Python numpy 模块，eye() 实例源码

def initialize_estimate(self, estimate_id, initial_state):
        """Initialize a state estimate with identity covariance.

        The initial estimate is saved in the `self.estimates` dictionary.
        The timestamp in the `self.estimate_times` is updated.

        Args:
             estimate_id (int): ID of the tracked target.
             initial_state (int): Initial state of the estimate.

        Returns:
             X (numpy.ndarray): Solution of equation.
        """
        x = initial_state
        P = self.initial_position_covariance * np.eye(6)
        P[3:6, 3:6] = 0
        estimate = UWBTracker.StateEstimate(x, P)
        self.estimates[estimate_id] = estimate
        self.estimate_times[estimate_id] = rospy.get_time()
        self.ikf_prev_outlier_flags[estimate_id] = False
        self.ikf_outlier_counts[estimate_id] = 0

def _compute_process_and_covariance_matrices(self, dt):
        """Computes the transition and covariance matrix of the process model and measurement model.

        Args:
             dt (float): Timestep of the discrete transition.

        Returns:
            F (numpy.ndarray): Transition matrix.
            Q (numpy.ndarray): Process covariance matrix.
            R (numpy.ndarray): Measurement covariance matrix.
        """
        F = np.array(np.bmat([[np.eye(3), dt * np.eye(3)], [np.zeros((3, 3)), np.eye(3)]]))
        self.process_matrix = F
        q_p = self.process_covariance_position
        q_v = self.process_covariance_velocity
        Q = np.diag([q_p, q_p, q_p, q_v, q_v, q_v]) ** 2 * dt
        r = self.measurement_covariance
        R = r * np.eye(4)
        self.process_covariance = Q
        self.measurement_covariance = R
        return F, Q, R

def get_covariance(self):
        """Compute data covariance with the generative model.
        ``cov = components_.T * S**2 * components_ + sigma2 * eye(n_features)``
        where  S**2 contains the explained variances.
        Returns
        -------
        cov : array, shape=(n_features, n_features)
            Estimated covariance of data.
        """
        components_ = self.components_
        exp_var = self.explained_variance_
        if self.whiten:
            components_ = components_ * np.sqrt(exp_var[:, np.newaxis])
        exp_var_diff = np.maximum(exp_var - self.noise_variance_, 0.)
        cov = np.dot(components_.T * exp_var_diff, components_)
        cov.flat[::len(cov) + 1] += self.noise_variance_  # modify diag inplace
        return cov

def infExact_scipy_post(self, K, covars, y, sig2e, fixedEffects):
        n = y.shape[0]

        #mean vector
        m = covars.dot(fixedEffects)

        if (K.shape[1] < K.shape[0]): K_true = K.dot(K.T)
        else: K_true = K

        if sig2e<1e-6:
            L = la.cholesky(K_true + sig2e*np.eye(n), overwrite_a=True, check_finite=False)      #Cholesky factor of covariance with noise
            sl =   1
            pL = -self.solveChol(L, np.eye(n))                                                   #L = -inv(K+inv(sW^2))
        else:
            L = la.cholesky(K_true/sig2e + np.eye(n), overwrite_a=True, check_finite=False)      #Cholesky factor of B
            sl = sig2e                     
            pL = L                                                                               #L = chol(eye(n)+sW*sW'.*K)
        alpha = self.solveChol(L, y-m, overwrite_b=False) / sl

        post = dict([]) 
        post['alpha'] = alpha                                                                   #return the posterior parameters
        post['sW'] = np.ones(n) / np.sqrt(sig2e)                                                #sqrt of noise precision vector
        post['L'] = pL
        return post

def BorthogonalityTest(B, U):
    """
    Test the frobenious norm of  U^TBU - I_k
    """
    BU = np.zeros(U.shape)
    Bu = Vector()
    u = Vector()
    B.init_vector(Bu,0)
    B.init_vector(u,1)

    nvec  = U.shape[1]
    for i in range(0,nvec):
        u.set_local(U[:,i])
        B.mult(u,Bu)
        BU[:,i] = Bu.get_local()

    UtBU = np.dot(U.T, BU)
    err = UtBU - np.eye(nvec, dtype=UtBU.dtype)
    print("|| UtBU - I ||_F = ", np.linalg.norm(err, 'fro') )

def KeyGen(**kwargs):
    '''
    Appendix B of BLISS paper
    m_bar = m + n

    o/p:    
    A: Public Key n x m' numpy array
    S: Secret Key m'x n numpy array
    '''
    q, n, m, alpha = kwargs['q'], kwargs['n'], kwargs['m'], kwargs['alpha']
    Aq_bar = util.crypt_secure_matrix(-(q-1)/2, (q-1)/2, n, m)
    S_bar = util.crypt_secure_matrix(-(2)**alpha, (2)**alpha, m, n) # alpha is small enough, we need not reduce (modq)
    S = np.vstack((S_bar, np.eye(n, dtype = int))) # dimension is m_bar x n, Elements are in Z mod(2q)
    A = np.hstack((2*Aq_bar, q * np.eye(n, dtype = int) - 2*np.matmul(Aq_bar,S_bar))) # dimension is n x m_bar , Elements are in Z mod(2q)
    #return util.matrix_to_Zq(A, 2*q), S, Aq_bar, S_bar
    return util.matrix_to_Zq(A, 2*q), S

def test():
    # Classical SIS parameters
    n, m, alpha, q = 128, 872, 1, 114356107
    kappa = 20

    #Discrete Gaussian Parameters
    sd = 300
    eta = 1.2

    A, S = KeyGen(q = q,n = n,m = m,alpha = alpha)
    #print np.array(np.matmul(A,S) - q*np.eye(n),dtype=float)/(2*q) #to test AS = q mod(2q)
    z, c = Sign(msg = "Hello Bob",A = A,S = S,m = m,n = n,sd = sd,q = q,M = 3.0,kappa = kappa)
    print z
    print c
    print Verify(msg = "Hello Bob", A=A, m=m, n=n, sd=sd, q=q, eta=eta, z=z, c=c, kappa = kappa)
    print Verify(msg = "Hello Robert", A=A, m=m, n=n, sd=sd, q=q, eta=eta, z=z, c=c, kappa = kappa)
    print Verify(msg = "Hello Roberto", A=A, m=m, n=n, sd=sd, q=q, eta=eta, z=z, c=c, kappa = kappa)
    print Verify(msg = "Hola Roberto", A=A, m=m, n=n, sd=sd, q=q, eta=eta, z=z, c=c, kappa = kappa)

def __init__(self, X, Y, R=None, t=None, s=None, sigma2=None, maxIterations=100, tolerance=0.001, w=0):
    if X.shape[1] != Y.shape[1]:
      raise 'Both point clouds must have the same number of dimensions!'

    self.X             = X
    self.Y             = Y
    self.TY            = Y
    (self.N, self.D)   = self.X.shape
    (self.M, _)        = self.Y.shape
    self.R             = np.eye(self.D) if R is None else R
    self.t             = np.atleast_2d(np.zeros((1, self.D))) if t is None else t
    self.s             = 1 if s is None else s
    self.sigma2        = sigma2
    self.iteration     = 0
    self.maxIterations = maxIterations
    self.tolerance     = tolerance
    self.w             = w
    self.q             = 0
    self.err           = 0

def __init__(self, X, Y, B=None, t=None, sigma2=None, maxIterations=100, tolerance=0.001, w=0):
    if X.shape[1] != Y.shape[1]:
        raise 'Both point clouds must have the same number of dimensions!'

    self.X             = X
    self.Y             = Y
    self.TY            = Y
    (self.N, self.D)   = self.X.shape
    (self.M, _)        = self.Y.shape
    self.B             = np.eye(self.D) if B is None else B
    self.t             = np.atleast_2d(np.zeros((1, self.D))) if t is None else t
    self.sigma2        = sigma2
    self.iteration     = 0
    self.maxIterations = maxIterations
    self.tolerance     = tolerance
    self.w             = w
    self.q             = 0
    self.err           = 0

def get_loss(pred, label, end_points, reg_weight=0.001):
    """ pred: B*NUM_CLASSES,
        label: B, """
    loss = tf.nn.sparse_softmax_cross_entropy_with_logits(logits=pred, labels=label)
    classify_loss = tf.reduce_mean(loss)
    tf.summary.scalar('classify loss', classify_loss)

    # Enforce the transformation as orthogonal matrix
    transform = end_points['transform'] # BxKxK
    K = transform.get_shape()[1].value
    mat_diff = tf.matmul(transform, tf.transpose(transform, perm=[0,2,1]))
    mat_diff -= tf.constant(np.eye(K), dtype=tf.float32)
    mat_diff_loss = tf.nn.l2_loss(mat_diff) 
    tf.summary.scalar('mat loss', mat_diff_loss)

    return classify_loss + mat_diff_loss * reg_weight

def get_loss(pred, label, end_points, reg_weight=0.001):
    """ pred: BxNxC,
        label: BxN, """
    loss = tf.nn.sparse_softmax_cross_entropy_with_logits(logits=pred, labels=label)
    classify_loss = tf.reduce_mean(loss)
    tf.scalar_summary('classify loss', classify_loss)

    # Enforce the transformation as orthogonal matrix
    transform = end_points['transform'] # BxKxK
    K = transform.get_shape()[1].value
    mat_diff = tf.matmul(transform, tf.transpose(transform, perm=[0,2,1]))
    mat_diff -= tf.constant(np.eye(K), dtype=tf.float32)
    mat_diff_loss = tf.nn.l2_loss(mat_diff) 
    tf.scalar_summary('mat_loss', mat_diff_loss)

    return classify_loss + mat_diff_loss * reg_weight

def get_loss(l_pred, seg_pred, label, seg, weight, end_points):
    per_instance_label_loss = tf.nn.sparse_softmax_cross_entropy_with_logits(logits=l_pred, labels=label)
    label_loss = tf.reduce_mean(per_instance_label_loss)

    # size of seg_pred is batch_size x point_num x part_cat_num
    # size of seg is batch_size x point_num
    per_instance_seg_loss = tf.reduce_mean(tf.nn.sparse_softmax_cross_entropy_with_logits(logits=seg_pred, labels=seg), axis=1)
    seg_loss = tf.reduce_mean(per_instance_seg_loss)

    per_instance_seg_pred_res = tf.argmax(seg_pred, 2)

    # Enforce the transformation as orthogonal matrix
    transform = end_points['transform'] # BxKxK
    K = transform.get_shape()[1].value
    mat_diff = tf.matmul(transform, tf.transpose(transform, perm=[0,2,1])) - tf.constant(np.eye(K), dtype=tf.float32)
    mat_diff_loss = tf.nn.l2_loss(mat_diff) 


    total_loss = weight * seg_loss + (1 - weight) * label_loss + mat_diff_loss * 1e-3

    return total_loss, label_loss, per_instance_label_loss, seg_loss, per_instance_seg_loss, per_instance_seg_pred_res

def refit_model(self):
        """Learns a new surrogate model using the data observed so far.

        """

        # only fit the model if there is data for it.
        if len(self.known_models) > 0:

            self._build_feature_maps(self.known_models, self.ngram_maxlen, self.thres)

            X = sp.vstack([ self._compute_features(mdl)
                    for mdl in self.known_models], "csr")
            y = np.array(self.known_scores, dtype='float64')

            #A = np.dot(X.T, X) + lamb * np.eye(X.shape[1])
            #b = np.dot(X.T, y)
            self.surr_model = lm.Ridge(self.lamb_ridge)
            self.surr_model.fit(X, y)


# NOTE: if the search space has holes, it break. needs try/except module.

def covariance_matrix(self):
        if self._debug:
            # return None
            ka = self.k_active
            if ka > 0:
                C = np.eye(self.N) + np.dot(self.V[:ka].T * self.S[:ka],
                                            self.V[:ka])
                C = (C * self.D).T * self.D
            else:
                C = np.diag(self.D**2)
            C *= self.sigma**2
        else:
            # Fake Covariance Matrix for Speed
            C = np.ones(1)
            self.B = np.ones(1)
        return C

def Minibatch_Discriminator(input, num_kernels=100, dim_per_kernel=5, init=False, name='MD'):
    num_inputs=df_dim*4
    theta = tf.get_variable(name+"/theta",[num_inputs, num_kernels, dim_per_kernel], initializer=tf.random_normal_initializer(stddev=0.05))
    log_weight_scale = tf.get_variable(name+"/lws",[num_kernels, dim_per_kernel], initializer=tf.constant_initializer(0.0))
    W = tf.mul(theta, tf.expand_dims(tf.exp(log_weight_scale)/tf.sqrt(tf.reduce_sum(tf.square(theta),0)),0))
    W = tf.reshape(W,[-1,num_kernels*dim_per_kernel])
    x = input
    x=tf.reshape(x, [batchsize,num_inputs])
    activation = tf.matmul(x, W)
    activation = tf.reshape(activation,[-1,num_kernels,dim_per_kernel])
    abs_dif = tf.mul(tf.reduce_sum(tf.abs(tf.sub(tf.expand_dims(activation,3),tf.expand_dims(tf.transpose(activation,[1,2,0]),0))),2),
                                                1-tf.expand_dims(tf.constant(np.eye(batchsize),dtype=np.float32),1))
    f = tf.reduce_sum(tf.exp(-abs_dif),2)/tf.reduce_sum(tf.exp(-abs_dif))
    print(f.get_shape())
    print(input.get_shape())
    return tf.concat(1,[x, f])

def _build_policy(env, predictor, epsilon):
    eye = np.eye(env.num_states)
    q_values = predictor.predict(
        {str(i): eye[i]
         for i in range(env.num_states)}
    )
    policy_vector = [
        env.ACTIONS[np.argmax([q_values[action][i] for action in env.ACTIONS])]
        for i in range(env.num_states)
    ]

    def policy(state) -> str:
        if np.random.random() < epsilon:
            return np.random.choice(env.ACTIONS)
        else:
            return policy_vector[state]

    return policy

def export_data(prediction_dir, nii_image_dir, tfrecords_dir, export_dir, transformation=None):

    for file_path in os.listdir(prediction_dir):
        name, prediction, probability = read_prediction_file(os.path.join(prediction_dir, file_path))

        if transformation:
            image, ground_truth = get_original_image(os.path.join(tfrecords_dir, name + '.tfrecord'), False)
            prediction = transformation.transform_image(prediction, probability, image)

        # build cv_predictions .nii image
        img = nib.Nifti1Image(prediction, np.eye(4))
        img.set_data_dtype(dtype=np.uint8)

        path = os.path.join(nii_image_dir, name)

        adc_name = next(l for l in os.listdir(path) if 'MR_ADC' in l)
        export_image = nib.load(os.path.join(nii_image_dir, name, adc_name, adc_name + '.nii'))

        i = export_image.get_data()
        i[:] = img.get_data()

        # set name to specification and export
        _id = next(l for l in os.listdir(path) if 'MR_MTT' in l).split('.')[-1]
        export_path = os.path.join(export_dir, 'SMIR.' + name + '.' + _id + '.nii')
        nib.save(export_image, os.path.join(export_path))

def eye(sites, ldim):
    """Returns a MPA representing the identity matrix

    :param sites: Number of sites
    :param ldim: Int-like local dimension or iterable of local dimensions
    :returns: Representation of the identity matrix as MPA

    >>> I = eye(4, 2)
    >>> I.ranks, I.shape
    ((1, 1, 1), ((2, 2), (2, 2), (2, 2), (2, 2)))
    >>> I = eye(3, (3, 4, 5))
    >>> I.shape
    ((3, 3), (4, 4), (5, 5))
    """
    if isinstance(ldim, collections.Iterable):
        ldim = tuple(ldim)
        assert len(ldim) == sites
    else:
        ldim = it.repeat(ldim, sites)
    return mp.MPArray.from_kron(map(np.eye, ldim))

def _embed_ltens_identity(mpa, embed_tensor=None):
    """Embed with identity matrices by default.

