AutoARIMA Model
Stepbystep guide on using the AutoARIMA Model
with Statsforecast
.
Table of Contents
 What is AutoArima with StatsForecast?
 Definition of the Arima model
 Advantages of using AutoArima
 Loading libraries and data
 Explore data with the plot method
 Split the data into training and testing
 Implementation of AutoARIMA with StatsForecast
 Crossvalidation
 Model evaluation
 References
What is AutoArima with StatsForecast?
An autoARIMA is a time series model that uses an automatic process to select the optimal ARIMA (Autoregressive Integrated Moving Average) model parameters for a given time series. ARIMA is a widely used statistical model for modeling and predicting time series.
The process of automatic parameter selection in an autoARIMA model is performed using statistical and optimization techniques, such as the Akaike Information Criterion (AIC) and crossvalidation, to identify optimal values for autoregression, integration, and moving average parameters. of the ARIMA model.
Automatic parameter selection is useful because it can be difficult to determine the optimal parameters of an ARIMA model for a given time series without a thorough understanding of the underlying stochastic process that generates the time series. The autoARIMA model automates the parameter selection process and can provide a fast and effective solution for time series modeling and forecasting.
The statsforecast.models
library brings the
AutoARIMA
function from Python provides an implementation of autoARIMA that allows
to automatically select the optimal parameters for an ARIMA model given
a time series.
Definition of the Arima model
An Arima model (autoregressive integrated moving average) process is the combination of an autoregressive process AR(p), integration I(d), and the moving average process MA(q).
Just like the ARMA process, the ARIMA process states that the present value is dependent on past values, coming from the AR(p) portion, and past errors, coming from the MA(q) portion. However, instead of using the original series, denoted as yt, the ARIMA process uses the differenced series, denoted as $y'_{t}$. Note that $y'_{t}$ can represent a series that has been differenced more than once.
Therefore, the mathematical expression of the ARIMA(p,d,q) process states that the present value of the differenced series $y'_{t}$ is equal to the sum of a constant $C$, past values of the differenced series $\phi_{p}y'_{tp}$, the mean of the differenced series $\mu$, past error terms $\theta_{q}\varepsilon_{tq}$, and a current error term $\varepsilon_{t}$, as shown in equation
where $y'_{t}$ is the differenced series (it may have been differenced more than once). The “predictors” on the right hand side include both lagged values of $y_{t}$ and lagged errors. We call this an ARIMA( p,d,q) model, where
p  order of the autoregressive part 
d  degree of first differencing involved 
q  order of the moving average part 
The same stationarity and invertibility conditions that are used for autoregressive and moving average models also apply to an ARIMA model.
Many of the models we have already discussed are special cases of the ARIMA model, as shown in Table
Model  p d q  Differenced  Method 

