Vignette EIEntropy
In a wide range of analyses, there is a common problem. The data is
usually not available at the desired spatial scale that researchers
need. Ecological inference allows us to recover incomplete or
unavailable data by inferring individual behaviours based on the
aggregated information. In some cases, the database containing the
variable of interest does not have it at the needed spatial level.
Entropy allows us to obtain the best solution according to the
information that we have. To learn more about the methodology, see
Fernández-Vázquez et al. (2020)
The package extrapolates the value of the variable of interest above
a data frame with more detailed geographical information, being
consistent with the aggregates available. This package contains two
functions, ei_gce() and ei_gme(). The method that the function ei_gce()
applies is framed between GCE (Generalized Cross entropy), minimising
the distance between the two distributions \(P\) and \(Q\), being this \(Q\) the prior information we can initially
have. If the user does not have prior information can choose ei_gme()
for computational reasons which will use GME (Generalized maximum
entropy) since our distribution a priori will be the uniform
distribution. These two kinds of problems will be explained in
the next sections.
The EIentropy package makes the process of assessing uncertainty to
estimate information at the desired level of disaggregation into one
function. It gathers all the steps making it easy to apply this
methodology to problems of absence of data. This document introduces you
to the functions ei_gme() and ei_gce(), which apply entropy to solve
issues of ecological inference, providing, consistent disaggregated
indicators with observable aggregated data and cross moments.
Once you’ve installed it, the vignette (“EIEntropy”) will allow you
to learn more. We will use the data included in this package to explore
its functions. This data is a straightforward application of this
methodology developed in…
This package is on CRAN and can be installed from within R.
1. Methodology
The problem assessed is the need to obtain a variable \(Y\) at a spatial level that is not
available. This variable \(Y\) can take
\(j\) values. The method aims to obtain
a matrix \(P\) with dimension \(n\) x \(J\) being \(n\) the number of observations. Matrix
\(P\) is compounded by the
probabilities associated with each \(j\) value for each observation. Taking
advantage of the information that we have, the method introduces the
cross-moments as a restriction to assure consistency.
Taking into account \(Y\) is the sum
of \(P\) and \(U\), being \(U\) the random noise. Once we have
estimated matrix \(P\) and the error
term, we can obtain \(Y\) as \(Y=P+U\). The error term is built as a
weighted mean of a support vector \(V\)
in which weights \(W\) are
estimated.
The support vector is the component of the noise defining the
flexibility of the estimation. It represents the maximum and minimum
error around a value in the center. By default, \(V\) is defined as (-var, 0, var) where
var represents the variance of the dependent variable. If the
estimation requires more flexibility \(V\) can be defined wider–with (-1,0,1) as
the recommended maximum.
We present two functions to apply ecological inference through
entropy. The general case minimizes the Kullback-Leibler divergence to
find the matrix of probabilities \(P\)
with the minimum divergence with a prior. This case is adequate even
when we have prior information on the distribution of the probabilities
of our variable of interest. This information refers to all existing
knowledge previous to the data that can influence the expectations of
the variable of interest.
\[\[
\text{{Min}}_{P} \text{{KL}}(P, Q) = \sum_{i=1}^{n} \sum_{j=1}^{J}
P_{ij} \log\left(\frac{P_{ij}}{Q_{ij}}\right)\text{{ s.t.}}
\]\]
\[
\frac{1}{n} \sum_{i=1}^{n} x_{s_{ik}} y_{ij} = \frac{1}{n}
\sum_{i=1}^{n} x_{c_{ik}} (p_{ij} + u_{ij}) =\frac{1}{n} \sum_{i=1}^{n}
x_{cik} [p_{ij} + w_{Lij} v_{l}];
j=1,\ldots,J
\]
\] \[\[
\sum_{j=1}^{J} p_{ij} = 1 \quad \text{for } i=1,\ldots,n
\]
\] \[\[
\sum_{l=1}^{L} w_{Lij} = 1 \quad \text{for } i=1,\ldots,n;
j=1,\ldots,J
\]
\]
This function minimizes the distance between \(P\) and \(Q\). Being \(Q\) the prior information that we already
have. If we have some information about the possible distribution of our
variable of interest we can include this information here. For example,
if you know your variable of interest can take two values but one of
them is more common than the other you can include it in your estimation
by using \(Q\).
If we don’t have any information then we will assume there is the
same probability for each possibility \(j\) of our variable of interest. In this
sense, we minimize the divergence with a uniform distribution.
Solving the optimization problem, we obtain estimations as \(\hat{Y}= \hat{P} + \hat{W}V\). The solution
should be consistent with the restrictions. In this case, the
restrictions will contain information from two data sources, datahp and
datahs. If there were divergences between the two sources of data, they
would be captured in the error term. This is one of the main advantages
of the methodology because it allows the user to use two databases with
divergences between them.
2. Data
To explore the functions we will use data included in the package. In
our example, we want to obtain microdata about poverty, in terms of
wealth, with an indication of location at regional level. We have the
variable of interest in one official database but without information
about location. At the same time, in other database, the variable of
interest is not available but the household location is.
