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Why Do Fools Fall in Love (song)
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"Why Do Fools Fall in Love"
Single by Frankie Lymon & The Teenagers
from the album
The Teenagers Featuring Frankie Lymon
B-side
"Please Be Mine"
Released
January 10, 1956
Recorded
November 1955
Genre
Doo-wop, rock and roll
Length
2:20
Label
Gee
Songwriter(s)
Frankie Lymon, Herman Santiago, Jimmy Merchant
Frankie Lymon & The Teenagers singles chronology
"Why Do Fools Fall in Love"
(1956) "I Want You to Be My Girl"
(1956)
"Why Do Fools Fall in Love"
(1956) "I Want You to Be My Girl"
(1956)
78 RPM record without Herman Santiago listed in the song writer credits.
"Why Do Fools Fall in Love" is a song that was originally a hit for early New York City-based rock and roll group Frankie Lymon & the Teenagers in January 1956. It reached No. 1 on the R&B chart,[1] No. 6 on Billboard's Pop Singles chart,[2] and number 1 on the UK Singles Chart in July.[3][4]
The song helped to make Frankie Lymon a household name and would make him a rock and roll pioneer.
The Canadian group The Diamonds also did a more traditional doo wop version that came out the week after Lymon's, in March 1956. This version spent 19 weeks on the Billboard chart, topping out at No. 12.[5]
The song was ranked #314 on the Rolling Stone magazine's list of The 500 Greatest Songs of All Time.[6]
The song was the first hit recording made in the Bell Sound Studio in New York City. It was recorded in November of 1955.
Contents
[hide]
1 Writing credits
1.1 Personnel
2 Diana Ross version
2.1 Music video
2.2 Personnel
2.3 Track listings
3 Chart performance
3.1 Weekly charts
3.2 Year-end charts
4 The Beach Boys version
5 Other charting versions
6 Other cover versions
7 Legacy
8 See also
9 References
10 External links
Writing credits[edit]
Original vinyl produced in December of 1955.
In late 1955, The Teenagers (at that time calling themselves The Premiers) auditioned a song called "Why do Birds Sing So Gay?" for George Goldner, recording producer and owner of Gee Records. Herman Santiago, tenor of the group, had written the song based on a line from some love letters given to the guys by a tenant in bassist Sherman Garnes' apartment building. One of them featured the words "Why do birds sing so gay?," which fit in with lyrics of other songs that Herman had been writing based on a 1-6-2-5 chord pattern. So Herman worked with it, creating a song for Herman Santiago to lead, and adjusting the harmony to take advantage of Frankie Lymon's high tenor/soprano. Along the way, Herman changed some of the lyrics. During the audition Frankie's voice stood out and, at Goldner's suggestion, the lead in subsequent recording sessions was given to Frankie. Frankie did some improvising and re-created the melody to match his own style. According to Jimmy Merchant, what happened at the recording session was a combination of "Frankie's singing ability coupled with George Goldner's special ability to bring out the best in Frankie."[7]
Although early vinyl single releases of "Why Do Fools Fall in Love" credit Frankie Lymon, Herman Santiago, and George Goldner as co-writers of the song, later releases and cover versions were attributed only to Lymon and record producer George Goldner. Goldner's name was later replaced by Morris Levy when Levy bought Goldner's interest in Gee Records, the Teenagers' record company.
After a lengthy court battle, song-writing credits were awarded to original Teenagers members Herman Santiago and Jimmy Merchant in December 1992.[8]
However, four years later in 1996, this ruling was overturned by the Court of Appeals for the 2nd Circuit under the statute of limitations and authorship, because Santiago and Merchant did not bring the case to court soon enough. This gave the song rights back to Lymon and Levy.[8] The current publisher of the song is EMI Music Publishing, which still lists these as the songwriters.
Personnel[edit]
Frankie Lymon – lead vocals
Herman Santiago – vocals
Jimmy Merchant – vocals
Joe Negroni – vocals
Sherman Garnes – bass vocals
Jimmy Wright – production, bandleader
instrumentalists unknown – feature double bass, drums, piano, saxophone and trombone
Preceded by
"The Great Pretender" by The Platters Billboard R&B Best Seller in Stores number-one single
March 17, 1956 - April 7, 1956 Succeeded by
"Long Tall Sally" by Little Richard and His Band
Diana Ross version[edit]
"Why Do Fools Fall in Love"
Single by Diana Ross
from the album Why Do Fools Fall in Love
B-side
"Think I'm in Love"
Released
September 25, 1981
Format
7" vinyl
Recorded
1980
Genre
R&B, soul
Length
2:51
Label
RCA Records
Songwriter(s)
Herman Santiago
Producer(s)
Diana Ross
Diana Ross singles chronology
"Endless Love"
(1981) "Why Do Fools Fall in Love"
(1981) "Mirror Mirror"
(1981)
"Endless Love"
(1981) "Why Do Fools Fall in Love"
(1981) "Mirror Mirror"
(1981)
Diana Ross released a cover version on the RCA label in 1981, peaking at No. 2 on the US adult contemporary chart, No. 4 in the UK Singles Chart, No. 6 on the US R&B chart, and No. 7 on Billboard's Pop Singles Chart,[9] and earning her a British Phonographic Industry silver disc award for sales in excess of 250,000 copies. It also reached #1 in Belgium and the Netherlands and climbed to the top 10 in Ireland, New Zealand and Switzerland as well as making the top 20 in Australia, Canada and Germany. Unlike the original, this version contains echoes of the soul of the 1960s. A reissue of Ross' cover peaked at #36 on the UK chart in July, 1994.[3]
Music video[edit]
The plot of the music video Diana Ross performs the song in a city full of casinos and clubs (Las Vegas).[10]
Personnel[edit]
Diana Ross – producer, lead vocals, backing vocals
Rob Mounsey – musical arrangements
Ray Chew – electric piano
Eric Gale – electric guitar
Yogi Horton – drums
Neil Jason – bass
Ralph MacDonald – percussion
Jeff Mironov – electric guitar
Ted Jensen at Sterling Sound, NYC – mastering
Track listings[edit]
7" Single
"Why Do Fools Fall in Love" - 2:51
"Think I'm In Love"
UK remix CD
"Why Do Fools Fall in Love" (159.0 bpm) - 2:53
"I'm Coming Out" (Joey Negro Extended 12", 109.7 bpm) - 6:05
"The Boss" (David Morales Club, taken from: Diana Extended/The Remixes, 124.0 bpm) - 6:29
"Love Hangover" (Joey Negro Hangover Symphony, 121.0 bpm) - 8:57
UK reissue 7"
"Why Do Fools Fall in Love"
"I'm Coming Out" (Joey Negro 7" Mix)
Chart performance[edit]
Weekly charts[edit]
Chart (1981–82)
Peak
position
Australia Kent Music Report[11] 15
Belgian Singles Chart 1
Canada RPM Top 100 17
Ireland (IRMA)[12] 9
Netherlands 1
German Singles Chart 17
New Zealand Singles Chart 3
Swiss Singles Chart 9
UK Singles Chart 4
US Billboard Hot 100 7
US Billboard Adult Contemporary 2
US Billboard Hot Black Singles 6
US Cash Box Top 100 7
Year-end charts[edit]
Chart (1981)
Rank
Australia[11] 166
UK[13] 72
US Cash Box[14] 64
Chart (1982)
Rank
US Billboard Hot 100[15] 54
Chart (1994)
Peak
position
UK Singles Chart 36
Preceded by
"Under Pressure" by Queen featuring David Bowie Dutch Top 40 number-one single
December 19, 1981 – January 8, 1982 Succeeded by
"One of Us" by ABBA
Preceded by
"Pretend" by Alvin Stardust Belgian Singles Chart number-one single
December 26, 1981 – January 1, 1982 Succeeded by
"One of Us" by ABBA
The Beach Boys version[edit]
"Why Do Fools Fall in Love"
Single by The Beach Boys
from the album Shut Down Volume 2
A-side
"Fun, Fun, Fun"
Released
February 3, 1964
Format
Vinyl
Recorded
January 1, 1964
Length
2:07
Label
Capitol
Songwriter(s)
Frankie Lymon / Morris Levy
Producer(s)
Brian Wilson
The Beach Boys singles chronology
"Little Saint Nick"
(1963) "Why Do Fools Fall in Love"
(1964) "I Get Around"
(1964)
"Little Saint Nick"
(1963) "Fun, Fun, Fun"
(1964) "I Get Around"
(1964)
The song was used as a B-side for The Beach Boys single "Fun, Fun, Fun", which reached #5 on the Billboard Hot 100 in 1964.[16] The Beach Boys version of the song charted at #120.[17] It was included on the Beach Boys 1964 album Shut Down Volume 2 and had only appeared in mono since the release of the single back in 1964. The single mix of the song was later found and used on the 2007 The Warmth of the Sun compilation and on the Original US Singles Collection The Capitol Years 1962–1965. This box set, released in 2008, also used a recently found mono single edit mix. In 2009, a new stereo mix was created with a newly discovered intro, thanks to the discovery of the original multitrack masters by Jon Stebbins and is featured on the band's compilation Summer Love Songs. The song was also performed as part of the band's 50th Anniversary Tour, usually during the first half of the shows. One of the performances was later included on the live album from the tour.
Other charting versions[edit]
Gale Storm released a version of the song as a single in 1956 that reached #9 on the Billboard pop chart.[18]
Gloria Mann released a version of the song as a single in 1956 that reached #59 on the Billboard pop chart.[19]
Alma Cogan released a version of the song as a single in 1956 that reached #25 in the UK.[20]
The Happenings released a version of the song as a single in 1967 that reached #41 on the Billboard pop chart.[21]
Ponderosa Twins Plus One released a version of the song as a single in 1972 that reached #40 on the US R&B chart and #102 on the Billboard pop chart.[22]
Joni Mitchell released a live version of the song as a single in 1980 that reached #102 on the Billboard pop chart.[23]
Other cover versions[edit]
Barry Frank
Kenny Rankin
Marion Ryan
Voices of Theory
Frankie Valli and the Four Seasons
Mud in 1979 as part of a medley with "The Book of Love"
Tatyana Ali performed the song as a contestant on Star Search in 1987. She also performed it in an introductory skit in Eddie Murphy Raw where she played Eddie's sister.
Gina Thompson and female rapper Mocha covered the song with production by Timbaland and Missy Elliott for the 1998 original soundtrack of the film of the same name.
The Overtones on their debut album Good Ol' Fashioned Love in November 2010.
Legacy[edit]
The recording by Frankie Lymon & The Teenagers features in the 1973 film American Graffiti in the scene where Richard Dreyfuss' character first notices the mysterious blonde girl. The same recording was also featured on the in-game radio station "Empire Central Radio" in the 2010 video game Mafia II. It is featured in the final episode of the first series of the 2012 BBC Drama Call the Midwife when the character Chummy played by Miranda Hart marries PC Noakes. In 1998, the legacy of the song and its author was the theme of the movie of the same name.
See also[edit]
List of UK Singles Chart number ones of the 1950s
References[edit]
Jump up ^ Whitburn, Joel (2004). Top R&B/Hip-Hop Singles: 1942-2004. Record Research. p. 368.
Jump up ^ The heart of rock & soul: the 1001 greatest singles ever made - Dave Marsh - Google Books. Books.google.com. Retrieved 2012-01-08.
^ Jump up to: a b EveryHit.com (search for songs called "Why Do Fools Fall In Love")
Jump up ^ Rice, Jo (1982). The Guinness Book of 500 Number One Hits (1st ed.). En-field, Middle-sex: Guinness Superlatives Ltd. pp. 26–7. ISBN 0-85112-250-7.
Jump up ^ "The Diamonds, "Why Do Fools Fall in Love" Chart Position". Retrieved December 8, 2016.
Jump up ^ "Rolling Stone's 500 Greatest Songs of All Time". Rolling Stone. April 7, 2011. Retrieved October 1, 2015.
Jump up ^ Marv Goldberg's R&B Notebooks, "The Teenagers", based on interviews with Jimmy Merchant.
^ Jump up to: a b "Frankie Lymon (1942 - 1968) - Find A Grave Memorial". Findagrave.com. Retrieved 2014-03-18.
Jump up ^ "Diana Ross, "Why Do Fools Fall in Love" Chart Positions". Retrieved December 8, 2016.
Jump up ^ "Diana Ross - Why Do Fools Fall In Love - Video Dailymotion". Dailymotion.com. 2007-04-28. Retrieved 2014-03-18.
^ Jump up to: a b Steffen Hung. "Forum - Top 100 End of Year AMR Charts - 1980s (ARIA Charts: Special Occasion Charts)". Australian-charts.com. Archived from the original on 2014-10-06. Retrieved 2016-08-22.
Jump up ^ "The Irish Charts – Search Results – Why Do Fools Fall in Love". Irish Singles Chart. Retrieved June 22, 2017.
Jump up ^ "Chart File Top 100" (PDF). Record Mirror. London: Spotlight Publications. December 26, 1981. p. 27. Retrieved June 13, 2016.
Jump up ^ "Archived copy". Archived from the original on 2016-03-04. Retrieved 2015-10-04.
Jump up ^ "Top 100 Hits for 1982". Longboredsurfer.com. Retrieved January 17, 2016.
Jump up ^ "The Beach Boys, "Fun, Fun, Fun" Chart Position". Retrieved December 8, 2016.
Jump up ^ "The Beach Boys, "Why Do Fools Fall in Love" Chart Position". Retrieved December 8, 2016.
Jump up ^ "Gale Storm, "Why Do Fools Fall in Love?" Chart Position". Retrieved December 8, 2016.
Jump up ^ "Gloria Mann, "Why Do Fools Fall in Love?" Chart Position". Retrieved December 8, 2016.
Jump up ^ "Alma Cogan, "Why Do Fools Fall in Love" Chart Position". Retrieved December 8, 2016.
Jump up ^ "The Happenings, "Why Do Fools Fall in Love" Chart Position". Retrieved December 8, 2016.
Jump up ^ "Ponderosa Twins Plus One, "Why Do Fools Fall in Love" Chart Positions". Retrieved December 8, 2016.
Jump up ^ "Joni Mitchell, "Why Do Fools Fall in Love" Chart Position". Retrieved December 8, 2016.
