How to use the lifetimes.utils.expected_cumulative_transactions function in Lifetimes

freq_multiplier = 1

    transactions_summary = utils.summary_data_from_transaction_data(
        cdnow_transactions, customer_id_col, datetime_col,
        datetime_format=datetime_format, freq=freq,
        freq_multiplier=freq_multiplier,
        observation_period_end=observation_period_end)

    transactions_summary = transactions_summary.reset_index()

    model = BetaGeoFitter()
    model.fit(transactions_summary['frequency'],
              transactions_summary['recency'],
              transactions_summary['T'])

    df_cum = utils.expected_cumulative_transactions(
        model, cdnow_transactions, datetime_col, customer_id_col, t,
        datetime_format, freq, set_index_date=True,
        freq_multiplier=freq_multiplier)

    dates = ['1997-01-11', '1997-01-12', '1997-01-13', '1997-01-14']
    actual_trans = [11, 12, 15, 19]
    expected_trans = [10.67, 12.67, 14.87, 17.24]

    date_index = df_cum.iloc[-4:].index.to_timestamp().astype(str)
    actual = df_cum['actual'].iloc[-4:].values
    predicted = df_cum['predicted'].iloc[-4:].values.round(2)

    assert all(dates == date_index)
    assert_allclose(actual, actual_trans)
    assert_allclose(predicted, expected_trans, atol=1e-2)

observation_period_end = '19970930'
    freq_multiplier = 7

    transactions_summary = utils.summary_data_from_transaction_data(
        cdnow_transactions, customer_id_col, datetime_col,
        datetime_format=datetime_format, freq=freq, freq_multiplier=freq_multiplier,
        observation_period_end=observation_period_end)

    transactions_summary = transactions_summary.reset_index()

    model = ParetoNBDFitter()
    model.fit(transactions_summary['frequency'],
              transactions_summary['recency'],
              transactions_summary['T'])

    df_cum = utils.expected_cumulative_transactions(
        model, cdnow_transactions, datetime_col, customer_id_col, t,
        datetime_format, freq, set_index_date=False, freq_multiplier=freq_multiplier)
    return df_cum

def test_expected_cumulative_transactions_dedups_inside_a_time_period(fitted_bg, example_transaction_data):
    by_week = utils.expected_cumulative_transactions(fitted_bg, example_transaction_data, 'date', 'id', 10, freq='W')
    by_day = utils.expected_cumulative_transactions(fitted_bg, example_transaction_data, 'date', 'id', 10, freq='D')
    assert (by_week['actual'] >= by_day['actual']).all()

def test_expected_cumulative_transactions_dedups_inside_a_time_period(fitted_bg, example_transaction_data):
    by_week = utils.expected_cumulative_transactions(fitted_bg, example_transaction_data, 'date', 'id', 10, freq='W')
    by_day = utils.expected_cumulative_transactions(fitted_bg, example_transaction_data, 'date', 'id', 10, freq='D')
    assert (by_week['actual'] >= by_day['actual']).all()

ax: matplotlib.AxesSubplot, optional
        Using user axes
    kwargs
        Passed into the pandas.DataFrame.plot command.

    Returns
    -------
    axes: matplotlib.AxesSubplot

    """
    from matplotlib import pyplot as plt

    if ax is None:
        ax = plt.subplot(111)

    df_cum_transactions = expected_cumulative_transactions(
        model,
        transactions,
        datetime_col,
        customer_id_col,
        t,
        datetime_format=datetime_format,
        freq=freq,
        set_index_date=set_index_date,
    )

    # get incremental from cumulative transactions
    df_cum_transactions = df_cum_transactions.apply(lambda x: x - x.shift(1))
    ax = df_cum_transactions.plot(ax=ax, title=title, **kwargs)

    if set_index_date:
        x_vline = df_cum_transactions.index[int(t_cal)]

ax: matplotlib.AxesSubplot, optional
        Using user axes
    kwargs
        Passed into the pandas.DataFrame.plot command.

    Returns
    -------
    axes: matplotlib.AxesSubplot

    """
    from matplotlib import pyplot as plt

    if ax is None:
        ax = plt.subplot(111)

    df_cum_transactions = expected_cumulative_transactions(
        model,
        transactions,
        datetime_col,
        customer_id_col,
        t,
        datetime_format=datetime_format,
        freq=freq,
        set_index_date=set_index_date,
    )

    ax = df_cum_transactions.plot(ax=ax, title=title, **kwargs)

    if set_index_date:
        x_vline = df_cum_transactions.index[int(t_cal)]
        xlabel = "date"
    else: