White noise are variations in your data that cannot be explained by any regression model.

白噪声是数据中的变化，任何回归模型都无法解释。

And yet, there happens to be a statistical model for white noise. It goes like this for time series data:

然而，碰巧有一个白噪声统计模型。 时间序列数据如下所示：

The observed value Y_i at time step i is the sum of the current level L_i and a random component N_i around the current level.

在时间步长i处的观测值Y_i是当前水平L_i与当前水平附近的随机分量N_i之和。

If the extent of random variation is proportional to the current level, then we have the following multiplicative version of the same model:

如果随机变化的程度与当前水平成正比，那么我们可以得到同一模型的以下乘性形式：

If the current level L_i is constant for all i, i.e. L_i = L for all i, then the noise will be seen to fluctuate around a fixed level.

如果当前水平L_i对于所有i都是恒定的，即L_i = L对于所有i ，则将看到噪声围绕固定水平波动。

It’s easy to generate a white noise data set. Here’s how to do it in Excel:

生成白噪声数据集很容易。 这是在Excel中执行的方法：

And here is the output plot of noise that is fluctuating around a constant level of 100:

这是噪声的输出图，它在100的恒定水平附近波动：

The current level L_i often changes in response to real world factors. For example, if L_i changes linearly in response to a set of regression variables X, then we get the following linear regression model:

当前水平L_i经常响应于现实世界因素而改变。 例如，如果L_i响应一组回归变量X线性变化，那么我们得到以下线性回归模型：

In the above equation, β is the vector of regression coefficients and X_i is a vector of regression variables.

在上式中， β是回归系数的向量， X_i是回归变量的向量。

为什么研究白噪声模型很重要？ (Why is it important to study the white noise model?)

There are three reasons why:

原因有以下三个：

1. If you discover using some techniques which I will describe soon, that your data is basically white noise around a fixed level, then the best that you can do is fit a model around that fixed level. It will be a waste of time to try to do anything better than that.如果您使用我将很快描述的一些技术发现，您的数据基本上是固定水平附近的白噪声，那么您可以做的最好的事情就是将模型固定在该水平附近。 尝试做任何比这更好的事情都是浪费时间。
2. Suppose you have already fitted a regression model to a data set. If you are able to show that the residual errors of the fitted model are white noise, it means your model has done a great job of explaining the variance in the dependent variable. There is nothing left to extract in the way of information and whatever is left is noise. You can pat yourself on the back for a job well done!假设您已经对数据集拟合了回归模型。 如果您能够证明拟合模型的残留误差是白噪声，则表明您的模型在解释因变量的方差方面做得很好。 没有什么可以提取信息的方式了，剩下的就是噪音。 您可以轻拍自己的背，以完成出色的工作！
3. Thirdly, the white noise model happens to be a stepping stone to another important and famous model in statistics called the Random Walk model which I will explain in the next section.第三，白噪声模型恰好是统计学中另一个重要且著名的模型(称为随机游走模型)的垫脚石，我将在下一部分中进行解释。

随机游走模型 (The Random Walk Model)

Let’s again look at the White Noise Model’s equation:

让我们再次看一下白噪声模型的方程：

If we make the level level L_i at time step i be the output value of the model from the previous time step (i-1), we get the Random Walk model, made famous in the popular literature by Burton Malkiel’s A Random Walk Down Wall Street.

如果我们将时间步长i处的水平L_i设为上一个时间步长(i-1)的模型的输出值， 则会得到随机游走模型，该模型在伯顿·马尔基尔(Burton Malkiel)的《随机游走的墙壁》一书中广受欢迎街 。

The Random Walk model is like the mirage of the Data Science dessert. It has lured many profit-thirsty investors into betting (and losing) their shirt on illusions of trends in stock price movements, movements that were in reality little more than a random walk.

