p2distance: Welfare's Synthetic Indicator function
In p2distance: Welfare's Synthetic Indicator

Description Usage Arguments Details Value Author(s) References Examples

This function calculates the P_{2} distance synthetic indicator for a set of variables.

1 2	p2distance(matriz, reference_vector = NULL, reference_vector_function = min, iterations = 20, umbral = 1e-04)

`matriz`	An object of matrix type with spatial entities in rows and variables in columns
`reference_vector`	Optionally. A reference vector defined for each partial indicator so as to compare different spatial entities
`reference_vector_function`	Optionally. Function to make the reference vector. Minimum es the default. Others common functions used: min, max, mean, median, etc. See `makeReferenceVector` for further details
`iterations`	Numbers of maximum iterations for the computational process until reach the convergence
`umbral`	The algorithm stop when the difference between iterations is lower than this umbral

This is the main function on package. It calculates the Pena distance indicator, also called DP2, which is used to measure welfare in quality-of-life applications, to create Environmental Quality Indexes, etc. (see references). It is a multidimensional indicator capable to aggregate various partial indicators (variables) in a unique measure to compare the state of different spatial entities. The P2 Distance from a spatial entity r is definied as

DP_{2}=∑^{n}_{i=1}≤ft\lbrace≤ft(\frac{d_{i}}{σ_{i}}\right)≤ft(1-R^{2}_{i,i-1,i-2,…,1}\right)\right\rbrace

with R^{2}_{1}=0; where d_{i}=|x_{ri}-x_{*i}| with the reference base X_{*}=≤ft(x_{*1},x_{*2},…,x_{*n}\right) where:

n is the number of variables
x_{ri}, is the value of the variable i in the spatial entity r
σ_{i} is the standard deviation of variable i
R^{2}_{i,i-1,…,1} is the coefficient of determination in the regression of X_{i} over X_{i-1}, X_{i-2}, …,X_{1} already included.

The numerical value of the DP2 index has no real meaning, but its is useful for comparing the state of different spatial entities in terms of welfare, environmental conditions, etc.

`discrimination.coefficient`	Vector of discrimination coefficients (DC) for each variable. The value of DC, defined by Ivanovic (1974) is DC_{i}=\frac{2}{m(m-1)}∑_{j,l>j}^{k_{i}}m_{ji}m_{li}≤ft\|\frac{x_{ji}-x_{li}}{\overline{X}_{i}}\right\| where m is the number of spatial entities and m_{ji} is the absolute frequency of x_{ji}. This measure ranges between 0 an 2. If a variable takes the same values for all spatial entities, DC equals zero, indicating zero discriminant power. By contrast, if a variable only has a value other than zero for one spatial entity and in the remainder m-1, is equal to zero, DC reaches its maximun value (2) and the variable has full discriminant power (see Zarzosa, 1996; Zarzosa and Somarriba, 2012). There is an alternative way of calculating the coefficient, by using the Gini index, DC_{i}=2\frac{m}{m-1}G where m is the number of spatial entities and G the Gini index
`p2distance`	Vector with the last P_{2} distance value for each spatial entity
`p2distances`	Array with vectors of P_{2} distances values resulting for each iteration
`diff_p2distances`	Array with differeces between two contiguous P_{2} distances
`iteration`	Number of calculated iterations
`umbral`	Threshold in difference for two contiguous P_{2} distances
`variables_sort`	Vector with the variable names by entrance order determined by last iteration
`correction_factors`	Correction Factors for each variable
`cor.coeff`	Correlation coefficient for each variable with the synthetic indicator (P_{2} distance) calculated
`partial.Indicators`	For each spatial entity the difference between the reference vector and the value of each variable divided by the standard deviation. For a spatial entity, the sum of all partial indicators is the Frechet Distance (DF), which is the maximun value that P_{2} distance can reach.

A.J. Perez-Luque; R. Moreno; R. Perez-Perez and F.J. Bonet

Ivanovic, B. (1974) Comment ètablir une liste des indicateurs de developpment. Revue de Statistique Apliquée, XXII(2), 37–50

Montero, J. M., Chasco, C. and Larraz, B. (2010). Building an environmental quality index for a big city: a spatial interpolation approach combined with a distance indicator. Journal of Geographical Systems, 12, 435–459.

