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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
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
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
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
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
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
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
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 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
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
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
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
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
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
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
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
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)
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
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)
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
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
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
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
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)
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
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
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)
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
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
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 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
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
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)
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
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
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
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
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
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
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
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
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)
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
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
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
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
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
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
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
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
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
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)
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
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
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
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
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
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
Kind of distributed data structure
defined by the membership vector m ( x ) {\displaystyle m(x)} . The membership vector is defined as an infinite random word over a fixed alphabet
Skip_graph
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
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
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
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
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
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)
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
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
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
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
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
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
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
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
Sum of elements on the main diagonal
hermitian inner product on the complex vector space of all complex matrices of a fixed size, by replacing B by its complex conjugate. The symmetry of the
Trace_(linear_algebra)
Machine learning paradigm
Subsymbolic machine learning algorithms Support vector machines Minimum complexity machines (MCM) Random forests Ensembles of classifiers Ordinal classification
Supervised_learning
Mathematical form
following sections. By the very definition of a vector space, one can form the product of any scalar with any vector, giving a map R × V → V {\displaystyle \mathbb
Product_(mathematics)
Machine learning methods using multiple input modalities
input images as a series of patches, turning them into vectors, and treating them like embedding vector of tokens in a standard transformer. Conformer and
Multimodal_learning
COMPLEX RANDOM-VECTOR
COMPLEX RANDOM-VECTOR
Male
English
Medieval form of English Randolf, RANDAL means "shield-wolf."
Surname or Lastname
English
English : patronymic from Rand 1.
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 American
Son of Rand.
Surname or Lastname
English
English : variant of Ransom.
Boy/Male
English
Son of Rand.
Male
Scandinavian
 Scandinavian form of Old Norse Randolfr, RANDOLF means "shield-wolf." Compare with another form of Randolf.
Surname or Lastname
English
English : variant of Brandon.
Male
English
 Variant spelling of Middle English Randulf, RANDOLF means "shield-wolf." Compare with other forms of Randolf.
Male
English
Pet form of English Randall and Randolph, both RANDY means "shield-wolf." Compare with feminine Randy.
Surname or Lastname
English (chiefly East Anglia)
English (chiefly East Anglia) : patronymic from the Middle English personal name Rand(e) (see Rand 1).
Female
English
Pet form of English Miranda, RANDY means "worthy of admiration."Â Compare with masculine Randy.Â
Male
Norwegian
 Norwegian form of Old Norse Arnþórr, ANDOR means "eagle of Thor." Compare with another form of Andor.
Surname or Lastname
English
English : variant of Rand 1, from the Old French oblique case.
Male
Hungarian
 Variant spelling of Hungarian András, ANDOR means "man; warrior." Compare with another form of Andor.
Female
English
Variant spelling of English Randy, RANDI means "worthy of admiration."
Female
English
Short form of English Miranda, RANDA means "worthy of admiration."Â
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
English : unexplained; perhaps a variant of Francom.
Surname or Lastname
English
English : variant spelling of Randall.Americanized spelling of Randel.
COMPLEX RANDOM-VECTOR
COMPLEX RANDOM-VECTOR
Boy/Male
British, Christian, English, French, German, Portuguese
Bear-spear
Girl/Female
Tamil
Lilavarti | லீலாவரà¯à®¤à¯€Â
Playful, Amusing, Charming
Boy/Male
Shakespearean
King Henry IV, Part 2' The play's presenter.
Male
Hebrew
(ש×ְלמׄה) Contracted form of Hebrew Shelomoh, SHLOMO means "peaceable."Â
Girl/Female
Indian, Sanskrit
Evolving
Boy/Male
Hindu, Indian
Joy
Girl/Female
American, Australian, French
Dear
Boy/Male
Tamil
Twentieth
Boy/Male
Arabic, Muslim
Life of the World
Boy/Male
Australian, French, Italian
Manly
COMPLEX RANDOM-VECTOR
COMPLEX RANDOM-VECTOR
COMPLEX RANDOM-VECTOR
COMPLEX RANDOM-VECTOR
COMPLEX RANDOM-VECTOR
n.
Composed of two or more parts; composite; not simple; as, a complex being; a complex idea.
n.
A complex; an aggregate of parts; a complication.
a.
Complex, complicated.
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.
imp. & p. p.
of Compile
adv.
In a complex manner; not simply.
n.
Random.
a.
Repeatedly compound; made up of complex constituents.
adv.
In a random manner.
n.
Distance to which a missile is cast; range; reach; as, the random of a rifle ball.
v. i.
To go or stray at random.
n.
To exact a ransom for, or a payment on.
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.
The release of a captive, or of captured property, by payment of a consideration; redemption; as, prisoners hopeless of ransom.
imp. & p. p.
of Couple
a.
Not complex; uncompounded; simple.
imp. & p. p.
of Comply
a.
Intricate; entangled; complicated; complex.
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.