## Gaussian Process Regression on the Boston Housing Dataset Using the scikit Library

A regression problem is one where the goal is to predict a single numeric value. For example, you might want to predict the median house price in a town near Boston based on the town’s crime rate, percentage of Black residents, school student-pupil ratio, and so on.

There are several completely different techniques for regression. The simplest is linear regression; it’s a technique most people are exposed to in an introductory statistics class. Other common techniques for regression include ridge regression and neural networks for regression.

A relatively rare technique for regression (among my colleagues at least) is Gaussian process regression (GPR). The bottom line is that when GPR works, it often works very well, but when GPR doesn’t work, it often fails spectacularly.

I put together a brief demo using the scikit GaussianProcessRegressor module. For my demo I used the Boston Area Housing Dataset. There are 506 source data items, each representing a town or village near Boston. There are 13 predictor variables (12 numeric and 1 Boolean). The target value to predict is the median house price (circa 1970 — the dataset is very old so most median house prices are about \$24,000).

As is usually the case, preparing the data was very time-consuming and took far longer than building the model. I used divide-by-k normalization on the 13 predictors and the target house price. The constants are 100, 100, 30, 1, 1, 10, 100, 20, 30, 800, 30, 400, 40, 50. I split the 506 data items into a 400-item training dataset and a 106-item dataset using a stratified technique because the Boolean predictor with value = 1 occurs in only 35 of the 506 data items.

Using scikit GPR is simultaneously easy and difficult. The key code that creates my demo model is:

```  print("print("Creating GPR ConstantKernel + RBF model ")

krnl = ConstantKernel(1.0, (1e-1, 1e3)) + RBF(1.0, (1e-3, 1e3))
model = GaussianProcessRegressor(kernel=krnl, normalize_y=True,
alpha=0.0, random_state=0)

print("Training model ")
model.fit(train_X, train_y)
print("Done ")
```

The code is deceptively easy because the main effort is spent on guessing which kernel functions to use and all their parameters. Using the ConstantKernel() + RBF() kernel gave fairly good results but other kernels, and combinations of kernels, and combinations of kernel parameters gave results that varied from OK to horrible.

Gaussian process egression is highly susceptible to model overfitting. There are two main ways to add noise to try and reduce overfitting. First, the alpha parameter in the GaussianProcessRegressor() constructor. Larger values add more noise. Second, using the WhiteKernel(), like:

```krnl = ConstantKernel(1.0, (1e-1, 1e3)) + \
RBF(1.0, (1e-3, 1e3)) + \
WhiteKernel(noise_level=1.0)
```

Gaussian process regression is very complex mathematically. There are dozens of blog posts and YouTube videos that explain GPR at many different levels. There’s not one best explanation of GPR — it all depends on what sort of background you have. Do you already understand kernels or not? Do you already understand multivariate Gaussian distributions or not? Do you already understand covariance matrices or not? And so on.

At an extremely high level of abstraction: GPR examines many possible functions that fit the training data. The functions are defined by the kernel function. The functions are averaged to produce a final prediction function.

GPR works with small datasets but doesn’t scale well to large datasets (lots of data items and/or lots of predictor variables).

An interesting characteristic of GPR compared to most other regression techniques is that GPR returns a predicted value but also a standard deviation which is a measure of uncertainty in the prediction. For example:

```  # 4. use model
X = test_X[0].reshape(1, -1)  # first test item
pred_y, pred_std = model.predict(X, return_std=True)
print("\nPredicting for X = ")
print(X)
print("\nPredicted y = %0.4f with std = %0.4f " \
% (pred_y, pred_std))
```

Good fun.

The complexity of creating a Gaussian process regression model is due to the vast number of combinations of kernels and their parameters. The game of chess is all about combinatorics. Easter was a few weeks ago. The Venice Carnival runs roughly the two weeks before Lent — the 40 days preceding Easter — and has featured beautiful masks and costumes since the 12th century. Here are examples of clever and beautiful chess-theme Venice Carnival costumes.

Demo code. Replace “lt” with Boolean operator symbol.

```# boston_gauss_process.py
# Gaussian process regression

# Anaconda3-2022.10  Python 3.9.13
# scikit 1.0.2
# Windows 10/11

import numpy as np
from sklearn.gaussian_process import GaussianProcessRegressor
from sklearn.gaussian_process.kernels import RBF
from sklearn.gaussian_process.kernels import DotProduct
from sklearn.gaussian_process.kernels import WhiteKernel
from sklearn.gaussian_process.kernels import ConstantKernel

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

def accuracy(model, data_X, data_y, pct_close):
# correct within pct of true income
n_correct = 0; n_wrong = 0

for i in range(len(data_X)):
X = data_X[i].reshape(1, -1)  # one-item batch
y = data_y[i]
pred = model.predict(X)       # predicted house price

if np.abs(pred - y) "lt" np.abs(pct_close * y):
n_correct += 1
else:
n_wrong += 1
acc = (n_correct * 1.0) / (n_correct + n_wrong)
return acc

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

def main():
# 0. prepare
print("\nBegin scikit Gaussian process regression ")
print("Predict Boston area house median price ")
np.random.seed(1)
np.set_printoptions(precision=4, suppress=True)

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

# 1. load data
train_file = ".\\Data\\boston_train.txt"
train_X = np.loadtxt(train_file, delimiter="\t",
usecols=[0,1,2,3,4,5,6,7,8,9,10,11,12],
train_y = np.loadtxt(train_file, delimiter="\t",

test_file = ".\\Data\\boston_test.txt"
test_X = np.loadtxt(test_file, delimiter="\t",
usecols=[0,1,2,3,4,5,6,7,8,9,10,11,12],
test_y = np.loadtxt(test_file, delimiter="\t",
print("Done ")

print("\nData: ")
print(train_X[0:4][:])
print(". . .")
print("\nActual prices: ")
print(train_y[0:4])
print(". . .")

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

# 2. create and train GPR model
print("\nCreating GPR ConstantKernel + RBF model ")

# GaussianProcessRegressor(kernel=None, *, alpha=1e-10,
#  optimizer='fmin_l_bfgs_b', n_restarts_optimizer=0,
#  normalize_y=False, copy_X_train=True, random_state=None)
#
# RBF(length_scale=1.0, length_scale_bounds=(1e-5,
#  100000.0))
# DotProduct(sigma_0=1.0, sigma_0_bounds=(1e-5, 100000.0))
# WhiteKernel(noise_level=1.0,
#  noise_level_bounds=(1e-5, 100000.0))

# krnl = RBF(1.0, length_scale_bounds=(1e-6, 100000.0))
# krnl = DotProduct() + WhiteKernel(noise_level=0.5)
# krnl = ConstantKernel(1.0, (1e-1, 1e3)) * \
#   RBF(1.0, (1e-3, 1e3))
krnl = ConstantKernel(1.0, (1e-1, 1e3)) + \
RBF(1.0, (1e-3, 1e3))
model = GaussianProcessRegressor(kernel=krnl, \
alpha=0.0, normalize_y=True, random_state=0)

print("\nTraining model ")
model.fit(train_X, train_y)
print("Done ")

# 3. compute model accuracy
print("\nComputing accuracy (within 0.15 of true price) ")
acc_train = accuracy(model, train_X, train_y, 0.15)
print("\nAccuracy on train data = %0.4f " % acc_train)
acc_test = accuracy(model, test_X, test_y, 0.15)
print("Accuracy on test data = %0.4f " % acc_test)

# 4. use model
# X = test_X[0].reshape(1, -1)  # first test item
# pred_y, pred_std = model.predict(X, return_std=True)
# print("\nPredicting for X = ")
# print(X)
# print("\nPredicted y = %0.4f with std = %0.4f " \
#   % (pred_y, pred_std))

print("\nEnd GPR demo ")

if __name__ == "__main__":
main()
```

