Quasi-polynomial

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 is calculated from dividing the range of the values exhibited in a portion of the time series by the standard deviation of the values over the same portion of the time series. For example, consider a time series {2, 5, 3, 7, 8, 12, 4, 2} which has a range, R, of 12 - 2 = 10. Its standard deviation, s, is 3.46, so the rescaled range is R/s = 2.71.

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 n. 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. This is referred to as long term memory. However this result is controversial and several studies using Lo's (Lo, 1991) modified rescaled range statistic have contradicted Peters' results.

Calculation

The Rescaled Range is calculated for a time series, $X=X_{1},X_{2},\dots ,X_{n}\,$, as follows:

1. Calculate the mean

   $m={\frac {1}{n}}\sum _{i=1}^{n}X_{i}\,$

2. Create a mean adjusted series

   $Y_{t}=X_{t}-m{\text{ for }}t=1,2,\dots ,n\,$

3. Calculate the cumulative deviate series Z;

   $Z_{t}=\sum _{i=1}^{t}Y_{i}{\text{ for }}t=1,2,\dots ,n\,$

4. Create a range series R;

   $R_{t}=\max \left(Z_{1},Z_{2},\dots ,Z_{t}\right)-\min \left(Z_{1},Z_{2},\dots ,Z_{t}\right){\text{ for }}t=1,2,\dots ,n\,$

5. Create a standard deviation series S;

   $S_{t}={\sqrt {{\frac {1}{t}}\sum _{i=1}^{t}\left(X_{i}-u\right)^{2}}}{\text{ for }}t=1,2,\dots ,n\,$

   Where u is the mean for the time series values $X_{1},X_{2},\dots ,X_{t}\,$

6. Calculate the rescaled range series (R/S)

   $\left(R/S\right)_{t}={\frac {R_{t}}{S_{t}}}{\text{ for }}t=1,2,\dots ,n\,$

Lo (1991) advocates adjusting the standard deviation $S$ for the expected increase in range $R$ resulting from short-range autocorrelation in the time series. This involves replacing $S$ by ${\hat {S}}$, which is the square root of

${\hat {S}}^{2}=S^{2}+2\sum _{j=1}^{q}\left(1-{\frac {j}{q+1}}\right)C(j),$

where $q$ is some maximum lag over which short-range autocorrelation might be substantial and $C(j)$ is the sample autocovariance at lag $j$. Using this adjusted rescaled range, he concludes that stock market return time series show no evidence of long-range memory.

See also

• Fractal
• Fractional Brownian motion
• Fat tail

References

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• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
  
  In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
  
  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
  Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
  
  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
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