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In probability theory, a nonlinear expectation is a nonlinear generalization of the expectation. Nonlinear expectations are useful in utility theory as
Nonlinear_expectation
Average value of a random variable
X Median – indicated by m {\displaystyle m} in a drawing above Nonlinear expectation – a generalization of the expected value Population mean Predicted
Expected_value
In probability theory, the g-expectation is a nonlinear expectation based on a backwards stochastic differential equation (BSDE) originally developed by
G-expectation
Topics referred to by the same term
Look up nonlinear or nonlinearity in Wiktionary, the free dictionary. Nonlinearity is a property of mathematical functions or data that cannot be graphed
Nonlinearity_(disambiguation)
Subadditive or superadditive integral
related methods in their formulation of cumulative prospect theory. Nonlinear expectation Superadditivity Subadditivity Choquet, G. (1953). "Theory of capacities"
Choquet_integral
Chinese mathematician (born 1947)
finance. A type of nonlinear expectation, called the g-expectation, was also derived from the theory of BSDEs. General theories of nonlinear expectations were
Peng_Shige
Quantum state with the lowest possible energy
tiny nonlinearity can be interpreted in terms of virtual pair production. A characteristic electric field strength for which the nonlinearities become
Quantum_vacuum_state
Identification of nonlinear systems
that there are very many different types of nonlinear systems. Historically, system identification for nonlinear systems has developed by focusing on specific
Nonlinear system identification
Nonlinear_system_identification
Nonlinear narrative is a storytelling technique in which the events are depicted, for example, out of chronological order, or in other ways where the
List of nonlinear narrative television series
List_of_nonlinear_narrative_television_series
relatively strong nonlinear optical properties due to delocalized electrons at π − π {\displaystyle \pi -\pi } * orbitals. This expectation explains the extensive
Organic nonlinear optical materials
Organic_nonlinear_optical_materials
Class of statistical models
approximation of the expectation-maximization algorithm gives an alternative approach for doing maximum-likelihood estimation. Nonlinear mixed-effects models
Nonlinear_mixed-effects_model
Generalization of the standard Boltzmann–Gibbs entropy
generalization of the standard Boltzmann–Gibbs entropy. It is proportional to the expectation of the q-logarithm of a distribution. The concept was introduced in 1988
Tsallis_entropy
Construct in quantum information theory
be linear or nonlinear functionals of the density matrix. If linear, then they can also be viewed as observables for which the expectation value of the
Entanglement_witness
Type of feedforward neural network
feedforward neural network consisting of fully connected neurons with nonlinear activation functions, organized in layers, notable for being able to distinguish
Multilayer_perceptron
Dirac equation for self-interacting fermions
Van der Waerden notation for the notation. In quantum field theory, the nonlinear Dirac equation is a model of self-interacting Dirac fermions. This model
Nonlinear_Dirac_equation
Property of differential equations describing physical phenomena
initial data can result in much larger errors in the answers. Problems in nonlinear complex systems (so-called chaotic systems) provide well-known examples
Well-posed_problem
Algorithm that estimates unknowns from a series of measurements over time
in the minimum mean-square-error sense, although there may be better nonlinear estimators. It is a common misconception (perpetuated in the literature)
Kalman_filter
Method of data analysis
and in particular to the DCT-II which is simply known as the "DCT". Nonlinear dimensionality reduction techniques tend to be more computationally demanding
Principal_component_analysis
Mathematical model for state estimation
implemented in a computer with finite memory. A finite dimensional approximated nonlinear filter may be more based on heuristics, such as the extended Kalman filter
Filtering problem (stochastic processes)
Filtering_problem_(stochastic_processes)
Amplifier that uses quantum mechanical methods
{x}})~} . Spatial coordinates do not appear in the solution. Denote the expectation value of the initial field as ⟨ a ^ ⟩ i n i t i a l {\displaystyle
Quantum_amplifier
Equation in statistical mechanics
work is negative, and those contribute enormously to the expectation, giving us an expectation that is exactly one. A question has been raised about who
Jarzynski_equality
smooth map and the noise are all learned from the training data using the expectation–maximization (EM) algorithm. GTM was introduced in 1996 in a paper by
Generative_topographic_map
Hypothesis about quantum and statistical mechanics
|\Psi (0)\rangle =\sum _{\alpha }c_{\alpha }|E_{\alpha }\rangle ,} the expectation value of any observable A ^ {\displaystyle {\hat {A}}} is ⟨ A ^ ⟩ t ≡
