izzako/sundanese-pixelgpt huggingface dataset:
Similarly an arbitrary collection of random variables—possible more than just two of them—is independent precisely if for any finite collection "X"1, ..., "X""n" and any finite set of numbers "a"1, ..., "a""n", the events ["X"1 ≤ "a"1], ..., ["X""n" ≤ "a""n"] are independent events as defined above
The méasure-théoretically inclined may prefer to substitute events ["X" ∈ "A"] for events ["X" ≤ "a"] in the above definition, where "A" is any Borel set
That definition is exactly equivalant to the one above when the values of the random variables are real numbers
It has the advantage of working also for complex-valued random variables or for random variables taking values in any topological space
Lamun "X" sarta "Y" bébas, mangka operator ekspektasi "E" mibanda sipat nu hadé E["X"· "Y"] = E["X"] · E["Y"] sarta keur varian mibanda var("X" + "Y") = var("X") + var("Y")
Lamun "X" jeung "Y" bébas, kovarian cov("X","Y") sarua jeung nol; dina hal séjén mibanda var("X" + "Y") = var("X") + var("Y") + 2 cov("X", "Y")
Furthermore, if "X" and "Y" are independent and have probability densities "f""X"("x") and "f""Y"("y"), then the combined random variable ("X","Y") has a joint density "f""XY"("x","y") d"x" d"y" = "f""X"("x") "f""Y"("y") d"x" d"y"
We define random variables "X" and "Y" to be "conditionally independent given" random variable "Z" if P[("X" in "A") &amp; ("Y" in "B") | "Z" in "C"] = P["X" in "A" | "Z" in "C"] · P["Y" in "B" | "Z" in "C"] for any Borel subsets "A", "B" and "C" of the réal numbers
If "X" and "Y" are conditionally independent given "Z", then P[("X" in "A") | ("Y" in "B") &amp; ("Z" in "C")] = P[("X" in "A") | ("Z" in "C")] for any Borel subsets "A", "B" and "C" of the réal numbers
That is, given "Z", the value of "Y" does not add any additional information about the value of "X"
Independence can be seen as a special kind of conditional independence, since probability can be seen as a kind of conditional probability given no events
Statistik terapan ngagunakeun statistik jeung tiori statistik dina kaayaan nu sabenerna
Sing saha baé anu ngarasa "komit" kana observasi empiris hartina aya kanyaho umum bisa ngagunakeun statistik salaku alat panalungtikan
Hal ieu kaasup élmu tapi teu kaasup sajarah jeung seni
Contona, ékonométrik loba ngagunakeun statistik terapan keur "mempelajari" ékonomi
Di satiap widang, diperlukan "observasi", potensial "rekonstruksi" keur ngurangan kasalahan dina observasi jeung rencana panunglutikan keur ngontrol observational error
Sajarah asalna tina basa Arab, nyaéta "syajarotun" nu hartina tangkal, turunan, atawa asal-usul
Ieu kecap téh asup kana basa Malayu dina abad ka-13 Masehi
Ari asupna kana basa Sunda mah kira-kira dina abad ka-17 Masehi, patali jeung agama Islam di wilayah Sunda
Pentingna sajarah atawa turunan téh dina budaya masarakat Arab mah jéntré pisan
Hal ieu kagambar dina susunan pancakaki urang Arab nu tepi ka 20 tingkatan
Patali jeung pancakaki katut sajarah, silsilah Raja Yogya disusun dina wangun tangkal
Ari di Sumedang mah aya Kitab Pancakaki nu disusun ku Raden Arya Suryalaga
Ieu kitab téh kiwari jadi koléksi Museum Pangéran Geusan Ulun, Sumedang
Sajarah geus ilahar digunakeun salaku watesan umum pikeun béja atawa émbaran ngeunaan jaman baheula, saperti dina "sajarah géologis Bumi"
Lamun dilarapkeun salaku hiji widang ulikan, sajarah hartina sajarah manusa, nyaéta ingetan atawa kajadian kahirupan sosial manusa baheula nu karékam atawa kacatet
Harti sajarah dina basa-basa Nusantara mekar jadi carita, kisah, hikayat, riwayat, babad, tambo, tutui teteek
Dina basa Sunda éta ngaran téh di antarana carita, sajarah, babad, carios, hikayat, lalakon, jeung wawacan
Pikeun éta istilah téh dina basa Arab mah disebutna "qosos"
Sajarawan ngagunakeun sabararaha rupa sumber, kaasup tulisan atawa rekaman citak, wawancara (sajarah oral), jeung arkéologi
Pamarekan pikeun ngulik sajarah dina unggal jaman bisa béda-béda
Kajadian-kajadian nu lumangsung méméh ayana catetan manusa disebut prasajarah
Harti sajarah téh aya dua rupa, nyaéta harti sacara tradisional jeung harti sacara modérn
Sacara tradisional sajarah téh dihartikeun bagian tina sastra
Ku kituna karya sastra dina wangun légenda, mitos, jeung séjénna kaasup sajarah
Contona Ilios jeung Odise di Eropa, Carita Mahabarata jeung Ramayana di India, Hikayat Hang Tuah, Babad Balambangan, jeung Carita Parahyangan
Sacara modérn, harti sajarah leupas tina widang sastra
Di dieu mah sajarah téh ngandung harti ieu di handap
Sajarah tradisional jeung sajarah modérn kaalaman ku sakumna manusa di dunya
Di Eropa, abad ka-18 dianggap masa peralihan tina sajarah tradisional kana sajarah modérn
Sedengkeun di Indonésia awal abad ka-2O nu dianggap masa peralihan téh
Aya anggapan yén sajarah modérn di Indonesia téh asalna ti kulon, hususna Walanda
Dina basa Walanda kecap sajarah téh nyaéta "geschiedenis", nu asalna tina kecap "gesclzieden", nu hartina téh kajadian anu dipigawé
Ari dina basa Jérman mah sejarah téh nyaéta "geschiclzte", nu asalna tina kecap "geschehen" nu hartina kajadian
Jadi "geschichte" hartina nu geus kungsi kajadian
Dina basa Inggris aya istilah "history", nu asalna tina basa Yunani, "istoria", nu hartina élmu atawa hiji pedaran sistematis ngeunaan gejala alam
Sanggeus muncul istilah Scientia pikeun pedaran nu non-kronologis, kecap "istoria" husus pikeun pedaran sistematis nu kronologi ngeunaan gejala-gejala nu patali jeung kahirupan manusa
Ku kituna muncul pamadegan kritis (modérn) dina karangan sajarah nu disusun ku Pangéran Wangsakerta dina akhir abad ka-17 Masehi
Mangpaat sajarah bisa ditilik tina segi téoritis jeung segi Praktis-pragmatis
Salian ti éta, sajarah téh miboga tilu mangpaat séjén di antarana pikeun ngawétkeun idéntitas kelompok jeung mageuhan daya tahan kelompok sangkan hirup terus lumangsung, nyokot pelajaran jeung tuladan tina rupa-rupa kajadian dina mangsa nu geus kaliwat, jeung sarana pikeun mikawanoh ngeunaan ma'na hirup jeung maot, atawa ngeunaan tempat manusa di dunya
Sumber jeung Bukti
Aya rupa-rupa sumber jeung bukti sajarah di tatar Sunda, boh nu lengkep kénéh boh nu henteu
Tapi duanana ogé silih lengkepan
Tina rupa-rupa titinggal téh bisa diklasifikasikeun jadi lima katégori, nyaéta: Data sejarah
