A function to calculate the Probability Integral Transform (PIT) values for any fitted model that assumes a normal distribution of the output.
Arguments
- ycal
Numeric vector representing the true observations (y-values) of the response variable from the calibration dataset.
- yhat
Numeric vector of predicted y-values on the calibration dataset.
- mse
Mean Squared Error calculated from the calibration dataset.
Details
This function is designed to work with models that is, even implicitly, assuming normal distribution of the response variable.
This includes, but is not limited to, linear models created using lm()
or neural networks utilizing Mean Squared Error as the loss function.
The OLS method is used to minimized residuals in these models. This mathematical optimization will also yield a probabilistic optimization
when normal distribution of the response variable is assumed, since OLS and maximum likelihood estimation are equivalent under normality.
Therefore, in order to render a probabilistic interpretation of the predictions, the model is intrinsically assuming a normal distribution of the response variable.
Examples
n <- 10000
split <- 0.8
# generating heterocedastic data
mu <- function(x1){
10 + 5*x1^2
}
sigma_v <- function(x1){
30*x1
}
x <- runif(n, 1, 10)
y <- rnorm(n, mu(x), sigma_v(x))
x_train <- x[1:(n*split)]
y_train <- y[1:(n*split)]
x_cal <- x[(n*split+1):n]
y_cal <- y[(n*split+1):n]
model <- lm(y_train ~ x_train)
y_hat <- predict(model, newdata=data.frame(x_train=x_cal))
MSE_cal <- mean((y_hat - y_cal)^2)
PIT_global(ycal=y_cal, yhat=y_hat, mse=MSE_cal)
#> [1] 3.209528e-01 2.542822e-01 5.848373e-01 5.522800e-01 4.941958e-01
#> [6] 2.841665e-01 4.765463e-01 5.241583e-01 4.617327e-01 6.711078e-01
#> [11] 4.987337e-01 9.814634e-01 6.186309e-01 7.613117e-01 9.443548e-01
#> [16] 1.639562e-01 4.994037e-01 5.379369e-01 4.949492e-01 5.750353e-01
#> [21] 7.634381e-01 2.578784e-01 7.400041e-01 6.948636e-01 2.356780e-01
#> [26] 1.426774e-01 3.843668e-01 5.461879e-01 5.221393e-01 6.838410e-01
#> [31] 2.170843e-01 2.533476e-01 4.953253e-01 6.200301e-01 2.582158e-01
#> [36] 5.597634e-01 6.762802e-01 9.554940e-01 5.049130e-04 4.145003e-01
#> [41] 4.177099e-01 9.110754e-01 2.208708e-02 6.678141e-01 3.025740e-01
#> [46] 1.699051e-02 4.342269e-03 7.429606e-01 9.814865e-01 8.575487e-01
#> [51] 9.387832e-01 6.878286e-01 1.687177e-01 6.729524e-01 6.847628e-01
#> [56] 4.265289e-01 5.879793e-02 5.246961e-01 6.578146e-01 6.057525e-01
#> [61] 4.365575e-01 5.393396e-01 6.937027e-01 8.371172e-01 7.577937e-01
#> [66] 5.828714e-01 8.861078e-01 6.996180e-01 6.012944e-02 4.528861e-01
