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. Sometimes, 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 estimates the variable of interest using a second data
frame that provides more detailed geographical information, ensuring
consistency with the aggregates available in the first database. 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
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 while 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 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
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 in
the optimization.
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 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.
$$\min_{P} \, \text{KL}(P, Q) = \sum_{i=1}^{n} \sum_{j=1}^{J} p_{ij}
\log\left(\frac{p_{ij}}{q_{ij}}\right)+ \sum_{l=1}^{L} \sum_{i=1}^{n}
\sum_{j=1}^{J} w_{ijl} \log\left(\frac{w_{ijl}}{wo_{ijl}}\right)\quad
\text{s.t.}$$
\[
\frac{1}{n} \sum_{i=1}^{n} x_{s_{ik}} y_{ij} = \frac{1}{n}
\sum_{i=1}^{n} x_{c_{ik}} (p_{ij} + u_{ij}) =\frac{1}{n} \sum_{i=1}^{n}
x_{cik} [p_{ij} + w_{Lij} v_{l}];
j=1,\ldots,J
\]
\] \[\[
\sum_{j=1}^{J} p_{ij} = 1 \quad \text{for } i=1,\ldots,n
\]
\] \[\[
\sum_{l=1}^{L} w_{Lij} = 1 \quad \text{for } i=1,\ldots,n;
j=1,\ldots,J
\]
\]
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, dataA and
dataB. 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 the regional level. We have
the variable of interest in one official database but without
information about location. At the same time, in another 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().
dataA <- financial()
dataB <- 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 <- dataA$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, the maximum number of
iterations allowed and the support vector. With this function, weights
can be used and included with weights. 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(0.2,0.8)
v <- matrix(c(1,0,-1),nrow=1)
In this example we assume a priori distribution of poverty equal to
0.2 for poor and 0.8 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,dataA,dataB,q=q,weights = "w",v=v)
result
## $estimations
## weights predictions_1 predictions_0 probabilities_1 probabilities_0
## 1 6.408614e-03 0.3526612 1.03997932 0.1856802 0.8143198
## 2 4.069074e-03 0.3611852 1.00659540 0.1925139 0.8074861
## 3 8.478890e-05 0.2018007 0.79389077 0.2012848 0.7987152
## 4 6.019554e-03 0.2482836 0.77140431 0.2126642 0.7873358
## 5 8.529959e-03 0.3377428 0.27358677 0.3258196 0.6741804
## 6 3.466777e-03 0.2798052 0.66129187 0.2370367 0.7629633
## 7 7.143169e-03 0.3733840 0.52415554 0.2796025 0.7203975
## 8 6.728993e-03 0.3756344 1.09117708 0.1809317 0.8190683
## 9 9.459067e-03 0.2877814 0.78702116 0.2166944 0.7833056
## 10 1.952733e-03 0.2226752 0.75146695 0.2117173 0.7882827
## 11 6.884772e-03 0.3555312 1.03582456 0.1868151 0.8131849
## 12 1.046638e-03 0.2106771 0.72337741 0.2144218 0.7855782
## 13 6.632154e-03 0.4639106 0.85053121 0.2368135 0.7631865
## 14 5.694566e-03 0.2609696 0.81452272 0.2076120 0.7923880
## 15 8.199805e-03 0.4555912 0.36432259 0.3283205 0.6716795
## 16 7.974757e-03 0.3397106 0.72431237 0.2364912 0.7635088
## 17 1.016052e-05 0.2003965 0.80005099 0.2000560 0.7999440
## 18 1.077733e-03 0.2337937 0.75041296 0.2137528 0.7862472
## 19 4.555596e-03 0.3224837 1.00715977 0.1862268 0.8137732
## 20 1.797040e-03 0.2095780 0.72479111 0.2139996 0.7860004
## 21 7.726297e-03 0.3498615 0.14997501 0.3584095 0.6415905
## 22 7.537852e-03 0.4983089 1.18914834 0.1841440 0.8158560
## 23 5.078895e-04 0.2066423 0.80453438 0.2003420 0.7996580
## 24 8.673474e-03 0.4659097 1.04261104 0.2039921 0.7960079
## 25 6.743411e-03 0.2848000 0.32400748 0.3043771 0.6956229
## 26 2.597985e-03 0.2216336 0.59251985 0.2391501 0.7608499
## 27 7.946911e-03 0.2965673 0.85155155 0.2073982 0.7926018
## 28 3.132542e-03 0.2671697 0.66162186 0.2347626 0.7652374
## 29 1.413325e-03 0.2339130 0.85377483 0.1967876 0.8032124
## 30 1.952905e-03 0.2747870 0.80538222 0.2114290 0.7885710
## 31 7.786354e-03 0.3641713 1.03627013 0.1881235 0.8118765
## 32 6.412179e-03 0.3141028 0.23559225 0.3302812 0.6697188
## 33 8.813903e-03 0.2427698 0.40473956 0.2792660 0.7207340
## 34 3.104395e-03 0.2259749 0.69449306 0.2219212 0.7780788
## 35 1.494452e-03 0.2359713 0.85715421 0.1965743 0.8034257
## 36 1.979537e-03 0.2505294 0.73628529 0.2189597 0.7810403
## 37 1.781772e-03 0.2445111 0.68548055 0.2266515 0.7733485
## 38 8.631740e-04 0.2199829 0.80551695 0.2023531 0.7976469
## 39 2.624354e-03 0.2540498 0.79849383 0.2091165 0.7908835
## 40 7.278409e-03 0.3803848 1.09029772 0.1818259 0.8181741
## 41 9.198636e-03 0.3721894 0.29280984 0.3279855 0.6720145
## 42 8.939710e-04 0.2045391 0.78882755 0.2025561 0.7974439
## 43 5.755561e-03 0.3538831 0.43337648 0.2940743 0.7059257
## 44 9.197598e-03 0.4221829 0.13263798 0.3767581 0.6232419
## 45 7.835590e-03 0.2497892 0.72658450 0.2204847 0.7795153
## 46 7.304276e-03 0.3784180 1.08466672 0.1824422 0.8175578
## 47 2.825585e-03 0.2473003 0.76871006 0.2129502 0.7870498
## 48 5.701607e-03 0.2950633 0.28273923 0.3156210 0.6843790
## 49 9.403839e-03 0.3027989 0.53329222 0.2647413 0.7352587
## 50 8.874280e-03 0.2153520 0.60381911 0.2360101 0.7639899
## 51 5.856153e-03 0.3093848 0.29888952 0.3146391 0.6853609
## 52 7.506319e-04 0.2278652 0.79982179 0.2045744 0.7954256
## 53 2.829871e-03 0.3079285 0.93203077 0.1960680 0.8039320
## 54 1.403796e-03 0.2254491 0.72701196 0.2162915 0.7837085
## 55 2.256222e-03 0.2659105 0.85121088 0.2023972 0.7976028
## 56 8.706739e-03 0.3780882 1.05008290 0.1880874 0.8119126
## 57 6.495227e-04 0.2271737 0.80808595 0.2031082 0.7968918
## 58 7.423215e-03 0.4947906 0.60169178 0.2880435 0.7119565
## 59 9.409367e-03 0.2688901 0.43967038 0.2769284 0.7230716
## 60 3.467645e-03 0.2922539 0.95545377 0.1897663 0.8102337
## 61 4.891585e-03 0.2119124 0.70263851 0.2181418 0.7818582
## 62 5.191919e-05 0.2009450 0.80117419 0.1999628 0.8000372
## 63 6.960881e-04 0.2251791 0.84905269 0.1961436 0.8038564
## 64 3.234067e-03 0.3093491 0.66643422 0.2413431 0.7586569
## 65 8.095405e-03 0.5170827 1.21099845 0.1834243 0.8165757
## 66 2.123864e-03 0.2349545 0.67860615 0.2262012 0.7737988
## 67 3.309518e-03 0.2716568 0.65532966 0.2366639 0.7633361
## 68 5.300329e-03 0.3787056 0.74454595 0.2398998 0.7601002
## 69 7.890242e-03 0.4749215 0.74281574 0.2580209 0.7419791
## 70 5.248723e-03 0.3735130 0.73513441 0.2406247 0.7593753
## 71 9.625057e-04 0.2301381 0.78463504 0.2074481 0.7925519
## 72 3.609097e-03 0.3434140 0.98412877 0.1933175 0.8066825
## 73 5.617329e-03 0.3155370 0.34811668 0.3048609 0.6951391
## 74 9.237887e-03 0.3340911 0.19039502 0.3451287 0.6548713
## 75 9.275353e-03 0.2707651 0.74512726 0.2208981 0.7791019
## 76 7.561670e-04 0.2153829 0.82142421 0.1990213 0.8009787
## 77 1.374263e-03 0.2227906 0.82514861 0.1996176 0.8003824
## 78 2.232129e-03 0.2006943 0.74323505 0.2094332 0.7905668
## 79 7.759625e-03 0.2454837 0.71683753 0.2214161 0.7785839
## 80 3.268253e-03 0.2485843 0.49296628 0.2626450 0.7373550
## 81 5.423895e-03 0.3468515 0.46127656 0.2870675 0.7129325
## 82 3.519797e-03 0.2595236 0.86757856 0.1986919 0.8013081
## 83 3.686690e-03 0.2955618 0.95862538 0.1897817 0.8102183
## 84 8.770650e-03 0.4690099 1.04597156 0.2039524 0.7960476
## 85 4.499834e-03 0.2341002 0.42601309 0.2733101 0.7266899
## 86 6.913635e-03 0.4683562 0.59760543 0.2836355 0.7163645
## 87 7.423989e-03 0.3468818 0.77757893 0.2284887 0.7715113
## 88 9.184997e-03 0.4987431 1.10650907 0.1986493 0.8013507
## 89 9.934938e-04 0.2367018 0.87195615 0.1943118 0.8056882
## 90 7.108557e-03 0.4622404 0.54914224 0.2917684 0.7082316
## 91 6.307975e-03 0.3911718 1.13991627 0.1753216 0.8246784
## 92 5.444530e-03 0.3998260 1.18037638 0.1699324 0.8300676
## 93 5.708904e-03 0.3052747 0.93315246 0.1954563 0.8045437
## 94 7.141561e-03 0.3061271 0.33806639 0.3052852 0.6947148
## 95 7.903713e-03 0.2472353 0.47049796 0.2667997 0.7332003
## 96 5.645807e-03 0.3552547 1.06499349 0.1820116 0.8179884
## 97 3.539554e-03 0.3141657 0.90981346 0.2007132 0.7992868
## 98 2.396317e-03 0.2912799 0.91136892 0.1967303 0.8032697
## 99 3.785424e-03 0.2417960 0.52237233 0.2557632 0.7442368
## 100 6.899727e-03 0.3156277 0.72045937 0.2329166 0.7670834
## 101 1.963958e-03 0.2087034 0.71289891 0.2158593 0.7841407
## 102 4.301069e-03 0.3645696 0.67341825 0.2500181 0.7499819
## 103 8.824290e-03 0.3261669 -0.03881251 0.4082048 0.5917952
## 104 9.245072e-03 0.4464166 1.21046119 0.1720583 0.8279417
## 105 7.604964e-03 0.3329462 0.36007955 0.3055547 0.6944453
## 106 7.759268e-04 0.2027814 0.76416679 0.2063128 0.7936872
## 107 6.426895e-03 0.4162794 0.51408413 0.2897314 0.7102686
## 108 5.150249e-03 0.3031295 0.37982774 0.2957041 0.7042959
## 109 5.158771e-03 0.2891811 0.74918109 0.2233589 0.7766411
## 110 7.247160e-03 0.4980240 0.63910815 0.2815973 0.7184027
## 111 7.664692e-03 0.3025904 0.27356754 0.3191574 0.6808426
## 112 4.033598e-03 0.2334386 0.78399010 0.2081005 0.7918995
## 113 8.610544e-03 0.3053493 0.18321237 0.3414836 0.6585164
## 114 9.173485e-03 0.2940720 0.81201873 0.2135588 0.7864412
## 115 2.312141e-03 0.2691141 0.85671036 0.2020226 0.7979774
## 116 5.785912e-03 0.4188237 1.06871442 0.1916173 0.8083827
## 117 3.133029e-03 0.2956236 0.88240503 0.2021624 0.7978376
## 118 3.027803e-04 0.2084002 0.79234649 0.2026119 