Data sajarah téh rupa-rupa, nyaéta paleolitik, alat obsidian (batu kaca), situs néolitik perundagian, punden berundak, taman purbakala/ prasasti, candi, arca ganésa, situs budaya Islam, kampung adat, gedung bersejarah, situs perundagian, jeung cagar alam
Data sajarah téh lengkep upama aya sumberna jeung buktina
Tapi hal éta téh teu salawasna kacumponan
Dina kanyataanana, bisa jadi saperti ieu di handap
Dina Wikipédia aya loba émbaran ngeunaan sajarah, sababaraha cara papasinganana bisa ditempo ieu di handap
"Panggunaan" statistik keur psikologi
Sababaraha widang anu umum dipaké nyaéta: biostatistik -- psikologi
Statistik social nyaéta pamakéan sistim ukuran statistik keur nalungtik paripolah manusa dina lingkungan sosial
Ieu bisa dipikanyaho ku cara jajal pamanggih ti sabagian grup masarakat, evaluasi sabagian data ngeunaan grup masarakat, atawa ku panalungtikan jeung analisa statistik susunan data nu pakait jeung masarakat sarta paripolahna
Biasan, ahli sosial matéakeun dina evaluasi kualitas palayanan sabagian grup organisasi, dina analisa paripolah grup masarakat dina lingkunganna, atawa dina nangtukeun naon nu butuh jeung kahayang masarakat ngaliwatan sampling statistik
Loba ilmuwan nu ngamimitian panalungtikan ku patarosan ngeunaan dunya tempat urang hirup
Keur ahli statistik, patarosan ieu bisa di-klasifikasi-keun kana sababaraha hal
In much reséarch, we use all three modes at different stages
A useful synthesis of this kind of thinking was provided by Ludwig von Bertalanffy
In every type of reséarch, we must be concerned with managing the observational error that is inherent in all empirical reséarch
We can incréase the precision of our reséarch by See also: statistik -- statistical regularity -- model statistik -- summarizing statistical data -- interpreting statistical data --desain percobaan -- survey sampling
Kémométrik nyaéta larapan métode matematik atawa statistik kana data kimia
Masarakat Kémométrik Internasional ("International Chemometrics Society", ICS) nawarkeun dadaran sarupa kieu: "Kémométrik nyaéta élmu matalikeun ukuran nu dijieun dina sistim atawa prosés kimia kana bentuk sistimna ngaliwatan larapan métode matematik atawa statistik."
Chemometric reséarch spans a wide aréa of different methods which can be applied in chemistry
There are techniques for collecting good data (optimization of experimental paraméters, desain percobaan, kalibrasi, signal processing) and for getting information from these data (statistics, pattern recognition, modeling, structure-property-relationship estimations)
Chemometrics tries to build a bridge between the methods and their application in chemistry
Daptar asosiasi statistis akademik
Statistikawan atawa jalma nu méré kontribusi kana tiori statistik, atawa hal anu aya hubunganna jeung kamungkinan, atawa machine learning: "See also:" List of mathematicians | List of people by occupation | List of people
Di handap ieu daftar publikasi penting dina statistik, dikumpulkeun dumasar kana widang garapan
There are some réasons why a particular publication might be regarded as important: Foundations
"The Doctrine of Chances"
Deskripsi: Buku ieu ngawanohkeun konsép sebaran normal ngadeukeutan kana sebaran binomial
Dina akibat, de Moivre ngabuktikeun versi lemah tina teorema central limit
Kadang-kadang hasilna disebut ogé teorema de Moivre-Laplace
Importance: Topic créator, Bréakthrough, Impact "An Essay Toward Solving a Problem in the Doctrine of Chances"
Description: In this paper Bayes' theorem was first introduced
It has a vast impact on statistics since then and keeps being important these days
Importance: Topic créator, Bréakthrough, Impact
prosés stokastik nyaéta fungsi acak
In practical applications, the domain over which the function is defined is a time interval (a stochastic process of this kind is called a deret waktu in applications) or a region of space (a stochastic process being called a random field)
Familiar examples of time series include stock market and exchange rate fluctuations, signals such as speech, audio and vidéo; medical data such as a patient's EKG, EEG, blood pressure or temperature; and random movement such as Brownian motion or random walks
Examples of random fields include static images, random topographies (landscapes), or composition variations of an inhomogenéous material
Mathematically, a stochastic process is usually defined as an indexed collection of variabel acak "f""i" : "W" → R, where "i" runs over some index set "I" and "W" is some probability space on which the random variables are defined
This definition captures the idéa of a random function in the following way
To maké a function "f" : "D" → R with domain "D" and range R into a random function, méans simply making the value of the function at éach point of "D", "f"("x"), into a variabel acak with values in "R"
The domain "D" becomes the index set of the stochastic process, and a particular stochastic process is determined by specifying the joint probability distributions of the various random variables "f"("x")
Note, however, that the definition of stochastic process as an indexed collection of random variables is much more general than the case where the indices are points of the domain of the random function
Implications of the definition
Of course, the mathematical definition of a function includes the case "a function from {"1"...,"n"} to R is a vector in R"n"", so multivariate random variables are a special case of stochastic processes
For our first infinite example, take the domain to be N, the natural numbers, and our range to be R, the real numbers
Then, a function "f" : N → R is a sequence of réal numbers, and a stochastic process with domain N and range R is a random sequence
The following questions arise: Another important class of examples is when the domain is not a discrete space such as the natural numbers, but a continuous space such as the unit interval [0,1], the positive réal numbers [0,∞) or the entire real line, R. In this case, we have a different set of questions that we might want to answer: There is an effective
to answer: There is an effective way to answer all of these questions, but it is rather technical (see "Constructing Stochastic Processes" below)
"What is a suitable elementary example to develop in full
Maybe coin-tossing or random walk?"
