Package 'BTYDplus' reference manual

Title:	Probabilistic Models for Assessing and Predicting your Customer Base
Description:	Provides advanced statistical methods to describe and predict customers' purchase behavior in a non-contractual setting. It uses historic transaction records to fit a probabilistic model, which then allows to compute quantities of managerial interest on a cohort- as well as on a customer level (Customer Lifetime Value, Customer Equity, P(alive), etc.). This package complements the BTYD package by providing several additional buy-till-you-die models, that have been published in the marketing literature, but whose implementation are complex and non-trivial. These models are: NBD [Ehrenberg (1959) <doi:10.2307/2985810>], MBG/NBD [Batislam et al (2007) <doi:10.1016/j.ijresmar.2006.12.005>], (M)BG/CNBD-k [Reutterer et al (2020) <doi:10.1016/j.ijresmar.2020.09.002>], Pareto/NBD (HB) [Abe (2009) <doi:10.1287/mksc.1090.0502>] and Pareto/GGG [Platzer and Reutterer (2016) <doi:10.1287/mksc.2015.0963>].
Authors:	Michael Platzer [aut, cre]
Maintainer:	Michael Platzer <[email protected]>
License:	GPL-3
Version:	1.2.0
Built:	2024-11-04 04:50:44 UTC
Source:	https://github.com/mplatzer/btydplus

Simulate data according to Pareto/NBD (Abe) model assumptions

Description

Simulate data according to Pareto/NBD (Abe) model assumptions

Usage

abe.GenerateData(
  n,
  T.cal,
  T.star,
  params,
  date.zero = "2000-01-01",
  covariates = NULL
)
abe.GenerateData(
  n,
  T.cal,
  T.star,
  params,
  date.zero = "2000-01-01",
  covariates = NULL
)

Arguments

`n`	Number of customers.
`T.cal`	Length of calibration period. If a vector is provided, then it is assumed that customers have different 'birth' dates, i.e. $max(T.cal)-T.cal$ .
`T.star`	Length of holdout period. This may be a vector.
`params`	A list of model parameters: `beta` and `gamma`.
`date.zero`	Initial date for cohort start. Can be of class character, Date or POSIXt.
`covariates`	Provide matrix of customer covariates. If NULL then random covariate values between [-1,1] are drawn.

Value

List of length 2:

`cbs`	A data.frame with a row for each customer and the summary statistic as columns.
`elog`	A data.frame with a row for each transaction, and columns `cust`, `date` and `t`.

Examples

# generate artificial Pareto/NBD (Abe) with 2 covariates
params <- list()
params$beta  <- matrix(c(0.18, -2.5, 0.5, -0.3, -0.2, 0.8), byrow = TRUE, ncol = 2)
params$gamma <- matrix(c(0.05, 0.1, 0.1, 0.2), ncol = 2)
data <- abe.GenerateData(n = 200, T.cal = 32, T.star = 32, params)
cbs <- data$cbs  # customer by sufficient summary statistic - one row per customer
elog <- data$elog  # Event log - one row per event/purchase
# generate artificial Pareto/NBD (Abe) with 2 covariates
params <- list()
params$beta  <- matrix(c(0.18, -2.5, 0.5, -0.3, -0.2, 0.8), byrow = TRUE, ncol = 2)
params$gamma <- matrix(c(0.05, 0.1, 0.1, 0.2), ncol = 2)
data <- abe.GenerateData(n = 200, T.cal = 32, T.star = 32, params)
cbs <- data$cbs  # customer by sufficient summary statistic - one row per customer
elog <- data$elog  # Event log - one row per event/purchase

Pareto/NBD (Abe) Parameter Draws

Description

Returns draws from the posterior distributions of the Pareto/NBD (Abe) parameters, on cohort as well as on customer level.

Usage

abe.mcmc.DrawParameters(
  cal.cbs,
  covariates = c(),
  mcmc = 2500,
  burnin = 500,
  thin = 50,
  chains = 2,
  mc.cores = NULL,
  trace = 100
)
abe.mcmc.DrawParameters(
  cal.cbs,
  covariates = c(),
  mcmc = 2500,
  burnin = 500,
  thin = 50,
  chains = 2,
  mc.cores = NULL,
  trace = 100
)

Arguments

`cal.cbs`	Calibration period customer-by-sufficient-statistic (CBS) data.frame. It must contain a row for each customer, and columns `x` for frequency, `t.x` for recency and `T.cal` for the total time observed. A correct format can be easily generated based on the complete event log of a customer cohort with `elog2cbs`.
`covariates`	A vector of columns of `cal.cbs` which contain customer-level covariates.
`mcmc`	Number of MCMC steps.
`burnin`	Number of initial MCMC steps which are discarded.
`thin`	Only every `thin`-th MCMC step will be returned.
`chains`	Number of MCMC chains to be run.
`mc.cores`	Number of cores to use in parallel (Unix only). Defaults to `min(chains, detectCores())`.
`trace`	Print logging statement every `trace`-th iteration. Not available for `mc.cores > 1`.

Details

See demo('pareto-abe') for how to apply this model.

Value

List of length 2:

`level_1`	list of `mcmc.list`s, one for each customer, with draws for customer-level parameters `k`, `lambda`, `tau`, `z`, `mu`
`level_2`	`mcmc.list`, with draws for cohort-level parameters

References

Abe, M. (2009). "Counting your customers" one by one: A hierarchical Bayes extension to the Pareto/NBD model. Marketing Science, 28(3), 541-553. doi:10.1287/mksc.1090.0502

Examples

data("groceryElog")
cbs <- elog2cbs(groceryElog, T.cal = "2006-12-31")
cbs$cov1 <- as.integer(cbs$cust) %% 2 # create dummy covariate
param.draws <- abe.mcmc.DrawParameters(cbs, c("cov1"),
  mcmc = 100, burnin = 50, thin = 10, chains = 1) # short MCMC to run demo fast

# cohort-level parameter draws
as.matrix(param.draws$level_2)
# customer-level parameter draws for customer with ID '4'
as.matrix(param.draws$level_1[["4"]])

# estimate future transactions
xstar.draws <- mcmc.DrawFutureTransactions(cbs, param.draws, cbs$T.star)
xstar.est <- apply(xstar.draws, 2, mean)
head(xstar.est)
data("groceryElog")
cbs <- elog2cbs(groceryElog, T.cal = "2006-12-31")
cbs$cov1 <- as.integer(cbs$cust) %% 2 # create dummy covariate
param.draws <- abe.mcmc.DrawParameters(cbs, c("cov1"),
  mcmc = 100, burnin = 50, thin = 10, chains = 1) # short MCMC to run demo fast

# cohort-level parameter draws
as.matrix(param.draws$level_2)
# customer-level parameter draws for customer with ID '4'
as.matrix(param.draws$level_1[["4"]])

# estimate future transactions
xstar.draws <- mcmc.DrawFutureTransactions(cbs, param.draws, cbs$T.star)
xstar.est <- apply(xstar.draws, 2, mean)
head(xstar.est)

Convert Event Log to customer-level summary statistic

Description

Efficient implementation for the conversion of an event log into a customer-by-sufficient-statistic (CBS) data.frame, with a row for each customer, which is the required data format for estimating model parameters.

Usage

elog2cbs(elog, units = "week", T.cal = NULL, T.tot = NULL)
elog2cbs(elog, units = "week", T.cal = NULL, T.tot = NULL)

Arguments

`elog`	Event log, a `data.frame` with field `cust` for the customer ID and field `date` for the date/time of the event, which should be of type `Date` or `POSIXt`. If a field `sales` is present, it will be aggregated as well.
`units`	Time unit, either `week`, `day`, `hour`, `min` or `sec`. See `difftime`.
`T.cal`	End date of calibration period. Defaults to `max(elog$date)`.
`T.tot`	End date of the observation period. Defaults to `max(elog$date)`.

