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Flexidraw is a 1985 graphics computer program published by Inkwell Systems. == Gameplay == Flexidraw is a graphics program that allows users to produce drawings using a light pen and print them. == Reception == Roy Wagner reviewed the product for Computer Gaming World, and stated that "Of the many graphics programs available Flexidraw is certainly the best supported by it's [sic] parent company."
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In statistics, the Huber loss is a loss function used in robust regression, that is less sensitive to outliers in data than the squared error loss. A variant for classification is also sometimes used. == Definition == The Huber loss function describes the penalty incurred by an estimation procedure f. Huber (1964) defines the loss function piecewise by L δ ( a ) = { 1 2 a 2 for | a | ≤ δ , δ ⋅ ( | a | − 1 2 δ ) , otherwise. {\displaystyle L_{\delta }(a)={\begin{cases}{\frac {1}{2}}{a^{2}}&{\text{for }}|a|\leq \delta ,\\[4pt]\delta \cdot \left(|a|-{\frac {1}{2}}\delta \right),&{\text{otherwise.}}\end{cases}}} This function is quadratic for small values of a, and linear for large values, with equal values and slopes of the different sections at the two points where | a | = δ {\displaystyle |a|=\delta } . The variable a often refers to the residuals, that is to the difference between the observed and predicted values a = y − f ( x ) {\displaystyle a=y-f(x)} , so the former can be expanded to L δ ( y , f ( x ) ) = { 1 2 ( y − f ( x ) ) 2 for | y − f ( x ) | ≤ δ , δ ⋅ ( | y − f ( x ) | − 1 2 δ ) , otherwise. {\displaystyle L_{\delta }(y,f(x))={\begin{cases}{\frac {1}{2}}{\left(y-f(x)\right)}^{2}&{\text{for }}\left|y-f(x)\right|\leq \delta ,\\[4pt]\delta \ \cdot \left(\left|y-f(x)\right|-{\frac {1}{2}}\delta \right),&{\text{otherwise.}}\end{cases}}} The Huber loss is the convolution of the absolute value function with the rectangular function, scaled and translated. Thus it "smoothens out" the former's corner at the origin. == Motivation == Two very commonly used loss functions are the squared loss, L ( a ) = a 2 {\displaystyle L(a)=a^{2}} , and the absolute loss, L ( a ) = | a | {\displaystyle L(a)=|a|} . The squared loss function results in an arithmetic mean-unbiased estimator, and the absolute-value loss function results in a median-unbiased estimator (in the one-dimensional case, and a geometric median-unbiased estimator for the multi-dimensional case). The squared loss has the disadvantage that it has the tendency to be dominated by outliers—when summing over a set of a {\displaystyle a} 's (as in ∑ i = 1 n L ( a i ) {\textstyle \sum _{i=1}^{n}L(a_{i})} ), the sample mean is influenced too much by a few particularly large a {\displaystyle a} -values when the distribution is heavy tailed: in terms of estimation theory, the asymptotic relative efficiency of the mean is poor for heavy-tailed distributions. As defined above, the Huber loss function is strongly convex in a uniform neighborhood of its minimum a = 0 {\displaystyle a=0} ; at the boundary of this uniform neighborhood, the Huber loss function has a differentiable extension to an affine function at points a = − δ {\displaystyle a=-\delta } and a = δ {\displaystyle a=\delta } . These properties allow it to combine much of the sensitivity of the mean-unbiased, minimum-variance estimator of the mean (using the quadratic loss function) and the robustness of the median-unbiased estimator (using the absolute value function). == Pseudo-Huber loss function == The Pseudo-Huber loss function can be used as a smooth approximation of the Huber loss function. It combines the best properties of L2 squared loss and L1 absolute loss by being strongly convex when close to the target/minimum and less steep for extreme values. The scale at which the Pseudo-Huber loss function transitions from L2 loss for values close to the minimum to L1 loss for extreme values and the steepness at extreme values can be controlled by the δ {\displaystyle \delta } value. The Pseudo-Huber loss function ensures that derivatives are continuous for all degrees. It is defined as L δ ( a ) = δ 2 ( 1 + ( a / δ ) 2 − 1 ) . {\displaystyle L_{\delta }(a)=\delta ^{2}\left({\sqrt {1+(a/\delta )^{2}}}-1\right).} As such, this function approximates a 2 / 2 {\displaystyle a^{2}/2} for small values of a {\displaystyle a} , and approximates a straight line with slope δ {\displaystyle \delta } for large values of a {\displaystyle a} . While the above is the most common form, other smooth approximations of the Huber loss function also exist. == Variant for classification == For classification purposes, a variant of the Huber loss called modified Huber is sometimes used. Given a prediction f ( x ) {\displaystyle f(x)} (a real-valued classifier score) and a true binary class label y ∈ { + 1 , − 1 } {\displaystyle y\in \{+1,-1\}} , the modified Huber loss is defined as L ( y , f ( x ) ) = { max ( 0 , 1 − y f ( x ) ) 2 for y f ( x ) > − 1 , − 4 y f ( x ) otherwise. {\displaystyle L(y,f(x))={\begin{cases}\max(0,1-y\,f(x))^{2}&{\text{for }}\,\,y\,f(x)>-1,\\[4pt]-4y\,f(x)&{\text{otherwise.}}\end{cases}}} The term max ( 0 , 1 − y f ( x ) ) {\displaystyle \max(0,1-y\,f(x))} is the hinge loss used by support vector machines; the quadratically smoothed hinge loss is a generalization of L {\displaystyle L} . == Applications == The Huber loss function is used in robust statistics, M-estimation and additive modelling.
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In the mathematical theory of probability, an absorbing Markov chain is a Markov chain in which every state can reach an absorbing state. An absorbing state is a state that, once entered, cannot be left. Like general Markov chains, there can be continuous-time absorbing Markov chains with an infinite state space. However, this article concentrates on the discrete-time discrete-state-space case. == Formal definition == A Markov chain is an absorbing chain if there is at least one absorbing state and it is possible to go from any state to at least one absorbing state in a finite number of steps. In an absorbing Markov chain, a state that is not absorbing is called transient. === Canonical form === Let an absorbing Markov chain with transition matrix P have t transient states and r absorbing states. The rows of P represent sources, while columns represent destinations. By ordering the transient states before the absorbing states, it can be assumed that P has the form P = [ Q R 0 I r ] , {\displaystyle P={\begin{bmatrix}Q&R\\\mathbf {0} &I_{r}\end{bmatrix}},} where Q is a t-by-t matrix, R is a nonzero t-by-r matrix, 0 is an r-by-t zero matrix, and Ir is the r-by-r identity matrix. Thus, Q describes the probability of transitioning from some transient state to another while R describes the probability of transitioning from some transient state to some absorbing state. The probability of transitioning from i to j in exactly k steps is the (i,j)-entry of Pk, further computed below. When considering only transient states, the probability is found in the upper left of Pk, the (i,j)-entry of Qk. == Fundamental matrix == === Expected number of visits to a transient state === A basic property about an absorbing Markov chain is the expected number of visits to a transient state j starting from a transient state i (before being absorbed). This can be established to be given by the (i, j) entry of so-called fundamental matrix N, obtained by summing Qk for all k (from 0 to ∞). It can be proven that N := ∑ k = 0 ∞ Q k = ( I t − Q ) − 1 , {\displaystyle N:=\sum _{k=0}^{\infty }Q^{k}=(I_{t}-Q)^{-1},} where It is the t-by-t identity matrix. The computation of this formula is the matrix equivalent of the geometric series of scalars, ∑ k = 0 ∞ q k = 1 1 − q {\displaystyle {\textstyle \sum }_{k=0}^{\infty }q^{k}={\tfrac {1}{1-q}}} . With the matrix N in hand, also other properties of the Markov chain are easy to obtain. === Expected number of steps before being absorbed === The expected number of steps before being absorbed in any absorbing state, when starting in transient state i can be computed via a sum over transient states. The value is given by the ith entry of the vector t := N 1 , {\displaystyle \mathbf {t} :=N\mathbf {1} ,} where 1 is a length-t column vector whose entries are all 1. === Absorbing probabilities === By induction, P k = [ Q k ( I t − Q k ) N R 0 I r ] . {\displaystyle P^{k}={\begin{bmatrix}Q^{k}&(I_{t}-Q^{k})NR\\\mathbf {0} &I_{r}\end{bmatrix}}.} The probability of eventually being absorbed in the absorbing state j when starting from transient state i is given by the (i,j)-entry of the matrix B := N R {\displaystyle B:=NR} . The number of columns of this matrix equals the number of absorbing states r. An approximation of those probabilities can also be obtained directly from the (i,j)-entry of P k {\displaystyle P^{k}} for a large enough value of k, when i is the index of a transient, and j the index of an absorbing state. This is because ( lim k → ∞ P k ) i , t + j = B i , j {\displaystyle \left(\lim _{k\to \infty }P^{k}\right)_{i,t+j}=B_{i,j}} . === Transient visiting probabilities === The probability of visiting transient state j when starting at a transient state i is the (i,j)-entry of the matrix H := ( N − I t ) ( N dg ) − 1 , {\displaystyle H:=(N-I_{t})(N_{\operatorname {dg} })^{-1},} where Ndg is the diagonal matrix with the same diagonal as N. === Variance on number of transient visits === The variance on the number of visits to a transient state j with starting at a transient state i (before being absorbed) is the (i,j)-entry of the matrix N 2 := N ( 2 N dg − I t ) − N sq , {\displaystyle N_{2}:=N(2N_{\operatorname {dg} }-I_{t})-N_{\operatorname {sq} },} where Nsq is the Hadamard product of N with itself (i.e. each entry of N is squared). === Variance on number of steps === The variance on the number of steps before being absorbed when starting in transient state i is the ith entry of the vector ( 2 N − I t ) t − t sq , {\displaystyle (2N-I_{t})\mathbf {t} -\mathbf {t} _{\operatorname {sq} },} where tsq is the Hadamard product of t with itself (i.e., as with Nsq, each entry of t is squared). == Examples == === String generation === Consider the process of repeatedly flipping a fair coin until the sequence (heads, tails, heads) appears. This process is modeled by an absorbing Markov chain with transition matrix P = [ 1 / 2 1 / 2 0 0 0 1 / 2 1 / 2 0 1 / 2 0 0 1 / 2 0 0 0 1 ] . {\displaystyle P={\begin{bmatrix}1/2&1/2&0&0\\0&1/2&1/2&0\\1/2&0&0&1/2\\0&0&0&1\end{bmatrix}}.