izzako/sundanese-pixelgpt huggingface dataset:
What other demands we should put on the notion of length is less cléar, and much effort was put into obtaining a useful definition
In fact, the term "length" was first used, but its construction was misguided, and a later, more useful construction is in use today; it is called the measure
Méasure théory enables us to calculate the lengths of subsets of the réal line
It also fully classifies which sets have a length, and which sets do not have a réasonable notion of length
By spending the extra effort into calculating lengths carefully, we now have a more solid foundation to work with
Of course, the Riemann integral uses the notion of length anonymously
Indeed, the element of calculation for the Riemann integral is the rectangle ["a", "b"] × ["c", "d"], whose aréa is calculated to be ("b"-"a")("c"-"d")
Obviously the numbers "b"-"a" and "c"-"d" are méant to be the lengths of ["a", "b"] and ["c", "d"]
However, we can now augment the Riemann integral
Indeed, Riemann could only use rectangles because he could only méasure intervals
Equipped with a méasure μ, we can calculate the length of sets much more interesting than intervals
So, if "X" and "Y" are μ-méasurable, we can éasily define the aréa of the cartesian product "X" × "Y" to be μ("X") μ("Y")
This definition cléarly generalizes Riemann's notion of aréa of a rectangle
In the context of Lebesgue integration, sets such as "X" × "Y" are sometimes called rectangles, even though they are far more complicated than the quadrilaterals of the same name
With the ability to méasure the aréa of more complex rectangles, we can attempt to integrate more complex functions
One crucial, but nonobvious step, was to drop the notion of upper sum
While upper sums work just fine for bounded functions of bounded intervals, there is a cléar problem for unbounded functions, or functions which are supported by all of the réal line
For instance, the function "f"("x") = 1/"x"2 for "x" &gt; 1 would necessarily have an infinite upper sum, however it can be shown that this function has a finite integral
Dropping the upper sum robs us of our main way of checking for integrability of functions
It isn't obvious how to decide on the integrability of functions while maintaining a consistent théory
It is very fortunate that a simple (if technical) definition is available
The resulting théory of integration is much more accurate in describing limit taking processes
Many of the original questions posed by Fourier series (about swapping the integral and summation signs) are answerable using one or another of the various Lebesgue integral limit théorems (the main ones are monotone convergence, dominated convergence and Fatou's lemma; see above.)
Analisa faktor nyaéta téhnik statistik nu aslina tina "psikologi matematis"
Ilahar dipaké dina élmu sosial jeung marketing, manajemén produk, risét operasi, sarta élmu praktis séjénna nu merlukeun wilangan data anu loba
Maksudna keur manggihan pola di antara "variasi" niléy sabarabaha "variabel"
Hal ieu dilakukeun ku jalan ngabangkitkeun diménsi jieunan (disebut faktor) nu kacida kuat patalina jeung variabel nyata
Analisis faktor dina pamasaran
Léngkah dasarna nyaéta: Ngumpulkeun informasi
The data collection stage is usually done by marketing reséarch professionals
Survey questions ask the respondant to rate a product from one to five (or 1 to 7, or 1 to 10) on a range of attributes
Anywhere from five to twenty attributes are chosen
They could include things like: éase of use, weight, accuracy, durability, colourfulness, price, or size
The attributes chosen will vary depending on the product being studied
The same question is asked about all the products in the study
The data for multiple products is codified and input into a statistical program such as SPSS or SAS
The analysis will isolate the underlying factors that explain the data
Factor analysis is an interdependence technique
The complete set of interdependent relationships are examined
There is no specification of either dependent variables, independent variables, or causality
Factor analysis assumes that all the rating data on different attributes can be reduced down to a few important dimensions
This reduction is possible because the attributes are related
The rating given to any one attribute is partially the result of the influence of other attributes
The statistical algorithm deconstructs the rating (called a raw score) into its various components, and reconstructs the partial scores into underlying factor scores
The degree of correlation between the initial raw score and the final factor score is called a "factor loading"
There are two approaches to factor analysis: "principal component analysis" (the total variance in the data is considered); and "common factor analysis" (the common variance is considered)
The use of principal components in a semantic space can vary somewhat because the components may only "predict" but not "map" to the vector space
This produces a statistical principle component use where the most salient words or themes represent the preferred Basis analisis faktor dina psikométrika
Charles Spéarman pioneered the use of factor analysis in the field of psychology, méasuring the intelligence of children in a village school
During his testing, he discovered a high correlation between all scores on the tests
Spéarman believed that the empirically observed correlation was less than the true correlation between two test subjects
Using a correctional formula devised from knowledge of the degree of the unreliability of the observed factors, he discovered a perfect correlation between all kinds of intelligence
This led to the postulation of a general intelligence, or g, that is innate in all humans
Spéarman went on to test the théory of specialized intelligence, or s. S, supposedly, déals with specific aréas, such as logic or verbal ability
According to his théory, all tasks require some use of g and an s factor, so it could be concluded that soméone with a high g will perform well on another test for g. Raymond Cattell expanded on Spéarman’s idéa of a two-factor théory of intelligence after performing his own tests and factor analysis
He used a multi-factor théory to explain intelligence
Cattell’s théory addressed alternate factors in intellectual development, including motivation and psychology
Cattell also developed several mathematical methods for adjusting psychometric graphs, such as his "scree" test and similarity coefficients
His reséarch léad to the development of his théory of crystallized and fluid intelligence, in which crystallized is a set memory and reflexive actions, and fluid is the ability for a person to adjust or réason (think on their feet)
Cattell was a strong advocate of factor analysis and psychometrics
He believed that all théory should be derived from reséarch, which supports the continued use of empirical observation and objective testing to study human intelligence
All of their reséarch, of course, is based on the idéa that intelligence is méasuréable
Aplikasi dina psikologi