    :param embed_tensor: If the MPAs do not have two physical legs or
        have non-square physical dimensions, you must provide an
        embedding tensor. The default is to use the square identity
        matrix, assuming that the size of the two physical legs is the
        same at each site.
    :returns: `embed_tensor` with one size-one virtual leg added at each
        end.

    """
    if embed_tensor is None:
        pdims = mpa.shape[0]
        assert len(pdims) == 2 and pdims[0] == pdims[1], (
            "For ndims != 2 or non-square shape, you must supply a tensor"
            "for embedding")
        embed_tensor = np.eye(pdims[0])
    embed_ltens = embed_tensor[None, ..., None]
    return embed_ltens

def test_povm_normalization_ic(dim):
    for name, constructor in ALL_POVMS.items():
        # Check that the POVM is normalized: elements must sum to the identity
        current_povm = constructor(dim)
        element_sum = sum(iter(current_povm))
        assert_array_almost_equal(element_sum, np.eye(dim))

        # Check that the attribute that says whether the POVM is IC is correct.
        linear_inversion_recons = np.dot(current_povm.linear_inversion_map,
                                         current_povm.probability_map)
        if current_povm.informationally_complete:
            assert_array_almost_equal(
                linear_inversion_recons, np.eye(dim**2),
                err_msg='POVM {} is not informationally complete'.format(name))
        else:
            assert np.abs(linear_inversion_recons - np.eye(dim**2)).max() > 0.1, \
                'POVM {} is informationally complete'.format(name)

def test_povm_ic_mpa(nr_sites, local_dim, rank, rgen):
    # Check that the tensor product of the PauliGen POVM is IC.
    paulis = povm.pauli_povm(local_dim)
    inv_map = mp_from_array_repeat(paulis.linear_inversion_map, nr_sites)
    probab_map = mp_from_array_repeat(paulis.probability_map, nr_sites)
    reconstruction_map = mp.dot(inv_map, probab_map)

    eye = factory.eye(nr_sites, local_dim**2)
    assert mp.norm(reconstruction_map - eye) < 1e-5

    # Check linear inversion for a particular example MPA.
    # Linear inversion works for arbitrary matrices, not only for states,
    # so we test it for an arbitrary MPA.
    # Normalize, otherwise the absolute error check below will not work.
    mpa = factory.random_mpa(nr_sites, local_dim**2, rank,
                             dtype=np.complex_, randstate=rgen, normalized=True)
    probabs = mp.dot(probab_map, mpa)
    recons = mp.dot(inv_map, probabs)
    assert mp.norm(recons - mpa) < 1e-6

def test_randomized_svd(rows, cols, rank, dtype, transpose, n_iter, target_gen,
                        rgen):
    rank = min(rows, cols) - 2 if rank is 'fullrank' else rank
    A = target_gen(rows, cols, rank=rank, randstate=rgen, dtype=dtype)

    U_ref, s_ref, V_ref = utils.truncated_svd(A, k=rank)
    U, s, V = em.randomized_svd(A, rank, transpose=transpose, randstate=rgen,
                                n_iter=n_iter)

    error_U = np.abs(U.conj().T.dot(U_ref)) - np.eye(rank)
    assert_allclose(np.linalg.norm(error_U), 0, atol=1e-3)
    error_V = np.abs(V.dot(V_ref.conj().T)) - np.eye(rank)
    assert_allclose(np.linalg.norm(error_V), 0, atol=1e-3)
    assert_allclose(s.ravel() - s_ref, 0, atol=1e-3)
    # Check that singular values are returned in descending order
    assert_array_equal(s, np.sort(s)[::-1])

def hessian(self, x, d=None):
        """
        Computes Hessian matrix
        """
        d = calc_distances(x) if d is None else d
        if d.ndim == 1: d = squareform(d)

        H = np.zeros((3*len(x), 3*len(x)))
        n = self.n

        for i in range(len(x)):
            for j in range(len(x)):

                if j == i: continue

                dx = x[i]-x[j]
                r  = d[i,j]
                h  = n / r**(0.5*n+2) * ((n+2) * np.multiply.outer(dx,dx) - np.eye(3) * r)

                H[3*i:3*(i+1), 3*j:3*(j+1)]  = -h 
                H[3*i:3*(i+1), 3*i:3*(i+1)] +=  h

        return H

def apply_gate(self,gate,on_qubit_name):
        on_qubit=self.qubits.get_quantum_register_containing(on_qubit_name)
        if len(on_qubit.get_noop()) > 0:
            print "NOTE this qubit has been measured previously, there should be no more gates allowed but we are reverting that measurement for consistency with IBM's language"
            on_qubit.set_state(on_qubit.get_noop())
            on_qubit.set_noop([])
        if not on_qubit.is_entangled():
            if on_qubit.get_num_qubits()!=1:
                raise Exception("This qubit is not marked as entangled but it has an entangled state")
            on_qubit.set_state(gate*on_qubit.get_state())
        else:
            if not on_qubit.get_num_qubits()>1:
                raise Exception("This qubit is marked as entangled but it does not have an entangled state")
            n_entangled=len(on_qubit.get_entangled())
            apply_gate_to_qubit_idx=[qb.name for qb in on_qubit.get_entangled()].index(on_qubit_name)
            if apply_gate_to_qubit_idx==0:
                entangled_gate=gate
            else:
                entangled_gate=Gate.eye
            for i in range(1,n_entangled):
                if apply_gate_to_qubit_idx==i:
                    entangled_gate=np.kron(entangled_gate,gate)
                else:
                    entangled_gate=np.kron(entangled_gate,Gate.eye)
            on_qubit.set_state(entangled_gate*on_qubit.get_state())