0.7973881
## 119 6.494774e-04 0.2093135 0.75999868 0.2080793 0.7919207
## 120 2.627321e-03 0.2861652 0.88500640 0.2001890 0.7998110
## 121 6.300118e-03 0.3336324 0.99310646 0.1902779 0.8097221
## 122 5.009005e-03 0.3685685 0.58307821 0.2674109 0.7325891
## 123 8.546357e-03 0.3704561 0.77785852 0.2325978 0.7674022
## 124 5.935042e-03 0.3940190 1.15535834 0.1732020 0.8267980
## 125 5.670056e-05 0.2014143 0.80063612 0.2001262 0.7998738
## 126 1.956694e-04 0.2032467 0.80358449 0.1999452 0.8000548
## 127 3.491569e-03 0.2794290 0.65760766 0.2376250 0.7623750
## 128 6.018842e-03 0.2627353 0.62510304 0.2404979 0.7595021
## 129 2.453778e-03 0.2259674 0.80548388 0.2033363 0.7966637
## 130 3.330195e-03 0.2683302 0.79726813 0.2117010 0.7882990
## 131 4.568058e-04 0.2111221 0.80439055 0.2010932 0.7989068
## 132 1.878873e-03 0.2477050 0.82329167 0.2039826 0.7960174
## 133 7.274750e-03 0.3405975 0.98709349 0.1923777 0.8076223
## 134 8.055270e-03 0.2745838 0.53634544 0.2590232 0.7409768
## 135 7.918719e-03 0.4353217 0.61834072 0.2733463 0.7266537
## 136 1.613960e-03 0.2144844 0.79665876 0.2029013 0.7970987
## 137 4.043005e-03 0.2879625 0.81027751 0.2128218 0.7871782
## 138 2.836492e-03 0.2538897 0.65611859 0.2334166 0.7665834
## 139 8.648536e-03 0.4846480 0.68513474 0.2704146 0.7295854
## 140 7.121030e-03 0.3961932 0.93421757 0.2103945 0.7896055
## 141 7.015624e-03 0.5058364 0.69956868 0.2718862 0.7281138
## 142 2.042604e-03 0.2551071 0.74232019 0.2187118 0.7812882
## 143 7.276373e-03 0.3018862 0.31256601 0.3101411 0.6898589
## 144 8.504937e-03 0.5003336 1.35088013 0.1555554 0.8444446
## 145 9.535921e-03 0.5385621 0.75287924 0.2685723 0.7314277
## 146 8.545091e-03 0.2764052 0.12028652 0.3523898 0.6476102
## 147 2.254398e-03 0.2121143 0.70556433 0.2176782 0.7823218
## 148 5.514651e-03 0.2562337 0.63694474 0.2372313 0.7627687
## 149 6.254560e-03 0.3619402 1.06753869 0.1826486 0.8173514
## 150 6.128050e-03 0.4379638 0.62340439 0.2729012 0.7270988
## 151 1.952048e-03 0.2101247 0.71743948 0.2153307 0.7846693
## 152 8.239200e-03 0.4406327 0.59736423 0.2783196 0.7216804
## 153 4.212187e-03 0.2279876 0.43988768 0.2693878 0.7306122
## 154 6.562633e-03 0.2531823 0.26134605 0.3127073 0.6872927
## 155 8.529452e-05 0.2030006 0.79954935 0.2005601 0.7994399
## 156 4.627415e-03 0.2576408 0.47604039 0.2676041 0.7323959
## 157 7.506022e-03 0.2401970 0.70922982 0.2218173 0.7781827
## 158 5.802968e-03 0.2936117 0.26531464 0.3193805 0.6806195
## 159 8.889093e-03 0.2602357 0.44721693 0.2738143 0.7261857
## 160 3.338891e-03 0.2617620 0.51924880 0.2599626 0.7400374
## 161 5.739428e-03 0.2712092 0.39425265 0.2867068 0.7132932
## 162 9.051019e-03 0.2815232 0.78083149 0.2166774 0.7833226
## 163 7.187216e-03 0.4751212 0.82312417 0.2437179 0.7562821
## 164 2.371262e-03 0.2282200 0.56049334 0.2461808 0.7538192
## 165 5.791201e-03 0.2845418 0.69915869 0.2312005 0.7687995
## 166 5.959915e-03 0.3025625 0.91951367 0.1972349 0.8027651
## 167 8.379680e-03 0.3201382 0.64428357 0.2472515 0.7527485
## 168 8.564844e-03 0.4032622 0.17058495 0.3633409 0.6366591
## 169 1.064177e-03 0.2397631 0.80037867 0.2064403 0.7935597
## 170 6.110630e-03 0.2528938 0.59245418 0.2446785 0.7553215
## 171 1.771706e-03 0.2798718 0.78408124 0.2158507 0.7841493
## 172 3.746941e-03 0.3272598 0.64516408 0.2483730 0.7516270
## 173 8.344937e-03 0.4199695 0.52340535 0.2886073 0.7113927
## 174 8.671874e-03 0.2596263 0.45848872 0.2714455 0.7285545
## 175 5.165876e-03 0.3075600 0.55592095 0.2613064 0.7386936
## 176 6.250491e-03 0.3127140 0.45204052 0.2825430 0.7174570
## 177 5.663535e-03 0.4123456 0.80308309 0.2357036 0.7642964
## 178 8.189466e-03 0.2759854 0.15405783 0.3433981 0.6566019
## 179 5.172787e-03 0.3452574 0.66206765 0.2485639 0.7514361
## 180 2.048906e-03 0.2901851 0.77536258 0.2190730 0.7809270
## 181 6.680525e-03 0.3810045 0.37593491 0.3113022 0.6886978
## 182 4.062967e-03 0.2784500 0.59382459 0.2489797 0.7510203
## 183 8.424910e-03 0.4398690 0.28689916 0.3424716 0.6575284
## 184 7.709358e-03 0.3922657 0.26015467 0.3394052 0.6605948
## 185 2.544983e-03 0.2403428 0.64941468 0.2322429 0.7677571
## 186 5.364966e-03 0.4269838 0.70328495 0.2561229 0.7438771
## 187 5.934410e-04 0.2193280 0.79261670 0.2043554 0.7956446
## 188 5.020772e-05 0.2020039 0.80257121 0.1999080 0.8000920
## 189 2.873655e-03 0.3291788 0.77215338 0.2263324 0.7736676
## 190 5.514606e-03 0.2547930 0.37431759 0.2878637 0.7121363
## 191 2.057597e-03 0.2227030 0.74558140 0.2127063 0.7872937
## 192 8.173496e-03 0.2669813 0.50831713 0.2630116 0.7369884
## 193 2.086392e-03 0.2521224 0.72960624 0.2203666 0.7796334
## 194 3.212343e-03 0.2666968 0.89458238 0.1954817 0.8045183
## 195 7.999160e-03 0.3703026 0.41959045 0.3000417 0.6999583
## 196 9.251449e-07 0.2000123 0.79998171 0.2000050 0.7999950
## 197 2.268155e-03 0.2405407 0.78202127 0.2096044 0.7903956
## 198 7.367501e-03 0.3409921 0.76424679 0.2297589 0.7702411
## 199 2.263759e-03 0.2265856 0.57070106 0.2440073 0.7559927
## 200 3.956566e-03 0.3566956 1.00071089 0.1927522 0.8072478
## errors_1 errors_0
## 1 1.669810e-01 2.256595e-01
## 2 1.686714e-01 1.991093e-01
## 3 5.159031e-04 -4.824456e-03
## 4 3.561938e-02 -1.593148e-02
## 5 1.192317e-02 -4.005936e-01
## 6 4.276853e-02 -1.016715e-01
## 7 9.378146e-02 -1.962420e-01
## 8 1.947027e-01 2.721088e-01
## 9 7.108698e-02 3.715604e-03
## 10 1.095793e-02 -3.681577e-02
## 11 1.687161e-01 2.226396e-01
## 12 -3.744729e-03 -6.220075e-02
## 13 2.270971e-01 8.734476e-02
## 14 5.335760e-02 2.213468e-02
## 15 1.272707e-01 -3.073569e-01
## 16 1.032194e-01 -3.919645e-02
## 17 3.404876e-04 1.070271e-04
## 18 2.004089e-02 -3.583426e-02
## 19 1.362569e-01 1.933865e-01
## 20 -4.421543e-03 -6.120934e-02
## 21 -8.547965e-03 -4.916155e-01
## 22 3.141649e-01 3.732923e-01
## 23 6.300331e-03 4.876363e-03
## 24 2.619176e-01 2.466031e-01
## 25 -1.957708e-02 -3.716154e-01
## 26 -1.751655e-02 -1.683300e-01
## 27 8.916904e-02 5.894978e-02
## 28 3.240708e-02 -1.036155e-01
## 29 3.712536e-02 5.056246e-02
## 30 6.335803e-02 1.681119e-02
## 31 1.760478e-01 2.243937e-01
## 32 -1.617845e-02 -4.341265e-01
## 33 -3.649623e-02 -3.159944e-01
## 34 4.053699e-03 -8.358571e-02
## 35 3.939703e-02 5.372847e-02
## 36 3.156971e-02 -4.475502e-02
## 37 1.785960e-02 -8.786797e-02
## 38 1.762980e-02 7.870039e-03
## 39 4.493328e-02 7.610342e-03
## 40 1.985589e-01 2.721236e-01
## 41 4.420394e-02 -3.792047e-01
## 42 1.983003e-03 -8.616400e-03
## 43 5.980881e-02 -2.725492e-01
## 44 4.542486e-02 -4.906039e-01
## 45 2.930445e-02 -5.293077e-02
## 46 1.959758e-01 2.671089e-01
## 47 3.435014e-02 -1.833975e-02
## 48 -2.055770e-02 -4.016397e-01
## 49 3.805753e-02 -2.019664e-01
## 50 -2.065807e-02 -1.601708e-01
## 51 -5.254278e-03 -3.864714e-01
## 52 2.329081e-02 4.396221e-03
## 53 1.118605e-01 1.280988e-01
## 54 9.157626e-03 -5.669652e-02
## 55 6.351329e-02 5.360805e-02
## 56 1.900008e-01 2.381703e-01
## 57 2.406550e-02 1.119412e-02
## 58 2.067472e-01 -1.102648e-01
## 59 -8.038289e-03 -2.834013e-01
## 60 1.024877e-01 1.452200e-01
## 61 -6.229375e-03 -7.921971e-02
## 62 9.821428e-04 1.137020e-03
## 63 2.903555e-02 4.519628e-02
## 64 6.800602e-02 -9.222271e-02
## 65 3.336584e-01 3.944227e-01
## 66 8.753335e-03 -9.519264e-02
## 67 3.499288e-02 -1.080064e-01
## 68 1.388058e-01 -1.555427e-02
## 69 2.169006e-01 8.366844e-04
## 70 1.328882e-01 -2.424088e-02
## 71 2.269006e-02 -7.916899e-03
## 72 1.500966e-01 1.774462e-01
## 73 1.067611e-02 -3.470224e-01
## 74 -1.103761e-02 -4.644763e-01
## 75 4.986696e-02 -3.397460e-02
## 76 1.636157e-02 2.044549e-02
## 77 2.317298e-02 2.476625e-02
## 78 -8.738872e-03 -4.733180e-02
## 79 2.406756e-02 -6.174635e-02
## 80 -1.406064e-02 -2.443887e-01
## 81 5.978399e-02 -2.516559e-01
## 82 6.083168e-02 6.627050e-02
## 83 1.057801e-01 1.484071e-01
## 84 2.650575e-01 2.499240e-01
## 85 -3.920990e-02 -3.006768e-01
## 86 1.847207e-01 -1.187591e-01
## 87 1.183930e-01 6.067645e-03
## 88 3.000939e-01 3.051583e-01
## 89 4.238995e-02 6.626796e-02
## 90 1.704720e-01 -1.590894e-01
## 91 2.158503e-01 3.152379e-01
## 92 2.298936e-01 3.503088e-01
## 93 1.098184e-01 1.286088e-01
## 94 8.418825e-04 -3.566484e-01
## 95 -1.956444e-02 -2.627024e-01
## 96 1.732431e-01 2.470051e-01
## 97 1.134525e-01 1.105267e-01
## 98 9.454955e-02 1.080992e-01
## 99 -1.396728e-02 -2.218644e-01
## 100 8.271109e-02 -4.662401e-02
## 101 -7.155890e-03 -7.124179e-02
## 102 1.145515e-01 -7.656362e-02
## 103 -8.203790e-02 -6.306077e-01
## 104 2.743583e-01 3.825194e-01
## 105 2.739151e-02 -3.343657e-01
## 106 -3.531368e-03 -2.952046e-02
## 107 1.265481e-01 -1.961845e-01
## 108 7.425400e-03 -3.244682e-01
## 109 6.582218e-02 -2.746003e-02
## 110 2.164267e-01 -7.929454e-02
## 111 -1.656696e-02 -4.072751e-01
## 112 2.533811e-02 -7.909438e-03
## 113 -3.613434e-02 -4.753040e-01
## 114 8.051320e-02 2.557757e-02
## 115 6.709150e-02 5.873300e-02
## 116 2.272063e-01 2.603318e-01
## 117 9.346121e-02 8.456744e-02
## 118 5.788276e-03 -5.041632e-03
## 119 1.234150e-03 -3.192199e-02
## 120 8.597619e-02 8.519541e-02
## 121 1.433545e-01 1.833844e-01
## 122 1.011576e-01 -1.495109e-01
## 123 1.378583e-01 1.045633e-02
## 124 2.208170e-01 3.285603e-01
## 125 1.288115e-03 7.623374e-04
## 126 3.301463e-03 3.529715e-03
## 127 4.180397e-02 -1.047673e-01
## 128 2.223747e-02 -1.343991e-01
## 129 2.263117e-02 8.820135e-03
## 130 5.662917e-02 8.969157e-03
## 131 1.002890e-02 5.483702e-03
## 132 4.372239e-02 2.727425e-02
## 133 1.482198e-01 1.794712e-01
## 134 1.556060e-02 -2.046313e-01
## 135 1.619754e-01 -1.083130e-01
## 136 1.158314e-02 -4.399418e-04
## 137 7.514075e-02 2.309929e-02
## 138 2.047307e-02 -1.104648e-01
## 139 2.142335e-01 -4.445071e-02
## 140 1.857988e-01 1.446121e-01
## 141 2.339502e-01 -2.854512e-02
## 142 3.639523e-02 -3.896799e-02
## 143 -8.254948e-03 -3.772929e-01
## 144 3.447782e-01 5.064356e-01
## 145 2.699898e-01 2.145156e-02
## 146 -7.598455e-02 -5.273237e-01
## 147 -5.563841e-03 -7.675751e-02
## 148 1.900240e-02 -1.258239e-01
## 149 1.792917e-01 2.501873e-01
## 150 1.650626e-01 -1.036944e-01
## 151 -5.206043e-03 -6.722982e-02
## 152 1.623131e-01 -1.243162e-01
## 153 -4.140015e-02 -2.907246e-01
## 154 -5.952496e-02 -4.259466e-01
## 155 2.440506e-03 1.094046e-04
## 156 -9.963320e-03 -2.563555e-01
## 157 1.837966e-02 -6.895286e-02
## 158 -2.576878e-02 -4.153049e-01
## 159 -1.357867e-02 -2.789687e-01
## 160 1.799375e-03 -2.207886e-01
## 161 -1.549752e-02 -3.190406e-01
## 162 6.484579e-02 -2.491135e-03
## 163 2.314033e-01 6.684205e-02
## 164 -1.796080e-02 -1.933258e-01
## 165 5.334131e-02 -6.964084e-02
## 166 1.053276e-01 1.167486e-01
## 167 7.288674e-02 -1.084649e-01
## 168 3.992137e-02 -4.660742e-01
## 169 3.332289e-02 6.818926e-03
## 170 8.215303e-03 -1.628673e-01
## 171 6.402109e-02 -6.807999e-05
## 172 7.888682e-02 -1.064630e-01
## 173 1.313622e-01 -1.879873e-01
## 174 -1.181920e-02 -2.700658e-01
## 175 4.625360e-02 -1.827727e-01
## 176 3.017096e-02 -2.654164e-01
## 177 1.766420e-01 3.878666e-02
## 178 -6.741271e-02 -5.025440e-01
## 179 9.669354e-02 -8.936845e-02
## 180 7.111207e-02 -5.564414e-03
## 181 6.970230e-02 -3.127629e-01
## 182 2.947025e-02 -1.571957e-01
## 183 9.739743e-02 -3.706292e-01
## 184 5.286048e-02 -4.004401e-01
## 185 8.099941e-03 -1.183424e-01
## 186 1.708608e-01 -4.059210e-02
## 187 1.497253e-02 -3.027878e-03
## 188 2.095865e-03 2.479219e-03
## 189 1.028464e-01 -1.514197e-03
## 190 -3.307066e-02 -3.378188e-01
## 191 9.996759e-03 -4.171234e-02
## 192 3.969631e-03 -2.286712e-01
## 193 3.175584e-02 -5.002716e-02
## 194 7.121516e-02 9.006407e-02
## 195 7.026087e-02 -2.803678e-01
## 196 7.376057e-06 -1.332560e-05
## 197 3.093626e-02 -8.374312e-03
## 198 1.112331e-01 -5.994262e-03
## 199 -1.742168e-02 -1.852916e-01
## 200 1.639433e-01 1.934631e-01
##
## $values
## $values$divergencekl
## [1] -10.87511
##
## $values$iterations
## [1] 66
##
## $values$message
## [1] "relative convergence (4)"
##
## $values$q
## [,1] [,2]
## [1,] 0.2 0.8
## [2,] 0.2 0.8
## [3,] 0.2 0.8
## [4,] 0.2 0.8
## [5,] 0.2 0.8
## [6,] 0.2 0.8
## [7,] 0.2 0.8
## [8,] 0.2 0.8
## [9,] 0.2 0.8
## [10,] 0.2 0.8
## [11,] 0.2 0.8
## [12,] 0.2 0.8
## [13,] 0.2 0.8
## [14,] 0.2 0.8
## [15,] 0.2 0.8
## [16,] 0.2 0.8
## [17,] 0.2 0.8
## [18,] 0.2 0.8
## [19,] 0.2 0.8
## [20,] 0.2 0.8
## [21,] 0.2 0.8
## [22,] 0.2 0.8
## [23,] 0.2 0.8
## [24,] 0.2 0.8
## [25,] 0.2 0.8
## [26,] 0.2 0.8
## [27,] 0.2 0.8
## [28,] 0.2 0.8
## [29,] 0.2 0.8
## [30,] 0.2 0.8
## [31,] 0.2 0.8
## [32,] 0.2 0.8
## [33,] 0.2 0.8
## [34,] 0.2 0.8
## [35,] 0.2 0.8
## [36,] 0.2 0.8
## [37,] 0.2 0.8
## [38,] 0.2 0.8
## [39,] 0.2 0.8
## [40,] 0.2 0.8
## [41,] 0.2 0.8
## [42,] 0.2 0.8
## [43,] 0.2 0.8
## [44,] 0.2 0.8
## [45,] 0.2 0.8
## [46,] 0.2 0.8
## [47,] 0.2 0.8
## [48,] 0.2 0.8
## [49,] 0.2 0.8
## [50,] 0.2 0.8
## [51,] 0.2 0.8
## [52,] 0.2 0.8
## [53,] 0.2 0.8
## [54,] 0.2 0.8
## [55,] 0.2 0.8
## [56,] 0.2 0.8
## [57,] 0.2 0.8
## [58,] 0.2 0.8
## [59,] 0.2 0.8
## [60,] 0.2 0.8
## [61,] 0.2 0.8
## [62,] 0.2 0.8
## [63,] 0.2 0.8
## [64,] 0.2 0.8
## [65,] 0.2 0.8
## [66,] 0.2 0.8
## [67,] 0.2 0.8
## [68,] 0.2 0.8
## [69,] 0.2 0.8
## [70,] 0.2 0.8
## [71,] 0.2 0.8
## [72,] 0.2 0.8
## [73,] 0.2 0.8
## [74,] 0.2 0.8
## [75,] 0.2 0.8
## [76,] 0.2 0.8
## [77,] 0.2 0.8
## [78,] 0.2 0.8
## [79,] 0.2 0.8
## [80,] 0.2 0.8
## [81,] 0.2 0.8
## [82,] 0.2 0.8
## [83,] 0.2 0.8
## [84,] 0.2 0.8
## [85,] 0.2 0.8
## [86,] 0.2 0.8
## [87,] 0.2 0.8
## [88,] 0.2 0.8
## [89,] 0.2 0.8
## [90,] 0.2 0.8
## [91,] 0.2 0.8
## [92,] 0.2 0.8
## [93,] 0.2 0.8
## [94,] 0.2 0.8
## [95,] 0.2 0.8
## [96,] 0.2 0.8
## [97,] 0.2 0.8
## [98,] 0.2 0.8
## [99,] 0.2 0.8
## [100,] 0.2 0.8
## [101,] 0.2 0.8
## [102,] 0.2 0.8
## [103,] 0.2 0.8
## [104,] 0.2 0.8
## [105,] 0.2 0.8
## [106,] 0.2 0.8
## [107,] 0.2 0.8
## [108,] 0.2 0.8
## [109,] 0.2 0.8
## [110,] 0.2 0.8
## [111,] 0.2 0.8
## [112,] 0.2 0.8
## [113,] 0.2 0.8
## [114,] 0.2 0.8
## [115,] 0.2 0.8
## [116,] 0.2 0.8
## [117,] 0.2 0.8
## [118,] 0.2 0.8
## [119,] 0.2 0.8
## [120,] 0.2 0.8
## [121,] 0.2 0.8
## [122,] 0.2 0.8
## [123,] 0.2 0.8
## [124,] 0.2 0.8
## [125,] 0.2 0.8
## [126,] 0.2 0.8
## [127,] 0.2 0.8
## [128,] 0.2 0.8
## [129,] 0.2 0.8
## [130,] 0.2 0.8
## [131,] 0.2 0.8
## [132,] 0.2 0.8
## [133,] 0.2 0.8
## [134,] 0.2 0.8
## [135,] 0.2 0.8
## [136,] 0.2 0.8
## [137,] 0.2 0.8
## [138,] 0.2 0.8
## [139,] 0.2 0.8
## [140,] 0.2 0.8
## [141,] 0.2 0.8
## [142,] 0.2 0.8
## [143,] 0.2 0.8
## [144,] 0.2 0.8
## [145,] 0.2 0.8
## [146,] 0.2 0.8
## [147,] 0.2 0.8
## [148,] 0.2 0.8
## [149,] 0.2 0.8
## [150,] 0.2 0.8
## [151,] 0.2 0.8
## [152,] 0.2 0.8
## [153,] 0.2 0.8
## [154,] 0.2 0.8
## [155,] 0.2 0.8
## [156,] 0.2 0.8
## [157,] 0.2 0.8
## [158,] 0.2 0.8
## [159,] 0.2 0.8
## [160,] 0.2 0.8
## [161,] 0.2 0.8
## [162,] 0.2 0.8
## [163,] 0.2 0.8
## [164,] 0.2 0.8
## [165,] 0.2 0.8
## [166,] 0.2 0.8
## [167,] 0.2 0.8
## [168,] 0.2 0.8
## [169,] 0.2 0.8
## [170,] 0.2 0.8
## [171,] 0.2 0.8
## [172,] 0.2 0.8
## [173,] 0.2 0.8
## [174,] 0.2 0.8
## [175,] 0.2 0.8
## [176,] 0.2 0.8
## [177,] 0.2 0.8
## [178,] 0.2 0.8
## [179,] 0.2 0.8
## [180,] 0.2 0.8
## [181,] 0.2 0.8
## [182,] 0.2 0.8
## [183,] 0.2 0.8
## [184,] 0.2 0.8
## [185,] 0.2 0.8
## [186,] 0.2 0.8
## [187,] 0.2 0.8
## [188,] 0.2 0.8
## [189,] 0.2 0.8
## [190,] 0.2 0.8
## [191,] 0.2 0.8
## [192,] 0.2 0.8
## [193,] 0.2 0.8
## [194,] 0.2 0.8
## [195,] 0.2 0.8
## [196,] 0.2 0.8
## [197,] 0.2 0.8
## [198,] 0.2 0.8
## [199,] 0.2 0.8
## [200,] 0.2 0.8
##
##
## $tol
## [1] 1e-10
##
## $v
## [,1] [,2] [,3]
## [1,] 1 0 -1
##
## $lambda
## (Intercept) Dcollege Totalincome Dunemp
## 1 57.03564 8.987007 -0.0001201035 -0.8332262
## 0 28.58571 75.368523 -0.0002268384 -25.0246657
##
## $checkrestrictions
## $checkrestrictions$g1
## [,1]
## [1,] 0
## [2,] 0
## [3,] 0
## [4,] 0
## [5,] 0
## [6,] 0
## [7,] 0
## [8,] 0
## [9,] 0
## [10,] 0
## [11,] 0
## [12,] 0
## [13,] 0
## [14,] 0
## [15,] 0
## [16,] 0
## [17,] 0
## [18,] 0
## [19,] 0
## [20,] 0
## [21,] 0
## [22,] 0
## [23,] 0
## [24,] 0
## [25,] 0
## [26,] 0
## [27,] 0
## [28,] 0
## [29,] 0
## [30,] 0
## [31,] 0
## [32,] 0
## [33,] 0
## [34,] 0
## [35,] 0
## [36,] 0
## [37,] 0
## [38,] 0
## [39,] 0
## [40,] 0
## [41,] 0
## [42,] 0
## [43,] 0
## [44,] 0
## [45,] 0
## [46,] 0
## [47,] 0
## [48,] 0
## [49,] 0
## [50,] 0
## [51,] 0
## [52,] 0
## [53,] 0
## [54,] 0
## [55,] 0
## [56,] 0
## [57,] 0
## [58,] 0
## [59,] 0
## [60,] 0
## [61,] 0
## [62,] 0
## [63,] 0
## [64,] 0
## [65,] 0
## [66,] 0
## [67,] 0
## [68,] 0
## [69,] 0
## [70,] 0
## [71,] 0
## [72,] 0
## [73,] 0
## [74,] 0
## [75,] 0
## [76,] 0
## [77,] 0
## [78,] 0
## [79,] 0
## [80,] 0
## [81,] 0
## [82,] 0
## [83,] 0
## [84,] 0
## [85,] 0
## [86,] 0
## [87,] 0
## [88,] 0
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##
## $checkrestrictions$g2
## [,1] [,2]
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##
## $checkrestrictions$g3
## y_factor1 y_factor0
## (Intercept) 0 0
## Dcollege 0 0
## Totalincome 0 0
## Dunemp 0 0
##
##
## $cross_moments_A
## 1 0
## (Intercept) 0.34 0.66
## Dcollege 0.17 0.44
## Totalincome 103036.63 189594.78
## Dunemp 0.16 0.29
##
## $cross_moments_B
## 1 0
## (Intercept) 0.34 0.66
## Dcollege 0.17 0.44
## Totalincome 103036.63 189594.78
## Dunemp 0.16 0.29
##
## $J
## [1] 2
##
## $fn
## dataA$poor_liq ~ Dcollege + Totalincome + Dunemp
##
## attr(,"class")
## [1] "kl"
The function will produce a data frame called estimations with the
following information:
- 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, a message
in case something went wrong and the prior included in the
optimization.
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 A
(the cross moments in dataA) and cross moments B (the cross moments in
dataB).
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.
suppressPackageStartupMessages(library(dplyr))
summary(result)
## Iterations[1] 66
## Kullback-Leibler divergence value[1] -10.87511
## [1] "mean_estimations"
## weights predictions_1 predictions_0 probabilities_1 probabilities_0
## 1 0.006608031 0.3377783 0.6622217 0.2507808 0.7492192
## errors_1 errors_0
## 1 0.08699743 -0.08699743
## [1] "lambda"
## (Intercept) Dcollege Totalincome Dunemp
## 1 57.03564 8.987007 -0.0001201035 -0.8332262
## 0 28.58571 75.368523 -0.0002268384 -25.0246657
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.
Notes: Arguments tol and iter can be defined by the user.
The default tolerance has been set in 1e-10 while the maximum number or
iterations by default is settled in 1000.
3.2 Without prior information
Suppose we do not have prior information about our variable of
interest. In that case, the process starting point will be the uniform
distribution as we do not have the information to think one
characteristic is more likely than the other.
The function ei_gce() will include by default the uniform
distribution as \(Q\) if the user does
not specify any other. This would be:
result2 <- ei_gce(fn,dataA,dataB,weights= "w")
result2
## $estimations
## weights predictions_1 predictions_0 probabilities_1 probabilities_0
## 1 6.408614e-03 0.37714125 1.0373453 0.17143080 0.8285692
## 2 4.069074e-03 0.46516838 0.9465387 0.26050109 0.7394989
## 3 8.478890e-05 0.49554427 0.4949818 0.50026475 0.4997352
## 4 6.019554e-03 0.29425456 0.7815640 0.27116773 0.7288323
## 5 8.529959e-03 0.24792086 0.3351759 0.45651989 0.5434801
## 6 3.466777e-03 0.37076597 0.4876873 0.44458711 0.5554129
## 7 7.143169e-03 0.25980107 0.5065021 0.38173083 0.6182692
## 8 6.728993e-03 0.36038533 1.0624024 0.14987220 0.8501278
## 9 9.459067e-03 0.24653141 0.9526972 0.16596553 0.8340345
## 10 1.952733e-03 0.42547709 0.5404905 0.44585291 0.5541471
## 11 6.884772e-03 0.36676136 1.0476604 0.16115983 0.8388402
## 12 1.046638e-03 0.42656461 0.4268691 0.49985561 0.5001444
## 13 6.632154e-03 0.45686702 0.9334005 0.26416701 0.7358330
## 14 5.694566e-03 0.34661343 0.8392470 0.26618187 0.7338181
## 15 8.199805e-03 0.29044879 0.3979352 0.44769091 0.5523091
## 16 7.974757e-03 0.34208516 0.8763006 0.24538694 0.7546131
## 17 1.016052e-05 0.50008404 0.5008871 0.49962202 0.5003780
## 18 1.077733e-03 0.47310069 0.4945723 0.48988963 0.5101104
## 19 4.555596e-03 0.43194367 0.9691883 0.23344952 0.7665505
## 20 1.797040e-03 0.40140327 0.4763705 0.46452977 0.5354702
## 21 7.726297e-03 0.36613331 0.2079584 0.57897919 0.4210208
## 22 7.537852e-03 0.33716479 1.0938416 0.12177907 0.8782209
## 23 5.078895e-04 0.48835958 0.5396067 0.47587229 0.5241277
## 24 8.673474e-03 0.34928903 1.0777078 0.13630039 0.8636996
## 25 6.743411e-03 0.28645585 0.2974432 0.49452584 0.5054742
## 26 2.597985e-03 0.34630227 0.3301959 0.50785849 0.4921415
## 27 7.946911e-03 0.32381881 0.9735060 0.18622722 0.8137728
## 28 3.132542e-03 0.37158327 0.4640120 0.45608396 0.5439160
## 29 1.413325e-03 0.50486840 0.6852096 0.41406952 0.5859305
## 30 1.952905e-03 0.50803523 0.6558099 0.42980542 0.5701946
## 31 7.786354e-03 0.34741236 1.0685414 0.14103857 0.8589614
## 32 6.412179e-03 0.36399171 0.2144078 0.57461752 0.4253825
## 33 8.813903e-03 0.14764168 0.4589503 0.34574856 0.6542514
## 34 3.104395e-03 0.35538687 0.5085653 0.42745368 0.5725463
## 35 1.494452e-03 0.50471084 0.6952247 0.40910466 0.5908953
## 36 1.979537e-03 0.43997262 0.5224387 0.46116813 0.5388319
## 37 1.781772e-03 0.42523291 0.4282943 0.49854806 0.5014519
## 38 8.631740e-04 0.49491102 0.5728497 0.46325832 0.5367417
## 39 2.624354e-03 0.46497183 0.6798734 0.39816837 0.6018316
## 40 7.278409e-03 0.34905418 1.0741074 0.13836260 0.8616374
## 41 9.198636e-03 0.21786461 0.3741713 0.42227208 0.5777279
## 42 8.939710e-04 0.45949760 0.5303565 0.46664821 0.5333518
## 43 5.755561e-03 0.32270787 0.3367562 0.49312040 0.5068796
## 44 9.197598e-03 0.32806665 0.2496263 0.53913535 0.4608646
## 45 7.835590e-03 0.20378551 0.7852619 0.22815796 0.7718420
## 46 7.304276e-03 0.34955434 1.0730092 0.13921843 0.8607816
## 47 2.825585e-03 0.43352583 0.6371530 0.40405005 0.5959499
## 48 5.701607e-03 0.36363400 0.2181770 0.57250290 0.4274971
## 49 9.403839e-03 0.14601797 0.6247015 0.27122818 0.7287718
## 50 8.874280e-03 0.08319604 0.6259964 0.23775918 0.7622408
## 51 5.856153e-03 0.35220680 0.2339843 0.55886956 0.4411304
## 52 7.506319e-04 0.50195984 0.5572225 0.47395691 0.5260431
## 53 2.829871e-03 0.51333910 0.8552131 0.33161744 0.6683826
## 54 1.403796e-03 0.42963844 0.4600167 0.48563963 0.5143604
## 55 2.256222e-03 0.50956486 0.7420473 0.38819982 0.6118002
## 56 8.706739e-03 0.32821719 1.0912941 0.11983046 0.8801695
## 57 6.495227e-04 0.51007566 0.5661375 0.47355883 0.5264412
## 58 7.423215e-03 0.46620774 0.7547749 0.36198157 0.6380184
## 59 9.409367e-03 0.12397261 0.5121636 0.30905232 0.6909477
## 60 3.467645e-03 0.47043120 0.8983506 0.28929940 0.7107006
## 61 4.891585e-03 0.25236016 0.5896630 0.34011455 0.6598855
## 62 5.191919e-05 0.49955438 0.5055161 0.49719380 0.5028062
## 63 6.960881e-04 0.52499287 0.6371782 0.44671371 0.5532863
## 64 3.234067e-03 0.43693106 0.5181213 0.46176054 0.5382395
## 65 8.095405e-03 0.32338768 1.1079813 0.10778777 0.8922122
## 66 2.123864e-03 0.38804726 0.4229340 0.48337200 0.5166280
## 67 3.309518e-03 0.36617689 0.4653322 0.45286639 0.5471336
## 68 5.300329e-03 0.45370052 0.7757856 0.34578402 0.6542160
## 69 7.890242e-03 0.43151836 0.9045737 0.26894812 0.7310519
## 70 5.248723e-03 0.44718763 0.7585647 0.35144311 0.6485569
## 71 9.625057e-04 0.48690886 0.5426170 0.47377134 0.5262287
## 72 3.609097e-03 0.48364078 0.9192295 0.28373016 0.7162698
## 73 5.617329e-03 0.34034003 0.2579076 0.54093438 0.4590656
## 74 9.237887e-03 0.26393393 0.3105135 0.47674372 0.5232563
## 75 9.275353e-03 0.20156879 0.8823507 0.18190526 0.8180947
## 76 7.561670e-04 0.49760860 0.5883611 0.45717689 0.5428231
## 77 1.374263e-03 0.48128816 0.6314624 0.42902088 0.5709791
## 78 2.232129e-03 0.37084550 0.5174122 0.43073033 0.5692697
## 79 7.759625e-03 0.19416138 0.7630443 0.23377581 0.7662242
## 80 3.268253e-03 0.35954595 0.2702605 0.54401485 0.4559852
## 81 5.423895e-03 0.32883027 0.3491396 0.49009389 0.5099061
## 82 3.519797e-03 0.44627564 0.8133954 0.32337015 0.6766299
## 83 3.686690e-03 0.46329523 0.9091087 0.28031363 0.7196864
## 84 8.770650e-03 0.34686168 1.0804290 0.13370219 0.8662978
## 85 4.499834e-03 0.32020854 0.2749703 0.52248058 0.4775194
## 86 6.913635e-03 0.45482457 0.7095721 0.37906307 0.6209369
## 87 7.423989e-03 0.38267173 0.9137446 0.24307454 0.7569255
## 88 9.184997e-03 0.32784976 1.1020921 0.11311844 0.8868816
## 89 9.934938e-04 0.53182183 0.6919862 0.42330355 0.5766964
## 90 7.108557e-03 0.41232918 0.6358960 0.39477027 0.6052297
## 91 6.307975e-03 0.35899760 1.0677309 0.14614703 0.8538530
## 92 5.444530e-03 0.36801098 1.0605664 0.15404711 0.8459529
## 93 5.708904e-03 0.40224861 0.9639683 0.22381773 0.7761823
## 94 7.141561e-03 0.26731227 0.3222444 0.47266687 0.5273331
## 95 7.903713e-03 0.16503977 0.4725605 0.34873889 0.6512611
## 96 5.645807e-03 0.39114729 1.0268658 0.18329921 0.8167008
## 97 3.539554e-03 0.49407716 0.8675386 0.31638400 0.6836160
## 98 2.396317e-03 0.52472961 0.8185412 0.35610946 0.6438905
## 99 3.785424e-03 0.31833636 0.3185355 0.49990197 0.5000980
## 100 6.899727e-03 0.34665495 0.8118087 0.27989298 0.7201070
## 101 1.963958e-03 0.38847654 0.4653266 0.46355677 0.5364432
## 102 4.301069e-03 0.46970567 0.6283352 0.42510995 0.5748901
## 103 8.824290e-03 0.44210578 0.1264227 0.65781858 0.3421814
## 104 9.245072e-03 0.29598217 1.1344334 0.08099393 0.9190061
## 105 7.604964e-03 0.24465134 0.3562680 0.44460827 0.5553917
## 106 7.759268e-04 0.45320405 0.4823024 0.48628343 0.5137166
## 107 6.426895e-03 0.36017770 0.5030301 0.43233607 0.5676639
## 108 5.150249e-03 0.34234848 0.2619044 0.53989153 0.4601085
## 109 5.158771e-03 0.38924139 0.7550982 0.32689679 0.6731032
## 110 7.247160e-03 0.48560880 0.7937026 0.35114912 0.6488509
## 111 7.664692e-03 0.27774103 0.3016060 0.48809770 0.5119023
## 112 4.033598e-03 0.35902332 0.7016212 0.33886187 0.6611381
## 113 8.610544e-03 0.29051865 0.2822042 0.50415282 0.4958472
## 114 9.173485e-03 0.27410838 0.9733603 0.16627370 0.8337263
## 115 2.312141e-03 0.51128192 0.7527716 0.38361563 0.6163844
## 116 5.785912e-03 0.40348498 1.0217407 0.19141205 0.8085879
## 117 3.133029e-03 0.50150724 0.8249308 0.34222256 0.6577774
## 118 3.027803e-04 0.49290501 0.5075815 0.49309179 0.5069082
## 119 6.494774e-04 0.46052427 0.4682541 0.49635504 0.5036450
## 120 2.627321e-03 0.51441103 0.8032533 0.35933654 0.6406635
## 121 6.300118e-03 0.38606931 1.0151756 0.18797894 0.8120211
## 122 5.009005e-03 0.39456016 0.5113977 0.44484480 0.5551552
## 123 8.546357e-03 0.36682710 0.9593040 0.21165629 0.7883437
## 124 5.935042e-03 0.36343500 1.0640665 0.15011602 0.8498840
## 125 5.670056e-05 0.49996438 0.5053768 0.49745234 0.5025477
## 126 1.956694e-04 0.49743772 0.5190521 0.48982517 0.5101748
## 127 3.491569e-03 0.36763750 0.4826404 0.44545031 0.5545497
## 128 6.018842e-03 0.25616411 0.5745524 0.34884899 0.6511510
## 129 2.453778e-03 0.42789201 0.6552214 0.39282715 0.6071729
## 130 3.330195e-03 0.45417305 0.7229733 0.37220846 0.6277915
## 131 4.568058e-04 0.49890141 0.5417206 0.47983381 0.5201662
## 132 1.878873e-03 0.49715241 0.6752413 0.41533263 0.5846674
## 133 7.274750e-03 0.36343791 1.0385279 0.16518574 0.8348143
## 134 8.055270e-03 0.17014299 0.5542960 0.31464697 0.6853530
## 135 7.918719e-03 0.35004369 0.7405532 0.31614416 0.6838558
## 136 1.613960e-03 0.44481836 0.5890183 0.43212130 0.5678787
## 137 4.043005e-03 0.45292971 0.7853299 0.34062783 0.6593722
## 138 2.836492e-03 0.36867919 0.4335517 0.46904473 0.5309553
## 139 8.648536e-03 0.40487932 0.8830264 0.26894496 0.7310550
## 140 7.121030e-03 0.40200465 1.0056064 0.20009289 0.7999071
## 141 7.015624e-03 0.50187230 0.8437294 0.33252515 0.6674748
## 142 2.042604e-03 0.44644732 0.5396343 0.45614015 0.5438598
## 143 7.276373e-03 0.27335984 0.3110208 0.48123851 0.5187615
## 144 8.504937e-03 0.28247517 1.1493785 0.06658550 0.9334145
## 145 9.535921e-03 0.40889049 0.9714828 0.22238905 0.7776110
## 146 8.545091e-03 0.32920843 0.2406706 0.54424966 0.4557503
## 147 2.254398e-03 0.37776176 0.4714298 0.45556209 0.5444379
## 148 5.514651e-03 0.27258801 0.5631967 0.36208360 0.6379164
## 149 6.254560e-03 0.37577593 1.0434209 0.16731376 0.8326862
## 150 6.128050e-03 0.46189360 0.6907313 0.39148614 0.6085139
## 151 1.952048e-03 0.39256313 0.4730384 0.46188233 0.5381177
## 152 8.239200e-03 0.33030142 0.7244077 0.31481580 0.6851842
## 153 4.212187e-03 0.32458168 0.2727708 0.52573386 0.4742661
## 154 6.562633e-03 0.32256037 0.2521837 0.53513537 0.4648646
## 155 8.529452e-05 0.49978515 0.5056391 0.49724449 0.5027555
## 156 4.627415e-03 0.30253610 0.3171597 0.49277405 0.5072260
## 157 7.506022e-03 0.19114462 0.7362700 0.24439886 0.7556011
## 158 5.802968e-03 0.36849401 0.2110728 0.57850615 0.4214939
## 159 8.889093e-03 0.13810196 0.4959950 0.32384848 0.6761515
## 160 3.338891e-03 0.35704129 0.2980884 0.52889689 0.4711031
## 161 5.739428e-03 0.29183905 0.3035596 0.49417589 0.5058241
## 162 9.051019e-03 0.24618977 0.9294581 0.17769374 0.8223063
## 163 7.187216e-03 0.44810276 0.9357911 0.25904136 0.7409586
## 164 2.371262e-03 0.37102330 0.2899472 0.53968911 0.4603109
## 165 5.791201e-03 0.33183351 0.7032194 0.32554578 0.6744542
## 166 5.959915e-03 0.39382659 0.9629482 0.22086087 0.7791391
## 167 8.379680e-03 0.25726600 0.7659295 0.26171938 0.7382806
## 168 8.564844e-03 0.32893640 0.2510774 0.53882508 0.4611749
## 169 1.064177e-03 0.50320864 0.5820375 0.46280363 0.5371964
## 170 6.110630e-03 0.23336254 0.5255237 0.35980767 0.6401923
## 171 1.771706e-03 0.52165766 0.6221197 0.45237445 0.5476256
## 172 3.746941e-03 0.42731665 0.5216065 0.45558329 0.5444167
## 173 8.344937e-03 0.25568851 0.5881502 0.34243333 0.6575667
## 174 8.671874e-03 0.14356028 0.4964644 0.32654345 0.6734566
## 175 5.165876e-03 0.29467774 0.4293383 0.43492991 0.5650701
## 176 6.250491e-03 0.26256158 0.3704982 0.44683128 0.5531687
## 177 5.663535e-03 0.47401131 0.8621297 0.31031897 0.6896810
## 178 8.189466e-03 0.32238324 0.2483320 0.53700277 0.4629972
## 179 5.172787e-03 0.37674556 0.6194582 0.38587109 0.6141289
## 180 2.048906e-03 0.51888456 0.6288527 0.44784838 0.5521516
## 181 6.680525e-03 0.31019463 0.3270606 0.49169947 0.5083005
## 182 4.062967e-03 0.32344524 0.4156342 0.45557096 0.5444290
## 183 8.424910e-03 0.28920352 0.3299449 0.47986599 0.5201340
## 184 7.709358e-03 0.31518486 0.2788500 0.51806096 0.4819390
## 185 2.544983e-03 0.36650207 0.4030769 0.48246229 0.5175377
## 186 5.364966e-03 0.50714408 0.7578390 0.37916478 0.6208352
## 187 5.934410e-04 0.49408196 0.5305317 0.48283973 0.5171603
## 188 5.020772e-05 0.50207164 0.5088920 0.49678951 0.5032105
## 189 2.873655e-03 0.52784565 0.6870685 0.42385754 0.5761425
## 190 5.514606e-03 0.30662476 0.2813498 0.51258088 0.4874191
## 191 2.057597e-03 0.41814279 0.5359942 0.44449887 0.5555011
## 192 8.173496e-03 0.15968125 0.5246355 0.32242309 0.6775769
## 193 2.086392e-03 0.43346769 0.5171342 0.46058870 0.5394113
## 194 3.212343e-03 0.46856269 0.8320486 0.32370141 0.6762986
## 195 7.999160e-03 0.22223376 0.4232661 0.40126814 0.5987319
## 196 9.251449e-07 0.49996937 0.5000287 0.49997206 0.5000279
## 197 2.268155e-03 0.45362675 0.6246519 0.41936308 0.5806369
## 198 7.367501e-03 0.37636981 0.8968114 0.24939839 0.7506016
## 199 2.263759e-03 0.37305021 0.2949578 0.53817011 0.4618299
## 200 3.956566e-03 0.46982353 0.9399072 0.26622578 0.7337742
## errors_1 errors_0
## 1 2.057104e-01 2.087761e-01
## 2 2.046673e-01 2.070398e-01
## 3 -4.720487e-03 -4.753428e-03
## 4 2.308683e-02 5.273171e-02
## 5 -2.085990e-01 -2.083042e-01
## 6 -7.382114e-02 -6.772560e-02
## 7 -1.219298e-01 -1.117670e-01
## 8 2.105131e-01 2.122746e-01
## 9 8.056588e-02 1.186627e-01
## 10 -2.037582e-02 -1.365657e-02
## 11 2.056015e-01 2.088203e-01
## 12 -7.329100e-02 -7.327534e-02
## 13 1.927000e-01 1.975675e-01
## 14 8.043156e-02 1.054288e-01
## 15 -1.572421e-01 -1.543739e-01
## 16 9.669821e-02 1.216876e-01
## 17 4.620245e-04 5.090774e-04
## 18 -1.678894e-02 -1.553809e-02
## 19 1.984941e-01 2.026378e-01
## 20 -6.312650e-02 -5.909976e-02
## 21 -2.128459e-01 -2.130624e-01
## 22 2.153857e-01 2.156207e-01
## 23 1.248730e-02 1.547897e-02
## 24 2.129886e-01 2.140081e-01
## 25 -2.080700e-01 -2.080310e-01
## 26 -1.615562e-01 -1.619456e-01
## 27 1.375916e-01 1.597333e-01
## 28 -8.450068e-02 -7.990403e-02
## 29 9.079888e-02 9.927907e-02
## 30 7.822981e-02 8.561535e-02
## 31 2.063738e-01 2.095800e-01
## 32 -2.106258e-01 -2.109747e-01
## 33 -1.981069e-01 -1.953012e-01
## 34 -7.206681e-02 -6.398106e-02
## 35 9.560617e-02 1.043293e-01
## 36 -2.119550e-02 -1.639318e-02
## 37 -7.331516e-02 -7.315760e-02
## 38 3.165269e-02 3.610801e-02
## 39 6.680346e-02 7.804174e-02
## 40 2.106916e-01 2.124700e-01
## 41 -2.044075e-01 -2.035567e-01
## 42 -7.150611e-03 -2.995334e-03
## 43 -1.704125e-01 -1.701234e-01
## 44 -2.110687e-01 -2.112383e-01
## 45 -2.437245e-02 1.341985e-02
## 46 2.103359e-01 2.122277e-01
## 47 2.947577e-02 4.120303e-02
## 48 -2.088689e-01 -2.093201e-01
## 49 -1.252102e-01 -1.040704e-01
## 50 -1.545631e-01 -1.362445e-01
## 51 -2.066628e-01 -2.071462e-01
## 52 2.800293e-02 3.117940e-02
## 53 1.817217e-01 1.868306e-01
## 54 -5.600119e-02 -5.434367e-02
## 55 1.213650e-01 1.302471e-01
## 56 2.083867e-01 2.111246e-01
## 57 3.651684e-02 3.969632e-02
## 58 1.042262e-01 1.167564e-01
## 59 -1.850797e-01 -1.787841e-01
## 60 1.811318e-01 1.876500e-01
## 61 -8.775439e-02 -7.022242e-02
## 62 2.360581e-03 2.709862e-03
## 63 7.827916e-02 8.389189e-02
## 64 -2.482948e-02 -2.011813e-02
## 65 2.155999e-01 2.157690e-01
## 66 -9.532473e-02 -9.369402e-02
## 67 -8.668950e-02 -8.180145e-02
## 68 1.079165e-01 1.215697e-01
## 69 1.625702e-01 1.735218e-01
## 70 9.574452e-02 1.100079e-01
## 71 1.313751e-02 1.638838e-02
## 72 1.999106e-01 2.029597e-01
## 73 -2.005943e-01 -2.011580e-01
## 74 -2.128098e-01 -2.127428e-01
## 75 1.966353e-02 6.425598e-02
## 76 4.043171e-02 4.553799e-02
## 77 5.226728e-02 6.048331e-02
## 78 -5.988483e-02 -5.185747e-02
## 79 -3.961444e-02 -3.179883e-03
## 80 -1.844689e-01 -1.857247e-01
## 81 -1.612636e-01 -1.607665e-01
## 82 1.229055e-01 1.367655e-01
## 83 1.829816e-01 1.894223e-01
## 84 2.131595e-01 2.141312e-01
## 85 -2.022720e-01 -2.025491e-01
## 86 7.576150e-02 8.863520e-02
## 87 1.395972e-01 1.568191e-01
## 88 2.147313e-01 2.152105e-01
## 89 1.085183e-01 1.152898e-01
## 90 1.755891e-02 3.066622e-02
## 91 2.128506e-01 2.138780e-01
## 92 2.139639e-01 2.146135e-01
## 93 1.784309e-01 1.877860e-01
## 94 -2.053546e-01 -2.050887e-01
## 95 -1.836991e-01 -1.787007e-01
## 96 2.078481e-01 2.101650e-01
## 97 1.776932e-01 1.839226e-01
## 98 1.686202e-01 1.746507e-01
## 99 -1.815656e-01 -1.815625e-01
## 100 6.676197e-02 9.170167e-02
## 101 -7.508023e-02 -7.111658e-02
## 102 4.459572e-02 5.344515e-02
## 103 -2.157128e-01 -2.157587e-01
## 104 2.149882e-01 2.154274e-01
## 105 -1.999569e-01 -1.991238e-01
## 106 -3.307938e-02 -3.141418e-02
## 107 -7.215837e-02 -6.463382e-02
## 108 -1.975430e-01 -1.982041e-01
## 109 6.234460e-02 8.199501e-02
## 110 1.344597e-01 1.448517e-01
## 111 -2.103567e-01 -2.102963e-01
## 112 2.016145e-02 4.048309e-02
## 113 -2.136342e-01 -2.136430e-01
## 114 1.078347e-01 1.396340e-01
## 115 1.276663e-01 1.363872e-01
## 116 2.120729e-01 2.131528e-01
## 117 1.592847e-01 1.671534e-01
## 118 -1.867798e-04 6.732463e-04
## 119 -3.583076e-02 -3.539085e-02
## 120 1.550745e-01 1.625898e-01
## 121 1.980904e-01 2.031546e-01
## 122 -5.028463e-02 -4.375748e-02
## 123 1.551708e-01 1.709603e-01
## 124 2.133190e-01 2.141825e-01
## 125 2.512032e-03 2.829127e-03
## 126 7.612545e-03 8.877262e-03
## 127 -7.781280e-02 -7.190929e-02
## 128 -9.268488e-02 -7.659865e-02
## 129 3.506486e-02 4.804852e-02
## 130 8.196459e-02 9.518179e-02
## 131 1.906760e-02 2.155438e-02
## 132 8.181978e-02 9.057391e-02
## 133 1.982522e-01 2.037136e-01
## 134 -1.445040e-01 -1.310570e-01
## 135 3.389953e-02 5.669736e-02
## 136 1.269706e-02 2.113961e-02
## 137 1.123019e-01 1.259577e-01
## 138 -1.003655e-01 -9.740352e-02
## 139 1.359344e-01 1.519713e-01
## 140 2.019118e-01 2.056993e-01
## 141 1.693471e-01 1.762546e-01
## 142 -9.692835e-03 -4.225555e-03
## 143 -2.078787e-01 -2.077407e-01
## 144 2.158897e-01 2.159640e-01
## 145 1.865014e-01 1.938718e-01
## 146 -2.150412e-01 -2.150798e-01
## 147 -7.780032e-02 -7.300813e-02
## 148 -8.949558e-02 -7.471968e-02
## 149 2.084622e-01 2.107346e-01
## 150 7.040746e-02 8.221743e-02
## 151 -6.931920e-02 -6.507926e-02
## 152 1.548562e-02 3.922351e-02
## 153 -2.011522e-01 -2.014953e-01
## 154 -2.125750e-01 -2.126809e-01
## 155 2.540660e-03 2.883625e-03
## 156 -1.902379e-01 -1.900663e-01
## 157 -5.325424e-02 -1.933118e-02
## 158 -2.100121e-01 -2.104210e-01
## 159 -1.857465e-01 -1.801565e-01
## 160 -1.718556e-01 -1.730147e-01
## 161 -2.023368e-01 -2.022645e-01
## 162 6.849603e-02 1.071519e-01
## 163 1.890614e-01 1.948324e-01
## 164 -1.686658e-01 -1.703637e-01
## 165 6.287725e-03 2.876523e-02
## 166 1.729657e-01 1.838091e-01
## 167 -4.453383e-03 2.764886e-02
## 168 -2.098887e-01 -2.100975e-01
## 169 4.040500e-02 4.484118e-02
## 170 -1.264451e-01 -1.146687e-01
## 171 6.928322e-02 7.449418e-02
## 172 -2.826664e-02 -2.281019e-02
## 173 -8.674482e-02 -6.941650e-02
## 174 -1.829832e-01 -1.769921e-01
## 175 -1.402522e-01 -1.357318e-01
## 176 -1.842697e-01 -1.826705e-01
## 177 1.636923e-01 1.724487e-01
## 178 -2.146195e-01 -2.146653e-01
## 179 -9.125533e-03 5.329327e-03
## 180 7.103619e-02 7.670110e-02
## 181 -1.815048e-01 -1.812400e-01
## 182 -1.321257e-01 -1.287948e-01
## 183 -1.906625e-01 -1.901891e-01
## 184 -2.028761e-01 -2.030891e-01
## 185 -1.159602e-01 -1.144608e-01
## 186 1.279793e-01 1.370038e-01
## 187 1.124223e-02 1.337146e-02
## 188 5.282130e-03 5.681506e-03
## 189 1.039881e-01 1.109260e-01
## 190 -2.059561e-01 -2.060693e-01
## 191 -2.635608e-02 -1.950693e-02
## 192 -1.627418e-01 -1.529414e-01
## 193 -2.712101e-02 -2.227712e-02
## 194 1.448613e-01 1.557500e-01
## 195 -1.790344e-01 -1.754657e-01
## 196 -2.691493e-06 7.862666e-07
## 197 3.426368e-02 4.401497e-02
## 198 1.269714e-01 1.462098e-01
## 199 -1.651199e-01 -1.668721e-01
## 200 2.035978e-01 2.061330e-01
##
## $values
## $values$divergencekl
## [1] -184.3254
##
## $values$iterations
## [1] 94
##
## $values$message
## [1] "relative convergence (4)"
##
## $values$q
## [,1] [,2]
