Propagation of uncertainty

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Propagation_of_uncertainty

In statistics, propagation of uncertainty (or propagation of error) is the effect of variables' uncertainties (or errors, more specifically random errors) on the uncertainty of a function based on them. When the variables are the values of experimental measurements they have uncertainties due to measurement limitations (e.g., instrument precision) which propagate due to the combination of variables in the function.

The uncertainty u can be expressed in a number of ways. It may be defined by the absolute error Δx. Uncertainties can also be defined by the relative error (Δx)/x, which is usually written as a percentage. Most commonly, the uncertainty on a quantity is quantified in terms of the standard deviation, σ, which is the positive square root of the variance. The value of a quantity and its error are then expressed as an interval x ± u. However, the most general way of characterizing uncertainty is by specifying its probability distribution. If the probability distribution of the variable is known or can be assumed, in theory it is possible to get any of its statistics. In particular, it is possible to derive confidence limits to describe the region within which the true value of the variable may be found. For example, the 68% confidence limits for a one-dimensional variable belonging to a normal distribution are approximately ± one standard deviation σ from the central value x, which means that the region x ± σ will cover the true value in roughly 68% of cases.

If the uncertainties are correlated then covariance must be taken into account. Correlation can arise from two different sources. First, the measurement errors may be correlated. Second, when the underlying values are correlated across a population, the uncertainties in the group averages will be correlated.

In a general context where a nonlinear function modifies the uncertain parameters (correlated or not), the standard tools to propagate uncertainty, and infer resulting quantity probability distribution/statistics, are sampling techniques from the Monte Carlo method family. For very expansive data or complex functions, the calculation of the error propagation may be very expansive so that a surrogate model or a parallel computing strategy may be necessary.

In some particular cases, the uncertainty propagation calculation can be done through simplistic algebraic procedures. Some of these scenarios are described below.

Linear combinations

Let ${\displaystyle \{f_k(x_1,x_2,\dots ,x_n)\}}$ be a set of m functions, which are linear combinations of ${\displaystyle n}$ variables ${\displaystyle x_1,x_2,\dots ,x_n}$ with combination coefficients ${\displaystyle A_{k1},A_{k2},\dots ,A_{kn},(k=1,\dots ,m)}$:

${\displaystyle f_k=\sum _{i=1}^n{A}_{ki}{x}_i,}$

or in matrix notation,

${\displaystyle \mathbf{f}=\mathbf{A}\mathbf{x}.}$

Also let the variance–covariance matrix of x = (x₁, ..., x_n) be denoted by ${\displaystyle {\mathbf{\Sigma }}^x}$ and let the mean value be denoted by ${\displaystyle \mu }$:

${\displaystyle {\mathbf{\Sigma }}^x=E[\mathbf{(}\mathbf{x}\mathbf{-}\mu \mathbf{)}\otimes \mathbf{(}\mathbf{x}\mathbf{-}\mu \mathbf{)}]=\left(\begin{array}{cccc}{\sigma }_1^2& {\sigma }_{12}& {\sigma }_{13}& \cdots \\ {\sigma }_{21}& {\sigma }_2^2& {\sigma }_{23}& \cdots \\ {\sigma }_{31}& {\sigma }_{32}& {\sigma }_3^2& \cdots \\ ⋮& ⋮& ⋮& \ddots \end{array}\right)=\left(\begin{array}{cccc}{\mathrm{\Sigma }}_{11}^x& {\mathrm{\Sigma }}_{12}^x& {\mathrm{\Sigma }}_{13}^x& \cdots \\ {\mathrm{\Sigma }}_{21}^x& {\mathrm{\Sigma }}_{22}^x& {\mathrm{\Sigma }}_{23}^x& \cdots \\ {\mathrm{\Sigma }}_{31}^x& {\mathrm{\Sigma }}_{32}^x& {\mathrm{\Sigma }}_{33}^x& \cdots \\ ⋮& ⋮& ⋮& \ddots \end{array}\right).}$

${\displaystyle \otimes }$ is the outer product.

Then, the variance–covariance matrix ${\displaystyle {\mathbf{\Sigma }}^f}$ of f is given by

${\displaystyle {\mathbf{\Sigma }}^f=E[(\mathbf{f}-E[\mathbf{f}])\otimes (\mathbf{f}-E[\mathbf{f}])]=E[\mathbf{A}\mathbf{(}\mathbf{x}\mathbf{-}\mu \mathbf{)}\otimes \mathbf{A}\mathbf{(}\mathbf{x}\mathbf{-}\mu \mathbf{)}]=\mathbf{A}E[\mathbf{(}\mathbf{x}\mathbf{-}\mu \mathbf{)}\otimes \mathbf{(}\mathbf{x}\mathbf{-}\mu \mathbf{)}]{\mathbf{A}}^{\mathrm{T}}=\mathbf{A}{\mathbf{\Sigma }}^x{\mathbf{A}}^{\mathrm{T}}}$

In component notation, the equation

${\displaystyle {\mathbf{\Sigma }}^f=\mathbf{A}{\mathbf{\Sigma }}^x{\mathbf{A}}^{\mathrm{T}}.}$

reads

${\displaystyle {\mathrm{\Sigma }}_{ij}^f=\sum _k^n\sum _l^n{A}_{ik}{\mathrm{\Sigma }}_{kl}^x{A}_{jl}.}$

This is the most general expression for the propagation of error from one set of variables onto another. When the errors on x are uncorrelated, the general expression simplifies to

${\displaystyle {\mathrm{\Sigma }}_{ij}^f=\sum _k^n{A}_{ik}{\mathrm{\Sigma }}_k^x{A}_{jk},}$

where ${\displaystyle {\mathrm{\Sigma }}_k^x={\sigma }_{x_k}^2}$ is the variance of k-th element of the x vector. Note that even though the errors on x may be uncorrelated, the errors on f are in general correlated; in other words, even if ${\displaystyle {\mathbf{\Sigma }}^x}$ is a diagonal matrix, ${\displaystyle {\mathbf{\Sigma }}^f}$ is in general a full matrix.