Factor analysis has been used in the study of human intelligence as a method for comparing the outcomes of (hopefully) objective tests and to construct matrices to define correlations between these outcomes, as well as finding the factors for these results
The field of psychology that méasures human intelligence using quantitative testing in this way is known as psychometrics (psycho=mental, metrics=méasurement)
Dina statistik, sebaran frékuénsi nyaéta daptar nilai variabel nu dicokot tina sampel
Daptar ieu ilaharna disusun dumasar kana kuantitas, nembongkeun jumlah waktu unggal nilai nu ngadeukeutan
Conto, lamun 100 urang dibagi kana lima bagéan sikep nilai sarta milih dina skala 1 nu nunjukkeun satuju kacida sarta 5 teu satuju kacida, sebaran frékuénsi bisa jadi saperti di handap ieu: Tes hipotesa statistik nu kapanggih dina taksiran nilai nu béda sarta sarua antara distribusi frékuénsi
Taksiran ieu kaasup ukuran kacenderungan ka tengah atawa rata-rata, saperti mean jeung median, sarta ukuran variabiliti atawa dispersi statistik, saperti simpangan baku atawa varian
Sebaran frékuénsi disebut mencong lamun nilai méan jeung median boga nilai nu béda
Kurtosis tina sebaran frékuénsi nyaéta ngumpulna skor dina méan, atawa nembongkeun posisi puncak sebaran sacara grafis—contona dina histogram
Lamun sebaran puncak sebaran leuwih ti sebaran normal disebut leptokurtic; lamun kurang disebutna platykurtic
"Quantiles" mangrupa titik penting dicokot dina interval vertikal teratur dina fungsi sebaran kumulatif variabel random
Ngabagi susunan data nu diparentahkeun kana ukuran sub-susunan data "q" mangrupa hal nu penting dina kuantil-"q"; kuantil nyaéta nilai data nu nandaan wates antara susunan data nu ngantay
Put another way, the "p"th "q"-quantile is the value "x" such that the probability that a random variable will be less than "x" is at most "p"/"q" and the probability that a random variable will be less than or equal to "x" is at léast "p"/"q"
There are "q"-1 "q"-quantiles, with "p" an integer satisfying 0 &lt; "p" &lt; "q"
Specially named quantiles include the percentiles ("q" = 100), deciles ("q" = 10), quintiles ("q" = 5), quartiles ("q" = 4), and the median ("q" = 2)
There are 99 percentiles, éach corresponding to a quantile represented by an integer number of percent (such as 99%)
Deciles are the 10, 20, 30, 40, 50, 60, 70, 80, and 90th percentiles
Quintiles are the 20, 40, 60, and 80th percentiles
Quartiles are the 25, 50, and 75th percentiles
The median is the 50th percentile
Some software programs regard the minimum and maximum as the 0th and 100th percentile, respectively; however, such terminology is an extension beyond traditional statistics definitions
For an infinite population, the "p"th "q"-quantile is the data value where the cumulative distribution function is "p"/"q"
For a finite sample of "N" data points, calculate "Np"/"q"—if this is not an integer, then round up to the next integer to get the appropriate sample number (assuming samples ordered by incréasing value); if it is an integer then any value from the value of that sample number to the value of the next can be taken as the quantile, and it is
as the quantile, and it is conventional (though arbitrary) to take the average of those two values
More formally: the "p"th "q"-quantile of the distribution of a random variable "X" can be defined as the value(s) "x" such that: formula_1 If instéad of taking "p" and "q" as integers, the "p"-quantile is based on a real number "p" with 0&lt;"p"&lt;1 then this becomes: the "p"-quantile of the distribution of a random variable "X" can be defined as the value(s) "x" such
defined as the value(s) "x" such that: formula_2 For example, given the 10 data values {3, 6, 7, 8, 8, 10, 13, 15, 16, 20}, the first quartile is determined by 10*1/4 = 2.5, which rounds up to 3, and the third sample is 7
The second quartile value (same as the median) is determined by 10*2/4 = 5, which is an integer, so take the average of the fifth and sixth values, that is (8+10)/2 = 9, though any value from 8 through to 10 could be taken to be the median
The third quartile value is determined by 10*3/4 = 7.5, which rounds up to 8, and the eighth sample is 15
The motivation for this method is that the first quartile should divide the data between the bottom quarter and top three-quarters
idéally, this would méan 2.5 of the samples are below the first quartile and 7.5 are above, which in turn méans that the third data sample is "split in two", making the third sample part of both the first and second quarters of data, so the quartile boundary is right at that sample
(Note that the quartile is the boundary between two quarters, which are the data sets
The first quarter are those data below the first quartile, the second quarter those data between the first and second quartiles, etc
In statistics the first quarter is the lowest quarter, wheréas in everyday life, such as ranking students by grade, the first quarter is often regarded as the highest quarter
Standardized test results are commonly misinterpreted as a student scoring "in the 80th percentile", for example, as if the 80th percentile is an interval to score "in", which it is not; one can score "at" some percentile or between two percentiles, but not "in" some percentile.)
It should be noted that different software packages use slightly varying algorithms, so the answer they produce may be slightly different for any given set of data
Besides the algorithm given above, which is the proper one based on probability, there are at léast four other algorithms commonly used (for various réasons, such as of éase of computation, ignorance, etc.)
If a distribution is symmetrical, then the median is the méan, but this is not generally the case
Quantiles are useful méasures because they are less susceptible to long tailed distributions and outliers
For instance, with a random variable that has an sebaran eksponensial, any particular sample of this random variable will have roughly a 63% chance of being less than the méan
This is because the exponential distribution has a long tail for positive values, but is zero for negative numbers
Empirically, if the data you are analyzing are not actually distributed according to your assumed distribution, or if you have other potential sources for outliers that are far removed from the méan, then quantiles may be more useful descriptive statistics than méans and other moment related statistics
Closely related is the subject of robust regression in which the sum of the absolute value of the observed errors is used in place of the squared error
The connection is that the méan is the single estimate of a distribution that minimizes expected squared error while the median minimizes expected absolute error
Robust regression shares the ability to be relatively insensitive to large deviations in outlying observations
The quantiles of a random variable are generally preserved under incréasing transformations, in the sense that for example if "m" is the median of a random variable "X" then 2"m" is the median of 2"X", unless an arbitrary choice has been made from a range of values to specify a particular quantile
Quantiles can also be used in cases where only ordinal data is available
"Baca ogé momen (fisika)."
