AI And Analytics Course

AI And Analytics Course — independent reviews, comparisons, pricing and step-by-step guides on Aizhi.

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  LanguageWare

  LanguageWare is a natural language processing (NLP) technology developed by IBM, which allows applications to process natural language text. It comprises a set of Java libraries that provide a range of NLP functions: language identification, text segmentation/tokenization, normalization, entity and relationship extraction, and semantic analysis and disambiguation. The analysis engine uses a finite-state machine approach at multiple levels, which aids its performance characteristics while maintaining a reasonably small footprint. The behaviour of the system is driven by a set of configurable lexico-semantic resources which describe the characteristics and domain of the processed language. A default set of resources comes as part of LanguageWare and these describe the native language characteristics, such as morphology, and the basic vocabulary for the language. Supplemental resources have been created that capture additional vocabularies, terminologies, rules and grammars, which may be generic to the language or specific to one or more domains. A set of Eclipse-based customization tooling, LanguageWare Resource Workbench, is available on IBM's alphaWorks site, and allows domain knowledge to be compiled into these resources and thereby incorporated into the analysis process. LanguageWare can be deployed as a set of UIMA-compliant annotators, Eclipse plug-ins or Web Services.

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  Count sketch

  Count sketch is a type of dimensionality reduction that is particularly efficient in statistics, machine learning and algorithms. It was invented by Moses Charikar, Kevin Chen and Martin Farach-Colton in an effort to speed up the AMS Sketch by Alon, Matias and Szegedy for approximating the frequency moments of streams (these calculations require counting of the number of occurrences for the distinct elements of the stream). The sketch is nearly identical to the Feature hashing algorithm by John Moody, but differs in its use of hash functions with low dependence, which makes it more practical. In order to still have a high probability of success, the median trick is used to aggregate multiple count sketches, rather than the mean. These properties allow use for explicit kernel methods, bilinear pooling in neural networks and is a cornerstone in many numerical linear algebra algorithms. == Intuitive explanation == The inventors of this data structure offer the following iterative explanation of its operation: at the simplest level, the output of a single hash function s mapping stream elements q into {+1, -1} is feeding a single up/down counter C. After a single pass over the data, the frequency n ( q ) {\displaystyle n(q)} of a stream element q can be approximated, although extremely poorly, by the expected value E [ C ⋅ s ( q ) ] {\displaystyle {\mathbf {E}}[C\cdot s(q)]} ; a straightforward way to improve the variance of the previous estimate is to use an array of different hash functions s i {\displaystyle s_{i}} , each connected to its own counter C i {\displaystyle C_{i}} . For each i, the E [ C i ⋅ s i ( q ) ] = n ( q ) {\displaystyle {\mathbf {E}}[C_{i}\cdot s_{i}(q)]=n(q)} still holds, so averaging across the i range will tighten the approximation; the previous construct still has a major deficiency: if a lower-frequency-but-still-important output element a exhibits a hash collision with a high-frequency element even for one of the s i {\displaystyle s_{i}} hashes, n ( a ) {\displaystyle n(a)} estimate can be significantly affected. Avoiding this requires reducing the frequency of collision counter updates between any two distinct elements. This is achieved by replacing each C i {\displaystyle C_{i}} in the previous construct with an array of m counters (making the counter set into a two-dimensional matrix C i , j {\displaystyle C_{i,j}} ), with index j of a particular counter to be incremented/decremented selected via another set of hash functions h i {\displaystyle h_{i}} that map element q into the range {1..m}. Since E [ C i , h i ( q ) ⋅ s i ( q ) ] = n ( q ) {\displaystyle {\mathbf {E}}[C_{i,h_{i}(q)}\cdot s_{i}(q)]=n(q)} , averaging across all values of i will work. == Mathematical definition == 1. For constants w {\displaystyle w} and t {\displaystyle t} (to be defined later) independently choose d = 2 t + 1 {\displaystyle d=2t+1} random hash functions h 1 , … , h d {\displaystyle h_{1},\dots ,h_{d}} and s 1 , … , s d {\displaystyle s_{1},\dots ,s_{d}} such that h i : [ n ] → [ w ] {\displaystyle h_{i}:[n]\to [w]} and s i : [ n ] → { ± 1 } {\displaystyle s_{i}:[n]\to \{\pm 1\}} . It is necessary that the hash families from which h i {\displaystyle h_{i}} and s i {\displaystyle s_{i}} are chosen be pairwise independent. 2. For each item q i {\displaystyle q_{i}} in the stream, add s j ( q i ) {\displaystyle s_{j}(q_{i})} to the h j ( q i ) {\displaystyle h_{j}(q_{i})} th bucket of the j {\displaystyle j} th hash. At the end of this process, one has w d {\displaystyle wd} sums ( C i j ) {\displaystyle (C_{ij})} where C i , j = ∑ h i ( k ) = j s i ( k ) . {\displaystyle C_{i,j}=\sum _{h_{i}(k)=j}s_{i}(k).} To estimate the count of q {\displaystyle q} s one computes the following value: r q = median i = 1 d s i ( q ) ⋅ C i , h i ( q ) . {\displaystyle r_{q}={\text{median}}_{i=1}^{d}\,s_{i}(q)\cdot C_{i,h_{i}(q)}.} The values s i ( q ) ⋅ C i , h i ( q ) {\displaystyle s_{i}(q)\cdot C_{i,h_{i}(q)}} are unbiased estimates of how many times q {\displaystyle q} has appeared in the stream. The estimate r q {\displaystyle r_{q}} has variance O ( m i n { m 1 2 / w 2 , m 2 2 / w } ) {\displaystyle O(\mathrm {min} \{m_{1}^{2}/w^{2},m_{2}^{2}/w\})} , where m 1 {\displaystyle m_{1}} is the length of the stream and m 2 2 {\displaystyle m_{2}^{2}} is ∑ q ( ∑ i [ q i = q ] ) 2 {\displaystyle \sum _{q}(\sum _{i}[q_{i}=q])^{2}} . Furthermore, r q {\displaystyle r_{q}} is guaranteed to never be more than 2 m 2 / w {\displaystyle 2m_{2}/{\sqrt {w}}} off from the true value, with probability 1 − e − O ( t ) {\displaystyle 1-e^{-O(t)}} . === Vector formulation === Alternatively Count-Sketch can be seen as a linear mapping with a non-linear reconstruction function. Let M ( i ∈ [ d ] ) ∈ { − 1 , 0 , 1 } w × n {\displaystyle M^{(i\in [d])}\in \{-1,0,1\}^{w\times n}} , be a collection of d = 2 t + 1 {\displaystyle d=2t+1} matrices, defined by M h i ( j ) , j ( i ) = s i ( j ) {\displaystyle M_{h_{i}(j),j}^{(i)}=s_{i}(j)} for j ∈ [ w ] {\displaystyle j\in [w]} and 0 everywhere else. Then a vector v ∈ R n {\displaystyle v\in \mathbb {R} ^{n}} is sketched by C ( i ) = M ( i ) v ∈ R w {\displaystyle C^{(i)}=M^{(i)}v\in \mathbb {R} ^{w}} . To reconstruct v {\displaystyle v} we take v j ∗ = median i C j ( i ) s i ( j ) {\displaystyle v_{j}^{}={\text{median}}_{i}C_{j}^{(i)}s_{i}(j)} . This gives the same guarantees as stated above, if we take m 1 = ‖ v ‖ 1 {\displaystyle m_{1}=\|v\|_{1}} and m 2 = ‖ v ‖ 2 {\displaystyle m_{2}=\|v\|_{2}} . == Relation to Tensor sketch == The count sketch projection of the outer product of two vectors is equivalent to the convolution of two component count sketches. The count sketch computes a vector convolution C ( 1 ) x ∗ C ( 2 ) x T {\displaystyle C^{(1)}x\ast C^{(2)}x^{T}} , where C ( 1 ) {\displaystyle C^{(1)}} and C ( 2 ) {\displaystyle C^{(2)}} are independent count sketch matrices. Pham and Pagh show that this equals C ( x ⊗ x T ) {\displaystyle C(x\otimes x^{T})} – a count sketch C {\displaystyle C} of the outer product of vectors, where ⊗ {\displaystyle \otimes } denotes Kronecker product. The fast Fourier transform can be used to do fast convolution of count sketches. By using the face-splitting product such structures can be computed much faster than normal matrices.

