关于高振动异方差离散时间序列适配AR/ARMA模型的可行性与预处理方法的技术问询

Great question—dealing with non-stationary, high-vibration time series with heteroskedasticity can be tricky, especially when standard preprocessing steps haven't worked. Let's break this down:

Can You Fit AR/ARMA Models to This Series? #

Short answer: Not directly in its current form, since AR/ARMA requires weak stationarity (constant mean, constant variance, autocovariance dependent only on lag length). Your series has heteroskedasticity (time-varying variance) and failed standard stationarization steps, so plain AR/ARMA would produce unreliable forecasts and parameter estimates.

That said, you can still use AR/ARMA as part of a hybrid approach (more on that later) if you first address the non-stationarity and heteroskedasticity with targeted preprocessing.

Targeted Preprocessing to Enable AR/ARMA #

If you're set on using AR/ARMA, try these additional preprocessing steps beyond what you've already tested:

• Detect and account for structural breaks: Your high vibration might stem from unobserved structural changes (e.g., regime shifts). Use tests like the Bai-Perron multiple break test to identify break points, then split the series into stationary segments and fit AR/ARMA to each segment separately. You can also use rolling-window AR/ARMA to adapt to time-varying patterns.
• Alternative variance-stabilizing transformations: Box-Cox works well for positive, right-skewed data, but try:
  • Yeo-Johnson transformation (supports zero/negative values, more flexible than Box-Cox)
  • Square-root transformation (effective for heteroskedasticity where variance scales with the mean, common in count or volatility data)
• Combine heteroskedasticity filtering with differencing: First model the heteroskedasticity with a simple GARCH filter to get standardized residuals, then test those residuals for stationarity. If stationary, you can fit AR/ARMA to the filtered residuals.
• Advanced detrending: Instead of plain differencing, use LOESS smoothing or the Hodrick-Prescott (HP) filter to separate the trend component from the cyclical/noise component. Fit AR/ARMA to the cyclical residuals, which may be stationary once the trend is removed.

Alternative Time Series Techniques If AR/ARMA Isn't Feasible #

If even targeted preprocessing doesn't yield a stationary series, these methods are better suited for your high-vibration, heteroskedastic data:

• ARMA-GARCH Hybrid Models: This is the gold standard for heteroskedastic time series. Fit an ARMA model to capture the mean (autocorrelated) component, then a GARCH (or EGARCH, TGARCH) model to capture the time-varying variance (heteroskedasticity). This handles both non-stationarity in the mean and volatility clustering.
• Nonlinear Autoregressive Models: Threshold Autoregressive (TAR) or Smooth Transition Autoregressive (STAR) models account for nonlinear regime shifts (e.g., sudden spikes in vibration). They let the autoregressive parameters change based on a threshold (e.g., past values exceeding a certain level), which fits high-vibration behavior.
• State Space Models with Kalman Filter: These models allow for time-varying parameters (mean, variance, autoregressive coefficients), making them ideal for non-stationary, heteroskedastic series. They can adapt to changing patterns in real time and handle missing data if needed.
• Recurrent Neural Networks (RNNs): LSTMs or GRUs don't require strict stationarity assumptions. They can learn complex, nonlinear temporal dependencies and capture heteroskedasticity directly from the data, especially useful if your series has high dimensionality or non-obvious patterns.

Practical Tips to Get Started #

• Double-check stationarity tests: Use both ADF (tests for unit root) and KPSS (tests for stationarity) to confirm whether your non-stationarity comes from a unit root (trend) or heteroskedasticity. This will guide your preprocessing choice.
• Diagnose heteroskedasticity: Plot the squared residuals of a simple AR model to see if volatility clusters (groups of high/low variance). If clusters exist, GARCH is your best bet; if variance increases/decreases with the mean, focus on variance-stabilizing transformations.

内容的提问来源于stack exchange，提问作者Eric94