} The first state represents the empty string, the second state the string "H", the third state the string "HT", and the fourth state the string "HTH". Although in reality, the coin flips cease after the string "HTH" is generated, the perspective of the absorbing Markov chain is that the process has transitioned into the absorbing state representing the string "HTH" and, therefore, cannot leave. For this absorbing Markov chain, the fundamental matrix is N = ( I − Q ) − 1 = ( [ 1 0 0 0 1 0 0 0 1 ] − [ 1 / 2 1 / 2 0 0 1 / 2 1 / 2 1 / 2 0 0 ] ) − 1 = [ 1 / 2 − 1 / 2 0 0 1 / 2 − 1 / 2 − 1 / 2 0 1 ] − 1 = [ 4 4 2 2 4 2 2 2 2 ] . {\displaystyle {\begin{aligned}N&=(I-Q)^{-1}=\left({\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}}-{\begin{bmatrix}1/2&1/2&0\\0&1/2&1/2\\1/2&0&0\end{bmatrix}}\right)^{-1}\\[4pt]&={\begin{bmatrix}1/2&-1/2&0\\0&1/2&-1/2\\-1/2&0&1\end{bmatrix}}^{-1}={\begin{bmatrix}4&4&2\\2&4&2\\2&2&2\end{bmatrix}}.\end{aligned}}} The expected number of steps starting from each of the transient states is t = N 1 = [ 4 4 2 2 4 2 2 2 2 ] [ 1 1 1 ] = [ 10 8 6 ] . {\displaystyle \mathbf {t} =N\mathbf {1} ={\begin{bmatrix}4&4&2\\2&4&2\\2&2&2\end{bmatrix}}{\begin{bmatrix}1\\1\\1\end{bmatrix}}={\begin{bmatrix}10\\8\\6\end{bmatrix}}.} Therefore, the expected number of coin flips before observing the sequence (heads, tails, heads) is 10, the entry for the state representing the empty string. === Games of chance === Games based entirely on chance can be modeled by an absorbing Markov chain. A classic example of this is the ancient Indian board game Snakes and Ladders. The graph on the left plots the probability mass in the lone absorbing state that represents the final square as the transition matrix is raised to larger and larger powers. To determine the expected number of turns to complete the game, compute the vector t as described above and examine tstart, which is approximately 39.2. === Infectious disease testing === Infectious disease testing, either of blood products or in medical clinics, is often taught as an example of an absorbing Markov chain. The public U.S. Centers for Disease Control and Prevention (CDC) model for HIV and for hepatitis B, for example, illustrates the property that absorbing Markov chains can lead to the detection of disease, versus the loss of detection through other means. In the standard CDC model, the Markov chain has five states, a state in which the individual is uninfected, then a state with infected but undetectable virus, a state with detectable virus, and absorbing states of having quit/been lost from the clinic, or of having been detected (the goal). The typical rates of transition between the Markov states are the probability p per unit time of being infected with the virus, w for the rate of window period removal (time until virus is detectable), q for quit/loss rate from the system, and d for detection, assuming a typical rate λ {\displaystyle \lambda } at which the health system administers tests of the blood product or patients in question. It follows that we can "walk along" the Markov model to identify the overall probability of detection for a person starting as undetected, by multiplying the probabilities of transition to each next state of the model as: p ( p + q ) w ( w + q ) d ( d + q ) {\displaystyle {\frac {p}{(p+q)}}{\frac {w}{(w+q)}}{\frac {d}{(d+q)}}} . The subsequent total absolute number of false negative tests—the primary CDC concern—would then be the rate of tests, multiplied by the probability of reaching the infected but undetectable state, times the duration of staying in the infected undetectable state: p ( p + q ) 1 ( w + q ) λ {\displaystyle {\frac {p}{(p+q)}}{\frac {1}{(w+q)}}\lambda } .
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In computational learning theory in mathematics, a concept over a domain X is a total Boolean function over X. A concept class is a class of concepts. Concept classes are a subject of computational learning theory. Concept class terminology frequently appears in model theory associated with probably approximately correct (PAC) learning. In this setting, if one takes a set Y as a set of (classifier output) labels, and X is a set of examples, the map c : X → Y {\displaystyle c:X\to Y} , i.e. from examples to classifier labels (where Y = { 0 , 1 } {\displaystyle Y=\{0,1\}} and where c is a subset of X), c is then said to be a concept. A concept class C {\displaystyle C} is then a collection of such concepts. Given a class of concepts C, a subclass D is reachable if there exists a sample s such that D contains exactly those concepts in C that are extensions to s. Not every subclass is reachable. == Background == A sample s {\displaystyle s} is a partial function from X {\displaystyle X} to { 0 , 1 } {\displaystyle \{0,1\}} . Identifying a concept with its characteristic function mapping X {\displaystyle X} to { 0 , 1 } {\displaystyle \{0,1\}} , it is a special case of a sample. Two samples are consistent if they agree on the intersection of their domains. A sample s ′ {\displaystyle s'} extends another sample s {\displaystyle s} if the two are consistent and the domain of s {\displaystyle s} is contained in the domain of s ′ {\displaystyle s'} . == Examples == Suppose that C = S + ( X ) {\displaystyle C=S^{+}(X)} . Then: the subclass { { x } } {\displaystyle \{\{x\}\}} is reachable with the sample s = { ( x , 1 ) } {\displaystyle s=\{(x,1)\}} ; the subclass S + ( Y ) {\displaystyle S^{+}(Y)} for Y ⊆ X {\displaystyle Y\subseteq X} are reachable with a sample that maps the elements of X − Y {\displaystyle X-Y} to zero; the subclass S ( X ) {\displaystyle S(X)} , which consists of the singleton sets, is not reachable. == Applications == Let C {\displaystyle C} be some concept class. For any concept c ∈ C {\displaystyle c\in C} , we call this concept 1 / d {\displaystyle 1/d} -good for a positive integer d {\displaystyle d} if, for all x ∈ X {\displaystyle x\in X} , at least 1 / d {\displaystyle 1/d} of the concepts in C {\displaystyle C} agree with c {\displaystyle c} on the classification of x {\displaystyle x} . The fingerprint dimension F D ( C ) {\displaystyle FD(C)} of the entire concept class C {\displaystyle C} is the least positive integer d {\displaystyle d} such that every reachable subclass C ′ ⊆ C {\displaystyle C'\subseteq C} contains a concept that is 1 / d {\displaystyle 1/d} -good for it. This quantity can be used to bound the minimum number of equivalence queries needed to learn a class of concepts according to the following inequality: F D ( C ) − 1 ≤ # E Q ( C ) ≤ ⌈ F D ( C ) ln ( | C | ) ⌉ {\textstyle FD(C)-1\leq \#EQ(C)\leq \lceil FD(C)\ln(|C|)\rceil } .
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Pixel binning, also known as binning, is a process image sensors of digital cameras use to combine adjacent pixels throughout an image, by summing or averaging their values, during or after readout. It improves low-light performance while still allowing for highly detailed photographs in good light. Charge from adjacent pixels in CCD or charge-coupled device image sensors and some other image sensors can be combined during readout, increasing the line rate or frame rate. In the context of image processing, binning is the procedure of combining clusters of adjacent pixels, throughout an image, into single pixels. For example, in 2×2 binning, an array of 4 pixels becomes a single larger pixel, reducing the number of pixels to 1/4 and halving the image resolution in each dimension. The result can be the sum, average, median, minimum, or maximum value of the cluster. Some systems use more advanced algorithms such as considering the values of nearby pixels, edge detection, self-claimed "AI", etc. to increase the perceived visual quality of the final downsized image. This aggregation, although associated with loss of information, reduces the amount of data to be processed, facilitating analysis. The binned image has lower resolution, but the relative noise level in each pixel is generally reduced. == History == Normally, an increase in megapixel count on a constant image sensor size would lead to a sacrifice of the surface size of the individual pixels, which would result in each pixel being able to catch less light in the same time, thus leading to a darker and/or noisier image in low light (given the same exposure time). In the past, camera manufacturers had to compromise between low-light performance and the amount of detail in good light, by dropping the megapixel count like HTC did in 2013 with their four-megapixel "UltraPixel" camera. However, this results in less detailed images in daylight where enough light is available. With pixel binning, the camera has "the best of both worlds", meaning both the benefit of high detail in good light and the benefit of high brightness in low light. In low light, the surfaces of four or more pixels can act as one large pixel that catches far more light. For example, some smartphones such as the Samsung Galaxy A15 are able to capture photographs with up to fifty megapixels in daylight. However, in low light, the individual pixels would be too small to capture the light needed for a bright image with the short exposure time available for handheld shooting. Therefore, with pixel binning activated, the 50-megapixel image sensor acts as a 12.5-megapixel image sensor, a quarter of its original resolution, with an accordingly larger surface area per pixel.
