to the China Inland Mission and lived in London at the training house for women, where she met author and missionary to China Mary Geraldine Guinness, who encouraged her to pursue missionary work. Carmichael was ready to sail for Asia, but it was determined that her health made her unfit for the work. She postponed her missionary career with the CIM and decided later to join the Church Missionary Society.


title: Amy Carmichael

text: Initially Carmichael traveled to Japan, staying for fifteen months, but got neuralgia and returned home. After a brief period of service in Ceylon (Sri Lanka), she went to Bangalore, India, for her health and found her lifelong vocation. She was commissioned by the Church of England Zenana Mission. Carmichael's most notable work was with girls and young women, some of whom were saved from customs that amounted to forced prostitution. Hindu temple children were primarily young girls dedicated to the gods, then usually forced into prostitution to earn money for the priests (i.e., Devadasi) Families often sold their children to the temples if they did not want them or if they needed extra money and fewer


title: Amy Carmichael

text: Carmichael founded the Dohnavur Fellowship in 1901 to continue her work, as she later wrote in "The Gold Cord" (1932). Dohnavur is situated in Tamil Nadu, thirty miles from India's southern tip. The name derives from Count Dohna, who initially funded German missionaries at the site in the early 19th century, on which the Rev. Thomas Walker then established a school. Carmichael's fellowship transformed Dohnavur into a sanctuary for over one thousand children who would otherwise have faced a bleak future. Carmichael often said that her ministry of rescuing temple children started with a girl named Preena. Having become a temple servant against her wishes, Preena managed to escape. Amy Carmichael provided her shelter and withstood the threats of those who insisted that the girl be returned either to the temple directly to continue her sexual assignments, or to her family for more indirect return to the temple. The number of such incidents soon grew, thus beginning Amy Carmichael's new ministry. When the children were asked what drew them to Amy, they most often replied, "It was love. Amma (Tamil for 'mother', referring to Amy) loved us."


title: Amy Carmichael

text: In order to respect Indian culture, members of the organization wore Indian dress and gave the rescued children Indian names. Carmichael herself dressed in Indian clothes, dyed her skin with dark coffee, and often travelled long distances on India's hot, dusty roads to save just one child from suffering. While serving in India, Carmichael received a letter from a young lady who was considering life as a missionary, asking, "What is missionary life like?" Carmichael wrote back, "Missionary life is simply a chance to die."


title: Amy Carmichael

text: In 1912, money and workers were available that helped fund a hospital at Dohnavur. By 1913, the Dohnavur Fellowship was serving 130 girls. In 1918, Dohnavur added a home for young boys, many born to the former temple prostitutes. Meanwhile, in 1916 Carmichael formed a Protestant religious order called Sisters of the Common Life.


title: Amy Carmichael

text: In 1931, a fall severely injured Carmichael, and she remained bedridden for much of her final two decades. However, she continued her inspirational writing, publishing 16 additional books (including "His Thoughts Said . . . His Father Said" (1951), "If" (1953), "Edges of His Ways" (1955) and "God's Missionary" (1957)), as well as revising others she had previously published. Biographers differ on the number of her published works, which may have reached 35 or as many as six dozen, although only a few remain in print today.


title: Amy Carmichael

text: Carmichael died in India in 1951 at the age of 83. She asked that no stone be put over her grave at Dohnavur. Instead, the children she had cared for put a bird bath over it with the single inscription "Amma", meaning "mother" in the Tamil language.


title: Amy Carmichael

text: Her example as a missionary inspired others to pursue a similar vocation. Many webpages include quotes from Carmichael's works, such as


title: Amy Carmichael

text: India outlawed temple prostitution in 1948. However, the Dohnavur Fellowship continues, now supporting approximately 500 people on 400 acres with 16 nurseries and a hospital. Rescued women can leave, join the community, or return for important occasions, including the Christmas season. The foundation is now run by Indians under the jurisdiction of the C.S.I. Tirunelveli Diocese, founded in 1896. Changed policies acknowledging Indian law require that all children born in or brought to Dohnavur be sent out for education in the 6th grade. Furthermore, since 1982, baby boys have been adopted out rather than remaining in the community.


title: Michael Morrison (actor)

text: Milton Owen Ingley (May 23, 1946 - December 22, 2006), also known as Michael Morrison, was an American pornographic actor, producer, and director.


title: Michael Morrison (actor)

text: Ingley performed in over 140 films during the 1970s and 1980s under various names including Michael Morrison.


title: Michael Morrison (actor)

text: He also produced and directed over 100 films with his own production company Chandler Studios during the 1980s and 1990s.


title: Michael Morrison (actor)

text: Ingley, as Morrison, was inducted into the X-Rated Critics Organization Hall of Fame in 1993 as a "Film Pioneer".


title: Michael Morrison (actor)

text: In 1997, Ingley obtained a copy of an intimate video tape stolen from the home of Tommy and Pamela Anderson Lee and sold copies from the web site www.pamlee.com. The Lees sued Ingley and obtained a court injunction ordering him to stop selling the video.


title: Michael Morrison (actor)

text: Ingley lived in Amsterdam from 1997 to 2004. He moved to Texas and then Arizona to be closer to his family, according to his friend Sharon Mitchell.


title: Michael Morrison (actor)

text: Milton Ingley died at his home in Mesa, Arizona on December 22, 2006 from complications of diabetes. He is buried at the National Memorial Cemetery of Arizona in Phoenix.


title: Expected utility hypothesis

text: The expected utility hypothesis is a popular concept in economics that serves as a reference guide for decisions when the payoff is uncertain. The theory recommends which option rational individuals should choose in a complex situation, based on their risk appetite and preferences.


