Abstract: State space models for time series forecasting have been dominated by the single source of error (SSOE) framework, most notably the ETS family of models. The idea of SSOE is to use the same error across all equations in the model. Multiple source of error (MSOE) models, by contrast, assign independent stochastic disturbances to each component – level, trend, and seasonality – offering a richer and more flexible representation of uncertainty, yet they lack a systematic, unifying taxonomy. This paper introduces the PTS taxonomy, a structured classification of MSOE state space models tailored to the specific properties of the MSOE setting. The taxonomy organises models along three dimensions: P (Power transform, based on the Box-Cox transformation), T (Trend, with options for none, local, global, or damped), and S (Seasonality, with options for none, discrete, or trigonometric), yielding up to 24 well-defined model variants. All models are cast within a general state space system and estimated via the Kalman filter using maximum likelihood, with model selection performed through standard information criteria. The framework also incorporates a robust outlier detection procedure covering additive outliers, level shifts, and slope changes, as well as natural handling of missing observations through the Kalman smoother. We illustrate the practical utility of the taxonomy through empirical experiments on the real life dataset, demonstrating that the PTS family is both theoretically coherent and empirically competitive with established alternatives.

Slides are here.
Package that implements PTS: muse

Message ISF2026: PTS Taxonomy of Multiple Source of Error State Space Models for Demand Forecasting first appeared on Open Forecast.

What’s greybox? It is a package for statistical modelling focusing on forecasting and time series analysis. I created it back in 2018 to split the static models (such as linear regression) from the dynamic ones that landed in the smooth package. Greybox has evolved since then, and now has linear regression (alm), regression selection (stepwise) and combinations (calm), a variety of tools for feature generation, diagnostics, forecast evaluation (e.g. rolling origin) etc. You can read more about it here. Originally, the package was available for R only, but Claude and I ported its main functions to Python back in February.

The Exploratory Data Analysis techniques for time series fit the package quite well, although I don’t have many of those yet. So, I’ve implemented the main idea of the STI of Hans Levenbach in a function called “stick” (Seasonal, Trend, Irregular Contribution Kit) in the greybox package for R/Python. The idea is straightforward: apply stick to a time series, it will use ANOVA, and give you the strength of each component. Here, for example, is how to apply the function to the AirPassengers data (everyone’s favourite toy time series) in R:

library(greybox)
stick(AirPassengers)

and in Python:

from fcompdata import AirPassengers
from greybox import stick

result = stick(AirPassengers.y, lags=12)
print(result)

which gives exactly the same result:

Strength of the components:
seasonal12      trend  irregular
    0.1061     0.8613     0.0326

So, trend dominates the time series, explaining 86.13% of its variability, meaning that if you capture it correctly, you solve a big chunk of the problem. This split also gives you a rough idea about the structure-versus-noise breakdown in the time series, although it assumes that the seasonal component does not evolve over time.

The function supports several seasonal components, and I might extend it to include external information (e.g. promotions) in the future if there is demand for it.

Message stick function for the EDA in time series first appeared on Open Forecast.

There are situations where non-normal distributions fit considerably better. The main candidate for substitution is the conditional distribution of the response variable. For example, sales of engines cannot follow the normal distribution by definition: they are intermittent and integer-based — you cannot sell 1.78 engines. More generally, while demand can be fractional, it cannot be negative. It is therefore only logical to use distributions that support positive values only in these situation. Examples include Log-Normal, Gamma, and Inverse Gaussian, among many others.

In my last paper with John Boylan (this one), we discussed how ETS can be extended to use these three distributions instead of the normal one. I implemented this functionality (together with the others, such as Laplace and Generalised Normal) in ADAM. It supports any of these distributions with any ETS/ARIMA/regression model, for both additive and multiplicative error terms. There is some maths involved, which you can find here and here.

Why bother? The main is in the predictive distribution. If the data is not normal, we may end up with poorly calibrated forecasts and misleading prediction intervals. Using a more appropriate distribution can resolve this.

But how do we choose the right distribution for our data?