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In statistics, the mean squared error (MSE) or mean squared deviation (MSD) of an estimator (of a procedure for estimating an unobserved quantity) measures the average of the squares of the errors—that is, the average squared difference between the estimated values and the true value. MSE is a risk function, corresponding to the expected value of the squared error loss. The fact that MSE is almost always strictly positive (and not zero) is because of randomness or because the estimator does not account for information that could produce a more accurate estimate. In machine learning, specifically empirical risk minimization, MSE may refer to the empirical risk (the average loss on an observed data set), as an estimate of the true MSE (the true risk: the average loss on the actual population distribution). The MSE is a measure of the quality of an estimator. As it is derived from the square of Euclidean distance, it is always a positive value that decreases as the error approaches zero. The MSE is the second moment (about the origin) of the error, and thus incorporates both the variance of the estimator (how widely spread the estimates are from one data sample to another) and its bias (how far off the average estimated value is from the true value). For an unbiased estimator, the MSE is the variance of the estimator. Like the variance, MSE has the same units of measurement as the square of the quantity being estimated. In an analogy to standard deviation, taking the square root of MSE yields the root-mean-square error or root-mean-square deviation (RMSE or RMSD), which has the same units as the quantity being estimated; for an unbiased estimator, the RMSE is the square root of the variance, known as the standard error. == Definition and basic properties == The MSE either assesses the quality of a predictor (i.e., a function mapping arbitrary inputs to a sample of values of some random variable), or of an estimator (i.e., a mathematical function mapping a sample of data to an estimate of a parameter of the population from which the data is sampled). In the context of prediction, understanding the prediction interval can also be useful as it provides a range within which a future observation will fall, with a certain probability. The definition of an MSE differs according to whether one is describing a predictor or an estimator. === Predictor === If a vector of n {\displaystyle n} predictions is generated from a sample of n {\displaystyle n} data points on all variables, and Y {\displaystyle Y} is the vector of observed values of the variable being predicted, with Y ^ {\displaystyle {\hat {Y}}} being the predicted values (e.g. as from a least-squares fit), then the within-sample MSE of the predictor is computed as MSE = 1 n ∑ i = 1 n ( Y i − Y i ^ ) 2 {\displaystyle \operatorname {MSE} ={\frac {1}{n}}\sum _{i=1}^{n}\left(Y_{i}-{\hat {Y_{i}}}\right)^{2}} In other words, the MSE is the mean ( 1 n ∑ i = 1 n ) {\textstyle \left({\frac {1}{n}}\sum _{i=1}^{n}\right)} of the squares of the errors ( Y i − Y i ^ ) 2 {\textstyle \left(Y_{i}-{\hat {Y_{i}}}\right)^{2}} . This is an easily computable quantity for a particular sample (and hence is sample-dependent). In matrix notation, MSE = 1 n ∑ i = 1 n ( e i ) 2 = 1 n e T e {\displaystyle \operatorname {MSE} ={\frac {1}{n}}\sum _{i=1}^{n}(e_{i})^{2}={\frac {1}{n}}\mathbf {e} ^{\mathsf {T}}\mathbf {e} } where e i {\displaystyle e_{i}} is Y i − Y i ^ {\displaystyle Y_{i}-{\hat {Y_{i}}}} and e {\displaystyle \mathbf {e} } is a n × 1 {\displaystyle n\times 1} column vector. The MSE can also be computed on q data points that were not used in estimating the model, either because they were held back for this purpose, or because these data have been newly obtained. Within this process, known as cross-validation, the MSE is often called the test MSE, and is computed as MSE = 1 q ∑ i = n + 1 n + q ( Y i − Y i ^ ) 2 {\displaystyle \operatorname {MSE} ={\frac {1}{q}}\sum _{i=n+1}^{n+q}\left(Y_{i}-{\hat {Y_{i}}}\right)^{2}} === Estimator === The MSE of an estimator θ ^ {\displaystyle {\hat {\theta }}} with respect to an unknown parameter θ {\displaystyle \theta } is defined as MSE ( θ ^ ) = E θ [ ( θ ^ − θ ) 2 ] . {\displaystyle \operatorname {MSE} ({\hat {\theta }})=\operatorname {E} _{\theta }\left[({\hat {\theta }}-\theta )^{2}\right].} This definition depends on the unknown parameter, therefore the MSE is a priori property of an estimator. The MSE could be a function of unknown parameters, in which case any estimator of the MSE based on estimates of these parameters would be a function of the data (and thus a random variable). If the estimator θ ^ {\displaystyle {\hat {\theta }}} is derived as a sample statistic and is used to estimate some population parameter, then the expectation is with respect to the sampling distribution of the sample statistic. The MSE can be written as the sum of the variance of the estimator and the squared bias of the estimator, providing a useful way to calculate the MSE and implying that in the case of unbiased estimators, the MSE and variance are equivalent. MSE ( θ ^ ) = Var θ ( θ ^ ) + Bias ( θ ^ , θ ) 2 . {\displaystyle \operatorname {MSE} ({\hat {\theta }})=\operatorname {Var} _{\theta }({\hat {\theta }})+\operatorname {Bias} ({\hat {\theta }},\theta )^{2}.} ==== Proof of variance and bias relationship ==== MSE ( θ ^ ) = E θ [ ( θ ^ − θ ) 2 ] = E θ [ ( θ ^ − E θ [ θ ^ ] + E θ [ θ ^ ] − θ ) 2 ] = E θ [ ( θ ^ − E θ [ θ ^ ] ) 2 + 2 ( θ ^ − E θ [ θ ^ ] ) ( E θ [ θ ^ ] − θ ) + ( E θ [ θ ^ ] − θ ) 2 ] = E θ [ ( θ ^ − E θ [ θ ^ ] ) 2 ] + E θ [ 2 ( θ ^ − E θ [ θ ^ ] ) ( E θ [ θ ^ ] − θ ) ] + E θ [ ( E θ [ θ ^ ] − θ ) 2 ] = E θ [ ( θ ^ − E θ [ θ ^ ] ) 2 ] + 2 ( E θ [ θ ^ ] − θ ) E θ [ θ ^ − E θ [ θ ^ ] ] + ( E θ [ θ ^ ] − θ ) 2 E θ [ θ ^ ] − θ = constant = E θ [ ( θ ^ − E θ [ θ ^ ] ) 2 ] + 2 ( E θ [ θ ^ ] − θ ) ( E θ [ θ ^ ] − E θ [ θ ^ ] ) + ( E θ [ θ ^ ] − θ ) 2 E θ [ θ ^ ] = constant = E θ [ ( θ ^ − E θ [ θ ^ ] ) 2 ] + ( E θ [ θ ^ ] − θ ) 2 = Var θ ( θ ^ ) + Bias θ ( θ ^ , θ ) 2 {\displaystyle {\begin{aligned}\operatorname {MSE} ({\hat {\theta }})&=\operatorname {E} _{\theta }\left[({\hat {\theta }}-\theta )^{2}\right]\\&=\operatorname {E} _{\theta }\left[\left({\hat {\theta }}-\operatorname {E} _{\theta }[{\hat {\theta }}]+\operatorname {E} _{\theta }[{\hat {\theta }}]-\theta \right)^{2}\right]\\&=\operatorname {E} _{\theta }\left[\left({\hat {\theta }}-\operatorname {E} _{\theta }[{\hat {\theta }}]\right)^{2}+2\left({\hat {\theta }}-\operatorname {E} _{\theta }[{\hat {\theta }}]\right)\left(\operatorname {E} _{\theta }[{\hat {\theta }}]-\theta \right)+\left(\operatorname {E} _{\theta }[{\hat {\theta }}]-\theta \right)^{2}\right]\\&=\operatorname {E} _{\theta }\left[\left({\hat {\theta }}-\operatorname {E} _{\theta }[{\hat {\theta }}]\right)^{2}\right]+\operatorname {E} _{\theta }\left[2\left({\hat {\theta }}-\operatorname {E} _{\theta }[{\hat {\theta }}]\right)\left(\operatorname {E} _{\theta }[{\hat {\theta }}]-\theta \right)\right]+\operatorname {E} _{\theta }\left[\left(\operatorname {E} _{\theta }[{\hat {\theta }}]-\theta \right)^{2}\right]\\&=\operatorname {E} _{\theta }\left[\left({\hat {\theta }}-\operatorname {E} _{\theta }[{\hat {\theta }}]\right)^{2}\right]+2\left(\operatorname {E} _{\theta }[{\hat {\theta }}]-\theta \right)\operatorname {E} _{\theta }\left[{\hat {\theta }}-\operatorname {E} _{\theta }[{\hat {\theta }}]\right]+\left(\operatorname {E} _{\theta }[{\hat {\theta }}]-\theta \right)^{2}&&\operatorname {E} _{\theta }[{\hat {\theta }}]-\theta ={\text{constant}}\\&=\operatorname {E} _{\theta }\left[\left({\hat {\theta }}-\operatorname {E} _{\theta }[{\hat {\theta }}]\right)^{2}\right]+2\left(\operatorname {E} _{\theta }[{\hat {\theta }}]-\theta \right)\left(\operatorname {E} _{\theta }[{\hat {\theta }}]-\operatorname {E} _{\theta }[{\hat {\theta }}]\right)+\left(\operatorname {E} _{\theta }[{\hat {\theta }}]-\theta \right)^{2}&&\operatorname {E} _{\theta }[{\hat {\theta }}]={\text{constant}}\\&=\operatorname {E} _{\theta }\left[\left({\hat {\theta }}-\operatorname {E} _{\theta }[{\hat {\theta }}]\right)^{2}\right]+\left(\operatorname {E} _{\theta }[{\hat {\theta }}]-\theta \right)^{2}\\&=\operatorname {Var} _{\theta }({\hat {\theta }})+\operatorname {Bias} _{\theta }({\hat {\theta }},\theta )^{2}\end{aligned}}} An even shorter proof can be achieved using the well-known formula that for a random variable X {\textstyle X} , E ( X 2 ) = Var ( X ) + ( E ( X ) ) 2 {\textstyle \mathbb {E} (X^{2})=\operatorname {Var} (X)+(\mathbb {E} (X))^{2}} . By substituting X {\textstyle X} with, θ ^ − θ {\textstyle {\hat {\theta }}-\theta } , we have MSE ( θ ^ ) = E [ ( θ ^ − θ ) 2 ] = Var ( θ ^ − θ ) + ( E [ θ ^ − θ ] ) 2 = Var ( θ ^ ) + Bias 2 ( θ ^ , θ ) {\displaystyle {\begin{aligned}\operatorname {MSE} ({\hat {\theta }})&=\mathbb {E} [({\hat {\theta }}-\theta )^{2}]\\&=\operator