Training data. Replace comma characters with tabs or modify data loading code.

```# boston_train.txt
# norm constants: 100, 100, 30, 1, 1, 10, 100, 20, 30, 800, 30, 400, 40, 50
#
# crime    zoning     indus      river      nox        rooms      oldness    dist       access     tax        pup_tch    blackness  low_stat   med_val
0.00027310,0.00000000,0.23566667,0.00000000,0.46900000,0.64210000,0.78900000,0.24835500,0.06666667,0.30250000,0.59333333,0.99225000,0.22850000,0.43200000
0.00027290,0.00000000,0.23566667,0.00000000,0.46900000,0.71850000,0.61100000,0.24835500,0.06666667,0.30250000,0.59333333,0.98207500,0.10075000,0.69400000
0.00032370,0.00000000,0.07266667,0.00000000,0.45800000,0.69980000,0.45800000,0.30311000,0.10000000,0.27750000,0.62333333,0.98657500,0.07350000,0.66800000
0.00069050,0.00000000,0.07266667,0.00000000,0.45800000,0.71470000,0.54200000,0.30311000,0.10000000,0.27750000,0.62333333,0.99225000,0.13325000,0.72400000
0.00088290,0.12500000,0.26233333,0.00000000,0.52400000,0.60120000,0.66600000,0.27802500,0.16666667,0.38875000,0.50666667,0.98900000,0.31075000,0.45800000
0.00144550,0.12500000,0.26233333,0.00000000,0.52400000,0.61720000,0.96100000,0.29752500,0.16666667,0.38875000,0.50666667,0.99225000,0.47875000,0.54200000
0.00211240,0.12500000,0.26233333,0.00000000,0.52400000,0.56310000,1.00000000,0.30410500,0.16666667,0.38875000,0.50666667,0.96657500,0.74825000,0.33000000
0.00170040,0.12500000,0.26233333,0.00000000,0.52400000,0.60040000,0.85900000,0.32960500,0.16666667,0.38875000,0.50666667,0.96677500,0.42750000,0.37800000
0.00117470,0.12500000,0.26233333,0.00000000,0.52400000,0.60090000,0.82900000,0.31133500,0.16666667,0.38875000,0.50666667,0.99225000,0.33175000,0.37800000
0.00093780,0.12500000,0.26233333,0.00000000,0.52400000,0.58890000,0.39000000,0.27254500,0.16666667,0.38875000,0.50666667,0.97625000,0.39275000,0.43400000
0.00629760,0.00000000,0.27133333,0.00000000,0.53800000,0.59490000,0.61800000,0.23537500,0.13333333,0.38375000,0.70000000,0.99225000,0.20650000,0.40800000
0.00637960,0.00000000,0.27133333,0.00000000,0.53800000,0.60960000,0.84500000,0.22309500,0.13333333,0.38375000,0.70000000,0.95005000,0.25650000,0.36400000
0.01053930,0.00000000,0.27133333,0.00000000,0.53800000,0.59350000,0.29300000,0.22493000,0.13333333,0.38375000,0.70000000,0.96712500,0.16450000,0.46200000
0.00784200,0.00000000,0.27133333,0.00000000,0.53800000,0.59900000,0.81700000,0.21289500,0.13333333,0.38375000,0.70000000,0.96687500,0.36675000,0.35000000
0.00802710,0.00000000,0.27133333,0.00000000,0.53800000,0.54560000,0.36600000,0.18982500,0.13333333,0.38375000,0.70000000,0.72247500,0.29225000,0.40400000
0.00725800,0.00000000,0.27133333,0.00000000,0.53800000,0.57270000,0.69500000,0.18982500,0.13333333,0.38375000,0.70000000,0.97737500,0.28200000,0.36400000
0.00852040,0.00000000,0.27133333,0.00000000,0.53800000,0.59650000,0.89200000,0.20061500,0.13333333,0.38375000,0.70000000,0.98132500,0.34575000,0.39200000
0.01232470,0.00000000,0.27133333,0.00000000,0.53800000,0.61420000,0.91700000,0.19884500,0.13333333,0.38375000,0.70000000,0.99225000,0.46800000,0.30400000
0.00988430,0.00000000,0.27133333,0.00000000,0.53800000,0.58130000,1.00000000,0.20476000,0.13333333,0.38375000,0.70000000,0.98635000,0.49700000,0.29000000
0.00750260,0.00000000,0.27133333,0.00000000,0.53800000,0.59240000,0.94100000,0.21998000,0.13333333,0.38375000,0.70000000,0.98582500,0.40750000,0.31200000
0.00671910,0.00000000,0.27133333,0.00000000,0.53800000,0.58130000,0.90300000,0.23410000,0.13333333,0.38375000,0.70000000,0.94220000,0.37025000,0.33200000
0.00955770,0.00000000,0.27133333,0.00000000,0.53800000,0.60470000,0.88800000,0.22267000,0.13333333,0.38375000,0.70000000,0.76595000,0.43200000,0.29600000
0.00772990,0.00000000,0.27133333,0.00000000,0.53800000,0.64950000,0.94400000,0.22273500,0.13333333,0.38375000,0.70000000,0.96985000,0.32000000,0.36800000
0.01002450,0.00000000,0.27133333,0.00000000,0.53800000,0.66740000,0.87300000,0.21195000,0.13333333,0.38375000,0.70000000,0.95057500,0.29950000,0.42000000
0.01354720,0.00000000,0.27133333,0.00000000,0.53800000,0.60720000,1.00000000,0.20875000,0.13333333,0.38375000,0.70000000,0.94182500,0.32600000,0.29000000
0.01387990,0.00000000,0.27133333,0.00000000,0.53800000,0.59500000,0.82000000,0.19950000,0.13333333,0.38375000,0.70000000,0.58150000,0.69275000,0.26400000
0.01151720,0.00000000,0.27133333,0.00000000,0.53800000,0.57010000,0.95000000,0.18936000,0.13333333,0.38375000,0.70000000,0.89692500,0.45875000,0.26200000
0.01612820,0.00000000,0.27133333,0.00000000,0.53800000,0.60960000,0.96900000,0.18799000,0.13333333,0.38375000,0.70000000,0.62077500,0.50850000,0.27000000
0.00097440,0.00000000,0.19866667,0.00000000,0.49900000,0.58410000,0.61400000,0.16889500,0.16666667,0.34875000,0.64000000,0.94390000,0.28525000,0.40000000
0.00080140,0.00000000,0.19866667,0.00000000,0.49900000,0.58500000,0.41500000,0.19671000,0.16666667,0.34875000,0.64000000,0.99225000,0.21925000,0.42000000
0.00175050,0.00000000,0.19866667,0.00000000,0.49900000,0.59660000,0.30200000,0.19236500,0.16666667,0.34875000,0.64000000,0.98357500,0.25325000,0.49400000
