Ripley′s指数的一个新变形--G(d)

A modified Ripley′s index

在对Ripley′s指数的物理背景进行分析的基础上提出了一个新的变形——G(d)指数。原始Ripley′s指数K(d)是一个半径为d(d为尺度)的圆的面积估计量,是有量纲的,且K(d)随着d的增大迅速增大,应用起来不太方便。Ripley′s指数的变形L(d)将K(d)换算成d的估计量,再减去d,使得L(d)在随机分布的假定下有数学期望0,这使得L(d)的应用比K(d)要方便。但L(d)还是一个具有长度单位的量,其上下包迹线呈明显的喇叭形状,对于应用还不是十分方便。提出的G(d)指数是一个比值,无量纲,数学期望也是0。文章给出了4个例子,这些例子说明,它保持了L(d)指数区分分布类型的能力,同时具有稳定这个良好特性,到了一定尺度以后,上下包迹线趋向常数。进一步分析得知,这些常数与单位面积的个体密度相关,呈对数关系,其相关系数r2达到0.9左右。这样在实际应用中,包迹线只要模拟到稳定点即可,余下部分可通过回归公式计算,从而节省计算工作量。

This paper proposes a new index （G（d） ） based on the Ripley＇s. The original Ripley＇s index K（d） （here d is a distance scale ） is an area estimate of a circle with radius d under the assumption that individuals have a Poisson distribution. The fact that an area estimate becomes larger when d gets larger makes it inconvenient for practical applications. The modified Ripley＇s index L（d） transforms K（d） to an estimate of d and then d is subtracted from the estimate. The difference has an expectation of zero under the assumption of Poisson distribution. Although the use of L（d） is more convenient than K（ d）, the L（d） is still a number with a distance measure and not very convenient for use because its upper and lower envelopes look like a bell-mouthed form. The G（d） index proposed in this paper is a ratio with an expectation of zero and with no dimension and is defined as G（d）=K（d）/（πd^2）-1 Three examples with simulations and one real application show that G（d） has the property of stability, which previous Ripley＇ s indices （ i. e. , L（d） and K（d） ） do not have and is able to distinguish different spatial patterns. When the scale d reaches a certain value, such as between 2 and 6 in most cases, the envelopes become constants. Further analyses show that there is a relationship between the constants and the density of individuals （ number of individuals per 100 m^2 ）. The relationship can be expressed as equations, which the stable values of upper - （SVU） and lower - （SVL） envelopes are as follows： SVU = 0. 152549 - 0. 0396694 in（density）, r2 = 0. 919 SVL = - 0.175449 ＋ 0.0610485 in（density）, r2= 0.883 The results provide a possibility that it is not necessary to calculate the envelopes beyond the stable point if G（d） is employed.