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In [[statistics]], the '''Pearson product-moment correlation coefficient''' (sometimes referred to as the '''PPMCC''' or '''PCC''',<ref>"The human disease network", Albert Barabasi et al., Plos.org</ref> or '''Pearson's ''r''''', and is typically denoted by ''r'') is a measure of the [[correlation]] (linear dependence) between two variables ''X'' and ''Y'', giving a value between +1 and −1 inclusive. It is widely used in the sciences as a measure of the strength of linear dependence between two variables. It was developed by [[Karl Pearson]] from a similar but slightly different idea introduced by [[Francis Galton]] in the 1880s.<ref name="thirteenways">J. L. Rodgers and W. A. Nicewander. [http://www.jstor.org/stable/2685263 Thirteen ways to look at the correlation coefficient]. The American Statistician, 42(1):59–66, February 1988.</ref><ref>{{Cite journal| doi = 10.1214/ss/1177012580| last = Stigler | first = Stephen M. | title = Francis Galton's Account of the Invention of Correlation | journal = Statistical Science | volume=4 | issue=2 | pages = 73–79 | year = 1989 | jstor=2245329}}</ref>
{{other uses}}


[[Image:Correlation examples2.svg|thumb|400px|right|Several sets of (''x'',&nbsp;''y'') points, with the correlation coefficient of ''x'' and ''y'' for each set. Note that the correlation reflects the non-linearity and direction of a linear relationship (top row), but not the slope of that relationship (middle), nor many aspects of nonlinear relationships (bottom). N.B.: the figure in the center has a slope of 0 but in that case the correlation coefficient is undefined because the variance of ''Y'' is zero.]]
[[Image:Cirrus clouds2.jpg|thumb|right|[[Cirrus clouds|Cirrus]] uncinus ice crystal plumes showing high level wind shear, with changes in wind speed and direction.]]


==Definition==
'''Wind shear''', sometimes referred to as '''windshear''' or [[wind gradient]], is a difference in [[wind]] [[wind speed|speed]] and [[wind direction|direction]] over a relatively short distance in the [[Earth's atmosphere|atmosphere]]. Wind shear can be broken down into vertical and horizontal components, with horizontal wind shear seen across [[Weather front|fronts]] and near the coast, and vertical shear typically near the surface, though also at higher levels in the atmosphere near upper level jets and frontal zones aloft.
Pearson's correlation coefficient between two variables is defined as the [[covariance]] of the two variables divided by the product of their [[standard deviations]]. The form of the definition involves a "product moment", that is, the mean (the first moment about the origin) of the product of the mean-adjusted random variables; hence the modifier ''product-moment'' in the name.


===For a population===
Wind shear itself is a [[microscale meteorology|microscale meteorological]] phenomenon occurring over a very small distance, but it can be associated with [[mesoscale meteorology|mesoscale]] or [[synoptic scale]] weather features such as squall lines and cold fronts. It is commonly observed near [[microburst]]s and [[downburst]]s caused by [[thunderstorm]]s, fronts, areas of locally higher low level winds referred to as low level jets, near [[mountain]]s, radiation inversions that occur due to clear skies and calm winds, buildings, wind turbines, and sailboats. Wind shear has a significant effect during take-off and landing of aircraft due to its effects on control of the aircraft, and it has been a sole or contributing cause of many aircraft accidents.
Pearson's correlation coefficient when applied to a population is commonly represented by the Greek letter ''ρ'' (rho) and may be referred to as the ''population correlation coefficient'' or the ''population Pearson correlation coefficient''. The formula for ''ρ'' is:


:<math> \rho_{X,Y}={\mathrm{cov}(X,Y) \over \sigma_X \sigma_Y} ={E[(X-\mu_X)(Y-\mu_Y)] \over \sigma_X\sigma_Y} </math>
Sound movement through the atmosphere is affected by wind shear, which can bend the wave front, causing sounds to be heard where they normally would not, or vice versa. Strong vertical wind shear within the [[troposphere]] also inhibits [[tropical cyclone]] development, but helps to organize individual thunderstorms into longer life cycles which can then produce [[severe weather]]. The [[thermal wind]] concept explains how differences in wind speed at different heights are dependent on horizontal temperature differences, and explains the existence of the [[jet stream]].<ref name="IP">{{cite web|url=http://www.tpub.com/weather3/6-15.htm|title=LOW-LEVEL WIND SHEAR.] Retrieved on 2007-11-25|last=Publishing|first=Integrated}}</ref>


===For a sample===
==Definition==
Pearson's correlation coefficient when applied to a sample is commonly represented by the letter ''r'' and may be referred to as the ''sample correlation coefficient'' or the ''sample Pearson correlation coefficient''. We can obtain a formula for ''r'' by substituting estimates of the covariances and variances based on a [[statistical sample|sample]] into the formula above. That formula for ''r'' is:
Wind shear refers to the variation of wind over either horizontal or vertical distances. Airplane pilots generally regard significant wind shear to be a horizontal change in airspeed of 30 [[knot (unit)|knots]] (15&nbsp;m/s) for light aircraft, and near 45 [[knot (unit)|knots]] (22&nbsp;m/s) for airliners at flight altitude.<ref>[[FAA]] [http://www.airweb.faa.gov/Regulatory_and_Guidance_Library/rgAdvisoryCircular.nsf/0/b3fb7dd636fb870b862569ba0068920b/$FILE/AC00-54.pdf FAA Advisory Circular Pilot Wind Shear Guide.] Retrieved on 2007-12-15.</ref> Vertical speed changes greater than 4.9&nbsp;knots (2.5&nbsp;m/s) also qualify as significant wind shear for aircraft. Low level wind shear can affect aircraft airspeed during take off and landing in disastrous ways and airliner pilots are trained to avoid all microburst wind shear (headwind loss in excess of 30 knots, Ibid.).<ref>{{cite web|title=NASA|url=http://oea.larc.nasa.gov/PAIS/Concept2Reality/wind_shear.html Wind Shear.|accessdate=2007-10-09}}</ref> The rationale for this additional caution includes (1) microburst intensity can double in a minute or less, (2) the winds can shift to excessive cross wind, (3) 40-50 knots is the threshold for survivability at some stages of low-altitude operations (Ibid.), and (4) several of the historical wind shear accidents involved 35-45 knot microbursts. Wind shear is also a key factor in the creation of severe thunderstorms. The additional hazard of [[turbulence]] is often associated with wind shear.
 
:<math>r = \frac{\sum ^n _{i=1}(X_i - \bar{X})(Y_i - \bar{Y})}{\sqrt{\sum ^n _{i=1}(X_i - \bar{X})^2} \sqrt{\sum ^n _{i=1}(Y_i - \bar{Y})^2}}</math>
 
An equivalent expression gives the correlation coefficient as the mean of the products of the [[standard score]]s. Based on a [[Statistical sample|sample]] of paired data (''X''<sub>''i''</sub>,&nbsp;''Y''<sub>''i''</sub>), the sample Pearson correlation coefficient is
 
:<math>r = \frac{1}{n-1} \sum ^n _{i=1} \left( \frac{X_i - \bar{X}}{s_X} \right) \left( \frac{Y_i - \bar{Y}}{s_Y} \right)</math>
 
where


:<math>\frac{X_i - \bar{X}}{s_X}, \bar{X}, \text{ and } s_X</math>
==Where and when it is strongly observed==
{{See also|Jet stream}}
[[Image:Microburstnasa.JPG|thumb|right|250px|[[Microburst]] schematic from NASA. Note the downward motion of the air until it hits ground level, then spreads outward in all directions. The wind regime in a microburst is completely opposite to a tornado.]]
Weather situations where shear is observed include:
*[[Weather fronts]]. Significant shear is observed when the temperature difference across the front is 5&nbsp;°C (9&nbsp;°F) or more, and the front moves at 30 [[Knot (unit)|knots]] or faster. Because fronts are three-dimensional phenomena, frontal shear can be observed at any altitude between surface and [[tropopause]], and therefore be seen both horizontally and vertically. Vertical wind shear above warm fronts is more of an aviation concern than near and behind cold fronts due to their greater duration.<ref name="IP"/>
*Upper-level jet streams. Associated with upper level jet streams is a phenomenon known as [[clear air turbulence]] (CAT), caused by vertical and horizontal wind shear connected to the wind gradient at the edge of the jet streams.<ref>{{cite web|title=BBC|url=http://www.bbc.co.uk/weather/features/understanding/jetstreams_uk.shtml Jet Streams in the UK.|accessdate=2008-05-08}} {{Dead link|date=April 2012|bot=H3llBot}}</ref> The CAT is strongest on the cold air side of the jet,<ref>M. P. de Villiers and J. van Heerden. [http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=73339 Clear air turbulence over South Africa.] Retrieved on 2008-05-08.</ref> usually next to or just below the axis of the jet.<ref>CLARK T. L., HALL W. D., KERR R. M., MIDDLETON D., RADKE L., RALPH F. M., NEIMAN P. J., LEVINSON D. [http://cat.inist.fr/?aModele=afficheN&cpsidt=1345004 Origins of aircraft-damaging clear-air turbulence during the 9 December 1992 Colorado downslope windstorm : Numerical simulations and comparison with observations.] Retrieved on 2008-05-08.</ref>
*Low-level jet streams. When a nocturnal low-level jet forms overnight above the Earth's surface ahead of a cold front, significant low level vertical wind shear can develop near the lower portion of the low level jet. This is also known as nonconvective wind shear since it is not due to nearby thunderstorms.<ref name="IP"/>
*Mountains. When winds blow over a mountain, vertical shear is observed on the [[Windward and leeward|lee]] side. If the flow is strong enough, turbulent [[Eddy (fluid dynamics)|eddies]] known as "rotors" associated with [[lee waves]] may form, which are dangerous to ascending and descending aircraft.<ref>National Center for Atmospheric Research. [http://www.ucar.edu/communications/quarterly/spring06/trex.jsp T-REX: Catching the Sierra’s waves and rotors] Retrieved on 2006-10-21.</ref>
*[[Inversion (meteorology)|Inversions]]. When on a clear and calm night, a radiation inversion is formed near the ground, the [[friction]] does not affect wind above the top of the inversion layer. The change in wind can be 90&nbsp;degrees in direction and 40&nbsp;kt in speed. Even a nocturnal (overnight) low level jet can sometimes be observed. It tends to be strongest towards sunrise. Density differences cause additional problems to aviation.<ref name="IP"/>
*[[Downburst]]s. When an outflow boundary forms due to a shallow layer of rain-cooled air spreading out near ground level from the parent thunderstorm, both speed and directional wind shear can result at the leading edge of the three dimensional boundary. The stronger the [[outflow boundary]] is, the stronger the resultant vertical wind shear will become.<ref>[[Ted Fujita|Fujita, T.T.]] (1985). "The Downburst, microburst and macroburst". SMRP Research Paper 210, 122 pp.</ref>


are the [[standard score]], sample [[mean]], and sample [[standard deviation]], respectively.
==Horizontal component==