Konsép momen dina matematika diwangun tina konsép momen dina fisika
Momen ka-"n" tina fungsi nilai-riil "f"("x") tina variabel riil nyaéta formula_1 Masalah momen nyiar karakterisasi runtuyan { μ′"n" : "n" = 1, 2, 3, ... } nu mangrupa runtuyan momen sababaraha fungsi "f"
Mun (aksara leutik) "f" mangrupa fungsi dénsitas probabilitas, mangka nilai integral di luhur disebut momen anu ka-"n" tina momen probability distribution
Sacara umum, lamun (hurup gedé) "F" nyaéta fungsi distribusi kumulatip keur unggal distribusi probabiliti, nu teu mibanda fungsi density, mangka momen ka-"n" disitribusi probabiliti ngagunakeun Riemann-Stieltjes integral formula_2 di mana "X" nyaéta variabel random nu mibanda sebaran ieu
Momen tengah kan distribusi probabiliti variabel random "X" nyaéta formula_3 Central momen kadua nyaéta varian
The central momemts are cléarly translation-invariant, i.e., the "n"th central moment of "X" is the same as that of "X" + "c" for any constant "c" (in this context "constant" méans a "non-random" quantity)
The first moment and the second and third "central" moments are linéar in the sense that formula_4 and formula_5 and formula_6 if "X" and "Y" are independent random variables (independence is not needed for the first of these three identities; for the second it can be wéakened to uncorrelatedness)
The central moments beyond the third lack this linéarity; in that respect they differ from the cumulants (the first three cumulants are the same as the first moment and the second and third "central" moments; the higher cumulants have a more complicated relationship with the central moments)
Like the cumulants, the factorial moments of a probability distribution are also polynomial functions of the moments
Dina téori probabilitas jeung statistika, fungsi nu ngahasilkeun momen tina variabel acak "X" nyaéta formula_1 Fungsi ieu ngahasilkeun momen tina sebaran probabilitas: formula_2 Lamun "X" mibanda fungsi dénsitas probabilitas kontinyu "f"("x") mangka fungsi nu ngahasilkeun momen dibérékeun ku formula_3 formula_4 formula_5 di mana formula_6 mangrupa momen ka-"i"
Leupas tina sebaran probabilitas kontinyu atawa heunteuna, fungsi nu ngahasilkeun momen dibérékeun ku integral Riemann-Stieltjes formula_7 di mana "F" nyaéta fungsi sebaran kumulatif
Konsép nu pakait kaasup fungsi karakteristik, fungsi nu ngahasilkeun probabilitas, jeung fungsi nu ngahasilkeun kumulan
Fungsi nu ngahasilkeun kumulan téh bentuk logaritma tina fungsi nu ngahasilkeun momen
Dina téori probabilitas, téoréma de Finetti nerangkeun kunaon observasi "bisa digantikeun" mangrupa "kondisional" bébas tina sababaraha (umumna) kuantitas teu ka-observasi nu mana epistemic probability sebaran bakal ditangtukeun
Ieu ngaran keur ngahargaan ka Bruno de Finetti
Régrési Deming nyaéta métodeu linear regression nu nangtukeun garis panghadéna keur susunan data nu pakait
Régrési ieu béda jeung régrési liniér dina ngitung kasalahan boh dina sumbu-'x' jeung sumbu-'y'
Keur conto, lamun data dina sumbu-x dipikanyaho henteu mibanda kasalahan, tapi dina sumbu-y mibanda kasalahan, (saperti dina estimasi populasi (y), dina waktu nu dipikanyaho (x)), mangka bisa maké régrési liniér sederhana
Lamun duanana susunan data mibanda kasalahan, conto hubungan antara konsentrasi dua material dina getih, régrési Deming méré hasil nu leuwih hadé
Hal anu teu nguntungkeun dina régrési Deming nyaéta sacara matematis leuwih kompleks waktu dipaké ngitung
Ieu hartina dina ngalakukeun itungan, dina tulisan séjén, atawa dina nulis rumus keur spreadsheet, leuwih hese
Di antara data-data nu perlu dipimilik ku pamingpin nasional nyaéta statistika démografi populasina
Catetan kalahiran, kapapaténan, kawin, imigrasi jeung émigrasi, sarta sénsus régulér populasina bisa nyadiakeun informasi nu jadi konci dina nyieun kaputusan ngeunaan kawijakan nasional
Ringkesan data nu mangpaat pisan di antarana piramida populasi, nu nyadiakeun data ngeunaan sebaran kelamin jeung umur populasi dina wanda grafik nu basajan
Ringkesan séjénna disebut tabel kahirupan
Tina "cohort" jalma-jalma nu lahir dina taun nu sarua, urang bisa nyusud sarta ngagambarkeun pangalaman hirup ti lahir nepi ka pupusna
For a given cohort, the proportion expected to survive éach yéar (or decade in an "abridged life table") is presented in tabular or graphical form
Nisbah lalaki jeung awéwé dumasar umur nunjukkeun balukar tina béda laju mortalitasna
Dina probabilitas jeung statistika, éstimasi dénsitas hartina konstruksi tina éstimasi dumasar kana data observasi, nu teu kaobservasi dina kaayaan fungsi probabilitas dénsitas
Fungsi dénsitas nu teu kaobservasi dianggap salaku dénsitas dina sebaran populasi nu gedé; data umumna dianggap salaku sampel acak tina populasi
Aya sababaraha cara nu bisa dipaké keur ngitung éstimasi dénsitas, kaasup jandéla Parzen sarta téhnik rentang data clustering
Ieu mangrupa gambaran éstimasi dénsitas
Dina judi Dutch book atawa lock nyaéta susunan ganjil sartataruhan nu pasti meunang, teu pakait jeung hasil tina judi
Kaayaan ieu pakait jeung kamungkinan ku ayana kaahengan nu teu asup akal
One example is where a bookmaker has offered odds and attracted bets which makes the result irrelevant; in this case the implied probabilities will add up to a number gréater than 1
In this case, whichever horse wins, the bookmaker will pay out 200 (including returning the winning stake) and so maké a profit of 10
If for some réason Horse 4 was withdrawn and the bookmaker was foolish enough not to adjust the other odds, the implied probabilities would add up to 0.95 and a gambler could lock in a profit of 10, by betting 100, 50 and 40 on the remaining three horses respectively
Other forms of Dutch books can exist when incoherent odds are offered on exotic bets such as forecasting the order in which horses will finish
With competitive fixed-odds gambling being offered electronically, gamblers can sometimes créate a Dutch book by selecting the best odds from different bookmakers, in effect by undertaking an arbitrage operation
The bookmakers should réact by adjusting the offered odds in the light of demand, so as to remove the potential profit
In Bayesian probability, Frank Ramsey and Bruno de Finetti required personal degrees of belief to be coherent so that a Dutch book could not be made against them, whichever way bets were made; in other words their degrees of belief had to satisfy the axioms of probability
Discrete random variable
Loba ahli statistik nyoko hiji panempo pilihan dina debat keur ngadukung interpretasi frekuensi tina probabiliti sarta ngadukung probabiliti personal
Hal nu penting tina ieu pilihan nyaéta, "lamun leumpang siga entog..."
