Models

Regularized Canonical Correlation Analysis and Partial Least Squares

Canonical Correlation Analysis

class cca_zoo.models.rcca.CCA(latent_dims=1, scale=True, centre=True, copy_data=True, random_state=None)

Bases: cca_zoo.models.rcca.rCCA

A class used to fit a simple CCA model

Implements CCA by inheriting regularised CCA with 0 regularisation

Maths:

\[ \begin{align}\begin{aligned}\begin{split}w_{opt}=\underset{w}{\mathrm{argmax}}\{ w_1^TX_1^TX_2w_2 \}\\\end{split}\\\text{subject to:}\\w_1^TX_1^TX_1w_1=n\\w_2^TX_2^TX_2w_2=n\end{aligned}\end{align} \]

Citation:

Hotelling, Harold. “Relations between two sets of variates.” Breakthroughs in statistics. Springer, New York, NY, 1992. 162-190.

Example:

>>> from cca_zoo.models import CCA
>>> import numpy as np
>>> rng=np.random.RandomState(0)
>>> X1 = rng.random((10,5))
>>> X2 = rng.random((10,5))
>>> model = CCA()
>>> model.fit((X1,X2)).score((X1,X2))
array([1.])

Constructor for CCA

Parameters:

• latent_dims (int) – number of latent dimensions to fit
• scale (bool) – normalize variance in each column before fitting
• centre – demean data by column before fitting (and before transforming out of sample
• copy_data – If True, X will be copied; else, it may be overwritten
• random_state – Pass for reproducible output across multiple function calls

correlations(views, **kwargs)

Predicts the correlations between each view for each dimension for the given data using the fit model

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• kwargs – any additional keyword arguments required by the given model

Returns:

all_corrs: an array of the pairwise correlations (k,k,self.latent_dims) where k is the number of views

fit(views, y=None, **kwargs)

Fits a regularised CCA (canonical ridge) model

Parameters:views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)

fit_transform(views, **kwargs)

Fits and then transforms the training data

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• kwargs – any additional keyword arguments required by the given model

get_loadings(views, normalize=False, **kwargs)

Returns the model loadings for each view for the given data

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• kwargs – any additional keyword arguments required by the given model
• normalize – scales loadings to ensure that they represent correlations between features and scores

score(views, y=None, **kwargs)

Returns average correlation in each dimension (averages over all pairs for multiview)

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• y – unused but needed to integrate with scikit-learn

transform(views, **kwargs)

Transforms data given a fit model

Parameters:

• views (Iterable[ndarray]) – numpy arrays with the same number of rows (samples) separated by commas
• kwargs – any additional keyword arguments required by the given model

Partial Least Squares

class cca_zoo.models.rcca.PLS(latent_dims=1, scale=True, centre=True, copy_data=True, random_state=None)

Bases: cca_zoo.models.rcca.rCCA

A class used to fit a simple PLS model

Implements PLS by inheriting regularised CCA with maximal regularisation

Maths:

\[ \begin{align}\begin{aligned}\begin{split}w_{opt}=\underset{w}{\mathrm{argmax}}\{ w_1^TX_1^TX_2w_2 \}\\\end{split}\\\text{subject to:}\\w_1^Tw_1=1\\w_2^Tw_2=1\end{aligned}\end{align} \]

Example:

>>> from cca_zoo.models import PLS
>>> import numpy as np
>>> rng=np.random.RandomState(0)
>>> X1 = rng.random((10,5))
>>> X2 = rng.random((10,5))
>>> model = PLS()
>>> model.fit((X1,X2)).score((X1,X2))
array([0.81796873])

Constructor for PLS

Parameters:

• latent_dims (int) – number of latent dimensions to fit
• scale (bool) – normalize variance in each column before fitting
• centre – demean data by column before fitting (and before transforming out of sample
• copy_data – If True, X will be copied; else, it may be overwritten
• random_state – Pass for reproducible output across multiple function calls

correlations(views, **kwargs)

Predicts the correlations between each view for each dimension for the given data using the fit model

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• kwargs – any additional keyword arguments required by the given model

Returns:

all_corrs: an array of the pairwise correlations (k,k,self.latent_dims) where k is the number of views

fit(views, y=None, **kwargs)

Fits a regularised CCA (canonical ridge) model

Parameters:views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)

fit_transform(views, **kwargs)

Fits and then transforms the training data

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• kwargs – any additional keyword arguments required by the given model

get_loadings(views, normalize=False, **kwargs)

Returns the model loadings for each view for the given data

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• kwargs – any additional keyword arguments required by the given model
• normalize – scales loadings to ensure that they represent correlations between features and scores

score(views, y=None, **kwargs)

Returns average correlation in each dimension (averages over all pairs for multiview)

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• y – unused but needed to integrate with scikit-learn

transform(views, **kwargs)

Transforms data given a fit model

Parameters:

• views (Iterable[ndarray]) – numpy arrays with the same number of rows (samples) separated by commas
• kwargs – any additional keyword arguments required by the given model

Ridge Regularized Canonical Correlation Analysis

class cca_zoo.models.rcca.rCCA(latent_dims=1, scale=True, centre=True, copy_data=True, random_state=None, c=None, eps=0.001, accept_sparse=None)

Bases: cca_zoo.models._cca_base._CCA_Base

A class used to fit Regularised CCA (canonical ridge) model. Uses PCA to perform the optimization efficiently for high dimensional data.

Maths:

\[ \begin{align}\begin{aligned}\begin{split}w_{opt}=\underset{w}{\mathrm{argmax}}\{ w_1^TX_1^TX_2w_2 \}\\\end{split}\\\text{subject to:}\\(1-c_1)w_1^TX_1^TX_1w_1+c_1w_1^Tw_1=n\\(1-c_2)w_2^TX_2^TX_2w_2+c_2w_2^Tw_2=n\end{aligned}\end{align} \]

Citation:

Vinod, Hrishikesh D. “Canonical ridge and econometrics of joint production.” Journal of econometrics 4.2 (1976): 147-166.

Example:

>>> from cca_zoo.models import rCCA
>>> import numpy as np
>>> rng=np.random.RandomState(0)
>>> X1 = rng.random((10,5))
>>> X2 = rng.random((10,5))
>>> model = rCCA(c=[0.1,0.1])
>>> model.fit((X1,X2)).score((X1,X2))
array([0.95222128])

Constructor for rCCA

Parameters:

• latent_dims (int) – number of latent dimensions to fit
• scale (bool) – normalize variance in each column before fitting
• centre – demean data by column before fitting (and before transforming out of sample
• copy_data – If True, X will be copied; else, it may be overwritten
• random_state – Pass for reproducible output across multiple function calls
• c (Union[Iterable[float], float, None]) – Iterable of regularisation parameters for each view (between 0:CCA and 1:PLS)
• eps – epsilon for stability
• accept_sparse – which forms are accepted for sparse data

fit(views, y=None, **kwargs)

Fits a regularised CCA (canonical ridge) model

Parameters:views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)

correlations(views, **kwargs)

Predicts the correlations between each view for each dimension for the given data using the fit model

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• kwargs – any additional keyword arguments required by the given model

Returns:

all_corrs: an array of the pairwise correlations (k,k,self.latent_dims) where k is the number of views

fit_transform(views, **kwargs)

Fits and then transforms the training data

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• kwargs – any additional keyword arguments required by the given model

get_loadings(views, normalize=False, **kwargs)

Returns the model loadings for each view for the given data

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• kwargs – any additional keyword arguments required by the given model
• normalize – scales loadings to ensure that they represent correlations between features and scores

score(views, y=None, **kwargs)

Returns average correlation in each dimension (averages over all pairs for multiview)

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• y – unused but needed to integrate with scikit-learn

transform(views, **kwargs)

Transforms data given a fit model

Parameters:

• views (Iterable[ndarray]) – numpy arrays with the same number of rows (samples) separated by commas
• kwargs – any additional keyword arguments required by the given model

GCCA and KGCCA

Generalized (MAXVAR) Canonical Correlation Analysis

class cca_zoo.models.gcca.GCCA(latent_dims=1, scale=True, centre=True, copy_data=True, random_state=None, c=None, view_weights=None)

Bases: cca_zoo.models.rcca.rCCA

A class used to fit GCCA model. For more than 2 views, GCCA optimizes the sum of correlations with a shared auxiliary vector

Maths:

\[ \begin{align}\begin{aligned}\begin{split}w_{opt}=\underset{w}{\mathrm{argmax}}\{ \sum_iw_i^TX_i^TT \}\\\end{split}\\\text{subject to:}\\T^TT=1\end{aligned}\end{align} \]

Citation:

Tenenhaus, Arthur, and Michel Tenenhaus. “Regularized generalized canonical correlation analysis.” Psychometrika 76.2 (2011): 257.