def _matrix_inverse(self, matrix):
        """
        Computes inverse of a matrix.
        """
        matrix = np.array(matrix)
        n_features = matrix.shape[0]
        rank = np.linalg.matrix_rank(matrix)

        if rank == n_features:
            return np.linalg.inv(matrix)
        else:
            # Matrix is not full rank, so use Hadi's technique to compute inverse
            # Reference: Ali S. Hadi (1992) "Identifying Multiple Outliers in Multivariate Data" eg. 2.3, 2.4
            eigenValues, eigenVectors = np.linalg.eig(matrix)
            eigenValues = np.abs(eigenValues)  # to deal with -0 values
            idx = eigenValues.argsort()[::-1]
            eigenValues = eigenValues[idx]
            eigenVectors = eigenVectors[:, idx]

            s = eigenValues[eigenValues != 0].min()
            w = [1 / max(e, s) for e in eigenValues]
            W = w * np.eye(n_features)

            return eigenVectors.dot(W).dot(eigenVectors.T)

def swap_affine(axes):
    """ Build a correction matrix, from the given orientation of axes to RAS.

    Parameters
    ----------
    axes: str (manadtory)
        the given orientation of the axes.

    Returns
    -------
    rotation: array (4, 4)
        the correction matrix.
    """
    rotation = numpy.eye(4)
    rotation[:3, 0] = CORRECTION_MATRIX_COLUMNS[axes[0]]
    rotation[:3, 1] = CORRECTION_MATRIX_COLUMNS[axes[1]]
    rotation[:3, 2] = CORRECTION_MATRIX_COLUMNS[axes[2]]
    return rotation

def distribution_parameters(self, parameter_name):
        if parameter_name=='trend':
            E = dot(dot(self.derivative_matrix.T,inv(diag(self.parameters.list['omega'].current_value))),self.derivative_matrix)
            mean = dot(inv(eye(self.size)+E),self.data)
            cov = (self.parameters.list['sigma2'].current_value)*inv(eye(self.size)+E)
            return {'mean' : mean, 'cov' : cov}
        elif parameter_name=='sigma2':
            E = dot(dot(self.derivative_matrix.T,inv(diag(self.parameters.list['omega'].current_value))),self.derivative_matrix)
            pos = self.size
            loc = 0
            scale = 0.5*dot((self.data-dot(eye(self.size),self.parameters.list['trend'].current_value)).T,(self.data-dot(eye(self.size),self.parameters.list['trend'].current_value)))+0.5*dot(dot(self.parameters.list['trend'].current_value.T,E),self.parameters.list['trend'].current_value)
        elif parameter_name=='lambda2':
            pos = self.size-self.total_variation_order-1+self.alpha
            loc = 0.5*(norm(dot(self.derivative_matrix,self.parameters.list['trend'].current_value),ord=1))/self.parameters.list['sigma2'].current_value+self.rho
            scale = 1
        elif parameter_name==str('omega'):
            pos = [sqrt(((self.parameters.list['lambda2'].current_value**2)*self.parameters.list['sigma2'].current_value)/(dj**2)) for dj in dot(self.derivative_matrix,self.parameters.list['trend'].current_value)]
            loc = 0
            scale = self.parameters.list['lambda2'].current_value**2
        return {'pos' : pos, 'loc' : loc, 'scale' : scale}

def estimate_time_constant(y, p=2, sn=None, lags=5, fudge_factor=1.):
    """
    Estimate AR model parameters through the autocovariance function

    Parameters
    ----------
    y : array, shape (T,)
        One dimensional array containing the fluorescence intensities with
        one entry per time-bin.
    p : positive integer
        order of AR system
    sn : float
        sn standard deviation, estimated if not provided.
    lags : positive integer
        number of additional lags where he autocovariance is computed
    fudge_factor : float (0< fudge_factor <= 1)
        shrinkage factor to reduce bias

    Returns
    -------
    g : estimated coefficients of the AR process
    """

    if sn is None:
        sn = GetSn(y)

    lags += p
    xc = axcov(y, lags)
    xc = xc[:, np.newaxis]

    A = scipy.linalg.toeplitz(xc[lags + np.arange(lags)],
                              xc[lags + np.arange(p)]) - sn**2 * np.eye(lags, p)
    g = np.linalg.lstsq(A, xc[lags + 1:])[0]
    gr = np.roots(np.concatenate([np.array([1]), -g.flatten()]))
    gr = (gr + gr.conjugate()) / 2.
    gr[gr > 1] = 0.95 + np.random.normal(0, 0.01, np.sum(gr > 1))
    gr[gr < 0] = 0.15 + np.random.normal(0, 0.01, np.sum(gr < 0))
    g = np.poly(fudge_factor * gr)
    g = -g[1:]

    return g.flatten()

def gaussian_function(y, dimension, ?, cov, log=False, standard=False):
            """??????????? y???(???) ??????(?????) cov??????,log????????,standard??????"""
            x = y - ?
            if standard:
                x = np.dot(x, np.linalg.inv(cov) ** 0.5)
                cov_ = np.eye(dimension)
            else:
                cov_ = cov
            np.seterr(all='ignore')  # ??????
            if log:
                func = - (dimension / 2) * np.log(2 * math.pi) - 0.5 * np.log(np.linalg.det(cov_))
                exp = -0.5 * np.dot(np.dot(x, np.linalg.inv(cov_)), x.T)
                return func + exp
            else:
                sigma = (2 * math.pi) ** (dimension / 2) * np.linalg.det(cov_) ** 0.5
                func = 1. / sigma
                exp = np.exp(-0.5 * np.dot(np.dot(x, np.linalg.inv(cov_)), x.T))
                return func * exp

def batch_works(k):
    if k == n_processes - 1:
        paths = all_paths[k * int(len(all_paths) / n_processes) : ]
    else:
        paths = all_paths[k * int(len(all_paths) / n_processes) : (k + 1) * int(len(all_paths) / n_processes)]

    for path in paths:
        probs = np.load(os.path.join(input_path, path))
        pred = np.argmax(probs, axis=3)
        fg_prob = 1 - probs[..., 0]
        pred = clean_contour(fg_prob, pred)
        seg = np.zeros(pred.shape, dtype=np.int16)
        seg[pred == 1] = 1
        seg[pred == 2] = 2
        seg[pred == 3] = 4
        img = nib.Nifti1Image(seg, np.eye(4))
        nib.save(img, os.path.join(output_path, path.replace('_probs.npy', '.nii.gz')))

def __prepare_controls_and_actions(self):
        self.__discrete_controls_to_net = np.array([i for i in range(len(self.__discrete_controls))
                                                    if i not in self.__discrete_controls_manual])
        self.__num_manual_controls = len(self.__discrete_controls_manual)
        self.__net_discrete_actions = []
        if not self.__opposite_button_pairs:
            for perm in it.product([False, True], repeat=len(self.__discrete_controls_to_net)):
                self.__net_discrete_actions.append(list(perm))
        else:
            for perm in it.product([False, True], repeat=len(self.__discrete_controls_to_net)):
                act = list(perm)
                valid = True
                for button_p in self.__opposite_button_pairs:
                    if act[button_p[0]] and act[button_p[1]]:
                        valid = False
                if valid:
                    self.__net_discrete_actions.append(act)

        self.__num_net_discrete_actions = len(self.__net_discrete_actions)
        self.__action_to_index = {tuple(val): ind for (ind, val) in enumerate(self.__net_discrete_actions)}
        self.__net_discrete_actions = np.array(self.__net_discrete_actions)
        self.__onehot_discrete_acitons = np.eye(self.__num_net_discrete_actions)

def initialize_match_matrix(self):
        """
        Construct the initial match matrix.

        Returns:
        --------
        match_matrix: array
            The match matrix

        """
        # TODO add possibility for slack

        match_matrix = np.zeros((self.reactants_elements.size, self.products_elements.size))

        # set sub blocks of the match matrix to one plus random pertubation
        # followed by column normalization
        for indices in self.element_type_subset_indices:
            match_matrix[indices] = 1 + 1e-3 * np.random.random(match_matrix[indices].shape)
            match_matrix[indices] /= match_matrix[indices].sum(0)

        #match_matrix = np.eye(match_matrix.shape[0])
        #for i,j in [(0,0),(0,4),(1,1),(1,3),(4,0),(4,4),(3,3),(3,1),(7,7),(7,11),(8,8),(8,10),(20,20),(20,24),(21,21),(21,23),(11,7),(11,11),(10,8),(10,10),(24,20),(24,24),(23,23),(23,21)]:
        #    match_matrix[i,j] = 0.5

        return match_matrix

def test_nonzero_twodim(self):
        x = np.array([[0, 1, 0], [2, 0, 3]])
        assert_equal(np.count_nonzero(x), 3)
        assert_equal(np.nonzero(x), ([0, 1, 1], [1, 0, 2]))

        x = np.eye(3)
        assert_equal(np.count_nonzero(x), 3)
        assert_equal(np.nonzero(x), ([0, 1, 2], [0, 1, 2]))

        x = np.array([[(0, 1), (0, 0), (1, 11)],
                   [(1, 1), (1, 0), (0, 0)],
                   [(0, 0), (1, 5), (0, 1)]], dtype=[('a', 'f4'), ('b', 'u1')])
        assert_equal(np.count_nonzero(x['a']), 4)
        assert_equal(np.count_nonzero(x['b']), 5)
        assert_equal(np.nonzero(x['a']), ([0, 1, 1, 2], [2, 0, 1, 1]))
        assert_equal(np.nonzero(x['b']), ([0, 0, 1, 2, 2], [0, 2, 0, 1, 2]))

        assert_(not x['a'].T.flags.aligned)
        assert_equal(np.count_nonzero(x['a'].T), 4)
        assert_equal(np.count_nonzero(x['b'].T), 5)
        assert_equal(np.nonzero(x['a'].T), ([0, 1, 1, 2], [1, 1, 2, 0]))
        assert_equal(np.nonzero(x['b'].T), ([0, 0, 1, 2, 2], [0, 1, 2, 0, 2]))

def test_matrix_rank(self):
        # Full rank matrix
        yield assert_equal, 4, matrix_rank(np.eye(4))
        # rank deficient matrix
        I = np.eye(4)
        I[-1, -1] = 0.
        yield assert_equal, matrix_rank(I), 3
        # All zeros - zero rank
        yield assert_equal, matrix_rank(np.zeros((4, 4))), 0
        # 1 dimension - rank 1 unless all 0
        yield assert_equal, matrix_rank([1, 0, 0, 0]), 1
        yield assert_equal, matrix_rank(np.zeros((4,))), 0
        # accepts array-like
        yield assert_equal, matrix_rank([1]), 1
        # greater than 2 dimensions raises error
        yield assert_raises, TypeError, matrix_rank, np.zeros((2, 2, 2))
        # works on scalar
        yield assert_equal, matrix_rank(1), 1

