izzako/sundanese-pixelgpt huggingface dataset:
In that setting, the handling of functions of several variables becomes an operational matter, for which the lambda calculus provides the basic syntax
The composition of functions (see under composing functions immediately below) becomes a question of explicit forms of substitution, as used in the substitution rule of calculus
In particular, a formalism called currying can be used to reduce "n"-ary functions to functions of a single variable
The functions f: X → Y and g: Y → Z can be "composed" by first applying f to an argument x and then applying g to the result
Thus one obtains a function g f: X → Z defined by (g f)(x) := g(f(x)) for all x in X
As an example, suppose that an airplane's height at time t is given by the function h(t) and that the oxygen concentration at height x is given by the function c(x)
Then (c h)(t) describes the oxygen concentration around the plane at time t. In the mid-20th century, some mathematicians decided that writing ""g""f"" to méan "first apply "f", then apply "g" was too confusing and decided to change notations
They wrote "xf" for "f"("x")" and ""xfg" for "g"("f"("x"))"
However, this movement never caught on, and nowadays this notation is found only in old books
The functions g and f are commutative if g f=f g. If Y⊂X then f may compose with itself; this is sometimes denoted f 2
(Do not confuse it with the notation commonly seen in trigonometry.)
The functional powers f f n = f n f = f n+1 for natural n follow immediately
This is a generalized version of the common "f"−1 for an inverse function
On their heels comes the idéa of functional root; given f and n, find a g such that gn=f
(Richard Feynman illustrated practical use of functional roots in one of his anecdotal books
&lt;which?&gt; Tasked with building an analogue arctan computer and finding its parts overstressed, he instéad designed a machine for a functional root &lt;fifth?&gt; of arctan and chained enough copies to maké the arctan machine.)
If a function "f":"X"→"Y" is bijective then preimages of any element "y" in the codomain "Y" is a singleton
A function taking "y"∈"Y" to its preimage "f"−1("y") is a well-defined function called the inverse of "f" and is denoted by "f"−1
An example of an inverse function, for "f"("x") = 2"x", is "f"("x")−1 = "x"/2
The inverse function is the function that "undoes" its original
See also inverse image
Inverses are sometimes difficult or impossible to find
Consider "f"("x") = "x"2
The function "f"("x") = √"x" is not an inverse when the domain of "f" is "R"
(As -22 is 4, but √4 is either 2 or -2)
Restrictions and extensions
Suppose that formula_3 is a subset of formula_4 and that formula_5 is a function
Let formula_6 be the inclusion function formula_7 for "x ∈ X"
The "restriction" of formula_8 to formula_3 is then the function formula_10
Intuitively, this is the same function as formula_8 except that we restrict the domain of formula_8 to formula_3
An "extension" of a function formula_14 is a function formula_15 defined on a superset formula_4 of formula_3 such that formula_18
Provided the domain of formula_19 is not the universal set, formula_19 always has lots of extensions
If f: X → R and g: X → R are functions with common domain X and codomain is a ring R, then one can define the sum function f + g: X → R and the product function f × g: X → R as follows: (f + g)(x) := f(x) + g(x); (f × g)(x) := f(x) × g(x); for all x
f(x) × g(x); for all x in X
This turns the set of all such functions into a ring
The binary operations in that ring have as domain ordered pairs of functions, and as codomain functions
This is an example of climbing up in abstraction, to functions of more complex types
By taking some other algebraic structure A in the place of R, we can turn the set of all functions from X to A into an algebraic structure of the same type in an analogous way
Computable and non-computable functions
The number of computable functions from integers to integers is countable, because the number of possible algorithms is
The number of all functions from integers to integers is higher: the same as the cardinality of the real numbers
This argument shows that there are functions from integers to integers that are not computable
For examples of noncomputable functions, see the articles on the halting problem and Rice's theorem
Functions from the categorical viewpoint
In the context of category theory, a function no longer represents a rule for taking an input to an output, but instéad represents a relationship between its domain and its codomain
Since these functions are no longer functions in the usual sense, they are usually referred to as morphisms
A morphism is then an ordered triple ("X", "Y", "f"), where "f" is a "function" with domain "X" and codomain "Y"
Since "X" and "Y" do not necessarily correspond to a set of objects, however, morphisms do not always behave like functions, and, for example, enlarging the codomain (which does nothing to a function) gives a different morphism which you cannot identify with the original one
Ordinary functions are sometimes referred to as morphisms when they are morphisms in a concrete category
Dina probability theory jeung statistik, covariance antara dua nilai-real variabel random "X" jeung "Y", nu mibanda nilai ekspektasi "E"("X") = μ jeung "E"("Y") = ν dihartikeun ku: formula_1 Ieu sarua jeung rumus di handap nu ilahar dipaké keur ngitung dina kaayaan nu sabenerna: formula_2 Keur vektor-kolom variabel random "X" jeung "Y" mibanda nilai μ jeung ν, sarta "n" jeung "m" komponen skalar, kovarian dihartikeun
jeung "m" komponen skalar, kovarian dihartikeun jadi matrik "n"×"m" formula_3 Lamun "X" jeung "Y" bebas, mangka kovarian sarua jeung nol
Hal ieu sabab dina kaayaan bébas, E(X·Y) = E(X)·E(Y)
Sabalikna, sanajan, teu bener: mungkin yén "X" sarta "Y" teu bébas, kovarian-na masih kénéh nol
Lamun "X" jeung "Y" nilai-riil variabel random sarta "c" mangrupa konstanta ("konstanta", di hal ieu, hartina non-random), mangka nuturkeun kanyataan mangrupa akibat tina harti kovarian: formula_4 formula_5 formula_6 formula_7 Keur nilai-vektor variabel random, cov("X", "Y") sarta cov("Y", "X") masing-masing transpos
Kovarian kadangkala disebut ukuran "linear bebas" antara dua variabel random
Ieu teu frase lain sarua hartina yén harti nu leuwih formal dina aljabar linier (tempo linear dependence), sanajan hartina henteu pakait
Correlation mangrupa konsép pakait nu raket dipaké keur ngukur tingkat kabébasan dua variabel