Example:

>>> from cca_zoo.models import GCCA
>>> import numpy as np
>>> rng=np.random.RandomState(0)
>>> X1 = rng.random((10,5))
>>> X2 = rng.random((10,5))
>>> X3 = rng.random((10,5))
>>> model = GCCA()
>>> model.fit((X1,X2,X3)).score((X1,X2,X3))
array([0.97229856])

Constructor for GCCA

Parameters:

• latent_dims (int) – number of latent dimensions to fit
• scale (bool) – normalize variance in each column before fitting
• centre – demean data by column before fitting (and before transforming out of sample
• copy_data – If True, X will be copied; else, it may be overwritten
• random_state – Pass for reproducible output across multiple function calls
• c (Union[Iterable[float], float, None]) – regularisation between 0 (CCA) and 1 (PLS)
• view_weights (Optional[Iterable[float]]) – list of weights of each view

correlations(views, **kwargs)

Predicts the correlations between each view for each dimension for the given data using the fit model

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• kwargs – any additional keyword arguments required by the given model

Returns:

all_corrs: an array of the pairwise correlations (k,k,self.latent_dims) where k is the number of views

fit(views, y=None, **kwargs)

Fits a regularised CCA (canonical ridge) model

Parameters:views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)

fit_transform(views, **kwargs)

Fits and then transforms the training data

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• kwargs – any additional keyword arguments required by the given model

get_loadings(views, normalize=False, **kwargs)

Returns the model loadings for each view for the given data

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• kwargs – any additional keyword arguments required by the given model
• normalize – scales loadings to ensure that they represent correlations between features and scores

score(views, y=None, **kwargs)

Returns average correlation in each dimension (averages over all pairs for multiview)

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• y – unused but needed to integrate with scikit-learn

transform(views, **kwargs)

Transforms data given a fit model

Parameters:

• views (Iterable[ndarray]) – numpy arrays with the same number of rows (samples) separated by commas
• kwargs – any additional keyword arguments required by the given model

Kernel Generalized (MAXVAR) Canonical Correlation Analysis

class cca_zoo.models.gcca.KGCCA(latent_dims=1, scale=True, centre=True, copy_data=True, random_state=None, c=None, eps=0.001, kernel=None, gamma=None, degree=None, coef0=None, kernel_params=None)

Bases: cca_zoo.models.gcca.GCCA

A class used to fit KGCCA model. For more than 2 views, KGCCA optimizes the sum of correlations with a shared auxiliary vector

Maths:

\[ \begin{align}\begin{aligned}\begin{split}w_{opt}=\underset{w}{\mathrm{argmax}}\{ \sum_i\alpha_i^TK_i^TT \}\\\end{split}\\\text{subject to:}\\T^TT=1\end{aligned}\end{align} \]

Citation:

Tenenhaus, Arthur, Cathy Philippe, and Vincent Frouin. “Kernel generalized canonical correlation analysis.” Computational Statistics & Data Analysis 90 (2015): 114-131.

Example:

>>> from cca_zoo.models import KGCCA
>>> import numpy as np
>>> rng=np.random.RandomState(0)
>>> X1 = rng.random((10,5))
>>> X2 = rng.random((10,5))
>>> X3 = rng.random((10,5))
>>> model = KGCCA()
>>> model.fit((X1,X2,X3)).score((X1,X2,X3))
array([0.97019284])

Constructor for PLS

Parameters:

• latent_dims (int) – number of latent dimensions to fit
• scale (bool) – normalize variance in each column before fitting
• centre – demean data by column before fitting (and before transforming out of sample
• copy_data – If True, X will be copied; else, it may be overwritten
• random_state – Pass for reproducible output across multiple function calls
• c (Union[Iterable[float], float, None]) – Iterable of regularisation parameters for each view (between 0:CCA and 1:PLS)
• eps – epsilon for stability
• kernel (Optional[Iterable[Union[float, <built-in function callable>]]]) – Iterable of kernel mappings used internally. This parameter is directly passed to pairwise_kernel. If element of kernel is a string, it must be one of the metrics in pairwise.PAIRWISE_KERNEL_FUNCTIONS. Alternatively, if element of kernel is a callable function, it is called on each pair of instances (rows) and the resulting value recorded. The callable should take two rows from X as input and return the corresponding kernel value as a single number. This means that callables from sklearn.metrics.pairwise are not allowed, as they operate on matrices, not single samples. Use the string identifying the kernel instead.
• gamma (Optional[Iterable[float]]) – Iterable of gamma parameters for the RBF, laplacian, polynomial, exponential chi2 and sigmoid kernels. Interpretation of the default value is left to the kernel; see the documentation for sklearn.metrics.pairwise. Ignored by other kernels.
• degree (Optional[Iterable[float]]) – Iterable of degree parameters of the polynomial kernel. Ignored by other kernels.
• coef0 (Optional[Iterable[float]]) – Iterable of zero coefficients for polynomial and sigmoid kernels. Ignored by other kernels.
• kernel_params (Optional[Iterable[dict]]) – Iterable of additional parameters (keyword arguments) for kernel function passed as callable object.
• eps – epsilon value to ensure stability of smallest eigenvalues

transform(views, y=None, **kwargs)

Transforms data given a fit KGCCA model

Parameters:

• views (ndarray) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• kwargs – any additional keyword arguments required by the given model

correlations(views, **kwargs)

Predicts the correlations between each view for each dimension for the given data using the fit model

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• kwargs – any additional keyword arguments required by the given model

Returns:

all_corrs: an array of the pairwise correlations (k,k,self.latent_dims) where k is the number of views

fit(views, y=None, **kwargs)

Fits a regularised CCA (canonical ridge) model

Parameters:views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)

fit_transform(views, **kwargs)

Fits and then transforms the training data

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• kwargs – any additional keyword arguments required by the given model

get_loadings(views, normalize=False, **kwargs)

Returns the model loadings for each view for the given data

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• kwargs – any additional keyword arguments required by the given model
• normalize – scales loadings to ensure that they represent correlations between features and scores

score(views, y=None, **kwargs)

Returns average correlation in each dimension (averages over all pairs for multiview)

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• y – unused but needed to integrate with scikit-learn

MCCA and KCCA

Multiset (SUMCOR) Canonical Correlation Analysis

class cca_zoo.models.mcca.MCCA(latent_dims=1, scale=True, centre=True, copy_data=True, random_state=None, c=None, eps=0.001)

Bases: cca_zoo.models.rcca.rCCA

A class used to fit MCCA model. For more than 2 views, MCCA optimizes the sum of pairwise correlations.

Maths:

\[ \begin{align}\begin{aligned}\begin{split}w_{opt}=\underset{w}{\mathrm{argmax}}\{\sum_i\sum_{j\neq i} w_i^TX_i^TX_jw_j \}\\\end{split}\\\text{subject to:}\\(1-c_i)w_i^TX_i^TX_iw_i+c_iw_i^Tw_i=1\end{aligned}\end{align} \]

Citation:

Kettenring, Jon R. “Canonical analysis of several sets of variables.” Biometrika 58.3 (1971): 433-451.

