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SARIMA: Robust Seasonal ARIMA

The SARIMA model (Seasonal AutoRegressive Integrated Moving Average) extends the robust forecasting capabilities of Gnostic ARIMA to data with seasonal patterns. It supports both non-seasonal \((p,d,q)\) and seasonal \((P,D,Q,s)\) components, estimated using robust Gnostic Weighted Least Squares.

Overview

Machine Gnostics SARIMA allows for robust modeling of complex time series that exhibit periodic behavior (seasonality), such as monthly sales or daily temperatures, while maintaining resilience against outliers.

  • Seasonal Integrated (D): Handles seasonal non-stationarity (e.g., year-over-year growth).
  • Seasonal AR/MA (P, Q): Models relationships at seasonal lags.
  • Robust Estimation: Down-weights outliers that might distort seasonal pattern detection.
  • Recursive Forecasting: Supports multi-step ahead prediction by unrolling the differencing layers.

Key Features

  • Robust SARIMA(p,d,q)x(P,D,Q,s) modeling
  • Automatic handling of seasonal differencing
  • Supports Constant ('c') and Constant+Linear ('ct') trends
  • Robust to outliers via Gnostic Weights
  • Recursive multi-step forecasting

Parameters

Parameter Type Default Description
order tuple (1, 0, 0) The non-seasonal \((p, d, q)\) order.
seasonal_order tuple (0, 0, 0, 0) The seasonal \((P, D, Q, s)\) order. \(s\) is the periodicity.
trend str 'c' Trend type: 'c' (bias), 'ct' (linear), 'n' (none).
optimize bool False (Experimental) Auto-select orders.
scale str | float 'auto' Scaling method for gnostic calculations.
max_iter int 100 Maximum reweighting iterations.
tolerance float 1e-3 Convergence tolerance.
learning_rate float 0.1 Learning rate for weight updates.
mg_loss str 'hi' Gnostic loss function ('hi' or 'hj').
early_stopping bool True Stop early if converged.
history bool True Record training history.

Attributes

  • weights: np.ndarray
    • The final robust weights assigned to the training samples.
  • coefficients: np.ndarray
    • The fitted model parameters.
  • training_data_diff_: np.ndarray
    • The fully stationarized series (after both differences) used internally.
  • training_residuals_: np.ndarray
    • Estimated residuals from the training phase.

Methods

fit(y, X=None)

Fit the SARIMA model to the time series y.

Parameters

  • y: array-like
    • Target time series.
  • X: Ignored
    • Not used.

Returns

  • self: SARIMA

predict(steps=1)

Forecast future values recursively.

This method handles the complex inverse transformation of both regular (\(d\)) and seasonal (\(D\)) differencing to return forecasts on the original scale.

Parameters

  • steps: int
    • Number of time steps to forecast.

Returns

  • forecast: np.ndarray
    • Predicted values for the next steps.

score(y, X=None)

Evaluate the model on the training time series y. Returns the Robust R² calculated on the stationary (fully differenced) series.

Parameters

  • y: array-like
    • Time series to evaluate.

Returns

  • score: float
    • Robust R² score.

save(path)

Saves the trained model to disk using joblib.

  • path: str Directory path to save the model.

load(path)

Loads a previously saved model from disk.

  • path: str Directory path where the model is saved.

Returns

Instance of SARIMA with loaded parameters.

Example Usage

import numpy as np
from machinegnostics.models import SARIMA
import matplotlib.pyplot as plt

# Generate synthetic Seasonal Data (Period=12)
np.random.seed(42)
t = np.arange(120)
seasonal = np.sin(2 * np.pi * t / 12)
trend = 0.05 * t
noise = np.random.normal(0, 0.2, 120)
y = seasonal + trend + noise

# Add seasonal outlier
y[13] += 5.0 # Spike in month 2 of year 2

# Initialize SARIMA(1,0,0)x(1,1,0,12)
# Simple AR(1) with Seasonal Integration and Seasonal AR(1)
model = SARIMA(
    order=(1, 0, 0),
    seasonal_order=(1, 1, 0, 12),
    trend='c',
    verbose=True
)

# Fit
model.fit(y)

# Forecast next 24 steps (2 seasons)
forecast = model.predict(steps=24)

# Plot
plt.figure(figsize=(10, 6))
plt.plot(np.arange(120), y, label='History')
plt.plot(np.arange(120, 144), forecast, label='Forecast', color='red')
plt.title("Gnostic SARIMA Forecast")
plt.legend()
plt.show()

SARIMA

Notes

  • Initial Residuals: For MA terms, initial residuals are estimated using a high-order AR model covering the seasonal period.
  • Differencing: The model applies seasonal differencing (\(D\)) first, followed by regular differencing (\(d\)). Forecasts reverse this order.
  • Complexity: Higher seasonal orders (large \(P, Q, s\)) significantly increase the number of parameters and required history length.

Author: Nirmal Parmar