Kest(spatstat)
Kest()所属R语言包：spatstat
K-function
K函数
译者：生物统计家园网 机器人LoveR
描述----------Description----------
Estimates Ripley's reduced second moment function K(r) from a point pattern in a window of arbitrary shape.
估计里普利的二阶矩功能降低，K(r)从一个点在一个窗口中任意形状的模式。
用法----------Usage----------
Kest(X, ..., r=NULL, breaks=NULL,
correction=c("border", "isotropic", "Ripley", "translate"),
nlarge=3000, domain=NULL, var.approx=FALSE, ratio=FALSE)
参数----------Arguments----------
参数：X
The observed point pattern, from which an estimate of K(r) will be computed. An object of class "ppp", or data in any format acceptable to as.ppp().
观测点的模式，从一个估算的K(r)将被计算。对象的类"ppp"，或任何格式的数据中接受的as.ppp()。
参数：...
Ignored.
忽略。
参数：r
Optional. Vector of values for the argument r at which K(r) should be evaluated. Users are advised not to specify this argument; there is a sensible default.
可选。向量参数的值r，K(r)应该进行评估。建议用户在不指定此参数，是一个明智的默认。
参数：breaks
Optional. An alternative to the argument r. Not normally invoked by the user. See the Details section.
可选。替代到的参数r。通常不是由用户调用。查看详细信息“一节。
参数：correction
Optional. A character vector containing any selection of the options "none", "border", "bord.modif", "isotropic", "Ripley", "translate", "none" or "best". It specifies the edge correction(s) to be applied.
可选。一个字符向量选择任何选项"none"，"border"，"bord.modif"，"isotropic"，"Ripley"，"translate"，"none"或"best"。指定，边缘校正（S）。
参数：nlarge
Optional. Efficiency threshold. If the number of points exceeds nlarge, then only the border correction will be computed (by default), using a fast algorithm.
可选。效率的阈值。如果点的数量超过nlarge，只有边界校正计算（默认情况下），使用的快速算法。
参数：domain
Optional. Calculations will be restricted to this subset of the window. See Details.
可选。计算将受到限制窗口的子集。查看详细信息。
参数：var.approx
Logical. If TRUE, the approximate variance of Kest(r) under CSR will also be computed.
逻辑。如果TRUE，Kest(r)的方差近似的下的企业社会责任也将被计算。
参数：ratio
Logical. If TRUE, the numerator and denominator of each edge-corrected estimate will also be saved, for use in analysing replicated point patterns.
逻辑。如果TRUE，分子和分母的每个边缘校正的估计也将被保存，用于在分析复制的点图案。
Details
详细信息----------Details----------
The K function (variously called “Ripley's K-function” and the “reduced second moment function”) of a stationary point process X is defined so that lambda K(r) equals the expected number of additional random points within a distance r of a typical random point of X. Here lambda is the intensity of the process, i.e. the expected number of points of X per unit area. The K function is determined by the second order moment properties of X.
K功能（各种所谓“Ripley的K-函数”和“减少二阶矩功能”）的一个固定的点过程中X的定义使lambda K(r)等于预期数额外的随机点的距离内r一个典型的随机点X。这是lambda过程的强度，即点X每单位面积的预期。是由二阶矩属性的K的X功能。
An estimate of K derived from a spatial point pattern dataset can be used in exploratory data analysis and formal inference about the pattern (Cressie, 1991; Diggle, 1983; Ripley, 1977, 1988). In exploratory analyses, the estimate of K is a useful statistic summarising aspects of inter-point “dependence” and “clustering”. For inferential purposes, the estimate of K is usually compared to the true value of K for a completely random (Poisson) point process, which is K(r) = pi * r^2. Deviations between the empirical and theoretical K curves may suggest spatial clustering or spatial regularity.