External links[edit]
Lyrics of this song at MetroLyrics
[show]
v
t
e
The Teenagers
Frankie Lymon
Jimmy Merchant
Herman Santiago
Joe Negroni
Sherman Garnes
Billy Lobrano
Howard Kenny Bobo
Studio albums
The Teenagers Featuring Frankie Lymon
Singles
"Why Do Fools Fall in Love"
"I Want You to Be My Girl"
"I Promise to Remember"/"Who Can Explain?"
"The ABC's of Love"
"I'm Not a Juvenile Delinquent"
"Teenage Love"
"Out in the Cold Again"
"Everything to Me"
"My Broken Heart"
"Can You Tell Me"
"The Lemon-Twist Dance"
[show]
v
t
e
Shut Down Volume 2
Songs
Side one
"Fun, Fun, Fun"
"Don't Worry Baby"
"In the Parkin' Lot"
""Cassius" Love vs. "Sonny" Wilson"
"The Warmth of the Sun"
"This Car of Mine"
Side two
"Why Do Fools Fall in Love"
"Pom, Pom Play Girl"
"Keep an Eye on Summer"
"Shut Down, Part II"
"Louie, Louie"
"Denny's Drums"
Related articles
"I Do"
Keep an Eye on Summer – The Beach Boys Sessions 1964
The Beach Boys
The Beach Boys discography
Surfin' Safari
Surfin' U.S.A.
Surfer Girl
Little Deuce Coupe
Shut Down Vol. 2
All Summer Long
Christmas Album
Today!
Summer Days (And Summer Nights!!)
Party
Pet Sounds
Smiley Smile
Wild Honey
Friends
20/20
Sunflower
Surf's Up
Carl and the Passions – "So Tough"
Holland
15 Big Ones
Love You
M.I.U. Album
L.A. (Light Album)
Keepin' the Summer Alive
The Beach Boys
Still Cruisin'
Summer in Paradise
Stars and Stripes Vol. 1
That's Why God Made the Radio
[show]
v
t
e
The Beach Boys singles discography
1960s
1961
"Surfin'" / "Luau"
1962
"Surfin' Safari" / "409"
"Ten Little Indians" / "County Fair"
1963
"Surfin' U.S.A." / "Shut Down"
"Surfer Girl" / "Little Deuce Coupe"
"Be True to Your School" / "In My Room"
"Little Saint Nick" / "The Lord's Prayer"
1964
"Fun, Fun, Fun" / "Why Do Fools Fall in Love"
"I Get Around" / "Don't Worry Baby"
"When I Grow Up (To Be a Man)" / "She Knows Me Too Well"
"Dance, Dance, Dance" / "The Warmth of the Sun"
"The Man with All the Toys" / "Blue Christmas"
1965
"Do You Want to Dance" / "Please Let Me Wonder"
"Help Me, Rhonda" / "Kiss Me, Baby"
"California Girls"/ "Let Him Run Wild"
"The Little Girl I Once Knew" / "There's No Other (Like My Baby)"
"Barbara Ann" / "Girl Don't Tell Me"
1966
"Sloop John B" / "You're So Good to Me"
"Wouldn't It Be Nice" / "God Only Knows"
"Good Vibrations" / "Let's Go Away for Awhile"
1967
"Then I Kissed Her" / "Mountain of Love"
"Heroes and Villains" / "You're Welcome"
"Wild Honey" / "Wind Chimes"
"Darlin'" / "Here Today"
1968
"Friends" / "Little Bird"
"Do It Again" / "Wake the World"
"Bluebirds over the Mountain" / "Never Learn Not to Love"
1969
"I Can Hear Music" / "All I Want to Do"
"Break Away" / "Celebrate the News"
1970s
1970
"Add Some Music to Your Day" / "Susie Cincinnati"
"Cottonfields" / "The Nearest Faraway Place"
"Slip On Through" / "This Whole World"
"Tears in the Morning" / "It's About Time"
1971
"Cool, Cool Water" / "Forever"
"Wouldn't It Be Nice" (live) / "The Times They Are a-Changin'" (by Merry Clayton)
"Long Promised Road" / "Deirdre"
"Long Promised Road" / "'Til I Die"
"Surf's Up" / "Don't Go Near the Water"
1972
"You Need a Mess of Help to Stand Alone" / "Cuddle Up"
"Marcella" / "Hold On Dear Brother"
1973
"Sail On, Sailor" / "Only with You"
"California Saga: California" / "Funky Pretty"
1974
"Surfin' U.S.A." / "The Warmth of the Sun"
"Child of Winter (Christmas Song)" / "Susie Cincinnati"
1975
"Sail On, Sailor" / "Only with You"
1976
"Rock and Roll Music" / "T M Song"
"It's O.K." / "Had to Phone Ya"
"Everyone's in Love with You" / "Susie Cincinnati"
1977
"Honkin' Down the Highway" / "Solar System"
1978
"Peggy Sue" / "Hey Little Tomboy"
1979
"Here Comes the Night" / "Baby Blue"
"Good Timin'" / "Love Surrounds Me"
"Lady Lynda" / "Full Sail"
"It's a Beautiful Day" / "Sumahama"
1980s
1980
"Goin' On" / "Endless Harmony"
"Livin' with a Heartache" / "Santa Ana Winds"
1981
"The Beach Boys Medley" / "God Only Knows"
"Come Go with Me" / "Don't Go Near the Water"
1985
"Getcha Back" / "Male Ego"
"It's Gettin' Late" / "It's O.K."
"She Believes in Love Again" / "It's Just a Matter of Time"
1986
"Rock 'n' Roll to the Rescue" / "Good Vibrations" (live)
"California Dreamin'" / "Lady Liberty"
1987
"Happy Endings" (with Little Richard) / "California Girls" (live)
1988
"Kokomo" / "Tutti Frutti" (Little Richard)
1989
"Still Cruisin'" / "Kokomo"
1990s
1990
"Somewhere Near Japan" / "Kokomo"
"Problem Child" / "Problem Child (Instrumental)"
1992
"Hot Fun in the Summertime" / "Summer of Love"
1996
"I Just Wasn't Made for These Times" / "Wouldn't It Be Nice" / "Here Today"
2010s
2011
"Good Vibrations" / "Heroes and Villains"
"Don't Fight the Sea" / "Friends"
"Cabinessence" / "Wonderful"
2012
"That's Why God Made the Radio" / "That's Why God Made the Radio" (instrumental)
"Isn't It Time" / Live in Chicago
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The Supreme Court of the United States is the highest federal court in the United States and the only court specifically established by the Constitution of the United States, implemented in 1789.
This article is concerned with the process and procedures used by the modern court. For general discussion of the court's jurisdiction and composition, see Supreme Court of the United States; for discussion of the court's present and historical accommodations, see United States Supreme Court building; for discussion of the history of the court, see History of the Supreme Court of the United States.
Contents
[hide]
1 Terms and sittings
2 The Court's authority to hear cases
2.1 Appellate jurisdiction
2.2 Original jurisdiction
3 Selection of cases
4 Hearing cases
4.1 Filing briefs
4.2 Oral arguments
5 Forming opinions
5.1 The conference: assignment of opinions
5.2 Circulating draft opinions and changing of views
5.2.1 Tied votes and lack of quorum
5.3 Announcement of opinions
6 Reporting and citation of cases
7 See also
8 Footnotes and references
9 External links
Terms and sittings[edit]
A term of the Supreme Court commences on the first Monday of each October, and continues until June or early July of the following year. Each term consists of alternating periods of approximately two weeks known as "sittings" and "recesses." Justices hear cases and deliver rulings during sittings; they discuss cases and write opinions during recesses.