The data can be loaded by calling the functions financial() and
social().
datahp <- financial()
datahs <- social()
In this example, we aim to obtain probabilities of being poor in
terms of wealth for each individual or observation in the survey with
detailed information about location. With this procedure, we will have
our variable of interest at the desired spatial scale. As it is known,
some variables, such as education level, income, or employment status
are related to wealth. Hence, we will use these variables as regressors
in the example. These variables are going to be our X. We have the same
variables in both data frames with the same name. For this example, the
financial survey includes 100 observations for the variables Dcollege,
Dunemp, total income and the variable of interest:poor_liq. They are a
dummy for college, a dummy for being unemployed, the household income
(in euros) and a dummy for being poor in terms of liquid assets
respectively.
The data called social has the same variables but instead of the
variable of interest, it has another variable with the region of each
household. In this database, we have 200 observations (households).
3. Examples
3.1 Generalized Cross entropy: ei_gce()
The function ei_gce() allows the user to introduce prior information
in the estimation. In this example, we keep using the databases included
in this package financial and social. Once we have
chosen the best function for our case note that we need to specify our
function :
fn <- datahp$poor_liq ~ Dcollege+Totalincome+Dunemp
Note that the same name in both datasets for the independent
variables is required.
This function’s arguments are the previously defined function, the
databases used, the weights, the tolerance and the method applied in the
optimization and the support vector. With this function, weights can be
used and included with w. If there are no weights the function assumes a
matrix of 1. Note that the weights used in this methodology are
normalised so analytics and sampling weights can be used without
distinction.
In this example the variable of interest (poor_liq) is defined with a
function in the argument fn (see previous section)
The arguments corresponding to the information a priori and the
support vector can be included as :
q <- c(0.4,0.6)
v <- matrix(c(-1,0,1))
In this example we assume a priori distribution of poverty equal to
0.4 for poor and 0.6 for non-poor. The support vector has been set to
the maximum (-1,0,1). Applying the ei_gce() we can solve the estimation
as:
result <- ei_gce(fn,datahp,datahs,q=q,w=w,tol=NULL,method="BFGS",v=v)
The function will produce a data frame called a table with the
following information:
Probabilities for each individual to each possibility \(j\) of the variable of interest \(Y\).
The errors are calculated to the \(j\) possibilities of \(Y\).
The prediction for each individual is the result of the sum of
the probability plus the error.
The function provides information about the optimization process, in
concrete:
- The value of entropy resulting from the optimization, the
iterations, which indicate the times the objective function and the
gradient have been evaluated during the optimization process and a
message in case something went wrong.
Lagrange multipliers \(\lambda\)
associated to each independent variables are also provided in the form
of a data frame. In addition, it provides an object with the
restrictions checked which should be approximately zero. 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
separately which are to be provided in the output as cross moments hp
(the cross moments in datahp) and cross moments hs (the cross moments in
datahs).
{r} result
To make the results more visual, this package includes a personalized
summary function, providing the means for each category \(j\) for the predictions, the probabilities
and the error.
##
## Adjuntando el paquete: 'dplyr'
## The following objects are masked from 'package:stats':
##
## filter, lag
## The following objects are masked from 'package:base':
##
## intersect, setdiff, setequal, union
## Iterationsfunction
## 1001
## Entropy value[1] -15.05311
## [1] "mean_estimations"
## n predictions_j1 predictions_j2 p_dual_j1 p_dual_j2 e_dual_j1 e_dual_j2
## 1 100.5 0.5874094 0.3825055 0.4873519 0.5126481 0.1000575 -0.1301426
## [1] "lambda"
## intercept Dcollege Totalincome Dunemp
## j1 63.037111 56.62013 -1.763225e-04 -3.627390
## j2 -8.711875 -23.03967 -4.508848e-05 -8.457388
Graphs included are generated with the plot function, showing the
averages of the predictions for each territorial unit and the 95%
confidence interval associated with each of them.
plot(x=result,datahs$reg)
Notes: Arguments tol and method can be defined by
the user. The default tolerance has been set in 1e-24 while the
by-default method applied in the optimization is BFGS. Other acceptable
methods are Nelder-Mead, CG, L-BFGS-B, SANN or Brent from the optim()
function.
3.2.1 Another option when we do not have prior information:
ei_gme()
In the case the user does not have prior information the function
ei_gme() is also available. This function is provided for cases where
the user may prefer using GME (Generalized maximum entropy) if the user
has a big volume of data, because it is a more efficient function in
computational terms than ei_gce() with the uniform distribution as
prior. Both functions should provide the same result.
ei_gme() applies the Shannon entropy function to the optimization. It
is adequate when you cannot assume anything about the distribution of
\(Y\). So the starting point is the
uniform distribution.
Using the same example, our variable of interest is the poverty rate
in terms of wealth and our independent variables are the income and
dummies for unemployment and college studies. In this case, we need to
define the function
result3 <- ei_gme (fn,datahp,datahs,w,tol=NULL,method="BFGS",v)
The function will produce the same output as ei_gce(), see the
previous section for further details.