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Policy gradient methods are a class of reinforcement learning algorithms and a sub-class of policy optimization methods. Unlike value-based methods which learn a value function to derive a policy, policy optimization methods directly learn a policy function π {\displaystyle \pi } that selects actions without consulting a value function. For policy gradient to apply, the policy function π θ {\displaystyle \pi _{\theta }} is parameterized by a differentiable parameter θ {\displaystyle \theta } . == Overview == In policy-based RL, the actor is a parameterized policy function π θ {\displaystyle \pi _{\theta }} , where θ {\displaystyle \theta } are the parameters of the actor. The actor takes as argument the state of the environment s {\displaystyle s} and produces a probability distribution π θ ( ⋅ ∣ s ) {\displaystyle \pi _{\theta }(\cdot \mid s)} . If the action space is discrete, then ∑ a π θ ( a ∣ s ) = 1 {\displaystyle \sum _{a}\pi _{\theta }(a\mid s)=1} . If the action space is continuous, then ∫ a π θ ( a ∣ s ) d a = 1 {\displaystyle \int _{a}\pi _{\theta }(a\mid s)\mathrm {d} a=1} . The goal of policy optimization is to find some θ {\displaystyle \theta } that maximizes the expected episodic reward J ( θ ) {\displaystyle J(\theta )} : J ( θ ) = E π θ [ ∑ t = 0 T γ t R t | S 0 = s 0 ] {\displaystyle J(\theta )=\mathbb {E} _{\pi _{\theta }}\left[\sum _{t=0}^{T}\gamma ^{t}R_{t}{\Big |}S_{0}=s_{0}\right]} where γ {\displaystyle \gamma } is the discount factor, R t {\displaystyle R_{t}} is the reward at step t {\displaystyle t} , s 0 {\displaystyle s_{0}} is the starting state, and T {\displaystyle T} is the time-horizon (which can be infinite). The policy gradient is defined as ∇ θ J ( θ ) {\displaystyle \nabla _{\theta }J(\theta )} . Different policy gradient methods stochastically estimate the policy gradient in different ways. The goal of any policy gradient method is to iteratively maximize J ( θ ) {\displaystyle J(\theta )} by gradient ascent. Since the key part of any policy gradient method is the stochastic estimation of the policy gradient, they are also studied under the title of "Monte Carlo gradient estimation". == REINFORCE == === Policy gradient === The REINFORCE algorithm, introduced by Ronald J. Williams in 1992, was the first policy gradient method. It is based on the identity for the policy gradient ∇ θ J ( θ ) = E π θ [ ∑ t = 0 T ∇ θ ln π θ ( A t ∣ S t ) ∑ t = 0 T ( γ t R t ) | S 0 = s 0 ] {\displaystyle \nabla _{\theta }J(\theta )=\mathbb {E} _{\pi _{\theta }}\left[\sum _{t=0}^{T}\nabla _{\theta }\ln \pi _{\theta }(A_{t}\mid S_{t})\;\sum _{t=0}^{T}(\gamma ^{t}R_{t}){\Big |}S_{0}=s_{0}\right]} which can be improved via the "causality trick" ∇ θ J ( θ ) = E π θ [ ∑ t = 0 T ∇ θ ln π θ ( A t ∣ S t ) ∑ τ = t T ( γ τ R τ ) | S 0 = s 0 ] {\displaystyle \nabla _{\theta }J(\theta )=\mathbb {E} _{\pi _{\theta }}\left[\sum _{t=0}^{T}\nabla _{\theta }\ln \pi _{\theta }(A_{t}\mid S_{t})\sum _{\tau =t}^{T}(\gamma ^{\tau }R_{\tau }){\Big |}S_{0}=s_{0}\right]} Thus, we have an unbiased estimator of the policy gradient: ∇ θ J ( θ ) ≈ 1 N ∑ n = 1 N [ ∑ t = 0 T ∇ θ ln π θ ( A t , n ∣ S t , n ) ∑ τ = t T ( γ τ − t R τ , n ) ] {\displaystyle \nabla _{\theta }J(\theta )\approx {\frac {1}{N}}\sum _{n=1}^{N}\left[\sum _{t=0}^{T}\nabla _{\theta }\ln \pi _{\theta }(A_{t,n}\mid S_{t,n})\sum _{\tau =t}^{T}(\gamma ^{\tau -t}R_{\tau ,n})\right]} where the index n {\displaystyle n} ranges over N {\displaystyle N} rollout trajectories using the policy π θ {\displaystyle \pi _{\theta }} . The score function ∇ θ ln π θ ( A t ∣ S t ) {\displaystyle \nabla _{\theta }\ln \pi _{\theta }(A_{t}\mid S_{t})} can be interpreted as the direction in the parameter space that increases the probability of taking action A t {\displaystyle A_{t}} in state S t {\displaystyle S_{t}} . The policy gradient, then, is a weighted average of all possible directions to increase the probability of taking any action in any state, but weighted by reward signals, so that if taking a certain action in a certain state is associated with high reward, then that direction would be highly reinforced, and vice versa. === Algorithm === The REINFORCE algorithm is a loop: Rollout N {\displaystyle N} trajectories in the environment, using π θ t {\displaystyle \pi _{\theta _{t}}} as the policy function. Compute the policy gradient estimation: g i ← 1 N ∑ n = 1 N [ ∑ t = 0 T ∇ θ t ln π θ ( A t , n ∣ S t , n ) ∑ τ = t T ( γ τ R τ , n ) ] {\displaystyle g_{i}\leftarrow {\frac {1}{N}}\sum _{n=1}^{N}\left[\sum _{t=0}^{T}\nabla _{\theta _{t}}\ln \pi _{\theta }(A_{t,n}\mid S_{t,n})\sum _{\tau =t}^{T}(\gamma ^{\tau }R_{\tau ,n})\right]} Update the policy by gradient ascent: θ i + 1 ← θ i + α i g i {\displaystyle \theta _{i+1}\leftarrow \theta _{i}+\alpha _{i}g_{i}} Here, α i {\displaystyle \alpha _{i}} is the learning rate at update step i {\displaystyle i} . == Variance reduction == REINFORCE is an on-policy algorithm, meaning that the trajectories used for the update must be sampled from the current policy π θ {\displaystyle \pi _{\theta }} . This can lead to high variance in the updates, as the returns R ( τ ) {\displaystyle R(\tau )} can vary significantly between trajectories. Many variants of REINFORCE have been introduced, under the title of variance reduction. === REINFORCE with baseline === A common way for reducing variance is the REINFORCE with baseline algorithm, based on the following identity: ∇ θ J ( θ ) = E π θ [ ∑ t = 0 T ∇ θ ln π θ ( A t | S t ) ( ∑ τ = t T ( γ τ R τ ) − b ( S t ) ) | S 0 = s 0 ] {\displaystyle \nabla _{\theta }J(\theta )=\mathbb {E} _{\pi _{\theta }}\left[\sum _{t=0}^{T}\nabla _{\theta }\ln \pi _{\theta }(A_{t}|S_{t})\left(\sum _{\tau =t}^{T}(\gamma ^{\tau }R_{\tau })-b(S_{t})\right){\Big |}S_{0}=s_{0}\right]} for any function b : States → R {\displaystyle b:{\text{States}}\to \mathbb {R} } . This can be proven by applying the previous lemma. The algorithm uses the modified gradient estimator g i ← 1 N ∑ n = 1 N [ ∑ t = 0 T ∇ θ t ln π θ ( A t , n | S t , n ) ( ∑ τ = t T ( γ τ R τ , n ) − b i ( S t , n ) ) ] {\displaystyle g_{i}\leftarrow {\frac {1}{N}}\sum _{n=1}^{N}\left[\sum _{t=0}^{T}\nabla _{\theta _{t}}\ln \pi _{\theta }(A_{t,n}|S_{t,n})\left(\sum _{\tau =t}^{T}(\gamma ^{\tau }R_{\tau ,n})-b_{i}(S_{t,n})\right)\right]} and the original REINFORCE algorithm is the special case where b i ≡ 0 {\displaystyle b_{i}\equiv 0} . === Actor-critic methods === If b i {\textstyle b_{i}} is chosen well, such that b i ( S t ) ≈ ∑ τ = t T ( γ τ R τ ) = γ t V π θ i ( S t ) {\textstyle b_{i}(S_{t})\approx \sum _{\tau =t}^{T}(\gamma ^{\tau }R_{\tau })=\gamma ^{t}V^{\pi _{\theta _{i}}}(S_{t})} , this could significantly decrease variance in the gradient estimation. That is, the baseline should be as close to the value function V π θ i ( S t ) {\displaystyle V^{\pi _{\theta _{i}}}(S_{t})} as possible, approaching the ideal of: ∇ θ J ( θ ) = E π θ [ ∑ t = 0 T ∇ θ ln π θ ( A t | S t ) ( ∑ τ = t T ( γ τ R τ ) − γ t V π θ ( S t ) ) | S 0 = s 0 ] {\displaystyle \nabla _{\theta }J(\theta )=\mathbb {E} _{\pi _{\theta }}\left[\sum _{t=0}^{T}\nabla _{\theta }\ln \pi _{\theta }(A_{t}|S_{t})\left(\sum _{\tau =t}^{T}(\gamma ^{\tau }R_{\tau })-\gamma ^{t}V^{\pi _{\theta }}(S_{t})\right){\Big |}S_{0}=s_{0}\right]} Note that, as the policy π θ t {\displaystyle \pi _{\theta _{t}}} updates, the value function V π θ i ( S t ) {\displaystyle V^{\pi _{\theta _{i}}}(S_{t})} updates as well, so the baseline should also be updated. One common approach is to train a separate function that estimates the value function, and use that as the baseline. This is one of the actor-critic methods, where the policy function is the actor and the value function is the critic. The Q-function Q π {\displaystyle Q^{\pi }} can also be used as the critic, since ∇ θ J ( θ ) = E π θ [ ∑ 0 ≤ t ≤ T γ t ∇ θ ln π θ ( A t | S t ) ⋅ Q π θ ( S t , A t ) | S 0 = s 0 ] {\displaystyle \nabla _{\theta }J(\theta )=E_{\pi _{\theta }}\left[\sum _{0\leq t\leq T}\gamma ^{t}\nabla _{\theta }\ln \pi _{\theta }(A_{t}|S_{t})\cdot Q^{\pi _{\theta }}(S_{t},A_{t}){\Big |}S_{0}=s_{0}\right]} by a similar argument using the tower law. Subtracting the value function as a baseline, we find that the advantage function A π ( S , A ) = Q π ( S , A ) − V π ( S ) {\displaystyle A^{\pi }(S,A)=Q^{\pi }(S,A)-V^{\pi }(S)} can be used as the critic as well: ∇ θ J ( θ ) = E π θ [ ∑ 0 ≤ t ≤ T γ t ∇ θ ln π θ ( A t | S t ) ⋅ A π θ ( S t , A t ) | S 0 = s 0 ] {\displaystyle \nabla _{\theta }J(\theta )=E_{\pi _{\theta }}\left[\sum _{0\leq t\leq T}\gamma ^{t}\nabla _{\theta }\ln \pi _{\theta }(A_{t}|S_{t})\cdot A^{\pi _{\theta }}(S_{t},A_{t}){\Big |}S_{0}=s_{0}\right]} In summary, there are many unbiased estimators for ∇ θ J θ {\textstyle \nabla _{\theta }J_{\theta }} , all in the form of: ∇ θ J ( θ ) = E π θ [ ∑ 0 ≤ t ≤ T ∇ θ ln π θ ( A t | S t ) ⋅ Ψ t | S 0 = s 0 ] {\displaystyle \nabla _{\theta }J(\theta )=E_{\pi _{\theta }}\left[\su