的一个估计K来自空间的点模式数据集可以使用在探索数据分析和正式推理有关的图案（经验Cressie，1991; Diggle，1983;里普利，1977年，1988年）。在探索性分析，估计K是一个有用的统计总结性的方面点间的“依赖”和“聚类”。推理的目的，估计K通常K，这是一个完全随机的（泊松）点的过程K(r) = pi * r^2的真正价值相比。的经验和理论K曲线之间的偏差可能会建议的空间聚类或空间的规律性。
This routine Kest estimates the K function of a stationary point process, given observation of the process inside a known, bounded window. The argument X is interpreted as a point pattern object (of class "ppp", see ppp.object) and can be supplied in any of the formats recognised by as.ppp().
这个例程Kest估计K功能的一个固定的点过程中，由于观察的过程在一个已知的，有限的窗口。参数X被解释为一个点图形对象（类"ppp"，看到ppp.object），并且可以在任何认可的as.ppp()的格式提供。
The estimation of K is hampered by edge effects arising from the unobservability of points of the random pattern outside the window. An edge correction is needed to reduce bias (Baddeley, 1998; Ripley, 1988). The corrections implemented here are
估计K阻碍了边缘效应所产生的不可观测点的随机模式窗外。边缘校正是必要的减少偏差（巴德雷，1998;里普利，1988）。在这里实施的更正
border the border method or “reduced sample” estimator (see Ripley, 1988). This is the least efficient (statistically) and the fastest to compute. It can be computed for a window of arbitrary shape.
毗邻边界的方法或“减少样本”估计（见里普利，1988年）。这是最有效的（统计），以最快的速度计算。它可以计算一个窗口的任意形状。
isotropic/Ripley Ripley's isotropic correction (see Ripley, 1988; Ohser, 1983). This is implemented for rectangular and polygonal windows (not for binary masks).
各向同性/ Ripley旅游Ripley的各向同性修正（见里普利，1988; Ohser，1983年）。这是实现矩形和多边形窗口（而不是二进制口罩）。
translate Translation correction (Ohser, 1983). Implemented for all window geometries, but slow for complex windows.
翻译的翻译的校正（Ohser，1983）。实现所有窗口的几何形状，但速度缓慢复杂的Windows。
Selects the best edge correction that is available for the geometry of the window. Currently this is Ripley's isotropic correction for a rectangular or polygonal window, and the translation correction for masks.
选择最佳的边缘的校正，是可用于窗口的几何形状。目前，这是雷普利各向同性校正为矩形或多边形的窗口，和翻译掩模校正。
Uncorrected estimate. An estimate of the K function without edge correction. (i.e. setting e[i,j] = 1 in the equation below. This estimate is biased and should not be used for data analysis, unless you have an extremely large point pattern (more than 100,000 points).
未校正的估计。 K函数的估计没有边缘校正。 （即设置e[i,j] = 1在下面的公式。这估计是带有偏见和不应该被用于数据分析，除非你有一个非常大的点模式（10万点以上）。
The estimates of K(r) are of the form
的估计K(r)的形式为
where a is the area of the window, n is the number of data points, and the sum is taken over all ordered pairs of points i and j in X. Here d[i,j] is the distance between the two points, and I(d[i,j] <= r) is the indicator that equals 1 if the distance is less than or equal to r. The term e[i,j] is the edge correction weight (which depends on the choice of edge correction listed above).
其中a是该区域的窗口，n是多少的数据点，被接管的总和下令对点i和j中 X>。这里X是在两个点之间的距离，和d[i,j]是指标等于1，如果该距离小于或等于I(d[i,j] <= r)。术语r是边缘校正重量（这取决于上面列出的边缘校正的选择）。
Note that this estimator assumes the process is stationary (spatially homogeneous). For inhomogeneous point patterns, see Kinhom.
请注意，此估计假设的过程是平稳的（空间均匀）。对于非均匀的点图案，请参阅Kinhom。
If the point pattern X contains more than about 3000 points, the isotropic and translation edge corrections can be computationally prohibitive. The computations for the border method are much faster, and are statistically efficient when there are large numbers of points. Accordingly, if the number of points in X exceeds the threshold nlarge, then only the border correction will be computed. Setting nlarge=Inf or correction="best" will prevent this from happening. Setting nlarge=0 is equivalent to selecting only the border correction with correction="border".
