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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 _customer_lifetime_value( transaction_prediction_model, frequency, recency, T, monetary_value, time=12, discount_rate=0.01, freq="D" ):
 """ Compute the average lifetime value for a group of one or more customers. This method computes the average lifetime value for a group of one or more customers. Parameters transaction_prediction_model: the model to predict future transactions frequency: array_like the frequency vector of customers' purchases (denoted x in literature). recency: array_like the recency vector of customers' purchases (denoted t_x in literature). T: array_like the vector of customers' age (time since first purchase) monetary_value: array_like the monetary value vector of customer's purchases (denoted m in literature). time: int, optional the lifetime expected for the user in months. Default: 12 discount_rate: float, optional the monthly adjusted discount rate. Default: 1 Returns ------- :obj: Series series with customer ids as index and the estimated customer lifetime values as values """

    df = pd.DataFrame(index=frequency.index)
    df["clv"] = 0  # initialize the clv column to zeros

    steps = np.arange(1, time + 1)
    factor = {"W": 4.345, "M": 1.0, "D": 30, "H": 30 * 24}[freq]

    for i in steps * factor:
        # since the prediction of number of transactions is cumulative, we have to subtract off the previous periods
        expected_number_of_transactions = transaction_prediction_model.predict(
            i, frequency, recency, T
        ) - transaction_prediction_model.predict(i - factor, frequency, recency, T)
        # sum up the CLV estimates of all of the periods
        df["clv"] += (monetary_value * expected_number_of_transactions) / (1 + discount_rate) ** (i / factor)

    return df["clv"]