Principal Component Analysis

Principal component analysis (PCA) is a mathematical procedure that uses an orthogonal transformation to convert a set of observations of possibly correlated variables into a set of values of linearly uncorrelated variables called principal components. This transformation is defined in such a way that the first principal component has the largest possible variance (i.e., accounts for as much of the variability in the data as possible), and each succeeding component in turn has the highest variance possible under the constraint that it be orthogonal to (i.e., uncorrelated with) the preceding components.

See the Technical Background for an introduction to principal component analysis.

- Training Function
- The training functions are slightly different for dense and sparse matrices. For dense matrices:
pca_train( source_table, out_table, row_id, components_param, grouping_cols, lanczos_iter, use_correlation, result_summary_table )

For sparse matrices:pca_sparse_train( source_table, out_table, row_id, col_id, -- Sparse matrices only val_id, -- Sparse matrices only row_dim, -- Sparse matrices only col_dim, -- Sparse matrices only components_param, grouping_cols, lanczos_iter, use_correlation, result_summary_table )

**Arguments**

- source_table
TEXT. Name of the input table containing the data for PCA training. The input data matrix should have \( N \) rows and \( M \) columns, where \( N \) is the number of data points, and \( M \) is the number of features for each data point.

A dense input table is expected to be in the one of the two standard MADlib dense matrix formats, and a sparse input table should be in the standard MADlib sparse matrix format.

The two standard MADlib dense matrix formats are:

{TABLE|VIEW}

*source_table*(*row_id*INTEGER,*row_vec*FLOAT8[], )and

{TABLE|VIEW}

*source_table*(*row_id*INTEGER,*col1*FLOAT8,*col2*FLOAT8, ... )Note that the column name

*row_id*is taken as an input parameter, and should contain a continguous list of row indices (starting at 1) for the input matrix.The input table for sparse PCA is expected to be in the form:

{TABLE|VIEW}

*source_table*( ...*row_id*INTEGER,*col_id*INTEGER,*val_id*FLOAT8, ... )The

*row_id*and*col_id*columns specify which entries in the matrix are nonzero, and the*val_id*column defines the values of the nonzero entries.Please refer to the Matrix Operations documentation for more details on defining matrices.

- out_table
TEXT. The name of the table that will contain the output. There are three possible output tables as described below.

The primary output table (

*out_table*) encodes the principal components with the*k*highest eigenvalues where*k*is either directly provided by the user or computed according to the proportion of variance. The table has the following columns:row_id Eigenvalue rank in descending order of the eigenvalue size. principal_components Vectors containing elements of the principal components. std_dev The standard deviation of each principal component. proportion The proportion of variance covered by the principal component. The table

*out_table_mean*contains the column means. This table has just one column:column_mean A vector containing the column means for the input matrix. The optional table

*result_summary_table*contains information about the performance of the PCA. The contents of this table are described under the*result_summary_table*argument.- row_id
TEXT. Column name containing the row IDs in the input source table. The column should be of type INT (or a type that can be cast to INT) and should only contain values between 1 and

*N*. For dense matrix format, it should contain all continguous integers from 1 to*N*describing the full matrix.- col_id
TEXT. Column name containing the column IDs in sparse matrix representation. The column should be of type INT (or a type that can be cast to INT) and should only contain values between 1 and

*M*.*This parameter applies to sparse matrices only.*- val_id
TEXT. Name of 'val_id' column in sparse matrix representation defining the values of the nonzero entries.

*This parameter applies to sparse matrices only.*- row_dim
INTEGER. The actual number of rows in the matrix. That is, if the matrix was transformed into dense format, this is the number of rows it would have.

*This parameter applies to sparse matrices only.*- col_dim
INTEGER. The actual number of columns in the matrix. That is, if the matrix was transformed into dense format, this is the number of columns it would have.

*This parameter applies to sparse matrices only.*- Note
- The parameters 'row_dim' and 'col_dim' could actually be inferred from the sparse matrix representation, so they will be removed in the future. For now they are maintained for backward compatability so you must enter them. Making 'row_dim' or 'col_dim' larger than the actual matrix has the effect of padding it with zeros, which is probably not useful.