The general expressions for a scalar-valued function f are a little simpler (here a is a row vector):

${\displaystyle f=\sum _i^n{a}_i{x}_i=\mathbf{a}\mathbf{x},}$

${\displaystyle {\sigma }_f^2=\sum _i^n\sum _j^n{a}_i{\mathrm{\Sigma }}_{ij}^x{a}_j=\mathbf{a}{\mathbf{\Sigma }}^x{\mathbf{a}}^{\mathrm{T}}.}$

Each covariance term ${\displaystyle {\sigma }_{ij}}$ can be expressed in terms of the correlation coefficient ${\displaystyle {\rho }_{ij}}$ by ${\displaystyle {\sigma }_{ij}={\rho }_{ij}{\sigma }_i{\sigma }_j}$, so that an alternative expression for the variance of f is

${\displaystyle {\sigma }_f^2=\sum _i^n{a}_i^2{\sigma }_i^2+\sum _i^n\sum _{j(j\ne i)}^n{a}_i{a}_j{\rho }_{ij}{\sigma }_i{\sigma }_j.}$

In the case that the variables in x are uncorrelated, this simplifies further to

${\displaystyle {\sigma }_f^2=\sum _i^n{a}_i^2{\sigma }_i^2.}$

In the simple case of identical coefficients and variances, we find

${\displaystyle {\sigma }_f=\sqrt{n}|a|\sigma .}$

For the arithmetic mean, ${\displaystyle a=1/n}$, the result is the standard error of the mean:

${\displaystyle {\sigma }_f=\sigma /\sqrt{n}.}$

Non-linear combinations

See also: Taylor expansions for the moments of functions of random variables

When f is a set of non-linear combination of the variables x, an interval propagation could be performed in order to compute intervals which contain all consistent values for the variables. In a probabilistic approach, the function f must usually be linearised by approximation to a first-order Taylor series expansion, though in some cases, exact formulae can be derived that do not depend on the expansion as is the case for the exact variance of products.^[7] The Taylor expansion would be:

${\displaystyle f_k\approx f_k^0+\sum _i^n\frac{\mathrm{\partial }{f}_k}{\mathrm{\partial }{x}_i}{x}_i}$

where ${\displaystyle \mathrm{\partial }{f}_k/\mathrm{\partial }{x}_i}$ denotes the partial derivative of f_k with respect to the i-th variable, evaluated at the mean value of all components of vector x. Or in matrix notation,

${\displaystyle \mathrm{f}\approx {\mathrm{f}}^0+\mathrm{J}\mathrm{x}}$

where J is the Jacobian matrix. Since f⁰ is a constant it does not contribute to the error on f. Therefore, the propagation of error follows the linear case, above, but replacing the linear coefficients, A_ki and A_kj by the partial derivatives, ${\displaystyle \frac{\mathrm{\partial }{f}_k}{\mathrm{\partial }{x}_i}}$ and ${\displaystyle \frac{\mathrm{\partial }{f}_k}{\mathrm{\partial }{x}_j}}$. In matrix notation,^[8]

${\displaystyle {\mathrm{\Sigma }}^{\mathrm{f}}=\mathrm{J}{\mathrm{\Sigma }}^{\mathrm{x}}{\mathrm{J}}^{\mathrm{\top }}.}$

That is, the Jacobian of the function is used to transform the rows and columns of the variance-covariance matrix of the argument. Note this is equivalent to the matrix expression for the linear case with ${\displaystyle \mathrm{J}=\mathrm{A}}$.

Simplification

Neglecting correlations or assuming independent variables yields a common formula among engineers and experimental scientists to calculate error propagation, the variance formula:^[9]

${\displaystyle s_f=\sqrt{{\left(\frac{\mathrm{\partial }f}{\mathrm{\partial }x}\right)}^2{s}_x^2+{\left(\frac{\mathrm{\partial }f}{\mathrm{\partial }y}\right)}^2{s}_y^2+{\left(\frac{\mathrm{\partial }f}{\mathrm{\partial }z}\right)}^2{s}_z^2+\cdots }}$

where ${\displaystyle s_f}$ represents the standard deviation of the function ${\displaystyle f}$, ${\displaystyle s_x}$ represents the standard deviation of ${\displaystyle x}$, ${\displaystyle s_y}$ represents the standard deviation of ${\displaystyle y}$, and so forth.

It is important to note that this formula is based on the linear characteristics of the gradient of ${\displaystyle f}$ and therefore it is a good estimation for the standard deviation of ${\displaystyle f}$ as long as ${\displaystyle s_x,s_y,s_z,\dots }$ are small enough. Specifically, the linear approximation of ${\displaystyle f}$ has to be close to ${\displaystyle f}$ inside a neighbourhood of radius ${\displaystyle s_x,s_y,s_z,\dots }$.^[10]

Example

Any non-linear differentiable function, ${\displaystyle f(a,b)}$, of two variables, ${\displaystyle a}$ and ${\displaystyle b}$, can be expanded as

${\displaystyle f\approx f^0+\frac{\mathrm{\partial }f}{\mathrm{\partial }a}a+\frac{\mathrm{\partial }f}{\mathrm{\partial }b}b}$

now, taking variance on both sides, and using the formula^[11] for variance of a linear combination of variables:

${\displaystyle Var(aX+bY)=a^{2}Var(X)+b^{2}Var(Y)+2ab\ast Cov(X,Y)}$

hence:

${\displaystyle {\sigma }_f^2\approx {\left|\frac{\mathrm{\partial }f}{\mathrm{\partial }a}\right|}^2{\sigma }_a^2+{\left|\frac{\mathrm{\partial }f}{\mathrm{\partial }b}\right|}^2{\sigma }_b^2+2\frac{\mathrm{\partial }f}{\mathrm{\partial }a}\frac{\mathrm{\partial }f}{\mathrm{\partial }b}{\sigma }_{ab}}$

where ${\displaystyle {\sigma }_f}$ is the standard deviation of the function ${\displaystyle f}$, ${\displaystyle {\sigma }_a}$ is the standard deviation of ${\displaystyle a}$, ${\displaystyle {\sigma }_b}$ is the standard deviation of ${\displaystyle b}$ and ${\displaystyle {\sigma }_{ab}={\sigma }_a{\sigma }_b{\rho }_{ab}}$ is the covariance between ${\displaystyle a}$ and ${\displaystyle b}$.

In the particular case that ${\displaystyle f=ab}$, ${\displaystyle \frac{\mathrm{\partial }f}{\mathrm{\partial }a}=b,\frac{\mathrm{\partial }f}{\mathrm{\partial }b}=a}$. Then

${\displaystyle {\sigma }_f^2\approx b^2{\sigma }_a^2+a^2{\sigma }_b^2+2ab{\sigma }_{ab}}$

or

${\displaystyle {\left(\frac{{\sigma }_f}{f}\right)}^2\approx {\left(\frac{{\sigma }_a}{a}\right)}^2+{\left(\frac{{\sigma }_b}{b}\right)}^2+2\left(\frac{{\sigma }_a}{a}\right)\left(\frac{{\sigma }_b}{b}\right){\rho }_{ab}}$

where ${\displaystyle {\rho }_{ab}}$ is the correlation between ${\displaystyle a}$ and ${\displaystyle b}$.

