---
title: "Convergence monitoring"
author: "<br>Federico M. Stefanini, Nedka D. Nikiforova, Chiara Litardi, Eleonora Peruffo and Massimiliano Mascherini"
date: "`r Sys.Date()` <br><br> Index:"
output:   
  rmarkdown::html_vignette:
    fig_caption: yes
    toc: true
    number_sections: true
vignette: >
  %\VignetteIndexEntry{Convergence monitoring}
  %\VignetteEncoding{UTF-8}
  \usepackage[utf8]{inputenc}
  % \VignetteDepends{ggplot2,dplyr,eurostat,purrr,tibble,tidyr,formattable,kableExtra,caTools,gridExtra,knitr,magrittr,readr,readxl,utf8}
  %\VignetteEngine{knitr::rmarkdown}
editor_options:
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---
  
  






```{r setup, include = FALSE}
library(ggplot2)
library(dplyr)
#library(tidyverse)
library(eurostat)
library(purrr)
library(tibble)
library(tidyr)
library(formattable) 
library(kableExtra)
library(caTools)

library(convergEU)

knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>",
  fig.width = 7,
  fig.height = 5
)

```


<br><br><br>
The evaluation of convergence is 
important not only for determining 
the dynamic of member states in the EU
but also as a support to policy makers. 

The R package *convergEU* is a suite of functions
to download, clean and analyze some convergence
features.



In this document,  the package  *convergEU* 
is described and the main functionalities illustrated.




# Datasets on EU member states

Two types of sources are considered:
data produced by Eurofound, available without
and active Internet connection,
and Eurostat data that can be downloaded
on the fly, upon necessity from this package.



##  Locally accessible datasets

Some datasets are accessible from package  *convergEU* 
using the R function *data()*, for example :
```{r,eval=FALSE}
data("emp_20_64_MS",package = "convergEU")
head(emp_20_64_MS)
```

Eurofound datasets are
locally available within the  *convergEU* package, see:
```{r,eval=FALSE}
data(package = "convergEU")
```



A description of the above data is available 
by the R help, for example:
```{r,eval=FALSE}
help(emp_20_64_MS)

```

Eurofond local data are considered below:
```{r}
data(dbEurofound)
head(dbEurofound)
```
where variable names are:
```{r}
names(dbEurofound)
```
and time ranges in the interval:
```{r}
c(min(dbEurofound$time), max(dbEurofound$time))
```
and the dataset is not complete in such a time range
for all considered countries.

Further details on Eurofound dataset are available 
as follows (metainformation):
```{r}
data(dbEUF2018meta)
print(dbEUF2018meta,n=20,width=100)
```
**NOTE: within convergeEU package, Eurofound data are statically stored**.
Please update this package to have the most recent version of Eurofound data. 


The first step of an analysis is data preparation.
This amounts to choose a time  interval, an indicator and a set of countries
(MS, Member States), for example:
```{r}
convergEU_glb()$EU12$memberStates$codeMS
```
thus, selecting "lifesatisf" from the column "Code\_in\_database"
```{r}
myTB <- extract_indicator_EUF(
    indicator_code = "lifesatisf", #Code_in_database
    fromTime=2003,
    toTime=2016,
    gender= c("Total","Females","Males")[2],
    countries= convergEU_glb()$EU12$memberStates$codeMS
    )
  
myTB
```
which results in a complete dataset ready for further analysis.
**IMPORTANT:**  the analysis of convergence is performed on clean and imputed data, i.e.
 a tidy dataset in the format years by countries.
This means that the dataset must always have these characteristics:

If missing values are present, then imputation is required,
as described in the next sections.


Another illustrative example follows.
```{r}
print(dbEUF2018meta,n=20,width=100)
 
names(convergEU_glb())
myTB <- extract_indicator_EUF(
    indicator_code = "JQIintensity_i", #Code_in_database
    fromTime= 1965,
    toTime=2016,
    gender= c("Total","Females","Males")[1],
    countries= convergEU_glb()$EU27_2020$memberStates$codeMS
    )
  
print(myTB$res,n=35,width=250)
```

Imputation must take place before doing any analysis:
```{r,out.width="100%"}
myTBinp <- impute_dataset(myTB$res, timeName = "time",
                          countries=convergEU_glb()$EU27_2020$memberStates$codeMS,
                          tailMiss = c("cut", "constant")[2],
                          headMiss = c("cut", "constant")[2]) 
print(myTBinp$res,n=35,width=250)
```












## Metaresults and  missing values check


Several functions in *convergEU* package
return a list with metainformation,
that is three components: *res, msg, err*.
The first list component, *res*, is the actual result, 
if computed.
The second component, *msg* is a message
decorating the computed result,
possibly a warning.
The third component, *err*, is
an error message or a list of errors
when a result is not computed.
Below this behavior is illustrated
for function *check_data*.


The structure of the standard dataset is
a time by countries rectangular table.
All variables are quantitative.
The following function check for such features:
```{r}
check_data(emp_20_64_MS)
```
where the list component *res* is TRUE,
that is all checks are passed.


In case of qualitative variable or missing data
checks fail, for example if time is qualitative:
```{r}
tmp <-  emp_20_64_MS
tmp <-  mutate(tmp, time=factor(emp_20_64_MS$time))
check_data(tmp)
```
the *err* component explains what went wrong.

