什么是线性关系？

What is a linear relationship?

<!-- @@@ -->

A linear relationship (or linear association) is a statistical term used to describe a straight-line relationship between two variables. A linear relationship can be expressed graphically, where the variable and constant are connected by a straight line, or mathematically, where the independent variable is multiplied by the slope coefficient, plus a constant, to determine the dependent variable.

Linear relationships can be contrasted with polynomial or nonlinear (curvilinear) relationships.

Key Points

• A linear relationship (or linear association) is a statistical term used to describe a straight-line relationship between two variables.
• A linear relationship can be represented graphically or as a mathematical equation of the form y = mx + b.
• Linear relationships are quite common in daily life.

Formula for linear relationship

Mathematically, a linear relationship satisfies the following equation:

<!-- @@@ -->

In this equation, "x" and "y" are two variables related by the parameters "m" and "b". Graphically, y = mx + b is plotted on the x-y plane as a straight line with slope "m" and y-intercept "b". The y-intercept "b" is the value of "y" when x=0. The slope "m" can be calculated from any two points (x1, y1) and (x2, y2):

<!-- @@@ -->

What does the linear relationship tell you?

To be legitimately defined as a linear relationship, an equation must meet three necessary sets of criteria: An equation expressing a linear relationship should not contain more than two variables, all variables in the equation must be of first order, and the graph of the equation must be a straight line.

A commonly used linear relationship is correlation, which describes the extent to which one variable changes linearly as another variable changes.

In econometrics, linear regression is a common method for generating linear relationships to explain various phenomena. It is often used to extrapolate future predictions from past events. However, not all relationships are linear. Some data describe curvilinear relationships (such as polynomial relationships), and other data cannot be parameterized.

Linear function

In mathematics, similar to a linear relationship is the concept of a linear function. In the case of one variable, a linear function can be written as:

<!-- @@@ -->

This is the same as the given linear relationship formula, except that f(x) is used instead of y. This substitution is to highlight the implication that x maps to f(x), while using y simply indicates that x and y are two quantities related through A and B.

<!-- @@@ -->

Example of linear relationship

Linear relationships are quite common in daily life. Taking the concept of speed as an example, the formula we use to calculate speed is as follows: Speed ​​rate is the ratio of distance to time. If a traveler travels 44.1 miles on California's Highway 99 from Sacramento to Marysville and the entire trip takes approximately 45 minutes, they are traveling nearly 60 mph.

Although there are multiple variables in this equation, it is still a linear equation because one of the variables is always a constant (distance).

A linear relationship can also be found in the equation distance = speed x time. Because distance is usually a positive number, this linear relationship will be represented in the upper right quadrant of the linear graph.

If a tandem travels at 30 miles per hour for 20 hours, the rider will have covered 600 miles. Graphically, distance is the Y-axis and time is the X-axis, so a line tracking distance over 20 hours would extend outward from the intersection of the X and Y axes.

When converting Celsius to Fahrenheit, or Fahrenheit to Celsius, you can use the following equations. These equations appear as linear relationships on the graph:

<!-- @@@ --> <!-- @@@ -->

Assume that the independent variable is the area of the house (in square feet), and the relationship between it and the market price of the house (the dependent variable) is: the area of the house multiplied by the slope coefficient 207.65, plus a constant

<!-- @@@ -->

10,500 = $270,062.50. Graphically and mathematically, it looks like this:

In this example, as the size of the house increases, the market value of the house increases linearly.

Linear relationships between certain objects can be called "proportional relationships." The relationship is expressed as:

<!-- @@@ -->

When analyzing behavioral data, there is rarely a perfectly linear relationship between variables. However, it is often possible to find trend lines in the data that form rough linear relationships. For example, you could observe the daily sales of ice cream and the daily maximum temperature as two variables that interact in a chart and find a rough linear relationship between the two.

What is a positive linear relationship?

A positive linear relationship appears as an upward straight line on a graph. This means that if one variable increases, the other variable will also increase. In contrast, a negative linear relationship appears as a downward straight line on the graph; if one variable increases, the other decreases.

What is a non-linear relationship?

A non-linear relationship can be represented by a scatter plot seen on a chart, which follows a pattern, but the pattern is not linear, meaning the representation is not a straight line.

Give an example of a linear relationship in statistics?

An example of a linear relationship is workers who are paid hourly wages. The more hours they work, the more money they make. The relationship is linear because each additional hour worked will correspond to an equivalent increase in income.

