【机器学习】线性回归

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 npimport matplotlib.pyplot as pltplt.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 pointsplt.scatter(x_train, y_train, marker='x', c='r')# Set the titleplt.title("Housing Prices")# Set the y-axis labelplt.ylabel('Price (in 1000s of dollars)')# Set the x-axis labelplt.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 = 100b = 100tmp_f_wb = compute_model_output(x_train, w, b,)# Plot our model predictionplt.plot(x_train, tmp_f_wb, c='b',label='Our Prediction')# Plot the data pointsplt.scatter(x_train, y_train, marker='x', c='r',label='Actual Values')# Set the titleplt.title("Housing Prices")# Set the y-axis labelplt.ylabel('Price (in 1000s of dollars)')# Set the x-axis labelplt.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 = 1.2cost_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：

# see https://matplotlib.org/stable/tutorials/introductory/customizing.htmllines.linewidth: 4lines.solid_capstyle: buttlegend.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 7030A0axes.prop_cycle: cycler('color', ['0096FF', 'FF9300', 'FF40FF', '7030A0', 'C00000'])#axes.facecolor: f0f0f0 # greyaxes.facecolor: ffffff  # whiteaxes.labelsize: largeaxes.axisbelow: trueaxes.grid: Falseaxes.edgecolor: f0f0f0axes.linewidth: 3.0axes.titlesize: x-largepatch.edgecolor: f0f0f0patch.linewidth: 0.5svg.fonttype: pathgrid.linestyle: -grid.linewidth: 1.0grid.color: cbcbcbxtick.major.size: 0xtick.minor.size: 0ytick.major.size: 0ytick.minor.size: 0savefig.edgecolor: f0f0f0savefig.facecolor: f0f0f0#figure.subplot.left: 0.08#figure.subplot.right: 0.95#figure.subplot.bottom: 0.07#figure.facecolor: f0f0f0  # greyfigure.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 smallerfont.family:  sans-seriffont.style:   normalfont.variant: normalfont.weight:  normalfont.stretch: normalfont.size:    8.0font.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, seriffont.sans-serif: Verdana, DejaVu Sans, Bitstream Vera Sans, Computer Modern Sans Serif, Lucida Grande, Geneva, Lucid, Arial, Helvetica, Avant Garde, sans-seriffont.cursive:    Apple Chancery, Textile, Zapf Chancery, Sand, Script MT, Felipa, Comic Neue, Comic Sans MS, cursivefont.fantasy:    Chicago, Charcoal, Impact, Western, Humor Sans, xkcd, fantasyfont.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

来源地址：https://blog.csdn.net/weixin_47936614/article/details/131994860