Similar errors are signaled if the dataset
is not complete:
```{r}
tmp <-  emp_20_64_MS 
tmp[3:6,1]<- NA
check_data(tmp)
```





## Imputation for artificially generated missing values in the  Eurofound database


Let's consider the following  indicator from the  Eurofound database:
```{r}

myTB <- extract_indicator_EUF(
    indicator_code = "exposdiscr_p", #Code_in_database
    fromTime=1966,
    toTime=2016,
    gender= c("Total","Females","Males")[1],
    countries= convergEU_glb()$EU12$memberStates$codeMS
    )
```
where missing value are  absent
```{r}
sapply(myTB$res,function(vx)sum(is.na(vx)))
```
thus an artificial dataset is built by  introducing
some missing values and  by taking further years for testing purposes:
```{r}
set.seed(1999)
myTB2 <- dplyr::bind_rows(myTB$res,myTB$res,myTB$res)
myTB2 <- dplyr::mutate(myTB2, time= seq(1975,2015,5))
for(aux in 3:14){
  myTB2[[aux]] <-   myTB2[[aux]] + c(runif(6,-2.5,2.5),0,0,0)
}
```


```{r}
myTB2[["BE"]][1:2] <-  NA
myTB2[["DE"]][8:9] <-  NA
myTB2[["IT"]][c(3,4, 6,7,8)] <-  NA
myTB2[["DK"]][6] <-  NA
myTB2
```

Now an imputation function may be called to prepare data for
calculations on convergence.
The two examples below differ about what to do with 
missing starting values.
```{r}
toBeProcessed <- c( "IT","BE", "DE", "DK","UK")
# debug(impute_dataset)

impute_dataset(myTB2, countries=toBeProcessed,
                            timeName = "time",
                            tailMiss = c("cut", "constant")[1],
                            headMiss = c("cut", "constant")[1]) 

impute_dataset(myTB2, countries=toBeProcessed,
                            timeName = "time",
                            tailMiss = c("cut", "constant")[2],
                            headMiss = c("cut", "constant")[1]) 

```
The above calculations passed numerical tests and comparisons.
If a country is processed but it has no missing, then no numerical value change.


# On Convergence
 
Several measures of convergence have been recently 
proposed by Eurofound 
(Eurofound (2018), Upward convergence in the EU: Concepts, measurements and indicators, Publications Office of the European Union, Luxembourg;
by: Massimiliano Mascherini, Martina Bisello, Hans Dubois and Franz Eiffe)

In this section each each measure is considered by one or more examples.  





## Beta-convergence

Let's assume we have a dataset (tibble) of sorted times by countries values.
The calculations are performed according to the  following linear model:
$$
 ln(y_{m,i,t+\tau})-ln(y_{m,i,t}) = \beta_0 + \beta_1 ln(y_{m,i,t}) +\epsilon_{m,i,t}
$$
where $m$ represent the member state of EU (country), $i$ refers to an indicator
of interest, $t$ is the reference time and $\tau \in \{1,2,\ldots\}$
the length of the time window (typically $1$ or more years).

In the simplest case, just two time values are considered, $t$ and $t+\tau$,
while in a more  general setup all observed times in set
$\{t,t+1,\ldots,t+\tau-1, t+\tau\}$ are included into regression.
<br>

<br> 
In this more general case,
the current implementation of beta-convergence function
always maintain the same reference time across different years and
it  divides the left hand side by the amount of time elasped as an option,
that is the alternative formula:
$$
 \tau^{-1}(ln(y_{m,i,t+\tau})-ln(y_{m,i,t})) = \beta_0 + \beta_1 ln(y_{m,i,t}) +\epsilon_{m,i,t}
$$
is available.
<br> 


The output of *beta_conv()* is a list
in which transformed data, the point estimate of $\beta_1$
and a standard two tails test is reported (p-value and adjusted R squared).
One tail test $H_0: \beta_1 \geq 0$ against $H_1: \beta1< 0$
might be of some interest, but it is not implemented.


Below an example on how to invoke the function:
```{r}
#library(ggplot2)
#library(dplyr)
#library(tibble)

testTB <- tribble(
  ~time, ~countryA ,  ~countryB,  ~countryC,
    2000,     0.8,   2.7,    3.9,
    2001,     1.2,   3.2,    4.2,
    2002,     0.9,   2.9,    4.1,
    2003,     1.3,   2.9,    4.0,
    2004,     1.2,   3.1,    4.1,
    2005,     1.2,   3.0,    4.0
  )
 
res <- beta_conv(tavDes = testTB, time_0 = 2002, time_t = 2004, 
                 all_within = TRUE, 
                 timeName = "time")
res
```
but note that this is not the common practice,
which considers the first and last time instead.


In order to consider just two times, starting and ending times,
the option *all_within = FALSE* must be specified
```{r}
res <- beta_conv(tavDes = testTB, time_0 = 2002, time_t = 2004, 
                 all_within = FALSE, 
                 timeName = "time")
res

```
Note that *all_within = FALSE* is the default.




## Sigma-convergence


The key concept in sigma-convergence is variability  with respect to the mean.
Let $Y_{m,i,t}$ be the value of indicator $i$ for member state $m$ at time $t$,
and $\overline{Y}_{A,i,t}$ the average over aggregation $A$, for example $A = EU27_2020$,
than:    

  * the average is  $\overline{Y}_{A,i,t} = n(A)^{-1}\sum_{m \in A} Y_{m,i,t}$,
    where $n(A)$ is the number of member states within aggregation $A$;       
  * the standard deviation is $s_{A,i,t} = \sqrt(n(A)^{-1} \sum_{m\in A} (Y_{m,i,t} - \overline{Y}_{A,i,t})^2)$;     
  * the coefficient of variation is $CV(A,i,t) = 100\cdot \frac{s_{A,i,t}}{\overline{Y}_{A,i,t}}$.     

For each year, the above summaries are calculated to quantify if a reduction in heterogeneity took place.



