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Generalized Variance
The generalized variance is a scalar value which generalizes variance for multivariate random variables. It was introduced by Samuel S. Wilks. The generalized variance is defined as the determinant of the covariance matrix, \det(\Sigma). It can be shown to be related to the multidimensional scatter of points around their mean. Minimizing the generalized variance gives the Kalman filter In statistics and control theory, Kalman filtering (also known as linear quadratic estimation) is an algorithm that uses a series of measurements observed over time, including statistical noise and other inaccuracies, to produce estimates of unk ... gain.Proof that the Kalman gain minimizes the generalized variance, Eviatar Bach https://arxiv.org/abs/2103.07275 References {{statistics-stub Covariance and correlation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Variance
In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers is spread out from their average value. It is the second central moment of a distribution, and the covariance of the random variable with itself, and it is often represented by \sigma^2, s^2, \operatorname(X), V(X), or \mathbb(X). An advantage of variance as a measure of dispersion is that it is more amenable to algebraic manipulation than other measures of dispersion such as the expected absolute deviation; for example, the variance of a sum of uncorrelated random variables is equal to the sum of their variances. A disadvantage of the variance for practical applications is that, unlike the standard deviation, its units differ from the random variable, which is why the standard devi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Multivariate Random Variable
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, either because the value has not yet occurred or because there is imperfect knowledge of its value. The individual variables in a random vector are grouped together because they are all part of a single mathematical system — often they represent different properties of an individual statistical unit. For example, while a given person has a specific age, height and weight, the representation of these features of ''an unspecified person'' from within a group would be a random vector. Normally each element of a random vector is a real number. Random vectors are often used as the underlying implementation of various types of aggregate random variables, e.g. a random matrix, random tree, random sequence, stochastic process, etc. Formally, a multivariate random variable is a column vector \mathbf = (X_1,\dots,X_n)^\mathsf ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Samuel S
Samuel is a figure who, in the narratives of the Hebrew Bible, plays a key role in the transition from the biblical judges to the United Kingdom of Israel under Saul, and again in the monarchy's transition from Saul to David. He is venerated as a prophet in Judaism, Christianity, and Islam. In addition to his role in the Bible, Samuel is mentioned in Jewish rabbinical literature, in the Christian New Testament, and in the second chapter of the Quran (although the text does not mention him by name). He is also treated in the fifth through seventh books of '' Antiquities of the Jews'', written by the Jewish scholar Josephus in the first century. He is first called "the Seer" in 1 Samuel 9:9. Biblical account Family Samuel's mother was Hannah and his father was Elkanah. Elkanah lived at Ramathaim in the district of Zuph. His genealogy is also found in a pedigree of the Kohathites (1 Chronicles 6:3–15) and in that of Heman the Ezrahite, apparently his grandson (1 Chron ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Determinant
In mathematics, the determinant is a Scalar (mathematics), scalar-valued function (mathematics), function of the entries of a square matrix. The determinant of a matrix is commonly denoted , , or . Its value characterizes some properties of the matrix and the linear map represented, on a given basis (linear algebra), basis, by the matrix. In particular, the determinant is nonzero if and only if the matrix is invertible matrix, invertible and the corresponding linear map is an linear isomorphism, isomorphism. However, if the determinant is zero, the matrix is referred to as singular, meaning it does not have an inverse. The determinant is completely determined by the two following properties: the determinant of a product of matrices is the product of their determinants, and the determinant of a triangular matrix is the product of its diagonal entries. The determinant of a matrix is :\begin a & b\\c & d \end=ad-bc, and the determinant of a matrix is : \begin a & b & c \\ d & e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Covariance Matrix
In probability theory and statistics, a covariance matrix (also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix) is a square matrix giving the covariance between each pair of elements of a given random vector. Intuitively, the covariance matrix generalizes the notion of variance to multiple dimensions. As an example, the variation in a collection of random points in two-dimensional space cannot be characterized fully by a single number, nor would the variances in the x and y directions contain all of the necessary information; a 2 \times 2 matrix would be necessary to fully characterize the two-dimensional variation. Any covariance matrix is symmetric and positive semi-definite and its main diagonal contains variances (i.e., the covariance of each element with itself). The covariance matrix of a random vector \mathbf is typically denoted by \operatorname_, \Sigma or S. Definition Throughout this article, boldfaced u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Kalman Filter
In statistics and control theory, Kalman filtering (also known as linear quadratic estimation) is an algorithm that uses a series of measurements observed over time, including statistical noise and other inaccuracies, to produce estimates of unknown variables that tend to be more accurate than those based on a single measurement, by estimating a joint probability distribution over the variables for each time-step. The filter is constructed as a mean squared error minimiser, but an alternative derivation of the filter is also provided showing how the filter relates to maximum likelihood statistics. The filter is named after Rudolf E. Kálmán. Kalman filtering has numerous technological applications. A common application is for guidance, navigation, and control of vehicles, particularly aircraft, spacecraft and ships Dynamic positioning, positioned dynamically. Furthermore, Kalman filtering is much applied in time series analysis tasks such as signal processing and econometrics. K ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |