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2014-06-01T18:24:43Z

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2014-06-01T15:35:57Z

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2.25.181.200: fix ref

[[File:Simple-indifference-curves.svg|thumb|240px|right|An example of an indifference map with three indifference curves represented]]

In [[microeconomic theory]], an '''indifference curve''' is a [[graph of a function|graph]] showing different bundles of [[good (economics)|goods]] between which a consumer is ''indifferent.'' That is, at each point on the curve, the consumer has no [[Preference (economics)|preference]] for one bundle over another. One can equivalently refer to each point on the indifference curve as rendering the same level of [[utility]] (satisfaction) for the consumer. In other words an indifference curve is the locus of various points showing different combinations of two goods providing equal utility to the consumer. Utility is then a device to represent [[preference]]s rather than something from which preferences come.<ref>Geanakoplis (1987), p. 117.</ref> The main use of indifference curves is in the [[Mathematical problem|representation]] of potentially observable [[demand]] patterns for individual consumers over commodity bundles.<ref>Böhm and Haller (1987), p. 785.</ref>

There are infinitely many indifference curves: one passes through each combination. A collection of (selected) indifference curves, illustrated graphically, is referred to as an '''indifference map'''.

== History ==
The theory of indifference curves was developed by [[Francis Ysidro Edgeworth]], who explained in his book "Mathematical Psychics: an Essay on the Application of Mathematics to the Moral Sciences,” 1881,<ref>http://onlinebooks.library.upenn.edu/webbin/book/lookupid?key=olbp34052</ref> the mathematics needed for its drawing; later on, [[Vilfredo Pareto]] was the first author to actually draw these curves, in his book "Manual of Political Economy," 1906;<ref>http://archive.org/details/manualedieconomi00pareuoft</ref><ref>http://www.policonomics.com/indifference-curves/</ref> and others in the first part of the 20th century. The theory can be derived from [[William Stanley Jevons]]'s [[ordinal utility]] theory, which posits that individuals can always rank any consumption bundles by order of preference.<ref>http://www.policonomics.com/william-stanley-jevons/</ref>

== Map and properties of indifference curves ==
[[Image:Indifference curve example.png|thumb|240px|right|An example of how indifference curves are obtained as the [[level curves]] of a utility function]]
A graph of indifference curves for an individual consumer associated with different utility levels is called an '''indifference map'''. Points yielding different utility levels are each associated with distinct indifference curves and these indifference curves on the indifference map are like contour lines on a topographical map. Each point on the curve represents the same elevation. If you move "off" an indifference curve traveling in a northeast direction (assuming positive marginal utility for the goods) you are essentially climbing a mound of utility. The higher you go the greater the level of utility. The non-satiation requirement means that you will never reach the "top," or a "[[bliss point]]," a consumption bundle that is preferred to all others

Indifference curves are typically represented to be:
# Defined only in the non-negative [[Cartesian coordinate system|quadrant]] of commodity quantities (i.e. the possibility of having negative quantities of any good is ignored).
# [[Inverse relationship|Negatively]] sloped. That is, as quantity consumed of one good (X) increases, total satisfaction would increase if not offset by a decrease in the quantity consumed of the other good (Y). Equivalently, [[Economic satiation|satiation]], such that more of either good (or both) is equally preferred to no increase, is excluded. (If utility ''U = f(x, y)'', ''U'', in the third dimension, does not have a [[Maxima and minima#Functions of more than one variable|local maximum]] for any ''x'' and ''y'' values.) The negative slope of the indifference curve reflects the assumption of the monotonicity of consumer's preferences, which generates monotonically increasing utility functions, and the assumption of non-satiation (marginal utility for all goods is always positive); an upward sloping indifference curve would imply that a consumer is indifferent between a bundle A and another bundle B because they lie on the same indifference curve, even in the case in which the quantity of both goods in bundle B is higher. Because of monotonicity of preferences and non-satiation, a bundle with more of both goods must be preferred to one with less of both, thus the first bundle must yield a higher utility, and lie on a different indifference curve at a higher utility level.
The negative slope of the indifference curve implies that the [[marginal rate of substitution]] is always positive;
# [[Total order|Complete]], such that all points on an indifference curve are ranked equally preferred and ranked either more or less preferred than every other point not on the curve. So, with (2), no two curves can intersect (otherwise non-satiation would be violated).
# [[Transitive relation#Examples|Transitive]] with respect to points on distinct indifference curves. That is, if each point on ''I2'' is (strictly) preferred to each point on ''I1'', and each point on ''I3'' is preferred to each point on ''I2'', each point on ''I3'' is preferred to each point on ''I1''. A negative slope and transitivity exclude indifference curves crossing, since straight lines from the origin on both sides of where they crossed would give opposite and intransitive preference rankings.
# (Strictly) [[convex function|convex]]. With (2), [[convex preferences]] imply that the indifference curves cannot be concave to the origin, i.e. they will either be straight lines or bulge toward the origin of the indifference curve. If the latter is the case, then as a consumer decreases consumption of one good in successive units, successively larger doses of the [[composite good|other good]] are required to keep satisfaction unchanged.

==Assumptions of consumer preference theory==
*Preferences are '''complete'''. The consumer has ranked all available alternative combinations of commodities in terms of the satisfaction they provide him.
**Assume that there are two consumption bundles A and B each containing two commodities x and y. A consumer can unambiguously determine that one and only one of the following is the case:
***A is preferred to B ⇒ A p B<ref name="Binger">Binger and Hoffman (1998). pp. 109-17.</ref>
***B is preferred to A ⇒ B p A<ref name="Binger"/>
***A is indifferent to B ⇒ A I B<ref name="Binger"/>
****Note that this axiom precludes the possibility that the consumer cannot decide,<ref name="Perloff 2008. p. 62">Perloff (2008). p. 62.</ref> and that a consumer is able to make this comparison with respect to every conceivable bundle of goods.<ref name="Binger"/>
*Preferences are '''reflexive'''
**Means that if A and B are identical in all respects the consumer will recognize this fact and be indifferent in comparing A and B
***A = B ⇒ A I B<ref name="Binger" />
*Preferences are '''transitive'''{{#tag:ref|The transitivity of weak preferences is sufficient for most IC analysis: If A is weakly preferred to B meaning that the consumer likes A ''at least as much'' as B and B is weakly preferred to C then A is weakly preferred to C.<ref name="Perloff 2008. p. 62"/>|group=nb}}
**If A p B and B p C then A p C.<ref name="Binger"/>
**Also A I B and B I C then A I C.<ref name="Binger"/>
***This is a consistency assumption.
*Preferences are '''continuous'''
**If A is preferred to B and C is infinitesimally close to B then A is preferred to C.
**A p B & C → B ⇒ A p C.
***"Continuous" means infinitely divisible - just like there are infinite numbers between 1 and 2 all bundles are infinitely divisible. This assumption makes indifference curves continuous.
*Preferences exhibit '''strong monotonicity'''
**if A has more of both x and y than B then A is preferred to B
***this assumption is commonly called the "more is better" assumption
***an alternative version of this assumption is strong monotonicity which requires that if A and B have the same quantity of one good, but A has more of the other, then A is preferred to B.
It also implies that the commodities are '''good''' rather than '''bad'''. Examples of '''bad''' commodities can be disease, pollution etc. because we always desire less of such things.
*Indifference curves exhibit '''diminishing marginal rates of substitution'''
**The marginal rate of substitution tells how much 'y' a person is willing to sacrifice to get one more unit of 'x'.
**This assumption assures that indifference curves are smooth and convex to the origin.
**This assumption also set the stage for using techniques of constrained optimization because the shape of the curve assures that the first derivative is negative and the second is positive.
**Another name for this assumption is the '''substitution assumption'''. It is the most critical assumption of consumer theory: Consumers are willing to give up or trade-off some of one good to get more of another. The fundamental assertion is that there is a maximum amount that "a consumer will give up, of one commodity, to get one unit of another good, in that amount which will leave the consumer indifferent between the new and old situations"<ref name="Silberberg">Silberberg and Suen (2000).</ref> The negative slope of the indifference curves represents the willingness of the consumer to make a trade off.<ref name="Silberberg"/>
*There are also many sub-assumptions:
**Irreflexivity - for no x is xpx
**Negative transitivity - if xnot-py then for any third commodity z, either xnot-pz or znot-py or both.

=== Application ===
[[File:Indifference curves showing budget line.svg|thumb|right|To maximise utility, a household should consume at (Qx, Qy). Assuming it does, a full demand schedule can be deduced as the price of one good fluctuates.]]

