In statistics, EM (expectation maximization) algorithm handles latent variables, while GMM is the Gaussian mixture model. == Background == In the picture below, are shown the red blood cell hemoglobin concentration and the red blood cell volume data of two groups of people, the Anemia group and the control group (i.e. the group of people without Anemia). As expected, people with Anemia have lower red blood cell volume and lower red blood cell hemoglobin concentration than those without Anemia. x {\displaystyle x} is a random vector such as x := ( red blood cell volume , red blood cell hemoglobin concentration ) {\displaystyle x:={\big (}{\text{red blood cell volume}},{\text{red blood cell hemoglobin concentration}}{\big )}} , and from medical studies it is known that x {\displaystyle x} are normally distributed in each group, i.e. x ∼ N ( μ , Σ ) {\displaystyle x\sim {\mathcal {N}}(\mu ,\Sigma )} . z {\displaystyle z} is denoted as the group where x {\displaystyle x} belongs, with z i = 0 {\displaystyle z_{i}=0} when x i {\displaystyle x_{i}} belongs to the Anemia group and z i = 1 {\displaystyle z_{i}=1} when x i {\displaystyle x_{i}} belongs to the control group. Also z ∼ Categorical ( k , ϕ ) {\displaystyle z\sim \operatorname {Categorical} (k,\phi )} where k = 2 {\displaystyle k=2} , ϕ j ≥ 0 , {\displaystyle \phi _{j}\geq 0,} and ∑ j = 1 k ϕ j = 1 {\displaystyle \sum _{j=1}^{k}\phi _{j}=1} . See Categorical distribution. The following procedure can be used to estimate ϕ , μ , Σ {\displaystyle \phi ,\mu ,\Sigma } . A maximum likelihood estimation can be applied: ℓ ( ϕ , μ , Σ ) = ∑ i = 1 m log ( p ( x ( i ) ; ϕ , μ , Σ ) ) = ∑ i = 1 m log ∑ z ( i ) = 1 k p ( x ( i ) ∣ z ( i ) ; μ , Σ ) p ( z ( i ) ; ϕ ) {\displaystyle \ell (\phi ,\mu ,\Sigma )=\sum _{i=1}^{m}\log(p(x^{(i)};\phi ,\mu ,\Sigma ))=\sum _{i=1}^{m}\log \sum _{z^{(i)}=1}^{k}p\left(x^{(i)}\mid z^{(i)};\mu ,\Sigma \right)p(z^{(i)};\phi )} As the z i {\displaystyle z_{i}} for each x i {\displaystyle x_{i}} are known, the log likelihood function can be simplified as below: ℓ ( ϕ , μ , Σ ) = ∑ i = 1 m log p ( x ( i ) ∣ z ( i ) ; μ , Σ ) + log p ( z ( i ) ; ϕ ) {\displaystyle \ell (\phi ,\mu ,\Sigma )=\sum _{i=1}^{m}\log p\left(x^{(i)}\mid z^{(i)};\mu ,\Sigma \right)+\log p\left(z^{(i)};\phi \right)} Now the likelihood function can be maximized by making partial derivative over μ , Σ , ϕ {\displaystyle \mu ,\Sigma ,\phi } , obtaining: ϕ j = 1 m ∑ i = 1 m 1 { z ( i ) = j } {\displaystyle \phi _{j}={\frac {1}{m}}\sum _{i=1}^{m}1\{z^{(i)}=j\}} μ j = ∑ i = 1 m 1 { z ( i ) = j } x ( i ) ∑ i = 1 m 1 { z ( i ) = j } {\displaystyle \mu _{j}={\frac {\sum _{i=1}^{m}1\{z^{(i)}=j\}x^{(i)}}{\sum _{i=1}^{m}1\left\{z^{(i)}=j\right\}}}} Σ j = ∑ i = 1 m 1 { z ( i ) = j } ( x ( i ) − μ j ) ( x ( i ) − μ j ) T ∑ i = 1 m 1 { z ( i ) = j } {\displaystyle \Sigma _{j}={\frac {\sum _{i=1}^{m}1\{z^{(i)}=j\}(x^{(i)}-\mu _{j})(x^{(i)}-\mu _{j})^{T}}{\sum _{i=1}^{m}1\{z^{(i)}=j\}}}} If z i {\displaystyle z_{i}} is known, the estimation of the parameters results to be quite simple with maximum likelihood estimation. But if z i {\displaystyle z_{i}} is unknown it is much more complicated. Being z {\displaystyle z} a