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== you and I shot four people at the same time ==
{{cleanup|date=June 2012|reason=Article is overly technical.}}


In [[spatial statistics]] the theoretical '''variogram''' <math>2\gamma(x,y)</math> is a function describing the degree of spatial dependence of a spatial [[random field]] or [[stochastic process]] <math>Z(x)</math>. It is defined as the [[variance]] of the difference between field values at two locations (<math>x</math> and <math>y</math>) across realizations of the field (Cressie 1993):
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:<math>2\gamma(x,y)=\text{var}(Z(x) - Z(y)) = E\left(|(Z(x)-\mu(x))-(Z(y) - \mu(y))|^2\right). </math>
== the other one is bound to be hit. ==


If the spatial random field has constant mean <math>\mu</math>, this is equivalent to the expectation for the squared increment of the values between locations <math>x</math> and <math>y</math> (Wackernagel 2003) (where <math>x</math> and <math>y</math> are not coordinates but points in space):
Xiao Yan is taking into account the two difficult position, [http://nrcil.net/sitemap.xml http://nrcil.net/sitemap.xml] if only anti Xiao [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-0.html casio 腕時計] Yan a, the other one is bound to be hit.<br><br>in numerous road gaze, as sculptural Xiao Yan, the two Quanfeng getting to that Yi Sha, stature and shoved shocked, everyone is stunned immediately saw two illusory shadow flotation [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-15.html カシオ 腕時計 gps] feet on both sides , while two unreal foot shadow, while Cliff and Lin Xiu Qing and Liu fist collision, but like turning [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-3.html カシオ 腕時計 電波 ソーラー] to [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-14.html casio 腕時計 デジタル] the substance of like, an instant burst of extremely terror forces.<br><br>first trick!<br><br>'bang!'<br><br>sounded muffled muffled presence, fury Effort at that moment, such as a flood vent out, so that was also a direct cliff and Liu Qing Lin Xiu figure Baotui nearly ten steps, every foot of the fall, will be in left deep marks on the floor in the foot.<br><br>'This guy, good speed and strength of terror,' Liu Qing stabilize stature, heart an idea just flashed, muddy
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:<math>2\gamma(x,y)=E\left(|Z(x)-Z(y)|^2\right) , </math>
== is sharp with a little overbearing ==


where <math>\gamma(x,y)</math> itself is called the '''semivariogram'''. In the case of a [[stationary process]], the variogram and semivariogram can be represented as a function <math>\gamma_s(h)=\gamma(0,0+h)</math> of the difference <math>h=y-x</math> between locations only, by the following relation (Cressie 1993):
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:<math>\gamma(x,y)=\gamma_s(y-x).</math>
== that he ==


If the process is furthermore [[isotropy|isotropic]], then the variogram and semivariogram can be represented by a function <math>\gamma_i(h):=\gamma_s(h e_1)</math> of the distance  <math>h=\|y-x\|</math> only (Cressie 1993):
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:<math>\gamma(x,y)=\gamma_i(h).</math>
== relying on a stone bench ==


The indexes <math>i</math> or <math>s</math> are typically not written. The terms are used for all three forms of the function. Moreover, the term "variogram" is sometimes used to denote the semivariogram, and the symbol <math>\gamma</math> is sometimes used for the variogram, which brings some confusion.
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==Properties==
== Xiao Yan also did not nonsense ==
According to (Cressie 1993, Chiles and Delfiner 1999, Wackernagel 2003) the theoretical variogram has the following properties:
* The semivariogram is nonnegative <math>\gamma(x,y)\geq 0</math>, since it is the expectation of a square.
* The semivariogram  <math>\gamma(x,x)=\gamma_i(0)=E\left((Z(x)-Z(x))^2\right)=0</math> at distance 0  is always 0, since <math>Z(x)-Z(x)=0</math>.
* A function is a semivariogram if and only if it is a conditionally negative definite function, i.e. for all weights <math>w_1,\ldots,w_N</math> subject to <math>\sum_{i=1}^N w_i=0</math> and locations <math>x_1,\ldots,x_N</math> it holds:<blockquote><math>\sum_{i=1}^N\sum_{j=1}^N w_{i}\gamma(x_i,x_j)w_j \leq 0</math></blockquote>which corresponds to the fact that the variance <math>var(X)</math> of <math>X=\sum_{i=1}^N w_i Z(x_i)</math> is given by the negative of this double sum and must be nonnegative.
* As a consequence the semivariogram might be non continuous only at the origin. The height of the jump at the origin is sometimes referred to as ''nugget'' or nugget effect.
* If the [[covariance function]] of a stationary process exists it is related to variogram by<blockquote><math>2\gamma(x,y)=C(x,x)+C(y,y)-2C(x,y)</math></blockquote>For a non-stationary process the square of the difference between expected values at both points must be added:<blockquote><math>2\gamma(x,y)=C(x,x)+C(y,y)-2C(x,y) + (E(Z(x))-E(Z(y)))^2</math></blockquote>
* If a stationary random field has no spatial dependence (i.e. <math>C(h)=0</math> if <math>h\not= 0</math>), the semivariogram is the constant <math>var(Z(x))</math> everywhere except at the origin, where it is zero. 
* <math>\gamma(x,y)=E(|Z(x)-Z(y)|^2)=\gamma(y,x)</math> is a symmetric function.
* Consequently, <math>\gamma_s(h)=\gamma_s(-h)</math> is an [[even function]].
* If the random field is [[stationary process|stationary]] and [[ergodic]], the <math>\lim_{h\to \infty} \gamma_s(h) = var(Z(x))</math> corresponds to the variance of the field. The limit of the semivariogram is also called its ''sill''.