Example:

>>> from cca_zoo.models import MCCA
>>> import numpy as np
>>> rng=np.random.RandomState(0)
>>> X1 = rng.random((10,5))
>>> X2 = rng.random((10,5))
>>> X3 = rng.random((10,5))
>>> model = MCCA()
>>> model.fit((X1,X2,X3)).score((X1,X2,X3))
array([0.97200847])

Constructor for MCCA

Parameters:

• latent_dims (int) – number of latent dimensions to fit
• scale (bool) – normalize variance in each column before fitting
• centre – demean data by column before fitting (and before transforming out of sample
• copy_data – If True, X will be copied; else, it may be overwritten
• random_state – Pass for reproducible output across multiple function calls
• c (Union[Iterable[float], float, None]) – Iterable of regularisation parameters for each view (between 0:CCA and 1:PLS)
• eps – epsilon for stability

correlations(views, **kwargs)

Predicts the correlations between each view for each dimension for the given data using the fit model

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• kwargs – any additional keyword arguments required by the given model

Returns:

all_corrs: an array of the pairwise correlations (k,k,self.latent_dims) where k is the number of views

fit(views, y=None, **kwargs)

Fits a regularised CCA (canonical ridge) model

Parameters:views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)

fit_transform(views, **kwargs)

Fits and then transforms the training data

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• kwargs – any additional keyword arguments required by the given model

get_loadings(views, normalize=False, **kwargs)

Returns the model loadings for each view for the given data

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• kwargs – any additional keyword arguments required by the given model
• normalize – scales loadings to ensure that they represent correlations between features and scores

score(views, y=None, **kwargs)

Returns average correlation in each dimension (averages over all pairs for multiview)

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• y – unused but needed to integrate with scikit-learn

transform(views, **kwargs)

Transforms data given a fit model

Parameters:

• views (Iterable[ndarray]) – numpy arrays with the same number of rows (samples) separated by commas
• kwargs – any additional keyword arguments required by the given model

Kernel Multiset (SUMCOR) Canonical Correlation Analysis

class cca_zoo.models.mcca.KCCA(latent_dims=1, scale=True, centre=True, copy_data=True, random_state=None, c=None, eps=0.001, kernel=None, gamma=None, degree=None, coef0=None, kernel_params=None)

Bases: cca_zoo.models.mcca.MCCA

A class used to fit KCCA model.

Maths:

\[ \begin{align}\begin{aligned}\begin{split}\alpha_{opt}=\underset{\alpha}{\mathrm{argmax}}\{\sum_i\sum_{j\neq i} \alpha_i^TK_i^TK_j\alpha_j \}\\\end{split}\\\text{subject to:}\\c_i\alpha_i^TK_i\alpha_i + (1-c_i)\alpha_i^TK_i^TK_i\alpha_i=1\end{aligned}\end{align} \]

Example:

>>> from cca_zoo.models import KCCA
>>> import numpy as np
>>> rng=np.random.RandomState(0)
>>> X1 = rng.random((10,5))
>>> X2 = rng.random((10,5))
>>> X3 = rng.random((10,5))
>>> model = KCCA()
>>> model.fit((X1,X2,X3)).score((X1,X2,X3))
array([0.96893666])

Parameters:

• latent_dims (int) – number of latent dimensions to fit
• scale (bool) – normalize variance in each column before fitting
• centre – demean data by column before fitting (and before transforming out of sample
• copy_data – If True, X will be copied; else, it may be overwritten
• random_state – Pass for reproducible output across multiple function calls
• c (Union[Iterable[float], float, None]) – Iterable of regularisation parameters for each view (between 0:CCA and 1:PLS)
• eps – epsilon for stability
• kernel (Optional[Iterable[Union[float, <built-in function callable>]]]) – Iterable of kernel mappings used internally. This parameter is directly passed to pairwise_kernel. If element of kernel is a string, it must be one of the metrics in pairwise.PAIRWISE_KERNEL_FUNCTIONS. Alternatively, if element of kernel is a callable function, it is called on each pair of instances (rows) and the resulting value recorded. The callable should take two rows from X as input and return the corresponding kernel value as a single number. This means that callables from sklearn.metrics.pairwise are not allowed, as they operate on matrices, not single samples. Use the string identifying the kernel instead.
• gamma (Optional[Iterable[float]]) – Iterable of gamma parameters for the RBF, laplacian, polynomial, exponential chi2 and sigmoid kernels. Interpretation of the default value is left to the kernel; see the documentation for sklearn.metrics.pairwise. Ignored by other kernels.
• degree (Optional[Iterable[float]]) – Iterable of degree parameters of the polynomial kernel. Ignored by other kernels.
• coef0 (Optional[Iterable[float]]) – Iterable of zero coefficients for polynomial and sigmoid kernels. Ignored by other kernels.
• kernel_params (Optional[Iterable[dict]]) – Iterable of additional parameters (keyword arguments) for kernel function passed as callable object.
• eps – epsilon value to ensure stability of smallest eigenvalues

transform(views, **kwargs)

Transforms data given a fit KCCA model

Parameters:

• views (ndarray) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• kwargs – any additional keyword arguments required by the given model

correlations(views, **kwargs)

Predicts the correlations between each view for each dimension for the given data using the fit model

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• kwargs – any additional keyword arguments required by the given model

Returns:

all_corrs: an array of the pairwise correlations (k,k,self.latent_dims) where k is the number of views

fit(views, y=None, **kwargs)

Fits a regularised CCA (canonical ridge) model

Parameters:views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)

fit_transform(views, **kwargs)

Fits and then transforms the training data

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• kwargs – any additional keyword arguments required by the given model

get_loadings(views, normalize=False, **kwargs)

Returns the model loadings for each view for the given data

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• kwargs – any additional keyword arguments required by the given model
• normalize – scales loadings to ensure that they represent correlations between features and scores

score(views, y=None, **kwargs)

Returns average correlation in each dimension (averages over all pairs for multiview)

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• y – unused but needed to integrate with scikit-learn

Tensor Canonical Correlation Analysis

Tensor Canonical Correlation Analysis

class cca_zoo.models.tcca.TCCA(latent_dims=1, scale=True, centre=True, copy_data=True, random_state=None, c=None)

Bases: cca_zoo.models._cca_base._CCA_Base

Fits a Tensor CCA model. Tensor CCA maximises higher order correlations

Maths:

\[ \begin{align}\begin{aligned}\begin{split}\alpha_{opt}=\underset{\alpha}{\mathrm{argmax}}\{\sum_i\sum_{j\neq i} \alpha_i^TK_i^TK_j\alpha_j \}\\\end{split}\\\text{subject to:}\\\alpha_i^TK_i^TK_i\alpha_i=1\end{aligned}\end{align} \]

Citation:

Kim, Tae-Kyun, Shu-Fai Wong, and Roberto Cipolla. “Tensor canonical correlation analysis for action classification.” 2007 IEEE Conference on Computer Vision and Pattern Recognition. IEEE, 2007 https://github.com/rciszek/mdr_tcca

Example:

>>> from cca_zoo.models import TCCA
>>> rng=np.random.RandomState(0)
>>> X1 = rng.random((10,5))
>>> X2 = rng.random((10,5))
>>> X3 = rng.random((10,5))
>>> model = TCCA()
>>> model.fit((X1,X2,X3)).score((X1,X2,X3))
array([1.14595755])

Constructor for TCCA

Parameters:

• latent_dims (int) – number of latent dimensions to fit
• scale – normalize variance in each column before fitting
• centre – demean data by column before fitting (and before transforming out of sample
• copy_data – If True, X will be copied; else, it may be overwritten
• random_state – Pass for reproducible output across multiple function calls
• c (Union[Iterable[float], float, None]) – Iterable of regularisation parameters for each view (between 0:CCA and 1:PLS)

fit(views, y=None, **kwargs)

Parameters:views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)

correlations(views, **kwargs)

Predicts the correlation for the given data using the fit model

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• kwargs – any additional keyword arguments required by the given model

score(views, **kwargs)

Returns the higher order correlations in each dimension

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• kwargs – any additional keyword arguments required by the given model

fit_transform(views, **kwargs)

Fits and then transforms the training data

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• kwargs – any additional keyword arguments required by the given model

get_loadings(views, normalize=False, **kwargs)

Returns the model loadings for each view for the given data

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• kwargs – any additional keyword arguments required by the given model
• normalize – scales loadings to ensure that they represent correlations between features and scores

transform(views, **kwargs)

Transforms data given a fit model

Parameters:

• views (Iterable[ndarray]) – numpy arrays with the same number of rows (samples) separated by commas
• kwargs – any additional keyword arguments required by the given model