==Mathematical properties==
===Weather fronts===
The absolute value of both the sample and population Pearson correlation coefficients are less than or equal to 1. Correlations equal to 1 or -1 correspond to data points lying exactly on a line (in the case of the sample correlation), or to a bivariate distribution entirely supported on a line (in the case of the population correlation). The Pearson correlation coefficient is symmetric: ''corr''(''X'',''Y'')&nbsp;=&nbsp;''corr''(''Y'',''X'').
{{main|Weather fronts}}
Weather fronts are boundaries between two masses of air of different [[density|densities]], or different temperature and moisture properties, which normally are [[convergence zone]]s in the wind field and are the principal cause of significant weather. Within surface weather analyses, they are depicted using various colored lines and symbols. The air masses usually differ in [[temperature]] and may also differ in [[humidity]]. Wind shear in the horizontal occurs near these boundaries.
[[Cold front]]s feature narrow bands of [[thunderstorm]]s and [[severe weather]], and may be preceded by [[squall line]]s and [[dry line]]s. Cold fronts are sharper surface boundaries with more significant horizontal wind shear than warm fronts. When a front becomes [[stationary front|stationary]], it can degenerate into a line which separates regions of differing wind speed, known as a [[shear line (meteorology)|shear line]], though the wind direction across the front normally remains constant. In the [[tropics]], [[tropical wave]]s move from east to west across the [[Atlantic basin|Atlantic]] and eastern [[Pacific basin]]s. Directional and speed shear can occur across the axis of stronger tropical waves, as northerly winds precede the wave axis and southeast winds are seen behind the wave axis. Horizontal wind shear can also occur along local land breeze and [[sea breeze]] boundaries.<ref name="DR">David M. Roth. Hydrometeorological Prediction Center. [http://www.wpc.ncep.noaa.gov/sfc/UASfcManualVersion1.pdf Unified Surface Analysis Manual.] Retrieved on 2006-10-22.</ref>


A key mathematical property of the Pearson correlation coefficient is that it is [[invariant estimator|invariant]] (up to a sign) to separate changes in location and scale in the two variables. That is, we may transform ''X'' to ''a''&nbsp;+&nbsp;''bX'' and transform ''Y'' to ''c''&nbsp;+&nbsp;''dY'', where ''a'', ''b'', ''c'', and ''d'' are constants, without changing the correlation coefficient (this fact holds for both the population and sample Pearson correlation coefficients). Note that more general linear transformations do change the correlation: see [[#Removing correlation|a later section]] for an application of this.
===Near coastlines===
The magnitude of winds offshore are nearly double the wind speed observed onshore. This is attributed to the differences in friction between land masses and offshore waters. Sometimes, there are even directional differences, particularly if local sea breezes change the wind on shore during daylight hours.<ref>Franklin B. Schwing and Jackson O. Blanton. [http://ams.allenpress.com/perlserv/?request=get-document&doi=10.1175%2F1520-0485(1984)014%3C0193%3ATUOLAS%3E2.0.CO%3B2&ct=1 The Use of Land and Sea Based Wind Data in a Simple Circulation Model.] Retrieved on 2007-10-03.</ref>


The Pearson correlation can be expressed in terms of uncentered moments.  Since ''μ''<sub>''X''</sub> = E(''X''), ''σ''<sub>''X''</sub><sup>2</sup> = E[(''X''&nbsp;−&nbsp;E(''X''))<sup>2</sup>] =&nbsp;E(''X''<sup>2</sup>)&nbsp;−&nbsp;E<sup>2</sup>(''X'') and
==Vertical component==
likewise for ''Y'', and since


: <math>E[(X-E(X))(Y-E(Y))]=E(XY)-E(X)E(Y),\,</math>
===Thermal wind===
{{main|Thermal wind}}
Thermal wind is a meteorological term not referring to an actual [[wind]], but a ''difference'' in the [[geostrophic wind]] between two [[pressure level]]s <math>p_1</math> and <math>p_0</math>, with <math>p_1< p_0</math>; in essence, wind shear. It is only present in an atmosphere with horizontal changes in [[temperature]] (or in an ocean with horizontal gradients of [[density]]), i.e. [[baroclinicity]]. In a [[barotropic]] atmosphere, where temperature is uniform, the geostrophic wind is independent of height. The name stems from the fact that this wind flows around areas of low (and high) temperature in the same manner as the [[geostrophic wind]] flows around areas of [[Low pressure area|low]] (and [[High pressure area|high]]) [[pressure]].<ref name="Holton">James R. Holton (2004). ''An Introduction to Dynamic Meteorology.'' ISBN 0-12-354015-1</ref>


the correlation can also be written as
The ''thermal wind equation'' is


:<math>\rho_{X,Y}=\frac{E(XY)-E(X)E(Y)}{\sqrt{E(X^2)-(E(X))^2}~\sqrt{E(Y^2)- (E(Y))^2}}.</math>
:<math>f \mathbf{v}_T = \mathbf{k} \times \nabla ( \phi_1 - \phi_0 )</math>,


Alternative formulae for the ''sample'' Pearson correlation coefficient are also available:
where the <math>\phi_x</math> are [[geopotential height]] fields with <math>\phi_1 > \phi_0</math>, <math>f</math> is the [[Coriolis parameter]], and <math>\mathbf{k}</math> is the upward-pointing [[unit vector]] in the [[vertical direction]]. The thermal wind equation does not determine the wind in the [[tropics]]. Since <math>f</math> is small or zero, such as near the equator, the equation reduces to stating that <math>\nabla ( \phi_1 - \phi_0 )</math> is small.<ref name="Holton"/>


:<math>
This equation basically describes the existence of the jet stream, a westerly current of air with maximum wind speeds close to the [[tropopause]] which is (even though other factors are also important) the result of the temperature contrast between equator and pole.
r_{xy}=\frac{\sum x_iy_i-n \bar{x} \bar{y}}{(n-1) s_x s_y}=\frac{n\sum x_iy_i-\sum x_i\sum y_i}
{\sqrt{n\sum x_i^2-(\sum x_i)^2}~\sqrt{n\sum y_i^2-(\sum y_i)^2}}.
</math>


The above formula suggests a convenient single-pass algorithm for calculating sample correlations, but, depending on the numbers involved, it can sometimes be [[numerical stability|numerically unstable]].
===Effects on tropical cyclones===
[[Image:Thunderhead.anvil.jpg|thumb|right|Strong wind shear in the high [[troposphere]] forms the anvil-shaped top of this mature [[cumulonimbus]] cloud, or thunderstorm.<ref>{{cite book | last = Mcilveen | first = J. | title = Fundamentals of Weather and Climate | publisher = Chapman & Hall | location = London | year = 1992 | isbn = 0-412-41160-1 | pages = 339}}</ref>]]
{{main|Tropical cyclogenesis}}
[[Tropical cyclone]]s are basically [[heat engine]]s that are fueled by the [[temperature gradient]] between the warm tropical ocean surface and the colder upper atmosphere. Tropical cyclone development requires relatively low values of vertical wind shear so that their warm core can remain above their surface circulation center, thereby promoting strengthening. Vertical wind shear tears up the "machinery" of the heat engine causing it to break down. Strongly sheared tropical cyclones weaken as the upper circulation is blown away from the low level center.<ref>University of Illinois. [http://ww2010.atmos.uiuc.edu/(Gh)/guides/mtr/hurr/grow/home.rxml Hurricanes.] Retrieved 2006-10-21.</ref>


==Interpretation==
===Effects on thunderstorms and severe weather===
The correlation coefficient ranges from −1 to 1. A value of 1 implies that a linear equation describes the relationship between ''X'' and ''Y'' perfectly, with all data points lying on a [[line (mathematics)|line]] for which ''Y'' increases as ''X'' increases. A value of −1 implies that all data points lie on a line for which ''Y'' decreases as ''X'' increases. A value of 0 implies that there is no linear correlation between the variables.
{{main|Severe thunderstorm}}
Severe thunderstorms, which can spawn [[tornado]]es and hailstorms, require wind shear to organize the storm in such a way as to maintain the [[thunderstorm]] for a longer period of time. This occurs as the storm's inflow becomes separated from its rain-cooled outflow. An increasing nocturnal, or overnight, low level jet can increase the severe weather potential by increasing the vertical wind shear through the troposphere. Thunderstorms in an atmosphere with virtually no vertical wind shear weaken as soon as they send out an outflow boundary in all directions, which then quickly cuts off its inflow of relatively warm, moist air and kills the thunderstorm.<ref>University of Illinois. [http://ww2010.atmos.uiuc.edu/(Gh)/guides/mtr/svr/comp/wind/home.rxml Vertical Wind Shear] Retrieved on 2006-10-21.</ref>


More generally, note that (''X''<sub>''i''</sub>&nbsp;&minus;&nbsp;<font style="text-decoration: overline;">''X''</font>)(''Y''<sub>''i''</sub>&nbsp;&minus;&nbsp;<font style="text-decoration: overline;">''Y''</font>) is positive if and only if ''X''<sub>''i''</sub> and ''Y''<sub>''i''</sub> lie on the same side of their respective means.  Thus the correlation coefficient is positive if ''X''<sub>''i''</sub> and ''Y''<sub>''i''</sub> tend to be simultaneously greater than, or simultaneously less than, their respective means.  The correlation coefficient is negative if ''X''<sub>''i''</sub> and ''Y''<sub>''i''</sub> tend to lie on opposite sides of their respective means.
{{clear}}


