Taylor series

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

As the degree of the Taylor polynomial rises, it approaches the correct function. This image shows sin x and its Taylor approximations by polynomials of degree 1, 3, 5, 7, 9, 11, and 13 at x = 0.

In mathematics, the Taylor series of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor series are named after Brook Taylor, who introduced them in 1715. If 0 is the point where the derivatives are considered, a Taylor series is also called a Maclaurin series, after Colin Maclaurin, who made extensive use of this special case of Taylor series in the mid 1700s.

The partial sum formed by the first n + 1 terms of a Taylor series is a polynomial of degree n that is called the nth Taylor polynomial of the function. Taylor polynomials are approximations of a function, which become generally better as n increases. Taylor's theorem gives quantitative estimates on the error introduced by the use of such approximations. If the Taylor series of a function is convergent, its sum is the limit of the infinite sequence of the Taylor polynomials. A function may differ from the sum of its Taylor series, even if its Taylor series is convergent. A function is analytic at a point x if it is equal to the sum of its Taylor series in some open interval (or open disk in the complex plane) containing x. This implies that the function is analytic at every point of the interval (or disk).

Definition

The Taylor series of a real or complex-valued function f (x) that is infinitely differentiable at a real or complex number a is the power series

${\displaystyle f(a)+{\frac {f'(a)}{1!}}(x-a)+{\frac {f''(a)}{2!}}(x-a)^{2}+{\frac {f'''(a)}{3!}}(x-a)^{3}+\cdots ,}$

where n! denotes the factorial of n. In the more compact sigma notation, this can be written as

${\displaystyle \sum _{n=0}^{\infty }{\frac {f^{(n)}(a)}{n!}}(x-a)^{n},}$

where f⁽ⁿ⁾(a) denotes the nth derivative of f evaluated at the point a. (The derivative of order zero of f is defined to be f itself and (x − a)⁰ and 0! are both defined to be 1.)

When a = 0, the series is also called a Maclaurin series.

Examples

The Taylor series of any polynomial is the polynomial itself.

The Maclaurin series of 1/1 − x is the geometric series

${\displaystyle 1+x+x^{2}+x^{3}+\cdots .}$

So, by substituting x for 1 − x, the Taylor series of 1/x at a = 1 is

${\displaystyle 1-(x-1)+(x-1)^{2}-(x-1)^{3}+\cdots .}$

By integrating the above Maclaurin series, we find the Maclaurin series of ln(1 − x), where ln denotes the natural logarithm:

${\displaystyle -x-{\tfrac {1}{2}}x^{2}-{\tfrac {1}{3}}x^{3}-{\tfrac {1}{4}}x^{4}-\cdots .}$

The corresponding Taylor series of ln x at a = 1 is

${\displaystyle (x-1)-{\tfrac {1}{2}}(x-1)^{2}+{\tfrac {1}{3}}(x-1)^{3}-{\tfrac {1}{4}}(x-1)^{4}+\cdots ,}$

and more generally, the corresponding Taylor series of ln x at an arbitrary nonzero point a is:

${\displaystyle \ln a+{\frac {1}{a}}(x-a)-{\frac {1}{a^{2}}}{\frac {\left(x-a\right)^{2}}{2}}+\cdots .}$

The Maclaurin series of the exponential function e^x is

${\displaystyle {\begin{aligned}\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}&={\frac {x^{0}}{0!}}+{\frac {x^{1}}{1!}}+{\frac {x^{2}}{2!}}+{\frac {x^{3}}{3!}}+{\frac {x^{4}}{4!}}+{\frac {x^{5}}{5!}}+\cdots \\&=1+x+{\frac {x^{2}}{2}}+{\frac {x^{3}}{6}}+{\frac {x^{4}}{24}}+{\frac {x^{5}}{120}}+\cdots .\end{aligned}}}$

The above expansion holds because the derivative of e^x with respect to x is also e^x, and e⁰ equals 1. This leaves the terms (x − 0)ⁿ in the numerator and n! in the denominator of each term in the infinite sum.

History

The Greek philosopher Zeno considered the problem of summing an infinite series to achieve a finite result, but rejected it as an impossibility; the result was Zeno's paradox. Later, Aristotle proposed a philosophical resolution of the paradox, but the mathematical content was apparently unresolved until taken up by Archimedes, as it had been prior to Aristotle by the Presocratic Atomist Democritus. It was through Archimedes's method of exhaustion that an infinite number of progressive subdivisions could be performed to achieve a finite result. Liu Hui independently employed a similar method a few centuries later.

In the 14th century, the earliest examples of the use of Taylor series and closely related methods were given by Madhava of Sangamagrama. Though no record of his work survives, writings of later Indian mathematicians suggest that he found a number of special cases of the Taylor series, including those for the trigonometric functions of sine, cosine, tangent, and arctangent. The Kerala School of Astronomy and Mathematics further expanded his works with various series expansions and rational approximations until the 16th century.

In the 17th century, James Gregory also worked in this area and published several Maclaurin series. It was not until 1715 however that a general method for constructing these series for all functions for which they exist was finally provided by Brook Taylor, after whom the series are now named.

The Maclaurin series was named after Colin Maclaurin, a professor in Edinburgh, who published the special case of the Taylor result in the mid 1700s.

Analytic functions

The function e^(−1/x2) is not analytic at x = 0: the Taylor series is identically 0, although the function is not.

Main article: Analytic function

If f (x) is given by a convergent power series in an open disk centred at b in the complex plane (or an interval in the real line), it is said to be analytic in this region. Thus for x in this region, f is given by a convergent power series

${\displaystyle f(x)=\sum _{n=0}^{\infty }a_{n}(x-b)^{n}.}$

Differentiating by x the above formula n times, then setting x = b gives:

${\displaystyle {\frac {f^{(n)}(b)}{n!}}=a_{n}}$

and so the power series expansion agrees with the Taylor series. Thus a function is analytic in an open disk centred at b if and only if its Taylor series converges to the value of the function at each point of the disk.

If f (x) is equal to the sum of its Taylor series for all x in the complex plane, it is called entire. The polynomials, exponential function e^x, and the trigonometric functions sine and cosine, are examples of entire functions. Examples of functions that are not entire include the square root, the logarithm, the trigonometric function tangent, and its inverse, arctan. For these functions the Taylor series do not converge if x is far from b. That is, the Taylor series diverges at x if the distance between x and b is larger than the radius of convergence. The Taylor series can be used to calculate the value of an entire function at every point, if the value of the function, and of all of its derivatives, are known at a single point.

Uses of the Taylor series for analytic functions include:

1. The partial sums (the Taylor polynomials) of the series can be used as approximations of the function. These approximations are good if sufficiently many terms are included.
2. Differentiation and integration of power series can be performed term by term and is hence particularly easy.
3. An analytic function is uniquely extended to a holomorphic function on an open disk in the complex plane. This makes the machinery of complex analysis available.
4. The (truncated) series can be used to compute function values numerically, (often by recasting the polynomial into the Chebyshev form and evaluating it with the Clenshaw algorithm).
5. Algebraic operations can be done readily on the power series representation; for instance, Euler's formula follows from Taylor series expansions for trigonometric and exponential functions. This result is of fundamental importance in such fields as harmonic analysis.
6. Approximations using the first few terms of a Taylor series can make otherwise unsolvable problems possible for a restricted domain; this approach is often used in physics.

Approximation error and convergence

Main article: Taylor's theorem

 

The sine function (blue) is closely approximated by its Taylor polynomial of degree 7 (pink) for a full period centered at the origin.

 

The Taylor polynomials for ln(1 + x) only provide accurate approximations in the range −1 < x ≤ 1. For x > 1, Taylor polynomials of higher degree provide worse approximations.

 

The Taylor approximations for ln(1 + x) (black). For x > 1, the approximations diverge.

Pictured is an accurate approximation of sin x around the point x = 0. The pink curve is a polynomial of degree seven:

${\displaystyle \sin \left(x\right)\approx x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}.\!}$

The error in this approximation is no more than |x|⁹ / 9!. For a full cycle centered at the origin (−π < x < π) the error is less than 0.08215. In particular, for −1 < x < 1, the error is less than 0.000003.

In contrast, also shown is a picture of the natural logarithm function ln(1 + x) and some of its Taylor polynomials around a = 0. These approximations converge to the function only in the region −1 < x ≤ 1; outside of this region the higher-degree Taylor polynomials are worse approximations for the function.

The error incurred in approximating a function by its nth-degree Taylor polynomial is called the remainder or residual and is denoted by the function R_n(x). Taylor's theorem can be used to obtain a bound on the size of the remainder.

In general, Taylor series need not be convergent at all. And in fact the set of functions with a convergent Taylor series is a meager set in the Fréchet space of smooth functions. And even if the Taylor series of a function f does converge, its limit need not in general be equal to the value of the function f (x). For example, the function

${\displaystyle f(x)={\begin{cases}e^{-1/x^{2}}&{\text{if }}x\neq 0\\[3mu]0&{\text{if }}x=0\end{cases}}}$

is infinitely differentiable at x = 0, and has all derivatives zero there. Consequently, the Taylor series of f (x) about x = 0 is identically zero. However, f (x) is not the zero function, so does not equal its Taylor series around the origin. Thus, f (x) is an example of a non-analytic smooth function.

