The fuzzy pay-off method for real option valuation (FPOM or pay-off method) is a method for valuing real options, developed by Mikael Collan, Robert Fullér, and József Mezei; and published in 2009. It is based on the use of fuzzy logic and fuzzy numbers for the creation of the possible pay-off distribution of a project (real option). The structure of the method is similar to the probability theory based Datar–Mathews method for real option valuation, but the method is not based on probability theory and uses fuzzy numbers and possibility theory in framing the real option valuation problem. == Method == The Fuzzy pay-off method derives the real option value from a pay-off distribution that is created by using three or four cash-flow scenarios (most often created by an expert or a group of experts). The pay-off distribution is created simply by assigning each of the three cash-flow scenarios a corresponding definition with regards to a fuzzy number (triangular fuzzy number for three scenarios and a trapezoidal fuzzy number for four scenarios). This means that the pay-off distribution is created without any simulation whatsoever. This makes the procedure easy and transparent. The scenarios used are a minimum possible scenario (the lowest possible outcome), the maximum possible scenario (the highest possible outcome) and a best estimate (most likely to happen scenario) that is mapped as a fully possible scenario with a full degree of membership in the set of possible outcomes, or in the case of four scenarios used - two best estimate scenarios that are the upper and lower limit of the interval that is assigned a full degree of membership in the set of possible outcomes. The main observations that lie behind the model for deriving the real option value are the following: The fuzzy NPV of a project is (equal to) the pay-off distribution of a project value that is calculated with fuzzy numbers. The mean value of the positive values of the fuzzy NPV is the "possibilistic" mean value of the positive fuzzy NPV values. Real option value, ROV, calculated from the fuzzy NPV is the "possibilistic" mean value of the positive fuzzy NPV values multiplied with the positive area of the fuzzy NPV over the total area of the fuzzy NPV. The real option formula can then be written simply as: R O V = A ( P o s ) A ( P o s ) + A ( N e g ) × E [ A + ] {\displaystyle \mathrm {ROV} ={\frac {A(\mathrm {Pos} )}{A(\mathrm {Pos} )+A(\mathrm {Neg} )}}\times E[A_{+}]} where A(Pos) is the area of the positive part of the fuzzy distribution, A(Neg) is the area of the negative part of the fuzzy distribution, and E[A+] is the mean value of the positive part of the distribution. It can be seen that when the distribution is totally positive, the real options value reduces to the expected (mean) value, E[A+]. As can be seen, the real option value can be derived directly from the fuzzy NPV, without simulation. At the same time, simulation is not an absolutely necessary step in the Datar–Mathews method, so the two methods are not very different in that respect. But what is totally different is that the Datar–Mathews method is based on probability theory and as such has a very different foundation from the pay-off method that is based on possibility theory: the way that the two models treat uncertainty is fundamentally different. == Use of the method == The pay-off method for real option valuation is very easy to use compared to the other real option valuation methods and it can be used with the most commonly used spreadsheet software without any add-ins. The method is useful in analyses for decision making regarding investments that have an uncertain future, and especially so if the underlying data is in the form of cash-flow scenarios. The method is less useful if optimal timing is the objective. The method is flexible and accommodates easily both one-stage investments and multi-stage investments (compound real options). The method has been taken into use in some large international industrial companies for the valuation of research and development projects and portfolios. In these analyses triangular fuzzy numbers are used. Other uses of the method so far are, for example, R&D project valuation IPR valuation, valuation of M&A targets and expected synergies, valuation and optimization of M&A strategies, valuation of area development (construction) projects, valuation of large industrial real investments. The use of the pay-off method is lately taught within the larger framework of real options, for example at the Lappeenranta University of Technology and at the Tampere University of Technology in Finland.
Ultra Hal
Ultra Hal is a chatbot intended to function as a virtual assistant. It was developed by Zabaware, Inc. Ultra Hal uses a natural language interface with animated characters using speech synthesis. Users can communicate with the chatterbot via typing or via a speech recognition engine. It utilizes the WordNet lexical dictionary. Its name is an allusion to HAL 9000, the artificial intelligence from the movie 2001: A Space Odyssey. Ultra Hal won the 2007 Loebner Prize for "most human" chatterbot.