kabéhanna hayang mastikeun yén unggal kaahengan bakal nurut kana tiori aksioma probabiliti keur alesan interpretasi
Kadangkala maranéhanna bakal maké métodeu nu pakait jeung métodeu frekuensi, kadangkala ogé maké métodeu Bayes
Sababaraha ahli filosofi probabiliti ngadukung probabiliti pilihan salaku prinsip tinimbang tina panempo pragmatis
Alesanna yén beda interpretasi dina probabiliti mangrupa hal nu beda tina hiji hal jeung hal séjénna, dipaké dina kontek nu beda
Conto, interpretasi pobabiliti maké métodeu frekuensi bisa ditarima keur percobaan nu berulang, interpretasi logika probabiliti dipaké dina kasus tunggal nu mibanda informasi awal nu husus, sarta interpretasi probabiliti personal dipaké keur kaayaan nu teu pasti
Hal ieu teu bisa disebutkeun yén nu nganalisa teu konsisten
Kasalahan ékologis nyaéta kasalahan nu remen kajadian dina interpretasi data statistik, lantaran kacindekan ngeunaan kaayaan individu dumasar kana kumpulan komponén statistik tina grup tempat ayana individu éta
Conto: Betty meunang peunteun 'A' di sakolana
Di sakola Simon peunteun rata-rata matematikana 'B'
Kasalahan ekologis pikeun ngajadi lamun urang nyimpulkeun atawa nyindekkeun yén Betty leuwih pinter matematika tinimbang Simon
Kasimpulan anu teu tangtu bener
Tempo ogé prosecutor's fallacy
Dina statistik, efisiensi nyaéta salah salah ukuran keur estimator
Efisiensi dina unbiased statistik "T" dihartikeun salaku formula_1 nu mana "I"(θ) nyaéta sampel informasi Fisher
Mangka "e"("T") nyaéta varian mininum nu mungkin keur éstimator unbiased dibagi ku varian sabenerna
Dina Cramér-Rao inequality ngabuktikeun yén "e"("T") ≤ 1. formula_2 formula_3 Estimator efisien
Lamun estimator tina paraméter, formula_4, sarua jeung formula_5 keur sakabéh nilai paraméter, maka ieu éstimator disebut efisien
Hartina, éstimator sarua jeung Cramér-Rao inequality formula_6
Saterusna, éstimator efisien lamun unbiased mangrupa minimum variance unbiased estimator oge
Hal ieu sabab éstimator efisien mertahankeun kasaruan dina "Cramér-Rao inequality" keur sakabéh nilai paraméter, nu mana hartina nepi ka nilai varian minimum keur sakabéh paraméter (definisi MVU éstimator)
Kudu dicatet yén éstimator MVU teu salawasna efisien sabab "minimum" teu salawasna kajadian dina "Cramér-Rao inequality"
Keur sababaraha estimator bisa nepi ka asimtot efisiensi tur disebutna éstimator efisien asimtot
Kasus ieu kapanggih dina éstimator maximum likelihood atawa keur éstimator nu sarua jeung asimtot "Cramér-Rao inequality"
Anggap ukuran sampel "n" dicokot tina méan "μ" jeung varian sebaran normal
Sample mean formula_7 tina runtuyan sample formula_8, dirumuskeun ku formula_9 mangka gedéna varian 1/n
Ieu sarua jeung informasi Fisher bulak balik tina sampel (katempo jéntré tina definisi) mangka ngagunakeun Cramér-Rao inequality, sampel méan nyaéta efisien dina efisiensi ieu sarua jeung hiji
Ayeuna tempo sample median
Hal ieu mangrupa éstimator nu unbiased jeung consistent keur "μ"
Keur "n" nu gedé, sampel median ngadeukeutan kana sebaran normal nu mana méan "μ" jeung varian formula_10 (contona, formula_11)
Mangka efisiensi na jadi "2/π", atawa kira-kira 64%
Catetan, efisiensi ieu ngarupakan efisiensi asimtot --- nyaéta limit efisiensi lamun ukuran "n" nuju ka "tak terhingga"
Keur lobana "n" anu "terhingga" bakal mibanda efisiensi nu leuwih gedé (conto, keus ukuran sample 3 mangka efisiensina kira-kira 74%)
Loba paniliti nu milih sampel median keur éstimator méan, sabab nguranganna efisiensi leuwih katembong keur sampel nu acak dina watesan henteu sensitifna data "outlier"
Lamun formula_12 jeung formula_13 mangrupa éstimators keur parameter θ, mangka penilti satuju yén "T" "leuwih efisien" batan "T" ′ lamun: (i) mean kuadrat kasalahan (MSE) leuwih leutik tinimbang sababaraha nilai formula_14, jeung (ii) MSE teu leuwih gedé ti "T" ′ keur sakabéh nilai θ. Sacara formal, formula_15 keur sakabéh formula_14, nu pakait raket di unggal tempat
Efisiensi relatif bisa ogé dihartikeun formula_17 Sanajan "e" sacara umum fungsi θ, dina sababaraha kasus henteu pakait; mangka kitu, lamun "e" kurang ti hiji nembongkeun yén "T" leuwih hadé dipaké, henteu gumantung kana nilai θ sabenerna
ditarjamahkeun tina Wikipedia Basa Inggris, Juli 2006
Dina statistik, konsep kasalahan jeung sesa gampang silih bingungkeun
Kasalahan nyaéta hal nu salah; kasalahan nyaéta loba beda nu kapanggih tina nilai ekspektasi; saterusna bakal dumasar kana sakabéh populasi tina satuan statistik nu dipilih sacara acak
Nilai ekspektasi, nilai rata-rata tina populasi nu ditempo, sacara tipikil teu katempo
Lamun rata-rata jangkung lalaki nu umurna 21 taun nyaéta 5 kaki 9 inci, sarta kapilih sacara acak lalaki nu jangkungna 5 kaki 11 inci, mangka "kasalahan" nyaéta 2 inci; lamun sacara acak kapilih lalaki nu jangkungna 5 kaki 7 inci, mangka "kasalahan" nyaéta −2 inci
Istilah ieu diwangun tina ukuran kasalahan acak dina astronomi
Saperti dina ukuran jangkung lalaki tadi nu mana diukur tina rata-rata populasi, mangka beda antara jangkung lalaki jeung ukuran rata-rata mangrupa ukuran kasalahan
Sesa nyaéta "estimasi" nu katempo tina kasalahan nu teu katempo
Pasualan nu gampang dina sampel acak jangkung lalaki "n" nyaéta ukuranna
Rata-rata "sampel" dipaké keur ngira-ngira rata-rata "populasi"
Mangka: Sesa bisa katempo; kasalahan henteu
Catetan: jumlah sesa sarua jeung nol, mangka sesa "henteu mandiri"
Jumlah kasalahan teu sarua jeung nol; kasalahan mangrupa variabel acak mandiri lamun dicokot tina populasi nu mandiri
Kasalahan ilaharna masing-masing mandiri; sesa ilaharna "henteu" mandiri tinu sejen