Kernel Tensor Canonical Correlation Analysis

class cca_zoo.models.tcca.KTCCA(latent_dims=1, scale=True, centre=True, copy_data=True, random_state=None, eps=0.001, c=None, kernel=None, gamma=None, degree=None, coef0=None, kernel_params=None)

Bases: cca_zoo.models.tcca.TCCA

Fits a Kernel Tensor CCA model. Tensor CCA maximises higher order correlations

Maths:

\[ \begin{align}\begin{aligned}\begin{split}\alpha_{opt}=\underset{\alpha}{\mathrm{argmax}}\{\sum_i\sum_{j\neq i} \alpha_i^TK_i^TK_j\alpha_j \}\\\end{split}\\\text{subject to:}\\\alpha_i^TK_i^TK_i\alpha_i=1\end{aligned}\end{align} \]

Citation:

Kim, Tae-Kyun, Shu-Fai Wong, and Roberto Cipolla. “Tensor canonical correlation analysis for action classification.” 2007 IEEE Conference on Computer Vision and Pattern Recognition. IEEE, 2007

Example:

>>> from cca_zoo.models import KTCCA
>>> rng=np.random.RandomState(0)
>>> X1 = rng.random((10,5))
>>> X2 = rng.random((10,5))
>>> X3 = rng.random((10,5))
>>> model = KTCCA()
>>> model.fit((X1,X2,X3)).score((X1,X2,X3))
array([1.69896269])

Constructor for TCCA

Parameters:

• latent_dims (int) – number of latent dimensions to fit
• scale (bool) – normalize variance in each column before fitting
• centre – demean data by column before fitting (and before transforming out of sample
• copy_data – If True, X will be copied; else, it may be overwritten
• random_state – Pass for reproducible output across multiple function calls
• c (Union[Iterable[float], float, None]) – Iterable of regularisation parameters for each view (between 0:CCA and 1:PLS)
• kernel (Optional[Iterable[Union[float, <built-in function callable>]]]) – Iterable of kernel mappings used internally. This parameter is directly passed to pairwise_kernel. If element of kernel is a string, it must be one of the metrics in pairwise.PAIRWISE_KERNEL_FUNCTIONS. Alternatively, if element of kernel is a callable function, it is called on each pair of instances (rows) and the resulting value recorded. The callable should take two rows from X as input and return the corresponding kernel value as a single number. This means that callables from sklearn.metrics.pairwise are not allowed, as they operate on matrices, not single samples. Use the string identifying the kernel instead.
• gamma (Optional[Iterable[float]]) – Iterable of gamma parameters for the RBF, laplacian, polynomial, exponential chi2 and sigmoid kernels. Interpretation of the default value is left to the kernel; see the documentation for sklearn.metrics.pairwise. Ignored by other kernels.
• degree (Optional[Iterable[float]]) – Iterable of degree parameters of the polynomial kernel. Ignored by other kernels.
• coef0 (Optional[Iterable[float]]) – Iterable of zero coefficients for polynomial and sigmoid kernels. Ignored by other kernels.
• kernel_params (Optional[Iterable[dict]]) – Iterable of additional parameters (keyword arguments) for kernel function passed as callable object.
• eps – epsilon value to ensure stability

transform(views, **kwargs)

Transforms data given a fit k=KCCA model

Parameters:

• views (ndarray) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• kwargs – any additional keyword arguments required by the given model

correlations(views, **kwargs)

Predicts the correlation for the given data using the fit model

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• kwargs – any additional keyword arguments required by the given model

fit(views, y=None, **kwargs)

Parameters:views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)

fit_transform(views, **kwargs)

Fits and then transforms the training data

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• kwargs – any additional keyword arguments required by the given model

get_loadings(views, normalize=False, **kwargs)

Returns the model loadings for each view for the given data

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• kwargs – any additional keyword arguments required by the given model
• normalize – scales loadings to ensure that they represent correlations between features and scores

score(views, **kwargs)

Returns the higher order correlations in each dimension

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• kwargs – any additional keyword arguments required by the given model

More Complex Regularisation using Iterative Models

Normal CCA and PLS by alternating least squares

Quicker and more memory efficient for very large data

CCA by Alternating Least Squares

class cca_zoo.models.CCA_ALS(latent_dims=1, scale=True, centre=True, copy_data=True, random_state=None, deflation='cca', max_iter=100, initialization='random', tol=1e-09, stochastic=True, positive=None)

Bases: cca_zoo.models.iterative.ElasticCCA

Fits a CCA model with CCA deflation by NIPALS algorithm. Implemented by ElasticCCA with no regularisation

Maths:

\[ \begin{align}\begin{aligned}\begin{split}w_{opt}=\underset{w}{\mathrm{argmax}}\{\sum_i\sum_{j\neq i} \|X_iw_i-X_jw_j\|^2\}\\\end{split}\\\text{subject to:}\\w_i^TX_i^TX_iw_i=n\end{aligned}\end{align} \]

Citation:

Golub, Gene H., and Hongyuan Zha. “The canonical correlations of matrix pairs and their numerical computation.” Linear algebra for signal processing. Springer, New York, NY, 1995. 27-49.

Example:

>>> from cca_zoo.models import CCA_ALS
>>> import numpy as np
>>> rng=np.random.RandomState(0)
>>> X1 = rng.random((10,3))
>>> X2 = rng.random((10,3))
>>> model = CCA_ALS(random_state=0)
>>> model.fit((X1,X2)).score((X1,X2))
array([0.858906])

Constructor for CCA_ALS

Parameters:

• latent_dims (int) – number of latent dimensions to fit
• scale (bool) – normalize variance in each column before fitting
• centre – demean data by column before fitting (and before transforming out of sample
• copy_data – If True, X will be copied; else, it may be overwritten
• random_state – Pass for reproducible output across multiple function calls
• max_iter (int) – the maximum number of iterations to perform in the inner optimization loop
• initialization (str) – either string from “pls”, “cca”, “random”, “uniform” or callable to initialize the score variables for iterative methods
• tol (float) – tolerance value used for early stopping
• stochastic – use stochastic regression optimisers for subproblems
• positive (Union[Iterable[bool], bool, None]) – constrain model weights to be positive

correlations(views, **kwargs)

Predicts the correlations between each view for each dimension for the given data using the fit model

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• kwargs – any additional keyword arguments required by the given model

Returns:

all_corrs: an array of the pairwise correlations (k,k,self.latent_dims) where k is the number of views

fit(views, y=None, **kwargs)

Fits the model by running an inner loop to convergence and then using either CCA or PLS deflation

Parameters:views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)

fit_transform(views, **kwargs)

Fits and then transforms the training data

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• kwargs – any additional keyword arguments required by the given model

get_loadings(views, normalize=False, **kwargs)

Returns the model loadings for each view for the given data

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• kwargs – any additional keyword arguments required by the given model
• normalize – scales loadings to ensure that they represent correlations between features and scores

score(views, y=None, **kwargs)

Returns average correlation in each dimension (averages over all pairs for multiview)

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• y – unused but needed to integrate with scikit-learn

transform(views, **kwargs)

Transforms data given a fit model

Parameters:

• views (Iterable[ndarray]) – numpy arrays with the same number of rows (samples) separated by commas
• kwargs – any additional keyword arguments required by the given model

PLS by Alternating Least Squares

class cca_zoo.models.PLS_ALS(latent_dims=1, scale=True, centre=True, copy_data=True, random_state=None, max_iter=100, initialization='random', tol=1e-09)

Bases: cca_zoo.models.iterative._Iterative

A class used to fit a PLS model

Fits a partial least squares model with CCA deflation by NIPALS algorithm

\[ \begin{align}\begin{aligned}\begin{split}w_{opt}=\underset{w}{\mathrm{argmax}}\{\sum_i\sum_{j\neq i} \|X_iw_i-X_jw_j\|^2\}\\\end{split}\\\text{subject to:}\\w_i^Tw_i=1\end{aligned}\end{align} \]

Example:

>>> from cca_zoo.models import PLS
>>> import numpy as np
>>> rng=np.random.RandomState(0)
>>> X1 = rng.random((10,5))
>>> X2 = rng.random((10,5))
>>> model = PLS_ALS(random_state=0)
>>> model.fit((X1,X2)).score((X1,X2))
array([0.81796854])