When the variables ${\displaystyle a}$ and ${\displaystyle b}$ are uncorrelated, ${\displaystyle {\rho }_{ab}=0}$. Then

${\displaystyle {\left(\frac{{\sigma }_f}{f}\right)}^2\approx {\left(\frac{{\sigma }_a}{a}\right)}^2+{\left(\frac{{\sigma }_b}{b}\right)}^2.}$

Caveats and warnings

Error estimates for non-linear functions are biased on account of using a truncated series expansion. The extent of this bias depends on the nature of the function. For example, the bias on the error calculated for log(1+x) increases as x increases, since the expansion to x is a good approximation only when x is near zero.

For highly non-linear functions, there exist five categories of probabilistic approaches for uncertainty propagation; see Uncertainty quantification for details.

Reciprocal and shifted reciprocal

Main article: Reciprocal normal distribution

In the special case of the inverse or reciprocal ${\displaystyle 1/B}$, where ${\displaystyle B=N(0,1)}$ follows a standard normal distribution, the resulting distribution is a reciprocal standard normal distribution, and there is no definable variance.

However, in the slightly more general case of a shifted reciprocal function ${\displaystyle 1/(p-B)}$ for ${\displaystyle B=N(\mu ,\sigma )}$ following a general normal distribution, then mean and variance statistics do exist in a principal value sense, if the difference between the pole ${\displaystyle p}$ and the mean ${\displaystyle \mu }$ is real-valued.

Ratios

Main article: Normal ratio distribution

Ratios are also problematic; normal approximations exist under certain conditions.

Example formulae

This table shows the variances and standard deviations of simple functions of the real variables ${\displaystyle A,B}$, with standard deviations ${\displaystyle {\sigma }_A,{\sigma }_B,}$ covariance ${\displaystyle {\sigma }_{AB}={\rho }_{AB}{\sigma }_A{\sigma }_B}$, and correlation ${\displaystyle {\rho }_{AB}}$. The real-valued coefficients ${\displaystyle a}$ and ${\displaystyle b}$ are assumed exactly known (deterministic), i.e., ${\displaystyle {\sigma }_a={\sigma }_b=0}$.

In the columns "Variance" and "Standard Deviation", A and B should be understood as expectation values (i.e. values around which we're estimating the uncertainty), and ${\displaystyle f}$ should be understood as the value of the function calculated at the expectation value of ${\displaystyle A,B}$.

Function	Variance	Standard Deviation
${\displaystyle f=aA}$	${\displaystyle {\sigma }_f^2=a^2{\sigma }_A^2}$	${\displaystyle {\sigma }_f=|a|{\sigma }_A}$
${\displaystyle f=aA+bB}$	${\displaystyle {\sigma }_f^2=a^2{\sigma }_A^2+b^2{\sigma }_B^2+2ab{\sigma }_{AB}}$	${\displaystyle {\sigma }_f=\sqrt{a^2{\sigma }_A^2+b^2{\sigma }_B^2+2ab{\sigma }_{AB}}}$
${\displaystyle f=aA-bB}$	${\displaystyle {\sigma }_f^2=a^2{\sigma }_A^2+b^2{\sigma }_B^2-2ab{\sigma }_{AB}}$	${\displaystyle {\sigma }_f=\sqrt{a^2{\sigma }_A^2+b^2{\sigma }_B^2-2ab{\sigma }_{AB}}}$
${\displaystyle f=A-B,}$	${\displaystyle {\sigma }_f^2={\sigma }_A^2+{\sigma }_B^2-2{\sigma }_{AB}}$	${\displaystyle {\sigma }_f=\sqrt{{\sigma }_A^2+{\sigma }_B^2-2{\sigma }_{AB}}}$
${\displaystyle f=AB}$	${\displaystyle {\sigma }_f^2\approx f^2\left[{\left(\frac{{\sigma }_A}{A}\right)}^2+{\left(\frac{{\sigma }_B}{B}\right)}^2+2\frac{{\sigma }_{AB}}{AB}\right]}$	${\displaystyle {\sigma }_f\approx \left|f\right|\sqrt{{\left(\frac{{\sigma }_A}{A}\right)}^2+{\left(\frac{{\sigma }_B}{B}\right)}^2+2\frac{{\sigma }_{AB}}{AB}}}$
${\displaystyle f=\frac{A}{B}}$	${\displaystyle {\sigma }_f^2\approx f^2\left[{\left(\frac{{\sigma }_A}{A}\right)}^2+{\left(\frac{{\sigma }_B}{B}\right)}^2-2\frac{{\sigma }_{AB}}{AB}\right]}$	${\displaystyle {\sigma }_f\approx \left|f\right|\sqrt{{\left(\frac{{\sigma }_A}{A}\right)}^2+{\left(\frac{{\sigma }_B}{B}\right)}^2-2\frac{{\sigma }_{AB}}{AB}}}$
${\displaystyle f=a{A}^b}$	${\displaystyle {\sigma }_f^2\approx {\left(ab{A}^{b-1}{\sigma }_A\right)}^2={\left(\frac{fb{\sigma }_A}{A}\right)}^2}$	${\displaystyle {\sigma }_f\approx \left|ab{A}^{b-1}{\sigma }_A\right|=\left|\frac{fb{\sigma }_A}{A}\right|}$
${\displaystyle f=a\ln (bA)}$	${\displaystyle {\sigma }_f^2\approx {\left(a\frac{{\sigma }_A}{A}\right)}^2}$	${\displaystyle {\sigma }_f\approx \left|a\frac{{\sigma }_A}{A}\right|}$
${\displaystyle f=a{\log }_{10}(bA)}$	${\displaystyle {\sigma }_f^2\approx {\left(a\frac{{\sigma }_A}{A\ln (10)}\right)}^2}$	${\displaystyle {\sigma }_f\approx \left|a\frac{{\sigma }_A}{A\ln (10)}\right|}$
${\displaystyle f=a{e}^{bA}}$	${\displaystyle {\sigma }_f^2\approx f^2{\left(b{\sigma }_A\right)}^2}$[19]	${\displaystyle {\sigma }_f\approx \left|f\right|\left|\left(b{\sigma }_A\right)\right|}$
${\displaystyle f=a^{bA}}$	${\displaystyle {\sigma }_f^2\approx f^2(b\ln (a){\sigma }_A{)}^2}$	${\displaystyle {\sigma }_f\approx \left|f\right|\left|(b\ln (a){\sigma }_A)\right|}$
${\displaystyle f=a\sin (bA)}$	${\displaystyle {\sigma }_f^2\approx {\left[ab\cos (bA){\sigma }_A\right]}^2}$	${\displaystyle {\sigma }_f\approx \left|ab\cos (bA){\sigma }_A\right|}$
${\displaystyle f=a\cos \left(bA\right)}$	${\displaystyle {\sigma }_f^2\approx {\left[ab\sin (bA){\sigma }_A\right]}^2}$	${\displaystyle {\sigma }_f\approx \left|ab\sin (bA){\sigma }_A\right|}$
${\displaystyle f=a\tan \left(bA\right)}$	${\displaystyle {\sigma }_f^2\approx {\left[ab{\sec }^2(bA){\sigma }_A\right]}^2}$	${\displaystyle {\sigma }_f\approx \left|ab{\sec }^2(bA){\sigma }_A\right|}$
${\displaystyle f=A^B}$	${\displaystyle {\sigma }_f^2\approx f^2\left[{\left(\frac{B}{A}{\sigma }_A\right)}^2+{\left(\ln (A){\sigma }_B\right)}^2+2\frac{B\ln (A)}{A}{\sigma }_{AB}\right]}$	${\displaystyle {\sigma }_f\approx \left|f\right|\sqrt{{\left(\frac{B}{A}{\sigma }_A\right)}^2+{\left(\ln (A){\sigma }_B\right)}^2+2\frac{B\ln (A)}{A}{\sigma }_{AB}}}$
${\displaystyle f=\sqrt{a{A}^2\pm b{B}^2}}$	${\displaystyle {\sigma }_f^2\approx {\left(\frac{A}{f}\right)}^2{a}^2{\sigma }_A^2+{\left(\frac{B}{f}\right)}^2{b}^2{\sigma }_B^2\pm 2ab\frac{AB}{f^2}{\sigma }_{AB}}$	${\displaystyle {\sigma }_f\approx \sqrt{{\left(\frac{A}{f}\right)}^2{a}^2{\sigma }_A^2+{\left(\frac{B}{f}\right)}^2{b}^2{\sigma }_B^2\pm 2ab\frac{AB}{f^2}{\sigma }_{AB}}}$