#> [71] 8.683483e-01 6.473983e-01 4.278458e-01 7.266505e-01 3.906435e-01
#> [76] 1.872083e-01 3.380191e-01 7.389251e-02 1.215886e-01 5.472933e-01
#> [81] 5.085622e-01 3.038262e-01 2.446219e-01 9.095126e-01 3.168781e-01
#> [86] 9.328806e-01 9.664716e-01 1.494852e-01 5.576187e-01 4.687967e-01
#> [91] 2.684255e-01 3.822580e-01 4.811633e-01 3.373678e-01 7.426035e-01
#> [96] 4.506117e-01 3.326797e-01 5.582804e-01 9.471109e-01 3.850320e-02
#> [101] 3.566103e-01 5.078086e-01 2.648426e-02 5.366813e-01 6.735404e-01
#> [106] 6.060324e-01 8.660170e-01 7.944941e-01 4.069576e-01 7.964448e-01
#> [111] 7.233757e-01 2.721450e-01 8.525928e-01 2.375411e-02 6.747939e-01
#> [116] 3.505730e-01 5.079863e-01 8.324804e-01 2.236901e-02 4.009909e-01
#> [121] 2.238197e-01 9.481683e-01 3.857298e-02 7.181686e-01 7.277364e-01
#> [126] 3.446236e-01 3.065754e-01 5.349311e-01 3.094180e-01 5.704436e-01
#> [131] 6.844809e-02 8.330340e-01 8.469795e-01 4.004108e-01 9.980528e-01
#> [136] 5.540584e-01 9.516905e-01 5.546023e-02 9.701903e-01 7.056137e-01
#> [141] 3.879256e-01 1.927386e-01 3.609027e-04 1.457337e-01 9.977579e-01
#> [146] 4.629744e-01 2.989980e-01 3.408646e-01 6.084183e-01 2.773938e-01
#> [151] 5.029694e-01 9.329656e-01 2.884244e-01 3.248259e-01 5.681085e-01
#> [156] 8.586596e-01 7.771373e-01 7.117091e-01 8.566002e-02 3.038173e-01
#> [161] 5.386503e-03 1.167573e-01 9.756200e-01 5.972398e-01 4.890154e-01
#> [166] 1.204637e-01 2.802092e-01 3.526749e-01 4.428582e-01 5.923338e-01
#> [171] 4.507230e-01 7.014757e-01 1.415888e-01 3.723933e-01 3.580957e-01
#> [176] 3.336864e-01 4.964990e-01 3.226519e-01 9.830101e-01 6.422924e-01
#> [181] 7.032791e-01 7.192926e-01 6.001618e-01 9.998183e-01 1.782125e-02
#> [186] 3.001945e-01 3.626527e-01 5.746659e-01 3.955776e-01 7.610542e-01
#> [191] 3.887457e-01 4.730374e-01 4.950890e-01 6.418405e-01 5.006575e-01
#> [196] 1.557800e-01 6.473796e-01 9.799437e-01 9.574561e-01 8.712224e-01
#> [201] 5.813606e-01 4.903663e-01 6.836948e-01 4.296365e-01 9.986650e-01
#> [206] 5.492456e-01 7.225288e-01 6.670911e-01 2.264972e-01 8.846454e-01
#> [211] 5.778253e-01 2.113359e-02 2.612052e-01 8.993121e-01 9.273065e-01
#> [216] 2.379286e-01 6.021698e-01 3.544872e-01 1.564039e-01 4.246967e-01
#> [221] 6.366790e-01 6.725571e-01 8.069578e-01 8.153741e-01 1.854190e-01
#> [226] 2.172898e-01 1.684592e-01 7.676812e-01 9.970012e-01 6.569859e-01
#> [231] 6.139197e-01 4.847418e-01 7.926957e-02 7.700962e-01 4.697670e-01
#> [236] 8.656109e-01 5.696149e-01 8.717138e-01 7.574000e-01 9.264827e-01
#> [241] 4.918852e-01 6.308719e-01 2.998063e-01 5.440481e-01 6.670092e-01
#> [246] 2.267068e-01 9.582720e-05 8.601971e-01 3.490073e-01 5.364982e-01