The Court's authority to hear cases[edit]
Appellate jurisdiction[edit]
In nearly all of the cases heard by the Supreme Court, the Court exercises the appellate jurisdiction granted it by Article III of the Constitution. This authority permits the Court to review—and affirm or overturn—decisions made by lower courts and tribunals. Procedures for bringing cases before the Supreme Court have changed significantly over time. Today, cases are brought before the Supreme Court by one of several methods, of which the first two account for the overwhelming majority of cases decided:
By petition for a writ of certiorari, filed by a party to a case that has been decided by one of the United States courts of appeals or by the United States Court of Appeals for the Armed Forces.
By petition for writ of certiorari with respect to a decision of one of the territorial or state courts, after all state appeals have been exhausted, where an issue of federal constitutional or statutory law is in question. The writ is usually issued to a state supreme court (including high courts of the District of Columbia, Puerto Rico, the U.S. Virgin Islands, Guam, the Northern Mariana Islands, and American Samoa), but is occasionally issued to a state's intermediate appellate court for cases where the state supreme court denied certiorari or review and thereby refused to hear the appeal.
By petition for certiorari before judgment, which permits the Court to expedite a case pending before a United States court of appeals by accepting the case for review before the appellate court has decided it. However, Supreme Court Rule 11 provides that a case may be taken by the Court before judgment in a lower court "only upon a showing that the case is of such imperative public importance as to justify deviation from normal appellate practice and to require immediate determination in this Court."
By appeal from certain decisions of United States district courts in certain cases involving redistricting of congressional or state legislative districts, or when specifically authorized in a particular statute.
By a certified question or proposition of law from one of the United States Courts of Appeals, meaning that the Court of Appeals requests the Supreme Court to instruct it on how to decide the case. This procedure was once common but is now rarely invoked; the last certificate accepted for review was in 1981.
By petition for an "extraordinary writ" such as mandamus, prohibition, or habeas corpus. These writs are rarely granted by the Supreme Court though they are more frequently granted by lower courts.
Original jurisdiction[edit]
Certain cases that have not been considered by a lower court may be heard by the Supreme Court in the first instance under what is termed original jurisdiction. The Supreme Court's authority in this respect is also derived from Article III of the Constitution, which states that the Supreme Court shall have original jurisdiction "in all cases affecting ambassadors, other public ministers and consuls, and those in which a state shall be party." The original jurisdiction of the Court is set forth in 28 U.S.C. § 1251. This statute provides further that, in the case of disputes between two or more states, the Supreme Court holds both original and exclusive jurisdiction and no lower court may hear such cases.
The number of original jurisdiction cases heard by the court is small; generally only one or two such cases are heard per term. Because the nine-member Supreme Court is not well-suited to conducting pretrial proceedings or trials, original jurisdiction cases accepted by the Court are typically referred to a well-qualified lawyer or lower-court judge to serve as special master, conduct the proceedings, and report recommendations to the Court. The Court then considers whether to accept the special master's report or whether to sustain any exceptions filed to the report.
Although jury trials are in theory possible in the Court's original jurisdiction cases, there has not been one since Georgia v. Brailsford in 1794.[1] In 1950, in the case United States v. Louisiana, the state of Louisiana moved for a jury trial, but the Court denied the motion, ruling that the suit was an equity action and not an action at law, and that therefore the Seventh Amendment guarantee of a jury trial did not apply. If a matter involving an action at law did come before the court, however, a jury would likely be empaneled and would hear the case alongside the justices of the Court.[2]
Selection of cases[edit]
Since the Judiciary Act of 1925 ("The Certiorari Act" in some texts), the majority of the Supreme Court's jurisdiction has been discretionary.[3] Each year, the court receives approximately 9,000–10,000 petitions for certiorari, of which less than 1% (approximately 80–100), are granted plenary review with oral arguments, and an additional 50 to 60 are disposed of without plenary review.[4][5]
The Court strictly enforces its requirements for the preparation and timely filing of certiorari petitions, in order to manage such a massive caseload. This occasionally results in harsh consequences, as Justice Thomas acknowledged in a 2007 opinion: "Just a few months ago, the Clerk, pursuant to this Court’s Rule 13.2, refused to accept a petition for certiorari submitted by Ryan Heath Dickson because it had been filed one day late ... Dickson was executed on April 26, 2007, without any Member of this Court having even seen his petition for certiorari. The rejected certiorari petition was Dickson’s first in this Court, and one can only speculate as to whether denial of that petition would have been a foregone conclusion."[6]
In theory, each Justice's clerks write a brief for the Justice outlining the questions presented, and offering a recommendation as to whether certiorari should be granted; in practice, most Justices (all of the current court, except Justices Alito and Gorsuch) have their clerks participate in the cert pool.[7][8][9]
During the Justices' regular conference, the Justices discuss the petitions, and grant certiorari in less than five percent of the cases filed. (During the 1980s and 1990s, the number of cases accepted and decided each term approached 150 per year; more recently, the number of cases granted has averaged well under 100 annually). Before each conference, the Chief Justice prepares a list of those petitions he believes have sufficient merit to warrant discussion. Any other Justice may also add a case to the "discuss list"; cases not designated for discussion by any Justice are automatically denied review. The Court or a Justice may also decide that a case be "re-listed" for discussion at a later conference; this occurs, for example, where the Court decides to request input from the Solicitor General of the United States on whether a petition should be granted.[10]
The votes of four Justices at Conference (see Rule of four) will suffice to grant certiorari and place the case on the court's calendar. If the Supreme Court grants certiorari (or the certified question or other extraordinary writ), then a briefing schedule is arranged for the parties to submit their briefs in favor of or against a particular form of relief. During this time, an individual or group having an interest in a case but not a party to the case may submit a motion to appear before the court as amicus curiae ("friend of the court"). Except for certain specific categories (such as lawyers for state and local governments) or where all parties to the case consent, it is in the Court's discretion whether such motions are granted.
The grant or denial of certiorari petitions by the Court are usually issued as one-sentence orders without explanation.
Cases that fall within the Court's original jurisdiction are initiated by filing a complaint directly with the Supreme Court, and normally are assigned to a special master appointed by the Court for the taking of evidence and making recommendations, after which the Court may accept briefs and hear oral arguments as in an appellate case.
Hearing cases[edit]
Filing briefs[edit]
Before oral arguments, the parties to a case file legal briefs outlining their arguments. An amicus curiae may also submit a brief in support of a particular outcome in the case if the Court grants it permission. Formal rules govern every aspect of these briefs; Chief Justice William Rehnquist described the rules thus:
The rules direct what information must be included in a brief, describe the size of paper and type of print, and limit the number of pages. Even the colors of the covers of the briefs are specified: the petitioner's brief must have a blue cover and the respondent's must have a red cover. The Court also often receives briefs from amici curiae (friends of the Court) in particular cases, and these must have a green cover. This color-coding comes in very handy when you have a stack of eight or ten briefs in a particular case and can locate the brief you want by its color without having to read the covers of each.[11]
Oral arguments[edit]
Thereafter, if the Court chooses to hold a hearing, each side has thirty minutes to present its case orally. In exceptional and controversial cases, however, the time limit may be extended. In the Court's early years, attorneys might argue a single case for hours or even days; but as the judicial workload increased, the time available for argument has been restricted. The late Chief Justice Rehnquist was noted for his especially strict enforcement of the argument time limits.
