@@ -23,13 +23,13 @@ def __init__(
2323 offset_covariance = 0.2 ,
2424 mu = 0.01 ,
2525 eps = np .finfo (np .float64 ).eps ,
26- ridge_param = np .finfo (np .float64 ).eps ,# for regularized ridge regression
26+ ridge_param = np .finfo (np .float64 ).eps ,
2727 gama = 0.2 ,
2828 weight = 0.02 ,
2929 basis_function = None ,
3030 ):
3131 self .eps = eps
32- self .ridge_param = ridge_param # for regularized ridge regression
32+ self .ridge_param = ridge_param
3333 self .mu = mu
3434 self .offset_covariance = offset_covariance
3535 self .max_lag = max_lag
@@ -54,6 +54,7 @@ def _validate_params(self):
5454 "ridge_param" : self .ridge_param ,
5555 "gama" : self .gama ,
5656 "weight" : self .weight ,
57+ "ridge_param" : self .ridge_param ,
5758 }
5859 for attribute , value in attributes .items ():
5960 if not isinstance (value , (np .integer , int , float )):
@@ -118,7 +119,7 @@ def least_squares(self, psi, y):
118119 theta = np .linalg .lstsq (psi , y , rcond = None )[0 ]
119120 return theta
120121
121- def ridge_regression (self , psi , y ):
122+ def ridge_regression_classic (self , psi , y ):
122123 """Estimate the model parameters using the regularized least squares method
123124 known as ridge regression. Based on the least_squares module and uses
124125 the same data format but you need to pass ridge_param in the call to
@@ -936,3 +937,47 @@ def least_mean_squares_mixed_norm(self, psi, y):
936937 theta [:, i ] = tmp_list .flatten ()
937938
938939 return theta [:, - 1 ].reshape (- 1 , 1 )
940+ 941+ def ridge_regression (self , psi , y ):
942+ """Estimate the model parameters using SVD and Ridge Regression method.
943+
944+ Parameters
945+ ----------
946+ psi : ndarray of floats
947+ The information matrix of the model.
948+ y : array-like of shape = y_training
949+ The data used to training the model.
950+
951+ Returns
952+ -------
953+ theta : array-like of shape = number_of_model_elements
954+ The estimated parameters of the model.
955+
956+ References
957+ ----------
958+ - Manuscript: Hoerl, A. E.; Kennard, R. W. Ridge regression:
959+ applications to nonorthogonal problems. Technometrics,
960+ Taylor & Francis, v. 12, n. 1, p. 69-82, 1970.
961+
962+ - StackExchange: whuber. The proof of shrinking coefficients using ridge
963+ regression through "spectral decomposition".
964+ Cross Validated, accessed 21 September 2023,
965+ https://stats.stackexchange.com/q/220324
966+ """
967+ self ._check_linear_dependence_rows (psi )
968+ 969+ y = y [self .max_lag :, 0 ].reshape (- 1 , 1 )
970+ 971+ try :
972+ U , d , Vt = np .linalg .svd (psi , full_matrices = False )
973+ D = np .diag (d )
974+ I = np .identity (len (D ))
975+ 976+ theta = Vt .T @ np .linalg .inv (D ** 2 + self .ridge_param * I ) @ D @ U .T @ y
977+ except :
978+ warnings .warn ("The SVD computation does not converge. Value calculated with the classic algorithm" ,
979+ stacklevel = 2 )
980+ 981+ theta = self .ridge_regression_classic (psi , y )
982+ 983+ return theta
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