## $estimations
## n predictions_j1 predictions_j2 p_dual_j1 p_dual_j2 e_dual_j1
## 1 1 0.6618706 0.32607778 0.5718150 0.4281850 9.005565e-02
## 2 2 0.5411147 0.58480211 0.4906148 0.5093852 5.049989e-02
## 3 3 0.4989333 0.49987851 0.4997974 0.5002026 -8.641873e-04
## 4 4 0.5983162 0.18177093 0.5896916 0.4103084 8.624687e-03
## 5 5 0.5626436 0.20232164 0.5777308 0.4222692 -1.508722e-02
## 6 6 0.5422113 0.42645192 0.5248037 0.4751963 1.740762e-02
## 7 7 0.5894027 0.35647775 0.5498886 0.4501114 3.951410e-02
## 8 8 0.6786707 0.34359141 0.5716670 0.4283330 1.070036e-01
## 9 9 0.6591332 0.02952256 0.6361413 0.3638587 2.299191e-02
## 10 10 0.4889157 0.44874183 0.5086138 0.4913862 -1.969814e-02
## 11 11 0.6691102 0.29957043 0.5790099 0.4209901 9.010037e-02
## 12 12 0.5054964 0.45614111 0.5105782 0.4894218 -5.081802e-03
## 13 13 0.6452125 0.55359745 0.5197544 0.4802456 1.254581e-01
## 14 14 0.6017768 0.22285204 0.5813991 0.4186009 2.037766e-02
## 15 15 0.4309197 0.58967130 0.4659945 0.5340055 -3.507476e-02
## 16 16 0.4803617 0.36753580 0.5242618 0.4757382 -4.390017e-02
## 17 17 0.5002174 0.50006363 0.5000329 0.4999671 1.844253e-04
## 18 18 0.4923101 0.51597537 0.4949289 0.5050711 -2.618736e-03
## 19 19 0.6228203 0.39834679 0.5480625 0.4519375 7.475782e-02
## 20 20 0.4834905 0.43402126 0.5106117 0.4893883 -2.712117e-02
## 21 21 0.3791572 0.41703664 0.4918287 0.5081713 -1.126715e-01
## 22 22 0.5805534 0.67002077 0.4806358 0.5193642 9.991758e-02
## 23 23 0.5098109 0.47695728 0.5070401 0.4929599 2.770807e-03
## 24 24 0.5465811 0.55831032 0.4974822 0.5025178 4.909895e-02
## 25 25 0.5370076 0.22700596 0.5669189 0.4330811 -2.991124e-02
## 26 26 0.5100419 0.38072899 0.5277551 0.4722449 -1.771318e-02
## 27 27 0.6476888 0.13579325 0.6100275 0.3899725 3.766131e-02
## 28 28 0.5354230 0.42543051 0.5235706 0.4764294 1.185240e-02
## 29 29 0.5358292 0.46136053 0.5159560 0.4840440 1.987314e-02
## 30 30 0.5408979 0.50964467 0.5066998 0.4933002 3.419816e-02
## 31 31 0.6845505 0.25506116 0.5917895 0.4082105 9.276097e-02
## 32 32 0.3956760 0.41934521 0.4949001 0.5050999 -9.922402e-02
## 33 33 0.4133952 0.16567445 0.5545815 0.4454185 -1.411863e-01
## 34 34 0.4765755 0.40131500 0.5161644 0.4838356 -3.958887e-02
## 35 35 0.5379471 0.45932977 0.5168448 0.4831552 2.110234e-02
## 36 36 0.5272205 0.46725380 0.5128493 0.4871507 1.437120e-02
## 37 37 0.4806397 0.50656707 0.4944440 0.5055560 -1.380438e-02
## 38 38 0.5006645 0.49394174 0.5014406 0.4985594 -7.760580e-04
## 39 39 0.4982605 0.47036197 0.5059791 0.4940209 -7.718630e-03
## 40 40 0.6878697 0.31543882 0.5796125 0.4203875 1.082573e-01
## 41 41 0.5812498 0.21851009 0.5780571 0.4219429 3.192695e-03
## 42 42 0.5133029 0.44763150 0.5140748 0.4859252 -7.719531e-04
## 43 43 0.4384496 0.52405261 0.4816547 0.5183453 -4.320507e-02
## 44 44 0.3800996 0.47320429 0.4799828 0.5200172 -9.988322e-02
## 45 45 0.6195057 0.07057342 0.6188697 0.3811303 6.360590e-04
## 46 46 0.6870533 0.31041675 0.5805070 0.4194930 1.065463e-01
## 47 47 0.4920137 0.44985832 0.5090379 0.4909621 -1.702420e-02
## 48 48 0.4045166 0.42026764 0.4966081 0.5033919 -9.209149e-02
## 49 49 0.4405209 0.23803502 0.5440915 0.4559085 -1.035705e-01
## 50 50 0.6104348 -0.04188454 0.6431120 0.3568880 -3.267722e-02
## 51 51 0.4087599 0.43861016 0.4935796 0.5064204 -8.481968e-02
## 52 52 0.5152489 0.50228641 0.5027778 0.4972222 1.247110e-02
## 53 53 0.5259218 0.55101338 0.4946182 0.5053818 3.130362e-02
## 54 54 0.5135431 0.45958700 0.5115628 0.4884372 1.980348e-03
## 55 55 0.5093622 0.50692759 0.5005217 0.4994783 8.840513e-03
## 56 56 0.7031336 0.21855675 0.6035124 0.3964876 9.962120e-02
## 57 57 0.5148976 0.50706170 0.5016792 0.4983208 1.321834e-02
## 58 58 0.4785499 0.70215903 0.4518725 