Details

The time unit for expressing t.x, T.cal and litt are determined via the argument units, which is passed forward to method difftime, and defaults to weeks.

Argument T.tot allows one to specify the end of the observation period, i.e. the last possible date of an event to still be included in the event log. If T.tot is not provided, then the date of the last recorded event will be assumed to coincide with the end of the observation period. If T.tot is provided, then any event that occurs after that date is discarded.

Argument T.cal allows one to split the summary statistics into a calibration and a holdout period. This can be useful for evaluating forecasting accuracy for a given dataset. If T.cal is not provided, then the whole observation period is considered, and is then subsequently used for for estimating model parameters. If it is provided, then the returned data.frame contains two additional fields, with x.star representing the number of repeat transactions during the holdout period of length T.star. And only those customers are contained, who have had at least one event during the calibration period.

Transactions with identical cust and date field are treated as a single transaction, with sales being summed up.

Value

data.frame with fields:

`cust`	Customer id (unique key).
`x`	Number of recurring events in calibration period.
`t.x`	Time between first and last event in calibration period.
`litt`	Sum of logarithmic intertransaction timings during calibration period.
`sales`	Sum of sales in calibration period, incl. initial transaction. Only if `elog$sales` is provided.
`sales.x`	Sum of sales in calibration period, excl. initial transaction. Only if `elog$sales` is provided.
`first`	Date of first transaction in calibration period.
`T.cal`	Time between first event and end of calibration period.
`T.star`	Length of holdout period. Only if `T.cal` is provided.
`x.star`	Number of events within holdout period. Only if `T.cal` is provided.
`sales.star`	Sum of sales within holdout period. Only if `T.cal` and `elog$sales` are provided.

Examples

data("groceryElog")
cbs <- elog2cbs(groceryElog, T.cal = "2006-12-31", T.tot = "2007-12-30")
head(cbs)
data("groceryElog")
cbs <- elog2cbs(groceryElog, T.cal = "2006-12-31", T.tot = "2007-12-30")
head(cbs)

Convert Event Log to Transaction Counts

Description

Aggregates an event log to either incremental or cumulative number of transactions. If first=TRUE then the initial transactions of each customer are included in the count as well.

Usage

elog2cum(elog, by = 7, first = FALSE)

elog2inc(elog, by = 7, first = FALSE)
elog2cum(elog, by = 7, first = FALSE)

elog2inc(elog, by = 7, first = FALSE)

Arguments

`elog`	Event log, a `data.frame` with columns `cust` and transaction time `t` or `date`.
`by`	Only return every `by`-th count Defaults to 7, and thus returns weekly numbers.
`first`	If TRUE, then the first transaction for each customer is being counted as well

Details

Duplicate transactions with identical cust and date (or t) field are counted only once.

Value

Numeric vector of transaction counts.

Examples

data("groceryElog")
cum <- elog2cum(groceryElog)
plot(cum, typ="l", frame = FALSE)
inc <- elog2inc(groceryElog)
plot(inc, typ="l", frame = FALSE)
data("groceryElog")
cum <- elog2cum(groceryElog)
plot(cum, typ="l", frame = FALSE)
inc <- elog2inc(groceryElog)
plot(inc, typ="l", frame = FALSE)

Estimate Regularity in Intertransaction Timings

Description

The models (M)BG/CNBD-k and Pareto/GGG are capable of leveraging regularity within transaction timings for improving forecast accuracy. This method provides a quick check for the degree of regularity in the event timings. A return value of close to 1 supports the assumption of exponentially distributed intertransaction times, whereas values significantly larger than 1 reveal the presence of regularity.

Usage

estimateRegularity(
  elog,
  method = "wheat",
  plot = FALSE,
  title = "",
  min = NULL
)
estimateRegularity(
  elog,
  method = "wheat",
  plot = FALSE,
  title = "",
  min = NULL
)

Arguments

`elog`	Event log, a `data.frame` with columns `cust` and transaction time `t` or `date`
`method`	Either `wheat`, `mle`, `mle-minka`, `mle-thom` or `cv`.
`plot`	If `TRUE` then an additional diagnostic plot is provided.
`title`	Plot title.
`min`	Minimum number of intertransaction times per customer. Customers with less than `min` intertransactions are not considered. Defaults to 2 for method 'wheat', and to 10 otherwise.

Details

Estimation is either done by 1) assuming the same degree of regularity across all customers (Wheat & Morrison (1990) via method = "wheat"), or 2) by estimating regularity for each customer separately, as the shape parameter of a fitted gamma distribution, and then return the median across estimates. The latter methods, though, require sufficient (>=min) transactions per customer.

Wheat & Morrison (1990)'s method calculates for each customer a statistic M based on her last two number of intertransaction times as ITT_1 / (ITT_1 + ITT_2). That measure is known to follow a Beta(k, k) distribution, and k can be estimated as (1-4*Var(M))/(8*Var(M)). The corresponding diagnostic plot (plot = TRUE) shows the actual distribution of M vs. the theoretical distribution for k = 1 and k = 2.

Value

Estimated real-valued regularity parameter.

References

Wheat, Rita D., and Donald G. Morrison. "Estimating purchase regularity with two interpurchase times." Journal of Marketing Research (1990): 87-93.

Dunn, Richard, Steven Reader, and Neil Wrigley. 'An investigation of the assumptions of the NBD model' Applied Statistics (1983): 249-259.

Wu, Couchen, and H-L. Chen. 'A consumer purchasing model with learning and departure behaviour.' Journal of the Operational Research Society (2000): 583-591.

https://tminka.github.io/papers/minka-gamma.pdf

Examples

data("groceryElog")
estimateRegularity(groceryElog, plot = TRUE, method = 'wheat')
estimateRegularity(groceryElog, plot = TRUE, method = 'mle-minka')
estimateRegularity(groceryElog, plot = TRUE, method = 'mle-thom')
estimateRegularity(groceryElog, plot = TRUE, method = 'cv')
data("groceryElog")
estimateRegularity(groceryElog, plot = TRUE, method = 'wheat')
estimateRegularity(groceryElog, plot = TRUE, method = 'mle-minka')
estimateRegularity(groceryElog, plot = TRUE, method = 'mle-thom')
estimateRegularity(groceryElog, plot = TRUE, method = 'cv')

Event log for customers of an online grocery store.

Description

These data came from an online retailer offering a broad range of grocery categories. The original data set spans four years, but lacked the customers' acquisition date. Therefore, we constructed a quasi cohort by limiting the provided data analysis to those customers who haven't purchased at all in the first two years, and had their first purchase in the first quarter of 2006. This resulted in 10483 transactions being recorded for 1525 customers during a period of two years (2006-2007).

Usage

groceryElog
groceryElog

Format

A data frame with 10483 rows and 2 variables:

cust: customer ID, factor vector
date: transaction date, Date vector

Source

Thomas Reutterer <[email protected]>

References

Platzer, M., & Reutterer, T. (2016). Ticking away the moments: Timing regularity helps to better predict customer activity. Marketing Science, 35(5), 779-799. doi:10.1287/mksc.2015.0963

(M)BG/CNBD-k Log-Likelihood

Description

Calculates the log-likelihood of the (M)BG/CNBD-k model.