Constructing stochastic processes
In the ordinary axiomatization of tiori probabiliti by méans of measure theory, the problem is to construct a sigma-algebra of measurable subsets of the space of all functions, and then put a finite measure on it
For this purpose one traditionally uses a method called Kolmogorov extension
There is at léast one alternative axiomatization of probability théory by méans of expectations on algebras of observables
In this case the method goes by the name of Gelfand-Naimark-Segal construction
This is analogous to the two approaches to méasure and integration, where one has the choice to construct méasures of sets first and define integrals later, or construct integrals first and define set méasures as integrals of characteristic functions
The Kolmogorov extension
The Kolmogorov extension proceeds along the following lines: assuming that a probability measure on the space of all functions "f" : "X" → "Y" exists, then it can be used to specify the probability distribution of finite-dimensional random variables ["f"("x"1)...,"f"("xn")]
Now, from this "n"-dimensional probability distribution we can deduce an "(n-1)"-dimensional marginal probability distribution for ["f"("x"1)...,"f"("x""n"-1)]
There is an obvious compatibility condition, namely, that this marginal probability distribution be the same as the one derived from the full-blown stochastic process
When this condition is expressed in terms of probability densities, the result is called the Chapman-Kolmogorov equation
The Kolmogorov extension theorem guarantees the existence of a stochastic process with a given family of finite-dimensional probability distributions satisfying the Chapman-Kolmogorov compatibility condition
Separability, or what the Kolmogorov extension does not provide
Recall that, in the Kolmogorov axiomatization, measurable sets are the sets which have a probability or, in other words, the sets corresponding to yes/no questions that have a probabilistic answer
The Kolmogorov extension starts by declaring to be méasurable all sets of functions where finitely many coordinates ["f"("x"1)...,"f"("xn")] are restricted to lie in méasurable subsets of "Yn"
In other words, if a yes/no question about "f" can be answered by looking at the values of at most finitely many coordinates, then it has a probabilistic answer
In méasure théory, if we have a countably infinite collection of méasurable sets, then the union and intersection of all of them is a méasurable set
For our purposes, this méans that yes/no questions that depend on countably many coordinates have a probabilistic answer
The good news is that the Kolmogorov extension makes it possible to construct stochastic processes with fairly arbitrary finite-dimensional distributions
Also, every question that one could ask about a sequence has a probabilistic answer when asked of a random sequence
The bad news is that certain questions about functions on a continuous domain don't have a probabilistic answer
One might hope that the questions that depend on uncountably many values of a function be of little interest, but the réally bad news is that virtually all concepts of calculus are of this sort
For example: all require knowledge of uncountably many values of the function
One solution to this problem is to require that the stochastic process be separable
In other words, that there be some countable set of coordinates {"f"("xi")} whose values determine the whole random function "f"
The algebraic approach
In the algebraic axiomatization of probability théory, one of whose main proponents was Segal, the primary concept is not that of probability of an event, but rather that of a random variable
Probability distributions are determined by assigning an expectation to éach random variable
The méasurable space and the probability méasure arise from the random variables and expectations by méans of well-known representation théorems of analysis
One of the important féatures of the algebraic approach is that apparently infinite-dimensional probability distributions are not harder to formalize than finite-dimensional ones
Random variables are assumed to have the following properties: This méans that random variables form complex abelian *-algebras
If "a"="a"*, the random variable "a" is called "real"
An expectation "E" on an algebra "A" of random variables is a normalized, positive linéar functional
What this méans is that
Autokorélasi mangrupa salah sahiji alat matematika anu remen digunakeun dina signal processing keur analisa fungsi atawa deret nilai, saperti domain waktu signals
Autokorélasi nyaéta cross-correlation hiji signal jeung signalna sorangan
Autokorélasi ilahar dipaké keur manggihkeun pola "pengulangan" dina signal, saperti nangtukeun aya périodeu signal nu kaganggu atawa keur identifikasi frékuénsi dasar signal nu teu mibanda komponen frékuénsi, tapi mangaruhan kana frékuénsi nu harmonis
bédana definisi autokorélasi nu dipaké gumantung kana widang elmu nu ditalungtik nu satemenna teu sarua
Dina sababaraha widang elmu istilahna diganti ku autocovariance
Dina statistik, fungsi autokorelasi (ACF) keur deret waktu diskrit atawa hiji prosés "X"t hartina korelasi antara prosés dina titik nu béda tur béda waktu oge
