The ARIMA model for time series analysis and forecasting can be tricky to configure. We can automate the process of evaluating a large number of hyperparameters for the ARIMA model by using a grid search procedure.
In this tutorial, you will discover how to tune the ARIMA model using a grid search of hyperparameters in Python.
After completing this tutorial, you will know:
- A general procedure that you can use to tune the ARIMA hyperparameters for a rolling one-step forecast.
- How to apply ARIMA hyperparameter optimization on a standard univariate time series dataset.
- Ideas for extending the procedure for more elaborate and robust models.
A. Grid Searching Method
We can automate the process of training and evaluating ARIMA models on different combinations of model hyperparameters. In machine learning this is called a grid search or model tuning.
The approach is broken down into two parts:
- Evaluate an ARIMA model.
- Evaluate sets of ARIMA parameters
B. Evaluate ARIMA Model
We can evaluate an ARIMA model by preparing it on a training dataset and evaluating predictions on a test dataset. This approach involves the following steps:
1. Split the dataset into training and test sets.
2. Walk the time steps in the test dataset.
(a) Train an ARIMA model.
(b) Make a one-step prediction.
(c) Store prediction; get and store actual observation.
3. Calculate error score for predictions compared to expected values
# evaluate an ARIMA model for a given order (p,d,q)
def evaluate_arima_model(X, arima_order):
# prepare training dataset
train_size = int(len(X) * 0.66)
train, test = X[0:train_size], X[train_size:]
history = [x for x in train]
# make predictions
predictions = list()
for t in range(len(test)):
model = ARIMA(history, order=arima_order)
model_fit = model.fit(disp=0)
yhat = model_fit.forecast()[0]
predictions.append(yhat)
history.append(test[t])
# calculate out-of-sample error
rmse = sqrt(mean_squared_error(test, predictions))
return rmse
C. Iterate ARIMA Parameters
Evaluating a suite of parameters is relatively straightforward. The user must specify a grid of p, d, and q ARIMA parameters to iterate. A model is created for each parameter and its performance evaluated by calling the evaluate_arima_model() function.
# evaluate combinations of p, d and q values for an ARIMA model
def evaluate_models(dataset, p_values, d_values, q_values):
dataset = dataset.astype('float32')
best_score, best_cfg = float("inf"), None
for p in p_values:
for d in d_values:
for q in q_values:
order = (p,d,q)
try:
rmse = evaluate_arima_model(dataset, order)
if rmse < best_score:
best_score, best_cfg = rmse, order
print('ARIMA%s RMSE=%.3f' % (order,rmse))
except:
continue
print('Best ARIMA%s RMSE=%.3f' % (best_cfg, best_score))
D. Shampoo Sales Case Study
# grid search ARIMA parameters for time series
import warnings
from math import sqrt
from pandas import read_csv
from pandas import datetime
from statsmodels.tsa.arima_model import ARIMA
from sklearn.metrics import mean_squared_error
# evaluate an ARIMA model for a given order (p,d,q)
def evaluate_arima_model(X, arima_order):
# prepare training dataset
train_size = int(len(X) * 0.66)
train, test = X[0:train_size], X[train_size:]
history = [x for x in train]
# make predictions
predictions = list()
for t in range(len(test)):
model = ARIMA(history, order=arima_order)
model_fit = model.fit(disp=0)
yhat = model_fit.forecast()[0]
predictions.append(yhat)
history.append(test[t])
# calculate out of sample error
rmse = sqrt(mean_squared_error(test, predictions))
return rmse
# evaluate combinations of p, d and q values for an ARIMA model
def evaluate_models(dataset, p_values, d_values, q_values):
dataset = dataset.astype('float32')
best_score, best_cfg = float("inf"), None
for p in p_values:
for d in d_values:
for q in q_values:
order = (p,d,q)
try:
rmse = evaluate_arima_model(dataset, order)
if rmse < best_score:
best_score, best_cfg = rmse, order
print('ARIMA%s RMSE=%.3f' % (order,rmse))
except:
continue
print('Best ARIMA%s RMSE=%.3f' % (best_cfg, best_score))
# load dataset
def parser(x):
return datetime.strptime('190'+x, '%Y-%m')
series = read_csv('shampoo-sales.csv', header=0, index_col=0, parse_dates=True, squeeze=True, date_parser=parser)
# evaluate parameters
p_values = [0, 1, 2, 4, 6, 8, 10]
d_values = range(0, 3)
q_values = range(0, 3)
warnings.filterwarnings("ignore")
evaluate_models(series.values, p_values, d_values, q_values)
-----Result-----
ARIMA(0, 0, 0) RMSE=228.966
ARIMA(0, 0, 1) RMSE=195.308
ARIMA(0, 0, 2) RMSE=154.886
ARIMA(0, 1, 0) RMSE=134.176
ARIMA(0, 1, 1) RMSE=97.767
ARIMA(0, 2, 0) RMSE=259.499
ARIMA(0, 2, 1) RMSE=135.363
ARIMA(1, 0, 0) RMSE=152.029
ARIMA(1, 1, 0) RMSE=84.388
ARIMA(1, 1, 1) RMSE=83.688
ARIMA(1, 2, 0) RMSE=136.411
ARIMA(2, 1, 0) RMSE=75.432
ARIMA(2, 1, 1) RMSE=88.089
ARIMA(2, 2, 0) RMSE=99.302
ARIMA(4, 1, 0) RMSE=81.545
ARIMA(4, 1, 1) RMSE=82.440
ARIMA(4, 2, 0) RMSE=87.157
ARIMA(4, 2, 1) RMSE=68.519
ARIMA(6, 1, 0) RMSE=82.523
ARIMA(6, 2, 0) RMSE=79.127
ARIMA(8, 0, 0) RMSE=85.182
ARIMA(8, 1, 0) RMSE=81.114
Best ARIMA(4, 2, 1) RMSE=68.519
The best parameters of ARIMA(4,2,1) are reported at the end of the run with a root mean squared error of 68.519 sales.