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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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Information extraction (IE) is the task of automatically extracting structured information from unstructured and/or semi-structured machine-readable documents and other electronically represented sources. Typically, this involves processing human language texts by means of natural language processing (NLP). Recent activities in multimedia document processing like automatic annotation and content extraction out of images/audio/video/documents could be seen as information extraction. Recent advances in NLP techniques have allowed for significantly improved performance compared to previous years. An example is the extraction from newswire reports of corporate mergers, such as denoted by the formal relation: MergerBetween ( c o m p a n y 1 , c o m p a n y 2 , d a t e ) {\displaystyle \operatorname {MergerBetween} (\mathrm {company} _{1},\mathrm {company} _{2},\mathrm {date} )} , from an online news sentence such as: "Yesterday, New York based Foo Inc. announced their acquisition of Bar Corp." A broad goal of IE is to allow computation to be done on the previously unstructured data. A more specific goal is to allow automated reasoning about the logical form of the input data. Structured data is semantically well-defined data from a chosen target domain, interpreted with respect to category and context. Information extraction is the part of a greater puzzle which deals with the problem of devising automatic methods for text management, beyond its transmission, storage and display. The discipline of information retrieval (IR) has developed automatic methods, typically of a statistical flavor, for indexing large document collections and classifying documents. Another complementary approach is that of natural language processing (NLP) which has solved the problem of modelling human language processing with considerable success when taking into account the magnitude of the task. In terms of both difficulty and emphasis, IE deals with tasks in between both IR and NLP. In terms of input, IE assumes the existence of a set of documents in which each document follows a template, i.e. describes one or more entities or events in a manner that is similar to those in other documents but differing in the details. An example, consider a group of newswire articles on Latin American terrorism with each article presumed to be based upon one or more terroristic acts. We also define for any given IE task a template, which is a(or a set of) case frame(s) to hold the information contained in a single document. For the terrorism example, a template would have slots corresponding to the perpetrator, victim, and weapon of the terroristic act, and the date on which the event happened. An IE system for this problem is required to "understand" an attack article only enough to find data corresponding to the slots in this template. == History == Information extraction dates back to the late 1970s in the early days of NLP. An early commercial system from the mid-1980s was JASPER built for Reuters by the Carnegie Group Inc with the aim of providing real-time financial news to financial traders. Beginning in 1987, IE was spurred by a series of Message Understanding Conferences. MUC is a competition-based conference that focused on the following domains: MUC-1 (1987), MUC-3 (1989): Naval operations messages. MUC-3 (1991), MUC-4 (1992): Terrorism in Latin American countries. MUC-5 (1993): Joint ventures and microelectronics domain. MUC-6 (1995): News articles on management changes. MUC-7 (1998): Satellite launch reports. Considerable support came from the U.S. Defense Advanced Research Projects Agency (DARPA), who wished to automate mundane tasks performed by government analysts, such as scanning newspapers for possible links to terrorism. == Present significance == The present significance of IE pertains to the growing amount of information available in unstructured form. Tim Berners-Lee, inventor of the World Wide Web, refers to the existing Internet as the web of documents and advocates that more of the content be made available as a web of data. Until this transpires, the web largely consists of unstructured documents lacking semantic metadata. Knowledge contained within these documents can be made more accessible for machine processing by means of transformation into relational form, or by marking-up with XML tags. An intelligent agent monitoring a news data feed requires IE to transform unstructured data into something that can be reasoned with. A typical application of IE is to scan a set of documents written in a natural language and populate a database with the information extracted. == Tasks and subtasks == Applying information extraction to text is linked to the problem of text simplification in order to create a structured view of the information present in free text. The overall goal being to create a more easily machine-readable text to process the sentences. Typical IE tasks and subtasks include: Template filling: Extracting a fixed set of fields from a document, e.g. extract perpetrators, victims, time, etc. from a newspaper article about a terrorist attack. Event extraction: Given an input document, output zero or more event templates. For instance, a newspaper article might describe multiple terrorist attacks. Knowledge Base Population: Fill a database of facts given a set of documents. Typically the database is in the form of triplets, (entity 1, relation, entity 2), e.g. (Barack Obama, Spouse, Michelle Obama) Named entity recognition: recognition of known entity names (for people and organizations), place names, temporal expressions, and certain types of numerical expressions, by employing existing knowledge of the domain or information extracted from other sentences. Typically the recognition task involves assigning a unique identifier to the extracted entity. A simpler task is named entity detection, which aims at detecting entities without having any existing knowledge about the entity instances. For example, in processing the sentence "M. Smith likes fishing", named entity detection would denote detecting that the phrase "M. Smith" does refer to a person, but without necessarily having (or using) any knowledge about a certain M. Smith who is (or, "might be") the specific person whom that sentence is talking about. Coreference resolution: detection of coreference and anaphoric links between text entities. In IE tasks, this is typically restricted to finding links between previously extracted named entities. For example, "International Business Machines" and "IBM" refer to the same real-world entity. If we take the two sentences "M. Smith likes fishing. But he doesn't like biking", it would be beneficial to detect that "he" is referring to the previously detected person "M. Smith". Relationship extraction: identification of relations between entities, such as: PERSON works for ORGANIZATION (extracted from the sentence "Bill works for IBM.") PERSON located in LOCATION (extracted from the sentence "Bill is in France.") Semi-structured information extraction which may refer to any IE that tries to restore some kind of information structure that has been lost through publication, such as: Table extraction: finding and extracting tables from documents. Table information extraction : extracting information in structured manner from the tables. This task is more complex than table extraction, as table extraction is only the first step, while understanding the roles of the cells, rows, columns, linking the information inside the table and understanding the information presented in the table are additional tasks necessary for table information extraction. Comments extraction : extracting comments from the actual content of articles in order to restore the link between authors of each of the sentences Language and vocabulary analysis Terminology extraction: finding the relevant terms for a given corpus Audio extraction Template-based music extraction: finding relevant characteristic in an audio signal taken from a given repertoire; for instance time indexes of occurrences of percussive sounds can be extracted in order to represent the essential rhythmic component of a music piece. Note that this list is not exhaustive and that the exact meaning of IE activities is not commonly accepted and that many approaches combine multiple sub-tasks of IE in order to achieve a wider goal. Machine learning, statistical analysis and/or natural language processing are often used in IE. IE on non-text documents is becoming an increasingly interesting topic in research, and information extracted from multimedia documents can now be expressed in a high level structure as it is done on text. This naturally leads to the fusion of extracted information from multiple kinds of documents and sources. == World Wide Web applications == IE has been the focus of the MUC conferences. The proliferation of the Web, however, intensified the need for developing IE systems that help people
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The sum of absolute transformed differences (SATD) is a block matching criterion widely used in fractional motion estimation for video compression. It works by taking a frequency transform, usually a Hadamard transform, of the differences between the pixels in the original block and the corresponding pixels in the block being used for comparison. The transform itself is often of a small block rather than the entire macroblock. For example, in x264, a series of 4×4 blocks are transformed rather than doing the more processor-intensive 16×16 transform. == Comparison to other metrics == SATD is slower than the sum of absolute differences (SAD), both due to its increased complexity and the fact that SAD-specific MMX and SSE2 instructions exist, while there are no such instructions for SATD. However, SATD can still be optimized considerably with SIMD instructions on most modern CPUs. The benefit of SATD is that it more accurately models the number of bits required to transmit the residual error signal. As such, it is often used in video compressors, either as a way to drive and estimate rate explicitly, such as in the Theora encoder (since 1.1 alpha2), as an optional metric used in wide motion searches, such as in the Microsoft VC-1 encoder, or as a metric used in sub-pixel refinement, such as in x264.