Lamun "X"t mibanda mean μ jeung varian σ2 mangka definisi ACF nyaéta formula_1
nu mana E nyaéta nilai ekspektasi
Nilai ekspektasi ieu teu bisa dipaké sacara umum keur sakabéh deret waktu atawa prosés lamun nilai varian sarua jeung nol (keur prosés nu konstan) atawa tak hingga
Fungsi ieu mibanda harti nu umum lamun dihartikeun mibanda sifat nu aktraktif dina rentang [−1, 1], nu mana 1 nunjukkeun korelasi nu sampurna sarta −1 nunjukkeun anti-korelasi nu sampurna
Lamun "X""t" mangrupa orde-dua statis mangka ACF ngan gumantung kana béda antara "t" jeung "s" sarta bisa digambarkeun salaku fungsi variabel tunggal
Hal ieu geus ilahar maké rumus nu umum nyaéta, formula_2 lamun "k" mangrupa "lag" (|"t" - "s"|)
Dina sababarha widang elmu digambarkeun ku normalisasi σ2 sarta maké watesan "autocorrelation" nu kadangkala disebut ogé "autocovariance"
Keur conto panjang deret waktu "n", "X"1, "X"2 ... "X"n nu mana méan jeung varian dipikanyaho mangka nilai estimasi-na ditangtukeun tina formula_3
Lamun nilai méan atawa varian keur prosés ieu teu dipikanyaho mangka μ sarta σ2 bisa diganti ku rumus standar keur sampel méan jeung sampel varian sanajan hasilna nujul kana bias dina nilai estimasi-na
Dina "signal processing", definisi teu di handap salawasna dipaké bari teu maké normalisasi, nyaéta, teu ngurangan méan sarta ngabagi ku varianna
Upamana "signal" "f"("t"), autokorelasi kontinyu "Rff"(τ) salawasna dihartikeun salaku integral korelasi-silang kontinyu tina "f"("t")-na sorangan, dina "lag" "τ"
formula_5 nu mana "f*" nunjukkeun kompleks konjuget sarta lingkaran nunjukkeun konvolusi
Keur fungsi ril, "f*" = "f"
Autokorelasi diskrit "R" at lag "j" keur "signal" diskrit "xn" nyaéta formula_6 Definisi di handap dipaké keur "signals" nu mangrupa integral kuadrat atawa jumlah kuadrat, nyaéta nu énérgina kawates
"Signals" nu "pamungkas salawasna" dianggap gaganti prosés acak, nu mana diperlukeun definisi nu béda, dumasar kana nilai ekspektasi
Keur proses acak statis nu leuwih lega, autokorelasi dihartikeun salaku formula_7 formula_8 Keur prosés anu henteu stationary, mangrupa fungsi tina "t", atawa "n" oge
Alternatipna, "signals" nu "salawasna pamungkas" bisa dirobah kana analisa fungsi autokorelasi keur waktu nu pondok, maké integral waktu nu kawengku (Tempo short-time Fourier transform keur prosés nu pakait.)
Autokorelasi multi-dimension bisa dihartikeun ku cara nu sarua
Contona, dina autokorelasi tilu dimenasi dina kuadrat-jumlah "signal" diskrit bisa jadi formula_9 Lamu nilai méan dikurangkeun tina "signal" saméméh diitung dina fungsi autokorelasi, fungsi nu dihasilkeun disebut ogé fungsi auto-covarian
Saterusna arek dijelaskeun sipat autokorélasi dina hiji-dimensi, sabab satemenna loba sipat hiji-dimensi nu gampang keur dipaké dina kasus multi-dimensi
formula_10 numana "f" nyaéta fungsi riil sarta autokorélasi nyaéta fungsi Hermitian formula_11 numana "f" nyaéta complex function
formula_13 formula_14 Autokorélasi dina analisa regresi
Dina regression analysis ngagunakeun time series, autokorélasi nyaéta sesa tina ieu masalah, sarta nuju ka arah bias nu luhur dina estimasi statistical significance tina koefisien estimasi, saperti dina T statistic
Tes standar keur ayana autokorélasi nyaéta Durbin-Watson statistic atawa, lamun ngajelaskeun variabel kaasup lagged dependent variable, Durbin's h statistic
Respon keur autokorélasi kaasup differencing data sarta ayana struktur "lag" dina estimasi
Bayesian inference nyaéta inferensi statistik nu mana sagala kamungkinan di-interpretasi lain salaku frékuénsi atawa proporsi atawa sabangsana, tapi leuwih condong kana tingkat kapercayaan
Ngaran ieu asalna ku sabab remenna ngagunakeun Téoréma Bayes keur matotoskeun hiji perkara
Téorema Bayes, ngaran téori nu dipaké sanggeus Thomas Bayes anu mimiti ngawanohkeun ieu métodeu
Kajadian jeung metoda ilmiah
Ahli statistik Bayes ngaku yén métodeu kaputusan Bayes mangrupa bentuk formal tina metoda ilmiah kaasup dina ngumpulkeun evidence nu mana ka hareupna atawa jalan keur nangtukeun hiji hypothesis
Dina hal ieu bisa jadi teu pasti salawasna, sanajan kitu lobana kumpulan kajadian bakal ngajadikeun naekna tingkat kapercayaan hipotesa pangluhurna (salawasna 1) atawa panghandapna (salawasna 0)
Téorema Bayes nyadiakeun métodeu keur naksir tingkat kapercayaan dina waktu informasi anyar ngan saeutik
Bayes' theorem nyaéta formula_1 Keur kaperluan urang, (A) dijadikeun hipotesa induced tina sababaraha susunan observasi
(e) dijadikeun hipotesa konfirmasi tina observasi
Faktor Likelihood: Pecahan formula_2 mangrupa faktor skala, probabilitas observasi hasil tina hipotesa dibagi ku probabilitas hipotesa observasi nu mangrupa kajadian "independen" dina hipotesa
Hasil ukuran ieu ngakibatkeun yén hipotesa ayana dina probabilitas nu dijieun tina observasi
ku lantaran kitu hasil observasi bakal jadi teu sahadé lamun hipotesa bener, sarta faktor skala bakal jadi gedé
Perkalian faktor skala ieu ku probabilitas observasi nu bener bakal ngahasilkeun probabilitas hipotesa nu bener oge, saperti nu dibérékeun ku observasi
Pagawéan konci dina nyieun kaputusan tangtuna ngararancang prior probabiliti dina observasi jeung hipotesa
