How to use the abcpy.statistics.Identity function in abcpy

def test(self):
        B1 = Binomial([10, 0.2])
        N1 = Normal([0.03, 0.01])
        N2 = Normal([0.1, N1])
        graph1 = Normal([B1, N2])
        graph2 = Normal([1, N2])

        statistics_calculator = Identity(degree=2, cross=False)
        distance_calculator = LogReg(statistics_calculator)
        backend = Backend()

        sampler = RejectionABC([graph1, graph2], [distance_calculator, distance_calculator], backend)

        rng = np.random.RandomState(1)

        sampler.sample_from_prior(rng=rng)

        y_sim = sampler.simulate(1, rng=rng)

        self.assertTrue(isinstance(y_sim, list))

        self.assertTrue(len(y_sim)==2)

        self.assertTrue(isinstance(y_sim[0][0], np.ndarray))

def setUp(self):
        # define prior and model
        sigma = Uniform([[10], [20]])
        mu = Normal([0, 1])
        self.Y = Normal([mu, sigma])

        # define backend
        self.backend = Backend()

        # define statistics
        self.statistics_cal = Identity(degree=3, cross=False)

        if has_torch:
            # Initialize statistics learning
            self.statisticslearning = TripletDistanceLearning([self.Y], self.statistics_cal, self.backend,
                                                              n_samples=100, n_samples_per_param=1, seed=1, n_epochs=10)

def infer_parameters():
    # define observation for true parameters mean=170, std=15
    height_obs = [160.82499176, 167.24266737, 185.71695756, 153.7045709, 163.40568812, 140.70658699, 169.59102084, 172.81041696, 187.38782738, 179.66358934, 176.63417241, 189.16082803, 181.98288443, 170.18565017, 183.78493886, 166.58387299, 161.9521899, 155.69213073, 156.17867343, 144.51580379, 170.29847515, 197.96767899, 153.36646527, 162.22710198, 158.70012047, 178.53470703, 170.77697743, 164.31392633, 165.88595994, 177.38083686, 146.67058471763457, 179.41946565658628, 238.02751620619537, 206.22458790620766, 220.89530574344568, 221.04082532837026, 142.25301427453394, 261.37656571434275, 171.63761180867033, 210.28121820385866, 237.29130237612236, 175.75558340169619, 224.54340549862235, 197.42448680731226, 165.88273684581381, 166.55094082844519, 229.54308602661584, 222.99844054358519, 185.30223966014586, 152.69149367593846, 206.94372818527413, 256.35498655339154, 165.43140916577741, 250.19273595481803, 148.87781549665536, 223.05547559193792, 230.03418198709608, 146.13611923127021, 138.24716809523139, 179.26755740864527, 141.21704876815426, 170.89587081800852, 222.96391329259626, 188.27229523693822, 202.67075179617672, 211.75963110985992, 217.45423324370509]
    # define prior
    from abcpy.continuousmodels import Uniform
    mu = Uniform([[150], [200]], )
    sigma = Uniform([[5], [25]], )

    # define the model
    height = Gaussian([mu, sigma], name='height')

    # define statistics
    from abcpy.statistics import Identity
    statistics_calculator = Identity(degree = 2, cross = False)

    # define distance
    from abcpy.distances import LogReg
    distance_calculator = LogReg(statistics_calculator)

    # define kernel
    from abcpy.perturbationkernel import DefaultKernel
    kernel = DefaultKernel([mu, sigma])

    # define backend
    # Note, the dummy backend does not parallelize the code!
    from abcpy.backends import BackendDummy as Backend
    backend = Backend()

    # define sampling scheme
    from abcpy.inferences import PMCABC

# The grade a student of a certain school receives
    final_grade = grade-class_size-background

    # The data corresponding to model_2 defined below
    y_obs_scholarship = [2.7179657436207805, 2.124647285937229, 3.07193407853297, 2.335024761813643, 2.871893855192, 3.4332002458233837, 3.649996835818173, 3.50292335102711, 2.815638168018455, 2.3581613289315992, 2.2794821846395568, 2.8725835459926503, 3.5588573782815685, 2.26053126526137, 1.8998143530749971, 2.101110815311782, 2.3482974964831573, 2.2707679029919206, 2.4624550491079225, 2.867017757972507, 3.204249152084959, 2.4489542437714213, 1.875415915801106, 2.5604889644872433, 3.891985093269989, 2.7233633223405205, 2.2861070389383533, 2.9758813233490082, 3.1183403287267755, 2.911814060853062, 2.60896794303205, 3.5717098647480316, 3.3355752461779824, 1.99172284546858, 2.339937680892163, 2.9835630207301636, 2.1684912355975774, 3.014847335983034, 2.7844122961916202, 2.752119871525148, 2.1567428931391635, 2.5803629307680644, 2.7326646074552103, 2.559237193255186, 3.13478196958166, 2.388760269933492, 3.2822443541491815, 2.0114405441787437, 3.0380056368041073, 2.4889680313769724, 2.821660164621084, 3.343985964873723, 3.1866861970287808, 4.4535037154856045, 3.0026333138006027, 2.0675706089352612, 2.3835301730913185, 2.584208398359566, 3.288077633446465, 2.6955853384148183, 2.918315169739928, 3.2464814419322985, 2.1601516779909433, 3.231003347780546, 1.0893224045062178, 0.8032302688764734, 2.868438615047827]

    # A quantity that determines whether a student will receive a scholarship
    scholarship = Normal([[2], [0.5]], )

    # A quantity determining whether a student receives a scholarship, including his social background
    final_scholarship = scholarship + 3*background

    # Define a summary statistics
    from abcpy.statistics import Identity
    statistics_calculator = Identity(degree = 2, cross = False)

    # Define a distance measure
    from abcpy.distances import Euclidean
    distance_calculator = Euclidean(statistics_calculator)

    # Define a backend
    from abcpy.backends import BackendDummy as Backend
    backend = Backend()

    # Define kernels
    from abcpy.perturbationkernel import MultivariateNormalKernel, MultivariateStudentTKernel
    kernel_1 = MultivariateNormalKernel([school_location, scholarship])
    kernel_2 = MultivariateStudentTKernel([class_size, background, grade], df=3)

    # Join the defined kernels
    from abcpy.perturbationkernel import JointPerturbationKernel

# The grade a student of a certain school receives
    final_grade = grade_without_additional_effects-class_size-background

    # The data corresponding to model_2 defined below
    scholarship_obs = [2.7179657436207805, 2.124647285937229, 3.07193407853297, 2.335024761813643, 2.871893855192, 3.4332002458233837, 3.649996835818173, 3.50292335102711, 2.815638168018455, 2.3581613289315992, 2.2794821846395568, 2.8725835459926503, 3.5588573782815685, 2.26053126526137, 1.8998143530749971, 2.101110815311782, 2.3482974964831573, 2.2707679029919206, 2.4624550491079225, 2.867017757972507, 3.204249152084959, 2.4489542437714213, 1.875415915801106, 2.5604889644872433, 3.891985093269989, 2.7233633223405205, 2.2861070389383533, 2.9758813233490082, 3.1183403287267755, 2.911814060853062, 2.60896794303205, 3.5717098647480316, 3.3355752461779824, 1.99172284546858, 2.339937680892163, 2.9835630207301636, 2.1684912355975774, 3.014847335983034, 2.7844122961916202, 2.752119871525148, 2.1567428931391635, 2.5803629307680644, 2.7326646074552103, 2.559237193255186, 3.13478196958166, 2.388760269933492, 3.2822443541491815, 2.0114405441787437, 3.0380056368041073, 2.4889680313769724, 2.821660164621084, 3.343985964873723, 3.1866861970287808, 4.4535037154856045, 3.0026333138006027, 2.0675706089352612, 2.3835301730913185, 2.584208398359566, 3.288077633446465, 2.6955853384148183, 2.918315169739928, 3.2464814419322985, 2.1601516779909433, 3.231003347780546, 1.0893224045062178, 0.8032302688764734, 2.868438615047827]

    # A quantity that determines whether a student will receive a scholarship
    scholarship_without_additional_effects = Normal([[2], [0.5]], )

    # A quantity determining whether a student receives a scholarship, including his social background
    final_scholarship = scholarship_without_additional_effects + 3*background

    # Define a summary statistics for final grade and final scholarship
    from abcpy.statistics import Identity
    statistics_calculator_final_grade = Identity(degree = 2, cross = False)
    statistics_calculator_final_scholarship = Identity(degree = 3, cross = False)

    # Define a distance measure for final grade and final scholarship
    from abcpy.approx_lhd import SynLiklihood
    approx_lhd_final_grade = SynLiklihood(statistics_calculator_final_grade)
    approx_lhd_final_scholarship = SynLiklihood(statistics_calculator_final_scholarship)

    # Define a backend
    # from abcpy.backends import BackendDummy as Backend
    # backend = Backend()

    setup_backend()

    # Define a perturbation kernel
    from abcpy.perturbationkernel import DefaultKernel
    kernel = DefaultKernel([school_location, class_size, grade_without_additional_effects, \
                            background, scholarship_without_additional_effects])

def infer_parameters_rsmcabc():
    # define observation for true parameters mean=170, 65
    rng = np.random.RandomState()
    y_obs = [np.array(rng.multivariate_normal([170, 65], np.eye(2), 1).reshape(2,))]

    # define prior
    from abcpy.continuousmodels import Uniform
    mu0 = Uniform([[150], [200]], )
    mu1 = Uniform([[25], [100]], )

    # define the model
    height_weight_model = NestedBivariateGaussian([mu0, mu1])

    # define statistics
    from abcpy.statistics import Identity
    statistics_calculator = Identity(degree = 2, cross = False)

    # define distance
    from abcpy.distances import Euclidean
    distance_calculator = Euclidean(statistics_calculator)

    # define sampling scheme
    from abcpy.inferences import RSMCABC
    sampler = RSMCABC([height_weight_model], [distance_calculator], backend, seed=1)
    print('sampling')
    steps, n_samples, n_samples_per_param, alpha, epsilon_init, epsilon_final = 2, 10, 1, 0.1, 10000, 500
    print('RSMCABC Inferring')
    journal = sampler.sample([y_obs], steps, n_samples, n_samples_per_param, alpha , epsilon_init, epsilon_final,full_output=1)

    return journal

# The grade a student of a certain school receives
    final_grade = grade_without_additional_effects-class_size-background

    # The data corresponding to model_2 defined below
    scholarship_obs = [2.7179657436207805, 2.124647285937229, 3.07193407853297, 2.335024761813643, 2.871893855192, 3.4332002458233837, 3.649996835818173, 3.50292335102711, 2.815638168018455, 2.3581613289315992, 2.2794821846395568, 2.8725835459926503, 3.5588573782815685, 2.26053126526137, 1.8998143530749971, 2.101110815311782, 2.3482974964831573, 2.2707679029919206, 2.4624550491079225, 2.867017757972507, 3.204249152084959, 2.4489542437714213, 1.875415915801106, 2.5604889644872433, 3.891985093269989, 2.7233633223405205, 2.2861070389383533, 2.9758813233490082, 3.1183403287267755, 2.911814060853062, 2.60896794303205, 3.5717098647480316, 3.3355752461779824, 1.99172284546858, 2.339937680892163, 2.9835630207301636, 2.1684912355975774, 3.014847335983034, 2.7844122961916202, 2.752119871525148, 2.1567428931391635, 2.5803629307680644, 2.7326646074552103, 2.559237193255186, 3.13478196958166, 2.388760269933492, 3.2822443541491815, 2.0114405441787437, 3.0380056368041073, 2.4889680313769724, 2.821660164621084, 3.343985964873723, 3.1866861970287808, 4.4535037154856045, 3.0026333138006027, 2.0675706089352612, 2.3835301730913185, 2.584208398359566, 3.288077633446465, 2.6955853384148183, 2.918315169739928, 3.2464814419322985, 2.1601516779909433, 3.231003347780546, 1.0893224045062178, 0.8032302688764734, 2.868438615047827]

    # A quantity that determines whether a student will receive a scholarship
    scholarship_without_additional_effects = Normal([[2], [0.5]], )

    # A quantity determining whether a student receives a scholarship, including his social background
    final_scholarship = scholarship_without_additional_effects + 3*background

    # Define a summary statistics for final grade and final scholarship
    from abcpy.statistics import Identity
    statistics_calculator_final_grade = Identity(degree = 2, cross = False)
    statistics_calculator_final_scholarship = Identity(degree = 3, cross = False)

    # Define a distance measure for final grade and final scholarship
    from abcpy.distances import Euclidean
    distance_calculator_final_grade = Euclidean(statistics_calculator_final_grade)
    distance_calculator_final_scholarship = Euclidean(statistics_calculator_final_scholarship)

    # Define a backend
    from abcpy.backends import BackendDummy as Backend
    backend = Backend()

    # Define a perturbation kernel
    from abcpy.perturbationkernel import DefaultKernel
    kernel = DefaultKernel([school_location, class_size, grade_without_additional_effects, \
                            background, scholarship_without_additional_effects])

def infer_parameters():
    # define observation for true parameters mean=170, std=15
    height_obs = [160.82499176, 167.24266737, 185.71695756, 153.7045709, 163.40568812, 140.70658699, 169.59102084, 172.81041696, 187.38782738, 179.66358934, 176.63417241, 189.16082803, 181.98288443, 170.18565017, 183.78493886, 166.58387299, 161.9521899, 155.69213073, 156.17867343, 144.51580379, 170.29847515, 197.96767899, 153.36646527, 162.22710198, 158.70012047, 178.53470703, 170.77697743, 164.31392633, 165.88595994, 177.38083686, 146.67058471763457, 179.41946565658628, 238.02751620619537, 206.22458790620766, 220.89530574344568, 221.04082532837026, 142.25301427453394, 261.37656571434275, 171.63761180867033, 210.28121820385866, 237.29130237612236, 175.75558340169619, 224.54340549862235, 197.42448680731226, 165.88273684581381, 166.55094082844519, 229.54308602661584, 222.99844054358519, 185.30223966014586, 152.69149367593846, 206.94372818527413, 256.35498655339154, 165.43140916577741, 250.19273595481803, 148.87781549665536, 223.05547559193792, 230.03418198709608, 146.13611923127021, 138.24716809523139, 179.26755740864527, 141.21704876815426, 170.89587081800852, 222.96391329259626, 188.27229523693822, 202.67075179617672, 211.75963110985992, 217.45423324370509]

    # define prior
    from abcpy.continuousmodels import Uniform
    mu = Uniform([[150], [200]], )
    sigma = Uniform([[5], [25]], )
    
    # define the model
    from abcpy.continuousmodels import Normal
    height = Normal([mu, sigma], )
    
    # define statistics
    from abcpy.statistics import Identity
    statistics_calculator = Identity(degree = 2, cross = False)
    
    # define distance
    from abcpy.distances import LogReg
    distance_calculator = LogReg(statistics_calculator)
    
    # define kernel
    from abcpy.perturbationkernel import DefaultKernel
    kernel = DefaultKernel([mu, sigma])

    # define backend
    # Note, the dummy backend does not parallelize the code!
    from abcpy.backends import BackendDummy as Backend
    backend = Backend()
    
    # define sampling scheme
    from abcpy.inferences import PMCABC

def infer_parameters_pmc():
    # define observation for true parameters mean=170, 65
    rng = np.random.RandomState()
    y_obs = [np.array(rng.multivariate_normal([170, 65], np.eye(2), 1).reshape(2,))]

    # define prior
    from abcpy.continuousmodels import Uniform
    mu0 = Uniform([[150], [200]], )
    mu1 = Uniform([[25], [100]], )

    # define the model
    height_weight_model = NestedBivariateGaussian([mu0, mu1])

    # define statistics
    from abcpy.statistics import Identity
    statistics_calculator = Identity(degree = 2, cross = False)

    from abcpy.approx_lhd import SynLikelihood
    approx_lhd = SynLikelihood(statistics_calculator)

    # define sampling scheme
    from abcpy.inferences import PMC
    sampler = PMC([height_weight_model], [approx_lhd], backend, seed=1)

    # sample from scheme
    T, n_sample, n_samples_per_param = 2, 10, 10

    journal = sampler.sample([y_obs],  T, n_sample, n_samples_per_param)

    return journal