An information-fusion method to identify pattern of spatial heterogeneity for improving the accuracy of estimation

Abstract

While spatial autocorrelation is used in spatial sampling survey to improve the precision of the feature’s estimate of a certain population at area units, spatial heterogeneity as the stratification frame in survey also often have a considerable effect upon the precision. Under the context of increasingly enriched spatiotemporal data, this paper suggests an information-fusion method to identify pattern of spatial heterogeneity, which can be used as an informative stratification for improving the estimation accuracy. Data mining is major analysis components in our method: multivariate statistics, association analysis, decision tree and rough set are used in data filter, identification of contributing factors, and examination of relationship; classification and clustering are used to identify pattern of spatial heterogeneity using the auxiliary variables relevant to the goal and thus to stratify the samples. These methods are illustrated and examined in the case study of the cultivable land survey in Shandong Province in China. Different from many stratification schemes which just uses the goal variable to stratify which is too simplified, information from multiple sources can be fused to identify pattern of spatial heterogeneity, thus stratifying samples at geographical units as an informative polygon map, and thereby to increase the precision of estimates in sampling survey, as demonstrated in our case research.

Appendices

Appendix 1: The equation of k-nearest neighbor for a nominal variable

$$ \hat{f}(x_{i} ) \leftarrow {\mathop {\arg \max }\limits_{v\; \in \;V} }{\sum\limits_{i\; = \;1}^k {\delta (v,f(s_{i} \left| {s_{i} \in N(x_{i} )} \right.))} } $$

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where $\hat{f}(x_{i} )$ is the estimate of the grid unit x _i , V is the finite set {v ₁,…,v _s } of the band variable, k is the number of the nearest neighbors, f(s _i |s _i  ∈ N(x _i )) is the categorical or discrete value of the ith nearest neighbor unit, s _i belonging to the neighborhood of x _i , N(x _i ), δ(a,b) = 1 if a = b and δ(a,b) = 0 otherwise.

Appendix 2: A brief introduction to rough set

The rough set assumes that an information system consists of the 4-tuples S = 〈U, B, V, f〉 where U is a finite set of objects, i.e. each a gird unit, x _i, in our dataset, B is a finite set of attributes, i.e. each a band in the dataset, $V = {\bigcup\limits_{b \in B} {V_{b} } },$ V _b is the value domain of the band attribute, b and f: U × B → V is a total function such that f(x _i , b) ∈ V _b for every b ∈ B, x _i  ∈ U, called information function. Any pair (b,v), b ∈ B, v ∈ V _q is called descriptor in S. In rough set, those attributes used in classification are called the conditional variables, in fact, auxiliary variables X _k (k is the order no. of band variables) and the classification variable is the decisive variable, Y. Conditional variables are used to classify the object x _i in the system. If two objects in the dataset U, x _i and x _j have the relation, f(x _i ,b) = f(x _j ,b) for every b ∈ B, we call x _i and x _j indiscernible and all of such indiscernible objects composes a class set of the goal variable. Each conditional (auxiliary) variable has different levels of classifying the objects and the level is called significance of attribute (SA) in terms of the decisive variable. For more, please refer to Aleksander (1999), Komorowski et al. (1999) and Wang (2001).

Appendix 3: Modeling for the stratification survey of the cultivatable land’s areal proportion

Given

N :

is the number of all the aerial photos that cover the whole study region;

n :

is the number of the photo units sampled;

L :

is the number of stratums;

N _h :

is the total number of units in the hth stratum;

n _h :

is the number of units sampled for analysis in the hth stratum;

β _ih :

is the areal proportion of the cultivable land of the ith aerial photo in the hth stratum;

• 1. If a sample unit of aerial photo is overlapped by several polygons within different strata the samples will be separated into smaller sub-sample units within the strata. The areal proportion of the cultivatable land in the sub-sample unit remains same and each sample unit’s weight in the stratum is proportional to the unit’s total area: w _ih  = S _ih /S _h where S _ih is the area of the sample unit in the hth stratum and S _h is the total area of all the units sampled in the hth stratum.