## [1,] 0.5 0.5
## [2,] 0.5 0.5
## [3,] 0.5 0.5
## [4,] 0.5 0.5
## [5,] 0.5 0.5
## [6,] 0.5 0.5
## [7,] 0.5 0.5
## [8,] 0.5 0.5
## [9,] 0.5 0.5
## [10,] 0.5 0.5
## [11,] 0.5 0.5
## [12,] 0.5 0.5
## [13,] 0.5 0.5
## [14,] 0.5 0.5
## [15,] 0.5 0.5
## [16,] 0.5 0.5
## [17,] 0.5 0.5
## [18,] 0.5 0.5
## [19,] 0.5 0.5
## [20,] 0.5 0.5
## [21,] 0.5 0.5
## [22,] 0.5 0.5
## [23,] 0.5 0.5
## [24,] 0.5 0.5
## [25,] 0.5 0.5
## [26,] 0.5 0.5
## [27,] 0.5 0.5
## [28,] 0.5 0.5
## [29,] 0.5 0.5
## [30,] 0.5 0.5
## [31,] 0.5 0.5
## [32,] 0.5 0.5
## [33,] 0.5 0.5
## [34,] 0.5 0.5
## [35,] 0.5 0.5
## [36,] 0.5 0.5
## [37,] 0.5 0.5
## [38,] 0.5 0.5
## [39,] 0.5 0.5
## [40,] 0.5 0.5
## [41,] 0.5 0.5
## [42,] 0.5 0.5
## [43,] 0.5 0.5
## [44,] 0.5 0.5
## [45,] 0.5 0.5
## [46,] 0.5 0.5
## [47,] 0.5 0.5
## [48,] 0.5 0.5
## [49,] 0.5 0.5
## [50,] 0.5 0.5
## [51,] 0.5 0.5
## [52,] 0.5 0.5
## [53,] 0.5 0.5
## [54,] 0.5 0.5
## [55,] 0.5 0.5
## [56,] 0.5 0.5
## [57,] 0.5 0.5
## [58,] 0.5 0.5
## [59,] 0.5 0.5
## [60,] 0.5 0.5
## [61,] 0.5 0.5
## [62,] 0.5 0.5
## [63,] 0.5 0.5
## [64,] 0.5 0.5
## [65,] 0.5 0.5
## [66,] 0.5 0.5
## [67,] 0.5 0.5
## [68,] 0.5 0.5
## [69,] 0.5 0.5
## [70,] 0.5 0.5
## [71,] 0.5 0.5
## [72,] 0.5 0.5
## [73,] 0.5 0.5
## [74,] 0.5 0.5
## [75,] 0.5 0.5
## [76,] 0.5 0.5
## [77,] 0.5 0.5
## [78,] 0.5 0.5
## [79,] 0.5 0.5
## [80,] 0.5 0.5
## [81,] 0.5 0.5
## [82,] 0.5 0.5
## [83,] 0.5 0.5
## [84,] 0.5 0.5
## [85,] 0.5 0.5
## [86,] 0.5 0.5
## [87,] 0.5 0.5
## [88,] 0.5 0.5
## [89,] 0.5 0.5
## [90,] 0.5 0.5
## [91,] 0.5 0.5
## [92,] 0.5 0.5
## [93,] 0.5 0.5
## [94,] 0.5 0.5
## [95,] 0.5 0.5
## [96,] 0.5 0.5
## [97,] 0.5 0.5
## [98,] 0.5 0.5
## [99,] 0.5 0.5
## [100,] 0.5 0.5
## [101,] 0.5 0.5
## [102,] 0.5 0.5
## [103,] 0.5 0.5
## [104,] 0.5 0.5
## [105,] 0.5 0.5
## [106,] 0.5 0.5
## [107,] 0.5 0.5
## [108,] 0.5 0.5
## [109,] 0.5 0.5
## [110,] 0.5 0.5
## [111,] 0.5 0.5
## [112,] 0.5 0.5
## [113,] 0.5 0.5
## [114,] 0.5 0.5
## [115,] 0.5 0.5
## [116,] 0.5 0.5
## [117,] 0.5 0.5
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## [119,] 0.5 0.5
## [120,] 0.5 0.5
## [121,] 0.5 0.5
## [122,] 0.5 0.5
## [123,] 0.5 0.5
## [124,] 0.5 0.5
## [125,] 0.5 0.5
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## [128,] 0.5 0.5
## [129,] 0.5 0.5
## [130,] 0.5 0.5
## [131,] 0.5 0.5
## [132,] 0.5 0.5
## [133,] 0.5 0.5
## [134,] 0.5 0.5
## [135,] 0.5 0.5
## [136,] 0.5 0.5
## [137,] 0.5 0.5
## [138,] 0.5 0.5
## [139,] 0.5 0.5
## [140,] 0.5 0.5
## [141,] 0.5 0.5
## [142,] 0.5 0.5
## [143,] 0.5 0.5
## [144,] 0.5 0.5
## [145,] 0.5 0.5
## [146,] 0.5 0.5
## [147,] 0.5 0.5
## [148,] 0.5 0.5
## [149,] 0.5 0.5
## [150,] 0.5 0.5
## [151,] 0.5 0.5
## [152,] 0.5 0.5
## [153,] 0.5 0.5
## [154,] 0.5 0.5
## [155,] 0.5 0.5
## [156,] 0.5 0.5
## [157,] 0.5 0.5
## [158,] 0.5 0.5
## [159,] 0.5 0.5
## [160,] 0.5 0.5
## [161,] 0.5 0.5
## [162,] 0.5 0.5
## [163,] 0.5 0.5
## [164,] 0.5 0.5
## [165,] 0.5 0.5
## [166,] 0.5 0.5
## [167,] 0.5 0.5
## [168,] 0.5 0.5
## [169,] 0.5 0.5
## [170,] 0.5 0.5
## [171,] 0.5 0.5
## [172,] 0.5 0.5
## [173,] 0.5 0.5
## [174,] 0.5 0.5
## [175,] 0.5 0.5
## [176,] 0.5 0.5
## [177,] 0.5 0.5
## [178,] 0.5 0.5
## [179,] 0.5 0.5
## [180,] 0.5 0.5
## [181,] 0.5 0.5
## [182,] 0.5 0.5
## [183,] 0.5 0.5
## [184,] 0.5 0.5
## [185,] 0.5 0.5
## [186,] 0.5 0.5
## [187,] 0.5 0.5
## [188,] 0.5 0.5
## [189,] 0.5 0.5
## [190,] 0.5 0.5
## [191,] 0.5 0.5
## [192,] 0.5 0.5
## [193,] 0.5 0.5
## [194,] 0.5 0.5
## [195,] 0.5 0.5
## [196,] 0.5 0.5
## [197,] 0.5 0.5
## [198,] 0.5 0.5
## [199,] 0.5 0.5
## [200,] 0.5 0.5
##
##
## $tol
## [1] 1e-10
##
## $v
## [,1] [,2] [,3]
## [1,] 0.2160606 0 -0.2160606
##
## $lambda
## (Intercept) Dcollege Totalincome Dunemp
## 1 1925.469 2117.91 -0.008238330 -485.6138
## 0 2092.312 2267.59 -0.008558383 -539.4989
##
## $checkrestrictions
## $checkrestrictions$g1
## [,1]
## [1,] 0
## [2,] 0
## [3,] 0
## [4,] 0
## [5,] 0
## [6,] 0
## [7,] 0
## [8,] 0
## [9,] 0
## [10,] 0
## [11,] 0
## [12,] 0
## [13,] 0
## [14,] 0
## [15,] 0
## [16,] 0
## [17,] 0
## [18,] 0
## [19,] 0
## [20,] 0
## [21,] 0
## [22,] 0
## [23,] 0
## [24,] 0
## [25,] 0
## [26,] 0
## [27,] 0
## [28,] 0
## [29,] 0
## [30,] 0
## [31,] 0
## [32,] 0
## [33,] 0
## [34,] 0
## [35,] 0
## [36,] 0
## [37,] 0
## [38,] 0
## [39,] 0
## [40,] 0
## [41,] 0
## [42,] 0
## [43,] 0
## [44,] 0
## [45,] 0
## [46,] 0
## [47,] 0
## [48,] 0
## [49,] 0
## [50,] 0
## [51,] 0
## [52,] 0
## [53,] 0
## [54,] 0
## [55,] 0
## [56,] 0
## [57,] 0
## [58,] 0
## [59,] 0
## [60,] 0
## [61,] 0
## [62,] 0
## [63,] 0
## [64,] 0
## [65,] 0
## [66,] 0
## [67,] 0
## [68,] 0
## [69,] 0
## [70,] 0
## [71,] 0
## [72,] 0
## [73,] 0
## [74,] 0
## [75,] 0
## [76,] 0
## [77,] 0
## [78,] 0
## [79,] 0
## [80,] 0
## [81,] 0
## [82,] 0
## [83,] 0
## [84,] 0
## [85,] 0
## [86,] 0
## [87,] 0
## [88,] 0
## [89,] 0
## [90,] 0
## [91,] 0
## [92,] 0
## [93,] 0
## [94,] 0
## [95,] 0
## [96,] 0
## [97,] 0
## [98,] 0
## [99,] 0
## [100,] 0
## [101,] 0
## [102,] 0
## [103,] 0
## [104,] 0
## [105,] 0
## [106,] 0
## [107,] 0
## [108,] 0
## [109,] 0
## [110,] 0
## [111,] 0
## [112,] 0
## [113,] 0
## [114,] 0
## [115,] 0
## [116,] 0
## [117,] 0
## [118,] 0
## [119,] 0
## [120,] 0
## [121,] 0
## [122,] 0
## [123,] 0
## [124,] 0
## [125,] 0
## [126,] 0
## [127,] 0
## [128,] 0
## [129,] 0
## [130,] 0
## [131,] 0
## [132,] 0
## [133,] 0
## [134,] 0
## [135,] 0
## [136,] 0
## [137,] 0
## [138,] 0
## [139,] 0
## [140,] 0
## [141,] 0
## [142,] 0
## [143,] 0
## [144,] 0
## [145,] 0
## [146,] 0
## [147,] 0
## [148,] 0
## [149,] 0
## [150,] 0
## [151,] 0
## [152,] 0
## [153,] 0
## [154,] 0
## [155,] 0
## [156,] 0
## [157,] 0
## [158,] 0
## [159,] 0
## [160,] 0
## [161,] 0
## [162,] 0
## [163,] 0
## [164,] 0
## [165,] 0
## [166,] 0
## [167,] 0
## [168,] 0
## [169,] 0
## [170,] 0
## [171,] 0
## [172,] 0
## [173,] 0
## [174,] 0
## [175,] 0
## [176,] 0
## [177,] 0
## [178,] 0
## [179,] 0
## [180,] 0
## [181,] 0
## [182,] 0
## [183,] 0
## [184,] 0
## [185,] 0
## [186,] 0
## [187,] 0
## [188,] 0
## [189,] 0
## [190,] 0
## [191,] 0
## [192,] 0
## [193,] 0
## [194,] 0
## [195,] 0
## [196,] 0
## [197,] 0
## [198,] 0
## [199,] 0
## [200,] 0
##
## $checkrestrictions$g2
## [,1] [,2]
## [1,] 0 0
## [2,] 0 0
## [3,] 0 0
## [4,] 0 0
## [5,] 0 0
## [6,] 0 0
## [7,] 0 0
## [8,] 0 0
## [9,] 0 0
## [10,] 0 0
## [11,] 0 0
## [12,] 0 0
## [13,] 0 0
## [14,] 0 0
## [15,] 0 0
## [16,] 0 0
## [17,] 0 0
## [18,] 0 0
## [19,] 0 0
## [20,] 0 0
## [21,] 0 0
## [22,] 0 0
## [23,] 0 0
## [24,] 0 0
## [25,] 0 0
## [26,] 0 0
## [27,] 0 0
## [28,] 0 0
## [29,] 0 0
## [30,] 0 0
## [31,] 0 0
## [32,] 0 0
## [33,] 0 0
## [34,] 0 0
## [35,] 0 0
## [36,] 0 0
## [37,] 0 0
## [38,] 0 0
## [39,] 0 0
## [40,] 0 0
## [41,] 0 0
## [42,] 0 0
## [43,] 0 0
## [44,] 0 0
## [45,] 0 0
## [46,] 0 0
## [47,] 0 0
## [48,] 0 0
## [49,] 0 0
## [50,] 0 0
## [51,] 0 0
## [52,] 0 0
## [53,] 0 0
## [54,] 0 0
## [55,] 0 0
## [56,] 0 0
## [57,] 0 0
## [58,] 0 0
## [59,] 0 0
## [60,] 0 0
## [61,] 0 0
## [62,] 0 0
## [63,] 0 0
## [64,] 0 0
## [65,] 0 0
## [66,] 0 0
## [67,] 0 0
## [68,] 0 0
## [69,] 0 0
## [70,] 0 0
## [71,] 0 0
## [72,] 0 0
## [73,] 0 0
## [74,] 0 0
## [75,] 0 0
## [76,] 0 0
## [77,] 0 0
## [78,] 0 0
## [79,] 0 0
## [80,] 0 0
## [81,] 0 0
## [82,] 0 0
## [83,] 0 0
## [84,] 0 0
## [85,] 0 0
## [86,] 0 0
## [87,] 0 0
## [88,] 0 0
## [89,] 0 0
## [90,] 0 0
## [91,] 0 0
## [92,] 0 0
## [93,] 0 0
## [94,] 0 0
## [95,] 0 0
## [96,] 0 0
## [97,] 0 0
## [98,] 0 0
## [99,] 0 0
## [100,] 0 0
## [101,] 0 0
## [102,] 0 0
## [103,] 0 0
## [104,] 0 0
## [105,] 0 0
## [106,] 0 0
## [107,] 0 0
## [108,] 0 0
## [109,] 0 0
## [110,] 0 0
## [111,] 0 0
## [112,] 0 0
## [113,] 0 0
## [114,] 0 0
## [115,] 0 0
## [116,] 0 0
## [117,] 0 0
## [118,] 0 0
## [119,] 0 0
## [120,] 0 0
## [121,] 0 0
## [122,] 0 0
## [123,] 0 0
## [124,] 0 0
## [125,] 0 0
## [126,] 0 0
## [127,] 0 0
## [128,] 0 0
## [129,] 0 0
## [130,] 0 0
## [131,] 0 0
## [132,] 0 0
## [133,] 0 0
## [134,] 0 0
## [135,] 0 0
## [136,] 0 0
## [137,] 0 0
## [138,] 0 0
## [139,] 0 0
## [140,] 0 0
## [141,] 0 0
## [142,] 0 0
## [143,] 0 0
## [144,] 0 0
## [145,] 0 0
## [146,] 0 0
## [147,] 0 0
## [148,] 0 0
## [149,] 0 0
## [150,] 0 0
## [151,] 0 0
## [152,] 0 0
## [153,] 0 0
## [154,] 0 0
## [155,] 0 0
## [156,] 0 0
## [157,] 0 0
## [158,] 0 0
## [159,] 0 0
## [160,] 0 0
## [161,] 0 0
## [162,] 0 0
## [163,] 0 0
## [164,] 0 0
## [165,] 0 0
## [166,] 0 0
## [167,] 0 0
## [168,] 0 0
## [169,] 0 0
## [170,] 0 0
## [171,] 0 0
## [172,] 0 0
## [173,] 0 0
## [174,] 0 0
## [175,] 0 0
## [176,] 0 0
## [177,] 0 0
## [178,] 0 0
## [179,] 0 0
## [180,] 0 0
## [181,] 0 0
## [182,] 0 0
## [183,] 0 0
## [184,] 0 0
## [185,] 0 0
## [186,] 0 0
## [187,] 0 0
## [188,] 0 0
## [189,] 0 0
## [190,] 0 0
## [191,] 0 0
## [192,] 0 0
## [193,] 0 0
## [194,] 0 0
## [195,] 0 0
## [196,] 0 0
## [197,] 0 0
## [198,] 0 0
## [199,] 0 0
## [200,] 0 0
##
## $checkrestrictions$g3
## y_factor1 y_factor0
## (Intercept) 0.000 0.000
## Dcollege 0.000 0.000
## Totalincome -0.155 0.029
## Dunemp 0.000 0.000
##
##
## $cross_moments_A
## 1 0
## (Intercept) 0.34 0.66
## Dcollege 0.17 0.44
## Totalincome 103036.63 189594.78
## Dunemp 0.16 0.29
##
## $cross_moments_B
## 1 0
## (Intercept) 0.34 0.66
## Dcollege 0.17 0.44
## Totalincome 103036.79 189594.75
## Dunemp 0.16 0.29
##
## $J
## [1] 2
##
## $fn
## dataA$poor_liq ~ Dcollege + Totalincome + Dunemp
##
## attr(,"class")
## [1] "kl"
The summary and the plot options are the same as in the previous
case.
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).
Specifically, 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,dataA,dataB,weights="w")
The function will produce the same output as ei_gce(), see the
previous section for further details.
