The idea behind the Simple Moving Average (SMA) is to take the average of the last few observations and use it as a forecast for the next several steps ahead. Very crude and very simple. In fact, it has Naive (from this post) as a special case if you take the average of one most recent observation. On the other hand, if you increase the order to include all the observations, you will end up with the Global Mean. And this simple method works quite well if you have level data, i.e. no apparent strong trend, no obvious seasonality, and no other important elements of structure.

The only thing that makes it a bit harder to use in practice is the choice of the order, i.e. the number of observations to average over. Unfortunately, there is no universal answer here. But people report that the order of 12 or 13 is fine for weekly data, although it’s not completely clear why. In the academic literature, Aris Syntetos & John Boylan (2005) found that SMA(13) performed quite well on intermittent demand data, which was unexpected given the nature of the data (lots of zeroes). And almost 10 years ago, Fotios Petropoulos and I proposed a model underlying SMA with automatic order selection. We showed that it outperforms other simple benchmarks on supply chain data.

There is also some evidence from the VN2 inventory competition by Nicolas Vandeput. The benchmark there was built around a 13-week moving average with a simple seasonal adjustment, and only 25 out of 180+ participants managed to beat it. Many sophisticated ML pipelines lost to a method that predates computers.

So, if you work, for example, in retail or in supply chain, SMA is a method to consider for your sanity-check pool of models. But don’t expect miracles from it! It is still a simple method that works for level time series. Use it as a stepping stone to find a better model that has more features.

Anyone else found SMA to be a strong contender? Leave a comment – it would be interesting to see how many of you have had the same experience.

And yes, we discuss it in our “Demand Forecasting Principles” training in more detail. The next one will be held online in November, with live sessions from 2pm to 4pm UK time. We still have a few places left, so register here.

Message The Menace of ML: Simple Moving Average first appeared on OpenForecast.

One of the definitions I see on the internet: “A measure of the degree to which something may be forecast with accuracy”. This definition is so disturbing that I cannot resist ranting about it. The thing that annoys me the most is the “with accuracy”, which is really arbitrary and can mean anything. Your Naive method produces a relative RMSE of 1.0. Is that accurate enough? Does this make the time series forecastable? What if ETS produces 1.2, while LightGBM delivers 0.8? Is the data forecastable now? And when do you say “this is fine”?

But let me breathe and take a step back. The thing is, any method of measuring forecastability has assumptions behind it. Here is my list of five (did I miss any?):

1. The models you use: each time series has its own characteristics, so you can only say that the data is “forecastable” given the set of models you have. The data with trend might look unforecastable for the model that only has the level component.
2. The features/indicators that are available: if you do not know when promotions happen, some observations will look unforecastable. It’s a similar argument to (1), but more about what you include in your model. If you spend more time on feature generation and transformation, then the series that looked very hard might become quite easily forecastable.
3. The statistic you focus on: this could be the conditional mean, the median, some quantile, or the whole predictive distribution. A model can do great in terms of point forecasts but very poorly when it comes to prediction intervals. Does that make the time series unforecastable? Which one do you care more about?
4. The forecast horizon: the longer it is, the less accurate the forecasts become. One-step-ahead forecasts are easy, multiple steps ahead are hard. Which one do you use to measure forecastability?
5. The error measures: I’ll just leave this here: https://openforecast.org/category/forecasting-theory/forecast-evaluation/

Do I hear someone mentioning the coefficient of variation (CoV) as a good measure of forecastability? This is, by the way, what is used in the conventional XYZ classification. Well, let’s check the five points above against it and see what it assumes: (1) the global mean is suitable and the variance is constant; (2) no features are available; (3) conditional mean as the statistic of interest; (4) 1-step-ahead point forecast; (5) Root Mean Squared Error (the core of the CoV). So, it is suitable for a very small set of time series. If you use it universally, you might decide that some time series with a very clear structure are not forecastable. And the same exercise works for the more sophisticated measures, entropy-based ones included: run them through the five questions and see what they silently assume.