title: Expected utility hypothesis

text: The expected utility hypothesis states an agent chooses between risky prospects by comparing expected utility values (i.e. the weighted sum of adding the respective utility values of payoffs multiplied by their probabilities). The summarised formula for expected utility is formula_1 where formula_2 is the probability that outcome indexed by formula_3 with payoff formula_4 is realized, and function "u" expresses the utility of each respective payoff. On a graph, the curvature of u will explain the agent's risk attitude.


title: Expected utility hypothesis

text: For example, if an agent derives 0 utils from 0 apples, 2 utils from one apple, and 3 utils from two apples, their expected utility for a 50–50 gamble between zero apples and two is 0.5"u"(0 apples) + 0.5"u"(2 apples) = 0.5(0 utils) + 0.5(3 utils) = 1.5 utils. Under the expected utility hypothesis, the consumer would prefer 1 apple (giving him 2 utils) to the gamble between zero and two.


title: Expected utility hypothesis

text: Standard utility functions represent ordinal preferences. The expected utility hypothesis imposes limitations on the utility function and makes utility cardinal (though still not comparable across individuals). In the example above, any function such that "u"(0) < "u"(1) < "u"(2) would represent the same preferences; we could specify "u"(0) = 0, "u"(1) = 2, and "u"(2) = 40, for example. Under the expected utility hypothesis, setting "u"(2) = 3 and assuming the agent is indifferent between one apple with certainty and a gamble with a 1/3 probability of no apple and a 2/3 probability of two apples, requires that the utility of one apple must be set to "u"(1) = 2. This is because it requires that (1/3)"u"(0) + (2/3)"u"(2) = "u"(1), and (1/3)(0) + (2/3)(3) = 2.


title: Expected utility hypothesis

text: Although the expected utility hypothesis is standard in economic modelling, it has been found to be violated in psychology experiments. For many years, psychologists and economic theorists have been developing new theories to explain these deficiencies. These include prospect theory, rank-dependent expected utility and cumulative prospect theory, and bounded rationality.


title: Expected utility hypothesis

text: In the early days of the calculus of probability, classic utilitarians believed that the option which has the greatest utility will produce more pleasure or happiness for the agent and therefore must be chosen The main problem with the expected value theory is that there might not be a unique correct way to quantify utility or to identify the best trade-offs. For example, some of the trade-offs may be intangible or qualitative. Rather than monetary incentives, other desirable ends can also be included in utility such as pleasure, knowledge, friendship, etc. Originally the total utility of the consumer was the sum of independent utilities of the goods. However, the expected value theory was dropped as it was considered too static and deterministic. The classical counter example to the expected value theory (where everyone makes the same "correct" choice) is the St. Petersburg Paradox. This paradox questioned if marginal utilities should be ranked differently as it proved that a “correct decision” for one person is not necessarily right for another person.


title: Expected utility hypothesis

text: The expected utility theory takes into account that individuals may be risk-averse, meaning that the individual would refuse a fair gamble (a fair gamble has an expected value of zero). Risk aversion implies that their utility functions are concave and show diminishing marginal wealth utility. The risk attitude is directly related to the curvature of the utility function: risk neutral individuals have linear utility functions, while risk seeking individuals have convex utility functions and risk averse individuals have concave utility functions. The degree of risk aversion can be measured by the curvature of the utility function.


title: Expected utility hypothesis

text: Since the risk attitudes are unchanged under affine transformations of "u", the second derivative "u<nowiki>"</nowiki>" is not an adequate measure of the risk aversion of a utility function. Instead, it needs to be normalized. This leads to the definition of the Arrow–Pratt measure of absolute risk aversion:


title: Expected utility hypothesis

text: Special classes of utility functions are the CRRA (constant relative risk aversion) functions, where RRA(w) is constant, and the CARA (constant absolute risk aversion) functions, where ARA(w) is constant. They are often used in economics for simplification.


title: Expected utility hypothesis

text: A decision that maximizes expected utility also maximizes the probability of the decision's consequences being preferable to some uncertain threshold. In the absence of uncertainty about the threshold, expected utility maximization simplifies to maximizing the probability of achieving some fixed target. If the uncertainty is uniformly distributed, then expected utility maximization becomes expected value maximization. Intermediate cases lead to increasing risk aversion above some fixed threshold and increasing risk seeking below a fixed threshold.


title: Expected utility hypothesis

text: The St. Petersburg paradox created by Daniel Bernoulli empirically established that the decisions of rational individuals sometimes violate the axioms of preferences. When a probability distribution function has an infinite expected value, it is expected that a rational person would pay an arbitrarily large finite amount to take this gamble. However, this experiment demonstrated that there is no upper bound on the potential rewards from very low probability events. In his experimental game, a person had to flip a coin as many times as possible until it was tails. The participant's prize will be determined by the number of times the coin was turned heads consecutively. For every time the coin comes up heads (1/2 probability), the participant's prize will be doubled. The game ends when the participant flips the coin and it comes out a tail. According to the axioms of preferences, a player should be willing to pay a high price to play because his entry cost will always be less than the expected value of the game, since he could potentially win an infinite payout. However, in reality, people don't do this. “Only a few of the participants were willing to pay a maximum of $25 to enter the game because many of them were risk averse and unwilling to bet on a very small possibility at a very high price.