latent variable (i.e. not observed), with unlabeled scenario, the expectation maximization algorithm is needed to estimate z {\displaystyle z} as well as other parameters. Generally, this problem is set as a GMM since the data in each group is normally distributed. In machine learning, the latent variable z {\displaystyle z} is considered as a latent pattern lying under the data, which the observer is not able to see very directly. x i {\displaystyle x_{i}} is the known data, while ϕ , μ , Σ {\displaystyle \phi ,\mu ,\Sigma } are the parameter of the model. With the EM algorithm, some underlying pattern z {\displaystyle z} in the data x i {\displaystyle x_{i}} can be found, along with the estimation of the parameters. The wide application of this circumstance in machine learning is what makes EM algorithm so important. == EM algorithm in GMM == The EM algorithm consists of two steps: the E-step and the M-step. Firstly, the model parameters and the z ( i ) {\displaystyle z^{(i)}} can be randomly initialized. In the E-step, the algorithm tries to guess the value of z ( i ) {\displaystyle z^{(i)}} based on the parameters, while in the M-step, the algorithm updates the value of the model parameters based on the guess of z ( i ) {\displaystyle z^{(i)}} of the E-step. These two steps are repeated until convergence is reached. The algorithm in GMM is: Repeat until convergence: 1. (E-step) For each i , j {\displaystyle i,j} , set w j ( i ) := p ( z ( i ) = j | x ( i ) ; ϕ , μ , Σ ) {\displaystyle w_{j}^{(i)}:=p\left(z^{(i)}=j|x^{(i)};\phi ,\mu ,\Sigma \right)} 2. (M-step) Update the parameters ϕ j := 1 m ∑ i = 1 m w j ( i ) {\displaystyle \phi _{j}:={\frac {1}{m}}\sum _{i=1}^{m}w_{j}^{(i)}} μ j := ∑ i = 1 m w j ( i ) x ( i ) ∑ i = 1 m w j ( i ) {\displaystyle \mu _{j}:={\frac {\sum _{i=1}^{m}w_{j}^{(i)}x^{(i)}}{\sum _{i=1}^{m}w_{j}^{(i)}}}} Σ j := ∑ i = 1 m w j ( i ) ( x ( i ) − μ j ) ( x ( i ) − μ j ) T ∑ i = 1 m w j ( i ) {\displaystyle \Sigma _{j}:={\frac {\sum _{i=1}^{m}w_{j}^{(i)}\left(x^{(i)}-\mu _{j}\right)\left(x^{(i)}-\mu _{j}\right)^{T}}{\sum _{i=1}^{m}w_{j}^{(i)}}}} With Bayes' rule, the following result is obtained by the E-step: p ( z ( i ) = j | x ( i ) ; ϕ , μ , Σ ) = p ( x ( i ) | z ( i ) = j ; μ , Σ ) p ( z ( i ) = j ; ϕ ) ∑ l = 1 k p ( x ( i ) | z ( i ) = l ; μ , Σ ) p ( z ( i ) = l ; ϕ ) {\displaystyle p\left(z^{(i)}=j|x^{(i)};\phi ,\mu ,\Sigma \right)={\frac {p\left(x^{(i)}|z^{(i)}=j;\mu ,\Sigma \right)p\left(z^{(i)}=j;\phi \right)}{\sum _{l=1}^{k}p\left(x^{(i)}|z^{(i)}=l;\mu ,\Sigma \right)p\left(z^{(i)}=l;\phi \right)}}} According to GMM setting, these following formulas are obtained: p ( x ( i ) | z ( i ) = j ; μ , Σ ) = 1 ( 2 π ) n / 2 | Σ j | 1 / 2 exp ( − 1 2 ( x ( i ) − μ j ) T Σ j − 1 ( x ( i ) − μ j ) ) {\displaystyle p\left(x^{(i)}|z^{(i)}=j;\mu ,\Sigma \right)={\frac {1}{(2\pi )^{n/2}\left|\Sigma _{j}\right|^{1/2}}}\exp \left(-{\frac {1}{2}}\left(x^{(i)}-\mu _{j}\right)^{T}\Sigma _{j}^{-1}\left(x^{(i)}-\mu _{j}\right)\right)} p ( z ( i ) = j ; ϕ ) = ϕ j {\displaystyle p\left(z^{(i)}=j;\phi \right)=\phi _{j}} In this way, a switch between the E-step and the M-step is possible, according to the randomly initialized parameters.
