FWEP2
We are interested to represent the inputs and outputs of the models in FUSED-Wind with information about their probability distribution and / or their uncertainty. The question that this extension proposal is trying to address is how could we complement the I/O definition dictionary with those additional information. How this information is then taken into account by the component is not the focus of this discussion.
The idea is that we would be extending the definition of a scalar value, e.g.
wind_speed:
name: wind_speed
desc: wind speed value
type: float
value: 5.0
min: 4.0
max: 25.0
units: m/sto definitions about the variables that represent a distribution instead, e.g.
wind_speed:
name: wind_speed
desc: wind speed distribution (truncated Weibull)
type: weibull
A: 4.0
k: 2.0
min: 4.0
max: 25.0
units: m/s- Gaussian
type: gaussian
mean: 10.0
sigma: 2.0- Weibull
type: weibull
A: 4.0
k: 2.0-
Beta...
-
Kernel Density Estimator
type: KDE
value: array([]) # size:n*3 array, with n the number of normals centers
``
* Gaussian process ?
```yaml
type: GP
value: array([]) # size:n*2 array, with n the number of points of the GP
covariance: cubic # type of covariance matrix used
``
#### Additive uncertainty
The additive uncertainty can be represented as an overall distribution. In this example we assume to have a time series
of wind direction measurement with an estimated overall uncertainty assumed to be normal.
```yaml
wind_directions
name: wind_directions
desc: wind directions with measurement uncertainty
type: array
value: array([])
uncertainty:
type: gaussian
mean: 0.0
sigma: 10.0
``
#### Distribution hyper-parameter uncertainty
What if we have an uncertainty in the parameters of the distribution? This information could come from the fit of the
distribution achieved over a dataset.
```yaml
wind_speed:
name: wind_speed
desc: wind speed truncated weibull distribution fit with parameter uncertainty
type: weibull
A:
type: gaussian
mean: 4.0
sigma: 1.0
k:
type: gaussian
mean: 2.0
sigma: 0.2
min: 4.0
max: 25.0
units: m/sAnother way of describing the uncertainty of the fit would be to estimate the overall error of the fit:
wind_speed
name: wind_speed
desc: wind speed weibull distribution with overall fit uncertainty
type: weibull
A: 4.0
k: 2.0
min: 4.0
max: 25.0
uncertainty:
type: gaussian
mean: 0.0
sigma: 2.0If for some reason we know that the uncertainty of the fit is higher for some region of the distribution (f.ex not enough data point in the measurement, due to a model uncertainty, or propagated uncertainty) we can represent that uncertainty using another distribution (e.g. here a Kernel Density Estimator, KDE).
wind_speed
name: wind_speed
desc: wind speed weibull distribution with distribution dependent fit uncertainty
type: weibull
A: 4.0
k: 2.0
min: 4.0
max: 25.0
uncertainty:
type: KDE # kernel density estimator
value: array([...]), # size:n*3 array, with n the number of normals centersJoint distribution combine several inputs together. For instance a wind rose combine wind speed and wind direction together
wind_rose
name: wind_rose
desc: local wind resource
type: binned_weibull
dimensions: ['wind_speed', 'wind_direction'],
wind_speed: weibull
wind_direction: cdf
values: np.array([]),
columns:['wind_direction', 'frequency', 'A', 'k']
``
#### More complicated examples:
A joint distribution with 4 dimensions built from the marginal distribution of each dimension and a copula correlating
them together
```yaml
wind_atlas
name: wind_atlas
desc: joint distribution built from a Copula for U,D,TI,S. U.
U is a truncated Weibull, TI is a lognormal and D&S are KDEs.
dimensions:['wind_speed', 'wind_direction','TI','stability']
type: copula
copula:
type: gaussian #gumble...
correlation: array([] #4x4 correlation on the cdf-1 of the variables (might be more useful to store the inversed and determinant
marginals:
wind_speed:
type: weibull
A: 12.0
k: 2.0
min: 4.0
max: 25.0
wind_direction:
type: KDE
value: array([...]), #n*3 array, with 1 dimension, n the number of normals centers
TI:
type: lognormal
mean: 5.0
sigma: 1.0
stability:
type: KDE
value:array([...]), #n*3 array, with 1 dimension, n the number of normals centersA joint 4-dimensional distribution built from a multi-dimensional KDE
wind_atlas
name: wind_atlas
desc: joint KDE distribution for U,D,TI,S with fixed gaussian uncertainty
type: KDE
value: array([...]), #d*n*3 array, with d nb dimensions, n the number of normals centers
dimensions:['wind_speed', 'wind_direction','TI','stability']
uncertainty:
type: multivariate_gaussian
mean: [0.0,...],#list(4)
covariance: array([]),#array 4x4A joint 4-dimensional distribution built from a multi-dimensional KDE with an additional KDE to model its uncertainty
wind_atlas
name: wind_atlas
desc: joint distribution for U,D,TI,S with a probability uncertainty function of the inputs
type: KDE
value: array([...]), #d*n*3 array, with d nb dimensions, n the number of normals centers
dimensions:['wind_speed', 'wind_direction','TI','stability']
uncertainty:
type: KDE
value: array([...]), #d*n*3 array, with d nb dimensions, n the number of normals centers
dimensions:['wind_speed', 'wind_direction','TI','stability']
``