In this section we assume that all member states
contributing to the unweighted mean are contained into the dataset,
for example:
```{r}
testTB <- tribble(
  ~time, ~countryA ,  ~countryB,  ~countryC,
    2000,     0.8,   2.7,    3.9,
    2001,     1.2,   3.2,    4.2,
    2002,     0.9,   2.9,    4.1,
    2003,     1.3,   2.9,    4.0,
    2004,     1.2,   3.1,    4.1,
    2005,     1.2,   3.0,    4.0
  )

sigma_conv(testTB,timeName="time")
```

It is possible to select a time window, as follows:
```{r}
sigma_conv(testTB,timeName="time",time_0 = 2002,time_t = 2004)
sigma_conv(testTB,time_0 = 2002,time_t = 2004)

```





More interesting calculations deal with an Eurofound
dataset *emp_20_64_MS*.
Note that all and only countries in EU28 are included,
those that contribute to the average:
```{r}
data(emp_20_64_MS)
mySTB <- sigma_conv(emp_20_64_MS)
mySTB
```


As a first step, the departure from the mean 
is characterized 
```{r}
res <- departure_mean(oriTB = emp_20_64_MS, sigmaTB = mySTB$res)
names(res$res)
res$res$departures
```
where $-1,0,1$ indicates values respectively below $-1$,
within the interval $(-1,1)$ and above $+1$.
Details  on the contribution of each MS to the variance 
at a given time $t$ is evaluate by the square of the difference
$(Y_{m,i,t} - \overline{Y}_{EU27,i,t})^2$ 
between the indicator $i$ of country $m$ at time $t$
and the unweighted average over member states, say EU27:
```{r}
res$res$squaredContrib
```


It is also possible to decompose the numerator of the variance, called deviance,
at each time in order to appreciate the  percentage of 
contribution provided by each member state to the  total deviance,
$$100 \cdot \frac{(Y_{m,i,t} - \overline{Y}_{EU27,i,t})^2}{
 \sum_{m}  (Y_{m,i,t} - \overline{Y}_{EU27,i,t})^2
}$$ 
for the indicator $i$ of country $m$ at time $t$.

```{r}
##  sigma_conv(testTB,timeName="time",time_0 = 2002,time_t = 2004)
res$res$devianceContrib
```
thus each row adds to $100$.


It is possible  to produce a graphical output about the
main features of country time series, as shown below:
```{r,eval=T,fig.width=7,fig.height=9}
myGG <- graph_departure(res$res$departures,
                timeName = "time",
                displace = 0.25,
                displaceh = 0.45,
                dimeFontNum = 4,
                myfont_scale = 1.35,
                x_angle = 45,
                color_rect = c("-1"='red1', "0"='gray80',"1"='lightskyblue1'),
                axis_name_y = "Countries",
                axis_name_x = "Time",
                alpha_color = 0.9
                )
myGG

```


Any selection of countries is feasible:
```{r,eval=T}
#myWW1<- warnings()
myGG <- graph_departure(res$res$departures[1:10],
                timeName = "time",
                displace = 0.25,
                displaceh = 0.45,
                dimeFontNum = 4,
                myfont_scale = 1.35,
                x_angle = 45,
                color_rect = c("-1"='red1', "0"='gray80',"1"='lightskyblue1'),
                axis_name_y = "Countries",
                axis_name_x = "Time",
                alpha_color = 0.29
                )

myGG
```

 

 
 
 
## Gamma-convergence

We now introduce gamma convergence by an index
based on ranks.


Let $y_{m,i,t}$ be the value of indicator $i$ for member state $m$ at time
$t=0,1,\ldots, T$,
and $\{ \tilde{y}_{m,i,t}: m \in A )$ the ranks for indicator $i$
over member states in the  reference set $A$, for example $A = EU27$,
at a given time $t$.
The sum of ranks within member state $m$ is:
$$
 \tilde{y}^{(s)}_{m,i} = \sum_{t=0}^T  \tilde{y}_{m,i,t}
$$
thus the variance of the sum of ranks over the given interval 
$$
Var\left[ \{\tilde{y}^{(s)}_{m,i}: m \in A \} \right]
$$
may be compared to the
variance of ranks in the reference time $t=0$:
$$
Var\left[ \{\tilde{y}_{m,i,0}: m \in A \} \right]
$$


The Kendall index KI, with respect to aggregation $A$ 
of member states for the indicator  $i$ over a
given time interval is:
$$
KI(A,i,T) =  \frac{Var\left[ \{\tilde{y}^{(s)}_{m,i}: m \in A \} \right]
      }{
      (T+1)^2  ~~Var\left[\{\tilde{y}_{m,i,0}: m \in A \}\right] }
$$



The measure of  gamma-convergence is obtained with the following function:
```{r}
gamma_conv(emp_20_64_MS,2002,2016)
```










Note the starting time is zero, the reference,
but first a copy of the dataset is performed.
```{r}
(timeCounTB <- testTB)
```

Now we move to ranks within time using *rank()*:
```{r}
tmp <- c( 3, 6, 9, 1, 12)
rank(tmp)
```
therefore with the above data:
```{r}
# debug(gamma_conv)
(gamma_conv(timeCounTB,ref=2000,last=2005,timeName = "time"))
(gamma_conv(timeCounTB,ref=2000,last=2004,timeName = "time"))
(gamma_conv(timeCounTB,ref=2000,last=2003,timeName = "time"))
(gamma_conv(timeCounTB,ref=2000,last=2002,timeName = "time"))
(gamma_conv(timeCounTB,ref=2000,last=2001,timeName = "time"))
```

and changing reference year:
```{r}
(gamma_conv(timeCounTB,ref=2001,last=2005,timeName = "time"))
(gamma_conv(timeCounTB,ref=2002,last=2004,timeName = "time"))
```