0.5481275 2.667742e-02
## 59 59 0.4207752 0.18141389 0.5525787 0.4474213 -1.318035e-01
## 60 60 0.5929126 0.42054435 0.5369179 0.4630821 5.599467e-02
## 61 61 0.5646979 0.19618188 0.5795155 0.4204845 -1.481762e-02
## 62 62 0.5011501 0.49808345 0.5006571 0.4993429 4.929624e-04
## 63 63 0.5223271 0.49493067 0.5058709 0.4941291 1.645623e-02
## 64 64 0.4804011 0.55827000 0.4833116 0.5166884 -2.910451e-03
## 65 65 0.5855653 0.67975800 0.4795882 0.5204118 1.059771e-01
## 66 66 0.5182645 0.43226622 0.5184331 0.4815669 -1.685588e-04
## 67 67 0.5377520 0.42220693 0.5247603 0.4752397 1.299175e-02
## 68 68 0.5967803 0.48411158 0.5241634 0.4758366 7.261695e-02
## 69 69 0.6492062 0.49286747 0.5335959 0.4664041 1.156102e-01
## 70 70 0.5938180 0.47818635 0.5247931 0.4752069 6.902488e-02
## 71 71 0.5164302 0.49360890 0.5048903 0.4951097 1.153988e-02
## 72 72 0.5365424 0.57544462 0.4916472 0.5083528 4.489518e-02
## 73 73 0.4192393 0.46145869 0.4909325 0.5090675 -7.169325e-02
## 74 74 0.5571798 0.14936815 0.5883665 0.4116335 -3.118677e-02
## 75 75 0.6472341 0.01521104 0.6370446 0.3629554 1.018953e-02
## 76 76 0.5176452 0.47476415 0.5091886 0.4908114 8.456638e-03
## 77 77 0.5292074 0.44563392 0.5179082 0.4820918 1.129918e-02
## 78 78 0.5271962 0.35197533 0.5376155 0.4623845 -1.041937e-02
## 79 79 0.6161820 0.06868122 0.6186269 0.3813731 -2.444931e-03
## 80 80 0.4441683 0.45126331 0.4984770 0.5015230 -5.430865e-02
## 81 81 0.4434805 0.52711745 0.4820775 0.5179225 -3.859700e-02
## 82 82 0.5753677 0.36312167 0.5454731 0.4545269 2.989452e-02
## 83 83 0.5973891 0.41143731 0.5398232 0.4601768 5.756594e-02
## 84 84 0.5472715 0.55946875 0.4973815 0.5026185 4.989002e-02
## 85 85 0.5134243 0.28326472 0.5496045 0.4503955 -3.618018e-02
## 86 86 0.4762569 0.67752946 0.4567360 0.5432640 1.952090e-02
## 87 87 0.4915158 0.40568059 0.5184222 0.4815778 -2.690637e-02
## 88 88 0.5605823 0.59536936 0.4925166 0.5074834 6.806579e-02
## 89 89 0.5322829 0.49401794 0.5082003 0.4917997 2.408256e-02
## 90 90 0.4668448 0.65686314 0.4592070 0.5407930 7.637786e-03
## 91 91 0.6824468 0.39792177 0.5609467 0.4390533 1.215001e-01
## 92 92 0.6768850 0.47012032 0.5444282 0.4555718 1.324568e-01
## 93 93 0.6269196 0.29370357 0.5713371 0.4286629 5.558254e-02
## 94 94 0.5485989 0.23816950 0.5668995 0.4331005 -1.830064e-02
## 95 95 0.4278086 0.21479446 0.5465662 0.4534338 -1.187576e-01
## 96 96 0.6542445 0.38084181 0.5585212 0.4414788 9.572333e-02
## 97 97 0.5208693 0.52932601 0.4981871 0.5018129 2.268216e-02
## 98 98 0.5218216 0.54281400 0.4954986 0.5045014 2.632297e-02
## 99 99 0.5196272 0.34149919 0.5382679 0.4617321 -1.864065e-02
## 100 100 0.4800783 0.37661115 0.5222408 0.4777592 -4.216253e-02
## 101 101 0.4809483 0.42489925 0.5120270 0.4879730 -3.107868e-02
## 102 102 0.4846222 0.60915609 0.4732836 0.5267164 1.133859e-02
## 103 103 0.3317776 0.31309996 0.5040850 0.4959150 -1.723074e-01
## 104 104 0.7490548 0.30014866 0.5958912 0.4041088 1.531636e-01
## 105 105 0.5633193 0.25473414 0.5663699 0.4336301 -3.050634e-03
## 106 106 0.4921636 0.46938254 0.5048828 0.4951172 -1.271914e-02
## 107 107 0.4575733 0.60519482 0.4683588 0.5316412 -1.078554e-02
## 108 108 0.4249129 0.46159360 0.4921237 0.5078763 -6.721085e-02
## 109 109 0.4868086 0.41268823 0.5159072 0.4840928 -2.909866e-02
## 110 110 0.4856726 0.71664423 0.4502240 0.5497760 3.544855e-02
## 111 111 0.5444285 0.19791847 0.5748998 0.4251002 -3.047126e-02
## 112 112 0.5668816 0.28656606 0.5601936 0.4398064 6.688013e-03
## 113 113 0.5420407 0.14092562 0.5871292 0.4128708 -4.508853e-02
## 114 114 0.6597339 0.05619730 0.6302223 0.3697777 2.951165e-02
## 115 115 0.5104727 0.50972188 0.5001609 0.4998391 1.031184e-02
## 116 116 0.5537756 0.60659970 0.4886341 