#> [251] 7.879091e-02 7.670277e-01 6.644234e-01 7.966409e-01 5.950357e-03
#> [256] 8.038596e-01 7.726666e-01 4.161262e-01 6.980985e-01 2.355243e-01
#> [261] 6.573912e-01 9.905392e-01 7.469150e-01 3.217261e-01 9.624244e-01
#> [266] 4.321638e-01 5.911148e-01 7.789989e-01 3.779006e-01 6.548775e-01
#> [271] 9.107512e-02 5.600097e-01 3.761029e-01 5.586919e-01 4.668294e-02
#> [276] 1.486918e-01 3.050985e-01 9.972142e-01 9.049245e-01 1.932052e-01
#> [281] 4.405279e-01 2.562198e-01 5.100040e-01 4.505432e-01 4.441820e-01
#> [286] 1.636154e-01 7.289623e-03 8.586065e-02 5.807717e-01 7.050007e-02
#> [291] 1.206648e-01 1.670050e-01 3.008639e-02 6.618554e-01 5.071989e-01
#> [296] 4.476212e-01 2.105572e-01 2.950113e-01 9.718952e-01 6.584970e-01
#> [301] 7.324243e-02 9.499303e-01 3.576326e-01 9.998428e-01 6.158061e-01
#> [306] 5.961305e-01 3.432620e-01 4.919524e-01 8.172159e-01 2.246073e-01
#> [311] 5.956185e-01 2.137342e-01 5.803162e-01 1.175724e-01 1.384600e-01
#> [316] 9.804188e-01 4.811496e-01 6.181114e-01 9.499617e-02 5.378143e-01
#> [321] 1.912286e-01 1.329244e-01 7.696357e-01 2.695591e-01 1.874722e-01
#> [326] 1.101252e-01 6.079470e-02 9.442265e-01 5.938891e-01 2.420370e-01
#> [331] 5.702224e-01 5.053205e-01 2.144031e-01 6.784454e-01 2.681311e-01
#> [336] 6.694944e-01 7.017074e-01 7.247929e-01 3.040990e-01 8.602378e-01
#> [341] 1.009735e-01 4.340843e-01 1.836635e-01 2.284394e-01 9.792969e-01
#> [346] 6.274403e-01 4.581579e-01 3.584369e-01 7.040197e-01 5.019358e-01
#> [351] 2.341444e-01 2.274056e-01 1.749485e-01 5.978402e-01 6.809092e-01
#> [356] 9.994246e-01 9.997210e-01 9.585073e-01 9.828154e-01 5.503842e-01
#> [361] 4.105724e-01 2.054716e-01 2.547407e-01 3.205622e-01 4.290653e-01
#> [366] 4.647279e-01 8.954724e-01 8.583492e-01 2.187015e-01 3.884022e-01
#> [371] 9.706440e-01 5.852712e-01 4.899380e-01 9.610651e-01 9.391375e-01
#> [376] 4.385380e-01 7.865182e-02 7.064022e-01 7.911827e-01 9.084079e-01
#> [381] 7.308023e-01 3.864131e-01 6.684720e-01 9.238401e-01 9.470780e-01
#> [386] 5.103842e-01 6.168363e-01 5.290780e-01 2.480770e-01 5.501589e-03
#> [391] 1.820832e-01 7.668523e-01 5.551558e-01 2.143299e-01 2.140224e-01
#> [396] 2.748238e-01 4.571362e-01 7.687427e-01 9.234780e-01 6.128225e-01
#> [401] 6.332560e-01 1.079530e-01 6.438584e-01 3.136279e-02 9.675936e-01
#> [406] 1.007531e-01 7.737288e-01 8.739403e-01 9.371710e-01 9.972147e-01
#> [411] 1.155770e-02 7.098008e-01 2.633398e-01 9.825497e-01 4.235528e-01
#> [416] 6.142647e-01 8.039558e-01 3.030510e-01 9.514572e-01 6.891465e-01
#> [421] 6.160320e-01 6.667731e-01 2.682230e-01 6.882917e-01 3.197736e-01
#> [426] 4.118900e-01 8.619762e-01 3.506029e-02 2.983166e-01 9.679655e-01
