There are three criteria to judge whether a sequence is stationary or not:

1. Mean is a constant independent of time t. The following figure (left) satisfies the condition of stationary series, and the following figure (right) is obviously time-dependent.

1. Variance is a constant independent of time t. This property is called homogeneity of variance. The following figure shows what is and is not variance alignment. (Note the different distribution along the way on the right-hand side.)
   
   
2. Covariance is a constant that depends only on time interval K, not time t. In the figure below (right), you can notice that the curve gets closer and closer over time. So the covariance of the red sequence is not constant.
   
   

This is the basic concept of time series. You’re probably familiar with this concept. However, many in industry still view the random walk as a stationary sequence. In this section, I’ll use some mathematical tools to help understand this concept. Let’s start with an example

example: Imagine a girl randomly moving around on a giant chessboard. Here, the next position only depends on the last position.

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(source: scifun.chem.wisc.edu/WOP/RandomW…).

Now imagine you’re in a closed room and you can’t see this girl. But you want to predict where this girl will be at different times. How can ability predict a bit more accurate? Of course you get worse and worse as time goes on. At time t=0, you must know where this girl is. At the next moment the girl moves to one of the eight squares in the enclosure, at which point the probability of your prediction is reduced to 1/8. Moving on, let’s formalize the sequence:

$X(t) = X(t-1) + Er(t)$ This is the randomness of the girl at each point in time.Copy the code

Now we recurse over all x time points, and we’ll end up with the following equation:

$X(t) = X(0) + Sum(Er(1),Er(2),Er(3)..... Er(t))$Copy the code

Now, let’s try to test the stationary hypothesis of the random walk: 1. Is the mean constant?

E[X(t)] = E[X(0)] + Sum(E[Er(1)],E[Er(2)],E[Er(3)]..... E[Er(t)])Copy the code

We know that since the expected value of the random interference term of the random process is 0. So far: E[X(t)] = E[X(0)] = constant 2. Is the variance constant?

Var[X(t)] = Var[X(0)] + Sum(Var[Er(1)],Var[Er(2)],Var[Er(3)]..... Var[Er(t)]) Var[X(t)] = t * Var(Error) = Time correlationCopy the code

Therefore, we infer that the random walk is not a stationary process because it has a time-varying variance. Furthermore, if we examine the covariance, we see that the covariance depends on time.

E[X(t)] = Rho *E[ X(t-1)]
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This formula makes sense. The next X(or time point t) is pulled to the value of the X above Rho*. For example, if x (t — 1) = 1, E[x (t)] = 0.5 (Rho= 0.5). Now, if you go from zero to any direction the next step you want to expect is zero. The only thing that can make expectations bigger is the error rate. What happens when Rho becomes 1? There is no possibility of any further decline.

In this section we will learn how to use R to process time series. Here we are just exploring time series, not modeling time series. The data used in this section is the built-in data in R: AirPassengers. This data set is the number of international air passengers per month from 1949 to 1960.

1 #The number of passengers are distributed across The spectrum 2 > plot(AirPassengers >abline(reg=lm(AirPassengers~time(AirPassengers)) 5 #Copy the code

1 > cycle(AirPassengers) 2 Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec 3 1949 1 2 3 4 5 6 7 8 9 10 11 12 4 1950 1 2 3 4 5 6 7 8 9 10 11 12 5 1951 12 3 4 5 6 7 8 9 10 11 12 6 1952 12 3 4 5 6 7 8 9 10 11 12 7 1953 12 3 4 5 6 7 8 9 10 11 12 7 10 10 10 11 12 8 1954 12 3 4 56 7 8 9 10 11 12 9 1955 12 3 4 56 7 8 9 10 11 12 10 1956 12 3 4 56 7 8 9 10 11 12 11 1957 12 12 3 4 5 6 7 8 9 10 11 12 13 1959 12 3 4 5 6 7 8 9 10 11 12 14 1960 12 3 4 5 6 7 8 9 10 11 12 14 16 > plot(aggregate(AirPassengers,FUN=mean)) Boxplot (AirPassengers~cycle(AirPassengers)Copy the code

ARMA is also called autoregressive moving average mixed model. ARMA models are often used in time series. In ARMA model, AR stands for autoregression and MA stands for moving average. If these terms sound complicated, don’t worry – we’ll brief you on these concepts in a few minutes. We will see the characteristics of these models now. Before you start, remember that AR or MA does not apply to non-stationary sequences. In practice you may have a non-stationary sequence, the first thing you need to do is to turn the sequence into a stationary sequence (by differential differentiation/transformation), and then choose the time series model you can use. First, this article introduces the two models (AR&MA) separately. Let’s take a look at the characteristics of these models.