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The Arabic Speech Corpus is a Modern Standard Arabic (MSA) speech corpus for speech synthesis. The corpus contains phonetic and orthographic transcriptions of more than 3.7 hours of MSA speech aligned with recorded speech on the phoneme level. The annotations include word stress marks on the individual phonemes. The Arabic Speech Corpus was built as part of a doctoral project by Nawar Halabi at the University of Southampton funded by MicroLinkPC who own an exclusive license to commercialise the corpus, but the corpus is available for strictly non-commercial purposes through the official Arabic Speech Corpus website. It is distributed under the Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. == Purpose == The corpus was mainly built for speech synthesis purposes, specifically Speech Synthesis, but the corpus has been used for building HMM based voices in Arabic. It was also used to automatically align other speech corpora with their phonetic transcript and could be used as part of a larger corpus for training speech recognition systems. == Contents == The package contains the following: 1813 .wav files containing spoken utterances. 1813 .lab files containing text utterances. 1813 .TextGrid files containing the phoneme labels with time stamps of the boundaries where these occur in the .wav files. phonetic-transcript.txt which has the form "[wav_filename]" "[Phoneme Sequence]" in every line. orthographic-transcript.txt which has the form "[wav_filename]" "[Orthographic Transcript]" in every line. Orthography is in Buckwalter Format which is friendlier where there is software that does not read Arabic script. It can be easily converted back to Arabic. There is an extra 18 minutes of fully annotated corpus (separate from above but with the same structure as above) which was used to evaluated the corpus (see PhD thesis). The corpus was also used to prove that using automatically extracted, orthography-based stress marks improve the quality of speech synthesis in MSA.
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The softmax function, also known as softargmax or normalized exponential function, converts a tuple of K real numbers into a probability distribution over K possible outcomes. It is a generalization of the logistic function to multiple dimensions, and is used in multinomial logistic regression. The softmax function is often used as the last activation function of a neural network to normalize the output of a network to a probability distribution over predicted output classes. == Definition == The softmax function takes as input a tuple z of K real numbers, and normalizes it into a probability distribution consisting of K probabilities proportional to the exponentials of the input numbers. That is, prior to applying softmax, some tuple components could be negative, or greater than one; and might not sum to 1; but after applying softmax, each component will be in the interval ( 0 , 1 ) {\displaystyle (0,1)} , and the components will add up to 1, so that they can be interpreted as probabilities. Furthermore, the larger input components will correspond to larger probabilities. Formally, the standard (unit) softmax function σ : R K → ( 0 , 1 ) K {\displaystyle \sigma :\mathbb {R} ^{K}\to (0,1)^{K}} , where K > 1 {\displaystyle K>1} , takes a tuple z = ( z 1 , … , z K ) ∈ R K {\displaystyle \mathbf {z} =(z_{1},\dotsc ,z_{K})\in \mathbb {R} ^{K}} and computes each component of vector σ ( z ) ∈ ( 0 , 1 ) K {\displaystyle \sigma (\mathbf {z} )\in (0,1)^{K}} with σ ( z ) i = e z i ∑ j = 1 K e z j . {\displaystyle \sigma (\mathbf {z} )_{i}={\frac {e^{z_{i}}}{\sum _{j=1}^{K}e^{z_{j}}}}\,.} In words, the softmax applies the standard exponential function to each element z i {\displaystyle z_{i}} of the input tuple z {\displaystyle \mathbf {z} } (consisting of K {\displaystyle K} real numbers), and normalizes these values by dividing by the sum of all these exponentials. The normalization ensures that the sum of the components of the output vector σ ( z ) {\displaystyle \sigma (\mathbf {z} )} is 1. The term "softmax" derives from the amplifying effects of the exponential on any maxima in the input tuple. For example, the standard softmax of ( 1 , 2 , 8 ) {\displaystyle (1,2,8)} is approximately ( 0.001 , 0.002 , 0.997 ) {\displaystyle (0.001,0.002,0.997)} , which amounts to assigning almost all of the total unit weight in the result to the position of the tuple's maximal element (of 8). In general, instead of e a different base b > 0 can be used. As above, if b > 1 then larger input components will result in larger output probabilities, and increasing the value of b will create probability distributions that are more concentrated around the positions of the largest input values. Conversely, if 0 < b < 1 then smaller input components will result in larger output probabilities, and decreasing the value of b will create probability distributions that are more concentrated around the positions of the smallest input values. Writing b = e β {\displaystyle b=e^{\beta }} or b = e − β {\displaystyle b=e^{-\beta }} (for real β) yields the expressions: σ ( z ) i = e β z i ∑ j = 1 K e β z j or σ ( z ) i = e − β z i ∑ j = 1 K e − β z j for i = 1 , … , K . {\displaystyle \sigma (\mathbf {z} )_{i}={\frac {e^{\beta z_{i}}}{\sum _{j=1}^{K}e^{\beta z_{j}}}}{\text{ or }}\sigma (\mathbf {z} )_{i}={\frac {e^{-\beta z_{i}}}{\sum _{j=1}^{K}e^{-\beta z_{j}}}}{\text{ for }}i=1,\dotsc ,K.} A value proportional to the reciprocal of β is sometimes referred to as the temperature: β = 1 / k T {\textstyle \beta =1/kT} , where k is typically 1 or the Boltzmann constant and T is the temperature. A higher temperature results in a more uniform output distribution (i.e. with higher entropy; it is "more random"), while a lower temperature results in a sharper output distribution, with one value dominating. In some fields, the base is fixed, corresponding to a fixed scale, while in others the parameter β (or T) is varied. The softmax function is a multiple-variable generalization of the logistic function. == Interpretations == === Smooth arg max === The Softmax function is a smooth approximation to the arg max function: the function whose value is the index of a tuple's largest element. The name "softmax" may be misleading. Softmax is not a smooth maximum (that is, a smooth approximation to the maximum function). The term "softmax" is also used for the closely related LogSumExp function, which is a smooth maximum. For this reason, some prefer the more accurate term "softargmax", though the term "softmax" is conventional in machine learning. This section uses the term "softargmax" for clarity. Formally, instead of considering the arg max as a function with categorical output 1 , … , n {\displaystyle 1,\dots ,n} (corresponding to the index), consider the arg max function with one-hot representation of the output (assuming there is a unique maximum arg): a r g m a x ( z 1 , … , z n ) = ( y 1 , … , y n ) = ( 0 , … , 0 , 1 , 0 , … , 0 ) , {\displaystyle \operatorname {arg\,max} (z_{1},\,\dots ,\,z_{n})=(y_{1},\,\dots ,\,y_{n})=(0,\,\dots ,\,0,\,1,\,0,\,\dots ,\,0),} where the output coordinate y i = 1 {\displaystyle y_{i}=1} if and only if i {\displaystyle i} is the arg max of ( z 1 , … , z n ) {\displaystyle (z_{1},\dots ,z_{n})} , meaning z i {\displaystyle z_{i}} is the unique maximum value of ( z 1 , … , z n ) {\displaystyle (z_{1},\,\dots ,\,z_{n})} . For example, in this encoding a r g m a x ( 1 , 5 , 10 ) = ( 0 , 0 , 1 ) , {\displaystyle \operatorname {arg\,max} (1,5,10)=(0,0,1),} since the third argument is the maximum. This can be generalized to multiple arg max values (multiple equal z i {\displaystyle z_{i}} being the maximum) by dividing the 1 between all max args; formally 1/k where k is the number of arguments assuming the maximum. For example, a r g m a x ( 1 , 5 , 5 ) = ( 0 , 1 / 2 , 1 / 2 ) , {\displaystyle \operatorname {arg\,max} (1,\,5,\,5)=(0,\,1/2,\,1/2),} since the second and third argument are both the maximum. In case all arguments are equal, this is simply a r g m a x ( z , … , z ) = ( 1 / n , … , 1 / n ) . {\displaystyle \operatorname {arg\,max} (z,\dots ,z)=(1/n,\dots ,1/n).} Points z with multiple arg max values are singular points (or singularities, and form the singular set) – these are the points where arg max is discontinuous (with a jump discontinuity) – while points with a single arg max are known as non-singular or regular points. With the last expression given in the introduction, softargmax is now a smooth approximation of arg max: as β → ∞ {\displaystyle \beta \to \infty } , softargmax converges to arg max. There are various notions of convergence of a function; softargmax converges to arg max pointwise, meaning for each fixed input z as β → ∞ {\displaystyle \beta \to \infty } , σ β ( z ) → a r g m a x ( z ) . {\displaystyle \sigma _{\beta }(\mathbf {z} )\to \operatorname {arg\,max} (\mathbf {z} ).} However, softargmax does not converge uniformly to arg max, meaning intuitively that different points converge at different rates, and may converge arbitrarily slowly. In fact, softargmax is continuous, but arg max is not continuous at the singular set where two coordinates are equal, while the uniform limit of continuous functions is continuous. The reason it fails to converge uniformly is that for inputs where two coordinates are almost equal (and one is the maximum), the arg max is the index of one or the other, so a small change in input yields a large change in output. For example, σ β ( 1 , 1.0001 ) → ( 0 , 1 ) , {\displaystyle \sigma _{\beta }(1,\,1.0001)\to (0,1),} but σ β ( 1 , 0.9999 ) → ( 1 , 0 ) , {\displaystyle \sigma _{\beta }(1,\,0.9999)\to (1,\,0),} and σ β ( 1 , 1 ) = 1 / 2 {\displaystyle \sigma _{\beta }(1,\,1)=1/2} for all inputs: the closer the points are to the singular set ( x , x ) {\displaystyle (x,x)} , the slower they converge. However, softargmax does converge compactly on the non-singular set. Conversely, as β → − ∞ {\displaystyle \beta \to -\infty } , softargmax converges to arg min in the same way, where here the singular set is points with two arg min values. In the language of tropical analysis, the softmax is a deformation or "quantization" of arg max and arg min, corresponding to using the log semiring instead of the max-plus semiring (respectively min-plus semiring), and recovering the arg max or arg min by taking the limit is called "tropicalization" or "dequantization". It is also the case that, for any fixed β, if one input z i {\displaystyle z_{i}} is much larger than the others relative to the temperature, T = 1 / β {\displaystyle T=1/\beta } , the output is approximately the arg max. For example, a difference of 10 is large relative to a temperature of 1: σ ( 0 , 10 ) := σ 1 ( 0 , 10 ) = ( 1 / ( 1 + e 10 ) , e 10 / ( 1 + e 10 ) ) ≈ ( 0.00005 , 0.99995 ) {\displaystyle \sigma (0,\,10):=\sigma _{1}(0,\,10)=\left(1/\left(1+e^{10}\right),\,e^{10}/\left(1+e^{10}\right)\right)\approx (0.00005
