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statistics, a complex random vector is typically a tuple of complex-valued random variables, and generally is a random variable taking values in a vector space
Complex_random_vector
Probability that random variable X is less than or equal to x
{(Z_{n})}\leq \Im {(z_{n})})\end{aligned}}} as definition for the CDS of a complex random vector Z = ( Z 1 , … , Z N ) T {\displaystyle \mathbf {Z} =(Z_{1},\ldots
Cumulative distribution function
Cumulative_distribution_function
Statistical distribution of complex random variables
that the complex random vector Z = X + i Y {\displaystyle \mathbf {Z} =\mathbf {X} +i\mathbf {Y} \,} is a complex normal random vector or a complex Gaussian
Complex_normal_distribution
Concept in probability theory and statistics
statistics, complex random variables are a generalization of real-valued random variables to complex numbers, i.e. the possible values a complex random variable
Complex_random_variable
Type of matrix in probability theory and statistics
covariance between the i-th element of a random vector and j-th element of another random vector. When the two random vectors are the same, the cross-covariance
Cross-covariance_matrix
Measure of covariance of components of a random vector
matrix giving the covariance between each pair of elements of a given random vector. Intuitively, the covariance matrix generalizes the notion of variance
Covariance_matrix
Random variable with multiple component dimensions
In probability and statistics, a multivariate random variable or random vector is a list or vector of mathematical variables each of whose value is unknown
Multivariate_random_variable
matrix of two random vectors is a matrix containing as elements the cross-correlations of all pairs of elements of the random vectors. The cross-correlation
Cross-correlation_matrix
Generalization of the one-dimensional normal distribution to higher dimensions
normal distribution to higher dimensions. One definition is that a random vector is said to be k-variate normally distributed if every linear combination
Multivariate normal distribution
Multivariate_normal_distribution
Broad concept generalizing scalars in mathematics and physics
In mathematics and physics, a vector is a generalization of a single number. It may denote a vector quantity, i.e., physical quantity that cannot be expressed
Vector (mathematics and physics)
Vector_(mathematics_and_physics)
Covariance and correlation
entries of two random vectors X {\displaystyle \mathbf {X} } and Y {\displaystyle \mathbf {Y} } , while the correlations of a random vector X {\displaystyle
Cross-correlation
Correlation of a signal with a time-shifted copy of itself, as a function of shift
autocorrelation matrix is a Hermitian matrix for complex random vectors and a symmetric matrix for real random vectors. The autocorrelation matrix is a positive
Autocorrelation
Vector space with generalized dot product
product space is a real or complex vector space endowed with an operation called an inner product. The inner product of two vectors in the space is a scalar
Inner_product_space
Fourier transform of the probability density function
of the complex number z {\textstyle z} , If X is a k-dimensional complex random vector, then for t ∈ Ck φ X ( t ) = E [ exp ( i Re ( t ∗ X ) )
Characteristic function (probability theory)
Characteristic_function_(probability_theory)
Concept in probability theory
{\displaystyle \operatorname {K} _{\mathbf {X} \mathbf {Y} }} is zero. Two complex random vectors Z {\displaystyle \mathbf {Z} } and W {\displaystyle \mathbf {W}
Uncorrelatedness (probability theory)
Uncorrelatedness_(probability_theory)
Set of vectors used to define coordinates
and frames of reference. A basis B of a vector space V over a field F (such as the real numbers R or the complex numbers C) is a linearly independent subset
Basis_(linear_algebra)
principle of 'gravity-vector-averaging'. RPM provides a functional volume which is 'exposed' to simulated microweight. The concept of 'random' positioning has
Random_positioning_machine
Type of display device
