# Irescale v0.2.6

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## Calculate and Scale Moran's I

Provides a scaling method to obtain a standardized Moran's I measure. Moran's I is a measure for the spatial autocorrelation of a data set, it gives a measure of similarity between data and its surrounding. The range of this value must be [-1,1], but this does not happen in practice. This package scale the Moran's I value and map it into the theoretical range of [-1,1]. Once the Moran's I value is rescaled, it facilitates the comparison between projects, for instance, a researcher can calculate Moran's I in a city in China, with a sample size of n1 and area of interest a1. Another researcher runs a similar experiment in a city in Mexico with different sample size, n2, and an area of interest a2. Due to the differences between the conditions, it is not possible to compare Moran's I in a straightforward way. In this version of the package, the spatial autocorrelation Moran's I is calculated as proposed in Chen(2009) <arXiv:1606.03658>.

Many Geographical Analysis utilizes spatial autocorrelation, that allows us to study the geographical evolution from different points of view. One measurement for spatial autocorrelation is Moran's I, that is based on Pearson’s correlation coefficient in general statistics

## Performing the Analysis

This package offers a straight fordward to perform the whole analisys by using the function rescaleI which requires an input file with a specific format you can see it at [Loading data] section

{r whole_analysis} library(Irescale) fileInput<-system.file("testdata", "chen.csv", package="Irescale") data<-loadFile(fileInput) scaledI<-rescaleI(data,samples=1000, scalingUpTo="MaxMin") fn = file.path(tempdir(),"output.csv",fsep = .Platform$file.sep) saveFile(fn,scaledI) if (file.exists(fn)) #Delete file if it exists file.remove(fn)  ## Analysis Step by Step The analysis can be done following the steps ### Loading data The input file^[The data used in this example is taken from [@chen2009].] should have the following format. - The first column represents an unique id for the record. - The second and third column represent the latitute and longitud of where the sample was taken - The fourth and beyond represents the different measured variables {r} fileInput<-system.file("testdata", "chen.csv", package="Irescale") head(read.csv(fileInput))  To load data to performe the analysis is quite simple. The function loadFile provides the interface to make it. loadFile returns a list with two variables, data and varOfInterest, the first one represents a vector with latitude and longitude; varOfInterest is a matrix with all the measurements from the field. library(Irescale) fileInput<-system.file("testdata", "chen.csv", package="Irescale") input<-loadFile(fileInput) head(input$data)
head(input$varOfInterest)  If the data has a chessboard shape,the file is organized in rows and columns, where the rows represent latitute and columns longitude, the measurements are in the cell. The function loadChessBoard can be used to load into the analysis. library(Irescale) fileInput<-"../inst/testdata/chessboard.csv" input<-loadChessBoard(fileInput) head(input$data)
head(input$varOfInterest)  ### Calculate Distance Once the data is loaded, The distance matrix, the distance between all the points might be calcualted. The distance can be calculated using calculateEuclideanDistance' if the points are taken in a geospatial location. library(Irescale) fileInput<-system.file("testdata", "chen.csv", package="Irescale") input<-loadFile(fileInput) distM<-calculateEuclideanDistance(input$data)
distM[1:5,1:5]


If the data is taken from a chessboard a like field, the Manhattan distance can be used.

library(Irescale)
fileInput<-"../inst/testdata/chessboard.csv"
distM<-calculateManhattanDistance(input$data) distM[1:5,1:5]  ### Calculate Weighted Distance Matrix The weighted distance matrix can be calculated it using the function calculateWeightedDistMatrix, however it is not required to do it, because 'calculateMoranI' does it. library(Irescale) fileInput<-system.file("testdata", "chen.csv", package="Irescale") input<-loadFile(fileInput) distM<-calculateEuclideanDistance(input$data)
distW<-calculateWeightedDistMatrix(distM)
distW[1:5,1:5]


### Moran's I

It is time to calculate the spatial autocorrelation statistic Morans' I. The function calcualteMoranI, which requires the distance matrix, and the variable you want are interested on.

library(Irescale)
fileInput<-system.file("testdata", "chen.csv", package="Irescale")
distM<-calculateEuclideanDistance(input$data) I<-calculateMoranI(distM = distM,varOfInterest = input$varOfInterest)
I