0.5113659 6.514148e-02
## 117 117 0.5153668 0.51668503 0.4997175 0.5002825 1.564930e-02
## 118 118 0.5045809 0.49623470 0.5017885 0.4982115 2.792442e-03
## 119 119 0.5049597 0.47739851 0.5059062 0.4940938 -9.464650e-04
## 120 120 0.5161851 0.52383897 0.4983595 0.5016405 1.782560e-02
## 121 121 0.6498476 0.30157816 0.5744988 0.4255012 7.534881e-02
## 122 122 0.4679312 0.58504929 0.4748979 0.5251021 -6.966784e-03
## 123 123 0.4910321 0.39385596 0.5208645 0.4791355 -2.983234e-02
## 124 124 0.6795618 0.42755345 0.5540417 0.4459583 1.255201e-01
## 125 125 0.5001002 0.49977080 0.5000706 0.4999294 2.960500e-05
## 126 126 0.5041604 0.49225751 0.5025506 0.4974494 1.609743e-03
## 127 127 0.5419587 0.42427823 0.5252158 0.4747842 1.674296e-02
## 128 128 0.4608946 0.32807966 0.5286567 0.4713433 -6.776213e-02
## 129 129 0.5439464 0.37882255 0.5354008 0.4645992 8.545637e-03
## 130 130 0.4976340 0.46192186 0.5076544 0.4923456 -1.002042e-02
## 131 131 0.5006555 0.49770091 0.5006331 0.4993669 2.240824e-05
## 132 132 0.5037854 0.49379494 0.5021408 0.4978592 1.644600e-03
## 133 133 0.6651801 0.24941189 0.5889280 0.4110720 7.625214e-02
## 134 134 0.4416449 0.25336734 0.5409325 0.4590675 -9.928760e-02
## 135 135 0.6248079 0.41752660 0.5443968 0.4556032 8.041106e-02
## 136 136 0.5274883 0.41590775 0.5239172 0.4760828 3.571059e-03
## 137 137 0.4999418 0.46217011 0.5080956 0.4919044 -8.153783e-03
## 138 138 0.5282111 0.42096378 0.5229865 0.4770135 5.224629e-03
## 139 139 0.6531569 0.46099971 0.5412365 0.4587635 1.119204e-01
## 140 140 0.5241047 0.50561998 0.5039615 0.4960385 2.014314e-02
## 141 141 0.4972872 0.74208616 0.4471098 0.5528902 5.017741e-02
## 142 142 0.5297521 0.47117870 0.5125507 0.4874493 1.720139e-02
## 143 143 0.5455059 0.22201535 0.5698018 0.4301982 -2.429592e-02
## 144 144 0.7720348 0.44703157 0.5700579 0.4299421 2.019769e-01
## 145 145 0.6845455 0.50543393 0.5385836 0.4614164 1.459619e-01
## 146 146 0.5240592 0.09569097 0.5936936 0.4063064 -6.963438e-02
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## 167 167 0.4638191 0.31509747 0.5321089 0.4678911 -6.828987e-02
## 168 168 0.3872756 0.47194949 0.4817999 0.5182001 -9.452428e-02
## 169 169 0.5217525 0.50364479 0.5038806 0.4961194 1.787187e-02
## 170 170 0.4540933 0.30726350 0.5317423 0.4682577 -7.764902e-02
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## 182 182 0.5404502 0.38710619 0.5328737 0.4671263 7.576539e-03
## 183 183 0.4143352 0.54828195 0.4712997 0.5287003 -5.696446e-02
## 184 184 0.4057887 0.49675317 0.4804806 0.5195194 -7.469192e-02
## 185 185 0.5208411 0.41577189 0.5225250 0.4774750 -1.683969e-03
## 186 186 0.4935472 0.67334425 0.4613205 0.5386795 3.222670e-02
## 187 187 0.5105575 0.49733432 0.5028336 0.4971664 7.723931e-03
## 188 188 0.5005044 0.50103803 0.4998857 0.5001143 6.187711e-04
## 189 189 0.5011835 0.60706801 0.4772721 0.5227279 2.391138e-02
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## 193 193 0.5280286 0.46351537 0.5138234 0.4861766 1.420516e-02
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## 200 200 0.5399088 0.58225243 0.4909048 0.5090952 4.900393e-02
## e_dual_j2
## 1 -1.021072e-01
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## 42 -3.829367e-02
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## 79 -3.126919e-01
## 80 -5.025972e-02
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## 82 -9.140520e-02
## 83 -4.873949e-02
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## 85 -1.671308e-01
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## 91 -4.113151e-02
## 92 1.454852e-02
## 93 -1.349593e-01
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## 99 -1.202329e-01
## 100 -1.011480e-01
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## 102 8.243971e-02
