|
|
| Line 1: |
Line 1: |
| {{Other uses}}
| | Singapore has elevated a tax on foreign property consumers as a part of new temporary measures to cool its residential housing market which has seen continued strong demand despite earlier efforts to curb prices.<br><br>We've lost depend of the variety of rounds of cooling measures the Government has unleashed on the property market to curb spiralling prices. The newest is the revised debt servicing framework introduced by the Financial Authority of Singapore (MAS) in June 2013. This basically imposes stricter guidelines on borrowers seeking a bank mortgage to fund their actual-property purchases. Every property has an AV. This AV of a property is estimated based mostly on market leases of similar or comparable properties. What this means is that in case you own a five-room flat in Toa Payoh, the Inland Income Authority of Singapore (IRAS) seems at similar 5-room flats in Toa Payoh and how much they are rented out at, to determine the Annual Value. pending for sale licence Approval Apr, 2013<br><br>What's the outlook for buyers in these areas, notably rental property traders? With few houses on the market and excessive costs, there might be an increased demand for rental properties. Unfortunately, there might be few on the market that will probably be suitable for cash stream investing. Nevertheless, the sharp investor who desires to put money into these markets will look to the perimeters. Rents will probably be high, something we like. The tenants can be forced to move outward from the middle and they'll pay greater rents the closer they'll find.<br><br>Look via the unit plans and resolve what unit type you need. A worth psf (per sq. foot) range is often provided so that you'd be capable of estimate a rough price for every unit sort. (You may need to use the mortgage calculator on the right of this page to determine the instalments payable for getting property in Singapore.) 2)People who would love upgrade from their current house to a rental but are not sure if they'll afford it Use our free charting tools to chart the transacted prices, transacted leases and rental yields of properties in Singapore. Foreigners of sure nationalities who fall within the scope of the respective FTAs shall be accorded with the same treatment as Singapore Citizens. District 25, ninety nine years LH Executive [http://www.tropeaperamore.eu/members/brookevgzjqtps/ Apartment In singapore] Kind of itemizing<br><br>The Council for Estate Brokers (CEA ) is a statutory board which administers the enhanced regulatory framework for the real estate company industry. Estate agents should register with the CEA, go an business examination and endure yearly training (persevering with skilled growth, CPD). CEA additionally engages in public training efforts, helps shoppers in property transactions and assists in disputes between brokers and shoppers. Checking the Property and the Seller's Credentials<br><br>One Balmoral – Ultra High End Freehold Condominium To Match Your Status! One Balmoral is a ultra high end freehold condominium situated at One Balmoral Street. One Balmoral A growth by Hong Leong Holdings Limited, consisting of ninety one units in Oceanfront Suites, irresistible pricing for a 946 leasehold property with magnificent sea view. Dreaming of basking and feeling the heat of pure sunlight is now only a click on away. Oceanfront Suites - Seaside residing not needs to stay an unattainable Ang Mo Kio Cluster House is launching tentatively in April 2013. Click here to register your interest for Ang Mo Kio Cluster Home now! Ang Mo Kio Cluster House is a rare freehold strata landed improvement along Ang Mo Kio Avenue Scholar's boarding house or hostel<br><br>Achievers are at all times the first to check new technologies & services that may assist them enhance their gross sales. When property guru first began, many brokers didn't consider in it until they started listening to other colleagues getting incredible results. Most brokers desires to see proof first, before they dare to take the first step in trying. These are often the late comers or late adopters. There is a reason why top achievers are heading the wave or heading the way. Just because they struggle new things ahead of others. The remaining simply comply with after! |
| {{refimprove|date=September 2013}}
| |
| [[File:MovingAverage.GIF|thumb]]
| |
| <!-- Deleted image removed: [[Image:MA Example.jpg|thumb|right|13-point moving average. {{Deletable image-caption|date=May 2012}}]] -->
| |
| In [[statistics]], a '''moving average''' ('''rolling average''' or '''running average''') is a calculation to analyze data points by creating a series of [[average]]s of different subsets of the full data set. It is also called a '''moving mean (MM)'''<ref>[http://www.waterboards.ca.gov/waterrights/water_issues/programs/bay_delta/docs/cmnt091412/sldmwa/booth_et_al_2006.pdf Hydrologic Variability of the Cosumnes River Floodplain] (Booth et. al., San Francisco Estuary and Watershed Science, Volume 4, Issue 2, 2006)</ref> or '''rolling mean''' and is a type of [[finite impulse response]] filter. Variations include: [[#Simple moving average|simple]], and [[#Cumulative moving average|cumulative]], or [[#weighted moving average|weighted]] forms (described below).
| |
| | |
| Given a series of numbers and a fixed subset size, the first element of the moving average is obtained by taking the average of the initial fixed subset of the number series. Then the subset is modified by "shifting forward"; that is, excluding the first number of the series and including the next number following the original subset in the series. This creates a new subset of numbers, which is averaged. This process is repeated over the entire data series. The plot line connecting all the (fixed) averages is the moving average. A moving average is a set of numbers, each of which is the [[average]] of the corresponding subset of a larger set of datum points. A moving average may also use unequal weights for each datum value in the subset to emphasize particular values in the subset.
| |
| | |
| A moving average is commonly used with [[time series]] data to smooth out short-term fluctuations and highlight longer-term trends or cycles. The threshold between short-term and long-term depends on the application, and the parameters of the moving average will be set accordingly. For example, it is often used in [[technical analysis]] of financial data, like stock [[price]]s, [[return (finance)|returns]] or trading volumes. It is also used in [[economics]] to examine gross domestic product, employment or other macroeconomic time series. Mathematically, a moving average is a type of [[convolution]] and so it can be viewed as an example of a [[low-pass filter]] used in [[signal processing]]. When used with non-time series data, a moving average filters higher frequency components without any specific connection to time, although typically some kind of ordering is implied. Viewed simplistically it can be regarded as smoothing the data.
| |
| | |
| ==Simple moving average==
| |
| [[File:Moving Average Types comparison - Simple and Exponential.png|thumb]]
| |
| In financial applications a '''simple moving average''' (SMA) is the unweighted [[arithmetic mean|mean]] of the previous ''n'' datum points. However, in science and engineering the mean is normally taken from an equal number of data on either side of a central value. This ensures that variations in the mean are aligned with the variations in the data rather than being shifted in time.
| |
| {{Anchor|algorithm}}
| |
| An example of a simple equally weighted running mean for a n-day sample of closing price is the mean of the previous ''n'' days' closing prices. If those prices are <math>p_M, p_{M-1},\dots,p_{M-(n-1)}</math> then the formula is
| |
| <!--
| |
| this used to be a hairy sigma style with i's and j's, but prefer dots since it's a lot easier for financial people with limited math -->
| |
| | |
| :<math>\textit{SMA} = { p_M + p_{M-1} + \cdots + p_{M-(n-1)} \over n }</math>
| |
| | |
| When calculating successive values, a new value comes into the sum and an old value drops out, meaning a full summation each time is unnecessary for this simple case,
| |
| | |
| :<math>\textit{SMA}_\mathrm{today} = \textit{SMA}_\mathrm{yesterday} - {p_{M-n} \over n} + {p_{M} \over n}</math>
| |
| | |
| The period selected depends on the type of movement of interest, such as short, intermediate, or long-term. In financial terms moving-average levels can be interpreted as [[support (technical analysis)|support]] in a rising market, or [[resistance (technical analysis)|resistance]] in a falling market.
| |
| | |
| If the data used are not centered around the mean, a simple moving average lags behind the latest datum point by half the sample width. An SMA can also be disproportionately influenced by old datum points dropping out or new data coming in. One characteristic of the SMA is that if the data have a periodic fluctuation, then applying an SMA of that period will eliminate that variation (the average always containing one complete cycle). But a perfectly regular cycle is rarely encountered.<ref>''Statistical Analysis'', Ya-lun Chou, Holt International, 1975, ISBN 0-03-089422-0, section 17.9.</ref>
| |
| | |
| For a number of applications, it is advantageous to avoid the shifting induced by using only 'past' data. Hence a '''central moving average''' can be computed, using data equally spaced on either side of the point in the series where the mean is calculated. This requires using an odd number of datum points in the sample window.
| |
| | |
| A major drawback of the SMA is that it lets though a significant amount of the signal shorter than the window length. Worse, it '''actually inverts it'''. This can lead to unexpected artifacts, such as peaks in the "smoothed" result appearing where there were troughs in the data. It also leads to the result being less "smooth" than expected since some of the higher frequencies are not properly removed.
| |
| | |
| The problem can be over-come by repeating the process three times with the window being shortened by a factor of 1.4303 at each step.<ref>http://climategrog.wordpress.com/2013/05/19/triple-running-mean-filters/</ref> This removes the negation effects and provides a well-behaved filter. This solution is often used in [[real-time computing|real-time]] audio filtering since it is computationally quicker than other comparable filters such as a [[gaussian function|gaussian]] kernel.
| |
| | |
| An example of inversion defect in SMA and the application of repeating SMA to avoid it can be illustrated here: http://www.woodfortrees.org/plot/rss/from:1980/plot/rss/from:1980/mean:60/plot/rss/from:1980/mean:30/mean:22/mean:17
| |
| | |
| ==Cumulative moving average==
| |
| In a '''cumulative moving average''', the data arrive in an ordered datum stream, and the user would like to get the average of all of the data up until the current datum point. For example, an investor may want the average price of all of the stock transactions for a particular stock up until the current time. As each new transaction occurs, the average price at the time of the transaction can be calculated for all of the transactions up to that point using the cumulative average, typically an equally weighted [[average]] of the sequence of ''i'' values ''x''<sub>1</sub>, ..., ''x<sub>i</sub>'' up to the current time:
| |
| | |
| :<math>CA_i = {{x_1 + \cdots + x_i} \over i}\,.</math>
| |
| | |
| The brute-force method to calculate this would be to store all of the data and calculate the sum and divide by the number of datum points every time a new datum point arrived. However, it is possible to simply update cumulative average as a new value, ''x''<sub>''i''+1</sub> becomes available, using the formula: | |
| | |
| :<math>CA_{i+1} = {{x_{i+1} + i CA_i} \over {i+1}}\,,</math>
| |
| | |
| where <math>CA_0</math> can be taken to be equal to 0.
| |
| | |
| Thus the current cumulative average for a new datum point is equal to the previous cumulative average, times ''i'', plus the latest datum point, all divided by the number of points received so far, ''i''+1. When all of the datum points arrive ({{nowrap|1=''i'' = ''N''}}), then the cumulative average will equal the final average.
| |
| | |
| The derivation of the cumulative average formula is straightforward. Using
| |
| | |
| :<math>x_1 + \cdots + x_i = iCA_i\,,</math>
| |
| | |
| and similarly for {{nowrap|''i'' + 1}}, it is seen that | |
| | |
| :<math>x_{i+1} = (x_1 + \cdots + x_{i+1}) - (x_1 + \cdots + x_i) = (i+1)CA_{i+1} - iCA_i\,.</math>
| |
| | |
| Solving this equation for ''CA''<sub>''i''+1</sub> results in:
| |
| | |
| :<math>CA_{i+1} = {(x_{i+1} + iCA_i) \over {i+1}} = {CA_i} + {{x_{i+1} - CA_i} \over {i+1}}\,.</math>
| |
| | |
| ==Weighted moving average==
| |
| A weighted average is any average that has multiplying factors to give different weights to data at different positions in the sample window. Mathematically, the moving average is the [[convolution]] of the datum points with a fixed weighting function. One application is removing [[pixelisation]] from a digital graphical image.
| |
| | |
| In [[technical analysis]] of financial data, a '''weighted moving average''' (WMA) has the specific meaning of weights that decrease in arithmetical progression.<ref>{{cite web|title=Weighted Moving Averages: The Basics |url=http://www.investopedia.com/articles/technical/060401.asp |publisher=Investopedia}}</ref> In an ''n''-day WMA the latest day has weight ''n'', the second latest ''n'' − 1, etc., down to one.
| |
| | |
| :<math>\text{WMA}_{M} = { n p_{M} + (n-1) p_{M-1} + \cdots + 2 p_{(M-n+2)} + p_{(M-n+1)} \over n + (n-1) + \cdots + 2 + 1}</math>
| |
| | |
| [[Image:Weighted moving average weights N=15.png|thumb|right|WMA weights ''n'' = 15]]
| |
| | |
| The denominator is a [[triangle number]] equal to <math>\frac{n(n+1)}{2}.</math> In the more general case the denominator will always be the sum of the individual weights.
| |
| | |
| When calculating the WMA across successive values, the difference between the numerators of WMA<sub>''M''+1</sub> and WMA<sub>M</sub> is ''np''<sub>''M''+1</sub> − ''p''<sub>M</sub> − ⋅⋅⋅ − ''p''<sub>''M''−n+1</sub>. If we denote the sum ''p''<sub>M</sub> + ⋅⋅⋅ + ''p''<sub>''M''−''n''+1</sub> by Total<sub>M</sub>, then
| |
| | |
| :<math>\text{Total}_{M+1} = Total_{M} + p_{M+1} - p_{M-n+1} \,</math>
| |
| | |
| :<math>\text{Numerator}_{M+1} = \text{Numerator}_M + n p_{M+1} - Total_M \,</math>
| |
| | |
| :<math>\text{WMA}_{M+1} = { \text{Numerator}_{M+1} \over n + (n-1) + \cdots + 2 + 1} \,</math>
| |
| | |
| The graph at the right shows how the weights decrease, from highest weight for the most recent datum points, down to zero. It can be compared to the weights in the exponential moving average which follows.
| |
| | |
| ==Exponential moving average==
| |
| {{further2|[[EWMA chart]]}}
| |
| [[Image:Exponential moving average weights N=15.png|thumb|right|EMA weights ''N''=15]]
| |
| | |
| An '''exponential moving average''' (EMA), also known as an '''exponentially weighted moving average''' (EWMA),<ref>http://lorien.ncl.ac.uk/ming/filter/filewma.htm</ref> is a type of [[infinite impulse response]] filter that applies weighting factors which decrease [[exponentiation|exponentially]]. The weighting for each older [[data|datum]] decreases exponentially, never reaching zero. The graph at right shows an example of the weight decrease.
| |
| | |
| The EMA for a series ''Y'' may be calculated recursively:
| |
| | |
| :<math>S_1 = Y_1</math>
| |
| for <math> t > 1,\ \ S_{t} = \alpha \cdot Y_{t-1} + (1-\alpha) \cdot S_{t-1}</math>
| |
| | |
| Where:
| |
| * The coefficient ''α'' represents the degree of weighting decrease, a constant smoothing factor between 0 and 1. A higher ''α'' discounts older observations faster.
| |
| * ''Y<sub>t</sub>'' is the value at a time period ''t''.
| |
| * ''S<sub>t</sub>'' is the value of the EMA at any time period ''t''.
| |
| | |
| ''S''<sub>1</sub> is undefined. ''S''<sub>1</sub> may be initialized in a number of different ways, most commonly by setting ''S''<sub>1</sub> to ''Y''<sub>1</sub>, though other techniques exist, such as setting ''S''<sub>1</sub> to an average of the first 4 or 5 observations. The importance of the ''S''<sub>1</sub> initialisations effect on the resultant moving average depends on ''α''; smaller ''α'' values make the choice of ''S''<sub>1</sub> relatively more important than larger ''α'' values, since a higher ''α'' discounts older observations faster.
| |
| | |
| Whatever is done for ''S''<sub>1</sub> it assumes something about values prior to the available data and is necessarily in error. In view of this the early results should be regarded as unreliable until the iterations have had time to [[Convergent series|converge]]. This is sometimes called a 'spin-up' interval. One way to assess when it can be regarded as reliable is consider the required accuracy of the result. For example, if 3% accuracy is required, initialising with ''Y''<sub>1</sub> and taking data after five time constants (defined above) will ensure that the calculation has converged to within 3% (only <3% of ''Y''<sub>1</sub> will remain in the result ). Sometimes with very small alpha, this can mean little of the result is useful. This is analogous to the problem of using a convolution filter (such as a weighted average) with a very long window.
| |
| | |
| This formulation is according to Hunter (1986).<ref>[http://www.itl.nist.gov/div898/handbook/pmc/section4/pmc431.htm NIST/SEMATECH e-Handbook of Statistical Methods: Single Exponential Smoothing] at the [[National Institute of Standards and Technology]]</ref> By repeated application of this formula for different times, we can eventually write ''S<sub>t</sub>'' as a weighted sum of the datum points ''Y<sub>t</sub>'', as:
| |
| | |
| :<math>S_{t} = \alpha \times (Y_{t-1} + (1-\alpha) \times Y_{t-2} + (1-\alpha)^2 \times Y_{t-3} + \cdots + (1-\alpha)^k \times Y_{t-(k+1)}) + (1-\alpha)^{k+1} \times S_{t-(k+1)}</math>
| |
| | |
| for any suitable k = 0, 1, 2, ... The weight of the general datum point <math>Y_{t-i}</math> is <math>\alpha(1-\alpha)^{i-1} </math>.
| |
| | |
| An alternate approach by Roberts (1959) uses ''Y<sub>t</sub>'' in lieu of ''Y''<sub>''t''−1</sub>:<ref>[http://www.itl.nist.gov/div898/handbook/pmc/section3/pmc324.htm NIST/SEMATECH e-Handbook of Statistical Methods: EWMA Control Charts] at the [[National Institute of Standards and Technology]]</ref>
| |
| | |
| :<math>S_{t,\text{ alternate}} = \alpha \cdot Y_t + (1-\alpha) \cdot S_{t-1}</math>
| |
| | |
| This formula can also be expressed in technical analysis terms as follows, showing how the EMA steps towards the latest datum point, but only by a proportion of the difference (each time):
| |
| :<math>\text{EMA}_{\text{today}} = \text{EMA}_{\text{yesterday}} + \alpha \times
| |
| | |
| (\text{price}_{\text{today}} - \text{EMA}_\text{yesterday})</math>
| |
| | |
| Expanding out <math>\text{EMA}_{\text{yesterday}}</math> each time results in the following power series, showing how the weighting factor on each datum point ''p''<sub>1</sub>, ''p''<sub>2</sub>, etc., decreases exponentially:
| |
| :<math>\text{EMA}_{\text{today}} = { \alpha \times (p_1 + (1-\alpha) p_2 + (1-\alpha)^2 p_3 + (1-\alpha)^3
| |
| p_4 + \cdots ) }</math>
| |
| | |
| where
| |
| * <math>p_1</math> is <math>\text{price}_{\text{today}}</math>
| |
| * <math>p_2</math> is <math>\text{price}_{\text{yesterday}}</math>
| |
| * and so on
| |
| | |
| <math>\text{EMA}_{\text{today}} = { p_1 + (1-\alpha) p_2 + (1-\alpha)^2 p_3 + (1-\alpha)^3 p_4 + \cdots \over 1 + (1-\alpha) + (1-\alpha)^2 + (1-\alpha)^3 + \cdots }</math>,
| |
| | |
| since <math>1/\alpha = 1+(1-\alpha)+(1-\alpha)^2+\cdots</math>.
