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Solve the following problem using Python, implementing the functions described below, one line at a time
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Description:
 def sample(self, initial_pos, num_samples, trajectory_length, stepsize=None, return_type='dataframe'):
 """ Method to return samples using Hamiltonian Monte Carlo Parameters initial_pos: A 1d array like object Vector representing values of parameter position, the starting state in markov chain. num_samples: int Number of samples to be generated trajectory_length: int or float Target trajectory length, stepsize * number of steps(L), where L is the number of steps taken per HMC iteration, and stepsize is step size for splitting time method. stepsize: float , defaults to None The stepsize for proposing new values of position and momentum in simulate_dynamics If None, then will be choosen suitably return_type: string (dataframe | recarray) Return type for samples, either of 'dataframe' or 'recarray'. Defaults to 'dataframe' Returns ------- sampled: A pandas.DataFrame or a numpy.recarray object depending upon return_type argument Examples -------- x y 0 1.000000e+00 1.000000e+00 1 1.592133e+00 1.152911e+00 2 1.608700e+00 1.315349e+00 3 1.608700e+00 1.315349e+00 4 6.843856e-01 6.237043e-01 array([[ 1.00795398, 0.71384233, 0.79802097], [ 0.71384233, 1.00633524, 0.21313767], [ 0.79802097, 0.21313767, 0.98519017]]) """


        self.accepted_proposals = 1.0
        initial_pos = _check_1d_array_object(initial_pos, 'initial_pos')
        _check_length_equal(initial_pos, self.model.variables, 'initial_pos', 'model.variables')

        if stepsize is None:
            stepsize = self._find_reasonable_stepsize(initial_pos)

        types = [(var_name, 'float') for var_name in self.model.variables]
        samples = np.zeros(num_samples, dtype=types).view(np.recarray)

        # Assigning after converting into tuple because value was being changed after assignment
        # Reason for this is unknown
        samples[0] = tuple(initial_pos)
        position_m = initial_pos

        lsteps = int(max(1, round(trajectory_length / stepsize, 0)))
        for i in range(1, num_samples):

            # Genrating sample
            position_m, _ = self._sample(position_m, trajectory_length, stepsize, lsteps)
            samples[i] = tuple(position_m)

        self.acceptance_rate = self.accepted_proposals / num_samples

        return _return_samples(return_type, samples)