Time series ARIMA model

Data stationarity and difference method

Stability ：

• Stationarity is the fitting curve obtained through the sample time series In the future, we can still follow the existing form “ inertia ” Underground continuation
• Stationarity requires that the mean and variance of the sequence do not change significantly

Strict stability and weak stability ：

• Yan pingwen ： The distribution of strict stationary representation does not change with time . Such as ： White noise （ normal ）, No matter how you take , It's all about expectations 0, The variance of 1
• Weakly stationary ： Expectation and correlation coefficient （ Dependence ） unchanged At some point in the future t Value Xt It depends on its past information , So we need dependence

The difference method ： Time series in t and t-1 Time difference

Autoregressive model （AR）

• Describe the relationship between current value and historical value , Use the historical time data of variables to predict themselves
• Autoregressive model must satisfy the requirement of stationarity
• p The formula definition of order autoregressive process ：$y_t=\mu +{\sum }_{i=1}^p{\gamma }_i{y}_{t-1}+{ϵ}_t$
• $y_t$ It's the current value $\mu$ It's a constant term ,$P$ It's order ${\gamma }_i$ It's an autocorrelation function ${ϵ}_t$ It's error

Limitations of autoregressive models

• Autoregressive model uses its own data to predict
• Must be stable
• Must have autocorrelation , If the autocorrelation coefficient (φi) Less than 0.5, ... should not be used
• Autoregression is only applicable to predict the phenomenon related to its own early stage

Moving average model （MA）

• Moving average model focuses on the accumulation of error terms in autoregressive model
• q The formula definition of order autoregressive process ：$y_t=\mu +{ϵ}_t+{\sum }_{i=1}^q{\theta }_i{ϵ}_{t-1}$
• The moving average method can effectively eliminate the random fluctuation in prediction

Autoregressive moving average model （ARMA）

• The combination of autoregressive and moving average
• Formula definition ：$y_t=\mu +{\sum }_{i=1}^p{\gamma }_i{y}_{t-1}+{ϵ}_t+{\sum }_{i=1}^q{\theta }_i{ϵ}_{t-1}$

ARIMA

• ARIMA(p,d,q) The model is called differential autoregressive moving average model (Autoregressive Integrated Moving Average Model, Brief notes ARIMA
• AR It's autoregression , p Is the autoregressive term ; MA For moving average q Is the moving average number of items ,d It is the difference number of times when the time series becomes stationary
• principle ： The non-stationary time series is transformed into stationary time series, and then the dependent variable The model established by regressing only its lag value and the present value and lag value of random error term

Autocorrelation function ACF(autocorrelation function)

• An ordered sequence of random variables is compared to itself The autocorrelation function reflects the correlation between the values of the same sequence in different sequences
• The formula ：$ACF(k)={\rho }_k=\frac{Cov(y_t,y_{t-k})}{Var(y_t)}$
• ${\rho }_k$ The value range of is [-1,1]

Partial autocorrelation function （PACF）(partial autocorrelation funtion)

• For a smooth AR§ Model , Find the lag k Autocorrelation coefficient p(k) when In fact, it's not x(t) And x(t-k) A simple correlation between
• x(t) At the same time, it will suffer from the middle k-1 Random variables x(t-1)、x(t-2)、……、x(t-k+1) Influence And this k-1 All random variables and x(t-k) There is a correlation So the autocorrelation coefficient p(k) It's actually adulterated with other variables x(t) And x(t-k) Influence
• Cut out the middle k-1 Random variables x(t-1)、x(t-2)、……、x(t-k+1) After the interference of x(t-k) Yes x(t) The relevance of the impact .
• ACF It also includes the effects of other variables The partial autocorrelation coefficient PACF Is strictly the correlation between these two variables

ARIMA（p,d,q） Order determination ：

truncation ： Fall within the confidence interval （95% All of the points conform to the rule ）

ARIMA Modeling process ：

• Smooth the sequence （ Determination by difference method d）
• p and q Order determination ：ACF And PACF
• ARIMA(p,d,q)

ARIMA example （ be based on python Realization ）

Model selection AIC And BIC: Choose a simpler model

• AIC: Red pool information criterion （Akaike Information Criterion,AIC）
  ??? = 2? − 2ln(?)
• BIC: Bayesian information rule （Bayesian Information Criterion,BIC）
  ??? = ??? ? − 2ln(?)
• k There are several parameters in the model ,n Is the number of samples ,L Is the likelihood function

Model residual test ：

• ARIMA Whether the residual error of the model is the average value 0 And the variance is a constant normal distribution
• QQ chart ： Linear or normal distribution

Model selection and residual examples