| |
| | |
| This is an [[series (mathematics)|infinite sum]] with decreasing terms.
| |
| | |
| The ''N'' periods in an ''N''-day EMA only specify the ''α'' factor. ''N'' is not a stopping point for the calculation in the way it is in an [[#Simple moving average|SMA]] or [[#Weighted moving average|WMA]]. For sufficiently large ''N'', the first ''N'' datum points in an EMA represent about 86% of the total weight in the calculation when <math> \alpha = 2 / (N + 1) </math>:<ref>The denominator on the left-hand side should be unity, and the numerator will become the right-hand side ([[geometric progression#Geometric series|geometric series]]),
| |
| <math>\alpha \left({1-(1-\alpha)^{N+1} \over 1-(1-\alpha)}\right)</math>.</ref>
| |
| | |
| :<math> {{\alpha \times \left(1+(1-\alpha)+(1-\alpha)^2+\cdots +(1-\alpha)^N \right)} \over {\alpha \times \left(1+(1-\alpha)+(1-\alpha)^2+\cdots +(1-\alpha)^\infty \right)}}= 1-{\left(1-{2 \over N+1}\right)}^{N+1}</math>
| |
| | |
| :i.e. <math> \lim_{N \to \infty} \left[1-{\left(1-{2 \over N+1}\right)}^{N+1} \right] </math> simplified,<ref>Because (1+''x''/''n'')<sup>''n''</sup> tends to the limit e<sup>''x''</sup> for large ''n''.</ref> tends to <math>1-\text{e}^{-2} \approx 0.8647</math>.
| |
| | |
| The above discussion requires a bit of clarification. The sum of the weights of all the terms (i.e., infinite number of terms) in an exponential moving average is 1. The sum of the weights of <math> N </math> terms is <math> 1 - (1 - \alpha)^{N+1} </math>. Both of these sums can be derived by using the formula for the sum of a geometric series. The weight omitted after <math> N </math> terms is given by subtracting this from 1, and you get <math> 1 - (1 - (1 - \alpha)^{N+1}) = (1 - \alpha)^{N+1} </math> (this is essentially the formula given below for the weight omitted). Note that there is no "accepted" value that should be chosen for <math> \alpha </math> although there are some recommended values based on the application. In the above discussion, we have substituted a commonly used value for <math> \alpha = 2 / (N + 1) </math> in the formula for the weight of <math> N </math> terms. This value for <math> \alpha </math> comes from setting the average age of the data from a SMA equal to the average age of the data from an EWA and solving for <math> \alpha </math>. Again, it is just a recommendation—not a requirement. If you make this substitution, and you make use of<ref>See the following [http://options-trading-notes.blogspot.com/2013/06/derivation-of-ea.html link] for a proof.</ref> <math> \lim_{n \to \infty} \left( 1 + {a \over {1 + n}} \right)^n = e^a </math>, then you get the 0.864 approximation. Intuitively, what this is telling us is that the weight after <math> N </math> terms of an ``<math> N </math>-period" exponential moving average converges to 0.864.
