AutoCES Model
Stepbystep guide on using the AutoCES Model
with Statsforecast
.
Table of Contents
 Introduction
 Complex Exponential Smoothing
 Loading libraries and data
 Explore data with the plot method
 Split the data into training and testing
 Implementation of AutoCES with StatsForecast
 Crossvalidation
 Model evaluation
 References
Introduction
Exponential smoothing has been one of the most popular forecasting methods used to support various decisions in organizations, in activities such as inventory management, scheduling, revenue management, and other areas. Although its relative simplicity and transparency have made it very attractive for research and practice, identifying the underlying trend remains challenging with significant impact on the resulting accuracy. This has resulted in the development of various modifications of trend models, introducing a model selection problem. With the aim of addressing this problem, we propose the complex exponential smoothing (CES), based on the theory of functions of complex variables. The basic CES approach involves only two parameters and does not require a model selection procedure. Despite these simplifications, CES proves to be competitive with, or even superior to existing methods. We show that CES has several advantages over conventional exponential smoothing models: it can model and forecast both stationary and nonstationary processes, and CES can capture both level and trend cases, as defined in the conventional exponential smoothing classification. CES is evaluated on several forecasting competition datasets, demonstrating better performance than established benchmarks. We conclude that CES has desirable features for time series modeling and opens new promising avenues for research.
Complex Exponential Smoothing
Method and model
Using the complex valued representation of time series, we propose the CES in analogy to the conventional exponential smoothing methods. Consider the simple exponential smoothing method:
where $\alpha$ is the smoothing parameter and ${\hat{y}}_t$ is the estimated value of series. The same method can be represented as a weighted average of previous actual observations if we substitute ${\hat{y}}_{t1}$ by the formula (1) with an index instead of (Brown, 1956):
The idea of this representation is to demonstrate how the weights $\alpha {\left(1\alpha \right)}^{j1}$ are distributed over time in our sample. If the smoothing parameter $\alpha \in \left(0,1\right)$ then the weights decline exponentially with the increase of . If it lies in the so called “admissible bounds” (Brenner et al., 1968), that is $\alpha \in \left(0,2\right)$ then the weights decline in oscillating manner. Both traditional and admissible bounds have been used efficiently in practice and in academic literature (for application of the latter see for example Gardner & DiazSaiz, 2008; Snyder et al., 2017). However, in real life the distribution of weights can be more complex, with harmonic rather than exponential decline, meaning that some of the past observation might have more importance than the recent ones. In order to implement such distribution of weights, we build upon (2) and introduce complex dynamic interactions by substituting the real variables with the complex ones in (2). First, we substitute ${y}_{tj}$ by the complex variable ${y}_{tj}+{ie}_{tj}$, where ${e}_t$ is the error term of the model and $i$ is the imaginary unit (which satisfies the equation ${i}^2=1$). The idea behind this is to have the impact of both actual values and the error on each observation in the past on the final forecast. Second, we substitute $\alpha$ with a complex variable ${\alpha}_0+i{\alpha}_1$ and 1 by $1+i$ to introduce the harmonically declining weights. Depending on the values of the complex smoothing parameter, the weights distribution will exhibit a variety of trajectories over time, including exponential, oscillating, and harmonic. Finally, the result of multiplication of two complex numbers will be another complex number, so we substitute ${\hat{y}}_{tj}$ with ${\hat{y}}_{tj}+i{\hat{e}}_{tj}$, where ${\hat{e}}_{tj}$ is the proxy for the error term. The CES obtained as a result of this can be written as:
Having arrived to the model with harmonically distributed weights, we can now move to the shorter form by substituting
$\begin{array}{cc}& {\hat{y}}_{t1}+i{\hat{e}}_{t1}\\ {}& \kern1em =\left({\alpha}_0+i{\alpha}_1\right)\sum \limits_{j=2}^{t1}{\left(1+i\left({\alpha}_0+i{\alpha}_1\right)\right)}^{j1}\left({y}_{tj}+{ie}_{tj}\right)\end{array}$
in (3) to get:
Note that ${\hat{e}}_t$ is not interesting for the time series analysis and forecasting purposes, but is used as a vessel containing the information about the previous errors of the method. Having the complex variables instead of the real ones in (4), allows taking the exponentially weighted values of both actuals and the forecast errors. By changing the value of ${\alpha}_0+i{\alpha}_1$ , we can regulate what proportions of the actual and the forecast error should be carried out to the future in order to produce forecasts.
Representing the complexvalued function as a system of two realvalued functions leads to:
CES introduces an interaction between the real and imaginary parts, and the equations in (6) are connected via the previous values of each other, causing interactions over time, defined by complex smoothing parameter value.
But the method itself is restrictive and does not allow easily producing prediction intervals and deriving the likelihood function. It is also important to understand what sort of statistical model underlies CES. This model can be written in the following state space form:
where ${\epsilon}_t$ is the white noise error term, ${l}_t$ is the level component and ${c}_t$ is the nonlinear trend component at observation . Observe that dependencies in time series have an interactive structure and no explicit trend component is present in the time series as this model does not need to artificially break the series into level and trend, as ETS does. Although we call the ${c}_t$ component as “nonlinear trend,” it does not correspond to the conventional trend component, because it contains the information of both previous ${c}_{t1}$ and the level ${l}_{t1}$. Also, note that we use ${\epsilon}_t$ instead of ${e}_t$ in (6), which means that the CES has (6) as an underlying statistical model only when there is no misspecification error. In the case of the estimation of this model, the ${\epsilon}_t$ will be substituted by ${e}_t$, which will then lead us to the original formulation (4).