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  Probably approximately correct learning

  In computational learning theory, probably approximately correct (PAC) learning is a framework for mathematical analysis of machine learning. It was proposed in 1984 by Leslie Valiant. In this framework, the learner receives samples and must select a generalization function (called the hypothesis) from a certain class of possible functions. The goal is that, with high probability (the "probably" part), the selected function will have low generalization error (the "approximately correct" part). The learner must be able to learn the concept given any arbitrary approximation ratio, probability of success, or distribution of the samples. The model was later extended to treat noise (misclassified samples). An important innovation of the PAC framework is the introduction of computational complexity theory concepts to machine learning. In particular, the learner is expected to find efficient functions (time and space requirements bounded to a polynomial of the example size), and the learner itself must implement an efficient procedure (requiring an example count bounded to a polynomial of the concept size, modified by the approximation and likelihood bounds). == Definitions and terminology == In order to give the definition for something that is PAC-learnable, we first have to introduce some terminology. For the following definitions, two examples will be used. The first is the problem of character recognition given an array of n {\displaystyle n} bits encoding a binary-valued image. The other example is the problem of finding an interval that will correctly classify points within the interval as positive and the points outside of the range as negative. Let X {\displaystyle X} be a set called the instance space or the encoding of all the samples. In the character recognition problem, the instance space is X = { 0 , 1 } n {\displaystyle X=\{0,1\}^{n}} . In the interval problem the instance space, X {\displaystyle X} , is the set of all bounded intervals in R {\displaystyle \mathbb {R} } , where R {\displaystyle \mathbb {R} } denotes the set of all real numbers. A concept is a subset c ⊂ X {\displaystyle c\subset X} . One concept is the set of all patterns of bits in X = { 0 , 1 } n {\displaystyle X=\{0,1\}^{n}} that encode a picture of the letter "P". An example concept from the second example is the set of open intervals, { ( a , b ) ∣ 0 ≤ a ≤ π / 2 , π ≤ b ≤ 13 } {\displaystyle \{(a,b)\mid 0\leq a\leq \pi /2,\pi \leq b\leq {\sqrt {13}}\}} , each of which contains only the positive points. A concept class C {\displaystyle C} is a collection of concepts over X {\displaystyle X} . This could be the set of all subsets of the array of bits that are skeletonized 4-connected (width of the font is 1). Let EX ⁡ ( c , D ) {\displaystyle \operatorname {EX} (c,D)} be a procedure that draws an example, x {\displaystyle x} , using a probability distribution D {\displaystyle D} and gives the correct label c ( x ) {\displaystyle c(x)} , that is 1 if x ∈ c {\displaystyle x\in c} and 0 otherwise. Now, given 0 < ϵ , δ < 1 {\displaystyle 0<\epsilon ,\delta <1} , assume there is an algorithm A {\displaystyle A} and a polynomial p {\displaystyle p} in 1 / ϵ , 1 / δ {\displaystyle 1/\epsilon ,1/\delta } (and other relevant parameters of the class C {\displaystyle C} ) such that, given a sample of size p {\displaystyle p} drawn according to EX ⁡ ( c , D ) {\displaystyle \operatorname {EX} (c,D)} , then, with probability of at least 1 − δ {\displaystyle 1-\delta } , A {\displaystyle A} outputs a hypothesis h ∈ C {\displaystyle h\in C} that has an average error less than or equal to ϵ {\displaystyle \epsilon } on X {\displaystyle X} with the same distribution D {\displaystyle D} . Further if the above statement for algorithm A {\displaystyle A} is true for every concept c ∈ C {\displaystyle c\in C} and for every distribution D {\displaystyle D} over X {\displaystyle X} , and for all 0 < ϵ , δ < 1 {\displaystyle 0<\epsilon ,\delta <1} then C {\displaystyle C} is (efficiently) PAC learnable (or distribution-free PAC learnable). We can also say that A {\displaystyle A} is a PAC learning algorithm for C {\displaystyle C} . == Equivalence == Under some regularity conditions these conditions are equivalent: The concept class C is PAC learnable. The VC dimension of C is finite. C is a uniformly Glivenko-Cantelli class. C is compressible in the sense of Littlestone and Warmuth

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  Dynamic time warping

  In time series analysis, dynamic time warping (DTW) is an algorithm for measuring similarity between two temporal sequences, which may vary in speed. For instance, similarities in walking could be detected using DTW, even if one person was walking faster than the other, or if there were accelerations and decelerations during the course of an observation. DTW has been applied to temporal sequences of video, audio, and graphics data — indeed, any data that can be turned into a one-dimensional sequence can be analyzed with DTW. A well-known application has been automatic speech recognition, to cope with different speaking speeds. Other applications include speaker recognition and online signature recognition. It can also be used in partial shape matching applications. In general, DTW is a method that calculates an optimal match between two given sequences (e.g. time series) with certain restriction and rules: Every index from the first sequence must be matched with one or more indices from the other sequence, and vice versa The first index from the first sequence must be matched with the first index from the other sequence (but it does not have to be its only match) The last index from the first sequence must be matched with the last index from the other sequence (but it does not have to be its only match) The mapping of the indices from the first sequence to indices from the other sequence must be monotonically increasing, and vice versa, i.e. if j > i {\displaystyle j>i} are indices from the first sequence, then there must not be two indices l > k {\displaystyle l>k} in the other sequence, such that index i {\displaystyle i} is matched with index l {\displaystyle l} and index j {\displaystyle j} is matched with index k {\displaystyle k} , and vice versa We can plot each match between the sequences 1 : M {\displaystyle 1:M} and 1 : N {\displaystyle 1:N} as a path in a M × N {\displaystyle M\times N} matrix from ( 1 , 1 ) {\displaystyle (1,1)} to ( M , N ) {\displaystyle (M,N)} , such that each step is one of ( 0 , 1 ) , ( 1 , 0 ) , ( 1 , 1 ) {\displaystyle (0,1),(1,0),(1,1)} . In this formulation, we see that the number of possible matches is the Delannoy number. The optimal match is denoted by the match that satisfies all the restrictions and the rules and that has the minimal cost, where the cost is computed as the sum of absolute differences, for each matched pair of indices, between their values. The sequences are "warped" non-linearly in the time dimension to determine a measure of their similarity independent of certain non-linear variations in the time dimension. This sequence alignment method is often used in time series classification. Although DTW measures a distance-like quantity between two given sequences, it doesn't guarantee the triangle inequality to hold. In addition to a similarity measure between the two sequences (a so called "warping path" is produced), by warping according to this path the two signals may be aligned in time. The signal with an original set of points X(original), Y(original) is transformed to X(warped), Y(warped). This finds applications in genetic sequence and audio synchronisation. In a related technique sequences of varying speed may be averaged using this technique see the average sequence section. This is conceptually very similar to the Needleman–Wunsch algorithm. == Implementation == This example illustrates the implementation of the dynamic time warping algorithm when the two sequences s and t are strings of discrete symbols. For two symbols x and y, d ( x , y ) {\displaystyle d(x,y)} is a distance between the symbols, e.g., d ( x , y ) = | x − y | {\displaystyle d(x,y)=|x-y|} . int DTWDistance(s: array [1..n], t: array [1..m]) { DTW := array [0..n, 0..m] for i := 0 to n for j := 0 to m DTW[i, j] := infinity DTW[0, 0] := 0 for i := 1 to n for j := 1 to m cost := d(s[i], t[j]) DTW[i, j] := cost + minimum(DTW[i-1, j ], // insertion DTW[i , j-1], // deletion DTW[i-1, j-1]) // match return DTW[n, m] } where DTW[i, j] is the distance between s[1:i] and t[1:j] with the best alignment. We sometimes want to add a locality constraint. That is, we require that if s[i] is matched with t[j], then | i − j | {\displaystyle |i-j|} is no larger than w, a window parameter. We can easily modify the above algorithm to add a locality constraint (differences marked). However, the above given modification works only if | n − m | {\displaystyle |n-m|} is no larger than w, i.e. the end point is within the window length from diagonal. In order to make the algorithm work, the window parameter w must be adapted so that | n − m | ≤ w {\displaystyle |n-m|\leq w} (see the line marked with () in the code). int DTWDistance(s: array [1..n], t: array [1..m], w: int) { DTW := array [0..n, 0..m] w := max(w, abs(n-m)) // adapt window size () for i := 0 to n for j:= 0 to m DTW[i, j] := infinity DTW[0, 0] := 0 for i := 1 to n for j := max(1, i-w) to min(m, i+w) DTW[i, j] := 0 for i := 1 to n for j := max(1, i-w) to min(m, i+w) cost := d(s[i], t[j]) DTW[i, j] := cost + minimum(DTW[i-1, j ], // insertion DTW[i , j-1], // deletion DTW[i-1, j-1]) // match return DTW[n, m] } == Warping properties == The DTW algorithm produces a discrete matching between existing elements of one series to another. In other words, it does not allow time-scaling of segments within the sequence. Other methods allow continuous warping. For example, Correlation Optimized Warping (COW) divides the sequence into uniform segments that are scaled in time using linear interpolation, to produce the best matching warping. The segment scaling causes potential creation of new elements, by time-scaling segments either down or up, and thus produces a more sensitive warping than DTW's discrete matching of raw elements. == Complexity == The time complexity of the DTW algorithm is O ( N M ) {\displaystyle O(NM)} , where N {\displaystyle N} and M {\displaystyle M} are the lengths of the two input sequences. The 50 years old quadratic time bound was broken in 2016: an algorithm due to Gold and Sharir enables computing DTW in O ( N 2 / log ⁡ log ⁡ N ) {\displaystyle O({N^{2}}/\log \log N)} time and space for two input sequences of length N {\displaystyle N} . This algorithm can also be adapted to sequences of different lengths. Despite this improvement, it was shown that a strongly subquadratic running time of the form O ( N 2 − ϵ ) {\displaystyle O(N^{2-\epsilon })} for some ϵ > 0 {\displaystyle \epsilon >0} cannot exist unless the Strong exponential time hypothesis fails. While the dynamic programming algorithm for DTW requires O ( N M ) {\displaystyle O(NM)} space in a naive implementation, the space consumption can be reduced to O ( min ( N , M ) ) {\displaystyle O(\min(N,M))} using Hirschberg's algorithm. == Fast computation == Fast techniques for computing DTW include PrunedDTW, SparseDTW, FastDTW, and the MultiscaleDTW. A common task, retrieval of similar time series, can be accelerated by using lower bounds such as LB_Keogh, LB_Improved, or LB_Petitjean. However, the Early Abandon and Pruned DTW algorithm reduces the degree of acceleration that lower bounding provides and sometimes renders it ineffective. In a survey, Wang et al. reported slightly better results with the LB_Improved lower bound than the LB_Keogh bound, and found that other techniques were inefficient. Subsequent to this survey, the LB_Enhanced bound was developed that is always tighter than LB_Keogh while also being more efficient to compute. LB_Petitjean is the tightest known lower bound that can be computed in linear time. == Average sequence == Averaging for dynamic time warping is the problem of finding an average sequence for a set of sequences. NLAAF is an exact method to average two sequences using DTW. For more than two sequences, the problem is related to that of multiple alignment and requires heuristics. DBA is currently a reference method to average a set of sequences consistently with DTW. COMASA efficiently randomizes the search for the average sequence, using DBA as a local optimization process. == Supervised learning == A nearest-neighbour classifier can achieve state-of-the-art performance when using dynamic time warping as a distance measure. == Amerced Dynamic Time Warping == Amerced Dynamic Time Warping (ADTW) is a variant of DTW designed to better control DTW's permissiveness in the alignments that it allows. The windows that classical DTW uses to constrain alignments introduce a step function. Any warping of the path is allowed within the window and none beyond it. In contrast, ADTW employs an additive penalty that is incurred each time that the path is warped. Any amount of warping is allowed, but each warping action incurs a direct penalty. ADTW significantly outperforms DTW with windowing when applied as a nearest neighbor classifier on a set of benchmark time series classification tasks. == Alternative approaches == In functional data analysis, time series are regarde