0.00027630,0.75000000,0.09833333,0.00000000,0.42800000,0.65950000,0.21800000,0.27005500,0.10000000,0.31500000,0.61000000,0.98907500,0.10800000,0.61600000
0.00127440,0.00000000,0.23033333,0.00000000,0.44800000,0.67700000,0.02900000,0.28604500,0.10000000,0.29125000,0.59666667,0.96352500,0.12100000,0.53200000
0.00141500,0.00000000,0.23033333,0.00000000,0.44800000,0.61690000,0.06600000,0.28604500,0.10000000,0.29125000,0.59666667,0.95842500,0.14525000,0.50600000
0.00159360,0.00000000,0.23033333,0.00000000,0.44800000,0.62110000,0.06500000,0.28604500,0.10000000,0.29125000,0.59666667,0.98615000,0.18600000,0.49400000
0.00122690,0.00000000,0.23033333,0.00000000,0.44800000,0.60690000,0.40000000,0.28604500,0.10000000,0.29125000,0.59666667,0.97347500,0.23875000,0.42400000
0.00188360,0.00000000,0.23033333,0.00000000,0.44800000,0.57860000,0.33300000,0.25502000,0.10000000,0.29125000,0.59666667,0.99225000,0.35375000,0.40000000
0.00229270,0.00000000,0.23033333,0.00000000,0.44800000,0.60300000,0.85500000,0.28447000,0.10000000,0.29125000,0.59666667,0.98185000,0.47000000,0.33200000
0.00253870,0.00000000,0.23033333,0.00000000,0.44800000,0.53990000,0.95300000,0.29350000,0.10000000,0.29125000,0.59666667,0.99225000,0.77025000,0.28800000
0.00219770,0.00000000,0.23033333,0.00000000,0.44800000,0.56020000,0.62000000,0.30438500,0.10000000,0.29125000,0.59666667,0.99225000,0.40500000,0.38800000
0.00043370,0.21000000,0.18800000,0.00000000,0.43900000,0.61150000,0.63000000,0.34073500,0.13333333,0.30375000,0.56000000,0.98492500,0.23575000,0.41000000
0.00053600,0.21000000,0.18800000,0.00000000,0.43900000,0.65110000,0.21100000,0.34073500,0.13333333,0.30375000,0.56000000,0.99225000,0.13200000,0.50000000
0.00049810,0.21000000,0.18800000,0.00000000,0.43900000,0.59980000,0.21400000,0.34073500,0.13333333,0.30375000,0.56000000,0.99225000,0.21075000,0.46800000
0.00013600,0.75000000,0.13333333,0.00000000,0.41000000,0.58880000,0.47600000,0.36598500,0.10000000,0.58625000,0.70333333,0.99225000,0.37000000,0.37800000
0.00020550,0.85000000,0.02466667,0.00000000,0.41000000,0.63830000,0.35700000,0.45938000,0.06666667,0.39125000,0.57666667,0.99225000,0.14425000,0.49400000
0.00014320,1.00000000,0.04400000,0.00000000,0.41100000,0.68160000,0.40500000,0.41624000,0.16666667,0.32000000,0.50333333,0.98225000,0.09875000,0.63200000
0.00154450,0.25000000,0.17100000,0.00000000,0.45300000,0.61450000,0.29200000,0.39074000,0.26666667,0.35500000,0.65666667,0.97670000,0.17150000,0.46600000
0.00103280,0.25000000,0.17100000,0.00000000,0.45300000,0.59270000,0.47200000,0.34660000,0.26666667,0.35500000,0.65666667,0.99225000,0.23050000,0.39200000
0.00171710,0.25000000,0.17100000,0.00000000,0.45300000,0.59660000,0.93400000,0.34092500,0.26666667,0.35500000,0.65666667,0.94520000,0.36100000,0.32000000
0.00110270,0.25000000,0.17100000,0.00000000,0.45300000,0.64560000,0.67800000,0.36127500,0.26666667,0.35500000,0.65666667,0.99225000,0.16825000,0.44400000
0.00126500,0.25000000,0.17100000,0.00000000,0.45300000,0.67620000,0.43400000,0.39904500,0.26666667,0.35500000,0.65666667,0.98895000,0.23750000,0.50000000
0.00019510,0.17500000,0.04600000,0.00000000,0.41610000,0.71040000,0.59500000,0.46114500,0.10000000,0.27000000,0.62000000,0.98310000,0.20125000,0.66000000
0.00043790,0.80000000,0.11233333,0.00000000,0.39800000,0.57870000,0.31100000,0.33057500,0.13333333,0.42125000,0.53666667,0.99225000,0.25600000,0.38800000
0.00057890,0.12500000,0.20233333,0.00000000,0.40900000,0.58780000,0.21400000,0.32490000,0.13333333,0.43125000,0.63000000,0.99052500,0.20250000,0.44000000
0.00135540,0.12500000,0.20233333,0.00000000,0.40900000,0.55940000,0.36800000,0.32490000,0.13333333,0.43125000,0.63000000,0.99225000,0.32725000,0.34800000
0.00128160,0.12500000,0.20233333,0.00000000,0.40900000,0.58850000,0.33000000,0.32490000,0.13333333,0.43125000,0.63000000,0.99225000,0.21975000,0.41800000
0.00158760,0.00000000,0.36033333,0.00000000,0.41300000,0.59610000,0.17500000,0.26436500,0.13333333,0.38125000,0.64000000,0.94235000,0.24700000,0.43400000
0.00091640,0.00000000,0.36033333,0.00000000,0.41300000,0.60650000,0.07800000,0.26436500,0.13333333,0.38125000,0.64000000,0.97727500,0.13800000,0.45600000
0.00195390,0.00000000,0.36033333,0.00000000,0.41300000,0.62450000,0.06200000,0.26436500,0.13333333,0.38125000,0.64000000,0.94292500,0.18850000,0.46800000
0.00078960,0.00000000,0.42766667,0.00000000,0.43700000,0.62730000,0.06000000,0.21257500,0.16666667,0.49750000,0.62333333,0.98730000,0.16950000,0.48200000
0.00101530,0.00000000,0.42766667,0.00000000,0.43700000,0.62790000,0.74500000,0.20261000,0.16666667,0.49750000,0.62333333,0.93415000,0.29925000,0.40000000
0.00087070,0.00000000,0.42766667,0.00000000,0.43700000,0.61400000,0.45800000,0.20452500,0.16666667,0.49750000,0.62333333,0.96740000,0.25675000,0.41600000
0.00056460,0.00000000,0.42766667,0.00000000,0.43700000,0.62320000,0.53700000,0.25070500,0.16666667,0.49750000,0.62333333,0.96600000,0.30850000,0.42400000
0.00083870,0.00000000,0.42766667,0.00000000,0.43700000,0.58740000,0.36600000,0.22513000,0.16666667,0.49750000,0.62333333,0.99015000,0.22750000,0.40600000