Usage

mbgcnbd.cbs.LL(params, cal.cbs)

mbgcnbd.LL(params, x, t.x, T.cal, litt)

bgcnbd.cbs.LL(params, cal.cbs)

bgcnbd.LL(params, x, t.x, T.cal, litt)
mbgcnbd.cbs.LL(params, cal.cbs)

mbgcnbd.LL(params, x, t.x, T.cal, litt)

bgcnbd.cbs.LL(params, cal.cbs)

bgcnbd.LL(params, x, t.x, T.cal, litt)

Arguments

`params`	A vector with model parameters `k`, `r`, `alpha`, `a` and `b`, in that order.
`cal.cbs`	Calibration period customer-by-sufficient-statistic (CBS) data.frame. It must contain a row for each customer, and columns `x` for frequency, `t.x` for recency , `T.cal` for the total time observed, as well as the sum over logarithmic intertransaction times `litt`. A correct format can be easily generated based on the complete event log of a customer cohort with `elog2cbs`.
`x`	frequency, i.e. number of re-purchases
`t.x`	recency, i.e. time elapsed from first purchase to last purchase
`T.cal`	total time of observation period
`litt`	sum of logarithmic interpurchase times

Value

For bgcnbd.cbs.LL, the total log-likelihood of the provided data. For bgcnbd.LL, a vector of log-likelihoods as long as the longest input vector (x, t.x, or T.cal).

References

(M)BG/CNBD-k: Reutterer, T., Platzer, M., & Schroeder, N. (2020). Leveraging purchase regularity for predicting customer behavior the easy way. International Journal of Research in Marketing. doi:10.1016/j.ijresmar.2020.09.002

(M)BG/CNBD-k Conditional Expected Transactions

Description

Uses (M)BG/CNBD-k model parameters and a customer's past transaction behavior to return the number of transactions they are expected to make in a given time period.

Usage

mbgcnbd.ConditionalExpectedTransactions(params, T.star, x, t.x, T.cal)

bgcnbd.ConditionalExpectedTransactions(params, T.star, x, t.x, T.cal)
mbgcnbd.ConditionalExpectedTransactions(params, T.star, x, t.x, T.cal)

bgcnbd.ConditionalExpectedTransactions(params, T.star, x, t.x, T.cal)

Arguments

`params`	A vector with model parameters `k`, `r`, `alpha`, `a` and `b`, in that order.
`T.star`	Length of time for which we are calculating the expected number of transactions.
`x`	Number of repeat transactions in the calibration period T.cal, or a vector of calibration period frequencies.
`t.x`	Recency, i.e. length between first and last transaction during calibration period.
`T.cal`	Length of calibration period, or a vector of calibration period lengths.

Value

Number of transactions a customer is expected to make in a time period of length t, conditional on their past behavior. If any of the input parameters has a length greater than 1, this will be a vector of expected number of transactions.

References

Examples

## Not run: 
data("groceryElog")
cbs <- elog2cbs(groceryElog)
params <- mbgcnbd.EstimateParameters(cbs, k = 2)
# estimate transactions for next 12 weeks
xstar.est <- mbgcnbd.ConditionalExpectedTransactions(params,
  T.star = 12, cbs$x, cbs$t.x, cbs$T.cal)
head(xstar.est) # expected number of transactions for first 6 customers
sum(xstar.est) # expected total number of transactions during holdout

## End(Not run)
## Not run: 
data("groceryElog")
cbs <- elog2cbs(groceryElog)
params <- mbgcnbd.EstimateParameters(cbs, k = 2)
# estimate transactions for next 12 weeks
xstar.est <- mbgcnbd.ConditionalExpectedTransactions(params,
  T.star = 12, cbs$x, cbs$t.x, cbs$T.cal)
head(xstar.est) # expected number of transactions for first 6 customers
sum(xstar.est) # expected total number of transactions during holdout

## End(Not run)

(M)BG/CNBD-k Parameter Estimation

Description

Estimates parameters for the (M)BG/CNBD-k model via Maximum Likelihood Estimation.

Usage

mbgcnbd.EstimateParameters(
  cal.cbs,
  k = NULL,
  par.start = c(1, 3, 1, 3),
  max.param.value = 10000,
  trace = 0
)

bgcnbd.EstimateParameters(
  cal.cbs,
  k = NULL,
  par.start = c(1, 3, 1, 3),
  max.param.value = 10000,
  trace = 0
)

mbgnbd.EstimateParameters(
  cal.cbs,
  par.start = c(1, 3, 1, 3),
  max.param.value = 10000,
  trace = 0
)
mbgcnbd.EstimateParameters(
  cal.cbs,
  k = NULL,
  par.start = c(1, 3, 1, 3),
  max.param.value = 10000,
  trace = 0
)

bgcnbd.EstimateParameters(
  cal.cbs,
  k = NULL,
  par.start = c(1, 3, 1, 3),
  max.param.value = 10000,
  trace = 0
)

mbgnbd.EstimateParameters(
  cal.cbs,
  par.start = c(1, 3, 1, 3),
  max.param.value = 10000,
  trace = 0
)

Arguments

`cal.cbs`	Calibration period customer-by-sufficient-statistic (CBS) data.frame. It must contain a row for each customer, and columns `x` for frequency, `t.x` for recency , `T.cal` for the total time observed, as well as the sum over logarithmic intertransaction times `litt`, in case that `k` is not provided. A correct format can be easily generated based on the complete event log of a customer cohort with `elog2cbs`.
`k`	Integer-valued degree of regularity for Erlang-k distributed interpurchase times. By default this `k` is not provided, and a grid search from 1 to 12 is performed in order to determine the best-fitting `k`. The grid search is stopped early, if the log-likelihood does not increase anymore when increasing k beyond 4.
`par.start`	Initial (M)BG/CNBD-k parameters. A vector with `r`, `alpha`, `a` and `b` in that order.
`max.param.value`	Upper bound on parameters.
`trace`	If larger than 0, then the parameter values are is printed every `trace`-step of the maximum likelihood estimation search.

Value

A vector of estimated parameters.

References

Batislam, E. P., Denizel, M., & Filiztekin, A. (2007). Empirical validation and comparison of models for customer base analysis. International Journal of Research in Marketing, 24(3), 201-209. doi:10.1016/j.ijresmar.2006.12.005

Examples

## Not run: 
data("groceryElog")
cbs <- elog2cbs(groceryElog)
(params <- mbgcnbd.EstimateParameters(cbs))

## End(Not run)
## Not run: 
data("groceryElog")
cbs <- elog2cbs(groceryElog)
(params <- mbgcnbd.EstimateParameters(cbs))

## End(Not run)

(M)BG/CNBD-k Expectation

Description

Returns the number of repeat transactions that a randomly chosen customer (for whom we have no prior information) is expected to make in a given time period, i.e. $E(X(t) | k, r, alpha, a, b)$ .

Usage

mbgcnbd.Expectation(params, t)

bgcnbd.Expectation(params, t)
mbgcnbd.Expectation(params, t)

bgcnbd.Expectation(params, t)

Arguments

`params`	A vector with model parameters `k`, `r`, `alpha`, `a` and `b`, in that order.
`t`	Length of time for which we are calculating the expected number of repeat transactions.

Details

Note: Computational time increases with the number of unique values of t.

Value

Number of repeat transactions a customer is expected to make in a time period of length t.

References

Examples

## Not run: 
data("groceryElog")
cbs <- elog2cbs(groceryElog)
params <- mbgcnbd.EstimateParameters(cbs)
mbgcnbd.Expectation(params, t = c(26, 52))

## End(Not run)
## Not run: 
data("groceryElog")
cbs <- elog2cbs(groceryElog)
params <- mbgcnbd.EstimateParameters(cbs)
mbgcnbd.Expectation(params, t = c(26, 52))

## End(Not run)

(M)BG/CNBD-k Expected Cumulative Transactions

Description

Calculates the expected cumulative total repeat transactions by all customers for the calibration and holdout periods.

