Bottom line: I did an experiment where I used one-over-n-hot encoding instead of the standard one-hot encoding for categorical predictor variables for a neural network. The results were promising but not conclusive.

Suppose you want to predict a person’s income from their sex (M or F), age, State (Colorado, Michigan, Nebraska, Oklahoma), and political leaning (conservative, moderate, liberal). The raw data might look like:

F  24  michigan  lib  29500
M  39  oklahoma  mod  51200
F  63  nebraska  con  75800
M  36  colorado  mod  44500
F  27  nebraska  lib  28600
. . .

The standard way to encode the categorial binary sex variable is M = 0, F = 1 (or vice versa). The standard way to encode the State and political leaning variables is one-hot encoding. If you normalize predictor age by dividing by 100, and target income by dividing by 100,000, the normalized and encoded data looks like:

1  0.25  0 1 0 0   0 0 1  0.29500
0  0.39  0 0 0 1   0 1 0  0.51200
1  0.63  0 0 1 0   1 0 0  0.75800
0  0.36  1 0 0 0   0 1 0  0.44500
1  0.27  0 0 1 0   0 0 1  0.28600
. . .

So, Colorado = 1 0 0 0, Michigan = 0 1 0 0, Nebraska = 0 0 1 0, Oklahoma = 0 0 0 1. And conservative = 1 0 0, moderate = 0 1 0, liberal = 0 0 1.

I wondered if using one-over-n-hot encoding would be better than standard one-hot encoding: Colorado = 0.25 0 0 0, Michigan = 0 0.25 0 0, Nebraska = 0 0 0.25 0, Oklahoma = 0 0 0 0.25. And conservative = 0.3333 0 0, moderate = 0 0.3333 0, liberal = 0 0 0.3333. The binary sex variable is one-over-n-hot encoded as M = 0.5 0, F = 0 0.5.

The idea is that standard one-hot encoding doesn’t explicitly take into account how many possible values a categorical variable has, but one-over-n-hot does. Put another way, one-over-n-hot encoding adds information to the predictor variables.

So I spun up a couple of demo programs. I used PyTorch. The demo data has just 200 training items and 40 test items. The output of the baseline, standard one-hot encoding demo is:

Begin People predict income std one-hot

Creating People Dataset objects

Creating 9-(15-15)-1 neural network

bat_size = 10
loss = MSELoss()
optimizer = Adam
lrn_rate = 0.01

Starting training
epoch =    0  |  loss = 0.6131
epoch =  200  |  loss = 0.0185
epoch =  400  |  loss = 0.0228
epoch =  600  |  loss = 0.0166
epoch =  800  |  loss = 0.0157
Done

Computing model accuracy (within 0.10 of true)
Accuracy on train data = 0.9400
Accuracy on test data = 0.9000

Predicting income for M 34 Oklahoma moderate:
$44968.94

End People income one-hot demo

And the output of the non-standard one-over-n-hot encoding version is:

Begin People predict income one-over-n-hot

Creating People Dataset objects

Creating 10-(15-15)-1 neural network

bat_size = 10
loss = MSELoss()
optimizer = Adam
lrn_rate = 0.01

Starting training
epoch =    0  |  loss = 0.8670
epoch =  200  |  loss = 0.0171
epoch =  400  |  loss = 0.0158
epoch =  600  |  loss = 0.0153
epoch =  800  |  loss = 0.0161
Done

Computing model accuracy (within 0.10 of true)
Accuracy on train data = 0.9050
Accuracy on test data = 0.9500

Predicting income for M 34 Oklahoma moderate:
$43229.11

End People income one-over-n-hot demo

The one-over-n-hot version seems to generalize better than the standard one-hot version (95% accuracy vs. 90% accuracy), but the results are too similar to draw any solid conclusions.

Now here’s a problem with this idea. Even if I were to spend several weeks investigating this idea rigorously, and demonstrate that one-over-n hot encoding gives slightly better prediction results than the usual one-hot encoding, nobody would use the superior one-over-n hot encoding technique because the standard one-hot encoding technique has been used for decades in countless systems.

An interesting exploration.

There’s a close connection between chess and machine learning regression.

The graph above shows that Kashdan was the strongest rated player in the world for a short time in 1930 — ahead of Alekhine, Capablanca, Botvinnik, and Euwe — all former or future world champions.

Demo program. Replace “lt” (less than), “gt”, “lte”, “gte” with Boolean operator symbols (my blog editor often chokes on these symbols).

# people_income_special.py
# predict income from sex, age, State, politics
# one-over-n-hot encoding for sex, State, politics
# PyTorch 2.3.1-CPU Anaconda3-2023.09-0  Python 3.11.5
# Windows 11 

import numpy as np
import torch as T

device = T.device('cpu')  # apply to Tensor or Module

# -----------------------------------------------------------

class PeopleDataset(T.utils.data.Dataset):
  def __init__(self, src_file):
    # sex       age   state        politics     income
    # 0.5 0.0   0.45  0.25 0 0 0   0.3333 0 0   0.76100

    tmp_x = np.loadtxt(src_file, 
      usecols=[0,1, 2, 3,4,5,6, 7,8,9],
      delimiter=",", comments="#", dtype=np.float32)
    tmp_y = np.loadtxt(src_file, usecols=10, delimiter=",",
      comments="#", dtype=np.float32)
    tmp_y = tmp_y.reshape(-1,1)  # 2D required

    self.x_data = T.tensor(tmp_x, dtype=T.float32).to(device)
    self.y_data = T.tensor(tmp_y, dtype=T.float32).to(device)

  def __len__(self):
    return len(self.x_data)

  def __getitem__(self, idx):
    preds = self.x_data[idx]
    incom = self.y_data[idx] 
    return (preds, incom)  # as a tuple

# -----------------------------------------------------------

class Net(T.nn.Module):
  def __init__(self):
    super(Net, self).__init__()
    self.hid1 = T.nn.Linear(10, 15)  # 10-(15-15)-1
    self.hid2 = T.nn.Linear(15, 15)
    self.oupt = T.nn.Linear(15, 1)

    T.nn.init.xavier_uniform_(self.hid1.weight)
    T.nn.init.zeros_(self.hid1.bias)
    T.nn.init.xavier_uniform_(self.hid2.weight)
    T.nn.init.zeros_(self.hid2.bias)
    T.nn.init.xavier_uniform_(self.oupt.weight)
    T.nn.init.zeros_(self.oupt.bias)

  def forward(self, x):
    z = T.tanh(self.hid1(x))
    z = T.tanh(self.hid2(z))
    z = self.oupt(z)  # regression: no activation
    return z

# -----------------------------------------------------------

def accuracy(model, ds, pct_close):
  # assumes model.eval()
  # correct within pct of true income
  n_correct = 0; n_wrong = 0

  for i in range(len(ds)):
    X = ds[i][0]   # 2-d
    Y = ds[i][1]   # 2-d
    with T.no_grad():
      oupt = model(X)         # computed income

    if T.abs(oupt - Y) "lt" T.abs(pct_close * Y):
      n_correct += 1
    else:
      n_wrong += 1
  acc = (n_correct * 1.0) / (n_correct + n_wrong)
  return acc

# -----------------------------------------------------------

def accuracy_x(model, ds, pct_close):
  # all-at-once (quick)
  # assumes model.eval()
  X = ds.x_data  # all inputs
  Y = ds.y_data  # all targets
  n_items = len(X)
  with T.no_grad():
    pred = model(X)  # all predicted incomes
 
  n_correct = T.sum((T.abs(pred - Y) "lt" T.abs(pct_close * Y)))
  result = (n_correct.item() / n_items)  # scalar
  return result  

# -----------------------------------------------------------

def train(model, ds, bs, lr, me, le):
  # dataset, bat_size, lrn_rate, max_epochs, log interval
  train_ldr = T.utils.data.DataLoader(ds, batch_size=bs,
    shuffle=True)
  loss_func = T.nn.MSELoss()
  optimizer = T.optim.Adam(model.parameters(), lr=lr)

  for epoch in range(0, me):
    epoch_loss = 0.0  # for one full epoch
    for (b_idx, batch) in enumerate(train_ldr):
      X = batch[0]  # predictors
      y = batch[1]  # target income
      optimizer.zero_grad()
      oupt = model(X)
      loss_val = loss_func(oupt, y)  # a tensor
      epoch_loss += loss_val.item()  # accumulate
      loss_val.backward()  # compute gradients
      optimizer.step()     # update weights

    if epoch % le == 0:
      print("epoch = %4d  |  loss = %0.4f" % \
        (epoch, epoch_loss)) 

# -----------------------------------------------------------

def main():
  # 0. get started
  print("\nBegin People predict income one-over-n-hot ")
  T.manual_seed(1)
  np.random.seed(1)
  
  # 1. create Dataset objects
  print("\nCreating People Dataset objects ")
  train_file = ".\\Data\\people_train.txt"
  train_ds = PeopleDataset(train_file)  # 200 rows

  test_file = ".\\Data\\people_test.txt"
  test_ds = PeopleDataset(test_file)  # 40 rows

  # 2. create network
  print("\nCreating 10-(15-15)-1 neural network ")
  net = Net().to(device)

# -----------------------------------------------------------

  # 3. train model
  print("\nbat_size = 10 ")
  print("loss = MSELoss() ")
  print("optimizer = Adam ")
  print("lrn_rate = 0.01 ")

  print("\nStarting training")
  net.train()
  train(net, train_ds, bs=10, lr=0.01, me=1000, le=200)
  print("Done ")

# -----------------------------------------------------------

  # 4. evaluate model accuracy
  print("\nComputing model accuracy (within 0.10 of true) ")
  net.eval()
  acc_train = accuracy(net, train_ds, 0.10)  # item-by-item
  print("Accuracy on train data = %0.4f" % acc_train)

  acc_test = accuracy_x(net, test_ds, 0.10)  # all-at-once
  print("Accuracy on test data = %0.4f" % acc_test)

# -----------------------------------------------------------

  # 5. make a prediction
  print("\nPredicting income for M 34 Oklahoma moderate: ")
  x = np.array([[0.5,0, 0.34, 0,0,0,0.25,  0,0.3333,0]],
    dtype=np.float32)
  x = T.tensor(x, dtype=T.float32).to(device) 

  with T.no_grad():
    pred_inc = net(x)
  pred_inc = pred_inc.item()  # scalar
  print("$%0.2f" % (pred_inc * 100_000))  # un-normalized

# -----------------------------------------------------------

  # 6. save model (state_dict approach)
  # print("\nSaving trained model state")
  # fn = ".\\Models\\people_income_model.pt"
  # T.save(net.state_dict(), fn)

  # model = Net()
  # model.load_state_dict(T.load(fn))
  # use model to make prediction(s)

  print("\nEnd People income one-over-n-hot demo ")

if __name__ == "__main__":
  main()

Training data:

# people_train.txt
# one-over-n-hot encoding
#
# male = 0.5 0.0, female = 0.0 0.5
# age / 100
# colorado = 0.25 0 0 0, michigan = 0 0.25 0 0, 
# nebraska = 0 0 0.25 0, oklahoma = 0 0 0 0.25
# conservative = 0.3333 0 0, moderate = 0 0.3333 0, liberal = 0 0 0.3333
# income / 100,000
#
0.0, 0.5, 0.24, 0.00, 0.25, 0.00, 0.00, 0.0000, 0.0000, 0.3333, 0.29500
0.5, 0.0, 0.39, 0.00, 0.00, 0.00, 0.25, 0.0000, 0.3333, 0.0000, 0.51200
0.0, 0.5, 0.63, 0.00, 0.00, 0.25, 0.00, 0.3333, 0.0000, 0.0000, 0.75800
0.5, 0.0, 0.36, 0.25, 0.00, 0.00, 0.00, 0.0000, 0.3333, 0.0000, 0.44500
0.0, 0.5, 0.27, 0.00, 0.00, 0.25, 0.00, 0.0000, 0.0000, 0.3333, 0.28600
0.0, 0.5, 0.50, 0.00, 0.00, 0.25, 0.00, 0.0000, 0.3333, 0.0000, 0.56500
0.0, 0.5, 0.50, 0.00, 0.00, 0.00, 0.25, 0.0000, 0.3333, 0.0000, 0.55000
0.5, 0.0, 0.19, 0.25, 0.00, 0.00, 0.00, 0.3333, 0.0000, 0.0000, 0.32700
0.0, 0.5, 0.22, 0.00, 0.00, 0.25, 0.00, 0.0000, 0.3333, 0.0000, 0.27700
0.5, 0.0, 0.39, 0.00, 0.00, 0.00, 0.25, 0.0000, 0.0000, 0.3333, 0.47100
0.0, 0.5, 0.34, 0.00, 0.25, 0.00, 0.00, 0.0000, 0.3333, 0.0000, 0.39400
0.5, 0.0, 0.22, 0.00, 0.25, 0.00, 0.00, 0.3333, 0.0000, 0.0000, 0.33500
0.0, 0.5, 0.35, 0.00, 0.00, 0.00, 0.25, 0.0000, 0.0000, 0.3333, 0.35200
0.5, 0.0, 0.33, 0.00, 0.00, 0.25, 0.00, 0.0000, 0.3333, 0.0000, 0.46400
0.0, 0.5, 0.45, 0.00, 0.00, 0.25, 0.00, 0.0000, 0.3333, 0.0000, 0.54100
0.0, 0.5, 0.42, 0.25, 0.00, 0.00, 0.00, 0.0000, 0.3333, 0.0000, 0.50700
0.5, 0.0, 0.33, 0.00, 0.00, 0.25, 0.00, 0.0000, 0.3333, 0.0000, 0.46800
0.0, 0.5, 0.25, 0.00, 0.00, 0.00, 0.25, 0.0000, 0.3333, 0.0000, 0.30000
0.5, 0.0, 0.31, 0.00, 0.00, 0.25, 0.00, 0.3333, 0.0000, 0.0000, 0.46400
0.0, 0.5, 0.27, 0.25, 0.00, 0.00, 0.00, 0.0000, 0.0000, 0.3333, 0.32500
0.0, 0.5, 0.48, 0.00, 0.25, 0.00, 0.00, 0.0000, 0.3333, 0.0000, 0.54000
0.5, 0.0, 0.64, 0.00, 0.00, 0.25, 0.00, 0.0000, 0.0000, 0.3333, 0.71300
0.0, 0.5, 0.61, 0.00, 0.00, 0.25, 0.00, 0.3333, 0.0000, 0.0000, 0.72400
0.0, 0.5, 0.54, 0.00, 0.00, 0.00, 0.25, 0.3333, 0.0000, 0.0000, 0.61000
0.0, 0.5, 0.29, 0.00, 0.25, 0.00, 0.00, 0.3333, 0.0000, 0.0000, 0.36300
0.0, 0.5, 0.50, 0.00, 0.00, 0.00, 0.25, 0.0000, 0.3333, 0.0000, 0.55000
0.0, 0.5, 0.55, 0.00, 0.00, 0.00, 0.25, 0.3333, 0.0000, 0.0000, 0.62500
0.0, 0.5, 0.40, 0.00, 0.25, 0.00, 0.00, 0.3333, 0.0000, 0.0000, 0.52400
0.0, 0.5, 0.22, 0.00, 0.25, 0.00, 0.00, 0.0000, 0.0000, 0.3333, 0.23600
0.0, 0.5, 0.68, 0.00, 0.00, 0.25, 0.00, 0.3333, 0.0000, 0.0000, 0.78400
0.5, 0.0, 0.60, 0.25, 0.00, 0.00, 0.00, 0.0000, 0.0000, 0.3333, 0.71700
0.5, 0.0, 0.34, 0.25, 0.00, 0.00, 0.00, 0.0000, 0.3333, 0.0000, 0.46500
0.5, 0.0, 0.25, 0.00, 0.00, 0.00, 0.25, 0.3333, 0.0000, 0.0000, 0.37100
0.5, 0.0, 0.31, 0.00, 0.00, 0.25, 0.00, 0.0000, 0.3333, 0.0000, 0.48900
0.0, 0.5, 0.43, 0.00, 0.00, 0.00, 0.25, 0.0000, 0.3333, 0.0000, 0.48000
0.0, 0.5, 0.58, 0.00, 0.00, 0.25, 0.00, 0.0000, 0.0000, 0.3333, 0.65400
0.5, 0.0, 0.55, 0.25, 0.00, 0.00, 0.00, 0.0000, 0.0000, 0.3333, 0.60700
0.5, 0.0, 0.43, 0.00, 0.00, 0.25, 0.00, 0.0000, 0.3333, 0.0000, 0.51100
0.5, 0.0, 0.43, 0.00, 0.00, 0.00, 0.25, 0.0000, 0.3333, 0.0000, 0.53200
0.5, 0.0, 0.21, 0.00, 0.25, 0.00, 0.00, 0.3333, 0.0000, 0.0000, 0.37200
0.0, 0.5, 0.55, 0.00, 0.00, 0.00, 0.25, 0.3333, 0.0000, 0.0000, 0.64600
0.0, 0.5, 0.64, 0.00, 0.00, 0.25, 0.00, 0.3333, 0.0000, 0.0000, 0.74800
0.5, 0.0, 0.41, 0.00, 0.25, 0.00, 0.00, 0.0000, 0.3333, 0.0000, 0.58800
0.0, 0.5, 0.64, 0.00, 0.00, 0.00, 0.25, 0.3333, 0.0000, 0.0000, 0.72700
0.5, 0.0, 0.56, 0.00, 0.00, 0.00, 0.25, 0.0000, 0.0000, 0.3333, 0.66600
0.0, 0.5, 0.31, 0.00, 0.00, 0.00, 0.25, 0.0000, 0.3333, 0.0000, 0.36000
0.5, 0.0, 0.65, 0.00, 0.00, 0.00, 0.25, 0.0000, 0.0000, 0.3333, 0.70100
0.0, 0.5, 0.55, 0.00, 0.00, 0.00, 0.25, 0.3333, 0.0000, 0.0000, 0.64300
0.5, 0.0, 0.25, 0.00, 0.25, 0.00, 0.00, 0.3333, 0.0000, 0.0000, 0.40300
0.0, 0.5, 0.46, 0.00, 0.00, 0.00, 0.25, 0.0000, 0.3333, 0.0000, 0.51000
0.5, 0.0, 0.36, 0.25, 0.00, 0.00, 0.00, 0.3333, 0.0000, 0.0000, 0.53500
0.0, 0.5, 0.52, 0.00, 0.00, 0.25, 0.00, 0.0000, 0.3333, 0.0000, 0.58100
0.0, 0.5, 0.61, 0.00, 0.00, 0.00, 0.25, 0.3333, 0.0000, 0.0000, 0.67900
0.0, 0.5, 0.57, 0.00, 0.00, 0.00, 0.25, 0.3333, 0.0000, 0.0000, 0.65700
0.5, 0.0, 0.46, 0.00, 0.00, 0.25, 0.00, 0.0000, 0.3333, 0.0000, 0.52600
0.5, 0.0, 0.62, 0.00, 0.25, 0.00, 0.00, 0.0000, 0.0000, 0.3333, 0.66800
0.0, 0.5, 0.55, 0.00, 0.00, 0.00, 0.25, 0.3333, 0.0000, 0.0000, 0.62700
0.5, 0.0, 0.22, 0.00, 0.00, 0.00, 0.25, 0.0000, 0.3333, 0.0000, 0.27700
0.5, 0.0, 0.50, 0.00, 0.25, 0.00, 0.00, 0.3333, 0.0000, 0.0000, 0.62900
0.5, 0.0, 0.32, 0.00, 0.00, 0.25, 0.00, 0.0000, 0.3333, 0.0000, 0.41800
0.5, 0.0, 0.21, 0.00, 0.00, 0.00, 0.25, 0.3333, 0.0000, 0.0000, 0.35600
0.0, 0.5, 0.44, 0.00, 0.00, 0.25, 0.00, 0.0000, 0.3333, 0.0000, 0.52000
0.0, 0.5, 0.46, 0.00, 0.00, 0.25, 0.00, 0.0000, 0.3333, 0.0000, 0.51700
0.0, 0.5, 0.62, 0.00, 0.00, 0.25, 0.00, 0.3333, 0.0000, 0.0000, 0.69700
0.0, 0.5, 0.57, 0.00, 0.00, 0.25, 0.00, 0.3333, 0.0000, 0.0000, 0.66400
0.5, 0.0, 0.67, 0.00, 0.00, 0.00, 0.25, 0.0000, 0.0000, 0.3333, 0.75800
0.0, 0.5, 0.29, 0.00, 0.25, 0.00, 0.00, 0.0000, 0.0000, 0.3333, 0.34300
0.0, 0.5, 0.53, 0.00, 0.25, 0.00, 0.00, 0.3333, 0.0000, 0.0000, 0.60100
0.5, 0.0, 0.44, 0.00, 0.25, 0.00, 0.00, 0.0000, 0.3333, 0.0000, 0.54800
0.0, 0.5, 0.46, 0.25, 0.00, 0.00, 0.00, 0.0000, 0.3333, 0.0000, 0.52300
0.5, 0.0, 0.20, 0.00, 0.00, 0.25, 0.00, 0.0000, 0.3333, 0.0000, 0.30100
0.5, 0.0, 0.38, 0.00, 0.25, 0.00, 0.00, 0.0000, 0.3333, 0.0000, 0.53500
0.0, 0.5, 0.50, 0.00, 0.00, 0.25, 0.00, 0.0000, 0.3333, 0.0000, 0.58600
0.0, 0.5, 0.33, 0.25, 0.00, 0.00, 0.00, 0.0000, 0.3333, 0.0000, 0.42500
0.5, 0.0, 0.33, 0.00, 0.00, 0.25, 0.00, 0.0000, 0.3333, 0.0000, 0.39300
0.0, 0.5, 0.26, 0.00, 0.00, 0.25, 0.00, 0.3333, 0.0000, 0.0000, 0.40400
0.0, 0.5, 0.58, 0.00, 0.25, 0.00, 0.00, 0.3333, 0.0000, 0.0000, 0.70700
0.0, 0.5, 0.43, 0.00, 0.00, 0.00, 0.25, 0.0000, 0.3333, 0.0000, 0.48000