title: Expected utility hypothesis

text: Nicolaus Bernoulli described the St. Petersburg paradox (involving infinite expected values) in 1713, prompting two Swiss mathematicians to develop expected utility theory as a solution. Bernoulli's paper was the first formalization of marginal utility, which has broad application in economics in addition to expected utility theory. He used this concept to formalize the idea that the same amount of additional money was less useful to an already-wealthy person than it would be to a poor person. The theory can also more accurately describe more realistic scenarios (where expected values are finite) than expected value alone. He proposed that a nonlinear function of utility of an outcome should be used instead of the expected value of an outcome, accounting for risk aversion, where the risk premium is higher for low-probability events than the difference between the payout level of a particular outcome and its expected value. Bernoulli further proposed that it was not the goal of the gambler to maximize his expected gain but to instead maximize the logarithm of his gain.


title: Expected utility hypothesis

text: Daniel Bernoulli drew attention to psychological and behavioral behind the individual's decision-making process and found that the utility of wealth has a diminishing marginal utility. For example, as someone gets wealthier, an extra dollar or an additional good is perceived as less valuable. In other words, he found that the desirability related with a financial gain depends not only on the gain itself but also on the wealth of the person. He suggested that people maximize "moral expectation" rather than expected monetary value. Bernoulli made a clear distinction between expected value and expected utility. Instead of using the weighted outcomes, he used the weighted utility multiplied by probabilities. He proved that the utility function used in real life means is finite, even when its expected value is infinite.


title: Expected utility hypothesis

text: Other experiments proposed that very low probability events are neglected by considering the finite resources of the participants. For example, it makes rational sense for a rich person, but not for a poor person to pay 10,000USD in exchange for a lottery ticket that yields a 50% chance of winning and a 50% chance of nothing.  Even though both individuals have the same chance at each monetary price, they will assign different values to the potential outcomes , according to their income levels. Bernoulli's paper was the first formalization of marginal utility, which has broad application in economics in addition to expected utility theory.


title: Expected utility hypothesis

text: In 1926, Frank Ramsey introduced the Ramsey's Representation Theorem. This representation theorem for expected utility assumed that preferences are defined over set of bets where each option has a different yield. Ramsey believed that we always choose decisions to receive the best expected outcome according to our personal preferences. This implies that if we are able to understand the priorities and personal preferences of an individual we can anticipate what choices they are going to take. In this model he defined numerical utilities for each option to exploit the richness of the space of prices. The outcome of each preference is exclusive from each other. For example, if you study, then you can't see your friends, however you will get a good grade in your course. In this scenario, if we analyze what are his personal preferences and beliefs we will be able to predict which he might choose. (e.g. if someone prioritizes their social life more than academic results, they will go out with their friends). Assuming that the decisions of a person are rational, according to this theorem we should be able to know the beliefs and utilities from a person just by looking the choices someone takes (which is wrong). Ramsey defines a proposition as “ethically neutral” when two possible outcome has an equal value. In other words, if the probability can be defined in terms of preference, each proposition should have ½ in order to be indifferent between both options.


title: Expected utility hypothesis

text: In the 1950s, Leonard Jimmie Savage, an American statistician, derived a framework for comprehending expected utility. At that point, it was considered the first and most thorough foundation to understanding the concept. Savage's framework involved proving that expected utility could be used to make an optimal choice among several acts through seven axioms. In his book, The Foundations of Statistics, Savage integrated a normative account of decision making under risk (when probabilities are known) and under uncertainty (when probabilities are not objectively known).  Savage concluded that people have neutral attitudes towards uncertainty and that observation is enough to predict the probabilities of uncertain events.   A crucial methodological aspect of Savage's framework is its focus on observable choices. Cognitive processes and other psychological aspects of decision making matter only to the extent that they have directly measurable implications on choice.


title: Expected utility hypothesis

text: The theory of subjective expected utility combines two concepts: first, a personal utility function, and second a personal probability distribution (usually based on Bayesian probability theory).  This theoretical model has been known for its clear and elegant structure and its considered for some researchers one of “the most brilliant axiomatic theory of utility ever developed”. Instead assuming the probability of an event, Savage defines it in terms of preferences over acts.  Savage used the states (something that is not in your control) to calculate the probability of an event. On the other hand, he used utility and intrinsic preferences to predict the outcome of the event. Savage assumed that each act and state are enough to uniquely determine an outcome. However, this assumption breaks in the cases where the individual doesn't have enough information about the event.


title: Expected utility hypothesis

text: Additionally, he believed that outcomes must have the same utility regardless of the state. For that reason, it is essential to correctly identify which statement is considered an outcome. For example, if someone says “I got the job” this affirmation is not considered an outcome, since the utility of the statement will be different on each person depending on intrinsic factors such as financial necessity or judgments about the company. For that reason, no state can rule out the performance of any act, only when the state and the act are evaluated simultaneously you will be able to determine an outcome with certainty.


title: Expected utility hypothesis

text: The Savage representation theorem (Savage, 1954) A preference < satisfies P1–P7 if and only if there is a finitely additive probability measure P and a function u : C → R such that for every pair of acts "f" and "g".


title: Expected utility hypothesis

text: <nowiki>*</nowiki>If and only if all the axioms are satisfied when can used the information to reduce the uncertainty about the events that are out of your control. Additionally the theorem ranks the outcome according to utility function that reflects the personal preferences.


title: Expected utility hypothesis

text: There are four axioms of the expected utility theory that define a "rational" decision maker: completeness; transitivity; independence of irrelevant alternatives; and continuity.


title: Expected utility hypothesis

text: "Completeness" assumes that an individual has well defined preferences and can always decide between any two alternatives.