0.00044620,0.25000000,0.16200000,0.00000000,0.42600000,0.66190000,0.70400000,0.27003500,0.13333333,0.35125000,0.63333333,0.98907500,0.18050000,0.47800000
0.00036590,0.25000000,0.16200000,0.00000000,0.42600000,0.63020000,0.32200000,0.27003500,0.13333333,0.35125000,0.63333333,0.99225000,0.16800000,0.49600000
0.00035510,0.25000000,0.16200000,0.00000000,0.42600000,0.61670000,0.46700000,0.27003500,0.13333333,0.35125000,0.63333333,0.97660000,0.18775000,0.45800000
0.00050590,0.00000000,0.14966667,0.00000000,0.44900000,0.63890000,0.48000000,0.23897000,0.10000000,0.30875000,0.61666667,0.99225000,0.24050000,0.47800000
0.00051880,0.00000000,0.14966667,0.00000000,0.44900000,0.60150000,0.45100000,0.22136000,0.10000000,0.30875000,0.61666667,0.98997500,0.32150000,0.45000000
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0.04422280,0.00000000,0.60333333,0.00000000,0.58400000,0.60030000,0.94500000,0.12701500,0.80000000,0.83250000,0.67333333,0.82822500,0.53300000,0.38200000
0.15575700,0.00000000,0.60333333,0.00000000,0.58000000,0.59260000,0.71000000,0.14542000,0.80000000,0.83250000,0.67333333,0.92185000,0.45325000,0.38200000
0.13075100,0.00000000,0.60333333,0.00000000,0.58000000,0.57130000,0.56700000,0.14118500,0.80000000,0.83250000,0.67333333,0.99225000,0.36900000,0.40200000
0.04038410,0.00000000,0.60333333,0.00000000,0.53200000,0.62290000,0.90700000,0.15496500,0.80000000,0.83250000,0.67333333,0.98832500,0.32175000,0.39200000
0.03568680,0.00000000,0.60333333,0.00000000,0.58000000,0.64370000,0.75000000,0.14482500,0.80000000,0.83250000,0.67333333,0.98342500,0.35900000,0.46400000
0.04646890,0.00000000,0.60333333,0.00000000,0.61400000,0.69800000,0.67600000,0.12664500,0.80000000,0.83250000,0.67333333,0.93670000,0.29150000,0.59600000
0.08055790,0.00000000,0.60333333,0.00000000,0.58400000,0.54270000,0.95400000,0.12149000,0.80000000,0.83250000,0.67333333,0.88145000,0.45350000,0.27600000
0.04871410,0.00000000,0.60333333,0.00000000,0.61400000,0.64840000,0.93600000,0.11526500,0.80000000,0.83250000,0.67333333,0.99052500,0.46700000,0.33400000
0.15023400,0.00000000,0.60333333,0.00000000,0.61400000,0.53040000,0.97300000,0.10503500,0.80000000,0.83250000,0.67333333,0.87370000,0.62275000,0.24000000
0.10233000,0.00000000,0.60333333,0.00000000,0.61400000,0.61850000,0.96700000,0.10852500,0.80000000,0.83250000,0.67333333,0.94925000,0.45075000,0.29200000
0.14333700,0.00000000,0.60333333,0.00000000,0.61400000,0.62290000,0.88000000,0.09756000,0.80000000,0.83250000,0.67333333,0.95830000,0.32775000,0.42800000
0.05708180,0.00000000,0.60333333,0.00000000,0.53200000,0.67500000,0.74900000,0.16658500,0.80000000,0.83250000,0.67333333,0.98267500,0.19350000,0.47400000
0.05731160,0.00000000,0.60333333,0.00000000,0.53200000,0.70610000,0.77000000,0.17053000,0.80000000,0.83250000,0.67333333,0.98820000,0.17525000,0.50000000
0.02818380,0.00000000,0.60333333,0.00000000,0.53200000,0.57620000,0.40300000,0.20491500,0.80000000,0.83250000,0.67333333,0.98230000,0.26050000,0.43600000
0.02378570,0.00000000,0.60333333,0.00000000,0.58300000,0.58710000,0.41900000,0.18620000,0.80000000,0.83250000,0.67333333,0.92682500,0.33350000,0.41200000
0.05691750,0.00000000,0.60333333,0.00000000,0.58300000,0.61140000,0.79800000,0.17729500,0.80000000,0.83250000,0.67333333,0.98170000,0.37450000,0.38200000
0.04835670,0.00000000,0.60333333,0.00000000,0.58300000,0.59050000,0.53200000,0.15761500,0.80000000,0.83250000,0.67333333,0.97055000,0.28625000,0.41200000
0.00150860,0.00000000,0.92466667,0.00000000,0.60900000,0.54540000,0.92700000,0.09104500,0.13333333,0.88875000,0.67000000,0.98772500,0.45150000,0.30400000
0.00183370,0.00000000,0.92466667,0.00000000,0.60900000,0.54140000,0.98300000,0.08777000,0.13333333,0.88875000,0.67000000,0.86012500,0.59925000,0.14000000
0.00105740,0.00000000,0.92466667,0.00000000,0.60900000,0.59830000,0.98800000,0.09340500,0.13333333,0.88875000,0.67000000,0.97527500,0.45175000,0.27200000
0.00111320,0.00000000,0.92466667,0.00000000,0.60900000,0.59830000,0.83500000,0.10549500,0.13333333,0.88875000,0.67000000,0.99225000,0.33375000,0.40200000
0.00173310,0.00000000,0.32300000,0.00000000,0.58500000,0.57070000,0.54000000,0.11908500,0.20000000,0.48875000,0.64000000,0.99225000,0.30025000,0.43600000
0.00279570,0.00000000,0.32300000,0.00000000,0.58500000,0.59260000,0.42600000,0.11908500,0.20000000,0.48875000,0.64000000,0.99225000,0.33975000,0.49000000
0.00289600,0.00000000,0.32300000,0.00000000,0.58500000,0.53900000,0.72900000,0.13993000,0.20000000,0.48875000,0.64000000,0.99225000,0.52850000,0.39400000
0.00268380,0.00000000,0.32300000,0.00000000,0.58500000,0.57940000,0.70600000,0.14463500,0.20000000,0.48875000,0.64000000,0.99225000,0.35250000,0.36600000
0.00239120,0.00000000,0.32300000,0.00000000,0.58500000,0.60190000,0.65300000,0.12045500,0.20000000,0.48875000,0.64000000,0.99225000,0.32300000,0.42400000
0.00177830,0.00000000,0.32300000,0.00000000,0.58500000,0.55690000,0.73500000,0.11999500,0.20000000,0.48875000,0.64000000,0.98942500,0.37750000,0.35000000