A possible solution (similar to selecting ETS components) is to fit models with different distributions and pick the one with the lowest information criterion. This is implemented in the ADAM function from the smooth package. We can do this manually, or use AutoADAM (called auto.adam in R) to select the most suitable distribution based on AICc automatically:

from fcompdata import AirPassengers
from smooth import AutoADAM

model = AutoADAM(lags=[12], h=12, holdout=True, orders=None, verbose=True)
model.fit(AirPassengers.y)
model.summary()

The orders=None line stops the function from trying different ARIMA orders – something we will come back to in a future post. For this example, the output is:

Model estimated using ADAM() function: ETS(MAM)
Response variable: y
Distribution used in the estimation: Normal
Loss function type: likelihood; Loss function value: 523.2756
Coefficients:
       Estimate  Std. Error  Lower 2.5%  Upper 97.5%   
alpha    0.7575      0.0895      0.5807       0.9343  *
beta     0.0000      0.0080      0.0000       0.0158   
gamma    0.0000      0.0503      0.0000       0.0994   
Error standard deviation: 0.0358
Sample size: 144
Number of estimated parameters: 4
Number of degrees of freedom: 140
Information criteria:
      AIC     AICc       BIC      BICc
1054.5512 1054.839 1066.4305 1067.1455

Boring… the function found that the Normal distribution has the lowest AICc among those tested – the Air Passengers data is too well-behaved.

Oh, and don’t forget to produce the forecasts:

model.predict(h=18, interval="prediction")

Smooth forecasting!

Install smooth: pip install smooth

Message smooth in python: Non-normal distributions in ETS/ARIMA first appeared on Open Forecast.

One of the solutions (old but gold) is the multiple seasonal ETS model, which was originally developed by James Taylor (2003) for the pure additive exponential smoothing. The idea was quite simple: to model multiple seasonal cycles, one can add multiple seasonal components, i.e. to capture the day-of-week (frequency 7) and the day-of-year (frequency 365) effects. While it worked fine for some examples, the main issue with it has been its computational speed (or rather slowness): the original ETS needs to estimate all smoothing parameters + all the initial values for seasonal indices and other components. Both ADAM and ES in the smooth package support multiple seasonalities and avoid the whole issue by using a different model initialisation called “backcasting”.

Here is a classical example from James’ paper on the half-hourly electricity demand (see the image in the post). It is clear that there is a half-hour-of-day and the day-of-week effects. In ES, this means that we need to provide the vector for the lags variable:

from smooth import ES
from fcompdata import taylor

# Fit ES with automatic ETS model selection
model = ES(lags=[48, 336], h=336, holdout=True)
model.fit(taylor.y)
model.predict(h=336)
print(model)

This is the output I get from the function:

Time elapsed: 2.03 seconds
Model estimated using ES() function: ETS(MNM)
With backcasting initialisation
Distribution assumed in the model: Normal
Loss function type: likelihood; Loss function value: 25391.1773
Persistence vector g:
 alpha gamma1 gamma2
0.2899 0.1283 0.5270
Sample size: 3696
Number of estimated parameters: 4
Number of degrees of freedom: 3692
Information criteria:
      AIC      AICc       BIC      BICc
50790.3546 50790.3654 50815.2146 50815.2591

Forecast errors:
ME: 829.1195; MAE: 942.1447; RMSE: 1065.1127
sCE: 941.5012%; Asymmetry: 9.2%; sMAE: 3.1841%; sMSE: 0.1296%
MASE: 1.4491; RMSSE: 1.1286; rMAE: 0.1408; rRMSE: 0.1300

The computational time on this data was only 2.03 second. In this time, the function tried several possible ETS models and selected the best one based on the AICc value. The resulting best model is ETS(M,N,M), which makes perfect sense for this data.