title: Expected utility hypothesis

text: This means that the individual prefers formula_9 to formula_10, formula_10 to formula_9 or is indifferent between formula_9 and formula_10.


title: Expected utility hypothesis

text: "Transitivity" assumes that, as an individual decides according to the completeness axiom, the individual also decides consistently.


title: Expected utility hypothesis

text: "Independence of irrelevant alternatives" pertains to well-defined preferences as well. It assumes that two gambles mixed with an irrelevant third one will maintain the same order of preference as when the two are presented independently of the third one. The independence axiom is the most controversial axiom..


title: Expected utility hypothesis

text: "Continuity" assumes that when there are three lotteries (formula_19 and formula_20) and the individual prefers formula_9 to formula_10 and formula_10 to formula_20, then there should be a possible combination of formula_9 and formula_20 in which the individual is then indifferent between this mix and the lottery formula_10.


title: Expected utility hypothesis

text: If all these axioms are satisfied, then the individual is said to be rational and the preferences can be represented by a utility function, i.e. one can assign numbers (utilities) to each outcome of the lottery such that choosing the best lottery according to the preference formula_44 amounts to choosing the lottery with the highest expected utility. This result is called the von Neumann–Morgenstern utility representation theorem.


title: Expected utility hypothesis

text: In other words, if an individual's behavior always satisfies the above axioms, then there is a utility function such that the individual will choose one gamble over another if and only if the expected utility of one exceeds that of the other. The expected utility of any gamble may be expressed as a linear combination of the utilities of the outcomes, with the weights being the respective probabilities. Utility functions are also normally continuous functions. Such utility functions are also referred to as von Neumann–Morgenstern (vNM) utility functions. This is a central theme of the expected utility hypothesis in which an individual chooses not the highest expected value, but rather the highest expected utility. The expected utility maximizing individual makes decisions rationally based on the axioms of the theory.


title: Expected utility hypothesis

text: The von Neumann–Morgenstern formulation is important in the application of set theory to economics because it was developed shortly after the Hicks–Allen "ordinal revolution" of the 1930s, and it revived the idea of cardinal utility in economic theory. However, while in this context the "utility function" is cardinal, in that implied behavior would be altered by a non-linear monotonic transformation of utility, the "expected utility function" is ordinal because any monotonic increasing transformation of expected utility gives the same behavior.


title: Expected utility hypothesis

text: The utility function formula_45 was originally suggested by Bernoulli (see above). It has relative risk aversion constant and equal to one, and is still sometimes assumed in economic analyses. The utility function


title: Expected utility hypothesis

text: exhibits constant absolute risk aversion, and for this reason is often avoided, although it has the advantage of offering substantial mathematical tractability when asset returns are normally distributed. Note that, as per the affine transformation property alluded to above, the utility function formula_47 gives exactly the same preferences orderings as does formula_48; thus it is irrelevant that the values of formula_48 and its expected value are always negative: what matters for preference ordering is which of two gambles gives the higher expected utility, not the numerical values of those expected utilities.


title: Expected utility hypothesis

text: The class of constant relative risk aversion utility functions contains three categories. Bernoulli's utility function


title: Expected utility hypothesis

text: When the entity formula_57 whose value formula_58 affects a person's utility takes on one of a set of discrete values, the formula for expected utility, which is assumed to be maximized, is


title: Expected utility hypothesis

text: where the left side is the subjective valuation of the gamble as a whole, formula_60 is the "i"th possible outcome, formula_61 is its valuation, and formula_62 is its probability. There could be either a finite set of possible values formula_63 in which case the right side of this equation has a finite number of terms; or there could be an infinite set of discrete values, in which case the right side has an infinite number of terms.


title: Expected utility hypothesis

text: Often people refer to "risk" in the sense of a potentially quantifiable entity. In the context of mean-variance analysis, variance is used as a risk measure for portfolio return; however, this is only valid if returns are normally distributed or otherwise jointly elliptically distributed, or in the unlikely case in which the utility function has a quadratic form. However, David E. Bell proposed a measure of risk which follows naturally from a certain class of von Neumann–Morgenstern utility functions. Let utility of wealth be given by


title: Expected utility hypothesis

text: Thus the risk measure is formula_70, which differs between two individuals if they have different values of the parameter formula_71 allowing different people to disagree about the degree of risk associated with any given portfolio. Individuals sharing a given risk measure (based on given value of "a") may choose different portfolios because they may have different values of "b". See also Entropic risk measure.


title: Expected utility hypothesis

text: For general utility functions, however, expected utility analysis does not permit the expression of preferences to be separated into two parameters with one representing the expected value of the variable in question and the other representing its risk.


title: Expected utility hypothesis

text: Expected utility theory is a theory about how to make optimal decisions under a given probability of risk. It has a normative interpretation which economists used to think applies in all situations to rational agents but now tend to regard as a useful and insightful first order approximation. In empirical applications, a number of violations have been proven to be systematic and these falsifications have deepened understanding of how people actually decide. Daniel Kahneman and Amos Tversky in 1979 presented their prospect theory which showed empirically, how preferences of individuals are inconsistent among the same choices, depending on how those choices are presented. This is mainly because people are different in terms of their preferences and parameters. Additionally, personal behaviors may be different between individuals even when they are facing the same choice problem.


title: Expected utility hypothesis

text: Like any mathematical model, expected utility theory is a simplification of reality. The mathematical correctness of expected utility theory and the salience of its primitive concepts do not guarantee that expected utility theory is a reliable guide to human behavior or optimal practice. The mathematical clarity of expected utility theory has helped scientists design experiments to test its adequacy, and to distinguish systematic departures from its predictions. This has led to the field of behavioral finance, which has produced deviations from expected utility theory to account for the empirical facts.


