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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_adapt, num_samples, stepsize=None, return_type='dataframe'):
 """ Returns samples using No U Turn Sampler with dual averaging Parameters initial_pos: A 1d array like object Vector representing values of parameter position, the starting state in markov chain. num_adapt: int The number of interations to run the adaptation of stepsize num_samples: int Number of samples to be generated 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 12.000000 -4.000000 1 11.864821 -3.696109 2 10.546986 -4.892169 3 8.526596 -21.555793 4 8.526596 -21.555793 5 11.343194 -6.353789 6 -1.583269 -12.802931 7 12.411957 -11.704859 8 13.253336 -20.169492 9 11.295901 -7.665058 """

        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)

        if num_adapt <= 1:
            return NoUTurnSampler(self.model, self.grad_log_pdf,
                                  self.simulate_dynamics).sample(initial_pos, num_samples, stepsize)

        mu = np.log(10.0 * stepsize)
        stepsize_bar = 1.0
        h_bar = 0.0

        types = [(var_name, 'float') for var_name in self.model.variables]
        samples = np.zeros(num_samples, dtype=types).view(np.recarray)
        samples[0] = tuple(initial_pos)
        position_m = initial_pos

        for i in range(1, num_samples):

            position_m, alpha, n_alpha = self._sample(position_m, stepsize)
            samples[i] = tuple(position_m)

            if i <= num_adapt:
                stepsize, stepsize_bar, h_bar = self._adapt_params(stepsize, stepsize_bar, h_bar, mu,
                                                                   i, alpha, n_alpha)
            else:
                stepsize = stepsize_bar

        return _return_samples(return_type, samples)