E. Daily Female Births Case Study
# grid search ARIMA parameters for time series
import warnings
from math import sqrt
from pandas import read_csv
from pandas import datetime
from statsmodels.tsa.arima_model import ARIMA
from sklearn.metrics import mean_squared_error
# evaluate an ARIMA model for a given order (p,d,q)
def evaluate_arima_model(X, arima_order):
# prepare training dataset
train_size = int(len(X) * 0.66)
train, test = X[0:train_size], X[train_size:]
history = [x for x in train]
# make predictions
predictions = list()
for t in range(len(test)):
model = ARIMA(history, order=arima_order)
model_fit = model.fit(disp=0)
yhat = model_fit.forecast()[0]
predictions.append(yhat)
history.append(test[t])
# calculate out of sample error
rmse = sqrt(mean_squared_error(test, predictions))
return rmse
# evaluate combinations of p, d and q values for an ARIMA model
def evaluate_models(dataset, p_values, d_values, q_values):
dataset = dataset.astype('float32')
best_score, best_cfg = float("inf"), None
for p in p_values:
for d in d_values:
for q in q_values:
order = (p,d,q)
try:
rmse = evaluate_arima_model(dataset, order)
if rmse < best_score:
best_score, best_cfg = rmse, order
print('ARIMA%s RMSE=%.3f' % (order,rmse))
except:
continue
print('Best ARIMA%s RMSE=%.3f' % (best_cfg, best_score))
series = read_csv('daily-total-female-births.csv', header=0, index_col=0, parse_dates=True, squeeze=True)
# evaluate parameters
p_values = [0, 1, 2, 4, 6, 8, 10]
d_values = range(0, 3)
q_values = range(0, 3)
warnings.filterwarnings("ignore")
evaluate_models(series.values, p_values, d_values, q_values)
-----Result-----
ARIMA(0, 0, 0) RMSE=8.189
ARIMA(0, 0, 1) RMSE=7.884
ARIMA(0, 0, 2) RMSE=7.771
ARIMA(0, 1, 0) RMSE=9.167
ARIMA(0, 1, 1) RMSE=7.527
ARIMA(0, 1, 2) RMSE=7.434
ARIMA(0, 2, 0) RMSE=15.698
ARIMA(0, 2, 1) RMSE=9.201
ARIMA(1, 0, 0) RMSE=7.802
ARIMA(1, 1, 0) RMSE=8.120
ARIMA(1, 1, 1) RMSE=7.425
ARIMA(1, 1, 2) RMSE=7.429
ARIMA(1, 2, 0) RMSE=11.990
ARIMA(2, 0, 0) RMSE=7.697
ARIMA(2, 1, 0) RMSE=7.713
ARIMA(2, 1, 1) RMSE=7.417
ARIMA(2, 2, 0) RMSE=10.373
ARIMA(4, 0, 0) RMSE=7.693
ARIMA(4, 1, 0) RMSE=7.578
ARIMA(4, 1, 1) RMSE=7.474
ARIMA(4, 2, 0) RMSE=8.956
ARIMA(6, 0, 0) RMSE=7.666
ARIMA(6, 1, 0) RMSE=7.293
ARIMA(6, 1, 1) RMSE=7.553
ARIMA(6, 2, 0) RMSE=8.352
ARIMA(8, 0, 0) RMSE=7.549
ARIMA(8, 1, 0) RMSE=7.569
ARIMA(8, 2, 0) RMSE=8.126
ARIMA(8, 2, 1) RMSE=7.608
ARIMA(10, 0, 0) RMSE=7.581
ARIMA(10, 1, 0) RMSE=7.574
ARIMA(10, 2, 0) RMSE=8.093
ARIMA(10, 2, 1) RMSE=7.608
ARIMA(10, 2, 2) RMSE=7.636
Best ARIMA(6, 1, 0) RMSE=7.293
The mean best parameters are reported as ARIMA(6,1,0) with a root mean squared error of 7.293 births.
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