Time Series SARIMA – Rainshtaryl chan-nel

May 28, 2024 | 14:24

Why differentiation works

For time series :
- $E[X_t] = \beta_0 +\beta_1t + E[W_t]$ and it is dependent on t
- So X is not stationary
Let and it is a MA(1) model
- So Y is stationary
By using a difference, the nonstationary shits can be trandformed to stationary shits
Formal shits:
- backshift operator
- $Βx_t:=x_{t-1}$ → $B^kx_t = x_{t-k}$
- difference of order d
- - $d=1: Y_t = X_t-X_{t-1} = (1-B)X_t$
  - $d=2: Y_t = (1-B)^2X_t = X_t-2X_{t-1}+X_{t-2}$
  - …

ARIMA model

steps:
- Assume $X_t$ is not stationary, and by using $Y_t = (1-B)^dX_t$ , it become stationary
- Fit an ARMA(p, q) on $Y_t$ for forcasting, which is equivalent to fitting an ARIMA(p,d,q) model to $X_t$
- Use forecasts from $Y_t$ with linear transofrmation to predict $X_t$ values
example:
- Suppose $Y_t=(1-B)X_t$
- Forecast $Y_t$ by using an ARMA model $\hat Y_{n+1}, \hat Y_{n+2}, ...$
- Predict $X_t$ as: $\hat X_{n+1} = X_n + \hat Y_{n+1}, \hat X_{n+2} = \hat X_n+1 + \hat Y_{n+2}, ...$
Ljung-Box test:
- H0: Model does not exihibit lack of fit
- Ha: Model exhibits lack of fit
- idea is to look at AC residuals (actual-predicted), if they exhibit correlation, then the model has not fully captured the underlying strutcure of the TS data
Forecasting accuracy metrics:
- MAE $= \frac {1}{m} \sum^m_{t=1}|x_{n+t} - \hat x_{n+t}|$
- MA Percentage E $= \frac {1}{m} \sum^m_{t=1}|\frac{x_{n+t} - \hat x_{n+t}}{x_{n+t}}|$ x 100%
- Root MSE

Box-Cox transofrmation

Use thid transformation to transofrm a non-normal time series into a near normal one
how
- $y = \large \frac {x^\lambda - 1}{\lambda}$ if $\lambda ≠0$
- $y = log(x)$ if $\lambda = 0$
- $\lambda$ is estimated by finding the value that maximizes the log-likelyhood for the transformed data
This is used to reduce heteroskedasticity
steps:
- Hetero time series X can be transformed into homo series Y by choosing a value lambda
- Fit ARIMA model on Y and get Y hat
- Reverse the transform to get X hat

SARIMA model

SARIMA(p,d,q)(P,D,Q)s
- seasonal AR(P):
- ACF tails off at lags S, 2S, …
- PACF cuts off after lag PS
- seasonal MA(Q):
- ACF cuts off after lag QS
- PACF tails off at lags S, 2S, …
- seasonal ARMA(P,Q):
- ACF tails off at lags S, 2S, …
- PACF tails off at lags S, 2S, …
- Then p,q are estimated within the lag interval btwn 1 and S

Hybrid SARIMA-regression model

Combine SARIMA with external variables by regressing over it and in some cases it kicks fucking ass oh yeah
For time series :
- $E[X_t] = \beta_0 +\beta_1t + \beta_2E[W_{t-1}]+ E[W_t]$
- This value is dependent on t, so $X_t$ is not stationary
$Y_t = X_t-X_{t-1} = \beta_0+\beta_1t+\beta_2W_{t-1}+W_t - \beta_0-\beta_1(t-1)-\beta_2W_{t-2}-W_{t-1}$ $= \beta_1+W_t+(\beta_2-1)W_{t-1}-\beta_2W_{t-2}$
By definition, $Y_t$ is a MA(2) model. MA models are always stationary.

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