PCA¶

Assume the historical instrument returns of the estimation universe is represented by a T x N matrix R. With singular value decomposition (SDV), the covariance matrix \(\hat{Q}\) is decomposed by its eigenvectors and eigenvalues.

\[ \hat{Q} = \frac{R^T R}{T} = VDV^T \]

where V is a matrix of eigenvalues (each column is an eigenvector) and D is a diagonal matrix with eigenvalues \(\lambda_i\) in the decreasing order on the diagonal.

The factor exposure matrix \(B\) is taken to be \(V_nD^{\frac{1}{2}}_n\), where n is the number of largest eigenvalues selected, in a dimension of \((n, N)\).

Factor \(F\) (in dimension \((T, n)\)) and residual returns \({\Gamma}\) (in dimension \((T, N)\)) can then be computed by either ordinary or weighted least-squares

\[\begin{split} & F = (B^T W B)^{-1} B^T W R \\ & {\Gamma} = R - B F \end{split}\]

where \(W\) is the weight matrix in regression, e.g. an identity matrix in ordinary weighted least-squares.

Module¶

class fpm_risk_model.statistical.pca.PCAConfig(*, show_all_instruments: bool = False, n_components: Union[int, float, str], demean: Optional[bool] = True, speedup: Optional[bool] = True)¶

PCA statistics model configuration class.

Parameters¶

n_componentsint: Number of components.
demeanOptional[bool]: Indicate whether to demean before fitting. Default is True.
speedup: Optional[bool]: Indicate whether to speed up the computation as much as possible. Default is True.

n_components: Union[int, float, str]¶

demean: Optional[bool]¶

speedup: Optional[bool]¶

class fpm_risk_model.statistical.pca.PCA(n_components: int, demean: Optional[bool] = True, speedup: Optional[bool] = True, **kwargs)¶

PCA statistics model.

ConfigClass¶: alias of PCAConfig

__init__(n_components: int, demean: Optional[bool] = True, speedup: Optional[bool] = True, **kwargs)¶

Constructor.

Parameters¶

n_componentsint: Number of components.
demeanOptional[bool]: Indicate whether to demean before fitting. Default is True.
speedup: Optional[bool]: Indicate whether to speed up the computation as much as possible. Default is True.

fit(X: Union[ndarray, DataFrame], weights: Optional[Union[ndarray, Series]] = None) → object¶

Fit the returns into the risk model.

Parameters¶

X: pandas.DataFrame or numpy.ndarray: Instrument returns where the columns are the instruments and the index is the date / time in ascending order. For example, if there are N instruments and T days of returns, the input is with the dimension of (T, N).

Returns¶

object: The object itself.