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[회귀분석] 선형회귀 구현

선형회귀 수식 구현하기. 수업에서 쪽지시험으로 진행했던 문제

문제

  1. read the data.txt and define first column as \(t\), second column as \(v\), third column as \(e\).

  2. use following formulas below to create python def LineFit(x, y) which defines \(y = b + Wx\).

    \[ \begin{align*} \overline{x} & = \frac{1}{n} \sum_{i=1}^{n}x_{i} \\ \\ \overline{y} & = \frac{1}{n} \sum_{i=1}^{n}y_{i} \\ \\ W & = \frac{\sum_{i=1}^{n}(x_{i} - \overline{x})y_{i}}{\sum_{i=1}^{n}(x_{i} - \overline{x})x_{i}} \\ \\ b & = \overline{y} - W\overline{x} \\ \\ \end{align*} \]

  3. in case of \(v = v_{0} + gt\), find values of \(v_{0}\), \(g\).

  4. plotting linear regression of \(t\) and \(v\).
data.txt
Velocity vs time data for a falling mass
time (s) velocity (m/s) uncertainty (m/s)
2.23 139 16
4.78 123 16
7.21 115 4
9.37 96 9
11.64 62 17
14.23 54 17
16.55 10 12
18.70 -3 15
21.05 -13 18
23.21 -55 10

풀이

1번 문제 풀이

import numpy as np
import matplotlib.pyplot as plt

data = np.loadtxt('./data.txt', skiprows=2)

t = data[:, 0]
v = data[:, 1]
e = data[:, 2]

2번 문제 풀이

numeric = list[int | float]


def bar(data: numeric) -> float:
    """returns expectation/sample mean"""

    return sum(data) / len(data)


def lineFit(x: numeric, y: numeric) -> tuple:
    """returns linear regression coefficient(weight) and intercept of two random variables"""

    x_bar, y_bar = bar(x), bar(y)
    w = sum((x - x_bar) * y for x, y in zip(x, y)) / sum((x - x_bar) * x for x in x)
    i = y_bar - (w * x_bar)
    return i, w

3번 문제 풀이

v0, g = lineFit(t, v)

print(f'v0 = {v0}\ng = {g}')

v0 = 171.89273556960842
g = -9.23414248039144

4번 문제 풀이

exog = np.linspace(0, 25, 251)
endog = v0 + (g * exog)

fig, axes = plt.subplots()

axes.scatter(
    x=t,
    y=v,
    label='data',
)

axes.plot(
    exog,
    endog,
    label='Regression',
    color='red',
)

axes.set_xlabel('Time (s)')
axes.set_ylabel('Velocity (m/s)')

axes.text(
    x=12,
    y=170,
    s='Regression Plotting',
    ha='center',
    va='top',
)

axes.set_title(
    label='Velocity vs time data for a falling mass',
    weight='heavy',
)

axes.axhline(
    color='black',
    linewidth=1.5,
    zorder=-1,
)

axes.legend()
plt.show()

simple_linear_regression