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<h1 class="title toc-ignore">Composite Index Construction</h1>
</div>
<div id="brief-explanation" class="section level2">
<h2>Brief Explanation</h2>
<p>An explanation of what indicators are and of the reason why they are widely used in companies and international organizations will be provided. The latest developments in methodology for the construction of indexes and indicators will be explained (economic activity, human development, poverty, finance, etc.). Special attention will be paid to their application in decision-making processes and, more precisely, to their usefulness in the context of forecasting. In particular, the methodology applied by international institutions in the construction of the most commonly known indexes will be explained, identifying and providing potential improvements. Students will solve a case study based on the real situation of an international institution using indicators.</p>
</div>
<div id="introduction" class="section level2">
<h2>Introduction</h2>
<p>A composite indicator measures multi-dimensional concepts (e.g. competitiveness, e-trade or environmental quality) which cannot be captured by a single indicator. Ideally, a composite indicator should be based on a theoretical framework / definition, which allows individual indicators / variables to be selected, combined and weighted in a manner which reflects the dimensions or structure of the phenomena being measure.</p>
<p><a href="https://www.youtube.com/watch?v=kz9g3d3xn6s">EU VIDEO</a></p>
</div>
<div id="some-examples" class="section level2">
<h2>Some Examples</h2>
<p>Some Composite Indicators:</p>
<ul>
<li>Global AgeWatch Index. <a href="compositeIndex/GAWI.pdf">GAWI.pdf</a></li>
<li>HDI. <a href="https://en.wikipedia.org/wiki/Human_Development_Index">Link Wiki</a> <a href="https://www.youtube.com/watch?v=w5wORaWcWPY">Australia VIDEO</a></li>
<li>IQ</li>
<li>Economic Risk Indicator. <a href="compositeIndex/Paper.pdf">Paper.pdf</a></li>
</ul>
<p>Another useful applications:</p>
<ul>
<li>Forecasting. <a href="compositeIndex/Forecast.png">Forecast.png</a></li>
<li>Stress Testing. Density Forecast.</li>
</ul>
<p>##Data <a href="compositeIndex/DataSet.pdf">DataSet.pdf</a></p>
<p><a href="compositeIndex/Income.csv">Income.csv</a></p>
<p><a href="compositeIndex/Index.csv">Index.csv</a></p>
</div>
<div id="missing-value-imputation" class="section level2">
<h2>Missing Value Imputation</h2>
<pre class="r"><code>options(warn = -1)
#Import Data
path <- "compositeIndex"
Data <- read.csv2(paste(path, "Income.csv", sep = "/"))
Data2 <- read.csv2(paste(path, "Index.csv", sep = "/"))
Data <- apply(Data, 2, as.numeric)
Data2 <- apply(Data2, 2, as.numeric)
Data_estimate<-data.frame(Data)
x1<-Data_estimate[,1]
x2<-Data_estimate[,2]
x3<-Data_estimate[,3]
x4<-Data_estimate[,4]
#
dif<-matrix(0,length(x3),1)
dif2<-matrix(0,length(x3),1)
Data_estimate<-data.frame(Data_estimate)
leaps<-regsubsets(x3~x1+x2+x4,data=Data_estimate,nbest=5)
plot(leaps,scale="r2")</code></pre>
<p><img src="compositeIndex_files/figure-html/unnamed-chunk-1-1.png" width="672" /></p>
<pre class="r"><code>fit <- lm(x3~x1+x2+x4)
for (i in 1:length(x3)) {
value_estimate=fit$coefficients[1]+fit$coefficients[2]*x2[i]+fit$coefficients[3]*x3[i]+fit$coefficients[4]*x4[i]
dif[i]<-abs(x3[i]-value_estimate)/x3[i]
}
for (i in 1:length(x3)) {
Data_estim<-Data
Data_estim[i,3]<-NA
Data_estimate<-data.frame(Data_estim)
a.out <- amelia(Data_estimate, m = 1, boot.type = "none")
unlist(a.out$imputations[[1]])
em_estimation<-as.matrix(a.out$imputations[[1]])
dif2[i]<-abs(x3[i]-em_estimation[i,3])/x3[i]
}
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Errors=cbind(dif,dif2)
ddf<-data.frame(Errors,GRP = c("Regression","EM"))
boxplot(Errors ~ GRP, main="Relative Welfare",data = ddf, lwd = 1, ylab = 'Errors')
stripchart(Errors~ GRP,vertical = TRUE, data = ddf,
method = "jitter", add = TRUE, pch = 20, col = c('blue','red'),cex=2)</code></pre>
<p><img src="compositeIndex_files/figure-html/unnamed-chunk-1-2.png" width="672" /></p>
<pre class="r"><code>check_value<-mean(dif[!is.na(dif)])
check_value2<-median(dif[!is.na(dif)])
check_value3<-sd(dif[!is.na(dif)])
CV<-c(check_value,check_value2,check_value3)
check_valueb<-mean(dif2[!is.na(dif2)])
check_valueb2<-median(dif2[!is.na(dif2)])
check_valueb3<-sd(dif2[!is.na(dif2)])
CVb<-matrix(c(check_valueb,check_valueb2,check_valueb3),1,3)
C<-matrix(cbind(CV,CVb),2,3)
barplot(C, main="Blue:Regression; Red: EM",
xlab="Mean, Median, StDesv", col=c("darkblue","red"),
beside=TRUE)</code></pre>
<p><img src="compositeIndex_files/figure-html/unnamed-chunk-1-3.png" width="672" /></p>
</div>
<div id="outlier-detection" class="section level2">
<h2>Outlier Detection</h2>
<pre class="r"><code>Data_estimate<-data.frame(Data)
x1<-Data_estimate[,1]
x2<-Data_estimate[,2]
x3<-Data_estimate[,3]
x4<-Data_estimate[,4]
#Univariate
outlier_values <- boxplot.stats(x3)$out # outlier values.