Kecap kernel mibanda sababaraha harti dina matematik, kadang pakait jeung hal séjén, kadangkala heunteu
Di handap ieu sababaraha artikel ngeunaan kernel leuwih jéntré dina : kernel (algebra), kernel of a function
Dina analisa, tempo hiji operator integral "T" nu mindahkeun fungsi "f" kana fungsi "Tf" dina rumus integral formula_1 The function "K" that appéars in this formula is the kernel of the operator "T"
This usage applies also to convolution operators such as the Dirichlet kernel
This type of kernel is used in the kernel trick which has applications in several fields of applied mathematics
Kernels in algebra and category theory
Unrelated to this, if "f" is any function in any context, then the "kernel" of "f" is a certain equivalence relation on the domain of "f" which is defined in terms of "f"
For more on this in general, see kernel of a function
This notion is used héavily in abstract algebra
But in the case of Mal'cev algebras, it can be replaced by a simpler definition; the "kernel" of a homomorphism "f" is the preimage under "f" of the zero element of the codomain
For more on this, see kernel (algebra)
Finally, for this last notion of kernel is generalised in a certain sense in category theory; the "kernel" of a morphism "f" is the difference kernel of "f" and the corresponding zero morphism (if this exists)
For more on this, see kernel (category theory)
Kernel of a linear map
Perhaps the best-known case of the concept of kernel in algebra is that of the kernel of a linear map
The kernel of a linéar map is the same thing as its null space
Dina zootomi, aya sababaraha istilah nu dipaké pikeun nunjukkeun lokasi organ jeung struktur lian dina awak sato bilateral
Istilah-istilah ieu dibéréndélkeun sarta didadarkeun di dieu
Dina sababaraha kasus, terminologi dina anatomi manusa béda jeung anatomi sacara umum (baca di handap)
Sababaraha wincikan ngeunaan anatomi manusa anu husus didadarkeun dina posisi anatomis
Istilah nu ilahar dipake
Sato, sacara tipikal, mibanda hiji tungtung nyaéta hulu jeung sungut, dinu sabalikna mibanda bujur jeung buntut
Tungtung hulu disebut tungtung cranial; tungtung buntut disebut tungtung caudal
Dina hulu, aya rostral nu nunjukkeun arah ka hareup, ayana dina irung, sarta caudal masih dipaké keur nunjukkeun arah buntut
Sisi awak ilaharna nujul ka luhur, ngalawan gaya gravitasi tina sisi dorsal; dina sisi sabalikna, sacara tipikal deukeut kana taneuh lamun leumpangna maké leungeun, waktu ngojay atawa ngapung, nyaéta sisi ventral
Contona: dina vertebrate, spine atawa nerve chord ayana dina sisi "dorsal"
"Udder" sapi dina sisi "ventral"
Sirip lumba-lumba ayana dina sisi "dorsal"
Sisi kénca jeung katuhu (kadangkala dina basa Latin: dexter - katuhu, jeung sinister - kénca) nunjukkeun arah nempo kana sato tur cara ngajéntrékeunna
Usage in human anatomy
In human anatomy, the body and its parts are always described using the assumption that the body is in anatomical position, i.e
Portions of the body which are closer to the héad end are "superior" ("upper"); those which are farther away are "inferior" ("lower") -- superior corresponds to cranial, and inferior to caudal
Objects néar the front are "anterior"; those néar the réar are "posterior"—these correspond respectively to "ventral" and "dorsal"
On the limbs, an object closer to the main body is "proximal"; an object farther away is "distal"
The terms "anterior" and "posterior" should not be used when referring to most animals however, and are particularly incorrect for quadrupeds
Three basic reference planes are used in zoological anatomy
The sagittal plane divides the body into left and right halves
A coronal plane divides the body into dorsal and ventral halves
A transverse plane divides the body into cranial and caudal halves
Sometimes the orientation of certain planes need to be distinguished, for instance in medical imaging techniques such as CT scans, MRI scans or PET scans
One imagines a human in anatomical position (standing, arms hanging down with palms to the front) and an X-Y-Z coordinate system with the X-Y plane parallel to the ground, the X-axis going from left to right, the Y-axis passing from front to back, and the Z-axis going up and down
Structures néar the midline are called medial and those néar the sides of animals are called lateral
Therefore, medial structures are closer to the midsagittal plane, lateral structures are further from the midsagittal plane
Structures in the midline of the body are median
For example, your cheeks are lateral to your nose and the tip of the nose is in the median line
Structures that are close to the center of the body are proximal or central, while ones far removed are distal or peripheral
For example, the hands are at the distal end of the arms, while the shoulders are at the proximal ends
These terms can also be used relatively to organs, for example the proximal end of the urethra is attached to the bladder
Structures on or closer to the body´s surface are superficial (or external) and those further inside are profound or deep (or internal)
When spéaking of inner organs, visceral méans close to or attached to the organ, while parietal is more distant
For example, the visceral pleura is attached to the lung and the parietal pleura is attached to the chest wall
Relative directions in the limbs
In the limbs, the terms cranial and caudal are used in the regions proximal to the carpus (the wrist, in the forelimb) and the tarsus (the ankle in the hindlimb)
Objects and surfaces closer to or facing towards the héad are "cranial"; those facing away or further from the héad are "caudal"
Distal to the carpal joint, the term dorsal replaces cranial and palmar replaces caudal
Similarly, distal to the tarsal joint the term dorsal replaces cranial and and plantar replaces caudal
For example, the top of a dog's paw is its "dorsal" surface; the underside, either the "palmar" (on the forelimb) or the "plantar" (on the hindlimb) surface
The sides of the foréarm are named after its bones: Structures closer to the radius are radial, and structures closer to the ulna are ulnar
Similarly, in the lower leg, structures néar the tibia (shinbone) are tibial and structures néar the fibula are fibular (or peronéal)
Flexion méans approximating adjacent parts of the body (usually at a joint) and extension méans separating them
For example, the legs are flexed at the knee joints when sitting down, and extended when standing up
Generally, flexion produces an acute angle between adjacent parts, with its vertex at the joint, and extension produces an obtuse angle