Now we exchange values and calculate gamma-convergence:
```{r}
timeCounTB2 <- timeCounTB
timeCounTB2[2,2:4] <-  timeCounTB[2,4:2]
timeCounTB2[4,2:4] <-  timeCounTB[4,c(4,2,3)]
timeCounTB2

gamma_conv(timeCounTB2,last=2005,ref=2000, timeName = "time",printRanks = T)
```
and after random permutation:
```{r}
timeCounTB3 <- cbind(timeCounTB[1],t(apply(timeCounTB,1,
                                        function(vet)vet[sample(2:4,3)])))


timeCounTB3
(gamma_conv(timeCounTB3,last=2005,ref=2000, timeName = "time",printRanks = T))
```















## Delta-convergence


Delta-convergence can be calculated as follows:

```{r,echo=FALSE,eval=FALSE}

timeCounTB <- tribble(
  ~time, ~countryA ,  ~countryB,  ~countryC,
    0,     0.8,   2.7,    3.9,
    1,     1.2,   3.2,    4.2,
    2,     0.9,   2.9,    4.1,
    3,     1.3,   2.9,    4.0,
    4,     1.2,   3.1,    4.1,
    5,     1.2,   3.0,    4.0
  )
timeCounTB
```

```{r}
delta_conv(timeCounTB)
```










## Absolute change

Absolute change as described in the reserved Eurofound Annex
is defined as:
$$
\Delta y_{m,i,t} = y_{m,i,t} - y_{m,i,t-1}
$$
for country $m$, indicator $i$ at time $t$.

The R function *abso_change* calculates  the above quantity,
for example in the *emp_20_64_MS* dataset
```{r}
data(emp_20_64_MS)
mySTB <- abso_change(emp_20_64_MS, 
                        time_0 = 2005, 
                        time_t = 2010,
                        all_within=TRUE,
                        timeName = "time")
names(mySTB$res)
```
thus the above equation results in:
```{r}
mySTB$res$abso_change
```
The sum of absolute values
$$
\sum_{t=t_0+1}^{} | \Delta y_{m,i,t}|  
$$
is:
```{r}
round(mySTB$res$sum_abs_change,4)
```
and such sum can be divided by the number of pair of years
so that the result is an average per pair of years:
```{r}
round(mySTB$res$average_abs_change,4)
```















## Convergence measures  on Eurofound lifesatisf indicator

Here we assume that larger the index, better the performance.

Let's load the Eurofound indicator *lifesatisf*:
```{r}
workDF <- extract_indicator_EUF(
  indicator_code ="lifesatisf", #Code_in_database
  fromTime=2000,
  toTime =2018,
  gender= c("Total","Females","Males")[1],
  countries =  convergEU_glb()$EU27_2020$memberStates$codeMS)
workDF

wDF <- workDF$res
```
then we ask if it is complete or some missing values are present:
```{r}
check_data(select(wDF,-sex),timeName="time")
```
thus at least one missing value is present.
In the next step, imputation of missing values is performed:
```{r}
wDFI <- impute_dataset(select(wDF,-sex),
               countries= names(select(wDF,-sex,-time)),
               timeName = "time",
               tailMiss = c("cut", "constant")[2],
               headMiss = c("cut", "constant")[1])
```
and some checking is done:
```{r}
check_data(wDFI$res,timeName="time")
```
which returns TRUE.


First, we calculate the EU unweighted average of emp:
```{r}
wwTB <- (wDFI$res %>%
   average_clust(timeName="time",cluster="EU27"))$res

wwTB$EU27
```

Time series can be plotted: 
```{r}
mini_EU <- min(wwTB$EU27)
maxi_EU <- max(wwTB$EU27)

qplot(time, EU27, data=wwTB,
      ylim=c(mini_EU,maxi_EU))+geom_line(colour="navy blue")+
      ylab("lifesatisf")
```



### Beta convergence

Now the beta-convergence is calculated for just two years:
```{r}
betaRes <- beta_conv(wDFI$res,time_0=2007, time_t=2011, all_within=FALSE)
betaRes 
```


<!-- 
now more years are considered
{r}
betaRes <- beta_conv(wDFI$res,time_0=2007, time_t=2016, all_within=TRUE)
-->




A plot of transformed data and the straight line may be useful:
```{r,out.width="100%"}
mybetaplot<-beta_conv_graph(betaRes,
                            indiName = 'Mean Life Satisfaction',
                            time_0 = 2007,
                            time_t = 2011)
mybetaplot
```
Note that label are replicated as many times as 
the number of included  subsequent years. 







### Sigma  convergence


Here we go with calculating the sigma-convergence:
```{r}
mysigmares<-sigma_conv(wwTB)
#mysigmares
```
It is also possible to obtain a graphical representation of the standard deviation and the coefficient of variation obtained for the Sigma convergence by invoking the *sigma_conv_graph* function as follows:

```{r,fig.width=5,fig.height=4,out.width="65%"}
mysigmaplot<-sigma_conv_graph(sigmaconvOut=mysigmares, 
         time_0 = 2007, 
         time_t = 2011,
        aggregation='EU27_2020')
mysigmaplot
```







### Gamma convergence


Let's reload Eurofound data:
```{r}
workDF <- extract_indicator_EUF(
  indicator_code ="lifesatisf", #Code_in_database
  fromTime=2000,
  toTime =2018,
  gender= c("Total","Females","Males")[1],
  countries =  convergEU_glb()$EU27_2020$memberStates$codeMS)
wDFI <- impute_dataset(select(workDF$res,-sex),
               countries= names(select(wDF,-sex,-time)),
               timeName = "time",
               tailMiss = c("cut", "constant")[2],
               headMiss = c("cut", "constant")[1])

check_data(wDFI$res,timeName="time")
```
 
 
 
 

Now  gamma-convergence is computed:
```{r}
gamma_conv(wDFI$res,ref=2003,last=2016,timeName = "time")
```
or equivalently:
```{r}
tmpRes <- gamma_conv(wDFI$res,ref=2007,last=2011,timeName = "time")
```


