# Post-processing¶

Functions:

 add_factor_metadata(cp_tensor, dataset) Adds the additional coordinates along each dataset dimension as new columns in the factor matrices. factor_matrix_to_tidy(factor_matrix[, ...]) Convert a factor matrix into a tidy dataset, for use with Plotly Express. label_cp_tensor(cp_tensor, dataset) Label the CP tensor by converting the factor matrices into DataFrames with a sensible index. postprocess(cp_tensor[, dataset, ...]) Standard postprocessing of a CP decomposition. resolve_cp_sign_indeterminacy(cp_tensor, dataset) Resolve the sign indeterminacy of CP models.
tlviz.postprocessing.add_factor_metadata(cp_tensor, dataset)[source]

Adds the additional coordinates along each dataset dimension as new columns in the factor matrices.

The coordinates of xarray DataArrays can contain metadata. For each dimension, there may be additional coordinates that are not used for indexing purposes. This function will iterate over all modes of a dataset and a labelled CP tensor and add the additional coordinates as new columns in the factor matrices.

Parameters:
cp_tensorlabelled CP Tensor
datasetxarray.DataArray
Returns:
tuple

CP-tensor like tuple where the factor matrices are augmented with additional metadata.

Examples

>>> from tlviz.data import load_oslo_city_bike
>>> from tlviz.postprocessing import postprocess, add_factor_metadata
>>> from tensorly.decomposition import parafac
>>> bikes = load_oslo_city_bike()
>>> bikes.coords
Coordinates:
* End station name  (End station name) object '7 Juni Plassen' ... 'Økernve...
lat               (End station name) float64 59.92 59.93 ... 59.93 59.92
lon               (End station name) float64 10.73 10.75 ... 10.8 10.78
* Hour              (Hour) int32 0 1 2 3 4 5 6 7 8 ... 16 17 18 19 20 21 22 23
* Month             (Month) int32 1 2 3 4 5 6 7 8 9 10 11 12
* Day of week       (Day of week) int32 0 1 2 3 4 5 6
* Year              (Year) int32 2020 2021


We see that the End station name dimension has two additional columns: lat and lon. These contain metadata about the end station coordinates, and it can be useful to have these columns also in the factor matrices. To do this, we first fit a PARAFAC model to the dataset, then we postprocess it to label the CP tensor and finally, we add the metadata information

>>> cp = parafac(bikes.data, 3, init="random")
>>> cp_labelled = postprocess(cp, bikes)
>>> print(cp_labelled[1][0].columns)
RangeIndex(start=0, stop=3, step=1)
>>> cp_with_metadata = add_factor_metadata(cp_labelled, bikes)
>>> print(cp_with_metadata[1][0].columns)
Index([0, 1, 2, 'lat', 'lon'], dtype='object')


We see that when we add the metadata, then the latitude and longitude columns are added to the dataframe.

tlviz.postprocessing.factor_matrix_to_tidy(factor_matrix, var_name='Component', value_name='Signal', id_vars=None)[source]

Convert a factor matrix into a tidy dataset, for use with Plotly Express.

If we convert a factor matrix into a tidy dataset (or long table), then we get a table on the form

Tidy format factor matrix

Index

Component

Signal

0

0

0.1

1

0

0.5

38

2

0.7

39

2

0.2

The component vectors are all stacked on top of each other, with a separate column that specifies which component each row belongs to. This function can also preserve metadata, which is signified by columns that have non-integer column names. For example, if we have a dataframe on the form

Factor matrix with metadata

Index

0

1

2

lat

lon

0

0.1

0.2

0.5

59.91273

10.74609

1

0.5

0.2

0.1

63.43049

10.39506

5

0.2

0.1

0.3

60.39299

5.32415

5

0.0

0.2

0.1

58.97005

5.73332

and convert it into a tidy format factor matrix, then we get a table on the form

Tidy format factor matrix with metadata

Index

lat

lon

Component

Signal

0

59.91273

10.74609

0

0.1

1

63.43049

10.39506

0

0.5

4

69.6489

18.95508

2

0.0

5

58.97005

5.73332

2

0.1

Parameters:
factor_matrixpd.DataFrame

A labelled factor matrix potentially with metadata columns

var_namestr

Name of the column that specifies which component each row belongs to

value_namestr

Name of the column that holds the magnitude of each component entry

id_varsiterable or None (default=None)

Which columns to interpret as metadata. The columns not specified here are considered as the components. If id_vars is None, then all columns with non-integer names are considered metadata columns.

