| Title: | Ecological Inference Applying Entropy |
|---|---|
| Description: | Implements two estimations related to the foundations of info metrics applied to ecological inference. These methodologies assess the lack of disaggregated data and provide an approach to obtaining disaggregated territorial-level data. For more details, see the following references: Fernández-Vázquez, E., Díaz-Dapena, A., Rubiera-Morollón, F. et al. (2020) "Spatial Disaggregation of Social Indicators: An Info-Metrics Approach." <doi:10.1007/s11205-020-02455-z>. Díaz-Dapena, A., Fernández-Vázquez, E., Rubiera-Morollón, F., & Vinuela, A. (2021) "Mapping poverty at the local level in Europe: A consistent spatial disaggregation of the AROPE indicator for France, Spain, Portugal and the United Kingdom." <doi:10.1111/rsp3.12379>. |
| Authors: | Alberto Díaz-Dapena [aut, cph], Esteban Fernández-Vázquez [aut, cph], Silvia María Franco Anaya [aut, cre, cph] |
| Maintainer: | Silvia María Franco Anaya <[email protected]> |
| License: | GPL-3 |
| Version: | 0.0.1.3 |
| Built: | 2024-11-05 06:44:01 UTC |
| Source: | CRAN |
The function ei_gce defines the Kullback-Leibler function which
minimises the distance between the distribution of probabilities P and the
distribution Q. The distribution Q is based on prior information that we have
of our variable of interest previous to the analysis. The function will set
the optimization parameters and, using the "nlminb"
function, an optimal solution is obtained.
The function defines the independent variables in the two databases
needed, which we call datahp with "n_hp" observations and datahs with "n_hs"
observations; and the function of the variable of interest y. Then the
weights of each observation for the two databases used are defined, if there
are not weights available it will be 1 by default. The errors are calculated
pondering the support vector of dimension var, 0, -var. This support vector
can be specified by the user. The default support vector is based on variance.
We recommend a wider interval with v(1,0,-1) as the maximum.
The restrictions are defined in order to guarantee consistency. The
minimization of Kullback_Leibler distance is solved with "nlminb" function
with maximum number of iterations 1000 and with tolerance
defined by the user. If the user did not define tolerance it will be 1e-10 by
default. For additional details about the methodology see Fernández-Vazquez, et al. (2020)
ei_gce(fn, datahp, datahs, q, weights = NULL, v, tol, iter)ei_gce(fn, datahp, datahs, q, weights = NULL, v, tol, iter)
fn |
is the formula that represents the dependent variable in the optimization. In the context of this function, 'fn' is used to define the dependent variable to be optimized by the Kullback-Leibler divergence function. Note: If the dependent variable is categorical the sorting criterion for the columns, and therefore for J, is alphabetical order. |
datahp |
The data where the variable of interest y is available and also the independent variables. Note: The variables and weights used as independent variables must have the same name in 'datahp' and in 'datahs' |
datahs |
The data with the information of the independent variables as a disaggregated level. Note: The variables and weights used as independent variables must have the same name in 'datahp' and in 'datahs'. The variables in both databases need to match up in content. |
q |
The prior distribution Q |
weights |
A character string specifying the column name to be used as weights in both |
v |
The support vector |
tol |
The tolerance to be applied in the optimization function. If the tolerance is not specified, the default tolerance has been set in 1e-10 |
iter |
The maximum number of iterations allowed for the optimization algorithm to run Increasing the number of iterations may improve the likelihood of finding an optimal solution, but can also increases computation time.If the maximum number of iterations is not specified, it will default to 1000 |
To solve the optimization upper and lower bounds for p and w are settled, specifically, p and w must be above 0 and lower than 1. In addition, the initial values of p are settled as the defined prior and the errors (w) as 1/L.
The function will provide you a dataframe called table with the next information:
probabilities Probabilities for each individual to each possibility j of the variable of interest y.
error dual Errors calculated to the j possibilities of y.
predictions The prediction for each individual is calculated as the sum of the probability plus the error. The function provides information about the optimization process as:
divergenceklThe Kullback-Leibler divergence value resulting from the optimization.
iterations Indicates the times the objective function and the gradient has been evaluated during the optimization process,if any.
message Indicates the message if it has been generated in the process of optimization.
tol Indicates the tolerance of the optimization process.
v Indicates the support vector used in the function. The function provides a dataframe containing the information about lambda:
lambda The estimated lambda values. It is provided an object with the restrictions checked which should be zero.
check restrictions Being g1 the restriction related to the unit probability constraint, g2 to the error unit sum constraint, and g3 to the consistency restriction that implies that the difference between the cross moment in both datasets must be zero.
The restriction g3 can be checked thoroughly with the objects by separate.
cross moments hp Cross moments in datahp.
cross moments hs Cross moments in datahs.