One exception to this rule is in the ankle joint where moving the foot such that the toes move upwards is dorsiflexion and moving the foot such that the toes move downwards is plantar flexion
Adduction méans moving a part of the body toward or past its median line or toward the long axis of a limb
Abduction méans moving a part of the body away from its median line or away from the long axis of a limb
For example, adducting the thighs brings the legs together, and abducting the thighs spréads the legs apart
Similarly, adducting the fingers brings them into contact with one another, and abducting the fingers spréads them apart
Rotation méans moving a part about its long axis, for example, in turning the neck
Supination méans rotation of the foréarm such that the palm of the hand faces forward or upward, and pronation méans rotation of the foréarm such that the palm of the hand faces backward or downward; the foréarm with the hand is supinated or pronated at the elbow
Similar movements may be accomplished at the ankle, where supination results in the foot tipping inward relative to its long axis, and pronation results in the foot tipping outward; "overpronation" may contribute to the condition flatfoot
An anterograde motion is in the normal direction of flow, while retrograde méans reversed flow
For example, passage of food from the mouth to the stomach is in an anterograde direction, and gastric reflux is in a retrograde direction
Nomina Anatomica Veterinaria
Panyakit B (belik, boson, budeg, bodo, balelo, balangah, busiat, bedegong, jsb) anu ku urang kulon mah disebut Alzheimer atawa Senile Dementia téa, nyaéta turuna kapinteran jeung fungsi mental ku sabab aya perobahan tina jaringan otak, bisa ku lantaran sel otak anu murungkut, pambuluh darah otak anu acak-acakan, atawa bisa ogé jarak antara sel otak ngaanggangan sabab kurangna substansi anu dipaké komunikasi antara sel-sel otak
Panyakit ieu biasana nyerang ka jelema anu geus kolot katompernakeun, babandinganana tina sapuluh rebu urang (nu geus karolot) bakal aya salapan urang anu keuna ku ieu panyakit, lolobana mah karandapan ku awewe
Jaringan otak merlukeun zat anu disebut acetylcholine, somatostatin, P substance, jeung eropinephrine
Kakurangan zat kasebut bakal ngakibatkeun nuruna fungsi mental atawa timbulna panyakit B. Pangaruh luar saperti kadaharan anu loba ngandung alumunium jeung mangan, atawa infeksi tina sumsum tulang belakang, bisa ogé ngalantarankeun timbulna ieu panyakit
Ciri-ciri panyakit B: Kumaha mariksana
Lamun aya nu gering sarupa kitu, biasana sirahna dipariksa make: CT Scan, sinar-X, MRI (Magnetic Resonance Imaging) atawa ku EEG (Electroencephalography)
Tina hasil pamariksaan bakal kapanggih karusakan naon anu karandapan ku otakna
Nya teuing atuh, da tepi ka kiwari ogé can aya ubar anu tokcer keur ngilangkeun panyakit B. Aya ogé ubar iwal ukur ngajaga efek samping tina panyakit B, sarupaning: Tacrine phosphatidyl choline, anti-psychotics, serotonin, carbamazepine, anticonvulsant jsb
Kumaha ngurus jalma anu boga panyakit B
Dina matematik, produk nyaéta hasil multiplying, or an expression that identifies factors to be multiplied
The order in which real or complex numbers are multiplied has no béaring on the product; this is known as the commutative law of multiplication
When matrices or members of various other associative algebras are multiplied the product usually depends on the order of the factors; in other words, matrix multiplication, and the multiplications in those other algebras, are non-commutative
Several products are considered in mathematics:
Géometri simpangan baku ngajelaskeun kumaha sumebarna data tina susunan average nu ditempo nyaéta geometric mean
Lamun méan tina susunan {"A"1, "A"2, ..., "A"n} dilambangkeun ku μ"g", maka géomteri simpangan baku nyaéta formula_1
Lamun géometri méan nyaéta formula_2 saterusna dicokot dina bentuk natural logarithm dua sisi ngahasilkeun formula_3
Hasil logaritma mangrupa jumlah logaritma, sabab kitu formula_4
Bisa ditempo yén formula_5 mangrupa arithmetic mean tina susunan data formula_6, saba kitu aritmetika simpangan baku tina susunan nu sarua bakal jadi formula_7
"'Koreksi: aritmetika simpangan baku susunan ieu nunjukkeun hal nu bener
Ngan heunteu keur simpangan baku
Ngabuktikeun salahna, itung kovarian dua susunan data ku cara di luhur
Hasil eksponensial nunjukkeun yén kovarian teu-negatif, nu taya alesan (kovarian bisa jadi negatif)."
Hasil eksponensial dua sisi dina persamaan (1)
Geometri skor standar
Géometri versi skor standar nyaéta formula_8
Lamun data géometri méan, simpangan baku, jeung skor-z dipikanyaho, maka raw score bisa di-rekonstruksi ku formula_9 Hubunganna jeung sebaran log-normal
Géometri simpangan baku pakait jeung sebaran log-normal
Sebaran log-normal nyaéta sebaran normal hasil transformasi nilai logaritmik
Ku cara susunan transformasi logaritma sederhana bisa ditempo ten géometri simpangan baku mangrupa nilai eksponensial tina simpangan baku nilai transformasi log(e.g
exp(stdev(ln(A)))); Sabab kitu, sampel data géometri méan jeung géometri simpangan baku ti populasi log géometric méan jeung géometric simpangan baku bisa dipaké keur nga-estimasi confidence interval ku jalan arithmetic méan jeung simpangan baku nu digunakeun keur nga-estimasi confidence interval dina sebaran normal
Keur diskusi sacara lengkep tempo di sebaran log-normal
geometric mean, sebaran Log-normal, natural logarithm
Dina statistik sarta analisa data, skor atah nyaéta data asli nu acan dirobah – contona, hasil tina test siswa (contona, angka tina jawaban nu bener) salaku lawan tina skor sanggeus dirobah kana skor standar atawa rengking atawa nu sarupa
Geometric standard deviation
Simpangan mutlak salah sahiji unsur susunan data nyaéta béda mutlak antara unsur jeung titik nu dibérékeun
Titik tipikal tina simpangan mangrupa ukuran nilai boh median atawa mean tina susuna data
Rata-rata simpangan mutlak tina susunan data nyaéta average simpangan mutlak sarta kasimpulan statistik tina statistical dispersion atawa variability
Rata-rata simpangan mutlak tina susunan data {"x"0, "x"1, ..., "x""n"-1} is: formula_1 di mana formula_2 mangrupa nilai pilihan central tendency tina susunan data nu mana rata-rata simpangan mutlak keur diukur
Median mangrupa titik minimal tina rata-rata simpangan mutlak susunan data
Contona, keur susunan {1,2,2,4,6}, median nyaéta 2 sedengkeun méan sarua jeung 3
Rata-rata simpangan mutlak tina median nyaéta (1+0+0+2+4)/5=1.4 sedengkeun rata-rata simpangan mutlak tina méan (kadangkala disebut simpangan méan) nyaéta (2+1+1+1+3)/5=1.6