```

Test data. Replace comma characters with tabs or modify data loading code.

```# boston_test.txt
# norm constants: 100, 100, 30, 1, 1, 10, 100, 20, 30, 800, 30, 400, 40, 50
#
# crime    zoning     indus      river      nox        rooms      oldness    dist       access     tax        pup_tch    blackness  low_stat   med_val
0.00006320,0.18000000,0.07700000,0.00000000,0.53800000,0.65750000,0.65200000,0.20450000,0.03333333,0.37000000,0.51000000,0.99225000,0.12450000,0.48000000
0.00029850,0.00000000,0.07266667,0.00000000,0.45800000,0.64300000,0.58700000,0.30311000,0.10000000,0.27750000,0.62333333,0.98530000,0.13025000,0.57400000
0.00224890,0.12500000,0.26233333,0.00000000,0.52400000,0.63770000,0.94300000,0.31733500,0.16666667,0.38875000,0.50666667,0.98130000,0.51125000,0.30000000
0.00627390,0.00000000,0.27133333,0.00000000,0.53800000,0.58340000,0.56500000,0.22493000,0.13333333,0.38375000,0.70000000,0.98905000,0.21175000,0.39800000
0.01251790,0.00000000,0.27133333,0.00000000,0.53800000,0.55700000,0.98100000,0.18989500,0.13333333,0.38375000,0.70000000,0.94142500,0.52550000,0.27200000
0.00840540,0.00000000,0.27133333,0.00000000,0.53800000,0.55990000,0.85700000,0.22273000,0.13333333,0.38375000,0.70000000,0.75855000,0.41275000,0.27800000
0.01130810,0.00000000,0.27133333,0.00000000,0.53800000,0.57130000,0.94100000,0.21165000,0.13333333,0.38375000,0.70000000,0.90042500,0.56500000,0.25400000
0.00064170,0.00000000,0.19866667,0.00000000,0.49900000,0.59330000,0.68200000,0.16801500,0.16666667,0.34875000,0.64000000,0.99225000,0.24200000,0.37800000
0.00033590,0.75000000,0.09833333,0.00000000,0.42800000,0.70240000,0.15800000,0.27005500,0.10000000,0.31500000,0.61000000,0.98905000,0.04950000,0.69800000
0.00171420,0.00000000,0.23033333,0.00000000,0.44800000,0.56820000,0.33800000,0.25502000,0.10000000,0.29125000,0.59666667,0.99225000,0.25525000,0.38600000
0.00088730,0.21000000,0.18800000,0.00000000,0.43900000,0.59630000,0.45700000,0.34073500,0.13333333,0.30375000,0.56000000,0.98890000,0.33625000,0.39400000
0.00013110,0.90000000,0.04066667,0.00000000,0.40300000,0.72490000,0.21900000,0.43483000,0.16666667,0.28250000,0.59666667,0.98982500,0.12025000,0.70800000
0.00149320,0.25000000,0.17100000,0.00000000,0.45300000,0.57410000,0.66200000,0.36127000,0.26666667,0.35500000,0.65666667,0.98777500,0.32875000,0.37400000
0.00035840,0.80000000,0.11233333,0.00000000,0.39800000,0.62900000,0.17800000,0.33057500,0.13333333,0.42125000,0.53666667,0.99225000,0.11675000,0.47000000
0.00088260,0.00000000,0.36033333,0.00000000,0.41300000,0.64170000,0.06600000,0.26436500,0.13333333,0.38125000,0.64000000,0.95932500,0.16800000,0.48400000
0.00095120,0.00000000,0.42766667,0.00000000,0.43700000,0.62860000,0.45000000,0.22513000,0.16666667,0.49750000,0.62333333,0.95807500,0.22350000,0.42800000
0.00041130,0.25000000,0.16200000,0.00000000,0.42600000,0.67270000,0.33500000,0.27003500,0.13333333,0.35125000,0.63333333,0.99225000,0.13225000,0.56000000
0.00057350,0.00000000,0.14966667,0.00000000,0.44900000,0.66300000,0.56100000,0.22188500,0.10000000,0.30875000,0.61666667,0.98075000,0.16325000,0.53200000
0.00046840,0.00000000,0.11366667,0.00000000,0.48900000,0.64170000,0.66100000,0.15461500,0.06666667,0.33750000,0.59333333,0.98045000,0.22025000,0.45200000
0.00122040,0.00000000,0.09633333,0.00000000,0.44500000,0.66250000,0.57800000,0.17476000,0.06666667,0.34500000,0.60000000,0.89495000,0.16625000,0.56800000
0.00148660,0.00000000,0.28533333,0.00000000,0.52000000,0.67270000,0.79900000,0.13889000,0.16666667,0.48000000,0.69666667,0.98690000,0.23550000,0.55000000
0.00132620,0.00000000,0.28533333,0.00000000,0.52000000,0.58510000,0.96700000,0.10534500,0.16666667,0.48000000,0.69666667,0.98512500,0.41175000,0.39000000
0.00107930,0.00000000,0.28533333,0.00000000,0.52000000,0.61950000,0.54400000,0.13889000,0.16666667,0.48000000,0.69666667,0.98372500,0.32500000,0.43400000
0.00171340,0.00000000,0.33366667,0.00000000,0.54700000,0.59280000,0.88200000,0.12315500,0.20000000,0.54000000,0.59333333,0.86227500,0.39400000,0.36600000
0.00068990,0.00000000,0.85500000,0.00000000,0.58100000,0.58700000,0.69700000,0.11288500,0.06666667,0.23500000,0.63666667,0.97287500,0.35925000,0.44000000