For uncorrelated variables (${\displaystyle {\rho }_{AB}=0}$, ${\displaystyle {\sigma }_{AB}=0}$) expressions for more complicated functions can be derived by combining simpler functions. For example, repeated multiplication, assuming no correlation, gives

${\displaystyle f=ABC;{\left(\frac{{\sigma }_f}{f}\right)}^2\approx {\left(\frac{{\sigma }_A}{A}\right)}^2+{\left(\frac{{\sigma }_B}{B}\right)}^2+{\left(\frac{{\sigma }_C}{C}\right)}^2.}$

For the case ${\displaystyle f=AB}$ we also have Goodman's expression for the exact variance: for the uncorrelated case it is

${\displaystyle V(XY)=E(X{)}^{2}V(Y)+E(Y{)}^{2}V(X)+E((X-E(X){)}^2(Y-E(Y){)}^2)}$

and therefore we have:

${\displaystyle {\sigma }_f^2=A^2{\sigma }_B^2+B^2{\sigma }_A^2+{\sigma }_A^2{\sigma }_B^2}$

Effect of correlation on differences

If A and B are uncorrelated, their difference A-B will have more variance than either of them. An increasing positive correlation (${\displaystyle {\rho }_{AB}\to 1}$) will decrease the variance of the difference, converging to zero variance for perfectly correlated variables with the same variance. On the other hand, a negative correlation (${\displaystyle {\rho }_{AB}\to -1}$) will further increase the variance of the difference, compared to the uncorrelated case.

For example, the self-subtraction f=A-A has zero variance ${\displaystyle {\sigma }_f^2=0}$ only if the variate is perfectly autocorrelated (${\displaystyle {\rho }_A=1}$). If A is uncorrelated, ${\displaystyle {\rho }_A=0}$, then the output variance is twice the input variance, ${\displaystyle {\sigma }_f^2=2{\sigma }_A^2}$. And if A is perfectly anticorrelated, ${\displaystyle {\rho }_A=-1}$, then the input variance is quadrupled in the output, ${\displaystyle {\sigma }_f^2=4{\sigma }_A^2}$ (notice ${\displaystyle 1-{\rho }_A=2}$ for f = aA - aA in the table above).

Example calculations

Inverse tangent function

We can calculate the uncertainty propagation for the inverse tangent function as an example of using partial derivatives to propagate error.

Define

${\displaystyle f(x)=\arctan (x),}$

where ${\displaystyle {\mathrm{\Delta }}_x}$ is the absolute uncertainty on our measurement of x. The derivative of f(x) with respect to x is

${\displaystyle \frac{df}{dx}=\frac{1}{1+x^2}.}$

Therefore, our propagated uncertainty is

${\displaystyle {\mathrm{\Delta }}_f\approx \frac{{\mathrm{\Delta }}_x}{1+x^2},}$

where ${\displaystyle {\mathrm{\Delta }}_f}$ is the absolute propagated uncertainty.

Resistance measurement

A practical application is an experiment in which one measures current, I, and voltage, V, on a resistor in order to determine the resistance, R, using Ohm's law, R = V / I.

Given the measured variables with uncertainties, I ± σ_I and V ± σ_V, and neglecting their possible correlation, the uncertainty in the computed quantity, σ_R, is:

${\displaystyle {\sigma }_R\approx \sqrt{{\sigma }_V^2{\left(\frac{1}{I}\right)}^2+{\sigma }_I^2{\left(\frac{-V}{I^2}\right)}^2}=R\sqrt{{\left(\frac{{\sigma }_V}{V}\right)}^2+{\left(\frac{{\sigma }_I}{I}\right)}^2}.}$

 

Volatility (finance)

From Wikipedia, the free encyclopedia

 

 

The VIX

In finance, volatility (usually denoted by σ) is the degree of variation of a trading price series over time, usually measured by the standard deviation of logarithmic returns.

Historic volatility measures a time series of past market prices. Implied volatility looks forward in time, being derived from the market price of a market-traded derivative (in particular, an option).