Usage

mbgcnbd.ExpectedCumulativeTransactions(params, T.cal, T.tot, n.periods.final)

bgcnbd.ExpectedCumulativeTransactions(params, T.cal, T.tot, n.periods.final)
mbgcnbd.ExpectedCumulativeTransactions(params, T.cal, T.tot, n.periods.final)

bgcnbd.ExpectedCumulativeTransactions(params, T.cal, T.tot, n.periods.final)

Arguments

`params`	A vector with model parameters `k`, `r`, `alpha`, `a` and `b`, in that order.
`T.cal`	A vector to represent customers' calibration period lengths.
`T.tot`	End of holdout period. Must be a single value, not a vector.
`n.periods.final`	Number of time periods in the calibration and holdout periods.

Details

Note: Computational time increases with the number of unique values of T.cal.

Value

Vector of length n.periods.final with expected cumulative total repeat transactions by all customers.

Examples

## Not run: 
data("groceryElog")
cbs <- elog2cbs(groceryElog)
params <- mbgcnbd.EstimateParameters(cbs, k = 2)
# Returns a vector containing expected cumulative repeat transactions for 104
# weeks, with every eigth week being reported.
mbgcnbd.ExpectedCumulativeTransactions(params,
  T.cal = cbs$T.cal,
  T.tot = 104,
  n.periods.final = 104 / 8)

## End(Not run)
## Not run: 
data("groceryElog")
cbs <- elog2cbs(groceryElog)
params <- mbgcnbd.EstimateParameters(cbs, k = 2)
# Returns a vector containing expected cumulative repeat transactions for 104
# weeks, with every eigth week being reported.
mbgcnbd.ExpectedCumulativeTransactions(params,
  T.cal = cbs$T.cal,
  T.tot = 104,
  n.periods.final = 104 / 8)

## End(Not run)

Simulate data according to (M)BG/CNBD-k model assumptions

Description

Simulate data according to (M)BG/CNBD-k model assumptions

Usage

mbgcnbd.GenerateData(n, T.cal, T.star = NULL, params, date.zero = "2000-01-01")

bgcnbd.GenerateData(n, T.cal, T.star = NULL, params, date.zero = "2000-01-01")
mbgcnbd.GenerateData(n, T.cal, T.star = NULL, params, date.zero = "2000-01-01")

bgcnbd.GenerateData(n, T.cal, T.star = NULL, params, date.zero = "2000-01-01")

Arguments

`n`	Number of customers.
`T.cal`	Length of calibration period. If a vector is provided, then it is assumed that customers have different 'birth' dates, i.e. $max(T.cal)-T.cal$ .
`T.star`	Length of holdout period. This may be a vector.
`params`	A vector with model parameters `k`, `r`, `alpha`, `a` and `b`, in that order.
`date.zero`	Initial date for cohort start. Can be of class character, Date or POSIXt.

Value

List of length 2:

`cbs`	A data.frame with a row for each customer and the summary statistic as columns.
`elog`	A data.frame with a row for each transaction, and columns `cust`, `date` and `t`.

References

Examples

params <- c(k = 3, r = 0.85, alpha = 1.45, a = 0.79, b = 2.42)
data <- mbgcnbd.GenerateData(n = 200, T.cal = 24, T.star = 32, params)

# customer by sufficient summary statistic - one row per customer
head(data$cbs)

# event log - one row per event/transaction
head(data$elog)
params <- c(k = 3, r = 0.85, alpha = 1.45, a = 0.79, b = 2.42)
data <- mbgcnbd.GenerateData(n = 200, T.cal = 24, T.star = 32, params)

# customer by sufficient summary statistic - one row per customer
head(data$cbs)

# event log - one row per event/transaction
head(data$elog)

(M)BG/CNBD-k P(alive)

Description

Uses (M)BG/CNBD-k model parameters and a customer's past transaction behavior to return the probability that they are still alive at the end of the calibration period.

Usage

mbgcnbd.PAlive(params, x, t.x, T.cal)

bgcnbd.PAlive(params, x, t.x, T.cal)
mbgcnbd.PAlive(params, x, t.x, T.cal)

bgcnbd.PAlive(params, x, t.x, T.cal)

Arguments

`params`	A vector with model parameters `k`, `r`, `alpha`, `a` and `b`, in that order.
`x`	Number of repeat transactions in the calibration period T.cal, or a vector of calibration period frequencies.
`t.x`	Recency, i.e. length between first and last transaction during calibration period.
`T.cal`	Length of calibration period, or a vector of calibration period lengths.

Value

Probability that the customer is still alive at the end of the calibration period.

References

Examples

## Not run: 
data("groceryElog")
cbs <- elog2cbs(groceryElog)
params <- mbgcnbd.EstimateParameters(cbs)
palive <- mbgcnbd.PAlive(params, cbs$x, cbs$t.x, cbs$T.cal)
head(palive) # Probability of being alive for first 6 customers
mean(palive) # Estimated share of customers to be still alive

## End(Not run)
## Not run: 
data("groceryElog")
cbs <- elog2cbs(groceryElog)
params <- mbgcnbd.EstimateParameters(cbs)
palive <- mbgcnbd.PAlive(params, cbs$x, cbs$t.x, cbs$T.cal)
head(palive) # Probability of being alive for first 6 customers
mean(palive) # Estimated share of customers to be still alive

## End(Not run)

(M)BG/CNBD-k Plot Frequency in Calibration Period

Description

Plots a histogram and returns a matrix comparing the actual and expected number of customers who made a certain number of repeat transactions in the calibration period, binned according to calibration period frequencies.

Usage

mbgcnbd.PlotFrequencyInCalibration(
  params,
  cal.cbs,
  censor = 7,
  xlab = "Calibration period transactions",
  ylab = "Customers",
  title = "Frequency of Repeat Transactions"
)

bgcnbd.PlotFrequencyInCalibration(
  params,
  cal.cbs,
  censor = 7,
  xlab = "Calibration period transactions",
  ylab = "Customers",
  title = "Frequency of Repeat Transactions"
)
mbgcnbd.PlotFrequencyInCalibration(
  params,
  cal.cbs,
  censor = 7,
  xlab = "Calibration period transactions",
  ylab = "Customers",
  title = "Frequency of Repeat Transactions"
)

bgcnbd.PlotFrequencyInCalibration(
  params,
  cal.cbs,
  censor = 7,
  xlab = "Calibration period transactions",
  ylab = "Customers",
  title = "Frequency of Repeat Transactions"
)

Arguments

`params`	A vector with model parameters `k`, `r`, `alpha`, `a` and `b`, in that order.
`cal.cbs`	Calibration period CBS (customer by sufficient statistic). It must contain columns for frequency ('x') and total time observed ('T.cal').
`censor`	Cutoff point for number of transactions in plot.
`xlab`	Descriptive label for the x axis.
`ylab`	Descriptive label for the y axis.
`title`	Title placed on the top-center of the plot.

Value

Calibration period repeat transaction frequency comparison matrix (actual vs. expected).

References

Examples

## Not run: 
data("groceryElog")
cbs <- elog2cbs(groceryElog)
params <- mbgcnbd.EstimateParameters(cbs)
mbgcnbd.PlotFrequencyInCalibration(params, cbs)

## End(Not run)
## Not run: 
data("groceryElog")
cbs <- elog2cbs(groceryElog)
params <- mbgcnbd.EstimateParameters(cbs)
mbgcnbd.PlotFrequencyInCalibration(params, cbs)

## End(Not run)

(M)BG/CNBD-k Plot Frequency vs. Conditional Expected Frequency

Description

Plots the actual and conditional expected number transactions made by customers in the holdout period, binned according to calibration period frequencies, and returns this comparison in a matrix.