Indeed there is the possibility of performing calculation
for each pair of subsequent years in the dataset,
that is, each year is the reference of the subsequent year:
```{r}
wDFI$res
```

```{r}
gamma_conv_msteps(wDFI$res,
                  startTime=2003, 
                  endTime=2016,
                  timeName = "time")

```





### Delta convergence


Let $y_{m,i,t}$ be the value of indicator $i$ for member state $m$ at time $t$,
and $y^{(M)}_{i,t}$ the maximum value  over member states in the 
reference set $A$, for example $A = EU27$:
$$
y^{(M)}_{i,t} = max(\{ y_{m,i,t}: m \in A\})
$$

The distance of a member state $m$ from the top performer at time $i$ is:
$$
y^{(M)}_{i,t} - y_{m,i,t}
$$
thus the overall distance at time $t$, called delta, is the sum of distances over the 
reference set  $A$ of MS:
$$
\delta_{i,t} = \sum_{m \in A} (y^{(M)}_{i,t} -  y_{m,i,t})
$$
for the considered indicator $i$.



The measure of  delta-convergence is obtained as follows:
```{r}
delta_conv(wwTB)
```


It must be noted that the *delta_conv* function allows to obtain also the declaration of convergence. To this end, the argument *extended* should be specified as *TRUE*. For example, for the *wwTB* indicator the syntax is as follows:
```{r}
delta_conv(wwTB,"time", extended=TRUE)
```

It is also useful to evaluate how much a collection of **MS** deviates from the **EU** mean for a given indicator and a period of time. In order to obtain this further information the *demea_change* function has been implemented in the *convergEU* package:
```{r}
res1<-demea_change(wwTB,
                   timeName="time",
                   time_0 = 2003,
                   time_t = 2016,
                   sele_countries= NA,
                   doplot=TRUE)
res1
```

To plot the calculated differences, the user should invoke the *plot* function as follows:
```{r,fig.width = 6,out.width="100%"}
plot(res1$res$res_graph)
```









# Support functions

There are several auxiliary functions that help to prepare
the tidy dataset time by member states (MS, that is countries
in EU),
which is needed in almost all computations.
Here the most important resources are described.  






## Summaries and clusters of countries

An important summary is obtained  
as  unweighted average of country values.
The cluster of considered countries may be specified
and is also stored within the function generating global
static objects and tables, called
*convergEU_glb()*.
The  illustration of this function exploits 
the *emp_20_64_MS* dataframe in *convergEU*
package.

First note that the EU area is made by the following
MS:
```{r}
convergEU_glb()$Eurozone
```
while labels representing the 28 MS are:
```{r}
convergEU_glb()$EU27_2020
```

The list of known MS labels is shown in the appendix.


For example, the unweighted average
in the *emp_20_64_MS* dataset is:
```{r}
testTB <- emp_20_64_MS
average_clust(testTB,timeName = "time",cluster = "EU27")$res[,c(1,30)]
```
while for EU12 is:
```{r}
average_clust(testTB,timeName = "time",cluster = "EU12")$res[,c(1,30)]
```



An unknown label, like "EUspirit", causes computation error:
```{r}
average_clust(testTB,timeName = "TTime",cluster = "EUspirit")
```





## Imputing missing values using a straight line

The basic imputation  method is deterministic, like the average of 
interval endpoints, but it assumes that a linear
change of an indicator  happened between the two 
observed time points flanking a chunk of missing values.

```{r,out.width="65%"} 
intervalTime <-  c(1999,2000,2001) 
intervalMeasure <- c( 66.5, NA,87.2) 
currentData <- tibble(time= intervalTime, veval= intervalMeasure) 
currentData 
resImputed <- impute_dataset(currentData,
                           countries = "veval",
                           timeName = "time",
                           tailMiss = c("cut", "constant")[2],
                           headMiss = c("cut", "constant")[2]) 
resImputed  
``` 
 
 
```{r,echo=FALSE,out.width="65%"} 
tmp <-  as.data.frame(currentData[ c(1,3),] )
tmp2 <- as.data.frame(resImputed$res[2,] )
 
myg <- ggplot(as.data.frame(resImputed$res),  mapping=aes(x=time,y=veval)) + 
  geom_point() + 
  geom_line(data=resImputed$res,col="red") + 
  geom_point(data=tmp,mapping=aes(x=time,y=veval), 
              size=4, 
              colour="blue")  + 
  geom_point(data= tmp2, 
             aes(x=time,y=veval),size=4,alpha=1/3,col="black") + 
  xlab("Time") + ylab("Measure / Index") +  
  ggtitle( "Blue points are observed values (grey ones are missing) \n") 
   
myg 
``` 
 
 
If several missing values are present in a  row 
```{r} 
intervalTime <-  c(1999,2000,2001,2002,2003) 
intervalMeasure <- c( 66.5, NA,NA,NA,87.2) 
currentData <- tibble(time= intervalTime, veval= intervalMeasure) 
currentData
resImputed <- impute_dataset(currentData,
                           countries = "veval",
                           timeName = "time",
                           tailMiss = c("cut", "constant")[2],
                           headMiss = c("cut", "constant")[2]) 
tmp <-  as.data.frame(currentData[ c(1,5),] )
tmp2 <- as.data.frame(resImputed$res[2:4,] )

resImputed  
``` 
 
 
```{r,echo=FALSE,out.width="65%"} 
myg <- ggplot(as.data.frame(resImputed$res),  mapping=aes(x=time,y=veval)) + 
  geom_point() + 
  geom_line(data=resImputed$res,col="red") + 
  geom_point(data=tmp,mapping=aes(x=time,y=veval), 
              size=4, 
              colour="blue")  + 
  geom_point(data= tmp2, 
             aes(x=time,y=veval),size=4,alpha=1/3,col="black") + 
  xlab("Time") + ylab("Measure / Index") +  
  ggtitle( "Blue points are observed values (grey ones are missing) \n") 
   
myg 

``` 
 
 
 
 







## Weighted average smoothing of a complete dataset


It may be of interest to assume that part of the variability 
observed in a country on a given index is **not structural**,
i.e. not due to causal determinants by to transient
fluctuations.
Furthermore, the interest here is not directed towards
prediction but on smoothing values observed in the whole
considered time interval.

