Get better fit on test data using Auto_Arima

Question

I am using the AirPassengers dataset to predict a timeseries. For the model I am using, I chosen to use auto_arima to forecast the predicted values. However, it seems that the chosen order by the auto_arima is unable to fit the model. The corresponding chart is produced.

What can I do to get a better fit?

My code for those that want to try:

import pandas as pd
import numpy as np
import matplotlib.pylab as plt
%matplotlib inline

from pmdarima import auto_arima

df = pd.read_csv("https://raw.githubusercontent.com/AileenNielsen/TimeSeriesAnalysisWithPython/master/data/AirPassengers.csv")
df = df.rename(columns={"#Passengers":"Passengers"})
df.Month = pd.to_datetime(df.Month)
df.set_index('Month',inplace=True)

train,test=df[:-24],df[-24:]

model = auto_arima(train,trace=True,error_action='ignore', suppress_warnings=True)
model.fit(train)

forecast = model.predict(n_periods=24)
forecast = pd.DataFrame(forecast,index = test.index,columns=['Prediction'])

plt.plot(train, label='Train')
plt.plot(test, label='Valid')
plt.plot(forecast, label='Prediction')
plt.show()

from sklearn.metrics import mean_squared_error
print(mean_squared_error(test['Passengers'],forecast['Prediction']))

Thank you for reading. Any advice is appreciated.

Answer 1

This series is not stationary, and no amount of differencing (notice that the amplitude of the variations keeps increasing) will make it so. However, transforming the data first by taking logs should do better (experiment shows that it does do better, but not what I would call well). Setting the seasonality (as I suggest in the comment by m=12, and taking logs produces this: which is essentially perfect.