This idea allows rewriting (6) in a shorter more generic way, resembling the general single source of error (SSOE) state space framework:
where ${\mathbf{v}}_t=\left(\begin{array}{c}{l}_t\\ {}{c}_t\end{array}\right)$ is the state vector
$\mathbf{F}=\left(\begin{array}{cc}1& \left(1{\alpha}_1\right)\\ {}1& 1{\alpha}_0\end{array}\right)$ is the transition matrix
$\mathbf{g}=\left(\begin{array}{c}{\alpha}_0{\alpha}_1\\ {}{\alpha}_0+{\alpha}_1\end{array}\right)$ is the persistence vector and
$\mathbf{w}=\left(\begin{array}{c}1\\ {}0\end{array}\right)$ is the measurement vector.
The state space form (7) permits extending CES in a similar ways to ETS to include additional states for seasonality or exogenous variables. The main difference between model (7) and the conventional ETS is that the transition matrix in (7) includes smoothing parameters which is not a standard feature of ETS models. Furthermore persistence vector includes the interaction of complex smoothing parameters, rather than smoothing parameters themselves.
The error term in (6) is additive, so the likelihood function for CES is trivial and is similar to the one in the additive exponential smoothing models (Hyndman et al., 2008, p. 68):
where ${\mathbf{v}}_0$ is the vector of initial states, ${\sigma}^2$ is the variance of the error term and $\mathbf{Y}$ is the vector of all the insample observations.
Stationarity and stability conditions for CES
In order to understand the properties of CES, we need to study its stationarity and stability conditions. The former holds for general exponential smoothing in the state space form (8) when all the eigenvalues of lie inside the unit circle (Hyndman et al., 2008, p. 38). CES can be either stationary or not, depending on the complex smoothing parameter value, in contrast to ETS models that are always nonstationary. Calculating eigenvalues of for CES gives the following roots:
If the absolute values of both roots are less than 1 then the estimated CES is stationary.
When ${\alpha}_1>1$ one of the eigenvalues will always be greater than one. In this case both eigenvalues will be real numbers and CES produces a nonstationary trajectory. When ${\alpha}_1=1$ CES becomes equivalent to ETS(A,N,N). Finally, the model becomes stationary when:
Note that we are not restricting CES with the conditions (10), we merely show, how the model will behave depending on the value of the complex smoothing parameter. This property of CES means that it is able to model either stationary or nonstationary processes, without the need to switch between them. The property of CES for each separate time series depends on the value of the smoothing parameters.
The other important property that arises from (7) is the stability condition for CES. With $\epsilon_t={y}_t{l}_{t1}$ the following is obtained:
The matrix $\mathbf{D}=\left(\begin{array}{cc}1{\alpha}_0+{\alpha}_1& \left(1{\alpha}_1\right)\\ {}1{\alpha}_0{\alpha}_1& 1{\alpha}_0\end{array}\right)$ is called the discount matrix and can be written in the general form:
The model is said to be stable if all the eigenvalues of (12) lie inside the unit circle. This is more important condition than the stationarity for the model, because it ensures that the complex weights decline over time and that the older observations have smaller weights than the new ones, which is one of the main features of the conventional ETS models. The eigenvalues are given by the following formula:
CES will be stable when the following system of inequalities is satisfied:
Both the stationarity and stability regions are shown in Figure 1. The stationarity region (10) corresponds to the triangle. All the combinations of smoothing parameters lying below the curve in the triangle will produce the stationary harmonic trajectories, while the rest lead to the exponential trajectories. The stability condition (14) corresponds to the dark region. The stability region intersects the stationarity region, but in general stable CES can produce both stationary and nonstationary forecasts
Conditional mean and variance of CES
The conditional mean of CES for $h$ steps ahead with known ${l}_t$ and ${c}_t$ can be calculated using the state space model (6):
where
$\mathrm{E}\left({y}_{t+h}\mid {\mathbf{v}}_t\right)={\hat{y}}_{t+h}$
while $\mathbf{F}$ and $\mathbf{w}$ re the matrices from (7).
The forecasting trajectories of (15) will differ depending on the values of ${l}_t, {c}_t$, and the complex smoothing parameter. The analysis of stationarity condition shows that there are several types of forecasting trajectories of CES depending on the particular value of the complex smoothing parameter:
 When ${\alpha}_1=1$ all the values of forecast will be equal to the last obtained forecast, which corresponds to a flat line. This trajectory is shown in Figure 2A.
 When ${\alpha}_1>1$ the model produces trajectory with exponential growth which is shown in Figure 2B.
 When $\frac{4{\alpha}_0^2}{4}<{\alpha}_1<1$ trajectory becomes stationary and CES produces exponential decline shown in Figure 2C.
 When $1{\alpha}_0<{\alpha}_1<\frac{4{\alpha}_0^2}{4}$ trajectory becomes harmonic and will converge to zero Figure 2D.
 Finally, when $0<{\alpha}_1<1{\alpha}_0$ the diverging harmonic trajectory is produced, the model becomes nonstationary. This trajectory is of no use in forecasting, that is why we do not show it on graphs.
Using (7) the conditional variance of CES for $h$ steps ahead with known ${l}_t$ and ${c}_t$ can be calculated similarly to the pure additive ETS models (Hyndman et al., 2008, p. 96).
Loading libraries and data
Tip
Statsforecast will be needed. To install, see instructions.
Next, we import plotting libraries and configure the plotting style.
Read Data
year  value  

