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时间序列线性回归区别_时间序列分析的完整介绍（带R）：：线性过程I

發布時間：2023/12/15 编程问答 26 豆豆

生活随笔收集整理的這篇文章主要介紹了时间序列线性回归区别_时间序列分析的完整介绍（带R）：：线性过程I 小編覺得挺不錯的,現在分享給大家,幫大家做個參考.

時間序列線性回歸區別

In the last tutorial , we saw how we could express the probabilistic form of the best linear predictor of a future observation based on the data at hand. We will see how to implement this in R later! In this article, we will study an important class of time series: linear processes. Let’s jump right into it!

在上一教程中，我們看到了如何基于手頭的數據來表達未來觀察的最佳線性預測變量的概率形式。稍后我們將看到如何在R中實現這一點！在本文中，我們將研究重要的時間序列類別： 線性過程。 讓我們跳進去吧！

q相關和強平穩性 (q-correlation and strong stationarity)

How can more rigorously define the stationarity of time series? How about “semi-stationarity”? We use the concept of q-correlation to do this.

如何更嚴格地定義時間序列的平穩性？ “半平穩性”怎么樣？我們使用q相關的概念來做到這一點。

A time-series process is called strictly stationary or strongly stationary if

時間序列過程稱為嚴格平穩或強平穩，如果

that is, if the joint distribution of the observations X_{1}, …, X_{n} is the same as the one of the h-lagged set of observations.

也就是說，如果觀測值X_ {1}，…，X_ {n}的聯合分布與h滯后觀測值之一相同。

Properties

物產

All elements of a strongly stationary time series are identically distributed (but not necessarily independent!)
高度平穩的時間序列的所有元素都是相同分布的 (但不一定是獨立的！)
The distribution of any subset of observations is the same as the h-lagged set
觀察的任何子集的分布與h滯后集相同

Finite second moment implies weakly stationarity
有限的第二時刻意味著平穩性較弱

For convenience, here’s once more the definition of weak stationarity:

為了方便起見，這里再次給出弱平穩性的定義：

Note that strong stationarity is a much stronger condition! There are two important things to notice:

請注意，強烈的平穩性是更強的條件！有兩件事要注意：

Strong stationarity implies weak stationarity, but the opposite is not true!

平穩性強意味著平穩性弱 ，但事實并非如此！

The I.I.D process is strongly stationary.

IID過程是非常固定的 。

The main question is: how do we construct / how can we characterize stationary processes? In particular, if we know that the IID is stationary, perhaps there is some mapping to more general sequences. In such a case, how can we determine their “level of stationarity”? The following concepts.

主要問題是： 我們如何構建/如何表征平穩過程？ 特別是，如果我們知道IID是固定的，則可能存在一些映射到更通用的序列。在這種情況下，我們如何確定他們的“平穩程度”？以下概念。

q相關性和q相關性 (q-dependence and q-correlation)

What the previous proposition is saying is that if we have an IID sequence, under certain conditions, we can construct a new series that is also strongly stationary, using some function g. Further, we can define q-dependence by saying that observations |t-s| lags apart are independent, but everything in between is dependent, and similarly for correlation.

先前的命題是說，如果我們有一個IID序列，則在某些條件下，我們可以使用某些函數g構造一個也非常平穩的新序列。此外，我們可以通過說| ts |來定義q-依賴性 。滯后是獨立的，但是介于兩者之間的所有事物都是相關的，并且類似地進行相關。

線性過程 (Linear Processes)

Now we finally get to one of the most important parts of Time Series Analysis: linear processes.

現在，我們終于可以了解時間序列分析中最重要的部分之一： 線性過程。

Example: MA(1)

示例：MA(1)

The MA(1) process

MA(1)過程

Example: AR(1)

示例：AR(1)

The AR(1) process

AR(1)流程

and we have that

而且我們有

If you are curious about why the coefficients are that way, you can attempt to solve the recursion by plugging back the definition of X_{t} substituting {t-1}. Note that we have to play the restriction on the AR(1) coefficients so that it satisfies the conditions of a linear process as we described above.

如果您對為什么采用這種系數感到好奇，則可以嘗試通過插入X_ {t}代替{t-1}的定義來解決遞歸問題。請注意，我們必須對AR(1)系數進行限制，以使其滿足如上所述的線性過程的條件。

關于后向運算符的線性過程 (Linear processes in terms of the backward operator)

Remember the backward shift operator that we saw back in the differencing section? We can also use it to represent linear processes.

還記得我們在微分部分中看到的向后移位運算符嗎？我們還可以使用它來表示線性過程。

That is, we define the Psi operator as the infinite polynomial above, in which each term is exponentiated accordingly. If we plug in the backward shift operator, we then have a concrete way to represent the linear process briefly! This will become very useful to represent a more complex process without having to write down everything explicitly. Another way to see it is that we can see a linear process as an operation applied to the noise at time t.

也就是說，我們將Psi運算符定義為上面的無窮多項式，其中每個項相應地取冪。如果我們插入后移運算符，那么我們就有一種具體的方法來簡要表示線性過程！這對于表示更復雜的過程而不必顯式寫下所有內容將非常有用。另一種看待它的方式是，我們可以將線性過程視為在時間t處應用于噪聲的運算。

下次 (Next time)

That’s it for today! Next time, we will continue studying some more properties of the linear process, along with some interesting propositions, and other useful operators commonly used when dealing with Time Series. Until then!

今天就這樣！下次，我們將繼續研究線性過程的更多屬性，以及一些有趣的主張，以及在處理時間序列時常用的其他有用的運算符。直到那時！

上次 (Last time)

Prediction 1 → Best Linear Predictors II

預測1→最佳線性預測器II

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翻譯自: https://medium.com/analytics-vidhya/a-complete-introduction-to-time-series-analysis-with-r-linear-processes-i-88a1b55db9ef