지도학습-비선형회귀

Introduction to Machine Learning with Python¶

Chapter 2. 지도학습¶

---

비선형 회귀¶

In [2]:

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from sklearn.model_selection import train_test_split

In [3]:

from sklearn.datasets import load_breast_cancer

cancer = load_breast_cancer()

In [4]:

col1 = 0
col2 = 6

plt.scatter(cancer.data[:,col1], cancer.data[:,col2], c= cancer.target, alpha=0.3)

Out[4]:

<matplotlib.collections.PathCollection at 0x213eec4f9e8>

선형회귀¶

In [5]:

from sklearn.linear_model import LinearRegression

model = LinearRegression()
model.fit(cancer.data[:,[col1]], cancer.data[:,col2])

model.coef_, model.intercept_

Out[5]:

(array([0.01530951]), -0.1274825984768875)

In [6]:

xs = np.arange(5,30,0.1)
ys = xs*model.coef_[0] + model.intercept_

plt.scatter(cancer.data[:,col1], cancer.data[:,col2], c= cancer.target, alpha=0.3)
plt.plot(xs,ys,'r-',lw=3)

Out[6]:

[<matplotlib.lines.Line2D at 0x16cccd074a8>]

2차식 회귀¶

In [11]:

X = np.c_[cancer.data[:,col1], (cancer.data[:,col1])**2]
y = cancer.data[:,col2]

model = LinearRegression()
model.fit(X, y)

model.coef_, model.intercept_

Out[11]:

(array([-0.00546943,  0.00065831]), 0.026519913885121926)

In [14]:

xs = np.arange(0,30,0.1)
ys = xs*model.coef_[0] + (xs**2)*model.coef_[1] + model.intercept_

plt.scatter(cancer.data[:,col1], cancer.data[:,col2], c= cancer.target, alpha=0.3)
plt.plot(xs,ys,'r-',lw=3)

Out[14]:

[<matplotlib.lines.Line2D at 0x16ccce7ada0>]

3차식 회귀¶

In [6]:

X = np.c_[cancer.data[:,col1], (cancer.data[:,col1])**2, (cancer.data[:,col1])**3]
y = cancer.data[:,col2]

model = LinearRegression()
model.fit(X, y)

model.coef_, model.intercept_

Out[6]:

(array([-5.53456968e-02,  3.81149612e-03, -6.31326376e-05]),
 0.27649793237737497)

In [8]:

xs = np.arange(-10,50,0.1)
ys = xs*model.coef_[0] + (xs**2)*model.coef_[1] + (xs**3)*model.coef_[2] + model.intercept_

plt.scatter(cancer.data[:,col1], cancer.data[:,col2], c= cancer.target, alpha=0.3)
plt.plot(xs,ys,'r-',lw=3)

Out[8]:

[<matplotlib.lines.Line2D at 0x213f0de64e0>]

지수함수 회귀 (Exponential Regression)¶

$$ y = exp(ax + b) $$ $$ log(y) = ax + b $$

In [17]:

X = cancer.data[:,[col1]]
y = np.log(cancer.data[:,col2]+0.01) # 값이 0인 데이터가 있어 0.01 을 더해줌

model = LinearRegression()
model.fit(X, y)

model.coef_, model.intercept_

Out[17]:

(array([0.1579534]), -4.875933569557782)

In [18]:

xs = np.arange(0,30,0.1)
ys = np.exp(xs*model.coef_[0] + model.intercept_) - 0.01

plt.scatter(cancer.data[:,col1], cancer.data[:,col2], c= cancer.target, alpha=0.3)
plt.plot(xs,ys,'r-',lw=3)

Out[18]:

[<matplotlib.lines.Line2D at 0x16cccf7e208>]

멱함수 회귀 (Power Law Regression)¶

$$ y = a \cdot x^n $$ $$ log(y) = n \cdot log(x) + log(a) $$

In [20]:

X = np.log(cancer.data[:,[col1]])
y = np.log(cancer.data[:,col2]+0.01) # 값이 0인 데이터가 있어 0.01 을 더해줌

model = LinearRegression()
model.fit(X, y)

model.coef_, model.intercept_

Out[20]:

(array([2.321263]), -8.724171466884766)

In [21]:

xs = np.arange(0,30,0.1)
ys = np.exp(model.intercept_) * (xs**model.coef_[0]) - 0.01

plt.scatter(cancer.data[:,col1], cancer.data[:,col2], c= cancer.target, alpha=0.3)
plt.plot(xs,ys,'r-',lw=3)

Out[21]:

[<matplotlib.lines.Line2D at 0x16cccfc5908>]

멱함수와 복잡계¶

• 샘플을 만들어 적용해 보자 (n = -3/4 인 경우)
• 지진강도 대비 어떤 지역의 지진이 일어나는 횟수로 생각해 보자

In [10]:

xs = np.random.uniform(0.1, 2, size=100)
ys = 10 * (xs**(-3/4)) + np.random.normal(0, 0.5, size=len(xs)) # n=-3/4, a = 10

In [11]:

plt.scatter(xs, ys, alpha=0.3)
plt.title('Earthquake report', fontsize=30)
plt.xlabel('Power')
plt.ylabel('Count')

Out[11]:

Text(0,0.5,'Count')

In [13]:

plt.scatter(np.log(xs), np.log(ys))
plt.axis('equal')

Out[13]:

(-2.4304567152063568,
 0.8418633354476694,
 1.5257218394123955,
 4.134815453302616)

In [14]:

X = np.log(xs).reshape(-1,1)
y = np.log(ys)

model = LinearRegression()
model.fit(X, y)

model.coef_, model.intercept_ # n = model.coef_[0], a = exp(model.intercept_)

Out[14]:

(array([-0.75247268]), 2.3058752643130114)

In [68]:

input_x = np.arange(0.1,2.2,0.05)
pred_y = np.exp(model.intercept_) * (input_x**model.coef_[0])

plt.scatter(xs, ys, alpha=0.3)
plt.plot(input_x,pred_y,'r-')

Out[68]:

[<matplotlib.lines.Line2D at 0x16cce8f8d30>]