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Variogram
In spatial statistics the theoretical variogram 2\gamma(\mathbf_1,\mathbf_2) is a function describing the degree of spatial dependence of a spatial random field or stochastic process Z(\mathbf). The semivariogram \gamma(\mathbf_1,\mathbf_2) is half the variogram. In the case of a concrete example from the field of gold mining, a variogram will give a measure of how much two samples taken from the mining area will vary in gold percentage depending on the distance between those samples. Samples taken far apart will vary more than samples taken close to each other. Definition The semivariogram \gamma(h) was first defined by Matheron (1963) as half the average squared difference between the values at points (\mathbf_1 and \mathbf_2) separated at distance h. Formally :\gamma(h)=\frac\iiint_V \left (M+h) - f(M) \right2dV, where M is a point in the geometric field V, and f(M) is the value at that point. The triple integral is over 3 dimensions. h is the separation distance (e.g., ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Covariance Function
In probability theory and statistics, the covariance function describes how much two random variables change together (their ''covariance'') with varying spatial or temporal separation. For a random field or stochastic process ''Z''(''x'') on a domain ''D'', a covariance function ''C''(''x'', ''y'') gives the covariance of the values of the random field at the two locations ''x'' and ''y'': :C(x,y):=\operatorname(Z(x),Z(y))=\mathbb\left \cdot\ \right\, The same ''C''(''x'', ''y'') is called the autocovariance function in two instances: in time series (to denote exactly the same concept except that ''x'' and ''y'' refer to locations in time rather than in space), and in multivariate random fields (to refer to the covariance of a variable with itself, as opposed to the cross covariance between two different variables at different locations, Cov(''Z''(''x''1), ''Y''(''x''2))). Admissibility For locations ''x''1, ''x''2, …, ''x''''N'' ∈ ''D'' the varianc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geostatistics
Geostatistics is a branch of statistics focusing on spatial or spatiotemporal datasets. Developed originally to predict probability distributions of ore grades for mining operations, it is currently applied in diverse disciplines including petroleum geology, hydrogeology, hydrology, meteorology, oceanography, geochemistry, geometallurgy, geography, forestry, environmental control, landscape ecology, soil science, and agriculture (esp. in precision farming). Geostatistics is applied in varied branches of geography, particularly those involving the spread of diseases (epidemiology), the practice of commerce and military planning (logistics), and the development of efficient spatial networks. Geostatistical algorithms are incorporated in many places, including geographic information systems (GIS). Background Geostatistics is intimately related to interpolation methods, but extends far beyond simple interpolation problems. Geostatistical techniques rely on statistical models ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gaussian Process
In probability theory and statistics, a Gaussian process is a stochastic process (a collection of random variables indexed by time or space), such that every finite collection of those random variables has a multivariate normal distribution, i.e. every finite linear combination of them is normally distributed. The distribution of a Gaussian process is the joint distribution of all those (infinitely many) random variables, and as such, it is a distribution over functions with a continuous domain, e.g. time or space. The concept of Gaussian processes is named after Carl Friedrich Gauss because it is based on the notion of the Gaussian distribution (normal distribution). Gaussian processes can be seen as an infinite-dimensional generalization of multivariate normal distributions. Gaussian processes are useful in statistical modelling, benefiting from properties inherited from the normal distribution. For example, if a random process is modelled as a Gaussian process, the distribu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Random Field
In physics and mathematics, a random field is a random function over an arbitrary domain (usually a multi-dimensional space such as \mathbb^n). That is, it is a function f(x) that takes on a random value at each point x \in \mathbb^n(or some other domain). It is also sometimes thought of as a synonym for a stochastic process with some restriction on its index set. That is, by modern definitions, a random field is a generalization of a stochastic process where the underlying parameter need no longer be real or integer valued "time" but can instead take values that are multidimensional vectors or points on some manifold. Formal definition Given a probability space (\Omega, \mathcal, P), an ''X''-valued random field is a collection of ''X''-valued random variables indexed by elements in a topological space ''T''. That is, a random field ''F'' is a collection : \ where each F_t is an ''X''-valued random variable. Examples In its discrete version, a random field is a list of r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Correlogram
In the analysis of data, a correlogram is a chart of correlation statistics. For example, in time series analysis, a plot of the sample autocorrelations r_h\, versus h\, (the time lags) is an autocorrelogram. If cross-correlation is plotted, the result is called a cross-correlogram. The correlogram is a commonly used tool for checking randomness in a data set. If random, autocorrelations should be near zero for any and all time-lag separations. If non-random, then one or more of the autocorrelations will be significantly non-zero. In addition, correlograms are used in the model identification stage for Box–Jenkins autoregressive moving average time series models. Autocorrelations should be near-zero for randomness; if the analyst does not check for randomness, then the validity of many of the statistical conclusions becomes suspect. The correlogram is an excellent way of checking for such randomness. In multivariate analysis, correlation matrices shown as color ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kriging
