# Normalization and Standardization ¶

Learning come to slow case, and prevent to use normalization and standardization.

Basically, Stochastic Gradient Descent method is used for minimizing the loss function, but when we don't use normalization or standrdization, it is possible that gradient direction is not appropriate.
In such a case, weight does not update towards the direction that we would like to.
But, nomarlization and standardization is one of the method that prevent the inappropriate update, and update the correct in terms of minimizing the loss function direction.
In practical, sometimes each feature have different scale, for example, one is kilogram[kg], another is miligram[mg].
In this time, we see the case that learning can be slow and effect of the normalization and standardization as bellow.

So, in this time, we use the standardization and see the how fast the loss decreases when we use the standardization.

## Required Libaries ¶

• matplotlib 2.0.2
• numpy 1.12.1
• scikit-learn 0.18.2
• pandas 0.20.3
In [1]:

from __future__ import division, print_function
import numpy as np
import pandas as pd

import renom as rm
from renom.cuda import set_cuda_active
from sklearn.preprocessing import LabelBinarizer, label_binarize
from sklearn.metrics import confusion_matrix, classification_report
from sklearn.model_selection import train_test_split
import matplotlib.pyplot as plt
# If you would like to use GPU computing, set this to True. Otherwise, False.
set_cuda_active(False)


## Make Data ¶

The reference of the dataset is below.

MAGIC Gamma Telescope Data Set, R. K. Bock Major Atmospheric Gamma Imaging Cherenkov Telescope project (MAGIC),
P. Savicky Institute of Computer Science, AS of CR Czech Republic.
In [2]:

df = pd.read_csv("magic04.data",header=None)
X = df.drop(10,axis=1).values.astype(float)
y = df.iloc[:,10].replace("g",1).replace("h",0).values.astype(float).reshape(-1,1)
print("X:{} y:{}".format(X.shape, y.shape))

X:(19020, 10) y:(19020, 1)

In [3]:

X_train, X_test, y_train, y_test = train_test_split(X,y,test_size=0.2)
print("X_train:{} y_train:{} X_test:{} y_test:{}".format(X_train.shape, y_train.shape, X_test.shape, y_test.shape))

X_train:(15216, 10) y_train:(15216, 1) X_test:(3804, 10) y_test:(3804, 1)


## No standardization to the input data ¶

In [4]:

sequential = rm.Sequential([
rm.Dense(64),
rm.Relu(),
rm.Dense(32),
rm.Relu(),
rm.Dense(1)
])

In [5]:

batch_size = 128
epoch = 15
N = len(X_train)
optimizer = Sgd(lr=0.01)
learning_curve = []
test_learning_curve = []

for i in range(epoch):
perm = np.random.permutation(N)
loss = 0
for j in range(0, N // batch_size):
train_batch = X_train[perm[j*batch_size : (j+1)*batch_size]]
response_batch = y_train[perm[j*batch_size : (j+1)*batch_size]]

with sequential.train():
z = sequential(train_batch)
l = rm.sigmoid_cross_entropy(z, response_batch)
loss += l.as_ndarray()
train_loss = loss / (N // batch_size)
z_test = sequential(X_test)
test_loss = rm.sigmoid_cross_entropy(z_test, y_test).as_ndarray()
test_learning_curve.append(test_loss)
learning_curve.append(train_loss)
print("epoch:{:03d}, train_loss:{:.4f}, test_loss:{:.4f}".format(i, float(train_loss), float(test_loss)))

epoch:000, train_loss:0.6906, test_loss:0.5011
epoch:001, train_loss:0.4930, test_loss:0.4914
epoch:002, train_loss:0.4805, test_loss:0.4932
epoch:003, train_loss:0.4707, test_loss:0.4717
epoch:004, train_loss:0.4669, test_loss:0.4788
epoch:005, train_loss:0.4612, test_loss:0.4841
epoch:006, train_loss:0.4542, test_loss:0.4646
epoch:007, train_loss:0.4510, test_loss:0.4687
epoch:008, train_loss:0.4503, test_loss:0.4518
epoch:009, train_loss:0.4425, test_loss:0.4446
epoch:010, train_loss:0.4415, test_loss:0.4811
epoch:011, train_loss:0.4392, test_loss:0.4473
epoch:012, train_loss:0.4357, test_loss:0.4441
epoch:013, train_loss:0.4341, test_loss:0.4414
epoch:014, train_loss:0.4351, test_loss:0.4362