In such a case a smoothing procedure remove sudden large changes 
showing a less variable time serie than the original.

Given that here short time series (panel data) are considered,
a three points weighted average is proposed.
The smoother substitutes an original raw value $y_{m,i,t}$ of country $m$
indicator $i$ at time $t$ with the weighted average
$$\check{y}_{m,i,t}  = y_{m,i,t-1} ~ (1-w)/2   +w ~y_{m,i,t} +y_{m,i,t+1} ~(1-w)/2$$
where $0< w \leq 1$. The special case $w=1$ corresponds to no smoothing.
In case of missing values an NA is returned. If the weight is outside
the interval $(0,1]$ then a NA is returned.
The first and last values are smoothed using weights $w$ and $1-w$.

After loading data, imputation takes place and finally smoothing is performed.
Now, countries IT and DE are considered to illustrate the procedure.
First check if missing values are present:
```{r}
workTB <- dplyr::select(emp_20_64_MS, time, IT,DE)
check_data(workTB)
```
thus checking is passed, so we go with the smoothing step
after deleting the time variable:
```{r}
resSM <- smoo_dataset(select(workTB,-time), leadW = 0.149, timeTB= select(workTB,time))
resSM
```
and for a comparison:
```{r}
tmpSM <- dplyr::rename(dplyr::select(resSM,-time),IT1=IT,DE1=DE)
compaTB <- dplyr::select(bind_cols(workTB, tmpSM), time,IT,IT1,DE,DE1)
compaTB
```

A graphical output shows changes for "IT", with original
index in blue and smoothed index in red:
```{r,out.width="70%"}
qplot(time,IT,data=compaTB) + 
  geom_line(colour="navyblue") +
  geom_line(aes(x=time,y=IT1),colour="red") +
  geom_point(aes(x=time,y=IT1),colour="red",shape=8)
``` 

Similarly for Germany, i.e. "DE":  

```{r,out.width="70%"}
qplot(time,DE,data=compaTB) + 
  geom_line(colour="navyblue") +
  geom_line(aes(x=time,y=DE1),colour="red") +
  geom_point(aes(x=time,y=DE1),colour="red",shape=8)

```

A weight equal to 1 leaves data unchanged:
```{r,out.width="70%"}
resSM <- smoo_dataset(dplyr::select(workTB,-time), leadW = 1,
                      timeTB= dplyr::select(workTB,time))
resSM <- dplyr::rename(resSM,IT1=IT, DE1=DE)
compaTB <- dplyr::select(dplyr::bind_cols(workTB, 
                     dplyr::select(resSM,-time)), time,IT,IT1,DE,DE1)
qplot(time,IT,data=compaTB) + 
  geom_line(colour="navyblue") +
  geom_line(aes(x=time,y=IT1),colour="red") +
  geom_point(aes(x=time,y=IT1),colour="red",shape=8)
```


**A time window larger than $3$ could be considered, but
deep thoughts  are recommended on how much economic and social changes
may happen in $5$ consecutive years.**









## Moving Average smoother 


Several alternative smoothing algorithm are available in R.
Classical *ma* smoothers are also available from the *caTools* package.


The emp_20_64_MS dataset is now chosen for example, first with
Italy and then
with Germany as member states of interest.

```{r}
data(emp_20_64_MS)
cuTB <- dplyr::tibble(ITori =emp_20_64_MS$IT)
cuTB <- dplyr::mutate(cuTB,time =emp_20_64_MS$time)
```

At the beginning and end of this series values are averages on
smaller and smaller number of observations on the tails:
```{r}

cuTB <-  dplyr:: mutate(cuTB, IT_k_3= caTools::runmean(emp_20_64_MS$IT, k=3, 
        alg=c("C", "R", "fast", "exact")[4],
        endrule=c("mean", "NA", "trim", "keep", "constant", "func")[4],
        align = c("center", "left", "right")[1]))

cuTB <-  dplyr:: mutate(cuTB, IT_k_5= caTools::runmean(emp_20_64_MS$IT, k=5, 
        alg=c("C", "R", "fast", "exact")[4],
        endrule=c("mean", "NA", "trim", "keep", "constant", "func")[4],
        align = c("center", "left", "right")[1]))

cuTB <-  dplyr:: mutate(cuTB, IT_k_7= caTools::runmean(emp_20_64_MS$IT, k=7, 
        alg=c("C", "R", "fast", "exact")[4],
        endrule=c("mean", "NA", "trim", "keep", "constant", "func")[4],
        align = c("center", "left", "right")[1]))

```


```{r}
myG <- ggplot(cuTB,aes(x=time,y=ITori))+geom_line()+geom_point()+
       geom_line(aes(x=time,y=IT_k_3),colour="red")+
       geom_point(aes(x=time,y=IT_k_3),colour="red")+
       #
       geom_line(aes(x=time,y=IT_k_5),colour="blue")+
       geom_point(aes(x=time,y=IT_k_5),colour="blue")+
       #
       geom_line(aes(x=time,y=IT_k_7),colour="orange")+
       geom_point(aes(x=time,y=IT_k_7),colour="orange")+
       theme(legend.position = c(.5, .5),
              legend.title = element_text(face = "bold"))

myG
```