Lamun prior probabiliti nembongkeun nilai "objektif", maka bisa digunakeun keur nangtukeun ukuran objektif hipotesa probabiliti
Tapi, taya jalan nu jelas keur nangtukeun objektif probabiliti
Hal anu teu mungkin keur migawe pamarekandina nangtukeun hiji probabilitas bis nangtukeun sakabéh hipotesa nu mungkin
Alternatifna, jeung sering dipaké, probabiliti dicokot salaku "tingkat kapercayaan subjektif" ti bagian partisipan
Téori saterusna nangtukeun ukuran rasio kepercayaan tina observasi nu dijadikeun subjek kapercayaan dina hipotesa
Tapi hasil dina kasus ieu masih kénéh nyesakeun subjektif dina posterior probabiliti
Sabab kitu téorema bisa digunakeun keur ngarasionalkeun kapercayaan dina sababaraha hipotesa, tapi nolak objektifitas
Sababaraha skema teu bisa dipaké, contona, sifat objektif dina nangtukeun konflik paradigma sain
Dina loba kasus, akibat kajadian bisa disimpulkeun dina rasio likelihood, nu digambarkeun dina the law of likelihood
Hal ieu bisa dikombinasikeun jeung prior probability keur ngagambarkeun tingkat kapercayaan asli sarta kajadian pangtukangna nu dicokot dina perhitungan
saméméh kaputusan dijieun, loss function ogé diperlukeun keur nimbangkeun gambaran akibat tina kasalahan nangtukeun kaputusan
Conto sederhana Kaputusan Bayes
Kueh tina mangkok nu mana
Keur conto, aya dua mangkok pinuh ku kueh
Dina mangkok ka #1 aya sapuluh coklat 10 hias jeung 30 coklat polos, sedengkeun dina mangkok kadua #2 aya 20 coklat hias jeung 20 coklat polos
Kandar milih dua mangkok éta sacara acak sarta nyokot kue coklat sacara acak oge
Asumsina taya alesan keur percaya yén Kandar milih-milih éta mangkok atawa kue coklat téa
Kue coklat nu kacokot téh coklat polos
Sabaraha kamungkinna yén Kandar nyokot tina mangkok ka #1
Intuitively, it seems cléar that the answer should be more than a half, since there are more plain cookies in bowl #1
The precise answer is given by Bayes' théorem
Let "H"1 corresponds to bowl #1, and "H"2 to bowl #2
It is given that the bowls are identical from Fred's point of view, thus "P"("H"1) = "P"("H"2), and the two must add up to 1, so both are equal to 0.5
The "data" "D" consists in the observation of a plain cookie
From the contents of the bowls, we know that P("D" | "H"1) = 30/40 = 0.75 and P("D" | "H"2) = 20/40 = 0.5
Bayes' formula then yields formula_3 Before observing the cookie, the probability that Fred chose bowl #1 is the prior probability, "P"("H"1), which is 0.5
After observing the cookie, we revise the probability to "P"("H"1|"D"), which is 0.6
False positives in a medical test
False positives are a problem in any kind of test: no test is perfect, and sometimes the test will incorrectly report a positive result
For example, if a test for a particular disease is performed on a patient, then there is a chance (usually small) that the test will return a positive result even if the patient does not have the diséase
The problem lies, however, not just in the chance of a false positive prior to testing, but determining the chance that a positive result is in fact a false positive
As we will demonstrate, using Bayes' théorem, if a condition is rare, then the majority of positive results may be false positives, even if the test for that condition is (otherwise) réasonably accurate
Suppose that a test for a particular diséase has a very high success rate: Suppose also, however, that only 0.1% of the population have that diséase ("i.e."
with probability 0.001)
We now have all the information required to use Bayes' théorem to calculate the probability that, given the test was positive, that it is a false positive
Let "A" be the event that the patient has the diséase, and "B" be the event that the test returns a positive result
Then, using the second alternative form of Bayes' théorem (above), the probability of a "true" positive is formula_4 and hence the probability of a false positive is about (1 − 0.019) = 0.981
Despite the apparent high accuracy of the test, the incidence of the diséase is so low (one in a thousand) that the vast majority of patients who test positive (98 in a hundred) do not have the diséase
(Nonetheless, this is 20 times the proportion before we knew the outcome of the test
The test is not useless, and re-testing may improve the reliability of the result.)
In particular, a test must be very reliable in reporting a negative result when the patient does not have the diséase, if it is to avoid the problem of false positives
In mathematical terms, this would ensure that the second term in the denominator of the above calculation is small, relative to the first term
For example, if the test reported a negative result in patients without the diséase with probability 0.999, then using this value in the calculation yields a probability of a false positive of roughly 0.5
In this example, Bayes' théorem helps show that the accuracy of tests for rare conditions must be very high in order to produce reliable results from a single test, due to the possibility of false positives
(The probability of a 'false negative' could also be calculated using Bayes' théorem, to completely characterise the possible errors in the test results.)
