Sharp tools make good work
--The Analects of Confucius Geodetector Software for measure and attribution of spatial stratified
heterogeneity (SSH) |
1.
Introduction
Spatial Stratified Heterogeneity (SSH) refers to the phenomena that the within strata are more similar than the between strata. Examples are landuse types and climate zones in spatial data, seasons and years in time series, occupations, age groups, incomes strata. SH offers windows for human beings to understand the universe since Aristotle (384–322 BC). Geodetector, i.e. Geographical
Detector, is a statistical tool to measure SH and to explore the determinants of SH (Fig. 1):
(1) measure and find SH among data; (2) test the coupling between two
variables Y and X, according to their SHs, without
assumption of linearity of the association; and (3) investigate interaction between two
explanatory variables X1
and X2 to a response
variable Y, without any specific
form of interaction such as the assumed product in econometrics (Fig. 2).
Each of the tasks can be accomplished by the Geodetector q-statistic:
Fig. 1. Principle of Geodetector (The bottom map, the
color indicates the values of a population Y. The top map, the population Y is stratified into strata {h};
the terms “stratification” and “partition” are equivalent, can be either
classification or zonation. Between the two maps is the equation q(Y|{h}), in which the numerator is the
summation of the within strata variance and the denominator is the pooled
variance.) where N
and s2 stand for the number
of units and the variance of Y in a
study area, respectively; the population Y
is composed of L strata (h = 1, 2, …, L). The strata of Y
(red polygons in Fig.1) are a
partition of Y, either by itself h(Y)
or by an explanatory variable X
which is a categorical h(X). X should be stratified if it is a numerical variable, the number
of strata L might be 2-10 or more,
according to prior knowledge or a classification algorithm. [(N-L)q]/[(L-1)(1-q)] ~ F(L-1, N-L, g), where g is a non central parameter (Wang et
al 2016). The strata of Y (red
polygons in Fig.1) are a partition
of Y, either by Y itself or by an explanatory variable
X. X is a categorical variable or should be stratified if it is a
numerical variable. The number of strata L
might be 2-10 or more, according to prior knowledge or a classification
algorithm. The terms “stratified heterogeneity (SH)”, “stratification”,
“classification” and “partition” are equivalent. SH
can be either spatial (spatial stratified heterogeneity, SSH) or aspatial
such as time and any attributes. Interpretation of q value (Fig.1). The value of q is strictly within [0, 1]. (1) If Y is stratified by Y itself,
then q = 0 indicates that Y is absent of SH; q = 1 indicates that Y is SH perfectly; 100q% measures the degree of SH of Y. (2) If Y is stratified by an explanatory variable X, then q = 0 indicates
that there is no coupling between Y
and X; q = 1 indicates that Y
is completely determined by X; X explains 100q% of Y. Please notice
that the q-statistic measures the
association between X and Y, both linearly and nonlinearly. Geodetector q
statistic helps understand spatial confounding, sample bias and overfitting. (1)
Confounding arises if a global model was applied to a SH
population, leading to statistical insignificance. The problem can be simply
avoided if SH is identified (by Geodetector q statistic) then modelling in the strata, separately. (2)
A sample would be biased if a population is SH
and the sample do not cover all strata. The problem can be solved if SH is
identified (by Geodetector q
statistic) then apply bias remedy models such as Heckman regression and
Bshade method. (3)
Local models aim to overcome heterogeneity but
often suffer overfitting and too many parameters to interpret. The problems
can be avoided if modelling in strata or stratifying the outputs of a local
model then interpreting the stratified parameters. Functions of Geodetector: (1)
The risk detector maps response variable in strata: Y(X); (2)
The factor detector q-statistic
measures the degree of SH of a variable Y;
and the determinant power of an explanatory variable X of Y; (3)
The ecological detector identifies the difference of the impacts
between two explanatory variables X1
~ X2; (4)
The interaction detector reveals whether the risk factors X1 and X2 (and more X) have an
interactive influence on a response variable Y (Fig.2).
Fig.
2. Interaction between
explanatory variables X1 and X2 impacting on a response variable Y: q(Y|X1 |
2.