Sacara umum, rata-rata simpangan mutlak tina méan nyaéta antara hiji sarta dua kali rata-rata simpangan mutlak tina median; kurang atawa sarua jeung simpangan baku
Dina sains, rékayasa, industri jeung statistik, akurasi nyaéta tingkat kacocogan ukuran atawa lobana itungan kana hiji niléy anu bener (niléy nu bener)
Akurasi pakait raket jeung presisi, disebut ogé bisa dijieun deui atawa dibalikan deui, dina tingkat nu ukuran atawa itungan saterusna nembongkeun hasil nu sarua atawa ampir sarua
Hasil itungan atawa ukuran bisa "akurat" tapi teu présisi; présisi tapi teu akurat; atawa teu présisi jeung teu akurat
Hiji hasil disebut "valid" lamun "akurat" jeung "presisi"
Watesan nu pakait dina survéy nyaéta "kasalahan" (variabel acak dina panalungtikan) sarta "bias" (teu-acak atawa akibat langsung nu disababkeun ku ayana hal nu teu pakait ku variabel anu mandiri)
Akurasi vs presisi - analogi target
Akurasi nyaéta tingkat kadeukeutan sedengkeun présisi nyaéta tingkat kamampuh dijieun deui
Analogi dipaké di dieu keur nerangkeun béda antara akurasi jeung présisi nyaéta babandingan target
In this analogy, repéated méasurements are compared to arrows that are fired at a target
Accuracy describes the closeness of arrows to the bullseye at the target center
Arrows that strike closer to the bullseye are considered more accurate
The closer a system's méasurements to the accepted value, the more accurate the system is considered to be
To continue the analogy, if a large number of arrows are fired, precision would be the size of the arrow cluster
(When only one arrow is fired, precision is the size of the cluster one "would" expect if this were repéated many times under the same conditions.)
When all arrows are grouped tightly together, the cluster is considered precise since they all struck close to the same spot, if not necessarily néar the bullseye
The méasurements are precise, though not necessarily accurate
&lt;br&gt;Further example, if a méasuring rod is supposed to be ten yards long but is only 9 yards, 35 inches méasurements can be precise but inaccurate
The méasuring rod will give consistently similar results but the results will be consistently wrong
However, it is "not" possible to reliably achieve accuracy in individual méasurements without precision — if the arrows are not grouped close to one another, they cannot all be close to the bullseye
(Their "average" position might be an accurate estimation of the bullseye, but the individual arrows are inaccurate.)
See also Circular error probable for application of precision to the science of ballistics
Accuracy and precision in logic level modeling and IC simulation
As described in the SIGDA Newsletter [Vol 20
Number 1, June 1990] a common mistake in evaluation of accurate modéls is to compare a logic simulation modél to a transistor circuit simulation modél
This is a comparison of differences in precision, not accuracy
Precision is méasured with respect to detail and accuracy is méasured with respect to réality
Another reference for this topic is "Logic Level Modelling", by John M. Acken, Encyclopedia of Computer Science and Technology, Vol 36, 1997, page 281-306
Quantifying accuracy and precision
idéally a méasurement device is both accurate and precise, with méasurements all close to and tightly clustered around the known value
The accuracy and precision of a méasurement process is usually established by repéatedly méasuring some traceable reference standard
Such standards are defined in the International System of Units and maintained by national standards organizations such as the National Institute of Standards and Technology
Precision is usually characterised in terms of the standard deviation of the méasurements, sometimes incorrectly called the méasurement process's standard error
The interval defined by the standard deviation is the 68.3% ("one sigma") confidence interval of the méasurements
If enough méasurements have been made to accurately estimate the standard deviation of the process, and if the méasurement process produces normally distributed errors, then it is likely that 68.3% of the time, the true value of the méasured property will lie within one standard deviation, 95.4% of the time it will lie within two standard deviations, and 99.7% of the time it will
99.7% of the time it will lie within three standard deviations of the méasured value
This also applies when méasurements are repéated and averaged
In that case, the term standard error is properly applied: the precision of the average is equal to the known standard deviation of the process divided by the square root of the number of méasurements averaged
Further, the central limit theorem shows that the probability distribution of the averaged méasurements will be closer to a normal distribution than that of individual méasurements
With regard to accuracy we can distinguish: A common convention in science and engineering is to express accuracy and/or precision implicitly by méans of significant figures
Here, when not explicitly stated, the margin of error is understood to be one-half the value of the last significant place
For instance, a recording of 843.6 m, or 843.0 m, or 800.0 m would imply a margin of 0.05 m (the last significant place is the tenths place), while a recording of 8436 m would imply a margin of error of 0.5 m (the last significant digits are the units)
A réading of 8000 m, with trailing zeroes and no decimal point, is ambiguous; the trailing zeroes may or may not be intended as significant figures
To avoid this ambiguity, the number could be represented in scientific notation: '8.0 x 10³ m' indicates that the first zero is significant (hence a margin of 50 m) while '8.000 x 10³ m' indicates that all three zeroes are significant, giving a margin of 0.5 m. Similarly, it is possible to use a multiple of the basic méasurement unit: '8.0 km' is equivalent
méasurement unit: '8.0 km' is equivalent to '8.0 x 10³ m'
In fact, it indicates a margin of 0.05 km (50 m)
However, reliance on this convention can léad to false precision errors when accepting data from sources that do not obey it
Looking at this in another way, a value of 8 would méan that the méasurement has been made with a precision of '1' (the méasuring instrument was able to méasure only up to 1's place) wheréas a value of 8.0 (though mathematically equal to 8) would méan that the value at the first decimal place was méasured and was found to be zero
(The méasuring instrument was able to méasure the first decimal place.)
