# TransportMaps.Maps.LinearSpanParametricTriangularComponentwiseMapBase¶

## Module Contents¶

### Classes¶

 LinearSpanParametricTriangularComponentwiseMap Triangular map $$T[{\bf a}_{1:d_y}]({\bf x})= [T_1[{\bf a}_1],\ldots,T_{d_y}[{\bf a}_{d_y}]]^\top$$ where :math:T_i[{bf a}_i](x_{1:i}) := {bf a}_i^top , Phi(x_{1:i}) . CommonBasisLinearSpanParametricTriangularComponentwiseMap Triangular map $$T[{\bf a}_{1:d_y}]({\bf x})= [T_1[{\bf a}_1],\ldots,T_{d_y}[{\bf a}_{d_y}]]^\top$$ where $$T_i[{\bf a}_i](x_{1:i}) := {\bf a}_i^\top \, \Phi(x_{1:i})$$, and for each dimension $$i$$, every component $$T_k$$ share the same basis type.
class TransportMaps.Maps.LinearSpanParametricTriangularComponentwiseMapBase.LinearSpanParametricTriangularComponentwiseMap(**kwargs)[source]

Triangular map $$T[{\bf a}_{1:d_y}]({\bf x})= [T_1[{\bf a}_1],\ldots,T_{d_y}[{\bf a}_{d_y}]]^\top$$ where :math:T_i[{bf a}_i](x_{1:i}) := {bf a}_i^top , Phi(x_{1:i}) .

get_identity_coeffs()[source]

Returns the coefficients corresponding to the identity map

Returns:

coefficients

Return type:

(ndarray [$$N$$])

class TransportMaps.Maps.LinearSpanParametricTriangularComponentwiseMapBase.CommonBasisLinearSpanParametricTriangularComponentwiseMap(**kwargs)[source]

Triangular map $$T[{\bf a}_{1:d_y}]({\bf x})= [T_1[{\bf a}_1],\ldots,T_{d_y}[{\bf a}_{d_y}]]^\top$$ where $$T_i[{\bf a}_i](x_{1:i}) := {\bf a}_i^\top \, \Phi(x_{1:i})$$, and for each dimension $$i$$, every component $$T_k$$ share the same basis type.

This is leads to some more efficient evaluation operations.

precomp_evaluate(x, precomp=None, precomp_type='uni')[source]

Precompute necessary structures for the evaluation of $$T({\bf x},{\bf a})$$

This returns a list of uni-variate Vandermonde matrices with order maximum among the components $$T_i$$.

Enriches the dictionaries in the precomp list if necessary.

Parameters:
• x (ndarray [$$m,d$$]) – evaluation points

• precomp (dict) – list of dictionaries of precomputed values

• precomp_type (str) – only option ‘uni’ is allowed for this TransportMap

Returns:

(dict of list [$$d$$]

ndarray) – necessary structures

Precompute necessary structures for the evaluation of $$\nabla_{\bf x}T({\bf x},{\bf a})$$

Enriches the dictionaries in the precomp list if necessary.

Parameters:
• x (ndarray [$$m,d$$]) – evaluation points

• precomp (dict) – list of dictionaries of precomputed values

• precomp_type (str) – only option ‘uni’ is allowed for this TransportMap

Returns:

(dict of list [$$d$$]

ndarray) – necessary structures

precomp_hess_x(x, precomp=None, precomp_type='uni')[source]

Precompute necessary structures for the evaluation of $$\nabla^2_{\bf x}T({\bf x},{\bf a})$$

Enriches the dictionaries in the precomp list if necessary.

Parameters:
• x (ndarray [$$m,d$$]) – evaluation points

• precomp (dict) – list of dictionaries of precomputed values

• precomp_type (str) – only option ‘uni’ is allowed for this TransportMap

Returns:

(dict of list [$$d$$]

ndarray) – necessary structures

precomp_nabla3_x(x, precomp=None, precomp_type='uni')[source]

Precompute necessary structures for the evaluation of $$\nabla^3_{\bf x}T({\bf x},{\bf a})$$

Enriches the dictionaries in the precomp list if necessary.

Parameters:
• x (ndarray [$$m,d$$]) – evaluation points

• precomp (dict) – list of dictionaries of precomputed values

• precomp_type (str) – only option ‘uni’ is allowed for this TransportMap

Returns:

(dict of list [$$d$$]

ndarray) – necessary structures

precomp_partial_xd(x, precomp=None, precomp_type='uni')[source]

Precompute necessary structures for the evaluation of $$\partial_{x_k}T_k({\bf x},{\bf a})$$ for $$k=1,\ldots,d$$

Enriches the dictionaries in the precomp list if necessary.

Parameters:
• x (ndarray [$$m,d$$]) – evaluation points

• precomp (dict) – list of dictionaries of precomputed values

• precomp_type (str) – only option ‘uni’ is allowed for this TransportMap

Returns:

(dict of list [$$d$$]

ndarray) – necessary structures

Precompute necessary structures for the evaluation of $$\nabla_{\bf x}\partial_{x_k}T_k({\bf x},{\bf a})$$ for $$k=1,\ldots,d$$

Enriches the dictionaries in the precomp list if necessary.

Parameters:
• x (ndarray [$$m,d$$]) – evaluation points

• precomp (dict) – list of dictionaries of precomputed values

• precomp_type (str) – only option ‘uni’ is allowed for this TransportMap

Returns:

(dict of list [$$d$$]

dict) – necessary structures

precomp_hess_x_partial_xd(x, precomp=None, precomp_type='uni')[source]

Precompute necessary structures for the evaluation of $$\nabla^2_{\bf x}\partial_{x_k}T_k({\bf x},{\bf a})$$ for $$k=1,\ldots,d$$

Parameters:
• x (ndarray [$$m,d$$]) – evaluation points

• precomp (dict) – list of dictionaries of precomputed values

• precomp_type (str) – only option ‘uni’ is allowed for this TransportMap

Returns:

(dict of list [$$d$$]

dict) – necessary structures