title: Expected utility hypothesis

text: Other critics argue applying expected utility to economic and policy decisions, has engendered inappropriate valuations, particularly in scenarios in which monetary units are used to scale the utility of nonmonetary outcomes, such as deaths.


title: Expected utility hypothesis

text: Psychologists have discovered systematic violations of probability calculations and behavior by humans. This have been evidenced with examples such as the Monty Hall problem where it was demonstrated that people do not revise their degrees on belief in line with experimented probabilities and also that probabilities cannot be applied to single cases. On the other hand, in updating probability distributions using evidence, a standard method uses conditional probability, namely the rule of Bayes. An experiment on belief revision has suggested that humans change their beliefs faster when using Bayesian methods than when using informal judgment.


title: Expected utility hypothesis

text: According to the empirical results there has been almost no recognition in decision theory of the distinction between the problem of justifying its theoretical claims regarding the properties of rational belief and desire. One of the main reasons is because people's basic tastes and preferences for losses cannot be represented with utility as they change under different scenarios.


title: Expected utility hypothesis

text: instances where people's choices deviate from those predicted by expected utility theory. These deviations are described as "irrational" because they can depend on the way the problem is presented, not on the actual costs, rewards, or probabilities involved. Particular theories include prospect theory, rank-dependent expected utility and cumulative prospect theory are considered insufficient to predict preferences and the expected utility. Additionally, experiments have shown systematic violations and generalizations based on the results of Savage and von Neumann–Morgenstern. This is because preferences and utility functions constructed under different contexts are significantly different. This is demonstrated in the contrast of individual preferences under the insurance and lottery context shows the degree of indeterminacy of the expected utility theory. Additionally, experiments have shown systematic violations and generalizations based on the results of Savage and von Neumann–Morgenstern.


title: Expected utility hypothesis

text: In practice there will be many situations where the probabilities are unknown, and one is operating under uncertainty. In economics, Knightian uncertainty or ambiguity may occur. Thus one must make assumptions about the probabilities, but then the expected values of various decisions can be very sensitive to the assumptions. This is particularly a problem when the expectation is dominated by rare extreme events, as in a long-tailed distribution. Alternative decision techniques are robust to uncertainty of probability of outcomes, either not depending on probabilities of outcomes and only requiring scenario analysis (as in minimax or minimax regret), or being less sensitive to assumptions.


title: Expected utility hypothesis

text: Bayesian approaches to probability treat it as a degree of belief and thus they do not draw a distinction between risk and a wider concept of uncertainty: they deny the existence of Knightian uncertainty. They would model uncertain probabilities with hierarchical models, i.e. where the uncertain probabilities are modelled as distributions whose parameters are themselves drawn from a higher-level distribution (hyperpriors).


title: Expected utility hypothesis

text: Starting with studies such as Lichtenstein & Slovic (1971), it was discovered that subjects sometimes exhibit signs of preference reversals with regard to their certainty equivalents of different lotteries. Specifically, when eliciting certainty equivalents, subjects tend to value "p bets" (lotteries with a high chance of winning a low prize) lower than "$ bets" (lotteries with a small chance of winning a large prize). When subjects are asked which lotteries they prefer in direct comparison, however, they frequently prefer the "p bets" over "$ bets". Many studies have examined this "preference reversal", from both an experimental (e.g., Plott & Grether, 1979) and theoretical (e.g., Holt, 1986) standpoint, indicating that this behavior can be brought into accordance with neoclassical economic theory under specific assumptions.


title: Expected utility hypothesis

text: Understanding utilities in term of personal preferences is really challenging as it face a challenge known as the Problem of Interpersonal Utility Comparisons or the Social Welfare Function. It is frequently pointed out that ordinary people usually make comparisons, however such comparisons are empirically meaningful because the interpersonal comparisons does not show the desire of strength which is extremely relevant to measure the expected utility of decision. In other words, beside we can know X and Y has similar or identical preferences (e.g. both love cars) we cannot determine which love it more or is willing to sacrifice more to get it.


title: Expected utility hypothesis

text: In conclusion Expected Utility theories such as Savage and von Neumann–Morgenstern have to be improved or replaced by more general representations theorems.


title: Expected utility hypothesis

text: There are three components in the psychology field that are seen as crucial to the development of a more accurate descriptive theory of decision under risks.


title: Expected utility hypothesis

text: In this model Conte (2011) found that behaviour differs between individuals and for the same individual at different times. Applying a Mixture Model fits the data significantly better than either of the two preference functionals individually. Additionally it helps to estimate preferences much more accurately than the old economic models because it takes heterogeneity into account. In other words, the model assumes that different agents in the population have different functionals. The model estimate the proportion of each group to consider all forms of heterogeneity.


title: Expected utility hypothesis

text: In this model, Caplin (2001) expanded the standard prize space to include anticipatory emotions such suspense and anxiety influence on preferences and decisions. The author have replaced the standard prize space with a space of "psychological states," In this research, they open up a variety of psychologically interesting phenomena to rational analysis. This model explained how time inconsistency arises naturally in the presence of anticipations and also how this preceded emotions may change the result of choices, For example, this model founds that anxiety is anticipatory and that the desire to reduce anxiety motivates many decisions. A better understanding of the psychologically relevant outcome space will facilitate theorists to develop richer theory of determinants.