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  Spherical basis

  In pure and applied mathematics, particularly quantum mechanics and computer graphics and their applications, a spherical basis is the basis used to express spherical tensors. The spherical basis closely relates to the description of angular momentum in quantum mechanics and spherical harmonic functions. While spherical polar coordinates are one orthogonal coordinate system for expressing vectors and tensors using polar and azimuthal angles and radial distance, the spherical basis are constructed from the standard basis and use complex numbers. == In three dimensions == A vector A in 3D Euclidean space R3 can be expressed in the familiar Cartesian coordinate system in the standard basis ex, ey, ez, and coordinates Ax, Ay, Az: or any other coordinate system with associated basis set of vectors. From this extend the scalars to allow multiplication by complex numbers, so that we are now working in C 3 {\displaystyle \mathbb {C} ^{3}} rather than R 3 {\displaystyle \mathbb {R} ^{3}} . === Basis definition === In the spherical bases denoted e+, e−, e0, and associated coordinates with respect to this basis, denoted A+, A−, A0, the vector A is: where the spherical basis vectors can be defined in terms of the Cartesian basis using complex-valued coefficients in the xy plane: in which i {\displaystyle i} denotes the imaginary unit, and one normal to the plane in the z direction: e 0 = e z {\displaystyle \mathbf {e} _{0}=\mathbf {e} _{z}} The inverse relations are: === Commutator definition === While giving a basis in a 3-dimensional space is a valid definition for a spherical tensor, it only covers the case for when the rank k {\displaystyle k} is 1. For higher ranks, one may use either the commutator, or rotation definition of a spherical tensor. The commutator definition is given below, any operator T q ( k ) {\displaystyle T_{q}^{(k)}} that satisfies the following relations is a spherical tensor: [ J ± , T q ( k ) ] = ℏ ( k ∓ q ) ( k ± q + 1 ) T q ± 1 ( k ) {\displaystyle [J_{\pm },T_{q}^{(k)}]=\hbar {\sqrt {(k\mp q)(k\pm q+1)}}T_{q\pm 1}^{(k)}} [ J z , T q ( k ) ] = ℏ q T q ( k ) {\displaystyle [J_{z},T_{q}^{(k)}]=\hbar qT_{q}^{(k)}} === Rotation definition === Analogously to how the spherical harmonics transform under a rotation, a general spherical tensor transforms as follows, when the states transform under the unitary Wigner D-matrix D ( R ) {\displaystyle {\mathcal {D}}(R)} , where R is a (3×3 rotation) group element in SO(3). That is, these matrices represent the rotation group elements. With the help of its Lie algebra, one can show these two definitions are equivalent. D ( R ) T q ( k ) D † ( R ) = ∑ q ′ = − k k T q ′ ( k ) D q ′ q ( k ) {\displaystyle {\mathcal {D}}(R)T_{q}^{(k)}{\mathcal {D}}^{\dagger }(R)=\sum _{q'=-k}^{k}T_{q'}^{(k)}{\mathcal {D}}_{q'q}^{(k)}} === Coordinate vectors === For the spherical basis, the coordinates are complex-valued numbers A+, A0, A−, and can be found by substitution of (3B) into (1), or directly calculated from the inner product ⟨, ⟩ (5): A 0 = ⟨ e 0 , A ⟩ = ⟨ e z , A ⟩ = A z {\displaystyle A_{0}=\left\langle \mathbf {e} _{0},\mathbf {A} \right\rangle =\left\langle \mathbf {e} _{z},\mathbf {A} \right\rangle =A_{z}} with inverse relations: In general, for two vectors with complex coefficients in the same real-valued orthonormal basis ei, with the property ei·ej = δij, the inner product is: where · is the usual dot product and the complex conjugate must be used to keep the magnitude (or "norm") of the vector positive definite. == Properties (three dimensions) == === Orthonormality === The spherical basis is an orthonormal basis, since the inner product ⟨, ⟩ (5) of every pair vanishes meaning the basis vectors are all mutually orthogonal: ⟨ e + , e − ⟩ = ⟨ e − , e 0 ⟩ = ⟨ e 0 , e + ⟩ = 0 {\displaystyle \left\langle \mathbf {e} _{+},\mathbf {e} _{-}\right\rangle =\left\langle \mathbf {e} _{-},\mathbf {e} _{0}\right\rangle =\left\langle \mathbf {e} _{0},\mathbf {e} _{+}\right\rangle =0} and each basis vector is a unit vector: ⟨ e + , e + ⟩ = ⟨ e − , e − ⟩ = ⟨ e 0 , e 0 ⟩ = 1 {\displaystyle \left\langle \mathbf {e} _{+},\mathbf {e} _{+}\right\rangle =\left\langle \mathbf {e} _{-},\mathbf {e} _{-}\right\rangle =\left\langle \mathbf {e} _{0},\mathbf {e} _{0}\right\rangle =1} hence the need for the normalizing factors of 1 / 2 {\displaystyle 1/\!{\sqrt {2}}} . === Change of basis matrix === The defining relations (3A) can be summarized by a transformation matrix U: ( e + e − e 0 ) = U ( e x e y e z ) , U = ( − 1 2 − i 2 0 + 1 2 − i 2 0 0 0 1 ) , {\displaystyle {\begin{pmatrix}\mathbf {e} _{+}\\\mathbf {e} _{-}\\\mathbf {e} _{0}\end{pmatrix}}=\mathbf {U} {\begin{pmatrix}\mathbf {e} _{x}\\\mathbf {e} _{y}\\\mathbf {e} _{z}\end{pmatrix}}\,,\quad \mathbf {U} ={\begin{pmatrix}-{\frac {1}{\sqrt {2}}}&-{\frac {i}{\sqrt {2}}}&0\\+{\frac {1}{\sqrt {2}}}&-{\frac {i}{\sqrt {2}}}&0\\0&0&1\end{pmatrix}}\,,} with inverse: ( e x e y e z ) = U − 1 ( e + e − e 0 ) , U − 1 = ( − 1 2 + 1 2 0 + i 2 + i 2 0 0 0 1 ) . {\displaystyle {\begin{pmatrix}\mathbf {e} _{x}\\\mathbf {e} _{y}\\\mathbf {e} _{z}\end{pmatrix}}=\mathbf {U} ^{-1}{\begin{pmatrix}\mathbf {e} _{+}\\\mathbf {e} _{-}\\\mathbf {e} _{0}\end{pmatrix}}\,,\quad \mathbf {U} ^{-1}={\begin{pmatrix}-{\frac {1}{\sqrt {2}}}&+{\frac {1}{\sqrt {2}}}&0\\+{\frac {i}{\sqrt {2}}}&+{\frac {i}{\sqrt {2}}}&0\\0&0&1\end{pmatrix}}\,.} It can be seen that U is a unitary matrix, in other words its Hermitian conjugate U† (complex conjugate and matrix transpose) is also the inverse matrix U−1. For the coordinates: ( A + A − A 0 ) = U ∗ ( A x A y A z ) , U ∗ = ( − 1 2 + i 2 0 + 1 2 + i 2 0 0 0 1 ) , {\displaystyle {\begin{pmatrix}A_{+}\\A_{-}\\A_{0}\end{pmatrix}}=\mathbf {U} ^{\mathrm {} }{\begin{pmatrix}A_{x}\\A_{y}\\A_{z}\end{pmatrix}}\,,\quad \mathbf {U} ^{\mathrm {} }={\begin{pmatrix}-{\frac {1}{\sqrt {2}}}&+{\frac {i}{\sqrt {2}}}&0\\+{\frac {1}{\sqrt {2}}}&+{\frac {i}{\sqrt {2}}}&0\\0&0&1\end{pmatrix}}\,,} and inverse: ( A x A y A z ) = ( U ∗ ) − 1 ( A + A − A 0 ) , ( U ∗ ) − 1 = ( − 1 2 + 1 2 0 − i 2 − i 2 0 0 0 1 ) . {\displaystyle {\begin{pmatrix}A_{x}\\A_{y}\\A_{z}\end{pmatrix}}=(\mathbf {U} ^{\mathrm {} })^{-1}{\begin{pmatrix}A_{+}\\A_{-}\\A_{0}\end{pmatrix}}\,,\quad (\mathbf {U} ^{\mathrm {} })^{-1}={\begin{pmatrix}-{\frac {1}{\sqrt {2}}}&+{\frac {1}{\sqrt {2}}}&0\\-{\frac {i}{\sqrt {2}}}&-{\frac {i}{\sqrt {2}}}&0\\0&0&1\end{pmatrix}}\,.} === Cross products === Taking cross products of the spherical basis vectors, we find an obvious relation: e q × e q = 0 {\displaystyle \mathbf {e} _{q}\times \mathbf {e} _{q}={\boldsymbol {0}}} where q is a placeholder for +, −, 0, and two less obvious relations: e ± × e ∓ = ± i e 0 {\displaystyle \mathbf {e} _{\pm }\times \mathbf {e} _{\mp }=\pm i\mathbf {e} _{0}} e ± × e 0 = ± i e ± {\displaystyle \mathbf {e} _{\pm }\times \mathbf {e} _{0}=\pm i\mathbf {e} _{\pm }} === Inner product in the spherical basis === The inner product between two vectors A and B in the spherical basis follows from the above definition of the inner product: ⟨ A , B ⟩ = A + B + ⋆ + A − B − ⋆ + A 0 B 0 ⋆ {\displaystyle \left\langle \mathbf {A} ,\mathbf {B} \right\rangle =A_{+}B_{+}^{\star }+A_{-}B_{-}^{\star }+A_{0}B_{0}^{\star }}