===Geometric interpretation===
===Planetary boundary layer===
[[File:Regression lines.png|thumb|upright=1.5|Regression lines for y=g<sub>x</sub>(x) [red] and x=g<sub>y</sub>(y) [blue]]]
[[File:PBLimage.jpg|thumb|right|Depiction of where the planetary boundary layer lies on a sunny day]]
{{see also|Ekman layer|Ekman spiral|Planetary boundary layer|Surface layer}}
The atmospheric effect of surface friction with winds aloft force surface winds to slow and back counterclockwise near the surface of the [[Earth]] blowing inward across isobars (lines of equal pressure), when compared to the winds in frictionless flow well above the Earth's surface. This layer where friction slows and changes the wind is known as the planetary boundary layer, sometimes the Ekman layer, and it is thickest during the day and thinnest at night. Daytime heating thickens the boundary layer as winds at the surface become increasingly mixed with winds aloft due to [[insolation]], or solar heating. Radiative cooling overnight further enhances wind decoupling between the winds at the surface and the winds above the boundary layer by calming the surface wind which increases wind shear. These wind changes force wind shear between the boundary layer and the wind aloft, and is most emphasized at night.<ref>Glossary of Meteorology. [http://amsglossary.allenpress.com/glossary/browse?s=e&p=14 E.] Retrieved on 2007-06-03.</ref>


For uncentered data, the correlation coefficient corresponds with the cosine of the angle <math>\varphi</math> between both possible [[regression line]]s y=g<sub>x</sub>(x) and x=g<sub>y</sub>(y).
====Effects on flight====
{{main|Aeronautics|Gliding}}


For centered data (i.e., data which have been shifted by the sample mean so as to have an average of zero), the correlation coefficient can also be viewed as the [[cosine]] of the [[angle]] <math>\ \theta</math> between the two [[Vector (geometry)|vectors]] of samples drawn from the two random variables (see below).
=====Gliding=====
[[Image:FAA-8083-13 Fig 7-20.PNG|thumb|right|Glider ground launch affected by wind shear.]]
In gliding, wind gradients just above the surface affect the takeoff and landing phases of flight of a [[Glider (sailplane)|glider]].
Wind gradient can have a noticeable effect on [[ground launch]]es, also known as winch launches or wire launches. If the wind gradient is significant or sudden,
or both, and the pilot maintains the same pitch attitude, the indicated airspeed will increase, possibly exceeding
the maximum ground launch tow speed. The pilot must adjust the airspeed to deal with the effect of the
gradient.<ref>{{cite book
| title = Glider Flying Handbook
| year = 2003
| publisher = U.S. Federal Aviation Administration
| location = U.S. Government Printing Office, Washington D.C.
| id = FAA-8083-13_GFH
| pages = 7–16
| url=http://www.faa.gov/library/manuals/aircraft/glider_handbook/
}}</ref>


Both the uncentered (non-Pearson-compliant) and centered correlation coefficients can be determined for a dataset. As an example, suppose five countries are found to have gross national products of 1, 2, 3, 5, and 8 billion dollars, respectively. Suppose these same five countries (in the same order) are found to have 11%, 12%, 13%, 15%, and 18% poverty. Then let '''x''' and '''y''' be ordered 5-element vectors containing the above data: '''x''' = (1, 2, 3, 5, 8) and '''y''' = (0.11, 0.12, 0.13, 0.15, 0.18).
When landing, wind shear is also a hazard, particularly when the winds are strong. As the glider descends through the wind gradient on final approach to landing, airspeed decreases while sink rate increases, and there is insufficient time to accelerate prior to ground contact. The pilot must anticipate the wind gradient and use a higher approach speed to compensate for it.<ref name=Piggott>{{cite book | last = Piggott | first = Derek | title = Gliding: a Handbook on Soaring Flight | publisher = Knauff & Grove | year = 1997 | isbn = 978-0-9605676-4-5 | pages = 85–86, 130–132}}</ref>


By the usual procedure for finding the angle <math>\ \theta</math> between two vectors (see [[dot product]]), the ''uncentered'' correlation coefficient is:
Wind shear is also a hazard for aircraft making steep turns near the ground. It is a particular problem for gliders which have a relatively long [[wingspan]], which exposes them to a greater wind speed difference for a given [[Roll (flight)|bank]] angle. The different airspeed experienced by each wing tip can result in an aerodynamic stall on one wing, causing a loss of control accident.<ref name=Piggott/><ref>{{cite book | last = Knauff | first = Thomas | title = Glider Basics from First Flight to Solo | publisher = Thomas Knauff | year = 1984 | isbn = 0-9605676-3-1 }}</ref>


<!-- cos theta = (X dot Y) / ||X|| ||Y|| = 2.93 / sqrt(103 * 0.0983) = 0.920814711. -->
=====Parachuting=====
:<math> \cos \theta = \frac { \bold{x} \cdot \bold{y} } { \left\| \bold{x} \right\| \left\| \bold{y} \right\| } = \frac { 2.93 } { \sqrt { 103 } \sqrt { 0.0983 } } = 0.920814711. </math>
Wind shear or wind gradients are a threat to parachutists, particularly to [[BASE jumping]] and [[wingsuit flying]]. Skydivers have been pushed off of their course by sudden shifts in wind direction and speed, and have collided with bridges, cliffsides, trees, other skydivers, the ground, and other obstacles.{{Citation needed|date=February 2012}} Skydivers routinely make adjustments to the position of their open canopies to compensate for changes in direction while making landings to prevent accidents such as canopy collisions and canopy inversion.


Note that the above data were deliberately chosen to be perfectly correlated: ''y'' = 0.10 + 0.01 ''x''. The Pearson correlation coefficient must therefore be exactly one. Centering the data (shifting '''x''' by E('''x''') = 3.8 and '''y''' by E('''y''') = 0.138) yields '''x''' = (−2.8, −1.8, −0.8, 1.2, 4.2) and '''y''' = (−0.028, −0.018, −0.008, 0.012, 0.042), from which
=====Soaring=====
Soaring related to wind shear, also called [[dynamic soaring]], is a technique used by [[soaring birds]] like [[albatross]]es, who can maintain flight without wing flapping. If the wind shear is of sufficient magnitude, a bird can climb into the wind gradient, trading ground speed for height, while maintaining airspeed.<ref>{{cite book | last = Alexander | first = R. | title = Principles of Animal Locomotion | publisher = Princeton University Press | location = Princeton | year = 2002 | pages = 206 | isbn = 0-691-08678-8 }}</ref> By then turning downwind, and diving through the wind gradient, they can also gain energy.<ref>{{cite book | last = Alerstam | first = Thomas | title = Bird Migration | publisher = Cambridge University Press | location = Cambridge | year = 1990 | pages = 275 | isbn = 0-521-44822-0 }}</ref> It has also been used by [[gliding|glider pilots]] on rare occasions.


<!-- cos theta = (X dot Y) / ||X|| ||Y|| = 0.308 / sqrt(30.8 * 0.00308) = 1. -->
Wind shear can also create [[lee wave|wave]]. This occurs when an [[atmospheric inversion]] separates two layers with a marked difference in wind direction. If the wind encounters distortions in the inversion layer caused by [[thermal]]s coming up from below, it will create significant shear waves that can be used for soaring.<ref>{{cite book
:<math> \cos \theta = \frac { \bold{x} \cdot \bold{y} } { \left\| \bold{x} \right\| \left\| \bold{y} \right\| } = \frac { 0.308 } { \sqrt { 30.8 } \sqrt { 0.00308 } } = 1 = \rho_{xy}, </math>
|last = Eckey
 
|first = Bernard
as expected.
|title = Advanced Soaring Made Easy
 
|publisher = Eqip Verbung & Verlag GmbH
===Interpretation of the size of a correlation===
|year = 2007
{|class="wikitable" align="right"
|ISBN = 3-9808838-2-5
|-
! Correlation !! Negative !! Positive
|-
| None || −0.09 to 0.0 || 0.0 to 0.09
|-
| Small || −0.3 to −0.1 || 0.1 to 0.3
|-
| Medium || −0.5 to −0.3 || 0.3 to 0.5
|-
|Strong || −1.0 to −0.5|| 0.5 to 1.0
|}
 
Several authors<ref name="Buda">A. Buda and A.Jarynowski (2010) ''Life-time of correlations and its applications vol.1'', Wydawnictwo Niezalezne: 5–21, December 2010, ISBN 978-83-915272-9-0</ref><ref name="Cohen88"/> have offered guidelines for the interpretation of a correlation coefficient. However, all such criteria are in some ways arbitrary and should not be observed too strictly.<ref name="Cohen88">Cohen, J. (1988). ''Statistical power analysis for the behavioral sciences'' (2nd ed.)</ref>  The interpretation of a correlation coefficient depends on the context and purposes.  A correlation of 0.9 may be very low if one is verifying a physical law using high-quality instruments, but may be regarded as very high in the social sciences where there may be a greater contribution from complicating factors.
 
===Pearson’s distance===
A distance metric for two variables X and Y known as ''Pearson's distance'' can be defined from their correlation coefficient as<ref>Fulekar (Ed.), M.H. (2009) ''Bioinformatics: Applications in Life and Environmental Sciences'', Springer (pp. 110) ISBN 1-4020-8879-5</ref>
:<math>d_{X,Y}=1-\rho_{X,Y}.</math>
Considering that the Pearson correlation coefficient falls between [-1, 1], the Pearson distance lies in [0, 2].
 
==Inference==
[[Image:correlation significance.svg|300px|right|thumb|A graph showing the minimum value of Pearson's correlation coefficient that is significantly different from zero at the 0.05 level, for a given sample size.]]Statistical inference based on Pearson's correlation coefficient often focuses on one of the following two aims: 
* One aim is to test the [[null hypothesis]] that the true correlation coefficient ''ρ'' is equal to 0, based on the value of the sample correlation coefficient ''r''.
* The other aim is to construct a [[confidence interval]] around ''r'' that has a given probability of containing ''ρ''.
We discuss methods of achieving one or both of these aims below.
 
===Use a permutation test===
[[Resampling (statistics)#Permutation tests|Permutation tests]] provide a direct approach to performing hypothesis tests and constructing confidence intervals.  A permutation test for Pearson's correlation coefficient involves the following two steps:
* (i) using the original paired data (''x''<sub>''i''</sub>,&nbsp;''y''<sub>''i''</sub>), randomly redefine the pairs to create a new data set (''x''<sub>''i''</sub>,&nbsp;''y''<sub>''i&prime;''</sub>), where the ''i&prime;'' are a permutation of the set {1,...,''n''}.  The permutation ''i&prime;'' is selected randomly, with equal probabilities placed on all ''n''! possible permutations.  This is equivalent to drawing the ''i&prime;'' randomly "without replacement" from the set {1,..., ''n''}.  A closely related and equally justified ([[Bootstrapping (statistics)|bootstrapping]]) approach is to separately draw the ''i'' and the ''i&prime;'' "with replacement" from {1,..., ''n''};
* (ii) Construct a correlation coefficient ''r'' from the randomized data.
To perform the permutation test, repeat (i) and (ii) a large number of times.  The [[p-value]] for the permutation test is the proportion of the ''r'' values generated in step (ii) that are larger than the Pearson correlation coefficient that was calculated from the original data. Here "larger" can mean either that the value is larger in magnitude, or larger in signed value, depending on whether a [[two-tailed test|two-sided]] or [[two-tailed test|one-sided]] test is desired.
 