Fernandez-Vazquez, E., Diaz-Dapena, A., Rubiera-Morollon, F., Viñuela, A., (2020) Spatial Disaggregation of Social Indicators: An Info-Metrics Approach. Social Indicators Research, 152(2), 809–821. https://doi.org/10.1007/s11205-020-02455-z.
#In this example we use the data of this package datahp <- financial() datahs <- social() # Setting up our function for the dependent variable. fn <- datahp$poor_liq ~ Dcollege+Totalincome+Dunemp #In this case we know that the mean probability of being poor is 0.35.With this function #we can add the information as information a priori. This information a priori correspond to the #Q distribution and in this function is called q for the sake of simplicity: q<- c(0.5,0.5) v<- matrix(c(0.2,0,-0.2)) #Applying the function ei_gce to our databases. In this case datahp is the # data where we have our variable of interest #datahs is the data where we have the information for the disaggregation. #w can be included if we have weights in both surveys result <- ei_gce(fn,datahp,datahs,q=q,weights="w",v=v)#In this example we use the data of this package datahp <- financial() datahs <- social() # Setting up our function for the dependent variable. fn <- datahp$poor_liq ~ Dcollege+Totalincome+Dunemp #In this case we know that the mean probability of being poor is 0.35.With this function #we can add the information as information a priori. This information a priori correspond to the #Q distribution and in this function is called q for the sake of simplicity: q<- c(0.5,0.5) v<- matrix(c(0.2,0,-0.2)) #Applying the function ei_gce to our databases. In this case datahp is the # data where we have our variable of interest #datahs is the data where we have the information for the disaggregation. #w can be included if we have weights in both surveys result <- ei_gce(fn,datahp,datahs,q=q,weights="w",v=v)
The function ei_gme defines the Shannon entropy function which takes a vector of probabilities as input and returns the negative
sum of p times the natural logarithm of p.The function will set the optimization parameters and using the "nlminb" function an optimal
solution is obtained.
The function defines the independent variables in the two databases needed, which we call datahp with "n_hp" observations and datahs
with "n_hs" observations; and the function of the binary variable of interest y. Then the weights of each observation for the two
databases used are defined, if there are no weights available it will be 1.
The errors are calculated pondering the support vector of dimension var, 0, -var. This support vector can be specified by the user.
The default support vector is based on variance.We recommend a wider interval with v(1,0,-1) as the maximum.
The restrictions are defined to guarantee consistency.
The optimization of the Shannon entropy function is solved with "nlminb" function
with maximum number of iterations 1000 and with tolerance
defined by the user.
ei_gme(fn, datahp, datahs, weights = NULL, tol, v, iter)ei_gme(fn, datahp, datahs, weights = NULL, tol, v, iter)
fn |
is the formula that represents the dependent variable in the optimization. In the context of this function, 'fn' is used to define the dependent variable to be optimized by the entropy function. Note: If the dependent variable is categorical the sorting criterion for the columns, and therefore for J, is alphabetical order. |
datahp |
The data where the variable of interest y is available and also the independent variables. Note: The variables and weights used as independent variables must have the same name in 'datahp' and in 'datahs' The variables in both databases need to match up in content. |
datahs |
The data with the information of the independent variables as a disaggregated level. Note: The variables and weights used as independent variables must be the same and must have the same name in 'datahp' and in 'datahs' |
weights |
A character string specifying the column name to be used as weights in both |
tol |
The tolerance to be applied in the optimization function. If the tolerance is not specified, the default tolerance has been set in 1e-10 |
v |
The support vector |
iter |
The maximum number of iterations allowed for the optimization algorithm to run Increasing the number of iterations may improve the likelihood of finding an optimal solution, but can also increases computation time.If the maximum number of iterations is not specified, it will default to 1000 |
To solve the optimization upper and lower bounds for p and w are settled, specifically, p and w must be above 0 and lower than 1. In addition, the initial values of p are settled as a uniform distribution and the errors (w) as 1/L.
The function will provide you a dataframe called table with the next information:
probabilities Probabilities for each individual to each possibility j of the variable of interest y.
error primal Errors calculated to the j possibilities of y.
predictions The prediction for each individual is calculated as the sum of the probability plus the error primal. The function provides information about the optimization process as :
value_of_entropy The value of entropy resulting from the optimization.
iterations Indicates the times the objective function and the gradient has been evaluated during the optimization process
message Indicates the message if it has been generated in the process of optimization
tol Indicates the tolerance used in the optimization
v Indicates the vector of support used in the function The function provides a dataframe containing the information about lambda:
lambda The estimated lambda values. It is provided an object with the restrictions checked which should be approximately zero.
check restrictions Being g1 the restriction related to the unit probability constraint, g2 to the error unit sum constraint, and g3 to the consistency restriction that implies that the difference between the cross moment in both datasets must be zero.
The restriction g3 can be checked thoroughly with the objects by separate.
cross moments hp Cross moments in datahp.
cross moments hs Cross moments in datahs.