A vector monitor, vector display, or calligraphic display is a display device used for computer graphics up through the 1970s. It is a type of CRT, similar
Vector_monitor
Graph generated by a random process
generalizes Gilbert's random graph model, is the random dot-product model. A random dot-product graph associates with each vertex a real vector. The probability
Random_graph
Probability distribution
practice.) A second example of the distribution arises in the case of random complex numbers whose real and imaginary components are independently and identically
Rayleigh_distribution
Variable representing a random phenomenon
can consider random elements of other sets E {\displaystyle E} , such as random Boolean values, categorical values, complex numbers, vectors, matrices,
Random_variable
probability space is usually referred to as a random variable (or random vector if it takes values in a vector space such as the Banach space B {\displaystyle
Weakly_measurable_function
Type of signal in signal processing
memory slightly. A random vector (that is, a random variable with values in Rn) is said to be a white noise vector or white random vector if its components
White_noise
Collection of random variables
collection of random variables is usually called a random field instead. The values of a stochastic process are not always numbers and can be vectors or other
Stochastic_process
Machine learning technique
papers. Kernel method Support vector machine Fourier transform Monte Carlo method Rahimi, Ali; Recht, Benjamin (2007). "Random Features for Large-Scale Kernel
Random_feature
Set of methods for supervised statistical learning
In machine learning, support vector machines (SVMs, also support vector networks) are supervised max-margin models with associated learning algorithms
Support_vector_machine
Measure of the joint variability
the mean and the covariance matrix of the random vector X {\displaystyle \textstyle \mathbf {X} } , a vector whose jth element ( j = 1 , … , K ) {\displaystyle
Covariance
Type of gradient noise in computer graphics
with it a fixed random n-dimensional unit-length gradient vector, except in the one dimensional case where the gradients are random scalars between −1
Perlin_noise
Tree-based ensemble machine learning methods
notice the link between random forest and kernel methods. He pointed out that random forests trained using i.i.d. random vectors in the tree construction
Random_forest
Concepts from linear algebra
factor λ {\displaystyle \lambda } (possibly a negative or complex number). Geometrically, vectors are multi-dimensional quantities with magnitude and direction
Eigenvalues_and_eigenvectors
Mathematical operation in linear algebra
represented by capital letters in bold, e.g. A; vectors in lowercase bold, e.g. a; and entries of vectors and matrices are italic (they are numbers from
Matrix_multiplication
Apparent lack of pattern or predictability in events
In common usage, randomness is the apparent or actual lack of definite patterns or predictability in information. A random sequence of events, symbols
Randomness
Optimization algorithm
{\displaystyle f(\mathbf {x} )} , where x {\displaystyle \mathbf {x} } is a vector of continuous and/or discrete values. At each iteration, hill climbing will
Hill_climbing
Second-smallest eigenvalue of a graph Laplacian
the Fiedler vector. The Fiedler vector can be used to partition a graph. For the example graph in the introductory section, the Fiedler vector is ( 0.415
Algebraic_connectivity
Number of values in the final calculation of a statistic that are free to vary
the domain of a random vector, or essentially the number of "free" components (how many components need to be known before the vector is fully determined)
Degrees of freedom (statistics)
Degrees_of_freedom_(statistics)
System for describing optical polarization
Jones vector of the incident light. Note that Jones calculus is only applicable to light that is already fully polarized. Light which is randomly polarized
Jones_calculus
Physical quantities taking values at each point in space and time
field or field quantity is a physical quantity – represented by a scalar, vector, spinor, or tensor – that has a value for each point in space and time.