title: Snow in Florida

text: It is very rare for snow to fall in the U.S. state of Florida, especially in the central and southern portions of the state. With the exception of the far northern areas of the state, most of the major cities in Florida have never recorded measurable snowfall, though trace amounts have been recorded, or flurries in the air observed few times each century. According to the National Weather Service, in the Florida Keys and Key West there is no known occurrence of snow flurries since the European colonization of the region more than 300 years ago. In Miami, Fort Lauderdale, and Palm Beach there has been only one known report of snow flurries observed in the air in more than 200 years; this occurred in January 1977. In any event, Miami, Fort Lauderdale, and Palm Beach have not seen snow flurries before or since this 1977 event.


title: Snow in Florida

text: Due to Florida's low latitude and subtropical climate, temperatures low enough to support significant snowfall are infrequent and their duration is fleeting. In general, frost is more common than snow, requiring temperatures of 32 °F (0 °C) or less at above sea level, a cloudless sky, and a relative humidity of 65% or more. Generally, for snow to occur, the polar jet stream must move southward through Texas and into the Gulf of Mexico, with a stalled cold front across the southern portion of the state curving northeastward to combine freezing air into the frontal clouds. While light snowfall occurs a few times each decade across the northern panhandle, most of the state is too far south of the cold continental air masses responsible for generating snowfall in the rest of the country. The mean maximum monthly snowfall in most parts of Florida is zero. The only other areas in the continental United States with this distinction are southern and southeast Texas (around McAllen and Houston) and parts of coastal southern California at low elevations.


title: Snow in Florida

text: Much of the known information on snow in Florida prior to 1900 is from climatological records provided by the National Weather Service meteorological station in Jacksonville, and information for other locations is sparse. The earliest recorded instance of snow in Florida occurred in 1774; being unaccustomed to snow, some Jacksonville residents called it "extraordinary white rain." The first White Christmas in northeastern Florida's history resulted from a snow event that occurred on December 23, 1989.


title: Snow in Florida

text: The vast majority of snow events in Florida occurred in north Florida and the Jacksonville area. According to the National Weather Service, the record snowfall for the city of Jacksonville is 1.9 inches (4.8 cm), which fell on February 12, 1899. Tampa has a record snowfall of 0.2 inches (5.08 mm) which occurred on January 18, 1977.


title: Snow in Florida

text: Due to larger populations and more advanced communication networks, snow events are witnessed and reported much more frequently in recent years than in historical eras. Interpretations of this timeline must therefore be made with caution, as observed patterns may not reflect actual climate-related trends in annual snowfall but rather improved reporting. Additionally, the presence of rime or sleet being mistaken for snowflakes should also be considered. Finally, many of the reports below are not "official" National Weather Service reports, many being compiled by the newspapers and media, personal observations, and stories passed down through the years.


title: Cooley–Tukey FFT algorithm

text: The Cooley–Tukey algorithm, named after J. W. Cooley and John Tukey, is the most common fast Fourier transform (FFT) algorithm. It re-expresses the discrete Fourier transform (DFT) of an arbitrary composite size formula_1 in terms of "N" smaller DFTs of sizes "N", recursively, to reduce the computation time to O("N" log "N") for highly composite "N" (smooth numbers). Because of the algorithm's importance, specific variants and implementation styles have become known by their own names, as described below.


title: Cooley–Tukey FFT algorithm

text: Because the Cooley–Tukey algorithm breaks the DFT into smaller DFTs, it can be combined arbitrarily with any other algorithm for the DFT. For example, Rader's or Bluestein's algorithm can be used to handle large prime factors that cannot be decomposed by Cooley–Tukey, or the prime-factor algorithm can be exploited for greater efficiency in separating out relatively prime factors.


title: Cooley–Tukey FFT algorithm

text: The algorithm, along with its recursive application, was invented by Carl Friedrich Gauss. Cooley and Tukey independently rediscovered and popularized it 160 years later.


title: Cooley–Tukey FFT algorithm

text: This algorithm, including its recursive application, was invented around 1805 by Carl Friedrich Gauss, who used it to interpolate the trajectories of the asteroids Pallas and Juno, but his work was not widely recognized (being published only posthumously and in neo-Latin). Gauss did not analyze the asymptotic computational time, however. Various limited forms were also rediscovered several times throughout the 19th and early 20th centuries. FFTs became popular after James Cooley of IBM and John Tukey of Princeton published a paper in 1965 reinventing the algorithm and describing how to perform it conveniently on a computer.


title: Cooley–Tukey FFT algorithm

text: Tukey reportedly came up with the idea during a meeting of President Kennedy's Science Advisory Committee discussing ways to detect nuclear-weapon tests in the Soviet Union by employing seismometers located outside the country. These sensors would generate seismological time series. However, analysis of this data would require fast algorithms for computing DFTs due to the number of sensors and length of time. This task was critical for the ratification of the proposed nuclear test ban so that any violations could be detected without need to visit Soviet facilities. Another participant at that meeting, Richard Garwin of IBM, recognized the potential of the method and put Tukey in touch with Cooley. However, Garwin made sure that Cooley did not know the original purpose. Instead, Cooley was told that this was needed to determine periodicities of the spin orientations in a 3-D crystal of helium-3. Cooley and Tukey subsequently published their joint paper, and wide adoption quickly followed due to the simultaneous development of Analog-to-digital converters capable of sampling at rates up to 300 kHz.


title: Cooley–Tukey FFT algorithm

text: The fact that Gauss had described the same algorithm (albeit without analyzing its asymptotic cost) was not realized until several years after Cooley and Tukey's 1965 paper. Their paper cited as inspiration only the work by I. J. Good on what is now called the prime-factor FFT algorithm (PFA); although Good's algorithm was initially thought to be equivalent to the Cooley–Tukey algorithm, it was quickly realized that PFA is a quite different algorithm (working only for sizes that have relatively prime factors and relying on the Chinese Remainder Theorem, unlike the support for any composite size in Cooley–Tukey).


