# Uses of Classdev.nm.analysis.function.rn2r1.AbstractRealScalarFunction

Packages that use AbstractRealScalarFunction
• ## Uses of AbstractRealScalarFunction in dev.nm.analysis.curvefit.interpolation

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Define a univariate function by linearly interpolating between adjacent points.
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Neville's algorithm is a polynomial interpolation algorithm.
• ## Uses of AbstractRealScalarFunction in dev.nm.analysis.differentiation

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Ridders' method computes the numerical derivative of a function.
• ## Uses of AbstractRealScalarFunction in dev.nm.analysis.differentiation.multivariate

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A partial derivative of a multivariate function is the derivative with respect to one of the variables with the others held constant.
• ## Uses of AbstractRealScalarFunction in dev.nm.analysis.differentiation.univariate

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This is the first order derivative function of the Beta function w.r.t x, $${\partial \over \partial x} \mathrm{B}(x, y)$$.
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This is the first order derivative function of the Regularized Incomplete Beta function, BetaRegularized, w.r.t the upper limit, x.
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This is the first order derivative function of the Error function, Erf.
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The first derivative is a measure of how a function changes as its input changes.
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This is the first order derivative function of the Gamma function, $${d \mathrm{\Gamma}(x) \over dx}$$.
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This is the first order derivative function of a Gaussian function, $${d \mathrm{\phi}(x) \over dx}$$.
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This is the first order derivative function of a Polynomial, which, again, is a polynomial.
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A finite difference (divided by a small increment) is an approximation of the derivative of a function.
• ## Uses of AbstractRealScalarFunction in dev.nm.analysis.function.polynomial

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The Cauchy's polynomial of a polynomial takes this form:
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A polynomial is a UnivariateRealFunction that represents a finite length expression constructed from variables and constants, using the operations of addition, subtraction, multiplication, and constant non-negative whole number exponents.
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A quadratic monomial has this form: x2 + ux + v.
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This constructs a scaled polynomial that has neither too big or too small coefficients, hence avoiding overflow or underflow.
• ## Uses of AbstractRealScalarFunction in dev.nm.analysis.function.rn2r1

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A bivariate real function takes two real arguments and outputs one real value.
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A trivariate real function takes three real arguments and outputs one real value.
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A quadratic function takes this form: $$f(x) = \frac{1}{2} \times x'Hx + x'p + c$$.
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Projection creates a real-valued function RealScalarFunction from a vector-valued function RealVectorFunction by taking only one of its coordinate components in the vector output.
• ## Uses of AbstractRealScalarFunction in dev.nm.analysis.function.rn2r1.univariate

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A univariate real function takes one real argument and outputs one real value.
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A continued fraction representation of a number has this form: $z = b_0 + \cfrac{a_1}{b_1 + \cfrac{a_2}{b_2 + \cfrac{a_3}{b_3 + \cfrac{a_4}{b_4 + \ddots\,}}}}$ ai and bi can be functions of x, which in turn makes z a function of x.
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A step function (or staircase function) is a finite linear combination of indicator functions of intervals.
• ## Uses of AbstractRealScalarFunction in dev.nm.analysis.function.special

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The Rastrigin function is a non-convex function used as a performance test problem for optimization algorithms.
• ## Uses of AbstractRealScalarFunction in dev.nm.analysis.function.special.beta

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The beta function defined as: $B(x,y) = \frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}= \int_0^1t^{x-1}(1-t)^{y-1}\,dt, x > 0, y > 0$
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The Regularized Incomplete Beta function is defined as: $I_x(p,q) = \frac{B(x;\,p,q)}{B(p,q)} = \frac{1}{B(p,q)} \int_0^x t^{p-1}\,(1-t)^{q-1}\,dt, p > 0, q > 0$
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The inverse of the Regularized Incomplete Beta function is defined at: $x = I^{-1}_{(p,q)}(u), 0 \le u \le 1$
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This class represents the log of Beta function log(B(x, y)).
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A multinomial Beta function is defined as: $\frac{\prod_{i=1}^K \Gamma(\alpha_i)}{\Gamma\left(\sum_{i=1}^K \alpha_i\right)},\qquad\boldsymbol{\alpha}=(\alpha_1,\cdots,\alpha_K)$
• ## Uses of AbstractRealScalarFunction in dev.nm.analysis.function.special.gamma