To file pleadings or to argue a case, an attorney must be a member of the bar of the Court. (The primary requirement for admission to the Bar is that the attorney must have been admitted to practice in the highest court of a state or territory for at least the past three years.) Justices are allowed to interrupt the attorney speaking in order to ask him or her questions, and particularly since the arrival of Justice Antonin Scalia in 1986, do so often.[12]
The Court began recording Oral Arguments in October 1955.[13] Beginning in October 2010, the Supreme Court began the practice of posting recordings and transcripts of the oral arguments made during the preceding week on Fridays on the Court's website.
In an interview for C-SPAN, former Justice Scalia, speaking for himself, noted that by the time the Justices hear oral arguments, having read the submissions by the parties and amici, it is "very rare, though not unheard of", for the discussion during the oral arguments to change his view of a case in which he has already made up his mind based on the submissions and his research about the case. However, he also made the point that it is "quite common" for him to go into Oral Arguments with his mind not made up yet, as the cases are usually very hard and difficult, and that in those situations a persuasive attorney can make the difference for him.[14]
Forming opinions[edit]
The conference: assignment of opinions[edit]
At the end of a week in which the Court has heard oral arguments, the Justices hold a conference to discuss the cases and vote on any new petitions of certiorari. The Justices discuss the points of law at issue in the cases. No clerks are permitted to be present, which would make it exceedingly difficult for a justice without a firm grasp of the matters at hand to participate.[15] At this conference, each justice—in order from most to least senior—states the basis on which he or she would decide the case, and a preliminary vote is taken.
Former Justice Scalia professed frustration that there is little substantive discussion,[16] while former Chief Justice Rehnquist wrote that this makes the conference more efficient.[17] The votes are tallied, and the responsibility for writing the opinion in the case is assigned to one of the justices; the most senior Justice voting in the majority (but always the Chief Justice if he is in the majority) makes the assignment, and can assign the responsibility to him- or herself.
Circulating draft opinions and changing of views[edit]
The justice writing the opinion for the court will produce and circulate a draft opinion to the other justices. Each justice's law clerks may be involved in this phase. In modern Supreme Court history only a few justices, such as former Justice Antonin Scalia, have regularly written their own first drafts.[18] Once the draft opinion has been reviewed, the remaining Justices may recommend changes to the opinion. Whether these changes are accommodated depends on the legal philosophy of the drafters as well as on how strong a majority the opinion garnered at conference. A justice may instead simply join the opinion at that point without comment.
Votes at conference are preliminary; while opinions are being circulated, it is not unheard of for a justice to change sides. A justice may be swayed by the persuasiveness (or lack thereof) of the opinion or dissent, or as a result of reflection and discussion on the points of law at issue.
The evolution of the justices' views during the circulation of draft opinions can change the outcome of the case; an opinion that begins as a majority opinion can become a dissenting opinion, and vice versa. At the conference for Planned Parenthood v. Casey, Justice Kennedy is said to have initially voted with Chief Justice Rehnquist, but then changed his mind, feeling unable to join Rehnquist's draft opinion.[19] While working for the Justice Department, present-day Chief Justice John Roberts—a former Rehnquist law clerk—wrote an analysis of Wallace v. Jaffree[20] in which he indicated his belief (based on the length and structure) that Rehnquist's dissent had started out as an opinion for the court, but lost its majority; similar speculation is often heard of Justice O'Connor's dissent in Kelo v. New London.[21] Justice Kennedy is known within the Court for changing his mind subsequent to the conference, and Justice Thomas is known for having the tendency to lose a majority.[22] Justices may change sides at any time prior to the handing down of the Court's opinion. Generally, the Court's decision is the opinion which a majority (five or more) of justices have joined. In rare instances, the Court will issue a plurality opinion in which four or fewer Justices agree on one opinion, but the others are so fractured that they cannot agree on a position. In this circumstance, in order to determine what the decision is lawyers and judges will analyze the opinions to determine on which points a majority agrees. An example of a case decided by a plurality opinion is Hamdi v. Rumsfeld.
A justice voting with the majority may write a concurring opinion; this is an opinion where the justice agrees with the majority holding itself, but where he or she wishes to express views on the legal elements of the case that are not encompassed in the majority opinion. Justices who do not agree with the decision made by the majority may also submit dissenting opinions, which may give alternative legal viewpoints. Dissenting opinions carry no legal weight or precedent, but they can set the argument for future cases. John Marshall Harlan's dissent in Plessy v. Ferguson set down for the majority opinion later in Brown v. Board of Education.
After granting a writ of certiorari and accepting a case for review, the justices may decide against further review of the case. For example, the Court may feel the case presented during oral arguments did not present the constitutional issues in a clear-cut way, and that adjudication of these issues is better deferred until a suitable case comes before the court. In this event the writ of certiorari is "dismissed as improvidently granted" (DIG)—saying, in effect that the Court should not have accepted the case. As with the granting or denial of cert, this dismissal is customarily made using a simple per curiam decision without explanation.[23]
Customarily, justices who were not seated at the time oral arguments were heard by the Supreme Court do not participate in the formulation of an opinion. Likewise, a justice leaving the Court prior to the handing down of an opinion does not take part in the Court's opinion. Should the composition of the Court materially affect the outcome of a pending case, the justices will likely elect to reschedule the case for rehearing.
Tied votes and lack of quorum[edit]
If not all of the nine justices vote on a case, or the Court has a vacancy, then a tied vote is possible. If this occurs, then the decision of the court below is affirmed, but the case is not considered to be binding precedent. The effect is a return to the status quo ante. No opinions (or voting alignments) are issued in such a case, only the one-sentence announcement that "[t]he judgment is affirmed by an equally divided Court." Omega S.A. v. Costco Wholesale Corp. is an example of such a case. The court tries to avoid such rulings when possible: After the retirement of Justice O'Connor in 2006 three cases would have ended with a tie. All cases were reargued to allow the newly appointed Samuel Alito to cast a decisive vote.
A quorum of justices to hear and decide a case is six. If, through recusals or vacancies, fewer than six justices can participate in a case, and a majority of qualified justices determines that the case cannot be heard in the next term, then the decision of the court below is affirmed as if the Court had been equally divided on the case. An exception exists when this situation arises in one of the now-rare cases brought directly to the Supreme Court on appeal from a United States District Court; in this situation, the case is referred to the U.S. Court of Appeals for the corresponding circuit for a final decision there by either the Court of Appeals sitting en banc, or a panel consisting of the three most senior active circuit judges.[24]
Announcement of opinions[edit]
Throughout the term, but mostly during the last months of the term—May, June, and, if necessary, July—the Court announces its opinions. The decision of the Court is subsequently published, first as a slip opinion, and subsequently in the United States Reports. In recent years, opinions have been available on the Supreme Court's website and other legal websites on the morning they are announced. Since recording devices have been banned inside the courtroom, the fastest way for decisions of landmark cases to reach the press is through the Running of the Interns.[25]
The opinion of the Court is usually signed by the author; occasionally, the Supreme Court may issue an unsigned opinion per curiam. The practice of issuing a single opinion of the Court was initiated during the tenure of Chief Justice John Marshall during the early 19th century. This custom replaced the previous practice under which each Justice, whether in the majority or the minority, issued a separate opinion. The older practice is still followed by appellate courts in many common law jurisdictions outside the United States.
Reporting and citation of cases[edit]
For more specific details on how cases are cited, see case citation.
Supreme Court decisions are typically cited as in the following example: "Roe v. Wade, 410 U.S. 113 (1973)." The court citation consists of the names of the opposing parties; the volume number; "U.S." (signifying United States Reports, the official reporter of Supreme Court decisions); the page number on which the decision begins; and the year in which the case was decided. The names of the opposing parties are listed in the format "Petitioner v. Respondent" or "Appellant v. Appellee." The Reporter of Decisions is responsible for publication of the Court's rulings. Two other widely used citation formats exist: the Supreme Court Reporter and the Lawyers' Edition, corresponding to two privately published collections of decisions. Citations to cases in the Supreme Court Reporter would be structured as follows: Snowden v. Hughes, 64 S. Ct. 397 (1944). Citations to cases in the Lawyers' Edition would be as follows: Snowden v. Hughes, 88 L. Ed. 497 (1944). Judicial opinions often use the citation from all three sources (the United States Reports, Supreme Court Reporter, and Lawyers' Edition), as seen here: Martin v. Texas, 200 U.S. 316, 26 S. Ct. 338, 50 L. Ed. 497 (1906). Since the 1930s, prior to publication of the decisions in these reporters, they are available from the United States Law Week (U.S.L.W.). In more recent years, opinions have been available electronically soon after they appeared on commercial sites such as Lexis or Westlaw, on Internet sites such as FindLaw and on the Court's own website.[26]
The Reporter of Decisions is the court official responsible for the publication of the Court's opinions and orders. The current Reporter of Decisions is Christine Luchok Fallon.
Decisions of the Supreme Court are precedents that bind all lower courts, both federal and state. The Supreme Court generally respects its own precedents, but has in some cases overturned them.
See also[edit]
Supreme Court Case Selections Act
Footnotes and references[edit]
Jump up ^ Georgia v. Brailsford, Powell & Hopton , 3 U.S. 1 (1794).
Jump up ^ United States v. Louisiana , 399 U.S. 699 (1950).
Jump up ^ Scalia, Antonin. "Originalism: The Lesser Evil". University of Cincinnati Law Review. 57: 849, 850. Archived from the original on February 21, 2006; Rehnquist, William H. (September 27, 2001). Remarks by Chief Justice William H. Rehnquist (Speech). Lecture at the Faculty of Law of the University of Guanajuato, Mexico.
Jump up ^ See, e.g., Lazarus, Edward (June 14, 1998). "Closed Chambers". Booknotes (Interview). Interview with Brian Lamb. C-SPAN.
Jump up ^ "The Justices' Caseload" (PDF). Supreme Court of the United States.
Jump up ^ Bowles v. Russell , 551 U.S. 205 (2007).
Jump up ^ Mauro, Tony (October 21, 2005). "Roberts Dips Toe Into Cert Pool". Law.com. Regarding the picking of cases, see generally, Rehnquist, William H. (2001). The Supreme Court (2nd ed.). New York: Knopf. pp. 224–238. ISBN 9780375409431.
Jump up ^ Liptak, Adam (September 25, 2008). "A Second Justice Opts Out of a Longtime Custom: The 'Cert. Pool'". The New York Times. Retrieved October 17, 2008.
Jump up ^ Liptak, Adam (May 1, 2017). "Gorsuch, in sign of independence, is out of Supreme Court's clerical pool". New York Times. Retrieved May 2, 2017.
Jump up ^ Thompson, David C.; Wachtell, Melanie F. (2009). "An Empirical Analysis of Supreme Court Certiorari Petition Procedures". George Mason University Law Review. 16 (2): 237, 241. SSRN 1377522  .
Jump up ^ Rehnquist, William H. (September 27, 2001). Remarks by Chief Justice William H. Rehnquist (Speech). Lecture at the Faculty of Law of the University of Guanajuato, Mexico; cf. "Part VII. Practice and Procedure. Rule 33. Document Preparation: Booklet Format; 8 1/2- by 11-Inch Paper Format". Rules of the Supreme Court of the United States. Supreme Court of the United States. April 19, 2013 – via Legal Information Institute, Cornell University.
Jump up ^ Talbot, Margaret (March 28, 2005). "Supreme Confidence: The Jurisprudence of Justice Antonin Scalia". The New Yorker. Archived from the original on September 3, 2005 – via New America Foundation. In his early years on the Court, Scalia's exuberant questioning was not well received by his colleagues ... In Scalia's first oral argument he asked so many questions that Powell whispered, 'Do you think he knows that the rest of us are here?' The other Justices have since caught up with Scalia. Now all of them—with the exception of Clarence Thomas—are garrulous Linder, Doug. "Justices Brennan & Scalia Debate Creation-Science". On a bench lined with solemn gray figures who often sat as silently as pigeons on a railing, Scalia stood out like a talking parrot.
Jump up ^ "About the Oyez Project". Oyez Project. Retrieved January 24, 2011.
Jump up ^ Scalia, Antonin (June 19, 2009). "Justice Antonin Scalia" (PDF) (Interview). Interview with Susan Swain. C-SPAN. Retrieved November 6, 2010. [I]t is probably quite rare, although not unheard of, that oral argument will change my mind. But it is quite common that I go in with my mind not made up. I mean a lot of these cases are very close, and you go in on the knife's edge. Persuasive counsel can make the difference. There are things you can do with oral argument that cannot be done in a brief. You can convey the relative importance of your various points.
Jump up ^ Rehnquist, William H. (2001). The Supreme Court (2nd ed.). New York: Knopf. p. 253. ISBN 9780375409431.
Jump up ^ Lazarus, Edward (2005). Closed Chambers. New York: Penguin. p. 285. ISBN 9780143035275. to call our discussion of a case a conference is really something of a misnomer. It's much more a statement of the views of each of the nine justices.
Jump up ^ Rehnquist, William H. (2001). The Supreme Court (2nd ed.). New York: Knopf. pp. 254–58. ISBN 9780375409431.
Jump up ^ Lazarus, Edward (2005). Closed Chambers. New York: Penguin. p. 271. ISBN 9780143035275.
Jump up ^ See, e.g., Zimmerman, Carol (March 18, 2004). "Justice Blackmun's Papers Reveal Near Overturn of Roe". Catholic Herald. Catholic News Service. Archived from the original on April 19, 2004.
Jump up ^ Roberts, memo to Fred F. Fielding, June 4, 1985.
Jump up ^ See, e.g., Claeys, Eric (June 24, 2005). "Did O'Connor Lose the Majority?". SCOTUSblog. Archived from the original on August 22, 2006.
Jump up ^ See Tushnet, Mark V. (2005). A Court Divided: The Rehnquist Court and the Future of Constitutional Law. New York: Norton. ISBN 9780393058680.