| |
| | |
| The [[power series|power formula]] above gives a starting value for a particular day, after which the successive days formula shown first can be applied. The question of how far back to go for an initial value depends, in the worst case, on the data. Large price values in old data will affect on the total even if their weighting is very small. If prices have small variations then just the weighting can be considered. The weight omitted by stopping after ''k'' terms is
| |
| | |
| :<math>\alpha \times \left( (1-\alpha)^k + (1-\alpha)^{k+1} + (1-\alpha)^{k+2} + \cdots \right),</math>
| |
| | |
| which is
| |
| | |
| :<math>\alpha \times (1-\alpha)^k \times \left(1 + (1-\alpha) + (1-\alpha)^2 + \cdots \right),</math>
| |
| | |
| i.e. a fraction
| |
| | |
| :<math>
| |
| {{\text{weight omitted by stopping after k terms}} \over {\text{total weight}}} = { { \alpha \times \left[ (1-\alpha)^k +(1-\alpha)^{k+1} +(1-\alpha)^{k+2} + \cdots \right] } \over { { \alpha \times \left[ 1 + (1-\alpha) +(1-\alpha)^{2} + \cdots \right] } } }
| |
| </math>
| |
| | |
| :<math>
| |
| = { {\alpha (1-\alpha)^k \times {{1} \over {1-(1-\alpha)}}} \over { { {\alpha} \over {1-(1-\alpha) } } } }
| |
| </math>
| |
| | |
| :<math>
| |
| = (1 - \alpha)^k
| |
| </math>
| |
| | |
| out of the total weight.
| |
| | |
| For example, to have 99.9% of the weight, set above ratio equal to 0.1% and solve for ''k'':
| |
| | |
| :<math>k={ \log (0.001) \over \log (1-\alpha)}</math>
| |
| | |
| terms should be used. Since <math>\log\,(1-\alpha)</math> approaches <math>-2 \over N+1</math> as N increases,<ref>It means <math>\alpha</math> -> 0, and the [[Taylor series]] of <math>\log(1-\alpha) = -\alpha -\alpha^2/2 - \cdots</math> is equivalent to <math>-\alpha</math>.</ref> this simplifies to approximately<ref>log<sub>e</sub>(0.001) / 2 = -3.45</ref>
| |
| | |
| :<math>k = 3.45(N+1) \,</math>
| |
| | |
| for this example (99.9% weight). | |
| | |
| ===Modified moving average===
| |
| A '''modified moving average''' (MMA), '''running moving average''' (RMA), or '''smoothed moving average''' is defined as:
| |
| | |
| :<math>\text{MMA}_{\text{today}} = {(N - 1) \times \text{MMA}_{\text{yesterday}} + \text{price} \over{N}}</math>
| |
| | |
| In short, this is an exponential moving average, with <math>\alpha=1/N</math>.
| |
| | |
| ===Application to measuring computer performance===
| |
| Some computer performance metrics, e.g. the average process queue length, or the average CPU utilization, use a form of exponential moving average.
| |
| | |
| :<math>S_n = \alpha(t_{n}-t_{n-1}) \times Y_n + (1-\alpha(t_n-t_{n-1})) \times
| |
| S_{n-1}.</math>
| |
| | |
| Here <math>\alpha</math> is defined as a function of time between two readings. An example of a coefficient giving bigger weight to the current reading, and smaller weight to the older readings is
| |
| | |
| :<math>\alpha(t_{n}-t_{n-1}) = 1-\exp\left({-{ {t_{n}-t_{n-1}} \over {W \times 60} }}\right)</math>
| |
| | |
| where ''exp()'' is the [[exponential function]], time for readings ''t''<sub>''n''</sub> is expressed in seconds, and <math>W</math> is the period of time in minutes over which the reading is said to be averaged (the mean lifetime of each reading in the average). Given the above definition of <math>\alpha</math>, the moving average can be expressed as
| |
| | |
| :<math>S_n = (1-\exp\left({-{ {t_n - t_{n-1}} \over {W \times 60}}}\right)) \times Y_n +
| |
| | |
| \exp\left({-{{t {n}-t_{n-1}} \over {W \times 60}}}\right) \times S_{n-1}</math>
| |
| | |
| For example, a 15-minute average ''L'' of a process queue length ''Q'', measured every 5 seconds (time difference is 5 seconds), is computed as
| |
| | |
| :<math>L_n = (1-\exp\left({-{5 \over {15 \times 60}}}\right)) \times Q_n + e^{-{5 \over {15 \times 60}}}
| |
| \times L_{n-1} = (1-\exp\left({-{1 \over {180}}}\right)) \times Q_n + e^{-1/180} \times L_{n-1} = Q_n + e^{-1/180} \times ( L_{n-1} - Q_n )</math>
| |
| | |
| ==Other weightings==
| |
| Other weighting systems are used occasionally – for example, in share trading a '''volume weighting''' will weight each time period in proportion to its trading volume.
