def test_basis_fpca_transform_result(self):
n_basis = 9
n_components = 3
fd_data = fetch_weather()['data'].coordinates[0]
fd_data = FDataGrid(np.squeeze(fd_data.data_matrix),
np.arange(0.5, 365, 1))
# initialize basis data
basis = Fourier(n_basis=n_basis, domain_range=(0, 365))
fd_basis = fd_data.to_basis(basis)
fpca = FPCA(n_components=n_components,
regularization=TikhonovRegularization(
LinearDifferentialOperator(2),
regularization_parameter=1e5))
fpca.fit(fd_basis)
scores = fpca.transform(fd_basis)
# results obtained using Ramsay's R package
results = [[-7.68307641e+01, 5.69034443e+01, -1.22440149e+01],
[-9.02873996e+01, 1.46262257e+01, -1.78574536e+01],
[-8.21155683e+01, 3.19159491e+01, -2.56212328e+01],
[-1.14163637e+02, 3.66425562e+01, -1.00810836e+01],
[-6.97263223e+01, 1.22817168e+01, -2.39417618e+01],
[-6.41886364e+01, -1.07261045e+01, -1.10587407e+01],
[1.35824412e+02, 2.03484658e+01, -9.04815324e+00],
[-1.46816399e+01, -2.66867491e+01, -1.20233465e+01],
[1.02507511e+00, -2.29840736e+01, -9.06081296e+00],
[-3.62936903e+01, -2.09520442e+01, -1.14799951e+01],
[-4.20649313e+01, -1.13618094e+01, -6.24909009e+00],
[-7.38115985e+01, -3.18423866e+01, -1.50298626e+01],
[-6.69822456e+01, -3.35518632e+01, -1.25167352e+01],
[-1.03534763e+02, -1.29513941e+01, -1.49103879e+01],
[-1.04542036e+02, -1.36794907e+01, -1.41555965e+01],
[-7.35863347e+00, -1.41171956e+01, -2.97562788e+00],
[7.28804530e+00, -5.34421830e+01, -3.39823418e+00],
[5.59974094e+01, -4.02154080e+01, 3.78800103e-01],
[1.80778702e+02, 1.87798201e+01, -1.99043247e+01],
[-3.69700617e+00, -4.19441020e+01, 6.45820740e+00],
[3.76527216e+01, -4.23056953e+01, 1.04221757e+01],
[1.23850646e+02, -4.24648130e+01, -2.22336786e-01],
[-7.23588457e+00, -1.20579536e+01, 2.07502089e+01],
[-4.96871011e+01, 8.88483448e+00, 2.02882768e+01],
[-1.36726355e+02, -1.86472599e+01, 1.89076217e+01],
[-1.83878661e+02, 4.12118550e+01, 1.78960356e+01],
[-1.81568820e+02, 5.20817910e+01, 2.01078870e+01],
[-5.08775852e+01, 1.34600555e+01, 3.18602712e+01],
[-1.37633866e+02, 7.50809631e+01, 2.42320782e+01],
[4.98276375e+01, 1.33401270e+00, 3.50611066e+01],
[1.51149934e+02, -5.47417776e+01, 3.97592325e+01],
[1.58366096e+02, -3.80762686e+01, -5.62415023e+00],
[2.17139548e+02, 6.34055987e+01, -1.98853635e+01],
[2.33615480e+02, -7.90787574e-02, 2.69069525e+00],
[3.45371437e+02, 9.58703622e+01, 8.47570770e+00]]
results = np.array(results)
# compare results
np.testing.assert_allclose(scores, results, atol=1e-7)
def test_grid_fpca_transform_result(self):
n_components = 1
fd_data = fetch_weather()['data'].coordinates[0]
fpca = FPCA(n_components=n_components, weights=[1] * 365)
fpca.fit(fd_data)
scores = fpca.transform(fd_data)
# results obtained
results = [[-77.05020176], [-90.56072204], [-82.39565947],
[-114.45375934], [-69.99735931], [-64.44894047],
[135.58336775], [-14.93460852], [0.75024737], [-36.4781038],
[-42.35637749], [-73.98910492], [-67.11253749],
[-103.68269798], [-104.65948079],
[-7.42817782], [7.48125036], [56.29792942], [181.00258791],
[-3.53294736], [37.94673912], [124.43819913], [-7.04274676],
[-49.61134859], [-136.86256785], [-184.03502398],
[-181.72835749], [-51.06323208], [-137.85606731],
[50.10941466], [151.68118097], [159.01360046],
[217.17981302], [234.40195237], [345.39374006]]
results = np.array(results)
np.testing.assert_allclose(scores, results, rtol=1e-6)
def test_basis_fpca_fit_result(self):
n_basis = 9
n_components = 3
fd_data = fetch_weather()['data'].coordinates[0]