[[Consumer theory]] uses indifference curves and budget constraints to generate [[Supply and demand|consumer demand curves]]. For a single consumer, this is a relatively simple process. First, let one good be an example market e.g., carrots, and let the other be a composite of all other goods. Budget constraints give a straight line on the indifference map showing all the possible distributions between the two goods; the point of maximum utility is then the point at which an indifference curve is tangent to the budget line (illustrated). This follows from common sense: if the market values a good more than the household, the household will sell it; if the market values a good less than the household, the household will buy it. The process then continues until the market's and household's marginal rates of substitution are equal.<ref name="Lipsey 1975. pp. 182-186">Lipsey (1975). pp. 182-186.</ref> Now, if the price of carrots were to change, and the price of all other goods were to remain constant, the gradient of the budget line would also change, leading to a different point of tangency and a different quantity demanded. These price / quantity combinations can then be used to deduce a full demand curve.<ref name="Lipsey 1975. pp. 182-186"/> A line connecting all points of tangency between the indifference curve and the [[budget constraint]] is called the [[expansion path]].<ref name="Salvatore">Salvatore, Dominick (1989). ''Schaum's outline of theory and problems of managerial economics,'' McGraw-Hill, ISBN 978-0-07-054513-7</ref>
{{clr}}

=== Examples of indifference curves ===
<center><gallery>
File:Simple-indifference-curves.svg|Figure 1: An example of an indifference map with three indifference curves represented
File:Indifference-curves-perfect-substitutes.svg|Figure 2: Three indifference curves where Goods X and Y are perfect substitutes. The gray line perpendicular to all curves indicates the curves are mutually parallel.
File:Indifference-curves-perfect-complements.svg|Figure 3: Indifference curves for perfect complements X and Y. The elbows of the curves are [[collinear]].
</gallery></center>

In Figure 1, the consumer would rather be on ''I3'' than ''I2'', and would rather be on ''I2'' than ''I1'', but does not care where he/she is on a given indifference curve. The slope of an indifference curve (in absolute value), known by economists as the [[marginal rate of substitution]], shows the rate at which consumers are willing to give up one good in exchange for more of the other good. For ''most'' goods the marginal rate of substitution is not constant so their indifference curves are curved. The curves are convex to the origin, describing the negative [[substitution effect]]. As price rises for a fixed money income, the consumer seeks less the expensive substitute at a lower indifference curve. The substitution effect is reinforced through the [[income effect]] of lower real income (Beattie-LaFrance). An example of a utility function that generates indifference curves of this kind is the Cobb-Douglas function <math>\scriptstyle U\left(x,y\right)=x^\alpha y^{1-\alpha }, 0 \leq \alpha \leq 1</math>. The negative slope of the indifference curve incorporates the willingness of the consumer to make trade offs.<ref>Id.</ref>

If two goods are [[substitute good|perfect substitutes]] then the indifference curves will have a constant slope since the consumer would be willing to switch between at a fixed ratio. The marginal rate of substitution between perfect substitutes is likewise constant. An example of a utility function that is associated with indifference curves like these would be <math>\scriptstyle U\left(x,y\right)=\alpha x + \beta y</math>.

If two goods are [[complement good|perfect complements]] then the indifference curves will be L-shaped. Examples of perfect complements include left shoes compared to right shoes: the consumer is no better off having several right shoes if she has only one left shoe - additional right shoes have zero marginal utility without more left shoes, so bundles of goods differing only in the number of right shoes they includes - however many - are equally preferred. The marginal rate of substitution is either zero or infinite. An example of the type of utility function that has an indifference map like that above is the Leontief function: <math>\scriptstyle U\left(x,y\right)= \min \{ \alpha x, \beta y \}</math>.

The different shapes of the curves imply different responses to a change in price as shown from demand analysis in [[consumer theory]]. The results will only be stated here. A price-budget-line change that kept a consumer in equilibrium on the same indifference curve:
:in Fig. 1 would reduce quantity demanded of a good smoothly as price rose relatively for that good.
:in Fig. 2 would have either no effect on quantity demanded of either good (at one end of the budget constraint) or would change quantity demanded from one end of the budget constraint to the other.
:in Fig. 3 would have no effect on equilibrium quantities demanded, since the budget line would rotate around the corner of the indifference curve.{{#tag:ref|Indifference curves can be used to derive the individual demand curve. However, the assumptions of consumer preference theory do not guarantee that the demand curve will have a negative slope.<ref>Binger and Hoffman (1998). p. 141-43.</ref>|group=nb}}

== Preference relations and utility ==
Choice theory formally represents consumers by a '''preference relation''', and use this representation to derive indifference curves showing combinations of equal preference to the consumer.

=== Preference relations ===
Let
:<math>A\;</math> be a set of mutually exclusive alternatives among which a consumer can choose.
:<math>a\;</math> and <math>b\;</math> be generic elements of <math>A\;</math>.
In the language of the example above, the set <math>A\;</math> is made of combinations of apples and bananas. The symbol <math>a\;</math> is one such combination, such as 1 apple and 4 bananas and <math>b\;</math> is another combination such as 2 apples and 2 bananas.

A preference relation, denoted <math>\succeq</math>, is a [[binary relation]] define on the set <math>A\;</math>.

The statement
:<math>a\succeq b\;</math>
is described as '<math>a\;</math> is weakly preferred to <math>b\;</math>.' That is, <math>a\;</math> is at least as good as <math>b\;</math> (in preference satisfaction).

The statement
:<math>a\sim b\;</math>
is described as '<math>a\;</math> is weakly preferred to <math>b\;</math>, and <math>b\;</math> is weakly preferred to <math>a\;</math>.' That is, one is ''indifferent'' to the choice of <math>a\;</math> or <math>b\;</math>, meaning not that they are unwanted but that they are equally good in satisfying preferences.

The statement
:<math>a\succ b\;</math>
is described as '<math>a\;</math> is weakly preferred to <math>b\;</math>, but <math>b\;</math> is not weakly preferred to <math>a\;</math>.' One says that '<math>a\;</math> is strictly preferred to <math>b\;</math>.'

The preference relation <math>\succeq</math> is '''complete''' if all pairs <math>a,b\;</math> can be ranked. The relation is a [[transitive relation]] if whenever <math>a\succeq b\;</math> and <math>b\succeq c,\;</math> then <math>a\succeq c\;</math>.

For any element <math>a \in A\;</math>, the corresponding indifference curve, <math>\mathcal{C}_a</math> is made up of all elements of <math>A\;</math> which are indifferent to <math>a</math>. Formally,

<math>\mathcal{C}_a=\{b \in A:b \sim a\}</math>.

=== Formal link to utility theory ===

In the example above, an element <math>a\;</math> of the set <math>A\;</math> is made of two numbers: The number of apples, call it <math>x,\;</math> and the number of bananas, call it <math>y.\;</math>

In [[utility]] theory, the [[utility function]] of an [[agent (economics)|agent]] is a function that ranks ''all'' pairs of consumption bundles by order of preference (''completeness'') such that any set of three or more bundles forms a [[transitive relation]]. This means that for each bundle <math>\left(x,y\right)</math> there is a unique relation, <math>U\left(x,y\right)</math>, representing the [[utility]] (satisfaction) relation associated with <math>\left(x,y\right)</math>. The relation <math>\left(x,y\right)\to U\left(x,y\right)</math> is called the [[utility function]]. The [[Range (mathematics)|range]] of the function is a set of [[real numbers]]. The actual values of the function have no importance. Only the ranking of those values has content for the theory. More precisely, if <math>U(x,y)\geq U(x',y')</math>, then the bundle <math>\left(x,y\right)</math> is described as at least as good as the bundle <math>\left(x',y'\right)</math>. If <math>U\left(x,y\right)>U\left(x',y'\right)</math>, the bundle <math>\left(x,y\right)</math> is described as strictly preferred to the bundle <math>\left(x',y'\right)</math>.

Consider a particular bundle <math>\left(x_0,y_0\right)</math> and take the [[total derivative]] of <math>U\left(x,y\right)</math> about this point:

:<math>dU\left(x_0,y_0\right)=U_1\left(x_0,y_0\right)dx+U_2\left(x_0,y_0\right)dy </math> or, without loss of generality,

:<math>\frac{dU\left(x_0,y_0\right)}{dx}= U_1(x_0,y_0).1+ U_2(x_0,y_0)\frac{dy}{dx}</math> '''(Eq. 1)'''

where <math>U_1\left(x,y\right)</math> is the partial derivative of <math>U\left(x,y\right)</math> with respect to its first argument, evaluated at <math>\left(x,y\right)</math>. (Likewise for <math>U_2\left(x,y\right).</math>)

The indifference curve through <math>\left(x_0,y_0\right)</math> must deliver at each bundle on the curve the same utility level as bundle <math>\left(x_0,y_0\right)</math>. That is, when preferences are represented by a utility function, the indifference curves are the [[level curve]]s of the utility function. Therefore, if one is to change the quantity of <math>x\,</math> by <math>dx\,</math>, without moving off the indifference curve, one must also change the quantity of <math>y\,</math> by an amount <math>dy\,</math> such that, in the end, there is no change in ''U'':
:<math>\frac{dU\left(x_0,y_0\right)}{dx}= 0</math>, or, substituting ''0'' into ''(Eq. 1)'' above to solve for ''dy/dx'':
:<math>\frac{dU\left(x_0,y_0\right)}{dx} = 0\Leftrightarrow\frac{dy}{dx}=-\frac{U_1(x_0,y_0)}{U_2(x_0,y_0)}</math>.
Thus, the ratio of marginal utilities gives the absolute value of the [[slope]] of the indifference curve at point <math>\left(x_0,y_0\right)</math>. This ratio is called the [[marginal rate of substitution]] between <math>x\,</math> and <math>y\,</math>.