Volatility terminology

Volatility as described here refers to the actual volatility, more specifically:

• actual current volatility of a financial instrument for a specified period (for example 30 days or 90 days), based on historical prices over the specified period with the last observation the most recent price.
• actual historical volatility which refers to the volatility of a financial instrument over a specified period but with the last observation on a date in the past
  • near synonymous is realized volatility, the square root of the realized variance, in turn calculated using the sum of squared returns divided by the number of observations.
• actual future volatility which refers to the volatility of a financial instrument over a specified period starting at the current time and ending at a future date (normally the expiry date of an option)

Now turning to implied volatility, we have:

• historical implied volatility which refers to the implied volatility observed from historical prices of the financial instrument (normally options)
• current implied volatility which refers to the implied volatility observed from current prices of the financial instrument
• future implied volatility which refers to the implied volatility observed from future prices of the financial instrument

For a financial instrument whose price follows a Gaussian random walk, or Wiener process, the width of the distribution increases as time increases. This is because there is an increasing probability that the instrument's price will be farther away from the initial price as time increases. However, rather than increase linearly, the volatility increases with the square-root of time as time increases, because some fluctuations are expected to cancel each other out, so the most likely deviation after twice the time will not be twice the distance from zero.

Since observed price changes do not follow Gaussian distributions, others such as the Lévy distribution are often used. These can capture attributes such as "fat tails". Volatility is a statistical measure of dispersion around the average of any random variable such as market parameters etc.

Mathematical definition

For any fund that evolves randomly with time, volatility is defined as the standard deviation of a sequence of random variables, each of which is the return of the fund over some corresponding sequence of (equally sized) times.

Thus, "annualized" volatility σ_annually is the standard deviation of an instrument's yearly logarithmic returns.

The generalized volatility σ_T for time horizon T in years is expressed as:

${\displaystyle {\sigma }_{\text{T}}={\sigma }_{\text{annually}}\sqrt{T}.}$

Therefore, if the daily logarithmic returns of a stock have a standard deviation of σ_daily and the time period of returns is P in trading days, the annualized volatility is

${\displaystyle {\sigma }_{\text{annually}}={\sigma }_{\text{daily}}\sqrt{P}.}$

so

${\displaystyle {\sigma }_{\text{T}}={\sigma }_{\text{daily}}\sqrt{PT}.}$

A common assumption is that P = 252 trading days in any given year. Then, if σ_daily = 0.01, the annualized volatility is

${\displaystyle {\sigma }_{\text{annually}}=0.01\sqrt{252}=\mathrm{0.1587.}}$

The monthly volatility (i.e. ${\displaystyle T=\frac{1}{12}}$ of a year) is

${\displaystyle {\sigma }_{\text{monthly}}=0.01\sqrt{\frac{252}{12}}=\mathrm{0.0458.}}$

The formulas used above to convert returns or volatility measures from one time period to another assume a particular underlying model or process. These formulas are accurate extrapolations of a random walk, or Wiener process, whose steps have finite variance. However, more generally, for natural stochastic processes, the precise relationship between volatility measures for different time periods is more complicated. Some use the Lévy stability exponent α to extrapolate natural processes:

${\displaystyle {\sigma }_T=T^{1/\alpha }\sigma .}$

If α = 2 the Wiener process scaling relation is obtained, but some people believe α < 2 for financial activities such as stocks, indexes and so on. This was discovered by Benoît Mandelbrot, who looked at cotton prices and found that they followed a Lévy alpha-stable distribution with α = 1.7. (See New Scientist, 19 April 1997.)

Volatility origin

Much research has been devoted to modeling and forecasting the volatility of financial returns, and yet few theoretical models explain how volatility comes to exist in the first place.

Roll (1984) shows that volatility is affected by market microstructure. Glosten and Milgrom (1985) shows that at least one source of volatility can be explained by the liquidity provision process. When market makers infer the possibility of adverse selection, they adjust their trading ranges, which in turn increases the band of price oscillation.

In September 2019, JPMorgan Chase determined the effect of US President Donald Trump's tweets, and called it the Volfefe index combining volatility and the covfefe meme.

Volatility for investors

Investors care about volatility for at least eight reasons:

1. The wider the swings in an investment's price, the harder emotionally it is to not worry;
2. Price volatility of a trading instrument can define position sizing in a portfolio;
3. When certain cash flows from selling a security are needed at a specific future date, higher volatility means a greater chance of a shortfall;
4. Higher volatility of returns while saving for retirement results in a wider distribution of possible final portfolio values;
5. Higher volatility of return when retired gives withdrawals a larger permanent impact on the portfolio's value;
6. Price volatility presents opportunities to buy assets cheaply and sell when overpriced;
7. Portfolio volatility has a negative impact on the compound annual growth rate (CAGR) of that portfolio
8. Volatility affects pricing of options, being a parameter of the Black–Scholes model.

Volatility versus direction

Volatility does not measure the direction of price changes, merely their dispersion. This is because when calculating standard deviation (or variance), all differences are squared, so that negative and positive differences are combined into one quantity. Two instruments with different volatilities may have the same expected return, but the instrument with higher volatility will have larger swings in values over a given period of time.

For example, a lower volatility stock may have an expected (average) return of 7%, with annual volatility of 5%. This would indicate returns from approximately negative 3% to positive 17% most of the time (19 times out of 20, or 95% via a two standard deviation rule). A higher volatility stock, with the same expected return of 7% but with annual volatility of 20%, would indicate returns from approximately negative 33% to positive 47% most of the time (19 times out of 20, or 95%). These estimates assume a normal distribution; in reality stocks are found to be leptokurtotic.

Volatility over time

Although the Black-Scholes equation assumes predictable constant volatility, this is not observed in real markets, and amongst the models are Emanuel Derman and Iraj Kani's and Bruno Dupire's local volatility, Poisson process where volatility jumps to new levels with a predictable frequency, and the increasingly popular Heston model of stochastic volatility.

It is common knowledge that types of assets experience periods of high and low volatility. That is, during some periods, prices go up and down quickly, while during other times they barely move at all. In foreign exchange market, price changes are seasonally heteroskedastic with periods of one day and one week.

Periods when prices fall quickly (a crash) are often followed by prices going down even more, or going up by an unusual amount. Also, a time when prices rise quickly (a possible bubble) may often be followed by prices going up even more, or going down by an unusual amount.

Most typically, extreme movements do not appear 'out of nowhere'; they are presaged by larger movements than usual. This is termed autoregressive conditional heteroskedasticity. Whether such large movements have the same direction, or the opposite, is more difficult to say. And an increase in volatility does not always presage a further increase—the volatility may simply go back down again.