Field_(physics)
Algorithm for supervised learning of binary classifiers
is a function that can decide whether or not an input, represented by a vector of numbers, belongs to some specific class. It is a type of linear classifier
Perceptron
Description of continuous random distribution
f_{X_{n}}(x_{n}).} If the joint probability density function of a vector of n random variables can be factored into a product of n functions of one variable
Probability_density_function
Statistical measure of how far values spread from their average
x^{*}} is the complex conjugate of x {\displaystyle x} . This variance is a real scalar. If X {\displaystyle X} is a vector-valued random variable, with
Variance
Process by which a quantum system takes on a definitive state
quantum mechanics, wave function collapse, also called reduction of the state vector, occurs when a wave function—initially in a superposition of several eigenstates—reduces
Wave_function_collapse
Statistical concept
will be a vector of V probabilities summing to 1. In addition, in a Bayesian setting, the mixture weights and parameters will themselves be random variables
Mixture_model
Mathematical function, in linear algebra
mapping) is a particular kind of function between vector spaces, which respects the basic operations of vector addition and scalar multiplication. A standard
Linear_map
Algorithm
a j ‖ 2 . {\displaystyle {\tfrac {a_{j}}{\|a_{j}\|^{2}}}.} Define a random vector Z whose values are the normals to all the equations of A x = b {\displaystyle
Kaczmarz_method
Process forming a path from many random steps
to the distribution associated to the vector R → {\displaystyle {\vec {R}}} that links the two ends of the random walk, in 3D. The variance associated
Random_walk
Algorithm for modelling sequential data
numerical representations called tokens, and each token is converted into a vector via lookup from a word embedding table. At each layer, each token is then
Transformer_(deep_learning)
Probabilistic problem-solving algorithm
application for random numbers in numerical simulation is in numerical optimization. The problem is to minimize (or maximize) functions of some vector that often
Monte_Carlo_method
Type of vector space in math
space C2 whose elements are pairs of complex numbers z = (z1, z2). Then an inner product of z with another such vector w = (w1, w2) is given by ⟨ z , w ⟩
Hilbert_space
Collection of statistical models
used to describe otherwise complex relations among variables. A dog show provides an example. A dog show is not a random sampling of the breed: it is
Analysis_of_variance
Cryptography algorithm
initialization vector (IV), for each encryption operation. The IV must be non-repeating, and for some modes must also be random. The initialization vector is used
Block cipher mode of operation
Block_cipher_mode_of_operation
Regularization method for artificial neural networks
artificial neural networks by preventing complex co-adaptations on training data. The technique consists of randomly setting the inputs and/or outputs of
Dropout_(neural_networks)
Type of function in linear algebra
sublinear function on a real vector space is symmetric if and only if it is a seminorm. A sublinear function on a real or complex vector space is a seminorm if
Sublinear_function
Vector quantization algorithm minimizing the sum of squared deviations
k-means clustering is a method of vector quantization, originally from signal processing, that aims to partition n observations into k clusters in which
K-means_clustering
Conversion of a matrix or a tensor to a vector
theory, the vectorization of a matrix is a linear transformation which converts the matrix into a vector. Specifically, the vectorization of a m × n matrix
Vectorization_(mathematics)
Matrix representing a Euclidean rotation
four-element vector where each element is a sampling of a normal distribution. Normalize its length and you have a uniformly sampled random unit quaternion
Rotation_matrix
Mathematical inequality relating inner products and norms
value of the inner product between two vectors in an inner product space in terms of the product of the vector norms. It is considered one of the most
Cauchy–Schwarz_inequality
Process of making something random
Randomization is a statistical process in which a random mechanism is employed to select a sample from a population or assign subjects to different groups
Randomization
Matrix-valued random variable
mathematical physics, a random matrix is a matrix-valued random variable—that is, a matrix in which some or all of its entries are sampled randomly from a probability
Random_matrix
Technique for the generative modeling of a continuous probability distribution
data as generated by a diffusion process, whereby a new datum performs a random walk with drift through the space of all possible data. A trained diffusion
Diffusion_model
Computer processor which works on arrays of several numbers at once