In real analysis, this example shows that there are infinitely differentiable functions f (x) whose Taylor series are not equal to f (x) even if they converge. By contrast, the holomorphic functions studied in complex analysis always possess a convergent Taylor series, and even the Taylor series of meromorphic functions, which might have singularities, never converge to a value different from the function itself. The complex function e^−1/z2, however, does not approach 0 when z approaches 0 along the imaginary axis, so it is not continuous in the complex plane and its Taylor series is undefined at 0.

More generally, every sequence of real or complex numbers can appear as coefficients in the Taylor series of an infinitely differentiable function defined on the real line, a consequence of Borel's lemma. As a result, the radius of convergence of a Taylor series can be zero. There are even infinitely differentiable functions defined on the real line whose Taylor series have a radius of convergence 0 everywhere.

A function cannot be written as a Taylor series centred at a singularity; in these cases, one can often still achieve a series expansion if one allows also negative powers of the variable x; see Laurent series. For example, f (x) = e^−1/x2 can be written as a Laurent series.

Generalization

There is, however, a generalization of the Taylor series that does converge to the value of the function itself for any bounded continuous function on (0,∞), using the calculus of finite differences. Specifically, one has the following theorem, due to Einar Hille, that for any t > 0,

${\displaystyle \lim _{h\to 0^{+}}\sum _{n=0}^{\infty }{\frac {t^{n}}{n!}}{\frac {\Delta _{h}^{n}f(a)}{h^{n}}}=f(a+t).}$

Here Δⁿ
_h is the nth finite difference operator with step size h. The series is precisely the Taylor series, except that divided differences appear in place of differentiation: the series is formally similar to the Newton series. When the function f is analytic at a, the terms in the series converge to the terms of the Taylor series, and in this sense generalizes the usual Taylor series.

In general, for any infinite sequence a_i, the following power series identity holds:

${\displaystyle \sum _{n=0}^{\infty }{\frac {u^{n}}{n!}}\Delta ^{n}a_{i}=e^{-u}\sum _{j=0}^{\infty }{\frac {u^{j}}{j!}}a_{i+j}.}$

So in particular,

${\displaystyle f(a+t)=\lim _{h\to 0^{+}}e^{-t/h}\sum _{j=0}^{\infty }f(a+jh){\frac {(t/h)^{j}}{j!}}.}$

The series on the right is the expectation value of f (a + X), where X is a Poisson-distributed random variable that takes the value jh with probability e^−t/h·(t/h)^j/j!. Hence,

${\displaystyle f(a+t)=\lim _{h\to 0^{+}}\int _{-\infty }^{\infty }f(a+x)dP_{t/h,h}(x).}$

The law of large numbers implies that the identity holds.

List of Maclaurin series of some common functions

See also: List of mathematical series

Several important Maclaurin series expansions follow. All these expansions are valid for complex arguments x.

Exponential function

The exponential function e^x (in blue), and the sum of the first n + 1 terms of its Taylor series at 0 (in red).

The exponential function ${\displaystyle e^{x}}$ (with base e) has Maclaurin series

${\displaystyle e^{x}=\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}=1+x+{\frac {x^{2}}{2!}}+{\frac {x^{3}}{3!}}+\cdots }$.

It converges for all x.

The exponential generating function of the Bell numbers is the exponential function of the predecessor of the exponential function:

${\displaystyle \exp[\exp(x)-1]=\sum _{n=0}^{\infty }{\frac {B_{n}}{n!}}x^{n}}$

Natural logarithm

Main article: Mercator series

The natural logarithm (with base e) has Maclaurin series

${\displaystyle {\begin{aligned}\ln(1-x)&=-\sum _{n=1}^{\infty }{\frac {x^{n}}{n}}=-x-{\frac {x^{2}}{2}}-{\frac {x^{3}}{3}}-\cdots ,\\\ln(1+x)&=\sum _{n=1}^{\infty }(-1)^{n+1}{\frac {x^{n}}{n}}=x-{\frac {x^{2}}{2}}+{\frac {x^{3}}{3}}-\cdots .\end{aligned}}}$

They converge for ${\displaystyle |x|<1}$. (In addition, the series for ln(1 − x) converges for x = −1, and the series for ln(1 + x) converges for x = 1.)

Geometric series

The geometric series and its derivatives have Maclaurin series

${\displaystyle {\begin{aligned}{\frac {1}{1-x}}&=\sum _{n=0}^{\infty }x^{n}\\{\frac {1}{(1-x)^{2}}}&=\sum _{n=1}^{\infty }nx^{n-1}\\{\frac {1}{(1-x)^{3}}}&=\sum _{n=2}^{\infty }{\frac {(n-1)n}{2}}x^{n-2}.\end{aligned}}}$

All are convergent for ${\displaystyle |x|<1}$. These are special cases of the binomial series given in the next section.

Binomial series

The binomial series is the power series

$${\displaystyle (1+x)^{\alpha }=\sum _{n=0}^{\infty }{\binom {\alpha }{n}}x^{n}}$$

whose coefficients are the generalized binomial coefficients

$${\displaystyle {\binom {\alpha }{n}}=\prod _{k=1}^{n}{\frac {\alpha -k+1}{k}}={\frac {\alpha (\alpha -1)\cdots (\alpha -n+1)}{n!}}.}$$

(If n = 0, this product is an empty product and has value 1.) It converges for ${\displaystyle |x|<1}$ for any real or complex number α.

When α = −1, this is essentially the infinite geometric series mentioned in the previous section. The special cases α = 1/2 and α = −1/2 give the square root function and its inverse:

$${\displaystyle {\begin{aligned}(1+x)^{\frac {1}{2}}&=1+{\tfrac {1}{2}}x-{\tfrac {1}{8}}x^{2}+{\tfrac {1}{16}}x^{3}-{\tfrac {5}{128}}x^{4}+{\tfrac {7}{256}}x^{5}-\cdots &&=\sum _{n=0}^{\infty }{\frac {(-1)^{n-1}(2n)!}{4^{n}(n!)^{2}(2n-1)}}x^{n},\\(1+x)^{-{\frac {1}{2}}}&=1-{\tfrac {1}{2}}x+{\tfrac {3}{8}}x^{2}-{\tfrac {5}{16}}x^{3}+{\tfrac {35}{128}}x^{4}-{\tfrac {63}{256}}x^{5}+\cdots &&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}(2n)!}{4^{n}(n!)^{2}}}x^{n}.\end{aligned}}}$$

When only the linear term is retained, this simplifies to the binomial approximation.

Trigonometric functions

The usual trigonometric functions and their inverses have the following Maclaurin series:

${\displaystyle {\begin{aligned}\sin x&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)!}}x^{2n+1}&&=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-\cdots &&{\text{for all }}x\\[6pt]\cos x&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n)!}}x^{2n}&&=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-\cdots &&{\text{for all }}x\\[6pt]\tan x&=\sum _{n=1}^{\infty }{\frac {B_{2n}(-4)^{n}\left(1-4^{n}\right)}{(2n)!}}x^{2n-1}&&=x+{\frac {x^{3}}{3}}+{\frac {2x^{5}}{15}}+\cdots &&{\text{for }}|x|<{\frac {\pi }{2}}\\[6pt]\sec x&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}E_{2n}}{(2n)!}}x^{2n}&&=1+{\frac {x^{2}}{2}}+{\frac {5x^{4}}{24}}+\cdots &&{\text{for }}|x|<{\frac {\pi }{2}}\\[6pt]\arcsin x&=\sum _{n=0}^{\infty }{\frac {(2n)!}{4^{n}(n!)^{2}(2n+1)}}x^{2n+1}&&=x+{\frac {x^{3}}{6}}+{\frac {3x^{5}}{40}}+\cdots &&{\text{for }}|x|\leq 1\\[6pt]\arccos x&={\frac {\pi }{2}}-\arcsin x\\&={\frac {\pi }{2}}-\sum _{n=0}^{\infty }{\frac {(2n)!}{4^{n}(n!)^{2}(2n+1)}}x^{2n+1}&&={\frac {\pi }{2}}-x-{\frac {x^{3}}{6}}-{\frac {3x^{5}}{40}}-\cdots &&{\text{for }}|x|\leq 1\\[6pt]\arctan x&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{2n+1}}x^{2n+1}&&=x-{\frac {x^{3}}{3}}+{\frac {x^{5}}{5}}-\cdots &&{\text{for }}|x|\leq 1,\ x\neq \pm i\end{aligned}}}$

All angles are expressed in radians. The numbers B_k appearing in the expansions of tan x are the Bernoulli numbers. The E_k in the expansion of sec x are Euler numbers.