Fernandez-Vazquez, E., Díaz-Dapena, A., Rubiera-Morollon, F., Viñuela, A., (2020) Spatial Disaggregation of Social Indicators: An Info-Metrics Approach. Social Indicators Research, 152(2), 809–821. https://doi.org/10.1007/s11205-020-02455-z.
#In this example we use the data of this package datahp <- financial() datahs <- social() # Setting up our function for the dependent variable. fn <- datahp$poor_liq ~ Dcollege+Totalincome+Dunemp #Applying the function ei_gme to our databases. In this case datahp #is the data where we have our variable of interest datahs is the data # where we have the information for the disaggregation. #w can be included if we have weights in both surveys #Tolerance in this example is fixed in 1e-10 and v will be (1,0,-1) v=matrix(c(1, 0, -1), nrow = 1) result <- ei_gme(fn=fn,datahp=datahp,datahs=datahs,weights="w",v=v)#In this example we use the data of this package datahp <- financial() datahs <- social() # Setting up our function for the dependent variable. fn <- datahp$poor_liq ~ Dcollege+Totalincome+Dunemp #Applying the function ei_gme to our databases. In this case datahp #is the data where we have our variable of interest datahs is the data # where we have the information for the disaggregation. #w can be included if we have weights in both surveys #Tolerance in this example is fixed in 1e-10 and v will be (1,0,-1) v=matrix(c(1, 0, -1), nrow = 1) result <- ei_gme(fn=fn,datahp=datahp,datahs=datahs,weights="w",v=v)
This dataset contains 100 observations of 6 variables. The data was generated randomly for the purpose of exemplifying a database that could potentially be used with this function.
financial()financial()
A data frame with 100 observations of 6 variables called financial
A data frame containing the loaded "financial" data from the .rds file.
Dcollege: Dummy variable indicating college education.
Dunemp: Dummy variable indicating unemployment.
Totalincome: Total income of each observation.
poor_liq: Dummy variable indicating liquid poverty.
w: Weights for each observation.
n: Identifier for observations.
data(financial) head(financial)data(financial) head(financial)
This function generates a descriptive plot using the results obtained in ei_gce. It illustrates the mean and the confidence interval by disaggregated territorial unit.
## S3 method for class 'kl' plot(x, reg, ...)## S3 method for class 'kl' plot(x, reg, ...)
x |
The output produced by ei_gce |
reg |
The data column containing the disaggregated territorial units |
... |
Additional arguments passed to the plotting function. |
This function provides a graph representing the weighted mean and confidence interval of each disaggregated territorial unit
This function generates a descriptive plot using the result obtained in ei_gme. It illustrates the mean and the confidence interval by disaggregated territorial unit.
## S3 method for class 'shannon' plot(x, reg, ...)## S3 method for class 'shannon' plot(x, reg, ...)
x |
The output produced by ei_gme |
reg |
The data column containing the disaggregated territorial units |
... |
Additional arguments passed to the plotting function. |
This function provides a graph representing the weighted mean and confidence interval of each disaggregated territorial unit
This function provides a summary of the output obtained with the function ei_gce.
## S3 method for class 'kl' summary(object, ...)## S3 method for class 'kl' summary(object, ...)
object |
The output obtained from ei_gce |
... |
Additional arguments passed to the summary function. |
This summary function returns the Kullback-Leibler divergence value and the last iteration in the optimization process. A dataframe with the means of the estimations for each characteristic j with the predictions the probabilities and the error estimated. A dataframe with the lambda estimated for each k.
Iterations:Indicates the times the objective function and the gradient has been evaluated during the optimization process
divergencekl value:The Kullback-Leibler divergence value resulting from the optimization.
mean_estimations:The weighted mean of predictions, p_dual, and the error for each category j of the variable y
lambda:The estimated lambda values.
This function provides a summary of the output obtained with the function ei_gme.
## S3 method for class 'shannon' summary(object, ...)## S3 method for class 'shannon' summary(object, ...)
object |
The output obtained from ei_gme |
... |
Additional arguments passed to the summary function. |
This summary function returns the entropy value and the last iteration in the optimization process. A dataframe with the means of the estimations for each characteristic j with the predictions the probabilities and the error estimated. A dataframe with the lambda estimated for each k.
Iterations:Indicates the times the objective function and the gradient has been evaluated during the optimization process
Entropy value:The value of entropy resulting from the optimization.
mean_estimations: The weighted mean of predictions, p_dual, and the error for each category j of the variable y
lambda:The estimated lambda values.
Randomly Generated Data
Description
This dataset contains 200 observations of 6 variables. The data was generated randomly for the purpose of exemplifying a database that could potentially be used with this function.
Usage
Format
A data frame with 200 observations of 6 variables called social
Value
A data frame containing the loaded "social" data from the .rds file.
Dcollege: Dummy variable indicating college education.Dunemp: Dummy variable indicating unemployment.Totalincome: Total income of each observation.reg:Variable indicating the region of the observation.w: Weights for each observation.n: Identifier for observations.#' @examples data(social) head(social)