print(outlier_values)
## [1] 0.5
boxplot(x2, boxwex=1)</code></pre>
<p><img src="compositeIndex_files/figure-html/unnamed-chunk-2-1.png" width="672" /></p>
<pre class="r"><code>#Multivariate
#Cook DIstance
mod <- lm(x2 ~ ., data=Data_estimate)
cooksd <- cooks.distance(mod)
plot(cooksd, pch="*", cex=2, main="Influential Obs by Cooks distance") # plot cook's distance
abline(h = 4*mean(cooksd, na.rm=T), col="red") # add cutoff line
text(x=1:length(cooksd)+1, y=cooksd, labels=ifelse(cooksd>4*mean(cooksd, na.rm=T),names(cooksd),""), col="red")</code></pre>
<p><img src="compositeIndex_files/figure-html/unnamed-chunk-2-2.png" width="672" /></p>
<pre class="r"><code>#Mahalanobis Distance
Data2<-Data_estimate[,c(1:3)]
n.outliers <- 2 # Mark as outliers the 2 most extreme points
m.dist.order <- order(mahalanobis(Data2, colMeans(Data2), cov(Data2)), decreasing=TRUE)
is.outlier <- rep(FALSE, nrow(Data2))
is.outlier[m.dist.order[1:n.outliers]] <- TRUE
pch <- is.outlier * 16
col<- is.outlier + 1
plot3d(Data2[,1], Data2[,2], Data2[,3], type="s", col=col,xlab = "Pension",ylab="Poverty",zlab="Welfare")
</code></pre>
</div>
<div id="normalization" class="section level2">
<h2>Normalization</h2>
<pre class="r"><code>Data2<-Data
for (i in 1:4) {
Data2[,i]<-scale(Data2[,i],TRUE,TRUE)
}
Data3<-Data
for (i in 1:4) {
Data3[,i]<-(Data3[,i]-min(Data3[,i]))/(max(Data3[,i])-min(Data3[,i]))
}
Data4<-Data
for (i in 1:4) {
Data4[,i]<-SoftMax( Data4[,i], lambda = 2, avg = mean( Data4[,i], na.rm = T), std = sd( Data4[,i], na.rm = T))
}
RR<-cor(Data)
RR2<-cor(Data2)
RR3<-cor(Data3)
RR4<-cor(Data4)
dif2<-RR-RR2
dif3<-RR-RR3
dif4<-RR-RR4
check_value<-mean(dif2[!is.na(dif2)])
check_value2<-mean(dif3[!is.na(dif3)])
check_value3<-mean(dif4[!is.na(dif4)])
C<-matrix(c(check_value,check_value2,check_value3),1,3)
barplot(C, main="Changes in Correlation Matrix Structure",
xlab="Z-Score,Max-Min,Softmax",col="blue")</code></pre>
<p><img src="compositeIndex_files/figure-html/unnamed-chunk-3-1.png" width="672" /></p>
<pre class="r"><code>
par(mfrow=c(2,2))
hist(Data[,2], col=rgb(0.1,0,1,0.5),xlab="Original",main="Poverty Rate")
hist(Data2[,2], col=rgb(0.1,0,1,0.5),xlab="Standardization",main="Poverty Rate")
hist(Data3[,2], col=rgb(0.1,0,1,0.5),xlab="Max.Min",main="Poverty Rate")
hist(Data3[,2], col=rgb(0.1,0,1,0.5),xlab="SoftMax",main="Poverty Rate")</code></pre>
<p><img src="compositeIndex_files/figure-html/unnamed-chunk-3-2.png" width="672" /></p>
<pre class="r"><code>
Original<-matrix(0,99,1)
Zscore<-matrix(0,99,1)
Max.Min<-matrix(0,99,1)
softmax<-matrix(0,99,1)
z<-matrix(seq(0.01,0.99,0.01),99,1)
for ( i in c(1:99)) {
Original[i]<-quantile(Data[,2],z[i])
Zscore[i]<-quantile(Data2[,2],z[i])
Max.Min[i]<-quantile(Data3[,2],z[i])
softmax[i]<-quantile(Data4[,2],z[i])
}
dat<-data.frame(Original,Zscore)
dat2<-data.frame(Original,Max.Min)
dat3<-data.frame(Original,softmax)
ggplot(dat, aes(x=Original, y=Zscore)) +
geom_point(shape=1) + # Use hollow circles
geom_smooth(method=lm,level=0.9) </code></pre>
<p><img src="compositeIndex_files/figure-html/unnamed-chunk-3-3.png" width="672" /></p>
<pre class="r"><code>
ggplot(dat2, aes(x=Original, y=Max.Min)) +
geom_point(shape=1) + # Use hollow circles
geom_smooth(method=lm,level=0.9) </code></pre>
<p><img src="compositeIndex_files/figure-html/unnamed-chunk-3-4.png" width="672" /></p>
<pre class="r"><code>
ggplot(dat3, aes(x=Original, y=softmax)) +
geom_point(shape=1) + # Use hollow circles
geom_smooth(method=lm,level=0.9) </code></pre>
<p><img src="compositeIndex_files/figure-html/unnamed-chunk-3-5.png" width="672" /></p>
</div>
</div>
</div>
</div>
<script>
// add bootstrap table styles to pandoc tables
function bootstrapStylePandocTables() {