Hyperbolic functions

The hyperbolic functions have Maclaurin series closely related to the series for the corresponding trigonometric functions:

${\displaystyle {\begin{aligned}\sinh x&=\sum _{n=0}^{\infty }{\frac {x^{2n+1}}{(2n+1)!}}&&=x+{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}+\cdots &&{\text{for all }}x\\[6pt]\cosh x&=\sum _{n=0}^{\infty }{\frac {x^{2n}}{(2n)!}}&&=1+{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}+\cdots &&{\text{for all }}x\\[6pt]\tanh x&=\sum _{n=1}^{\infty }{\frac {B_{2n}4^{n}\left(4^{n}-1\right)}{(2n)!}}x^{2n-1}&&=x-{\frac {x^{3}}{3}}+{\frac {2x^{5}}{15}}-{\frac {17x^{7}}{315}}+\cdots &&{\text{for }}|x|<{\frac {\pi }{2}}\\[6pt]\operatorname {arsinh} x&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}(2n)!}{4^{n}(n!)^{2}(2n+1)}}x^{2n+1}&&=x-{\frac {x^{3}}{6}}+{\frac {3x^{5}}{40}}-\cdots &&{\text{for }}|x|\leq 1\\[6pt]\operatorname {artanh} x&=\sum _{n=0}^{\infty }{\frac {x^{2n+1}}{2n+1}}&&=x+{\frac {x^{3}}{3}}+{\frac {x^{5}}{5}}+\cdots &&{\text{for }}|x|\leq 1,\ x\neq \pm 1\end{aligned}}}$

The numbers B_k appearing in the series for tanh x are the Bernoulli numbers.

Polylogarithmic functions

The polylogarithms have these defining identities:

${\displaystyle {\text{Li}}_{2}(x)=\sum _{n=1}^{\infty }{\frac {1}{n^{2}}}x^{n}}$

${\displaystyle {\text{Li}}_{3}(x)=\sum _{n=1}^{\infty }{\frac {1}{n^{3}}}x^{n}}$

The Legendre chi functions are defined as follows:

${\displaystyle \chi _{2}(x)=\sum _{n=0}^{\infty }{\frac {1}{(2n+1)^{2}}}x^{2n+1}}$

${\displaystyle \chi _{3}(x)=\sum _{n=0}^{\infty }{\frac {1}{(2n+1)^{3}}}x^{2n+1}}$

And the formulas presented below are called inverse tangent integrals:

${\displaystyle {\text{Ti}}_{2}(x)=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)^{2}}}x^{2n+1}}$

${\displaystyle {\text{Ti}}_{3}(x)=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)^{3}}}x^{2n+1}}$

In statistical thermodynamics these formulas are of great importance.

Elliptic functions

The complete elliptic Integrals of first kind K and of second kind E can be defined as follows:

${\displaystyle {\frac {2}{\pi }}K(x)=\sum _{n=0}^{\infty }{\frac {[(2n)!]^{2}}{16^{n}(n!)^{4}}}x^{2n}}$

${\displaystyle {\frac {2}{\pi }}E(x)=\sum _{n=0}^{\infty }{\frac {[(2n)!]^{2}}{(1-2n)16^{n}(n!)^{4}}}x^{2n}}$

The Jacobi theta functions describe the world of the elliptic modular functions and they have these Taylor series:

${\displaystyle \vartheta _{00}(x)=1+2\sum _{n=1}^{\infty }x^{n^{2}}}$

${\displaystyle \vartheta _{01}(x)=1+2\sum _{n=1}^{\infty }(-1)^{n}x^{n^{2}}}$

The regular partition number sequence P(n) has this generating function:

${\displaystyle \vartheta _{00}(x)^{-1/6}\vartheta _{01}(x)^{-2/3}{\biggl [}{\frac {\vartheta _{00}(x)^{4}-\vartheta _{01}(x)^{4}}{16\,x}}{\biggr ]}^{-1/24}=\sum _{n=0}^{\infty }P(n)x^{n}=\prod _{k=1}^{\infty }{\frac {1}{1-x^{k}}}}$

The strict partition number sequence Q(n) has that generating function:

${\displaystyle \vartheta _{00}(x)^{1/6}\vartheta _{01}(x)^{-1/3}{\biggl [}{\frac {\vartheta _{00}(x)^{4}-\vartheta _{01}(x)^{4}}{16\,x}}{\biggr ]}^{1/24}=\sum _{n=0}^{\infty }Q(n)x^{n}=\prod _{k=1}^{\infty }{\frac {1}{1-x^{2k-1}}}}$

Calculation of Taylor series

Several methods exist for the calculation of Taylor series of a large number of functions. One can attempt to use the definition of the Taylor series, though this often requires generalizing the form of the coefficients according to a readily apparent pattern. Alternatively, one can use manipulations such as substitution, multiplication or division, addition or subtraction of standard Taylor series to construct the Taylor series of a function, by virtue of Taylor series being power series. In some cases, one can also derive the Taylor series by repeatedly applying integration by parts. Particularly convenient is the use of computer algebra systems to calculate Taylor series.

First example

In order to compute the 7th degree Maclaurin polynomial for the function

${\displaystyle f(x)=\ln(\cos x),\quad x\in \left(-{\frac {\pi }{2}},{\frac {\pi }{2}}\right)}$ ,

one may first rewrite the function as

${\displaystyle f(x)=\ln {\bigl (}1+(\cos x-1){\bigr )}\!}$.

The Taylor series for the natural logarithm is (using the big O notation)

${\displaystyle \ln(1+x)=x-{\frac {x^{2}}{2}}+{\frac {x^{3}}{3}}+{O}\left(x^{4}\right)\!}$

and for the cosine function

${\displaystyle \cos x-1=-{\frac {x^{2}}{2}}+{\frac {x^{4}}{24}}-{\frac {x^{6}}{720}}+{O}\left(x^{8}\right)\!}$.

The latter series expansion has a zero constant term, which enables us to substitute the second series into the first one and to easily omit terms of higher order than the 7th degree by using the big O notation:

${\displaystyle {\begin{aligned}f(x)&=\ln {\bigl (}1+(\cos x-1){\bigr )}\\&=(\cos x-1)-{\tfrac {1}{2}}(\cos x-1)^{2}+{\tfrac {1}{3}}(\cos x-1)^{3}+{O}\left((\cos x-1)^{4}\right)\\&=\left(-{\frac {x^{2}}{2}}+{\frac {x^{4}}{24}}-{\frac {x^{6}}{720}}+{O}\left(x^{8}\right)\right)-{\frac {1}{2}}\left(-{\frac {x^{2}}{2}}+{\frac {x^{4}}{24}}+{O}\left(x^{6}\right)\right)^{2}+{\frac {1}{3}}\left(-{\frac {x^{2}}{2}}+O\left(x^{4}\right)\right)^{3}+{O}\left(x^{8}\right)\\&=-{\frac {x^{2}}{2}}+{\frac {x^{4}}{24}}-{\frac {x^{6}}{720}}-{\frac {x^{4}}{8}}+{\frac {x^{6}}{48}}-{\frac {x^{6}}{24}}+O\left(x^{8}\right)\\&=-{\frac {x^{2}}{2}}-{\frac {x^{4}}{12}}-{\frac {x^{6}}{45}}+O\left(x^{8}\right).\end{aligned}}\!}$

Since the cosine is an even function, the coefficients for all the odd powers x, x³, x⁵, x⁷, ... have to be zero.

Second example

Suppose we want the Taylor series at 0 of the function

${\displaystyle g(x)={\frac {e^{x}}{\cos x}}.\!}$

We have for the exponential function

${\displaystyle e^{x}=\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}=1+x+{\frac {x^{2}}{2!}}+{\frac {x^{3}}{3!}}+{\frac {x^{4}}{4!}}+\cdots \!}$

and, as in the first example,

${\displaystyle \cos x=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-\cdots \!}$

Assume the power series is

${\displaystyle {\frac {e^{x}}{\cos x}}=c_{0}+c_{1}x+c_{2}x^{2}+c_{3}x^{3}+\cdots \!}$

Then multiplication with the denominator and substitution of the series of the cosine yields

${\displaystyle {\begin{aligned}e^{x}&=\left(c_{0}+c_{1}x+c_{2}x^{2}+c_{3}x^{3}+\cdots \right)\cos x\\&=\left(c_{0}+c_{1}x+c_{2}x^{2}+c_{3}x^{3}+c_{4}x^{4}+\cdots \right)\left(1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-\cdots \right)\\&=c_{0}-{\frac {c_{0}}{2}}x^{2}+{\frac {c_{0}}{4!}}x^{4}+c_{1}x-{\frac {c_{1}}{2}}x^{3}+{\frac {c_{1}}{4!}}x^{5}+c_{2}x^{2}-{\frac {c_{2}}{2}}x^{4}+{\frac {c_{2}}{4!}}x^{6}+c_{3}x^{3}-{\frac {c_{3}}{2}}x^{5}+{\frac {c_{3}}{4!}}x^{7}+c_{4}x^{4}+\cdots \end{aligned}}\!}$

Collecting the terms up to fourth order yields

${\displaystyle e^{x}=c_{0}+c_{1}x+\left(c_{2}-{\frac {c_{0}}{2}}\right)x^{2}+\left(c_{3}-{\frac {c_{1}}{2}}\right)x^{3}+\left(c_{4}-{\frac {c_{2}}{2}}+{\frac {c_{0}}{4!}}\right)x^{4}+\cdots \!}$

The values of ${\displaystyle c_{i}}$ can be found by comparison of coefficients with the top expression for ${\displaystyle e^{x}}$, yielding:

${\displaystyle {\frac {e^{x}}{\cos x}}=1+x+x^{2}+{\frac {2x^{3}}{3}}+{\frac {x^{4}}{2}}+\cdots .\!}$

Third example

Here we employ a method called "indirect expansion" to expand the given function. This method uses the known Taylor expansion of the exponential function. In order to expand (1 + x)e^x as a Taylor series in x, we use the known Taylor series of function e^x:

${\displaystyle e^{x}=\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}=1+x+{\frac {x^{2}}{2!}}+{\frac {x^{3}}{3!}}+{\frac {x^{4}}{4!}}+\cdots .}$

Thus,

${\displaystyle {\begin{aligned}(1+x)e^{x}&=e^{x}+xe^{x}=\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}+\sum _{n=0}^{\infty }{\frac {x^{n+1}}{n!}}=1+\sum _{n=1}^{\infty }{\frac {x^{n}}{n!}}+\sum _{n=0}^{\infty }{\frac {x^{n+1}}{n!}}\\&=1+\sum _{n=1}^{\infty }{\frac {x^{n}}{n!}}+\sum _{n=1}^{\infty }{\frac {x^{n}}{(n-1)!}}=1+\sum _{n=1}^{\infty }\left({\frac {1}{n!}}+{\frac {1}{(n-1)!}}\right)x^{n}\\&=1+\sum _{n=1}^{\infty }{\frac {n+1}{n!}}x^{n}\\&=\sum _{n=0}^{\infty }{\frac {n+1}{n!}}x^{n}.\end{aligned}}}$

Taylor series as definitions

Classically, algebraic functions are defined by an algebraic equation, and transcendental functions (including those discussed above) are defined by some property that holds for them, such as a differential equation. For example, the exponential function is the function which is equal to its own derivative everywhere, and assumes the value 1 at the origin. However, one may equally well define an analytic function by its Taylor series.

Taylor series are used to define functions and "operators" in diverse areas of mathematics. In particular, this is true in areas where the classical definitions of functions break down. For example, using Taylor series, one may extend analytic functions to sets of matrices and operators, such as the matrix exponential or matrix logarithm.

In other areas, such as formal analysis, it is more convenient to work directly with the power series themselves. Thus one may define a solution of a differential equation as a power series which, one hopes to prove, is the Taylor series of the desired solution.

Taylor series in several variables

The Taylor series may also be generalized to functions of more than one variable with

${\displaystyle {\begin{aligned}T(x_{1},\ldots ,x_{d})&=\sum _{n_{1}=0}^{\infty }\cdots \sum _{n_{d}=0}^{\infty }{\frac {(x_{1}-a_{1})^{n_{1}}\cdots (x_{d}-a_{d})^{n_{d}}}{n_{1}!\cdots n_{d}!}}\,\left({\frac {\partial ^{n_{1}+\cdots +n_{d}}f}{\partial x_{1}^{n_{1}}\cdots \partial x_{d}^{n_{d}}}}\right)(a_{1},\ldots ,a_{d})\\&=f(a_{1},\ldots ,a_{d})+\sum _{j=1}^{d}{\frac {\partial f(a_{1},\ldots ,a_{d})}{\partial x_{j}}}(x_{j}-a_{j})+{\frac {1}{2!}}\sum _{j=1}^{d}\sum _{k=1}^{d}{\frac {\partial ^{2}f(a_{1},\ldots ,a_{d})}{\partial x_{j}\partial x_{k}}}(x_{j}-a_{j})(x_{k}-a_{k})\\&\qquad \qquad +{\frac {1}{3!}}\sum _{j=1}^{d}\sum _{k=1}^{d}\sum _{l=1}^{d}{\frac {\partial ^{3}f(a_{1},\ldots ,a_{d})}{\partial x_{j}\partial x_{k}\partial x_{l}}}(x_{j}-a_{j})(x_{k}-a_{k})(x_{l}-a_{l})+\cdots \end{aligned}}}$

For example, for a function ${\displaystyle f(x,y)}$ that depends on two variables, x and y, the Taylor series to second order about the point (a, b) is

${\displaystyle f(a,b)+(x-a)f_{x}(a,b)+(y-b)f_{y}(a,b)+{\frac {1}{2!}}{\Big (}(x-a)^{2}f_{xx}(a,b)+2(x-a)(y-b)f_{xy}(a,b)+(y-b)^{2}f_{yy}(a,b){\Big )}}$

where the subscripts denote the respective partial derivatives.

A second-order Taylor series expansion of a scalar-valued function of more than one variable can be written compactly as

${\displaystyle T(\mathbf {x} )=f(\mathbf {a} )+(\mathbf {x} -\mathbf {a} )^{\mathsf {T}}Df(\mathbf {a} )+{\frac {1}{2!}}(\mathbf {x} -\mathbf {a} )^{\mathsf {T}}\left\{D^{2}f(\mathbf {a} )\right\}(\mathbf {x} -\mathbf {a} )+\cdots ,}$

where D f (a) is the gradient of f evaluated at x = a and D² f (a) is the Hessian matrix. Applying the multi-index notation the Taylor series for several variables becomes

${\displaystyle T(\mathbf {x} )=\sum _{|\alpha |\geq 0}{\frac {(\mathbf {x} -\mathbf {a} )^{\alpha }}{\alpha !}}\left({\mathrm {\partial } ^{\alpha }}f\right)(\mathbf {a} ),}$

which is to be understood as a still more abbreviated multi-index version of the first equation of this paragraph, with a full analogy to the single variable case.

Example

Second-order Taylor series approximation (in orange) of a function f (x,y) = e^x ln(1 + y) around the origin.

In order to compute a second-order Taylor series expansion around point (a, b) = (0, 0) of the function

${\displaystyle f(x,y)=e^{x}\ln(1+y),}$

one first computes all the necessary partial derivatives:

${\displaystyle {\begin{aligned}f_{x}&=e^{x}\ln(1+y)\\[6pt]f_{y}&={\frac {e^{x}}{1+y}}\\[6pt]f_{xx}&=e^{x}\ln(1+y)\\[6pt]f_{yy}&=-{\frac {e^{x}}{(1+y)^{2}}}\\[6pt]f_{xy}&=f_{yx}={\frac {e^{x}}{1+y}}.\end{aligned}}}$

Evaluating these derivatives at the origin gives the Taylor coefficients

${\displaystyle {\begin{aligned}f_{x}(0,0)&=0\\f_{y}(0,0)&=1\\f_{xx}(0,0)&=0\\f_{yy}(0,0)&=-1\\f_{xy}(0,0)&=f_{yx}(0,0)=1.\end{aligned}}}$

Substituting these values in to the general formula

${\displaystyle {\begin{aligned}T(x,y)=&f(a,b)+(x-a)f_{x}(a,b)+(y-b)f_{y}(a,b)\\&{}+{\frac {1}{2!}}\left((x-a)^{2}f_{xx}(a,b)+2(x-a)(y-b)f_{xy}(a,b)+(y-b)^{2}f_{yy}(a,b)\right)+\cdots \end{aligned}}}$

produces

${\displaystyle {\begin{aligned}T(x,y)&=0+0(x-0)+1(y-0)+{\frac {1}{2}}{\Big (}0(x-0)^{2}+2(x-0)(y-0)+(-1)(y-0)^{2}{\Big )}+\cdots \\&=y+xy-{\frac {y^{2}}{2}}+\cdots \end{aligned}}}$

Since ln(1 + y) is analytic in |y| < 1, we have

${\displaystyle e^{x}\ln(1+y)=y+xy-{\frac {y^{2}}{2}}+\cdots ,\qquad |y|<1.}$

Comparison with Fourier series

Main article: Fourier series

The trigonometric Fourier series enables one to express a periodic function (or a function defined on a closed interval [a,b]) as an infinite sum of trigonometric functions (sines and cosines). In this sense, the Fourier series is analogous to Taylor series, since the latter allows one to express a function as an infinite sum of powers. Nevertheless, the two series differ from each other in several relevant issues:

• The finite truncations of the Taylor series of f (x) about the point x = a are all exactly equal to f at a. In contrast, the Fourier series is computed by integrating over an entire interval, so there is generally no such point where all the finite truncations of the series are exact.
• The computation of Taylor series requires the knowledge of the function on an arbitrary small neighbourhood of a point, whereas the computation of the Fourier series requires knowing the function on its whole domain interval. In a certain sense one could say that the Taylor series is "local" and the Fourier series is "global".
• The Taylor series is defined for a function which has infinitely many derivatives at a single point, whereas the Fourier series is defined for any integrable function. In particular, the function could be nowhere differentiable. (For example, f (x) could be a Weierstrass function.)
• The convergence of both series has very different properties. Even if the Taylor series has positive convergence radius, the resulting series may not coincide with the function; but if the function is analytic then the series converges pointwise to the function, and uniformly on every compact subset of the convergence interval. Concerning the Fourier series, if the function is square-integrable then the series converges in quadratic mean, but additional requirements are needed to ensure the pointwise or uniform convergence (for instance, if the function is periodic and of class C¹ then the convergence is uniform).
• Finally, in practice one wants to approximate the function with a finite number of terms, say with a Taylor polynomial or a partial sum of the trigonometric series, respectively. In the case of the Taylor series the error is very small in a neighbourhood of the point where it is computed, while it may be very large at a distant point. In the case of the Fourier series the error is distributed along the domain of the function.

Algorithmic bias

From Wikipedia, the free encyclopedia

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

A flow chart showing the decisions made by a recommendation engine, circa 2001.

Algorithmic bias describes systematic and repeatable errors in a computer system that create "unfair" outcomes, such as "privileging" one category over another in ways different from the intended function of the algorithm.

Bias can emerge from many factors, including but not limited to the design of the algorithm or the unintended or unanticipated use or decisions relating to the way data is coded, collected, selected or used to train the algorithm. For example, algorithmic bias has been observed in search engine results and social media platforms. This bias can have impacts ranging from inadvertent privacy violations to reinforcing social biases of race, gender, sexuality, and ethnicity. The study of algorithmic bias is most concerned with algorithms that reflect "systematic and unfair" discrimination. This bias has only recently been addressed in legal frameworks, such as the European Union's General Data Protection Regulation (2018) and the proposed Artificial Intelligence Act (2021).

As algorithms expand their ability to organize society, politics, institutions, and behavior, sociologists have become concerned with the ways in which unanticipated output and manipulation of data can impact the physical world. Because algorithms are often considered to be neutral and unbiased, they can inaccurately project greater authority than human expertise (in part due to the psychological phenomenon of automation bias), and in some cases, reliance on algorithms can displace human responsibility for their outcomes. Bias can enter into algorithmic systems as a result of pre-existing cultural, social, or institutional expectations; because of technical limitations of their design; or by being used in unanticipated contexts or by audiences who are not considered in the software's initial design.

Algorithmic bias has been cited in cases ranging from election outcomes to the spread of online hate speech. It has also arisen in criminal justice, healthcare, and hiring, compounding existing racial, socioeconomic, and gender biases. The relative inability of facial recognition technology to accurately identify darker-skinned faces has been linked to multiple wrongful arrests of black men, an issue stemming from imbalanced datasets. Problems in understanding, researching, and discovering algorithmic bias persist due to the proprietary nature of algorithms, which are typically treated as trade secrets. Even when full transparency is provided, the complexity of certain algorithms poses a barrier to understanding their functioning. Furthermore, algorithms may change, or respond to input or output in ways that cannot be anticipated or easily reproduced for analysis. In many cases, even within a single website or application, there is no single "algorithm" to examine, but a network of many interrelated programs and data inputs, even between users of the same service.

Definitions

A 1969 diagram for how a simple computer program makes decisions, illustrating a very simple algorithm.

Algorithms are difficult to define, but may be generally understood as lists of instructions that determine how programs read, collect, process, and analyze data to generate output. For a rigorous technical introduction, see Algorithms. Advances in computer hardware have led to an increased ability to process, store and transmit data. This has in turn boosted the design and adoption of technologies such as machine learning and artificial intelligence. By analyzing and processing data, algorithms are the backbone of search engines, social media websites, recommendation engines, online retail, online advertising, and more.

Contemporary social scientists are concerned with algorithmic processes embedded into hardware and software applications because of their political and social impact, and question the underlying assumptions of an algorithm's neutrality. The term algorithmic bias describes systematic and repeatable errors that create unfair outcomes, such as privileging one arbitrary group of users over others. For example, a credit score algorithm may deny a loan without being unfair, if it is consistently weighing relevant financial criteria. If the algorithm recommends loans to one group of users, but denies loans to another set of nearly identical users based on unrelated criteria, and if this behavior can be repeated across multiple occurrences, an algorithm can be described as biased. This bias may be intentional or unintentional (for example, it can come from biased data obtained from a worker that previously did the job the algorithm is going to do from now on).

Methods

Bias can be introduced to an algorithm in several ways. During the assemblage of a dataset, data may be collected, digitized, adapted, and entered into a database according to human-designed cataloging criteria. Next, programmers assign priorities, or hierarchies, for how a program assesses and sorts that data. This requires human decisions about how data is categorized, and which data is included or discarded. Some algorithms collect their own data based on human-selected criteria, which can also reflect the bias of human designers. Other algorithms may reinforce stereotypes and preferences as they process and display "relevant" data for human users, for example, by selecting information based on previous choices of a similar user or group of users.

Beyond assembling and processing data, bias can emerge as a result of design. For example, algorithms that determine the allocation of resources or scrutiny (such as determining school placements) may inadvertently discriminate against a category when determining risk based on similar users (as in credit scores). Meanwhile, recommendation engines that work by associating users with similar users, or that make use of inferred marketing traits, might rely on inaccurate associations that reflect broad ethnic, gender, socio-economic, or racial stereotypes. Another example comes from determining criteria for what is included and excluded from results. This criteria could present unanticipated outcomes for search results, such as with flight-recommendation software that omits flights that do not follow the sponsoring airline's flight paths. Algorithms may also display an uncertainty bias, offering more confident assessments when larger data sets are available. This can skew algorithmic processes toward results that more closely correspond with larger samples, which may disregard data from underrepresented populations.

History

Early critiques

This card was used to load software into an old mainframe computer. Each byte (the letter 'A', for example) is entered by punching holes. Though contemporary computers are more complex, they reflect this human decision-making process in collecting and processing data.

The earliest computer programs were designed to mimic human reasoning and deductions, and were deemed to be functioning when they successfully and consistently reproduced that human logic. In his 1976 book Computer Power and Human Reason, artificial intelligence pioneer Joseph Weizenbaum suggested that bias could arise both from the data used in a program, but also from the way a program is coded.

Weizenbaum wrote that programs are a sequence of rules created by humans for a computer to follow. By following those rules consistently, such programs "embody law", that is, enforce a specific way to solve problems. The rules a computer follows are based on the assumptions of a computer programmer for how these problems might be solved. That means the code could incorporate the programmer's imagination of how the world works, including their biases and expectations. While a computer program can incorporate bias in this way, Weizenbaum also noted that any data fed to a machine additionally reflects "human decisionmaking processes" as data is being selected.

Finally, he noted that machines might also transfer good information with unintended consequences if users are unclear about how to interpret the results. Weizenbaum warned against trusting decisions made by computer programs that a user doesn't understand, comparing such faith to a tourist who can find his way to a hotel room exclusively by turning left or right on a coin toss. Crucially, the tourist has no basis of understanding how or why he arrived at his destination, and a successful arrival does not mean the process is accurate or reliable.

An early example of algorithmic bias resulted in as many as 60 women and ethnic minorities denied entry to St. George's Hospital Medical School per year from 1982 to 1986, based on implementation of a new computer-guidance assessment system that denied entry to women and men with "foreign-sounding names" based on historical trends in admissions. While many schools at the time employed similar biases in their selection process, St. George was most notable for automating said bias through the use of an algorithm, thus gaining the attention of people on a much wider scale.

In recent years, when more algorithms started to use machine learning methods on real world data, algorithmic bias can be found more often due to the bias existing in the data.

Contemporary critiques and responses

Though well-designed algorithms frequently determine outcomes that are equally (or more) equitable than the decisions of human beings, cases of bias still occur, and are difficult to predict and analyze. The complexity of analyzing algorithmic bias has grown alongside the complexity of programs and their design. Decisions made by one designer, or team of designers, may be obscured among the many pieces of code created for a single program; over time these decisions and their collective impact on the program's output may be forgotten. In theory, these biases may create new patterns of behavior, or "scripts", in relationship to specific technologies as the code interacts with other elements of society. Biases may also impact how society shapes itself around the data points that algorithms require. For example, if data shows a high number of arrests in a particular area, an algorithm may assign more police patrols to that area, which could lead to more arrests.

The decisions of algorithmic programs can be seen as more authoritative than the decisions of the human beings they are meant to assist, a process described by author Clay Shirky as "algorithmic authority". Shirky uses the term to describe "the decision to regard as authoritative an unmanaged process of extracting value from diverse, untrustworthy sources", such as search results. This neutrality can also be misrepresented by the language used by experts and the media when results are presented to the public. For example, a list of news items selected and presented as "trending" or "popular" may be created based on significantly wider criteria than just their popularity.

Because of their convenience and authority, algorithms are theorized as a means of delegating responsibility away from humans. This can have the effect of reducing alternative options, compromises, or flexibility. Sociologist Scott Lash has critiqued algorithms as a new form of "generative power", in that they are a virtual means of generating actual ends. Where previously human behavior generated data to be collected and studied, powerful algorithms increasingly could shape and define human behaviors.

Concerns over the impact of algorithms on society have led to the creation of working groups in organizations such as Google and Microsoft, which have co-created a working group named Fairness, Accountability, and Transparency in Machine Learning. Ideas from Google have included community groups that patrol the outcomes of algorithms and vote to control or restrict outputs they deem to have negative consequences. In recent years, the study of the Fairness, Accountability, and Transparency (FAT) of algorithms has emerged as its own interdisciplinary research area with an annual conference called FAccT. Critics have suggested that FAT initiatives cannot serve effectively as independent watchdogs when many are funded by corporations building the systems being studied.

Types

Pre-existing

Pre-existing bias in an algorithm is a consequence of underlying social and institutional ideologies. Such ideas may influence or create personal biases within individual designers or programmers. Such prejudices can be explicit and conscious, or implicit and unconscious. Poorly selected input data, or simply data from a biased source, will influence the outcomes created by machines. Encoding pre-existing bias into software can preserve social and institutional bias, and, without correction, could be replicated in all future uses of that algorithm.