Eduard Hovy
Eduard Hovy is a Research Professor in the Language Technologies Institute at Carnegie Mellon University. He is one of the original 17 Fellows of the Association for Computational Linguistics. == Biography == Eduard Hovy received M.S. (December 1982) and Ph.D. (May 1987) degrees in Computer Science from Yale University. He was awarded honorary doctorates from the National University of Distance Education (UNED) in Madrid in 2013 and the University of Antwerp in 2015.
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YaDICs
YaDICs is a program written to perform digital image correlation on 2D and 3D tomographic images. The program was designed to be both modular, by its plugin strategy and efficient, by it multithreading strategy. It incorporates different transformations (Global, Elastic, Local), optimizing strategy (Gauss-Newton, Steepest descent), Global and/or local shape functions (Rigid-body motions, homogeneous dilatations, flexural and Brazilian test models)... == Theoretical background == === Context === In solid mechanics, digital image correlation is a tool that allows to identify the displacement field to register a reference image (called herein fixed image) to images during an experiment (mobile image). For example, it is possible to observe the face of a specimen with a painted speckle on it in order to determine its displacement fields during a tensile test. Before the appearance of such methods, researchers usually used strain gauges to measure the mechanical state of the material but strain gauges only measure the strain on a point and don't allow to understand material with an heterogeneous behavior. One can obtain a full in plane strain tensor by derivation of the displacement fields. Many methods are based upon the optical flow. In fluid mechanics a similar method is used, called Particle Image Velocimetry (PIV); the algorithms are similar to those of DIC but it is impossible to ensure that the optical flow is conserved so a vast majority of the software used the normalized cross correlation metric. In mechanics the displacement or velocity fields are the only concern, registering images is just a side effect. There is another process called image registration using the same algorithms (on monomodal images) but where the goal is to register images and thereby identifying the displacement field is just a side effect. YaDICs uses the general principle of image registration with a particular attention to the displacement fields basis. === Image registration principle === YaDICs can be explained using the classical image registration framework: === Image registration general scheme === The common idea of image registration and digital image correlation is to find the transformation between a fixed image and a moving one for a given metric using an optimization scheme. While there are many methods to achieve such a goal, Yadics focuses on registering images with the same modality. The idea behind the creation of this software is to be able to process data that comes from a μ-tomograph; i.e.: data cube over 10003 voxels. With such a size it is not possible to use naive approach usually used in a two-dimensional context. In order to get sufficient performances OpenMP parallelism is used and data are not globally stored in memory. As an extensive description of the different algorithms is given in. === Sampling === Contrary to image registration, Digital Image Correlation targets the transformation, one wants to extracted the most accurate transformation from the two images and not just match the images. Yadics uses the whole image as a sampling grid: it is thus a total sampling. === Interpolator === It is possible to choose between bilinear interpolation and bicubic interpolation for the grey level evaluation at non integer coordinates. The bi-cubic interpolation is the recommended one. === Metrics === ==== Sum of squared differences (SSD) ==== The SSD is also known as mean squared error. The equation below defines the SSD