Let’s do another example of moving averages. It is easy to understand that a company generates a certain type of package. As a competitive market, the sales of bags increased from zero. So one day he did an experiment to design and make different bags that weren’t available for purchase. So let’s say the total demand in the market is 1,000 of these bags. On one day, demand for the bag was so high that inventory soon ran out. At the end of the day there are 100 bags left. Let’s call this error point in time. Several customers still bought the bag in the following days. Here is a simple formula to describe this scenario:

x(t) = beta *  error(t-1) + error (t)
Copy the code

Try to draw this, it looks like this:



Notice the difference between the MA and AR models? In the MA model, the noise/impact rapidity is small. In the AR model it will be affected for a long time.

Once we have a stationary time series. We have to answer two of the most important questions; Q1: Is this an AR or MA process? Q2: What is the order of AR or MA processes that we need to utilize?

To solve these two problems, we use two coefficients:

Time series X (t) lag k order sample autocorrelation coefficient (ACF) and lag K period under the case of sample partial autocorrelation coefficient (PACF). The formula is omitted.

ACF and PACF of AR model:

The calculation proves that:

The ACF of -AR is a trailing sequence, that is, no matter how large the lag period k is, the calculated value of ACF is related to the autocorrelation function of order 1 to P hysteresis.

PACF of -AR is truncated sequence, that is, PACF=0 when k> P in lag period.



The blue line above shows values that are significantly different from 0. Obviously, the PACF diagram above shows truncation at the second lag, which means that this is an AR (2) process.

ACF and PACF of MA model:

The ACF of -MA is a truncated sequence, that is, PACF=0 when k> P in lag period.

The PACF of -AR is a trailing sequence, that is, no matter how large the lag period k is, the calculated value of ACF is related to the autocorrelation function of order 1 to P hysteresis.



Obviously, the ACF diagram above truncates the second lag, which means that this is an MA (2) process.

So far, this paper has introduced the use of ACF&PACF diagrams to identify stationary sequence types. Now, I’ll introduce the overall framework of a time series model. In addition, the practical application of the time series model will be discussed.

The frame below shows how a step-by-step”Do a time series analysis“



The first three steps were discussed in the previous section. However, a quick note is needed:

Once we know the pattern, the trend, the cycle. We can check if the sequence is stationary. The Dicky-Fuller test is a popular one. This method of testing is introduced in the first part of the opinion. It’s not over here! What if the sequence is found to be nonstationary? There are three commonly used techniques for stabilizing a time series. 1. Trend elimination: Here we simply delete trend components in time series. For example, the equation for my time series is:

x(t) = (mean + trend * t) + error
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Here I simply delete trend*t from the formula above and establish x(t)=mean+error model 2 difference: this technique is often used to eliminate non-stationarity. Here we are modeling the difference results of the sequence rather than the actual sequence. Such as:

X (t) -- x(t-1) = ARMA (p, q)Copy the code

This difference is also part of ARIMA. Now we have three parameters:

p:AR
d:I
q:MA
Copy the code

3. Seasonality: Seasonality is directly incorporated into ARIMA model. We’ll talk more about this in the applications section below.

Having found these parameters, we can now try the resume ARIMA model. The value found from the previous step may only be an approximate estimate, and we need to explore more combinations of (p, D,q). The minimum BIC and AIC model parameters are what we want. We can also try some seasonal ingredients. Here, we’ll notice something seasonal in the ACF/PACF diagram.

Here we use the previous example. Use this time series to make predictions. We encourage you to look at this data before taking the next step.

It is clear that the ACF is falling very slowly, which means that passenger numbers are not flat. As we have discussed earlier, we are now going to do regression on the logarithmic difference of the sequence rather than directly on the logarithmic difference of the data entropy of the sequence. Let’s look at the ACF and PACF curves after difference.

> acf(diff(log(AirPassengers)))
> pacf(diff(log(AirPassengers)))
Copy the code

Obviously ACF ends with the first lag, so we know that p should be 0 and q should be 1 or 2. After several iterations, we found that AIC and BIC were the smallest when (p,d,q) was set to (0,1,1).

1 > fit <- arima(log(AirPassengers), c(0, 1, 1),seasonal = list(order = c(0, 1, 1), P < -predict (fit, n.ahead = 10*12) 3 ts.plot(AirPassengers,2.718^pred$pred, log = "y", Lty = c (1, 3))Copy the code

Chapter 8 Time Series Analysis