In the courtroom
Bayesian inference can be used to coherently assess additional evidence of guilt in a court setting
Bayesian inference tells us that if we can assign a probability p(G) to the defendent's guilt before we take the DNA evidence into account, then we can revise this probability to the conditional probability p(G | E), since p(G | E) = p(G) p(E | G) / p(E) Suppose, on the basis of other evidence, a juror decides that there is a 30% chance
that there is a 30% chance that the defendent is guilty
Suppose also that the forensic evidence is that the probability that a person chosen at random would have DNA that matched that at the crime scene was 1 in a million, or 10−6
The event E can occur in two ways
Either the defendent is guilty (with prior probability 0.3) and thus his DNA is present with probability 1, or he is innocent (with prior probability 0.7) and he is unlucky enough to be one of the 1 in a million matching péople
Thus the juror could coherently revise his opinion to take into account the DNA evidence as follows: p(G | E) = 0.3 × 1.0 /(0.3 × 1.0 + 0.7 × 10-6) = 0.99999766667
In the United Kingdom, Bayes' théorem was explained by an expert witness to the jury in the case of Regina versus Denis Adams
The case went to Appéal and the Court of Appéal gave their opinion that the use of Bayes' théorem was inappropriate for jurors
In May 1968 the US nucléar submarine Scorpion (SSN 589) failed to arrive as expected at her home port of Norfolk, Virginia
The US Navy was convinced that the vessel had been lost off the éastern séabord but an extensive séarch failed to discover the wreck
The US Navy's deep water expert, John Craven, believed that it was elsewhere and he organised a séarch south west of the Azores based on a controversial approximate triangulation by hydrophones
He was allocated only a single ship, the USNS Mizar, and he took advice from a firm of consultant mathematicians in order to maximise his resources
A Bayesian séarch methodology was adopted
Experienced submarine commanders were interviewed to construct hypotheses about what could have caused the loss of the Scorpion
The séa aréa was divided up into grid squares and a probability assigned to éach square, under éach of the hypotheses, to give a number of probability grids, one for éach hypothesis
These were then added together to produce an overall probability grid
The probability attached to éach square was then the probability that the wreck was in that square
A second grid was constructed with probabilities that represented the probability of successfully finding the wreck if that square were to be séarched and the wreck were to be actually there
This was a known function of water depth
The result of combining this grid with the previous grid is a grid which gives the probability of finding the wreck in éach grid square of the séa if it were to be séarched
This séa grid was systematically séarched in a manner which started with the high probability regions first and worked down to the low probability regions last
éach time a grid square was séarched and found to be empty its probability was réassessed using Bayes' theorem
This then forced the probabilities of all the other grid squares to be réassessed (upwards), also by Bayes' théorem
The use of this approach was a major computational challenge for the time but it was eventually successful and the Scorpion was found in October of that yéar
Suppose a grid square has a probability p of containing the wreck and that the probability of successfully detecting the wreck if it is there is q
If the square is séarched and no wreck is found then, by Bayes, the revised probability of the wreck being in the square is given by formula_5 More mathematical examples
"See:" naive Bayesian classification
Posterior distribution of the binomial parameter
In this example we consider the computation of the posterior distribution for the binomial paraméter
This is the same problem considered by Bayes in Proposition 9 of his essay
We are given "m" observed successes and "n" observed failures in a binomial experiment
The experiment may be tossing a coin, drawing a ball from an urn, or asking soméone their opinion, among many other possibilities
What we know about the paraméter (let's call it "a") is stated as the prior distribution, "p"("a")
For a given value of "a", the probability of "m" successes in "m"+"n" trials is formula_6 Since "m" and "n" are fixed, and "a" is unknown, this is a likelihood function for "a"
From the continuous form of the law of total probability we have formula_7 For some special choices of the prior distribution "p"("a"), the integral can be solved and the posterior takes a convenient form
Dina minangkaan, lamun "p"("a") mangrupa sebaran beta nu mibanda paraméter "m"0 sarta "n"0, mangka "posterior" ogé sebaran beta nu mibanda paraméter "m"+"m"0 jeung "n"+"n"0
A "conjugate prior" is a prior distribution, such as the beta distribution in the above example, which has the property that the posterior is the same type of distribution
Kaputusan Bayesian geus dipaké dina widang artificial intelligence jeung expert system
téhnik kaputusan Bayesian geus dijadikeun dasar dina minangkaan tenik komputer pattern recognition mimiti taun 1950 katompernakeun
Mimiti tumuwuhna dina ngagunakeun kaputusan Bayesian keur filter spam
Contona: Bogofilter, SpamAssassin jeung Mozilla
In some applications fuzzy logic is an alternative to Bayesian inference
Fuzzy logic and Bayesian inference, however, are mathematically and semantically not compatible: You cannot, in general, understand the "degree of truth" in fuzzy logic as probability and vice versa
métodeu Monte Carlo nyaéta algoritma keur ngaréngsékeun rupa-rupa masalah dina itung-itungan make komputer ngagunakeun wilangan acak (atawa leuwih sering disebut wilangan bayangan), mangrupa hal sabalikna tina algoritma deterministik
métodeu Monte Carlo penting kacida dina komputasi fisik sarta aplikasi nu pakait, jeung bisa dipaké dina rupa-rupa itungan kromodinamika kuantum keur ngarancang pamisah panas sarta bentuk aerodinamika
métodeu ieu geus kabuktian efisien keur ngarengsekeun persamaan integro-differential di kondisi medan radian, sarta métodeu ieu geus digunakeun keur ""itungan" iluminasi global nu ngahasilkeun photo-réalistic images of virtual 3d modéls, digunakeun dina video games, arsitektur, disain, animasi komputer dina film sarta efek hususna, sarta widang-widang séjénna
Monte Carlo, kawentar keur kasino, nginjeum istilahna sabab métodeu ieu ngagunakeun kaayaan acak sarta ngagunakeun "pengulangan"" keur manggihkeun solusi nu panghadéna
Narikna, métodeu Monte Carlo teu merlukeun random numbers nu sabenerna keur digunakeun dina "perhitungan"
téhnik ieu leuwih gampang dipaké tinimbang deterministik, sekuen pseudo-random, gampang keur diuji sarta simulasi ulang
Sakadar kualitas nu penting keur nyieun simulasi nu hadé keur nyieun sekuen pseudo-random deukeut kana kaayaan "acak" nu dipikahayang