• 2. Within each stratum, the units are randomly sampled according to the principle of SRS. However, the estimation equation with spatial proportion sampling is derived from Ripley (1981) and Wang et al. (2002).

  The sampling proportion:

  $$ f_{h} = n_{h} /N_{h} ; $$

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  The number of units sampled:

  $$ a = f_{h} N_{h} ; $$

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  The proportion:

  $$ \hat{\beta }_{h} (a) = {\sum\limits_{i = 1}^{n_{h} } {\beta _{{ih}} w_{{ih}} } } $$

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  The variance

  $$ \begin{aligned}{} \hat{\sigma }_{{\hat{\beta }_{h} (a)}} (n_{h} )^{2} & = E_{h} {\left[ {\hat{\beta }_{h} (a) - \beta _{h} (a)} \right]}^{2} = E{\left[ {\frac{1} {{n_{h} }}{\sum\limits_{a = 1}^{n_{h} } {n_{h} } }\beta _{h} (a)w_{h} (a) - \frac{1} {{N_{h} }}{\int\limits_{N_{h} } {(n_{h} \beta _{h} (a)w_{h} (a)\;{\text{d}}a} }} \right]}^{2} \\ {\text{ }} & = \frac{1} {{n_{h} }}\{ 1 - E_{h} [r(a - {a}\ifmmode{'}\else$'$\fi)]\} \hat{\sigma }^{2}_{{\hat{\beta }_{h} (N_{h} )}} = F(n_{h} )\hat{\sigma }^{2}_{{\hat{\beta }_{h} (N_{h} )}} \\ \end{aligned} $$

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  where β _h (a) = β _ah , w _h (a) = w _ah , and $F(n_{h} ) = (1/n_{h} )\{ 1 - E_{p} [r(a - {a}\ifmmode{'}\else$'$\fi)\left| R \right.]\} ;\,E_{h} [r(a - {a}\ifmmode{'}\else$'$\fi)\left| R \right.]$ is the expected value of the spatial correlation structure of the target variable in the study region, R (Rodriguez-Iturbe and Mejia 1974; Ripley 1981):

  $$ \hat{\sigma }_{{\hat{\beta }_{h} (a)}} (N_{h} ) \equiv {\sum\limits_{a = 1}^{N_{h} } {{\left\{ {{\left[ {\beta _{h} (a)w_{h} (a) - {\sum\limits_{a = 1}^{N_{h} } {\beta _{h} (a)w_{h} (a)} }} \right]}^{2} w_{h} (a)} \right\}}} } $$

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• 3. For the estimation of the population’s mean, Cochran’s equation is mainly referred to. The estimate of $\ifmmode\expandafter\bar\else\expandafter\=\fi{\beta }_{{{\text{STR}}}}$ is

  $$ \hat{\ifmmode\expandafter\bar\else\expandafter\=\fi{\beta }}_{{{\text{STR}}}} = \frac{{{\sum\limits_h^L {n_{h} \hat{\ifmmode\expandafter\bar\else\expandafter\=\fi{\beta }}_{h} } }}} {n} = {\sum\limits_{h = 1}^L {w_{h} \hat{\ifmmode\expandafter\bar\else\expandafter\=\fi{\beta }}_{h} } }\quad {\text{where}}\;n = n_{1} + n_{2} + \cdots + n_{L} $$

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  The sampling proportion in each stratum:

  $$ f_{h} = \frac{{n_{h} }} {n} = f = \frac{{{\sum\nolimits_{h = 1}^{n_{L} } {n_{h} } }}} {N} $$

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  The variance:

  $$ \hat{V}(\hat{\ifmmode\expandafter\bar\else\expandafter\=\fi{\beta }}) = {\sum\limits_{h = 1}^L {w^{2}_{h} \hat{V}_{h} (\hat{\ifmmode\expandafter\bar\else\expandafter\=\fi{\beta }})} } $$

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  where $\hat{V}_{h} (\hat{\ifmmode\expandafter\bar\else\expandafter\=\fi{\beta }})$ is the variance of the stratum h.