Jump up ^ Savage, David G. (January 2002). "Adarand Didn't Add Up: Lawyers Now Wonder Which Affirmative Action Case Court Will Take". ABA Journal. American Bar Association. 88: 26. Retrieved January 5, 2013.
Jump up ^ U.S.C. Title 28, Section 1; U.S.C. Title 28, Section 2109.
Jump up ^ Kessler, Robert. "Why Aren't Cameras Allowed at the Supreme Court Again?". The Atlantic. Retrieved 24 March 2017.
Jump up ^ "Opinions". Supreme Court of the United States. Retrieved 25 October 2014.
External links[edit]
Court Rules from the U.S. Supreme Court website
2010 rules from the U.S. Supreme Court website
2013 rules from the U.S. Supreme Court website
2013 rule revisions from the U.S. Supreme Court website (with "clerk comments" that clarify what's changing for 2013 and why)
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Determining the number of clusters in a data set
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Determining the number of clusters in a data set, a quantity often labelled k as in the k-means algorithm, is a frequent problem in data clustering, and is a distinct issue from the process of actually solving the clustering problem.
For a certain class of clustering algorithms (in particular k-means, k-medoids and expectation–maximization algorithm), there is a parameter commonly referred to as k that specifies the number of clusters to detect. Other algorithms such as DBSCAN and OPTICS algorithm do not require the specification of this parameter; hierarchical clustering avoids the problem altogether.
The correct choice of k is often ambiguous, with interpretations depending on the shape and scale of the distribution of points in a data set and the desired clustering resolution of the user. In addition, increasing k without penalty will always reduce the amount of error in the resulting clustering, to the extreme case of zero error if each data point is considered its own cluster (i.e., when k equals the number of data points, n). Intuitively then, the optimal choice of k will strike a balance between maximum compression of the data using a single cluster, and maximum accuracy by assigning each data point to its own cluster. If an appropriate value of k is not apparent from prior knowledge of the properties of the data set, it must be chosen somehow. There are several categories of methods for making this decision.
Contents
1 The elbow method
2 X-means clustering
3 Information criterion approach
4 An information–theoretic approach
5 The silhouette method
6 Cross-validation
7 Finding number of clusters in text databases
8 Analyzing the kernel matrix
9 External links
10 Bibliography
The elbow method[edit]
Explained Variance. The "elbow" is indicated by the red circle. The number of clusters chosen should therefore be 4.
The elbow method looks at the percentage of variance explained as a function of the number of clusters: One should choose a number of clusters so that adding another cluster doesn't give much better modeling of the data. More precisely, if one plots the percentage of variance explained by the clusters against the number of clusters, the first clusters will add much information (explain a lot of variance), but at some point the marginal gain will drop, giving an angle in the graph. The number of clusters is chosen at this point, hence the "elbow criterion". This "elbow" cannot always be unambiguously identified.[1] Percentage of variance explained is the ratio of the between-group variance to the total variance, also known as an F-test. A slight variation of this method plots the curvature of the within group variance.[2]
The method can be traced to speculation by Robert L. Thorndike in 1953.[3]
X-means clustering[edit]
In statistics and data mining, X-means clustering is a variation of k-means clustering that refines cluster assignments by repeatedly attempting subdivision, and keeping the best resulting splits, until a criterion such as the Akaike information criterion (AIC) or Bayesian information criterion (BIC) is reached.[4]
Information criterion approach[edit]
Another set of methods for determining the number of clusters are information criteria, such as the Akaike information criterion (AIC), Bayesian information criterion (BIC), or the Deviance information criterion (DIC) — if it is possible to make a likelihood function for the clustering model. For example: The k-means model is "almost" a Gaussian mixture model and one can construct a likelihood for the Gaussian mixture model and thus also determine information criterion values.[5]
An information–theoretic approach[edit]
Rate distortion theory has been applied to choosing k called the "jump" method, which determines the number of clusters that maximizes efficiency while minimizing error by information-theoretic standards.[6] The strategy of the algorithm is to generate a distortion curve for the input data by running a standard clustering algorithm such as k-means for all values of k between 1 and n, and computing the distortion (described below) of the resulting clustering. The distortion curve is then transformed by a negative power chosen based on the dimensionality of the data. Jumps in the resulting values then signify reasonable choices for k, with the largest jump representing the best choice.
The distortion of a clustering of some input data is formally defined as follows: Let the data set be modeled as a p-dimensional random variable, X, consisting of a mixture distribution of G components with common covariance, Γ. If we let c 1 … c K {\displaystyle c_{1}\ldots c_{K}} be a set of K cluster centers, with c X {\displaystyle c_{X}} the closest center to a given sample of X, then the minimum average distortion per dimension when fitting the K centers to the data is:
d K = 1 p min c 1 … c K E [ ( X − c X ) T Γ − 1 ( X − c X ) ] {\displaystyle d_{K}={\frac {1}{p}}\min _{c_{1}\ldots c_{K}}{E[(X-c_{X})^{T}\Gamma ^{-1}(X-c_{X})]}}
This is also the average Mahalanobis distance per dimension between X and the set of cluster centers C. Because the minimization over all possible sets of cluster centers is prohibitively complex, the distortion is computed in practice by generating a set of cluster centers using a standard clustering algorithm and computing the distortion using the result. The pseudo-code for the jump method with an input set of p-dimensional data points X is:
JumpMethod(X):
    Let Y = (p/2) Init a list D, of size n+1 Let D[0] = 0 For k = 1 ... n: Cluster X with k clusters (e.g., with k-means) Let d = Distortion of the resulting clustering D[k] = d^(-Y) Define J(i) = D[i] - D[i-1] Return the k between 1 and n that maximizes J(k)

The choice of the transform power Y = ( p / 2 ) {\displaystyle Y=(p/2)} is motivated by asymptotic reasoning using results from rate distortion theory. Let the data X have a single, arbitrarily p-dimensional Gaussian distribution, and let fixed K = ⌊ α p ⌋ {\displaystyle K=\lfloor \alpha ^{p}\rfloor } , for some α greater than zero. Then the distortion of a clustering of K clusters in the limit as p goes to infinity is α − 2 {\displaystyle \alpha ^{-2}} . It can be seen that asymptotically, the distortion of a clustering to the power ( − p / 2 ) {\displaystyle (-p/2)} is proportional to α p {\displaystyle \alpha ^{p}} , which by definition is approximately the number of clusters K. In other words, for a single Gaussian distribution, increasing K beyond the true number of clusters, which should be one, causes a linear growth in distortion. This behavior is important in the general case of a mixture of multiple distribution components.
Let X be a mixture of G p-dimensional Gaussian distributions with common covariance. Then for any fixed K less than G, the distortion of a clustering as p goes to infinity is infinite. Intuitively, this means that a clustering of less than the correct number of clusters is unable to describe asymptotically high-dimensional data, causing the distortion to increase without limit. If, as described above, K is made an increasing function of p, namely, K = ⌊ α p ⌋ {\displaystyle K=\lfloor \alpha ^{p}\rfloor } , the same result as above is achieved, with the value of the distortion in the limit as p goes to infinity being equal to α − 2 {\displaystyle \alpha ^{-2}} . Correspondingly, there is the same proportional relationship between the transformed distortion and the number of clusters, K.