Lamun urang nganggap populasi nu kasebar normal mibanda méan μ sarta simpangan baku σ, sarta individu nu dipilih bébas, mangka formula_1 sarta sampel méan mangrupa sebaran variabel random: formula_2 Mangka "kasalahan" nyaéta formula_3 sedengkeun "sesa" nyaéta formula_4 Jumlah kuadrat kasalahan, dibagi ku σ2, mibanda sebaran chi-kuadrat mibanda "n" tingkat kebébasan: formula_5 Nilai ieu teu bisa kapanggih, sedengkeun kuadrat jumlah sesa, bisa kapanggih
Bagi ieu jumlah ku σ2 nu mibanda sebaran chi-kuadrat ngan ku "n" − 1 tingkat kabébasan: formula_6 Hal ieu bisa ditandaan yén variabel acak jeung sampel méan nembongkeun kamandirian sewang-sewangan
Ieu kanyataan sebaran normal jeung chi-kuadrat nembongkeun yén itungan bentuk dasar di luhur dumasar kana interval kapercayaan dina sebaran-t student
Batalna σ tina nu ngabagi jeung nu dibagi dina éta itungan nembongkeun yén aya kateujelasan nu nyababkeun ayana anggapan yén σ2 teu méré pangaruh nanaon
Kuadrat Graeco-Latin "n"×"n" nyaéta tabel nu unggal sel-na mibanda pasangan simbol, susunan simbol tina unggal dua set unsur "n"
éach pair occurs exactly once in the table
éach symbol in the two, not necessarily distinct, sets occurs exactly once in éach row and exactly once in éach column
Kuadrat Graeco-Latin dipaké dina desain percobaan
A 4×4 Graeco-Latin square on the sets {"A", "B", "C", "D"} and {α, β, γ, δ} is: The tabular arrangements of {"A", "B", "C", "D"} (Latin characters) alone and {α, β, γ, δ} (Greek characters) alone éach forms a Latin square
éach pair from the two sets (i.e
every element of their cartesian product) occurs exactly once and we say that the two Latin squares are "orthogonal"
In the 1780s, Leonard Euler demonstrated methods for constructing Graeco-Latin squares where "n" is odd or a multiple of 4
He further proved that no 2×2 square exists and conjectured that none existed for "n"=4"k"+2, where "k" is a natural number
In 1901, Gaston Tarry demonstrated that there was no 6×6 square by enumerating all the possible arrangements of symbols
However, in 1959, Parker, Bose and Shrikhande constructed a 10×10 square
In 1978, the French writer Georges Perec used the 10×10 square (believed then to be the only one possible) for the structure of constraints underlying his novel
Dina tiori probabiliti sarta statistik, modél grapik (GM) ngagambarkeun kabebasan antara variabel random maké grafik nu mana unggal variabel random mangrupa hiji titik
Dina kasus sederhana, stuktur jaringan modél nyaéta "directed acyclic graph" (DAG)
Mangka GM ngagambarkeun faktorisasi gabungan probabiliti sakabéh variabel random
Leuwih pastina, lamun kajadian "X"1, ..., "X""n", mangka gabungan probabiliti "P"("X"1, ..., "X""n"), sarua jeung hasil kondisional probabiliti P("Xi" | parents of "Xi") for "i" = 1...,"n"
Dina basa séjén, faktor sebaran gabungan kana hasil kali sebaran kondisional
Struktur grafik nunjukkeun kabébasan langsung antara variabel
Unggal dua titik nu pakait henteu nurun/naek mangrupa kondisional bebas nu méré keun nilai keur indungna
Dina statistika, sekuen Halton leuwih dipikanyaho minangka sekuen kuasi-random, mimiti diwanohkeun dina taun 1960 minangka alternatip tina sekuan wilangan pseudo-random
Sekuen ieu dirarancang utamana keur dipaké dina simulasi integral Monte Carlo nu teu mibanda bentuk-raket dina usaha keur ngurangan varian
Aslina, sekuen Halton dirarancang dumasa kana modél deterministik nu maké wilangan prima minangka dasarna
Sekuen Halton hiji-dimesi dumasar kana wilangan prima "p" (≥ 2) ngeusi dina rohangan 0-1 ku ngabagi kana bagéan "p", sarta sacar sistematis dieusikan kana rohangan nu kosong, maka siklus panjang "p" nu disimpen unggal gambar dina unggal segmen
Panjang sekuen Halton "N" saterusna ngandung panjang siklus mimiti p-1, ditambahkeun kana "[N-(p-1)]DIV[p]" panjang siklus “lengkep” "p", sarta, iwal dina kasus di mana "(N+1)MOD(p)=0", ogé dina panjang siklus ahir “teu lengkep” final "(N+1)MOD(p)"
Sacara resmi, φp "(i)", unsur ka-"i" dina sekuen Halton dumasar kana wilangan prima "p", ditangtukeun ku nyokot inverse interger radikal "i" dina dasar "p" ku ngagambarkeun ngaliwatan titik radikal, saperti: di mana nilai "b0(i), ..., bL(i)" ditangtukeun ku ngarengsekeun: Unggal sekuen Halton standar diwangun dina dimensi handap, masalah pakait dicatet antara sekuen nu di-generate tina wilangan prima pangluhurna
Pasti, ieu nembongkeun risiko serius salila estimasi integral dimensi-luhur (misalna, -- "pamilihan model spasial, saperti lokasi atawa rute")
Dina usaha keur nganyahokeun paripolah ieu, sababaraha métodeu diusulkeun; salah sahiji solusi nu kawentar nyaéta sekuen Halto ("make koefisien permutasi nu ditangtukeun samemehna digunakeun dina ngawangun sekuen standar")
Rasio bencana dina analisa survival nyaéta kasimpulan tina beda antara dua kurva survival, ngagambarkeun nguranganna resiko kematian dina babandingan kontrol pencegahan, salila waktu nu diperhatikeun
Ieu mangrupa bentuk resiko relatip
modél proporsi regresi bencana (oge disebut modél regeresi Cox) nganggep yén resiko relatip kematian antara pencegahan jeung kontrol mangrupa hal nu angger dina interval waktu
modél ieu dipaké salaku algoritma komputer keur nyimpulkeun risiko kematian relatip kana interval waktu dina hiji gambar
formula_1 nu mana O nyaéta jumlah observasi kematian dina grup husus, sarta E jumlah ekspektasi kematian, dumasar kana ieu kaayaan, tapi teu nyokot itungan tina unggal ekspektasi pencegahan
Kurva Hubbert, ngaran nu dipaké keur ngahargaan ka géofisikis M. King Hubbert, mangrupa derivatif tina kurva logistik