fd_data = FDataGrid(np.squeeze(fd_data.data_matrix),
np.arange(0.5, 365, 1))
# initialize basis data
basis = Fourier(n_basis=n_basis, domain_range=(0, 365))
fd_basis = fd_data.to_basis(basis)
fpca = FPCA(n_components=n_components,
regularization=TikhonovRegularization(
LinearDifferentialOperator(2),
regularization_parameter=1e5))
fpca.fit(fd_basis)
# results obtained using Ramsay's R package
results = [[
0.92407552, 0.13544888, 0.35399023, 0.00805966, -0.02148108,
-0.01709549, -0.00208469, -0.00297439, -0.00308224
],
[
-0.33314436, -0.05116842, 0.89443418, 0.14673902,
0.21559073, 0.02046924, 0.02203431, -0.00787185,
0.00247492
],
[
-0.14241092, 0.92131899, 0.00514715, 0.23391411,
-0.19497613, 0.09800817, 0.01754439, -0.00205874,
0.01438185
]]
results = np.array(results)
# compare results obtained using this library. There are slight
# variations due to the fact that we are in two different packages
for i in range(n_components):
if np.sign(fpca.components_.coefficients[i][0]) != np.sign(
results[i][0]):
results[i, :] *= -1
np.testing.assert_allclose(fpca.components_.coefficients,
results,
atol=1e-7)
def test_basis_fpca_regularization_fit_result(self):
n_basis = 9
n_components = 3
fd_data = fetch_weather()['data'].coordinates[0]
fd_data = FDataGrid(np.squeeze(fd_data.data_matrix),
np.arange(0.5, 365, 1))
# initialize basis data
basis = Fourier(n_basis=n_basis, domain_range=(0, 365))
fd_basis = fd_data.to_basis(basis)
fpca = FPCA(n_components=n_components)
fpca.fit(fd_basis)
# results obtained using Ramsay's R package
results = [[
0.9231551, 0.1364966, 0.3569451, 0.0092012, -0.0244525,
-0.02923873, -0.003566887, -0.009654571, -0.0100063
],
[
-0.3315211, -0.0508643, 0.89218521, 0.1669182,
0.2453900, 0.03548997, 0.037938051, -0.025777507,
0.008416904
],
[
-0.1379108, 0.9125089, 0.00142045, 0.2657423,
-0.2146497, 0.16833314, 0.031509179, -0.006768189,
0.047306718
]]
results = np.array(results)
# compare results obtained using this library. There are slight
# variations due to the fact that we are in two different packages
for i in range(n_components):
if np.sign(fpca.components_.coefficients[i][0]) != np.sign(
results[i][0]):
results[i, :] *= -1
np.testing.assert_allclose(fpca.components_.coefficients,
results,
atol=1e-7)
def test_discretized_fpca_fit_attributes(self):
fpca = FPCA()
with self.assertRaises(AttributeError):
fpca.fit(None)
# check that if n_components is bigger than the number of samples then
# an exception should be thrown
fd = FDataGrid([[0.5], [0.1]], grid_points=[0])
with self.assertRaises(AttributeError):
fpca.fit(fd)
# check that n_components must be smaller than the number of attributes
# in the FDataGrid object
fd = FDataGrid([[0.9], [0.7], [0.5]], grid_points=[0])
with self.assertRaises(AttributeError):
fpca.fit(fd)
def test_basis_fpca_fit_attributes(self):
fpca = FPCA()
with self.assertRaises(AttributeError):
fpca.fit(None)
basis = Fourier(n_basis=1)
# check that if n_components is bigger than the number of samples then
# an exception should be thrown
fd = FDataBasis(basis, [[0.9]])
with self.assertRaises(AttributeError):
fpca.fit(fd)
# check that n_components must be smaller than the number of elements
# of target basis
fd = FDataBasis(basis, [[0.9], [0.7], [0.5]])
with self.assertRaises(AttributeError):
fpca.fit(fd)
def test_grid_fpca_regularization_fit_result(self):
n_components = 1
fd_data = fetch_weather()['data'].coordinates[0]
fd_data = FDataGrid(np.squeeze(fd_data.data_matrix),
np.arange(0.5, 365, 1))
fpca = FPCA(n_components=n_components,