#> [431] 6.462806e-01 7.165370e-01 8.885618e-01 5.260310e-01 4.671826e-01
#> [436] 9.536811e-01 7.236465e-01 4.576799e-01 8.349414e-01 5.634648e-01
#> [441] 5.482312e-01 3.470745e-01 6.540868e-01 7.979455e-01 6.108402e-01
#> [446] 7.189960e-01 4.901169e-01 5.330378e-01 3.444335e-01 2.217240e-01
#> [451] 8.156262e-01 6.148483e-01 1.263090e-01 6.989804e-01 5.330237e-01
#> [456] 9.976841e-01 2.834725e-01 9.852655e-01 4.176053e-01 3.251531e-01
#> [461] 7.024314e-01 3.565683e-01 2.673699e-01 7.186028e-01 3.957805e-01
#> [466] 7.169461e-01 3.940913e-01 7.070158e-01 5.635226e-01 7.710455e-01
#> [471] 6.479522e-01 3.466959e-01 1.172514e-01 4.054321e-01 9.230608e-02
#> [476] 5.263395e-01 1.666226e-01 5.470581e-01 5.297409e-01 6.538907e-01
#> [481] 4.146060e-01 7.270154e-01 6.428663e-02 5.885199e-01 2.707255e-01
#> [486] 8.544356e-02 2.375469e-01 2.094058e-01 5.681658e-01 2.549011e-01
#> [491] 2.404293e-02 3.159765e-01 5.178230e-01 3.274941e-01 9.099284e-01
#> [496] 6.199290e-01 6.978089e-02 8.414753e-01 7.930515e-01 3.488702e-01
#> [501] 3.210221e-01 3.192779e-01 2.504349e-01 6.718586e-01 6.966701e-01
#> [506] 7.640630e-01 9.765603e-01 7.787838e-02 3.494088e-01 6.230563e-01
#> [511] 3.931817e-01 4.749640e-01 7.466248e-01 2.907655e-01 9.408910e-01
#> [516] 2.825798e-01 6.432390e-01 3.557594e-01 6.454840e-01 1.857779e-01
#> [521] 6.410991e-01 4.846269e-01 3.128575e-01 3.276353e-01 6.575424e-01
#> [526] 2.964029e-01 6.770094e-01 6.859469e-01 7.581518e-01 3.386725e-01
#> [531] 5.701291e-01 7.903824e-01 8.162299e-01 9.745326e-01 7.077387e-02
#> [536] 7.938761e-01 4.229804e-01 8.328790e-01 9.277912e-01 6.948243e-01
#> [541] 5.178250e-01 3.578075e-01 2.994095e-02 3.558267e-01 3.613691e-01
#> [546] 7.083694e-01 2.287719e-01 5.703143e-01 5.262076e-02 4.700596e-01
#> [551] 5.492869e-01 3.364416e-01 9.139081e-01 4.783198e-01 9.600436e-02
#> [556] 4.014401e-01 6.715411e-01 6.110722e-01 8.955153e-01 2.625834e-01
#> [561] 4.869918e-01 5.358802e-01 1.938409e-01 8.982452e-01 2.247116e-01
#> [566] 6.916782e-01 7.677548e-01 4.690357e-01 5.442777e-01 1.174217e-01
#> [571] 6.332673e-01 8.716147e-02 4.444707e-01 5.362228e-01 5.524600e-01
#> [576] 4.160168e-01 4.651853e-01 5.639764e-01 8.292627e-01 7.750013e-01
#> [581] 5.226119e-01 3.800509e-01 3.560815e-01 9.058208e-01 2.904927e-01
#> [586] 1.819415e-01 7.790951e-01 3.024187e-01 5.787625e-01 8.920494e-01
#> [591] 1.602084e-01 5.098210e-01 6.531631e-03 5.749816e-01 6.033913e-01
#> [596] 8.731474e-01 5.682103e-01 6.889114e-01 4.986541e-01 5.659835e-01
#> [601] 5.858882e-01 6.012657e-01 6.410998e-01 7.902950e-01 6.376906e-01