Pena, J. B. (1977). Problemas de la medición del bienestar y conceptos afines (una aplicación al caso Español). Madrid: INE.

Pena, J. B. (2009). La medición del bienestar social: una revisión crítica. Estudios de Economía Aplicada, 27(2), 299–324.

Zarzosa, P. (1996). Aproximación a la medición del Bienestar social. Valladolid: University of Valladolid. 248 pp.

Zarzosa, P. and Somarriba, N. (2012). An assessment of social welfare in Spain: Territorial analysis using a synthetic welfare indicator. Social Indicators Research, doi: http://dx.doi.org/10.1007/s11205-012-0005-0

## Calculate a welfare indicator for 27 countries of Europe 
data(welfare) 

## Convert welfare dataframe to matrix object 
welfare <- as.matrix(welfare)

## Calculate P2 Distance 
ind <- p2distance(matriz=welfare, reference_vector_function = min, 
        iterations = 20)

## Examine the results
# P2 distance
ind$p2distance

# Iterations to achieve convergence
ind$iteration 

# Order of entry of variables resulting the last iteration
ind$variables_sort

# Correction factors of each variable
ind$correction_factors

# Correlations between P2 distance indicator and variables
ind$cor.coeff

# Discrimination coefficient of each variable
ind$discrimination.coefficient

## Plot of P2 Distance Indicator for European countries
barplot(ind$p2distance, beside=TRUE, col="white", space=.3, ylab="P2 distance", 
      ylim=c(0,20), names.arg=rownames(ind$p2distance), las=3, cex.names=0.8)

[1] "Iteration 1"
[1] "Iteration 2"
[1] "Iteration 3"
[1] "Iteration 4"
              p2distance.4
Austria          14.243429
Belgium          14.205152
Bulgaria          3.300577
Cyprus           14.196170
CzechRepublic    10.595075
Germany          12.882661
Denmark          17.932001
Estonia          10.014157
Greece            9.467627
Spain            12.653989
Finland          16.014650
France           14.106968
Hungary           6.157913
Ireland          13.186726
Italy            10.822846
Lithuania         6.728374
Luxembourg       15.608905
Latvia            5.881641
Malta            14.124929
Netherlands      15.096630
Poland            9.072606
Portugal          9.927800
Romania           7.855658
Sweden           16.225990
Slovenia         12.005987
Slovakia          9.584544
UnitedKingdom    13.817885
[1] 4
 [1] "standard"    "social"      "life.satis"  "home"        "happiness"  
 [6] "family"      "night"       "area"        "life.0"      "life.65"    
[11] "job"         "judicial"    "education"   "employement" "people"     
[16] "health"      "inequality"  "stress"      "hobbies"     "dist.school"
   standard      social  life.satis        home   happiness      family 
 1.00000000  0.26994486  0.15647574  0.19019368  0.11374336  0.16464658 
      night        area      life.0     life.65         job    judicial 
 0.22771776  0.26803770  0.23019934  0.05377976  0.39866003  0.35480094 
  education employement      people      health  inequality      stress 
 0.25202059  0.28405573  0.19859718  0.42516392  0.08949479  0.18883919 
    hobbies dist.school 
 0.07721116  0.15296173 
            p2distance.4
happiness      0.8923932
life.satis     0.9032185
judicial       0.7240456
night          0.8259018
social         0.9193612
people         0.5994948
family         0.8366196
health         0.5648329
life.65        0.7758931
life.0         0.8013540
inequality    -0.4733350
hobbies       -0.3985391
education      0.6609428
standard       0.9572833
dist.school    0.3876915
area           0.8066489
home           0.8945082
stress        -0.4232577
employement    0.6416367
job            0.7744487
  happiness  life.satis    judicial       night      social      people 
 0.08033682  0.22114042  0.37154869  0.21365413  0.19085162  0.21602518 
     family      health     life.65      life.0  inequality     hobbies 
 0.07274169  0.56579365  0.09533417  0.04596938  0.29771635  0.16633727 
  education    standard dist.school        area        home      stress 
 0.09708026  0.14761252  0.06577282  0.07862064  0.12099290  0.31190294 
employement         job 
 0.10930471  0.13470473