## 103 -1.828151e-01
## 104 -1.039601e-01
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## 113 -2.719451e-01
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## 128 -1.432636e-01
## 129 -8.577666e-02
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## 131 -1.665957e-03
## 132 -4.064248e-03
## 133 -1.616601e-01
## 134 -2.057002e-01
## 135 -3.807658e-02
## 136 -6.017502e-02
## 137 -2.973429e-02
## 138 -5.604972e-02
## 139 2.236169e-03
## 140 9.581526e-03
## 141 1.891959e-01
## 142 -1.627062e-02
## 143 -2.081829e-01
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## 145 4.401753e-02
## 146 -3.106155e-01
## 147 -6.943131e-02
## 148 -1.329255e-01
## 149 -8.113299e-02
## 150 1.201753e-01
## 151 -6.057762e-02
## 152 -4.658918e-02
## 153 -1.617401e-01
## 154 -2.433848e-01
## 155 3.453417e-04
## 156 -1.388421e-01
## 157 -3.080909e-01
## 158 -8.994929e-02
## 159 -2.582082e-01
## 160 -3.337329e-02
## 161 -1.749194e-01
## 162 -3.242222e-01
## 163 6.305880e-02
## 164 -4.511094e-02
## 165 -1.028642e-01
## 166 -1.510279e-01
## 167 -1.527936e-01
## 168 -4.625060e-02
## 169 7.525387e-03
## 170 -1.609942e-01
## 171 5.259370e-02
## 172 4.832492e-02
## 173 -8.467875e-02
## 174 -2.507281e-01
## 175 -9.188228e-02
## 176 -1.392743e-01
## 177 4.150899e-02
## 178 -2.931036e-01
## 179 -3.582214e-02
## 180 5.757836e-02
## 181 6.976276e-03
## 182 -8.002014e-02
## 183 1.958164e-02
## 184 -2.276619e-02
## 185 -6.170308e-02
## 186 1.346647e-01
## 187 1.678745e-04
## 188 9.236849e-04
## 189 8.434009e-02
## 190 -1.877609e-01
## 191 -4.665337e-02
## 192 -2.211067e-01
## 193 -2.266122e-02
## 194 -6.602388e-02
## 195 -1.430517e-01
## 196 -1.778412e-05
## 197 -2.926160e-02
## 198 -8.235980e-02
## 199 -4.337447e-02
## 200 7.315726e-02
##
## $values
## $values$entropy
## [1] 573.5472
##
## $values$iterations
## function gradient
## 1001 NA
##
## $values$message
## NULL
##
## $values$method
## [1] "BFGS"
##
##
## $tol
## [1] 1e-24
##
## $v
## [,1]
## [1,] -1
## [2,] 0
## [3,] 1
##
## $lambda
## X1 X2 X3 X4
## 1 31.42586 6.169519 -7.450156e-05 -17.51268
## 2 21.79873 -16.245905 -1.337929e-04 24.90991
##
## $checkrestrictions
## $checkrestrictions$g1
## [,1]
## 1 -1.110223e-16
## 2 0.000000e+00
## 3 0.000000e+00
## 4 0.000000e+00
## 5 0.000000e+00
## 6 0.000000e+00
## 7 0.000000e+00
## 8 0.000000e+00
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## 10 0.000000e+00
## 11 0.000000e+00
## 12 -1.110223e-16
## 13 0.000000e+00
## 14 0.000000e+00
## 15 -2.220446e-16
## 16 0.000000e+00
## 17 0.000000e+00
## 18 0.000000e+00
## 19 0.000000e+00
## 20 -1.110223e-16
## 21 0.000000e+00
## 22 0.000000e+00
## 23 0.000000e+00
## 24 0.000000e+00
## 25 0.000000e+00
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## 27 0.000000e+00
## 28 0.000000e+00
## 29 0.000000e+00
## 30 0.000000e+00
## 31 0.000000e+00
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## 103 -1.110223e-16
## 104 0.000000e+00
## 105 0.000000e+00
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## 110 -1.110223e-16
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## 112 0.000000e+00
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## 114 0.000000e+00
## 115 0.000000e+00
## 116 -2.220446e-16
## 117 -1.110223e-16
## 118 0.000000e+00
## 119 0.000000e+00
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## 126 0.000000e+00
## 127 -1.110223e-16
## 128 0.000000e+00
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## 131 0.000000e+00
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## 135 2.220446e-16
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## 141 -1.110223e-16
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## 144 2.220446e-16