Eigenstate thermalization hypothesis
Eigenstate_thermalization_hypothesis
Approximation method in statistics
problems fall into two categories: linear or ordinary least squares and nonlinear least squares, depending on whether or not the model functions are linear
Least_squares
Paradox involving a game with repeated coin flipping
"Although the standard calculation shows that the value of [the player's] expectation is infinitely great, it has ... to be admitted that any fairly reasonable
St._Petersburg_paradox
Australian and American mathematician (born 1975)
existence and smoothness problem must take into account the specific nonlinear structure of the equations. In particular, certain previously proposed
Terence_Tao
Field theory of scalar fields
sometimes said to be interacting, because the Euler–Lagrange equation is now nonlinear, implying a self-interaction. The action for the most general such theory
Scalar_field_theory
Statistical theory
data. IFT extends such known filter formula to situations with nonlinear physics, nonlinear devices, non-Gaussian field or noise statistics, dependence of
Information_field_theory
Type of Monte Carlo algorithms for signal processing and statistical inference
algorithms used to find approximate solutions for filtering problems for nonlinear state-space systems, such as signal processing and Bayesian statistical
Particle_filter
Concept in economics
expected values are finite) than expected value alone. He proposed that a nonlinear function of the utility of an outcome should be used instead of the expected
Expected_utility_hypothesis
Numerical optimization algorithm
search method (based on function comparison) and is often applied to nonlinear optimization problems for which derivatives may not be known. However
Nelder–Mead_method
Statistics concept
variable y is modeled as a polynomial in x. Polynomial regression fits a nonlinear relationship between the value of x and the corresponding conditional
Polynomial_regression
Introductory article
has been distorted by a coordinate transformation, so that there is a nonlinear relationship between the old (x, y) coordinates and the new ones. Einstein's
Introduction_to_gauge_theory
Theorem related to ordinary least squares
the linear regression model are uncorrelated, have equal variances and expectation value of zero. The errors do not need to be normal, nor do they need
Gauss–Markov_theorem
Set of statistical processes for estimating the relationships among variables
regression), this allows the researcher to estimate the conditional expectation (or population average value) of the dependent variable when the independent
Regression_analysis
Law of electrical current and voltage
but in the turbulent flow region the pressure–flow relations become nonlinear. The hydraulic analogy to Ohm's law has been used, for example, to approximate
Ohm's_law
High-temperature expansion in statistical mechanics
quantum-optical properties, all measurable expectation values can be expressed in the form of an N-particle expectation value ⟨ N ^ ⟩ ≡ ⟨ B ^ 1 † ⋯ B ^ K †
Cluster_expansion
Statistical model used in time series analysis
various generalizations of ARMA. Nonlinear AR (NAR), nonlinear MA (NMA) and nonlinear ARMA (NARMA) model nonlinear dependence on past values and error
Autoregressive moving-average model
Autoregressive_moving-average_model
Nonlinear modification of the Schrödinger equation
referred to as the Newton–Schrödinger or Schrödinger–Poisson equation, is a nonlinear modification of the Schrödinger equation with a Newtonian gravitational
Schrödinger–Newton_equation
Statistical model containing both fixed effects and random effects
linear mixed-effects models rather than generalized linear mixed models or nonlinear mixed-effects models. Linear mixed models (LMMs) are statistical models
Mixed_model
Set of methods for supervised statistical learning
nonlinear and the transformed space high-dimensional; although the classifier is a hyperplane in the transformed feature space, it may be nonlinear in
Support_vector_machine
Covariance and correlation
between the input and output of a system with nonlinear dynamics can be completely blind to certain nonlinear effects. This problem arises because some quadratic
Cross-correlation
Optimization algorithm
are preferred. Gradient descent can also be used to solve a system of nonlinear equations. Below is an example that shows how to use the gradient descent
Gradient_descent
Technique for improving the efficiency of estimators in conditional moment models
generate conditional expectation functions. To estimate parameters of a conditional moment model, the statistician can derive an expectation function (defining
Optimal_instruments
Mathematical methods used in Bayesian inference and machine learning
conceptually similar to the expectation–maximization algorithm. (Using the KL-divergence in the other way produces the expectation propagation algorithm.)