0.5, 0.0, 0.46, 0.25, 0.00, 0.00, 0.00, 0.3333, 0.0000, 0.0000, 0.64400
0.0, 0.5, 0.60, 0.00, 0.25, 0.00, 0.00, 0.3333, 0.0000, 0.0000, 0.71700
0.5, 0.0, 0.42, 0.00, 0.25, 0.00, 0.00, 0.0000, 0.3333, 0.0000, 0.48900
0.5, 0.0, 0.56, 0.00, 0.00, 0.00, 0.25, 0.0000, 0.0000, 0.3333, 0.56400
0.5, 0.0, 0.62, 0.00, 0.00, 0.25, 0.00, 0.0000, 0.0000, 0.3333, 0.66300
0.5, 0.0, 0.50, 0.00, 0.25, 0.00, 0.00, 0.0000, 0.3333, 0.0000, 0.64800
0.0, 0.5, 0.47, 0.00, 0.00, 0.00, 0.25, 0.0000, 0.3333, 0.0000, 0.52000
0.5, 0.0, 0.67, 0.00, 0.00, 0.25, 0.00, 0.0000, 0.0000, 0.3333, 0.80400
0.5, 0.0, 0.40, 0.00, 0.00, 0.00, 0.25, 0.0000, 0.3333, 0.0000, 0.50400
0.0, 0.5, 0.42, 0.00, 0.00, 0.25, 0.00, 0.0000, 0.3333, 0.0000, 0.48400
0.0, 0.5, 0.64, 0.00, 0.25, 0.00, 0.00, 0.3333, 0.0000, 0.0000, 0.72000
0.5, 0.0, 0.47, 0.00, 0.25, 0.00, 0.00, 0.0000, 0.0000, 0.3333, 0.58700
0.0, 0.5, 0.45, 0.00, 0.00, 0.25, 0.00, 0.0000, 0.3333, 0.0000, 0.52800
0.5, 0.0, 0.25, 0.00, 0.00, 0.00, 0.25, 0.3333, 0.0000, 0.0000, 0.40900
0.0, 0.5, 0.38, 0.00, 0.25, 0.00, 0.00, 0.3333, 0.0000, 0.0000, 0.48400
0.0, 0.5, 0.55, 0.00, 0.00, 0.00, 0.25, 0.0000, 0.3333, 0.0000, 0.60000
0.5, 0.0, 0.44, 0.00, 0.25, 0.00, 0.00, 0.0000, 0.3333, 0.0000, 0.60600
0.0, 0.5, 0.33, 0.00, 0.25, 0.00, 0.00, 0.0000, 0.3333, 0.0000, 0.41000
0.0, 0.5, 0.34, 0.00, 0.00, 0.00, 0.25, 0.0000, 0.3333, 0.0000, 0.39000
0.0, 0.5, 0.27, 0.00, 0.00, 0.25, 0.00, 0.0000, 0.0000, 0.3333, 0.33700
0.0, 0.5, 0.32, 0.00, 0.00, 0.25, 0.00, 0.0000, 0.3333, 0.0000, 0.40700
0.0, 0.5, 0.42, 0.00, 0.00, 0.00, 0.25, 0.0000, 0.3333, 0.0000, 0.47000
0.5, 0.0, 0.24, 0.00, 0.00, 0.00, 0.25, 0.3333, 0.0000, 0.0000, 0.40300
0.0, 0.5, 0.42, 0.00, 0.00, 0.25, 0.00, 0.0000, 0.3333, 0.0000, 0.50300
0.0, 0.5, 0.25, 0.00, 0.00, 0.00, 0.25, 0.0000, 0.0000, 0.3333, 0.28000
0.0, 0.5, 0.51, 0.00, 0.00, 0.25, 0.00, 0.0000, 0.3333, 0.0000, 0.58000
0.5, 0.0, 0.55, 0.00, 0.00, 0.25, 0.00, 0.0000, 0.0000, 0.3333, 0.63500
0.0, 0.5, 0.44, 0.25, 0.00, 0.00, 0.00, 0.0000, 0.0000, 0.3333, 0.47800
0.5, 0.0, 0.18, 0.00, 0.25, 0.00, 0.00, 0.3333, 0.0000, 0.0000, 0.39800
0.5, 0.0, 0.67, 0.25, 0.00, 0.00, 0.00, 0.0000, 0.0000, 0.3333, 0.71600
0.0, 0.5, 0.45, 0.00, 0.00, 0.00, 0.25, 0.0000, 0.3333, 0.0000, 0.50000
0.0, 0.5, 0.48, 0.00, 0.25, 0.00, 0.00, 0.0000, 0.3333, 0.0000, 0.55800
0.5, 0.0, 0.25, 0.25, 0.00, 0.00, 0.00, 0.0000, 0.3333, 0.0000, 0.39000
0.5, 0.0, 0.67, 0.25, 0.00, 0.00, 0.00, 0.0000, 0.3333, 0.0000, 0.78300
0.0, 0.5, 0.37, 0.00, 0.00, 0.00, 0.25, 0.0000, 0.3333, 0.0000, 0.42000
0.5, 0.0, 0.32, 0.00, 0.25, 0.00, 0.00, 0.0000, 0.3333, 0.0000, 0.42700
0.0, 0.5, 0.48, 0.00, 0.25, 0.00, 0.00, 0.0000, 0.3333, 0.0000, 0.57000
0.5, 0.0, 0.66, 0.00, 0.00, 0.00, 0.25, 0.0000, 0.0000, 0.3333, 0.75000
0.0, 0.5, 0.61, 0.00, 0.25, 0.00, 0.00, 0.3333, 0.0000, 0.0000, 0.70000
0.5, 0.0, 0.58, 0.00, 0.00, 0.00, 0.25, 0.0000, 0.3333, 0.0000, 0.68900