Is there a way to improve this model? Yes! Taylor mentions that adding AR(1) to the cocktail tends to improve the accuracy in case of multiple seasonal series. We can try that if we switch to ADAM:

from smooth import ADAM

# Fit ADAM ETS(MNM)+AR(1) model
model = ADAM(model="MNM", ar_orders=1, lags=[48, 336], h=336, holdout=True)
model.fit(taylor.y)
print(model)
model.plot(7)

Here is the output:

Time elapsed: 1.04 seconds
Model estimated using ADAM() function: ETS(MNM)+ARIMA(1,0,0)
With backcasting initialisation
Distribution assumed in the model: Gamma
Loss function type: likelihood; Loss function value: 24157.2473
Persistence vector g:
 alpha gamma1 gamma2
0.1097 0.2225 0.3481
ARMA parameters of the model:
             Lag 1
AR(1)       0.6852
Sample size: 3696
Number of estimated parameters: 5
Number of degrees of freedom: 3691
Information criteria:
      AIC      AICc       BIC      BICc
48324.4947 48324.5109 48355.5697 48355.6365

Forecast errors:
ME: 276.4061; MAE: 462.5092; RMSE: 588.5957
sCE: 313.8711%; Asymmetry: 2.1%; sMAE: 1.5631%; sMSE: 0.0396%
MASE: 0.7114; RMSSE: 0.6237; rMAE: 0.0691; rRMSE: 0.0719

The resulting model has lower AICc, but also produces more accurate point forecasts (compare RMSSE values) for the holdout set. The following image shows the data and the point forecasts for it:

Double seasonal ETS(M,N,M) applied to the half-hourly electricity demand data

What else can we do here? Actually, quite a lot: multistep losses, seasonal ARIMA, explanatory variables – things can get only more complicated from here. Have a look at this.

Do I hear someone shouting “TBATS”? TBATS is the exponential smoothing with additional bells and whistles (ETS + adapted Fourier terms + ARMA errors). I don’t have it as a separate function in the smooth just yet, but you can reproduce it, for example, like this.

So, what are you waiting for? Dive in and see how it works for yourself!

Install smooth: pip install smooth

Message smooth in python: multiple seasonal ETS first appeared on Open Forecast.

We all know that in real life sales don’t just evolve over time on their own. Any univariate model, such as ARIMA or ETS is just a way to approximate a complex reality. In practice, there are many factors affecting the demand for your product. What would happen if the price on your product increases? What if you run a promotion (e.g. “Buy One, Get One Free”)? Your competitor’s strategy impacts the demand for your product as well… There’s lots of different factors, and some of them can be quite useful in demand forecasting. But can we join the dynamic univariate models with regression?

Yes, we can! Although ETS is thought as a pure univariate model, it is easy to extend to include explanatory variables. There are several great papers showing how it works (e.g. Kourentzes & Petropoulos, 2016), and in fact the es() function from the smooth package for R was used as a benchmark in the M5 competition.

So, consider a situation where you have weekly sales of a product with some recorded promotions (encoded as dummy variables). We will use a time series from the fcompdata package for Python. The first image shows how the series looks, the vertical lines show when promotions happen. The series itself seems to be seasonal, roughly repeating peaks and troughs every 52 observations (every year). Also, we see that there are two types of promotions, and when they happen sales tend to increase. So, including them should improve the model fit, and if the company decides to run promotions again, the model will forecast demand better. I will start by fitting the ETS(M,N,M) to the data:

from smooth import ES
from fcompdata import PromoData

y = PromoData.y

model = ES(model="MNM", lags=52, holdout=True, h=13)
model.fit(y)
model.predict(h=13)
model.plot(7)

NOTE: PromoData has a specific structure with several attributes. PromoData.x contains the in-sample data, PromoData.xx has the holdout – this is consistent with the Mcomp package for R. The new features in python are:

• PromoData.y – concatenated training and test sets,
• PromoData.xregx – matrix of explanatory variables for the training set,
• PromoData.xregxx – matrix of explanatory variables for the test set,
• PromoData.xreg – the full (concatenated) matrix of explanatory variables.