===Use a bootstrap===
The [[bootstrapping (statistics)|bootstrap]] can be used to construct confidence intervals for Pearson's correlation coefficient.  In the "non-parametric" bootstrap, ''n'' pairs (''x''<sub>''i''</sub>,&nbsp;''y''<sub>''i''</sub>) are resampled "with replacement" from the observed set of ''n'' pairs, and the correlation coefficient ''r'' is calculated based on the resampled data.  This process is repeated a large number of times, and the empirical distribution of the resampled ''r'' values are used to approximate the [[sampling distribution]] of the statistic.  A 95% [[confidence interval]] for ''ρ'' can be defined as the interval spanning from the 2.5<sup>''th''</sup> to the 97.5<sup>''th''</sup> [[percentile]] of the resampled ''r'' values.
 
===Testing using Student's t-distribution===
For pairs from an uncorrelated [[bivariate normal distribution]], the [[sampling distribution]] of Pearson's correlation coefficient follows [[Student's t-distribution]] with degrees of freedom ''n''&nbsp;&minus;&nbsp;2.  Specifically, if the underlying variables have a bivariate normal distribution, the variable
 
:<math>t = r\sqrt{\frac{n-2}{1 - r^2}}</math>
 
has a Student's t-distribution in the null case (zero correlation).<ref>Rahman, N.A. (1968) ''A Course in Theoretical Statistics'', Charles Griffin and Company, 1968</ref> This also holds approximately even if the observed values are non-normal, provided sample sizes are not very small.<ref>Kendall, M.G., Stuart, A. (1973) ''The Advanced Theory of Statistics, Volume 2: Inference and Relationship'', Griffin. ISBN 0-85264-215-6 (Section 31.19)</ref><ref>For deviation on small sample sizes, see for example http://www.neustats.com/neu-da-documentation/whats-that-pearson-p-value-in-the-tables/</ref>  For determining the critical values for ''r'' the inverse of this transformation is also needed:
 
:<math>r = \frac{t}{\sqrt{n - 2 + t^2}}.</math>
 
Alternatively, large sample approaches can be used.
 
Early work on the distribution of the sample correlation coefficient was carried out by [[R. A. Fisher]]<ref>{{Cite journal
| last = Fisher | first = R.A.
| authorlink = R. A. Fisher
| title = Frequency distribution of the values of the correlation coefficient in samples from an indefinitely large population
| journal = [[Biometrika]]
| volume = 10
| issue = 4
| pages = 507–521
| year = 1915
| doi = 10.1093/biomet/10.4.507
}}</ref><ref>{{Cite journal
| last = Fisher | first = R.A.
| authorlink = R. A. Fisher
| title = On the probable error of a coefficient of correlation deduced from a small sample
| journal = [[Metron (journal)|Metron]]
| year = 1921
| volume = 1
| issue = 4
| pages = 3–32
| url = http://hdl.handle.net/2440/15169
| accessdate = 2009-03-25
| format = [[PDF]]
}}</ref>
}}</ref>
and A. K. Gayen.<ref>{{Cite journal
| first = A.K. | last = Gayen
| title = The frequency distribution of the product moment correlation coefficient in random samples of any size draw from non-normal universes
| journal = [[Biometrika]]
| year = 1951
| volume = 38
| pages = 219–247
| doi = 10.1093/biomet/38.1-2.219
}}</ref>
Another early paper<ref>Soper, H.E., Young, A.W., Cave, B.M., Lee, A., Pearson, K. (1917). "On the distribution of the correlation coefficient in small samples. Appendix II to the papers of "Student" and R. A. Fisher. A co-operative study", ''[[Biometrika]]'', 11, 328-413. {{doi|10.1093/biomet/11.4.328}}</ref> provides graphs and tables for general values of ''ρ'', for small sample sizes, and discusses computational approaches.
===Use the exact distribution===
For data that follows a [[bivariate normal distribution]], the exact density function for the sample correlation of a normal bivariate is<ref>Kenney, J. F. and Keeping, E. S., ''Mathematics of Statistics'', Pt. 2, 2nd ed. Princeton, NJ: Van Nostrand, 1951.</ref><ref>[http://mathworld.wolfram.com/CorrelationCoefficientBivariateNormalDistribution.html Correlation Coefficient - Bivariate Normal Distribution]</ref>
:<math>f\left(r\right) = \frac{\left(n - 2\right)\, \mathbf{\Gamma}\left(n - 1\right) \left(1 - \rho^2\right)^{\frac{n - 1}{2}} \left(1 - r^2\right)^{\frac{n - 4}{2}}}{\sqrt{2\pi}\, \mathbf{\Gamma}\left(n - \frac{1}{2}\right) \left(1 - \rho r\right)^{n - \frac{3}{2}}} \,\mathbf{_2F_1}\left(\frac{1}{2}, \frac{1}{2}; \frac{2n - 1}{2}; \frac{\rho r + 1}{2}\right)</math>
where <math>\mathbf{\Gamma}</math> is the [[gamma function]], <math>\,\mathbf{_2F_1}(a,b;c;z)</math> is the [[hypergeometric function|Gaussian hypergeometric function]]. In the special case when <math>\,\rho = 0</math>, the density can be written as:
:<math>f\left(r\right) = \frac{\left(1 - r^2\right)^{\frac{n - 4}{2}}}{\mathbf{B}\left(\frac{1}{2}, \frac{n - 2}{2}\right)},</math>
where <math>\mathbf{B}</math> is the [[beta function]], which is one way of writing the density of a Student's t-distribution, as above.
Note that{{citation needed|date=April 2012}} <math>E\left(r\right) = \rho - \frac{\rho \left(1 - \rho^2\right)}{2 \left(n - 1\right)} + \cdots </math>, therefore ''r'' is a biased estimator of <math>\,\rho</math>. An approximately unbiased estimator can be obtained by solving the equation <math>r = E\left(r\right) = \rho - \frac{\rho \left(1 - \rho^2\right)}{2 \left(n - 1\right)}</math> for <math>\,\rho</math>. However, the solution, <math>\breve{\rho} = r \left[1 + \frac{1 - r^2}{2\left(n - 1\right)}\right]</math>,{{citation needed|date=April 2012}} is suboptimal.{{citation needed|date=April 2012}} An approximately unbiased estimator,{{citation needed|date=April 2012}} with minimum variance for large values of ''n'', with a bias of order <math>\frac{1}{n - 1}</math>, can be obtained by maximizing <math>\log{f\left(r\right)}</math>, i.e. <math>\hat{\rho} = r \left[1 - \frac{1 - r^2}{2\left(n - 1\right)}\right]</math>.{{citation needed|date=April 2012}}
===Use the Fisher transformation===
In practice, [[confidence intervals]] and [[hypothesis test]]s relating to ρ are usually carried out using the [[Fisher transformation]]:
: <math>F(r) = {1 \over 2}\ln{1 + r \over 1 - r} = \operatorname{artanh}(r).</math>
If ''F''(''r'') is the Fisher transformation of ''r'', and ''n'' is the sample size, then ''F''(''r'') approximately follows a [[normal distribution]] with
:<math>\text{mean} = F(\rho) = \operatorname{artanh}(\rho)</math>&nbsp;&nbsp;&nbsp;&nbsp;and standard error&nbsp;&nbsp;&nbsp;&nbsp;<math>\text{SE} = \frac{1}{\sqrt{n - 3}}.</math>
Thus, a [[standard score|z-score]] is
:<math>z = \frac{x - \text{mean}}{\text{SE}} = [F(r) - F(\rho_0)]\sqrt{n - 3}</math>
under the [[null hypothesis]] of that <math>\rho = \rho_0</math>, given the assumption that the sample pairs are [[independent and identically distributed]] and follow a [[bivariate normal distribution]].  Thus an approximate [[p-value]] can be obtained from a normal probability table.  For example, if ''z''&nbsp;=&nbsp;2.2 is observed and a two-sided p-value is desired to test the null hypothesis that <math>\rho = 0</math>, the p-value is 2·Φ(−2.2) = 0.028, where Φ is the standard normal [[cumulative distribution function]].
To obtain a confidence interval for ρ, we first compute a confidence interval for ''F''(''<math>\rho</math>''):
<br />
<br />
:<math>100(1 - \alpha)%\text{CI}: \operatorname{artanh}(\rho) \in [\operatorname{artanh}(r) \pm z_{\alpha/2}SE]</math>
<br />
The inverse Fisher transformation bring the interval back to the correlation scale.
<br />
<br />
:<math>100(1 - \alpha)%\text{CI}: \rho \in [\operatorname{tanh}(\operatorname{artanh}(r) - z_{\alpha/2}SE), \operatorname{tanh}(\operatorname{artanh}(r) + z_{\alpha/2}SE)]</math>
<br />
For example, suppose we observe ''r''&nbsp;=&nbsp;0.3 with a sample size of ''n''=50, and we wish to obtain a 95% confidence interval for ρ.  The transformed value is artanh(''r'')&nbsp;=&nbsp;0.30952, so the confidence interval on the transformed scale is 0.30952 ± 1.96/√47, or (0.023624,&nbsp;0.595415).  Converting back to the correlation scale yields (0.024,&nbsp;0.534).
==Pearson's correlation and least squares regression analysis==
The square of the sample correlation coefficient, which is also known as the [[coefficient of determination]], estimates the fraction of the variance in ''Y'' that is explained by ''X'' in a [[simple linear regression]]. As a starting point, the total variation in the ''Y''<sub>''i''</sub> around their average value can be decomposed as follows
:<math>
\sum_i (Y_i - \bar{Y})^2 = \sum_i (Y_i-\hat{Y}_i)^2 + \sum_i (\hat{Y}_i-\bar{Y})^2,
</math>
where the <math>\hat{Y}_i</math> are the fitted values from the regression analysis.  This can be rearranged to give
:<math>
1 = \frac{\sum_i (Y_i-\hat{Y}_i)^2}{\sum_i (Y_i - \bar{Y})^2} + \frac{\sum_i (\hat{Y}_i-\bar{Y})^2}{\sum_i (Y_i - \bar{Y})^2}.
</math>
The two summands above are the fraction of variance in ''Y'' that is explained by ''X'' (right) and that is unexplained by ''X'' (left).
Next, we apply a property of least square regression models, that the sample covariance between <math>\hat{Y}_i</math> and <math>Y_i-\hat{Y}_i</math> is zero.  Thus, the sample correlation coefficient between the observed and fitted response values in the regression can be written
<math>
\begin{align}
r(Y,\hat{Y}) &= \frac{\sum_i(Y_i-\bar{Y})(\hat{Y}_i-\bar{Y})}{\sqrt{\sum_i(Y_i-\bar{Y})^2\cdot \sum_i(\hat{Y}_i-\bar{Y})^2}}\\
&= \frac{\sum_i(Y_i-\hat{Y}_i+\hat{Y}_i-\bar{Y})(\hat{Y}_i-\bar{Y})}{\sqrt{\sum_i(Y_i-\bar{Y})^2\cdot \sum_i(\hat{Y}_i-\bar{Y})^2}}\\
&= \frac{ \sum_i [(Y_i-\hat{Y}_i)(\hat{Y}_i-\bar{Y}) +(\hat{Y}_i-\bar{Y})^2 ]}{\sqrt{\sum_i(Y_i-\bar{Y})^2\cdot \sum_i(\hat{Y}_i-\bar{Y})^2}}\\
&= \frac{ \sum_i (\hat{Y}_i-\bar{Y})^2 }{\sqrt{\sum_i(Y_i-\bar{Y})^2\cdot \sum_i(\hat{Y}_i-\bar{Y})^2}}\\
&= \sqrt{\frac{\sum_i(\hat{Y}_i-\bar{Y})^2}{\sum_i(Y_i-\bar{Y})^2}}.
\end{align}
</math>
Thus
:<math>
r(Y,\hat{Y})^2 = \frac{\sum_i(\hat{Y}_i-\bar{Y})^2}{\sum_i(Y_i-\bar{Y})^2}
</math>
is the proportion of variance in ''Y'' explained by a linear function of ''X''.
==Sensitivity to the data distribution==
===Existence===
The population Pearson correlation coefficient is defined in terms of [[moment (mathematics)|moments]], and therefore exists for any bivariate [[probability distribution]] for which the [[statistical population|population]] [[covariance]] is defined and the [[marginal distribution|marginal]] [[population variance]]s are defined and are non-zero.  Some probability distributions such as the [[Cauchy distribution]] have undefined variance and hence ρ is not defined if ''X'' or ''Y'' follows such a distribution.  In some practical applications, such as those involving data suspected to follow a [[heavy-tailed distribution]], this is an important consideration.  However, the existence of the correlation coefficient is usually not a concern; for instance, if the range of the distribution is bounded, ρ is always defined.
===Large sample properties===
In the case of the bivariate [[normal distribution]] the population Pearson correlation coefficient characterizes the joint distribution as long as the marginal means and variances are known.  For most other bivariate distributions this is not true.  Nevertheless, the correlation coefficient is highly informative about the degree of linear dependence between two random quantities regardless of whether their joint distribution is normal.<ref name="thirteenways"/>
The sample correlation coefficient is the [[maximum likelihood estimate]] of the population correlation coefficient for bivariate normal data, and is [[asymptotic distribution|asymptotically]] [[bias of an estimator|unbiased]] and [[efficiency (statistics)|efficient]], which roughly means that it is impossible to construct a more accurate estimate than the sample correlation coefficient if the data are normal and the sample size is moderate or large. For non-normal populations, the sample correlation coefficient remains approximately unbiased, but may not be efficient.  The sample correlation coefficient is  a [[consistent estimator]] of the population correlation coefficient as long as the sample means, variances, and covariance are consistent (which is guaranteed when the [[law of large numbers]] can be applied).
===Robustness===
Like many commonly used statistics, the sample statistic ''r'' is not [[robust statistics|robust]],<ref name="wilcox">{{Cite book| title=Introduction to robust estimation and hypothesis testing | last = Wilcox | first = Rand R. | publisher= Academic Press | year=2005}}</ref> so its value can be misleading if [[outlier]]s are present.<ref>{{Cite journal| title= Robust Estimation and Outlier Detection with Correlation Coefficients | last= Devlin | first = Susan J | coauthors = Gnanadesikan, R; Kettenring J.R. | journal= Biometrika | volume= 62 |  issue= 3 |year=1975 | pages=531–545 | doi= 10.1093/biomet/62.3.531 | jstor=2335508}}</ref><ref>{{Cite book| title=Robust Statistics | last = Huber | first = Peter. J.| publisher= Wiley | year=2004}}{{Page needed|date=September 2010}}</ref> Specifically, the PMCC is neither distributionally robust,{{Citation needed|date=November 2009}} nor outlier resistant<ref name="wilcox"/> (see [[Robust statistics#Definition]]). Inspection of the [[scatterplot]] between ''X'' and ''Y'' will typically reveal a situation where lack of robustness might be an issue, and in such cases it may be advisable to use a robust measure of association.  Note however that while most robust estimators of association measure [[statistical dependence]] in some way, they are generally not interpretable on the same scale as the Pearson correlation coefficient.
Statistical inference for Pearson's correlation coefficient is sensitive to the data distribution.  Exact tests, and asymptotic tests based on the [[Fisher transformation]] can be applied if the data are approximately normally distributed, but may be misleading otherwise.  In some situations, the [[bootstrapping (statistics)|bootstrap]] can be applied to construct confidence intervals, and [[resampling (statistics)|permutation tests]] can be applied to carry out hypothesis tests.  These [[non-parametric statistics|non-parametric]] approaches may give more meaningful results in some situations where bivariate normality does not hold.  However the standard versions of these approaches rely on [[exchangeable random variables|exchangeability]] of the data, meaning that there is no ordering or grouping of the data pairs being analyzed that might affect the behavior of the correlation estimate.
A stratified analysis is one way to either accommodate a lack of bivariate normality, or to isolate the correlation resulting from one factor while controlling for another.  If ''W'' represents cluster membership or another factor that it is desirable to control, we can stratify the data based on the value of ''W'', then calculate a correlation coefficient within each stratum.  The stratum-level estimates can then be combined to estimate the overall correlation while controlling for ''W''.<ref>Katz., Mitchell H. (2006) ''Multivariable Analysis - A Practical Guide for Clinicians''. 2nd Edition.  Cambridge University Press. ISBN 978-0-521-54985-1. ISBN 0-521-54985-X {{DOI|10.2277/052154985X}}</ref>
==Calculating a weighted correlation==
Suppose observations to be correlated have differing degrees of importance that can be expressed with a weight vector ''w''. To calculate the correlation between vectors ''x'' and ''y'' with the weight vector ''w'' (all of length&nbsp;''n''),<ref>http://sci.tech-archive.net/Archive/sci.stat.math/2006-02/msg00171.html</ref><ref>[http://www.mathworks.com/matlabcentral/fileexchange/20846 A MATLAB Toolbox for computing Weighted Correlation Coefficients]</ref>
* Weighted mean:
:: <math>\operatorname{m}(x; w) = {\sum_i w_i x_i \over \sum_i w_i}.</math>
* Weighted covariance
:: <math>\operatorname{cov}(x,y;w) = {\sum_i w_i (x_i - \operatorname{m}(x; w)) (y_i - \operatorname{m}(y; w)) \over \sum_i w_i }.</math>
* Weighted correlation
:: <math>\operatorname{corr}(x,y;w) = {\operatorname{cov}(x,y;w) \over \sqrt{\operatorname{cov}(x,x;w) \operatorname{cov}(y,y;w)}}.</math>
==Removing correlation==
It is always possible to remove the correlation between random variables with a linear transformation, even if the relationship between the variables is nonlinear. A presentation of this result for population distributions is given by Cox & Hinkley.<ref>Cox, D.R., Hinkley, D.V. (1974) ''Theoretical Statistics'', Chapman & Hall (Appendix 3) ISBN 0-412-12420-3</ref>
A corresponding result exists for sample correlations, in which the sample correlation is reduced to zero. Suppose a vector of ''n'' random variables is sampled ''m'' times.  Let ''X'' be a matrix where <math>X_{i,j}</math> is the ''j''th variable of sample ''i''.  Let <math>Z_{m,m}</math> be an ''m'' by ''m'' square matrix with every element 1.  Then ''D'' is the data transformed so every random variable has zero mean, and ''T'' is the data transformed so all variables have zero mean and zero correlation with all other variables - the moment matrix of ''T'' will be the identity matrix. This has to be further divided by the standard deviation to get unit variance. The transformed variables will be uncorrelated, even though they may not be [[Statistical independence|independent]].
:<math>D = X -\frac{1}{m} Z_{m,m} X</math>
<!-- extra blank line between two lines of "displayed" [[TeX]], for legibility -->
:<math>T = D (D^T D)^{-\frac{1}{2}}</math>
where an exponent of -1/2 represents the [[matrix square root]] of the [[matrix inverse|inverse]] of a matrix.  The covariance matrix of ''T'' will be the identity matrix.  If a new data sample ''x'' is a row vector of ''n'' elements, then the same transform can be applied to ''x'' to get the transformed vectors ''d'' and ''t'':
:<math>d = x - \frac{1}{m} Z_{1,m} X</math>
<!-- extra blank line between two lines of "displayed" [[TeX]], for legibility -->
:<math>t = d (D^T D)^{-\frac{1}{2}}</math>
This decorrelation is related to [[Principal Components Analysis]] for multivariate data.
==Reflective correlation==
The reflective correlation is a variant of Pearson's correlation in which the data are not centered around their mean values.{{Citation needed|date=January 2011}} The population reflective correlation is
:<math>
\text{Corr}_r(X,Y) = \frac{E[XY]}{\sqrt{EX^2\cdot EY^2}}.
</math>
The reflective correlation is symmetric, but it is not invariant under translation:
:<math>
\text{Corr}_r(X, Y) = \text{Corr}_r(Y, X) = \text{Corr}_r(X, bY) \neq \text{Corr}_r(X, a + b Y), \quad a \neq 0, b > 0.
</math>
The sample reflective correlation is
:<math>
rr_{xy} = \frac{\sum x_i y_i}{\sqrt{(\sum x_i^2)(\sum y_i^2)}}.
</math>
The weighted version of the sample reflective correlation is
:<math>
rr_{xy, w} = \frac{\sum w_i x_i y_i}{\sqrt{(\sum w_i x_i^2)(\sum w_i y_i^2)}}.
</math>
==Scaled correlation==
{{Main|Scaled correlation}}