one-dimensional arrays of data called vectors. When integrated as a hardware component the vector processor is often called a vector processing unit (VPU). This
Vector_processor
Probability distribution
the negative elements reset to 0. Complex normal distribution deals with the complex normal vectors. A complex vector X ∈ Ck is said to be normal if both
Normal_distribution
Mathematical function for the probability a given outcome occurs in an experiment
set of all possible outcomes of a random phenomenon being observed. The sample space may be any set of numbers, vectors, labels, or whatever else. For example
Probability_distribution
Array of numbers
the complex conjugate of A. By the spectral theorem, real symmetric matrices and complex Hermitian matrices have an eigenbasis; that is, every vector is
Matrix_(mathematics)
Method of data analysis
second principal component, etc. Let X be a d-dimensional random vector expressed as column vector. Without loss of generality, assume X has zero mean. We
Principal_component_analysis
Approximate nearest neighbor search algorithm
searching vector data. In these systems, an item such as a document, image, song, or user profile is represented by a list of numbers called a vector. Items
Hierarchical navigable small world
Hierarchical_navigable_small_world
probabilistic one, in which the question is of the distribution of these random vectors, and what can be said knowing nothing more about the vi. Littlewood
Littlewood–Offord_problem
Measure of joint variability in statistics
used in order to distinguish this concept from the covariance of a random vector X {\displaystyle \mathbf {X} } , which is understood to be the matrix
Cross-covariance
Family of distributions that generalize the multivariate normal distribution
distributions for real random-vectors has been extended to accommodate random vectors in Euclidean spaces over the field of complex numbers, so facilitating
Elliptical_distribution
Law of electrical current and voltage
also used to refer to various generalizations of the law; for example the vector form of the law used in electromagnetics and material science: J = σ E
Ohm's_law
Statistical models for network analysis
efficiently representing complex structures, and linking local-level processes to global-level properties. Degree-preserving randomization, for example, is a
Exponential family random graph models
Exponential_family_random_graph_models
Database of handwritten digits
"Mehrad Mahmoudian / MNIST with RandomForest". Decoste, Dennis; Schölkopf, Bernhard (2002). "Training Invariant Support Vector Machines". Machine Learning
MNIST_database
Inequalities in probability theory
{\displaystyle A} is a complex Hermitian matrix and a {\displaystyle a} is complex vector of size N {\displaystyle N} . The vector g ∼ C N ( 0 , I ) {\displaystyle
Bernstein inequalities (probability theory)
Bernstein_inequalities_(probability_theory)
Mathematical entity to describe the probability of each possible measurement on a system
in a complex Hilbert space H {\displaystyle H} can be obtained from another vector by multiplying by some non-zero complex number, the two vectors in H
Quantum_state
Real square matrix whose columns and rows are orthogonal unit vectors
matrices of complex numbers that leads instead to the unitary requirement. Orthogonal matrices preserve the dot product, so, for vectors u and v in an
Orthogonal_matrix
C standard library header file
calculation) CORE-MATH, correctly rounded for single and double precision. SIMD (vectorized) math libraries include SLEEF, Yeppp! Archived 2020-07-14 at the Wayback
C_mathematical_functions
Matrix of inner products of vectors
case that the vector coordinates are complex numbers, which simplifies to X ⊤ X {\displaystyle X^{\top }X} for the case that the vector coordinates are
Gram_matrix
Set of values that describe the polarization state of electromagnetic radiation
expectation values. The light can be viewed as a random variable taking values in the space C2 of Jones vectors ( E 1 , E 2 ) {\displaystyle (E_{1},E_{2})}
Stokes_parameters
Kind of probability distribution
where D is a diagonal matrix and where x represents a vector of uncorrelated standard normal random variables. A generalized chi-square variable or distribution
Generalized chi-squared distribution
Generalized_chi-squared_distribution
Signal representation
correct position for that symbol. An electronic test instrument called a vector signal analyzer can display the constellation diagram of a digital signal
Constellation_diagram
Probability distribution
N ( 0 , Σ ) {\displaystyle X,Y\sim N(0,\Sigma )} and any random p {\displaystyle p} -vector w {\displaystyle w} independent of X {\displaystyle X} and
Cauchy_distribution
Probability distribution
proportions converge almost surely and in mean to a limiting random vector. To see that this limiting vector has the above Dirichlet distribution, check that all
Dirichlet_distribution
Field of physics that studies polymers