TAChart
TAChart is a component for the Lazarus IDE that provides charting services. Similar to Tchart and Teechart for Delphi it supports a collection of different chart types including bar charts, pie charts, line charts and point series. Apart from a screen canvas, output is possible in form of SVG, OpenGL, printer, WMF, and other formats. TAChart is bundled with the Lazarus Component Library. Although not intended to be a TChart clone, why its usage differs in certain points, its basic functionality is very similar and some source code written for TeeChart may be reused. == History == The first version of TAChart was developed by Philippe Martinole for the TeleAuto project, a program for automation of astronomic observations. Later functionality was introduced by Luis Rodrigues while porting the Epanet application from Delphi to Lazarus. In the ensuing years the code has extensively rewritten, expanded and is now maintained by Alexander Klenin. == Data sources == TAChart is able to use input from various sources. Examples include lists of real values, user defined buffers in the computer's memory, vectors of random values, fields in databases, calculated values provided by pre-defined functions and results of embedded code written in Pascal Script
Emma Brunskill
Emma Patricia Brunskill is an American computer scientist. Her research combines machine learning with human–computer interaction by studying the effects of AI systems in human-centered applications including educational software and healthcare, and the theory of reinforcement learning in situations where mistakes impose high risks or costs. She is an associate professor of computer science at Stanford University, where she also holds a courtesy appointment in the Stanford Graduate School of Education and is an affiliate of the King Center on Global Development. == Education and career == Brunskill grew up in Seattle and Edmonds, Washington, and entered the University of Washington at age 15. She graduated magna cum laude in 2000, with a bachelor's degree in computer engineering and physics. A Rhodes Scholarship took her to Magdalen College, Oxford in England, where she received a master's degree in neuroscience in 2002. After a summer working in Rwanda, she became a graduate student of computer science at the Massachusetts Institute of Technology, where she completed her Ph.D. in 2009. Her doctoral dissertation, Compact parametric models for efficient sequential decision making in high-dimensional, uncertain domains, was supervised by Nicholas Roy. After working as an NSF Postdoctoral Research Fellow at the University of California, Berkeley, she joined Carnegie Mellon University (CMU) in 2011 as an assistant professor of computer science. She moved from CMU to Stanford University in 2017. == Recognition == Brunskill was a 2014 recipient of the National Science Foundation CAREER Award and a 2015 recipient of the Office of Naval Research Young Investigator Award. She was one of two alumni of the University of Washington's Paul G. Allen School of Computer Science and Engineering to be honored in 2020 by the school's Alumni Impact Awards. She was elected as a Fellow of the Association for the Advancement of Artificial Intelligence in 2025, "for significant contributions to the field of reinforcement learning, and applications for societal benefit, in particular AI for education".
AI Blog Writers Reviews: What Actually Works in 2026
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Armin B. Cremers
Armin Bernd Cremers (born June 7, 1946) is a German mathematician and computer scientist. He is a professor in the computer science institute at the University of Bonn, Germany. He is most notable for his contributions to several fields of discrete mathematics including formal languages and automata theory. In more recent years he has been recognized for his work in artificial intelligence, machine learning and robotics as well as in geoinformatics and deductive databases. == Life and work == Armin B. Cremers studied mathematics and physics at the University of Karlsruhe, Germany. After his graduate diploma (1971) and PhD (1972), both in mathematics, both summa cum laude, he received his academic lectureship qualification for computer science (1974), all from the University of Karlsruhe. Following an invitation by Seymour Ginsburg, he joined the University of Southern California (USC), Los Angeles, in 1973 where he worked until 1976 as an assistant professor of electrical engineering and computer science. With Ginsburg he initiated Grammar Forms, a new formalism for grammatical families. In 1976 A. B. Cremers returned to Germany and was appointed to full professor of computer science at the University of Dortmund, where he remained until 1990, holding the chair for information systems. During the same time he continued working as a visiting research professor at USC, where together with Thomas N. Hibbard he developed the concept of Data Spaces, a comprehensive computational model, in theory and applications. At the University of Dortmund A. B. Cremers served as chairman of the computer science department and, since early 1985, as vice president for Research and Junior Scientific Staff. In this position he was liaison for the development of the Technology Center Dortmund Archived 2021-05-09 at the Wayback Machine. He was the initiator and founding director of the Center for Expert Systems Dortmund (ZEDO) and the NRW State Research Collaborative in Artificial Intelligence (KI-NRW). From 1988 to 1996 he was also a member of the supervisory board of the German National Research Center for Mathematics and Data Processing (GMD). Since 1990 A. B. Cremers has been professor and director of computer science and head of the research group in artificial intelligence at the University of Bonn. From Bonn he has contributed fundamentally to artificial intelligence and robotics (with Wolfram Burgard, Dieter Fox, Sebastian Thrun among his students), and to the development of software engineering, particularly in civil engineering, and information systems, particularly in the geosciences. The paper "The Interactive Museum Tour-Guide Robot" won the AAAI Classic Paper award of 2016. Together with Matthias Jarke A. B. Cremers established the Bonn-Aachen International Center for Information Technology (B-IT) in 2001 and led this as Founding Scientific Director from the University of Bonn side until his retirement from teaching in 2014. From 2004 to 2008 Cremers was Dean of the School of Mathematics and Natural Sciences, and from April 2009 to July 2014 University Vice President for Planning and Finance. He is member of advisory boards, e.g., as well as Chairman of the University Council of the University of Koblenz-Landau.