title: Cooley–Tukey FFT algorithm

text: A radix-2 decimation-in-time (DIT) FFT is the simplest and most common form of the Cooley–Tukey algorithm, although highly optimized Cooley–Tukey implementations typically use other forms of the algorithm as described below. Radix-2 DIT divides a DFT of size "N" into two interleaved DFTs (hence the name "radix-2") of size "N"/2 with each recursive stage.


title: Cooley–Tukey FFT algorithm

text: and of the odd-indexed inputs formula_6, and then combines those two results to produce the DFT of the whole sequence. This idea can then be performed recursively to reduce the overall runtime to O("N" log "N"). This simplified form assumes that "N" is a power of two; since the number of sample points "N" can usually be chosen freely by the application (e.g. by changing the sample rate or window, zero-padding, etcetera), this is often not an important restriction.


title: Cooley–Tukey FFT algorithm

text: The radix-2 DIT algorithm rearranges the DFT of the function formula_7 into two parts: a sum over the even-numbered indices formula_8 and a sum over the odd-numbered indices formula_9:


title: Cooley–Tukey FFT algorithm

text: One can factor a common multiplier formula_11 out of the second sum, as shown in the equation below. It is then clear that the two sums are the DFT of the even-indexed part formula_12 and the DFT of odd-indexed part formula_13 of the function formula_7. Denote the DFT of the Even-indexed inputs formula_12 by formula_16 and the DFT of the Odd-indexed inputs formula_17 by formula_18 and we obtain:


title: Cooley–Tukey FFT algorithm

text: Thanks to the periodicity of the complex exponential, formula_24 is also obtained from formula_16 and formula_18:


title: Cooley–Tukey FFT algorithm

text: This result, expressing the DFT of length "N" recursively in terms of two DFTs of size "N"/2, is the core of the radix-2 DIT fast Fourier transform. The algorithm gains its speed by re-using the results of intermediate computations to compute multiple DFT outputs. Note that final outputs are obtained by a +/− combination of formula_16 and formula_32, which is simply a size-2 DFT (sometimes called a butterfly in this context); when this is generalized to larger radices below, the size-2 DFT is replaced by a larger DFT (which itself can be evaluated with an FFT).


title: Cooley–Tukey FFT algorithm

text: This process is an example of the general technique of divide and conquer algorithms; in many conventional implementations, however, the explicit recursion is avoided, and instead one traverses the computational tree in breadth-first fashion.


title: Cooley–Tukey FFT algorithm

text: The above re-expression of a size-"N" DFT as two size-"N"/2 DFTs is sometimes called the Danielson–Lanczos lemma, since the identity was noted by those two authors in 1942 (influenced by Runge's 1903 work). They applied their lemma in a "backwards" recursive fashion, repeatedly "doubling" the DFT size until the transform spectrum converged (although they apparently didn't realize the linearithmic [i.e., order "N" log "N"] asymptotic complexity they had achieved). The Danielson–Lanczos work predated widespread availability of mechanical or electronic computers and required manual calculation (possibly with mechanical aids such as adding machines); they reported a computation time of 140 minutes for a size-64 DFT operating on real inputs to 3–5 significant digits. Cooley and Tukey's 1965 paper reported a running time of 0.02 minutes for a size-2048 complex DFT on an IBM 7094 (probably in 36-bit single precision, ~8 digits). Rescaling the time by the number of operations, this corresponds roughly to a speedup factor of around 800,000. (To put the time for the hand calculation in perspective, 140 minutes for size 64 corresponds to an average of at most 16 seconds per floating-point operation, around 20% of which are multiplications.)


title: Cooley–Tukey FFT algorithm

text: Here, codice_1("x","N",1), computes "X"=DFT("x") out-of-place by a radix-2 DIT FFT, where "N" is an integer power of 2 and "s"=1 is the stride of the input "x" array. "x"+"s" denotes the array starting with "x".


title: Cooley–Tukey FFT algorithm

text: High-performance FFT implementations make many modifications to the implementation of such an algorithm compared to this simple pseudocode. For example, one can use a larger base case than "N"=1 to amortize the overhead of recursion, the twiddle factors formula_33 can be precomputed, and larger radices are often used for cache reasons; these and other optimizations together can improve the performance by an order of magnitude or more. (In many textbook implementations the depth-first recursion is eliminated in favor of a nonrecursive breadth-first approach, although depth-first recursion has been argued to have better memory locality.) Several of these ideas are described in further detail below.


title: Cooley–Tukey FFT algorithm

text: More generally, Cooley–Tukey algorithms recursively re-express a DFT of a composite size "N" = "N""N" as:


title: Cooley–Tukey FFT algorithm

text: Typically, either "N" or "N" is a small factor ("not" necessarily prime), called the radix (which can differ between stages of the recursion). If "N" is the radix, it is called a decimation in time (DIT) algorithm, whereas if "N" is the radix, it is decimation in frequency (DIF, also called the Sande–Tukey algorithm). The version presented above was a radix-2 DIT algorithm; in the final expression, the phase multiplying the odd transform is the twiddle factor, and the +/- combination ("butterfly") of the even and odd transforms is a size-2 DFT. (The radix's small DFT is sometimes known as a butterfly, so-called because of the shape of the dataflow diagram for the radix-2 case.)