And then we come to the final point: so what? Let’s say you split your data into the “forecastable/not” categories. What are you going to do with that? The intention behind this is legitimate — you cannot babysit ten thousand SKUs equally, so you want to know where to spend your time. But the label “unforecastable” answers the wrong question. The right question is why the series is hard to forecast: missing promotion information? Wrong model? Genuinely random demand? Each of these implies a different action. It might well be that your “unforecastable” time series just require more time and effort to become forecastable again.

We don’t teach “forecastability” in our Demand Forecasting Principles course — because, as you can see, I don’t believe in it. What we do teach is everything in the list above: models, their assumptions, features, horizons, and how to evaluate forecasts properly. See details about the next course here.

This is my personal view. Happy to hear what others have to say.

Message Five assumptions behind “forecastability” first appeared on OpenForecast.

In the conventional ARIMA, taking differences is treated as a pre-processing step, where we switch from, for example, the sales of a product to the sales increase/decrease. If you had a sample of 100 observations, after first differences you are left with 99 – the first observation has nothing to be subtracted from. So yes, with the conventional approach, ARIMA(1,0,1) and ARIMA(1,1,1) are estimated on different sample sizes, and their information criteria are not directly comparable.

But! MSARIMA (and ADAM in general) is a state space model, and this pre-processing step is not needed. The differences are embedded in the model as additional components: ARIMA(0,1,1) has one, ARIMA(0,2,1) has two. The sample stays the same; what changes is the number of estimated parameters – each new component needs its initial value, and the information criterion penalises that. So the comparison stays fair: same sample, same likelihood basis, different number of parameters. I have several examples of how ARIMA is formulated in the ADAM SSOE framework.

What this implies: any ARIMA model of any orders (non-seasonal, seasonal, or multiseasonal) can be compared with other ARIMA models directly via information criteria, on the same sample of data.

This is not a new idea, by the way. I used this approach in the ARIMA algorithm I developed for the DemandWorks company (now part of Netstock) more than ten years ago, and John Boylan and I later built the State Space ARIMA on the same principle (the paper).

So, yes: if you use the conventional ARIMA (auto_arima from pmdarima, ARIMA from aeon or statsforecast in Python; stats or forecast implementations in R), you NEED to make sure that you compare models with the same order of differencing. But if you use the smooth implementation, you don’t need to worry – everything is already taken care of for you. This comes as a bonus from the specific formulation of the model.

Message On differencing of ARIMA in state space first appeared on OpenForecast.

It is called “Seasonal Naïve”. While the simple Naïve copies the last observed actual into the future as a forecast, the seasonal one copies the whole seasonal shape of the data and uses it as a forecast. The logic is straightforward: if you see an increase in sales every January, why not use the actual sales of January 2025 as the forecast for January 2026? Simple, easy to do, and hard to beat in some cases, especially when your demand does not have high variability.

What this method doesn’t do is filter out the noise in the data. This means that if you had a promotion-driven spike this February, Seasonal Naïve will happily copy it into next February’s forecast. So, if you have some distinct components in your time series and/or effects of explanatory variables on sales, Seasonal Naïve might not be a good choice. But it remains an essential benchmark for any seasonal data.

I actually have an anecdote related to the Seasonal Naïve. Yves Sagaert and I were working on a paper, and we decided to apply our new method to data with multiple seasonality. It worked great, better than the double seasonal exponential smoothing and ARIMA. I was really hyped and was ready to celebrate, when Yves suggested trying the Seasonal Naïve as well. It’s good that he did, because it turned out that Seasonal Naïve beat them all, including our new method, without even blinking. This was a great demonstration of a principle I had been preaching to others: if you have seasonal data, always use Seasonal Naïve as a benchmark.

In the Demand Forecasting Principles course in November, we cover simple methods like this one properly, including when to stop trusting them. Details and booking can be found here.