def test_byteorder_check():
    # Byte order check should pass for native order
    if sys.byteorder == 'little':
        native = '<'
    else:
        native = '>'

    for dtt in (np.float32, np.float64):
        arr = np.eye(4, dtype=dtt)
        n_arr = arr.newbyteorder(native)
        sw_arr = arr.newbyteorder('S').byteswap()
        assert_equal(arr.dtype.byteorder, '=')
        for routine in (linalg.inv, linalg.det, linalg.pinv):
            # Normal call
            res = routine(arr)
            # Native but not '='
            assert_array_equal(res, routine(n_arr))
            # Swapped
            assert_array_equal(res, routine(sw_arr))

def test_100(self):
        x, w = leg.leggauss(100)

        # test orthogonality. Note that the results need to be normalized,
        # otherwise the huge values that can arise from fast growing
        # functions like Laguerre can be very confusing.
        v = leg.legvander(x, 99)
        vv = np.dot(v.T * w, v)
        vd = 1/np.sqrt(vv.diagonal())
        vv = vd[:, None] * vv * vd
        assert_almost_equal(vv, np.eye(100))

        # check that the integral of 1 is correct
        tgt = 2.0
        assert_almost_equal(w.sum(), tgt)

def _fwlinear(self, args, output_size, scope=None):
    if args is None or (nest.is_sequence(args) and not args):
      raise ValueError("`args` must be specified")
    if not nest.is_sequence(args):
      args = [args]
    assert len(args) == 2
    assert args[0].get_shape().as_list()[1] == output_size

    dtype = [a.dtype for a in args][0]

    with vs.variable_scope(scope or "Linear"):
      matrixW = vs.get_variable(
        "MatrixW", dtype=dtype, initializer=tf.convert_to_tensor(np.eye(output_size, dtype=np.float32) * .05))

      matrixC = vs.get_variable(
        "MatrixC", [args[1].get_shape().as_list()[1], output_size], dtype=dtype)

      res = tf.matmul(args[0], matrixW) + tf.matmul(args[1], matrixC)
      return res

def getValuesFromPose(self, P):
        '''return the virtual values of the pots corresponding to the pose P'''
        vals = []
        grads = []
        for i, r, l, placement, attach_p in zip(range(3), self.rs, self.ls, self.placements, self.attach_ps):
            #first pot axis
            a = placement.rot * col([1, 0, 0])
            #second pot axis
            b = placement.rot * col([0, 1, 0])
            #string axis
            c = placement.rot * col([0, 0, 1])

            #attach point on the joystick
            p_joystick = P * attach_p
            v = p_joystick - placement.trans
            va = v - dot(v, a)*a
            vb = v - dot(v, b)*b
            #angles of the pots
            alpha = math.atan2(dot(vb, a), dot(vb, c))
            beta = math.atan2(dot(va, b), dot(va, c))
            vals.append(alpha)
            vals.append(beta)

            #calculation of the derivatives
            dv = np.bmat([-P.rot.mat() * quat.skew(attach_p), P.rot.mat()])
            dva = (np.eye(3) - a*a.T) * dv
            dvb = (np.eye(3) - b*b.T) * dv
            dalpha = (1/dot(vb,vb)) * (dot(vb,c) * a.T - dot(vb,a) * c.T) * dvb
            dbeta = (1/dot(va,va)) * (dot(va,c) * b.T - dot(va,b) * c.T) * dva
            grads.append(dalpha)
            grads.append(dbeta)
        return (col(vals), np.bmat([[grads]]))

def prepare_cholesky(N=100, dtype=np.double):
    N = int(N*2)
    A = np.asarray(np.random.rand(N, N), dtype=dtype)
    return ( A*A.transpose() + N*np.eye(N), )
    #return toc/trials, N*N*N/3.0*1e-9, times

#inv:    return toc/trials, 2*N*N*N*1e-9, times


##################################################################################

def update_filter(self, timestep, estimate, ranges):
        """Update position filter.

        Args:
             timestep (float): Time elapsed since last update.
             estimate (StateEstimate): Position estimate to update.
             ranges (list of floats): Range measurements.

        Returns:
            new_estimate (StateEstimate): Updated position estimate.
            outlier_flag (bool): Flag indicating whether the measurement was rejected as an outlier.
        """
        num_of_units = len(ranges)
        x = estimate.state
        P = estimate.covariance
        # Compute process matrix and covariance matrices
        F, Q, R = self._compute_process_and_covariance_matrices(timestep)
        # rospy.logdebug('F: {}'.format(F))
        # rospy.logdebug('Q: {}'.format(Q))
        # rospy.logdebug('R: {}'.format(R))
        # Prediction
        x = np.dot(F, x)
        P = np.dot(F, np.dot(P, F.T)) + Q
        # Update
        n = np.copy(x)
        H = np.zeros((num_of_units, x.size))
        z = np.zeros((num_of_units, 1))
        h = np.zeros((num_of_units, 1))
        for i in xrange(self.ikf_iterations):
            n, K, outlier_flag = self._ikf_iteration(x, n, ranges, h, H, z, estimate, R)
        if outlier_flag:
            new_estimate = estimate
        else:
            new_state = n
            new_covariance = np.dot((np.eye(6) - np.dot(K, H)), P)
            new_estimate = UWBTracker.StateEstimate(new_state, new_covariance)
        return new_estimate, outlier_flag

def get_precision(self):
        """Compute data precision matrix with the generative model.
        Equals the inverse of the covariance but computed with
        the matrix inversion lemma for efficiency.
        Returns
        -------
        precision : array, shape=(n_features, n_features)
            Estimated precision of data.
        """
        n_features = self.components_.shape[1]