Returns:
pd.DataFrame

Tidy format factor matrix

Examples

>>> import numpy as np
>>> import pandas as pd
>>> from tlviz.postprocessing import factor_matrix_to_tidy
>>> rng = np.random.default_rng(0)
>>> factor_matrix = pd.DataFrame(rng.uniform(size=(10, 3)))
>>> factor_matrix.head()
0         1         2
0  0.636962  0.269787  0.040974
1  0.016528  0.813270  0.912756
2  0.606636  0.729497  0.543625
3  0.935072  0.815854  0.002739
4  0.857404  0.033586  0.729655
>>> tidy_factor_matrix = factor_matrix_to_tidy(factor_matrix)
>>> tidy_factor_matrix.head()
index Component    Signal
0      0         0  0.636962
1      1         0  0.016528
2      2         0  0.606636
3      3         0  0.935072
4      4         0  0.857404
>>> factor_matrix_with_metadata = factor_matrix.copy()
>>> factor_matrix_with_metadata["Metadata"] = rng.uniform(size=10)
>>> factor_matrix_with_metadata.head()
0         1         2  Metadata
0  0.636962  0.269787  0.040974  0.688447
1  0.016528  0.813270  0.912756  0.388921
2  0.606636  0.729497  0.543625  0.135097
3  0.935072  0.815854  0.002739  0.721488
4  0.857404  0.033586  0.729655  0.525354
>>> tidy_factor_matrix_with_metadata = factor_matrix_to_tidy(factor_matrix_with_metadata)
>>> tidy_factor_matrix_with_metadata.head()
Metadata  index Component    Signal
0  0.688447      0         0  0.636962
1  0.388921      1         0  0.016528
2  0.135097      2         0  0.606636
3  0.721488      3         0  0.935072
4  0.525354      4         0  0.857404

tlviz.postprocessing.label_cp_tensor(cp_tensor, dataset)[source]

Label the CP tensor by converting the factor matrices into DataFrames with a sensible index.

Convert the factor matrices into Pandas DataFrames where the DataFrame indices are given by the coordinate names of an xarray DataArray. If the dataset has only two modes, then it can also be a pandas DataFrame.

Parameters:
cp_tensorCPTensor

CP Tensor whose factor matrices should be labelled

datasetxarray.DataArray of pandas.DataFrame

Dataset used to label the factor matrices

Returns:
CPTensor

Tuple on the CPTensor format, except that the factor matrices are DataFrames.

tlviz.postprocessing.postprocess(cp_tensor, dataset=None, reference_cp_tensor=None, permute=True, resolve_mode=None, unresolved_mode=-1, flip_method='transpose', weight_behaviour='normalise', weight_mode=0, allow_smaller_rank=False, include_metadata=False)[source]

Standard postprocessing of a CP decomposition.

This function will perform standard postprocessing of a CP decomposition. If a reference CP tensor is provided, then the columns of cp_tensor’s factor matrices are aligned with the columns of reference_cp_tensor’s factor matrices.

Next, the factor matrices of the CP tensor are scaled according the the specified weight behaviour (default is normalised).

If a dataset is provided, then the sign indeterminacy is resolved and if the dataset is labelled (i.e. is an xarray or a dataframe), then the factor matrices of the CP tensor is labelled too.

This function is equivalent to calling

• permute_cp_tensor

• distribute_weights

• resolve_cp_sign_indeterminacy

• label_cp_tensor

Parameters:
cp_tensorCPTensor or tuple

CPTensor to postprocess

datasetndarray or xarray (optional)

Dataset the CP tensor represents

reference_cp_tensorCPTensor or tuple (optional)

If provided, then the tensor whose factors we align the CP tensor’s columns with.

permutebool

If True, then the factors are permuted in descending weight order if reference_cp_tensor is None.

resolve_modeint, iterable or None

Mode(s) whose factor matrix should be aligned with the data. If None, then the sign should be corrected for all modes except the unresolved_mode.

unresolved_modeint

Mode used to correct the sign indeterminacy in other mode(s). The factor matrix in this mode may not be aligned with the data.

method“transpose” or “positive_coord”

Which method to use when computing the signs. Use "transpose" for the method in [BAK08, BLJ13], and "positive_coord" for the method corrected for non-orthogonal factor matrices described above.

weight_behaviour{“ignore”, “normalise”, “evenly”, “one_mode”} (default=”normalise”)

How to handle the component weights.