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Dynamic epistemic logic (DEL) is a logical framework dealing with knowledge and information change. Typically, DEL focuses on situations involving multiple agents and studies how their knowledge changes when events occur. These events can change factual properties of the actual world (they are called ontic events): for example a red card is painted in blue. They can also bring about changes of knowledge without changing factual properties of the world (they are called epistemic events): for example, a card is revealed publicly (or privately) to be red. Originally, DEL focused on epistemic events. Only some of the basic ideas are present in this entry of the original DEL framework; more details about DEL in general can be found in the references. Due to the nature of its object of study and its abstract approach, DEL is related and has applications to numerous research areas, such as computer science (artificial intelligence), philosophy (formal epistemology), economics (game theory) and cognitive science. In computer science, DEL is for example very much related to multi-agent systems, which are systems where multiple intelligent agents interact and exchange information. As a combination of dynamic logic and epistemic logic, dynamic epistemic logic is a young field of research. It really started in 1989 with Plaza's logic of public announcement. Independently, Gerbrandy and Groeneveld proposed a system dealing moreover with private announcement and that was inspired by the work of Veltman. Another system was proposed by van Ditmarsch whose main inspiration was the Cluedo game. But the most influential and original system was the system proposed by Baltag, Moss and Solecki. This system can deal with all the types of situations studied in the works above and its underlying methodology is conceptually grounded. This entry will present some of its basic ideas. Formally, DEL extends ordinary epistemic logic by the inclusion of event models to describe actions, and a product update operator that defines how epistemic models are updated as the consequence of executing actions described through event models. Epistemic logic will first be recalled. Then, actions and events will enter into the picture and we will introduce the DEL framework. == Epistemic logic == Epistemic logic is a modal logic dealing with the notions of knowledge and belief. As a logic, it is concerned with understanding the process of reasoning about knowledge and belief: which principles relating the notions of knowledge and belief are intuitively plausible? Like epistemology, it stems from the Greek word ϵ π ι σ τ η μ η {\displaystyle \epsilon \pi \iota \sigma \tau \eta \mu \eta } or ‘episteme’ meaning knowledge. Epistemology is nevertheless more concerned with analyzing the very nature and scope of knowledge, addressing questions such as “What is the definition of knowledge?” or “How is knowledge acquired?”. In fact, epistemic logic grew out of epistemology in the Middle Ages thanks to the efforts of Burley and Ockham. The formal work, based on modal logic, that inaugurated contemporary research into epistemic logic dates back only to 1962 and is due to Hintikka. It then sparked in the 1960s discussions about the principles of knowledge and belief and many axioms for these notions were proposed and discussed. For example, the interaction axioms K p → B p {\displaystyle Kp\rightarrow Bp} and B p → K B p {\displaystyle Bp\rightarrow KBp} are often considered to be intuitive principles: if an agent Knows p {\displaystyle p} then (s)he also Believes p {\displaystyle p} , or if an agent Believes p {\displaystyle p} , then (s)he Knows that (s)he Believes p {\displaystyle p} . More recently, these kinds of philosophical theories were taken up by researchers in economics, artificial intelligence and theoretical computer science where reasoning about knowledge is a central topic. Due to the new setting in which epistemic logic was used, new perspectives and new features such as computability issues were then added to the research agenda of epistemic logic. === Syntax === In the sequel, A G T S = { 1 , … , n } {\displaystyle AGTS=\{1,\ldots ,n\}} is a finite set whose elements are called agents and P R O P {\displaystyle PROP} is a set of propositional letters. The epistemic language is an extension of the basic multi-modal language of modal logic with a common knowledge operator C A {\displaystyle C_{A}} and a distributed knowledge operator D A {\displaystyle D_{A}} . Formally, the epistemic language L EL C {\displaystyle {\mathcal {L}}_{\textsf {EL}}^{C}} is defined inductively by the following grammar in BNF: L EL C : ϕ ::= p ∣ ¬ ϕ ∣ ( ϕ ∧ ϕ ) ∣ K j ϕ ∣ C A ϕ ∣ D A ϕ {\displaystyle {\mathcal {L}}_{\textsf {EL}}^{C}:\phi ~~::=~~p~\mid ~\neg \phi ~\mid ~(\phi \land \phi )~\mid ~K_{j}\phi ~\mid ~C_{A}\phi ~\mid ~D_{A}\phi } where p ∈ P R O P {\displaystyle p\in PROP} , j ∈ A G T S {\displaystyle j\in {AGTS}} and A ⊆ A G T S {\displaystyle A\subseteq {AGTS}} . The basic epistemic language L E L {\displaystyle {\mathcal {L}}_{EL}} is the language L E L C {\displaystyle {\mathcal {L}}_{EL}^{C}} without the common knowledge and distributed knowledge operators. The formula ⊥ {\displaystyle \bot } is an abbreviation for ¬ p ∧ p {\displaystyle \neg p\land p} (for a given p ∈ P R O P {\displaystyle p\in PROP} ), ⟨ K j ⟩ ϕ {\displaystyle \langle K_{j}\rangle \phi } is an abbreviation for ¬ K j ¬ ϕ {\displaystyle \neg K_{j}\neg \phi } , E A ϕ {\displaystyle E_{A}\phi } is an abbreviation for ⋀ j ∈ A K j ϕ {\displaystyle \bigwedge \limits _{j\in A}K_{j}\phi } and C ϕ {\displaystyle C\phi } an abbreviation for C A G T S ϕ {\displaystyle C_{AGTS}\phi } . Group notions: general, common and distributed knowledge. In a multi-agent setting there are three important epistemic concepts: general knowledge, distributed knowledge and common knowledge. The notion of common knowledge was first studied by Lewis in the context of conventions. It was then applied to distributed systems and to game theory, where it allows to express that the rationality of the players, the rules of the game and the set of players are commonly known. General knowledge. General knowledge of ϕ {\displaystyle \phi } means that everybody in the group of agents A G T S {\displaystyle {AGTS}} knows that ϕ {\displaystyle \phi } . Formally, this corresponds to the following formula: E ϕ := ⋀ j ∈ A G T S K j ϕ . {\displaystyle E\phi :={\underset {j\in {AGTS}}{\bigwedge }}K_{j}\phi .} Common knowledge. Common knowledge of ϕ {\displaystyle \phi } means that everybody knows ϕ {\displaystyle \phi } but also that everybody knows that everybody knows ϕ {\displaystyle \phi } , that everybody knows that everybody knows that everybody knows ϕ {\displaystyle \phi } , and so on ad infinitum. Formally, this corresponds to the following formula C ϕ := E ϕ ∧ E E ϕ ∧ E E E ϕ ∧ … {\displaystyle C\phi :=E\phi \land EE\phi \land EEE\phi \land \ldots } As we do not allow infinite conjunction the notion of common knowledge will have to be introduced as a primitive in our language. Before defining the language with this new operator, we are going to give an example introduced by Lewis that illustrates the difference between the notions of general knowledge and common knowledge. Lewis wanted to know what kind of knowledge is needed so that the statement p {\displaystyle p} : “every driver must drive on the right” be a convention among a group of agents. In other words, he wanted to know what kind of knowledge is needed so that everybody feels safe to drive on the right. Suppose there are only two agents i {\displaystyle i} and j {\displaystyle j} . Then everybody knowing p {\displaystyle p} (formally E p {\displaystyle Ep} ) is not enough. Indeed, it might still be possible that the agent i {\displaystyle i} considers possible that the agent j {\displaystyle j} does not know p {\displaystyle p} (formally ¬ K i K j p {\displaystyle \neg K_{i}K_{j}p} ). In that case the agent i {\displaystyle i} will not feel safe to drive on the right because he might consider that the agent j {\displaystyle j} , not knowing p {\displaystyle p} , could drive on the left. To avoid this problem, we could then assume that everybody knows that everybody knows that p {\displaystyle p} (formally E E p {\displaystyle EEp} ). This is again not enough to ensure that everybody feels safe to drive on the right. Indeed, it might still be possible that agent i {\displaystyle i} considers possible that agent j {\displaystyle j} considers possible that agent i {\displaystyle i} does not know p {\displaystyle p} (formally ¬ K i K j K i p {\displaystyle \neg K_{i}K_{j}K_{i}p} ). In that case and from i {\displaystyle i} ’s point of view, j {\displaystyle j} considers possible that i {\displaystyle i} , not knowing p {\displaystyle p} , will drive on the left. So from i {\displaystyle i} ’s point of view, j {\displaystyle j} might drive on the left as well (by the same argument as abov