Scaled correlation is a variant of Pearson's correlation in which the range of the data is restricted intentionally and in a controlled manner to reveal correlations between fast components in time series.<ref name = "Nikolicetal">Nikolić D, Muresan RC, Feng W, Singer W (2012) Scaled correlation analysis: a better way to compute a cross-correlogram. ''European Journal of Neuroscience'', pp. 1–21, {{doi|10.1111/j.1460-9568.2011.07987.x}} http://www.danko-nikolic.com/wp-content/uploads/2012/03/Scaled-correlation-analysis.pdf</ref> Scaled correlation is defined as average correlation across short segments of data.
=====Impact on passenger aircraft=====
[[Image:Windshearaircraftnasa.gif|thumb|right|200 px|Effect of wind shear on aircraft trajectory. Note how merely correcting for the initial gust front can have dire consequences.]]
Strong outflow from thunderstorms causes rapid changes in the three-dimensional wind velocity just above ground level. Initially, this outflow causes a headwind that increases airspeed, which normally causes a pilot to reduce engine power if they are unaware of the wind shear. As the aircraft passes into the region of the downdraft, the localized headwind diminishes, reducing the aircraft's airspeed and increasing its sink rate. Then, when the aircraft passes through the other side of the downdraft, the headwind becomes a tailwind, reducing lift generated by the wings, and leaving the aircraft in a low-power, low-speed descent. This can lead to an accident if the aircraft is too low to effect a recovery before ground contact.
[[File:Delta 191 wreckage.jpg|thumb|left|Wreckage of [[Delta Air Lines Flight 191]] tail section after a microburst slammed the aircraft into the ground. Another aircraft can be seen flying in the background past the crash scene.]]
As the result of the accidents in the 1970s and 1980s, most notably following the 1985 crash of [[Delta Air Lines Flight 191]], in 1988 the U.S. [[Federal Aviation Administration]] mandated that all commercial aircraft have [[Airborne wind shear detection and alert system|on-board wind shear detection systems]] by 1993. Between 1964 and 1985, wind shear directly caused or contributed to 26 major civil transport aircraft accidents in the U.S. that led to 620 deaths and 200 injuries.<ref>{{cite web |author=National Aeronautics and Space Administration, [[Langley Research Center]] |url=http://web.archive.org/web/20100329221032/http://oea.larc.nasa.gov/PAIS/Windshear.html|title=Making the Skies Safer From Windshear|date=June 1992|accessdate=2012-11-16}}</ref> Since 1995, the number of major civil aircraft accidents caused by wind shear has dropped to approximately one every ten years, due to the mandated on-board detection as well as the addition of Doppler [[weather radar]] units on the ground ([[NEXRAD]]).{{fact|date=February 2014}} The installation of high-resolution [[Terminal Doppler Weather Radar]] stations at many U.S. airports that are commonly affected by wind shear has further aided the ability of pilots and ground controllers to avoid wind shear conditions.<ref name=tdwr-nws>{{cite web |url=http://www.erh.noaa.gov/gsp/tdwr/info/specs.html
|title=Terminal Doppler Weather Radar Information |accessdate=4 August 2009 |publisher=National Weather Service }}</ref>


Let <math>K</math> be the number of segments that can fit into the total length of the signal <math>T</math> for a given scale <math>s</math>:
====Sailing====
Wind shear affects [[sailboats]] in motion by presenting a different wind speed and direction at different heights along the [[mast (sailing)|mast]]. The effect of low level wind shear can be factored into the selection of [[sail twist]] in the sail design, but this can be difficult to predict since wind shear may vary widely in different weather conditions. [[Sailors]] may also adjust the trim of the sail to account for low level wind shear, for example using a [[boom vang]].<ref>{{cite book | last = Garrett | first = Ross | title = The Symmetry of Sailing | publisher = Sheridan House | location = Dobbs Ferry | year = 1996 | pages = 97–99 | isbn = 1-57409-000-3 }}</ref>


:<math>K = \operatorname{round}\left(\frac{T}{s}\right).</math>
====Sound propagation====
{{see also|Speed of sound}}
Wind shear can have a pronounced effect upon sound propagation in the lower atmosphere, where waves can be "bent" by [[refraction]] phenomenon. The audibility of sounds from distant sources, such as [[thunder]] or [[gunshot]]s, is very dependent on the amount of shear. The result of these differing sound levels is key in [[noise pollution]] considerations, for example from [[roadway noise]] and [[aircraft noise]], and must be considered in the design of [[noise barrier]]s.<ref>{{cite journal
| publisher = Washington State Department of Transportation.
| url = http://www.wsdot.wa.gov/Research/Reports/000/033.1.htm
| title = Ground Plane Wind Shear Interaction on Acoustic Transmission
| accessdate = 2007-05-30
| version = WA-RD 033.1
| author = Foss, Rene N.
| date = June 1978
}}</ref> This phenomenon was first applied to the field of [[noise pollution]] study in the 1960s, contributing to the design of urban highways as well as [[noise barrier]]s.<ref>{{cite journal|url=http://www.springerlink.com/content/x1707075n815g604/|title= C. Michael Hogan, '' Analysis of highway noise'', Journal of Water, Air, & Soil Pollution, Volume 2, Number 3, Biomedical and Life Sciences and Earth and Environmental Science Issue, Pages 387-392, September, 1973, Springer Verlag, Netherlands ISSN 0049-6979}}</ref>


The scaled correlation across the entire signals <math>\bar{r}_s</math> is then computed as
[[Image:Hodographe NOAA.PNG|thumb|[[Hodograph]] plot of wind vectors at various heights in the troposphere. Meteorologists can use this plot to evaluate vertical wind shear in weather forecasting. (Source: [[NOAA]])]]


:<math>\bar{r}_s = \frac{1}{K} \sum\limits_{k=1}^K r_k,</math>
The [[speed of sound]] varies with temperature. Since temperature and sound velocity normally decrease with increasing altitude, sound is [[refraction|refracted]] upward, away from listeners on the ground, creating an [[acoustic shadow]] at some distance from the source.<ref>{{cite book | last = Everest | first = F. | title = The Master Handbook of Acoustics | publisher = McGraw-Hill | location = New York | year = 2001 | isbn = 0-07-136097-2 | pages = 262–263 }}</ref> In the 1862, during the [[American Civil War]] [[Battle of Iuka]], an acoustic shadow, believed to have been enhanced by a northeast wind, kept two divisions of Union soldiers out of the battle,<ref>{{cite book | last = Cornwall | first = Sir | title = Grant as Military Commander | publisher = Barnes & Noble Inc | year = 1996 | isbn = 1-56619-913-1 | page = 92}}</ref> because they could not hear the sounds of battle only six miles downwind.<ref>{{cite book | last = Cozzens | first = Peter | title = The Darkest Days of the War: the Battles of Iuka and Corinth | publisher = The University of North Carolina Press | location = Chapel Hill | year = 2006 | isbn = 0-8078-5783-1 }}</ref>


where <math>r_k</math> is Pearson's coefficient of correlation for segment <math>k</math>.
====Effects on architecture====
[[Wind engineering]] is a field of [[engineering]] devoted to the analysis of [[wind]] effects on the natural and [[built environment]]. It includes strong winds which may cause discomfort as well as extreme winds such as [[tornadoes]], [[hurricanes]] and storms which may cause widespread destruction. Wind engineering draws upon [[meteorology]], [[aerodynamics]] and a number of specialist [[engineering]] disciplines. The tools used include climate models, atmospheric boundary layer wind tunnels and numerical models. It involves, among other topics, how wind impacting buildings must be accounted for in engineering.<ref>Professor John Twidell. [http://www.multi-science.co.uk/windeng.htm Wind Engineering.] Retrieved on 2007-11-25.</ref>


By choosing the parameter <math>s</math>, the range of values is reduced and the correlations on long time scale are filtered out, only the correlations on short time scales being revealed. Thus, the contributions of slow components are removed and those of fast components are retained.
<!-- Deleted image removed: [[Image:EU Windmill.jpg|thumb|right|[[Wind turbine]]s in [[Douglas]], [[Central Scotland]]]] -->
[[Wind turbine]]s are affected by wind shear. Vertical wind-speed profiles result in different wind speeds at the blades nearest to the ground level compared to those at the top of blade travel, and this in turn affects the turbine operation.<ref name=Heier>{{cite book | last = Heier | first = Siegfried | title = Grid Integration of Wind Energy Conversion Systems | publisher = John Wiley & Sons | location = Chichester | year = 2005 | isbn = 0-470-86899-6 | pages = 45}}</ref> This low level wind shear can create a large bending moment in the shaft of a two bladed turbine when the blades are vertical.<ref>{{cite book | last = Harrison | first = Robert | title = Large Wind Turbines | publisher = John Wiley & Sons | location = Chichester | year = 2001 | isbn = 0-471-49456-9 | pages = 30}}</ref> The reduced wind shear over water means shorter and less expensive wind turbine towers can be used in shallow seas.<ref name=Lubosny>{{cite book | last = Lubosny | first = Zbigniew | title = Wind Turbine Operation in Electric Power Systems: Advanced Modeling | publisher = Springer | location = Berlin | year = 2003 | isbn = 3-540-40340-X | pages = 17}}</ref>


==See also==
==See also==
{{Portal|Statistics}}
*[[Air safety]]
{{Wikiversity|Linear correlation}}
*[[Low level windshear alert system]]
* [[Correlation and dependence]]
*[[Sailing]]
* [[Spearman's rank correlation coefficient]]
* [[Association (statistics)]]
* [[Disattenuation]]
* [[Maximal information coefficient]]
* [[Normally distributed and uncorrelated does not imply independent]]


==References==
==References==
{{Reflist}}
{{reflist|2}}


{{Statistics}}
==External links==
{{Use dmy dates|date=September 2010}}
*[http://education.arm.gov/nsdl/Library/glossary.shtml#Wind_shear National Science Digital Library - Wind shear]


{{DEFAULTSORT:Pearson Product-Moment Correlation Coefficient}}
{{good article}}
[[Category:Covariance and correlation]]
[[Category:Parametric statistics]]
[[Category:Statistical ratios]]


[[ca:Coeficient de correlació de Pearson]]
{{DEFAULTSORT:Wind Shear}}
[[de:Korrelationskoeffizient]]
[[Category:Wind]]
[[et:Lineaarne korrelatsioonikordaja]]
[[Category:Weather hazards to aircraft]]
[[es:Coeficiente de correlación de Pearson]]
[[Category:Microscale meteorology]]
[[eu:Korrelazio-koefiziente]]
[[it:Indice di correlazione di Pearson]]
[[he:מתאם פירסון]]
[[nl:Correlatiecoëfficiënt]]
[[ja:相関係数]]
[[no:Pearsons produkt-moment korrelasjonskoeffisient]]
[[pl:Współczynnik korelacji Pearsona]]
[[pt:Coeficiente de correlação de Pearson]]
[[ru:Корреляция#Линейный коэффициент корреляции]]
[[sk:Bravaisov-Pearsonov korelačný koeficient]]
[[sl:Pearsonov koeficient korelacije]]
[[zh:皮尔逊积矩相关系数]]

Revision as of 20:20, 10 August 2014

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Cirrus uncinus ice crystal plumes showing high level wind shear, with changes in wind speed and direction.

Wind shear, sometimes referred to as windshear or wind gradient, is a difference in wind speed and direction over a relatively short distance in the atmosphere. Wind shear can be broken down into vertical and horizontal components, with horizontal wind shear seen across fronts and near the coast, and vertical shear typically near the surface, though also at higher levels in the atmosphere near upper level jets and frontal zones aloft.

Wind shear itself is a microscale meteorological phenomenon occurring over a very small distance, but it can be associated with mesoscale or synoptic scale weather features such as squall lines and cold fronts. It is commonly observed near microbursts and downbursts caused by thunderstorms, fronts, areas of locally higher low level winds referred to as low level jets, near mountains, radiation inversions that occur due to clear skies and calm winds, buildings, wind turbines, and sailboats. Wind shear has a significant effect during take-off and landing of aircraft due to its effects on control of the aircraft, and it has been a sole or contributing cause of many aircraft accidents.

Sound movement through the atmosphere is affected by wind shear, which can bend the wave front, causing sounds to be heard where they normally would not, or vice versa. Strong vertical wind shear within the troposphere also inhibits tropical cyclone development, but helps to organize individual thunderstorms into longer life cycles which can then produce severe weather. The thermal wind concept explains how differences in wind speed at different heights are dependent on horizontal temperature differences, and explains the existence of the jet stream.[1]

Definition

Wind shear refers to the variation of wind over either horizontal or vertical distances. Airplane pilots generally regard significant wind shear to be a horizontal change in airspeed of 30 knots (15 m/s) for light aircraft, and near 45 knots (22 m/s) for airliners at flight altitude.[2] Vertical speed changes greater than 4.9 knots (2.5 m/s) also qualify as significant wind shear for aircraft. Low level wind shear can affect aircraft airspeed during take off and landing in disastrous ways and airliner pilots are trained to avoid all microburst wind shear (headwind loss in excess of 30 knots, Ibid.).[3] The rationale for this additional caution includes (1) microburst intensity can double in a minute or less, (2) the winds can shift to excessive cross wind, (3) 40-50 knots is the threshold for survivability at some stages of low-altitude operations (Ibid.), and (4) several of the historical wind shear accidents involved 35-45 knot microbursts. Wind shear is also a key factor in the creation of severe thunderstorms. The additional hazard of turbulence is often associated with wind shear.

Where and when it is strongly observed

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Microburst schematic from NASA. Note the downward motion of the air until it hits ground level, then spreads outward in all directions. The wind regime in a microburst is completely opposite to a tornado.

Weather situations where shear is observed include:

  • Weather fronts. Significant shear is observed when the temperature difference across the front is 5 °C (9 °F) or more, and the front moves at 30 knots or faster. Because fronts are three-dimensional phenomena, frontal shear can be observed at any altitude between surface and tropopause, and therefore be seen both horizontally and vertically. Vertical wind shear above warm fronts is more of an aviation concern than near and behind cold fronts due to their greater duration.[1]
  • Upper-level jet streams. Associated with upper level jet streams is a phenomenon known as clear air turbulence (CAT), caused by vertical and horizontal wind shear connected to the wind gradient at the edge of the jet streams.[4] The CAT is strongest on the cold air side of the jet,[5] usually next to or just below the axis of the jet.[6]
  • Low-level jet streams. When a nocturnal low-level jet forms overnight above the Earth's surface ahead of a cold front, significant low level vertical wind shear can develop near the lower portion of the low level jet. This is also known as nonconvective wind shear since it is not due to nearby thunderstorms.[1]
  • Mountains. When winds blow over a mountain, vertical shear is observed on the lee side. If the flow is strong enough, turbulent eddies known as "rotors" associated with lee waves may form, which are dangerous to ascending and descending aircraft.[7]
  • Inversions. When on a clear and calm night, a radiation inversion is formed near the ground, the friction does not affect wind above the top of the inversion layer. The change in wind can be 90 degrees in direction and 40 kt in speed. Even a nocturnal (overnight) low level jet can sometimes be observed. It tends to be strongest towards sunrise. Density differences cause additional problems to aviation.[1]
  • Downbursts. When an outflow boundary forms due to a shallow layer of rain-cooled air spreading out near ground level from the parent thunderstorm, both speed and directional wind shear can result at the leading edge of the three dimensional boundary. The stronger the outflow boundary is, the stronger the resultant vertical wind shear will become.[8]

Horizontal component

Weather fronts

Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. Weather fronts are boundaries between two masses of air of different densities, or different temperature and moisture properties, which normally are convergence zones in the wind field and are the principal cause of significant weather. Within surface weather analyses, they are depicted using various colored lines and symbols. The air masses usually differ in temperature and may also differ in humidity. Wind shear in the horizontal occurs near these boundaries. Cold fronts feature narrow bands of thunderstorms and severe weather, and may be preceded by squall lines and dry lines. Cold fronts are sharper surface boundaries with more significant horizontal wind shear than warm fronts. When a front becomes stationary, it can degenerate into a line which separates regions of differing wind speed, known as a shear line, though the wind direction across the front normally remains constant. In the tropics, tropical waves move from east to west across the Atlantic and eastern Pacific basins. Directional and speed shear can occur across the axis of stronger tropical waves, as northerly winds precede the wave axis and southeast winds are seen behind the wave axis. Horizontal wind shear can also occur along local land breeze and sea breeze boundaries.[9]

Near coastlines

The magnitude of winds offshore are nearly double the wind speed observed onshore. This is attributed to the differences in friction between land masses and offshore waters. Sometimes, there are even directional differences, particularly if local sea breezes change the wind on shore during daylight hours.[10]

Vertical component

Thermal wind

Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. Thermal wind is a meteorological term not referring to an actual wind, but a difference in the geostrophic wind between two pressure levels and , with ; in essence, wind shear. It is only present in an atmosphere with horizontal changes in temperature (or in an ocean with horizontal gradients of density), i.e. baroclinicity. In a barotropic atmosphere, where temperature is uniform, the geostrophic wind is independent of height. The name stems from the fact that this wind flows around areas of low (and high) temperature in the same manner as the geostrophic wind flows around areas of low (and high) pressure.[11]

The thermal wind equation is

,

where the are geopotential height fields with , is the Coriolis parameter, and is the upward-pointing unit vector in the vertical direction. The thermal wind equation does not determine the wind in the tropics. Since is small or zero, such as near the equator, the equation reduces to stating that is small.[11]

This equation basically describes the existence of the jet stream, a westerly current of air with maximum wind speeds close to the tropopause which is (even though other factors are also important) the result of the temperature contrast between equator and pole.

Effects on tropical cyclones

Strong wind shear in the high troposphere forms the anvil-shaped top of this mature cumulonimbus cloud, or thunderstorm.[12]

Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. Tropical cyclones are basically heat engines that are fueled by the temperature gradient between the warm tropical ocean surface and the colder upper atmosphere. Tropical cyclone development requires relatively low values of vertical wind shear so that their warm core can remain above their surface circulation center, thereby promoting strengthening. Vertical wind shear tears up the "machinery" of the heat engine causing it to break down. Strongly sheared tropical cyclones weaken as the upper circulation is blown away from the low level center.[13]

Effects on thunderstorms and severe weather

Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. Severe thunderstorms, which can spawn tornadoes and hailstorms, require wind shear to organize the storm in such a way as to maintain the thunderstorm for a longer period of time. This occurs as the storm's inflow becomes separated from its rain-cooled outflow. An increasing nocturnal, or overnight, low level jet can increase the severe weather potential by increasing the vertical wind shear through the troposphere. Thunderstorms in an atmosphere with virtually no vertical wind shear weaken as soon as they send out an outflow boundary in all directions, which then quickly cuts off its inflow of relatively warm, moist air and kills the thunderstorm.[14]

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Planetary boundary layer

Depiction of where the planetary boundary layer lies on a sunny day

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is changing into a rising pattern among Singaporeans as the standard of living is increasing over time and more Singaporeans have abundance of capital to invest on properties. Investing in the personal properties in Singapore I would like to applaud you for arising with such a book which covers the secrets and techniques and tips of among the profitable Singapore property buyers. I believe many novice investors will profit quite a bit from studying and making use of some of the tips shared by the gurus." – Woo Chee Hoe Special bonus for consumers of Secrets of Singapore Property Gurus Actually, I can't consider one other resource on the market that teaches you all the points above about Singapore property at such a low value. Can you? Condominium For Sale (D09) – Yong An Park For Lease

In 12 months 2013, c ommercial retails, shoebox residences and mass market properties continued to be the celebrities of the property market. Models are snapped up in report time and at document breaking prices. Builders are having fun with overwhelming demand and patrons need more. We feel that these segments of the property market are booming is a repercussion of the property cooling measures no.6 and no. 7. With additional buyer's stamp responsibility imposed on residential properties, buyers change their focus to commercial and industrial properties. I imagine every property purchasers need their property funding to understand in value. The atmospheric effect of surface friction with winds aloft force surface winds to slow and back counterclockwise near the surface of the Earth blowing inward across isobars (lines of equal pressure), when compared to the winds in frictionless flow well above the Earth's surface. This layer where friction slows and changes the wind is known as the planetary boundary layer, sometimes the Ekman layer, and it is thickest during the day and thinnest at night. Daytime heating thickens the boundary layer as winds at the surface become increasingly mixed with winds aloft due to insolation, or solar heating. Radiative cooling overnight further enhances wind decoupling between the winds at the surface and the winds above the boundary layer by calming the surface wind which increases wind shear. These wind changes force wind shear between the boundary layer and the wind aloft, and is most emphasized at night.[15]

Effects on flight

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Gliding
Glider ground launch affected by wind shear.