## $estimations
## weights predictions_1 predictions_0 probabilities_1 probabilities_0
## 1 6.408614e-03 0.37714100 1.0373456 0.17143056 0.8285694
## 2 4.069074e-03 0.46516844 0.9465387 0.26050116 0.7394988
## 3 8.478890e-05 0.49554426 0.4949818 0.50026475 0.4997352
## 4 6.019554e-03 0.29425398 0.7815639 0.27116751 0.7288325
## 5 8.529959e-03 0.24791986 0.3351769 0.45651891 0.5434811
## 6 3.466777e-03 0.37076550 0.4876877 0.44458668 0.5554133
## 7 7.143169e-03 0.25980016 0.5065030 0.38172997 0.6182700
## 8 6.728993e-03 0.36038509 1.0624027 0.14987197 0.8501280
## 9 9.459067e-03 0.24653071 0.9526971 0.16596528 0.8340347
## 10 1.952733e-03 0.42547701 0.5404902 0.44585302 0.5541470
## 11 6.884772e-03 0.36676112 1.0476607 0.16115959 0.8388404
## 12 1.046638e-03 0.42656444 0.4268691 0.49985549 0.5001445
## 13 6.632154e-03 0.45686631 0.9334013 0.26416625 0.7358337
## 14 5.694566e-03 0.34661296 0.8392470 0.26618165 0.7338183
## 15 8.199805e-03 0.29044842 0.3979353 0.44769069 0.5523093
## 16 7.974757e-03 0.34208495 0.8762999 0.24538724 0.7546128
## 17 1.016052e-05 0.50008404 0.5008871 0.49962202 0.5003780
## 18 1.077733e-03 0.47310062 0.4945723 0.48988960 0.5101104
## 19 4.555596e-03 0.43194344 0.9691885 0.23344930 0.7665507
## 20 1.797040e-03 0.40140320 0.4763702 0.46452988 0.5354701
## 21 7.726297e-03 0.36613320 0.2079584 0.57897910 0.4210209
## 22 7.537852e-03 0.33716485 1.0938415 0.12177914 0.8782209
## 23 5.078895e-04 0.48835953 0.5396067 0.47587226 0.5241277
## 24 8.673474e-03 0.34928915 1.0777076 0.13630052 0.8636995
## 25 6.743411e-03 0.28645508 0.2974439 0.49452509 0.5054749
## 26 2.597985e-03 0.34630193 0.3301962 0.50785821 0.4921418
## 27 7.946911e-03 0.32381835 0.9735062 0.18622698 0.8137730
## 28 3.132542e-03 0.37158284 0.4640124 0.45608357 0.5439164
## 29 1.413325e-03 0.50486829 0.6852096 0.41406943 0.5859306
## 30 1.952905e-03 0.50803501 0.6558103 0.42980514 0.5701949
## 31 7.786354e-03 0.34741212 1.0685416 0.14103834 0.8589617
## 32 6.412179e-03 0.36399162 0.2144079 0.57461746 0.4253825
## 33 8.813903e-03 0.14764207 0.4589496 0.34574908 0.6542509
## 34 3.104395e-03 0.35538676 0.5085648 0.42745386 0.5725461
## 35 1.494452e-03 0.50471073 0.6952247 0.40910457 0.5908954
## 36 1.979537e-03 0.43997236 0.5224390 0.46116787 0.5388321
## 37 1.781772e-03 0.42523278 0.4282943 0.49854802 0.5014520
## 38 8.631740e-04 0.49491100 0.5728496 0.46325836 0.5367416
## 39 2.624354e-03 0.46497177 0.6798731 0.39816848 0.6018315
## 40 7.278409e-03 0.34905395 1.0741076 0.13836237 0.8616376
## 41 9.198636e-03 0.21786352 0.3741723 0.42227101 0.5777290
## 42 8.939710e-04 0.45949750 0.5303564 0.46664817 0.5333518
## 43 5.755561e-03 0.32270764 0.3367562 0.49312027 0.5068797
## 44 9.197598e-03 0.32806646 0.2496265 0.53913519 0.4608648
## 45 7.835590e-03 0.20378477 0.7852617 0.22815772 0.7718423
## 46 7.304276e-03 0.34955410 1.0730095 0.13921821 0.8607818
## 47 2.825585e-03 0.43352574 0.6371526 0.40405019 0.5959498
## 48 5.701607e-03 0.36363393 0.2181771 0.57250285 0.4274972
## 49 9.403839e-03 0.14601780 0.6247003 0.27122861 0.7287714
## 50 8.874280e-03 0.08319548 0.6259962 0.23775894 0.7622411
## 51 5.856153e-03 0.35220670 0.2339843 0.55886949 0.4411305
## 52 7.506319e-04 0.50195976 0.5572226 0.47395680 0.5260432
## 53 2.829871e-03 0.51333914 0.8552131 0.33161750 0.6683825
## 54 1.403796e-03 0.42963824 0.4600169 0.48563946 0.5143605
## 55 2.256222e-03 0.50956487 0.7420472 0.38819990 0.6118001
## 56 8.706739e-03 0.32821696 1.0912943 0.11983024 0.8801698
## 57 6.495227e-04 0.51007559 0.5661376 0.47355873 0.5264413
## 58 7.423215e-03 0.46620740 0.7547751 0.36198130 0.6380187
## 59 9.409367e-03 0.12397287 0.5121628 0.30905282 0.6909472
## 60 3.467645e-03 0.47043100 0.8983508 0.28929920 0.7107008
## 61 4.891585e-03 0.25235968 0.5896629 0.34011437 0.6598856
## 62 5.191919e-05 0.49955438 0.5055161 0.49719380 0.5028062
## 63 6.960881e-04 0.52499283 0.6371783 0.44671365 0.5532863
## 64 3.234067e-03 0.43693087 0.5181214 0.46176043 0.5382396
## 65 8.095405e-03 0.32338775 1.1079812 0.10778784 0.8922122
## 66 2.123864e-03 0.38804696 0.4229342 0.48337175 0.5166283
## 67 3.309518e-03 0.36617643 0.4653326 0.45286598 0.5471340
## 68 5.300329e-03 0.45369992 0.7757864 0.34578335 0.6542166
## 69 7.890242e-03 0.43151755 0.9045747 0.26894725 0.7310527
## 70 5.248723e-03 0.44718704 0.7585655 0.35144245 0.6485576
## 71 9.625057e-04 0.48690874 0.5426172 0.47377121 0.5262288
## 72 3.609097e-03 0.48364084 0.9192295 0.28373023 0.7162698
## 73 5.617329e-03 0.34033991 0.2579077 0.54093430 0.4590657
## 74 9.237887e-03 0.26393288 0.3105145 0.47674268 0.5232573
## 75 9.275353e-03 0.20156798 0.8823505 0.18190501 0.8180950
## 76 7.561670e-04 0.49760854 0.5883611 0.45717684 0.5428232
## 77 1.374263e-03 0.48128804 0.6314625 0.42902081 0.5709792
## 78 2.232129e-03 0.37084525 0.5174121 0.43073025 0.5692698
## 79 7.759625e-03 0.19416064 0.7630441 0.23377557 0.7662244
## 80 3.268253e-03 0.35954585 0.2702604 0.54401482 0.4559852
## 81 5.423895e-03 0.32883003 0.3491396 0.49009377 0.5099062
## 82 3.519797e-03 0.44627539 0.8133955 0.32336997 0.6766300
## 83 3.686690e-03 0.46329502 0.9091089 0.28031343 0.7196866
## 84 8.770650e-03 0.34686180 1.0804289 0.13370232 0.8662977
## 85 4.499834e-03 0.32020803 0.2749708 0.52248010 0.4775199
## 86 6.913635e-03 0.45482424 0.7095723 0.37906282 0.6209372
## 87 7.423989e-03 0.38267169 0.9137441 0.24307479 0.7569252
## 88 9.184997e-03 0.32784987 1.1020920 0.11311855 0.8868814
## 89 9.934938e-04 0.53182177 0.6919863 0.42330347 0.5766965
## 90 7.108557e-03 0.41232879 0.6358961 0.39477003 0.6052300
## 91 6.307975e-03 0.35899737 1.0677312 0.14614680 0.8538532
## 92 5.444530e-03 0.36801075 1.0605667 0.15404688 0.8459531
## 93 5.708904e-03 0.40224834 0.9639685 0.22381750 0.7761825
## 94 7.141561e-03 0.26731143 0.3222452 0.47266606 0.5273339
## 95 7.903713e-03 0.16504002 0.4725597 0.34873935 0.6512606
## 96 5.645807e-03 0.39114706 1.0268661 0.18329898 0.8167010
## 97 3.539554e-03 0.49407722 0.8675384 0.31638410 0.6836159
## 98 2.396317e-03 0.52472964 0.8185411 0.35610951 0.6438905
## 99 3.785424e-03 0.31833590 0.3185359 0.49990155 0.5000984
## 100 6.899727e-03 0.34665472 0.8118079 0.27989326 0.7201067
## 101 1.963958e-03 0.38847647 0.4653263 0.46355690 0.5364431
## 102 4.301069e-03 0.46970545 0.6283353 0.42510979 0.5748902
## 103 8.824290e-03 0.44210574 0.1264228 0.65781855 0.3421815
## 104 9.245072e-03 0.29598198 1.1344336 0.08099374 0.9190063
## 105 7.604964e-03 0.24465042 0.3562688 0.44460738 0.5553926
## 106 7.759268e-04 0.45320401 0.4823022 0.48628348 0.5137165
## 107 6.426895e-03 0.36017732 0.5030302 0.43233587 0.5676641
## 108 5.150249e-03 0.34234836 0.2619044 0.53989146 0.4601085
## 109 5.158771e-03 0.38924124 0.7550976 0.32689702 0.6731030
## 110 7.247160e-03 0.48560850 0.7937028 0.35114885 0.6488512
## 111 7.664692e-03 0.27774016 0.3016068 0.48809683 0.5119032
## 112 4.033598e-03 0.35902292 0.7016212 0.33886170 0.6611383
## 113 8.610544e-03 0.29051769 0.2822052 0.50415187 0.4958481
## 114 9.173485e-03 0.27410778 0.9733603 0.16627345 0.8337265
## 115 2.312141e-03 0.51128193 0.7527714 0.38361570 0.6163843
## 116 5.785912e-03 0.40348506 1.0217406 0.19141214 0.8085879
## 117 3.133029e-03 0.50150728 0.8249307 0.34222266 0.6577773
## 118 3.027803e-04 0.49290497 0.5075815 0.49309175 0.5069082
## 119 6.494774e-04 0.46052418 0.4682542 0.49635496 0.5036450
## 120 2.627321e-03 0.51441106 0.8032531 0.35933661 0.6406634
## 121 6.300118e-03 0.38606905 1.0151758 0.18797870 0.8120213
## 122 5.009005e-03 0.39455986 0.5113978 0.44484464 0.5551554
## 123 8.546357e-03 0.36682709 0.9593036 0.21165655 0.7883435
## 124 5.935042e-03 0.36343477 1.0640667 0.15011579 0.8498842
## 125 5.670056e-05 0.49996437 0.5053768 0.49745235 0.5025477
## 126 1.956694e-04 0.49743770 0.5190521 0.48982516 0.5101748
## 127 3.491569e-03 0.36763703 0.4826408 0.44544988 0.5545501
## 128 6.018842e-03 0.25616395 0.5745515 0.34884931 0.6511507
## 129 2.453778e-03 0.42789177 0.6552214 0.39282704 0.6071730
## 130 3.330195e-03 0.45417300 0.7229730 0.37220860 0.6277914
## 131 4.568058e-04 0.49890140 0.5417205 0.47983383 0.5201662
## 132 1.878873e-03 0.49715239 0.6752411 0.41533270 0.5846673
## 133 7.274750e-03 0.36343765 1.0385281 0.16518550 0.8348145
## 134 8.055270e-03 0.17014297 0.5542951 0.31464738 0.6853526
## 135 7.918719e-03 0.35004282 0.7405543 0.31614325 0.6838568
## 136 1.613960e-03 0.44481819 0.5890183 0.43212122 0.5678788
## 137 4.043005e-03 0.45292968 0.7853295 0.34062799 0.6593720
## 138 2.836492e-03 0.36867880 0.4335521 0.46904439 0.5309556
## 139 8.648536e-03 0.40487846 0.8830274 0.26894402 0.7310560
## 140 7.121030e-03 0.40200478 1.0056062 0.20009305 0.7999069
## 141 7.015624e-03 0.50187202 0.8437297 0.33252488 0.6674751
## 142 2.042604e-03 0.44644705 0.5396346 0.45613989 0.5438601
## 143 7.276373e-03 0.27335900 0.3110216 0.48123769 0.5187623
## 144 8.504937e-03 0.28247500 1.1493787 0.06658533 0.9334147
## 145 9.535921e-03 0.40888960 0.9714838 0.22238811 0.7776119
## 146 8.545091e-03 0.32920752 0.2406715 0.54424876 0.4557512
## 147 2.254398e-03 0.37776169 0.4714294 0.45556223 0.5444378
## 148 5.514651e-03 0.27258786 0.5631959 0.36208390 0.6379161
## 149 6.254560e-03 0.37577569 1.0434211 0.16731353 0.8326865
## 150 6.128050e-03 0.46189330 0.6907315 0.39148591 0.6085141
## 151 1.952048e-03 0.39256306 0.4730381 0.46188246 0.5381175
## 152 8.239200e-03 0.33030051 0.7244088 0.31481486 0.6851851
## 153 4.212187e-03 0.32458121 0.2727712 0.52573342 0.4742666
## 154 6.562633e-03 0.32255966 0.2521844 0.53513468 0.4648653
## 155 8.529452e-05 0.49978514 0.5056392 0.49724448 0.5027555
## 156 4.627415e-03 0.30253554 0.3171601 0.49277353 0.5072265
## 157 7.506022e-03 0.19114391 0.7362697 0.24439862 0.7556014
## 158 5.802968e-03 0.36849394 0.2110729 0.57850610 0.4214939
## 159 8.889093e-03 0.13810223 0.4959942 0.32384897 0.6761510
## 160 3.338891e-03 0.35704116 0.2980883 0.52889685 0.4711032
## 161 5.739428e-03 0.29183838 0.3035602 0.49417525 0.5058248
## 162 9.051019e-03 0.24618906 0.9294580 0.17769349 0.8223065
## 163 7.187216e-03 0.44810200 0.9357919 0.25904055 0.7409594
## 164 2.371262e-03 0.37102321 0.2899472 0.53968910 0.4603109
## 165 5.791201e-03 0.33183326 0.7032186 0.32554605 0.6744539
## 166 5.959915e-03 0.39382630 0.9629484 0.22086063 0.7791394
## 167 8.379680e-03 0.25726560 0.7659283 0.26171973 0.7382803
## 168 8.564844e-03 0.32893623 0.2510775 0.53882493 0.4611751
## 169 1.064177e-03 0.50320852 0.5820377 0.46280348 0.5371965
## 170 6.110630e-03 0.23336248 0.5255228 0.35980801 0.6401920
## 171 1.771706e-03 0.52165759 0.6221198 0.45237437 0.5476256
## 172 3.746941e-03 0.42731642 0.5216066 0.45558316 0.5444168
## 173 8.344937e-03 0.25568751 0.5881512 0.34243236 0.6575676
## 174 8.671874e-03 0.14356052 0.4964636 0.32654393 0.6734561
## 175 5.165876e-03 0.29467706 0.4293389 0.43492929 