- components_param
INTEGER or FLOAT. The parameter to control the number of principal components to calculate from the input data. If 'components_param' is INTEGER, it is used to denote the number of principal components (

*k*) to compute. If 'components_param' is FLOAT, the algorithm will return enough principal vectors so that the ratio of the sum of the eigenvalues collected thus far to the sum of all eigenvalues is greater than this parameter (proportion of variance). The value of 'components_param' must be either a positive INTEGER or a FLOAT in the range (0.0,1.0]- Note
- The difference in interpretation between INTEGER and FLOAT was introduced to maintain backward campatibility after the proportion of variance feature was introduced. A special case to be aware of: 'components_param' = 1 (INTEGER) will return 1 principal component, but 'components_param' = 1.0 (FLOAT) will return all principal components, i.e., proportion of variance of 100%.

Also, please note that the number of principal components (*k*) is global, even in the case where grouping is used (see 'grouping_cols' below). In the case of grouping, proportion of variance might be a better choice; this could result in different numbers of principal components for different groups.

- grouping_cols (optional)
TEXT, default: NULL. A comma-separated list of column names, with the source data grouped using the combination of all the columns. An independent PCA model will be computed for each combination of the grouping columns.

- Note
- Dense matrices can be different sizes for different groups if desired. Sparse matrices cannot be different sizes for different groups, because the 'row_dim' and 'col_dim' parameters used for sparse matrices are global across all groups.

- lanczos_iter (optional)
INTEGER, default: minimum of {

*k+40*, smallest matrix dimension} where*k*is the number of principal components specified in the parameter 'components_param'. This parameter defines the number of Lanczos iterations for the SVD calculation. The Lanczos iteration number roughly corresponds to the accuracy of the SVD calculation, and a higher number of iterations corresponds to greater accuracy but longer computation time. The number of iterations must be at least as large as the value of*k*, but no larger than the smallest dimension of the matrix. If the number of iterations is set to zero, then the default number of iterations will be used.- Note
- If both 'lanczos_iter' and proportion of variance (via the 'components_param' parameter) are defined, 'lanczos_iter' will take precedence in determining the number of principal components (i.e. the number of principal components will not be greater than 'lanczos_iter' even if the target proportion had not been reached).

- use_correlation (optional)
BOOLEAN, default FALSE. Whether to use the correlation matrix for calculating the principal components instead of the covariance matrix. Currently

*use_correlation*is a placeholder for forward compatibility, so this value must be set to false.- result_summary_table (optional)
TEXT, default NULL. Name of the optional summary table. When NULL, no summary table is generated.

This sumary table has the following columns:

rows_used INTEGER. Number of data points in the input. exec_time (ms) FLOAT8. Number of milliseconds for the PCA calculation to run. iter INTEGER. Number of iterations used in the SVD calculation. recon_error FLOAT8. The absolute error in the SVD approximation. relative_recon_error FLOAT8. The relative error in the SVD approximation. use_correlation BOOLEAN. Indicates if the correlation matrix was used.

- Examples

- View online help for the PCA training functions:
SELECT madlib.pca_train(); or SELECT madlib.pca_sparse_train();

- Create sample data in dense matrix form:
DROP TABLE IF EXISTS mat; CREATE TABLE mat (id integer, row_vec double precision[] ); INSERT INTO mat VALUES (1, '{1,2,3}'), (2, '{2,1,2}'), (3, '{3,2,1}');

- Run the PCA function for a specified number of principal components and view the results:
DROP TABLE IF EXISTS result_table, result_table_mean; SELECT madlib.pca_train('mat', -- Source table 'result_table', -- Output table 'id', -- Row id of source table 2); -- Number of principal components SELECT * FROM result_table ORDER BY row_id;

row_id | principal_components | std_dev | proportion --------+--------------------------------------------------------------+-------------------+------------------- 1 | {0.707106781186547,-6.93889390390723e-18,-0.707106781186548} | 1.41421356237309 | 0.857142857142244 2 | {0,1,0} | 0.577350269189626 | 0.142857142857041 (2 rows)

- Run the PCA function for a specified proportion of variance and view the results:
%sql DROP TABLE IF EXISTS result_table, result_table_mean; SELECT madlib.pca_train('mat', -- Source table 'result_table', -- Output table 'id', -- Row id of source table 0.9); -- Proportion of variance SELECT * FROM result_table ORDER BY row_id;

row_id | principal_components | std_dev | proportion --------+--------------------------------------------------------------+-------------------+------------------- 1 | {0.707106781186548,-2.77555756156289e-17,-0.707106781186548} | 1.4142135623731 | 0.857142857142245 2 | {-1.11022302462516e-16,-1,0} | 0.577350269189626 | 0.142857142857041 (2 rows)