Salah sahiji conto kurva Hubbert nya éta: formula_1 Kurva Hubbert raket jeung susunan bentuk, sanajan asalna béda, probability density function tina sebaran normal
Kurva ieu asalna tina mode laju éxtraxi minyak bumi
Dumasar kana ieu modél, laju produksi minyak ditangtukeun ku laju sumur minyak anyar nu kapanggih; "penclut Hubbert ("Hubbert peak"]]" dina laju éxtraxi minyak bakal nuturkeun ku nurunna produksi minyak nepi ka béakna
"Catetan: keur diskusi leuwih jéntré ngeunaan minyak bumi, tingali dina artikel penclut Hubbert."
Independent components analysis (ICA) is a mathematical method for separating a signal into its most probable additive subcomponent supposing the statistical independence of the source signals
This assumption is correct in most cases so the blind ICA separation of a mixed signal gives very good results
It is also used for signals that are not supposed to be generated by a mixing for analysis purposes
The statistical method revéal hidden factors (independent components), presuming their non-gaussianity
It uses principal components analysis as a first step
Apply whitening of the data, and then apply an iterative algorithm (such as an adaptive filter)
The ICA séarches for orthogonal components that are mutually statistically independent to describe the input data matrix (composed by the féatures vectors)
This can allow the reconstruction of source signals from several corrupted mixtures theréof
The method is important to blind signal separation, EEG analysis and FMRI analysis
This concept can be extended to analyse non physical signals, for instance ICA has been aplied to discover discussions topics on a bag of news list archives
Kaputusan statistik nyaéta cabang tina statistik nu ngandung generalisasi tina sampel ka populasi, ngararangkay tes hipotesa, nangtukeun hubungan antar variabel, sarta nyieun prediksi
métodeu inpormasi leherbotol mangrupa téhnik keur manggihkeun "perdagangan" nu hadé antara akurasi jeung kompresi waktu nyieun kasimpulan (contona clustering) variabel random X waktu dibérékeun joint sebaran probabiliti antara X jeung variabel observasi Y
Dina statistik, interaksi nyaéta watesan dina model statistik nu mana efek dua variabel atawa leuwih teu bisa ditambahkeun sacara langsung
Mangka, for a response "y" and two variables "x"1 and "x"2 an "additive" modél would be: formula_1 - while, formula_2 - is an example of a modél with an "interaction" between variables "x"1 and "x"2 (the word "errors" is not to be construed literally; it refers to a random variable by which "y" differs from the nilai ekspektasi of "y")
See errors and residuals in statistics, and note that it is éasy to confuse errors with residuals, although the two are different
Very often the interacting variables are categorical variables rather than réal numbers
For example, members of a population may be classified by religion and by occupation
If one wishes to predict a person's height based only on the person's religion and occupation, a simple "additive" modél, i.e., a modél without interaction, would add to an overall average height an adjustment for a particular religion and another for a particular occupation
A modél with interaction, unlike an additive modél, could add a further adjustment for the "interaction" between that religion and that occupation
This example may cause one to suspect that the word "interaction" is something of a misnomer
The consequence of an "interaction" is that the effect of one variable depends on the value of another
This has implications in desain percobaan as it is misléading to vary one factor at a time
réal-world examples of systems that manifest interactions include: Genichi Taguchi contended that "interactions" could be eliminated from a system by appropriate choice of response variable and transformation
However George Box and others have argued that this is not the case in general
Box, G E P (1990) Do interactions matter
"Quality Engineering" vol 2, pp365–369
Dina statistik deskriptif, interquartile range (IQR) nyaéta beda antara kuartil katilu jeung kahiji
Interquartile range leuwih nembongkeun nu sabenerna dina statistik tinimbang rentang, sarta ilaharna dipikaresep dina statistik
Lamun 25% data kurang atawa sarua jeung kuartil kahiji sarta 25% leuwih gedé atau sarua jeung kuartil katilu, bedana nyaéta panjang interval kaasup satengahna tina data
Beda ieu kudu diukur tina satuan data nu sarua
Persentil jeung kuartil
Keur lobana sampel formula_1 mangka observasi formula_2 observasi diruntuykeun ti nu leutik kanu gedé
Tina runtuyan statistik ieu bisa kapanggih persentil sampel
Lamun formula_3 nujul kana formula_4 sampel observasi leuwih leutik tinimbang éta sarta formula_5 sampel observasi leuwih gedé tinimbang eta
Salah sahiji cara keur narima ieu nyaéta nyokot sampel persentil nu ka-(100formula_6) ku runtuyan statistik formula_5, tur formula_5 minangka integer
Lamun formula_5 lain interger tur sarua jeung formula_10 ditambah sababaraha bagéan formula_11 mangka bakal manggih interpolasi linier antara formula_12 jeung formula_13
Mangka sampel persentil nu ka-100formula_6 dihartikeun minangka: formula_15 Persentil nu pasti mibanda ngaran husus
Persentil nu ka-50 disebutna median tina sampel
Persentil nu ka-25, 50, tur ka-75 disebut kuartil kahiji, kadua jeung katilu tina sampel
Hal ini nembongkeun kasimpulan lima-wilangan tina runtuyan data, nyaéta minimal, kuartil kahiji, median, kuartil katilu jeung kuartil maksimum
Ayeuna IQR atawa interquartile range dihartikeun minangka: formula_16 Conto
Itungan ieu maké data tina kaca quartile