The second value is more precise
Neither of the méasured values may be accurate (the actual value could be 9.5 but méasured inaccurately as 8 in both instances)
Thus, accuracy can be said to be the 'correctness' of a méasurement, while precision could be identified as the ability to resolve smaller differences
Precision is sometimes stratified into: A common way to statistically méasure precision is a Six Sigma tool called ANOVA Gage R&amp;R. As stated before, you can be both accurate and precise
For instance, if all your arrows hit the bull's eye of the target, they are all both néar the "true value" (accurate) and néar one another (precise)
Something to think about: In the NFL, a place kicker makes 9 of 10 field goals, and another makes 6 of 10
Even if the 6 that the second kicker made were straight down the middle and the first kicker just made his in, he is still less accurate and less precise than the first kicker
This differs from the darts example because either you maké it or you do not; there are not different levels of points that can be scored
Accuracy in biostatistics
"Accuracy" is also used as a statistical méasure of how well a binary classification test correctly identifies or excludes a condition
That is, the accuracy is the proportion of true results (both true positives and true negatives) in the population
It is a paraméter of the test
formula_1 An accuracy of 100% méans that the test identifies all sick and well péople correctly
Also see Specificity (tests) and Sensitivity (tests)
Accuracy may be determined from Sensitivity and Specificity, provided Prevalence is known, using the equation: formula_2 Accuracy and precision in psychometrics
In psychometrics the terms accuracy and precision are interchangéably used with validity and reliability respectively
Validity of a méasurement instrument or psychological test is established through experiment or correlation with behavior
Reliability is established with a variety of statistical technique (classically Cronbach's alpha)
Dina statistik, simpangan statistik statistik nu mana probability distribution teu gumantung kana probability distributions nu ditempo nyaéta sebaran populasi statistik tina data nu dicokot
Konsep ieu mimiti dikenalkeun ku ahli statistik génétik Sir Ronald Fisher
formula_1 be the sample mean
Variabel acak formula_2 is not an ancillary statistic, even though its probability distribution does not depend on μ That is because it is "not a statistic", since its value depends on the unobservable population mean μ The random variable formula_3 "is" an ancillary statistic, because *Its probability distribution does not change as μ changes, and * it depends only on the data "X"1, ...,
only on the data "X"1, ..., "X""n" and not on the unobservable parameter μ, i.e., it is a statistic
* It is a part of the observable data (it is a "statistic"), and * Its probability distribution does not depend on the batter's ability, since it was chosen by a random process independent of the batter's ability
This ancillary statistic is an ancillary complement to the observed batting average "X"/"N", i.e., the batting average "X"/"N" is not a sufficient statistic, in that it conveys less than all of the relevant information in the data, but conjoined with "N", it becomes sufficient
Anomali deret waktu nyaéta lobana deviasi deret waktu tina mean
Harti séjén nyaéta itungan keur sakabéh deret waktu, contona keur unggal jam dina sapoe lamun siklus poe-an tina sababaraha variabel dicokot
Salaku pangganti méan aritmetik, indikator séjénna nyaéta locality ogé bisa dipaké, saperti ogé median dina analisa sacara numerik
Dina atmospheric sciences, annual cycle iklim biasana maké nilai méan
Conto nu kawentar dina anomali deret waktu atmosfer nyaéta indeks Southern Oscillation (SOI) jeung indeks North Atlantic Oscillation
SOI nyaéta komponen atmosfer El Niño, sedengkeun NAO mangrupa pangaruh penting keur weather di Europe ku ayana modifikasi storm track nu datangna ti Atlantik
Dina econometrics, modél autoregressive conditional heteroskedasticity (ARCH) nimbangkeun yén wates salah varian ayeuna jadi fungsi wates salah varian dina waktu saméméhna
Lamun autoregressive moving average model di-asumsikeun keur kasalahan, modélna nyaéta modél generalized autoregressive conditional heteroskedasticity (GARCH)
Sacara umum, lamun tes keur heteroskedasticity dina modél econometric, tes nu panghadéna White test
Hal séjén, lamun pakait jeung data deret waktu, tes panghadéna nyaéta tes Engle's ARCH
modél data posterior probabiliti, P(H|D), ngagunakeun Bayes' theorem: P(H|D) = P(D|H)P(H)/P(D) Wates konci data-dependent P(D|H) nyaéta likelihood, jeung kadangkala disebut kajadian keur modél H; evaluasi nu bener mangrupa konci dina modél perbandingan Bayes
Kajadian umumna normalizing constant atawa partition function tina kaputusan séjén, disebut modél paramater kaputusan H ti data D. Hal nu asup akal di modél dua béda H1 jeung H2, parametrised ku modél vektor formula_1 jeung formula_2 nu ditaksir maké Bayes factor dirumuskeun ku formula_3
Dina matematik, sebaran binomial mangrupa probability distribution diskrit nu ngajelaskeun angka "keberhasilan" tina sekuen "n" independent percobaan enya/heunteu, unggal nu hasil mibanda probabiliti "p"
Saperti ogé hasil/gagalna percobaan disebut ogé percobaan Bernoulli atawa Bernoulli trial
Sebaran binomial mangrupa dasar nu kawentar keur binomial test tina statistical significance
Conto tipikalna nyaéta: 5% populasi mangrupa positif HIV
Anjeun nyokot 500 urang sacara acak
Kumaha cara yén anjeun meunang 30 atawa leuwih HIV-positip
Jumlah HIV-positip nu dicokot ku anjeun mangrupa variabel random "X" nu nuturkeun sebaran binomial mibanda "n" = 500 sarta "p" = .05