Usage

mbgcnbd.PlotFreqVsConditionalExpectedFrequency(
  params,
  T.star,
  cal.cbs,
  x.star,
  censor,
  xlab = "Calibration period transactions",
  ylab = "Holdout period transactions",
  xticklab = NULL,
  title = "Conditional Expectation"
)

bgcnbd.PlotFreqVsConditionalExpectedFrequency(
  params,
  T.star,
  cal.cbs,
  x.star,
  censor,
  xlab = "Calibration period transactions",
  ylab = "Holdout period transactions",
  xticklab = NULL,
  title = "Conditional Expectation"
)
mbgcnbd.PlotFreqVsConditionalExpectedFrequency(
  params,
  T.star,
  cal.cbs,
  x.star,
  censor,
  xlab = "Calibration period transactions",
  ylab = "Holdout period transactions",
  xticklab = NULL,
  title = "Conditional Expectation"
)

bgcnbd.PlotFreqVsConditionalExpectedFrequency(
  params,
  T.star,
  cal.cbs,
  x.star,
  censor,
  xlab = "Calibration period transactions",
  ylab = "Holdout period transactions",
  xticklab = NULL,
  title = "Conditional Expectation"
)

Arguments

`params`	A vector with model parameters `k`, `r`, `alpha`, `a` and `b`, in that order.
`T.star`	Length of the holdout period.
`cal.cbs`	Calibration period CBS (customer by sufficient statistic). It must contain columns for frequency ('x'), recency ('t.x') and total time observed ('T.cal').
`x.star`	Vector of transactions made by each customer in the holdout period.
`censor`	Cutoff point for number of transactions in plot.
`xlab`	Descriptive label for the x axis.
`ylab`	Descriptive label for the x axis.
`xticklab`	A vector containing a label for each tick mark on the x axis.
`title`	Title placed on the top-center of the plot.

Value

Holdout period transaction frequency comparison matrix (actual vs. expected).

Examples

## Not run: 
data("groceryElog")
cbs <- elog2cbs(groceryElog, T.cal = "2006-09-30")
params <- mbgcnbd.EstimateParameters(cbs, k=2)
mbgcnbd.PlotFreqVsConditionalExpectedFrequency(params, T.star=52, cbs, cbs$x.star, censor=7)

## End(Not run)
## Not run: 
data("groceryElog")
cbs <- elog2cbs(groceryElog, T.cal = "2006-09-30")
params <- mbgcnbd.EstimateParameters(cbs, k=2)
mbgcnbd.PlotFreqVsConditionalExpectedFrequency(params, T.star=52, cbs, cbs$x.star, censor=7)

## End(Not run)

(M)BG/CNBD-k Plot Actual vs. Conditional Expected Frequency by Recency

Description

Plots the actual and conditional expected number transactions made by customers in the holdout period, binned according to calibration period recencies, and returns this comparison in a matrix.

Usage

mbgcnbd.PlotRecVsConditionalExpectedFrequency(
  params,
  cal.cbs,
  T.star,
  x.star,
  xlab = "Calibration period recency",
  ylab = "Holdout period transactions",
  xticklab = NULL,
  title = "Actual vs. Conditional Expected Transactions by Recency"
)

bgcnbd.PlotRecVsConditionalExpectedFrequency(
  params,
  cal.cbs,
  T.star,
  x.star,
  xlab = "Calibration period recency",
  ylab = "Holdout period transactions",
  xticklab = NULL,
  title = "Actual vs. Conditional Expected Transactions by Recency"
)
mbgcnbd.PlotRecVsConditionalExpectedFrequency(
  params,
  cal.cbs,
  T.star,
  x.star,
  xlab = "Calibration period recency",
  ylab = "Holdout period transactions",
  xticklab = NULL,
  title = "Actual vs. Conditional Expected Transactions by Recency"
)

bgcnbd.PlotRecVsConditionalExpectedFrequency(
  params,
  cal.cbs,
  T.star,
  x.star,
  xlab = "Calibration period recency",
  ylab = "Holdout period transactions",
  xticklab = NULL,
  title = "Actual vs. Conditional Expected Transactions by Recency"
)

Arguments

`params`	A vector with model parameters `k`, `r`, `alpha`, `a` and `b`, in that order.
`cal.cbs`	Calibration period CBS (customer by sufficient statistic). It must contain columns for frequency ('x'), recency ('t.x') and total time observed ('T.cal').
`T.star`	Length of the holdout period.
`x.star`	Vector of transactions made by each customer in the holdout period.
`xlab`	Descriptive label for the x axis.
`ylab`	Descriptive label for the x axis.
`xticklab`	A vector containing a label for each tick mark on the x axis.
`title`	Title placed on the top-center of the plot.

Value

Matrix comparing actual and conditional expected transactions in the holdout period.

Examples

## Not run: 
data("groceryElog")
cbs <- elog2cbs(groceryElog, T.cal = "2006-09-30")
params <- mbgcnbd.EstimateParameters(cbs, k=2)
mbgcnbd.PlotRecVsConditionalExpectedFrequency(params, cbs, T.star=52, cbs$x.star)

## End(Not run)
## Not run: 
data("groceryElog")
cbs <- elog2cbs(groceryElog, T.cal = "2006-09-30")
params <- mbgcnbd.EstimateParameters(cbs, k=2)
mbgcnbd.PlotRecVsConditionalExpectedFrequency(params, cbs, T.star=52, cbs$x.star)

## End(Not run)

(M)BG/CNBD-k Tracking Cumulative Transactions Plot

Description

Plots the actual and expected cumulative total repeat transactions by all customers for the calibration and holdout periods, and returns this comparison in a matrix.

Usage

mbgcnbd.PlotTrackingCum(
  params,
  T.cal,
  T.tot,
  actual.cu.tracking.data,
  xlab = "Week",
  ylab = "Cumulative Transactions",
  xticklab = NULL,
  title = "Tracking Cumulative Transactions",
  ymax = NULL,
  legend = c("Actual", "Model")
)

bgcnbd.PlotTrackingCum(
  params,
  T.cal,
  T.tot,
  actual.cu.tracking.data,
  xlab = "Week",
  ylab = "Cumulative Transactions",
  xticklab = NULL,
  title = "Tracking Cumulative Transactions",
  ymax = NULL,
  legend = c("Actual", "Model")
)
mbgcnbd.PlotTrackingCum(
  params,
  T.cal,
  T.tot,
  actual.cu.tracking.data,
  xlab = "Week",
  ylab = "Cumulative Transactions",
  xticklab = NULL,
  title = "Tracking Cumulative Transactions",
  ymax = NULL,
  legend = c("Actual", "Model")
)

bgcnbd.PlotTrackingCum(
  params,
  T.cal,
  T.tot,
  actual.cu.tracking.data,
  xlab = "Week",
  ylab = "Cumulative Transactions",
  xticklab = NULL,
  title = "Tracking Cumulative Transactions",
  ymax = NULL,
  legend = c("Actual", "Model")
)

Arguments

`params`	A vector with model parameters `k`, `r`, `alpha`, `a` and `b`, in that order.
`T.cal`	A vector to represent customers' calibration period lengths.
`T.tot`	End of holdout period. Must be a single value, not a vector.
`actual.cu.tracking.data`	A vector containing the cumulative number of repeat transactions made by customers for each period in the total time period (both calibration and holdout periods).
`xlab`	Descriptive label for the x axis.
`ylab`	Descriptive label for the y axis.
`xticklab`	A vector containing a label for each tick mark on the x axis.
`title`	Title placed on the top-center of the plot.
`ymax`	Upper boundary for y axis.
`legend`	plot legend, defaults to 'Actual' and 'Model'.

Details

Note: Computational time increases with the number of unique values of T.cal.

Value

Matrix containing actual and expected cumulative repeat transactions.

Examples

## Not run: 
data("groceryElog")
groceryElog <- groceryElog[groceryElog$date < "2006-06-30", ]
cbs <- elog2cbs(groceryElog, T.cal = "2006-04-30")
cum <- elog2cum(groceryElog)
params <- mbgcnbd.EstimateParameters(cbs, k = 2)
mbgcnbd.PlotTrackingCum(params, cbs$T.cal,
  T.tot = max(cbs$T.cal + cbs$T.star), cum)

## End(Not run)
## Not run: 
data("groceryElog")
groceryElog <- groceryElog[groceryElog$date < "2006-06-30", ]
cbs <- elog2cbs(groceryElog, T.cal = "2006-04-30")
cum <- elog2cum(groceryElog)
params <- mbgcnbd.EstimateParameters(cbs, k = 2)
mbgcnbd.PlotTrackingCum(params, cbs$T.cal,
  T.tot = max(cbs$T.cal + cbs$T.star), cum)

## End(Not run)

(M)BG/CNBD-k Tracking Incremental Transactions Comparison

Description

Plots the actual and expected incremental total repeat transactions by all customers for the calibration and holdout periods, and returns this comparison in a matrix.

Usage

mbgcnbd.PlotTrackingInc(
  params,
  T.cal,
  T.tot,
  actual.inc.tracking.data,
  xlab = "Week",
  ylab = "Transactions",
  xticklab = NULL,
  title = "Tracking Weekly Transactions",
  ymax = NULL,
  legend = c("Actual", "Model")
)

bgcnbd.PlotTrackingInc(
  params,
  T.cal,
  T.tot,
  actual.inc.tracking.data,
  xlab = "Week",
  ylab = "Transactions",
  xticklab = NULL,
  title = "Tracking Weekly Transactions",
  ymax = NULL,
  legend = c("Actual", "Model")
)
mbgcnbd.PlotTrackingInc(
  params,
  T.cal,
  T.tot,
  actual.inc.tracking.data,
  xlab = "Week",
  ylab = "Transactions",
  xticklab = NULL,
  title = "Tracking Weekly Transactions",
  ymax = NULL,
  legend = c("Actual", "Model")
)

bgcnbd.PlotTrackingInc(
  params,
  T.cal,
  T.tot,
  actual.inc.tracking.data,
  xlab = "Week",
  ylab = "Transactions",
  xticklab = NULL,
  title = "Tracking Weekly Transactions",
  ymax = NULL,
  legend = c("Actual", "Model")
)

Arguments

`params`	A vector with model parameters `k`, `r`, `alpha`, `a` and `b`, in that order.
`T.cal`	A vector to represent customers' calibration period lengths.
`T.tot`	End of holdout period. Must be a single value, not a vector.
`actual.inc.tracking.data`	A vector containing the incremental number of repeat transactions made by customers for each period in the total time period (both calibration and holdout periods).
`xlab`	Descriptive label for the x axis.
`ylab`	Descriptive label for the y axis.
`xticklab`	A vector containing a label for each tick mark on the x axis.
`title`	Title placed on the top-center of the plot.
`ymax`	Upper boundary for y axis.
`legend`	plot legend, defaults to 'Actual' and 'Model'.

Details

Note: Computational time increases with the number of unique values of T.cal.

Value

Matrix containing actual and expected incremental repeat transactions.

Examples

## Not run: 
data("groceryElog")
groceryElog <- groceryElog[groceryElog$date < "2006-06-30", ]
cbs <- elog2cbs(groceryElog, T.cal = "2006-04-30")
inc <- elog2inc(groceryElog)
params <- mbgcnbd.EstimateParameters(cbs, k = 2)
mbgcnbd.PlotTrackingInc(params, cbs$T.cal,
  T.tot = max(cbs$T.cal + cbs$T.star), inc)

## End(Not run)
## Not run: 
data("groceryElog")
groceryElog <- groceryElog[groceryElog$date < "2006-06-30", ]
cbs <- elog2cbs(groceryElog, T.cal = "2006-04-30")
inc <- elog2inc(groceryElog)
params <- mbgcnbd.EstimateParameters(cbs, k = 2)
mbgcnbd.PlotTrackingInc(params, cbs$T.cal,
  T.tot = max(cbs$T.cal + cbs$T.star), inc)

## End(Not run)

(M)BG/CNBD-k Probability Mass Function

Description

Uses (M)BG/CNBD-k model parameters to return the probability distribution of purchase frequencies for a random customer in a given time period, i.e. $P(X(t)=x|r,alpha,a,b)$ .

Usage

mbgcnbd.pmf(params, t, x)

bgcnbd.pmf(params, t, x)
mbgcnbd.pmf(params, t, x)

bgcnbd.pmf(params, t, x)

Arguments

`params`	A vector with model parameters `k`, `r`, `alpha`, `a` and `b`, in that order.
`t`	Length end of time period for which probability is being computed. May also be a vector.
`x`	Number of repeat transactions for which probability is calculated. May also be a vector.

Value

$P(X(t)=x|r,alpha,a,b)$ . If either t or x is a vector, then the output will be a vector as well. If both are vectors, the output will be a matrix.

References

Examples

## Not run: 
data("groceryElog")
cbs <- elog2cbs(groceryElog)
params <- mbgcnbd.EstimateParameters(cbs)
mbgcnbd.pmf(params, t = 52, x = 0:6)
mbgcnbd.pmf(params, t = c(26, 52), x = 0:6)

## End(Not run)
## Not run: 
data("groceryElog")
cbs <- elog2cbs(groceryElog)
params <- mbgcnbd.EstimateParameters(cbs)
mbgcnbd.pmf(params, t = 52, x = 0:6)
mbgcnbd.pmf(params, t = c(26, 52), x = 0:6)

## End(Not run)

Draws number of future transactions based on MCMC parameter draws

Description

For each customer and each provided MCMC parameter draw this method will sample the number of transactions during the holdout period T.star. If argument size is provided then it returns a flexible number of draws, whereas for each customer and each draw it will first make a draw from the parameter draws.

Usage

mcmc.DrawFutureTransactions(
  cal.cbs,
  draws,
  T.star = cal.cbs$T.star,
  sample_size = NULL
)
mcmc.DrawFutureTransactions(
  cal.cbs,
  draws,
  T.star = cal.cbs$T.star,
  sample_size = NULL
)

Arguments

`cal.cbs`	Calibration period customer-by-sufficient-statistic (CBS) data.frame.
`draws`	MCMC draws as returned by `*.mcmc.DrawParameters`
`T.star`	Length of period for which future transactions are counted.
`sample_size`	Number of samples to draw. Defaults to the same number of parameter draws that are passed to `draws`.

Value

2-dim matrix [draw x customer] with sampled future transactions.