0.00169020,0.00000000,0.85500000,0.00000000,0.58100000,0.59860000,0.88400000,0.09964500,0.06666667,0.23500000,0.63666667,0.96255000,0.37025000,0.42800000
0.00340060,0.00000000,0.72966667,0.00000000,0.62400000,0.64580000,0.98900000,0.10592500,0.13333333,0.54625000,0.70666667,0.98760000,0.31500000,0.38400000
0.00557780,0.00000000,0.72966667,0.00000000,0.62400000,0.63350000,0.98200000,0.10553500,0.13333333,0.54625000,0.70666667,0.98667500,0.42400000,0.36200000
0.00290900,0.00000000,0.72966667,0.00000000,0.62400000,0.61740000,0.93600000,0.08059500,0.13333333,0.54625000,0.70666667,0.97020000,0.60400000,0.28000000
0.02379340,0.00000000,0.65266667,0.00000000,0.87100000,0.61300000,1.00000000,0.07095500,0.16666667,0.50375000,0.49000000,0.43227500,0.69500000,0.27600000
0.01656600,0.00000000,0.65266667,0.00000000,0.87100000,0.61220000,0.97300000,0.08090000,0.16666667,0.50375000,0.49000000,0.93200000,0.35250000,0.43000000
0.03535010,0.00000000,0.65266667,1.00000000,0.87100000,0.61520000,0.82600000,0.08727500,0.16666667,0.50375000,0.49000000,0.22002500,0.37550000,0.31200000
0.01273460,0.00000000,0.65266667,1.00000000,0.60500000,0.62500000,0.92600000,0.08992000,0.16666667,0.50375000,0.49000000,0.84730000,0.13750000,0.54000000
0.02924000,0.00000000,0.65266667,0.00000000,0.60500000,0.61010000,0.93000000,0.11417000,0.16666667,0.50375000,0.49000000,0.60040000,0.24525000,0.50000000
0.01207420,0.00000000,0.65266667,0.00000000,0.60500000,0.58750000,0.94600000,0.12129500,0.16666667,0.50375000,0.49000000,0.73072500,0.36075000,0.34800000
0.00066640,0.00000000,0.13500000,0.00000000,0.51000000,0.65460000,0.33100000,0.15661500,0.16666667,0.37000000,0.55333333,0.97740000,0.13325000,0.58800000
0.00065880,0.00000000,0.08200000,0.00000000,0.48800000,0.77650000,0.83300000,0.13705000,0.10000000,0.24125000,0.59333333,0.98890000,0.18900000,0.79600000
0.00060470,0.00000000,0.08200000,0.00000000,0.48800000,0.61530000,0.68800000,0.16398500,0.10000000,0.24125000,0.59333333,0.96777500,0.32875000,0.59200000
0.00090680,0.45000000,0.11466667,0.00000000,0.43700000,0.69510000,0.21500000,0.32399000,0.16666667,0.49750000,0.50666667,0.94420000,0.12750000,0.74000000
0.00013810,0.80000000,0.01533333,0.00000000,0.42200000,0.78750000,0.32000000,0.28242000,0.13333333,0.31875000,0.48000000,0.98557500,0.07425000,1.00000000
0.00017780,0.95000000,0.04900000,0.00000000,0.40300000,0.71350000,0.13900000,0.38267000,0.10000000,0.50250000,0.56666667,0.96075000,0.11125000,0.65800000
0.00136420,0.00000000,0.35300000,0.00000000,0.48900000,0.58910000,0.22300000,0.19727000,0.13333333,0.34625000,0.62000000,0.99225000,0.27175000,0.45200000
0.00174460,0.00000000,0.35300000,1.00000000,0.48900000,0.59600000,0.92100000,0.19385500,0.13333333,0.34625000,0.62000000,0.98312500,0.43175000,0.43400000
0.00198020,0.00000000,0.35300000,0.00000000,0.48900000,0.61820000,0.42400000,0.19727000,0.13333333,0.34625000,0.62000000,0.98407500,0.23675000,0.50000000
0.00358090,0.00000000,0.20666667,1.00000000,0.50700000,0.69510000,0.88500000,0.14308500,0.26666667,0.38375000,0.58000000,0.97925000,0.24275000,0.53400000
0.00526930,0.00000000,0.20666667,0.00000000,0.50400000,0.87250000,0.83000000,0.14472000,0.26666667,0.38375000,0.58000000,0.95500000,0.11575000,1.00000000
0.00537000,0.00000000,0.20666667,0.00000000,0.50400000,0.59810000,0.68100000,0.18357500,0.26666667,0.38375000,0.58000000,0.94587500,0.29125000,0.48600000
0.00330450,0.00000000,0.20666667,0.00000000,0.50700000,0.60860000,0.61500000,0.18259500,0.26666667,0.38375000,0.58000000,0.94187500,0.27200000,0.48000000
0.00113290,0.30000000,0.16433333,0.00000000,0.42800000,0.68970000,0.54300000,0.31680500,0.20000000,0.37500000,0.55333333,0.97812500,0.28450000,0.44000000
0.00191330,0.22000000,0.19533333,0.00000000,0.43100000,0.56050000,0.70200000,0.39774500,0.23333333,0.41250000,0.63666667,0.97282500,0.46150000,0.37000000
0.00140300,0.22000000,0.19533333,0.00000000,0.43100000,0.64870000,0.13000000,0.36983500,0.23333333,0.41250000,0.63666667,0.99070000,0.14750000,0.48800000
0.00035480,0.80000000,0.12133333,0.00000000,0.39200000,0.58760000,0.19100000,0.46101500,0.03333333,0.39375000,0.54666667,0.98795000,0.23125000,0.41800000