Arima(0,0,0)  0 0 0  $y_t=Y_t$  White noise 
ARIMA (0,1,0)  0 1 0  $y_t = Y_t  Y_{t1}$  Random walk 
ARIMA (0,2,0)  0 2 0  $y_t = Y_t  2Y_{t1} + Y_{t2}$  Constant 
ARIMA (1,0,0)  1 0 0  $\hat Y_t = \mu + \Phi_1 Y_{t1} + \epsilon$  AR(1): AR(1): Firstorder regression model 
ARIMA (2, 0, 0)  2 0 0  $\hat Y_t = \Phi_0 + \Phi_1 Y_{t1} + \Phi_2 Y_{t2} + \epsilon$  AR(2): Secondorder regression model 
ARIMA (1, 1, 0)  1 1 0  $\hat Y_t = \mu + Y_{t1} + \Phi_1 (Y_{t1} Y_{t2})$  Differenced firstorder autoregressive model 
ARIMA (0, 1, 1)  0 1 1  $\hat Y_t = Y_{t1}  \Phi_1 e^{t1}$  Simple exponential smoothing 
ARIMA (0, 0, 1)  0 0 1  $\hat Y_t = \mu_0+ \epsilon_t  \omega_1 \epsilon_{t1}$  MA(1): Firstorder regression model 
ARIMA (0, 0, 2)  0 0 2  $\hat Y_t = \mu_0+ \epsilon_t  \omega_1 \epsilon_{t1}  \omega_2 \epsilon_{t2}$  MA(2): Secondorder regression model 
ARIMA (1, 0, 1)  1 0 1  $\hat Y_t = \Phi_0 + \Phi_1 Y_{t1}+ \epsilon_t  \omega_1 \epsilon_{t1}$  ARMA model 
ARIMA (1, 1, 1)  1 1 1  $\Delta Y_t = \Phi_1 Y_{t1} + \epsilon_t  \omega_1 \epsilon_{t1}$  ARIMA model 
ARIMA (1, 1, 2)  1 1 2  $\hat Y_t = Y_{t1} + \Phi_1 (Y_{t1}  Y_{t2} ) \Theta_1 e_{t1}  \Theta_1 e_{t1}$  Dampedtrend linear Exponential smoothing 
ARIMA (0, 2, 1) OR (0,2,2)  0 2 1  $\hat Y_t = 2 Y_{t1}  Y_{t2}  \Theta_1 e_{t1}  \Theta_2 e_{t2}$  Linear exponential smoothing 
Once we start combining components in this way to form more complicated models, it is much easier to work with the backshift notation. For example, Equation (1) can be written in backshift notation as:
Selecting appropriate values for p, d and q can be difficult. However,
the AutoARIMA()
function from statsforecast
will do it for you
automatically.
For more information here
Loading libraries and data
Using an AutoARIMA()
model to model and predict time series has
several advantages, including:

Automation of the parameter selection process: The
AutoARIMA()
function automates the ARIMA model parameter selection process, which can save the user time and effort by eliminating the need to manually try different combinations of parameters. 
Reduction of prediction error: By automatically selecting optimal parameters, the ARIMA model can improve the accuracy of predictions compared to manually selected ARIMA models.

Identification of complex patterns: The
AutoARIMA()
function can identify complex patterns in the data that may be difficult to detect visually or with other time series modeling techniques. 
Flexibility in the choice of the parameter selection methodology: The ARIMA Model can use different methodologies to select the optimal parameters, such as the Akaike Information Criterion (AIC), crossvalidation and others, which allows the user to choose the methodology that best suits their needs.
In general, using the AutoARIMA()
function can help improve the
efficiency and accuracy of time series modeling and forecasting,
especially for users who are inexperienced with manual parameter
selection for ARIMA models.
Main results
We compared accuracy and speed against
pmdarima, Rob Hyndman’s
forecast package and
Facebook’s Prophet. We used the
Daily
, Hourly
and Weekly
data from the M4
competition.
The following table summarizes the results. As can be seen, our
auto_arima
is the best model in accuracy (measured by the MASE
loss)
and time, even compared with the original implementation in R.
dataset  metric  auto_arima_nixtla  auto_arima_pmdarima [1]  auto_arima_r  prophet 

Daily  MASE  3.26  3.35  4.46  14.26 
Daily  time  1.41  27.61  1.81  514.33 
Hourly  MASE  0.92  —  1.02  1.78 
Hourly  time  12.92  —  23.95  17.27 
Weekly  MASE  2.34  2.47  2.58  7.29 
Weekly  time  0.42  2.92  0.22  19.82 
[1] The model auto_arima
from pmdarima
had a problem with Hourly
data. An issue was opened.
The following table summarizes the data details.
group  n_series  mean_length  std_length  min_length  max_length 

Daily  4,227  2,371  1,756  107  9,933 
Hourly  414  901  127  748  1,008 
Weekly  359  1,035  707  93  2,610 
Loading libraries and data
Tip
Statsforecast will be needed. To install, see instructions.
Next, we import plotting libraries and configure the plotting style.
Loading Data
observation_date  IPG3113N  

0  19720101  85.6945 
1  19720201  71.8200 
2  19720301  66.0229 
3  19720401  64.5645 
4  19720501  65.0100 
The input to StatsForecast is always a data frame in long format with three columns: unique_id, ds and y:

The
unique_id
(string, int or category) represents an identifier for the series. 
The
ds
(datestamp) column should be of a format expected by Pandas, ideally YYYYMMDD for a date or YYYYMMDD HH:MM:SS for a timestamp. 
The
y
(numeric) represents the measurement we wish to forecast.
ds  y  unique_id  

0  19720101  85.6945  1 
1  19720201  71.8200  1 
2  19720301  66.0229  1 
3  19720401  64.5645  1 
4  19720501  65.0100  1 
We need to convert ds
from the object
type to datetime.
Explore data with the plot method
Plot a series using the plot method from the StatsForecast class. This method prints a random series from the dataset and is useful for basic EDA.
Autocorrelation plots
Decomposition of the time series
How to decompose a time series and why?
In time series analysis to forecast new values, it is very important to know past data. More formally, we can say that it is very important to know the patterns that values follow over time. There can be many reasons that cause our forecast values to fall in the wrong direction. Basically, a time series consists of four components. The variation of those components causes the change in the pattern of the time series. These components are:
 Level: This is the primary value that averages over time.
 Trend: The trend is the value that causes increasing or decreasing patterns in a time series.
 Seasonality: This is a cyclical event that occurs in a time series for a short time and causes shortterm increasing or decreasing patterns in a time series.
 Residual/Noise: These are the random variations in the time series.
Combining these components over time leads to the formation of a time series. Most time series consist of level and noise/residual and trend or seasonality are optional values.
If seasonality and trend are part of the time series, then there will be effects on the forecast value. As the pattern of the forecasted time series may be different from the previous time series.
The combination of the components in time series can be of two types: * Additive * multiplicative
Additive time series
If the components of the time series are added to make the time series. Then the time series is called the additive time series. By visualization, we can say that the time series is additive if the increasing or decreasing pattern of the time series is similar throughout the series. The mathematical function of any additive time series can be represented by: $y(t) = level + Trend + seasonality + noise$
Multiplicative time series
If the components of the time series are multiplicative together, then the time series is called a multiplicative time series. For visualization, if the time series is having exponential growth or decline with time, then the time series can be considered as the multiplicative time series. The mathematical function of the multiplicative time series can be represented as.
$y(t) = Level * Trend * seasonality * Noise$
Split the data into training and testing
Let’s divide our data into sets 1. Data to train our AutoArima
model
2. Data to test our model
For the test data we will use the last 12 months to test and evaluate the performance of our model.
Now let’s plot the training data and the test data.
Implementation of AutoArima with StatsForecast
To also know more about the parameters of the functions of the
AutoARIMA Model
, they are listed below. For more information, visit
the
documentation
Load libraries
Instantiating Model
Import and instantiate the models. Setting the argument is sometimes tricky. This article on Seasonal periods) by the master, Rob Hyndmann, can be useful.season_length
We fit the models by instantiating a new StatsForecast object with the following parameters:
models: a list of models. Select the models you want from models and import them.

freq:
a string indicating the frequency of the data. (See panda’s available frequencies.) 
n_jobs:
n_jobs: int, number of jobs used in the parallel processing, use 1 for all cores. 
fallback_model:
a model to be used if a model fails.
Any settings are passed into the constructor. Then you call its fit method and pass in the historical data frame.
Fit the Model
Once we have entered our model, we can use the
arima_string
function to see the parameters that the model has found.
The automation process gave us that the best model found is a model of
the form ARIMA(1,0,0)(0,1,2)[12]
, this means that our model contains
$p=1$ , that is, it has a nonseasonal autogressive element, on the
other hand, our model contains a seasonal part, which has an order of
$D=1$, that is, it has a seasonal differential, and $q=2$ that contains
2 moving average element.
To know the values of the terms of our model, we can use the following statement to know all the result of the model made.
Let us now visualize the residuals of our models.
As we can see, the result obtained above has an output in a dictionary,
to extract each element from the dictionary we are going to use the
.get()
function to extract the element and then we are going to save
it in a pd.DataFrame()
.
residual Model  