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## 155 -1.110223e-16
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## 162 0.000000e+00
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## 165 0.000000e+00
## 166 -1.110223e-16
## 167 0.000000e+00
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## 171 0.000000e+00
## 172 -1.110223e-16
## 173 0.000000e+00
## 174 0.000000e+00
## 175 0.000000e+00
## 176 0.000000e+00
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## 180 0.000000e+00
## 181 -1.110223e-16
## 182 0.000000e+00
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## 184 -1.110223e-16
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## 187 -1.110223e-16
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## 200 0.000000e+00
##
## $checkrestrictions$g2
## [,1] [,2]
## 1 0.000000e+00 -1.110223e-16
## 2 0.000000e+00 0.000000e+00
## 3 0.000000e+00 0.000000e+00
## 4 0.000000e+00 0.000000e+00
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## 14 0.000000e+00 0.000000e+00
## 15 0.000000e+00 0.000000e+00
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## 18 0.000000e+00 0.000000e+00
## 19 0.000000e+00 0.000000e+00
## 20 0.000000e+00 0.000000e+00
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## 25 0.000000e+00 0.000000e+00
## 26 0.000000e+00 0.000000e+00
## 27 0.000000e+00 -1.110223e-16
## 28 0.000000e+00 0.000000e+00
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## 30 -1.110223e-16 0.000000e+00
## 31 -1.110223e-16 0.000000e+00
## 32 -1.110223e-16 -1.110223e-16
## 33 0.000000e+00 0.000000e+00
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## 35 0.000000e+00 0.000000e+00
## 36 0.000000e+00 0.000000e+00
## 37 0.000000e+00 -1.110223e-16
## 38 0.000000e+00 0.000000e+00
## 39 0.000000e+00 0.000000e+00
## 40 0.000000e+00 0.000000e+00
## 41 2.220446e-16 -1.110223e-16
## 42 0.000000e+00 0.000000e+00
## 43 0.000000e+00 0.000000e+00
## 44 2.220446e-16 0.000000e+00
## 45 0.000000e+00 0.000000e+00
## 46 0.000000e+00 0.000000e+00
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## 51 0.000000e+00 -1.110223e-16
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## 90 0.000000e+00 0.000000e+00
## 91 0.000000e+00 -2.220446e-16
## 92 2.220446e-16 0.000000e+00
## 93 0.000000e+00 0.000000e+00
## 94 0.000000e+00 -1.110223e-16
## 95 0.000000e+00 0.000000e+00
## 96 0.000000e+00 -1.110223e-16
## 97 0.000000e+00 0.000000e+00
## 98 0.000000e+00 0.000000e+00
## 99 -1.110223e-16 0.000000e+00
## 100 -1.110223e-16 0.000000e+00
## 101 0.000000e+00 -1.110223e-16
## 102 0.000000e+00 0.000000e+00
## 103 0.000000e+00 0.000000e+00
## 104 0.000000e+00 0.000000e+00
## 105 0.000000e+00 0.000000e+00
## 106 0.000000e+00 0.000000e+00
## 107 -2.220446e-16 0.000000e+00
## 108 -1.110223e-16 0.000000e+00
## 109 0.000000e+00 0.000000e+00
## 110 0.000000e+00 0.000000e+00
## 111 0.000000e+00 0.000000e+00
## 112 2.220446e-16 0.000000e+00
## 113 0.000000e+00 0.000000e+00
## 114 0.000000e+00 0.000000e+00
## 115 0.000000e+00 0.000000e+00
## 116 0.000000e+00 0.000000e+00
## 117 0.000000e+00 0.000000e+00
## 118 0.000000e+00 -1.110223e-16
## 119 0.000000e+00 -1.110223e-16
## 120 0.000000e+00 0.000000e+00
## 121 0.000000e+00 0.000000e+00
## 122 0.000000e+00 0.000000e+00
## 123 0.000000e+00 0.000000e+00
## 124 2.220446e-16 0.000000e+00
## 125 0.000000e+00 0.000000e+00
## 126 0.000000e+00 -1.110223e-16
## 127 0.000000e+00 0.000000e+00
## 128 0.000000e+00 0.000000e+00
## 129 0.000000e+00 2.220446e-16
## 130 0.000000e+00 0.000000e+00
## 131 0.000000e+00 -1.110223e-16
## 132 0.000000e+00 -2.220446e-16
## 133 0.000000e+00 0.000000e+00
## 134 -2.220446e-16 0.000000e+00
## 135 0.000000e+00 0.000000e+00
## 136 0.000000e+00 0.000000e+00
## 137 2.220446e-16 0.000000e+00