Message Another important Naïve method first appeared on OpenForecast.

Naïve is a very simple forecasting method: the forecast equals to the last observed value. So, for example, if the room temperature at 12pm was 20 degrees Celsius, then Naive will forecast exactly the same temperature for 1pm, and it will usually be roughly right. That is what makes it dangerous to sophisticated models: it is often surprisingly hard to beat. If your model cannot beat Naïve, it is not worth deploying, no matter how sophisticated and beautiful it is.

This is why there is a well-established rule in forecasting: any proper evaluation should include simple benchmarks, and Naïve is the easiest one to implement, because it does not have any parameters to estimate and can work with the sample of one observation. This is why you will find it in every decent forecasting competition.

This is not a new finding, by the way. Back in 1979, Spyros Makridakis and Michèle Hibon published a paper, showing on a set of 111 real time series that simple methods were often at least as accurate as the sophisticated statistical ones. The audience did not take it well: the discussion that followed was openly hostile, with eminent statisticians suggesting that the results said more about the authors’ skills than about the methods. Spyros’ response was to test the claim at a much larger scale, which is how the M-competitions were born. I wrote a post about that some time ago. But the lesson survived the criticism: always compare your approach with the simple forecasting methods.

In our training course on Demand Forecasting Principles, we discuss this and other simple methods in more detail, showing where they work and where they fail. They are all building blocks for understanding applied forecasting.

The next course runs online in October, over Zoom. Details and booking can be found here.

P.S. There is one case where Naïve is not a good benchmark, read this post.

Message Why Naïve is popular and important first appeared on OpenForecast.

Since 2024, Kandrika Pritularga and I have run a demand forecasting principles course at the Centre for Marketing Analytics and Forecasting. We ran it three times, and it has received good reviews from the participants. They liked that the material was motivated by real-life problems, that we explained it in detail, answered all the questions, and had helpful workshops. So, having discussed this with Kandrika and Nikos, we decided to continue the course, but now under the umbrella of OpenForecast, and this time Nikos Kourentzes and I will be teaching it together.

Demand Forecasting Principles is a four-week online course running in October 2026. We plan to have eight live Zoom sessions (two per week), with hands-on workshops in both R and Python between them. All sessions are recorded and shared with the participants for their reference.

The course will cover all the essentials of forecasting, starting from the forecasting principles, moving to the simple methods, ETS, intermittent demand, forecast evaluation, judgement, and finishing with forecast combinations.

Who is this for?

• demand planners,
• data scientists,
• business analysts
• anyone who works with forecasts and wants to understand what happens behind the numbers.

What do you need to know?

You don’t need prior knowledge of forecasting or statistics – we explain everything as we go, in as much detail as you need. Yet, experienced practitioners will still find plenty that is new: some of our previous participants were seasoned forecasters and said exactly that.

The details, programme, prices and booking are available here. Students get a discount.

Exact October dates will be announced next week, but you can already book your place.

If you are not sure whether the course fits you or your team, get in touch.

Message Demand Forecasting Principles course, October 2026 first appeared on OpenForecast.

1. the forecast,
2. the variability of sales,
3. the target service level,
4. the lead time,
5. and the assumed demand distribution.

Get any of them wrong, and you either overstock or end up with stockouts.

I have written up a longer answer to this question, explaining how these five elements impact the safety stock formula, and provided a short example of an abstract calculation for an FMCG product.

Message Five things that determine your safety stock first appeared on OpenForecast.

Here is one of the recent examples, a screenshot of the post of Fotios Petropoulos about the effect.

In the original paper, Kruger and Dunning (1999) ran experiments with undergraduates on humour, logical reasoning, and grammar. Participants completed a test and estimated their percentile rank. The authors then sorted participants into four quartiles by actual performance and computed averages for actual and self-assessed performance for each quartile. The plots in their paper – the real Dunning–Kruger effect – are just four data points per line, not a smooth curve over a learning journey (second image).