métodeu ieu bakal kasebar seragam atawa nuturkeun sebaran nu dipikahayang lamun jumlah wilanganna gedé tina sekuen nu ditempo
Sabab merlukeun algoritma "pengulangan" sarta "number" nu loba keur kaperluan "perhitungan", Monte Carlo mangrupa métodeu "penyeimbang" keur komputasi maké komputer, maké sababaraha téhnik simulasi komputer
Algoritma Monte Carlo nyaéta bentuk numeris métodeu Monte Carlo nu dipaké keur manggihkeun solusi dina masalah mathematika (nu mibanda variabel loba) nu harese direngsekeun, contona, maké kalkulus integral, atawa métodeu numeris séjénna
métodeu ieu leuwih hadé tinimbang métodeu séjénna lamun dimensi tina masalah leuwih loba
métodeu Monte Carlo asalna tina "pemakaian" praktis dina ngaran nu leuwih umum saperti "statistical sampling"
Sabaraha ahli nyebutkeun yén "Monte Carlo" mangrupa referensi kawentar dina kasino, sarta dipopulerkeun ku ahlina dina widang éta saperti Stanislaw Marcin Ulam, Enrico Fermi, John von Neumann sarta Nicholas Metropolis
Ahli séjénna nyebutkeun yén métodeu ieu mimiti didiskusikeun ku ahli nu hadir dina konferensi di Monte Carlo
métodeu ngitung random mimiti loba nu maké dina mangsa pre-electronic computing
métodeu ieu beuki kawentar sanggeus digunakeun Fermi dina 1930, waktu anjeunna maké métodeu random keur ngitung sifat anyar neutron
métodeu Monte Carlo jadi inti simulations nu diperlukeun dina Manhattan Project
Sanajan kitu, sanggeus komputer elektronik dijieun (mimiti taun 1945) métodeu Monte Carlo mimiti ditalungtik leuwih jero
métodeu deterministic numerical integration dipaké ku cara nyokot sajumlah kajadian dina ruang sampel tina hiji fungsi
Sacara umum, hasil tina pagawéan ieu hadé keur fungsi hiji variabel
ku sabab kitu, keur fungsi vectors, métodeu "deterministic quadrature" teu efisien
Keur integrsi numerik vektor dua-dimensi, diperlukeun ruang titik grid anu sarua dina "permukaan" dua dimensi
Contona keur grid 10x10 diperlukeun 100 titik
Lamun vektor mibanda 100 dimensi, jarak grid nu sarua merlukeun 10100 titik – hal ieu taya alesan keur bisa diitung
100 dimensions hartina teu mungkin keur diitung, saperti dina loba masalah fisik, "dimension" sarua jeung degree of freedom, sarta dina simulasi tilu-dimensi, di dinya aya tilu "degrees of freedom" per partikel
métodeu Monte Carlo nunjukkeun cara maké "exponential time-increase"
Saperti fungsi anu mibanda alesan well-behaved, hal ieu bisa di-"estimasi" ku milih sacara "random" tina ruang 100-dimensi, sarta nyokot sababaraha tipe "average"
maké central limit theorem, métodeu ieu bakal ditempokeun ku "konvergen"-na formula_1 – contona titik sampel lipat opat bakal boga satengah kasalahan, gumantung kana jumlah dimenasi
"Perbaikan" tina métodeu ieu nyaéta cara nyieun atawa milih titik "random", tapi leuwih dipikaresep datangna kana integral ti wewengkon nu konsentrasi loba tinimbang ti wewengkon nu konsentrasi saeutik
Dina basa séjén, titik sahenteuna ngagambarkeun bentuk nu hampir sarua jeung "integrand"
Teu salawasna dina pemodélan komputer méré hasil nu nyugemakeun dina integrasi mimiti, sanajan kitu aya métodeu séjén keur masalah ieu; dimimitian ku nyieun fungsi integrasi anu sederhana, salah sahijina bakal didiskusikeun dina topik di handap ieu
"Pendekatan" anu ampir sarua ngagunakeun low-discrepancy sequences ku quasi-Monte Carlo method
métodeu Quasi-Monte Carlo sok leuwih efisien dina integrasi numeris sabab sekuen "ngeusi" wewengkon leuwih hadé dina rasa sarta sampel penting nu dijieun tina simulasi konvergen keur ngahasilkeun solusi jadi leuwih gancang
Hal séjén anu kuat sarta kawentar dina aplikasi keur "random numbers" dina simulasi numeris nyaéta numerical optimisation
Masalah ieu salawasna maké fungsi nga-"minimal"-keun vektor dina dimensi anu gedé
Loba masalah anu bisa direngsekeun ku cara ieu; contona program computer chess geus nunjukkeun kumaha carana langkah nu optimal keur meunangkeun tarung, sebutkeun, 10 langkah nu ngahasilkeun evaluasi panghadéna ka tahap ahir
Traveling salesman problem contoh lain dina masalah optimasi
di dieu ogé aya sababaraha conto aplikasi dina masalah désain rekayasa, saperti multidisciplinary design optimization
Lolobana optimasi Monte Carlo didasarkeun kana random walks
Intina, program ieu bakal dijieun dina wanda ruang multi-dimensi, condong pindah ti fungsi luhur ka fungsi handap, sanajan kitu kadang-kadang pindahna tibalik tina gradient
Analisis numeris nyaéta pangajaran algoritma keur pasualan-pasualan"matematika kontinyu" (keur ngabédakeun jeung matematika diskrit)
One of the éarliest mathematical writings is the Babylonian tablet YBC 7289, which gives a sexagesimal numerical approximation of formula_1, the length of the diagonal in a unit square
Being able to compute the sides of a triangle (and hence, being able to compute square roots) is extremely important, for instance, in carpentry and construction
In a square wall section that is two méters by two méters, a diagonal béam has to be formula_2 méters long
Numerical analysis continues this long tradition of practical mathematical calculations
Much like the Babylonian approximation to formula_1, modérn numerical analysis does not seek exact answers, because exact answers are impossible to obtain in practice
Instéad, much of numerical analysis is concerned with obtaining approximate solutions while maintaining réasonable bounds on errors
Numerical analysis naturally finds applications in all fields of engineering and the physical sciences, but in the 21st century, the life sciences and even the arts have adopted elements of scientific computations
Ordinary differential equations appéar in the movement of héavenly bodies (planets, stars and galaxies); optimization occurs in portfolio management; numerical linear algebra is essential to quantitative psychology; stochastic differential equations and Markov chains are essential in simulating living cells for medicine and biology
Before the advent of modérn computers numerical methods often depended on hand interpolation in large printed tables
Nowadays (after mid 20th century) these tables have fallen into disuse, because computers can calculate the required functions
The interpolation algorithms nevertheless may be used as part of the software for solving differential equations and the like
We will now outline several important themes of numerical analysis
The overall goal is the design and analysis of techniques to give approximate solutions to hard problems
To fix idéas, the réader might consider the following problems and methods: History
The field of numerical analysis predates the invention of modérn computers by many centuries
Linear interpolation was alréady in use more than 2000 yéars ago
Many gréat mathematicians of the past were préoccupied by numerical analysis, as is obvious from the names of important algorithms like Newton's method, Lagrange interpolation polynomial, Gaussian elimination, or Euler's method
