ei_gce: Ecologic Inference applying entropy

Description

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)

Usage

ei_gce(fn, datahp, datahs, q, weights = NULL, v, tol, iter)

Value

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.

Arguments

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 datahp and datahs datasets. If weights is provided and exists in both datasets, each dataset's weights will be normalized by the sum of the weights in that dataset. If weights is NULL or the specified column does not exist in both datasets, equal weights are applied across all observations.
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

Details

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.

References

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.

Examples

Run this code

#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)

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