=== Examples ===

==== Linear utility====

If the utility function is of the form <math>U\left(x,y\right)=\alpha x+\beta y</math> then the marginal utility of <math>x\,</math> is <math>U_1\left(x,y\right)=\alpha</math> and the marginal utility of <math>y\,</math> is <math>U_2\left(x,y\right)=\beta</math>. The slope of the indifference curve is, therefore,
:<math>\frac{dx}{dy}=-\frac{\beta}{\alpha}.</math>
Observe that the slope does not depend on <math>x\,</math> or <math>y\,</math>: the indifference curves are straight lines.

====Cobb-Douglas utility====

If the utility function is of the form <math>U\left(x,y\right)=x^\alpha y^{1-\alpha}</math> the marginal utility of <math>x\,</math> is <math>U_1\left(x,y\right)=\alpha \left(x/y\right)^{\alpha-1}</math> and the marginal utility of <math>y\,</math> is <math>U_2\left(x,y\right)=(1-\alpha) \left(x/y\right)^{\alpha}</math>.Where <math>\alpha<1</math>. The [[slope]] of the indifference curve, and therefore the negative of the [[marginal rate of substitution]], is then
:<math>\frac{dx}{dy}=-\frac{1-\alpha}{\alpha}\left(\frac{x}{y}\right).</math>

====CES utility====

A general CES ([[Constant Elasticity of Substitution]]) form is
:<math>U(x,y)=\left(\alpha x^\rho +(1-\alpha)y^\rho\right)^{1/\rho}</math>
where <math>\alpha\in(0,1)</math> and <math>\rho\leq 1</math>. (The [[Cobb-Douglas]] is a special case of the CES utility, with <math>\rho=0\,</math>.) The marginal utilities are given by
:<math>U_1(x,y)=\alpha \left(\alpha x^\rho +(1-\alpha)y^\rho\right)^{\left(1/\rho\right)-1} x^{\rho-1}</math>
and
:<math>U_2(x,y)=(1-\alpha)\left(\alpha x^\rho +(1-\alpha)y^\rho\right)^{\left(1/\rho\right)-1} y^{\rho-1}.</math>
Therefore, along an indifference curve,
:<math>\frac{dx}{dy}=-\frac{1-\alpha}{\alpha}\left(\frac{x}{y}\right)^{\rho-1}.</math>
These examples might be useful for [[model (economics)|modelling]] individual or aggregate demand.

====Biology====
As used in [[Biology]], the indifference curve is a model for how animals 'decide' whether to perform a particular behavior, based on changes in two variables which can increase in intensity, one along the x-axis and the other along the y-axis. For example, the x-axis may measure the quantity of food available while the y-axis measures the risk involved in obtaining it. The indifference curve is drawn to predict the animal's behavior at various levels of risk and food availability.

== See also ==
* [[Budget constraint]]
* [[Community indifference curve]]
* [[Consumer theory]]
* [[Convex preferences]]
* [[Endowment effect]]
* [[Indifference price]]
* [[Level curve]]
* [[Microeconomics]]
* [[Rationality]]
* [[Utility–possibility frontier]]

==Footnotes==
{{Reflist|group=nb}}

== Notes ==
{{reflist}}

== References ==
*{{cite journal |first=Bruce R. |last=Beattie |first2=Jeffrey T. |last2=LaFrance |title=The Law of Demand versus Diminishing Marginal Utility |year=2006 |journal=[[Applied Economic Perspectives and Policy|Appl. Econ. Perspect. Pol.]] |volume=28 |issue=2 |pages=263–271 |doi=10.1111/j.1467-9353.2006.00286.x }}
*{{cite book |last=Binger |last2=Hoffman |year=1998 |title=Microeconomics with Calculus |edition=2nd |publisher=Addison-Wesley |location=Reading |isbn=0321012259 }}
*{{cite book |first=Volker |last=Böhm |first2=Hans |last2=Haller |year=1987 |chapter=Demand theory |title=The [[New Palgrave: A Dictionary of Economics]] |volume=1 |pages=785–792 |isbn= }}
*{{cite book |first=John |last=Geanakoplos |year=1987 |chapter=Arrow-Debreu model of general equilibrium |title=The New Palgrave: A Dictionary of Economics |volume=1 |pages=116–124 |isbn= }}
*{{cite book |last=Perloff |first=Jeffrey M. |year=2008 |title=Microeconomics: Theory & Applications with Calculus |publisher=Addison-Wesley |location=Boston |isbn=9780321277947 }}
*{{cite book |last=Silberberg |last2=Suen |year=2000 |title=The Structure of Economics: A Mathematical Analysis |edition=3rd |publisher=McGraw-Hill |location=Boston |isbn=0071181369 }}
*{{Cite book|title=An introduction to positive economics|edition=fourth|pages=214–7|first=Richard G.|last=Lipsey|year=1975|publisher=[[Weidenfeld & Nicolson]] | isbn=0-297-76899-9}}

== External links ==
*[http://www2.hawaii.edu/~fuleky/anatomy/anatomy.html Anatomy of Cobb-Douglas Type Utility Functions in 3D]
*[http://www2.hawaii.edu/~fuleky/anatomy/anatomy2.html Anatomy of CES Type Utility Functions in 3D]

[[Category:Consumer theory]]
[[Category:Household behavior and family economics]]
[[Category:Microeconomics]]
[[Category:Economics curves]]
[[Category:Utility]]

5040 (number)

2014-01-30T17:19:20Z

2.25.131.166:

In [[statistics]], a '''linear probability model''' is a special case of a [[binomial regression]] model. Here the [[dependent and independent variables|observed variable]] for each observation takes values which are either 0 or 1. The probability of observing a 0 or 1 in any one case is treated as depending on one or more [[dependent and independent variables|explanatory variables]]. For the "linear probability model", this relationship is a particularly simple one, and allows the model to be fitted by [[simple linear regression]].

The model assumes that, for a binary outcome ([[Bernoulli trial]]), <math>Y</math>, and its associated vector of explanatory variables, <math>X</math>,<ref name=Cox>{{cite book |last=Cox |first=D. R. |year=1970 |title=Analysis of Binary Data |location=London |publisher=Methuen |isbn=0-416-10400-2 |chapter=Simple Regression |pages=33–42 }}</ref>

: <math> \Pr(Y=1 | X=x) = x'\beta . </math>

For this model,
:<math> E[Y|X] = \Pr(Y=1|X) =x'\beta,</math>
and hence the vector of parameters β can be estimated using [[least squares]]. This method of fitting would be [[Efficiency (statistics)|inefficient]].<ref name=Cox /> This method of fitting can be improved by adopting an iterative scheme based on [[weighted least squares]],<ref name=Cox/> in which the model from the previous iteration is used to supply estimates of the conditional variances, <math>Var(Y|X=x)</math>, which would vary between observations. This approach can be related to fitting the model by [[maximum likelihood]].<ref name=Cox/>

A drawback of this model for the parameter of the [[Bernoulli distribution]] is that, unless restrictions are placed on <math> \beta </math>, the estimated coefficients can imply probabilities outside the [[unit interval]] <math> [0,1] </math>. For this reason, models such as the [[logit model]] or the [[probit model]] are more commonly used.

== References ==
{{reflist}}

== Further reading ==
* {{cite book |last=Wooldridge |first=Jeffrey M. |year=2013 |title=Introductory Econometrics: A Modern Approach |location=Mason, OH |publisher=South-Western |edition=5th international |chapter=A Binary Dependent Variable: The Linear Probability Model |pages=238–243 |isbn=978-1-111-53439-4 }}

{{DEFAULTSORT:Linear Probability Model}}
[[Category:Generalized linear models]]

Mixed radix

2014-01-30T16:29:03Z

2.25.131.166: /* Examples */

{{Evolutionary biology}}
The '''molecular clock ''' (based on the '''molecular clock hypothesis''' ('''MCH''')) is a technique in [[molecular evolution]] that uses fossil constraints and rates of molecular change to deduce the time in [[Geologic time scale|geologic history]] when two [[species]] or other [[Taxon|taxa]] [[Genetic divergence|diverged]]. It is used to estimate the time of occurrence of events called [[speciation]] or [[Evolutionary radiation|radiation]]. The molecular data used for such calculations is usually [[nucleotide]] [[DNA sequence|sequences]] for [[DNA]] or [[amino acid]] sequences for [[protein]]s. It is sometimes called a '''gene clock''' or '''evolutionary clock'''.