## 138 0.000000e+00 -1.110223e-16
## 139 0.000000e+00 0.000000e+00
## 140 0.000000e+00 0.000000e+00
## 141 0.000000e+00 0.000000e+00
## 142 -1.110223e-16 0.000000e+00
## 143 0.000000e+00 2.220446e-16
## 144 0.000000e+00 0.000000e+00
## 145 0.000000e+00 0.000000e+00
## 146 -1.110223e-16 0.000000e+00
## 147 0.000000e+00 0.000000e+00
## 148 0.000000e+00 -1.110223e-16
## 149 0.000000e+00 0.000000e+00
## 150 0.000000e+00 0.000000e+00
## 151 0.000000e+00 0.000000e+00
## 152 -1.110223e-16 0.000000e+00
## 153 0.000000e+00 -2.220446e-16
## 154 0.000000e+00 0.000000e+00
## 155 2.220446e-16 0.000000e+00
## 156 0.000000e+00 0.000000e+00
## 157 0.000000e+00 0.000000e+00
## 158 0.000000e+00 -1.110223e-16
## 159 0.000000e+00 0.000000e+00
## 160 0.000000e+00 0.000000e+00
## 161 0.000000e+00 0.000000e+00
## 162 0.000000e+00 -1.110223e-16
## 163 0.000000e+00 0.000000e+00
## 164 0.000000e+00 0.000000e+00
## 165 -2.220446e-16 0.000000e+00
## 166 0.000000e+00 0.000000e+00
## 167 0.000000e+00 0.000000e+00
## 168 -2.220446e-16 0.000000e+00
## 169 0.000000e+00 0.000000e+00
## 170 0.000000e+00 0.000000e+00
## 171 0.000000e+00 0.000000e+00
## 172 -1.110223e-16 0.000000e+00
## 173 0.000000e+00 0.000000e+00
## 174 0.000000e+00 -1.110223e-16
## 175 0.000000e+00 0.000000e+00
## 176 0.000000e+00 0.000000e+00
## 177 -1.110223e-16 0.000000e+00
## 178 -1.110223e-16 0.000000e+00
## 179 0.000000e+00 -1.110223e-16
## 180 0.000000e+00 0.000000e+00
## 181 0.000000e+00 0.000000e+00
## 182 0.000000e+00 0.000000e+00
## 183 0.000000e+00 0.000000e+00
## 184 0.000000e+00 0.000000e+00
## 185 0.000000e+00 0.000000e+00
## 186 0.000000e+00 0.000000e+00
## 187 0.000000e+00 0.000000e+00
## 188 0.000000e+00 0.000000e+00
## 189 0.000000e+00 0.000000e+00
## 190 0.000000e+00 0.000000e+00
## 191 0.000000e+00 -2.220446e-16
## 192 0.000000e+00 0.000000e+00
## 193 0.000000e+00 0.000000e+00
## 194 0.000000e+00 2.220446e-16
## 195 -1.110223e-16 0.000000e+00
## 196 0.000000e+00 0.000000e+00
## 197 -2.220446e-16 -2.220446e-16
## 198 0.000000e+00 0.000000e+00
## 199 0.000000e+00 0.000000e+00
## 200 0.000000e+00 2.220446e-16
##
## $checkrestrictions$g3
## factor(y)0 factor(y)1
## (Intercept) 1.214107e-01 -0.03389694
## Dcollege 1.440984e-01 -0.00305705
## Totalincome 1.421231e+04 -751.50348277
## Dunemp 6.863777e-02 -0.05350674
##
##
## $cross_moments_hp
## factor(y)0 factor(y)1
## (Intercept) 6.622217e-01 3.377783e-01
## Dcollege 4.376594e-01 1.663265e-01
## Totalincome 1.895948e+05 1.030366e+05
## Dunemp 2.859345e-01 1.622465e-01
##
## $cross_moments_hs
## [,1] [,2]
## (Intercept) 5.408111e-01 3.716752e-01
## Dcollege 2.935610e-01 1.693835e-01
## Totalincome 1.753825e+05 1.037881e+05
## Dunemp 2.172968e-01 2.157533e-01
##
## $J
## [1] 2
##
## $fn
## datahp$poor_liq ~ Dcollege + Totalincome + Dunemp
##
## attr(,"class")
## [1] "shannon"
To make the results more visual, this package includes a personalized
summary function which will resume the main results, providing the means
for each category \(j\) for the
predictions, the probabilities and the error.
## Iterationsfunction
## 1001
## Entropy value[1] 573.5472
## [1] "mean_estimations"
## n predictions_j1 predictions_j2 p_dual_j1 p_dual_j2 e_dual_j1
## 1 100.5 0.5311341 0.4081335 0.5265115 0.4734885 0.004622612
## e_dual_j2
## 1 -0.06535502
## [1] "lambda"
## intercept Dcollege Totalincome Dunemp
## j1 31.42586 6.169519 -7.450156e-05 -17.51268
## j2 21.79873 -16.245905 -1.337929e-04 24.90991
Graphs are generated with the plot function, showing the averages of
the predictions for each territorial unit and the 95% confidence
interval associated with each of them.
plot(x=result3, reg=datahs$reg)
References
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.