SARIMA models

Modelli ARIMA in Python

James Fulton

Climate informatics researcher

The SARIMA model

Seasonal ARIMA = SARIMA

Non-seasonal orders
- p: autoregressive order
- d: differencing order
- q: moving average order

SARIMA(p,d,q)(P,D,Q)$_S$

Seasonal Orders
- P: seasonal autoregressive order
- D: seasonal differencing order
- Q: seasonal moving average order
- S: number of time steps per cycle

The SARIMA model

ARIMA(2,0,1) model : $$y_t = a_1 y_{t-1} + a_2 y_{t-2} + m_1 \epsilon_{t-1} + \epsilon_t$$

SARIMA(0,0,0)(2,0,1)$_7$ model: $$y_t = a_7 y_{t-7} + a_{14} y_{t-14} + m_7 \epsilon_{t-7} + \epsilon_t$$

Fitting a SARIMA model

# Imports
statsmodels.tsa.statespace.sarimax import SARIMAX

# Instantiate model
model = SARIMAX(df, order=(p,d,q), seasonal_order=(P,D,Q,S))

# Fit model
results = model.fit()

Seasonal differencing

Subtract the time series value of one season ago

$$\Delta y_t = y_t - y_{t-S}$$

# Take the seasonal difference
df_diff = df.diff(S)

Differencing for SARIMA models

Time series

Differencing for SARIMA models

First difference of time series

Differencing for SARIMA models

First difference and first seasonal difference of time series

Finding p and q

Finding P and Q

Plotting seasonal ACF and PACF

# Create figure
fig, (ax1, ax2) = plt.subplots(2,1)

# Plot seasonal ACF
plot_acf(df_diff,  lags=[12,24,36,48,60,72], ax=ax1)

# Plot seasonal PACF
plot_pacf(df_diff, lags=[12,24,36,48,60,72], ax=ax2)

plt.show()

Let's practice!

Modelli ARIMA in Python