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  Minimum Population Search

  In evolutionary computation, Minimum Population Search (MPS) is a computational method that optimizes a problem by iteratively trying to improve a set of candidate solutions with regard to a given measure of quality. It solves a problem by evolving a small population of candidate solutions by means of relatively simple arithmetical operations. MPS is a metaheuristic as it makes few or no assumptions about the problem being optimized and can search very large spaces of candidate solutions. For problems where finding the precise global optimum is less important than finding an acceptable local optimum in a fixed amount of time, using a metaheuristic such as MPS may be preferable to alternatives such as brute-force search or gradient descent. MPS is used for multidimensional real-valued functions but does not use the gradient of the problem being optimized, which means MPS does not require for the optimization problem to be differentiable as is required by classic optimization methods such as gradient descent and quasi-newton methods. MPS can therefore also be used on optimization problems that are not even continuous, are noisy, change over time, etc. == Background == In a similar way to Differential evolution, MPS uses difference vectors between the members of the population in order to generate new solutions. It attempts to provide an efficient use of function evaluations by maintaining a small population size. If the population size is smaller than the dimensionality of the search space, then the solutions generated through difference vectors will be constrained to the n − 1 {\displaystyle n-1} dimensional hyperplane. A smaller population size will lead to a more restricted subspace. With a population size equal to the dimensionality of the problem ( n = d ) {\displaystyle (n=d)} , the “line/hyperplane points” in MPS will be generated within a d − 1 {\displaystyle d-1} dimensional hyperplane. Taking a step orthogonal to this hyperplane will allow the search process to cover all the dimensions of the search space. Population size is a fundamental parameter in the performance of population-based heuristics. Larger populations promote exploration, but they also allow fewer generations, and this can reduce the chance of convergence. Searching with a small population can increase the chances of convergence and the efficient use of function evaluations, but it can also induce the risk of premature convergence. If the risk of premature convergence can be avoided, then a population-based heuristic could benefit from the efficiency and faster convergence rate of a smaller population. To avoid premature convergence, it is important to have a diversified population. By including techniques for explicitly increasing diversity and exploration, it is possible to have smaller populations with less risk of premature convergence. === Thresheld Convergence === Thresheld Convergence (TC) is a diversification technique which attempts to separate the processes of exploration and exploitation. TC uses a “threshold” function to establish a minimum search step, and managing this step makes it possible to influence the transition from exploration to exploitation, convergence is thus “held” back until the last stages of the search process. The goal of a controlled transition is to avoid an early concentration of the population around a few search regions and avoid the loss of diversity which can cause premature convergence. Thresheld Convergence has been successfully applied to several population-based metaheuristics such as Particle Swarm Optimization, Differential evolution, Evolution strategies, Simulated annealing and Estimation of Distribution Algorithms. The ideal case for Thresheld Convergence is to have one sample solution from each attraction basin, and for each sample solution to have the same relative fitness with respect to its local optimum. Enforcing a minimum step aims to achieve this ideal case. In MPS Thresheld Convergence is specifically used to preserve diversity and avoid premature convergence by establishing a minimum search step. By disallowing new solutions which are too close to members of the current population, TC forces a strong exploration during the early stages of the search while preserving the diversity of the (small) population. == Algorithm == A basic variant of the MPS algorithm works by having a population of size equal to the dimension of the problem. New solutions are generated by exploring the hyperplane defined by the current solutions (by means of difference vectors) and performing an additional orthogonal step in order to avoid getting caught in this hyperplane. The step sizes are controlled by the Thresheld Convergence technique, which gradually reduces step sizes as the search process advances. An outline for the algorithm is given below: Generate the first initial population. Allowing these solutions to lie near the bounds of the search space generally gives good results: s k = ( r s 1 ∗ b o u n d 1 / 2 , r s 2 ∗ b o u n d 2 / 2 , . . . , r s n ∗ b o u n d n / 2 ) {\displaystyle s_{k}=(rs_{1}bound_{1}/2,rs_{2}bound_{2}/2,...,rs_{n}bound_{n}/2)} where s k {\displaystyle s_{k}} is the k {\displaystyle k} -th population member, r s i {\displaystyle rs_{i}} are random numbers which can be −1 or 1, and the b o u n d i {\displaystyle bound_{i}} are the lower and upper bounds on each dimension. While a stop condition is not reached: Update threshold convergence values ( m i n _ s t e p {\displaystyle min\_step} and m a x _ s t e p {\displaystyle max\_step} ) Calculate the centroid of the current population ( x c {\displaystyle x_{c}} ) For each member of the population ( x i {\displaystyle x_{i}} ), generate a new offspring as follows: Uniformly generate a scaling factor ( F i {\displaystyle F_{i}} ) between − m a x _ s t e p {\displaystyle -max\_step} and m a x _ s t e p {\displaystyle max\_step} Generate a vector ( x o {\displaystyle x_{o}} ) orthogonal to the difference vector between x i {\displaystyle x_{i}} and x c {\displaystyle x_{c}} Calculate a scaling factor for the orthogonal vector: m i n _ o r t h = s q r t ( m a x ( m i n _ s t e p 2 − F i 2 , 0 ) ) {\displaystyle min\_orth=sqrt(max(min\_step^{2}-F_{i}^{2},0))} m a x _ o r t h = s q r t ( m a x ( m a x _ s t e p 2 − F i 2 , 0 ) ) {\displaystyle max\_orth=sqrt(max(max\_step^{2}-F_{i}^{2},0))} o r t h _ s t e p = u n i f o r m ( m i n _ o r t h , m a x _ o r t h ) {\displaystyle orth\_step=uniform(min\_orth,max\_orth)} Generate the new solution by adding the difference and the orthogonal vectors to the original solution n e w _ s o l u t i o n = x i + F i ∗ ( x i − x c ) ∗ o r t h _ s t e p ∗ x o {\displaystyle new\_solution=x_{i}+F_{i}(x_{i}-x_{c})orth\_stepx_{o}} Pick the best members between the old population and the new one by discarding the least fit members. Return the single best solution or the best population found as the final result.

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  Genotypic and phenotypic repair

  Genotypic and phenotypic repair are optional components of an evolutionary algorithm (EA). An EA reproduces essential elements of biological evolution as a computer algorithm in order to solve demanding optimization or planning tasks, at least approximately. A candidate solution is represented by a - usually linear - data structure that plays the role of an individual's chromosome. New solution candidates are generated by mutation and crossover operators following the example of biology. These offspring may be defective, which is corrected or compensated for by genotypic or phenotypic repair. == Description == Genotypic repair, also known as genetic repair, is the removal or correction of impermissible entries in the chromosome that violate restrictions. In phenotypic repair, the corrections are only made in the genotype-phenotype mapping and the chromosome remains unchanged. Michalewicz wrote about the importance of restrictions in real-world applications: "In general, constraints are an integral part of the formulation of any problem". Restriction violations are application-specific and therefore it depends on the current problem whether and which type of repair is useful. They can usually also be treated by a correspondingly extended evaluation and it depends on the problem which measures are possible and which is the most suitable. If a phenotypic repair is feasible, then it is usually the most efficient compared to the other measures. A survey on repair methods used as constraint handling techniques can be found in. Violations of the range limits of genes should be avoided as far as possible by the formulation of the genome. If this is not possible or if restrictions within the search space defined by the genome are involved, their violations are usually handled by the evaluation. This can be done, for example, by penalty functions that lower the fitness. Repair is often also required for combinatorial tasks. The application of a 1- or n-point crossover operator can, for example, lead to genes being missing in one of the child genomes that are present in duplicate in the other. In this case, a suitable genotypic repair measure is to move the surplus genes to the other genome in a positional manner. The use of the aforementioned operators in combinatorial tasks has also proven to be useful in combination with crossover types specially developed for permutations, at least for certain problems. Particularly in combinatorial problems, it has been observed that genotypic repair can promote premature convergence to a suboptimum, but can also significantly accelerate a successful search. Studies on various tasks have shown that this is application-dependent. An effective measure to avoid premature convergence is generally the use of structured populations instead of the usual panmictic ones. Sequence restrictions play a role in many scheduling tasks, for example when it comes to planning workflows. If, for example, it is specified that step A must be carried out before step B and the gene of step B is located before the gene of A in the chromosome, then there is an impermissible gene sequence. This is because the scheduling operation of step B requires the planned end of step A for correct scheduling, but this is not yet scheduled at the time gene B is processed. The problem can be solved in two ways: The scheduling operation of step B is postponed until the gene from step A has been processed. The genome remains unchanged and the repair only influences the genotype-phenotype mapping. Since only the phenotype is changed, this is referred to as phenotypic repair. If, on the other hand, the gene of step B is moved behind the gene of step A, this is a genotypic repair. The same applies to the alternative shift of gene A in front of gene B. In this case, genotypic repair has the disadvantage that it prevents a meaningful restructuring of the gene sequence in the chromosome if this requires several intermediate steps (mutations) that at least partially violate restrictions.