### Resampling Method for I

The scaling process is made using Monte Carlo resampling method. The idea is to shuffle the values and recalculate I for at least 1000 times. In the code below, after resampling the value of I, a set of statistics are calculated for that generated vector.

library(Irescale)
fileInput<-system.file("testdata", "chen.csv", package="Irescale")
distM<-calculateEuclideanDistance(input$data) I<-calculateMoranI(distM = distM,varOfInterest = input$varOfInterest)
vI<-resamplingI(1000,distM, input$varOfInterest) # This is the permutation statsVI<-summaryVector(vI) statsVI  #### Plotting Distribution (Optional) To see how the value of I is distribuited, the method plotHistogramOverlayNormal provides the functionality to get a histogram of the vector generated by resampling with a theorical normal distribution overlay. library(Irescale) fileInput<-system.file("testdata", "chen.csv", package="Irescale") input<-loadFile(fileInput) distM<-calculateEuclideanDistance(input$data)
I<-calculateMoranI(distM = distM,varOfInterest = input$varOfInterest) vI<-resamplingI(1000,distM, input$varOfInterest) # This is the permutation
statsVI<-summaryVector(vI)
plotHistogramOverlayNormal(vI,statsVI, main=colnames(input$varOfInterest))  ### Rescaling I Once we have calculated the null distribution via resampling, you need to scale by centering and streching. The method iCorrection, return an object with the resampling vector rescaled, and all the summary for this vector, the new value of I is returned in a variable named newI library(Irescale) fileInput<-system.file("testdata", "chen.csv", package="Irescale") input<-loadFile(fileInput) distM<-calculateEuclideanDistance(input$data)
I<-calculateMoranI(distM = distM,varOfInterest = input$varOfInterest) vI<-resamplingI(1000,distM, input$varOfInterest) # This is the permutation
statsVI<-summaryVector(vI)
corrections<-iCorrection(I,vI)
corrections$newI  ## Calculate P-value In order to provide a significance to this new value, you can calculate the pvalue using the method calculatePvalue. This method requires the scaled vector, you get this vector,scaledData, the scaled I, newI and the mean of the scaledData. library(Irescale) fileInput<-system.file("testdata", "chen.csv", package="Irescale") input<-loadFile(fileInput) distM<-calculateEuclideanDistance(input$data)
I<-calculateMoranI(distM = distM,varOfInterest = input$varOfInterest) vI<-resamplingI(1000,distM, input$varOfInterest) # This is the permutation
statsVI<-summaryVector(vI)
corrections<-iCorrection(I,vI)
pvalueIscaled<-calculatePvalue(corrections$scaledData,corrections$newI,corrections$summaryScaledD$mean)
pvalueIscaled


## Stability Analysis

In order to determine how many iterations it is necessary to run the resampling method, it is possible to run a stability analysis. This function draw a chart in log scale (10^x) of the number of interations needed to achieve the stability in the Monte Carlo simulation.

fileInput<-system.file("testdata", "chen.csv", package="Irescale")
resultsChen<-buildStabilityTable(data=input, times=100, samples=1000, plots=TRUE)


## Functions in Irescale

 Name Description calculateWeightedDistMatrix Calculates a weighted representation of the distance matrix. standardizedByColumn Scales a matrix by column. saveFile Saves a report with important statistics to describe the sample. rescaleI Performs the rescale for all the variables in an input file. summaryVector Calculates statistic for the received vector. loadChessBoard Loads a chessboard or matrix alike input file. loadFile Loads a file with latitude, longitude and variable of interest calculateEuclideanDistance Given a 2D data structure, it calculates the euclidean distance among all the points. coor Transforms a x,y position in a cartesian plane into a position in a 1D array. calculateMoranI Calculates the Moran's I using the algorithm proposed by Chen chen2009Irescale. calculateManhattanDistance Calculates the manhattan distance. plotHistogramOverlayNormal Creates an overlay of the histogram of the data and the theorical normal distribution. calculatePvalue p-value calculation. resamplingI Calculates n permutations of the variable of interest to calculate n different I in order to create the $Null$ distribution. iCorrection Scaling process for Moran's I. convexHull Plots the convexhull polygon from the data (latitude, longitude), and calculates the center of the convexhull and its area. buildStabilityTable Finds how many iterations are necessary to achieve stability in resampling method. calculateDistMatrixFromBoard Calculates the distance in a chessboard-alike structure. No Results!