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The digamma function is defined as the logarithmic derivative of the gamma function.
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The Gergo Nemes' algorithm is very simple and quick to compute the Gamma function, if accuracy is not critical.
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Lanczos approximation provides a way to compute the Gamma function such that the accuracy can be made arbitrarily precise.
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Lanczos approximation, computations are done in double.
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The Lower Incomplete Gamma function is defined as: $\gamma(s,x) = \int_0^x t^{s-1}\,e^{-t}\,{\rm d}t = P(s,x)\Gamma(s)$ P(s,x) is the Regularized Incomplete Gamma P function.
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The Regularized Incomplete Gamma P function is defined as: $P(s,x) = \frac{\gamma(s,x)}{\Gamma(s)} = 1 - Q(s,x), s \geq 0, x \geq 0$
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The inverse of the Regularized Incomplete Gamma P function is defined as: $x = P^{-1}(s,u), 0 \geq u \geq 1$ When s > 1, we use the asymptotic inversion method. When s <= 1, we use an approximation of P(s,x) together with a higher-order Newton like method. In both cases, the estimated value is then improved using Halley's method, c.f., HalleyRoot.
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The Regularized Incomplete Gamma Q function is defined as: $Q(s,x)=\frac{\Gamma(s,x)}{\Gamma(s)}=1-P(s,x), s \geq 0, x \geq 0$ The algorithm used for computing the regularized incomplete Gamma Q function depends on the values of s and x.
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The Upper Incomplete Gamma function is defined as: $\Gamma(s,x) = \int_x^{\infty} t^{s-1}\,e^{-t}\,{\rm d}t = Q(s,x) \times \Gamma(s)$ The integrand has the same form as the Gamma function, but the lower limit of the integration is a variable.
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The log-Gamma function, $$\log (\Gamma(z))$$, for positive real numbers, is the log of the Gamma function.
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The trigamma function is defined as the logarithmic derivative of the digamma function.
• ## Uses of AbstractRealScalarFunction in dev.nm.analysis.function.special.gaussian

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Hastings algorithm is faster but less accurate way to compute the cumulative standard Normal.
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The inverse of the cumulative standard Normal distribution function is defined as: $N^{-1}(u) /] class Marsaglia is about 3 times slower but is more accurate to compute the cumulative standard Normal. class The Error function is defined as: \[ \operatorname{erf}(x) = \frac{2}{\sqrt{\pi}}\int_{0}^x e^{-t^2} dt$
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This complementary Error function is defined as: $\operatorname{erfc}(x) = 1-\operatorname{erf}(x) = \frac{2}{\sqrt{\pi}} \int_x^{\infty} e^{-t^2}\,dt$
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The inverse of the Error function is defined as: $\operatorname{erf}^{-1}(x)$
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The Gaussian function is defined as: $f(x) = a e^{- { \frac{(x-b)^2 }{ 2 c^2} } }$
• ## Uses of AbstractRealScalarFunction in dev.nm.solver.multivariate.constrained.convex.sdp.socp.qp.problem

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A quadratic programming problem with only equality constraints can be converted into a equivalent quadratic programming problem without constraints, hence a mere quadratic function.
• ## Uses of AbstractRealScalarFunction in dev.nm.stat.evt.evd.univariate.fitting.acer

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The ACER (Average Conditional Exceedance Rate) function $$\epsilon_k(\eta)$$ approximates the probability $\epsilon_k(\eta) = Pr(X_k > \eta | X_1 \le \eta, X_2 \le \eta, ..., X_{k-1} \le \eta)$ for a sequence of stochastic process observations $$X_i$$ with a k-step memory.
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The inverse of the ACER function.
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The ACER function in log scale (base e), i.e., $$log(\epsilon_k(\eta))$$.
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Given an ACER function, compute the return level $$\eta$$ for a given return period $$R$$.
• ## Uses of AbstractRealScalarFunction in dev.nm.stat.evt.function

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Given a GEV distribution of a random variable $$X$$, the return level $$\eta$$ is the value that is expected to be exceeded on average once every interval of time $$T$$, with a probability of $$1 / T$$.
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The return period $$R$$ of a level $$\eta$$ for a random variable $$X$$ is the mean number of trials that must be done for $$X$$ to exceed $$\eta$$.
• ## Uses of AbstractRealScalarFunction in dev.nm.stat.stochasticprocess.univariate.filtration

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This is a FiltrationFunction that returns $$B(t_i)$$, the Brownian motion value at the i-th time point.
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This represents a function of this integral $I = \int_{0}^{1} B(t)dt$
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This represents a function of this integral $\int_{0}^{1} (t - 0.5) * B(t) dt$
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A filtration function, parameterized by a fixed filtration, is a function of time, $$f(\mathfrak{F_{t_i}})$$.
• ## Uses of AbstractRealScalarFunction in dev.nm.stat.timeseries.linear.univariate

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This is the auto-correlation function of a univariate time series {xt}.
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This is the auto-covariance function of a univariate time series {xt}.
• ## Uses of AbstractRealScalarFunction in dev.nm.stat.timeseries.linear.univariate.sample

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This is the sample Auto-Correlation Function (ACF) for a univariate data set.
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This is the sample Auto-Covariance Function (ACVF) for a univariate data set.
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This is the sample partial Auto-Correlation Function (PACF) for a univariate data set.
• ## Uses of AbstractRealScalarFunction in dev.nm.stat.timeseries.linear.univariate.stationaryprocess.arma

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Compute the Auto-Correlation Function (ACF) for an AutoRegressive Moving Average (ARMA) model, assuming that EXt = 0.
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Computes the Auto-CoVariance Function (ACVF) for an AutoRegressive Moving Average (ARMA) model by recursion.
• ## Uses of AbstractRealScalarFunction in tech.nmfin.portfoliooptimization.lai2010.ceta

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The function C(η) to be maximized (Eq.
• ## Uses of AbstractRealScalarFunction in tech.nmfin.portfoliooptimization.lai2010.ceta.maximizer

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