随机游走模型就像数据科学甜点的海市rage楼。 它已经吸引了许多渴求利润的投资者，将其押注(输掉)他们的衬衫，以幻想股价走势的错觉，实际上，这些走势只是随意走动而已。

Here’s a plot of data that was generated using the Random Walk model:

这是使用随机游走模型生成的数据图：

Just tell me you don’t see any trends in this plot!

告诉我，您在该图中看不到任何趋势！

If you are not completely convinced that the above data can be generated by a purely random process, let’s puff away any remaining illusions by showing how to generate this data in Excel:

如果您不完全相信上面的数据可以通过纯随机过程生成，那么让我们通过展示如何在Excel中生成此数据来消除任何剩余的幻想：

Let’s look at how we can make use of our knowledge of white noise and random walks to try to detect their presence in time series data.

让我们看看如何利用我们对白噪声和随机游走的知识来尝试检测时间序列数据中它们的存在。

如何在时间序列数据集中检测白噪声 (How to detect white noise in a time series data set)

We’ll look at 3 tests to determine whether your time series is in reality, just white noise:

我们将通过3种测试来确定您的时间序列是否真实，只是白噪声：

1. Auto-correlation plots自相关图
2. The Box-Pierce testBox-Pierce检验
3. The Ljung-Box testLjung-Box测试

使用自相关图测试白噪声 (Testing for white noise using auto-correlation plots)

When two variables move up or down in unison (or if one value goes up, the other one goes down), they are said to be positively (or negatively) correlated. The correlation coefficient can be used to measure the degree of linear correlation between two such variables:

当两个变量一致地上下移动(或者一个值上升时，另一个变量下降)时，它们被认为是正相关的(或负相关的)。 相关系数可用于测量两个此类变量之间的线性相关程度：

In the above formula, E(X) and E(Y) are the expected (i.e. mean) values of X and Y. σ_X and σ_Y are the standard deviations of X and Y.

在上式中， E( X )和E( Y )是X和Y的预期(即平均值)值。 σ_X和σ_Y是X和Y的标准偏差。

In time series data, correlations often exist between the current value and values that are 1 time step or more older than the current value, i.e. between Y_i and Y_(i-1), between Y_i and Y_(i-2) and so on. Stock price changes often show such patterns of positive and negative correlations (and beware, so do data containing random walks!).

在时间序列数据中，当前值和比当前值早1个时间步或更早的值之间通常存在相关性，即Y_i和Y_(i-1)之间， Y_i和Y_(i-2)之间等等。 。 股票价格变化通常显示出正相关和负相关的模式(请注意，包含随机游走的数据也是如此！)。

Because the values are correlated with past versions of themselves, we call them auto, meaning self correlated.

由于这些值与自身的过去版本相关，因此我们将其称为“自动”，即自相关。

Here is the formula for calculating the auto-correlation coefficient between Y_i and Y_(i-k):

这是用于计算Y_i和Y_(ik)之间的自相关系数的公式：

Before we can show how this auto-correlation coefficient r_k can be used to detect white noise, we need to take a short and pleasant side-trip into the land of random variables. I’ll explain why r_k is a normally distributed random variable and how this property of r_k can be used to detect white noise.

在我们展示如何使用该自相关系数r_k来检测白噪声之前，我们需要对随机变量进行短暂而愉快的旁通 。 我将解释为什么r_k是正态分布的随机变量，以及r_k的此属性如何用于检测白噪声。

LAG-k自相关系数 r_k的分布 (Distribution of the LAG-k auto-correlation coefficient r_k)

For any lag k, r_k is a normally distributed random variable with some mean µ_k and variance σ²_k.