In statistics, originally in geostatistics, kriging or Kriging, also known as Gaussian process regression, is a method of interpolation based on Gaussian process governed by prior covariances. Under suitable assumptions of the prior, kriging gives the best linear unbiased prediction (BLUP) at unsampled locations. Interpolating methods based on other criteria such as smoothness (e.g., smoothing spline) may not yield the BLUP. The method is widely used in the domain of spatial analysis and computer experiments. The technique is also known as Wiener–Kolmogorov prediction, after Norbert Wiener and Andrey Kolmogorov. The theoretical basis for the method was developed by the French mathematician Georges Matheron in 1960, based on the master's thesis of Danie G. Krige, the pioneering plotter of distance-weighted average gold grades at the Witwatersrand reef complex in South Africa. Krige sought to estimate the most likely distribution of gold based on samples from a few borehole ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geostatistics
Geostatistics is a branch of statistics focusing on spatial or spatiotemporal datasets. Developed originally to predict probability distributions of ore grades for mining operations, it is currently applied in diverse disciplines including petroleum geology, hydrogeology, hydrology, meteorology, oceanography, geochemistry, geometallurgy, geography, forestry, environmental control, landscape ecology, soil science, and agriculture (esp. in precision farming). Geostatistics is applied in varied branches of geography, particularly those involving the spread of diseases (epidemiology), the practice of commerce and military planning (logistics), and the development of efficient spatial networks. Geostatistical algorithms are incorporated in many places, including geographic information systems (GIS). Background Geostatistics is intimately related to interpolation methods, but extends far beyond simple interpolation problems. Geostatistical techniques rely on statistical models ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Absolute Difference
The absolute difference of two real numbers x and y is given by , x-y, , the absolute value of their difference. It describes the distance on the real line between the points corresponding to x and y. It is a special case of the Lp distance for all 1\le p\le\infty and is the standard metric used for both the set of rational numbers \Q and their completion, the set of real numbers \R. As with any metric, the metric properties hold: * , x-y, \ge 0, since absolute value is always non-negative. * , x-y, = 0 if and only if x=y. * , x-y, =, y-x, ('' symmetry'' or '' commutativity''). * , x-z, \le, x-y, +, y-z, (''triangle inequality''); in the case of the absolute difference, equality holds if and only if x\le y\le z or x\ge y\ge z. By contrast, simple subtraction is not non-negative or commutative, but it does obey the second and fourth properties above, since x-y=0 if and only if x=y, and x-z=(x-y)+(y-z). The absolute diff ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Seismic Risk
Seismic risk refers to the risk of damage from earthquake to a building, system, or other entity. Seismic risk has been defined, for most management purposes, as the potential economic, social and environmental consequences of hazardous events that may occur in a specified period of time. A building located in a region of high seismic hazard is at lower risk if it is built to sound seismic engineering principles. On the other hand, a building located in a region with a history of minor seismicity, in a brick building located on fill subject to liquefaction can be as high or higher risk. A special subset is urban seismic risk which looks at the specific issues of cities. Risk determination and emergency response can also be determined through the use of an Earthquake scenario. Determination of seismic risk The determination of seismic risk is the foundation for risk mitigation decision-making, a key step in risk management. Large corporations and other enterprises (e.g., local ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spatial Statistics
Spatial analysis or spatial statistics includes any of the formal techniques which studies entities using their topological, geometric, or geographic properties. Spatial analysis includes a variety of techniques, many still in their early development, using different analytic approaches and applied in fields as diverse as astronomy, with its studies of the placement of galaxies in the cosmos, to chip fabrication engineering, with its use of "place and route" algorithms to build complex wiring structures. In a more restricted sense, spatial analysis is the technique applied to structures at the human scale, most notably in the analysis of geographic data or transcriptomics data. Complex issues arise in spatial analysis, many of which are neither clearly defined nor completely resolved, but form the basis for current research. The most fundamental of these is the problem of defining the spatial location of the entities being studied. Classification of the techniques of spatial an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Root
In mathematics, a square root of a number is a number such that ; in other words, a number whose ''square'' (the result of multiplying the number by itself, or ⋅ ) is . For example, 4 and −4 are square roots of 16, because . Every nonnegative real number has a unique nonnegative square root, called the ''principal square root'', which is denoted by \sqrt, where the symbol \sqrt is called the '' radical sign'' or ''radix''. For example, to express the fact that the principal square root of 9 is 3, we write \sqrt = 3. The term (or number) whose square root is being considered is known as the ''radicand''. The radicand is the number or expression underneath the radical sign, in this case 9. For nonnegative , the principal square root can also be written in exponent notation, as . Every positive number has two square roots: \sqrt, which is positive, and -\sqrt, which is negative. The two roots can be written more concisely using the ± sign as \plusmn\sq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