Not only the volatility depends on the period when it is measured but also on the selected time resolution. The effect is observed due to the fact that the information flow between short-term and long-term traders is asymmetric. As a result, volatility measured with high resolution contains information that is not covered by low resolution volatility and vice versa.

The risk parity weighted volatility of the three assets Gold, Treasury bonds and Nasdaq acting as proxy for the Marketportfolio seems to have a low point at 4% after turning upwards for the 8th time since 1974 at this reading in the summer of 2014.

Alternative measures of volatility

Some authors point out that realized volatility and implied volatility are backward and forward looking measures, and do not reflect current volatility. To address that issue an alternative, ensemble measures of volatility were suggested. One of the measures is defined as the standard deviation of ensemble returns instead of time series of returns. Another considers the regular sequence of directional-changes as the proxy for the instantaneous volatility.

Implied volatility parametrisation

There exist several known parametrisations of the implied volatility surface, Schonbucher, SVI and gSVI.

Crude volatility estimation

Using a simplification of the above formula it is possible to estimate annualized volatility based solely on approximate observations. Suppose you notice that a market price index, which has a current value near 10,000, has moved about 100 points a day, on average, for many days. This would constitute a 1% daily movement, up or down.

To annualize this, you can use the "rule of 16", that is, multiply by 16 to get 16% as the annual volatility. The rationale for this is that 16 is the square root of 256, which is approximately the number of trading days in a year (252). This also uses the fact that the standard deviation of the sum of n independent variables (with equal standard deviations) is √n times the standard deviation of the individual variables.

The average magnitude of the observations is merely an approximation of the standard deviation of the market index. Assuming that the market index daily changes are normally distributed with mean zero and standard deviation σ, the expected value of the magnitude of the observations is √(2/π)σ = 0.798σ. The net effect is that this crude approach underestimates the true volatility by about 20%.

Estimate of compound annual growth rate (CAGR)

Consider the Taylor series:

${\displaystyle \log (1+y)=y-\frac{1}{2}{y}^2+\frac{1}{3}{y}^3-\frac{1}{4}{y}^4+\cdots }$

Taking only the first two terms one has:

${\displaystyle \mathrm{C}\mathrm{A}\mathrm{G}\mathrm{R}\approx \mathrm{A}\mathrm{R}-\frac{1}{2}{\sigma }^2}$

Volatility thus mathematically represents a drag on the CAGR (formalized as the "volatility tax"). Realistically, most financial assets have negative skewness and leptokurtosis, so this formula tends to be over-optimistic. Some people use the formula:

${\displaystyle \mathrm{C}\mathrm{A}\mathrm{G}\mathrm{R}\approx \mathrm{A}\mathrm{R}-\frac{1}{2}k{\sigma }^2}$

for a rough estimate, where k is an empirical factor (typically five to ten).

Criticisms of volatility forecasting models

Performance of VIX (left) compared to past volatility (right) as 30-day volatility predictors, for the period of Jan 1990-Sep 2009. Volatility is measured as the standard deviation of S&P500 one-day returns over a month's period. The blue lines indicate linear regressions, resulting in the correlation coefficients r shown. Note that VIX has virtually the same predictive power as past volatility, insofar as the shown correlation coefficients are nearly identical.

Despite the sophisticated composition of most volatility forecasting models, critics claim that their predictive power is similar to that of plain-vanilla measures, such as simple past volatility especially out-of-sample, where different data are used to estimate the models and to test them. Other works have agreed, but claim critics failed to correctly implement the more complicated models. Some practitioners and portfolio managers seem to completely ignore or dismiss volatility forecasting models. For example, Nassim Taleb famously titled one of his Journal of Portfolio Management papers "We Don't Quite Know What We are Talking About When We Talk About Volatility". In a similar note, Emanuel Derman expressed his disillusion with the enormous supply of empirical models unsupported by theory. He argues that, while "theories are attempts to uncover the hidden principles underpinning the world around us, as Albert Einstein did with his theory of relativity", we should remember that "models are metaphors – analogies that describe one thing relative to another".

Community wind energy

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Community_wind_energy 

 

Wind turbines at Findhorn Ecovillage make the community a net exporter of electricity.

Community wind projects are locally owned by farmers, investors, businesses, schools, utilities, or other public or private entities who utilize wind energy to support and reduce energy costs to the local community. The key feature is that local community members have a significant, direct financial stake in the project beyond land lease payments and tax revenue. Projects may be used for on-site power or to generate wholesale power for sale, usually on a commercial-scale greater than 100 kW.

Community wind farms

Australia

Hepburn Wind Farm

See also: Wind power in Australia

The Hepburn Wind Project is a wind farm at Leonards Hill near Daylesford, Victoria, north-west of Melbourne, Victoria. It comprises two 2MW wind turbines which produce enough power for 2,300 households.

This is the first Australian community-owned wind farm. The initiative has emerged because the community felt that the state and federal governments were not doing enough to address climate change.

Telecommunication towers will be repowered with small wind turbines under a new project led by a Newcastle startup. Ten small wind turbines will be installed at ten remote Australian communication sites as part of a new project to boost the uptake of the technology.

Canada

See also: WindShare

Community wind power is in its infancy in Canada but there are reasons for optimism. One such reason is the launch of a new Feed-in Tariff (FIT) program in the Province of Ontario . A number of community wind projects are in development in Ontario but the first project that is likely to obtain a FIT contract and connect to the grid is the Pukwis Community Wind Park. Pukwis will be unique in that it is a joint Aboriginal/Community wind project that will be majority-owned by the Chippewas of Georgina Island First Nation, with a local renewable energy co-operative (the Pukwis Energy Co-operative) owning the remainder of the project.

Denmark

Middelgrunden offshore wind park

 

See also: Wind power in Denmark and Middelgrunden wind farm

In Denmark, families were offered a tax exemption for generating their own electricity within their own or an adjoining commune. By 2001 over 100,000 families belonged to wind turbine cooperatives, which had installed 86% of all the wind turbines in Denmark, a world leader in wind power. Wind power has gained very high social acceptance in Denmark, with the development of community wind farms playing a major role.

In 1997, Samsø won a government competition to become a model renewable energy community. An offshore wind farm comprising 10 turbines (making a total of 21 altogether including land-based windmills), was completed, funded by the islanders. Now 100% of its electricity comes from wind power and 75% of its heat comes from solar power and biomass energy. An Energy Academy has opened in Ballen, with a visitor education center.

