Modifier and Type | Class and Description |
---|---|
class |
LinearInterpolator
Define a univariate function by linearly interpolating between adjacent points.
|
class |
NevilleTable
Neville's algorithm is a polynomial interpolation algorithm.
|
Modifier and Type | Class and Description |
---|---|
class |
DBetaRegularized
This is the first order derivative function of the Regularized Incomplete Beta function,
BetaRegularized , w.r.t the upper limit, x. |
class |
DErf
This is the first order derivative function of the Error function,
Erf . |
class |
Dfdx
The first derivative is a measure of how a function changes as its input changes.
|
class |
DGamma
This is the first order derivative function of the Gamma function, \({d \mathrm{\Gamma}(x) \over dx}\).
|
class |
DGaussian
This is the first order derivative function of a
Gaussian function, \({d \mathrm{\phi}(x) \over dx}\). |
class |
DPolynomial
This is the first order derivative function of a
Polynomial , which, again, is a polynomial. |
class |
FiniteDifference
A finite difference (divided by a small increment) is an approximation of the
derivative of a function.
|
Modifier and Type | Class and Description |
---|---|
class |
CauchyPolynomial
The Cauchy's polynomial of a polynomial takes this form:
|
class |
Polynomial
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. |
class |
QuadraticMonomial
A quadratic monomial has this form: x2 + ux + v.
|
class |
ScaledPolynomial
This constructs a scaled polynomial that has neither too big or too small coefficients,
hence avoiding overflow or underflow.
|
Modifier and Type | Class and Description |
---|---|
class |
ContinuedFraction
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.
|
class |
StepFunction
A step function (or staircase function) is a finite linear combination of indicator functions of
intervals.
|
Modifier and Type | Class and Description |
---|---|
class |
BetaRegularized
The Regularized Incomplete Beta function is defined as:
\[
I_x(p,q) = \frac{B(x;\,p,q)}{B(p,q)} = \frac{1}{B(p,q)} \int_0^x t^{p-1}\,(1-t)^{q-1}\,dt, p > 0, q > 0
\]
The R equivalent function is
pbeta . |
class |
BetaRegularizedInverse
The inverse of the Regularized Incomplete Beta function is defined at:
\[
x = I^{-1}_{(p,q)}(u), 0 \le u \le 1
\]
The R equivalent function is
qbeta . |
Modifier and Type | Class and Description |
---|---|
class |
Digamma
The digamma function is defined as the logarithmic derivative of the gamma function.
|
class |
GammaGergoNemes
The Gergo Nemes' algorithm is very simple and quick to compute the Gamma function, if accuracy is not critical.
|
class |
GammaLanczos
Lanczos approximation provides a way to compute the Gamma function such that the accuracy can be made arbitrarily precise.
|
class |
GammaLanczosQuick
Lanczos approximation, computations are done in
double . |
class |
LogGamma
The log-Gamma function, \(\log (\Gamma(z))\), for positive real numbers, is the log of the Gamma function.
|
class |
Trigamma
The trigamma function is defined as the logarithmic derivative of the digamma function.
|
Modifier and Type | Class and Description |
---|---|
class |
CumulativeNormalHastings
Hastings algorithm is faster but less accurate way to compute the cumulative standard Normal.
|
class |
CumulativeNormalInverse
The inverse of the cumulative standard Normal distribution function is defined as:
\[
N^{-1}(u)
/]
This implementation uses the Beasley-Springer-Moro algorithm.
|
class |
CumulativeNormalMarsaglia
Marsaglia is about 3 times slower but is more accurate to compute the cumulative standard Normal.
|
class |
Erf
The Error function is defined as:
\[
\operatorname{erf}(x) = \frac{2}{\sqrt{\pi}}\int_{0}^x e^{-t^2} dt
\]
|
class |
Erfc
This complementary Error function is defined as:
\[
\operatorname{erfc}(x)
= 1-\operatorname{erf}(x)
= \frac{2}{\sqrt{\pi}} \int_x^{\infty} e^{-t^2}\,dt
\]
|
class |
ErfInverse
The inverse of the Error function is defined as:
\[
\operatorname{erf}^{-1}(x)
\]
|
class |
Gaussian
The Gaussian function is defined as:
\[
f(x) = a e^{- { \frac{(x-b)^2 }{ 2 c^2} } }
\]
|
Modifier and Type | Class and Description |
---|---|
class |
ACERFunction
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.
|
class |
ACERInverseFunction
The inverse of the ACER function.
|
class |
ACERLogFunction
The ACER function in log scale (base e), i.e., \(log(\epsilon_k(\eta))\).
|
class |
ACERReturnLevel
Given an ACER function, compute the return level \(\eta\) for a given return period \(R\).
|
Modifier and Type | Class and Description |
---|---|
class |
ReturnLevel
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
\(1 / T\).
|
class |
ReturnPeriod
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 \(\eta\).
|
Modifier and Type | Class and Description |
---|---|
class |
Bt
This is a
FiltrationFunction that returns \(B(t_i)\),
the Brownian motion value at the i-th time point. |
class |
F_Sum_BtDt
This represents a function of this integral
\[
I = \int_{0}^{1} B(t)dt
\]
|
class |
F_Sum_tBtDt
This represents a function of this integral
\[
\int_{0}^{1} (t - 0.5) * B(t) dt
\]
|
class |
FiltrationFunction
A filtration function, parameterized by a fixed filtration, is a function of time,
\(f(\mathfrak{F_{t_i}})\).
|
Modifier and Type | Class and Description |
---|---|
class |
Ceta
The function C(η) to be maximized (Eq.
|
Modifier and Type | Class and Description |
---|---|
static class |
CetaMaximizer.NegCetaFunction |
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