0  0.085694 
1  0.071820 
2  0.066023 
…  … 
533  1.258873 
534  1.585062 
535  6.199166 
To generate forecasts we only have to use the predict method specifying the forecast horizon (h). In addition, to calculate prediction intervals associated to the forecasts, we can include the parameter level that receives a list of levels of the prediction intervals we want to build. In this case we will only calculate the 90% forecast interval (level=[90]).
Forecast Method
If you want to gain speed in productive settings where you have multiple
series or models we recommend using the
StatsForecast.forecast
method instead of .fit
and .predict
.
The main difference is that the .forecast
doest not store the fitted
values and is highly scalable in distributed environments.
The forecast method takes two arguments: forecasts next h
(horizon)
and level
.

h (int):
represents the forecast h steps into the future. In this case, 12 months ahead. 
level (list of floats):
this optional parameter is used for probabilistic forecasting. Set the level (or confidence percentile) of your prediction interval. For example,level=[90]
means that the model expects the real value to be inside that interval 90% of the times.
The forecast object here is a new data frame that includes a column with
the name of the model and the y hat values, as well as columns for the
uncertainty intervals. Depending on your computer, this step should take
around 1min. (If you want to speed things up to a couple of seconds,
remove the AutoModels like
ARIMA
and
Theta
)
ds  AutoARIMA  

unique_id  
1  20160901  109.955437 
1  20161001  121.920509 
1  20161101  122.458389 
1  20161201  120.562027 
1  20170101  106.864670 
ds  y  AutoARIMA  

unique_id  
1  19720101  85.694504  85.608803 
1  19720201  71.820000  71.748177 
1  19720301  66.022903  65.956879 
…  …  …  … 
1  20160601  102.404404  101.145523 
1  20160701  102.951202  101.366135 
1  20160801  104.697701  110.896866 
Adding 95% confidence interval with the forecast method
ds  AutoARIMA  AutoARIMAlo95  AutoARIMAhi95  

unique_id  
1  20160901  109.955437  102.116188  117.794685 
1  20161001  121.920509  112.380608  131.460403 
1  20161101  122.458389  112.200500  132.716278 
…  …  …  …  … 
1  20170601  96.751160  85.873802  107.628525 
1  20170701  97.451607  86.572372  108.330833 
1  20170801  103.420616  92.540489  114.300743 
unique_id  ds  AutoARIMA  

0  1  20160901  109.955437 
1  1  20161001  121.920509 
2  1  20161101  122.458389 
…  …  …  … 
9  1  20170601  96.751160 
10  1  20170701  97.451607 
11  1  20170801  103.420616 
Predict method with confidence interval
To generate forecasts use the predict method.
The predict method takes two arguments: forecasts the next h
(for
horizon) and level
.

h (int):
represents the forecast h steps into the future. In this case, 12 months ahead. 
level (list of floats):
this optional parameter is used for probabilistic forecasting. Set the level (or confidence percentile) of your prediction interval. For example,level=[95]
means that the model expects the real value to be inside that interval 95% of the times.
The forecast object here is a new data frame that includes a column with the name of the model and the y hat values, as well as columns for the uncertainty intervals.
This step should take less than 1 second.
ds  AutoARIMA  

unique_id  
1  20160901  109.955437 
1  20161001  121.920509 
1  20161101  122.458389 
…  …  … 
1  20170601  96.751160 
1  20170701  97.451607 
1  20170801  103.420616 
ds  AutoARIMA  AutoARIMAlo95  AutoARIMAlo80  AutoARIMAhi80  AutoARIMAhi95  

unique_id  
1  20160901  109.955437  102.116188  104.829628  115.081245  117.794685 
1  20161001  121.920509  112.380608  115.682701  128.158310  131.460403 
1  20161101  122.458389  112.200500  115.751114  129.165665  132.716278 
…  …  …  …  …  …  … 
1  20170601  96.751160  85.873802  89.638840  103.863487  107.628525 
1  20170701  97.451607  86.572372  90.338058  104.565147  108.330833 
1  20170801  103.420616  92.540489  96.306480  110.534752  114.300743 
We can join the forecast result with the historical data using the
pandas function pd.concat()
, and then be able to use this result for
graphing.
y  unique_id  AutoARIMA  AutoARIMAlo95  AutoARIMAlo80  AutoARIMAhi80  AutoARIMAhi95  