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  Constructing skill trees

  Constructing skill trees (CST) is a hierarchical reinforcement learning algorithm which can build skill trees from a set of sample solution trajectories obtained from demonstration. CST uses an incremental MAP (maximum a posteriori) change point detection algorithm to segment each demonstration trajectory into skills and integrate the results into a skill tree. CST was introduced by George Konidaris, Scott Kuindersma, Andrew Barto and Roderic Grupen in 2010. == Algorithm == CST consists of mainly three parts;change point detection, alignment and merging. The main focus of CST is online change-point detection. The change-point detection algorithm is used to segment data into skills and uses the sum of discounted reward R t {\displaystyle R_{t}} as the target regression variable. Each skill is assigned an appropriate abstraction. A particle filter is used to control the computational complexity of CST. The change point detection algorithm is implemented as follows. The data for times t ∈ T {\displaystyle t\in T} and models Q with prior p ( q ∈ Q ) {\displaystyle p(q\in Q)} are given. The algorithm is assumed to be able to fit a segment from time j + 1 {\displaystyle j+1} to t using model q with the fit probability P ( j , t , q ) {\displaystyle P(j,t,q)_{}^{}} . A linear regression model with Gaussian noise is used to compute P ( j , t , q ) {\displaystyle P(j,t,q)} . The Gaussian noise prior has mean zero, and variance which follows I n v e r s e G a m m a ( v 2 , u 2 ) {\displaystyle \mathrm {InverseGamma} \left({\frac {v}{2}},{\frac {u}{2}}\right)} . The prior for each weight follows N o r m a l ( 0 , σ 2 δ ) {\displaystyle \mathrm {Normal} (0,\sigma ^{2}\delta )} . The fit probability P ( j , t , q ) {\displaystyle P(j,t,q)} is computed by the following equation. P ( j , t , q ) = π − n 2 δ m | ( A + D ) − 1 | 1 2 u v 2 ( y + u ) u + v 2 Γ ( n + v 2 ) Γ ( v 2 ) {\displaystyle P(j,t,q)={\frac {\pi ^{-{\frac {n}{2}}}}{\delta ^{m}}}\left|(A+D)^{-1}\right|^{\frac {1}{2}}{\frac {u^{\frac {v}{2}}}{(y+u)^{\frac {u+v}{2}}}}{\frac {\Gamma ({\frac {n+v}{2}})}{\Gamma ({\frac {v}{2}})}}} Then, CST compute the probability of the changepoint at time j with model q, P t ( j , q ) {\displaystyle P_{t}(j,q)} and P j MAP {\displaystyle P_{j}^{\text{MAP}}} using a Viterbi algorithm. P t ( j , q ) = ( 1 − G ( t − j − 1 ) ) P ( j , t , q ) p ( q ) P j MAP {\displaystyle P_{t}(j,q)=(1-G(t-j-1))P(j,t,q)p(q)P_{j}^{\text{MAP}}} P j MAP = max i , q P j ( i , q ) g ( j − i ) 1 − G ( j − i − 1 ) , ∀ j < t {\displaystyle P_{j}^{\text{MAP}}=\max _{i,q}{\frac {P_{j}(i,q)g(j-i)}{1-G(j-i-1)}},\forall j Read more →

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  SAP NetWeaver Visual Composer

  SAP NetWeaver Visual Composer is SAP’s web-based software modelling tool. It enables business process specialists and developers to create business application components, without coding. Visual Composer produces applications in a declarative form, enabling code-free execution mode for multiple runtime environments. It provides application lifecycle support by maintaining the connection between an application and its model throughout its lifecycle. Visual Composer is designed with an open architecture, which enables developers to extend its design-time environment and modelling language, as well as to integrate external data services. The tool aims to increase productivity by reducing development effort time, and narrowing the gap between application definition and implementation. Starting with a blank canvas, the Visual Composer user, typically a business process specialist, draws the application in Visual Composer Storyboard (workspace), without writing code, to prototype, design and produce applications. A typical workflow for creating, deploying and running an application using Visual Composer is: Create a model Discover data services and add them to the model Select necessary UI elements and add them to the model Connect model elements to define the model logic and data flow Edit the layout Arranging the UI elements and the controls of the application on forms and tables. Deploy the model This step includes compilation, validation and deployment to a selected environment. Run the application The application can run using different runtime environment (such as Adobe Flex and HTML). In 2014 a runtime environment was introduced that is utilizing HTML5 capabilities of SAPUI5.

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  XGBoost

  XGBoost (eXtreme Gradient Boosting) is an open-source software library which provides a regularizing gradient boosting framework for C++, Java, Python, R, Julia, Perl, and Scala. It works on Linux, Microsoft Windows, and macOS. From the project description, it aims to provide a "Scalable, Portable and Distributed Gradient Boosting (GBM, GBRT, GBDT) Library". It runs on a single machine, as well as the distributed processing frameworks Apache Hadoop, Apache Spark, Apache Flink, and Dask. XGBoost gained much popularity and attention in the mid-2010s as the algorithm of choice for many winning teams of machine learning competitions. == History == XGBoost initially started as a research project by Tianqi Chen as part of the Distributed (Deep) Machine Learning Community (DMLC) group at the University of Washington. Initially, it began as a terminal application which could be configured using a libsvm configuration file. It became well known in the ML competition circles after its use in the winning solution of the Higgs Machine Learning Challenge. Soon after, the Python and R packages were built, and XGBoost now has package implementations for Java, Scala, Julia, Perl, and other languages. This brought the library to more developers and contributed to its popularity among the Kaggle community, where it has been used for a large number of competitions. It was soon integrated with a number of other packages making it easier to use in their respective communities. It has now been integrated with scikit-learn for Python users and with the caret package for R users. It can also be integrated into Data Flow frameworks like Apache Spark, Apache Hadoop, and Apache Flink using the abstracted Rabit and XGBoost4J. XGBoost is also available on OpenCL for FPGAs. An efficient, scalable implementation of XGBoost has been published by Tianqi Chen and Carlos Guestrin. While the XGBoost model often achieves higher accuracy than a single decision tree, it sacrifices the intrinsic interpretability of decision trees. For example, following the path that a decision tree takes to make its decision is trivial and self-explained, but following the paths of hundreds or thousands of trees is much harder. == Features == Salient features of XGBoost which make it different from other gradient boosting algorithms include: Clever penalization of trees A proportional shrinking of leaf nodes Newton Boosting Extra randomization parameter Implementation on single, distributed systems and out-of-core computation Automatic feature selection Theoretically justified weighted quantile sketching for efficient computation Parallel tree structure boosting with sparsity Efficient cacheable block structure for decision tree training == The algorithm == XGBoost works as Newton–Raphson in function space unlike gradient boosting that works as gradient descent in function space, a second order Taylor approximation is used in the loss function to make the connection to Newton–Raphson method. A generic unregularized XGBoost algorithm is: == Parameters == XGBoost has parameters which can be specified to affect how it functions and performs. Some parameters include: Learning rate (also known as the "step size" or “"shrinkage"), it is a number between 0 and 1 (default is 0.3), which determines the rate the algorithm learns from each iteration. n_estimators, sets the number of trees to be built in the ensemble, where more trees generally increases the complexity of the model, but can lead to overfitting with too many trees. Gamma (also known as Lagrange multiplier or the minimum loss reduction parameter), controls the minimum amount of loss reduction required to make a further split on a leaf node of the tree. The default in XGBoost is 0 . max_depth, represents how deeply each tree in the boosting process can grow during training, where the default is 6. == Awards == John Chambers Award (2016) High Energy Physics meets Machine Learning award (HEP meets ML) (2016)

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  Synaptic weight

  In neuroscience and computer science, synaptic weight refers to the strength or amplitude of a connection between two nodes, corresponding in biology to the amount of influence the firing of one neuron has on another. The term is typically used in artificial and biological neural network research. == Computation == In a computational neural network, a vector or set of inputs x {\displaystyle {\textbf {x}}} and outputs y {\displaystyle {\textbf {y}}} , or pre- and post-synaptic neurons respectively, are interconnected with synaptic weights represented by the matrix w {\displaystyle w} , where for a linear neuron y j = ∑ i w i j x i or y = w x {\displaystyle y_{j}=\sum _{i}w_{ij}x_{i}~~{\textrm {or}}~~{\textbf {y}}=w{\textbf {x}}} . where the rows of the synaptic matrix represent the vector of synaptic weights for the output indexed by j {\displaystyle j} . The synaptic weight is changed by using a learning rule, the most basic of which is Hebb's rule, which is usually stated in biological terms as Neurons that fire together, wire together. Computationally, this means that if a large signal from one of the input neurons results in a large signal from one of the output neurons, then the synaptic weight between those two neurons will increase. The rule is unstable, however, and is typically modified using such variations as Oja's rule, radial basis functions or the backpropagation algorithm. == Biology == For biological networks, the effect of synaptic weights is not as simple as for linear neurons or Hebbian learning. However, biophysical models such as BCM theory have seen some success in mathematically describing these networks. In the mammalian central nervous system, signal transmission is carried out by interconnected networks of nerve cells, or neurons. For the basic pyramidal neuron, the input signal is carried by the axon, which releases neurotransmitter chemicals into the synapse which is picked up by the dendrites of the next neuron, which can then generate an action potential which is analogous to the output signal in the computational case. The synaptic weight in this process is determined by several variable factors: How well the input signal propagates through the axon (see myelination), The amount of neurotransmitter released into the synapse and the amount that can be absorbed in the following cell (determined by the number of AMPA and NMDA receptors on the cell membrane and the amount of intracellular calcium and other ions), The number of such connections made by the axon to the dendrites, How well the signal propagates and integrates in the postsynaptic cell. The changes in synaptic weight that occur is known as synaptic plasticity, and the process behind long-term changes (long-term potentiation and depression) is still poorly understood. Hebb's original learning rule was originally applied to biological systems, but has had to undergo many modifications as a number of theoretical and experimental problems came to light.