is the same as that found for random walks in time. Assuming, as stated, that that distribution of end-to-end vectors for a very large number of identical
Polymer_physics
Method in machine learning
This results in a random forest, which possesses numerous benefits over a single decision tree generated without randomness. In a random forest, each tree
Bootstrap_aggregating
Signal processing computational method
vector form as x = ∑ k = 1 n s k a k {\displaystyle {\boldsymbol {x}}=\sum _{k=1}^{n}s_{k}{\boldsymbol {a}}_{k}} , where the observed random vector x
Independent component analysis
Independent_component_analysis
Generalization of gamma distribution to multiple dimensions
conjugate prior of the inverse covariance-matrix of a multivariate-normal random vector. Suppose G is a p × n matrix, each column of which is independently
Wishart_distribution
Type of large language model
process and generate text, images and audio. To improve performance on complex tasks, some GPTs, such as OpenAI o3, allocate more computation time analyzing
Generative pre-trained transformer
Generative_pre-trained_transformer
of constant pressure and temperature a vector potential, in electromagnetics it can refer to the magnetic vector potential with a subscript, an alternating
Latin letters used in mathematics, science, and engineering
Latin_letters_used_in_mathematics,_science,_and_engineering
Space with topology generated by convex sets
mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed
Locally convex topological vector space
Locally_convex_topological_vector_space
Matrix representation of a graph
matrix of a random walker on the graph, assuming non-negative weights. For example, let e i {\textstyle e_{i}} denote the i-th standard basis vector. Then x
Laplacian_matrix
Subset of artificial intelligence
compressor C(.) we define an associated vector space ℵ, such that C(.) maps an input string x, corresponding to the vector norm ||~x||. An exhaustive examination
Machine_learning
Selection of data points in statistics
determine if a production lot of material meets the governing specifications. Random sampling by using lots is an old idea, mentioned several times in the Bible
Sampling_(statistics)
Manifold of all orthonormal k-frames in n-dimensional Euclidean space
k-tuples of vectors in R n . {\displaystyle \mathbb {R} ^{n}.} It is named after Swiss mathematician Eduard Stiefel. Likewise one can define the complex Stiefel
Stiefel_manifold
Branch of mathematics concerning probability
single occurrences or evolve over time in a random fashion). Although it is not possible to perfectly predict random events, much can be said about their behavior
Probability_theory
Statistics concept
{R} }\mathbf {B} ({\hat {\mathbf {R} }})} . For complex Gaussian random variables, this bias vector field can be shown to equal B ( R ^ ) = − β ( p
Estimation of covariance matrices
Estimation_of_covariance_matrices
exists for all complex s with re(s) ≥ σ. The Laplace–Stieltjes transform appears naturally in the following context. If X is a random variable with cumulative
Laplace–Stieltjes_transform
Theorem in probability theory
{\displaystyle (Z_{1},\dots ,Z_{n})} is a zero-mean complex-valued multivariate normal random vector, then the formula still holds. The expression on the
Isserlis's_theorem
Machine learning technique
auto-regressively generate the corresponding response y {\displaystyle y} when given a random prompt x {\displaystyle x} . The original paper recommends to SFT for only
Reinforcement learning from human feedback
Reinforcement_learning_from_human_feedback
2025 multimodal model by OpenAI
Altman had previously criticized the manual model picker for being overly complex, suggesting a need for unification. GPT-5 also includes agentic functionality
GPT-5
When the occurrence of one event does not affect the likelihood of another
Two random variables X {\displaystyle X} and Y {\displaystyle Y} are independent if and only if the characteristic function of the random vector ( X
Independence (probability theory)
Independence_(probability_theory)
Optimized math routines developed by Intel
algebra: BLAS routines are vector-vector (Level 1), matrix-vector (Level 2) and matrix-matrix (Level 3) operations for real and complex single and double precision
Math_Kernel_Library
COMPLEX RANDOM-VECTOR
COMPLEX RANDOM-VECTOR
Surname or Lastname
English
English : unexplained; perhaps a variant of Francom.
Male
Scandinavian
 Scandinavian form of Old Norse Randolfr, RANDOLF means "shield-wolf." Compare with another form of Randolf.
Female
English
Variant spelling of English Randy, RANDI means "worthy of admiration."
Surname or Lastname
English
English : probably a variant of Crandon, a habitational name from Crandon in Somerset or Crandean in Falmer, Sussex. Compare Grandin.
Boy/Male
English
Son of Rand.
Male
English
Pet form of English Randall and Randolph, both RANDY means "shield-wolf." Compare with feminine Randy.
Surname or Lastname
English
English : variant of Ransom.