The following image shows the model fit and the point forecasts from the ETS(M,N,M):

ETS(M,N,M) fit and forecast for the promotional data example

As expected, because the model does not take promotions into account, it fits the data as best as it can and produces forecasts that are oblivious of the potential external effects on sales. We can improve it by including the promotional dummies:

X_train = PromoData.xreg
X_test =  PromoData.xregxx

model = ES(model="MNM", lags=52, holdout=True, h=13)
model.fit(y, X_train)
model.predict(h=13, X=X_test)
model.plot(7)

ETS(M,N,M) with explanatory variables

The image above shows the fit and the point forecasts from the ETSX(M,N,M) model that now takes the promotions into account. This is quite an improvement in comparison with the previous one. Furthermore, if we can control when to have promotions and what types of promotions to run, we can change the values in the `X_test` matrix and see what demand to expected in that situation. So, this gives an analyst a tool for a more advanced sensitivity analysis.

Read more about the ETSX here.
Install smooth: pip install smooth
ETSX wiki on github.

Message smooth in python: ETS with explanatory variables first appeared on Open Forecast.

Why bother?

There is a vast body of literature on forecast combinations (read this great review). The main idea is that you should not put all your eggs in one basket — the safer strategy is to combine forecasts from different models instead of selecting just one. Yes, it is more computationally expensive, but the trade-off is higher accuracy on average.

For ETS, a great solution was proposed by Stephan Kolassa in his 2011 paper: extract AIC values, calculate AIC weights (giving the highest weight to the best-performing model and lower ones to the rest), then combine the forecasts. The resulting forecasts tend to be more robust, because in practice it might be hard to tell the difference between, for example, ETS(M,A,M) and ETS(M,Md,M). So why choose one when you can have all? I implemented this mechanism in the smooth package for R years ago, and now it is also available in Python.

Here is how it works on an example using an M3 time series. I picked this specific one because it is seasonal, but the trend is not very well pronounced. The series is shown in the first image.

from smooth import ES
from fcompdata import M3

series = M3[1687]
y = series.y
freq = series.period

# Fit ETS models, combine forecasts
model = ES(model="CXC", lags=series.period, h=18, holdout=True)
model.fit(y)
model.predict(h=18)

The code above tells ES to fit all ETS models with additive and no trend (“X” in the middle), calculate AIC weights, produce forecasts from each one of them, and then combine them. The resulting point forecast is the weighted combination of the individual forecasts. If a prediction interval is required, the specific quantiles are combined directly (see the paper by Lichtendahl et al., 2013). This is inevitably slower than the default model selection mechanism, but is a safer approach. The point forecast and the prediction interval (grey lines) are shown in the attached image.

Note that the user can regulate the pool of combined models via the “model” parameter of the function. This wiki explains all the accepted options.

So why not go ahead and try it yourself, and see how it works for your data?

🔗 Install smooth: pip install smooth
📖 More on forecasts combination in ADAM.

Message smooth in python: ETS forecast combination first appeared on Open Forecast.

ETS stands for the “Error-Trend-Seasonal” model or ExponenTial Smoothing. It is a statistical model that relies on time series decomposition and updates the unobserved states (level/trend/seasonal) based on the mistakes it makes. In a way, you can call it an adaptive model that changes its forecast based on the most recent available information. It is relatively simple to explain and work with, and it has performed well in a variety of competitions (M3, M4, M5, for example).

The smooth package implements an advanced form of ETS in the ADAM and a more basic one in the ES classes. In fact, ES is just a wrapper of ADAM, it is the conventional model, with just some tuning. Both support all 30 ETS models, have automated model selection and forecast combination, allow producing point forecasts and a variety of prediction intervals types. In fact, if you want a straightforward robust implementation of ETS, give ES a try.