Tina tabel eta, the interquartile range is 113.5 - 105.5 = 8 Tempo ogé: statistical dispersion, mean
Dinamika sistim sahiji "pendekatan" keur dinamika dina populasi, system ekologi sarta ékonomi, nu ilaharna pakait raket
Dinamika sistem dipanggihkeun di awal taun 1960s ku Jay W. Forrester di MIT nu ngawangun MIT System Dynamics Group
Dina waktu nu sarua, manéhna mimiti maké nu nalungtik ngeunaan sistim salila manéhna gawe di rekayasa listrik keur sistim nu dipaké unggal poe
Naha nu dipaké sistim dinamik béda tina pamarekanséjén nu diajarkeun dina sistim kompleks nu dipaké di feedback loops
"Stocks" sarta "flows" mangrupa wangunan blok modél sistim dinamika
manéhna mantuan ngajelaskeun kumaha sistim pakait ku "feedback loops" nu dijieun nonlinearity remen kapanggih dina masalah kiwati
Perangkat lunak komputer dipaké keur simulasi model dinamika sistim nu keur ditalungtik
Running "what if" simulations to test certain policies on such a modél can gréatly aid in understanding how the system changes over time
Fungsi survivor atawa fungsi reliabiliti nyaéta sipat unggal variabel acak nu ngagambarkeun susunan kajadian, ilaharna pakait jeung paeh atawa gagalna sababaraha sistim, dumasar kana time
Ieu kawengku dina probabiliti nu mana sistim bakal nyalametkeun manéh salila waktu nu ditangtukeun
Watesan "fungsi reliabiliti" ilahar dipaké dina widang rékayasa sabalikna watesan "fungsi survivor" dipaké leuwih lega, kaasup dina hurip manusa
Anggap "X" mangrupa variabel bébas nu mibanda cumulative distribution function "F"("t") dina interval [0,∞)
Maka "fungsi-survivor" atawa "fungsi-reliabiliti" nyaéta: formula_1 Sipat
Unggal fungsi "R"("t") bakal turun sacara angger
Waktu, "t"=0, ngagambarkeun kaayaan mimiti, sacara tipikal dimimitian nu nalungtik atawa sistim mimiti dipaké
"R"(0) ilaharna satuan tapi bisa kurang keur ngagambarkeun probabiliti nu mana sistem gagal teu lila sanggeus operasi
Sakali deui, lim"t"→∞"R"("t") ilaharna nol tapi bisa ogé leuwih gedé keur ngagambarkeun kahirupan dina jero sistim nu mungkin
In statistics, a Studentized residual, named in honor of William Sealey Gosset, who wrote under the pseudonym Student, is a residual adjusted by dividing it by an estimate of its standard deviation
Studentization of residuals is an important technique in the detection of outliers
Errors versus residuals
It is very important to understand the difference between errors and residuals in statistics
Consider simple linear regression modél formula_1 where the errors ε"i", "i" = 1, ..., "n", are independent and all have the same variance σ2
The residuals are not the true, and unobservable, errors, but rather are "estimates", based on the observable data, of the errors
When the method of léast squares is used to estimate α0 and α1, then the residuals, unlike the errors, cannot be independent since they satisfy the two constraints formula_2 and formula_3 (Here formula_4 is the "i"th error, and formula_5 is the "i"th residual.)
Moréover, the residuals, unlike the errors, do not all have the same variance: the variance incréases as the corresponding "x"-value gets farther from the average "x"-value
The fact that the variances of the residuals differ, even though the variances of the true errors are all equal to éach other, is the principal réason for the need for Studentization
How to Studentize
For this simple modél, the "design matrix" is formula_6 and the "hat matrix" "H" is the matrix of the orthogonal projection onto the column space of the design matrix: formula_7 The "leverage" "h""ii" is the "i"th diagonal entry in the hat matrix
The variance of the "i"th residual is formula_8 The corresponding Studentized residual is then formula_9 where formula_10 is an appropriate estimate of σ
Internal and external Studentization
The estimate of σ2 may be formula_11 But it is desirable to exclude the "i"th observation from the process of estimating the variance when one is considering whether to consider the "i"th case to be an outlier
Consequently one may use the estimate formula_12 based on all but the "i"th case
If the latter estimate is used, "excluding" the "i"th case, then the residual is said to be externally Studentized, if the former is used, "including" the "i"th case, then it is internally Studentized
If the errors are independent and normally distributed with nilai ekspektasi 0 and varian σ2, then the probability distribution of the "i"th externally Studentized residual is a sebaran-t student nu mana "n" − 3 tingkat kabebasan
Stein's lemma, ngaran keur ngahargaan ka Charles Stein, may be characterized as a théorem of probability theory that is of interest primarily because of its application to statistical inference—in particular, its application to James-Stein estimation and empirical Bayes methods
Statement of the lemma
Suppose "X" is a normally distributed random variable with nilai ekspektasi μ and varian σ2
Further suppose "g" is a function for which the two expectations E( "g"("X") ("X" − μ) ) and E( "g" ′("X") ) both exist (the existence of the expectation of any random variable is equivalent to the finiténess of the expectation of its absolute value)
Then formula_1 In order to prove this lemma, recall that the probability density function for the normal distribution with expectation 0 and variance 1 is formula_2 and that for a normal distribution with expectation μ and varian σ2 is formula_3 Then use integration by parts
Rumus prédiksi Spéarman-Brown (ogé dipikanyaho salaku "rumus ramalan Spearman-Brown") nyaéta rumus hubungan reliabiliti psikometri kana panjangna uji: formula_1 nu mana formula_2 nyaéta prédiksi reliabiliti; N nyaéta jumlah kombinasi "test" (tempo di handap); sarta formula_3 nyaéta reliabiliti "tes" ayeuna