#> [606] 5.527858e-01 9.964609e-01 4.436712e-01 4.392464e-01 4.455992e-01
#> [611] 7.331340e-01 4.785280e-02 7.961712e-01 6.002810e-01 9.523661e-01
#> [616] 5.928954e-01 5.685669e-01 4.444605e-01 8.282959e-02 6.321323e-01
#> [621] 5.170730e-01 3.477551e-01 4.680966e-01 8.427991e-01 4.278012e-01
#> [626] 5.241894e-01 8.358471e-01 3.892099e-01 8.961772e-01 2.992590e-01
#> [631] 5.611537e-01 9.755196e-02 4.193524e-01 8.056997e-01 3.269908e-01
#> [636] 4.265333e-01 9.226137e-01 6.053548e-01 9.628666e-01 2.094042e-01
#> [641] 9.423321e-01 2.594233e-01 1.601184e-01 3.782354e-01 4.162561e-01
#> [646] 3.628194e-01 5.673326e-01 3.620361e-01 1.256250e-02 5.237216e-01
#> [651] 9.102740e-01 1.376892e-01 7.734860e-01 4.727682e-01 8.112410e-01
#> [656] 7.300581e-01 2.146690e-01 9.996660e-01 5.994690e-01 4.887470e-01
#> [661] 2.890383e-01 5.429336e-01 2.794233e-01 5.567439e-01 3.661317e-01
#> [666] 3.625172e-01 3.528648e-01 9.458979e-01 1.966751e-01 6.035613e-01
#> [671] 3.059878e-01 5.909469e-01 4.552182e-01 7.235039e-01 8.391237e-01
#> [676] 5.188315e-01 3.732818e-01 7.056301e-01 4.985309e-01 7.821846e-01
#> [681] 6.027216e-01 1.745843e-01 1.984140e-01 8.998867e-01 5.318924e-01
#> [686] 5.850282e-01 7.865049e-01 4.224303e-01 3.151797e-01 4.125238e-01
#> [691] 5.842601e-01 5.337467e-02 4.128630e-01 5.238856e-01 2.429520e-01
#> [696] 1.650808e-01 5.850224e-01 3.559747e-02 3.277866e-01 1.957318e-01
#> [701] 6.774947e-01 4.797203e-01 8.947257e-01 2.559951e-02 2.353754e-01
#> [706] 5.486971e-02 7.792366e-01 8.890431e-01 6.375221e-01 5.617598e-01
#> [711] 6.617409e-01 4.723919e-01 4.964649e-01 6.218685e-01 4.989846e-01
#> [716] 5.016222e-01 5.851027e-01 4.612707e-01 5.836774e-01 3.627334e-01
#> [721] 6.903366e-01 6.065970e-01 5.694647e-01 2.287998e-01 4.085515e-01
#> [726] 5.857110e-01 5.833096e-01 2.565479e-01 5.929932e-01 5.851739e-02
#> [731] 9.433526e-01 5.274391e-01 1.179636e-01 5.464003e-01 3.605678e-02
#> [736] 9.038981e-01 2.536475e-04 5.689570e-01 6.891879e-01 9.943144e-01
#> [741] 4.863624e-01 5.878660e-01 6.319555e-01 4.038334e-01 7.580725e-02
#> [746] 5.542493e-01 5.527634e-01 6.658606e-01 6.241884e-01 4.037474e-01
#> [751] 5.826828e-01 6.476843e-01 2.660224e-01 2.174323e-02 3.719546e-01
#> [756] 3.524815e-01 1.521963e-01 4.078132e-01 2.029477e-01 8.006332e-03
#> [761] 4.534977e-01 6.763963e-03 5.526746e-01 1.647759e-01 4.093808e-01
#> [766] 7.321555e-01 4.956426e-01 6.801390e-01 1.281679e-01 3.191613e-01
#> [771] 8.773691e-01 5.290820e-01 6.244100e-02 6.687448e-01 3.290061e-01
#> [776] 1.205455e-01 9.998865e-01 6.278460e-02 2.823239e-01 5.248586e-01
#> [781] 8.388841e-02 3.698869e-01 6.584592e-01 3.189524e-01 4.290356e-01