- Now we use grouping in dense form to learn different models for different groups. First, we create sample data in dense matrix form with a grouping column. Note we actually have different matrix sizes for the different groups, which is allowed for dense:
DROP TABLE IF EXISTS mat_group; CREATE TABLE mat_group ( id integer, row_vec double precision[], matrix_id integer ); INSERT INTO mat_group VALUES (1, '{1,2,3}', 1), (2, '{2,1,2}', 1), (3, '{3,2,1}', 1), (4, '{1,2,3,4,5}', 2), (5, '{2,5,2,4,1}', 2), (6, '{5,4,3,2,1}', 2);

- Run the PCA function with grouping for a specified proportion of variance and view the results:
DROP TABLE IF EXISTS result_table_group, result_table_group_mean; SELECT madlib.pca_train('mat_group', -- Source table 'result_table_group', -- Output table 'id', -- Row id of source table 0.8, -- Proportion of variance 'matrix_id'); -- Grouping column SELECT * FROM result_table_group ORDER BY matrix_id, row_id;

row_id | principal_components | std_dev | proportion | matrix_id --------+------------------------------------------------------------------------------------------------+-----------------+-------------------+----------- 1 | {0.707106781186548,0,-0.707106781186547} | 1.4142135623731 | 0.857142857142245 | 1 1 | {-0.555378486712784,-0.388303582074091,0.0442457354870796,0.255566375612852,0.688115693174023} | 3.2315220311722 | 0.764102534485173 | 2 2 | {0.587384101786277,-0.485138064894743,0.311532046315153,-0.449458074050715,0.347212037159181} | 1.795531127192 | 0.235897465516047 | 2 (3 rows)

- Now let's look at sparse matrices. Create sample data in sparse matrix form:
DROP TABLE IF EXISTS mat_sparse; CREATE TABLE mat_sparse ( row_id integer, col_id integer, value double precision ); INSERT INTO mat_sparse VALUES (1, 1, 1.0), (2, 2, 2.0), (3, 3, 3.0), (4, 4, 4.0), (1, 5, 5.0), (2, 4, 6.0), (3, 2, 7.0), (4, 3, 8.0);

As an aside, this is what the sparse matrix above looks like when put in dense form:DROP TABLE IF EXISTS mat_dense; SELECT madlib.matrix_densify('mat_sparse', 'row=row_id, col=col_id, val=value', 'mat_dense'); SELECT * FROM mat_dense ORDER BY row_id;

row_id | value --------+------------- 1 | {1,0,0,0,5} 2 | {0,2,0,6,0} 3 | {0,7,3,0,0} 4 | {0,0,8,4,0} (4 rows)

- Run the PCA sparse function for a specified number of principal components and view the results:
DROP TABLE IF EXISTS result_table, result_table_mean; SELECT madlib.pca_sparse_train( 'mat_sparse', -- Source table 'result_table', -- Output table 'row_id', -- Row id of source table 'col_id', -- Column id of source table 'value', -- Value of matrix at row_id, col_id 4, -- Actual number of rows in the matrix 5, -- Actual number of columns in the matrix 3); -- Number of principal components SELECT * FROM result_table ORDER BY row_id;

Result (with principal components truncated for readability):row_id | principal_components | std_dev | proportion --------+----------------------------------------------+------------------+------------------- 1 | {-0.0876046030186158,-0.0968983772909994,... | 4.21362803829554 | 0.436590030617467 2 | {-0.0647272661608605,0.877639526308692,... | 3.68408023747461 | 0.333748701544697 3 | {-0.0780380267884855,0.177956517174911,... | 3.05606908060098 | 0.229661267837836 (3 rows)

- Now we use grouping in sparse form to learn different models for different groups. First, we create sample data in sparse matrix form with a grouping column:
DROP TABLE IF EXISTS mat_sparse_group; CREATE TABLE mat_sparse_group ( row_id integer, col_id integer, value double precision, matrix_id integer); INSERT INTO mat_sparse_group VALUES (1, 1, 1.0, 1), (2, 2, 2.0, 1), (3, 3, 3.0, 1), (4, 4, 4.0, 1), (1, 5, 5.0, 1), (2, 4, 6.0, 2), (3, 2, 7.0, 2), (4, 3, 8.0, 2);