==Early discovery and genetic equidistance==
The notion of the existence of a so-called "molecular clock" was first attributed to [[Emile Zuckerkandl]] and [[Linus Pauling]] who, in 1962, noticed that the number of [[amino acid]] differences in [[hemoglobin]] between different lineages changes roughly [[Linear function|linearly]] with time, as estimated from fossil evidence.<ref name=Zuckerkand62>{{cite book | author = [[Emile Zuckerkandl|Zuckerkandl, E.]] and [[Linus Pauling|Pauling, L.B.]] | year = 1962 | title = Horizons in Biochemistry | chapter = Molecular disease, evolution, and genic heterogeneity | editor = Kasha, M. and Pullman, B (editors) | pages = 189–225 | publisher = Academic Press, New York}}</ref> They generalized this observation to assert that the rate of [[evolution]]ary change of any specified [[protein]] was approximately constant over time and over different lineages.

The '''genetic equidistance''' phenomenon was first noted in 1963 by E. Margoliash, who wrote: "It appears that the number of residue differences between [[cytochrome C]] of any two species is mostly conditioned by the time elapsed since the lines of evolution leading to these two species originally diverged. If this is correct, the cytochrome c of all mammals should be equally different from the cytochrome c of all birds. Since fish diverges from the main stem of vertebrate evolution earlier than either birds or mammals, the cytochrome c of both mammals and birds should be equally different from the cytochrome c of fish. Similarly, all vertebrate cytochrome c should be equally different from the yeast protein."<ref>{{cite journal |doi=10.1073/pnas.50.4.672 |author=Margoliash E |title=PRIMARY STRUCTURE AND EVOLUTION OF CYTOCHROME C |journal=Proc. Natl. Acad. Sci. U.S.A. |volume=50 |issue= 4|pages=672–9 |date=October 1963 |pmid=14077496 |pmc=221244}}</ref> For example, the difference between the cytochrome C of a carp and a frog, turtle, chicken, rabbit, and horse is a very constant 13% to 14%. Similarly, the difference between the cytochrome C of a bacterium and yeast, wheat, moth, tuna, pigeon, and horse ranges from 64% to 69%. Together with the work of Emile Zuckerkandl and Linus Pauling, the genetic equidistance result directly led to the formal postulation of the molecular clock hypothesis in the early 1960s.<ref>{{cite journal |author=Kumar S |title=Molecular clocks: four decades of evolution |journal=Nat. Rev. Genet. |volume=6 |issue=8 |pages=654–62 |date=August 2005 |pmid=16136655 |doi=10.1038/nrg1659}}</ref> Genetic equidistance has often been used to infer equal time of separation of different sister species from an outgroup.<ref>{{cite journal |doi=10.1007/PL00006487 |author=Pesole G, Gissi C, De Chirico A, Saccone C |title=Nucleotide substitution rate of mammalian mitochondrial genomes |journal=J. Mol. Evol. |volume=48 |issue=4 |pages=427–34 |date=April 1999 |pmid=10079281}}</ref><ref>Huang, S. (2008) The genetic equidistance result of molecular evolution is independent of mutation rates. J. Comp. Sci. Syst. Biol., 1: 92-102. http://omicsonline.com/ArchiveJCSB/Ab01/JCSB1.092.html</ref>

Later [[Allan Wilson]] and [[Vincent Sarich]] built upon this work.

==Relationship with neutral theory==

The observation of a clock-like rate of molecular change was originally purely [[Phenomenology (science)|phenomenological]]. Later, the work of [[Motoo Kimura]]<ref name=Kimura68>{{cite journal |last=Kimura |first=Motoo |year=1968 |title= Evolutionary rate at the molecular level |journal= Nature |volume=217 |pages=624–626 |doi=10.1038/217624a0 |pmid=5637732 |issue=5129}}</ref> developed the [[neutral theory of molecular evolution]], which predicted a molecular clock. Let there be N individuals, and to keep this calculation simple, let the individuals be [[Ploidy|haploid]] (i.e. have one copy of each gene). Let the rate of neutral [[mutation]]s (i.e. mutations with no effect on [[Fitness (biology)|fitness]]) in a new individual be <math>\mu</math>. The probability that this new mutation will become [[Fixation (population genetics)|fixed]] in the population is then 1/N, since each copy of the gene is as good as any other. Every generation, each individual can have new mutations, so there are <math>\mu</math>N new neutral mutations in the population as a whole. That means that each generation, <math>\mu</math> new neutral mutations will become fixed. If most changes seen during [[molecular evolution]] are neutral, then [[Fixation (population genetics)|fixations]] in a population will accumulate at a clock-rate that is equal to the rate of neutral [[mutation]]s in an individual.

==Calibration==
The molecular clock alone can only say that one time period is twice as long as another: it cannot assign concrete dates. To achieve this, the molecular clock must first be [[calibration|calibrated]] against independent evidence about dates, such as the [[fossil]] record.<ref name=Benton01>{{cite journal |author=benton, M. J. and Donoghue, P. C. J. |year=200t |title= Paleontological evidence to date the Tree of Life|journal=Molecular Biology & Evolution|volume= 24|pages=26–53 |doi=10.1093/molbev/msl150 |pmid=17047029 |issue=1}}</ref> Alternatively, for viral phylogenetics and [[ancient DNA]] studies, two areas of evolutionary biology where it is possible to sample sequences over an evolutionary timescale, the dates of the samples themselves can be used to calibrate the molecular clock.

==Non-constant rate of molecular clock==
Sometimes only a single divergence date can be estimated from fossils, with all other dates inferred from that. Other sets of species have abundant fossils available, allowing the MCH of constant divergence rates to be tested. DNA sequences experiencing low levels of [[negative selection (natural selection)|negative selection]] showed divergence rates of 0.7-0.8% per [[Myr]] in bacteria, mammals, invertebrates, and plants.<ref name=Ochman87>{{cite journal |author=Ochman H, Wilson AC. |journal=J Mol Evol.| year=1987| pages=74–86.|title=Evolution in bacteria: evidence for a universal substitution rate in cellular genomes | volume=26|doi=10.1007/BF02111283 |pmid=3125340 |issue=1–2}}</ref> In the same study, genomic regions experiencing very high negative or purifying selection (encoding rRNA) were considerably slower (1% per 50 Myr).

In addition to such variation in rate with genomic position, since the early 1990s, variation among taxa has proven fertile ground for research too,<ref name=Douzery03>{{cite journal |author=Douzery, E.J.P., Delsuc, F., Stanhope, M.J. and Huchon, D. |year=2003 |title= Local molecular clocks in three nuclear genes: divergence times for rodents and other mammals, and incompatibility among fossil calibrations|journal=Journal of Molecular Evolution |volume= 57|pages=S201–S213|url= |doi=10.1007/s00239-003-0028-x |pmid=15008417}}</ref> even over comparatively short periods of evolutionary time (for example [[mockingbird]]s<ref name=Hunt01>{{cite journal |author=Hunt, J.S., Bermingham, E., and Ricklefs, R.E. |year=2001 |title=Molecular systematics and biogeography of Antillean thrashers, tremblers, and mockingbirds (Aves: Mimidae) |journal=[[Auk (journal)|Auk]] |volume=118 |issue=1 |pages=35–55|url=http://findarticles.com/p/articles/mi_qa3793/is_200101/ai_n8930531 |doi=10.1642/0004-8038(2001)118[0035:MSABOA]2.0.CO;2 |issn=0004-8038 }}</ref>). [[Procellariiformes|Tube-nosed seabirds]] have molecular clocks that on average run at half speed of many other birds,<ref name=Rheindt05>{{cite journal |author=Rheindt, F. E. and Austin, J. |year=2005 |title=Major analytical and conceptual shortcomings in a recent taxonomic revision of the Procellariiformes - A reply to Penhallurick and Wink (2004) |journal=[[Emu (journal)|Emu]] |volume=105 |issue=2 |pages=181–186|url=http://www.publish.csiro.au/?act=view_file&file_id=MU04039.pdf |doi=10.1071/MU04039 }}</ref> possibly due to long [[generation]] times, and many turtles have a molecular clock running at one-eighth the speed it does in small mammals or even slower.<ref name=Avise92>{{cite journal |author= Avise, J.C., Bowen, W., Lamb, T., Meylan, A.B. and Bermingham, E. |date=1 May 1992|title= Mitochondrial DNA Evolution at a Turtle's Pace: Evidence for Low Genetic Variability and Reduced Microevolutionary Rate in the Testudines |journal= Molecular Biology and Evolution |volume= 9|issue= 3|pages=457–473|url= http://mbe.oxfordjournals.org/cgi/reprint/9/3/457 |pmid= 1584014 }}</ref> Effects of [[small population size]] are also likely to confound molecular clock analyses; [[cheetah]]s for example, having gone through at least 2 [[population bottleneck]]s, could not be adequately studied based on a molecular clock model alone.{{Citation needed|date=January 2008}} Researchers such as Francisco Ayala have more fundamentally challenged the molecular clock hypothesis.<ref name=Ayala99>{{cite journal |author= Ayala, F.J.|year=1999 |title=Molecular clock mirages |journal=[[BioEssays]] |volume=21|issue=1 |pages=71–75|url=http://www3.interscience.wiley.com/cgi-bin/abstract/60000186/ABSTRACT?CRETRY=1&SRETRY=0 |doi=10.1002/(SICI)1521-1878(199901)21:1<71::AID-BIES9>3.0.CO;2-B |pmid=10070256}}</ref><ref name=Schwartz06>{{cite journal |author=Schwartz, J. H. and Maresca, B. |year=2006 |title=Do Molecular Clocks Run at All? A Critique of Molecular Systematics |journal=Biological Theory |volume=1 |pages=357–371|url= |doi=10.1162/biot.2006.1.4.357|laysummary=http://www.sciencedaily.com/releases/2007/02/070210170623.htm|laysource=[[Science Daily]] |issue=4}}</ref> According to Ayala's 1999 study, 5 factors combine to limit the application of molecular clock models:

* Changing generation times (If the rate of new mutations depends at least partly on the number of generations rather than the number of years)
* Population size ([[Genetic drift]] is stronger in small populations, and so more mutations are effectively neutral)
* Species-specific differences (due to differing metabolism, ecology, evolutionary history,...)
* Change in function of the protein studied (can be avoided in closely related species by utilizing [[non-coding DNA]] sequences or emphasizing [[silent mutation]]s)
* Changes in the intensity of natural selection.