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  Autoencoder

  An autoencoder is a type of artificial neural network used to learn efficient codings of unlabeled data (unsupervised learning). An autoencoder learns two functions: an encoding function that transforms the input data, and a decoding function that recreates the input data from the encoded representation. The autoencoder learns an efficient representation (encoding) for a set of data, typically for dimensionality reduction, to generate lower-dimensional embeddings for subsequent use by other machine learning algorithms. Variants exist which aim to make the learned representations assume useful properties. Examples are regularized autoencoders (sparse, denoising and contractive autoencoders), which are effective in learning representations for subsequent classification tasks, and variational autoencoders, which can be used as generative models. Autoencoders are applied to many problems, including facial recognition, feature detection, anomaly detection, and learning the meaning of words. In terms of data synthesis, autoencoders can also be used to randomly generate new data that is similar to the input (training) data. == Mathematical principles == === Definition === An autoencoder is defined by the following components: Two sets: the space of encoded messages Z {\displaystyle {\mathcal {Z}}} ; the space of decoded messages X {\displaystyle {\mathcal {X}}} . Typically X {\displaystyle {\mathcal {X}}} and Z {\displaystyle {\mathcal {Z}}} are Euclidean spaces, that is, X = R m , Z = R n {\displaystyle {\mathcal {X}}=\mathbb {R} ^{m},{\mathcal {Z}}=\mathbb {R} ^{n}} with m > n . {\displaystyle m>n.} Two parametrized families of functions: the encoder family E ϕ : X → Z {\displaystyle E_{\phi }:{\mathcal {X}}\rightarrow {\mathcal {Z}}} , parametrized by ϕ {\displaystyle \phi } ; the decoder family D θ : Z → X {\displaystyle D_{\theta }:{\mathcal {Z}}\rightarrow {\mathcal {X}}} , parametrized by θ {\displaystyle \theta } .For any x ∈ X {\displaystyle x\in {\mathcal {X}}} , we usually write z = E ϕ ( x ) {\displaystyle z=E_{\phi }(x)} , and refer to it as the code, the latent variable, latent representation, latent vector, etc. Conversely, for any z ∈ Z {\displaystyle z\in {\mathcal {Z}}} , we usually write x ′ = D θ ( z ) {\displaystyle x'=D_{\theta }(z)} , and refer to it as the (decoded) message. Usually, both the encoder and the decoder are defined as multilayer perceptrons (MLPs). For example, a one-layer-MLP encoder E ϕ {\displaystyle E_{\phi }} is: E ϕ ( x ) = σ ( W x + b ) {\displaystyle E_{\phi }(\mathbf {x} )=\sigma (Wx+b)} where σ {\displaystyle \sigma } is an element-wise activation function, W {\displaystyle W} is a "weight" matrix, and b {\displaystyle b} is a "bias" vector. === Training an autoencoder === An autoencoder, by itself, is simply a tuple of two functions. To judge its quality, we need a task. A task is defined by a reference probability distribution μ r e f {\displaystyle \mu _{ref}} over X {\displaystyle {\mathcal {X}}} , and a "reconstruction quality" function d : X × X → [ 0 , ∞ ] {\displaystyle d:{\mathcal {X}}\times {\mathcal {X}}\to [0,\infty ]} , such that d ( x , x ′ ) {\displaystyle d(x,x')} measures how much x ′ {\displaystyle x'} differs from x {\displaystyle x} . With those, we can define the loss function for the autoencoder as L ( θ , ϕ ) := E x ∼ μ r e f [ d ( x , D θ ( E ϕ ( x ) ) ) ] {\displaystyle L(\theta ,\phi ):=\mathbb {\mathbb {E} } _{x\sim \mu _{ref}}[d(x,D_{\theta }(E_{\phi }(x)))]} The optimal autoencoder for the given task ( μ r e f , d ) {\displaystyle (\mu _{ref},d)} is then arg ⁡ min θ , ϕ L ( θ , ϕ ) {\displaystyle \arg \min _{\theta ,\phi }L(\theta ,\phi )} . The search for the optimal autoencoder can be accomplished by any mathematical optimization technique, but usually by gradient descent. This search process is referred to as "training the autoencoder". In most situations, the reference distribution is just the empirical distribution given by a dataset { x 1 , . . . , x N } ⊂ X {\displaystyle \{x_{1},...,x_{N}\}\subset {\mathcal {X}}} , so that μ r e f = 1 N ∑ i = 1 N δ x i {\displaystyle \mu _{ref}={\frac {1}{N}}\sum _{i=1}^{N}\delta _{x_{i}}} where δ x i {\displaystyle \delta _{x_{i}}} is the Dirac measure, the quality function is just L 2 {\displaystyle L^{2}} loss: d ( x , x ′ ) = ‖ x − x ′ ‖ 2 2 {\displaystyle d(x,x')=\|x-x'\|_{2}^{2}} , and ‖ ⋅ ‖ 2 {\displaystyle \|\cdot \|_{2}} is the Euclidean norm. Then the problem of searching for the optimal autoencoder is just a least-squares optimization: min θ , ϕ L ( θ , ϕ ) , where L ( θ , ϕ ) = 1 N ∑ i = 1 N ‖ x i − D θ ( E ϕ ( x i ) ) ‖ 2 2 {\displaystyle \min _{\theta ,\phi }L(\theta ,\phi ),\qquad {\text{where }}L(\theta ,\phi )={\frac {1}{N}}\sum _{i=1}^{N}\|x_{i}-D_{\theta }(E_{\phi }(x_{i}))\|_{2}^{2}} === Interpretation === An autoencoder has two main parts: an encoder that maps the message to a code, and a decoder that reconstructs the message from the code. An optimal autoencoder would perform as close to perfect reconstruction as possible, with "close to perfect" defined by the reconstruction quality function d {\displaystyle d} . The simplest way to perform the copying task perfectly would be to duplicate the signal. To suppress this behavior, the code space Z {\displaystyle {\mathcal {Z}}} usually has fewer dimensions than the message space X {\displaystyle {\mathcal {X}}} . Such an autoencoder is called undercomplete. It can be interpreted as compressing the message, or reducing its dimensionality. At the limit of an ideal undercomplete autoencoder, every possible code z {\displaystyle z} in the code space is used to encode a message x {\displaystyle x} that really appears in the distribution μ r e f {\displaystyle \mu _{ref}} , and the decoder is also perfect: D θ ( E ϕ ( x ) ) = x {\displaystyle D_{\theta }(E_{\phi }(x))=x} . This ideal autoencoder can then be used to generate messages indistinguishable from real messages, by feeding its decoder arbitrary code z {\displaystyle z} and obtaining D θ ( z ) {\displaystyle D_{\theta }(z)} , which is a message that really appears in the distribution μ r e f {\displaystyle \mu _{ref}} . If the code space Z {\displaystyle {\mathcal {Z}}} has dimension larger than (overcomplete), or equal to, the message space X {\displaystyle {\mathcal {X}}} , or the hidden units are given enough capacity, an autoencoder can learn the identity function and become useless. However, experimental results found that overcomplete autoencoders might still learn useful features. In the ideal setting, the code dimension and the model capacity could be set on the basis of the complexity of the data distribution to be modeled. A standard way to do so is to add modifications to the basic autoencoder, to be detailed below. == Variations == === Variational autoencoder (VAE) === Variational autoencoders (VAEs) belong to the families of variational Bayesian methods. Despite the architectural similarities with basic autoencoders, VAEs are architected with different goals and have a different mathematical formulation. The latent space is, in this case, composed of a mixture of distributions instead of fixed vectors. Given an input dataset x {\displaystyle x} characterized by an unknown probability function P ( x ) {\displaystyle P(x)} and a multivariate latent encoding vector z {\displaystyle z} , the objective is to model the data as a distribution p θ ( x ) {\displaystyle p_{\theta }(x)} , with θ {\displaystyle \theta } defined as the set of the network parameters so that p θ ( x ) = ∫ z p θ ( x , z ) d z {\displaystyle p_{\theta }(x)=\int _{z}p_{\theta }(x,z)dz} . === Sparse autoencoder (SAE) === Inspired by the sparse coding hypothesis in neuroscience, sparse autoencoders (SAE) are variants of autoencoders, such that the codes E ϕ ( x ) {\displaystyle E_{\phi }(x)} for messages tend to be sparse codes, that is, E ϕ ( x ) {\displaystyle E_{\phi }(x)} is close to zero in most entries. Sparse autoencoders may include more (rather than fewer) hidden units than inputs, but only a small number of the hidden units are allowed to be active at the same time. Encouraging sparsity improves performance on classification tasks. There are two main ways to enforce sparsity. One way is to simply clamp all but the highest-k activations of the latent code to zero. This is the k-sparse autoencoder. The k-sparse autoencoder inserts the following "k-sparse function" in the latent layer of a standard autoencoder: f k ( x 1 , . . . , x n ) = ( x 1 b 1 , . . . , x n b n ) {\displaystyle f_{k}(x_{1},...,x_{n})=(x_{1}b_{1},...,x_{n}b_{n})} where b i = 1 {\displaystyle b_{i}=1} if | x i | {\displaystyle |x_{i}|} ranks in the top k, and 0 otherwise. Backpropagating through f k {\displaystyle f_{k}} is simple: set gradient to 0 for b i = 0 {\displaystyle b_{i}=0} entries, and keep gradient for b i = 1 {\displaystyle b_{i}=1} entries. This is essentially a generalized ReLU function. The other way is a relaxed version of the k-