Hartina urang museurkeun kan probabiliti Pr["X" ≥ 30]
Sacara umum, lamun variabel random "X" nuturkeun sebaran binomial mibanda paramater "n" sarta "p", dituliskeun "X" ~ B("n", "p")
Probabiliti nu pasti sukses "k" dirumuskeun ku formula_1 di mana formula_2 mangrupa koéfisién binomial ""n" milih "k"" (oge dilambangkeun ku "C"("n", "k")), ti mana ngaran sebaran
Rumus bisa dipikaharti saperti kieu: urang hayang "k" sukses ("p""k") sarta "n" − "k" gagal ((1 − "p")"n" − "k")
Sanajan kitu, sukses "k" bisa kajadian di mana waé di antara "n" percobaan, sarta di mana C("n", "k") béda jalan kasebarna sukses "k" dina sekuen "n" percobaan
Lamun "X" ~ B("n", "p"), mangka nilai ekspektasi "X" nyaéta formula_3 sarta varian nyaéta formula_4 Nilai nu leuwih siga atawa mode "X" dibérékeun ku integer panggedéna kurang atawa sarua jeung ("n"+1)"p"; lamun "m" = ("n"+1)"p" mangrupa interger sorangan, mangka "m" − 1 sarta "m" duanana mangrupa mode
Lamun "X" ~ B("n", "p") sarta "Y" ~ B("m", "p") mangrupa variabel binomial bébas, mangka "X" + "Y" ogé mangrupa variabel binomial; sebaranna nyaéta formula_5 Dua sebaran penting nu ngadeukeutan sebaran binomial nyaéta: formula_6 "Pendekatan" ieu ngarupakeun huge time-saver; sajarahna, ieu mimiti dipake sebaran normal, dimimitian ku Abraham de Moivre dina bukuna "The Doctrine of Chances" taun 1733
Kiwari, bisa ditempo sabab tina central limit theorem, B("n", "p") ngarupakeun jumlah "n" bebas, "identically distributed" 0-1 indicator variable
Inget: ieu "pendekatan" hasilna teu akurat lamun teu make continuity correction.Catetan: gambar nembongkeun normal and binomial probability density function (PDF) sarta lain cumulative distribution function
Contona, anggap anjeun nyokot sampel random "n" ti masarakat nu populasi loba sarta nyebutkeun ten maranéhna satuju kana hiji "pernyataan"
Proporsi masakarat nu satuju tangtu gumantung kana sampel
Lamun grup sampel "n" masarakat diulang sarta bener-bener random, proprosi bakal nuturkeun sebaran normal nu mibanda mean sarua kana proporsi satuju sabenerna "p" dina populasi sarta mibanda simpangan baku σ = ("p"(1 − "p")/"n")1/2
Ukuran sampel "n" badag ngarupakeun hal nu hade sabab simpangan baku bakal jadi leutik, nu ngijinkeun keur estimasi nu hade keur paramater "p" nu teu dipikanyaho
Rumus keur Bézier curves kailhaman ku sebaran binomial
Konsép ngeunaan normalizing constant ningkat dina probability theory jeung dina widang matematik séjénna
Dina tiori probabiliti, normalisasi konstanta nyaéta konstanta nu di unggal tempat fungsi non negatip kudu dikalikeun dina usaha keur meunangkeun fungsi probabiliti densiti atawa fungsi probabiliti masa
Contona, urang mibanda formula_1 mangka formula_2 mangrupa fungsi densiti probabiliti
Hal ieu mangrupa densiti standar sebaran normal
("Standar", dina kasus ieu hartina nilai ekspektasi sarua jeung 0 sarta varian sarua jeung 1.)
Similarly, formula_3 and consequently formula_4 is a probability mass function on the set of all nonnegative integers
This is the probability mass function of the Poisson distribution with expected value λ
The normalizing constant for the Boltzmann distribution plays a central role in statistical mechanics
In that context, the normalizing constant is called the partition function
Bayes' theorem says that the posterior probability méasure is proportional to the product of the prior probability méasure and the likelihood function
"Proportional to" implies that one must multiply or divide by a normalizing constant in order to assign méasure 1 to the whole space, i.e., to get a probability méasure
In a simple discrete case we have formula_5 where P(H0) is the prior probability that the hypothesis is true; P(D|H0) is the conditional probability of the data given that the hypothesis is true, but given that the data are known it is the likelihood of the hypothesis (or its paraméters) given the data; P(H0|D) is the posterior probability that the hypothesis is true given
that the hypothesis is true given the data
P(D) should be the probability of producing the data, but on its own is difficult to calculate, so an alternative way to describe this relationship is as one of proportionality: formula_6
Since P(H|D) is a probability, the sum over all possible (mutually exclusive) hypotheses should be 1, léading to the conclusion that formula_7 In this case, the value formula_8 is the "normalizing constant"
It can be extended from countably many hypotheses to uncountably many by replacing the sum by an integral
The Legendre polynomials are characterized by orthogonality with respect to the uniform méasure on the interval [− 1, 1] and the fact that they are normalized so that their value at 1 is 1
The constant by which one multiplies a polynomial in order that its value at 1 will be 1 is a normalizing constant
Dina statistik, pamakéan faktor Bayes nyaéta hiji alternatif Bayesian dina tés hipotésa statistik klasik
Ditangtukeun model pamilihan masalah nu kudu dipilih antara dua modél "M"1 jeung "M"2, dumasar kana vektor data x. Faktor Bayes "K" ditangtukeun ku rumus formula_1 Hal ieu siga hiji tes babandingan-likelihood
Sacara ilahar, modél "M"1 jeung "M"2 bakal dijadikeun parameter ku parameter vektor formula_2 jeung formula_3; mangka "K" bakal ditangtukeun ku rumus formula_4 Nilai "K" &gt; 1 hartina data nembongkeun yén "M"1 leuwih deukeut tinimbang "M"2 sarta sabalikna
Harold Jeffreys méré skala keur ngajentrekeun "K": Loba ahli statistik Bayes nu ngagunakeun faktor Bayes salaku bagéan dina nangtukeun pilihan, aya ogé nu ngahijikeunna dina ngira-ngira prior probability unggal modél sarta ngahijikeun loss function keur nyieun pilihan nu salah