An example of this form of bias is the British Nationality Act Program, designed to automate the evaluation of new British citizens after the 1981 British Nationality Act. The program accurately reflected the tenets of the law, which stated that "a man is the father of only his legitimate children, whereas a woman is the mother of all her children, legitimate or not." In its attempt to transfer a particular logic into an algorithmic process, the BNAP inscribed the logic of the British Nationality Act into its algorithm, which would perpetuate it even if the act was eventually repealed.

Technical

Facial recognition software used in conjunction with surveillance cameras was found to display bias in recognizing Asian and black faces over white faces.

Technical bias emerges through limitations of a program, computational power, its design, or other constraint on the system. Such bias can also be a restraint of design, for example, a search engine that shows three results per screen can be understood to privilege the top three results slightly more than the next three, as in an airline price display. Another case is software that relies on randomness for fair distributions of results. If the random number generation mechanism is not truly random, it can introduce bias, for example, by skewing selections toward items at the end or beginning of a list.

A decontextualized algorithm uses unrelated information to sort results, for example, a flight-pricing algorithm that sorts results by alphabetical order would be biased in favor of American Airlines over United Airlines. The opposite may also apply, in which results are evaluated in contexts different from which they are collected. Data may be collected without crucial external context: for example, when facial recognition software is used by surveillance cameras, but evaluated by remote staff in another country or region, or evaluated by non-human algorithms with no awareness of what takes place beyond the camera's field of vision. This could create an incomplete understanding of a crime scene, for example, potentially mistaking bystanders for those who commit the crime.

Lastly, technical bias can be created by attempting to formalize decisions into concrete steps on the assumption that human behavior works in the same way. For example, software weighs data points to determine whether a defendant should accept a plea bargain, while ignoring the impact of emotion on a jury. Another unintended result of this form of bias was found in the plagiarism-detection software Turnitin, which compares student-written texts to information found online and returns a probability score that the student's work is copied. Because the software compares long strings of text, it is more likely to identify non-native speakers of English than native speakers, as the latter group might be better able to change individual words, break up strings of plagiarized text, or obscure copied passages through synonyms. Because it is easier for native speakers to evade detection as a result of the technical constraints of the software, this creates a scenario where Turnitin identifies foreign-speakers of English for plagiarism while allowing more native-speakers to evade detection.

Emergent

Emergent bias is the result of the use and reliance on algorithms across new or unanticipated contexts. Algorithms may not have been adjusted to consider new forms of knowledge, such as new drugs or medical breakthroughs, new laws, business models, or shifting cultural norms. This may exclude groups through technology, without providing clear outlines to understand who is responsible for their exclusion. Similarly, problems may emerge when training data (the samples "fed" to a machine, by which it models certain conclusions) do not align with contexts that an algorithm encounters in the real world.

In 1990, an example of emergent bias was identified in the software used to place US medical students into residencies, the National Residency Match Program (NRMP). The algorithm was designed at a time when few married couples would seek residencies together. As more women entered medical schools, more students were likely to request a residency alongside their partners. The process called for each applicant to provide a list of preferences for placement across the US, which was then sorted and assigned when a hospital and an applicant both agreed to a match. In the case of married couples where both sought residencies, the algorithm weighed the location choices of the higher-rated partner first. The result was a frequent assignment of highly preferred schools to the first partner and lower-preferred schools to the second partner, rather than sorting for compromises in placement preference.

Additional emergent biases include:

Correlations

Unpredictable correlations can emerge when large data sets are compared to each other. For example, data collected about web-browsing patterns may align with signals marking sensitive data (such as race or sexual orientation). By selecting according to certain behavior or browsing patterns, the end effect would be almost identical to discrimination through the use of direct race or sexual orientation data. In other cases, the algorithm draws conclusions from correlations, without being able to understand those correlations. For example, one triage program gave lower priority to asthmatics who had pneumonia than asthmatics who did not have pneumonia. The program algorithm did this because it simply compared survival rates: asthmatics with pneumonia are at the highest risk. Historically, for this same reason, hospitals typically give such asthmatics the best and most immediate care.

Unanticipated uses

Emergent bias can occur when an algorithm is used by unanticipated audiences. For example, machines may require that users can read, write, or understand numbers, or relate to an interface using metaphors that they do not understand. These exclusions can become compounded, as biased or exclusionary technology is more deeply integrated into society.

Apart from exclusion, unanticipated uses may emerge from the end user relying on the software rather than their own knowledge. In one example, an unanticipated user group led to algorithmic bias in the UK, when the British National Act Program was created as a proof-of-concept by computer scientists and immigration lawyers to evaluate suitability for British citizenship. The designers had access to legal expertise beyond the end users in immigration offices, whose understanding of both software and immigration law would likely have been unsophisticated. The agents administering the questions relied entirely on the software, which excluded alternative pathways to citizenship, and used the software even after new case laws and legal interpretations led the algorithm to become outdated. As a result of designing an algorithm for users assumed to be legally savvy on immigration law, the software's algorithm indirectly led to bias in favor of applicants who fit a very narrow set of legal criteria set by the algorithm, rather than by the more broader criteria of British immigration law.

Feedback loops

Emergent bias may also create a feedback loop, or recursion, if data collected for an algorithm results in real-world responses which are fed back into the algorithm. For example, simulations of the predictive policing software (PredPol), deployed in Oakland, California, suggested an increased police presence in black neighborhoods based on crime data reported by the public. The simulation showed that the public reported crime based on the sight of police cars, regardless of what police were doing. The simulation interpreted police car sightings in modeling its predictions of crime, and would in turn assign an even larger increase of police presence within those neighborhoods. The Human Rights Data Analysis Group, which conducted the simulation, warned that in places where racial discrimination is a factor in arrests, such feedback loops could reinforce and perpetuate racial discrimination in policing. Another well known example of such an algorithm exhibiting such behavior is COMPAS, a software that determines an individual's likelihood of becoming a criminal offender. The software is often criticized for labeling Black individuals as criminals much more likely than others, and then feeds the data back into itself in the event individuals become registered criminals, further enforcing the bias created by the dataset the algorithm is acting on.

Recommender systems such as those used to recommend online videos or news articles can create feedback loops. When users click on content that is suggested by algorithms, it influences the next set of suggestions. Over time this may lead to users entering a filter bubble and being unaware of important or useful content.

Impact

Commercial influences

Corporate algorithms could be skewed to invisibly favor financial arrangements or agreements between companies, without the knowledge of a user who may mistake the algorithm as being impartial. For example, American Airlines created a flight-finding algorithm in the 1980s. The software presented a range of flights from various airlines to customers, but weighed factors that boosted its own flights, regardless of price or convenience. In testimony to the United States Congress, the president of the airline stated outright that the system was created with the intention of gaining competitive advantage through preferential treatment.

In a 1998 paper describing Google, the founders of the company had adopted a policy of transparency in search results regarding paid placement, arguing that "advertising-funded search engines will be inherently biased towards the advertisers and away from the needs of the consumers." This bias would be an "invisible" manipulation of the user.

Voting behavior

A series of studies about undecided voters in the US and in India found that search engine results were able to shift voting outcomes by about 20%. The researchers concluded that candidates have "no means of competing" if an algorithm, with or without intent, boosted page listings for a rival candidate. Facebook users who saw messages related to voting were more likely to vote. A 2010 randomized trial of Facebook users showed a 20% increase (340,000 votes) among users who saw messages encouraging voting, as well as images of their friends who had voted. Legal scholar Jonathan Zittrain has warned that this could create a "digital gerrymandering" effect in elections, "the selective presentation of information by an intermediary to meet its agenda, rather than to serve its users", if intentionally manipulated.

Gender discrimination

In 2016, the professional networking site LinkedIn was discovered to recommend male variations of women's names in response to search queries. The site did not make similar recommendations in searches for male names. For example, "Andrea" would bring up a prompt asking if users meant "Andrew", but queries for "Andrew" did not ask if users meant to find "Andrea". The company said this was the result of an analysis of users' interactions with the site.

In 2012, the department store franchise Target was cited for gathering data points to infer when women customers were pregnant, even if they had not announced it, and then sharing that information with marketing partners. Because the data had been predicted, rather than directly observed or reported, the company had no legal obligation to protect the privacy of those customers.

Web search algorithms have also been accused of bias. Google's results may prioritize pornographic content in search terms related to sexuality, for example, "lesbian". This bias extends to the search engine showing popular but sexualized content in neutral searches. For example, "Top 25 Sexiest Women Athletes" articles displayed as first-page results in searches for "women athletes". In 2017, Google adjusted these results along with others that surfaced hate groups, racist views, child abuse and pornography, and other upsetting and offensive content. Other examples include the display of higher-paying jobs to male applicants on job search websites. Researchers have also identified that machine translation exhibits a strong tendency towards male defaults. In particular, this is observed in fields linked to unbalanced gender distribution, including STEM occupations. In fact, current machine translation systems fail to reproduce the real world distribution of female workers.

In 2015, Amazon.com turned off an AI system it developed to screen job applications when they realized it was biased against women. The recruitment tool excluded applicants who attended all-women's colleges and resumes that included the word "women's". A similar problem emerged with music streaming services—In 2019, it was discovered that the recommender system algorithm used by Spotify was biased against women artists. Spotify's song recommendations suggested more male artists over women artists.

