Rescaled range

Key takeaways

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

Reference excerpt

The rescaled range is a statistical measure of the variability of a time series introduced by the British hydrologist Harold Edwin Hurst (1880–1978). Its purpose is to provide an assessment of how the apparent variability of a series changes with the length of the time-period being considered. The rescaled range of time series is calculated from dividing the range of its mean adjusted cumulative deviate series (see § Calculation) by the standard deviation of the time series itself. For example, consider a time series {1,3,1,0,2,5}, which has a mean m = 2 and standard deviation S = 1.79. Subtracting m from each value of the series gives mean adjusted series {-1,1,-1,-2,0,3}. To calculate cumulative deviate series we take the first value -1, then sum of the first two values -1+1=0, then sum of the first three values and so on to get {-1,0,-1,-3,-3,0}, range of which is R = 3, so the rescaled range is R/S = 1.68. If we consider the same time series, but increase the number of observations of it, the rescaled range will generally also increase. The increase of the rescaled range can be characterized by making a plot of the logarithm of R/S vs. the logarithm of the number of samples. The slope of this line gives the Hurst exponent, H. If the time series is generated by a random walk (or a Brownian motion process) it has the value of H = 1/2. Many physical phenomena that have a long time series suitable for analysis exhibit a Hurst exponent greater than 1/2. For example, observations of the height of the Nile River measured annually over many years gives a value of H = 0.77. Several researchers (including Peters, 1991) have found that the prices of many financial instruments (such as currency exchange rates, stock values, etc.) also have H > 1/2. This means that they have a behavior that is distinct from a random walk, and therefore the time series is not generated by a stochastic process that has the nth value independent of all of the values before this. According to a model of Fractional Brownian motion this is referred to as long memory of positive linear autocorrelation. However it has been shown that this measure is correct only for linear evaluation: complex nonlinear processes with memory need additional descriptive parameters. Several studies using Lo's modified rescaled range statistic have contradicted Peters' results as well.

Calculation The Rescaled Range is calculated for a time series, X = X 1 , X 2 , … , X n {\displaystyle X=X_{1},X_{2},\dots ,X_{n}\,} , as follows: Calculate the mean

m = 1 n ∑ i = 1 n X i {\displaystyle m={\frac {1}{n}}\sum _{i=1}^{n}X_{i}\,}

Create a mean adjusted series

Y t = X t − m for t = 1 , 2 , … , n {\displaystyle Y_{t}=X_{t}-m{\text{ for }}t=1,2,\dots ,n\,}

Calculate the cumulative deviate series Z;

Z t = ∑ i = 1 t Y i for t = 1 , 2 , … , n {\displaystyle Z_{t}=\sum _{i=1}^{t}Y_{i}{\text{ for }}t=1,2,\dots ,n\,}

Create a range series R;

R t = max ( Z 1 , Z 2 , … , Z t ) − min ( Z 1 , Z 2 , … , Z t ) for t = 1 , 2 , … , n {\displaystyle R_{t}=\max \left(Z_{1},Z_{2},\dots ,Z_{t}\right)-\min \left(Z_{1},Z_{2},\dots ,Z_{t}\right){\text{ for }}t=1,2,\dots ,n\,}

Create a standard deviation series S;

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Rescaled range

Start with the simplest possible case. Write down what Rescaled range 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 Rescaled range 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 Rescaled range 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 Rescaled range

In research: Rescaled range 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 Rescaled range 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: Rescaled range is common in secondary-school and first-year university syllabi. It links to neighbouring topics Autocorrelation, Independence (probability theory), Statistical deviation and dispersion, so understanding it makes those chapters shorter.
In everyday life: Look for Rescaled range 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 Rescaled range in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Rescaled range 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 Rescaled range out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Rescaled range in simple terms?

The rescaled range is a statistical measure of the variability of a time series introduced by the British hydrologist Harold Edwin Hurst (1880–1978). Its purpose is to provide an assessment of how the apparent variability of a series changes with the length of the time-period being considered.

Why does Rescaled range 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 Rescaled range?

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 Rescaled range.