Constructor for PLS

Parameters:

• latent_dims (int) – number of latent dimensions to fit
• scale (bool) – normalize variance in each column before fitting
• centre – demean data by column before fitting (and before transforming out of sample
• copy_data – If True, X will be copied; else, it may be overwritten
• random_state – Pass for reproducible output across multiple function calls
• max_iter (int) – the maximum number of iterations to perform in the inner optimization loop
• initialization (Union[str, <built-in function callable>]) – either string from “pls”, “cca”, “random”, “uniform” or callable to initialize the score variables for iterative methods
• tol (float) – tolerance value used for early stopping

correlations(views, **kwargs)

Predicts the correlations between each view for each dimension for the given data using the fit model

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• kwargs – any additional keyword arguments required by the given model

Returns:

all_corrs: an array of the pairwise correlations (k,k,self.latent_dims) where k is the number of views

fit(views, y=None, **kwargs)

Fits the model by running an inner loop to convergence and then using either CCA or PLS deflation

Parameters:views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)

fit_transform(views, **kwargs)

Fits and then transforms the training data

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• kwargs – any additional keyword arguments required by the given model

get_loadings(views, normalize=False, **kwargs)

Returns the model loadings for each view for the given data

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• kwargs – any additional keyword arguments required by the given model
• normalize – scales loadings to ensure that they represent correlations between features and scores

score(views, y=None, **kwargs)

Returns average correlation in each dimension (averages over all pairs for multiview)

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• y – unused but needed to integrate with scikit-learn

transform(views, **kwargs)

Transforms data given a fit model

Parameters:

• views (Iterable[ndarray]) – numpy arrays with the same number of rows (samples) separated by commas
• kwargs – any additional keyword arguments required by the given model

Sparsity Inducing Models

Penalized Matrix Decomposition (Sparse PLS)

class cca_zoo.models.PMD(latent_dims=1, scale=True, centre=True, copy_data=True, random_state=None, deflation='cca', c=None, max_iter=100, initialization='pls', tol=1e-09, positive=None)

Bases: cca_zoo.models.iterative._Iterative

Fits a Sparse CCA (Penalized Matrix Decomposition) model.

Maths:

\[ \begin{align}\begin{aligned}\begin{split}w_{opt}=\underset{w}{\mathrm{argmax}}\{ w_1^TX_1^TX_2w_2 \}\\\end{split}\\\text{subject to:}\\w_i^Tw_i=1\\\|w_i\|<=c_i\end{aligned}\end{align} \]

Citation:

Witten, Daniela M., Robert Tibshirani, and Trevor Hastie. “A penalized matrix decomposition, with applications to sparse principal components and canonical correlation analysis.” Biostatistics 10.3 (2009): 515-534.

Example:

>>> from cca_zoo.models import PMD
>>> import numpy as np
>>> rng=np.random.RandomState(0)
>>> X1 = rng.random((10,5))
>>> X2 = rng.random((10,5))
>>> model = PMD(c=[1,1],random_state=0)
>>> model.fit((X1,X2)).score((X1,X2))
array([0.81796873])

Constructor for PMD

Parameters:

• latent_dims (int) – number of latent dimensions to fit
• scale (bool) – normalize variance in each column before fitting
• centre – demean data by column before fitting (and before transforming out of sample
• copy_data – If True, X will be copied; else, it may be overwritten
• random_state – Pass for reproducible output across multiple function calls
• c (Union[Iterable[float], float, None]) – l1 regularisation parameter between 1 and sqrt(number of features) for each view
• max_iter (int) – the maximum number of iterations to perform in the inner optimization loop
• initialization (Union[str, <built-in function callable>]) – either string from “pls”, “cca”, “random”, “uniform” or callable to initialize the score variables for iterative methods
• tol (float) – tolerance value used for early stopping
• positive (Union[Iterable[bool], bool, None]) – constrain model weights to be positive

correlations(views, **kwargs)

Predicts the correlations between each view for each dimension for the given data using the fit model

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• kwargs – any additional keyword arguments required by the given model

Returns:

all_corrs: an array of the pairwise correlations (k,k,self.latent_dims) where k is the number of views

fit(views, y=None, **kwargs)

Fits the model by running an inner loop to convergence and then using either CCA or PLS deflation

Parameters:views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)

fit_transform(views, **kwargs)

Fits and then transforms the training data

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• kwargs – any additional keyword arguments required by the given model

get_loadings(views, normalize=False, **kwargs)

Returns the model loadings for each view for the given data

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• kwargs – any additional keyword arguments required by the given model
• normalize – scales loadings to ensure that they represent correlations between features and scores

score(views, y=None, **kwargs)

Returns average correlation in each dimension (averages over all pairs for multiview)

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• y – unused but needed to integrate with scikit-learn

transform(views, **kwargs)

Transforms data given a fit model

Parameters:

• views (Iterable[ndarray]) – numpy arrays with the same number of rows (samples) separated by commas
• kwargs – any additional keyword arguments required by the given model

Sparse CCA by iterative lasso regression

class cca_zoo.models.SCCA(latent_dims=1, scale=True, centre=True, copy_data=True, random_state=None, deflation='cca', c=None, max_iter=100, maxvar=False, initialization='pls', tol=1e-09, stochastic=False, positive=None)

Bases: cca_zoo.models.iterative.ElasticCCA

Fits a sparse CCA model by iterative rescaled lasso regression. Implemented by ElasticCCA with l1 ratio=1

For default maxvar=False, the optimisation is given by:

Maths:

\[ \begin{align}\begin{aligned}\begin{split}w_{opt}=\underset{w}{\mathrm{argmax}}\{\sum_i\sum_{j\neq i} \|X_iw_i-X_jw_j\|^2 + \text{l1_ratio}\|w_i\|_1\}\\\end{split}\\\text{subject to:}\\w_i^TX_i^TX_iw_i=n\end{aligned}\end{align} \]

Citation:

Mai, Qing, and Xin Zhang. “An iterative penalized least squares approach to sparse canonical correlation analysis.” Biometrics 75.3 (2019): 734-744.

For maxvar=True, the optimisation is given by the ElasticCCA problem with no l2 regularisation:

Maths:

\[ \begin{align}\begin{aligned}\begin{split}w_{opt}, t_{opt}=\underset{w,t}{\mathrm{argmax}}\{\sum_i \|X_iw_i-t\|^2 + \text{l1_ratio}\|w_i\|_1\}\\\end{split}\\\text{subject to:}\\t^Tt=n\end{aligned}\end{align} \]

Citation:

Fu, Xiao, et al. “Scalable and flexible multiview MAX-VAR canonical correlation analysis.” IEEE Transactions on Signal Processing 65.16 (2017): 4150-4165.

Example:

>>> from cca_zoo.models import SCCA
>>> import numpy as np
>>> rng=np.random.RandomState(0)
>>> X1 = rng.random((10,5))
>>> X2 = rng.random((10,5))
>>> model = SCCA(c=[0.001,0.001], random_state=0)
>>> model.fit((X1,X2)).score((X1,X2))
array([0.99998761])

Constructor for SCCA

Parameters:

• latent_dims (int) – number of latent dimensions to fit
• scale (bool) – normalize variance in each column before fitting
• centre – demean data by column before fitting (and before transforming out of sample
• copy_data – If True, X will be copied; else, it may be overwritten
• random_state – Pass for reproducible output across multiple function calls
• max_iter (int) – the maximum number of iterations to perform in the inner optimization loop
• maxvar (bool) – use auxiliary variable “maxvar” form
• initialization (Union[str, <built-in function callable>]) – either string from “pls”, “cca”, “random”, “uniform” or callable to initialize the score variables for iterative methods
• tol (float) – tolerance value used for early stopping
• c (Union[Iterable[float], float, None]) – lasso alpha
• stochastic – use stochastic regression optimisers for subproblems
• positive (Union[Iterable[bool], bool, None]) – constrain model weights to be positive

correlations(views, **kwargs)

Predicts the correlations between each view for each dimension for the given data using the fit model