title: Cooley–Tukey FFT algorithm

text: There are many other variations on the Cooley–Tukey algorithm. Mixed-radix implementations handle composite sizes with a variety of (typically small) factors in addition to two, usually (but not always) employing the O("N") algorithm for the prime base cases of the recursion <nowiki>(</nowiki>it is also possible to employ an "N" log "N" algorithm for the prime base cases, such as Rader's or Bluestein's algorithm<nowiki>)</nowiki>. Split radix merges radices 2 and 4, exploiting the fact that the first transform of radix 2 requires no twiddle factor, in order to achieve what was long the lowest known arithmetic operation count for power-of-two sizes, although recent variations achieve an even lower count. (On present-day computers, performance is determined more by cache and CPU pipeline considerations than by strict operation counts; well-optimized FFT implementations often employ larger radices and/or hard-coded base-case transforms of significant size.).


title: Cooley–Tukey FFT algorithm

text: Another way of looking at the Cooley–Tukey algorithm is that it re-expresses a size "N" one-dimensional DFT as an "N" by "N" two-dimensional DFT (plus twiddles), where the output matrix is transposed. The net result of all of these transpositions, for a radix-2 algorithm, corresponds to a bit reversal of the input (DIF) or output (DIT) indices. If, instead of using a small radix, one employs a radix of roughly and explicit input/output matrix transpositions, it is called a four-step algorithm (or "six-step", depending on the number of transpositions), initially proposed to improve memory locality, e.g. for cache optimization or out-of-core operation, and was later shown to be an optimal cache-oblivious algorithm.


title: Cooley–Tukey FFT algorithm

text: The general Cooley–Tukey factorization rewrites the indices "k" and "n" as formula_34 and formula_35, respectively, where the indices "k" and "n" run from 0.."N"-1 (for "a" of 1 or 2). That is, it re-indexes the input ("n") and output ("k") as "N" by "N" two-dimensional arrays in column-major and row-major order, respectively; the difference between these indexings is a transposition, as mentioned above. When this re-indexing is substituted into the DFT formula for "nk", the formula_36 cross term vanishes (its exponential is unity), and the remaining terms give


title: Cooley–Tukey FFT algorithm

text: where each inner sum is a DFT of size "N", each outer sum is a DFT of size "N", and the <nowiki>[...]</nowiki> bracketed term is the twiddle factor.


title: Cooley–Tukey FFT algorithm

text: An arbitrary radix "r" (as well as mixed radices) can be employed, as was shown by both Cooley and Tukey as well as Gauss (who gave examples of radix-3 and radix-6 steps). Cooley and Tukey originally assumed that the radix butterfly required O("r") work and hence reckoned the complexity for a radix "r" to be O("r" "N"/"r" log"N") = O("N" log("N") "r"/log"r"); from calculation of values of "r"/log"r" for integer values of "r" from 2 to 12 the optimal radix is found to be 3 (the closest integer to "e", which minimizes "r"/log"r"). This analysis was erroneous, however: the radix-butterfly is also a DFT and can be performed via an FFT algorithm in O("r" log "r") operations, hence the radix "r" actually cancels in the complexity O("r" log("r") "N"/"r" log"N"), and the optimal "r" is determined by more complicated considerations. In practice, quite large "r" (32 or 64) are important in order to effectively exploit e.g. the large number of processor registers on modern processors, and even an unbounded radix "r"= also achieves O("N" log "N") complexity and has theoretical and practical advantages for large "N" as mentioned above.


title: Cooley–Tukey FFT algorithm

text: Although the abstract Cooley–Tukey factorization of the DFT, above, applies in some form to all implementations of the algorithm, much greater diversity exists in the techniques for ordering and accessing the data at each stage of the FFT. Of special interest is the problem of devising an in-place algorithm that overwrites its input with its output data using only O(1) auxiliary storage.


title: Cooley–Tukey FFT algorithm

text: The most well-known reordering technique involves explicit bit reversal for in-place radix-2 algorithms. Bit reversal is the permutation where the data at an index "n", written in binary with digits "b""b""b""b""b" (e.g. 5 digits for "N"=32 inputs), is transferred to the index with reversed digits "b""b""b""b""b" . Consider the last stage of a radix-2 DIT algorithm like the one presented above, where the output is written in-place over the input: when formula_16 and formula_18 are combined with a size-2 DFT, those two values are overwritten by the outputs. However, the two output values should go in the first and second "halves" of the output array, corresponding to the "most" significant bit "b" (for "N"=32); whereas the two inputs formula_16 and formula_18 are interleaved in the even and odd elements, corresponding to the "least" significant bit "b". Thus, in order to get the output in the correct place, "b" should take the place of "b" and the index becomes "b""b""b""b""b". And for next recursive stage, those 4 least significant bits will become "b""b""b""b", If you include all of the recursive stages of a radix-2 DIT algorithm, "all" the bits must be reversed and thus one must pre-process the input ("or" post-process the output) with a bit reversal to get in-order output. (If each size-"N"/2 subtransform is to operate on contiguous data, the DIT "input" is pre-processed by bit-reversal.) Correspondingly, if you perform all of the steps in reverse order, you obtain a radix-2 DIF algorithm with bit reversal in post-processing (or pre-processing, respectively).


title: Cooley–Tukey FFT algorithm

text: The following is pseudocode for iterative radix-2 FFT algorithm implemented using bit-reversal permutation.


title: Cooley–Tukey FFT algorithm

text: Alternatively, some applications (such as convolution) work equally well on bit-reversed data, so one can perform forward transforms, processing, and then inverse transforms all without bit reversal to produce final results in the natural order.