Examples

data("groceryElog")
cbs <- elog2cbs(groceryElog, T.cal = "2006-12-31")
param.draws <- pnbd.mcmc.DrawParameters(cbs,
  mcmc = 100, burnin = 50, thin = 10, chains = 1) # short MCMC to run demo fast
xstar.draws <- mcmc.DrawFutureTransactions(cbs, param.draws)
cbs$xstar.est <- apply(xstar.draws, 2, mean)
cbs$pactive <- mcmc.PActive(xstar.draws)
head(cbs)
data("groceryElog")
cbs <- elog2cbs(groceryElog, T.cal = "2006-12-31")
param.draws <- pnbd.mcmc.DrawParameters(cbs,
  mcmc = 100, burnin = 50, thin = 10, chains = 1) # short MCMC to run demo fast
xstar.draws <- mcmc.DrawFutureTransactions(cbs, param.draws)
cbs$xstar.est <- apply(xstar.draws, 2, mean)
cbs$pactive <- mcmc.PActive(xstar.draws)
head(cbs)

Unconditional Expectation for Pareto/GGG, Pareto/NBD (HB) and Pareto/NBD (Abe)

Description

Uses model parameter draws to return the expected number of repeat transactions that a randomly chosen customer (for whom we have no prior information) is expected to make in a given time period.

$E(X(t))$

Usage

mcmc.Expectation(draws, t, sample_size = 10000)
mcmc.Expectation(draws, t, sample_size = 10000)

Arguments

`draws`	MCMC draws as returned by `*.mcmc.DrawParameters`
`t`	Length of time for which we are calculating the expected number of transactions. May also be a vector.
`sample_size`	Sample size for estimating the probability distribution.

Details

The expected transactions need to be sampled. Due to this sampling, the return result varies from one call to another. Larger values of sample_size will generate more stable results.

Value

Number of repeat transactions a customer is expected to make in a time period of length t.

Examples

data("groceryElog")
cbs <- elog2cbs(groceryElog)
param.draws <- pnbd.mcmc.DrawParameters(cbs,
  mcmc = 100, burnin = 50, thin = 10, chains = 1) # short MCMC to run demo fast
mcmc.Expectation(param.draws, t = c(26, 52))
data("groceryElog")
cbs <- elog2cbs(groceryElog)
param.draws <- pnbd.mcmc.DrawParameters(cbs,
  mcmc = 100, burnin = 50, thin = 10, chains = 1) # short MCMC to run demo fast
mcmc.Expectation(param.draws, t = c(26, 52))

Expected Cumulative Transactions for Pareto/GGG, Pareto/NBD (HB) and Pareto/NBD (Abe)

Description

Uses model parameter draws to return the expected number of repeat transactions that a randomly chosen customer (for whom we have no prior information) is expected to make in a given time period.

Usage

mcmc.ExpectedCumulativeTransactions(
  draws,
  T.cal,
  T.tot,
  n.periods.final,
  sample_size = 10000,
  covariates = NULL
)
mcmc.ExpectedCumulativeTransactions(
  draws,
  T.cal,
  T.tot,
  n.periods.final,
  sample_size = 10000,
  covariates = NULL
)

Arguments

`draws`	MCMC draws as returned by `*.mcmc.DrawParameters`
`T.cal`	A vector to represent customers' calibration period lengths (in other words, the `T.cal` column from a customer-by-sufficient-statistic matrix). Considering rounding in order to speed up calculations.
`T.tot`	End of holdout period. Must be a single value, not a vector.
`n.periods.final`	Number of time periods in the calibration and holdout periods.
`sample_size`	Sample size for estimating the probability distribution.
`covariates`	(optional) Matrix of covariates, for Pareto/NBD (Abe) model, passed to `abe.GenerateData` for simulating data.

Details

The expected transactions need to be sampled. Due to this sampling, the return result varies from one call to another. Larger values of sample_size will generate more stable results.

Value

Numeric vector of expected cumulative total repeat transactions by all customers.

Examples

data("groceryElog")
cbs <- elog2cbs(groceryElog)
param.draws <- pnbd.mcmc.DrawParameters(cbs,
  mcmc = 100, burnin = 50, thin = 10, chains = 1) # short MCMC to run demo fast
# Returns a vector containing expected cumulative repeat transactions for 104
# weeks, with every eigth week being reported.
mcmc.ExpectedCumulativeTransactions(param.draws,
  T.cal = cbs$T.cal, T.tot = 104, n.periods.final = 104/8, sample_size = 1000)
data("groceryElog")
cbs <- elog2cbs(groceryElog)
param.draws <- pnbd.mcmc.DrawParameters(cbs,
  mcmc = 100, burnin = 50, thin = 10, chains = 1) # short MCMC to run demo fast
# Returns a vector containing expected cumulative repeat transactions for 104
# weeks, with every eigth week being reported.
mcmc.ExpectedCumulativeTransactions(param.draws,
  T.cal = cbs$T.cal, T.tot = 104, n.periods.final = 104/8, sample_size = 1000)

Calculates P(active) based on drawn future transactions.

Description

Calculates P(active) based on drawn future transactions.

Usage

mcmc.PActive(xstar)
mcmc.PActive(xstar)

Arguments

xstar

Future transaction draws as returned by mcmc.DrawFutureTransactions.

Value

numeric A vector with the customers' probabilities of being active during the holdout period.

Examples

data("groceryElog")
cbs <- elog2cbs(groceryElog, T.cal = "2006-12-31")
param.draws <- pnbd.mcmc.DrawParameters(cbs,
  mcmc = 100, burnin = 50, thin = 10, chains = 1) # short MCMC to run demo fast
xstar.draws <- mcmc.DrawFutureTransactions(cbs, param.draws)
cbs$pactive <- mcmc.PActive(xstar.draws)
head(cbs)
data("groceryElog")
cbs <- elog2cbs(groceryElog, T.cal = "2006-12-31")
param.draws <- pnbd.mcmc.DrawParameters(cbs,
  mcmc = 100, burnin = 50, thin = 10, chains = 1) # short MCMC to run demo fast
xstar.draws <- mcmc.DrawFutureTransactions(cbs, param.draws)
cbs$pactive <- mcmc.PActive(xstar.draws)
head(cbs)

Calculates P(alive) based on MCMC parameter draws

Description

Calculates P(alive) based on MCMC parameter draws

Usage

mcmc.PAlive(draws)
mcmc.PAlive(draws)

Arguments

draws

MCMC draws as returned by *.mcmc.DrawParameters

Value

Numeric vector with the customers' probabilities of being still alive at end of calibration period

Examples

data("groceryElog")
cbs <- elog2cbs(groceryElog)
param.draws <- pnbd.mcmc.DrawParameters(cbs,
  mcmc = 100, burnin = 50, thin = 10, chains = 1) # short MCMC to run demo fast
palive <- mcmc.PAlive(param.draws)
head(palive)
mean(palive)
data("groceryElog")
cbs <- elog2cbs(groceryElog)
param.draws <- pnbd.mcmc.DrawParameters(cbs,
  mcmc = 100, burnin = 50, thin = 10, chains = 1) # short MCMC to run demo fast
palive <- mcmc.PAlive(param.draws)
head(palive)
mean(palive)

Frequency in Calibration Period for Pareto/GGG, Pareto/NBD (HB) and Pareto/NBD (Abe)

Description

Usage

mcmc.PlotFrequencyInCalibration(
  draws,
  cal.cbs,
  censor = 7,
  xlab = "Calibration period transactions",
  ylab = "Customers",
  title = "Frequency of Repeat Transactions",
  sample_size = 1000
)
mcmc.PlotFrequencyInCalibration(
  draws,
  cal.cbs,
  censor = 7,
  xlab = "Calibration period transactions",
  ylab = "Customers",
  title = "Frequency of Repeat Transactions",
  sample_size = 1000
)

Arguments

`draws`	MCMC draws as returned by `*.mcmc.DrawParameters`
`cal.cbs`	Calibration period customer-by-sufficient-statistic (CBS) data.frame. It must contain columns for frequency ('x') and total time observed ('T.cal').
`censor`	Cutoff point for number of transactions in plot.
`xlab`	Descriptive label for the x axis.
`ylab`	Descriptive label for the y axis.
`title`	Title placed on the top-center of the plot.
`sample_size`	Sample size for estimating the probability distribution. See `mcmc.pmf`.

