Time series ARIMA model (three)

First look at the following picture:

This is the monthly price of crude oil from 1986 to 2006. It can be seen that after 2001 years, crude oil prices have a significant climb, then it is not reasonable to assume that the mean is a fixed value (constant), that is, the second stationary model in this case is too applicable. This is why we have this talk today.

To deal with this non-stationary data (for example, the mean in the above image is not a constant), a non-stationary model is required: sum autoregressive sliding average (autoregressive integrated moving Average, ARIMA). Next, let's look at a processed oil price:

is not familiar. Right, after a simple processing, the original uneven data, immediately became a smooth data. This can be done simply by using the model of the previous chapter. This approach is called differential (it's going to be, don't worry). The Arima model can be easily understood as: differential + stationary model . You see, this chapter of the content is not much simpler, so-called non-stationary model is not the differential processing of data, and then change the smooth data, and then use a smooth model to deal with it ~ indeed Jiangzi ~.

That immediately raises a question (or you have a question already): what is differential. Differential is a very important tool to deal with time series, which is widely used in econometrics and financial mathematics. 3.1 differential Operation 3.1.1 Differential Operation

Or using crude oil prices (monthly data) as an example, the so-called first-order difference is the difference between the two adjacent monthly. (So simple: Yes, that's it.

Normalize: set {Xt, t=±1,±2,...} for a time sequence, then:
δxt=xt−xt−1 t=±1,±2,...
Called time sequence {Xt, t=±1,±2,...}

First-order difference operation.

For example, 2005 crude oil prices are: 46.84, 48.15, 54.19, 52.98, 49.83, 56.35, 58.99, 64.98, 65.59, 62.26, 58.32, 59.41. The first-order difference in this set of data is 1.31, 6.04,-1.21,-3.15, 6.52, 2.64, 5.99, 0.61,-3.33,-3.94, which is the previous number minus the next number.

So if I want to go to the difference after the data to make a difference again. That is the second-order difference (that is, the easy~): The first differential data above is 1.31, 6.04,-1.21,-3.15, 6.52, 2.64, 5.99, 0.61,-3.33, 3.94, once again differential: 4.73,-7.25,-1.94, 9.67,-3.88, 3.35,-5.38,-3.94,-0.61.

If you want to do P-times (You're enough), that's called P-order differential, which is defined as:
Δpxt=δp−1xt−δp−1xt−1

is not very simple. 3.1.2 K-Step differential

If it is not the difference between adjacent data, for example, I would like to make a few data calls, such as I want to use crude oil data July data and January bad. No problem, your needs I must be satisfied, this is called K-step differential :
Δkxt=xt−xt−k

For example, the price difference between July 2005 and January is: 58.99-46.84 = 12.15. Also known as six-step differential. 3.1.3 Differential Selection

Focus on the!!! Sequence contains a significant linear trend, the 1-order difference can achieve a stable trend, such as the above price of crude oil (however, in fact, there is a pretreatment, one will say ~). The sequence contains the curve trend, usually low order (2 order or 3 order) difference can take the influence of the proposed curve trend. For a fixed-period sequence, it is usually better to extract the period information by the difference of the period length of the step.

PS: Although the difference is good, but also do not offs Oh, remember that any processing of information will only cause loss of information. 3.1.3* delay operator

This section as understanding, do not like can _ skip _, do not affect the understanding drip ~.

A delay operator, like a time pointer, multiplies the current sequence value by a delay operator, which is equivalent to pulling the time of the current sequence value to the past. (PS: operator is the mapping, that is, the relationship is the transformation [3]) 3.1.3.1 Definition and Nature

Note B is a delay operator, with: Xt−1=bxt,xt−2=b2xt,..., xt−p=bpxt−p

The delay operator has the following properties: B0 = 1 if C is a constant, then there is B (C∙XT) =CB (XT) =c∙xt−1
{XT},{YT} for any two sequences, with B (XT±YT) =xt−1±yt−1
Bnxt=xt−n
where (1−b) n=∑ni=0 (−1) PC