【机器学习】Linear Regression

1、问题描述

一套 1000 平方英尺 (sqft) 的房屋售价为300,000美元，一套 2000 平方英尺的房屋售价为500,000美元。这两点将构成我们的数据或训练集。面积单位为 1000 平方英尺，价格单位为 1000 美元。

希望通过这两个点拟合线性回归模型，以便可以预测其他房屋的价格。例如，面积为 1200 平方英尺的房屋价格是多少。

首先导入所需要的库

import numpy as np
import matplotlib.pyplot as plt
plt.style.use('./deeplearning.mplstyle')

以下代码来创建x_train和y_train变量。数据存储在一维 NumPy 数组中。

# x_train is the input variable (size in 1000 square feet)
# y_train is the target (price in 1000s of dollars)
x_train = np.array([1.0, 2.0])
y_train = np.array([300.0, 500.0])
print(f"x_train = {x_train}")
print(f"y_train = {y_train}")

2、表示说明

使用 m 来表示训练样本的数量。 (x${}^{(i)}$, y${}^{(i)}$) 表示第 i 个训练样本。由于 Python 是零索引的，(x${}^{(0)}$, y${}^{(0)}$) 是 (1.0, 300.0) ， (x${}^{(1)}$, y${}^{(1)}$) 是 (2.0, 500.0).

3、数据绘图

使用 matplotlib 库中的scatter()函数绘制这两个点。 其中，函数参数marker 和 c 将点显示为红叉（默认为蓝点）。使用matplotlib库中的其他函数来设置要显示的标题和标签。

# Plot the data points
plt.scatter(x_train, y_train, marker='x', c='r')
# Set the title
plt.title("Housing Prices")
# Set the y-axis label
plt.ylabel('Price (in 1000s of dollars)')
# Set the x-axis label
plt.xlabel('Size (1000 sqft)')
plt.show()

4、模型函数

线性回归的模型函数（这是一个从 x 映射到 y 的函数）可以表示为 $$f_{w,b}(x^{(i)})=w{x}^{(i)}+b\text{\hspace{1em}(1)}$$

计算$f_{w,b}(x^{(i)})$ 的值，可以将每个数据点显示地写为：

对于 $x^{(0)}$, f_wb = w * x[0] + b
对于 $x^{(1)}$, f_wb = w * x[1] + b

对于大量的数据点，这可能会变得笨拙且重复。 因此，可以在for 循环中计算输出，如下面的函数compute_model_output 所示。

def compute_model_output(x, w, b):
    """
    Computes the prediction of a linear model
    Args:
      x (ndarray (m,)): Data, m examples 
      w,b (scalar)    : model parameters  
    Returns
      y (ndarray (m,)): target values
    """
    m = x.shape[0]
    f_wb = np.zeros(m)
    for i in range(m):
        f_wb[i] = w * x[i] + b
        
    return f_wb

调用 compute_model_output 函数并绘制输出

w = 100
b = 100

tmp_f_wb = compute_model_output(x_train, w, b,)

# Plot our model prediction
plt.plot(x_train, tmp_f_wb, c='b',label='Our Prediction')

# Plot the data points
plt.scatter(x_train, y_train, marker='x', c='r',label='Actual Values')

# Set the title
plt.title("Housing Prices")
# Set the y-axis label
plt.ylabel('Price (in 1000s of dollars)')
# Set the x-axis label
plt.xlabel('Size (1000 sqft)')
plt.legend()
plt.show()

很明显， $w=100$ 和 $b=100$ 不会产生适合数据的直线。

根据学过的数学知识，可以容易求出 $w=200$ 和 $b=100$

5、预测

现在我们已经有了一个模型，可以用它来做出房屋价格的预测。来预测一下 1200 平方英尺的房子的价格。由于面积单位为 1000 平方英尺，所以 $x$ 是1.2。

w = 200                         
b = 100    
x_i = 1.2
cost_1200sqft = w * x_i + b    

print(f"${cost_1200sqft:.0f} thousand dollars")

输出的结果是：$340 thousand dollars

总结

• 线性回归建立一个特征和目标之间关系的模型
  • 在上面的例子中，特征是房屋面积，目标是房价。
  • 对于简单线性回归，模型有两个参数 $w$ 和 $b$ ，其值使用训练数据进行拟合。
  • 一旦确定了模型的参数，该模型就可以用于对新数据进行预测。