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An alternating decision tree (ADTree) is a machine learning method for classification. It generalizes decision trees and has connections to boosting. An ADTree consists of an alternation of decision nodes, which specify a predicate condition, and prediction nodes, which contain a single number. An instance is classified by an ADTree by following all paths for which all decision nodes are true, and summing any prediction nodes that are traversed. == History == ADTrees were introduced by Yoav Freund and Llew Mason. However, the algorithm as presented had several typographical errors. Clarifications and optimizations were later presented by Bernhard Pfahringer, Geoffrey Holmes and Richard Kirkby. Implementations are available in Weka and JBoost. == Motivation == Original boosting algorithms typically used either decision stumps or decision trees as weak hypotheses. As an example, boosting decision stumps creates a set of T {\displaystyle T} weighted decision stumps (where T {\displaystyle T} is the number of boosting iterations), which then vote on the final classification according to their weights. Individual decision stumps are weighted according to their ability to classify the data. Boosting a simple learner results in an unstructured set of T {\displaystyle T} hypotheses, making it difficult to infer correlations between attributes. Alternating decision trees introduce structure to the set of hypotheses by requiring that they build off a hypothesis that was produced in an earlier iteration. The resulting set of hypotheses can be visualized in a tree based on the relationship between a hypothesis and its "parent." Another important feature of boosted algorithms is that the data is given a different distribution at each iteration. Instances that are misclassified are given a larger weight while accurately classified instances are given reduced weight. == Alternating decision tree structure == An alternating decision tree consists of decision nodes and prediction nodes. Decision nodes specify a predicate condition. Prediction nodes contain a single number. ADTrees always have prediction nodes as both root and leaves. An instance is classified by an ADTree by following all paths for which all decision nodes are true and summing any prediction nodes that are traversed. This is different from binary classification trees such as CART (Classification and regression tree) or C4.5 in which an instance follows only one path through the tree. === Example === The following tree was constructed using JBoost on the spambase dataset (available from the UCI Machine Learning Repository). In this example, spam is coded as 1 and regular email is coded as −1. The following table contains part of the information for a single instance. The instance is scored by summing all of the prediction nodes through which it passes. In the case of the instance above, the score is calculated as The final score of 0.657 is positive, so the instance is classified as spam. The magnitude of the value is a measure of confidence in the prediction. The original authors list three potential levels of interpretation for the set of attributes identified by an ADTree: Individual nodes can be evaluated for their own predictive ability. Sets of nodes on the same path may be interpreted as having a joint effect The tree can be interpreted as a whole. Care must be taken when interpreting individual nodes as the scores reflect a re weighting of the data in each iteration. == Description of the algorithm == The inputs to the alternating decision tree algorithm are: A set of inputs ( x 1 , y 1 ) , … , ( x m , y m ) {\displaystyle (x_{1},y_{1}),\ldots ,(x_{m},y_{m})} where x i {\displaystyle x_{i}} is a vector of attributes and y i {\displaystyle y_{i}} is either -1 or 1. Inputs are also called instances. A set of weights w i {\displaystyle w_{i}} corresponding to each instance. The fundamental element of the ADTree algorithm is the rule. A single rule consists of a precondition, a condition, and two scores. A condition is a predicate of the form "attribute
Multilinear subspace learning is an approach for disentangling the causal factor of data formation and performing dimensionality reduction. The Dimensionality reduction can be performed on a data tensor that contains a collection of observations that have been vectorized, or observations that are treated as matrices and concatenated into a data tensor. Here are some examples of data tensors whose observations are vectorized or whose observations are matrices concatenated into data tensor images (2D/3D), video sequences (3D/4D), and hyperspectral cubes (3D/4D). The mapping from a high-dimensional vector space to a set of lower dimensional vector spaces is a multilinear projection. When observations are retained in the same organizational structure as matrices or higher order tensors, their representations are computed by performing linear projections into the column space, row space and fiber space. Multilinear subspace learning algorithms are higher-order generalizations of linear subspace learning methods such as principal component analysis (PCA), independent component analysis (ICA), linear discriminant analysis (LDA) and canonical correlation analysis (CCA). == Background == Multilinear methods may be causal in nature and perform causal inference, or they may be simple regression methods from which no causal conclusion are drawn. Linear subspace learning algorithms are traditional dimensionality reduction techniques that are well suited for datasets that are the result of varying a single causal factor. Unfortunately, they often become inadequate when dealing with datasets that are the result of multiple causal factors. . Multilinear subspace learning can be applied to observations whose measurements were vectorized and organized into a data tensor for causally aware dimensionality reduction. These methods may also be employed in reducing horizontal and vertical redundancies irrespective of the causal factors when the observations are treated as a "matrix" (ie. a collection of independent column/row observations) and concatenated into a tensor. == Algorithms == === Multilinear principal component analysis === Historically, multilinear principal component analysis has been referred to as "M-mode PCA", a terminology which was coined by Peter Kroonenberg. In 2005, Vasilescu and Terzopoulos introduced the Multilinear PCA terminology as a way to better differentiate between multilinear tensor decompositions that computed 2nd order statistics associated with each data tensor mode, and subsequent work on Multilinear Independent Component Analysis that computed higher order statistics for each tensor mode. MPCA is an extension of PCA. === Multilinear independent component analysis === Multilinear independent component analysis is an extension of ICA. === Multilinear linear discriminant analysis === Multilinear extension of LDA TTP-based: Discriminant Analysis with Tensor Representation (DATER) TTP-based: General tensor discriminant analysis (GTDA) TVP-based: Uncorrelated Multilinear Discriminant Analysis (UMLDA) === Multilinear canonical correlation analysis === Multilinear extension of CCA TTP-based: Tensor Canonical Correlation Analysis (TCCA) TVP-based: Multilinear Canonical Correlation Analysis (MCCA) TVP-based: Bayesian Multilinear Canonical Correlation Analysis (BMTF) A TTP is a direct projection of a high-dimensional tensor to a low-dimensional tensor of the same order, using N projection matrices for an Nth-order tensor. It can be performed in N steps with each step performing a tensor-matrix multiplication (product). The N steps are exchangeable. This projection is an extension of the higher-order singular value decomposition (HOSVD) to subspace learning. Hence, its origin is traced back to the Tucker decomposition in 1960s. A TVP is a direct projection of a high-dimensional tensor to a low-dimensional vector, which is also referred to as the rank-one projections. As TVP projects a tensor to a vector, it can be viewed as multiple projections from a tensor to a scalar. Thus, the TVP of a tensor to a P-dimensional vector consists of P projections from the tensor to a scalar. The projection from a tensor to a scalar is an elementary multilinear projection (EMP). In EMP, a tensor is projected to a point through N unit projection vectors. It is the projection of a tensor on a single line (resulting a scalar), with one projection vector in each mode. Thus, the TVP of a tensor object to a vector in a P-dimensional vector space consists of P EMPs. This projection is an extension of the canonical decomposition, also known as the parallel factors (PARAFAC) decomposition. === Typical approach in MSL === There are N sets of parameters to be solved, one in each mode. The solution to one set often depends on the other sets (except when N=1, the linear case). Therefore, the suboptimal iterative procedure in is followed. Initialization of the projections in each mode For each mode, fixing the projection in all the other mode, and solve for the projection in the current mode. Do the mode-wise optimization for a few iterations or until convergence. This is originated from the alternating least square method for multi-way data analysis. == Code == MATLAB Tensor Toolbox by Sandia National Laboratories. The MPCA algorithm written in Matlab (MPCA+LDA included). The UMPCA algorithm written in Matlab (data included). The UMLDA algorithm written in Matlab (data included). == Tensor data sets == 3D gait data (third-order tensors): 128x88x20(21.2M); 64x44x20(9.9M); 32x22x10(3.2M);
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