0.0, 0.5, 0.19, 0.00, 0.25, 0.00, 0.00, 0.0000, 0.0000, 0.3333, 0.24000
0.0, 0.5, 0.38, 0.00, 0.00, 0.00, 0.25, 0.0000, 0.3333, 0.0000, 0.43000
0.5, 0.0, 0.27, 0.00, 0.25, 0.00, 0.00, 0.0000, 0.3333, 0.0000, 0.36400
0.0, 0.5, 0.42, 0.00, 0.25, 0.00, 0.00, 0.0000, 0.3333, 0.0000, 0.48000
0.0, 0.5, 0.60, 0.00, 0.25, 0.00, 0.00, 0.3333, 0.0000, 0.0000, 0.71300
0.5, 0.0, 0.27, 0.00, 0.00, 0.00, 0.25, 0.3333, 0.0000, 0.0000, 0.34800
0.0, 0.5, 0.29, 0.00, 0.00, 0.25, 0.00, 0.3333, 0.0000, 0.0000, 0.37100
0.5, 0.0, 0.43, 0.25, 0.00, 0.00, 0.00, 0.0000, 0.3333, 0.0000, 0.56700
0.0, 0.5, 0.48, 0.00, 0.25, 0.00, 0.00, 0.0000, 0.3333, 0.0000, 0.56700
0.0, 0.5, 0.27, 0.00, 0.00, 0.00, 0.25, 0.0000, 0.0000, 0.3333, 0.29400
0.5, 0.0, 0.44, 0.00, 0.25, 0.00, 0.00, 0.3333, 0.0000, 0.0000, 0.55200
0.0, 0.5, 0.23, 0.25, 0.00, 0.00, 0.00, 0.0000, 0.0000, 0.3333, 0.26300
0.5, 0.0, 0.36, 0.00, 0.00, 0.25, 0.00, 0.0000, 0.0000, 0.3333, 0.53000
0.0, 0.5, 0.64, 0.00, 0.00, 0.00, 0.25, 0.3333, 0.0000, 0.0000, 0.72500
0.0, 0.5, 0.29, 0.00, 0.00, 0.00, 0.25, 0.0000, 0.0000, 0.3333, 0.30000
0.5, 0.0, 0.33, 0.00, 0.25, 0.00, 0.00, 0.0000, 0.3333, 0.0000, 0.49300
0.5, 0.0, 0.66, 0.00, 0.00, 0.25, 0.00, 0.0000, 0.0000, 0.3333, 0.75000
0.5, 0.0, 0.21, 0.00, 0.00, 0.00, 0.25, 0.3333, 0.0000, 0.0000, 0.34300
0.0, 0.5, 0.27, 0.00, 0.25, 0.00, 0.00, 0.0000, 0.0000, 0.3333, 0.32700
0.0, 0.5, 0.29, 0.00, 0.25, 0.00, 0.00, 0.0000, 0.0000, 0.3333, 0.31800
0.5, 0.0, 0.31, 0.00, 0.25, 0.00, 0.00, 0.0000, 0.3333, 0.0000, 0.48600
0.0, 0.5, 0.36, 0.00, 0.00, 0.00, 0.25, 0.0000, 0.3333, 0.0000, 0.41000
0.0, 0.5, 0.49, 0.00, 0.00, 0.25, 0.00, 0.0000, 0.3333, 0.0000, 0.55700
0.5, 0.0, 0.28, 0.00, 0.25, 0.00, 0.00, 0.3333, 0.0000, 0.0000, 0.38400
0.5, 0.0, 0.43, 0.00, 0.00, 0.00, 0.25, 0.0000, 0.3333, 0.0000, 0.56600
0.5, 0.0, 0.46, 0.00, 0.00, 0.25, 0.00, 0.0000, 0.3333, 0.0000, 0.58800
0.0, 0.5, 0.57, 0.00, 0.25, 0.00, 0.00, 0.3333, 0.0000, 0.0000, 0.69800
0.5, 0.0, 0.52, 0.00, 0.00, 0.00, 0.25, 0.0000, 0.3333, 0.0000, 0.59400
0.5, 0.0, 0.31, 0.00, 0.00, 0.00, 0.25, 0.0000, 0.3333, 0.0000, 0.43500
0.5, 0.0, 0.55, 0.00, 0.25, 0.00, 0.00, 0.0000, 0.0000, 0.3333, 0.62000
0.0, 0.5, 0.50, 0.00, 0.25, 0.00, 0.00, 0.0000, 0.3333, 0.0000, 0.56400
0.0, 0.5, 0.48, 0.00, 0.00, 0.25, 0.00, 0.0000, 0.3333, 0.0000, 0.55900
0.5, 0.0, 0.22, 0.25, 0.00, 0.00, 0.00, 0.3333, 0.0000, 0.0000, 0.34500
0.0, 0.5, 0.59, 0.00, 0.00, 0.00, 0.25, 0.3333, 0.0000, 0.0000, 0.66700
0.0, 0.5, 0.34, 0.00, 0.25, 0.00, 0.00, 0.0000, 0.0000, 0.3333, 0.42800
0.5, 0.0, 0.64, 0.00, 0.25, 0.00, 0.00, 0.0000, 0.0000, 0.3333, 0.77200
0.0, 0.5, 0.29, 0.00, 0.00, 0.00, 0.25, 0.0000, 0.0000, 0.3333, 0.33500
0.5, 0.0, 0.34, 0.00, 0.00, 0.25, 0.00, 0.0000, 0.3333, 0.0000, 0.43200
0.5, 0.0, 0.61, 0.00, 0.25, 0.00, 0.00, 0.0000, 0.0000, 0.3333, 0.75000
0.0, 0.5, 0.64, 0.00, 0.00, 0.00, 0.25, 0.3333, 0.0000, 0.0000, 0.71100
0.5, 0.0, 0.29, 0.00, 0.25, 0.00, 0.00, 0.3333, 0.0000, 0.0000, 0.41300