#> [786] 6.076814e-01 7.198238e-01 4.588289e-01 5.230503e-01 5.717004e-01
#> [791] 4.786573e-01 3.315674e-01 1.508549e-01 4.314851e-01 4.162416e-01
#> [796] 7.768009e-01 5.550853e-01 7.196970e-01 5.693347e-01 7.171248e-01
#> [801] 8.739105e-01 8.693778e-01 5.940658e-01 7.224398e-01 2.673465e-01
#> [806] 5.219425e-01 4.309676e-01 5.599067e-01 8.773491e-01 7.274320e-01
#> [811] 7.953347e-01 2.903813e-01 6.978498e-01 7.830920e-01 5.600471e-01
#> [816] 9.599369e-01 4.472307e-02 5.094630e-01 9.781615e-01 2.761436e-01
#> [821] 9.240027e-01 8.509987e-01 1.259784e-01 3.254186e-01 7.407602e-01
#> [826] 6.840154e-01 2.753709e-01 7.105855e-01 6.755479e-01 4.318736e-01
#> [831] 7.955869e-01 4.004440e-01 3.547024e-01 3.904591e-01 6.784898e-01
#> [836] 2.130456e-01 5.658298e-01 6.957480e-01 2.214241e-01 2.871698e-01
#> [841] 8.225068e-01 9.767322e-01 5.217971e-01 3.797647e-01 9.213658e-01
#> [846] 3.886637e-01 3.890143e-01 3.863363e-01 4.200584e-01 7.741113e-01
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#> [1716] 3.407863e-01 3.064204e-01 1.117669e-01 7.903382e-02 5.718648e-01
#> [1721] 1.749660e-02 2.184884e-01 2.039105e-02 7.351064e-01 9.392714e-01
#> [1726] 5.496263e-01 9.396409e-01 4.518385e-01 6.407353e-01 6.832618e-01
#> [1731] 2.156719e-02 4.547112e-02 1.032958e-01 1.010101e-01 6.012912e-01
#> [1736] 7.393891e-01 8.340310e-01 5.694919e-01 8.863026e-01 7.476427e-01
#> [1741] 9.162117e-01 3.021176e-01 6.822338e-01 4.973812e-01 3.913400e-01
#> [1746] 8.695274e-01 5.716337e-01 4.054399e-03 6.787327e-01 3.019045e-01
#> [1751] 3.411203e-01 3.209316e-01 6.721530e-01 2.390198e-01 8.787920e-01
#> [1756] 5.817333e-01 2.982527e-01 7.364411e-01 8.098504e-01 4.799467e-01
#> [1761] 6.387274e-01 2.064130e-01 6.722309e-01 2.593375e-01 3.851455e-01
#> [1766] 6.793444e-01 5.431186e-01 5.629742e-01 3.344066e-03 4.314044e-01
#> [1771] 5.860999e-01 7.045471e-01 4.360370e-01 8.791629e-01 3.140622e-01
#> [1776] 4.180700e-01 7.489595e-01 2.491531e-02 7.967895e-01 6.998083e-01
#> [1781] 9.719999e-01 9.982542e-01 5.017160e-01 3.282477e-01 2.954772e-01
#> [1786] 7.716683e-01 3.966822e-01 4.611353e-01 4.389913e-01 3.555861e-02
#> [1791] 2.794121e-01 7.275591e-01 3.710198e-02 2.000496e-01 2.789230e-01
#> [1796] 6.789118e-01 6.515143e-01 5.940844e-01 4.332465e-01 7.579573e-01
#> [1801] 5.750565e-01 5.233543e-01 3.929975e-01 2.594591e-01 2.366918e-03
#> [1806] 4.735079e-01 6.031430e-01 6.372081e-01 8.946001e-02 7.066084e-01
#> [1811] 7.148262e-01 8.838530e-01 4.825109e-01 4.175559e-01 5.167493e-01
#> [1816] 6.983707e-01 5.150429e-01 4.234627e-03 6.835199e-01 2.085124e-01
#> [1821] 5.916931e-01 1.830303e-01 7.212672e-03 7.204078e-01 3.734913e-01