ds  
20000501  108.7202  1  NaN  NaN  NaN  NaN  NaN 
20000601  114.2071  1  NaN  NaN  NaN  NaN  NaN 
20000701  111.8737  1  NaN  NaN  NaN  NaN  NaN 
…  …  …  …  …  …  …  … 
20170601  NaN  NaN  96.751160  85.873802  89.638840  103.863487  107.628525 
20170701  NaN  NaN  97.451607  86.572372  90.338058  104.565147  108.330833 
20170801  NaN  NaN  103.420616  92.540489  96.306480  110.534752  114.300743 
Now let’s visualize the result of our forecast and the historical data of our time series, also let’s draw the confidence interval that we have obtained when making the prediction with 95% confidence.
Let’s plot the same graph using the plot function that comes in
Statsforecast
, as shown below.
Crossvalidation
In previous steps, we’ve taken our historical data to predict the future. However, to asses its accuracy we would also like to know how the model would have performed in the past. To assess the accuracy and robustness of your models on your data perform CrossValidation.
With time series data, Cross Validation is done by defining a sliding window across the historical data and predicting the period following it. This form of crossvalidation allows us to arrive at a better estimation of our model’s predictive abilities across a wider range of temporal instances while also keeping the data in the training set contiguous as is required by our models.
The following graph depicts such a Cross Validation Strategy:
Perform time series crossvalidation
Crossvalidation of time series models is considered a best practice but most implementations are very slow. The statsforecast library implements crossvalidation as a distributed operation, making the process less timeconsuming to perform. If you have big datasets you can also perform Cross Validation in a distributed cluster using Ray, Dask or Spark.
In this case, we want to evaluate the performance of each model for the
last 5 months (n_windows=5)
, forecasting every second months
(step_size=12)
. Depending on your computer, this step should take
around 1 min.
The cross_validation method from the StatsForecast class takes the following arguments.

df:
training data frame 
h (int):
represents h steps into the future that are being forecasted. In this case, 12 months ahead. 
step_size (int):
step size between each window. In other words: how often do you want to run the forecasting processes. 
n_windows(int):
number of windows used for cross validation. In other words: what number of forecasting processes in the past do you want to evaluate.
The crossvaldation_df object is a new data frame that includes the following columns:
unique_id:
index. If you dont like working with index just run crossvalidation_df.resetindex()ds:
datestamp or temporal indexcutoff:
the last datestamp or temporal index for the n_windows.y:
true value"model":
columns with the model’s name and fitted value.
ds  cutoff  y  AutoARIMA  

unique_id  
1  20110901  20110801  93.906197  104.758850 
1  20111001  20110801  116.763397  118.705879 
1  20111101  20110801  116.825798  116.834129 
1  20111201  20110801  114.956299  117.070084 
1  20120101  20110801  99.966202  103.552246 
Model Evaluation
We can now compute the accuracy of the forecast using an appropiate
accuracy metric. Here we’ll use the Root Mean Squared Error (RMSE). To
do this, we first need to install datasetsforecast
, a Python library
developed by Nixtla that includes a function to compute the RMSE.
The function to compute the RMSE takes two arguments:
 The actual values.
 The forecasts, in this case, AutoArima.
As you have noticed, we have used the cross validation results to perform the evaluation of our model.
Now we are going to evaluate our model with the results of the predictions, we will use different types of metrics MAE, MAPE, MASE, RMSE, SMAPE to evaluate the accuracy.
mae  mape  mase  rmse  smape  

AutoARIMA  5.26042  4.794312  1.015379  6.021264  4.915602 
Acknowledgements
We would like to thank Naren Castellon for writing this tutorial.