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  Virtual intelligence

  Virtual intelligence (VI) is the term given to artificial intelligence that exists within a virtual world. Many virtual worlds have options for persistent avatars that provide information, training, role-playing, and social interactions. The immersion in virtual worlds provides a platform for VI beyond the traditional paradigm of past user interfaces (UIs). What Alan Turing established as a benchmark for telling the difference between human and computerized intelligence was devoid of visual influences. With today's VI bots, virtual intelligence has evolved past the constraints of past testing into a new level of the machine's ability to demonstrate intelligence. The immersive features of these environments provide nonverbal elements that affect the realism provided by virtually intelligent agents. Virtual intelligence is the intersection of these two technologies: Virtual environments: Immersive 3D spaces provide for collaboration, simulations, and role-playing interactions for training. Many of these virtual environments are currently being used for government and academic projects, including Second Life, VastPark, Olive, OpenSim, Outerra, Oracle's Open Wonderland, Duke University's Open Cobalt, and many others. Some of the commercial virtual worlds are also taking this technology into new directions, including the high-definition virtual world Blue Mars. Artificial intelligence (AI): AI is a branch of computer science that aims to create intelligent machines capable of performing tasks that typically require human intelligence. VI is a type of AI that operates within virtual environments to simulate human-like interactions and responses. == Applications == Cutlass Bomb Disposal Robot: Northrop Grumman developed a virtual training opportunity because of the prohibitive real-world cost and dangers associated with bomb disposal. By replicating a complicated system without having to learn advanced code, the virtual robot has no risk of damage, trainee safety hazards, or accessibility constraints. MyCyberTwin: NASA is among the companies that have used the MyCyberTwin AI technologies. They used it for the Phoenix rover in the virtual world Second Life. Their MyCyberTwin used a programmed profile to relay information about what the Phoenix rover was doing and its purpose. Second China: The University of Florida developed the "Second China" project as an immersive training experience for learning how to interact with the culture and language in a foreign country. Students are immersed in an environment that provides role-playing challenges coupled with language and cultural sensitivities magnified during country-level diplomatic missions or during times of potential conflict or regional destabilization. The virtual training provides participants with opportunities to access information, take part in guided learning scenarios, communicate, collaborate, and role-play. While China was the country for the prototype, this model can be modified for use with any culture to help better understand social and cultural interactions and see how other people think and what their actions imply. Duke School of Nursing Training Simulation: Extreme Reality developed virtual training to test critical thinking with a nurse performing trained procedures to identify critical data to make decisions and performing the correct steps for intervention. Bots are programmed to respond to the nurse's actions as the patient with their conditions improving if the nurse performs the correct actions.

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  T-distributed stochastic neighbor embedding

  t-distributed stochastic neighbor embedding (t-SNE) is a statistical method for visualizing high-dimensional data by giving each datapoint a location in a two or three-dimensional map. It is based on Stochastic Neighbor Embedding originally developed by Geoffrey Hinton and Sam Roweis, where Laurens van der Maaten and Hinton proposed the t-distributed variant. It is a nonlinear dimensionality reduction technique for embedding high-dimensional data for visualization in a low-dimensional space of two or three dimensions. Specifically, it models each high-dimensional object by a two- or three-dimensional point in such a way that similar objects are modeled by nearby points and dissimilar objects are modeled by distant points with high probability. The t-SNE algorithm comprises two main stages. First, t-SNE constructs a probability distribution over pairs of high-dimensional objects in such a way that similar objects are assigned a higher probability while dissimilar points are assigned a lower probability. Second, t-SNE defines a similar probability distribution over the points in the low-dimensional map, and it minimizes the Kullback–Leibler divergence (KL divergence) between the two distributions with respect to the locations of the points in the map. While the original algorithm uses the Euclidean distance between objects as the base of its similarity metric, this can be changed as appropriate. A Riemannian variant is UMAP. t-SNE has been used for visualization in a wide range of applications, including genomics, computer security research, natural language processing, music analysis, cancer research, bioinformatics, geological domain interpretation, and biomedical signal processing. For a data set with n {\displaystyle n} elements, t-SNE runs in O ( n 2 ) {\displaystyle O(n^{2})} time and requires O ( n 2 ) {\displaystyle O(n^{2})} space. == Details == Given a set of N {\displaystyle N} high-dimensional objects x 1 , … , x N {\displaystyle \mathbf {x} _{1},\dots ,\mathbf {x} _{N}} , t-SNE first computes probabilities p i j {\displaystyle p_{ij}} that are proportional to the similarity of objects x i {\displaystyle \mathbf {x} _{i}} and x j {\displaystyle \mathbf {x} _{j}} , as follows. For i ≠ j {\displaystyle i\neq j} , define p j ∣ i = exp ⁡ ( − ‖ x i − x j ‖ 2 / 2 σ i 2 ) ∑ k ≠ i exp ⁡ ( − ‖ x i − x k ‖ 2 / 2 σ i 2 ) {\displaystyle p_{j\mid i}={\frac {\exp(-\lVert \mathbf {x} _{i}-\mathbf {x} _{j}\rVert ^{2}/2\sigma _{i}^{2})}{\sum _{k\neq i}\exp(-\lVert \mathbf {x} _{i}-\mathbf {x} _{k}\rVert ^{2}/2\sigma _{i}^{2})}}} and set p i ∣ i = 0 {\displaystyle p_{i\mid i}=0} . Note the above denominator ensures ∑ j p j ∣ i = 1 {\displaystyle \sum _{j}p_{j\mid i}=1} for all i {\displaystyle i} . As van der Maaten and Hinton explained: "The similarity of datapoint x j {\displaystyle x_{j}} to datapoint x i {\displaystyle x_{i}} is the conditional probability, p j | i {\displaystyle p_{j|i}} , that x i {\displaystyle x_{i}} would pick x j {\displaystyle x_{j}} as its neighbor if neighbors were picked in proportion to their probability density under a Gaussian centered at x i {\displaystyle x_{i}} ." Now define p i j = p j ∣ i + p i ∣ j 2 N {\displaystyle p_{ij}={\frac {p_{j\mid i}+p_{i\mid j}}{2N}}} This is motivated because p i {\displaystyle p_{i}} and p j {\displaystyle p_{j}} from the N samples are estimated as 1/N, so the conditional probability can be written as p i ∣ j = N p i j {\displaystyle p_{i\mid j}=Np_{ij}} and p j ∣ i = N p j i {\displaystyle p_{j\mid i}=Np_{ji}} . Since p i j = p j i {\displaystyle p_{ij}=p_{ji}} , you can obtain previous formula. Also note that p i i = 0 {\displaystyle p_{ii}=0} and ∑ i , j p i j = 1 {\displaystyle \sum _{i,j}p_{ij}=1} . The bandwidth of the Gaussian kernels σ i {\displaystyle \sigma _{i}} is set in such a way that the entropy of the conditional distribution equals a predefined entropy using the bisection method. As a result, the bandwidth is adapted to the density of the data: smaller values of σ i {\displaystyle \sigma _{i}} are used in denser parts of the data space. The entropy increases with the perplexity of this distribution P i {\displaystyle P_{i}} ; this relation is seen as P e r p ( P i ) = 2 H ( P i ) {\displaystyle Perp(P_{i})=2^{H(P_{i})}} where H ( P i ) {\displaystyle H(P_{i})} is the Shannon entropy H ( P i ) = − ∑ j p j | i log 2 ⁡ p j | i . {\displaystyle H(P_{i})=-\sum _{j}p_{j|i}\log _{2}p_{j|i}.} The perplexity is a hand-chosen parameter of t-SNE, and as the authors state, "perplexity can be interpreted as a smooth measure of the effective number of neighbors. The performance of SNE is fairly robust to changes in the perplexity, and typical values are between 5 and 50.". Since the Gaussian kernel uses the Euclidean distance ‖ x i − x j ‖ {\displaystyle \lVert x_{i}-x_{j}\rVert } , it is affected by the curse of dimensionality, and in high dimensional data when distances lose the ability to discriminate, the p i j {\displaystyle p_{ij}} become too similar (asymptotically, they would converge to a constant). It has been proposed to adjust the distances with a power transform, based on the intrinsic dimension of each point, to alleviate this. t-SNE aims to learn a d {\displaystyle d} -dimensional map y 1 , … , y N {\displaystyle \mathbf {y} _{1},\dots ,\mathbf {y} _{N}} (with y i ∈ R d {\displaystyle \mathbf {y} _{i}\in \mathbb {R} ^{d}} and d {\displaystyle d} typically chosen as 2 or 3) that reflects the similarities p i j {\displaystyle p_{ij}} as well as possible. To this end, it measures similarities q i j {\displaystyle q_{ij}} between two points in the map y i {\displaystyle \mathbf {y} _{i}} and y j {\displaystyle \mathbf {y} _{j}} , using a very similar approach. Specifically, for i ≠ j {\displaystyle i\neq j} , define q i j {\displaystyle q_{ij}} as q i j = ( 1 + ‖ y i − y j ‖ 2 ) − 1 ∑ k ∑ l ≠ k ( 1 + ‖ y k − y l ‖ 2 ) − 1 {\displaystyle q_{ij}={\frac {(1+\lVert \mathbf {y} _{i}-\mathbf {y} _{j}\rVert ^{2})^{-1}}{\sum _{k}\sum _{l\neq k}(1+\lVert \mathbf {y} _{k}-\mathbf {y} _{l}\rVert ^{2})^{-1}}}} and set q i i = 0 {\displaystyle q_{ii}=0} . Herein a heavy-tailed Student t-distribution (with one-degree of freedom, which is the same as a Cauchy distribution) is used to measure similarities between low-dimensional points in order to allow dissimilar objects to be modeled far apart in the map. The locations of the points y i {\displaystyle \mathbf {y} _{i}} in the map are determined by minimizing the (non-symmetric) Kullback–Leibler divergence of the distribution P {\displaystyle P} from the distribution Q {\displaystyle Q} , that is: K L ( P ∥ Q ) = ∑ i ≠ j p i j log ⁡ p i j q i j {\displaystyle \mathrm {KL} \left(P\parallel Q\right)=\sum _{i\neq j}p_{ij}\log {\frac {p_{ij}}{q_{ij}}}} The minimization of the Kullback–Leibler divergence with respect to the points y i {\displaystyle \mathbf {y} _{i}} is performed using gradient descent. The result of this optimization is a map that reflects the similarities between the high-dimensional inputs. == Output == While t-SNE plots often seem to display clusters, the visual clusters can be strongly influenced by the chosen parameterization (especially the perplexity) and so a good understanding of the parameters for t-SNE is needed. Such "clusters" can be shown to even appear in structured data with no clear clustering, and so may be false findings. Similarly, the size of clusters produced by t-SNE is not informative, and neither is the distance between clusters. Thus, interactive exploration may be needed to choose parameters and validate results. It has been shown that t-SNE can often recover well-separated clusters, and with special parameter choices, approximates a simple form of spectral clustering. == Software == A C++ implementation of Barnes-Hut is available on the github account of one of the original authors. The R package Rtsne implements t-SNE in R. ELKI contains tSNE, also with Barnes-Hut approximation scikit-learn, a popular machine learning library in Python implements t-SNE with both exact solutions and the Barnes-Hut approximation. Tensorboard, the visualization kit associated with TensorFlow, also implements t-SNE (online version) The Julia package TSne implements t-SNE