Molecular clock users have developed workaround solutions using a number of statistical approaches including [[maximum likelihood]] techniques and later [[Bayesian statistics|Bayesian modeling]]. In particular, models that take into account rate variation across lineages have been proposed in order to obtain better estimates of divergence times. These models are called '''relaxed molecular clocks'''<ref name=Drummond06>{{cite journal |author=Drummond, A.J., Ho, S.Y.W., Phillips, M.J. and Rambaut A. |year=2006 |title= Relaxed Phylogenetics and Dating with Confidence|journal= [[Public Library of Science|PLoS Biology]] |volume= 4|issue=5 |pages=e88|doi= 10.1371/journal.pbio.0040088 |pmid=16683862 |pmc=1395354}}</ref> because they represent an intermediate position between the 'strict' molecular clock hypothesis and Felsenstein's many-rates model and are made possible through [[Markov chain Monte Carlo|MCMC]] techniques that explore a weighted range of tree topologies and simultaneously estimate parameters of the chosen substitution model. It must be remembered that divergence dates inferred using a molecular clock are based on statistical [[inference]] and not on direct [[evidence]].

The molecular clock runs into particular challenges at very short and very long timescales. At long timescales, the problem is [[Saturation (genetic)|saturation]]. When enough time has passed, many sites have undergone more than one change, but it is impossible to detect more than one. This means that the observed number of changes is no longer [[Linear function|linear]] with time, but instead flattens out.

At very short time scales, many differences between samples do not represent [[Fixation (population genetics)|fixation]] of different sequences in the different populations. Instead, they represent alternative [[alleles]] that were both present as part of a polymorphism in the common ancestor. The inclusion of differences that have not yet become [[Fixation (population genetics)|fixed]]
leads to a potentially dramatic inflation of the apparent rate of the molecular clock at very short timescales.<ref>{{cite journal |doi=10.1093/molbev/msi145 |author=Ho SYW, Phillips MJ, Cooper A, Drummond AJ |year=2005 |title= Time dependency of molecular rate estimates and systematic overestimation of recent divergence times |journal= Molecular Biology & Evolution |volume= 22 |issue=7 |pages=1561–1568 |pmid=15814826}}</ref><ref>{{cite journal |author=Peterson GI, Masel J| year=2009 |title= Quantitative Prediction of Molecular Clock and Ka/Ks at Short Timescales |journal= Molecular Biology & Evolution |pmid=19661199 |volume= 26 |issue=11 |pmc=2912466 |pages=2595–2603| doi=10.1093/molbev/msp175}}</ref>

== Methods ==

Bayesian methods can provide more appropriate estimates of divergence times, especially if large datasets - such as those yielded by [[phylogenomics]] - are employed.<ref name="Dos Reis2012">{{cite doi|10.1098/rspb.2012.0683}}</ref>

==Uses==
The molecular clock technique is an important tool in [[molecular systematics]], the use of [[molecular genetics]] information to determine the correct [[scientific classification]] of organisms or to study variation in selective forces.

Knowledge of approximately constant rate of molecular evolution in particular sets of lineages also facilitates establishing the dates of [[phylogeny|phylogenetic]] events, including those not documented by [[fossils]], such as the divergence of living [[taxon|taxa]] and the formation of the [[phylogenetic tree]]. But in these cases — especially over long stretches of time — the limitations of MCH (above) must be considered;  such estimates may be off by 50% or more.

==See also==
* [[Gene orders]]
* [[Human mitochondrial molecular clock]]
* [[Mitochondrial Eve]] and [[Y-chromosomal Adam]]
* [[Models of DNA evolution]]
* [[Molecular evolution]]
* [[Neutral theory of molecular evolution]]

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

==Further reading==
* {{Cite journal |author=Morgan, G.J. |year=1998 |title=Emile Zuckerkandl, Linus Pauling, and the Molecular Evolutionary Clock, 1959-1965 |journal=Journal of the History of Biology |volume=31 |issue=2 |pages=155–178 |doi=10.1023/A:1004394418084|url=http://www.springerlink.com/content/l1723tq0q2751215/fulltext.pdf |pmid=11620303}}
* {{Cite book | author = [[Emile Zuckerkandl|Zuckerkandl, E.]] and [[Linus Pauling|Pauling, L.B.]] | year = 1965 | title = Evolving Genes and Proteins | chapter = Evolutionary divergence and convergence in proteins | editor = Bryson, V.and Vogel, H.J. (editors) | pages = 97–166 | publisher = Academic Press, New York}}
*{{cite journal
| last = San Mauro
| first = D.
| title = Molecular systematics: a synthesis of the common methods and the state of knowledge
| journal = Cellular & Molecular Biology Letters
| volume = 15
| pages = 311–341
| year = 2010
| coauthors = Agorreta, A.
| doi=10.2478/s11658-010-0010-8
| issue = 2
}}

== External links ==
* [http://web.archive.org/web/20100525101859/http://awcmee.massey.ac.nz/aw.htm Allan Wilson and the molecular clock]
* [http://web.archive.org/web/20090213150149/http://rtis.com/nat/user/elsberry/evobio/evc/argresp/sequence.html Molecular clock explanation of the molecular equidistance phenomenon]
* [http://www.fossilrecord.net/dateaclade/index.html Date-a-Clade service for the molecular tree of life]

{{Chronology}}
{{Use dmy dates|date=August 2010}}

{{DEFAULTSORT:Molecular Clock}}
[[Category:Molecular evolution]]
[[Category:Molecular genetics]]
[[Category:Phylogenetics]]

Non-standard positional numeral systems

2014-01-30T16:27:35Z

2.25.131.166: /* Mixed bases */

In the mathematical discipline of [[descriptive set theory]], a '''coanalytic set''' is a [[Set (mathematics)|set]] (typically a set of [[real number]]s or more generally a subset of a [[Polish space]]) that is the complement of an [[analytic set]] (Kechris 1994:87). Coanalytic sets are also referred to as <math>\scriptstyle\boldsymbol{\Pi}^1_1</math> sets (see [[projective hierarchy]]).
==References==
* {{citation | authorlink=Alexander Kechris | author=Kechris, Alexander S. | title=Classical Descriptive Set Theory | publisher=Springer-Verlag | year=1994 | isbn=0-387-94374-9}}

[[Category:Descriptive set theory]]

{{settheory-stub|date=March 2006}}

Kaprekar number

2014-01-30T15:53:29Z

2.25.131.166: redundant, as the example of 297 is explained in the next sentence

[[Image:SubtangentDiagram.svg|thumb|300px|right|Subtangent and related concepts for a curve ('''black''') at a given point ''P''. The tangent and normal lines are shown in green and blue respectively. The distances shown are the '''ordinate''' (''AP''), '''tangent''' (''TP''), '''subtangent''' (''TA''), '''normal''' (''PN''), and '''subnormal''' (''AN''). The angle φ is the angle of inclination of the tangent line or the tangential angle.]]
In [[geometry]], the '''subtangent''' and related terms are certain line segments defined using the line [[tangent]] to a curve at a given point and the [[Cartesian coordinate system|coordinate axes]]. The terms are somewhat archaic today but were in common use until the early part of the 20th century.

==Definitions==
Let ''P'' = (''x'', ''y'') be a point on a given curve with ''A'' = (''x'', 0) its projection onto the ''x''-axis. Draw the tangent to the curve at ''P'' and let ''T'' be the point where this line intersects the ''x''-axis. Then ''TA'' is defined to be the '''subtangent''' at ''P''. Similarly, if normal to the curve at ''P'' intersects the ''x''-axis at ''N'' then ''AN'' is called the '''subnormal'''. In this context, the lengths ''PT'' and ''PN'' are called the '''tangent''' and '''normal''', not to be confused with the [[Tangent|tangent line]] and the normal line which are also called the tangent and normal.

