> ## Documentation Index > Fetch the complete documentation index at: https://nixtlaverse.nixtla.io/llms.txt > Use this file to discover all available pages before exploring further. # SARIMA SARIMA — Seasonal AutoRegressive Integrated Moving Average — is the workhorse linear model for series with autocorrelation, trend, and seasonality. Generating from a *known* `(p, d, q)(P, D, Q)` specification lets you confirm a model recovers the order you put in, or build panels with a precise, well-understood dependence structure. > **The model** > > $\phi(B)\,\Phi(B^s)\,(1-B)^d (1-B^s)^D\, y_t = \theta(B)\,\Theta(B^s)\, \varepsilon_t$ > > An ARIMA(p, d, q) process combines `p` autoregressive lags, `d` orders > of differencing (for trend / unit roots), and `q` moving-average lags; > the seasonal part `(P, D, Q)` repeats that structure at the seasonal > period. Set the orders and coefficients to dial in anything from white > noise to a strongly seasonal, integrated series. ```python theme={null} import polars as pl import matplotlib.pyplot as plt from synforecast.generators import SARIMAGenerator ``` ## 1. Basic SARIMA(2,1,1)(1,1,1)\_7 A SARIMA model with weekly seasonality, suitable for daily data. ```python theme={null} params = { "min_length": 200, "max_length": 200, "freq": "D", "p": 2, "d": 1, "q": 1, "P": 1, "D": 1, "Q": 1, "seasonal_period": 7, "noise_std": 2.0, "drift": 0.5, "seed": 42, } generator = SARIMAGenerator(engine="polars", **params) ``` ### Model information Inspect the auto-generated model parameters and polynomial structure. ```python theme={null} model_info = generator.get_model_info() print(f"Model: {model_info['model']}") print(f"AR parameters: {model_info['ar_params']}") print(f"MA parameters: {model_info['ma_params']}") print(f"Seasonal AR parameters: {model_info['seasonal_ar_params']}") print(f"Seasonal MA parameters: {model_info['seasonal_ma_params']}") print(f"Drift: {model_info['drift']}") print(f"Burn-in: {model_info['burn_in']}") print(f"\nExpanded AR polynomial lags: {model_info['full_ar_polynomial_lags']}") print(f"Expanded AR polynomial coeffs: {model_info['full_ar_polynomial_coeffs']}") ``` ```text theme={null} Model: SARIMA(2,1,1)(1,1,1)[7] AR parameters: [0.2298813963661166, -0.04889724819835817] MA parameters: [0.35859791991138246] Seasonal AR parameters: [0.15789442324749114] Seasonal MA parameters: [-0.40582265211235047] Drift: 0.5 Burn-in: 100 Expanded AR polynomial lags: [1, 2, 3, 4, 5, 6, 7, 8, 9] Expanded AR polynomial coeffs: [0.2298813963661166, -0.04889724819835817, -0.0, -0.0, -0.0, -0.0, 0.15789442324749114, -0.036296990494555884, 0.007720602802669188] ``` ### Generate and inspect data ```python theme={null} df = generator.generate(n_series=3) print(f"Generated {df['unique_id'].n_unique()} time series") print(f"Total observations: {len(df)}") df.head(10) ``` ```text theme={null} Generated 3 time series Total observations: 600 ``` | unique\_id | ds | y | | ---------- | ------------------- | ---------- | | cat | datetime\[ns] | f64 | | "0" | 2000-01-01 00:00:00 | 840.568595 | | "0" | 2000-01-02 00:00:00 | 849.409081 | | "0" | 2000-01-03 00:00:00 | 852.256521 | | "0" | 2000-01-04 00:00:00 | 864.365821 | | "0" | 2000-01-05 00:00:00 | 890.874551 | | "0" | 2000-01-06 00:00:00 | 904.417479 | | "0" | 2000-01-07 00:00:00 | 912.95809 | | "0" | 2000-01-08 00:00:00 | 927.189939 | | "0" | 2000-01-09 00:00:00 | 941.521277 | | "0" | 2000-01-10 00:00:00 | 946.445261 | ```python theme={null} fig, ax = plt.subplots(figsize=(12, 4)) for uid in df["unique_id"].unique().to_list(): series = df.filter(pl.col("unique_id") == uid) ax.plot(series["ds"].to_list(), series["y"].to_list(), label=uid, alpha=0.8) ax.set_xlabel("Timestamp") ax.set_ylabel("Value") ax.set_title("SARIMA(2,1,1)(1,1,1)_7 series") ax.legend() plt.tight_layout() plt.show() ``` ### Statistics by series ```python theme={null} df.group_by("unique_id").agg( [ pl.col("y").count().alias("count"), pl.col("y").min().alias("min_value"), pl.col("y").max().alias("max_value"), pl.col("y").mean().alias("mean_value"), pl.col("y").std().alias("std_value"), ] ).sort("unique_id") ``` | unique\_id | count | min\_value | max\_value | mean\_value | std\_value | | ---------- | ----- | ---------- | ----------- | ----------- | ----------- | | cat | u32 | f64 | f64 | f64 | f64 | | "0" | 200 | 840.568595 | 4716.719654 | 2557.619461 | 1132.631603 | | "1" | 200 | 628.393 | 4135.85017 | 2221.617457 | 1020.787936 | | "2" | 200 | 452.878919 | 3437.538082 | 1778.004889 | 857.699315 | ## 2. Stationary ARMA(1,1) with custom parameters A stationary model with no differencing and explicit AR/MA coefficients. ```python theme={null} params_arma = { "min_length": 200, "max_length": 200, "freq": "D", "p": 1, "d": 0, "q": 1, "P": 0, "D": 0, "Q": 0, "ar_params": [0.7], "ma_params": [0.3], "mean": 50.0, "noise_std": 1.0, "seed": 123, } generator_arma = SARIMAGenerator(engine="polars", **params_arma) df_arma = generator_arma.generate(n_series=1) print(f"Model: {generator_arma.get_model_info()['model']}") print(f"Mean: {generator_arma.get_model_info()['mean']}") print(f"Series mean: {df_arma['y'].mean():.2f}") print(f"Series std: {df_arma['y'].std():.2f}") ``` ```text theme={null} Model: ARIMA(1,0,1) Mean: 50.0 Series mean: 50.35 Series std: 1.61 ``` ```python theme={null} fig, ax = plt.subplots(figsize=(12, 4)) for uid in df_arma["unique_id"].unique().to_list(): series = df_arma.filter(pl.col("unique_id") == uid) ax.plot(series["ds"].to_list(), series["y"].to_list(), label=uid, alpha=0.8) ax.set_xlabel("Timestamp") ax.set_ylabel("Value") ax.set_title("Stationary ARMA(1,1) series") ax.legend() plt.tight_layout() plt.show() ``` ## 3. Pure seasonal ARIMA(0,0,0)(1,1,1)\_12 A purely seasonal model with monthly frequency and 12-month seasonality. ```python theme={null} params_seasonal = { "min_length": 100, "max_length": 100, "freq": "MS", "p": 0, "d": 0, "q": 0, "P": 1, "D": 1, "Q": 1, "seasonal_period": 12, "seasonal_ar_params": [0.5], "seasonal_ma_params": [0.3], "noise_std": 1.0, "seed": 456, } generator_seasonal = SARIMAGenerator(engine="polars", **params_seasonal) df_seasonal = generator_seasonal.generate(n_series=1) print(f"Model: {generator_seasonal.get_model_info()['model']}") print("\nFirst 24 months:") df_seasonal.head(24) ``` ```text theme={null} Model: SARIMA(0,0,0)(1,1,1)[12] First 24 months: ``` | unique\_id | ds | y | | ---------- | ------------------- | --------- | | cat | datetime\[ns] | f64 | | "0" | 2000-01-01 00:00:00 | -0.846697 | | "0" | 2000-02-01 00:00:00 | 9.129227 | | "0" | 2000-03-01 00:00:00 | 7.281973 | | "0" | 2000-04-01 00:00:00 | -1.550024 | | "0" | 2000-05-01 00:00:00 | 4.399888 | | … | … | … | | "0" | 2001-08-01 00:00:00 | -4.021058 | | "0" | 2001-09-01 00:00:00 | 0.429695 | | "0" | 2001-10-01 00:00:00 | -3.538732 | | "0" | 2001-11-01 00:00:00 | -5.016018 | | "0" | 2001-12-01 00:00:00 | 9.350066 | ```python theme={null} fig, ax = plt.subplots(figsize=(12, 4)) for uid in df_seasonal["unique_id"].unique().to_list(): series = df_seasonal.filter(pl.col("unique_id") == uid) ax.plot(series["ds"].to_list(), series["y"].to_list(), label=uid, alpha=0.8) ax.set_xlabel("Timestamp") ax.set_ylabel("Value") ax.set_title("Pure seasonal ARIMA(0,0,0)(1,1,1)_12 series") ax.legend() plt.tight_layout() plt.show() ``` > **Related generators** > > * [ETS](ets) — the exponential-smoothing counterpart for trend and > seasonality. > * [Seasonal](seasonal) — a simpler fixed seasonal wave without ARMA > dynamics. > * [Random walk](random_walk) — the special case ARIMA(0, 1, 0). > > All orders and coefficients are documented in the [generator > reference](https://github.com/Nixtla/synforecast/blob/main/GENERATORS.md).