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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
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
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
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
Concept in digital signal processing
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
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
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
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
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)
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
Concept in probability theory
if and only if their cross-covariance matrix K𝐗𝐘 is zero. Two complex random vectors 𝐙 and 𝐖 are called uncorrelated if their cross-covariance matrix
Uncorrelatedness
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)
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
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)
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
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
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 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
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
Set of methods for supervised statistical learning
In machine learning, a support vector machine (SVM) or support vector network is a supervised max-margin model with associated learning algorithms that
Support_vector_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
Mathematical inequality relating inner products and norms
for all vectors 𝐮 and 𝐯 of an inner product space where ⟨⋅,⋅⟩ is the inner product. Examples of inner products include the real and complex dot product;
Cauchy–Schwarz_inequality
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
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
Mathematical operation in linear algebra
column vector), which is a matrix with only one column. So, a column vector represents both a coordinate vector, and a vector of the original vector space
Matrix_multiplication
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
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
Algorithm for supervised learning of binary classifiers
conceptual foundations of the support-vector machine. The α-perceptron further used a pre-processing layer of fixed random weights, with thresholded output
Perceptron
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
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
Matrix of inner products of vectors
If the vectors v1,…,vn are the columns of matrix X then the Gram matrix is X†X in the general case that the vector coordinates are complex numbers,
Gram_matrix
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)
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
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
C standard library header file
standard library provides another header, <random>, for its own random number generation facilities, in particular random number engines and probability distributions
C_mathematical_functions
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
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)
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
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
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)
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
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. If the ε i
Littlewood–Offord_problem
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
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)
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
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
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
aggregated indices randomization method (AIRM) is a modification of a well-known aggregated indices method,[citation needed] targeting complex objects subjected
Aggregated indices randomization method
Aggregated_indices_randomization_method
Algorithm for modelling sequential data
numerical representations called tokens, and each token is converted into a vector through an embedding layer. At each layer, each token is then contextualized
Transformer_(deep_learning)
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
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
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
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
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
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
Matrix representing a Euclidean rotation
other words, a vector rotation corresponds to multiplication on a complex number (corresponding to the vector being rotated) by a complex number of modulus
Rotation_matrix
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
Endomorphism preserving the inner product
isomorphism that preserves the inner product: the inner product of two vectors before the transformation is equal to their inner product after the transformation
Unitary_transformation
Mathematical function, in linear algebra
{\displaystyle V} and W {\displaystyle W} be vector spaces over the same field K {\displaystyle K} , such as the real or complex numbers. A function f : V → W {\displaystyle
Linear_map
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
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)
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)
Signal processing computational method
in vector form as 𝒙=∑k=1nsk𝒂k, where the observed random vector 𝒙 is represented by the basis vectors 𝒂k=(𝒂1,k,…,𝒂m,k)T. The basis vectors 𝒂k
Independent component analysis
Independent_component_analysis
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
Space with topology generated by convex sets
be a vector space X along with a family 𝒫 of seminorms on X. Suppose that X is a vector space over 𝕂, where 𝕂 is either the real or complex numbers
Locally convex topological vector space
Locally_convex_topological_vector_space
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
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
Matrix decomposition
corresponding left-singular vector and right-singular vector are unique up to multiplication by a phase factor (a unit complex number), or, in the real case
Singular_value_decomposition
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
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
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
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
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
Matrix equal to its conjugate-transpose
have to be real. The Hermitian complex n × n {\displaystyle n{\times }n} matrices do not form a vector space over the complex numbers, C {\displaystyle
Hermitian_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
Matrix of second derivatives
or ∇∇ or ∇2 or ∇⊗∇ or D2. Suppose f:ℝn→ℝ is a function taking as input a vector 𝐱∈ℝn and outputting a scalar f(𝐱)∈ℝ. If all second-order partial derivatives
Hessian_matrix
in Gaussian random Beltrami fields. Beltrami flow Complex lamellar vector field Conservative vector field Aris, Rutherford (1989), Vectors, tensors, and
Beltrami_vector_field
Subset of artificial intelligence
colleagues introduced word2vec, techniques for efficiently learning distributed vector representations of words from large text corpora. In 2014, Ian Goodfellow
Machine_learning
Machine learning paradigm
Subsymbolic machine learning algorithms Support vector machines Minimum complexity machines (MCM) Random forests Ensembles of classifiers Ordinal classification
Supervised_learning
Measurable property or characteristic
vector and a vector of weights, qualifying those observations whose result exceeds a threshold. Algorithms for classification from a feature vector include
Feature_(machine_learning)
Mathematical concept
{d} +\mathbf {\nu } } , where ν {\displaystyle \mathbf {\nu } } is a vector of random variables such that E ( ν ) = 0 {\displaystyle \operatorname {E}
Constrained_least_squares
Numerical integration process
procedure is through random shifting. Let {x1,...,xN} be the point set from the low discrepancy sequence. We sample s-dimensional random vector U and mix it with
Quasi-Monte_Carlo_method
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
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
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)
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
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
Instantaneous rate of change (mathematics)
the object's jerk. A vector-valued function 𝐲 of a real variable sends real numbers to vectors in some vector space ℝn. A vector-valued function can
Derivative
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
Mathematical model used for classification or regression
(meta-algorithm) Conditional random fields Linear regression Computer vision Random forests k-nearest neighbors algorithm Support Vector Machines Decision Tree
Discriminative_model
Array data structure that compactly stores bits
A bit array (also known as bit map, bit set, bit string, or bit vector) is an array data structure that compactly stores bits. It can be used to implement
Bit_array
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
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
Design for testing technique for integrated circuits
vectors to detect a particular fault through the space of all possible vector sequences. Even a simple stuck-at fault requires a sequence of vectors for
Scan_chain
Vector graphics editor
based on vector objects. Tools allow manipulating primitive vector shapes: simple ones like rectangles, ellipses, and arcs, and more complex ones like
Inkscape
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COMPLEX RANDOM-VECTOR
COMPLEX RANDOM-VECTOR
COMPLEX RANDOM-VECTOR
COMPLEX RANDOM-VECTOR
COMPLEX RANDOM-VECTOR
COMPLEX RANDOM-VECTOR
COMPLEX RANDOM-VECTOR
COMPLEX RANDOM-VECTOR
COMPLEX RANDOM-VECTOR
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