Tutorial
The Geodetector software was developed using
Excel and R, respectively. The tools are free of charge, freely downloadable,
and easy to use, and were designed without any GIS plug-in components and
with “one click” execution. Users can run the following demo, then simply
replace the demo data in the software using your own data, click Run and you
get results ! We henceforth describe Excel Geodetector software. R users can
download the R Geodetector software in the following section “Download of Geodetector Software and
Example Datasets”. As a demo, neural-tube birth defects (NTD) Y and suspected risk factors or their proxies Xs in villages are provided, including
data for the health effect layers “NTD prevalence” and environmental factor
layers, “elevation”, “soil type”, and “watershed”. Their field names are
defined as Y and X1, X2, X3 respectively. Step 1. Download the software and input your
data in Excel (1) Download the Excel
Geodetector software (In the following section “Software and Examples Data
Download”), one click to download any one of the three Examples, unzip the
downloaded file, you will find an Excel file (this is Geodetector software
with an Example dataset!) and double click the Excel file, Fig. 3 and Fig. 5 appear. Fig. 3
is the format of the input data for the Geodetector: each row denotes a
sample unit (e.g. a village); the 1st column record the response
variable Y; the 2nd and
following columns denote partitions of Y
or factors X, the latter were
partitioned according to the similarity within strata. (2) Input your data into
the Excel Geodetector software in the format of Fig. 3. Then go to Step 2.
Fig. 3. Input data in Excel and the execution interface (Note: Y is numerical; X MUST
be categorical, e.g. landuse types, seasons. If X is numerical it should be transformed to be categorical, e.g.
GDP per capita is stratified into 5 strata) (3)
If your data is in GIS format, as Fig. 4, please transform the GIS data into Excel data as Fig. 3.
Fig. 4. Data in GIS format Step 2. Run Geodetector
software Only one operation interface was designed (Fig. 5). The function of the “Read
Data” button is to load data; thus, when the button is clicked, all variables
are listed in the “variables” list box. Then, disease and partition of Y or environmental factor variables
are selected into their corresponding list boxes Y and X on the right of the interface.
Finally, Geodetector is executed by clicking the “Run” button.
Fig. 5. User interface for Geodetector back to the top ||
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3.
Output
Geodetector outputs
results from the risk detector, factor detector, ecological detector, and
interaction detector in four Excel spreadsheets (Fig. 6).
Fig. 6. Interface for Geodetector results In the “Risk detector”
sheet (Fig. 7), result information
for each environmental risk factor is presented in two tables. The first
table gives the average disease incidence in each stratum of a
risk factor, the name of which is written at the top left of the table. The
second table gives the statistically significant difference in the average
disease incidence between two strata; if there is a significant difference,
the corresponding value is “Y”, else it is “N”.
Fig. 7. Results of risk detector The
Fig. 8 shows the output format of
the q values for each environmental
risk factor, as given in the “Factor detector” sheet. The table header gives
the names of the environmental risk factors, while the associated q values (q1, q2,
…, qn) and their
corresponding p values are
presented in the row below.
Fig. 8. Results of factor detector In the “Ecological
detector” sheet (Fig. 9), results of the statistically
significant differences between two environmental risk factors are presented.
If Y(X1) (risk factor names in row) was significantly
bigger than Y(X2) (risk factor names in column), the associated
value is “Y”, while “N” expresses the opposite meaning.
Fig. 9. Results of ecological detector The format of the
results for the interaction detector is shown in Fig. 10. “Interaction
relationships” below the table represent the interaction relationship for
the two factors. The relationship is
defined in a coordinate axis. It has 5 intervals, including “(-∞，min(q(x), q(y)))”,“(min(q(x), q(y)),
max(q(x), q(y)))”, “(max(q(x), q(y)),
q(x) + q(y))”,“q(x)
+ q(y)”,“( q(x)
+ q(y),+∞)”, and the interaction
relationship is determined by the location of q(xÇy) in the 5 intervals (see Table 1).
Fig. 10. Results of interaction detector Tab.
1. Interaction between Explanatory Variables (Xs)
Legend
|
4.
Download, with Datasets
The software was developed using Excel 2007 and
R, respectively. It is completely free. You can click any one of the
following links to download the Geodetector software. The first three are
Geodetector software in Excel: (1) click one and unzip the file, an Excel
file appears; (2) click the Excel file to start the Geodetector, you may
exercise the demo data; then (3) input your own data to get your own results.
1:
Geodetector Software in Excel, enclosed an Example of a Disease Dataset 2: Geodetector Software in Excel, enclosed an
Example of a Toy Dataset 3: Geodetector
Software in Excel, enclosed an Example of a NDVI Dataset |
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