- Run the PCA function with grouping for a specified proportion of variance and view the results:
DROP TABLE IF EXISTS result_table_group, result_table_group_mean; SELECT madlib.pca_sparse_train( 'mat_sparse_group', -- Source table 'result_table_group', -- Output table 'row_id', -- Row id of source table 'col_id', -- Column id of source table 'value', -- Value of matrix at row_id, col_id 4, -- Actual number of rows in the matrix 5, -- Actual number of columns in the matrix 0.8, -- Proportion of variance 'matrix_id'); SELECT * FROM result_table_group ORDER BY matrix_id, row_id;

Result (with principal components truncated for readability):row_id | principal_components | std_dev | proportion | matrix_id --------+--------------------------------------------+------------------+-------------------+----------- 1 | {-0.17805696611353,0.0681313257646983,... | 2.73659933165925 | 0.544652792875481 | 1 2 | {-0.0492086814863993,0.149371585357526,... | 2.06058314533194 | 0.308800210823714 | 1 1 | {0,-0.479486114660443,... | 4.40325305087975 | 0.520500333693473 | 2 2 | {0,0.689230898585949,... | 3.7435566458567 | 0.376220573442628 | 2 (4 rows)

- Notes

- Table names can be optionally schema qualified (current_schemas() would be searched if a schema name is not provided) and all table and column names should follow case-sensitivity and quoting rules per the database. (For instance, 'mytable' and 'MyTable' both resolve to the same entity, i.e. 'mytable'. If mixed-case or multi-byte characters are desired for entity names then the string should be double-quoted; in this case the input would be '"MyTable"').
- Because of the centering step in PCA (see Technical Background), sparse matrices almost always become dense during the training process. Since this implementation automatically densifies sparse matrix input, there should be no expected performance improvement in using sparse matrix input over dense matrix input.
- For the parameter 'components_param', INTEGER and FLOAT are interpreted differently. A special case to be aware of: 'components_param' = 1 (INTEGER) will return 1 principal component, but 'components_param' = 1.0 (FLOAT) will return all principal components, i.e., proportion of variance of 100%.
- If both 'lanczos_iter' and proportion of variance (via the 'components_param' parameter) are defined, 'lanczos_iter' will take precedence in determining the number of principal components (i.e. the number of principal components will not be greater than 'lanczos_iter' even if the target proportion had not been reached).

- Technical Background

The PCA implemented here uses a distributed SVD decomposition implementation to recover the principal components (as opposed to the directly computing the eigenvectors of the covariance matrix). Let \( \boldsymbol X \) be the data matrix, and let \( \hat{x} \) be a vector of the column averages of \( \boldsymbol{X}\). PCA computes the matrix \( \hat{\boldsymbol X} \) as

\[ \hat{\boldsymbol X} = {\boldsymbol X} - \vec{e} \hat{x}^T \]

where \( \vec{e} \) is the vector of all ones.

PCA then computes the SVD matrix factorization

\[ \hat{\boldsymbol X} = {\boldsymbol U}{\boldsymbol \Sigma}{\boldsymbol V}^T \]

where \( {\boldsymbol \Sigma} \) is a diagonal matrix. The eigenvalues are recovered as the entries of \( {\boldsymbol \Sigma}/(\sqrt{(N-1)} \), and the principal components are the rows of \( {\boldsymbol V} \). The reasoning behind using N − 1 instead of N to calculate the covariance is Bessel's correction.

- Note
- It is important to note that this PCA implementation assumes that the user will use only the principal components that have non-zero eigenvalues. The SVD calculation is done with the Lanczos method, which does not guarantee correctness for singular vectors with zero-valued eigenvalues. Consequently, principal components with zero-valued eigenvalues are not guaranteed to be correct. Generally, this will not be problem unless the user wants to use the principal components for the entire eigenspectrum.

- Literature

[1] Principal Component Analysis. http://en.wikipedia.org/wiki/Principal_component_analysis

[2] Shlens, Jonathon (2009), A Tutorial on Principal Component Analysis

- Related Topics

File pca.sql_in documenting the SQL functions