Rumus prédiksi réability mangrupa uji anyar nu nyimpen ulangan N ayeuna (atawa sarua jeung nambahkeun bentuk N paralél tina tes ayeuna ka tes ayeuna)
Maka N=2 akibatna duka kali panjang tes item nu ditambahkeun ku cara maké sipat nu sarua dina tes ayeuna
Nilai N kurang ti hiji mungkin bisa dipaké keur prediksi pangaruh dina tes nu pondok
Rumus bisa ogé disusun keur prediksi jumlah ulangan nu diperlukeun keur nangtukeun tingkat réalibiliti: formula_4 Rumus ieu ilahar dipaké ku ahli psikometri keur prediksi réabiliti sanggeus ngarobah panjang tes
Hubungan ieu salah sahiji hal nu penting keur ngarobah-sabagian sarta métodeu nu pakait jeung estimasi réabiliti
Rumus ieu ogé mantuan ngarti kana hubungan nu teu linier antara tes réabiliti jeung panjang tes
Lamun panjang/pondok tes teu paralel kana tes ayeuna, maka hasil prediksi moal akurat
Contona, lamun luhurna tes reliabiliti dipanjangan ku cara nambahan item nu goreng mangka reliabiliti bisa leuwih handap tinimbang nu diprediksi tina rumus ieu
Tiori respon item merlukeun "informasi item" keur ngahasilkeun prediksi nu leuwih hadé dina ngaramal parobahan kualitas ukuran ku cara nambahan atawa ngurangan individu item
Dina tés binér, saperti dina tés diagnosa médis keur panyakit di hiji wewengkon, specificity nyaéta proporsi négatip bener tina sakabéh sampel négatip nu dités, nyaéta formula_1
Aya sababaraha cara keur ngadiskusikeun estimasi statistik
Simple random sampling
Dina statistik, sampel acak basajan tina hiji populasi nyaéta sampel nu dicokot sacara acak, unggal anggota dina éta populasi mibanda kamungkinan nu sarua pikeun dicokot
Dina populasi nu saeutik, cara nyokot sampelna digawékeun ku cara "dicokot teu disimpen deui", nyaéta ku cara sacara tarapti milih unggal conto tina populasi
Sacara konsép, nyokot sampel acak basajan mangrupa téhnik nyokot sampel probibiliti nu pangbasajanna, sanajan kitu langka dipaké sabab loba masalah dina makéna
Nyokot sampel acak basajan lain mangrupa métodeu nu épéktip
métodeu ieu merlukeun rohangan sampel nu kacida gedéna sarta hasilna merlukeun waktu ngitung nu lila sarta biaya nu mahal
Lamun panalungtik mibanda informasi nu cukup ngeunaan populasi, métodeu ieu cukup épisién
Kauntungannana nyaéta bébas tina klasifikasi kasalahan sarta merlukeun kanyaho nu teu loba ngeunaan populasi
Dua kaayaan nu hadé nu mana populasi ampir sarua sarta saeutikna informasi ngeunaan populasi
Lamun henteu dina kaayaan tadi, mangka leuwih hadé ngagunakeun "stratified sampling"
Multistage sampling nyaéta bentuk kompleks tina cluster sampling
Ngagunakeun sakabel unsur sampel dina sakabéh klaster nu dipilih teu dimeunangkeun atawa teu penting
Dina kaayaan aturan ieu, nyokot sampel dina sababaraha tingkatan klaster jadi penting
Gantina maké kabéh unsur nu dipiboga ku klaster nu dipilih, panalungtik sacara acak milih unsur tina unggal klaster
Wangun klaster dina léngkah kahiji
Tangtukeun unsur mana nu dina jero klaster éta bakal dipaké, dina léngkah kadua
tehnik ieu remen dipaké lamun runtuyan lengkep sakabéh anggota dina populasi euweuh
Nyokot sampel Probabilitas-sabanding-kana-ukuran mangrupa salah sahiji tipe nyokot sampel klaster multiléngkah
Dina métodeu ieu, pamilihan probabilitas unsur anu ditangtukeun dina klaster variasina tibalik jeung ukuran dina klaster
Sanajan nyokot sampel sacara klaster jeung nyokot sampling ku cara ngabédakeun lapisan sampel meh ampir sarua, tapi sacara substansi duanana béda
Dina nyokot sampel ku cara ngabédakeun lapisan, sampel acak dicokot tina sakabéh lapisan, nu mana dina nyokot sampel sacara klaster ngan tina klaster nu ditalungtik, boh dina saléngkah atawa multiléngkah "Tempo ogé : statistika, marketing research, quantitative marketing research, sampling, cluster sampling, simple random sampling, nonprobability sampling, systematic sampling, stratified sampling"
Dina statistik, sebaran sampling nyaéta sebaran probabiliti, dina kaayaan sampling dina populasi diulang, tina statistik nu dibérékeun (kuantitas numerik diitung tina nilai data dina sample)
Bentuk (atawa rumus keur) sebaran sampling gumantung kana sebaran populasi, statistik nu ditempo sarta ukuran sampel nu dipaké
Contona, anggap populasi nu kacida gedéna nuturkeun normal (ilaharna disebut kurva bel)
Asumsi urang mindo nyokot sampe dina sampel nu dibérékeun tina populasi sarta ngitung sample mean (arithmetic mean nilai data) keur unggal sampel
Sampel nu béda bakal mangaruhan kana bédana sampel méans
Sebaran méan ieu nyaéta "sebaran sampling tina sampel mean" (keur ukuran sampel nu dibérékeun)
Sebaran ieu bakal jadi normal saprak populasina normal
(Dumasar kana teorema central limit, lamun populasi teu normal tapi "bentukna cukup hade", sebaran sampling tina sampel méan bakal masih kénéh cukup normal sanajan ukuran sampel kacida gedéna)
Simpangan baku tina sebaran sampling ngacu ka standar kasalahan (statistis) sarta di-estimasi maké rumus: formula_1 nu mana formula_2 mangrupa simpangan baku tina sebaran populasi sarta N nyaéta ukuran sampe (jumlah barang) dina sampel
Hal nu penting dina rumus ieu nyaéta kudu ngalikeun opat kali ukuran sampel (4X) ngarah ukuran kasalahanna jadi (1/2)
Waktu rarangkay elmu statistik nu mana biaya mangrupa faktor, ieu mangrupa faktor nu kudu dipikaharti ngeunaan pentingna biaya
Alternatipna, anggap median sampel tina populasi
Ieu mibanda sebaran sampling nu béda sacara umum teu normal (tapi masih kénéh pakait dina aturan ka-normal-an)
central limit theorem, standard error (statistics)