metric: S S D ( μ , I F , I M ) = 1 | Ω F | ∑ x i ∈ Ω F ( I F ( x i ) − I M ( T μ ( x i ) ) ) 2 , {\displaystyle SSD(\mu ,{\mathcal {I_{F}}},{\mathcal {I_{M}}})={\dfrac {1}{\left|\Omega _{F}\right|}}\sum _{x_{i}\in \Omega _{F}}\left({\mathcal {I_{F}}}(x_{i})-{\mathcal {I_{M}}}({T}_{\mu }(x_{i}))\right)^{2},} where I F {\displaystyle {\mathcal {I_{F}}}} is the fixed image, I M {\displaystyle {\mathcal {I_{M}}}} the moving one, Ω F {\displaystyle \Omega _{F}} the integration area | Ω F | {\displaystyle \left|\Omega _{F}\right|} the number of pi(vo)xels (cardinal) and T μ {\displaystyle {T}_{\mu }} the transformation parametrized by μ The transformation can be written as: T μ ( x ) = x + { Φ ( x ) } t { μ } . {\displaystyle T_{\mu }(x)=x+\left\{\Phi (x)\right\}^{t}\left\{\mu \right\}.} This metric is the main one used in the YaDICs as it works well with same modality images. One has to find the minimum of this metric ==== Normalized cross-correlation ==== The normalized cross-correlation (NCC) is used when one cannot assure the optical flow conservation; it happens in case of change of lighting or if particles disappear from the scene can occur in particle images velocimetry (PIV). The NCC is defined by: N C C ( μ , I F , I M ) = ∑ x i ∈ Ω F ( I F ( x i ) − I F ¯ ) ( I M ( T μ ( x i ) ) − I M ¯ ) ∑ x i ∈ Ω F ( I F ( x i ) − I F ¯ ) 2 ∑ x i ∈ Ω F ( I M ( T μ ( x i ) ) − I M ¯ ) 2 , {\displaystyle NCC(\mu ,{\mathcal {I_{F}}},{\mathcal {I_{M}}})={\dfrac {\sum _{x_{i}\in \Omega _{F}}\left({\mathcal {I_{F}}}(x_{i})-{\overline {\mathcal {I_{F}}}}\right)\left({\mathcal {I_{M}}}({T}_{\mu }(x_{i}))-{\overline {\mathcal {I_{M}}}}\right)}{\sqrt {\sum _{x_{i}\in \Omega _{F}}\left({\mathcal {I_{F}}}(x_{i})-{\overline {\mathcal {I_{F}}}}\right)^{2}\sum _{x_{i}\in \Omega _{F}}\left({\mathcal {I_{M}}}({T}_{\mu }(x_{i}))-{\overline {\mathcal {I_{M}}}}\right)^{2}}}},} where I F ¯ {\displaystyle {\overline {\mathcal {I_{F}}}}} and I M ¯ {\displaystyle {\overline {\mathcal {I_{M}}}}} are the mean values of the fixed and mobile images. This metric is only used to find local translation in Yadics. This metric with translation transform can be solved using cross-correlation methods, which are non iterative and can be accelerated using Fast Fourier Transform . === Classification of transformations === There are three categories of parametrization: elastic, global and local transformation. The elastic transformations respect the partition of unity, there are no holes created or surfaces counted several times. This is commonly used in Image Registration by the use of B-Spline functions and in solid mechanics with finite element basis. The global transformations are defined on the whole picture using rigid body or affine transformation (which is equivalent to homogeneous strain transformation). More complex transformations can be defined such as mechanically based one. These transformations have been used for stress intensity factor identification by and for rod strain by. The local transformation can be considered as the same global transformation defined on several Zone Of Interest (ZOI) of the fixed image. ==== Global ==== Several global transforms have been implemented: Rigid and homogeneous (Tx,Ty,Rz in 2D; Tx,Ty,Tz,Rx,Ry,Rz,Exx,Eyy,Ezz,Eyz,Exz,Exy in 3D) Brazilian (Only in 2D), Dynamic Flexion, ==== Elastic ==== First-order quadrangular finite elements Q4P1 are used in Yadics. ===== Local ===== Every global transform can be used on a local mesh. === Optimization === The YaDICs optimization process follows a gradient descent scheme. The first step is to compute the gradient of the metric regarding the transform parameters ∂ S S D ( μ , I F , I M ) ∂ μ = 2 | Ω F | ∑ x i ∈ Ω F ( I F ( x i ) − I M ( T μ ( x i ) ) ) ∂ I M ( T μ ( x i ) ∂ μ = 2 | Ω F | ∑ x i ∈ Ω F ( I F ( x i ) − I M ( T μ ( x i ) ) ) ( ∂ T μ ( x i ) ∂ μ ) t ∂ I M ( T μ ( x i ) ) ∂ x {\displaystyle {\begin{array}{lcl}{\dfrac {\partial SSD(\mu ,{\mathcal {I_{F}}},{\mathcal {I_{M}}})}{\partial \mu }}&=&{\dfrac {2}{\left|\Omega _{F}\right|}}\sum _{x_{i}\in \Omega _{F}}\left({\mathcal {I_{F}}}(x_{i})-{\mathcal {I_{M}}}({T}_{\mu }(x_{i}))\right){\dfrac {\partial {\mathcal {I_{M}}}({T}_{\mu }(x_{i})}{\partial \mu }}\\&=&{\dfrac {2}{\left|\Omega _{F}\right|}}\sum _{x_{i}\in \Omega _{F}}\left({\mathcal {I_{F}}}(x_{i})-{\mathcal {I_{M}}}({T}_{\mu }(x_{i}))\right)\left({\dfrac {\partial {T}_{\mu }(x_{i})}{\partial \mu }}\right)^{t}{\dfrac {\partial {\mathcal {I_{M}}}({T}_{\mu }(x_{i}))}{\partial x}}\\\end{array}}} ==== Gradient method ==== Once the metric gradient has been computed, one has to find an optimization strategy The gradient method principle is explained below: μ k + 1 = μ k + α k d k {\displaystyle \mu _{k+1}=\mu _{k}+\alpha _{k}d_{k}} The gradient step can be constant or updated at every iteration. d k = − γ k ∂ C ( μ , I F , I M ) ∂ μ {\displaystyle d_{k}=-\gamma _{k}{\dfrac {\partial {\mathcal {C}}(\mu ,{\mathcal {I_{F}}},{\mathcal {I_{M}}})}{\partial \mu }}} , γ k {\displaystyle \gamma _{k}} allows one to choose between the following methods : γ k {\displaystyle \gamma _{k}} ⟹ {\displaystyle \Longrightarrow } steepest descent, γ k = [ ∂ C ( μ , I F , I M ) ∂ μ ∂ C ( μ , I F , I M ) ∂ μ t ] − 1 {\displaystyle \gamma _{k}=\left[{\dfrac {\partial {\mathcal {C}}(\mu ,{\mathcal {I_{F}}},{\mathcal {I_{M}}})}{\partial \mu }}{\dfrac {\partial {\mathcal {C}}(\mu ,{\mathcal {I_{F}}},{\mathcal {I_{M}}})}{\partial \mu }}^{t}\right]^{-1}} ⟹ {\displaystyle \Longrightarrow } Gauss-Newto
Generalized nondeterministic finite automaton
In the theory of computation, a generalized nondeterministic finite automaton (GNFA), also known as an expression automaton or a generalized nondeterministic finite state machine, is a variation of a nondeterministic finite automaton (NFA) where each transition is labeled with any regular expression. The GNFA reads blocks of symbols from the input which constitute a string as defined by the regular expression on the transition. There are several differences between a standard finite state machine and a generalized nondeterministic finite state machine. A GNFA must have only one start state and one accept state, and these cannot be the same state, whereas an NFA or DFA both may have several accept states, and the start state can be an accept state. A GNFA must have only one transition between any two states, whereas a NFA or DFA both allow for numerous transitions between states. In a GNFA, a state has a single transition to every state in the machine, although often it is a convention to ignore the transitions that are labelled with the empty set when drawing generalized nondeterministic finite state machines. == Formal definition == A GNFA can be defined as a 5-tuple, (S, Σ, T, s, a), consisting of a finite set of states (S); a finite set called the alphabet (Σ); a transition function (T : (S ∖ {\displaystyle \setminus } {a}) × (S ∖ {\displaystyle \setminus } {s}) → R); a start state (s ∈ S); an accept state (a ∈ S); where R is the collection of all regular expressions over the alphabet Σ. The transition function takes as its argument a pair of two states and outputs a regular expression (the label of the transition). This differs from other finite state machines, which take as input a single state and an input from the alphabet (or the empty string in the case of nondeterministic finite state machines) and outputs the next state (or the set of possible states in the case of nondeterministic finite state machines). A DFA or NFA can easily be converted into a GNFA and then the GNFA can be easily converted into a regular expression by repeatedly collapsing parts of it to single edges until S = {s, a}. Similarly, GNFAs can be reduced to NFAs by changing regular expression operators into new edges until each edge is labelled with a regular expression matching a single string of length at most 1. NFAs, in turn, can be reduced to DFAs using the powerset construction. This shows that GNFAs recognize the same set of formal languages as DFAs and NFAs.