• "ignore" - Do nothing

• "normalise" - Normalise all factor matrices

• "evenly" - All factor matrices have equal norm

• "one_mode" - The weight is allocated in one mode, all other factor matrices have unit norm columns.

weight_modeint (optional)

Which mode to have the component weights in (only used if weight_behaviour="one_mode")

allow_smaller_rankbool (default=False)

If True, then a low rank decomposition can be permuted against one with higher rank. The “missing columns” are padded by nan values

include_metadatabool (default=True)

If True, then the factor metadata will be added as columns in the factor matrices.

Returns:
CPTensor

The post processed CPTensor.

Examples

Here is an example were we use postprocess on a decomposition of aminoacid data

>>> import tlviz
>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from tensorly.decomposition import parafac
>>> dataset = tlviz.data.load_aminoacids()
Loading Aminoacids dataset from:
Bro, R, PARAFAC: Tutorial and applications, Chemometrics and Intelligent Laboratory Systems, 1997, 38, 149-171


The dataset is an xarray DataArray and it contains relevant side information

>>> print(type(dataset))
<class 'xarray.core.dataarray.DataArray'>


We see that after postprocessing, the cp_tensor contains pandas DataFrames

>>> cp_tensor = parafac(dataset.data, 3, init="random", random_state=0)
>>> cp_tensor_postprocessed = tlviz.postprocessing.postprocess(cp_tensor, dataset)
>>> print(type(cp_tensor[1][0]))
<class 'numpy.ndarray'>
>>> print(type(cp_tensor_postprocessed[1][0]))
<class 'pandas.core.frame.DataFrame'>


We see that after postprocessing, the factor matrix has unit norm

>>> print(np.linalg.norm(cp_tensor[1][0], axis=0))
[160.82985402 182.37338941 125.3689186 ]
>>> print(np.linalg.norm(cp_tensor_postprocessed[1][0], axis=0))
[1. 1. 1.]


When we construct a dense tensor from a postprocessed cp_tensor it is constructed as an xarray DataArray

>>> print(type(tlviz.utils.cp_to_tensor(cp_tensor)))
<class 'numpy.ndarray'>
>>> print(type(tlviz.utils.cp_to_tensor(cp_tensor_postprocessed)))
<class 'xarray.core.dataarray.DataArray'>


The visualisation of the postprocessed cp_tensor shows that the scaling and sign indeterminacy is taken care of and x-xaxis has correct labels and ticks

>>> fig, ax = tlviz.visualisation.components_plot(cp_tensor)
>>> plt.show()

>>> fig, ax = tlviz.visualisation.components_plot(cp_tensor_postprocessed)
>>> plt.show()


(png, hires.png, pdf)

tlviz.postprocessing.resolve_cp_sign_indeterminacy(cp_tensor, dataset, resolve_mode=None, unresolved_mode=-1, method='transpose')[source]

Resolve the sign indeterminacy of CP models.

Tensor factorisations have a sign indeterminacy that allows any change in the sign of the component vectors in one mode, under the condition that the sign of a component vector in another mode changes as well. This means that we can “flip” any component vector so long as the corresponding component vector in another mode is also flipped. This flipping can hurt the model’s interpretability. For example, if a factor represents a chemical spectrum, then this flipping may lead to it being negative instead of positive.

To illustrate the sign indeterminacy, we start with the SVD, which is on the form

$\mathbf{X} = \mathbf{U} \mathbf{S} \mathbf{V}^\mathsf{T}.$

The factorisation above is equivalent with the following factorisation:

$\mathbf{X} = (\mathbf{U} \text{diag}(\mathbf{f})) \mathbf{S} (\mathbf{V} \text{diag}(\mathbf{f}))^\mathsf{T},$

where $$\mathbf{f}$$ is a vector containing only ones or negative ones. Similarly, a CP factorisation with factor matrices $$\mathbf{A}, \mathbf{B}$$ and $$\mathbf{C}$$ is equivalent to the CP factorisations with the following factor matrices:

• $$(\mathbf{A} \text{diag}(\mathbf{f})), (\mathbf{B} \text{diag}(\mathbf{f}))$$ and $$\mathbf{C}$$

• $$(\mathbf{A} \text{diag}(\mathbf{f})), \mathbf{B}$$ and $$(\mathbf{C} \text{diag}(\mathbf{f}))$$

• $$\mathbf{A}, (\mathbf{B} \text{diag}(\mathbf{f}))$$ and $$(\mathbf{C} \text{diag}(\mathbf{f}))$$

One way to circumvent the sign indeterminacy is by imposing non-negativity. However, that is not always a reasonable choice (e.g. if the data also contains negative entries). When we don’t want to impose non-negativity constraints, then we need some other way to resolve the sign indeterminacy (which this function provides). The idea is easiest described in the two-way (matrix) case.