0.0, 0.5, 0.63, 0.00, 0.00, 0.25, 0.00, 0.3333, 0.0000, 0.0000, 0.70600
0.5, 0.0, 0.29, 0.00, 0.00, 0.25, 0.00, 0.3333, 0.0000, 0.0000, 0.40000
0.5, 0.0, 0.51, 0.00, 0.25, 0.00, 0.00, 0.0000, 0.3333, 0.0000, 0.62700
0.5, 0.0, 0.24, 0.00, 0.00, 0.00, 0.25, 0.3333, 0.0000, 0.0000, 0.37700
0.0, 0.5, 0.48, 0.00, 0.00, 0.25, 0.00, 0.0000, 0.3333, 0.0000, 0.57500
0.0, 0.5, 0.18, 0.00, 0.25, 0.00, 0.00, 0.3333, 0.0000, 0.0000, 0.27400
0.0, 0.5, 0.18, 0.00, 0.25, 0.00, 0.00, 0.0000, 0.0000, 0.3333, 0.20300
0.0, 0.5, 0.33, 0.00, 0.00, 0.25, 0.00, 0.0000, 0.0000, 0.3333, 0.38200
0.5, 0.0, 0.20, 0.00, 0.00, 0.00, 0.25, 0.3333, 0.0000, 0.0000, 0.34800
0.0, 0.5, 0.29, 0.25, 0.00, 0.00, 0.00, 0.0000, 0.0000, 0.3333, 0.33000
0.5, 0.0, 0.44, 0.00, 0.00, 0.00, 0.25, 0.3333, 0.0000, 0.0000, 0.63000
0.5, 0.0, 0.65, 0.25, 0.00, 0.00, 0.00, 0.3333, 0.0000, 0.0000, 0.81800
0.5, 0.0, 0.56, 0.00, 0.25, 0.00, 0.00, 0.0000, 0.0000, 0.3333, 0.63700
0.5, 0.0, 0.52, 0.00, 0.00, 0.00, 0.25, 0.0000, 0.3333, 0.0000, 0.58400
0.5, 0.0, 0.29, 0.00, 0.00, 0.25, 0.00, 0.3333, 0.0000, 0.0000, 0.48600
0.5, 0.0, 0.47, 0.00, 0.00, 0.25, 0.00, 0.0000, 0.3333, 0.0000, 0.58900
0.0, 0.5, 0.68, 0.00, 0.25, 0.00, 0.00, 0.0000, 0.0000, 0.3333, 0.72600
0.0, 0.5, 0.31, 0.00, 0.00, 0.00, 0.25, 0.0000, 0.3333, 0.0000, 0.36000
0.0, 0.5, 0.61, 0.25, 0.00, 0.00, 0.00, 0.0000, 0.0000, 0.3333, 0.62500
0.0, 0.5, 0.19, 0.00, 0.00, 0.25, 0.00, 0.0000, 0.0000, 0.3333, 0.21500
0.0, 0.5, 0.38, 0.00, 0.00, 0.00, 0.25, 0.0000, 0.3333, 0.0000, 0.43000
0.5, 0.0, 0.26, 0.00, 0.25, 0.00, 0.00, 0.3333, 0.0000, 0.0000, 0.42300
0.0, 0.5, 0.61, 0.00, 0.00, 0.25, 0.00, 0.3333, 0.0000, 0.0000, 0.67400
0.0, 0.5, 0.40, 0.00, 0.25, 0.00, 0.00, 0.0000, 0.3333, 0.0000, 0.46500
0.5, 0.0, 0.49, 0.00, 0.25, 0.00, 0.00, 0.0000, 0.3333, 0.0000, 0.65200
0.0, 0.5, 0.56, 0.00, 0.25, 0.00, 0.00, 0.3333, 0.0000, 0.0000, 0.67500
0.5, 0.0, 0.48, 0.00, 0.00, 0.25, 0.00, 0.0000, 0.3333, 0.0000, 0.66000
0.0, 0.5, 0.52, 0.00, 0.25, 0.00, 0.00, 0.0000, 0.0000, 0.3333, 0.56300
0.5, 0.0, 0.18, 0.00, 0.25, 0.00, 0.00, 0.3333, 0.0000, 0.0000, 0.29800
0.5, 0.0, 0.56, 0.00, 0.00, 0.00, 0.25, 0.0000, 0.0000, 0.3333, 0.59300
0.5, 0.0, 0.52, 0.00, 0.00, 0.25, 0.00, 0.0000, 0.3333, 0.0000, 0.64400
0.5, 0.0, 0.18, 0.00, 0.00, 0.25, 0.00, 0.0000, 0.3333, 0.0000, 0.28600
0.5, 0.0, 0.58, 0.00, 0.25, 0.00, 0.00, 0.0000, 0.0000, 0.3333, 0.66200
0.5, 0.0, 0.39, 0.00, 0.00, 0.25, 0.00, 0.0000, 0.3333, 0.0000, 0.55100
0.5, 0.0, 0.46, 0.00, 0.25, 0.00, 0.00, 0.0000, 0.3333, 0.0000, 0.62900
0.5, 0.0, 0.40, 0.00, 0.00, 0.25, 0.00, 0.0000, 0.3333, 0.0000, 0.46200
0.5, 0.0, 0.60, 0.00, 0.25, 0.00, 0.00, 0.0000, 0.0000, 0.3333, 0.72700
0.0, 0.5, 0.36, 0.00, 0.00, 0.25, 0.00, 0.0000, 0.0000, 0.3333, 0.40700
0.0, 0.5, 0.44, 0.00, 0.25, 0.00, 0.00, 0.0000, 0.3333, 0.0000, 0.52300
0.0, 0.5, 0.28, 0.00, 0.25, 0.00, 0.00, 0.0000, 0.0000, 0.3333, 0.31300
0.0, 0.5, 0.54, 0.00, 0.00, 0.00, 0.25, 0.3333, 0.0000, 0.0000, 0.62600