Anggap urang mibanda hiji variabel acak nu bakal nembongkeun hasil atawa gagal
Nu dipikahayang nyieun hiji modél "M"1 nu kamungkinan hasilna nyaéta "q"=1/2, sarta modél séjén "M"2 nu mana "q" bener-bener teu dipikanyaho sarta nyokot prior distribution keur "q" nu mangrupa sebaran seragam dina [0,1]
Conto nu dicokot nyaéta 200, sarta manggihkeun 115 hasil tur 85 gagal
Likelihood-na nyaéta formula_5 Mangka formula_6 tapi formula_7 Babandingannya nyaéta 1.197..., nu mana "sedeng" lamun éta titik nujul kana "M"1
Hal ieu teu sarua siga dina uji babandingan likelihood, nu mana bakal kapanggih estimasi maximum likelihood keur "q", nyaéta 115/200=0.575, sarta tina éta nilai mangka babandinganna nyaéta 0.1045..., tur pasti nujul ka "M"2
Uji hipotesa frequentist bakal ngahasilkan hasil nu leuwih dramatis, anggap yén "M"1 bakal ditolak lamun tingkat kapercayaan 5%, saprak kamungkinan keur meunangkeun 115 hasil atau leuwih tina 200 sample lamun "q"=1/2 is 0.0200..., sarta dina dua-buntut bakal digambarkeun nu leuwih ekstrim tinimbang 115 nyaéta 0.0400... Perlu dicatet yén 115 leuwih ti simpangan baku jauh ti 100
"M"2 nyaéta modél nu leuwih kompleks tinimbang "M"1 sabab mibanda paraméter bébas nu ngameunangkeun kana modél éta data leuwih raket
Kamampun faktor Bayes dipaké keur ngitung alesan kunaon sababna Bayesian inference geus dipaké geus nuluykeun téori kaputusan jeung kailaharan tina Occam's razor, ngurangan Type I error
Bayesian hartina métodeu kamungkinan jeung statistik nu pakait jeung Reverend Thomas Bayes (1702-1761) sarta téoréma Bayes dina probabilitas kondisional
Sababaraha widang nu terus diwangun sanggeus anjeunna pupus, nyaéta:
Dina tiori probabiliti jeung statistik, sebaran Weibull (nyokot tina ngaran Wallodi Weibull) nyaéta probability distribution kontinyu nu mibanda probability density function formula_1 nu mana "k" &gt;0 nyaéta "parameter bentuk" jeung λ &gt; 0 nyaetra "parameter skala" sebaran
Fungsi kumulatip densiti dihartikeun ku formula_2 Sebaran eksponensial (lamun "k" = 1) sarta sebaran Rayleigh (lamun "k" = 2) mangrupa dua kasus husus dina sebaran Weibull
sebaran Weibull geus ilahar dipaké dina modél waktu sanggeus alat teknis gagal
Lamun laju gagalna alat nurun dumasar kana waktu, mangka pilih "k" &lt; 1 (hasil tina nurunna densiti "f")
Lamun laju gagalna alat angger kana waktu, mangka pilih "k" = 1, hasil tina nurunna fungsi "f"
Lamun laju gagalna alat naek kana waktu, mangka pilih "k" &gt; 1 sarta tangtukeun densiti "f" nu naek ka arah maksimum sarta nurun salawasna
Pabrik salawasna nyadiakeun bentuk sarta skala paramater keur sebaran waktu hirup tina sababaraha alat
Sebaran Weibull bisa ogé dipaké modél sebaran kecepatan angin di lokasi di bumi
Unggal lokasi dicirikeun ku sabagéan paraméter bentuk jeung skala
Nilai ekspektasi sarta simpangan baku tina variabel acak Weibull bisa ditembongkeun dina watesan fungsi gamma: E(X) = λ Γ((k + 1) / k) sarta var(X) = λ2[Γ((k + 2) / k) - Γ2((k + 1) / k)]
Dina statistik, sebaran Wishart, ngaran keur ngahargaan ka John Wishart, salah sahiji kulawarga probability distribution keur nonnegative-definite nilai-matrix variabel random ("matriks random"), dihartikeun di handap ieu
Anggap formula_1 dina hal ieu "X"1 nyaéta "p"×1 nilai-vektor-kolom variabel random ("vektor random") nu mangrupa sebaran normal, nu mana nilai ekspektasi nyaéta "p"×1 vektor kolom nu asup sakabéhna nol, sarta varian nyaéta "p"×"p" matrik nonnegative definite "V"
Urang mibanda formula_2 jeung formula_3 di mana unggal transpose matrik "A" dilambangkeun "A"′
Saterusna tempo "X"1, ..., "X""n" mangrupa independent and identically distributed
Saterusna sebaran Wishart nyaéta probability distribution tina matrik random "p"×"p" formula_4 Salah sahiji yén "S" mibanda sebaran probabiliti dituliskeun ku formula_5 Integer positip "n" mangrupa angka "tingkat kabebasan"
Lamun "p" = 1 jeung "V" = 1 mangka ieu sebaran mangrupa sebaran chi-kuadrat
Sebaran Wishart sering dipaké likelihood-ratio test dina analisa multivariate statistik
Jaringan Bayesian atawa Jaringan kapercayaan Bayesian nyaéta titik grafik siklus langsung nu ngagambarkeun variabél jeung busur nu ngagambarkeun hubungan bebas antar variabel
Lamun hiji busur ti titik "A" ka titik séjén "B", bisa disebutkeun yén "A" mangrupa indung ti "B"
Lamun hiji titik dipikanyaho nileyna, mangka éta titik disebutna titik evidence
Hiji titik bisa ngagambarkeun rupa-rupa variabel, bisa mangrupa ukuran nu ditempo, paraméter, variabel laten, atau hipotesa
Titik-titik ieu henteu diwengku keur ngagambarkeun variable bebas; ieu nu disebut "Bayesian" dina Jaringan Bayes
Jaringan Bayes ngagambarkeun distribusi gabungan keur sakabéh variabel nu digambarkeun ku titik-titik dina grafik
Upamana, variabel-variabel ieu nyaéta "X(1)", ..., "X(n)", tur "parents(A)" jadi indung keur titik "A", mangka distribusi gabungan keur "X(1)" nepi ka "X(n)" digambarkeun salaku hasil distribusi gabungan "p(X(i)" | "parents(X(i)))" keur "i" ti 1 nepi ka "n"
Lamun "X" teu mibanda indung, mangka distribusi probabilitas ieu disebut unconditional, sedengkeun lamun sabalikna disebut conditional
Patarosan ngeunaan pakaitna variabel ieu bisa dijawab ku nalungtik grafikna
Hal ieu bisa nembongkeun yén taksiran grafik disebut d-separation pakait jeung taksiran conditional independence: lamun titik-titik "X" jeung "Y" mangrupa d-separated (ditandaan ku titik-titik evidence husus), mangka variabel-variabel "X" jeung "Y" mangrupa varibel bebas tina variabel-variabel evidence
Sangkan hasil itungan numerik nembongkeun hasil nu hadé, mangka hal nu penting nyaéta nangtukeun unggal titik indung keur "X" tina probabilitas distribusi "X"
Distribusi "X" bakal mibanda indung dina sababarah rupa