weights=[1] * 365,
regularization=TikhonovRegularization(
LinearDifferentialOperator(2)))
fpca.fit(fd_data)
# results obtained using fda.usc for the first component
results = [[
-0.06961236, -0.07027042, -0.07090496, -0.07138247, -0.07162215,
-0.07202264, -0.07264893, -0.07279174, -0.07274672, -0.07300075,
-0.07365471, -0.07489002, -0.07617455, -0.07658708, -0.07551923,
-0.07375128, -0.0723776, -0.07138373, -0.07080555, -0.07111745,
-0.0721514, -0.07395427, -0.07558341, -0.07650959, -0.0766541,
-0.07641352, -0.07660864, -0.07669081, -0.0765396, -0.07640671,
-0.07634668, -0.07626304, -0.07603638, -0.07549114, -0.07410347,
-0.07181791, -0.06955356, -0.06824034, -0.06834077, -0.06944125,
-0.07133598, -0.07341109, -0.07471501, -0.07568844, -0.07631904,
-0.07647264, -0.07629453, -0.07598431, -0.07628157, -0.07654062,
-0.07616026, -0.07527189, -0.07426683, -0.07267961, -0.07079998,
-0.06927394, -0.068412, -0.06838534, -0.06888439, -0.0695309,
-0.07005508, -0.07066637, -0.07167196, -0.07266978, -0.07275299,
-0.07235183, -0.07207819, -0.07159814, -0.07077697, -0.06977026,
-0.0691952, -0.06965756, -0.07058327, -0.07075751, -0.07025415,
-0.06954233, -0.06899785, -0.06891026, -0.06887079, -0.06862183,
-0.06830082, -0.06777765, -0.06700202, -0.06639394, -0.06582435,
-0.06514987, -0.06467236, -0.06425272, -0.06359187, -0.062922,
-0.06300068, -0.06325494, -0.06316979, -0.06296254, -0.06246343,
-0.06136836, -0.0600936, -0.05910688, -0.05840872, -0.0576547,
-0.05655684, -0.05546518, -0.05484433, -0.05465746, -0.05449286,
-0.05397004, -0.05300742, -0.05196686, -0.05133129, -0.05064617,
-0.04973418, -0.04855687, -0.04714356, -0.04588103, -0.04547284,
-0.04571493, -0.04580704, -0.04523509, -0.04457293, -0.04405309,
-0.04338468, -0.04243512, -0.04137278, -0.04047946, -0.03984531,
-0.03931376, -0.0388847, -0.03888507, -0.03908662, -0.03877577,
-0.03830952, -0.03802713, -0.03773521, -0.03752388, -0.03743759,
-0.03714113, -0.03668387, -0.0363703, -0.03642288, -0.03633051,
-0.03574618, -0.03486536, -0.03357797, -0.03209969, -0.0306837,
-0.02963987, -0.029102, -0.0291513, -0.02932013, -0.02912619,
-0.02869407, -0.02801974, -0.02732363, -0.02690451, -0.02676622,
-0.0267323, -0.02664896, -0.02661708, -0.02637166, -0.02577496,
-0.02490428, -0.02410813, -0.02340367, -0.02283356, -0.02246305,
-0.0224229, -0.0225435, -0.02295603, -0.02324663, -0.02310005,
-0.02266893, -0.02221522, -0.02168056, -0.02129419, -0.02064909,
-0.02007801, -0.01979083, -0.01979541, -0.01978879, -0.01954269,
-0.0191623, -0.01879572, -0.01849678, -0.01810297, -0.01769666,
-0.01753802, -0.01794351, -0.01871307, -0.01930005, -0.01933,
-0.01901017, -0.01873486, -0.01861838, -0.01870777, -0.01879,
-0.01904219, -0.01945078, -0.0200607, -0.02076936, -0.02100213,
-0.02071439, -0.02052113, -0.02076313, -0.02128468, -0.02175631,
-0.02206387, -0.02201054, -0.02172142, -0.02143092, -0.02133647,
-0.02144956, -0.02176286, -0.02212579, -0.02243861, -0.02278316,
-0.02304113, -0.02313356, -0.02349275, -0.02417028, -0.0245954,
-0.0244062, -0.02388557, -0.02374682, -0.02401071, -0.02431126,
-0.02433125, -0.02427656, -0.02430442, -0.02424977, -0.02401619,
-0.02402294, -0.02415424, -0.02413262, -0.02404076, -0.02397651,
-0.0243893, -0.0253322, -0.02664395, -0.0278802, -0.02877936,
-0.02927182, -0.02937318, -0.02926277, -0.02931632, -0.02957945,
-0.02982133, -0.03023224, -0.03060406, -0.03066011, -0.03070932,
-0.03116429, -0.03179009, -0.03198094, -0.03149462, -0.03082037,
-0.03041594, -0.0303307, -0.03028465, -0.03052841, -0.0311837,
-0.03199307, -0.03262025, -0.03345083, -0.03442665, -0.03521313,
-0.0356433, -0.03606037, -0.03677406, -0.03735165, -0.03746578,
-0.03744154, -0.03752143, -0.03780898, -0.03837639, -0.03903232,
-0.03911629, -0.03857567, -0.03816592, -0.03819285, -0.03818405,
-0.03801684, -0.03788493, -0.03823232, -0.03906142, -0.04023251,
-0.04112434, -0.04188011, -0.04254759, -0.043, -0.04340181,
-0.04412687, -0.04484482, -0.04577669, -0.04700832, -0.04781373,
-0.04842662, -0.04923723, -0.05007637, -0.05037817, -0.05009794,
-0.04994083, -0.05012712, -0.05094001, -0.05216065, -0.05350458,
-0.05469781, -0.05566309, -0.05641011, -0.05688106, -0.05730818,
-0.05759156, -0.05763771, -0.05760073, -0.05766117, -0.05794587,
-0.05816696, -0.0584046, -0.05905105, -0.06014331, -0.06142231,
-0.06270788, -0.06388225, -0.06426245, -0.06386721, -0.0634656,
-0.06358049, -0.06442514, -0.06570047, -0.06694328, -0.0682621,
-0.06897846, -0.06896583, -0.06854621, -0.06797142, -0.06763755,
-0.06784024, -0.06844314, -0.06918567, -0.07021928, -0.07148473,
-0.07232504, -0.07272276, -0.07287021, -0.07289836, -0.07271531,
-0.07239956, -0.07214086, -0.07170078, -0.07081195, -0.06955202,
-0.06825156, -0.06690167, -0.06617102, -0.06683291, -0.06887539,
-0.07089424, -0.07174837, -0.07150888, -0.07070378, -0.06960066,
-0.06842496, -0.06777666, -0.06728403, -0.06681262, -0.06679066
]]
results = np.array(results)
# compare results obtained using this library. There are slight
# variations due to the fact that we are in two different packages
for i in range(n_components):
if np.sign(fpca.components_.data_matrix[i][0]) != np.sign(
results[i][0]):
results[i, :] *= -1
np.testing.assert_allclose(fpca.components_.data_matrix.reshape(
fpca.components_.data_matrix.shape[:-1]),
results,
rtol=1e-2)
def test_grid_fpca_fit_result(self):
n_components = 1
fd_data = fetch_weather()['data'].coordinates[0]
fpca = FPCA(n_components=n_components, weights=[1] * 365)
fpca.fit(fd_data)
# results obtained using fda.usc for the first component
results = [[
-0.06958281, -0.07015412, -0.07095115, -0.07185632, -0.07128256,
-0.07124209, -0.07364828, -0.07297663, -0.07235438, -0.07307498,
-0.07293423, -0.07449293, -0.07647909, -0.07796823, -0.07582476,
-0.07263243, -0.07241871, -0.0718136, -0.07015477, -0.07132331,
-0.0711527, -0.07435933, -0.07602666, -0.0769783, -0.07707199,
-0.07503802, -0.0770302, -0.07705581, -0.07633515, -0.07624817,
-0.07631568, -0.07619913, -0.07568, -0.07595155, -0.07506939,
-0.07181941, -0.06907624, -0.06735476, -0.06853985, -0.06902363,
-0.07098882, -0.07479412, -0.07425241, -0.07555835, -0.0765903,
-0.07651853, -0.07682536, -0.07458996, -0.07631711, -0.07726509,
-0.07641246, -0.0744066, -0.07501397, -0.07302722, -0.07045571,
-0.06912529, -0.06792186, -0.06830739, -0.06898433, -0.07000192,
-0.07014513, -0.06994886, -0.07115909, -0.073999, -0.07292669,
-0.07139879, -0.07226865, -0.07187915, -0.07122995, -0.06975022,
-0.06800613, -0.06900793, -0.07186378, -0.07114479, -0.07015252,
-0.06944782, -0.068291, -0.06905348, -0.06925773, -0.06834624,
-0.06837319, -0.06824067, -0.06644614, -0.06637313, -0.06626312,
-0.06470209, -0.0645058, -0.06477729, -0.06411049, -0.06158499,
-0.06305197, -0.06398006, -0.06277579, -0.06282124, -0.06317684,
-0.0614125, -0.05961922, -0.05875443, -0.05845781, -0.05828608,
-0.05666474, -0.05495706, -0.05446301, -0.05468254, -0.05478609,
-0.05440798, -0.05312339, -0.05102368, -0.05160285, -0.05077954,