==Empirical variogram==
Shadow Shanlue out from the room, the last neatly arranged [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-14.html カシオ 時計 電波 ソーラー] in four lived before XiaoYan small pavilion.<br><br>'squish'<br>After<br>long while, a small pavilion door slowly opened, Xiao Yan [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-11.html 腕時計 メンズ casio] slowly out of four, standing in the doorway, looking at those high momentum, his face excited Pan Gate members, could not help but laugh, though these members are now quite [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-6.html casio 腕時計 メンズ] weak, can the potential value, but it is not extremely weak, just give them enough [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-4.html カシオ 腕時計 バンド] time, Xiao Yan believes that sooner or later they can become truly strong.<br><br>'go!'<br><br>staff has completed the collection, Xiao Yan also did not nonsense, waved his hand, that is, with the bong three quick walk to take the lead in facing the outside, followed by forty to freshmen follow closely.<br><br>day on the way to the tower burning Lianqi journey to and from the staff looked at from the side Hula Bay had a large group of people, could not help [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-7.html カシオ 掛け時計] but are somewhat stunned however, and when the people saw some sharp-eyed leader
 
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For observations <math>z_i,\;i=1,\ldots,k</math> at locations <math>x_1,\ldots,x_k</math> the empirical variogram <math>\hat{\gamma}(h)</math> is defined as (Cressie 1993):
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<math>\hat{\gamma}(h):=\frac{1}{|N(h)|}\sum_{(i,j)\in N(h)} |z_i-z_j|^2</math>
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where <math>N(h)</math> denotes the set of pairs of observations <math>i,\;j</math> such that <math>|x_i-x_j| = h</math>, and <math>|N(h)|</math> is the number of pairs in the set. (Generally an  "approximate distance" <math>h</math> is used, implemented using a certain tolerance.) 
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The ''empirical variogram'' is used in [[geostatistics]] as a first estimate of the (theoretical) variogram needed for spatial interpolation by [[kriging]].
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According (Cressie 1993) for observations <math>z_i=Z(x_i)</math> from a [[Stationary process|stationary]] [[random field]] <math>Z(x)</math> the empirical variogram with lag tolerance 0 is an unbiased estimator of the theoretical variogram, due to:
</ul>
 
<math>E[\hat{\gamma}(h)]=\frac{1}{2|N(h)|}\sum_{(i,j)\in N(h)}E[|Z(x_i)-Z(x_j)|^2]=\frac{1}{2|N(h)|}\sum_{(i,j)\in N(h)}2\gamma(x_j-x_i)=\frac{2|N(h)|}{2|N(h)|}\gamma(h)</math>
 
==Variogram parameters==
 
The following parameters are often used to describe variograms:
 
* ''nugget'' <math>n</math>: The height of the jump of the semivariogram at the discontinuity at the origin.  
* ''sill'' <math>s</math>: Limit of the variogram tending to infinity lag distances. 
* ''range'' <math>r</math>: The distance in which the difference of the variogram from the sill becomes negligible. In models with a fixed sill, it is the distance at which this is first reached; for models with an asymptotic sill, it is conventionally taken to be the distance when the semivariance first reaches 95% of the sill.
 
==Variogram models==
 
The empirical variogram cannot be computed at every lag distance <math>h</math> and due to variation in the estimation it is not ensured that it is a valid variogram, as defined above. However some [[geostatistics|Geostatistical]] methods such as [[kriging]] need valid semivariograms. In applied geostatistics the empirical variograms are thus often approximated by model function ensuring validity (Chiles&Delfiner 1999). Some important models are (Chiles&Delfiner 1999, Cressie 1993):
 
* The exponential variogram model
 
:: <math>\gamma(h)=(s-n)(1-\exp(-h/(ra)))+n 1_{(0,\infty)}(h)</math>
 
* The spherical variogram model
 
:: <math>\gamma(h)=(s-n)\left(\left(\frac{3h}{2r}-\frac{h^3}{2r^3}\right)1_{(0,r)}(h)+1_{[r,\infty)}(h)\right)+n1_{(0,\infty)}(h)</math>
 
* The Gaussian variogram model
 
:: <math>\gamma(h)=(s-n)\left(1-\exp\left(-\frac{h^2}{r^2a}\right)\right) + n1_{(0,\infty)}(h)</math>
 
The parameter <math>a</math> has different values in different references, due to the ambiguity in the definition of the range. E.g. <math>a=1/3</math> is the value used in (Chiles&Delfiner 1999). The <math>1_A(h)</math> function is 1 if <math>h\in A</math> and 0 otherwise.
 