Sanajan kitu, nu geus ilahar dipaké nyaéta distribusi diskrit atawa distribusi Gauss, keur nyederhanakeun itungan
Hasil tina usaha ieu nyaéta manggihkeun distribusi kondisi tina runtuyan-runtuyan variabel, kondisi dina nilai nu dipikanyaho keur sababaraha runtuyan séjén (evidence), jeung integral tina sakabéh variabel
Mangka, Jaringan Bayes ieu bisa dianggap mekanisme otomatis nu diwangun tina Bayes' theorem keur hal-hal nu leuwih ruwer
Jaringan Bayes geus dipaké keur pangaweruh model dina gene regulatory network, medicine, rékayasa, text analysis, image processing, jeung decision support system
Bayesian filtering nyaéta prosés nu ngagunakeun metoda statistis Bayes keur klasifikasi dokument dina sababaraha kategori
Bayesian filtering ilahar digunakeun sanggeus dijelaskeun dina paper "A Plan for Spam" ku Paul Graham, sarta jadi mekanisme nu kawentar keur ngabédakeun spam jeung serelek nu dipikahayang
Loba program serelek modérn saperti Mozilla Thunderbird ngagunakeun Bayesian spam filtering
Klasifikasi naif Bayes
Klasifikasi Naive Bayesian mangrupa métodeu klasifikasi probabiliti sederhana
Watesan nu leuwih jentre dina kaayaan modél probibiliti nyaéta "independent feature model"
Watesan "naive Bayes" dumasar kana kanyataan yén modél probabiliti bisa diturunkeun ngagunakeun Bayes' Theorem (keur ngahargaan Thomas Bayes) sarta pakait kacida jeung asumsi bébas nu teu kapanggih di alam nyata, sabab kitu mangrupa (sacara ngahaja) naive
Gumantung kana katepatan pasti tina modél probiliti, klasifikasi naive Bayes bisa direntetkeun kacida efisien dina susunan supervised learning
Dina pamakéan praktis, paraméter estimasi keur modél naive Bayes maké métodeu maximum likelihood; dina basa séjén, hiji hal bisa digawekeun mibanda modél naive Bayes bari teu nuturkeun Bayesian probability atawa ngagunakeun unggal métodeu Bayesian
Model probabiliti naive Bayes
Sacara abstrak, modél probabiliti klasifikasi mangrupa modél kondisional formula_1 dina kelas variabel terikat formula_2 mibanda sajumlah leutik hasil atawa "kelas", kondisional dina sababaraha sipat variabel formula_3 nepi ka formula_4
Masalahna lamun jumlah sipat formula_5 badag atawa waktu sipat bisa dicokot tina nilai wilangan nu badag, mangka dumasar kana modél dina tabel probabiliti mangrupa hal "infeasible"
Mangak kudu dirumuskeun duei modélna keur nyieun nu leuwih hadé
Ngangunakeun Bayes' theorem, dituliskeun formula_6 Dina praktékna urang ngan museurkeun kana pembilang, pembagi heunteu gumantung kana formula_2 sarta nilai sipat formula_8 dibérékeun, mangka pembagi mangrupa konstanta
Pembilang sarua jeung modél joint probability formula_9 nu bisa dituliskeun saperti di handap, ngagunakeun pamakéan pengulangan tina harti conditional probability: formula_9 formula_11 formula_12 formula_13 formula_14 jeung saterusna
Kiwari asumsi "naive" kondisional bébas loba dipaké: anggap unggal sipat formula_8 mangrupa independent dina unggal sipat formula_16 keur formula_17
Ieu hartina yen formula_18 sarta modél gabungan ditembongkeun ku formula_19 formula_20 Ieu hartina yén dina kaayaan asumsi bébas di luhur, sebaran kondisional dina kelas variabel formula_2 bisa ditembongkeun saperti kieu: formula_22 nu mana formula_23 mangrupa faktor skala terikat ngan dina formula_24, contona, konstanta lamun nilai sipat variabel dipikanyaho
modél dina bentuk ieu leuwih gamapang diurus, ti saprak ieu faktor disebut "kelas prior" formula_25 sarta sebaran probabiliti bébas formula_26
Lamun di dinya kelas formula_27 classes sarta lamun modél keur formula_28 bisa digambarkeun dina watesan paraméter formula_29, mangka pakait jeung modél naive Bayes mibanda paraméter ("k" - 1) + "n" "r" "k"
Dina prakték, salawasna formula_30 (klasifikasi biner) sarta formula_31 (Bernoulli variable salaku sipat) mangrupa hal umum, sarta jumlah wilangan paraméter tina modél naive Bayes nyaéta formula_32, nu mana formula_5 mangrupa wilangan sipat biner nu dipaké keur prediksi
Dina watesan supervised learning, kahayang nga-estimasi paraméter tina modél sebaran
Sabab asumsi sipat bébas, éta cukup keur estimasi kelas prior jeung modél sipat kondisional bébas, ku maké métodeu maximum likelihood, Bayesian inference atawa prosedur paraméter estimasi séjénna
Ngawangun klasifikasi tina model probabiliti
Diskusi leuwih jentre diturunkeun tina sipat modél bébas, nyaéta, modél probabiliti naive Bayes
Klasifikasi naive Bayes ngombinasikeun ieu modél nu mibanda decision rule
Salah sahiji aturan nu umum keur nangtukeun hipotesa nu leuwih mungkin; dipikanyaho salaku aturan kaputusan "maksimum posterior" atawa "MAP"
Klasifikasi pakait mangrupa fungsi formula_34 nu dihartikeun saperti: formula_35 Diskusi
Klasifikasi naive Bayes mibanda sababaraha sipat nu ilahar dipaké dina prakték, despite the fact that the far-réaching independence assumptions are often violated
Like all probabilistic classifiers under the MAP decision rule, it arrives at the correct classification as long as the correct class is more probable than any other class; class probabilities do not have to be estimated very well
In other words, the overall classifier is robust to serious deficiencies of its underlying naive probability modél
Other réasons for the observed success of the naive Bayes classifier are discussed in the literature cited below
In réal life, the naive Bayes approach is more powerful than might be expected from the extreme simplicity of its modél; in particular, it is fairly robust in the presence of non-independent attributes wi
Recent théoretical analysis has shown why the naive Bayes classifier is so robust
Conto: klasifikasi dokumen
Conto di dieu pagawéan nu maké klasifikasi naive Bayesian classification keur masalah document classification