==Equations==
Let ''φ'' be the angle of inclination of the tangent with respect to the ''x''-axis; this is also known as the [[tangential angle]]. Then
:<math>\tan\varphi=\frac{dy}{dx}=\frac{AP}{TA}=\frac{AN}{AP}.</math>
So the subtangent is
:<math>y\cot\varphi=\frac{y}{\tfrac{dy}{dx}},</math>
and the subnormal is
:<math>y\tan\varphi=y\frac{dy}{dx}.</math>
The normal is given by
:<math>y\sec\varphi=y\sqrt{1+\left(\frac{dy}{dx}\right)^2},</math>
and the tangent is given by
:<math>y\csc\varphi=\frac{y}{\tfrac{dy}{dx}}\sqrt{1+\left(\frac{dy}{dx}\right)^2}.</math>

==Polar definitions==
[[Image:PolarSubtangentDiagram.svg|thumb|250px|right|Polar subtangent and related concepts for a curve ('''black''') at a given point ''P''. The tangent and normal lines are shown in green and blue respectively. The distances shown are the '''radius''' (''OP''), '''polar subtangent''' (''OT''), and '''polar subnormal''' (''ON''). The angle θ is the radial angle and the angle ψ of inclination of the tangent to the radius or the polar tangential angle.]]
Let ''P'' = (''r'', θ) be a point on a given curve defined by [[polar coordinates]] and let ''O'' denote the origin. Draw a line through ''O'' which is perpendicular to ''OP'' and let ''T'' now be the point where this line intersects the tangent to the curve at ''P''. Similarly, let ''N'' now be the point where the normal to the curve intersects the line. Then ''OT'' and ''ON'' are, respectively, called the '''polar subtangent''' and '''polar subnormal''' of the curve at ''P''.

==Polar equations==
Let ''ψ'' be the angle between the tangent and the ray ''OP''; this is also known as the polar tangential angle. Then
:<math>\tan\psi=\frac{r}{\tfrac{dr}{d\theta}}=\frac{OP}{ON}=\frac{OT}{OP}.</math>
So the polar subtangent is
:<math>r\tan\psi=\frac{r^2}{\tfrac{dr}{d\theta}},</math>
and the subnormal is
:<math>r\cot\psi=\frac{dr}{d\theta}.</math>

==References==
*{{cite book | author=J. Edwards | title=Differential Calculus
| publisher= MacMillan and Co.| location=London | pages=150, 154| year=1892
|url=http://books.google.com/books?id=unltAAAAMAAJ&pg=PA150#v=onepage&q&f=false}}

* B. Williamson "Subtangent and Subnormal" and "Polar Subtangent and Polar Subnormal" in ''An elementary treatise on the differential calculus'' (1899) p 215, 223 [http://books.google.com/books?id=vYkAAAAAMAAJ&pg=PA215 Google books]

[[Category:Curves]]

Triacontagon

2014-01-14T15:59:35Z

2.25.134.14:

In [[mathematics]], the '''Kantorovich inequality''' is a particular case of the [[Cauchy-Schwarz inequality]], which is itself a generalization of the [[triangle inequality]].

The triangle inequality states that the length of two sides of any triangle, added together, will be equal to or greater than the length of the third side. In simplest terms, the Kantorovich inequality translates the basic idea of the triangle inequality into the terms and notational conventions of [[linear programming]]. (See [[vector space]], [[inner product]], and [[normed vector space]] for other examples of how the basic ideas inherent in the triangle inequality--line segment and distance--can be generalized into a broader context.)

More formally, the Kantorovich inequality can be expressed this way:

:Let

:: <math>p_i \geq 0,\quad 0 < a \leq x_i \leq b\text{ for }i=1, \dots ,n.</math>

:Let <math>A_n=\{1,2,\dots ,n\}.</math>

:Then

:: <math>
\begin{align}
& {} \qquad \left( \sum_{i=1}^n p_ix_i \right ) \left (\sum_{i=1}^n \frac{p_i}{x_i} \right) \\
& \leq \frac{(a+b)^2}{4ab} \left (\sum_{i=1}^n p_i \right )^2
-\frac{(a-b)^2}{4ab} \cdot \min \left\{ \left (\sum_{i \in X}p_i-\sum_{j \in Y}p_j \right )^2\,:\, {X \cup Y=A_n},{X \cap Y=\varnothing} \right\}.
\end{align}
</math>

The Kantorovich inequality is used in [[convergence analysis]]; it bounds the convergence rate of Cauchy's [[steepest descent]].

Equivalents of the Kantorovich inequality have arisen in a number of different fields. For instance, the [[Bunyakovsky inequality]], the [[Wielandt inequality]], and the [[Cauchy–Schwarz inequality]] are equivalent to the Kantorovich inequality and all of these are, in turn, special cases of the [[Hölder inequality]].

The Kantorovich inequality is named after Soviet economist, mathematician, and [[Nobel Prize]] winner [[Leonid Kantorovich]], a pioneer in the field of [[linear programming]].

==References==
* {{MathWorld|urlname=KantorovichInequality|title=Kantorovich Inequality}}

* {{PlanetMath|urlname=KantorovichInequality|title=Cauchy-Schwarz inequality}}

* [http://carbon.cudenver.edu/~hgreenbe/glossary/index.php?page=K.html Mathematical Programming Glossary entry on "Kantorovich inequality"]

==External links==
*[http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Kantorovich.html Biography of Leonid Vitalyevich Kantorovich]

[[Category:Mathematical analysis]]
[[Category:Inequalities]]

Natural logarithm of 2

2013-11-25T20:36:12Z

2.25.120.161: ln should be used for natural logarithm

[[File:Bulk BTO PFM scan.png|thumb|alt=Piezoresponse Force Microscopy image of BaTiO3 domains|PFM of [[barium titanate|BaTiO3]] single crystal showing simultaneously acquired topography (top) and domain structure (bottom). The scale bar is 10 μm]]
'''Piezoresponse force microscopy''' (PFM) is a variant of [[atomic force microscopy]] (AFM) that allows imaging and manipulation of [[ferroelectric]] domains. This is achieved by bringing a sharp conductive probe into contact with a ferroelectric surface (or [[piezoelectric]] material) and applying an [[alternating current]] (AC) bias to the probe tip in order to excite deformation of the sample through the converse [[piezoelectric effect]] (CPE). The resulting deflection of the probe cantilever is detected through standard split [[photodiode]] detector methods and then demodulated by use of a [[lock-in amplifier]] (LiA). In this way topography and ferroelectric domains can be imaged simultaneously with high resolution.

==Basic principles==

===General overview===
Piezoresponse force microscopy is a technique which since its inception and first implementation<ref>{{cite journal|last1=Gruverman|first1=A.|doi=10.1116/1.587909|title=Domain structure and polarization reversal in ferroelectrics studied by atomic force microscopy|year=1995|pages=1095|volume=13|journal=Journal of Vacuum Science and Technology B|url=http://digitalcommons.unl.edu/cgi/viewcontent.cgi?article=1034&context=physicsgruverman&sei-redir=1#search=%22Domain+structure+and+polarization+reversal+in+ferroelectrics+studied+by+atomic+force+microscopy%22|bibcode = 1995JVSTB..13.1095G }}</ref> has steadily attracted more and more interest. This is due in large part to the many benefits and few drawbacks that PFM offers researchers in varying fields from ferroelectrics, semiconductors and even biology.<ref>{{cite journal|last1=Rodriguez|first1=B.J.|last2=Kalinin|first2=S.V.|last3=Shin|first3=J.|last4=Jesse|first4=S.|last5=Grichko|first5=V.|last6=Thundat|first6=T.|last7=Baddorf|first7=A.P.|last8=Gruverman|first8=A.|doi=10.1016/j.jsb.2005.10.008|title=Electromechanical imaging of biomaterials by scanning probe microscopy|pmid=16403652|year=2006|pages=151|issue=2|volume=153|journal=Journal of Structural Biology|url=http://physics.unl.edu/~agruverman/pubs/AG%207%20%28JSB%29.pdf}}</ref> In its most common format PFM allows for identification of domains from relatively large scale e.g. 100×100 µm2 scans right down to the nanoscale with the added advantage of simultaneous imaging of sample surface topography. Also possible is the ability to switch regions of ferroelectric domains with the application of a sufficiently high bias to the probe which opens up the opportunity of investigating domain formation on nanometre length scales with nanosecond time resolution.<ref>{{cite journal|last1=Kalinin|first1=Sergei V|last2=Morozovska|first2=Anna N|last3=Chen|first3=Long Qing|last4=Rodriguez|first4=Brian J|title=Local polarization dynamics in ferroelectric materials|journal=Reports on Progress in Physics|volume=73|pages=056502|year=2010|doi=10.1088/0034-4885/73/5/056502|bibcode = 2010RPPh...73e6502K }}</ref> Many recent advances have expanded the list of applications for PFM and further increased this powerful technique. Indeed what started as a user modified AFM has now attracted the attention of the major SPM manufacturers so much so that in fact many now supply ‘ready-made’ systems specifically for PFM each with novel features for research. This is testament to the growth of the field and reflects the numbers of users throughout the scientific world who are at the forefront of scientific research.