Consider a data matrix, $$\mathbf{X}$$ whose columns represent samples and rows represent measurements. Then, we want the measurement-mode component-vectors to be mostly aligned with the data matrix. The components should describe what the data is, not what it is not. For example, if the data is non-negative, then the measurement-mode component vectors should be mostly non-negative. With the SVD, we can compute whether we should flip the $$r$$-th column of $$\mathbf{U}$$ by computing

$f_r = \sum_{i=1^I} v_{ir}^2 \text{sign}{v_{ir}}$

if $$f_r$$ is negative, then we should flip the sign of the $$r-th$$ column of $$\mathbf{U}$$ and $$\mathbf{V}$$ [BAK08].

The methodology above works well in practice, and is rooted in the fact that the $$i$$-th row of $$\mathbf{V}$$ can be interpreted as the coordinates of the $$i$$-th row of $$\mathbf{X}$$ in a vector space spanned by the columns of $$\mathbf{U}$$. Then, the above equation will give us component vectors where the data points is mainly located in the non-negative orthant.

The above interpretation is correct under the assumption: $$\mathbf{U}^\mathsf{T}\mathbf{U} = \mathbf{I}$$. However, the heuristic still works well when this is not the case [BLJ13]. Still, we also include a modification of the above scheme where the same interpretation holds with non-orthogonal factors.

$f_r = \sum_{i=1}^I h_{ir}^2 \text{sign}(h_{ir}),$

where $$\mathbf{H} = \mathbf{U}(\mathbf{U}^\mathsf{T}\mathbf{U})^{-1} \mathbf{X}$$. That is the rows of $$\mathbf{H}$$ represent the rows of $$\mathbf{X}$$ as described by the column basis of $$\mathbf{U}$$.

In the multiway case, when $$\mathcal{X}$$ is a tensor instead of a matrix, we can apply the same logic [BLJ13]. If we have the factor matrices $$\mathbf{A}, \mathbf{B}$$ and $$\mathbf{C}$$, then we flip the sign of any factor matrix (e.g. $$\mathbf{A}$$) by computing

$f_r^{(\mathbf{A})} = \sum_{i=1}^I {h_{ir}^{(\mathbf{A})}}^2 \text{sign}({h_{ir}^{(\mathbf{A})}}),$

where $$\mathbf{H}^{(\mathbf{A})} = \mathbf{A}^\mathsf{T} \mathbf{X}_{(0)}$$ or $$\mathbf{H}^{(\mathbf{A})} = \mathbf{A}(\mathbf{A}^\mathsf{T}\mathbf{A})^{-1} \mathbf{X}_{(0)}$$, depending on whether the scheme based on the SVD scheme [BLJ13] or the corrected scheme. $$\mathbf{X}_{(0)} \in \mathbb{R}^{I \times JK}$$ is the tensor, $$\mathcal{X}$$, unfolded along the first mode. We can then correct the sign of $$\mathbf{A}$$ by multiplying and one of the other factor matrices by $$\text{diag}(\mathbf{f}^{(\mathbf{A})})$$. By using this procedure, we can align all factor matrices except for one (the unresolved mode) with the “direction of the data”.

Note that this sign indeterminacy comes as a direct consequence of the scaling indeterminacy of component models, since $$\text{diag}(\mathbf{f})^{-1} = \text{diag}(\mathbf{f})$$.

Parameters:
cp_tensorCPTensor or tuple

TensorLy-style CPTensor object or tuple with weights as first argument and a tuple of components as second argument.

resolve_modeint, iterable or None

Mode(s) whose factor matrix should be aligned with the data. If None, then the sign should be corrected for all modes except the unresolved_mode.

unresolved_modeint

Mode used to correct the sign indeterminacy in other mode(s). The factor matrix in this mode may not be aligned with the data.

method“transpose” or “positive_coord”

Which method to use when computing the signs. Use "transpose" for the method in [BAK08, BLJ13], and "positive_coord" for the method corrected for non-orthogonal factor matrices described above.

Returns:
CPTensor or tuple

The CP tensor after correcting the signs.

Raises:
ValueError

If unresolved_mode is not between -dataset.ndim and dataset.ndim-1.

ValueError

If unresolved_mode is in resolve_mode

Notes

For more information, see [BAK08, BLJ13]