For Germany, a similar implementation provides the following result:

```{r}
cuTB <- dplyr::mutate(cuTB, DEori =emp_20_64_MS$DE)

cuTB <-  dplyr:: mutate(cuTB, DE_k_3= runmean(emp_20_64_MS$DE, k=3, 
        alg=c("C", "R", "fast", "exact")[4],
        endrule=c("mean", "NA", "trim", "keep", "constant", "func")[4],
        align = c("center", "left", "right")[1]))

cuTB <-  dplyr:: mutate(cuTB, DE_k_5= runmean(emp_20_64_MS$DE, k=5, 
        alg=c("C", "R", "fast", "exact")[4],
        endrule=c("mean", "NA", "trim", "keep", "constant", "func")[4],
        align = c("center", "left", "right")[1]))

cuTB <-  dplyr:: mutate(cuTB, DE_k_7= runmean(emp_20_64_MS$DE, k=7, 
        alg=c("C", "R", "fast", "exact")[4],
        endrule=c("mean", "NA", "trim", "keep", "constant", "func")[4],
        align = c("center", "left", "right")[1]))

```


```{r}
myG <- ggplot(cuTB,aes(x=time,y=DEori))+geom_line()+geom_point()+
       geom_line(aes(x=time,y=DE_k_3),colour="red")+
       geom_point(aes(x=time,y=DE_k_3),colour="red")+
       #
       geom_line(aes(x=time,y=DE_k_5),colour="blue")+
       geom_point(aes(x=time,y=DE_k_5),colour="blue")+
       #
       geom_line(aes(x=time,y=DE_k_7),colour="orange")+
       geom_point(aes(x=time,y=DE_k_7),colour="orange")+
       theme(legend.position = c(.5, .5),
              legend.title = element_text(face = "bold"))

myG
```

The time serie is so short that at $k=7$ a lot of observations are smoothed with 
different number of observations (shorter at start and end).


The above calculations are performed by a function in the *convergEU* package:
```{r}
cuTB <-  emp_20_64_MS[,c("time","IT","DE")]

ma_dataset(cuTB, kappa=3, timeName= "time")
```
that is a bit less flexible but it produced standard results.










# Scoreboards

The basis of scoreboard are raw values of  an indicator (level, $y_{m,i,t}$)
for MS $m$ at time $t$ for indicator $i$.
Differences among subsequent years (change) are as well important, namely
$$
y_{m,i,t} - y_{m,i,t-1}
$$
thus a function  to calculate these values may be exploited.

Let's consider the dataset *emp_20_64_MS*,
to calculate such quantities we do the following:
```{r}
data(emp_20_64_MS)
resTB <- scoreb_yrs(emp_20_64_MS,timeName = "time")
resTB
```
where the result is a list of three components:
the summary statistics, the numerical labels to indicate 
the interval of the partition a level  belongs to,
the interval of the partition a change  belongs to.

Numerical labels are assigned as follows (see 
DRAFT JOINT EMPLOYMENT REPORT FROM THE COMMISSION AND THE COUNCIL):    
* value $-1$ if a the original level or change is   $y \leq m -1 \cdot s$;   
* value $-0.5$ if a the original level or change is  $m -1\cdot s < y \leq m - 0.5\cdot s$;   
* value $0$ if a the original level or change is  $m - 0.5\cdot s< y \leq m +0.5\cdot s$;   
* value $+0.5$ if a the original level or change is  $m +0.5\cdot s< y \leq m + 1\cdot s$;   
* value $1$ if a the original level or change is  $y > m +1\cdot s$.      


We note that there is the possibility of representing the above summaries as
coloured plots (TO DO) into scoreboards.


For the comparison of a country with the EU average,
the following steps are recommended, from raw data:

```{r}
# library(ggplot2)
data(emp_20_64_MS)
selectedCountry <- "IT"
timeName <-  "time"
myx_angle <-  45

outSig <- sigma_conv(emp_20_64_MS, timeName = timeName,
           time_0=2002,time_t=2016)
miniY <- min(emp_20_64_MS[,- which(names(emp_20_64_MS) == timeName )])
maxiY <-  max(emp_20_64_MS[,- which(names(emp_20_64_MS) == timeName )])
estrattore<-  emp_20_64_MS[[timeName]] >= 2002  &  emp_20_64_MS[[timeName]] <= 2016
ttmp <- cbind(outSig$res, dplyr::select(emp_20_64_MS[estrattore,], -contains(timeName)))

myG2 <- 
  ggplot(ttmp) + ggtitle(
  paste("EU average (black, solid) and country",selectedCountry ," (red, dotted)") )+
  geom_line(aes(x=ttmp[,timeName], y =ttmp[,"mean"]),colour="black") +
  geom_point(aes(x=ttmp[,timeName],y =ttmp[,"mean"]),colour="black") +
#        geom_line()+geom_point()+
    ylim(c(miniY,maxiY)) + xlab("Year") +ylab("Indicator") +
  theme(legend.position = "none")+
  # add countries
  geom_line( aes(x=ttmp[,timeName], y = ttmp[,"IT"],colour="red"),linetype="dotted") + 
  geom_point( aes(x=ttmp[,timeName], y = ttmp[,"IT"],colour="red")) +
  ggplot2::scale_x_continuous(breaks = ttmp[,timeName],
                     labels = ttmp[,timeName]) +
   ggplot2::theme(
         axis.text.x=ggplot2::element_text(
         #size = ggplot2::rel(myfont_scale ),
         angle = myx_angle 
         #vjust = 1,
         #hjust=1
         ))
  
myG2
```




It is also possible to graphically show departures  in terms of  the above 
defined partition:
```{r,fig.height=11}
obe_lvl <- scoreb_yrs(emp_20_64_MS,timeName = timeName)$res$sco_level_num
# select subset of time
estrattore <- obe_lvl[[timeName]] >= 2009 & obe_lvl[[timeName]] <= 2016  
scobelvl <- obe_lvl[estrattore,]

my_MSstd <- ms_dynam( scobelvl,
                timeName = "time",
                displace = 0.25,
                displaceh = 0.45,
                dimeFontNum = 3,
                myfont_scale = 1.35,
                x_angle = 45,
                axis_name_y = "Countries",
                axis_name_x = "Time",
                alpha_color = 0.9
                )   

my_MSstd
```











<br><br>

# Country fiche


The **counvergEU** package provides a function that
automatically prepares one or more country fiches.
This function is able to create a directory along an existing path
and to copy the rmarkdown file representing the template within it.
The rmarkdown file is parameterized so that passing different parameters
the compilation takes place with different data, say different 
indicators and countries.