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• kwargs – any additional keyword arguments required by the given model

Returns:

all_corrs: an array of the pairwise correlations (k,k,self.latent_dims) where k is the number of views

fit(views, y=None, **kwargs)

Fits the model by running an inner loop to convergence and then using either CCA or PLS deflation

Parameters:views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)

fit_transform(views, **kwargs)

Fits and then transforms the training data

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• kwargs – any additional keyword arguments required by the given model

get_loadings(views, normalize=False, **kwargs)

Returns the model loadings for each view for the given data

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• kwargs – any additional keyword arguments required by the given model
• normalize – scales loadings to ensure that they represent correlations between features and scores

score(views, y=None, **kwargs)

Returns average correlation in each dimension (averages over all pairs for multiview)

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• y – unused but needed to integrate with scikit-learn

transform(views, **kwargs)

Transforms data given a fit model

Parameters:

• views (Iterable[ndarray]) – numpy arrays with the same number of rows (samples) separated by commas
• kwargs – any additional keyword arguments required by the given model

Elastic CCA by MAXVAR

class cca_zoo.models.ElasticCCA(latent_dims=1, scale=True, centre=True, copy_data=True, random_state=None, deflation='cca', max_iter=100, initialization='pls', tol=1e-09, c=None, l1_ratio=None, maxvar=True, stochastic=False, positive=None)

Bases: cca_zoo.models.iterative._Iterative

Fits an elastic CCA by iterating elastic net regressions.

By default, ElasticCCA uses CCA with an auxiliary variable target i.e. MAXVAR configuration

\[ \begin{align}\begin{aligned}\begin{split}w_{opt}, t_{opt}=\underset{w,t}{\mathrm{argmax}}\{\sum_i \|X_iw_i-t\|^2 + c\|w_i\|^2_2 + \text{l1_ratio}\|w_i\|_1\}\\\end{split}\\\text{subject to:}\\t^Tt=n\end{aligned}\end{align} \]

Citation:

Fu, Xiao, et al. “Scalable and flexible multiview MAX-VAR canonical correlation analysis.” IEEE Transactions on Signal Processing 65.16 (2017): 4150-4165.

But we can force it to attempt to use the SUMCOR form which will approximate a solution to the problem:

\[ \begin{align}\begin{aligned}\begin{split}w_{opt}=\underset{w}{\mathrm{argmax}}\{\sum_i\sum_{j\neq i} \|X_iw_i-X_jw_j\|^2 + c\|w_i\|^2_2 + \text{l1_ratio}\|w_i\|_1\}\\\end{split}\\\text{subject to:}\\w_i^TX_i^TX_iw_i=n\end{aligned}\end{align} \]

Example:

>>> from cca_zoo.models import ElasticCCA
>>> import numpy as np
>>> rng=np.random.RandomState(0)
>>> X1 = rng.random((10,5))
>>> X2 = rng.random((10,5))
>>> model = ElasticCCA(c=[1e-1,1e-1],l1_ratio=[0.5,0.5], random_state=0)
>>> model.fit((X1,X2)).score((X1,X2))
array([0.9316638])

Constructor for ElasticCCA

Parameters:

• latent_dims (int) – number of latent dimensions to fit
• scale (bool) – normalize variance in each column before fitting
• centre – demean data by column before fitting (and before transforming out of sample
• copy_data – If True, X will be copied; else, it may be overwritten
• random_state – Pass for reproducible output across multiple function calls
• deflation – the type of deflation.
• max_iter (int) – the maximum number of iterations to perform in the inner optimization loop
• initialization (Union[str, <built-in function callable>]) – either string from “pls”, “cca”, “random”, “uniform” or callable to initialize the score variables for iterative methods
• tol (float) – tolerance value used for early stopping
• c (Union[Iterable[float], float, None]) – lasso alpha
• l1_ratio (Union[Iterable[float], float, None]) – l1 ratio in lasso subproblems
• maxvar (bool) – use auxiliary variable “maxvar” formulation
• stochastic – use stochastic regression optimisers for subproblems
• positive (Union[Iterable[bool], bool, None]) – constrain model weights to be positive

correlations(views, **kwargs)

Predicts the correlations between each view for each dimension for the given data using the fit model

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• kwargs – any additional keyword arguments required by the given model

Returns:

all_corrs: an array of the pairwise correlations (k,k,self.latent_dims) where k is the number of views

fit(views, y=None, **kwargs)

Fits the model by running an inner loop to convergence and then using either CCA or PLS deflation

Parameters:views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)

fit_transform(views, **kwargs)

Fits and then transforms the training data

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• kwargs – any additional keyword arguments required by the given model

get_loadings(views, normalize=False, **kwargs)

Returns the model loadings for each view for the given data

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• kwargs – any additional keyword arguments required by the given model
• normalize – scales loadings to ensure that they represent correlations between features and scores

score(views, y=None, **kwargs)

Returns average correlation in each dimension (averages over all pairs for multiview)

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• y – unused but needed to integrate with scikit-learn

transform(views, **kwargs)

Transforms data given a fit model

Parameters:

• views (Iterable[ndarray]) – numpy arrays with the same number of rows (samples) separated by commas
• kwargs – any additional keyword arguments required by the given model

Span CCA

class cca_zoo.models.SpanCCA(latent_dims=1, scale=True, centre=True, copy_data=True, max_iter=100, initialization='uniform', tol=1e-09, regularisation='l0', c=None, rank=1, positive=None, random_state=None, deflation='cca')

Bases: cca_zoo.models.iterative._Iterative

Fits a Sparse CCA model using SpanCCA.

\[ \begin{align}\begin{aligned}\begin{split}w_{opt}=\underset{w}{\mathrm{argmax}}\{\sum_i\sum_{j\neq i} \|X_iw_i-X_jw_j\|^2 + \text{l1_ratio}\|w_i\|_1\}\\\end{split}\\\text{subject to:}\\w_i^TX_i^TX_iw_i=1\end{aligned}\end{align} \]

Citation:

Asteris, Megasthenis, et al. “A simple and provable algorithm for sparse diagonal CCA.” International Conference on Machine Learning. PMLR, 2016.

Example:

>>> from cca_zoo.models import SpanCCA
>>> import numpy as np
>>> rng=np.random.RandomState(0)
>>> X1 = rng.random((10,5))
>>> X2 = rng.random((10,5))
>>> model = SpanCCA(regularisation="l0", c=[2, 2])
>>> model.fit((X1,X2)).score((X1,X2))
array([0.84556666])

Parameters:

• latent_dims (int) – number of latent dimensions to fit
• scale (bool) – normalize variance in each column before fitting
• centre – demean data by column before fitting (and before transforming out of sample
• copy_data – If True, X will be copied; else, it may be overwritten
• random_state – Pass for reproducible output across multiple function calls
• max_iter (int) – the maximum number of iterations to perform in the inner optimization loop
• initialization (str) – either string from “pls”, “cca”, “random”, “uniform” or callable to initialize the score variables for iterative methods
• tol (float) – tolerance value used for early stopping
• regularisation –
• c (Union[Iterable[Union[float, int]], float, int, None]) – regularisation parameter
• rank – rank of the approximation
• positive (Union[Iterable[bool], bool, None]) – constrain weights to be positive

correlations(views, **kwargs)

Predicts the correlations between each view for each dimension for the given data using the fit model

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• kwargs – any additional keyword arguments required by the given model

Returns:

all_corrs: an array of the pairwise correlations (k,k,self.latent_dims) where k is the number of views

fit(views, y=None, **kwargs)

Fits the model by running an inner loop to convergence and then using either CCA or PLS deflation

Parameters:views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)

fit_transform(views, **kwargs)

Fits and then transforms the training data

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• kwargs – any additional keyword arguments required by the given model

get_loadings(views, normalize=False, **kwargs)

Returns the model loadings for each view for the given data

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• kwargs – any additional keyword arguments required by the given model
• normalize – scales loadings to ensure that they represent correlations between features and scores

score(views, y=None, **kwargs)

Returns average correlation in each dimension (averages over all pairs for multiview)

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• y – unused but needed to integrate with scikit-learn

transform(views, **kwargs)

Transforms data given a fit model

Parameters:

• views (Iterable[ndarray]) – numpy arrays with the same number of rows (samples) separated by commas
• kwargs – any additional keyword arguments required by the given model

Parkhomenko (penalized) CCA

class cca_zoo.models.ParkhomenkoCCA(latent_dims=1, scale=True, centre=True, copy_data=True, random_state=None, deflation='cca', c=None, max_iter=100, initialization='pls', tol=1e-09)

Bases: cca_zoo.models.iterative._Iterative

Fits a sparse CCA (penalized CCA) model

\[ \begin{align}\begin{aligned}\begin{split}w_{opt}=\underset{w}{\mathrm{argmax}}\{ w_1^TX_1^TX_2w_2 \} + c_i\|w_i\|\\\end{split}\\\text{subject to:}\\w_i^Tw_i=1\end{aligned}\end{align} \]

Citation:

Parkhomenko, Elena, David Tritchler, and Joseph Beyene. “Sparse canonical correlation analysis with application to genomic data integration.” Statistical applications in genetics and molecular biology 8.1 (2009).