-0.04979648, -0.04890853, -0.04745462, -0.04496763, -0.0448713,
-0.04599596, -0.04688998, -0.04488872, -0.04404507, -0.04420729,
-0.04368153, -0.04254381, -0.0411764, -0.04022811, -0.03999746,
-0.03963634, -0.03832502, -0.0383956, -0.04015374, -0.0387544,
-0.03777315, -0.03830728, -0.03768616, -0.03714081, -0.03781918,
-0.03739374, -0.03659894, -0.03563342, -0.03658407, -0.03686991,
-0.03543746, -0.03518799, -0.03361226, -0.0321534, -0.03050438,
-0.02958411, -0.02855023, -0.02913402, -0.02992464, -0.02899548,
-0.02891629, -0.02809554, -0.02702642, -0.02672194, -0.02678648,
-0.02698471, -0.02628085, -0.02674285, -0.02658515, -0.02604447,
-0.0245711, -0.02413174, -0.02342496, -0.022898, -0.02216152,
-0.02272283, -0.02199741, -0.02305362, -0.02371371, -0.02320865,
-0.02234777, -0.0225018, -0.02104359, -0.02203346, -0.02052545,
-0.01987457, -0.01947911, -0.01986949, -0.02012196, -0.01958515,
-0.01906753, -0.01857869, -0.01874101, -0.01827973, -0.017752,
-0.01702056, -0.01759611, -0.01888485, -0.01988159, -0.01951675,
-0.01872967, -0.01866667, -0.0183576, -0.01909758, -0.018599,
-0.01910036, -0.01930315, -0.01958856, -0.02129936, -0.0216614,
-0.0204397, -0.02002368, -0.02058828, -0.02149915, -0.02167326,
-0.02238569, -0.02211907, -0.02168336, -0.02124387, -0.02131655,
-0.02130508, -0.02181227, -0.02230632, -0.02223732, -0.0228216,
-0.02355137, -0.02275145, -0.02286893, -0.02437776, -0.02523897,
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]]
results = np.array(results)
# compare results obtained using this library. There are slight
# variations due to the fact that we are in two different packages
for i in range(n_components):
if np.sign(fpca.components_.data_matrix[i][0]) != np.sign(
results[i][0]):
results[i, :] *= -1
np.testing.assert_allclose(fpca.components_.data_matrix.reshape(
fpca.components_.data_matrix.shape[:-1]),
results,
rtol=1e-6)
# same for each individual. To better understand the data we plot it. dataset = skfda.datasets.fetch_growth() fd = dataset['data'] y = dataset['target'] fd.plot() ############################################################################## # FPCA can be done in two ways. The first way is to operate directly with the # raw data. We call it discretized FPCA as the functional data in this case # consists in finite values dispersed over points in a domain range. # We initialize and setup the FPCADiscretized object and run the fit method to # obtain the first two components. By default, if we do not specify the number # of components, it's 3. Other parameters are weights and centering. For more # information please visit the documentation. fpca_discretized = FPCA(n_components=2) fpca_discretized.fit(fd) fpca_discretized.components_.plot() ############################################################################## # In the second case, the data is first converted to use a basis representation # and the FPCA is done with the basis representation of the original data. # We obtain the same dataset again and transform the data to a basis # representation. This is because the FPCA module modifies the original data. # We also plot the data for better visual representation. dataset = fetch_growth() fd = dataset['data'] basis = skfda.representation.basis.BSpline(n_basis=7) basis_fd = fd.to_basis(basis) basis_fd.plot() ##############################################################################
class BasisRegression(object):
"""Class implementing functional linear models with basis functions for vector-valued covariates.