Variational_Bayesian_methods
Artificial neural network node function
problems can be solved using only a few nodes if the activation function is nonlinear. Modern activation functions include the logistic (sigmoid) function used
Activation_function
Numerical technique for solving quantum Hamiltonians
used to obtain expectation values of observables. For example, if O {\displaystyle {\mathcal {O}}} is an observable, its thermal expectation value is ⟨ O
Exact_diagonalization
Mathematical technique
algebra for random variables, apart from the elementary symbolic algebra: Expectation algebra, Variance algebra, Covariance algebra, Moment algebra, etc. Considering
Algebra_of_random_variables
Iterative optimization method
optimization algorithms that follow the same construction pattern. The expectation–maximization algorithm can be treated as a special case of the MM algorithm
MM_algorithm
Derivation of the conjugate gradient method Nonlinear conjugate gradient method — generalization for nonlinear optimization problems Biconjugate gradient
List of numerical analysis topics
List_of_numerical_analysis_topics
Equations modelling predator–prey cycles
as the Lotka–Volterra predator–prey model, are a pair of first-order nonlinear differential equations, frequently used to describe the dynamics of biological
Lotka–Volterra_equations
Generalized filtering is a generic Bayesian filtering scheme for nonlinear state-space models. It is based on a variational principle of least action
Generalized_filtering
Type of massless subatomic particle
and are characterized by the quantum numbers of these. They transform nonlinearly (shift) under the action of these generators, and can thus be excited
Goldstone_boson
Geometric algorithms for signal processing
statistics, used to find approximate solutions for filtering problems for nonlinear state-space systems. The filtering problem consists of estimating the
Projection_filters
Moving average and polynomial regression method for smoothing data
LOESS and LOWESS thus build on "classical" methods, such as linear and nonlinear least squares regression. They address situations in which the classical
Local_regression
Elastic maps provide a tool for nonlinear dimensionality reduction. By their construction, they are a system of elastic springs embedded in the data space
Elastic_map
Yang–Mills theory in two dimensions with a well-defined measure
Fine used the formal Yang–Mills functional integral to compute loop expectation values. Other approaches include that of Klimek and Kondracki and Ashtekar
Two-dimensional Yang–Mills theory
Two-dimensional_Yang–Mills_theory
Force resulting from the quantisation of a field
interfaces, such as electrical conductors and dielectrics, alters the vacuum expectation value of the energy of the second-quantized electromagnetic field. Since
Casimir_effect
Mathematical decision rule
the posterior expected loss). Equivalently, it maximizes the posterior expectation of a utility function. An alternative way of formulating an estimator
Bayes_estimator
Probability theory term
to be a linear function, but in general it is nonlinear.) One may also treat the conditional expectation as a random variable, — a function of the random
Conditioning_(probability)
Approximation or recovery of classical mechanics in certain theories
the ordinary position and momentum in classical mechanics. The quantum expectation values satisfy the Ehrenfest theorem. For a one-dimensional quantum particle
Classical_limit
Interpretation of probability
propensity of some phenomenon, probability is interpreted as reasonable expectation representing a state of knowledge or as quantification of a personal
Bayesian_probability
Type of probability distribution in economics
scenario analysis or stress testing. hard-to-compute expectation A subtler issue is that expectation is very sensitive to assumptions about probability:
Taleb_distribution
Machine learning technique
x^{(2)}\mapsto \cdots } where each network module can be a linear transform, a nonlinear activation function, a convolution, etc. x ( 0 ) {\displaystyle x^{(0)}}
Normalization (machine learning)
Normalization_(machine_learning)
\tau =0,1,\ldots } , where E t − 1 {\displaystyle E_{t-1}} refers to an expectation conditional on values observed no later than time t–1; any temporary
Hysteresis_(economics)
Class of algorithms for pattern analysis
machine (SVM). These methods involve using linear classifiers to solve nonlinear problems. The general task of pattern analysis is to find and study general
