plans.stochastic.frequency#

Exceedance probabilities, return periods and the Gumbel model.

Formulas use \(r\) for the rank (1 is the largest value), \(n\) for the sample size, \(s\) for the standard deviation and \(T\) for the return period.

Functions

`empirical_px`(ranks)	Get the empirical exceedance probability P(X)
`get_cdf_gumbel`(df, varfield)	Get the Gumbel CDF for the data (it assumes that is a maxima dataset).
`get_cdf_weibull`(data)	Get the Weibull model for the data.
`get_frequency`(data)	Get data frequencies.
`get_tx`(fx)	Simple function for computing the Return Period from the CDF
`gringorten_px`(ranks)	Get the Gringorten exceedance probability P(X)
`gumbel_freqfactor`([tx])	Gumbel Frequency Factor K(T)
`gumbel_fx`(x, a, b)	Gumbel probability distribution F(X)
`gumbel_se`(std_sample, n_sample, tx)	Gumbel Standard Error for the MM fitted Gumbel function
`gumbel_tx`(x, a, b)	Gumbel return period distribution T(X)
`weibull_px`(ranks)	Get the Weibull exceedance probability P(X)

plans.stochastic.frequency.get_tx(fx)[source]#

Simple function for computing the Return Period from the CDF

\[T = \frac{1}{1 - F}\]

Parameters:: fx (float | numpy.ndarray) – CDF (FX)
Returns:: Return Period
Return type:: float | numpy.ndarray

plans.stochastic.frequency.gumbel_fx(x, a, b)[source]#

Gumbel probability distribution F(X)

\[F(x) = \exp\left[ -\exp\left( -\frac{x - a}{b} \right) \right]\]

Parameters:

x (float | numpy.ndarray) – function inputs
a (float) – distribution parameter a
b (float) – distribution parameter b

Returns:

function output

Return type:

float | numpy.ndarray

plans.stochastic.frequency.gumbel_tx(x, a, b)[source]#

Gumbel return period distribution T(X)

\[T(x) = \frac{1}{1 - F(x)}\]

Parameters:

x (float | numpy.ndarray) – function inputs
a (float) – distribution parameter a
b (float) – distribution parameter b

Returns:

function output

Return type:

float | numpy.ndarray

plans.stochastic.frequency.gumbel_freqfactor(tx=2)[source]#

Gumbel Frequency Factor K(T)

\[K_T = -\frac{\sqrt{6}}{\pi} \left[ 0.5772 + \ln \ln \left( \frac{T}{T - 1} \right) \right]\]

Parameters:: tx (float | numpy.ndarray) – return period T
Returns:: function output
Return type:: float | numpy.ndarray

plans.stochastic.frequency.gumbel_se(std_sample, n_sample, tx)[source]#

Gumbel Standard Error for the MM fitted Gumbel function

\[S_e = \frac{s}{\sqrt{n}} \sqrt{1 + 1.14 K_T + 1.1 K_T^2}\]

Parameters:

std_sample (float) – sample standard deviation
n_sample (int) – sample size
tx (float | numpy.ndarray) – return period T

Returns:

function output

Return type:

float | numpy.ndarray

plans.stochastic.frequency.empirical_px(ranks)[source]#

Get the empirical exceedance probability P(X)

\[P(X) = \frac{r}{n}\]

Parameters:: ranks (numpy.ndarray) – vector of ranks
Returns:: empirical exceedance probability P(X)
Return type:: numpy.ndarray

plans.stochastic.frequency.weibull_px(ranks)[source]#

Get the Weibull exceedance probability P(X)

\[P(X) = \frac{r}{n + 1}\]

Parameters:: ranks (numpy.ndarray) – vector of ranks
Returns:: Weibull exceedance probability P(X)
Return type:: numpy.ndarray

plans.stochastic.frequency.gringorten_px(ranks)[source]#

Get the Gringorten exceedance probability P(X)

\[P(X) = \frac{r - 0.44}{n + 0.12}\]

Parameters:: ranks (numpy.ndarray) – vector of ranks
Returns:: Gringorten exceedance probability P(X)
Return type:: numpy.ndarray

plans.stochastic.frequency.get_frequency(data)[source]#

Get data frequencies.

Parameters:: data (numpy.ndarray) – vector of data without nan values
Returns:: result dataframe
Return type:: pandas.DataFrame

plans.stochastic.frequency.get_cdf_weibull(data)[source]#

Get the Weibull model for the data.

\[P(X) = \frac{r}{n + 1}, \qquad F(X) = 1 - P(X)\]

Parameters:: data (numpy.ndarray) – vector of data without nan values
Returns:: model dataframe
Return type:: pandas.DataFrame

plans.stochastic.frequency.get_cdf_gumbel(df, varfield)[source]#

Get the Gumbel CDF for the data (it assumes that is a maxima dataset).

Stages by return period use the frequency factor and the standard error, with Student’s t bands at confidence \(c\):

\[x_T = \bar{x} + K_T s, \qquad x_T \pm t_{(1 + c) / 2, \, n - 1} \, S_e\]

Parameters:

df (pandas.DataFrame) – dataframe without nan values. Other columns are kept in the output.
varfield (str) – name of the variable column

Returns:

multiple results from the assessment. Keys: Data, Data_T(X), Data_QQ, Metadata

Return type:

dict