It is very important to prepare complete data in a tibble (dataset)
made by a time variable and as many other variables as countries
that enter into the calculation of the time average.
Failing to satisfy this requisite causes the use of a wrong mean value
at each year.
Nevertheless one key country is specified and some other countries of 
interest may be listed to better decorate graphs and compare performances.

Below, a call to the  function *go_ms_fish()* illustrates the syntax:  

```r
go_ms_fish(
    workDF ='myTB',
    countryRef ='DE',
    otherCountries = "c('IT','UK','FR')",
    time_0 = 2002,
    time_t = 2016,
    tName = 'time',
    indiType = "highBest",
    aggregation= 'EU27_2020',
    x_angle=  45,
    dataNow=  Sys.time(),
    author = 'A.Student',
    outFile = 'Germany-up2-2016', 
    outDir = "tt-fish",
    indiName= 'emp_20_64_MS'
)
  
```
but it is very important to emphasize some constraints
and unusual ways to pass parameters to such a function.
In fact, note that the first argument is the working dataset
which is passed not as an R object but as a string, the name
of the dataset that must be available in the R workspace
before invoking *go_ms_fish*.    
The second argument *countryRef* is a string with the short name of
a member country that will be shown in one-country plots.
Less obvious, argument *indiType = "lowBest"* specifies if
the considered indicator is built so that a low value is good for
a country or if a high value is good (*indiType = "highBest"*).  

Of particular importance the argument *outFile* that 
can be a string indicating the name of the output file.
Similarly *outDir* is the path (unit and folders) in which the
final compiled html will be  stored. 
The syntax of the path depend on the operating system;
for example *outDir='F:/analysis/IT2018'*  indicates that
in the usb disk called 'F', within the folder 'analysis'
is located folder 'IT2018' where R will write the country fiche.
Note that a disk called 'F' must exist and also folder 'analysis'
must exist in such unit, while on the contrary folder 'IT2018'
is created by the function if it does not already exist.

Within the above mentioned output directory, besides the compiled html,
it is also stored a file called like specified by  *outFile* 
but with added the string '-workspace.RData'
that contains data and plots produced 
during the compilation of the country fiche for further
subsequent use in other technical reports.





# Indicator fiches


An auxiliary function *go_indica_fish()*
is provided in the  R package  *convergEU*
to produce an indicator fiches, where the output is an html file.
At this purpose, an output directory must be also specified.
Note that some arguments are passed as strings instead of objects,
as described in the last section above.   


An example of syntax to invoke the procedure is:  

```
go_indica_fish(
    time_0 = 2005,
    time_t = 2010,
    timeName = 'time',
    workingDF = 'emp_20_64_MS' ,
    indicaT = 'emp_20_64',
    indiType = c('highBest','lowBest')[1],
    seleMeasure = 'all',
    seleAggre = 'EU27_2020',
    x_angle =  45,
    data_res_download =  FALSE,
    auth = 'A.Student',
    dataNow =  '2019/05/16',
    outFile = "test_IT-emp_20_64_MS",
    outDir = "tt-fish"
  )
```






<br> 

# References  

The following reference may be consulted for details:    

  * Brussels, 21.11.2018, COM(2018) 761 final,  DRAFT JOINT EMPLOYMENT REPORT FROM THE COMMISSION AND THE COUNCIL, accompanying the Communication from the Commission on the Annual Growth Survey 2019.     
  
  * Eurofound (2018), Upward convergence in the EU: Concepts, measurements and indicators, Publications Office of the European Union, Luxembourg; by: Massimiliano Mascherini, Martina Bisello, Hans Dubois and Franz Eiffe.    
  
  * Tuszynski, J. (2015). **caTools**: Tools: moving window statistics, GIF, Base64, ROC AUC, etc.
R package version 1.17.1.2, URL https://CRAN.R-project.org/package=caTools.       

  * Nedka D. Nikiforova, Federico M. Stefanini, Chiara Litardi, Eleonora Peruffo and Massimiliano Mascherini (2020) Tutorial: analysis of convergence with the convergEU package.
  Package vignette
  URL https://unimi2013-my.sharepoint.com/:u:/g/personal/federico_stefanini_unimi_it/EW0cVSIgbtZAvLPNbqcxdX8Bfn5VGSRHfAH88hQwc_RIEQ?e=MgtSZu
  
<br><br>  
  
  












# Appendix: clusters over time of EU MS


In this appendix several  lists of member states are  defined
as follows:
```{r}
setupConvergEU <- convergEU_glb()
names(setupConvergEU)
```
and, with more details:
```{r}
print(setupConvergEU$EUcodes,n=30)
print(setupConvergEU$Eurozone)
setupConvergEU$EU12
setupConvergEU$EU15
```

```{r}
print(setupConvergEU$EU25$dates)
print(setupConvergEU$EU25$memberStates,n=30)

print(setupConvergEU$EU27$dates)
print(setupConvergEU$EU27$memberStates,n=30)

print(setupConvergEU$EU27_2020$dates)
print(setupConvergEU$EU27_2020$memberStates,n=30)
```





