gemseo.problems.uncertainty.ishigami.statistics module#

Statistics associated with the Ishigami use case.

MEAN: Final[float] = 3.5#: The expectation of the output.

\[\mathbb{E}[Y] = \frac{a}{2}\]

SOBOL_1: Final[float] = 0.31390519114781146#: The first-order Sobol' index of \(X_1\).

\[S_1 = \frac{(1+b\frac{pi^4}{5})^2}{2\mathbb{V}[Y]}\]

SOBOL_12: Final[float] = 0.0#: The second-order Sobol' index of \(X_1\) and \(X_2\).

\[S_{1,2} = 0\]

SOBOL_123: Final[float] = 0.0#: The second-order Sobol' index of \(X_1\), \(X_2\) and \(X_3\).

\[S_{1,2,3} = 0\]

SOBOL_13: Final[float] = 0.24368366406214773#: The second-order Sobol' index of \(X_1\) and \(X_3\).

\[S_{1,3} = \frac{8b^2\pi^8}{225\mathbb{V}[Y]}\]

SOBOL_2: Final[float] = 0.4424111447900409#: The first-order Sobol' index of \(X_2\).

\[S_2 = \frac{a^2}{8\mathbb{V}[Y]}\]

SOBOL_23: Final[float] = 0.0#: The second-order Sobol' index of \(X_2\) and \(X_3\).

\[S_{2,3} = 0\]

SOBOL_3: Final[float] = 0.0#: The first-order Sobol' index of \(X_3\).

\[S_3 = 0\]

TOTAL_SOBOL_1: Final[float] = 0.5575888552099592#: The total Sobol' index of \(X_1\).

\[S_1^T = S_1 + S_{1,3}\]

TOTAL_SOBOL_2: Final[float] = 0.4424111447900409#: The total Sobol' index of \(X_2\).

\[S_2^T = S_2\]

TOTAL_SOBOL_3: Final[float] = 0.24368366406214773#: The total Sobol' index of \(X_3\).

\[S_3^T = S_{1,3}\]

VARIANCE: Final[float] = 13.844587940719254#: The variance of the output.

\[\mathbb{V}[Y] = \frac{1}{2} + \frac{a^2}{8} + \frac{b^2\pi^8}{18} + \frac{b\pi^4}{5}\]