Germany

See also: Wind power in Germany and Paderborn Wind Farm

In Germany, hundreds of thousands of people have invested in citizens' wind farms across the country and thousands of small and medium-sized enterprises are running successful businesses in a new sector that in 2008 employed 90,000 people and generated 8 percent of Germany's electricity. Wind power has gained very high social acceptance in Germany, with the development of community wind farms playing a major role.

In the German district of North Frisia there are more than 60 wind farms with a capacity of about 700 MW, and 90 percent are community-owned. North Frisia is seen to be a model location for community wind, leading the way for other regions, especially in southern Germany.

India

Starting in 2006, a village panchayat (local self-governing body) in Tamil Nadu state has become completely self-sufficient in energy by using renewable sources like wind, solar and biogas.

The Odanthurai village panchayat near Coimbatore city comprises 11 villages and has a population of about 8,000. By 2009, it had set up its own 350 kW windfarm to meet its energy needs. The windmill was set up at Malwadi near Udumalpet and generates about 8 lakh (800,000) units annually. The power requirement for Odanthurai stands at about 4.5 lakh (450,000) units, and the local panchayat body is now selling the surplus power to the state grid. This gives the panchayat an annual income of 19 lakh rupees.

The village cooperative is also using other sources of renewable energy. It has 65 solar streetlights in two hamlets and a nine-KW (kilowatt) biomass gasifier to pump drinking water from the river to the overhead tanks. Doing so, Odanthurai became the first local body in India to utilize the remunerative enterprises' scheme of the state government.

The Netherlands

See also: Wind power in the Netherlands

Sixty-three farmers in "De Zuidlob", the southern part of the municipality of Zeewolde, have entered into a cooperative agreement that aims to develop a wind farm of at least 108 MW. The project will include the installation of three phases of 12 wind turbines with capacities of 3 to 4.5 MW each. The aim is to put the wind farm into service in 2012.

The Netherlands has an active community of wind cooperatives. They build and operate wind parks in all regions of the Netherlands. This started in the 1980s with the first Lagerweij turbines. Back then, these turbines could be financed by the members of the cooperatives. Today, the cooperatives build larger wind parks, but not as large as commercial parties do. Some still operate self-sufficiently, others partner with larger commercial wind park developers.

Because of the very unproductive state policies for financing wind parks in the Netherlands, the cooperatives have developed a new financing model, where members of a cooperative do not have to pay taxes for the electricity they generate with their community wind park. In this construction the Zelfleveringsmodel the cooperative operates the wind park, and a traditional energy company only acts as a service provider, for billing and energy balance on the public grid. This is the new role for energy companies in the future, where production is largely decentralized.

In 2012 a new company launched a new business model for community energy, Windcentrale. The wind turbine is sold in physical shares to families. Every share does not give financial gains, but real power, 500 kWh per year, average. A power company, part of the model, subtracts the generated amount of power, from the yearly power bill. Owners only have to pay for the power they used in excess of the amount their share generated. The Windcentrale started with 2 existing turbines that were sold in about 3 months. 8 months later they sold a turbine in a single evening. By the end of 2016 they were a community of about 17.000 members with 10 turbines and about 15 MW rated power. Every turbine is owned by a separate cooperative, with the Windcentrale doing all organizational work in the cooperative. In three years they grew to the same size, in members, than older wind cooperatives with the average age of 25 years. Two of these older wind cooperatives, DeltaWind and Zeeuwind are run as a business and are building a 100 MW wind farm in Krammer

United Kingdom

See also: Wind power in the United Kingdom and Wind power in Scotland

 

Westmill Wind Farm

As of 2012, there are 43 communities that are in the process of or already producing renewable energy through co-operative structures in the UK. They are set up and run by everyday people, mostly local residents, who are investing their time and money and together installing large wind turbines, solar panels, or hydro-electric power for their local communities.

Baywind Energy Co-operative was the first co-operative to own wind turbines in the United Kingdom. Baywind was modeled on the similar wind turbine cooperatives and other renewable energy co-operatives that are common in Scandinavia, and was founded as an industrial and provident society in 1996. It grew to exceed 1,300 members, each with one vote.

A proportion of the profits is invested in local community environmental initiatives through the Baywind Energy Conservation Trust. As of 2006, Baywind owns a 2.5 megawatt five-turbine wind farm at Harlock Hill near Ulverston, Cumbria (operational since 29 January 1997), and one of the 600 kilowatt turbines at the Haverigg II wind farm near Millom, Cumbria.

Community-owned schemes in Scotland include schemes Harris in the Outer Hebrides and on the Isle of Gigha. The Heritage Trust set up Gigha Renewable Energy to buy and operate three Vestas V27 wind turbines, known locally as The Dancing Ladies or Creideas, Dòchas is Carthannas (Gaelic for Faith, Hope and Charity). They were commissioned on 21 January 2005 and are capable of generating up to 675 kW of power. Revenue is produced by selling the electricity to the grid via an intermediary called Green Energy UK. Gigha residents control the whole project and profits are reinvested in the community. The North Harris Trust has installed several turbines on Harris.

Another community-owned wind farm, Westmill Wind Farm Cooperative, opened in May 2008 in the Oxfordshire village of Watchfield. It consists of five 1.3 megawatt turbines, and is described by its promoters as the UK's largest community-owned wind farm. It was structured as a cooperative, whose shares and loan stock were sold to the local community. Other businesses, such as Midcounties Co-operative, also invested, and the Co-operative Bank provided a loan.

Community Energy Scotland is an independent Scottish charity established in 2008 that provides advice and financial support for renewable energy projects developed by community groups in Scotland. The stated aim of Community Energy Scotland is 'to build confidence, resilience and wealth at community level in Scotland through sustainable energy development'.

Findhorn Ecovillage has four Vestas wind turbines that can generate up to 750 kW. These make the community net exporters of renewable-generated electricity. Most of the generation is used on-site with any surplus exported to the National Grid.

Boyndie Wind Farm Co-operative is part of the Energy4All group, which promotes community ownership. A number of other schemes supported by Highlands and Islands Community Energy Company are in the pipeline.

Community Renewable Energy (CoRE) has worked with Berwick Community Development Trust who agreed on the installation of a 500 kW Enercon turbine near the A1. The Trust now has an income of £60,000 a year (increasing) after the turbine was installed in 2014. CoRE supported Oakenshaw Community Association setting up a 500 kW wind turbine near Durham. The turbine begun operating in 2014 and the Association now receives substantial yearly income.

Unity Wind Ltd is an industrial and provident society that intends to install two 2MW wind turbines at North Walsham in North Norfolk. Its key aim is community wind turbines and run by community investment and for financial benefit to the community.