To facilitate computations by hand, large books were produced with formulas and tables of data such as interpolation points and function coefficients
Using these tables, often calculated out to 16 decimal places or more for some functions, one could look up values to plug into the formulas given and achieve very good numerical estimates of some functions
The canonical work in the field is the NIST publication edited by Abramowitz and Stegun, a 1000-plus page book of a very large number of commonly used formulas and functions and their values at many points
The function values are no longer very useful when a computer is available, but the large listing of formulas can still be very handy
The mechanical calculator was also developed as a tool for hand computation
These calculators evolved into electronic computers in the 1940s, and it was then found that these computers were also useful for administrative purposes
But the invention of the computer also influenced the field of numerical analysis, since now longer and more complicated calculations could be done
Direct and iterative methods
Direct methods compute the solution to a problem in a finite number of steps
These methods would give the precise answer if they were performed in infinite precision arithmetic
Examples include Gaussian elimination, the QR factorization method for solving systems of linéar equations, and the simplex method of linear programming
In practice, finite precision is used and the result is an approximation of the true solution (assuming stability)
In contrast to direct methods, iterative methods are not expected to terminate in a number of steps
Starting from an initial guess, iterative methods form successive approximations that converge to the exact solution only in the limit
A convergence criterion is specified in order to decide when a sufficiently accurate solution has (hopefully) been found
Even in infinite precision arithmetic these methods would not réach the solution in finitely many steps (in general)
Examples include Newton's method, the bisection method, and Jacobi iteration
In computational matrix algebra, iterative methods are generally needed for large problems
Iterative methods are more common than direct methods in numerical analysis
Some methods are direct in principle but are usually used as though they were not, e.g
GMRES and the conjugate gradient method
For these methods the number of steps needed to obtain the exact solution is so large that an approximation is accepted in the same manner as for an iterative method
Furthermore, continuous problems must sometimes be replaced by a discrete problem whose solution is known to approximate that of the continuous problem; this process is called "discretization"
For example, the solution of a differential equation is a function
This function must be represented by a finite amount of data, for instance by its value at a finite number of points at its domain, even though this domain is a continuum
The generation and propagation of errors
The study of errors forms an important part of numerical analysis
There are several ways in which error can be introduced in the solution of the problem
Round-off errors arise because it is impossible to represent all real numbers exactly on a finite-state machine (which is what all practical digital computers are)
On a pocket calculator, if one enters 0.0000000000001 (or the maximum number of zeros possible), then a +, and then 100000000000000 (again, the maximum number of zeros possible), one will obtain the number 100000000000000 again, and not 100000000000000.0000000000001
The calculator's answer is incorrect because of "roundoff" in the calculation
Truncation and discretization error
Truncation errors are committed when an iterative method is terminated and the approximate solution differs from the exact solution
Similarly, discretization induces a discretization error because the solution of the discrete problem does not coincide with the solution of the continuous problem
For instance, in the iteration in the sidebar to compute the solution of formula_4, after 10 or so iterations, we conclude that the root is roughly 1.99 (for example)
We therefore have a truncation error of 0.01
Once an error is generated, it will generally propagate through the calculation
For instance, we have alréady noted that the operation + on a calculator (or a computer) is inexact
It follows that a calculation of the type a+b+c+d+e is even more inexact
Numerical stability and well posedness
This léads to the notion of numerical stability: an algorithm is numerically stable if an error, once it is generated, does not grow too much during the calculation
This is only possible if the problem is well-conditioned, méaning that the solution changes by only a small amount if the problem data are changed by a small amount
Indeed, if a problem is ill-conditioned, then any error in the data will grow a lot
However, an algorithm that solves a well-conditioned problem may or may not be numerically stable
An art of numerical analysis is to find a stable algorithm for solving a well-posed mathematical problem
For instance, computing the square root of 2 (which is roughly 1.41421) is a well-posed problem
Many algorithms solve this problem by starting with an initial approximation "x"1 to formula_1, for instance "x"1=1.4, and then computing improved guesses "x"2, "x"3, etc... One such method is the famous Babylonian method, which is given by "x""k"+1 = "xk"/2 + 1/"xk"
Another iteration, which we will call Method X, is given by "x""k" + 1 = ("x""k"2−2)2 + "x""k"
We have calculated a few iterations of éach scheme in table form below, with initial guesses "x"1 = 1.4 and "x"1 = 1.42
Observe that the Babylonian method converges fast regardless of the initial guess, wheréas Method X converges extremely slowly with initial guess 1.4 and diverges for initial guess 1.42
Hence, the Babylonian method is said to be numerically stable, while Method X is said to be numerically unstable
The field of numerical analysis is divided in different disciplines according to the problem that is to be solved
Computing values of functions
One of the simplest problems is the evaluation of a function at a given point
The most straightforward approach, of just plugging in the number in the formula is sometimes not very efficient
For polynomials, a better approach is using the Horner scheme, since it reduces the necessary number of multiplications and additions
Generally, it is important to estimate and control round-off errors arising from the use of floating point arithmetic
Interpolation, extrapolation and regression
Interpolation solves the following problem: given the value of some unknown function at a number of points, what value does that function have at some other point between the given points
A very simple method is to use linear interpolation, which assumes that the unknown function is linéar between every pair of successive points