0  19600101  69.123902 
1  19610101  69.760244 
2  19620101  69.149756 
3  19630101  69.248049 
4  19640101  70.311707 
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  19600101  69.123902  1 
1  19610101  69.760244  1 
2  19620101  69.149756  1 
3  19630101  69.248049  1 
4  19640101  70.311707  1 
Now, let’s now check the last few rows of our time series using the
.tail()
function.
We need to convert the ds
from object
type to datetime.
Explore data with the plot method
Plot some 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
 Data to train our model.
 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 AutoCES with StatsForecast
To also know more about the parameters of the functions of the
AutoCES Model
, they are listed below. For more information, visit the
documentation
Load libraries
Instantiate 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
Note
Automatically selects the best Complex Exponential Smoothing model
using an information criterion. Default is Akaike Information Criterion
(AICc), while particular models are estimated using maximum likelihood.
The statespace equations can be determined based on their $S$ simple,
$P$ parial, $Z$ optimized or $N$ ommited components. The model string
parameter defines the kind of CES model: $N$ for simple CES (withous
seasonality), $S$ for simple seasonality (lagged CES), $P$ for partial
seasonality (without complex part), $F$ for full seasonality (lagged CES
with real and complex seasonal parts).
If the component is selected as $Z$, it operates as a placeholder to ask the AutoCES model to figure out the best parameter.
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 pandas’ 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
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  0.727729 
1  0.144552 
2  0.762086 
…  … 
51  0.073258 
52  0.234578 
53  0.109990 
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  CES  