Summary

A linear relationship in statistics shows a straight line relationship between two variables. It usually shows the correlation between two variables. Although no perfect behavioral relationship yields a true linear relationship, trends can often be found in the data to hypothesize their existence.

About TheQuantHackers

TheQuantHackers is a cutting-edge community composed of a group of people from the world's top universities and quantitative finance practitioners, dedicated to exploring the infinite possibilities in the fields of artificial intelligence (AI) and quantification (Quant). Our team members come from world-renowned universities such as the University of Cambridge, Oxford University, Harvard University, ETH Zurich, Peking University, and the University of Science and Technology of China. Our external consultants come from first-class companies such as Microsoft, HSBC, Citadel, Man Group, Citi, Jump Trading, and top domestic private equity companies.$\begin{aligned} &y = mx + b \\ &\textbf{其中:}\\ &m=\text{斜率}\\ &b=\text{y轴截距}\\ \end{aligned}$

在这个方程中，“x”和“y”是由参数“m”和“b”相关联的两个变量。从图形上看，y = mx + b 在 x-y 平面上绘制为与斜率“m”和y轴截距“b”的直线。y轴截距“b”是当 x=0 时“y”的值。斜率“m”可以通过任意两个点 (x1, y1) 和 (x2, y2) 计算得出：

$m = \frac{(y_2 - y_1)}{(x_2 - x_1)}$

线性关系（或线性关联）是一个统计术语，用于描述两个变量之间的直线关系。线性关系可以以图形形式表达，变量和常数通过一条直线相连，或者以数学形式表达，其中自变量乘以斜率系数，再加上一个常数，从而确定因变量。

线性关系可以与多项式或非线性（曲线）关系相对比。

关键要点

• 线性关系（或线性关联）是一个统计术语，用于描述两个变量之间的直线关系。
• 线性关系可以以图形形式或数学方程的形式表示，形式为 y = mx + b。
• 线性关系在日常生活中相当常见。

线性关系的公式

在数学上，线性关系满足以下方程：

$\begin{aligned} &f(x) = mx + b \\ &\textbf{其中:}\\ &m=\text{斜率}\\ &b=\text{y轴截距}\\ \end{aligned}$

这与给定的线性关系公式相同，只是用 f(x) 替代了 y。这个替代是为了突出 x 映射到 f(x) 的含义，而使用 y 仅仅表明 x 和 y 是两个通过 A 和 B 相关的量。

$c \times f(A +B) = c \times f(A) + c \times f(B)$

在这个方程中，“x”和“y”是由参数“m”和“b”相关联的两个变量。从图形上看，y = mx + b 在 x-y 平面上绘制为与斜率“m”和y轴截距“b”的直线。y轴截距“b”是当 x=0 时“y”的值。斜率“m”可以通过任意两个点 (x1, y1) 和 (x2, y2) 计算得出：

$\degree C = \frac{5}{9}(\degree F - 32)$

$\degree F = \frac{9}{5}\degree C + 32$

线性关系告诉你什么？

要合理定义为线性关系，方程必须满足三组必要的标准：表达线性关系的方程不应包含超过两个变量，方程中的所有变量必须都是一阶的，并且该方程的图形必须为一条直线。

一个常用的线性关系是相关性，它描述了一个变量随着另一个变量变化而线性变化的程度。

在计量经济学中，线性回归是一种生成线性关系以解释各种现象的常用方法。它通常用于从过去的事件推断未来的预测。然而，并非所有关系都是线性的。有些数据描述的是曲线关系（例如多项式关系），还有一些数据则无法参数化。

线性函数

在数学上，与线性关系相似的是线性函数的概念。在一个变量的情况下，线性函数可以写成：

$10,500。如果一所房屋的面积为1250平方英尺，则其市场价值为 (1,250 x 207.65) +$10,500 = $270,062.50。在图形和数学上，它看起来如下：

在这个例子中，随着房屋面积的增加，房屋的市场价值线性增加。

某些对象之间的线性关系可以称为“成比例关系”。该关系表现为：

$\begin{aligned} &Y = k \times X \\ &\textbf{其中:}\\ &k=\text{常数}\\ &Y, X=\text{成比例的量}\\ \end{aligned}$

这与给定的线性关系公式相同，只是用 f(x) 替代了 y。这个替代是为了突出 x 映射到 f(x) 的含义，而使用 y 仅仅表明 x 和 y 是两个通过 A 和 B 相关的量。