==Discussion==
 
Three functions are used in [[geostatistics]] for describing the spatial or the temporal correlation of observations: these are the [[correlogram]], the [[covariance]] and the '''semivariogram'''. The last is also more simply called '''variogram'''. The [[sampling variogram]], unlike the semivariogram and the variogram, shows where a significant degree of spatial dependence in the sample space or sampling unit dissipates into randomness when the variance terms of a temporally or ''in-situ'' ordered set are plotted against the variance of the set and the lower limits of its 99% and 95% confidence ranges.
 
The variogram is the key function in [[geostatistics]] as it will be used to fit a model of the temporal/[[spatial correlation]] of the observed phenomenon. One is thus making a distinction between the ''experimental variogram'' that is a visualisation of a possible spatial/temporal correlation and the ''variogram model'' that is further used to define the weights of the [[kriging]] function. Note that the experimental variogram is an empirical estimate of the [[covariance]] of a [[Gaussian process]]. As such, it may not be [[positive definite]] and hence not directly usable in [[kriging]], without constraints or further processing. This explains why only a limited number of variogram models are used: most commonly, the linear, the spherical, the gaussian and the exponential models.  
 
When a variogram is used to describe the correlation of different variables it is called ''cross-variogram''. Cross-variograms are used in [[co-kriging]].
Should the variable be binary or represent classes of values, one is then talking about ''indicator variograms''. Indicator variogram is used in [[indicator kriging]].
 
== See also ==
* [[Covariance function]]
* [[Semivariance]]
 
==References==
{{Reflist}}
# Cressie, N., 1993, Statistics for spatial data, Wiley Interscience
# Chiles, J. P., P. Delfiner, 1999, Geostatististics, Modelling Spatial Uncertainty, Wiley-Interscience
# Wackernagel, H., 2003, Multivariate Geostatistics, Springer
# Burrough, P A and McDonnell, R A, 1998, Principles of Geographical Information Systems
# [http://www.kriging.com/pg1979_download.html Isobel Clark, 1979, Practical Geostatistics, Applied Science Publishers]
 
==External links==
* [http://www.ai-geostats.org/ AI-GEOSTATS: an educational resource about geostatistics and spatial statistics]
* [http://www.kriging.com/PG1979/ Practical Geostatistics 1979 by Isobel Clark : an introduction to geostatistics]
* [http://www.statistik.tuwien.ac.at/public/dutt/vorles/geost_05/geo.html Geostatistics: Lecture by Rudolf Dutter at the Technical University of Vienna]
 
[[Category:Geostatistics]]
[[Category:Statistical deviation and dispersion]]
[[Category:Spatial processes]]

Latest revision as of 22:23, 9 January 2015

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The dense seat, at the moment, in these seats are like a 'boom' like water poured countless dense mass of the head, this time with Han Hung family confrontation, attracting the attention of the entire day north of the city.

day in both directions east and north of the top of the stone, with some obvious exceptionally gorgeous seats, those seats, only a day just カシオ 時計 電波 ソーラー north of the city are eligible to mind some forces seat, where the eyes of the whole can be カシオの時計 condescending Square income, and North seat at the moment, is already filled with a lot of shadows, where most people are a red robe, in their chest, wearing a pretty カシオ 時計 価格 badge that symbolizes the casio 腕時計 メンズ Hung family

on the Hung family seat, the day Hung Chen Xiao Yan had ever seen, being arms Baoxiong, relying on a stone bench, his face looked red-hot Plaza export orientation, that カシオの時計 road was so haunted by him Hunmeng shadows, has been branded deep in their hearts, in his mind, has determined that this woman must 相关的主题文章：

Xiao Yan also did not nonsense

Shadow Shanlue out from the room, the last neatly arranged カシオ 時計 電波 ソーラー in four lived before XiaoYan small pavilion.

'squish'
After
long while, a small pavilion door slowly opened, Xiao Yan 腕時計 メンズ casio slowly out of four, standing in the doorway, looking at those high momentum, his face excited Pan Gate members, could not help but laugh, though these members are now quite casio 腕時計 メンズ weak, can the potential value, but it is not extremely weak, just give them enough カシオ 腕時計 バンド time, Xiao Yan believes that sooner or later they can become truly strong.

'go!'

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