Putting the results above together, it can be seen that for sufficiently high values of p, the transformed distortion d K − p / 2 {\displaystyle d_{K}^{-p/2}} is approximately zero for K < G, then jumps suddenly and begins increasing linearly for K ≥ G. The jump algorithm for choosing K makes use of these behaviors to identify the most likely value for the true number of clusters.
Although the mathematical support for the method is given in terms of asymptotic results, the algorithm has been empirically verified to work well in a variety of data sets with reasonable dimensionality. In addition to the localized jump method described above, there exists a second algorithm for choosing K using the same transformed distortion values known as the broken line method. The broken line method identifies the jump point in the graph of the transformed distortion by doing a simple least squares error line fit of two line segments, which in theory will fall along the x-axis for K < G, and along the linearly increasing phase of the transformed distortion plot for K ≥ G. The broken line method is more robust than the jump method in that its decision is global rather than local, but it also relies on the assumption of Gaussian mixture components, whereas the jump method is fully non-parametric and has been shown to be viable for general mixture distributions.
The silhouette method[edit]
The average silhouette of the data is another useful criterion for assessing the natural number of clusters. The silhouette of a data instance is a measure of how closely it is matched to data within its cluster and how loosely it is matched to data of the neighbouring cluster, i.e. the cluster whose average distance from the datum is lowest.[7] A silhouette close to 1 implies the datum is in an appropriate cluster, while a silhouette close to −1 implies the datum is in the wrong cluster. Optimization techniques such as genetic algorithms are useful in determining the number of clusters that gives rise to the largest silhouette.[8] It is also possible to re-scale the data in such a way that the silhouette is more likely to be maximised at the correct number of clusters.[9]
Cross-validation[edit]
One can also use the process of cross-validation to analyze the number of clusters. In this process, the data is partitioned into v parts. Each of the parts is then set aside at turn as a test set, a clustering model computed on the other v − 1 training sets, and the value of the objective function (for example, the sum of the squared distances to the centroids for k-means) calculated for the test set. These v values are calculated and averaged for each alternative number of clusters, and the cluster number selected such that further increase in number of clusters leads to only a small reduction in the objective function. [10]
Finding number of clusters in text databases[edit]
In text databases, a document collection defined by a document by term D matrix (of size m by n, m: number of documents, n: number of terms) number of clusters can roughly be estimated by the following formula m n t {\displaystyle {\tfrac {mn}{t}}} where t is the number of non-zero entries in D. Note that in D each row and each column must contain at least one non-zero element. [11]
Analyzing the kernel matrix[edit]
Kernel matrix defines the proximity of the input information. For example, in Gaussian Radial basis function, determines the dot product of the inputs in a higher-dimensional space, called feature space. It is believed that the data become more linearly separable in the feature space, and hence, linear algorithms can be applied on the data with a higher success.
The kernel matrix can thus be analyzed in order to find the optimal number of clusters .[12] The method proceeds by the eigenvalue decomposition of the kernel matrix. It will then analyze the eigenvalues and eigenvectors to obtain a measure of the compactness of the input distribution. Finally, a plot will be drawn, where the elbow of that plot indicates the optimal number of clusters in the data set. Unlike previous methods, this technique does not need to perform any clustering a-priori. It directly finds the number of clusters from the data.
External links[edit]
Clustergram – cluster diagnostic plot – for visual diagnostics of choosing the number of (k) clusters (R code)
Eight methods for determining an optimal k value for k-means analysis – Answer on stackoverflow containing R code for several methods of computing an optimal value of k for k-means cluster analysis
Bibliography[edit]
Jump up ^ See, e.g., David J. Ketchen Jr; Christopher L. Shook (1996). "The application of cluster analysis in Strategic Management Research: An analysis and critique". Strategic Management Journal. 17 (6): 441–458. doi:10.1002/(SICI)1097-0266(199606)17:6<441::AID-SMJ819>3.0.CO;2-G.
Jump up ^ See, e.g., Figure 6 in
Cyril Goutte, Peter Toft, Egill Rostrup, Finn Årup Nielsen, Lars Kai Hansen (March 1999). "On Clustering fMRI Time Series". NeuroImage. 9 (3): 298–310. doi:10.1006/nimg.1998.0391. PMID 10075900. CS1 maint: Multiple names: authors list (link)
Jump up ^ Robert L. Thorndike (December 1953). "Who Belongs in the Family?". Psychometrika. 18 (4): 267–276. doi:10.1007/BF02289263.
Jump up ^ D. Pelleg; AW Moore. X-means: Extending K-means with Efficient Estimation of the Number of Clusters (PDF). Proceedings of the Seventeenth International Conference on Machine Learning (ICML 2000). Retrieved 2016-08-16.
Jump up ^ Cyril Goutte, Lars Kai Hansen, Matthew G. Liptrot & Egill Rostrup (2001). "Feature-Space Clustering for fMRI Meta-Analysis". Human Brain Mapping. 13 (3): 165–183. doi:10.1002/hbm.1031. PMID 11376501. CS1 maint: Multiple names: authors list (link) see especially Figure 14 and appendix.
Jump up ^ Catherine A. Sugar; Gareth M. James (2003). "Finding the number of clusters in a data set: An information-theoretic approach". Journal of the American Statistical Association. 98 (January): 750–763. doi:10.1198/016214503000000666.
Jump up ^ Peter J. Rousseuw (1987). "Silhouettes: a Graphical Aid to the Interpretation and Validation of Cluster Analysis". Computational and Applied Mathematics. 20: 53–65. doi:10.1016/0377-0427(87)90125-7.
Jump up ^ R. Lleti; M.C. Ortiz; L.A. Sarabia; M.S. Sánchez (2004). "Selecting Variables for k-Means Cluster Analysis by Using a Genetic Algorithm that Optimises the Silhouettes". Analytica Chimica Acta. 515: 87–100. doi:10.1016/j.aca.2003.12.020.
Jump up ^ R.C. de Amorim & C. Hennig (2015). "Recovering the number of clusters in data sets with noise features using feature rescaling factors". Information Sciences. 324: 126–145. arXiv:1602.06989  . doi:10.1016/j.ins.2015.06.039.
Jump up ^ See e.g. "Finding the Right Number of Clusters in k-Means and EM Clustering: v-Fold Cross-Validation". Electronic Statistics Textbook. StatSoft. 2010. Retrieved 2010-05-03.
Jump up ^ Can, F.; Ozkarahan, E. A. (1990). "Concepts and effectiveness of the cover-coefficient-based clustering methodology for text databases". ACM Transactions on Database Systems. 15 (4): 483. doi:10.1145/99935.99938. especially see Section 2.7.
Jump up ^ Honarkhah, M; Caers, J (2010). "Stochastic Simulation of Patterns Using Distance-Based Pattern Modeling". Mathematical Geosciences. 42 (5): 487–517. doi:10.1007/s11004-010-9276-7.
Ralf Wagner, Sören W. Scholz, Reinhold Decker (2005): The Number of Clusters in Market Segmentation, in: Daniel Baier, Reinhold Decker; Lars Schmidt-Thieme (Eds.): Data Analysis and Decision Support, Berlin, Springer, 157–176.
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