## Standardization to the input data ¶

We change the input data to mean 0 shifting and unit variance scaling as bellow.

$$X\_new = \frac{X\_old - X\_mean}{X\_std}$$
In [6]:

X_train_mean = np.mean(X_train,axis=0)
X_train_std = np.std(X_train, axis=0)
X_train = (X_train - X_train_mean) / X_train_std

X_test = (X_test - X_train_mean) / X_train_std
print(X_train)

[[-0.58201867 -0.1658305  -0.18455665 ..., -0.59053373  2.37721221
-1.23456824]
[ 0.13326953  0.22006933  1.50925445 ..., -0.84979314 -0.98602078
0.17031323]
[-0.75331084 -0.42985235 -0.35030862 ..., -0.16949744 -1.02206549
0.58497995]
...,
[-0.78272801 -0.80924514 -1.47088532 ...,  0.2453504  -0.67826979
-1.19999152]
[-0.15077038  0.77937076  1.84861092 ...,  0.67049787 -0.88432171
-1.11888926]
[-0.76934393 -0.36479821 -0.26011069 ..., -0.43940831 -0.01727761
-1.95602149]]

In [7]:

sequential = rm.Sequential([
rm.Dense(64),
rm.Relu(),
rm.Dense(32),
rm.Relu(),
rm.Dense(1)
])

In [8]:

batch_size = 128
epoch = 15
N = len(X_train)
optimizer = Sgd(lr=0.01)
learning_curve = []
test_learning_curve = []

for i in range(epoch):
perm = np.random.permutation(N)
loss = 0
for j in range(0, N // batch_size):
train_batch = X_train[perm[j*batch_size : (j+1)*batch_size]]
response_batch = y_train[perm[j*batch_size : (j+1)*batch_size]]

with sequential.train():
z = sequential(train_batch)
l = rm.sigmoid_cross_entropy(z, response_batch)
loss += l.as_ndarray()
train_loss = loss / (N // batch_size)
z_test = sequential(X_test)
test_loss = rm.sigmoid_cross_entropy(z_test, y_test).as_ndarray()
test_learning_curve.append(test_loss)
learning_curve.append(train_loss)
print("epoch:{:03d}, train_loss:{:.4f}, test_loss:{:.4f}".format(i, float(train_loss), float(test_loss)))

epoch:000, train_loss:0.6376, test_loss:0.5432
epoch:001, train_loss:0.5090, test_loss:0.4764
epoch:002, train_loss:0.4617, test_loss:0.4439
epoch:003, train_loss:0.4365, test_loss:0.4244
epoch:004, train_loss:0.4204, test_loss:0.4116
epoch:005, train_loss:0.4090, test_loss:0.4021
epoch:006, train_loss:0.3999, test_loss:0.3945
epoch:007, train_loss:0.3930, test_loss:0.3881
epoch:008, train_loss:0.3872, test_loss:0.3831
epoch:009, train_loss:0.3817, test_loss:0.3784
epoch:010, train_loss:0.3761, test_loss:0.3747
epoch:011, train_loss:0.3717, test_loss:0.3709
epoch:012, train_loss:0.3687, test_loss:0.3675
epoch:013, train_loss:0.3648, test_loss:0.3647
epoch:014, train_loss:0.3614, test_loss:0.3618

Compare to the two case, when we use the standardization, the loss value decreases little faster.
It might standardization change the shape of loss function from elongated to like circle, so the we can avoid the learning go slow.