Kernel_method
Machine learning technique
also be trained by the expectation-maximization algorithm, just like gaussian mixture models. Specifically, during the expectation step, the "burden" for
Mixture_of_experts
Mathematical function, in linear algebra
E [ a X ] = a E [ X ] {\displaystyle E[aX]=aE[X]} . The conditional expectation is as well. But the variance of a random variable is not linear, because
Linear_map
Properties of the operation of a secure cipher
the permutation layer to consist of linear Boolean functions, although nonlinear functions can be used, too. In Shannon's original definitions, confusion
Confusion_and_diffusion
Concept in statistical mechanics
1975-1976 Feigenbaum, M. J. (1983). "Universal behavior in nonlinear systems". Physica D: Nonlinear Phenomena. 7 (1–3): 16–39. Bibcode:1983PhyD....7...16F
Universality (dynamical systems)
Universality_(dynamical_systems)
Type of "good" decision rule in Bayesian statistics
which is the negative of the loss.) Define the risk function as the expectation R ( θ , δ ) = E F ( x ∣ θ ) [ L ( θ , δ ( x ) ) ] . {\displaystyle
Admissible_decision_rule
Book by Jordan Ellenberg
Chapter 1, Less Like Sweden: Ellenberg encourages his readers to think nonlinearly, and know that "where you should go depends on where you are". To develop
How_Not_to_Be_Wrong
2017 video game
encourages exploration and experimentation; the story can be completed in a nonlinear fashion. The five-year development commenced after the release of The
The Legend of Zelda: Breath of the Wild
The_Legend_of_Zelda:_Breath_of_the_Wild
Mathematical device used in statistical mechanics
983. Kawasaki, K. (1973). "Simple derivations of generalized linear and nonlinear Langevin equations". J. Phys. A: Math. Nucl. Gen. 6 (9): 1289–1295. Bibcode:1973JPhA
Zwanzig_projection_operator
Algorithm for supervised learning of binary classifiers
function or the underlying process being modeled by the perceptron is nonlinear, alternative learning algorithms such as the delta rule can be used as
Perceptron
Statistical theorem
estimator of a parameter θ {\displaystyle \theta } , then the conditional expectation of δ ( X ) {\displaystyle \delta (X)} given T ( X ) {\displaystyle T(X)}
Rao–Blackwell_theorem
Lowest possible energy of a quantum system or field
materials that have nonlinear properties that enhance time-varying gravitational fields. Such an effect would be analogous to the nonlinear electromagnetic
Zero-point_energy
Banking valuation adjustments
default on its unrealized gain). This CVA is the discounted risk-neutral expectation value of the loss expected due to the counterparty not paying in accordance
XVA
Overview of and topical guide to regression analysis
estimation Ridge regression Polynomial regression Segmented regression Nonlinear regression Generalized linear models Logistic regression Multinomial logit
Outline of regression analysis
Outline_of_regression_analysis
Dramatic principle
long-winded anecdote designed to lure the audience into a false sense of expectation, only to disappoint them with an anticlimactic ending or punchline. Deus
Chekhov's_gun
2018 video game
is not very effective against heavy armor. Quests are intended to be nonlinear, with multiple ways to complete objectives to allow multiple character
Kingdom_Come:_Deliverance
{1}{n}}\sum _{i=1}^{n}\rho (r_{i}/s)=K} , where K {\displaystyle K} is the expectation value of ρ {\displaystyle \rho } for a standard normal distribution.
S-estimator
problem can be overcome by taking instead the expectation of the harmonic mean (1/x). The expectation and variance of 1/x are E [ 1 x ] = 1 m {\displaystyle
Contraharmonic_mean
Swiss mathematician (1655–1705)
problem of determining the isochrone is equivalent to solving a first-order nonlinear differential equation. The isochrone, or curve of constant descent, is
Jacob_Bernoulli
Study of uncertainty in the output of a mathematical model or system
regression, can inaccurately measure sensitivity when the model response is nonlinear with respect to its inputs. In such cases, variance-based measures are
Sensitivity_analysis
Image reconstruction algorithms
likelihood-based approaches: Statistical, likelihood-based iterative expectation-maximization algorithms are now the preferred method of reconstruction
Iterative_reconstruction
Error in statistical reasoning with groups
An information systems illustration. International Journal of Applied Nonlinear Science, 2(3), 200–234. Rogier A. Kievit, Willem E. Frankenhuis, Lourens