如果点模式X包含超过3000点左右，各向同性和翻译的边缘改正，可以计算望而却步。的边界的计算方法快得多，而且在统计上有效的，当有大量的点。因此，如果点的数量在X超过阈值nlarge，那么只有边界校正将被计算。设置nlarge=Inf或correction="best"防止这种情况的发生。设置nlarge=0是相当于只选择边界校正correction="border"。
If X contains more than about 100,000 points, even the border correction is time-consuming. You may want to consider setting correction="none" in this case. There is an even faster algorithm for the uncorrected estimate.
如果X包含超过约100,000点，甚至边界校正是很费时的。设置correction="none"在这种情况下，您可能要考虑。有一个更快的算法的裸预算。
Approximations to the variance of Kest(r) are available, for the case of the isotropic edge correction estimator, assuming complete spatial randomness (Ripley, 1988; Lotwick and Silverman, 1982; Diggle, 2003, pp 51-53). If var.approx=TRUE, then the result of Kest also has a column named rip values of Ripley's (1988) approximation to var(Kest(r)), and (if the window is a rectangle) a column named ls giving values of Lotwick and Silverman's (1982) approximation.
逼近到方差Kest(r)可用，各向同性的边缘校正估计的情况下，假设完整的空间随机性（雷普利，1988; Lotwick和Silverman，1982; Diggle，2003年51-53页）。如果var.approx=TRUE，那么结果Kest也有一列名为rip值里普利（1988）的近似var(Kest(r))，（如果窗口是一个矩形）列名为lsLotwick和Silverman（1982）近似的值。
If the argument domain is given, the calculations will be restricted to a subset of the data. In the formula for K(r) above, the first point i will be restricted to lie inside domain. The result is an approximately unbiased estimate of K(r) based on pairs of points in which the first point lies inside domain and the second point is unrestricted. This is useful in bootstrap techniques. The argument domain should be a window (object of class "owin") or something acceptable to as.owin. It must be a subset of the window of the point pattern X.
如果参数domain给出，计算将被限制到的数据的一个子集。在公式为K(r)以上，第一点i将受到限制躺在里面domain。其结果是一个近似无偏估计K(r)根据对点，其中，第一点位于内侧domain和第二点是不受限制。自举技术，这是非常有用的。参数domain应该是一个窗口（对象的类"owin"）或一些可以接受的as.owin。它必须是一个子集的窗口的点图案X。
The estimator Kest ignores marks. Its counterparts for multitype point patterns are Kcross, Kdot, and for general marked point patterns see Kmulti.
估计Kest忽略的痕迹。其对应下的多点模式Kcross，Kdot，和一般的标记点模式看Kmulti。
Some writers, particularly Stoyan (1994, 1995) advocate the use of the “pair correlation function”
一些作家，特别是斯托扬（1994年，1995年）主张“对相关功能的使用”
where K'(r) is the derivative of K(r). See pcf on how to estimate this function.
K'(r)是衍生工具的K(r)。见pcf了如何估计这个功能。
值----------Value----------
An object of class "fv", see fv.object, which can be plotted directly using plot.fv.
类的一个对象"fv"，fv.object，它可以绘制直接使用plot.fv的。
Essentially a data frame containing columns
本质上是一个数据框包含多个列
参数：r
the vector of values of the argument r at which the function K has been estimated
的参数的值的矢量r在哪些函数K已估计
参数：theo
the theoretical value K(r) = pi * r^2 for a stationary Poisson process
理论值K(r) = pi * r^2平稳泊松过程
together with columns named "border", "bord.modif", "iso" and/or "trans", according to the selected edge corrections. These columns contain estimates of the function K(r) obtained by the edge corrections named.