        # handle corner cases first
        if self.n_components_ == 0:
            return np.eye(n_features) / self.noise_variance_
        if self.n_components_ == n_features:
            return linalg.inv(self.get_covariance())

        # Get precision using matrix inversion lemma
        components_ = self.components_
        exp_var = self.explained_variance_
        exp_var_diff = np.maximum(exp_var - self.noise_variance_, 0.)
        precision = np.dot(components_, components_.T) / self.noise_variance_
        precision.flat[::len(precision) + 1] += 1. / exp_var_diff
        precision = np.dot(components_.T,
                           np.dot(linalg.inv(precision), components_))
        precision /= -(self.noise_variance_ ** 2)
        precision.flat[::len(precision) + 1] += 1. / self.noise_variance_
        return precision

def getTrainKernel(self, params):
        self.checkParams(params)
        return np.eye(self.n)

def AorthogonalityCheck(A, U, d):
    """
    Test the frobenious norm of  D^{-1}(U^TAU) - I_k
    """
    V = np.zeros(U.shape)
    AV = np.zeros(U.shape)
    Av = Vector()
    v = Vector()
    A.init_vector(Av,0)
    A.init_vector(v,1)

    nvec  = U.shape[1]
    for i in range(0,nvec):
        v.set_local(U[:,i])
        v *= 1./math.sqrt(d[i])
        A.mult(v,Av)
        AV[:,i] = Av.get_local()
        V[:,i] = v.get_local()

    VtAV = np.dot(V.T, AV)    
    err = VtAV - np.eye(nvec, dtype=VtAV.dtype)

#    plt.imshow(np.abs(err))
#    plt.colorbar()
#    plt.show()

    print("i, ||Vt(i,:)AV(:,i) - I_i||_F, V[:,i] = 1/sqrt(lambda_i) U[:,i]")
    for i in range(1,nvec+1):
        print(i, np.linalg.norm(err[0:i,0:i], 'fro') )

def chol_inv(B, lower=True):
    """
    Returns the inverse of matrix A, where A = B*B.T,
    ie B is the Cholesky decomposition of A.

    Solves Ax = I
    given B is the cholesky factorization of A.
    """
    return cho_solve((B, lower), np.eye(B.shape[0]))

def test_theta_0():
    rng.seed(0)
    n_samples = 100
    Y = rng.randn(n_samples, 5)
    X = rng.randn(n_samples, 5)

    sgcrf = SparseGaussianCRF(lamL=0.01, lamT=0.01)
    sgcrf.fit(X, Y)

    assert np.allclose(sgcrf.Lam, np.eye(5), .1, .2)

def init_layers(X, Z, dims, final_mean_function):
    M = Z.shape[0]
    q_mus, q_sqrts, mean_functions, Zs = [], [], [], []
    X_running, Z_running = X.copy(), Z.copy()

    for dim_in, dim_out in zip(dims[:-2], dims[1:-1]):
        if dim_in == dim_out: # identity for same dims
            W = np.eye(dim_in)
        elif dim_in > dim_out: # use PCA mf for stepping down
            _, _, V = np.linalg.svd(X_running, full_matrices=False)
            W = V[:dim_out, :].T
        elif dim_in < dim_out: # identity + pad with zeros for stepping up
            I = np.eye(dim_in)
            zeros = np.zeros((dim_in, dim_out - dim_in))
            W = np.concatenate([I, zeros], 1)

        mean_functions.append(Linear(A=W))
        Zs.append(Z_running.copy())
        q_mus.append(np.zeros((M, dim_out)))
        q_sqrts.append(np.eye(M)[:, :, None] * np.ones((1, 1, dim_out)))

        Z_running = Z_running.dot(W)
        X_running = X_running.dot(W)

    # final layer (as before but no mean function)
    mean_functions.append(final_mean_function)
    Zs.append(Z_running.copy())
    q_mus.append(np.zeros((M, dims[-1])))
    q_sqrts.append(np.eye(M)[:, :, None] * np.ones((1, 1, dims[-1])))

    return q_mus, q_sqrts, Zs, mean_functions

def eye_mask(self, shape):
        """
        Build a mask using np.eye.
        """
        return ~np.eye(*shape, dtype=bool)

def gl_update_joint_matrices(self,node,parent_joint=None,parent_joint_matrix=numpy.eye(3,4,dtype=numpy.float32)):
        for child in node.children:
            if child.node_type == j3d.inf1.NodeType.JOINT:
                joint = self.gl_joints[child.index]
                joint_matrix = self.gl_joint_matrices[child.index]
                joint_matrix[:] = joint.create_matrix(parent_joint,parent_joint_matrix)
                self.gl_update_joint_matrices(child,joint,joint_matrix)
            else:
                self.gl_update_joint_matrices(child,parent_joint,parent_joint_matrix)