Example:

>>> from cca_zoo.models import ParkhomenkoCCA
>>> import numpy as np
>>> rng=np.random.RandomState(0)
>>> X1 = rng.random((10,5))
>>> X2 = rng.random((10,5))
>>> model = ParkhomenkoCCA(c=[0.001,0.001],random_state=0)
>>> model.fit((X1,X2)).score((X1,X2))
array([0.81803527])

Constructor for ParkhomenkoCCA

Parameters:

• latent_dims (int) – number of latent dimensions to fit
• scale (bool) – normalize variance in each column before fitting
• centre – demean data by column before fitting (and before transforming out of sample
• copy_data – If True, X will be copied; else, it may be overwritten
• random_state – Pass for reproducible output across multiple function calls
• c (Union[Iterable[float], float, None]) – l1 regularisation parameter
• max_iter (int) – the maximum number of iterations to perform in the inner optimization loop
• initialization (Union[str, <built-in function callable>]) – either string from “pls”, “cca”, “random”, “uniform” or callable to initialize the score variables for iterative methods
• tol (float) – tolerance value used for early stopping

correlations(views, **kwargs)

Predicts the correlations between each view for each dimension for the given data using the fit model

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• kwargs – any additional keyword arguments required by the given model

Returns:

all_corrs: an array of the pairwise correlations (k,k,self.latent_dims) where k is the number of views

fit(views, y=None, **kwargs)

Fits the model by running an inner loop to convergence and then using either CCA or PLS deflation

Parameters:views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)

fit_transform(views, **kwargs)

Fits and then transforms the training data

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• kwargs – any additional keyword arguments required by the given model

get_loadings(views, normalize=False, **kwargs)

Returns the model loadings for each view for the given data

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• kwargs – any additional keyword arguments required by the given model
• normalize – scales loadings to ensure that they represent correlations between features and scores

score(views, y=None, **kwargs)

Returns average correlation in each dimension (averages over all pairs for multiview)

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• y – unused but needed to integrate with scikit-learn

transform(views, **kwargs)

Transforms data given a fit model

Parameters:

• views (Iterable[ndarray]) – numpy arrays with the same number of rows (samples) separated by commas
• kwargs – any additional keyword arguments required by the given model

Sparse CCA by ADMM

class cca_zoo.models.SCCA_ADMM(latent_dims=1, scale=True, centre=True, copy_data=True, random_state=None, deflation='cca', c=None, mu=None, lam=None, eta=None, max_iter=100, initialization='pls', tol=1e-09)

Bases: cca_zoo.models.iterative._Iterative

Fits a sparse CCA model by alternating ADMM

\[ \begin{align}\begin{aligned}\begin{split}w_{opt}=\underset{w}{\mathrm{argmax}}\{\sum_i\sum_{j\neq i} \|X_iw_i-X_jw_j\|^2 + \text{l1_ratio}\|w_i\|_1\}\\\end{split}\\\text{subject to:}\\w_i^TX_i^TX_iw_i=1\end{aligned}\end{align} \]

Citation:

Suo, Xiaotong, et al. “Sparse canonical correlation analysis.” arXiv preprint arXiv:1705.10865 (2017).

Example:

>>> from cca_zoo.models import SCCA_ADMM
>>> import numpy as np
>>> rng=np.random.RandomState(0)
>>> X1 = rng.random((10,5))
>>> X2 = rng.random((10,5))
>>> model = SCCA_ADMM(random_state=0,c=[1e-1,1e-1])
>>> model.fit((X1,X2)).score((X1,X2))
array([0.84348183])

Constructor for SCCA_ADMM

Parameters:

• latent_dims (int) – number of latent dimensions to fit
• scale (bool) – normalize variance in each column before fitting
• centre – demean data by column before fitting (and before transforming out of sample
• copy_data – If True, X will be copied; else, it may be overwritten
• random_state – Pass for reproducible output across multiple function calls
• c (Union[Iterable[float], float, None]) – l1 regularisation parameter
• max_iter (int) – the maximum number of iterations to perform in the inner optimization loop
• initialization (Union[str, <built-in function callable>]) – either string from “pls”, “cca”, “random”, “uniform” or callable to initialize the score variables for iterative methods
• tol (float) – tolerance value used for early stopping
• mu (Union[Iterable[float], float, None]) –
• lam (Union[Iterable[float], float, None]) –

Param:

eta:

correlations(views, **kwargs)

Predicts the correlations between each view for each dimension for the given data using the fit model

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• kwargs – any additional keyword arguments required by the given model

Returns:

all_corrs: an array of the pairwise correlations (k,k,self.latent_dims) where k is the number of views

fit(views, y=None, **kwargs)

Fits the model by running an inner loop to convergence and then using either CCA or PLS deflation

Parameters:views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)

fit_transform(views, **kwargs)

Fits and then transforms the training data

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• kwargs – any additional keyword arguments required by the given model

get_loadings(views, normalize=False, **kwargs)

Returns the model loadings for each view for the given data

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• kwargs – any additional keyword arguments required by the given model
• normalize – scales loadings to ensure that they represent correlations between features and scores

score(views, y=None, **kwargs)

Returns average correlation in each dimension (averages over all pairs for multiview)

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• y – unused but needed to integrate with scikit-learn

transform(views, **kwargs)

Transforms data given a fit model

Parameters:

• views (Iterable[ndarray]) – numpy arrays with the same number of rows (samples) separated by commas
• kwargs – any additional keyword arguments required by the given model

Miscellaneous

Nonparametric CCA

class cca_zoo.models.NCCA(latent_dims=1, scale=True, centre=True, copy_data=True, accept_sparse=False, random_state=None, nearest_neighbors=None, gamma=None)

Bases: cca_zoo.models._cca_base._CCA_Base

A class used to fit nonparametric (NCCA) model.

Citation:

Michaeli, Tomer, Weiran Wang, and Karen Livescu. “Nonparametric canonical correlation analysis.” International conference on machine learning. PMLR, 2016.

Example:

>>> from cca_zoo.models import NCCA
>>> X1 = np.random.rand(10,5)
>>> X2 = np.random.rand(10,5)
>>> model = NCCA()
>>> model.fit((X1,X2)).score((X1,X2))
array([1.])