Male
Norwegian
 Norwegian form of Old Norse Arnþórr, ANDOR means "eagle of Thor." Compare with another form of Andor.
Boy/Male
English American
Son of Rand.
Surname or Lastname
English
English : habitational name, probably from Comley in Shropshire or Combley on the Isle of Wight; both are named with Old English cumb ‘valley’ + lēah ‘woodland clearing’.
Surname or Lastname
English (chiefly East Anglia)
English (chiefly East Anglia) : patronymic from the Middle English personal name Rand(e) (see Rand 1).
Surname or Lastname
English
English : patronymic from Rand 1.
Surname or Lastname
English
English : variant of Brandon.
Male
Hungarian
 Variant spelling of Hungarian András, ANDOR means "man; warrior." Compare with another form of Andor.
Male
English
Medieval form of English Randolf, RANDAL means "shield-wolf."
Female
English
Short form of English Miranda, RANDA means "worthy of admiration."Â
Male
English
 Variant spelling of Middle English Randulf, RANDOLF means "shield-wolf." Compare with other forms of Randolf.
Surname or Lastname
English
English : variant spelling of Randall.Americanized spelling of Randel.
Surname or Lastname
English
English : variant of Rand 1, from the Old French oblique case.
Female
English
Pet form of English Miranda, RANDY means "worthy of admiration."Â Compare with masculine Randy.Â
COMPLEX RANDOM-VECTOR
COMPLEX RANDOM-VECTOR
Boy/Male
Teutonic French Shakespearean
Archer.
Surname or Lastname
English
English : habitational name from a place in Norfolk named in Old English with brīosa ‘gadfly’ + lēah ‘woodland clearing’.
Male
Icelandic
Icelandic form of Greek Petros, PÉTUR means "rock, stone."Â
Boy/Male
Indian
Clever
Girl/Female
Irish
Beautiful. Dear child.
Surname or Lastname
Scottish
Scottish : variant of Annan. In the Middle Ages, the place name and surname were most often recorded as Anand or Anant. The final d or t is a common feature of early Scottish spelling. Compare Donald, from Donall.English : from Anand, an East Anglian personal name, from Scandinavian Anund, recorded in Domesday Book and elsewhere.
Boy/Male
Muslim/Islamic
An authority on genealogy and the Quran
Boy/Male
Anglo, British, English
Home Loving Wolf
Boy/Male
Hindu, Indian, Punjabi, Sikh
One who Attains the Elixir of Enlightenment
Boy/Male
Hindu
Slayer of the famous Ravana
COMPLEX RANDOM-VECTOR
COMPLEX RANDOM-VECTOR
COMPLEX RANDOM-VECTOR
COMPLEX RANDOM-VECTOR
COMPLEX RANDOM-VECTOR
a.
Not complex; uncompounded; simple.
n.
Random.
imp. & p. p.
of Compile
v. i.
To go or stray at random.
n.
To exact a ransom for, or a payment on.
n.
A complex; an aggregate of parts; a complication.
imp. & p. p.
of Couple
adv.
In a complex manner; not simply.
a.
Repeatedly compound; made up of complex constituents.
adv.
In a random manner.
a.
Going at random or by chance; done or made at hazard, or without settled direction, aim, or purpose; hazarded without previous calculation; left to chance; haphazard; as, a random guess.
n.
The release of a captive, or of captured property, by payment of a consideration; redemption; as, prisoners hopeless of ransom.
a.
Complex, complicated.
n.
Distance to which a missile is cast; range; reach; as, the random of a rifle ball.
a.
Intricate; entangled; complicated; complex.
n.
A roving motion; course without definite direction; want of direction, rule, or method; hazard; chance; -- commonly used in the phrase at random, that is, without a settled point of direction; at hazard.
n.
Composed of two or more parts; composite; not simple; as, a complex being; a complex idea.
imp. & p. p.
of Comply
n.
To redeem from captivity, servitude, punishment, or forfeit, by paying a price; to buy out of servitude or penalty; to rescue; to deliver; as, to ransom prisoners from an enemy.