0.5650707
## 176 6.250491e-03 0.26256080 0.3704989 0.44683054 0.5531695
## 177 5.663535e-03 0.47401068 0.8621305 0.31031827 0.6896817
## 178 8.189466e-03 0.32238236 0.2483328 0.53700190 0.4629981
## 179 5.172787e-03 0.37674492 0.6194590 0.38587044 0.6141296
## 180 2.048906e-03 0.51888448 0.6288528 0.44784828 0.5521517
## 181 6.680525e-03 0.31019439 0.3270606 0.49169932 0.5083007
## 182 4.062967e-03 0.32344469 0.4156347 0.45557047 0.5444295
## 183 8.424910e-03 0.28920324 0.3299450 0.47986579 0.5201342
## 184 7.709358e-03 0.31518467 0.2788501 0.51806082 0.4819392
## 185 2.544983e-03 0.36650172 0.4030771 0.48246199 0.5175380
## 186 5.364966e-03 0.50714385 0.7578392 0.37916456 0.6208354
## 187 5.934410e-04 0.49408188 0.5305318 0.48283964 0.5171604
## 188 5.020772e-05 0.50207164 0.5088920 0.49678951 0.5032105
## 189 2.873655e-03 0.52784553 0.6870686 0.42385741 0.5761426
## 190 5.514606e-03 0.30662414 0.2813504 0.51258028 0.4874197
## 191 2.057597e-03 0.41814271 0.5359939 0.44449899 0.5555010
## 192 8.173496e-03 0.15968134 0.5246346 0.32242353 0.6775765
## 193 2.086392e-03 0.43346741 0.5171345 0.46058843 0.5394116
## 194 3.212343e-03 0.46856248 0.8320487 0.32370124 0.6762988
## 195 7.999160e-03 0.22223277 0.4232671 0.40126720 0.5987328
## 196 9.251449e-07 0.49996937 0.5000287 0.49997206 0.5000279
## 197 2.268155e-03 0.45362669 0.6246516 0.41936319 0.5806368
## 198 7.367501e-03 0.37636973 0.8968109 0.24939865 0.7506013
## 199 2.263759e-03 0.37305012 0.2949577 0.53817009 0.4618299
## 200 3.956566e-03 0.46982359 0.9399072 0.26622585 0.7337742
## errors_1 errors_0
## 1 2.057104e-01 2.087761e-01
## 2 2.046673e-01 2.070398e-01
## 3 -4.720492e-03 -4.753433e-03
## 4 2.308647e-02 5.273141e-02
## 5 -2.085990e-01 -2.083042e-01
## 6 -7.382118e-02 -6.772559e-02
## 7 -1.219298e-01 -1.117670e-01
## 8 2.105131e-01 2.122746e-01
## 9 8.056542e-02 1.186624e-01
## 10 -2.037601e-02 -1.365677e-02
## 11 2.056015e-01 2.088203e-01
## 12 -7.329105e-02 -7.327537e-02
## 13 1.927001e-01 1.975676e-01
## 14 8.043131e-02 1.054286e-01
## 15 -1.572423e-01 -1.543740e-01
## 16 9.669772e-02 1.216871e-01
## 17 4.620248e-04 5.090779e-04
## 18 -1.678897e-02 -1.553812e-02
## 19 1.984941e-01 2.026378e-01
## 20 -6.312668e-02 -5.909996e-02
## 21 -2.128459e-01 -2.130625e-01
## 22 2.153857e-01 2.156207e-01
## 23 1.248727e-02 1.547896e-02
## 24 2.129886e-01 2.140081e-01
## 25 -2.080700e-01 -2.080310e-01
## 26 -1.615563e-01 -1.619456e-01
## 27 1.375914e-01 1.597331e-01
## 28 -8.450073e-02 -7.990404e-02
## 29 9.079887e-02 9.927906e-02
## 30 7.822987e-02 8.561544e-02
## 31 2.063738e-01 2.095800e-01
## 32 -2.106258e-01 -2.109747e-01
## 33 -1.981070e-01 -1.953013e-01
## 34 -7.206710e-02 -6.398138e-02
## 35 9.560616e-02 1.043293e-01
## 36 -2.119551e-02 -1.639315e-02
## 37 -7.331524e-02 -7.315768e-02
## 38 3.165264e-02 3.610795e-02
## 39 6.680329e-02 7.804157e-02
## 40 2.106916e-01 2.124700e-01
## 41 -2.044075e-01 -2.035567e-01
## 42 -7.150672e-03 -2.995391e-03
## 43 -1.704126e-01 -1.701235e-01
## 44 -2.110687e-01 -2.112383e-01
## 45 -2.437296e-02 1.341938e-02
## 46 2.103359e-01 2.122277e-01
## 47 2.947555e-02 4.120279e-02
## 48 -2.088689e-01 -2.093201e-01
## 49 -1.252108e-01 -1.040711e-01
## 50 -1.545635e-01 -1.362448e-01
## 51 -2.066628e-01 -2.071462e-01
## 52 2.800295e-02 3.117943e-02
## 53 1.817216e-01 1.868306e-01
## 54 -5.600122e-02 -5.434368e-02
## 55 1.213650e-01 1.302471e-01
## 56 2.083867e-01 2.111246e-01
## 57 3.651686e-02 3.969636e-02
## 58 1.042261e-01 1.167564e-01
## 59 -1.850800e-01 -1.787844e-01
## 60 1.811318e-01 1.876500e-01
## 61 -8.775469e-02 -7.022273e-02
## 62 2.360579e-03 2.709861e-03
## 63 7.827917e-02 8.389191e-02
## 64 -2.482957e-02 -2.011820e-02
## 65 2.155999e-01 2.157690e-01
## 66 -9.532478e-02 -9.369404e-02
## 67 -8.668955e-02 -8.180145e-02
## 68 1.079166e-01 1.215698e-01
## 69 1.625703e-01 1.735219e-01
## 70 9.574459e-02 1.100080e-01
## 71 1.313752e-02 1.638841e-02
## 72 1.999106e-01 2.029597e-01
## 73 -2.005944e-01 -2.011580e-01
## 74 -2.128098e-01 -2.127428e-01
## 75 1.966296e-02 6.425551e-02
## 76 4.043169e-02 4.553798e-02
## 77 5.226723e-02 6.048327e-02
## 78 -5.988499e-02 -5.185763e-02
## 79 -3.961493e-02 -3.180358e-03
## 80 -1.844690e-01 -1.857248e-01
## 81 -1.612637e-01 -1.607666e-01
## 82 1.229054e-01 1.367655e-01
## 83 1.829816e-01 1.894223e-01
## 84 2.131595e-01 2.141312e-01
## 85 -2.022721e-01 -2.025491e-01
## 86 7.576142e-02 8.863514e-02
## 87 1.395969e-01 1.568188e-01
## 88 2.147313e-01 2.152105e-01
## 89 1.085183e-01 1.152898e-01
## 90 1.755876e-02 3.066611e-02
## 91 2.128506e-01 2.138780e-01
## 92 2.139639e-01 2.146135e-01
## 93 1.784308e-01 1.877860e-01
## 94 -2.053546e-01 -2.050887e-01
## 95 -1.836993e-01 -1.787009e-01
## 96 2.078481e-01 2.101650e-01
## 97 1.776931e-01 1.839225e-01
## 98 1.686201e-01 1.746506e-01
## 99 -1.815657e-01 -1.815625e-01
## 100 6.676146e-02 9.170119e-02
## 101 -7.508043e-02 -7.111679e-02
## 102 4.459566e-02 5.344510e-02
## 103 -2.157128e-01 -2.157587e-01
## 104 2.149882e-01 2.154274e-01
## 105 -1.999570e-01 -1.991238e-01
## 106 -3.307947e-02 -3.141428e-02
## 107 -7.215855e-02 -6.463398e-02
## 108 -1.975431e-01 -1.982041e-01
## 109 6.234423e-02 8.199463e-02
## 110 1.344596e-01 1.448517e-01
## 111 -2.103567e-01 -2.102963e-01
## 112 2.016121e-02 4.048288e-02
## 113 -2.136342e-01 -2.136430e-01
## 114 1.078343e-01 1.396337e-01
## 115 1.276662e-01 1.363871e-01
## 116 2.120729e-01 2.131528e-01
## 117 1.592846e-01 1.671533e-01
## 118 -1.867789e-04 6.732522e-04
## 119 -3.583079e-02 -3.539087e-02
## 120 1.550744e-01 1.625898e-01
## 121 1.980903e-01 2.031546e-01
## 122 -5.028477e-02 -4.375760e-02
## 123 1.551705e-01 1.709601e-01
## 124 2.133190e-01 2.141825e-01
## 125 2.512029e-03 2.829124e-03
## 126 7.612538e-03 8.877257e-03
## 127 -7.781284e-02 -7.190928e-02
## 128 -9.268536e-02 -7.659921e-02
## 129 3.506474e-02 4.804841e-02
## 130 8.196439e-02 9.518159e-02
## 131 1.906758e-02 2.155435e-02
## 132 8.181969e-02 9.057382e-02
## 133 1.982521e-01 2.037136e-01
## 134 -1.445044e-01 -1.310576e-01
## 135 3.389958e-02 5.669753e-02
## 136 1.269697e-02 2.113953e-02
## 137 1.123017e-01 1.259575e-01
## 138 -1.003656e-01 -9.740354e-02
## 139 1.359344e-01 1.519715e-01
## 140 2.019117e-01 2.056993e-01
## 141 1.693471e-01 1.762546e-01
## 142 -9.692835e-03 -4.225522e-03
## 143 -2.078787e-01 -2.077407e-01
## 144 2.158897e-01 2.159640e-01
## 145 1.865015e-01 1.938719e-01
## 146 -2.150412e-01 -2.150798e-01
## 147 -7.780054e-02 -7.300837e-02
## 148 -8.949604e-02 -7.472020e-02
## 149 2.084622e-01 2.107346e-01
## 150 7.040738e-02 8.221738e-02
## 151 -6.931940e-02 -6.507947e-02
## 152 1.548565e-02 3.922368e-02
## 153 -2.011522e-01 -2.014954e-01
## 154 -2.125750e-01 -2.126809e-01
## 155 2.540662e-03 2.883628e-03
## 156 -1.902380e-01 -1.900663e-01
## 157 -5.325471e-02 -1.933165e-02
## 158 -2.100122e-01 -2.104210e-01
## 159 -1.857467e-01 -1.801568e-01
## 160 -1.718557e-01 -1.730148e-01
## 161 -2.023369e-01 -2.022645e-01
## 162 6.849557e-02 1.071515e-01
## 163 1.890615e-01 1.948325e-01
## 164 -1.686659e-01 -1.703637e-01
## 165 6.287213e-03 2.876469e-02
## 166 1.729657e-01 1.838091e-01
## 167 -4.454133e-03 2.764807e-02
## 168 -2.098887e-01 -2.100975e-01
## 169 4.040504e-02 4.484123e-02
## 170 -1.264455e-01 -1.146692e-01
## 171 6.928322e-02 7.449420e-02
## 172 -2.826674e-02 -2.281027e-02
## 173 -8.674485e-02 -6.941641e-02
## 174 -1.829834e-01 -1.769924e-01
## 175 -1.402522e-01 -1.357318e-01
## 176 -1.842697e-01 -1.826705e-01
## 177 1.636924e-01 1.724487e-01
## 178 -2.146195e-01 -2.146653e-01
## 179 -9.125522e-03 5.329423e-03
## 180 7.103619e-02 7.670112e-02
## 181 -1.815049e-01 -1.812400e-01
## 182 -1.321258e-01 -1.287948e-01
## 183 -1.906626e-01 -1.901892e-01
## 184 -2.028761e-01 -2.030891e-01
## 185 -1.159603e-01 -1.144609e-01
## 186 1.279793e-01 1.370038e-01
## 187 1.124224e-02 1.337148e-02
## 188 5.282129e-03 5.681505e-03
## 189 1.039881e-01 1.109260e-01
## 190 -2.059561e-01 -2.060693e-01
## 191 -2.635628e-02 -1.950714e-02
## 192 -1.627422e-01 -1.529419e-01
## 193 -2.712102e-02 -2.227710e-02
## 194 1.448612e-01 1.557500e-01
## 195 -1.790344e-01 -1.754657e-01
## 196 -2.691577e-06 7.861760e-07
## 197 3.426350e-02 4.401479e-02
## 198 1.269711e-01 1.462095e-01
## 199 -1.651200e-01 -1.668722e-01
## 200 2.035977e-01 2.061330e-01
##
## $values
## $values$entropy
## [1] 393.749
##
## $values$iterations
## [1] 96
##
## $values$message
## [1] "relative convergence (4)"
##
##
## $tol
## [1] 1e-10
##
## $v
## [,1] [,2] [,3]
## [1,] 0.2160606 0 -0.2160606
##
## $lambda
## (Intercept) Dcollege Totalincome Dunemp
## 1 1925.471 2117.909 -0.008238336 -485.6150
## 0 2092.314 2267.589 -0.008558389 -539.5005
##
## $checkrestrictions
## $checkrestrictions$g1
## [,1]
## [1,] 0
## [2,] 0
## [3,] 0
## [4,] 0
## [5,] 0
## [6,] 0
## [7,] 0
## [8,] 0
## [9,] 0
## [10,] 0
## [11,] 0
## [12,] 0
## [13,] 0
## [14,] 0
## [15,] 0
## [16,] 0
## [17,] 0
## [18,] 0
## [19,] 0
## [20,] 0
## [21,] 0
## [22,] 0
## [23,] 0
## [24,] 0
## [25,] 0
## [26,] 0
## [27,] 0
## [28,] 0
## [29,] 0
## [30,] 0
## [31,] 0
## [32,] 0
## [33,] 0
## [34,] 0
## [35,] 0
## [36,] 0
## [37,] 0
## [38,] 0
## [39,] 0
## [40,] 0
## [41,] 0
## [42,] 0
## [43,] 0
## [44,] 0
## [45,] 0
## [46,] 0
## [47,] 0
## [48,] 0
## [49,] 0
## [50,] 0
## [51,] 0
## [52,] 0
## [53,] 0
## [54,] 0
## [55,] 0
## [56,] 0
## [57,] 0
## [58,] 0
## [59,] 0
## [60,] 0
## [61,] 0
## [62,] 0
## [63,] 0
## [64,] 0
## [65,] 0
## [66,] 0
## [67,] 0
## [68,] 0
## [69,] 0
## [70,] 0
## [71,] 0
## [72,] 0
## [73,] 0
## [74,] 0
## [75,] 0
## [76,] 0
## [77,] 0
## [78,] 0
## [79,] 0
## [80,] 0
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##
## $checkrestrictions$g2
## [,1] [,2]
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##
## $checkrestrictions$g3
## 1 0
## (Intercept) 0.000 0.00
## Dcollege 0.000 0.00
## Totalincome -0.031 0.01
## Dunemp 0.000 0.00
##
##
## $cross_moments_A
## 1 0
## (Intercept) 0.34 0.66
## Dcollege 0.17 0.44
## Totalincome 103036.63 189594.78
## Dunemp 0.16 0.29
##
## $cross_moments_B
## 1 0
## (Intercept) 0.34 0.66
## Dcollege 0.17 0.44
## Totalincome 103036.67 189594.77
## Dunemp 0.16 0.29
##
## $J
## [1] 2
##
## $fn
## dataA$poor_liq ~ Dcollege + Totalincome + Dunemp
##
## $divergencekl
## [1] 13.52074
##
## 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.
## Iterations[1] 96
## Entropy value[1] 393.749
## [1] 13.52074
## [1] "mean_estimations"
## weights predictions_1 predictions_0 probabilities_1 probabilities_0
## 1 0.006608031 0.3377784 0.6622217 0.3420553 0.6579447
## errors_1 errors_0
## 1 -0.004276907 0.004276958
## [1] "lambda"
## (Intercept) Dcollege Totalincome Dunemp
## 1 1925.471 2117.909 -0.008238336 -485.6150
## 0 2092.314 2267.589 -0.008558389 -539.5005
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=dataB$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.