一起列名为"border"，"bord.modif"，"iso"和/或"trans"，根据选定的边修正。这些列包含的功能K(r)命名的边缘修正的估计。
If var.approx=TRUE then the return value also has columns rip and ls containing approximations to the variance of Kest(r) under CSR.
如果var.approx=TRUE，则返回值也列rip和ls包含近似的Kest(r)根据“公务员事务规例”的差异。
If ratio=TRUE then the return value also has two attributes called "numerator" and "denominator" which are "fv" objects containing the numerators and denominators of each estimate of K(r).
如果ratio=TRUE，则返回值也有两个属性，称为"numerator"和"denominator""fv"含有的分子和分母的每一个估计的K(r)的对象。
信封，意义乐队和置信区间----------Envelopes, significance bands and confidence intervals----------
To compute simulation envelopes for the K-function under CSR, use envelope.
计算模拟的K功能下的企业社会责任的信封，使用envelope。
To compute a confidence interval for the true K-function, use varblock or lohboot.
要计算的置信区间为真正的K功能，使用varblock或lohboot。
警告----------Warnings----------
The estimator of K(r) is approximately unbiased for each fixed r. Bias increases with r and depends on the window geometry. For a rectangular window it is prudent to restrict the r values to a maximum of 1/4 of the smaller side length of the rectangle. Bias may become appreciable for point patterns consisting of fewer than 15 points.
对每一个固定K(r)r估计约为公正。偏置增加r和依赖于窗口的尺寸。对于一个矩形窗口，它是审慎限制r值到最大1/4的小的矩形的边长。少于15个点的点模式的偏差可能成为明显。
While K(r) is always a non-decreasing function, the estimator of K is not guaranteed to be non-decreasing. This is rarely a problem in practice.
而K(r)始终是一个非递减的函数，估计K但不保证非递减。这是很少在实践中的问题。
（作者）----------Author(s)----------
Adrian Baddeley
<a href="mailto:Adrian.Baddeley@csiro.au">Adrian.Baddeley@csiro.au</a>
<a href="http://www.maths.uwa.edu.au/~adrian/">http://www.maths.uwa.edu.au/~adrian/</a>
and Rolf Turner
<a href="mailto:r.turner@auckland.ac.nz">r.turner@auckland.ac.nz</a>
参考文献----------References----------
In O.E. Barndorff-Nielsen, W.S. Kendall and M.N.M. van Lieshout (eds) Stochastic Geometry: Likelihood and Computation. Chapman and Hall, 1998. Chapter 2, pages 37–78.
John Wiley and Sons, 1991.
Academic Press, 1983.
On estimators for the reduced second moment measure of point processes. Mathematische Operationsforschung und Statistik, series Statistics, 14, 63 – 71.
Modelling spatial patterns (with discussion). Journal of the Royal Statistical Society, Series B, 39, 172 – 212.
Cambridge University Press, 1988.
Stochastic geometry and its applications. 2nd edition. Springer Verlag.
Fractals, random shapes and point fields: methods of geometrical statistics. John Wiley and Sons.
参见----------See Also----------
localK to extract individual summands in the K function.
localK提取个别加数，，在K功能。
pcf for the pair correlation.
pcf对相关。
Fest, Gest, Jest for alternative summary functions.
Fest，Gest，Jest替代的汇总函数。
Kcross, Kdot, Kinhom, Kmulti for counterparts of the K function for multitype point patterns.
Kcross，Kdot，Kinhom，KmultiK功能下的多点模式的同行。
reduced.sample for the calculation of reduced sample estimators.
reduced.sample减少样本估计的计算。
实例----------Examples----------
pp <- runifpoint(50)
K <- Kest(pp)
data(cells)
K <- Kest(cells, correction="isotropic")
plot(K)
plot(K, main="K function for cells")
# plot the L function[绘制L功能]
plot(K, sqrt(iso/pi) ~ r)
plot(K, sqrt(./pi) ~ r, ylab="L(r)", main="L function for cells")
转载请注明:出自 生物统计家园网(http://www.biostatistic.net)。
注：
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发表于 2012-9-30 13:39:41
