Adapt Evaluation Plots from `lifetimes`

[ColtAllen]

lifetimes contains a variety of plotting functions for model evaluation:

• [ ] plot_period_transactions
• [ ] plot_calibration_purchases_vs_holdout_purchases
• [x] plot_frequency_recency_matrix
• [x] plot_probability_alive_matrix
• [ ] plot_expected_repeat_purchases
• [ ] plot_history_alive
• [ ] plot_cumulative_transactions
• [ ] plot_incremental_transactions
• [ ] plot_transaction_rate_heterogeneity
• [ ] plot_dropout_rate_heterogeneity

This notebook provides some examples of their use.

@larryshamalama already added the matrix plots, but the others would need to be modified for posterior confidence intervals. Some of them also require utility functions that I'll create PRs for soon. I don't consider plot_history_alive to be all that useful, but if there's interest it's something to consider.

It's also worth reviewing the research papers for additional ideas of plots to add.

[zwelitunyiswa]

I have also found tracking plots incredibly useful for explaining to stakeholders. CLVTools has some very nice implementations:

https://www.clvtools.com/articles/CLVTools.html (scroll down)

On Thu, Jul 13, 2023 at 3:24 PM Colt Allen @.***> wrote:

lifetimes contains a variety of plotting functions for model evaluation:

• plot_period_transactions
• plot_calibration_purchases_vs_holdout_purchases
• plot_frequency_recency_matrix
• plot_probability_alive_matrix
• plot_expected_repeat_purchases
• plot_history_alive
• plot_cumulative_transactions
• plot_incremental_transactions
• plot_transaction_rate_heterogeneity
• plot_dropout_rate_heterogeneity

This notebook https://github.com/ColtAllen/marketing-case-study/blob/main/case-study.ipynb provides some examples of their use.

@larryshamalama https://github.com/larryshamalama already added the matrix plots, but the others would need to be modified for posterior confidence intervals. Some of them also require utility functions that I'll create PRs for soon. I don't consider plot_history_alive to be all that useful, but if there's interest it's something to consider.

It's also worth reviewing the research papers https://github.com/pymc-labs/pymc-marketing/issues/91 for additional ideas of plots to add.

— Reply to this email directly, view it on GitHub https://github.com/pymc-labs/pymc-marketing/issues/326, or unsubscribe https://github.com/notifications/unsubscribe-auth/AH3QQV7R2Z43XJLC3ZMCPZDXQBDQVANCNFSM6AAAAAA2JMSLJA . You are receiving this because you are subscribed to this thread.Message ID: @.***>

[ColtAllen]

I have also found tracking plots incredibly useful for explaining to stakeholders. CLVTools has some very nice implementations:

I think plot_incremental_transactions is the equivalent in lifetimes. Here's an example:

[ColtAllen]

plot_period_transactions is essentially plotting a posterior predictive check for customer purchase frequency:

Unless I'm mistaken, this means we can't use a Potential to define the likelihood. GammaGammaModel is fine because this plotting method doesn't apply, but BetaGeoModel would need its own distribution block per https://github.com/pymc-labs/pymc-marketing/issues/128.

[zwelitunyiswa]

I have also found tracking plots incredibly useful for explaining to stakeholders. CLVTools has some very nice implementations:

I think plot_incremental_transactions is the equivalent in lifetimes. Here's an example:

@ColtAllen I did not realize that these were added/in there. That is very cool. Thanks for pointing them out.

[ColtAllen]

Some WIP ideas for adapting plot_period_transactions in this issue: https://github.com/pymc-labs/pymc-marketing/issues/278

[wd60622]

I have a PR to add ax argument to the current CLV plotting functions. Might be a good standard here in order to give additional flexibility to the user

[ColtAllen]

I also want to add a parameter to return the formatted data instead of the plot, in case anyone wants to build the plots in something other than matplotlib (this has come up in my day job).

In the case of the plot_cumulative_transactions and plot_period_transactions functions, returning the data also enables use of the Wasserstein Distance function from scipy for concept drift monitoring, which will be helpful if the model is registered to mlflow.

[wd60622]

Love that idea. Separate the data transformation from plotting. Total sense to me

[ColtAllen]

The ArviZ library also has a lot of useful plotting options to compliment whatever we add in pymc-marketing, and examples would be a great addition to the tutorial notebooks. I've played around with a few of them in the dev notebook for ParetoNBDModel:

https://github.com/pymc-labs/pymc-marketing/blob/main/docs/source/notebooks/clv/dev/pareto_nbd.ipynb

[jcfisher]

Hi all, I was trying to create the histogram comparing observed and expected purchase counts for the BG/NBD model for myself when I came across this issue. This would be the histogram that a plot_period_transactions function would create for BetaGeoModel. I put together a function that computes the values based on the Fader, Hardie, and Lee (2007) note, and thought I would share it here in case it's useful.

Based on that note, they calculated that histogram by estimating $Pr(X(t) = x | r, \alpha, a, b)$ for $x = 1, 2, \ldots, 7+$ using each value of $t$ in the original data. Then they summed over the values of $t$ to get the expected number of repeat purchases (i.e., the expected number of people with $x = 1, 2, \ldots, 7+$) for the original cohort ($E(f_x)$ in their notation).

They calculated this in Excel. I essentially transcribed their Excel functions into Python, making the following function:

import numpy as np
from scipy.special import betaln, gammaln

def calculate_pmf(
    t_val: np.array, 
    max_x: int,
    # defaults are values from lifetimes file
    r: float = 0.242594150135163,
    alpha: float = 4.41359212062015,
    a: float = 0.792919156875463,
    b: float = 2.4258950494563
) -> np.array:

    # First calculate values that don't vary with x or t
    log_beta_ab = betaln(a, b)
    log_gamma_r = gammaln(r)

    # Create an output matrix
    out = np.empty((len(t_val), max_x + 1))

    # Next loop through all the values of t and calculate the pmf for each one
    for idx, t in enumerate(t_val):
        log_alpha_r = r * (np.log(alpha) - np.log(alpha + t))

        # Create a range of values for x.  This will allow us to use
        # vectorized functions to loop
        x = np.arange(max_x)

        # For each value of the output range, calculate the part that depends on x
        log_term1 = betaln(a, b + x) - log_beta_ab \
            + gammaln(r + x) - log_gamma_r - gammaln(x + 1) \
            + log_alpha_r \
            + (x * (np.log(t) - np.log(alpha + t)))

        log_term2 = np.where(
            x > 0,
            betaln(a + 1, b + x - 1) - log_beta_ab,
            0
        )

        # To calculate the sum, we're going to calculate each term and apply cumsum
        # Two notes:
        # 1. You need to exponentiate before summing
        # 2. For a given value of x, we need the (x - 1)th term of the sum
        #    We'll take care of that on the fly when creating the output

        individual_sum_terms = np.cumsum(np.exp(
                gammaln(r + x) - gammaln(x + 1) \
                + (x * (np.log(t) - np.log(alpha + t)))
                ))

        # Log and add the terms that you factored out back in
        log_term3 = log_alpha_r - log_gamma_r + np.log(individual_sum_terms)

        # Get the rest of term 3
        term3 = 1 - np.exp(log_term3)

        # Now create the output
        out1 = np.exp(log_term1) + np.where(
            x > 0, 
            np.exp(log_term2) * term3[x - 1],
            0
            ) 
        
        # Append the remainder as the final term and add it to the output matrix
        out[idx, :] = np.append(out1, 1 - np.sum(out1))

    return out

This version takes just one value for each of the estimated parameters. So, either you could take the MAP values and plug them in, or calculate it once for each draw from the posterior, which would give you a posterior distribution. To do the latter, this function would need to be modified to allow the r, alpha, a, and b, arguments to be vectors.

Hopefully this is helpful. If you want me to create pull request to add this in somewhere, let me know, and I'll try to figure out how to do that.

[ricardoV94]

This version takes just one value for each of the estimated parameters. So, either you could take the MAP values and plug them in, or calculate it once for each draw from the posterior, which would give you a posterior distribution. To do the latter, this function would need to be modified to allow the r, alpha, a, and b, arguments to be vectors.

If you use xarray you shouldn't need much looping or worry about chains/draws. It will take of vectorizing all operations automatically. Something like that is done here: https://github.com/pymc-labs/pymc-marketing/blob/4c9f4ca163f6e34c4b555b51f14f89e8851a0a9b/pymc_marketing/clv/models/beta_geo.py#L223

By the way how does your function differ from expected_num_purchases or expected_num_purchases_new_customer? And can't you use the logp function defined inside build_model (we could extract it out)?

Hopefully this is helpful. If you want me to create pull request to add this in somewhere, let me know, and I'll try to figure out how to do that.

That would be much appreciated! Let's just first agree on the details before you jump in

CC @ColtAllen and @larryshamalama

[jcfisher]

Thank you for taking a look! I'll see if I can rewrite this to use xarray. (I am new to pymc and xarray, so that is a helpful pointer.)

Here is my understanding of how this function differs from expected_num_purchases and expected_num_purchases_new_customer. In short, those functions return the expected number of purchases for a customer, and this function returns the probability that a given customer has 1, 2, 3, ... purchases.

• For a given value of $t$, this function returns $Pr(X(t) = x | r, \alpha, a, b)$ for $x = 1, 2, \ldots,$ max_x. That is, it returns a vector of probabilities for each value of $t$ and $x$. This is equation (8) from Fader et al. (2005).
• expected_num_purchases_new_customer returns the expected number of purchases, which is $E(X(t) | r, \alpha, a, b) = \sum_{x} x * Pr(X(t) = x | r, \alpha, a, b)$. That is, this function averages over the values of $x$ for each value of $t$ to get the expected purchase count for each time interval $t$. This is equation (9) from Fader el. al (2005).
• expected num_purchases returns the expected number of additional purchases, conditional on the data (a given number of previous purchases) and parameters. So expected_num_purchases_new_customer covers the interval $(0, T]$, and expected_num_purchases covers the interval $(T, T + t]$. This is equation (10) from Fader et al. (2005)

The histogram from the paper averages equation (8) over all values of $t$ observed in the original data, to get expected counts of customers who have made 1, 2, 3, ... purchases. Letting $i$ index customers and $t_i$ represent a customer's age, the histogram calculates $E(f_x) = \sum_i Pr(X(t_i) = x | r, \alpha, a, b)$, which is a function of $x$. (In the note, they calculate $Pr((X(t) = x)$ for the unique values of $t$ and then multiply by the number of customers who have that value of $t$.)