In gliding, wind gradients just above the surface affect the takeoff and landing phases of flight of a glider. Wind gradient can have a noticeable effect on ground launches, also known as winch launches or wire launches. If the wind gradient is significant or sudden, or both, and the pilot maintains the same pitch attitude, the indicated airspeed will increase, possibly exceeding the maximum ground launch tow speed. The pilot must adjust the airspeed to deal with the effect of the gradient.[16]

When landing, wind shear is also a hazard, particularly when the winds are strong. As the glider descends through the wind gradient on final approach to landing, airspeed decreases while sink rate increases, and there is insufficient time to accelerate prior to ground contact. The pilot must anticipate the wind gradient and use a higher approach speed to compensate for it.[17]

Wind shear is also a hazard for aircraft making steep turns near the ground. It is a particular problem for gliders which have a relatively long wingspan, which exposes them to a greater wind speed difference for a given bank angle. The different airspeed experienced by each wing tip can result in an aerodynamic stall on one wing, causing a loss of control accident.[17][18]

Parachuting

Wind shear or wind gradients are a threat to parachutists, particularly to BASE jumping and wingsuit flying. Skydivers have been pushed off of their course by sudden shifts in wind direction and speed, and have collided with bridges, cliffsides, trees, other skydivers, the ground, and other obstacles.Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park. Skydivers routinely make adjustments to the position of their open canopies to compensate for changes in direction while making landings to prevent accidents such as canopy collisions and canopy inversion.

Soaring

Soaring related to wind shear, also called dynamic soaring, is a technique used by soaring birds like albatrosses, who can maintain flight without wing flapping. If the wind shear is of sufficient magnitude, a bird can climb into the wind gradient, trading ground speed for height, while maintaining airspeed.[19] By then turning downwind, and diving through the wind gradient, they can also gain energy.[20] It has also been used by glider pilots on rare occasions.

Wind shear can also create wave. This occurs when an atmospheric inversion separates two layers with a marked difference in wind direction. If the wind encounters distortions in the inversion layer caused by thermals coming up from below, it will create significant shear waves that can be used for soaring.[21]

Impact on passenger aircraft
Effect of wind shear on aircraft trajectory. Note how merely correcting for the initial gust front can have dire consequences.

Strong outflow from thunderstorms causes rapid changes in the three-dimensional wind velocity just above ground level. Initially, this outflow causes a headwind that increases airspeed, which normally causes a pilot to reduce engine power if they are unaware of the wind shear. As the aircraft passes into the region of the downdraft, the localized headwind diminishes, reducing the aircraft's airspeed and increasing its sink rate. Then, when the aircraft passes through the other side of the downdraft, the headwind becomes a tailwind, reducing lift generated by the wings, and leaving the aircraft in a low-power, low-speed descent. This can lead to an accident if the aircraft is too low to effect a recovery before ground contact.

Wreckage of Delta Air Lines Flight 191 tail section after a microburst slammed the aircraft into the ground. Another aircraft can be seen flying in the background past the crash scene.

As the result of the accidents in the 1970s and 1980s, most notably following the 1985 crash of Delta Air Lines Flight 191, in 1988 the U.S. Federal Aviation Administration mandated that all commercial aircraft have on-board wind shear detection systems by 1993. Between 1964 and 1985, wind shear directly caused or contributed to 26 major civil transport aircraft accidents in the U.S. that led to 620 deaths and 200 injuries.[22] Since 1995, the number of major civil aircraft accidents caused by wind shear has dropped to approximately one every ten years, due to the mandated on-board detection as well as the addition of Doppler weather radar units on the ground (NEXRAD).Template:Fact The installation of high-resolution Terminal Doppler Weather Radar stations at many U.S. airports that are commonly affected by wind shear has further aided the ability of pilots and ground controllers to avoid wind shear conditions.[23]

Sailing

Wind shear affects sailboats in motion by presenting a different wind speed and direction at different heights along the mast. The effect of low level wind shear can be factored into the selection of sail twist in the sail design, but this can be difficult to predict since wind shear may vary widely in different weather conditions. Sailors may also adjust the trim of the sail to account for low level wind shear, for example using a boom vang.[24]

Sound propagation

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Singapore has been beckoning to traders ever since the value of properties in Singapore started sky rocketing just a few years again. Many businesses have their places of work in Singapore and prefer to own their own workplace area within the country once they decide to have a everlasting office. Rentals in Singapore in the corporate sector can make sense for some time until a business has discovered a agency footing. Finding Commercial Property Singapore takes a variety of time and effort but might be very rewarding in the long term.

is changing into a rising pattern among Singaporeans as the standard of living is increasing over time and more Singaporeans have abundance of capital to invest on properties. Investing in the personal properties in Singapore I would like to applaud you for arising with such a book which covers the secrets and techniques and tips of among the profitable Singapore property buyers. I believe many novice investors will profit quite a bit from studying and making use of some of the tips shared by the gurus." – Woo Chee Hoe Special bonus for consumers of Secrets of Singapore Property Gurus Actually, I can't consider one other resource on the market that teaches you all the points above about Singapore property at such a low value. Can you? Condominium For Sale (D09) – Yong An Park For Lease

In 12 months 2013, c ommercial retails, shoebox residences and mass market properties continued to be the celebrities of the property market. Models are snapped up in report time and at document breaking prices. Builders are having fun with overwhelming demand and patrons need more. We feel that these segments of the property market are booming is a repercussion of the property cooling measures no.6 and no. 7. With additional buyer's stamp responsibility imposed on residential properties, buyers change their focus to commercial and industrial properties. I imagine every property purchasers need their property funding to understand in value. Wind shear can have a pronounced effect upon sound propagation in the lower atmosphere, where waves can be "bent" by refraction phenomenon. The audibility of sounds from distant sources, such as thunder or gunshots, is very dependent on the amount of shear. The result of these differing sound levels is key in noise pollution considerations, for example from roadway noise and aircraft noise, and must be considered in the design of noise barriers.[25] This phenomenon was first applied to the field of noise pollution study in the 1960s, contributing to the design of urban highways as well as noise barriers.[26]

Hodograph plot of wind vectors at various heights in the troposphere. Meteorologists can use this plot to evaluate vertical wind shear in weather forecasting. (Source: NOAA)

The speed of sound varies with temperature. Since temperature and sound velocity normally decrease with increasing altitude, sound is refracted upward, away from listeners on the ground, creating an acoustic shadow at some distance from the source.[27] In the 1862, during the American Civil War Battle of Iuka, an acoustic shadow, believed to have been enhanced by a northeast wind, kept two divisions of Union soldiers out of the battle,[28] because they could not hear the sounds of battle only six miles downwind.[29]

Effects on architecture

Wind engineering is a field of engineering devoted to the analysis of wind effects on the natural and built environment. It includes strong winds which may cause discomfort as well as extreme winds such as tornadoes, hurricanes and storms which may cause widespread destruction. Wind engineering draws upon meteorology, aerodynamics and a number of specialist engineering disciplines. The tools used include climate models, atmospheric boundary layer wind tunnels and numerical models. It involves, among other topics, how wind impacting buildings must be accounted for in engineering.[30]

Wind turbines are affected by wind shear. Vertical wind-speed profiles result in different wind speeds at the blades nearest to the ground level compared to those at the top of blade travel, and this in turn affects the turbine operation.[31] This low level wind shear can create a large bending moment in the shaft of a two bladed turbine when the blades are vertical.[32] The reduced wind shear over water means shorter and less expensive wind turbine towers can be used in shallow seas.[33]

See also

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

External links

Template:Good article

  1. ↑ 1.0 1.1 1.2 1.3 Template:Cite web
  2. ↑ FAA FAA Advisory Circular Pilot Wind Shear Guide. Retrieved on 2007-12-15.
  3. ↑ Template:Cite web
  4. ↑ Template:Cite web Template:Dead link
  5. ↑ M. P. de Villiers and J. van Heerden. Clear air turbulence over South Africa. Retrieved on 2008-05-08.
  6. ↑ CLARK T. L., HALL W. D., KERR R. M., MIDDLETON D., RADKE L., RALPH F. M., NEIMAN P. J., LEVINSON D. Origins of aircraft-damaging clear-air turbulence during the 9 December 1992 Colorado downslope windstorm : Numerical simulations and comparison with observations. Retrieved on 2008-05-08.
  7. ↑ National Center for Atmospheric Research. T-REX: Catching the Sierra’s waves and rotors Retrieved on 2006-10-21.
  8. ↑ Fujita, T.T. (1985). "The Downburst, microburst and macroburst". SMRP Research Paper 210, 122 pp.
  9. ↑ David M. Roth. Hydrometeorological Prediction Center. Unified Surface Analysis Manual. Retrieved on 2006-10-22.
  10. ↑ Franklin B. Schwing and Jackson O. Blanton. The Use of Land and Sea Based Wind Data in a Simple Circulation Model. Retrieved on 2007-10-03.
  11. ↑ 11.0 11.1 James R. Holton (2004). An Introduction to Dynamic Meteorology. ISBN 0-12-354015-1
  12. ↑ 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
  13. ↑ University of Illinois. Hurricanes. Retrieved 2006-10-21.
  14. ↑ University of Illinois. Vertical Wind Shear Retrieved on 2006-10-21.
  15. ↑ Glossary of Meteorology. E. Retrieved on 2007-06-03.
  16. ↑ 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
  17. ↑ 17.0 17.1 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
  18. ↑ 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
  19. ↑ 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
  20. ↑ 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
  21. ↑ 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
  22. ↑ Template:Cite web
  23. ↑ Template:Cite web
  24. ↑ 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
  25. ↑ 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
  26. ↑ 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
  27. ↑ 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
  28. ↑ 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
  29. ↑ 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
  30. ↑ Professor John Twidell. Wind Engineering. Retrieved on 2007-11-25.
  31. ↑ 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
  32. ↑ 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
  33. ↑ 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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