0.00540110,0.20000000,0.13233333,0.00000000,0.64700000,0.72030000,0.81800000,0.10560500,0.16666667,0.33000000,0.43333333,0.98200000,0.23975000,0.67600000
0.00761620,0.20000000,0.13233333,0.00000000,0.64700000,0.55600000,0.62800000,0.09932500,0.16666667,0.33000000,0.43333333,0.98100000,0.26125000,0.45600000
0.00299160,0.20000000,0.23200000,0.00000000,0.46400000,0.58560000,0.42100000,0.22145000,0.10000000,0.27875000,0.62000000,0.97162500,0.32500000,0.42200000
0.00096040,0.40000000,0.21366667,0.00000000,0.44700000,0.68540000,0.42800000,0.21336500,0.13333333,0.31750000,0.58666667,0.99225000,0.07450000,0.64000000
0.00035780,0.20000000,0.11100000,0.00000000,0.44290000,0.78200000,0.64500000,0.23473500,0.16666667,0.27000000,0.49666667,0.96827500,0.09400000,0.90800000
0.00010960,0.55000000,0.07500000,0.00000000,0.38900000,0.64530000,0.31900000,0.36536500,0.03333333,0.37500000,0.51000000,0.98680000,0.20575000,0.44000000
0.00035020,0.80000000,0.16500000,0.00000000,0.41100000,0.68610000,0.27900000,0.25583500,0.13333333,0.30625000,0.64000000,0.99225000,0.08325000,0.57000000
0.00129320,0.00000000,0.46400000,0.00000000,0.43700000,0.66780000,0.31100000,0.29802000,0.13333333,0.36125000,0.53333333,0.99225000,0.15675000,0.57200000
0.00044170,0.70000000,0.07466667,0.00000000,0.40000000,0.68710000,0.47400000,0.39139000,0.16666667,0.44750000,0.49333333,0.97715000,0.15175000,0.49600000
0.00054790,0.33000000,0.07266667,0.00000000,0.47200000,0.66160000,0.58100000,0.16850000,0.23333333,0.27750000,0.61333333,0.98340000,0.22325000,0.56800000
0.02635480,0.00000000,0.33000000,0.00000000,0.54400000,0.49730000,0.37800000,0.12597000,0.13333333,0.38000000,0.61333333,0.87612500,0.31600000,0.32200000
0.00253560,0.00000000,0.33000000,0.00000000,0.54400000,0.57050000,0.77700000,0.19725000,0.13333333,0.38000000,0.61333333,0.99105000,0.28750000,0.32400000
0.00167600,0.00000000,0.24600000,0.00000000,0.49300000,0.64260000,0.52300000,0.22702000,0.16666667,0.35875000,0.65333333,0.99225000,0.18000000,0.47600000
0.00191860,0.00000000,0.24600000,0.00000000,0.49300000,0.64310000,0.14700000,0.27079500,0.16666667,0.35875000,0.65333333,0.98420000,0.12700000,0.49200000
0.00045440,0.00000000,0.10800000,0.00000000,0.46000000,0.61440000,0.32200000,0.29368000,0.13333333,0.53750000,0.56333333,0.92142500,0.22725000,0.39600000
0.00039610,0.00000000,0.17300000,0.00000000,0.51500000,0.60370000,0.34500000,0.29926500,0.16666667,0.28000000,0.67333333,0.99225000,0.20025000,0.42200000
0.00061510,0.00000000,0.17300000,0.00000000,0.51500000,0.59680000,0.58500000,0.24061000,0.16666667,0.28000000,0.67333333,0.99225000,0.23225000,0.37400000
0.00031130,0.00000000,0.14633333,0.00000000,0.44200000,0.60140000,0.48500000,0.40068000,0.10000000,0.44000000,0.62666667,0.96410000,0.26325000,0.35000000
0.00062110,0.40000000,0.04166667,0.00000000,0.42900000,0.64900000,0.44400000,0.43960500,0.03333333,0.41875000,0.65666667,0.99225000,0.14950000,0.45800000
0.00106590,0.80000000,0.06366667,0.00000000,0.41300000,0.59360000,0.19500000,0.52928500,0.13333333,0.41750000,0.73333333,0.94010000,0.13925000,0.41200000
0.04541920,0.00000000,0.60333333,0.00000000,0.77000000,0.63980000,0.88000000,0.12591000,0.80000000,0.83250000,0.67333333,0.93640000,0.19475000,0.50000000
0.04555870,0.00000000,0.60333333,0.00000000,0.71800000,0.35610000,0.87900000,0.08066000,0.80000000,0.83250000,0.67333333,0.88675000,0.17800000,0.55000000
0.06538760,0.00000000,0.60333333,1.00000000,0.63100000,0.70160000,0.97500000,0.06012000,0.80000000,0.83250000,0.67333333,0.98012500,0.07400000,1.00000000
0.19609100,0.00000000,0.60333333,0.00000000,0.67100000,0.73130000,0.97900000,0.06581500,0.80000000,0.83250000,0.67333333,0.99225000,0.33600000,0.30000000
0.88976200,0.00000000,0.60333333,0.00000000,0.67100000,0.69680000,0.91900000,0.07082500,0.80000000,0.83250000,0.67333333,0.99225000,0.43025000,0.20800000
0.16811800,0.00000000,0.60333333,0.00000000,0.70000000,0.52770000,0.98100000,0.07130500,0.80000000,0.83250000,0.67333333,0.99225000,0.77025000,0.14400000
0.06962150,0.00000000,0.60333333,0.00000000,0.70000000,0.57130000,0.97000000,0.09632500,0.80000000,0.83250000,0.67333333,0.98607500,0.42775000,0.30200000