[[File:Piezoresponse of parallel and antiparallel domains.png|thumb|550px|left|alt=Diagram showing cantilever movements from mechanical deformation of piezoelectric domains|Top line shows an in-phase piezoresponse to the driving voltage and the bottom line shows a 180° out-of-phase piezoresponse to driving voltage. Alignment of electric field and polarisation orientation (top right) results in an expansion of the domain, giving a positive deflection as measured by the photodiode. When the bias is negative the domain contracts giving a negative deflection as measured by the photodiode meaning that the piezoresponse will always be in-phase with the driving voltage. For anti-alignment of electric field and polarisation orientation (bottom right) a positive bias results in a contraction of the domain and so gives a negative deflection as measured by the photodiode therefore the piezoresponse is 180° out-of-phase with the driving voltage. In this way the orientation of polarisation within a domain can be observed.]]

Consider that a static or DC voltage applied to a piezoelectric surface will produce a displacement but as applied fields are quite low and the piezoelectric tensor coefficients are relatively small then the physical displacement will also be small such that it is below the level of possible detection of the system. Take as an example, the ''d33'' piezoelectric tensor coefficient of BaTiO3, it has a value of 85.6 [[picometre|pm]][[Volt|V−1]] meaning that applying 1 V across the material results in a displacement of 85.6 pm or 0.0856 [[nanometre|nm]], a minute cantilever displacement even for the high precision of AFM deflection detection. In order to separate this low level signal from random noise a lock-in technique is used wherein a modulated voltage reference signal,

: <math>V(\boldsymbol\omega)=V_{\mathrm{ac}}\cos(\boldsymbol\omega t) \;</math>

of frequency ''ω'' and amplitude ''Vac'' is applied to the tip giving rise to an oscillatory deformation of the sample surface,

: <math>d=d_{\mathrm{0}}+D\cos(\boldsymbol\omega t + \boldsymbol\varphi) \;</math>

from the equilibrium position ''d0'' with amplitude ''D'', and an associated phase difference ''φ''. The resulting movement of the cantilever is detected by the photodiode and so an oscillating surface displacement is converted into an oscillating voltage. A lock-in-amplifier (LiA) is then able to retrieve the amplitude and phase of the CPE induced surface deformation by the process outlined below.

===Converse piezoelectric effect===
The converse piezoelectric effect (CPE) describes how an applied [[electric field]] will create a resultant [[Deformation (mechanics)|strain]] which in turn leads to a physical deformation of the material. This effect can be described through the constitutive equations.<ref>Rosen, C.Z., Hiremath, B.V., Newnham, R. (ed) "Piezoelectricity" American Institute of Physics, Key Papers in Physics, No 5, 227–283 (1992)</ref> The CPE can be written as

: <math>X_{\mathrm{i}}=d_{\mathrm{ki}}E_{\mathrm{k}} \;</math>

where ''Xi'' is the strain tensor, ''dki'' is the piezoelectric tensor, and ''Ek'' is the electric field. If the piezoelectric tensor is considered to be that of the tetragonal crystal system (that of BaTiO3) then it is

:<math>
\begin{bmatrix} X_1 \\ X_2 \\ X_3 \\ X_4 \\ X_5 \\ X_6 \end{bmatrix}
=

\begin{bmatrix} 0 & 0 & d_{31} \\
0 & 0 & d_{31} \\
0 & 0 & d_{33} \\
0 & d_{15} & 0 \\
d_{15} & 0 & 0 \\
0 & 0 & 0 \end{bmatrix}
\begin{bmatrix} E_1 \\ E_2 \\ E_3 \end{bmatrix}
</math>

such that the equation will lead to the strain components for an applied field. If the field is applied exclusively in one direction i.e. ''E3'' for example, then the resulting strain components are: ''d31E3'', ''d31E3'', ''d33E3''

Thus for an electric field applied along the c-axis of BaTiO3 i.e. ''E3'', then the resulting deformation of the crystal will be an elongation along the c-axis and an axially symmetric contraction along the other orthogonal directions. PFM uses the effect of this deformation to detect domains and also to determine their orientation.

===Conductive probe===
The most important property of the probe for use in PFM is that it should be conducting. This is generally required in order to provide a means of applying a bias to the sample, and can be achieved through manufacturing standard silicon probes and coating them in a conductive material. Common coatings are [[platinum]], [[gold]], [[tungsten]] and even [[diamond|conductive diamond]].

[[File:Conductive probe SEM images.png|thumb|center|700px|alt=Scanning electron microscope images of increasing magnification of a conductive coated scanning probe| [[Scanning electron microscopy]] images of a PtIr5 coated scanning probe. From left to right shows images of increasing magnification where the scale bar in the first image is 50 μm and in the third is 200 nm. The first image shows the substrate, cantilever and the tip whereas the second image shows the tip geometry whilst the last image shows the tip apex and demonstrates the fine point that is achieved e.g. radius of curvature of less than 40 nm.]]

===Lock-in amplifier===
In the general case a LiA ‘compares’ an ''input signal'' against that of a ''reference signal'' (either generated internally or supplied by an external function generator) in order to separate the information contained in the input signal at the frequency of the reference signal. This is called ''[[demodulation]]'' and is done in a number of easy steps. The reference signal <math>\scriptstyle V_\mathrm{ref}=B\cos(\boldsymbol\omega t)</math>, and input signal, <math>\scriptstyle V_\mathrm{in}=A\cos(\boldsymbol\omega t + \boldsymbol\varphi)</math>, are multiplied together to give the ''demodulator output'',

<math>V_\mathrm{out}=\frac{1}{2}AB\cos(\boldsymbol\varphi)+ \frac{1}{2}AB\cos(2\boldsymbol\omega t +\boldsymbol\varphi)</math>

where ''A'' is the input signal Amplitude and ''B'' is the reference signal Amplitude, ''ω'' is the frequency of both the reference and input signals, and ''φ'' is any phase shift between the two signals.

The above equation has an AC component at twice the frequency of the original signals (second term) and a DC component (first term) whose value is related to both the amplitude and phase of the input signal. The demodulator output is sent through a low-pass filter to remove the 2''ω'' component and leave the DC component then the signal is integrated over a period of time defined as the ''Time Constant'', ''τLiA'' which is a user-definable parameter. Several different outputs are commonly available from a LiA: ''X'' output is the demodulator output and ''Y'' is the second demodulator output which is shifted by 90° in reference to the first output, together they hold both the phase, ''θ'', and magnitude, ''R'', information and are given by

<math>X=\frac{1}{2}AB\cos(\boldsymbol\theta)</math> and

<math>Y=\frac{1}{2}AB\cos(\boldsymbol\theta +\frac{\pi}{2})=\frac{1}{2}AB\sin(\boldsymbol\theta) </math>

However, phase and amplitude of the input signal can also be calculated and outputted from the LiA if desired, so that the full amount of information is available. The phase output can be determined from the following equation:

<math>\boldsymbol\theta=\tan^{-1}\frac{Y}{X}</math>

The magnitude is then given by:

<math>R=\sqrt{X^2+Y^2}</math>

This allows ''R'' to be calculated even if the input signal differs in phase from the reference signal.

===Differentiating vertical and lateral PFM signals===
[[File:Cantilever movements and optical deflections.png|thumb|300px|alt=Diagram showing cantilever dynamics and the optical detection through AFM split photodiode detector|Diagrams showing the effect of cantilever movement with the photodetector represented by the square with quadrants labelled A, B, C and D. Torsional bending of the cantilever (left) leads to a change in lateral deflection and (right) vertical displacement of the cantilever leads to a change in vertical deflection]]

A basic interpretation of PFM (which is generally accepted) identifies that two modes of imaging are possible, one that is sensitive to out-of-plane and one to in-plane piezoresponse, termed, vertical and lateral PFM (VPFM and LPFM) respectively.<ref>{{cite journal|last1=Kalinin|first1=SV|last2=Rodriguez|first2=BJ|last3=Jesse|first3=S|last4=Shin|first4=J|last5=Baddorf|first5=AP|last6=Gupta|first6=P|last7=Jain|first7=H|last8=Williams|first8=DB|last9=Gruverman|first9=A|title=Vector piezoresponse force microscopy|journal=Microscopy and microanalysis : the official journal of Microscopy Society of America, Microbeam Analysis Society, Microscopical Society of Canada|volume=12|issue=3|pages=206–20|year=2006|pmid=17481357|doi=10.1017/S1431927606060156|bibcode = 2006MiMic..12..206K }}</ref> The separation of these components is possible through the use of a split photodiode detector, standard to all optical detection AFM systems. In this setup the detector is split into quadrants, nominally A, B, C and D. The centre of the entire detector outputs 0 V but as the laser spot moves a radial distance from this centre point the magnitude of the outputted voltage will increase linearly. A vertical deflection can be defined as {(A+B)-(C+D)}/(ABCD) so that now positive and negative voltages are ascribed to positive and negative cantilever vertical displacements. Similarly a lateral deflection is defined as {(B+D)-(A+C)}/(ABCD) to describe positive and negative torsional movements of the cantilever. So VPFM will utilise the vertical deflection signal from the photodiode detector so will only be sensitive to out-of-plane polar components and LPFM will utilise the lateral deflection signal from the photodiode and will only be sensitive to in-plane polar components.