United States

In 2009, the National Renewable Energy Laboratory published a report that identified three different types of community wind projects in the United States. The first model describes a project owned by a municipal utility, such as the Hull Wind Project in Massachusetts. The second model is a wind project that is jointly owned by local community members, such as the MinWind Projects near Luverne, Minnesota. The third type is a flip-style ownership. This model allows local investors to partner with a corporation in order to take advantage of Production Tax Credit federal incentives. Flip projects have been built in Minnesota and Texas.

Business models

Community shared ownership

In a community-based model, the developer/manager of a wind farm shares ownership of the project with area landowners and other community members. Property owners whose land was used for the wind farm are generally given a choice between a monthly cash lease and ownership units in the development.

Cooperative

A wind turbine cooperative, also known as a wind energy cooperative, is a jointly owned and democratically controlled enterprise that follows the cooperative model, investing in wind turbines or wind farms. The cooperative model was developed in Denmark. The model has also spread to Germany, the Netherlands, Australia and United Kingdom, with isolated examples elsewhere. At a European level, REScoop.eu advocates for renewable energy cooperatives to have fair access to the market, linking individual cooperatives and federations under its umbrella, representing around 1,000,000 citizens and 1,500 cooperatives.

Municipal

Some places have enacted policies to encourage development of municipally owned and operated wind turbines on town land. These projects are publicly owned and tax exempt. An example is the Hull Wind One project in Massachusetts' Boston Harbor in 2001. A 660 kW wind turbine was installed, and is still a great example of small scale commercial wind.

Impacts of community wind energy

Economic

Once a wind farm project is established in a community, jobs are needed for: manufacturing the materials needed to build the project, transportation of supplies to the project area, and construction of the project as well as building roads leading to the project. After the project is complete, jobs will be needed to maintain and operate the facility. According to a study by the New York State Energy Research and Development Authority, wind energy produces 27% more jobs per kilowatt-hour than coal plants and 66% more jobs than natural gas plants. 3. Landowners will also collect revenues for hosting turbines on their property. Given a typical wind turbine spacing requirements, a 250-acre farm could increase annual farm income by $14,000 per year with little effect on their normal farming and ranching operations. 4. Community wind energy projects increase local property tax revenue because there was very little to be taxed previously due to the sparse population and vast farm land. Once the wind turbines are in service they are taxed, creating much needed revenue for the local community.

Social

The Midwest and the Great Plains regions in the United States are ideal areas for community wind energy projects; they are also often prone to drought. Fossil fuel plants use large amounts of water for cooling purposes which is detrimental to communities' water supply if there is a drought. Wind turbines do not use any water since there is no considerable amount of heat produced during energy generation. Wind energy adds power to the electric grid which decreases the amount of oil needed to generate a community's electricity. Local land owners, who produce the wind energy, can also control the amount of energy produced, which expands the regional energy mix. Overall community wind energy reduces the local community's dependence on oil but, because of the subsidies involved, can greatly increase their costs for electricity.

Environmental

Main article: Environmental impact of wind power

 

Livestock ignore wind turbines, and continue to graze as they did before wind turbines were installed.

Compared to the environmental impact of traditional energy sources, the environmental impact of wind power is relatively minor. Wind power consumes no fuel, and emits no air pollution, unlike fossil fuel power sources. The energy consumed to manufacture and transport the materials used to build a wind power plant is equal to the new energy produced by the plant within a few months. While a wind farm may cover a large area of land, many land uses such as agriculture are compatible, with only small areas of turbine foundations and infrastructure made unavailable for use.

There are reports of bird and bat mortality at wind turbines as there are around other artificial structures. The scale of the ecological impact may or may not be significant, depending on specific circumstances. Prevention and mitigation of wildlife fatalities, and protection of peat bogs, affect the siting and operation of wind turbines.

There are anecdotal reports of negative effects from noise on people who live very close to wind turbines. Peer-reviewed research has generally not supported these statements.

Policy, issues, and legislation

In 1992, the renewable energy production tax credit of 2.1 cents per kilowatt-hour was established. In February 2009, through the American Recovery and Reinvestment Act, Congress acted to provide a three-year extension of the PTC through December 31, 2012. Wind projects that were up and running in 2009 and 2010 can choose to receive a 30% investment tax credit instead of the PTC. The investment tax credit is also an option for wind projects that are in service before 2013 if the final construction is complete before the end of 2010. Smaller wind farms (100 kW or less) can receive a credit for 30% towards the cost of installment of the system. The ITC, written into law through the Emergency Economic Stabilization Act of 2008, is available for equipment installed from October 3, 2008 through December 31, 2016. The value of the credit is now uncapped, through the American Recovery and Reinvestment Act of 2009.

In order to ensure wind energy's future in the energy market, the renewable electricity standard (RES) is a policy in which market mechanisms guarantee a growing percentage of electricity produced comes from renewable sources, like wind energy. The RES exists in 28 states (not at a national level). An example is the Obama-Biden New Energy for America plan, which sets future goals of rapid renewable energy production at 10% by 2012.

A pressing issue of concern is the lack of a modern interstate transmission grid which delivers carbon free electricity to customers. Currently the US Senate and the Natural Resources Committee have reported the bill out of committee on June 17, 2009. A combined energy and climate bill is expected to be considered by the full Senate this fall. In the US House of Representatives the House Energy and Commerce Committee approved a comprehensive energy and climate bill on May 21, 2010.

The clean air and climate change policy is goal to switch from fossil fuel energy sources to renewable carbon-free energy sources for electricity production. Generating 20% of U.S. electricity from wind would be the climate equivalent of removing 140 million vehicles from the roadways. Currently the US Senate Committee on Environmental and Public Works has control over the legislation and will begin to complete a markup by September 25, 2009. The House of Representatives passed the American Clean Energy and Security Act on June 26, 2009, comprising a provision to reduce carbon dioxide emissions 17% below 2005 levels by 2020 and 83% below 2005 levels by 2050. It also allocates a portion of the allowances given away for free to energy efficiency and renewable energy. However, the allowances flow through state governments rather than directly to renewable generators.

Overall federal funding for community wind research and development is insufficient and even more so when compared to other fuels and energy sources. In 2009 the US Department of Energy (DOE) received $118 million from the American Recovery and Reinvestment Act for wind energy research and development. In 2010 the Senate passed a bill granting the DOE $85 million for the DOE wind program. For the same purpose, the House of Representatives allowed the DOE $70 million.