Racial and ethnic discrimination

Algorithms have been criticized as a method for obscuring racial prejudices in decision-making. Because of how certain races and ethnic groups were treated in the past, data can often contain hidden biases. For example, black people are likely to receive longer sentences than white people who committed the same crime. This could potentially mean that a system amplifies the original biases in the data.

In 2015, Google apologized when black users complained that an image-identification algorithm in its Photos application identified them as gorillas. In 2010, Nikon cameras were criticized when image-recognition algorithms consistently asked Asian users if they were blinking. Such examples are the product of bias in biometric data sets. Biometric data is drawn from aspects of the body, including racial features either observed or inferred, which can then be transferred into data points. Speech recognition technology can have different accuracies depending on the user's accent. This may be caused by the a lack of training data for speakers of that accent.

Biometric data about race may also be inferred, rather than observed. For example, a 2012 study showed that names commonly associated with blacks were more likely to yield search results implying arrest records, regardless of whether there is any police record of that individual's name. A 2015 study also found that Black and Asian people are assumed to have lesser functioning lungs due to racial and occupational exposure data not being incorporated into the prediction algorithm's model of lung function.

In 2019, a research study revealed that a healthcare algorithm sold by Optum favored white patients over sicker black patients. The algorithm predicts how much patients would cost the health-care system in the future. However, cost is not race-neutral, as black patients incurred about $1,800 less in medical costs per year than white patients with the same number of chronic conditions, which led to the algorithm scoring white patients as equally at risk of future health problems as black patients who suffered from significantly more diseases.

A study conducted by researchers at UC Berkeley in November 2019 revealed that mortgage algorithms have been discriminatory towards Latino and African Americans which discriminated against minorities based on "creditworthiness" which is rooted in the U.S. fair-lending law which allows lenders to use measures of identification to determine if an individual is worthy of receiving loans. These particular algorithms were present in FinTech companies and were shown to discriminate against minorities.

Law enforcement and legal proceedings

Algorithms already have numerous applications in legal systems. An example of this is COMPAS, a commercial program widely used by U.S. courts to assess the likelihood of a defendant becoming a recidivist. ProPublica claims that the average COMPAS-assigned recidivism risk level of black defendants is significantly higher than the average COMPAS-assigned risk level of white defendants, and that black defendants are twice as likely to be erroneously assigned the label "high-risk" as white defendants.

One example is the use of risk assessments in criminal sentencing in the United States and parole hearings, judges were presented with an algorithmically generated score intended to reflect the risk that a prisoner will repeat a crime. For the time period starting in 1920 and ending in 1970, the nationality of a criminal's father was a consideration in those risk assessment scores. Today, these scores are shared with judges in Arizona, Colorado, Delaware, Kentucky, Louisiana, Oklahoma, Virginia, Washington, and Wisconsin. An independent investigation by ProPublica found that the scores were inaccurate 80% of the time, and disproportionately skewed to suggest blacks to be at risk of relapse, 77% more often than whites.

One study that set out to examine "Risk, Race, & Recidivism: Predictive Bias and Disparate Impact" alleges a two-fold (45 percent vs. 23 percent) adverse likelihood for black vs. Caucasian defendants to be misclassified as imposing a higher risk despite having objectively remained without any documented recidivism over a two-year period of observation.

In the pretrial detention context, a law review article argues that algorithmic risk assessments violate 14th Amendment Equal Protection rights on the basis of race, since the algorithms are argued to be facially discriminatory, to result in disparate treatment, and to not be narrowly tailored.

Online hate speech

In 2017 a Facebook algorithm designed to remove online hate speech was found to advantage white men over black children when assessing objectionable content, according to internal Facebook documents. The algorithm, which is a combination of computer programs and human content reviewers, was created to protect broad categories rather than specific subsets of categories. For example, posts denouncing "Muslims" would be blocked, while posts denouncing "Radical Muslims" would be allowed. An unanticipated outcome of the algorithm is to allow hate speech against black children, because they denounce the "children" subset of blacks, rather than "all blacks", whereas "all white men" would trigger a block, because whites and males are not considered subsets. Facebook was also found to allow ad purchasers to target "Jew haters" as a category of users, which the company said was an inadvertent outcome of algorithms used in assessing and categorizing data. The company's design also allowed ad buyers to block African-Americans from seeing housing ads.

While algorithms are used to track and block hate speech, some were found to be 1.5 times more likely to flag information posted by Black users and 2.2 times likely to flag information as hate speech if written in African American English. Without context for slurs and epithets, even when used by communities which have re-appropriated them, were flagged.

Surveillance

Surveillance camera software may be considered inherently political because it requires algorithms to distinguish normal from abnormal behaviors, and to determine who belongs in certain locations at certain times. The ability of such algorithms to recognize faces across a racial spectrum has been shown to be limited by the racial diversity of images in its training database; if the majority of photos belong to one race or gender, the software is better at recognizing other members of that race or gender. However, even audits of these image-recognition systems are ethically fraught, and some scholars have suggested the technology's context will always have a disproportionate impact on communities whose actions are over-surveilled. For example, a 2002 analysis of software used to identify individuals in CCTV images found several examples of bias when run against criminal databases. The software was assessed as identifying men more frequently than women, older people more frequently than the young, and identified Asians, African-Americans and other races more often than whites. Additional studies of facial recognition software have found the opposite to be true when trained on non-criminal databases, with the software being the least accurate in identifying darker-skinned females.

Sexual discrimination

In 2011, users of the gay hookup application Grindr reported that the Android store's recommendation algorithm was linking Grindr to applications designed to find sex offenders, which critics said inaccurately related homosexuality with pedophilia. Writer Mike Ananny criticized this association in The Atlantic, arguing that such associations further stigmatized gay men. In 2009, online retailer Amazon de-listed 57,000 books after an algorithmic change expanded its "adult content" blacklist to include any book addressing sexuality or gay themes, such as the critically acclaimed novel Brokeback Mountain.

In 2019, it was found that on Facebook, searches for "photos of my female friends" yielded suggestions such as "in bikinis" or "at the beach". In contrast, searches for "photos of my male friends" yielded no results.

Facial recognition technology has been seen to cause problems for transgender individuals. In 2018, there were reports of uber drivers who were transgender or transitioning experiencing difficulty with the facial recognition software that Uber implements as a built-in security measure. As a result of this, some of the accounts of trans uber drivers were suspended which cost them fares and potentially cost them a job, all due to the facial recognition software experiencing difficulties with recognizing the face of a trans driver who was transitioning. Although the solution to this issue would appear to be including trans individuals in training sets for machine learning models, an instance of trans YouTube videos that were collected to be used in training data did not receive consent from the trans individuals that were included in the videos, which created an issue of violation of privacy.

There has also been a study that was conducted at Stanford University in 2017 that tested algorithms in a machine learning system that was said to be able to detect an individuals sexual orientation based on their facial images. The model in the study predicted a correct distinction between gay and straight men 81% of the time, and a correct distinction between gay and straight women 74% of the time. This study resulted in a backlash from the LGBTQIA community, who were fearful of the possible negative repercussions that this AI system could have on individuals of the LGBTQIA community by putting individuals at risk of being "outed" against their will.

Google Search

While users generate results that are "completed" automatically, Google has failed to remove sexist and racist autocompletion text. For example, Algorithms of Oppression: How Search Engines Reinforce Racism Safiya Noble notes an example of the search for "black girls", which was reported to result in pornographic images. Google claimed it was unable to erase those pages unless they were considered unlawful.

Obstacles to research

Several problems impede the study of large-scale algorithmic bias, hindering the application of academically rigorous studies and public understanding.

Defining fairness

Main article: Fairness (machine learning)

Literature on algorithmic bias has focused on the remedy of fairness, but definitions of fairness are often incompatible with each other and the realities of machine learning optimization. For example, defining fairness as an "equality of outcomes" may simply refer to a system producing the same result for all people, while fairness defined as "equality of treatment" might explicitly consider differences between individuals. As a result, fairness is sometimes described as being in conflict with the accuracy of a model, suggesting innate tensions between the priorities of social welfare and the priorities of the vendors designing these systems. In response to this tension, researchers have suggested more care to the design and use of systems that draw on potentially biased algorithms, with "fairness" defined for specific applications and contexts.

Complexity

Algorithmic processes are complex, often exceeding the understanding of the people who use them. Large-scale operations may not be understood even by those involved in creating them. The methods and processes of contemporary programs are often obscured by the inability to know every permutation of a code's input or output. Social scientist Bruno Latour has identified this process as blackboxing, a process in which "scientific and technical work is made invisible by its own success. When a machine runs efficiently, when a matter of fact is settled, one need focus only on its inputs and outputs and not on its internal complexity. Thus, paradoxically, the more science and technology succeed, the more opaque and obscure they become." Others have critiqued the black box metaphor, suggesting that current algorithms are not one black box, but a network of interconnected ones.

An example of this complexity can be found in the range of inputs into customizing feedback. The social media site Facebook factored in at least 100,000 data points to determine the layout of a user's social media feed in 2013. Furthermore, large teams of programmers may operate in relative isolation from one another, and be unaware of the cumulative effects of small decisions within connected, elaborate algorithms. Not all code is original, and may be borrowed from other libraries, creating a complicated set of relationships between data processing and data input systems.

