Mean absolute scaled error

Key takeaways

Mean absolute scaled error belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Mean absolute scaled error to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Mean absolute scaled error from memory before moving on to harder problems.

Reference excerpt

In statistics, the mean absolute scaled error (MASE) is a measure of the accuracy of forecasts. It is the mean absolute error of the forecast values, divided by the mean absolute error of the in-sample one-step naive forecast. It was proposed in 2005 by statistician Rob J. Hyndman and decision scientist Anne B. Koehler, who described it as a "generally applicable measurement of forecast accuracy without the problems seen in the other measurements." The mean absolute scaled error has favorable properties when compared to other methods for calculating forecast errors, such as root-mean-square-deviation, and is therefore recommended for determining comparative accuracy of forecasts.

Rationale The mean absolute scaled error has the following desirable properties:

Scale invariance: The mean absolute scaled error is independent of the scale of the data, so can be used to compare forecasts across data sets with different scales. Predictable behavior as y t → 0 {\displaystyle y_{t}\rightarrow 0} : Percentage forecast accuracy measures such as the Mean absolute percentage error (MAPE) rely on division of y t {\displaystyle y_{t}} , skewing the distribution of the MAPE for values of y t {\displaystyle y_{t}} near or equal to 0. This is especially problematic for data sets whose scales do not have a meaningful 0, such as temperature in Celsius or Fahrenheit, and for intermittent demand data sets, where y t = 0 {\displaystyle y_{t}=0} occurs frequently. Symmetry: The mean absolute scaled error penalizes positive and negative forecast errors equally, and penalizes errors in large forecasts and small forecasts equally. In contrast, the MAPE and median absolute percentage error (MdAPE) fail both of these criteria, while the "symmetric" sMAPE and sMdAPE fail the second criterion. Interpretability: The mean absolute scaled error can be easily interpreted, as values greater than one indicate that in-sample one-step forecasts from the naïve method perform better than the forecast values under consideration. Asymptotic normality of the MASE: The Diebold-Mariano test for one-step forecasts is used to test the statistical significance of the difference between two sets of forecasts. To perform hypothesis testing with the Diebold-Mariano test statistic, it is desirable for D M ∼ N ( 0 , 1 ) {\displaystyle DM\sim N(0,1)} , where D M {\displaystyle DM} is the value of the test statistic. The DM statistic for the MASE has been empirically shown to approximate this distribution, while the mean relative absolute error (MRAE), MAPE and sMAPE do not.

Non seasonal time series For a non-seasonal time series, the mean absolute scaled error is estimated by

M A S E = m e a n ( | e j | 1 T − 1 ∑ t = 2 T | Y t − Y t − 1 | ) = 1 J ∑ j | e j | 1 T − 1 ∑ t = 2 T | Y t − Y t − 1 | {\displaystyle \mathrm {MASE} =\mathrm {mean} \left({\frac {\left|e_{j}\right|}{{\frac {1}{T-1}}\sum _{t=2}^{T}\left|Y_{t}-Y_{t-1}\right|}}\right)={\frac {{\frac {1}{J}}\sum _{j}\left|e_{j}\right|}{{\frac {1}{T-1}}\sum _{t=2}^{T}\left|Y_{t}-Y_{t-1}\right|}}}

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Worked examples

Example 1 — a first encounter with Mean absolute scaled error

Start with the simplest possible case. Write down what Mean absolute scaled error claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Mean absolute scaled error before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Mean absolute scaled error ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Mean absolute scaled error

In research: Mean absolute scaled error appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Mean absolute scaled error in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Mean absolute scaled error is common in secondary-school and first-year university syllabi. It links to neighbouring topics Point estimation performance, Statistical deviation and dispersion, Time series, so understanding it makes those chapters shorter.
In everyday life: Look for Mean absolute scaled error outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Mean absolute scaled error in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Mean absolute scaled error means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Mean absolute scaled error out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Mean absolute scaled error in simple terms?

In statistics, the mean absolute scaled error (MASE) is a measure of the accuracy of forecasts. It is the mean absolute error of the forecast values, divided by the mean absolute error of the in-sample one-step naive forecast.

Why does Mean absolute scaled error matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Mean absolute scaled error?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Mean absolute scaled error.