任何滞后K，r_k是与一些均值μ_K和方差σ²_K正态分布的随机变量。

To understand why, consider this thought experiment:

要了解原因，请考虑以下思想实验：

1. Take a time series data set containing 100,000 time points.取得包含100,000个时间点的时间序列数据集。

2. Draw 5000 randomly selected samples from this data set. Suppose each sample is of length 100 continuous time points.

   从该数据集中抽取5000个随机选择的样本。 假设每个样本的长度为100个连续时间点。

3. For each sample, calculate the LAG-1 auto-correlation coefficient r_1 using the above formula for r_k.

   对于每个样本，使用上述r_k公式计算LAG-1自相关系数r_1 。

4. One can see that each time, r_1 will come out to be some value between 0 and 1 for each sample of 100 time points. So we end up with 5000 values of r_1, each a number between 0 and 1. Thus r_1 is a random variable for which we have measured 5000 values.

   可以看到，对于100个时间点的每个样本， r_1每次都会得出介于0和1之间的某个值。 因此，我们得到r_1的5000个值，每个值在0到1之间。因此r_1是一个随机变量，我们已经为它测量了5000个值。

5. By appealing to the Limit Theorems of statistics, it can be shown r_1 is a normally distributed random variable, and the distribution of r_1 is centered at some population mean, we’ll call it µ_1, and some variance, we’ll call it σ²_1. In practice, the observed mean and variance of r_1 will be somewhere close to the mean of the 5000 values of r_1 which we measured.

   通过利用统计的极限定理， 可以证明r_1是正态分布的随机变量，并且r_1的分布以某个总体平均值为中心，我们将其称为µ_1，将某些方差称为σ²_1 。 实际上，观察到的r_1的均值和方差将接近我们测量的r_1的5000个值的均值。

6. By repeating the above experiment for all lags k, it can be shown that auto-correlation coefficients for all lags are normally distributed random variables with mean µ_k and variance σ²_k.

   通过对所有滞后k重复上述实验，可以证明所有滞后的自相关系数都是均值μ_k和方差σ²_k的 正态分布随机变量 。

Symbolically:

象征性地：

检测白噪声的含义 (Implications for detecting white noise)

If the time series is white noise, then in theory, its current value T_i ought not be correlated at all with past values T_(i-1), T_(i-2) etc, and the corresponding auto-correlation coefficients r_1, r_2,…etc. will be zero or close to zero.

如果时间序列是白噪声，那么从理论上讲，它的当前值T_i根本不应该与过去的值T_(i-1)，T_(i-2)等以及相应的自相关系数r_1，r_2相关，…等将为零或接近零。

i.e.when the time series is white noise, r_k is 0 for all k = 1, 2, 3,…

即，当时间序列是白噪声时，对于所有k = 1、2、3 ... ， r_k为0

But we have just seen that r_k is a N(µ_k, σ²_k) random variable.

但是我们刚刚看到r_k是一个N(µ_k，σ²_k)随机变量。

Putting the above two facts together, we arrive at the following first important implication:

综合以上两个事实，我们得出以下第一个重要含义：

If the time series is white noise, then the auto-correlation coefficient r_k for all lags k will have a zero mean and some variance σ²_k.

如果时间序列是白噪声，则所有滞后k的自相关系数r_k将具有零均值和一些方差σ²_k。

Symbolically:

象征性地：

But what about the variance σ²_k of the coefficients r_k?

但是关于系数r_k的方差σ²_k什么？

Anderson, Bartlett and Quenouille have shown that under white noise conditions, the standard deviation σ_k is as follows:

Anderson ， Bartlett和Quenouille证明，在白噪声条件下，标准偏差σ_k如下：

σ_k = 1/sqrt(n)

σ_k= 1 /平方根(n)

Where n is the same size. Recollect that in our thought experiment, n was 100.