0.08716750,0.00000000,0.60333333,0.00000000,0.69300000,0.64710000,0.98800000,0.08628500,0.80000000,0.83250000,0.67333333,0.97995000,0.42800000,0.26200000
0.25046100,0.00000000,0.60333333,0.00000000,0.69300000,0.59870000,1.00000000,0.07944000,0.80000000,0.83250000,0.67333333,0.99225000,0.66925000,0.11200000
0.67920800,0.00000000,0.60333333,0.00000000,0.69300000,0.56830000,1.00000000,0.07127000,0.80000000,0.83250000,0.67333333,0.96242500,0.57450000,0.10000000
0.51135800,0.00000000,0.60333333,0.00000000,0.59700000,0.57570000,1.00000000,0.07065000,0.80000000,0.83250000,0.67333333,0.00650000,0.25275000,0.30000000
0.18084600,0.00000000,0.60333333,0.00000000,0.67900000,0.64340000,1.00000000,0.09173500,0.80000000,0.83250000,0.67333333,0.06812500,0.72625000,0.14400000
0.11087400,0.00000000,0.60333333,0.00000000,0.71800000,0.64110000,1.00000000,0.09294500,0.80000000,0.83250000,0.67333333,0.79687500,0.37550000,0.33400000
0.15860300,0.00000000,0.60333333,0.00000000,0.67900000,0.58960000,0.95400000,0.09548000,0.80000000,0.83250000,0.67333333,0.01920000,0.60975000,0.16600000
0.08492130,0.00000000,0.60333333,0.00000000,0.58400000,0.63480000,0.86100000,0.10263500,0.80000000,0.83250000,0.67333333,0.20862500,0.44100000,0.29000000
0.11160400,0.00000000,0.60333333,0.00000000,0.74000000,0.66290000,0.94600000,0.10623500,0.80000000,0.83250000,0.67333333,0.27462500,0.58175000,0.26800000
0.22051100,0.00000000,0.60333333,0.00000000,0.74000000,0.58180000,0.92400000,0.09331000,0.80000000,0.83250000,0.67333333,0.97862500,0.55275000,0.21000000
0.10671800,0.00000000,0.60333333,0.00000000,0.74000000,0.64590000,0.94800000,0.09939500,0.80000000,0.83250000,0.67333333,0.10765000,0.59950000,0.23600000
0.06717720,0.00000000,0.60333333,0.00000000,0.71300000,0.67490000,0.92600000,0.11618000,0.80000000,0.83250000,0.67333333,0.00080000,0.43600000,0.26800000
0.04752370,0.00000000,0.60333333,0.00000000,0.71300000,0.65250000,0.86500000,0.12179000,0.80000000,0.83250000,0.67333333,0.12730000,0.45325000,0.28200000
0.04812130,0.00000000,0.60333333,0.00000000,0.71300000,0.67010000,0.90000000,0.12987500,0.80000000,0.83250000,0.67333333,0.63807500,0.41050000,0.32800000
0.03163600,0.00000000,0.60333333,0.00000000,0.65500000,0.57590000,0.48200000,0.15332500,0.80000000,0.83250000,0.67333333,0.83600000,0.35325000,0.39800000
0.04348790,0.00000000,0.60333333,0.00000000,0.58000000,0.61670000,0.84000000,0.15167000,0.80000000,0.83250000,0.67333333,0.99225000,0.40725000,0.39800000
0.06393120,0.00000000,0.60333333,0.00000000,0.58400000,0.61620000,0.97400000,0.11030000,0.80000000,0.83250000,0.67333333,0.75690000,0.60250000,0.26600000
0.05824010,0.00000000,0.60333333,0.00000000,0.53200000,0.62420000,0.64700000,0.17121000,0.80000000,0.83250000,0.67333333,0.99225000,0.26850000,0.46000000
0.03673670,0.00000000,0.60333333,0.00000000,0.58300000,0.63120000,0.51900000,0.19958500,0.80000000,0.83250000,0.67333333,0.97155000,0.26450000,0.42400000
0.00207460,0.00000000,0.92466667,0.00000000,0.60900000,0.50930000,0.98000000,0.09113000,0.13333333,0.88875000,0.67000000,0.79607500,0.74200000,0.16200000
0.00178990,0.00000000,0.32300000,0.00000000,0.58500000,0.56700000,0.28800000,0.13993000,0.20000000,0.48875000,0.64000000,0.98322500,0.44000000,0.46200000
0.00224380,0.00000000,0.32300000,0.00000000,0.58500000,0.60270000,0.79700000,0.12491000,0.20000000,0.48875000,0.64000000,0.99225000,0.35825000,0.33600000
0.00062630,0.00000000,0.39766667,0.00000000,0.57300000,0.65930000,0.69100000,0.12393000,0.03333333,0.34125000,0.70000000,0.97997500,0.24175000,0.44800000
0.00045270,0.00000000,0.39766667,0.00000000,0.57300000,0.61200000,0.76700000,0.11437500,0.03333333,0.34125000,0.70000000,0.99225000,0.22700000,0.41200000
0.00060760,0.00000000,0.39766667,0.00000000,0.57300000,0.69760000,0.91000000,0.10837500,0.03333333,0.34125000,0.70000000,0.99225000,0.14100000,0.47800000
0.00109590,0.00000000,0.39766667,0.00000000,0.57300000,0.67940000,0.89300000,0.11944500,0.03333333,0.34125000,0.70000000,0.98362500,0.16200000,0.44000000
0.00047410,0.00000000,0.39766667,0.00000000,0.57300000,0.60300000,0.80800000,0.12525000,0.03333333,0.34125000,0.70000000,0.99225000,0.19700000,0.23800000
```
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