For polar components orientated such that they are parallel to the electric field the resulting oscillating movement will be entirely in-phase with the modulated electric field but for an anti-parallel alignment the motion will be 180° out-of-phase. In this way it is possible to determine the orientation of the vertical components of polarisation from analysis of the phase information, ''φ'', contained in the input signal, readily available after demodulation in the LiA, when using the VPFM mode. In a similar sense the orientations of in-plane polar components can also be determined from the phase difference when using the LPFM mode. The amplitude of the piezoresponse of either VPFM or LPFM is also given by the LiA, in the form of the magnitude, ''R''.

==Examples of PFM imaging==
[[File:180 degree ferroelectric domains with profiles.png|thumb|left|alt=180° ferroelectric domains as imaged by PFM|500px|180° ferroelectric domains in [[potassium titanyl phosphate|KTP]] as imaged by PFM. Below are the associated line profiles across the domains]]

The image shows periodically poled 180° domains in [[potassium titanyl phosphate]] (KTP) as imaged by VPFM. In the image piezoresponse amplitude can be seen where dark areas represent the zero amplitude that is expected at domain boundaries where the unit cell is cubic i.e. centrosymmetric and so therefore not ferroelectric. On the left hand side piezoresponse phase can be seen where the measured phase changes to show the out-of-plane components that are pointing out of the screen, white areas, and into the screen, dark areas. The scan area is 20×10 µm2. Below each scan is the relevant cross-section that shows in arbitrary units the PR amplitude and phase.

===PFM applied to biological materials===

PFM has been successfully applied to a range of biological materials such as teeth,<ref>{{cite journal|last1=Kalinin|first1=Sergei V.|last2=Rodriguez|first2=B. J.|last3=Jesse|first3=S.|last4=Thundat|first4=T.|last5=Gruverman|first5=A.|title=Electromechanical imaging of biological systems with sub-10 nm resolution|doi=10.1063/1.2006984|year=2005|pages=053901|volume=87|journal=Applied Physics Letters|url=http://digitalcommons.unl.edu/cgi/viewcontent.cgi?article=1017&context=physicsgruverman&sei-redir=1#search=%22Electromechanical+Imaging+of+Biological+Systems+with+Sub-10+nm+Resolution%22|arxiv = cond-mat/0504232 |bibcode = 2005ApPhL..87e3901K }}</ref> bone, and single collagen fibrils.<ref>{{cite journal|last1=Minary-Jolandan|first1=Majid|last2=Yu|first2=Min-Feng|doi=10.1021/nn900472n|title=Uncovering Nanoscale Electromechanical Heterogeneity in the Subfibrillar Structure of Collagen Fibrils Responsible for the Piezoelectricity of Bone|year=2009|pages=1859|volume=3|journal=ACS Nano|url=https://netfiles.uiuc.edu/mfyu/www/Documents/Publications/ACSnano-1-2009.pdf}}</ref> It has been hypothesized that the endogenous piezoelectricity in these materials may be relevant in their mechanobiology. For example, using PFM it has been shown that a single collagen fibril as small as 100 nm behaves predominantly as a shear piezoelectric materials with an effective piezoelectric constant of ~1 pm/V.

==Advanced PFM modes==
Several additions have been made to PFM that substantially increase the flexibility of the technique to probe nanoscale features.

===Stroboscopic PFM===
Stroboscopic PFM allows for time resolved imaging of switching in pseudo real-time.<ref>{{cite journal|last1=Gruverman|first1=A.|last2=Rodriguez|first2=B. J.|last3=Dehoff|first3=C.|last4=Waldrep|first4=J. D.|last5=Kingon|first5=A. I.|last6=Nemanich|first6=R. J.|last7=Cross|first7=J. S.|doi=10.1063/1.2010605|title=Direct studies of domain switching dynamics in thin film ferroelectric capacitors|year=2005|pages=082902|volume=87|journal=Applied Physics Letters|url=http://digitalcommons.unl.edu/cgi/viewcontent.cgi?article=1015&context=physicsgruverman&sei-redir=1#search=%22Direct+studies+of+domain+switching+dynamics+in+thin+film+ferroelectric+capacitors%22|bibcode = 2005ApPhL..87h2902G }}</ref> A voltage pulse of amplitude much higher than the coercive voltage of the sample but shorter in duration than the characteristic switching time is applied to the sample and subsequently imaged. Further pulses with the same amplitude but longer in time are then applied with regular PFM imaging at the intervals. In this way a series of images showing the switching of the sample can be obtained. Typical pulses are of tens of [[nanoseconds]] in duration and are therefore capable of resolving the first nucleation sites of domain reversal and then observing how these sites evolve.

===Contact resonance PFM===
Remembering that in PFM an AC bias of a certain frequency causes a deformation of the sample material at that same frequency the system can be considered as a [[harmonic oscillator|driven harmonic oscillator]]. As such there exists a resonance as a function of driving frequency. This effect has been exploited in PFM to provide an enhancement in the PR signal, thus allowing for a higher [[signal-to-noise]] or similar signal-to-noise at lower driving bias amplitude.<ref>{{cite journal|last1=Harnagea|first1=C.|last2=Alexe|first2=M.|last3=Hesse|first3=D.|last4=Pignolet|first4=A.|doi=10.1063/1.1592307|title=Contact resonances in voltage-modulated force microscopy|year=2003|pages=338|volume=83|journal=Applied Physics Letters|url=http://www.mpi-halle.mpg.de/mpi/publi/pdf/4367_03.pdf|bibcode = 2003ApPhL..83..338H }}</ref> Typically this contact resonance is in the kilo- to mega-[[hertz]] range which is several times higher in frequency than the first free harmonic in air of the cantilever used. However a drawback is that the contact resonance is dependent not only on the dynamic response of the cantilever but also on the [[elastic modulus]] of the sample material immediately in contact with the probe tip and so therefore can change during scanning over different areas. This leads to a change in the measured PR amplitude and so is undesirable. One method of bypassing the inherent disadvantages of contact resonance PFM is to change the driving frequency in order to shadow or track the changes in the frequency of the contact resonance. This feature as developed by Asylum Research called Dual AC™ Resonance Tracking (DART) uses two limit frequencies on either side of the contact resonance peak and so can sense changes in the peak position. It is then possible to adapt the AC bias driving frequency correspondingly in order to maintain the signal boost that results from the contact resonance.

===Switching spectroscopy (SS) PFM===
In this technique the area underneath the PFM tip is switched with simultaneous acquisition of a [[hysteresis]] loop that can be analysed to obtain information about the sample properties.<ref>{{cite journal|last1=Rodriguez|first1=Brian J|last2=Jesse|first2=Stephen|last3=Baddorf|first3=Arthur P|last4=Zhao|first4=T|last5=Chu|first5=Y H|last6=Ramesh|first6=R|last7=Eliseev|first7=Eugene A|last8=Morozovska|first8=Anna N|last9=Kalinin|first9=Sergei V|title=Spatially resolved mapping of ferroelectric switching behavior in self-assembled multiferroic nanostructures: strain, size, and interface effects|journal=Nanotechnology|volume=18|pages=405701|year=2007|doi=10.1088/0957-4484/18/40/405701|bibcode = 2007Nanot..18N5701R }}</ref> A series of hysteresis loops are acquired across the sample surface in order to map the switching characteristics as a function of position. In this way an image representing switching properties such as coercive voltage, remnant polarisation, imprint and work of switching amongst others can be displayed in which each pixel displays the desired data from the hysteresis loop acquired at that point. This allows spatial analysis of switching properties to be compared with sample topography.

==Advantages and disadvantages==

===Advantages===
* High resolution on the nanometre scale
* Simultaneous acquisition of topography and piezoresponse
* Allows manipulation of ferroelectric domains
* Non-destructive
* Little sample preparation required

===Disadvantages===
* Scans can be slow, e.g. tens of minutes
* Tip wear changes surface interaction and can affect contrast
* Limited to lateral range of AFM i.e. approximately 100×100 µm2

==References==

{{Reflist|30em}}

==External links==
* [http://www.asylumresearch.com/Applications/PFMAppNote/PFMAppNote.shtml Asylum Research PFM Application Note]
[http://www.asylumresearch.com/Applications/ContactResonance/ContactResonance.shtml Contact Resonance Application Note]
* [http://www.ntmdt.com/spm-principles/view/piezoresponse-force-microscopy NT-MDT website with animation]
* [http://cp.literature.agilent.com/litweb/pdf/5989-7611EN.pdf Agilent Technologies PFM Application Note]
* [http://www.jpk.com/jpk-tech-piezoresponse-force-microscopy.download.c09f1c29f9ab908fa15897f36f87c510 JPK Instruments PFM Technical Report]
* [http://www.veeco.com Veeco website]

{{DEFAULTSORT:Piezoresponse Force Microscopy}}
[[Category:Scanning probe microscopy]]

Perfect measure

2012-08-11T23:11:24Z

2.25.175.128: /* Definition */

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