unique_id  
1  20140101  82.906075 
1  20150101  83.166687 
1  20160101  83.424744 
1  20170101  83.685760 
1  20180101  83.946213 
1  20190101  84.208359 
ds  y  CES  

unique_id  
1  19600101  69.123901  69.851631 
1  19610101  69.760246  69.615692 
1  19620101  69.149757  69.911842 
1  19630101  69.248047  69.657822 
1  19640101  70.311707  69.601196 
Adding 95% confidence interval with the forecast method
ds  CES  CESlo95  CEShi95  

unique_id  
1  20140101  82.906075  82.342484  83.454018 
1  20150101  83.166687  82.604027  83.717270 
1  20160101  83.424744  82.858574  83.975868 
1  20170101  83.685760  83.118942  84.239578 
1  20180101  83.946213  83.376907  84.501137 
1  20190101  84.208359  83.637741  84.765411 
unique_id  ds  CES  

0  1  20140101  82.906075 
1  1  20150101  83.166687 
2  1  20160101  83.424744 
3  1  20170101  83.685760 
4  1  20180101  83.946213 
5  1  20190101  84.208359 
ds  y  unique_id  CES  

0  20140101  83.090244  1  82.906075 
1  20150101  82.543902  1  83.166687 
2  20160101  83.243902  1  83.424744 
3  20170101  82.946341  1  83.685760 
4  20180101  83.346341  1  83.946213 
5  20190101  83.197561  1  84.208359 
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  CES  

unique_id  
1  20140101  82.906075 
1  20150101  83.166687 
1  20160101  83.424744 
1  20170101  83.685760 
1  20180101  83.946213 
1  20190101  84.208359 
ds  CES  CESlo95  CEShi95  

unique_id  
1  20140101  82.906075  82.342484  83.454018 
1  20150101  83.166687  82.604027  83.717270 
1  20160101  83.424744  82.858574  83.975868 
1  20170101  83.685760  83.118942  84.239578 
1  20180101  83.946213  83.376907  84.501137 
1  20190101  84.208359  83.637741  84.765411 
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  CES  CESlo95  CEShi95  

ds  
19600101  69.123902  1  NaN  NaN  NaN 
19610101  69.760244  1  NaN  NaN  NaN 
19620101  69.149756  1  NaN  NaN  NaN 
…  …  …  …  …  … 
20170101  NaN  NaN  83.685760  83.118942  84.239578 
20180101  NaN  NaN  83.946213  83.376907  84.501137 
20190101  NaN  NaN  84.208359  83.637741  84.765411 
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  CES  

unique_id  
1  19840101  19830101  75.389511  74.952705 
1  19850101  19830101  75.470734  75.161736 
1  19860101  19830101  75.770729  75.377945 
1  19870101  19830101  76.219513  75.590378 
1  19880101  19830101  76.370735  75.806343 
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, AutoCES.
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  

CES  0.556314  0.669916  0.087037  0.630183  0.667133 
Acknowledgements
We would like to thank Naren Castellon for writing this tutorial.