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  Ensemble learning

  In statistics and machine learning, ensemble methods use multiple learning algorithms to obtain better predictive performance than could be obtained from any of the constituent learning algorithms alone. Unlike a statistical ensemble in statistical mechanics, which is usually infinite, a machine learning ensemble consists of only a concrete finite set of alternative models, but typically allows for much more flexible structure to exist among those alternatives. == Overview == Supervised learning algorithms search through a hypothesis space to find a suitable hypothesis that will make good predictions with a particular problem. Even if this space contains hypotheses that are very well-suited for a particular problem, it may be very difficult to find a good one. Ensembles combine multiple hypotheses to form one which should be theoretically better. Ensemble learning trains two or more machine learning algorithms on a specific classification or regression task. The algorithms within the ensemble model are generally referred as "base models", "base learners", or "weak learners" in literature. These base models can be constructed using a single modelling algorithm, or several different algorithms. The idea is to train a diverse set of weak models on the same modelling task, such that the outputs of each weak learner have poor predictive ability (i.e., high bias), and among all weak learners, the outcome and error values exhibit high variance. Fundamentally, an ensemble learning model trains at least two high-bias (weak) and high-variance (diverse) models to be combined into a better-performing model. The set of weak models — which would not produce satisfactory predictive results individually — are combined or averaged to produce a single, high performing, accurate, and low-variance model to fit the task as required. Ensemble learning typically refers to bagging (bootstrap aggregating), boosting or stacking/blending techniques to induce high variance among the base models. Bagging creates diversity by generating random samples from the training observations and fitting the same model to each different sample — also known as homogeneous parallel ensembles. Boosting follows an iterative process by sequentially training each base model on the up-weighted errors of the previous base model, producing an additive model to reduce the final model errors — also known as sequential ensemble learning. Stacking or blending consists of different base models, each trained independently (i.e. diverse/high variance) to be combined into the ensemble model — producing a heterogeneous parallel ensemble. Common applications of ensemble learning include random forests (an extension of bagging), Boosted Tree models, and Gradient Boosted Tree Models. Models in applications of stacking are generally more task-specific — such as combining clustering techniques with other parametric and/or non-parametric techniques. Evaluating the prediction of an ensemble typically requires more computation than evaluating the prediction of a single model. In one sense, ensemble learning may be thought of as a way to compensate for poor learning algorithms by performing a lot of extra computation. On the other hand, the alternative is to do a lot more learning with one non-ensemble model. An ensemble may be more efficient at improving overall accuracy for the same increase in compute, storage, or communication resources by using that increase on two or more methods, than would have been improved by increasing resource use for a single method. Fast algorithms such as decision trees are commonly used in ensemble methods (e.g., random forests), although slower algorithms can benefit from ensemble techniques as well. By analogy, ensemble techniques have been used also in unsupervised learning scenarios, for example in consensus clustering or in anomaly detection. == Ensemble theory == Empirically, ensembles tend to yield better results when there is a significant diversity among the models. Many ensemble methods, therefore, seek to promote diversity among the models they combine. Although perhaps non-intuitive, more random algorithms (like random decision trees) can be used to produce a stronger ensemble than very deliberate algorithms (like entropy-reducing decision trees). Using a variety of strong learning algorithms, however, has been shown to be more effective than using techniques that attempt to dumb-down the models in order to promote diversity. It is possible to increase diversity in the training stage of the model using correlation for regression tasks or using information measures such as cross entropy for classification tasks. Theoretically, one can justify the diversity concept because the lower bound of the error rate of an ensemble system can be decomposed into accuracy, diversity, and the other term. === The geometric framework === Ensemble learning, including both regression and classification tasks, can be explained using a geometric framework. Within this framework, the output of each individual classifier or regressor for the entire dataset can be viewed as a point in a multi-dimensional space. Additionally, the target result is also represented as a point in this space, referred to as the "ideal point." The Euclidean distance is used as the metric to measure both the performance of a single classifier or regressor (the distance between its point and the ideal point) and the dissimilarity between two classifiers or regressors (the distance between their respective points). This perspective transforms ensemble learning into a deterministic problem. For example, within this geometric framework, it can be proved that the averaging of the outputs (scores) of all base classifiers or regressors can lead to equal or better results than the average of all the individual models. It can also be proved that if the optimal weighting scheme is used, then a weighted averaging approach can outperform any of the individual classifiers or regressors that make up the ensemble or as good as the best performer at least. == Ensemble size == While the number of component classifiers of an ensemble has a great impact on the accuracy of prediction, there is a limited number of studies addressing this problem. A priori determining of ensemble size and the volume and velocity of big data streams make this even more crucial for online ensemble classifiers. Mostly statistical tests were used for determining the proper number of components. More recently, a theoretical framework suggested that there is an ideal number of component classifiers for an ensemble such that having more or less than this number of classifiers would deteriorate the accuracy. It is called "the law of diminishing returns in ensemble construction." Their theoretical framework shows that using the same number of independent component classifiers as class labels gives the highest accuracy. == Common types of ensembles == === Bayes optimal classifier === The Bayes optimal classifier is a classification technique. It is an ensemble of all the hypotheses in the hypothesis space. On average, no other ensemble can outperform it. The Naive Bayes classifier is a version of this that assumes that the data is conditionally independent on the class and makes the computation more feasible. Each hypothesis is given a vote proportional to the likelihood that the training dataset would be sampled from a system if that hypothesis were true. To facilitate training data of finite size, the vote of each hypothesis is also multiplied by the prior probability of that hypothesis. The Bayes optimal classifier can be expressed with the following equation: y = a r g m a x c j ∈ C ∑ h i ∈ H P ( c j | h i ) P ( T | h i ) P ( h i ) {\displaystyle y={\underset {c_{j}\in C}{\mathrm {argmax} }}\sum _{h_{i}\in H}{P(c_{j}|h_{i})P(T|h_{i})P(h_{i})}} where y {\displaystyle y} is the predicted class, C {\displaystyle C} is the set of all possible classes, H {\displaystyle H} is the hypothesis space, P {\displaystyle P} refers to a probability, and T {\displaystyle T} is the training data. As an ensemble, the Bayes optimal classifier represents a hypothesis that is not necessarily in H {\displaystyle H} . The hypothesis represented by the Bayes optimal classifier, however, is the optimal hypothesis in ensemble space (the space of all possible ensembles consisting only of hypotheses in H {\displaystyle H} ). This formula can be restated using Bayes' theorem, which says that the posterior is proportional to the likelihood times the prior: P ( h i | T ) ∝ P ( T | h i ) P ( h i ) {\displaystyle P(h_{i}|T)\propto P(T|h_{i})P(h_{i})} hence, y = a r g m a x c j ∈ C ∑ h i ∈ H P ( c j | h i ) P ( h i | T ) {\displaystyle y={\underset {c_{j}\in C}{\mathrm {argmax} }}\sum _{h_{i}\in H}{P(c_{j}|h_{i})P(h_{i}|T)}} === Bootstrap aggregating (bagging) === Bootstrap aggregation (bagging) involves training an ensemble on bootstrapped data sets. A bootstrapped set is cr

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