Additional complexity occurs through machine learning and the personalization of algorithms based on user interactions such as clicks, time spent on site, and other metrics. These personal adjustments can confuse general attempts to understand algorithms. One unidentified streaming radio service reported that it used five unique music-selection algorithms it selected for its users, based on their behavior. This creates different experiences of the same streaming services between different users, making it harder to understand what these algorithms do. Companies also run frequent A/B tests to fine-tune algorithms based on user response. For example, the search engine Bing can run up to ten million subtle variations of its service per day, creating different experiences of the service between each use and/or user.

Lack of transparency

Commercial algorithms are proprietary, and may be treated as trade secrets. Treating algorithms as trade secrets protects companies, such as search engines, where a transparent algorithm might reveal tactics to manipulate search rankings. This makes it difficult for researchers to conduct interviews or analysis to discover how algorithms function. Critics suggest that such secrecy can also obscure possible unethical methods used in producing or processing algorithmic output. Other critics, such as lawyer and activist Katarzyna Szymielewicz, have suggested that the lack of transparency is often disguised as a result of algorithmic complexity, shielding companies from disclosing or investigating its own algorithmic processes.

Lack of data about sensitive categories

A significant barrier to understanding the tackling of bias in practice is that categories, such as demographics of individuals protected by anti-discrimination law, are often not explicitly considered when collecting and processing data. In some cases, there is little opportunity to collect this data explicitly, such as in device fingerprinting, ubiquitous computing and the Internet of Things. In other cases, the data controller may not wish to collect such data for reputational reasons, or because it represents a heightened liability and security risk. It may also be the case that, at least in relation to the European Union's General Data Protection Regulation, such data falls under the 'special category' provisions (Article 9), and therefore comes with more restrictions on potential collection and processing.

Some practitioners have tried to estimate and impute these missing sensitive categorisations in order to allow bias mitigation, for example building systems to infer ethnicity from names, however this can introduce other forms of bias if not undertaken with care. Machine learning researchers have drawn upon cryptographic privacy-enhancing technologies such as secure multi-party computation to propose methods whereby algorithmic bias can be assessed or mitigated without these data ever being available to modellers in cleartext.

Algorithmic bias does not only include protected categories, but can also concerns characteristics less easily observable or codifiable, such as political viewpoints. In these cases, there is rarely an easily accessible or non-controversial ground truth, and removing the bias from such a system is more difficult. Furthermore, false and accidental correlations can emerge from a lack of understanding of protected categories, for example, insurance rates based on historical data of car accidents which may overlap, strictly by coincidence, with residential clusters of ethnic minorities.

Solutions

A study of 84 policy guidelines on ethical AI found that fairness and "mitigation of unwanted bias" was a common point of concern, and were addressed through a blend of technical solutions, transparency and monitoring, right to remedy and increased oversight, and diversity and inclusion efforts.

Technical

Further information: Fairness (machine learning)

There have been several attempts to create methods and tools that can detect and observe biases within an algorithm. These emergent fields focus on tools which are typically applied to the (training) data used by the program rather than the algorithm's internal processes. These methods may also analyze a program's output and its usefulness and therefore may involve the analysis of its confusion matrix (or table of confusion). Explainable AI to detect algorithm Bias is a suggested way to detect the existence of bias in an algorithm or learning model. Using machine learning to detect bias is called, "conducting an AI audit", where the "auditor" is an algorithm that goes through the AI model and the training data to identify biases. Ensuring that an AI tool such as a classifier is free from bias is more difficult than just removing the sensitive information from its input signals, because this is typically implicit in other signals. For example, the hobbies, sports and schools attended by a job candidate might reveal their gender to the software, even when this is removed from the analysis. Solutions to this problem involve ensuring that the intelligent agent does not have any information that could be used to reconstruct the protected and sensitive information about the subject, as first demonstrated in where a deep learning network was simultaneously trained to learn a task while at the same time being completely agnostic about the protected feature. A simpler method was proposed in the context of word embeddings, and involves removing information that is correlated with the protected characteristic.

Currently, a new IEEE standard is being drafted that aims to specify methodologies which help creators of algorithms eliminate issues of bias and articulate transparency (i.e. to authorities or end users) about the function and possible effects of their algorithms. The project was approved February 2017 and is sponsored by the Software & Systems Engineering Standards Committee, a committee chartered by the IEEE Computer Society. A draft of the standard is expected to be submitted for balloting in June 2019.

Transparency and monitoring

Further information: Algorithmic transparency

Ethics guidelines on AI point to the need for accountability, recommending that steps be taken to improve the interpretability of results. Such solutions include the consideration of the "right to understanding" in machine learning algorithms, and to resist deployment of machine learning in situations where the decisions could not be explained or reviewed. Toward this end, a movement for "Explainable AI" is already underway within organizations such as DARPA, for reasons that go beyond the remedy of bias. Price Waterhouse Coopers, for example, also suggests that monitoring output means designing systems in such a way as to ensure that solitary components of the system can be isolated and shut down if they skew results.

An initial approach towards transparency included the open-sourcing of algorithms. Software code can be looked into and improvements can be proposed through source-code-hosting facilities. However, this approach doesn't necessarily produce the intended effects. Companies and organizations can share all possible documentation and code, but this does not establish transparency if the audience doesn't understand the information given. Therefore, the role of an interested critical audience is worth exploring in relation to transparency. Algorithms cannot be held accountable without a critical audience.

Right to remedy

From a regulatory perspective, the Toronto Declaration calls for applying a human rights framework to harms caused by algorithmic bias. This includes legislating expectations of due diligence on behalf of designers of these algorithms, and creating accountability when private actors fail to protect the public interest, noting that such rights may be obscured by the complexity of determining responsibility within a web of complex, intertwining processes. Others propose the need for clear liability insurance mechanisms.

Diversity and inclusion

Amid concerns that the design of AI systems is primarily the domain of white, male engineers, a number of scholars have suggested that algorithmic bias may be minimized by expanding inclusion in the ranks of those designing AI systems. For example, just 12% of machine learning engineers are women, with black AI leaders pointing to a "diversity crisis" in the field. Groups like Black in AI and Queer in AI are attempting to create more inclusive spaces in the AI community and work against the often harmful desires of corporations that control the trajectory of AI research. Critiques of simple inclusivity efforts suggest that diversity programs can not address overlapping forms of inequality, and have called for applying a more deliberate lens of intersectionality to the design of algorithms. Researchers at the University of Cambridge have argued that addressing racial diversity is hampered by the "whiteness" of the culture of AI.

Regulation

Europe

The General Data Protection Regulation (GDPR), the European Union's revised data protection regime that was implemented in 2018, addresses "Automated individual decision-making, including profiling" in Article 22. These rules prohibit "solely" automated decisions which have a "significant" or "legal" effect on an individual, unless they are explicitly authorised by consent, contract, or member state law. Where they are permitted, there must be safeguards in place, such as a right to a human-in-the-loop, and a non-binding right to an explanation of decisions reached. While these regulations are commonly considered to be new, nearly identical provisions have existed across Europe since 1995, in Article 15 of the Data Protection Directive. The original automated decision rules and safeguards found in French law since the late 1970s.

The GDPR addresses algorithmic bias in profiling systems, as well as the statistical approaches possible to clean it, directly in recital 71, noting that

the controller should use appropriate mathematical or statistical procedures for the profiling, implement technical and organisational measures appropriate ... that prevents, inter alia, discriminatory effects on natural persons on the basis of racial or ethnic origin, political opinion, religion or beliefs, trade union membership, genetic or health status or sexual orientation, or that result in measures having such an effect.

Like the non-binding right to an explanation in recital 71, the problem is the non-binding nature of recitals. While it has been treated as a requirement by the Article 29 Working Party that advised on the implementation of data protection law, its practical dimensions are unclear. It has been argued that the Data Protection Impact Assessments for high risk data profiling (alongside other pre-emptive measures within data protection) may be a better way to tackle issues of algorithmic discrimination, as it restricts the actions of those deploying algorithms, rather than requiring consumers to file complaints or request changes.

United States

The United States has no general legislation controlling algorithmic bias, approaching the problem through various state and federal laws that might vary by industry, sector, and by how an algorithm is used. Many policies are self-enforced or controlled by the Federal Trade Commission. In 2016, the Obama administration released the National Artificial Intelligence Research and Development Strategic Plan, which was intended to guide policymakers toward a critical assessment of algorithms. It recommended researchers to "design these systems so that their actions and decision-making are transparent and easily interpretable by humans, and thus can be examined for any bias they may contain, rather than just learning and repeating these biases". Intended only as guidance, the report did not create any legal precedent.

In 2017, New York City passed the first algorithmic accountability bill in the United States. The bill, which went into effect on January 1, 2018, required "the creation of a task force that provides recommendations on how information on agency automated decision systems may be shared with the public, and how agencies may address instances where people are harmed by agency automated decision systems." The task force is required to present findings and recommendations for further regulatory action in 2019.

India

On July 31, 2018, a draft of the Personal Data Bill was presented. The draft proposes standards for the storage, processing and transmission of data. While it does not use the term algorithm, it makes for provisions for "harm resulting from any processing or any kind of processing undertaken by the fiduciary". It defines "any denial or withdrawal of a service, benefit or good resulting from an evaluative decision about the data principal" or "any discriminatory treatment" as a source of harm that could arise from improper use of data. It also makes special provisions for people of "Intersex status".