#> [1826] 5.094491e-01 4.507095e-01 3.793425e-01 8.586045e-02 7.738846e-01
#> [1831] 6.023812e-01 2.240659e-01 5.277852e-01 3.815685e-01 1.690874e-01
#> [1836] 1.507372e-01 4.189307e-01 4.146400e-01 8.480710e-01 4.742373e-02
#> [1841] 5.923090e-01 3.558568e-01 4.176485e-01 6.044230e-01 1.408504e-01
#> [1846] 2.268343e-01 1.127294e-01 2.210104e-01 5.604168e-01 3.823551e-02
#> [1851] 9.531479e-01 6.225686e-01 1.055704e-01 4.036839e-01 3.062785e-01
#> [1856] 6.781945e-01 5.023901e-01 7.397493e-01 6.602551e-01 1.609746e-01
#> [1861] 7.788966e-01 6.957264e-01 5.959645e-01 6.768573e-01 7.342269e-03
#> [1866] 6.782820e-01 5.920132e-01 5.399067e-01 2.091592e-01 6.146463e-01
#> [1871] 6.186586e-01 7.939086e-01 7.323228e-01 5.236885e-01 8.272626e-01
#> [1876] 1.466087e-01 5.266806e-01 2.823292e-01 3.045121e-01 6.686728e-01
#> [1881] 3.380545e-01 6.499160e-01 1.343901e-01 4.125626e-01 2.268452e-01
#> [1886] 5.077862e-01 1.076238e-01 2.488840e-01 1.330912e-01 5.385482e-01
#> [1891] 7.963307e-01 4.301725e-01 6.171397e-01 2.052950e-03 6.174852e-01
#> [1896] 1.903783e-01 9.862635e-01 8.991319e-01 1.877129e-01 4.707806e-02
#> [1901] 6.487581e-01 5.953579e-01 5.997223e-01 2.211870e-01 7.723163e-01
#> [1906] 6.765108e-01 6.257692e-01 2.958351e-01 5.929265e-01 5.345449e-01
#> [1911] 4.427641e-01 3.365980e-01 1.252136e-01 1.732857e-01 4.242970e-01
#> [1916] 5.217419e-01 1.740722e-01 2.627098e-01 4.946286e-01 3.284404e-01
#> [1921] 2.033060e-02 1.513034e-01 3.017813e-01 5.268998e-02 4.456686e-01
#> [1926] 4.930353e-01 9.408174e-01 4.893942e-01 5.788809e-01 2.823301e-01
#> [1931] 5.218593e-01 8.169109e-01 2.494612e-01 1.852622e-01 3.436880e-01
#> [1936] 3.150536e-01 7.406442e-01 4.413533e-01 9.912757e-01 4.860385e-01
#> [1941] 7.240070e-01 5.114848e-01 6.271777e-01 4.877836e-01 4.850605e-02
#> [1946] 6.113550e-01 3.314411e-01 9.123951e-02 9.840932e-01 7.795874e-01
#> [1951] 6.484160e-01 1.135360e-01 9.699098e-01 5.671421e-01 4.264446e-01
#> [1956] 3.120284e-01 3.138598e-01 4.212018e-01 3.757032e-01 5.576084e-01
#> [1961] 7.341078e-01 5.245742e-01 6.470752e-01 5.993320e-01 6.921317e-01
#> [1966] 5.343384e-01 3.385299e-01 5.293995e-01 4.242423e-01 8.601112e-01
#> [1971] 2.166673e-02 6.763705e-01 4.147982e-01 4.521916e-01 4.613753e-01
#> [1976] 4.940971e-01 7.690830e-01 5.384153e-01 8.519834e-01 7.491064e-01
#> [1981] 4.524284e-01 4.738410e-01 4.818030e-01 6.554952e-01 5.644121e-01
#> [1986] 7.057145e-01 4.614571e-01 4.019925e-01 3.476020e-01 4.817820e-01
#> [1991] 8.662857e-01 4.901267e-01 6.770929e-01 8.565970e-01 5.469539e-01
#> [1996] 4.842959e-01 3.267069e-01 3.968082e-01 5.267576e-02 6.669564e-02