其中n是相同的大小。 回忆一下我们的思想实验中， n为100。

Thus, we know that r_k under white noise conditions has the following distribution:

因此，我们知道白噪声条件下的r_k具有以下分布：

An important property of the normal distribution is that approximately 95% of it lies within 1.96 standard deviations from the mean. In our case, the mean is 0 and standard deviation is 1/sqrt(n), so we get the following 95% confidence interval for the auto-correlation coefficients:

正态分布的一个重要属性是大约95％的分布在均值的1.96标准偏差之内。 在我们的情况下，平均值为0，标准偏差为1 / sqrt(n) ，因此对于自相关系数，我们得到以下95％的置信区间：

These results yield the following procedure for conducting the white noise test using the auto-correlation coefficients r_k:

这些结果得出以下使用自相关系数r_k进行白噪声测试的过程 ：

1. Calculate the first k auto-correlation coefficients r_k. k can be set to some high enough value depending on the length n of the time series data set.

   计算前k个自相关系数r_k 。 可以将k设置为足够高的值，具体取决于时间序列数据集的长度n 。

2. Calculate the 95% confidence interval [ — 1.96/sqrt(n), +1.96/sqrt(n)].

   计算95％置信区间[-1.96 / sqrt(n)，+ 1.96 / sqrt(n)]。

3. If for all k, if r_k lies within the above confidence interval, conclude at a 95% confidence level that the time series is in reality, possibly just white noise. We say possibly because if we experiment with larger sample sizes, i.e. larger n, the size of the confidence interval will shrink, and values of r_k that were previously inside the 95% bounds will now lie outside the 95% bounds.

   如果对于所有k ，如果r_k都在上述置信区间内，则以95％的置信度推断该时间序列实际上是现实的， 可能只是白噪声。 我们之所以说是可能的，是因为如果我们尝试使用更大的样本量(即更大的n) ，则置信区间的大小将缩小，并且先前在95％范围内的r_k值现在将在95％范围之外。

4. If any of the r_k lie outside the confidence interval, then the time series possibly has information in it.

   如果r_k中的任何一个位于置信区间之外，则时间序列中可能包含信息。

示例：使用Python检测白噪声 (Example: White noise detection using Python)

Let’s illustrate the above procedure using a real world time series of 5000 decibel level measurements taken at a restaurant using the Google Science Journal app.

让我们通过使用Google Science Journal应用程序在餐厅进行的5000分贝水平的真实世界时间序列说明上述过程。

The data set can be downloaded from here.

数据集可从此处下载。

We’ll use the pandas library to load the data set from the csv file and plot it:

我们将使用pandas库从csv文件加载数据集并进行绘制：

import pandas as pdimport numpy as npfrom matplotlib import pyplot as pltdf = pd.read_csv('restaurant_decibel_level.csv', header=0, index_col=[0])

Let’s print the top 10 rows:

让我们打印前十行：

df.head(10)           DecibelTimeIndex0          55.93132340         57.77926080         62.956952140        65.158100180        60.325242220        45.411725262        55.958807300        62.021807340        62.222563380        56.156684

Let’s plot all 5000 values in the series:

让我们绘制该系列中的所有5000个值：

Let’s fetch and plot the auto-correlation coefficients for the first 40 lags. We’ll the statsmodels library to do that.

让我们获取并绘制前40个滞后的自相关系数。 我们将使用statsmodels库来执行此操作。

import statsmodels.graphics.tsaplots as tsatsa.plot_acf(df['Decibel'], lags=40, alpha=0.05, title='Auto-correlation coefficients for lags 1 through 40')

The alpha=0.05 tells statsmodels to also plot the 95% confidence interval region. We get the following plot:

alpha = 0.05指示statsmodels也绘制95％置信区间区域。 我们得到以下图：

As we can see, the time series contains significant auto-correlations up through lags 17. Incidentally, the auto-correlation at lag 0 is always 1.0 as a value is always perfectly correlated with itself.

正如我们所看到的，时间序列在滞后17之前包含大量的自相关。顺便说一下，滞后0处的自相关始终为1.0，因为值始终与自身完全相关。

There is wave-like pattern in the auto-correlation plot that indicates that there could be some seasonality contained in the data. We can try to identify and isolate the seasonality by decomposing the time series into the trend, seasonality and noise components.