Parameters
----------
nbasis: int
Number of basis functions.
basis_type: str, default='bspline'
Type of basis used, possible values are 'bspline', 'fourier' and 'fPCA'
Attributes
----------
reg: object
Instance of skfda.ml.regression.LinearRegression
coef: array, default=None
Regression coefficients
fpca_basis: object
If basis_type='fPCA', instance of skfda.preprocessing.dim_reduction.projection.FPCA().
"""
def __init__(self, nbasis, basis_type='bspline'):
self.nbasis = nbasis
self.reg = LinearRegression()
self.basis_type = basis_type
self.coef = None
if self.basis_type == 'fPCA':
self.fpca_basis = FPCA(self.nbasis)
def data_to_basis(self, X, fit_fPCA=True):
"""Project the data to basis functions.
Parameters
----------
X: array, shape (n,n_points,d)
Array of paths. It is a 3-dimensional array, containing the coordinates in R^d of n piecewise linear paths,
each composed of n_points.
fit_fPCA: boolean, default=True
If n_basis='fPCA' and fit_fPCA=True, the basis functions are fitted to be the functional principal
components of X.
Returns
-------
fd_basis: object
Instance of skfda.representation.basis.FDataBasis, the basis representation of X, where the type of basis is
determined by self.n_basis.
"""
grid_points = np.linspace(0, 1, X.shape[1])
fd = FDataGrid(X, grid_points)
basis_vec = []
for i in range(X.shape[2]):
if self.basis_type == 'bspline':
basis_vec.append(BSpline(n_basis=self.nbasis))
elif self.basis_type == 'fourier':
basis_vec.append(Fourier(n_basis=self.nbasis))
elif self.basis_type == 'fPCA':
basis_vec.append(BSpline(n_basis=7))
basis = VectorValued(basis_vec)
fd_basis = fd.to_basis(basis)
if self.basis_type == 'fPCA':
if fit_fPCA:
self.fpca_basis = self.fpca_basis.fit(fd_basis)
fd_basis = self.fpca_basis.transform(fd_basis)
return fd_basis
def fit(self, X, Y):
"""Fit the functional linear model to X and Y
Parameters
----------
X: array, shape (n,n_points,d)
Array of training paths. It is a 3-dimensional array, containing the coordinates in R^d of n piecewise
linear paths, each composed of n_points.
Y: array, shape (n)
Array of target values.
Returns
-------
reg: object
Instance of skfda.ml.regression.LinearRegression
"""
fd_basis = self.data_to_basis(X)
self.reg.fit(fd_basis, Y)
self.coef = self.reg.coef_
return self.reg
def predict(self, X):
"""Predict the output of self.reg for X.
Parameters
----------
X: array, shape (n,n_points,d)
Array of training paths. It is a 3-dimensional array, containing the coordinates in R^d of n piecewise
linear paths, each composed of n_points.
Returns
-------
Ypred: array, shape (n)
Array of predicted values.
"""
fd_basis = self.data_to_basis(X, fit_fPCA=False)
return self.reg.predict(fd_basis)
def get_loss(self, X, Y, plot=False):
"""Computes the empirical squared loss obtained with the functional linear model
Parameters
----------
X: array, shape (n,n_points,d)
Array of training paths. It is a 3-dimensional array, containing the coordinates in R^d of n piecewise
linear paths, each composed of n_points.
Y: array, shape (n)
Array of target values.
plot: boolean, default=False
If True, plots the regression coefficients and a scatter plot of the target values Y against its predicted
values Ypred to assess the quality of the fit.
Returns
-------
hatL: float
The squared loss, that is the sum of the squares of Y-Ypred, where Ypred are the fitted values of the Ridge
regression of Y against signatures of X truncated at k.
"""
Ypred = self.predict(X)
if plot:
plt.scatter(Y, Ypred)
plt.plot([0.9 * np.min(Y), 1.1 * np.max(Y)],
[0.9 * np.min(Y), 1.1 * np.max(Y)],
'--',
color='black')
plt.title("Ypred against Y")
plt.show()
return np.mean((Y - Ypred)**2)
