CS250: Python for Data Science

When looking to fit time series data with a seasonal ARIMA model, our first goal is to find the values of ARIMA(p,d,q)(P,D,Q)s that optimize a metric of interest. There are many guidelines and best practices to achieve this goal, yet the correct parametrization of ARIMA models can be a painstaking manual process that requires domain expertise and time. Other statistical programming languages such as R provide automated ways to solve this issue, but those have yet to be ported over to Python. In this section, we will resolve this issue by writing Python code to programmatically select the optimal parameter values for our ARIMA(p,d,q)(P,D,Q)s time series model.

We will use a "grid search" to iteratively explore different combinations of parameters. For each combination of parameters, we fit a new seasonal ARIMA model with the SARIMAX() function from the statsmodels module and assess its overall quality. Once we have explored the entire landscape of parameters, our optimal set of parameters will be the one that yields the best performance for our criteria of interest. Let's begin by generating the various combination of parameters that we wish to assess:

# Define the p, d and q parameters to take any value between 0 and 2
p = d = q = range(0, 2)

# Generate all different combinations of p, q and q triplets
pdq = list(itertools.product(p, d, q))

# Generate all different combinations of seasonal p, q and q triplets
seasonal_pdq = [(x[0], x[1], x[2], 12) for x in list(itertools.product(p, d, q))]

print('Examples of parameter combinations for Seasonal ARIMA...')
print('SARIMAX: {} x {}'.format(pdq[1], seasonal_pdq[1]))
print('SARIMAX: {} x {}'.format(pdq[1], seasonal_pdq[2]))
print('SARIMAX: {} x {}'.format(pdq[2], seasonal_pdq[3]))
print('SARIMAX: {} x {}'.format(pdq[2], seasonal_pdq[4]))

 
Output
Examples of parameter combinations for Seasonal ARIMA...
SARIMAX: (0, 0, 1) x (0, 0, 1, 12)
SARIMAX: (0, 0, 1) x (0, 1, 0, 12)
SARIMAX: (0, 1, 0) x (0, 1, 1, 12)
SARIMAX: (0, 1, 0) x (1, 0, 0, 12)

We can now use the triplets of parameters defined above to automate the process of training and evaluating ARIMA models on different combinations. In Statistics and Machine Learning, this process is known as grid search (or hyperparameter optimization) for model selection.

When evaluating and comparing statistical models fitted with different parameters, each can be ranked against one another based on how well it fits the data or its ability to accurately predict future data points. We will use the AIC (Akaike Information Criterion) value, which is conveniently returned with ARIMA models fitted using statsmodels. The AIC measures how well a model fits the data while taking into account the overall complexity of the model. A model that fits the data very well while using lots of features will be assigned a larger AIC score than a model that uses fewer features to achieve the same goodness of fit. Therefore, we are interested in finding the model that yields the lowest AIC value.

The code chunk below iterates through combinations of parameters and uses the SARIMAX function from statsmodels to fit the corresponding Seasonal ARIMA model. Here, the order argument specifies the (p, d, q) parameters, while the seasonal_order argument specifies the (P, D, Q, S) seasonal component of the Seasonal ARIMA model. After fitting each SARIMAX()model, the code prints out its respective AIC score.

warnings.filterwarnings("ignore") # specify to ignore warning messages

for param in pdq:
    for param_seasonal in seasonal_pdq:
        try:
            mod = sm.tsa.statespace.SARIMAX(y,
                                            order=param,
                                            seasonal_order=param_seasonal,
                                            enforce_stationarity=False,
                                            enforce_invertibility=False)

            results = mod.fit()

            print('ARIMA{}x{}12 - AIC:{}'.format(param, param_seasonal, results.aic))
        except:
            continue

Because some parameter combinations may lead to numerical misspecifications, we explicitly disabled warning messages in order to avoid an overload of warning messages. These misspecifications can also lead to errors and throw an exception, so we make sure to catch these exceptions and ignore the parameter combinations that cause these issues.

The code above should yield the following results, this may take some time:

Output
SARIMAX(0, 0, 0)x(0, 0, 1, 12) - AIC:6787.3436240402125
SARIMAX(0, 0, 0)x(0, 1, 1, 12) - AIC:1596.711172764114
SARIMAX(0, 0, 0)x(1, 0, 0, 12) - AIC:1058.9388921320026
SARIMAX(0, 0, 0)x(1, 0, 1, 12) - AIC:1056.2878315690562
SARIMAX(0, 0, 0)x(1, 1, 0, 12) - AIC:1361.6578978064144
SARIMAX(0, 0, 0)x(1, 1, 1, 12) - AIC:1044.7647912940095
...
...
...
SARIMAX(1, 1, 1)x(1, 0, 0, 12) - AIC:576.8647112294245
SARIMAX(1, 1, 1)x(1, 0, 1, 12) - AIC:327.9049123596742
SARIMAX(1, 1, 1)x(1, 1, 0, 12) - AIC:444.12436865161305
SARIMAX(1, 1, 1)x(1, 1, 1, 12) - AIC:277.7801413828764

The output of our code suggests that SARIMAX(1, 1, 1)x(1, 1, 1, 12) yields the lowest AIC value of 277.78. We should therefore consider this to be the optimal option out of all the models we have considered.