Test data:

# people_test.txt
# one-over-n-hot encoding
#
0.5, 0.0, 0.51, 0.00, 0.25, 0.00, 0.00, 0.0000, 0.3333, 0.0000, 0.61200
0.5, 0.0, 0.32, 0.00, 0.00, 0.25, 0.00, 0.0000, 0.3333, 0.0000, 0.46100
0.0, 0.5, 0.55, 0.00, 0.25, 0.00, 0.00, 0.3333, 0.0000, 0.0000, 0.62700
0.0, 0.5, 0.25, 0.25, 0.00, 0.00, 0.00, 0.0000, 0.0000, 0.3333, 0.26200
0.0, 0.5, 0.33, 0.00, 0.00, 0.00, 0.25, 0.0000, 0.0000, 0.3333, 0.37300
0.5, 0.0, 0.29, 0.00, 0.00, 0.25, 0.00, 0.3333, 0.0000, 0.0000, 0.46200
0.0, 0.5, 0.65, 0.00, 0.25, 0.00, 0.00, 0.3333, 0.0000, 0.0000, 0.72700
0.5, 0.0, 0.43, 0.00, 0.00, 0.25, 0.00, 0.0000, 0.3333, 0.0000, 0.51400
0.5, 0.0, 0.54, 0.25, 0.00, 0.00, 0.00, 0.0000, 0.0000, 0.3333, 0.64800
0.0, 0.5, 0.61, 0.00, 0.00, 0.25, 0.00, 0.3333, 0.0000, 0.0000, 0.72700
0.0, 0.5, 0.52, 0.00, 0.00, 0.25, 0.00, 0.3333, 0.0000, 0.0000, 0.63600
0.0, 0.5, 0.30, 0.00, 0.00, 0.25, 0.00, 0.0000, 0.0000, 0.3333, 0.33500
0.0, 0.5, 0.29, 0.25, 0.00, 0.00, 0.00, 0.0000, 0.0000, 0.3333, 0.31400
0.5, 0.0, 0.47, 0.00, 0.00, 0.00, 0.25, 0.0000, 0.3333, 0.0000, 0.59400
0.0, 0.5, 0.39, 0.25, 0.00, 0.00, 0.00, 0.0000, 0.3333, 0.0000, 0.47800
0.0, 0.5, 0.47, 0.00, 0.00, 0.00, 0.25, 0.0000, 0.3333, 0.0000, 0.52000
0.5, 0.0, 0.49, 0.00, 0.25, 0.00, 0.00, 0.0000, 0.3333, 0.0000, 0.58600
0.5, 0.0, 0.63, 0.00, 0.00, 0.00, 0.25, 0.0000, 0.0000, 0.3333, 0.67400
0.5, 0.0, 0.30, 0.00, 0.25, 0.00, 0.00, 0.3333, 0.0000, 0.0000, 0.39200
0.5, 0.0, 0.61, 0.00, 0.00, 0.00, 0.25, 0.0000, 0.0000, 0.3333, 0.69600
0.5, 0.0, 0.47, 0.00, 0.00, 0.00, 0.25, 0.0000, 0.3333, 0.0000, 0.58700
0.0, 0.5, 0.30, 0.25, 0.00, 0.00, 0.00, 0.0000, 0.0000, 0.3333, 0.34500
0.5, 0.0, 0.51, 0.00, 0.00, 0.00, 0.25, 0.0000, 0.3333, 0.0000, 0.58000
0.5, 0.0, 0.24, 0.25, 0.00, 0.00, 0.00, 0.0000, 0.3333, 0.0000, 0.38800
0.5, 0.0, 0.49, 0.00, 0.25, 0.00, 0.00, 0.0000, 0.3333, 0.0000, 0.64500
0.0, 0.5, 0.66, 0.00, 0.00, 0.00, 0.25, 0.3333, 0.0000, 0.0000, 0.74500
0.5, 0.0, 0.65, 0.00, 0.25, 0.00, 0.00, 0.3333, 0.0000, 0.0000, 0.76900
0.5, 0.0, 0.46, 0.25, 0.00, 0.00, 0.00, 0.3333, 0.0000, 0.0000, 0.58000
0.5, 0.0, 0.45, 0.00, 0.00, 0.00, 0.25, 0.0000, 0.3333, 0.0000, 0.51800
0.5, 0.0, 0.47, 0.00, 0.25, 0.00, 0.00, 0.3333, 0.0000, 0.0000, 0.63600
0.5, 0.0, 0.29, 0.00, 0.25, 0.00, 0.00, 0.3333, 0.0000, 0.0000, 0.44800
0.5, 0.0, 0.57, 0.00, 0.00, 0.00, 0.25, 0.0000, 0.0000, 0.3333, 0.69300
0.5, 0.0, 0.20, 0.00, 0.25, 0.00, 0.00, 0.0000, 0.0000, 0.3333, 0.28700
0.5, 0.0, 0.35, 0.25, 0.00, 0.00, 0.00, 0.0000, 0.3333, 0.0000, 0.43400
0.5, 0.0, 0.61, 0.00, 0.00, 0.00, 0.25, 0.0000, 0.0000, 0.3333, 0.67000
0.5, 0.0, 0.31, 0.25, 0.00, 0.00, 0.00, 0.0000, 0.3333, 0.0000, 0.37300
0.0, 0.5, 0.18, 0.00, 0.25, 0.00, 0.00, 0.0000, 0.0000, 0.3333, 0.20800
0.0, 0.5, 0.26, 0.00, 0.00, 0.00, 0.25, 0.0000, 0.0000, 0.3333, 0.29200
0.5, 0.0, 0.28, 0.00, 0.25, 0.00, 0.00, 0.0000, 0.0000, 0.3333, 0.36400
0.5, 0.0, 0.59, 0.00, 0.00, 0.00, 0.25, 0.0000, 0.0000, 0.3333, 0.69400