Simpson's_paradox
Measure of linear correlation
{\displaystyle \operatorname {cov} (X,Y)} can be expressed in terms of mean and expectation. Since cov ( X , Y ) = E [ ( X − μ X ) ( Y − μ Y ) ] , {\displaystyle
Pearson correlation coefficient
Pearson_correlation_coefficient
symmetry breaking. Asymptotically safe weak interactions based on some nonlinear sigma models. Preon and models inspired by preons such as Ribbon model
Alternatives to the Standard Higgs Model
Alternatives_to_the_Standard_Higgs_Model
Empirical law on the variance of species in a habitat
{\displaystyle {\frac {\sum x}{n}}>3} where x is an individual sample value. The expectation of the index is equal to n and it is distributed as the chi-square distribution
Taylor's_law
Bet sizing formula for long-term growth
"Evaluating gambles using dynamics", Chaos: An Interdisciplinary Journal of Nonlinear Science, 26 (2): 023103, arXiv:1405.0585, doi:10.1063/1.4940236, PMID 26931584
Kelly_criterion
Principle in Bayesian statistics
with a given set of constraints (such as normalization or specified expectation values), the distribution that maximizes Shannon entropy should be selected
Principle_of_maximum_entropy
Quantum
A]=\sup _{B}F[B,\theta ].} The quantum Fisher information equals the expectation value of L ϱ 2 {\displaystyle L_{\varrho }^{2}} , where L ϱ {\displaystyle
Quantum_Fisher_information
Type of kernel induced by artificial neural networks
NTK is not specific to neural networks and can be observed in generic nonlinear models, usually by a suitable scaling. Let f ( x ; θ ) {\displaystyle
Neural_tangent_kernel
Describe the optical response of semiconductors
approach. At operator level, the microscopic polarization is defined by an expectation value for a single electronic transition between a valence and a conduction
Semiconductor_Bloch_equations
decentralized information and showed that for this system, there exist nonlinear control laws that outperform all linear laws. The problem of finding the
Witsenhausen's_counterexample
Type of artificial neural network
assigned and never updated (i.e. they are random projection but with nonlinear transforms), or can be inherited from their ancestors without being changed
Extreme_learning_machine
Quantum algorithm for solving systems of linear equations
algorithm. Solving nonlinear differential equations Two groups proposed efficient algorithms for numerically integrating dissipative nonlinear ordinary differential
HHL_algorithm
Convex shape with one stable and one unstable position of equilibrium
mono-monostatic object does not exist in two dimensions. Whereas a common expectation was that a three-dimensional body should have at least four extrema,
Gömböc
Laws in physics about force and motion
rule is used to calculate the expectation values of a position measurement or a momentum measurement. These expectation values will generally change over
Newton's_laws_of_motion
NONLINEAR EXPECTATION
NONLINEAR EXPECTATION
Girl/Female
Muslim
Hope, Expectation, Wish
Girl/Female
Tamil
Apekshaa | அபேகà¯à®·à®¾
Expected, Expectation
Apekshaa | அபேகà¯à®·à®¾
Boy/Male
Tamil
Hope, Expectation, Wish, Desire, Trust, Greed
Girl/Female
Tamil
Asha Rani | ஆஷா ராணீ Â
Hope, Aspiration, Expectation
Asha Rani | ஆஷா ராணீ Â
Girl/Female
Tamil
Pratikshya | பà¯à®°à®¤à¯€à®•à¯à®·à¯à®¯à®¾
To wait, Expectation
Pratikshya | பà¯à®°à®¤à¯€à®•à¯à®·à¯à®¯à®¾
Boy/Male
Indian
Bright, Clean, Pure, Hope, Expectation
Girl/Female
Indian
Expected, Expectation
Boy/Male
Tamil
Pratheesh | பà¯à®°à®¤à®¿à®·
Hope, Expectation, Pre-eminence
Pratheesh | பà¯à®°à®¤à®¿à®·
Girl/Female
Indian
Hope, Aspiration, Expectation
Girl/Female
Tamil
Expectation
Girl/Female
Indian
Hope, Aspiration, Expectation
Girl/Female
Indian
Expectation
Girl/Female
Tamil
Protyasha | பà¯à®°à¯‹à®¤à¯à®¯à®¾à®·à®¾
Expectation
Protyasha | பà¯à®°à¯‹à®¤à¯à®¯à®¾à®·à®¾
Girl/Female
Tamil
Apeksha | அபேகà¯à®·à®¾
Expected, Expectation
Apeksha | அபேகà¯à®·à®¾
Boy/Male
Tamil
Hope, Expectation, Pre-eminence
Boy/Male
Tamil
Hope, Expectation, Pre-eminence
Girl/Female
Tamil
Ashavathi | அஷாவதீ
Hope, Aspiration, Expectation
Ashavathi | அஷாவதீ
Girl/Female
Tamil
Samprathiksha | ஸமà¯à®ªà¯à®°à®¤à¯€à®•à¯à®·à®¾
Expectation, Hope
Samprathiksha | ஸமà¯à®ªà¯à®°à®¤à¯€à®•à¯à®·à®¾
Girl/Female
Tamil
Niriksha | நிரீகà¯à®·à®¾
Unseen, Expectation, Hope
Niriksha | நிரீகà¯à®·à®¾
Girl/Female
Indian
Expected, Expectation
NONLINEAR EXPECTATION
NONLINEAR EXPECTATION
NONLINEAR EXPECTATION
NONLINEAR EXPECTATION
NONLINEAR EXPECTATION
NONLINEAR EXPECTATION
NONLINEAR EXPECTATION