附录

deeplearning.mplstyle 源码：文章来源地址https://www.toymoban.com/news/detail-624117.html

# see https://matplotlib.org/stable/tutorials/introductory/customizing.html
lines.linewidth: 4
lines.solid_capstyle: butt

legend.fancybox: true

# Verdana" for non-math text,
# Cambria Math

#Blue (Crayon-Aqua) 0096FF
#Dark Red C00000
#Orange (Apple Orange) FF9300
#Black 000000
#Magenta FF40FF
#Purple 7030A0

axes.prop_cycle: cycler('color', ['0096FF', 'FF9300', 'FF40FF', '7030A0', 'C00000'])
#axes.facecolor: f0f0f0 # grey
axes.facecolor: ffffff  # white
axes.labelsize: large
axes.axisbelow: true
axes.grid: False
axes.edgecolor: f0f0f0
axes.linewidth: 3.0
axes.titlesize: x-large

patch.edgecolor: f0f0f0
patch.linewidth: 0.5

svg.fonttype: path

grid.linestyle: -
grid.linewidth: 1.0
grid.color: cbcbcb

xtick.major.size: 0
xtick.minor.size: 0
ytick.major.size: 0
ytick.minor.size: 0

savefig.edgecolor: f0f0f0
savefig.facecolor: f0f0f0

#figure.subplot.left: 0.08
#figure.subplot.right: 0.95
#figure.subplot.bottom: 0.07

#figure.facecolor: f0f0f0  # grey
figure.facecolor: ffffff  # white

## ***************************************************************************
## * FONT                                                                    *
## ***************************************************************************
## The font properties used by `text.Text`.
## See https://matplotlib.org/api/font_manager_api.html for more information
## on font properties.  The 6 font properties used for font matching are
## given below with their default values.
##
## The font.family property can take either a concrete font name (not supported
## when rendering text with usetex), or one of the following five generic
## values:
##     - 'serif' (e.g., Times),
##     - 'sans-serif' (e.g., Helvetica),
##     - 'cursive' (e.g., Zapf-Chancery),
##     - 'fantasy' (e.g., Western), and
##     - 'monospace' (e.g., Courier).
## Each of these values has a corresponding default list of font names
## (font.serif, etc.); the first available font in the list is used.  Note that
## for font.serif, font.sans-serif, and font.monospace, the first element of
## the list (a DejaVu font) will always be used because DejaVu is shipped with
## Matplotlib and is thus guaranteed to be available; the other entries are
## left as examples of other possible values.
##
## The font.style property has three values: normal (or roman), italic
## or oblique.  The oblique style will be used for italic, if it is not
## present.
##
## The font.variant property has two values: normal or small-caps.  For
## TrueType fonts, which are scalable fonts, small-caps is equivalent
## to using a font size of 'smaller', or about 83%% of the current font
## size.
##
## The font.weight property has effectively 13 values: normal, bold,
## bolder, lighter, 100, 200, 300, ..., 900.  Normal is the same as
## 400, and bold is 700.  bolder and lighter are relative values with
## respect to the current weight.
##
## The font.stretch property has 11 values: ultra-condensed,
## extra-condensed, condensed, semi-condensed, normal, semi-expanded,
## expanded, extra-expanded, ultra-expanded, wider, and narrower.  This
## property is not currently implemented.
##
## The font.size property is the default font size for text, given in points.
## 10 pt is the standard value.
##
## Note that font.size controls default text sizes.  To configure
## special text sizes tick labels, axes, labels, title, etc., see the rc
## settings for axes and ticks.  Special text sizes can be defined
## relative to font.size, using the following values: xx-small, x-small,
## small, medium, large, x-large, xx-large, larger, or smaller


font.family:  sans-serif
font.style:   normal
font.variant: normal
font.weight:  normal
font.stretch: normal
font.size:    8.0

font.serif:      DejaVu Serif, Bitstream Vera Serif, Computer Modern Roman, New Century Schoolbook, Century Schoolbook L, Utopia, ITC Bookman, Bookman, Nimbus Roman No9 L, Times New Roman, Times, Palatino, Charter, serif
font.sans-serif: Verdana, DejaVu Sans, Bitstream Vera Sans, Computer Modern Sans Serif, Lucida Grande, Geneva, Lucid, Arial, Helvetica, Avant Garde, sans-serif
font.cursive:    Apple Chancery, Textile, Zapf Chancery, Sand, Script MT, Felipa, Comic Neue, Comic Sans MS, cursive
font.fantasy:    Chicago, Charcoal, Impact, Western, Humor Sans, xkcd, fantasy
font.monospace:  DejaVu Sans Mono, Bitstream Vera Sans Mono, Computer Modern Typewriter, Andale Mono, Nimbus Mono L, Courier New, Courier, Fixed, Terminal, monospace


## ***************************************************************************
## * TEXT                                                                    *
## ***************************************************************************
## The text properties used by `text.Text`.
## See https://matplotlib.org/api/artist_api.html#module-matplotlib.text
## for more information on text properties
#text.color: black

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