Constructor for NCCA

Parameters:

• latent_dims (int) – number of latent dimensions to fit
• scale – normalize variance in each column before fitting
• centre – demean data by column before fitting (and before transforming out of sample
• copy_data – If True, X will be copied; else, it may be overwritten
• accept_sparse – Whether model can take sparse data as input
• random_state (Union[int, RandomState, None]) – Pass for reproducible output across multiple function calls
• nearest_neighbors – Number of nearest neighbors (l2 distance) to consider when constructing affinity
• gamma (Optional[Iterable[float]]) – Bandwidth parameter for rbf kernel

fit(views, y=None, **kwargs)

Fits a given model

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• y – unused but needed to integrate with scikit-learn

transform(views, **kwargs)

Transforms data given a fit model

Parameters:

• views (Iterable[ndarray]) – numpy arrays with the same number of rows (samples) separated by commas
• kwargs – any additional keyword arguments required by the given model

correlations(views, **kwargs)

Predicts the correlations between each view for each dimension for the given data using the fit model

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• kwargs – any additional keyword arguments required by the given model

Returns:

all_corrs: an array of the pairwise correlations (k,k,self.latent_dims) where k is the number of views

fit_transform(views, **kwargs)

Fits and then transforms the training data

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• kwargs – any additional keyword arguments required by the given model

get_loadings(views, normalize=False, **kwargs)

Returns the model loadings for each view for the given data

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• kwargs – any additional keyword arguments required by the given model
• normalize – scales loadings to ensure that they represent correlations between features and scores

score(views, y=None, **kwargs)

Returns average correlation in each dimension (averages over all pairs for multiview)

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• y – unused but needed to integrate with scikit-learn

Partial CCA

class cca_zoo.models.PartialCCA(latent_dims=1, scale=True, centre=True, copy_data=True, random_state=None, c=None, eps=0.001)

Bases: cca_zoo.models.mcca.MCCA

A class used to fit a partial cca model. The key difference between this and a vanilla CCA or MCCA is that the canonical score vectors must be orthogonal to the supplied confounding variables.

Citation:

Rao, B. Raja. “Partial canonical correlations.” Trabajos de estadistica y de investigación operativa 20.2-3 (1969): 211-219.

Example:

>>> from cca_zoo.models import PartialCCA
>>> X1 = np.random.rand(10,5)
>>> X2 = np.random.rand(10,5)
>>> partials = np.random.rand(10,3)
>>> model = PartialCCA()
>>> model.fit((X1,X2),partials=partials).score((X1,X2),partials=partials)
array([0.99993046])

Constructor for Partial CCA

Parameters:

• latent_dims (int) – number of latent dimensions to fit
• scale (bool) – normalize variance in each column before fitting
• centre – demean data by column before fitting (and before transforming out of sample
• copy_data – If True, X will be copied; else, it may be overwritten
• random_state – Pass for reproducible output across multiple function calls
• c (Union[Iterable[float], float, None]) – Iterable of regularisation parameters for each view (between 0:CCA and 1:PLS)
• eps – epsilon for stability

transform(views, partials=None, **kwargs)

Transforms data given a fit model

Parameters:views (Iterable[ndarray]) – numpy arrays with the same number of rows (samples) separated by commas

correlations(views, **kwargs)

Predicts the correlations between each view for each dimension for the given data using the fit model

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• kwargs – any additional keyword arguments required by the given model

Returns:

all_corrs: an array of the pairwise correlations (k,k,self.latent_dims) where k is the number of views

fit(views, y=None, **kwargs)

Fits a regularised CCA (canonical ridge) model

Parameters:views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)

fit_transform(views, **kwargs)

Fits and then transforms the training data

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• kwargs – any additional keyword arguments required by the given model

get_loadings(views, normalize=False, **kwargs)

Returns the model loadings for each view for the given data

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• kwargs – any additional keyword arguments required by the given model
• normalize – scales loadings to ensure that they represent correlations between features and scores

score(views, y=None, **kwargs)

Returns average correlation in each dimension (averages over all pairs for multiview)

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• y – unused but needed to integrate with scikit-learn

Sparse Weighted CCA

class cca_zoo.models.SWCCA(latent_dims=1, scale=True, centre=True, copy_data=True, random_state=None, max_iter=500, initialization='uniform', tol=1e-09, regularisation='l0', c=None, sample_support=None, positive=False)

Bases: cca_zoo.models.iterative._Iterative

A class used to fit SWCCA model

Citation:

Example:

>>> from cca_zoo.models import SWCCA
>>> import numpy as np
>>> rng=np.random.RandomState(0)
>>> X1 = rng.random((10,5))
>>> X2 = rng.random((10,5))
>>> model = SWCCA(regularisation='l0',c=[2, 2], sample_support=5, random_state=0)
>>> model.fit((X1,X2)).score((X1,X2))
array([0.61620969])

Parameters:

• latent_dims (int) – number of latent dimensions to fit
• scale (bool) – normalize variance in each column before fitting
• centre – demean data by column before fitting (and before transforming out of sample
• copy_data – If True, X will be copied; else, it may be overwritten
• random_state – Pass for reproducible output across multiple function calls
• max_iter (int) – the maximum number of iterations to perform in the inner optimization loop
• initialization (str) – either string from “pls”, “cca”, “random”, “uniform” or callable to initialize the score variables for iterative methods
• tol (float) – tolerance value used for early stopping
• regularisation – the type of regularisation on the weights either ‘l0’ or ‘l1’
• c (Union[Iterable[Union[float, int]], float, int, None]) – regularisation parameter
• sample_support – the l0 norm of the sample weights
• positive – constrain weights to be positive

correlations(views, **kwargs)

Predicts the correlations between each view for each dimension for the given data using the fit model

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• kwargs – any additional keyword arguments required by the given model

Returns:

all_corrs: an array of the pairwise correlations (k,k,self.latent_dims) where k is the number of views

fit(views, y=None, **kwargs)

Fits the model by running an inner loop to convergence and then using either CCA or PLS deflation

Parameters:views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)

fit_transform(views, **kwargs)

Fits and then transforms the training data

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• kwargs – any additional keyword arguments required by the given model

get_loadings(views, normalize=False, **kwargs)

Returns the model loadings for each view for the given data

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• kwargs – any additional keyword arguments required by the given model
• normalize – scales loadings to ensure that they represent correlations between features and scores

score(views, y=None, **kwargs)

Returns average correlation in each dimension (averages over all pairs for multiview)

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• y – unused but needed to integrate with scikit-learn

transform(views, **kwargs)

Transforms data given a fit model

Parameters:

• views (Iterable[ndarray]) – numpy arrays with the same number of rows (samples) separated by commas
• kwargs – any additional keyword arguments required by the given model

Base Class

class cca_zoo.models._cca_base._CCA_Base(latent_dims=1, scale=True, centre=True, copy_data=True, accept_sparse=False, random_state=None)

Bases: sklearn.base.BaseEstimator, sklearn.base.MultiOutputMixin, sklearn.base.RegressorMixin

A class used as the base for methods in the package. Allows methods to inherit fit_transform, predict_corr, and gridsearch_fit when only fit (and transform where it is different to the default) is provided.

weights

Type:list of weights for each view

Constructor for _CCA_Base

Parameters:

• latent_dims (int) – number of latent dimensions to fit
• scale – normalize variance in each column before fitting
• centre – demean data by column before fitting (and before transforming out of sample
• copy_data – If True, X will be copied; else, it may be overwritten
• accept_sparse – Whether model can take sparse data as input
• random_state (Union[int, RandomState, None]) – Pass for reproducible output across multiple function calls

fit(views, y=None, **kwargs)

Fits a given model

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• y – unused but needed to integrate with scikit-learn

transform(views, **kwargs)

Transforms data given a fit model

Parameters:

• views (Iterable[ndarray]) – numpy arrays with the same number of rows (samples) separated by commas
• kwargs – any additional keyword arguments required by the given model

fit_transform(views, **kwargs)

Fits and then transforms the training data

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• kwargs – any additional keyword arguments required by the given model

get_loadings(views, normalize=False, **kwargs)

Returns the model loadings for each view for the given data

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• kwargs – any additional keyword arguments required by the given model
• normalize – scales loadings to ensure that they represent correlations between features and scores

correlations(views, **kwargs)

Predicts the correlations between each view for each dimension for the given data using the fit model

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• kwargs – any additional keyword arguments required by the given model

Returns:

all_corrs: an array of the pairwise correlations (k,k,self.latent_dims) where k is the number of views

score(views, y=None, **kwargs)

Returns average correlation in each dimension (averages over all pairs for multiview)

Parameters:

• views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)
• y – unused but needed to integrate with scikit-learn

_centre_scale(views)

Removes the mean of the training data and standardizes for each view and stores mean and standard deviation during training

Parameters:views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples) Returns:train_views: the demeaned numpy arrays to be used to fit the model

_centre_scale_transform(views)

Removes the mean and standardizes each view based on the mean and standard deviation of the training data

Parameters:views (Iterable[ndarray]) – list/tuple of numpy arrays or array likes with the same number of rows (samples)