What did they find? People in the bottom quartile substantially overestimated their performance, often believing they were average or above. Top performers slightly underestimated their standing. The key finding is an asymmetry in miscalibration: low performers overestimate, high performers slightly underestimate.

This has almost nothing to do with the popular “experience vs. confidence” image. The original X‑axis is performance quartile at a single point in time; the meme’s X‑axis is a vague notion of “experience” through time. The original Y‑axis is the assessed test percentile; the meme’s is a free‑floating “confidence” construct. In the actual data, perceived performance increases with actual performance – there is no early spike, no “valley of despair,” no “slope of enlightenment.” That swooping curve is an internet-era graphic never reported by Kruger and Dunning, and it misleadingly frames the effect as a personal development trajectory the paper never studied.

There is also a serious critique of the original paper from statistical point of view. For example, Gignac and Zajenkowski (2020) showed that sorting people into quartiles and plotting average self-assessment against average performance can, by itself, generate the characteristic pattern – purely as a statistical artefact. In their own empirical data, miscalibration was roughly constant across ability levels, consistent with measurement noise rather than a special cognitive deficit in low performers. You can actually reproduce the pattern using two random uncorrelated variables. Here is a simple example in R:

set.seed(41)

x <- rnorm(10000, 100, 10)
y <- rnorm(10000, 100, 10)
plot(x,y)
xQ <- quantile(x)
yQ <- quantile(y)

yMeans <- xMeans <- vector("numeric",4)

for(i in 1:4){
    xMeans[i] <- mean(x[xxQ[i]])
    yMeans[i] <- mean(y[xxQ[i]])
}

plot(1:4, xMeans, type="b", ylim=range(xMeans,yMeans),
     xlab="Real performance", ylab="Assessed performance",
     lwd=2)
lines(yMeans, lwd=2, lty=2)
points(yMeans, lwd=2)
legend("topleft",
       legend=c("Actual performance", "Assessed performance"),
       lwd=2, lty=c(1,2), pch=1)

Which produces the image like this:

If you introduce a correlation between the two variables, the images starts looking even more similar to the ones from the original paper.

So there might be a real effect – many follow-up studies have measured it with more rigorous tools – but Dunning and Kruger’s method was not the right one to establish it. And that image with experience vs confidence is just a meme and a serious misconception that should not be used.

P.S. If you wonder who the “leading expert” that Fotios Petropoulos refers to in his post is – it’s me. Not sure why he doesn’t tag me properly.

Message The real Dunning-Kruger effect first appeared on OpenForecast.

In many contexts, “deterministic” has a precise meaning: no randomness, no uncertainty. A deterministic solution to an optimisation problem (e.g. linear programming) implies that there are no random inputs or outputs once the model and its parameters are fixed. Forecasting is different. As Chatfield and many others have pointed out, forecasting has multiple sources of uncertainty, and there is essentially zero chance that the future will unfold exactly as any single number suggests.

Yes, some people use “deterministic” as a synonym for “point forecast”. But that label is still misleading, because a point forecast is not uncertainty-free – it is just one summary of a predictive distribution (often the conditional mean, sometimes the median or another functional).

Here’s a quick reality check you can do yourself. Take a dataset, apply your model, and write down the point forecast for the next few observations. Now add one new observation, re-estimate, and forecast again (the image in this post depicts exactly that, but with 50 forecasts produced on different subsamples of data). The point forecast will change unless you are dealing with an exotic situation with non-random data (e.g. every day, you sell exactly 100 units). So, which of the two was the “deterministic” forecast? If forecasts were truly deterministic in the strict sense, you would not get multiple plausible values from small, reasonable changes in the sample.

This happens because any forecasting method (statistical or ML) depends on data and on modelling choices: parameter estimation, feature selection, splitting rules, tuning, even decisions like “use α=0.1”. Those choices can be fixed across samples of data, but fixing them does not remove uncertainty – it only hides it. The randomness is still there in the data and in the fact that we only observe a sample of it.