Here’s how to use it in Python:

from smooth import ES
from fcompdata import M3

# Pick a series from the M3 competition for demonstration
series = M3[2568]
y = series.x
freq = series.period

# Fit ES with the automatic model selection
model = ES(lags=freq, h=18, holdout=True)
model.fit(y)
print(model)

Running this produces output similar to this:

Time elapsed: 0.4 seconds
Model estimated using ES() function: ETS(MAM)
With backcasting initialisation
Distribution assumed in the model: Normal
Loss function type: likelihood; Loss function value: 724.8524
Persistence vector g:
 alpha   beta  gamma
0.0065 0.0000 0.0000
Sample size: 98
Number of estimated parameters: 4
Number of degrees of freedom: 94
Information criteria:
      AIC      AICc       BIC      BICc
1457.7047 1458.1348 1468.0446 1469.0306

Forecast errors:
ME: -580.9985; MAE: 604.0204; RMSE: 710.5457
sCE: -149.9347%; Asymmetry: -2.5%; sMAE: 8.6598%; sMSE: 1.0378%
MASE: 0.2653; RMSSE: 0.2452; rMAE: 0.2555; rRMSE: 0.2163

A few things worth noting from the output:

• ES automatically selected ETS(MAM) based on the AICc value – a multiplicative error, additive trend, multiplicative seasonality model – as the best fit
• It used backcasting for the model initialisation (default), which speeds up the process and requires fewer parameters to estimate
• It kept the last 18 observation for the holdout, produced autoforecasts for it and calculated several forecast errors. This is handy if you want to directly compare different smooth models on a time series.

But why are we here? We want to forecast! So, here it is:

model.predict(h=18, interval="prediction")
model.plot(7)

This should produce an image similar to the one attached to the post. As simple as that.

Now it’s your turn! :)

🔗 Install smooth: pip install smooth
📖 smooth wiki

Message smooth in python: ETS with model selection first appeared on Open Forecast.

When I test my functions in R, I usually use the M1, M3, and tourism competition datasets because they are diverse enough, containing seasonal, non-seasonal, trended, and non-trended time series of different frequencies (yearly, quarterly, monthly). The total number of these series is 5,315, which is large enough but not too heavy for my PC. So, when I run something on those datasets, it becomes like a stress test for the forecasting approach, and I can see where it fails and how it can be improved. I consider this type of test a toy experiment — something to do before applying anything to real-world data.

In R, there are the Mcomp and Tcomp packages that contain these datasets, and I like how they are organised. You can do something like this:

series <- Mcomp::M3[[2568]]
ourModel <- adam(series$x)
ourForecast <- forecast(model, h=series$h)
ourError <- series$xx - ourForecast$mean

Each series from the dataset contains all the necessary attributes to run the experiment without trouble. This is easy and straightforward. Plus, I don’t need to download or organise any data — I just use the installed package.

When I started vibe coding in Python, I realised that I missed this functionality. So, with the help of Claude AI, I created a Python script to download the data from the Monash repository and organise it the way I liked. But then I realised two things, which motivated me to package it:

1. I needed to drag this script with me to every project I worked on. It would be much easier to just run "pip install fcompdata" and forget about everything else.
2. Some series in the Monash repository differ from those in the R package.

Wait, what?! Really?

Yes. The difference is tiny — it’s a matter of rounding. For example, series N350 from the M1 competition data (T169 from the quarterly data subset) has three digits in the R package and only two if downloaded from the Monash repository (Zenodo website).

Who cares?! It's just one digit difference, right?

Well, if you want to reproduce results across different languages, this tiny difference might become your nightmare. So, I care (and probably nobody else in the world), and I decided to create a proper Python package. You can now do this in Python and relax:

pip install fcompdata

from fcompdata import M1, M3, Tourism
series = M3[2568]

The "series" object is now an instance of the MCompSeries class that has the same attributes as in R: series.x, series.h, series.xx, etc.

As simple as that!

One more thing: I’ve added support for the M4 competition data, which — when imported — will be downloaded and formatted properly. The dataset is large (100k time series), and I personally don’t like it. I even wrote a post about it back in 2020. But if I want the package to be useful to a wider audience, I shouldn’t impose my personal preferences — you should decide for yourselves whether to use it or not.

P.S. Submitting to PyPI gave me a good understanding of the submission process for Python and why it can be such a mess. My package was published just a few seconds after submission — nobody looked at it, nobody ran any tests. CRAN does a variety of checks to ensure you don’t submit garbage. PyPI doesn’t care. So, I’ve gained more respect for CRAN after submitting this package to PyPI.

Message Forecasting Competitions Datasets in Python first appeared on Open Forecast.