| |
| | |
| A further weighting, used by actuaries, is Spencer's 15-Point Moving Average<ref>[http://mathworld.wolfram.com/Spencers15-PointMovingAverage.html Spencer's 15-Point Moving Average — from Wolfram MathWorld<!-- Bot generated title -->]</ref> (a central moving average). The symmetric weight coefficients are −3, −6, −5, 3, 21, 46, 67, 74, 67, 46, 21, 3, −5, −6, −3.
| |
| | |
| Outside the world of finance, weighted running means have many forms and applications. Each weighting function or "kernel" has its own characteristics. In engineering and science the frequency and phase response of the filter is often of primary importance in understanding the desired and undesired distortions that a particular filter will apply to the data.
| |
| | |
| A mean does not just "smooth" the data. A mean is a form of low-pass filter. The effects of the particular filter used should be understood in order to make an appropriate choice. On this point,
| |
| the French version of this article discusses the spectral effects of 3 kinds of means (cumulative, exponential, Gaussian).
| |
| | |
| ==Moving median==
| |
| From a statistical point of view, the moving average, when used to estimate the underlying trend in a time series, is susceptible to rare events such as rapid shocks or other anomalies. A more robust estimate of the trend is the '''simple moving median''' over ''n'' time points:
| |
| | |
| :<math>\textit{SMM} = \text{Median}( p_M, p_{M-1}, \ldots, p_{M-n+1} )</math>
| |
| | |
| where the [[median]] is found by, for example, sorting the values inside the brackets and finding the value in the middle. For larger values of ''n'', the median can be efficiently computed by updating an [[Skip list#Indexable skiplist|indexable skiplist]].<ref>http://code.activestate.com/recipes/576930/</ref>
| |
| | |
| Statistically, the moving average is optimal for recovering the underlying trend of the time series when the fluctuations about the trend are [[normal distribution|normally distributed]]. However, the normal distribution does not place high probability on very large deviations from the trend which explains why such deviations will have a disproportionately large effect on the trend estimate. It can be shown that if the fluctuations are instead assumed to be [[Laplace distribution|Laplace distributed]], then the moving median is statistically optimal.<ref>G.R. Arce, "Nonlinear Signal Processing: A Statistical Approach", Wiley:New Jersey, USA, 2005.</ref> For a given variance, the Laplace distribution places higher probability on rare events than does the normal, which explains why the moving median tolerates shocks better than the moving mean.
| |
| | |
| When the simple moving median above is central, the smoothing is identical to the [[median filter]] which has applications in, for example, image signal processing.
| |
| | |
| ==See also==
| |
| *[[Exponential smoothing]]
| |
| *[[MACD|Moving average convergence/divergence indicator]]
| |
| *[[Window function]]
| |
| *[[Moving average crossover]]
| |
| *[[Rising moving average]]
| |
| *[[Running total]]
| |
| {{More footnotes|date=February 2010}}
| |
| | |
| ==Notes and references==
| |
| {{reflist}}
| |
| | |
| {{statistics}}
| |
| {{technical analysis|state=collapsed}}
| |
| | |
| {{DEFAULTSORT:Moving Average}}
| |
| [[Category:Statistical charts and diagrams]]
| |
| [[Category:Time series analysis]]
| |
| [[Category:Mathematical finance]]
| |
| [[Category:Chart overlays]]
| |