Statistical shape analysis
Statistical shape analysis is an analysis of the geometrical properties of some given set of shapes by statistical methods. For instance, it could be used to quantify differences between male and female gorilla skull shapes, normal and pathological bone shapes, leaf outlines with and without herbivory by insects, etc. Important aspects of shape analysis are to obtain a measure of distance between shapes, to estimate mean shapes from (possibly random) samples, to estimate shape variability within samples, to perform clustering and to test for differences between shapes. One of the main methods used is principal component analysis (PCA). Statistical shape analysis has applications in various fields, including medical imaging, computer vision, computational anatomy, sensor measurement, and geographical profiling. == Landmark-based techniques == In the point distribution model, a shape is determined by a finite set of coordinate points, known as landmark points. These landmark points often correspond to important identifiable features such as the corners of the eyes. Once the points are collected some form of registration is undertaken. This can be a baseline methods used by Fred Bookstein for geometric morphometrics in anthropology. Or an approach like Procrustes analysis which finds an average shape. David George Kendall investigated the statistical distribution of the shape of triangles, and represented each triangle by a point on a sphere. He used this distribution on the sphere to investigate ley lines and whether three stones were more likely to be co-linear than might be expected. Statistical distribution like the Kent distribution can be used to analyse the distribution of such spaces. Alternatively, shapes can be represented by curves or surfaces representing their contours, by the spatial region they occupy. == Shape deformations == Differences between shapes can be quantified by investigating deformations transforming one shape into another. In particular a diffeomorphism preserves smoothness in the deformation. This was pioneered in D'Arcy Thompson's On Growth and Form before the advent of computers. Deformations can be interpreted as resulting from a force applied to the shape. Mathematically, a deformation is defined as a mapping from a shape x to a shape y by a transformation function Φ {\displaystyle \Phi } , i.e., y = Φ ( x ) {\displaystyle y=\Phi (x)} . Given a notion of size of deformations, the distance between two shapes can be defined as the size of the smallest deformation between these shapes. Diffeomorphometry is the focus on comparison of shapes and forms with a metric structure based on diffeomorphisms, and is central to the field of Computational anatomy. Diffeomorphic registration, introduced in the 90's, is now an important player with existing codes bases organized around ANTS, DARTEL, DEMONS, LDDMM, StationaryLDDMM, and FastLDDMM are examples of actively used computational codes for constructing correspondences between coordinate systems based on sparse features and dense images. Voxel-based morphometry (VBM) is an important technology built on many of these principles. Methods based on diffeomorphic flows are also used. For example, deformations could be diffeomorphisms of the ambient space, resulting in the LDDMM (Large Deformation Diffeomorphic Metric Mapping) framework for shape comparison.
Edward Stabler
Edward Stabler is a Professor of Linguistics at the University of California, Los Angeles. His primary areas of research are (1) Natural Language Processing (NLP), (2) Parsing and formal language theory, and (3) Philosophy of Logic and Language. He was a member of the faculty at UCLA from 1984 to 2016. His work involves the production of software for minimalist grammars (MGs) and related systems. == Early life and education == Stabler received his Ph.D. from the Department of Linguistics and Philosophy at MIT in 1981. == Recent publications == Edward Stabler (2011) Computational perspectives on minimalism. Revised version in C. Boeckx, ed, Oxford Handbook of Linguistic Minimalism, pp. 617–642. Edward Stabler (2010) A defense of this perspective against the Evans&Levinson critique appears here, with revised version in Lingua 120(12): 2680-2685. Edward Stabler (2010) After GB. Revised version in J. van Benthem & A. ter Meulen, eds, Handbook of Logic and Language, pp. 395–414. Edward Stabler (2010) Recursion in grammar and performance. Presented at the 2009 UMass recursion conference. Edward Stabler (2009) Computational models of language universals. Revised version appears in M. H. Christiansen, C. Collins, and S. Edelman, eds., Language Universals, Oxford: Oxford University Press, pages 200-223. Edward Stabler (2008) Tupled pregroup grammars. Revised version appears in P. Casadio and J. Lambek, eds., Computational Algebraic Approaches to Natural Language, Milan: Polimetrica, pages 23–52. Edward Stabler (2006) Sidewards without copying. Proceedings of the 11th Conference on Formal Grammar, edited by P. Monachesi, G. Penn, G. Satta, and S. Wintner. Stanford: CSLI Publications, 2006, pages 133-146.