So when you see someone mentioning “deterministic forecast”, it’s worth translating it mentally to: “a point forecast, probably a conditional mean”. If you care about decisions and risk, you should know that there is an uncertainty associated with this so called “deterministic forecast”, and that it should not be ignored. But this is a topic for another discussion in another post.

Message There’s no such thing as “deterministic forecast” first appeared on OpenForecast.

In forecasting literature, since early days of the area, there has been a general consensus that the forecast errors from the individual time series should not be analysed and aggregated as is. This is because you can have very different time series capturing dynamics of very different processes.

Indeed, if you forecast sales of apples in kilograms, your actual value would be apples in kilograms, and your point forecast would also be in the same units. Subtracting one from another tells us how many kilograms of apples we missed with the forecast we produced. But if we then take the average between forecast errors for apples and beer, we would be aggregating things in different units, which contradicts some basic aggregating principles.

Furthermore, if the company sells thousands of kilograms of apples and jet engines, aggregating forecast errors on those (e.g. 3000 vs 3) might introduce all types of issues, because the models performance on apples might mask the performance of the model on jet engines. Still, the jet engines are much more expensive than apples and getting them accurately might be more important for the company than forecasting apples.

So, forecasting literature has agreed that the forecast errors need to be somehow scaled to make the errors unitless and not to distort performance of models on time series with different volumes. There are several ways of doing that, including the poor ones and reasonable ones. The state of the art at the moment is to divide error measures by some in-sample statistics to avoid potential holdout-sample distortion. Using mean absolute differences (MAD) for this (thus ending up with MASE or RMSSE) is considered as a standard. A couple of years ago, I have written a post about advantages and disadvantages of several scaling methods.

But there is one method that I haven’t looked at and which is not very well discussed in the forecasting literature. It relies on the monetary value of forecasts. We could multiply each individual forecast error “e” by the price of the product “p” (thus moving to the missed income per product) and then divide everything by the overall income (price times quantity) from different products. This can be written as:

\begin{equation}
\text{monetary Mean Error} = \frac{\sum_{j=1}^n (p_j \times e_j)} {\sum_{j=1}^n (p_j \times q_j)}
\end{equation}

(the above formula can be modified to have squares or absolute values of the error). This way we switch from the original units to the monetary values and each error would tell you the percentage of the missed income in the overall one. This is a useful measure because it connects models performance with some managerial decisions and it takes the value of product into account (thus we do not mask the expensive jet engines with cheap apples).

However, it might have a potential issue similar to what the MAE/Mean or wMAPE has: if the sales of the product are not stationary, the denominator would change, thus driving the proportion either up or down, irrespective of how good the forecast is. I am not sure whether this needs to be addressed, because there is an argument that if the income from a product has increased and the error hasn’t changed, then this means that the proportion of the missed income decreased, which makes sense. But if we need to address this, we can switch to the MAD multiplied by price in the denominator to address this issue. In fact, this was sort of done in M5 competition that used a weighted RMSSE, relying on the income from each product over the last 4 weeks of data.

But here is one more interesting thing about this error measure. If we assume that prices for all products are exactly the same, they will disappear from the numerator and the denominator, leaving us with just sum of errors divided by the overall sales of all products. This still maintains the original idea of the proportion of the missed income, but now has a very strong assumption, which is probably not correct in the real life (apples and engines for the same price?). Furthermore, this would mask the performance of the model for the expensive products again. I personally don’t like this measure and find the assumption unrealistic and potentially misleading. Having said that, I can see some cases where this could still be acceptable and useful (e.g. similar products with similar dynamics and similar prices).

Summarising:

1. If you are conducting a forecasting experiment without a specific context, I’d recommend using RMSSE or some other similar measure with scaling.
2. If you have prices of products, income-based scaling might be more informative.
3. Setting all prices to the same value does not sound appealing to me, but I understand that there is a context where this might work.

Message Scaling of error measures first appeared on OpenForecast.