自相关图上有一个波状图案，表明数据中可能包含一些季节性。 我们可以尝试通过将时间序列分解为趋势，季节性和噪声成分来识别和隔离季节性。

Related read: What is time series decomposition and how does it work

For now we’ll focus on the noise portion. The bottom line is that this time series, in its current form, does not appear to be pure white noise.

现在，我们将集中讨论噪声部分。 最重要的是，此时间序列以其当前形式似乎不是纯白噪声。

Next, we’ll two more tests on the time series to confirm this.

接下来，我们将在时间序列上再进行两次测试以确认这一点。

卡方检验用于白噪声检测 (The Chi-squared test for white noise detection)

The Chi-squared test is based on this powerful result in statistics: the sum of squares of k identical standard normal random variables is a Chi-squared distributed random variable with k degrees of freedom.

卡方检验基于此强大的统计结果： k个相同的标准正态随机变量的平方和是具有k个自由度的卡方分布随机变量。

The actual test is called Box-Pierce test and it’s test statistic is called the Q statistic. Its formula is as follows:

实际测试称为Box-Pierce测试，其测试统计量称为Q统计量。 其公式如下：

It can be shown that if the underlying data set is white noise, the expected value of the Q statistic is zero.

可以证明，如果基础数据集是白噪声，则Q统计量的期望值为零。

For any given time series, one can check if the value of Q deviates from zero in a statistically significant way looking up the p-value of the test statistic in the Chi-square tables for k degrees of freedom. Usually, a p-value of less than 0.05 indicates a significant auto-correlation that cannot be attributed to chance.

对于任何给定的时间序列，可以检查Q值是否以统计学上显着的方式偏离零，从而在卡方表中针对k个自由度查找测试统计量的p值。 通常，小于0.05的p值表示无法归因于偶然性的显着自相关。

Ljung-Box测试以检测白噪声 (The Ljung-Box test for white noise detection)

The Ljung-Box test improves upon the Box-Pierce test to obtain a test statistic having a distribution that is closer to the Chi-square distribution than the Q statistic. The test statistic of the Ljung-Box test is calculated as follows, and it is also Chi-square(k) distributed:

Ljung-Box检验在Box-Pierce检验的基础上进行了改进，从而获得了一个检验统计量，其分布比Q统计量更接近卡方分布。 Ljung-Box检验的检验统计量计算如下，并且也是卡方(k)分布：

Here, n is the number of data points in the time series and k is the number of time lags to be considered. As with the Box-Pierce test, if the underlying data set is white noise, the expected value of this Chi-square distributed random variable is zero. Again, a p-value of less than 0.05 indicates a significant auto-correlation that cannot be attributed to chance.

此处， n是时间序列中的数据点数，而k是要考虑的时间延迟数。 与Box-Pierce检验一样，如果基础数据集是白噪声，则此卡方分布随机变量的期望值为零。 再次，小于0.05的p值表示显着的自相关，不能将其归因于机会。

示例：使用Python中的Ljung-Box测试测试白噪声 (Example: Testing for white noise using the Ljung-Box test in Python)

Let’s run the Ljung-Box test on the restaurant decibel level data set. We will test upto 40 lags and we’ll ask the test to also run the Box-Pierce test.

让我们在餐厅分贝级别的数据集上运行Ljung-Box测试。 我们将测试多达40个延迟，然后要求该测试也运行Box-Pierce测试。

import statsmodels.stats.diagnostic as diagdiag.acorr_ljungbox(df['Decibel'], lags=[40], boxpierce=True, model_df=0, period=None, return_df=None)

We get the following output:

我们得到以下输出：

(array([13172.80554476]), array([0.]), array([13156.42074648]), array([0.]))

The value 13172.80554476 is the value of the test statistic for the Ljung-Box test and 0.0 is its p-value as per the Chi-square(k=40) table.

根据卡方(k = 40)表，值13172.80554476是Ljung-Box测试的测试统计值，而0.0是其p值。

The value 13156.42074648 is the test statistic of the Box-Pierce test and 0.0 is its p-value as per the Chi-square(k=40) tables.