Regarding using logp, good question. I think logp is equation (6) in the Fader et al. (2005) paper. I think the difference between equation (6) and equation (8) is that equation (6) is the joint probability of purchase count $x$, recency $t_x$, and customer age $T$ given the parameters, $Pr(X = x, t_x, T | r, \alpha, a, b) = \mathcal{L}(r, \alpha, a, b | X = x, t_x, T)$, while equation (8) is just the probability of a purchase count $x$ given the parameters as a function of the length of the time interval, $Pr(X(t) = x | r, \alpha, a, b)$. (That is, conditional on the parameters, equation (8) doesn't use recency or frequency.) I could definitely be wrong about that though.

[ColtAllen]

Hey @jcfisher,

There is already a PR open for the method you are proposing:

https://github.com/pymc-labs/pymc-marketing/pull/216

It was abandoned by its original author quite some time ago though, so it may be prudent to open another one.

As to using this method to generate plot_period_transactions, I am curious how it would compare to a backend posterior predictive check, but the latter approach is my preference as it can be used across all transaction models.

[jcfisher]

@ColtAllen Thanks! Happy to modify that one or create a new one, whichever is easier. I revised the function above to use xarray, so now it should work with a vector of draws for each parameter.

I tried to find a sample_posterior_predictive function for the BetaGeoModel for comparison, but I'm having trouble getting the one from pymc to work. Is there are standard way of getting draws from the posterior predictive distribution? (Apologies if the answer to this question is obvious.) Without seeing it, I'd assume they would be similar, and agree that if they are, just using a posterior predictive check would be preferable.

[wd60622]

I tried to find a sample_posterior_predictive function for the BetaGeoModel for comparison, but I'm having trouble getting the one from pymc to work. Is there are standard way of getting draws from the posterior predictive distribution? (Apologies if the answer to this question is obvious.) Without seeing it, I'd assume they would be similar, and agree that if they are, just using a posterior predictive check would be preferable.

for the BetaGeoModel, the parameters can be sampled like from within this method. You should just be able to pass self.fit_results or model.fit_results into pm.sample_posterior_predictive. If you need access to the pm.Model instance, it'd be in the model attribute of the BetaGeoModel instance

[jcfisher]

Got it, thanks. I was having trouble because I wasn't using a with model: block (same as this issue).

Just to double check that I'm understanding this correctly, to generate draws from the posterior predictive distribution, we'd need a random number generator function for the custom likelihood, right? I'm thinking something like this:

# Random number generator function for BG-NBD model
# Returns the original data sampled from the data generating process, given the parameters
# Original data are:
# 1. Frequency (number of repeat purchases, i.e., number of purchases - 1)
# 2. Recency (time of last purchase relative to now)
# 3. Age (time of first purchase relative to now)
# As inputs, we use:
# * T: "now" time
# * r, alpha, a, b: draw from the posterior parameter values

def beta_geo_rng(T, r, alpha, a, b):

    # Draw from the gamma and beta distributions to get a value of lambda and p
    base_rng = np.random.default_rng()
    lam_draw = base_rng.gamma(shape = r, scale = 1 / alpha)
    p_draw = base_rng.beta(a, b)

    # Time of first purchase
    t = first_purchase_time = np.random.exponential(scale = lam_draw)
    x = 0  # purchases are 0 indexed

    while t < T:
        # Drop out with probability p
        if base_rng.binomial(n = 1, p = p_draw) == 1:
            return (x, T - t, T - first_purchase_time)
        else:
            # If not dropping out, add a new purchase after exponential waiting time
            t += np.random.exponential(scale = lam_draw)
            x += 1
    
    # Return tuple of frequency, recency, age
    return (x, T - t, T - first_purchase_time)

If this doesn't look right, please let me know. I'll work on putting together a comparison of this with the $Pr(X(t) = x | \ldots)$ function above.

Also, hope I'm not hijacking the conversation on this issue. If I should take this discussion somewhere else (or just post a gist with these functions and let you all take it from there), let me know.

[ricardoV94]

@jcfisher I think we already have something like that in here:

https://github.com/pymc-labs/pymc-marketing/blob/32098cc1ba3adc0043bb15aa1f5fd2ced6b1de14/pymc_marketing/clv/distributions.py#L151-L186

That was rewritten recently, it used to look more like your implementation: https://github.com/pymc-labs/pymc-marketing/blob/05f97cfb8de15d014e0833ba85d11243dffbb110/pymc_marketing/clv/distributions.py#L158-L201

[ColtAllen]

@wd60622 this is the utility function which will need to be adapted from lifetimes to plot cumulative transactions over time:

https://github.com/CamDavidsonPilon/lifetimes/blob/master/lifetimes/utils.py#L506