值13156.42074648是Box-Pierce检验的检验统计量，0.0是按照卡方(k = 40)表的p值。

As we can see, both p-values are less than 0.01 and so we can say with 99% confidence that the restaurant decibel level time series is not pure white noise.

如我们所见，两个p值均小于0.01，因此我们可以有99％的把握说餐厅分贝级时间序列不是纯白噪声。

Earlier on, we introduced Random Walks as a special case of the White Noise model and pointed out how easy it is to mistake them for a pattern or trend that can be predicted.

早些时候，我们引入了随机游走作为白噪声模型的特例，并指出将它们误认为可预测的模式或趋势是多么容易。

We’ll look at how to avoid making this mistake by applying a technique that will bring out the true random nature of the Random Walk.

我们将研究如何通过应用一种能够展现出随机游走的真正随机性的技术来避免犯此错误。

检测随机游走 (Detecting Random Walks)

Random walks are often highly correlated. In fact, they are auto-correlated white noise!

随机游走通常是高度相关的。 实际上，它们是自动相关的白噪声！

The white noise detection tests presented above will latch on these auto-correlations, causing them to conclude that the time series is not white noise.

上面介绍的白噪声检测测试将锁定这些自相关，使他们得出时间序列不是白噪声的结论。

The remedy is to take the first difference of the time series that is suspected to be a random walk, and run the white noise tests on the differenced series.

补救措施是采取怀疑是随机游走的时间序列的第一个差异，并对差异序列进行白噪声测试。

If the original time series is a random walk, its first difference is pure white noise.

如果原始时间序列是随机游走，则其第一个差异是纯白噪声。

Let’s illustrate this:

让我们说明一下：

We’ll start by loading a data set that is suspected to be a random walk. The data set can be downloaded from here.

我们将从加载怀疑是随机游走的数据集开始。 数据集可从此处下载。

df = pd.read_csv('random_walk.csv', header=0, index_col=[0])

Let’s plot it to see how it looks like:

让我们对其进行绘图以查看其外观：

df.plot()plt.show()

Let’s run the Ljung-Box white noise test on this data:

让我们对这些数据运行Ljung-Box白噪声测试：

diag.acorr_ljungbox(df['Y_i'], lags=[40], boxpierce=True)

We get the following result:

我们得到以下结果：

(array([393833.91252517]), array([0.]), array([392952.07675659]), array([0.]))

The p value of 0.0 indicates that we must strongly reject the null hypothesis that the data is white noise. Both Ljung-Box and Box-Pierce tests think that this data set has not been generated by a pure random process.

p值为0.0表示我们必须强烈拒绝数据为白噪声的零假设。 Ljung-Box和Box-Pierce测试都认为此数据集 不是 由纯随机过程生成的。

This is obviously a false result.

这显然是错误的结果。

Let’s see if things change after we take the first difference of the data, i.e. we create a new data set with Y = Y_i —Y_(i-1) :

让我们看看在获取数据的第一个差异之后情况是否发生了变化，即我们创建了一个新的数据集，其中Y = Y_i —Y_(i-1) ：

diff_Y_i = df['Y_i'].diff()#drop the NAN in the first rowdiff_Y_i = diff_Y_i.dropna()

Let’s plot the diff-ed data set:

让我们绘制差异数据集：

diff_Y_i.plot()plt.show()

We now see a very different picture:

现在，我们看到了非常不同的图片：

Here is the zoomed in view:

这是放大的视图：

Let’s run the Ljung-Box test on the differenced data set:

让我们对不同的数据集运行Ljung-Box测试：

diag.acorr_ljungbox(diff_Y_i, lags=[40], boxpierce=True)

We get the following output:

我们得到以下输出：

(array([32.93405364]), array([0.77822417]), array([32.85051846]), array([0.78137548]))

Notice that this time the test statistic’s value 32.934 reported by Ljung-Box, and 32.850 reported by Box-Pierce tests is much smaller. And the corresponding p-values detected on the Chi-square(k=40) tables are 0.778 and 0.781 respectively, which are well above 0.05. This is easily enough to support the null hypothesis that the data (i.e. the differenced time series) is pure white noise.

请注意，这一次的检验统计量的值32.934报道Ljung的盒，以及由32.850箱皮尔斯测试报告的要小得多。 卡方(k = 40)表上检测到的相应p值分别为0.778和0.781 ，远高于0.05。 这足够容易地支持零假设，即数据(即时间序列不同)是纯白噪声。

The conclusion to be drawn from this exercise is that one should not fit anything except the White Noise model on this data.

从该练习中得出的结论是，除此数据上的白噪声模型外，其他任何条件都不适合。

摘要 (Summary)

• The white noise model can be used to represent the nature of noise in a data set.白噪声模型可用于表示数据集中噪声的性质。
• Testing for white noise is one of the first things that a data scientist should do so as to avoid spending time on fitting models on data sets that offer no meaningfully extract-able information.测试白噪声是数据科学家应该做的第一件事，以避免花时间在不提供有意义的可提取信息的数据集的拟合模型上。
• If a data set is not white noise, then after fitting a model to the data, one should run a white noise test on the residual errors to get a sense for how much information the model has been able to extract from the data.如果数据集不是白噪声，则在将模型拟合到数据之后，应该对残差进行白噪声测试，以了解模型能够从数据中提取多少信息。
• For time series data, auto-correlation plots and the Ljung-Box test offer two useful techniques for determining if the time series is in reality, just white noise.对于时间序列数据，自相关图和Ljung-Box测试提供了两种有用的技术来确定时间序列是否真实，只是白噪声。

参考，引用和版权 (References, Citations and Copyrights)

Data set of restaurant decibel levels is Copyright Sachin Date under CC-BY-NC-SA.

餐厅分贝级别的数据集为CC-BY-NC-SA下的版权Sachin日期 。

Amgen stock price chart is from stockcharts.com under these terms of use.

根据这些使用条款， Amgen股票价格图表来自stockcharts.com 。

Paper link: Anderson, R. L., Distribution of the Serial Correlation Coefficient, Annals of Mathematical Statistics, Volume 13, Number 1 (1942), 1–13.

论文链接：Anderson，RL， 串行相关系数的分布 ，《数学统计年鉴》，第13卷，第1期(1942)，1-13。

Paper link: Bartlett, M. S., On the Theoretical Specification and Sampling Properties of Autocorrelated Time-Series, Supplement to the Journal of the Royal Statistical Society, Vol. 8, №1 (1946), pp. 27–41.

论文链接：Bartlett，MS， 《自相关时间序列的理论规范和采样特性》，《皇家统计学会杂志》增刊 ，第1卷。 8，№1(1946)，第27-41页。

Paper link: Quenouille, M. H., The Joint Distribution of Serial Correlation Coefficients, The Annals of Mathematical Statistics, Vol. 20, №4 (Dec., 1949), pp. 561–571

论文链接：Quenouille，MH， 《序列相关系数的联合分布》 ，《数学统计年鉴》，第1卷。 20，№4(1949年12月)，第561–571页

Book link: Hyndman, R. J., Athanasopoulos, G., Forecasting: Principles and Practice, OTexts

图书链接：Hyndman，RJ，Athanasopoulos，G。，《 预测：原理与实践》 ，OTexts

All images in this article are copyright Sachin Date under CC-BY-NC-SA, unless a different source and copyright are mentioned underneath the image.

本文中的所有图像均为CC-BY-NC-SA下的版权Sachin Date ，除非在图像下方提及其他来源和版权。

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谢谢阅读！ 如果您喜欢本文，请 关注我 以获取有关回归和时间序列分析的提示，操作方法和编程建议。