Details

The method mcmc.pmf is called to calculate the expected numbers based on the corresponding model.

Value

Calibration period repeat transaction frequency comparison matrix (actual vs. expected).

Examples

## Not run: 
data("groceryElog")
cbs <- elog2cbs(groceryElog, T.cal = "2006-12-31")
param.draws <- pnbd.mcmc.DrawParameters(cbs,
  mcmc = 100, burnin = 50, thin = 10, chains = 1) # short MCMC to run demo fast
mcmc.PlotFrequencyInCalibration(param.draws, cbs, sample_size = 100)

## End(Not run)
## Not run: 
data("groceryElog")
cbs <- elog2cbs(groceryElog, T.cal = "2006-12-31")
param.draws <- pnbd.mcmc.DrawParameters(cbs,
  mcmc = 100, burnin = 50, thin = 10, chains = 1) # short MCMC to run demo fast
mcmc.PlotFrequencyInCalibration(param.draws, cbs, sample_size = 100)

## End(Not run)

Draw diagnostic plot to inspect error in P(active).

Description

Draw diagnostic plot to inspect error in P(active).

Usage

mcmc.plotPActiveDiagnostic(cbs, xstar, title = "Diagnostic Plot for P(active)")
mcmc.plotPActiveDiagnostic(cbs, xstar, title = "Diagnostic Plot for P(active)")

Arguments

`cbs`	A data.frame with column `x` and `x.star`.
`xstar`	Future transaction draws as returned by `mcmc.DrawFutureTransactions`.
`title`	Plot title.

Examples

data("groceryElog")
cbs <- elog2cbs(groceryElog, T.cal = "2006-12-31")
param.draws <- pnbd.mcmc.DrawParameters(cbs,
  mcmc = 100, burnin = 50, thin = 10, chains = 1) # short MCMC to run demo fast
xstar.draws <- mcmc.DrawFutureTransactions(cbs, param.draws)
mcmc.plotPActiveDiagnostic(cbs, xstar.draws)
data("groceryElog")
cbs <- elog2cbs(groceryElog, T.cal = "2006-12-31")
param.draws <- pnbd.mcmc.DrawParameters(cbs,
  mcmc = 100, burnin = 50, thin = 10, chains = 1) # short MCMC to run demo fast
xstar.draws <- mcmc.DrawFutureTransactions(cbs, param.draws)
mcmc.plotPActiveDiagnostic(cbs, xstar.draws)

Tracking Cumulative Transactions Plot for Pareto/GGG, Pareto/NBD (HB) and Pareto/NBD (Abe)

Description

Plots the actual and expected cumulative total repeat transactions by all customers for the calibration and holdout periods, and returns this comparison in a matrix.

Usage

mcmc.PlotTrackingCum(
  draws,
  T.cal,
  T.tot,
  actual.cu.tracking.data,
  xlab = "Week",
  ylab = "Cumulative Transactions",
  xticklab = NULL,
  title = "Tracking Cumulative Transactions",
  ymax = NULL,
  sample_size = 10000,
  covariates = NULL,
  legend = c("Actual", "Model")
)
mcmc.PlotTrackingCum(
  draws,
  T.cal,
  T.tot,
  actual.cu.tracking.data,
  xlab = "Week",
  ylab = "Cumulative Transactions",
  xticklab = NULL,
  title = "Tracking Cumulative Transactions",
  ymax = NULL,
  sample_size = 10000,
  covariates = NULL,
  legend = c("Actual", "Model")
)

Arguments

`draws`	MCMC draws as returned by `*.mcmc.DrawParameters`
`T.cal`	A vector to represent customers' calibration period lengths (in other words, the `T.cal` column from a customer-by-sufficient-statistic matrix). Considering rounding in order to speed up calculations.
`T.tot`	End of holdout period. Must be a single value, not a vector.
`actual.cu.tracking.data`	A vector containing the cumulative number of repeat transactions made by customers for each period in the total time period (both calibration and holdout periods).
`xlab`	Descriptive label for the x axis.
`ylab`	Descriptive label for the y axis.
`xticklab`	A vector containing a label for each tick mark on the x axis.
`title`	Title placed on the top-center of the plot.
`ymax`	Upper boundary for y axis.
`sample_size`	Sample size for estimating the probability distribution. See `mcmc.ExpectedCumulativeTransactions`.
`covariates`	(optional) Matrix of covariates, for Pareto/NBD (Abe) model, passed to `abe.GenerateData` for simulating data.
`legend`	plot legend, defaults to 'Actual' and 'Model'.

Details

The expected transactions need to be sampled. Due to this sampling, the return result varies from one call to another. Larger values of sample_size will generate more stable results.

Value

Matrix containing actual and expected cumulative repeat transactions.

Examples

## Not run: 
data("groceryElog")
cbs <- elog2cbs(groceryElog, T.cal = "2006-12-31")
cum <- elog2cum(groceryElog)
param.draws <- pnbd.mcmc.DrawParameters(cbs)
mat <- mcmc.PlotTrackingCum(param.draws,
  T.cal = cbs$T.cal,
  T.tot = max(cbs$T.cal + cbs$T.star),
  actual.cu.tracking.data = cum)

## End(Not run)
## Not run: 
data("groceryElog")
cbs <- elog2cbs(groceryElog, T.cal = "2006-12-31")
cum <- elog2cum(groceryElog)
param.draws <- pnbd.mcmc.DrawParameters(cbs)
mat <- mcmc.PlotTrackingCum(param.draws,
  T.cal = cbs$T.cal,
  T.tot = max(cbs$T.cal + cbs$T.star),
  actual.cu.tracking.data = cum)

## End(Not run)

Tracking Incremental Transactions Plot for Pareto/GGG, Pareto/NBD (HB) and Pareto/NBD (Abe)

Description

Plots the actual and expected incremental total repeat transactions by all customers for the calibration and holdout periods, and returns this comparison in a matrix.

Usage

mcmc.PlotTrackingInc(
  draws,
  T.cal,
  T.tot,
  actual.inc.tracking.data,
  xlab = "Week",
  ylab = "Transactions",
  xticklab = NULL,
  title = "Tracking Weekly Transactions",
  ymax = NULL,
  sample_size = 10000,
  covariates = NULL,
  legend = c("Actual", "Model")
)
mcmc.PlotTrackingInc(
  draws,
  T.cal,
  T.tot,
  actual.inc.tracking.data,
  xlab = "Week",
  ylab = "Transactions",
  xticklab = NULL,
  title = "Tracking Weekly Transactions",
  ymax = NULL,
  sample_size = 10000,
  covariates = NULL,
  legend = c("Actual", "Model")
)

Arguments

`draws`	MCMC draws as returned by `*.mcmc.DrawParameters`
`T.cal`	A vector to represent customers' calibration period lengths (in other words, the `T.cal` column from a customer-by-sufficient-statistic matrix). Considering rounding in order to speed up calculations.
`T.tot`	End of holdout period. Must be a single value, not a vector.
`actual.inc.tracking.data`	A vector containing the incremental number of repeat transactions made by customers for each period in the total time period (both calibration and holdout periods).
`xlab`	Descriptive label for the x axis.
`ylab`	Descriptive label for the y axis.
`xticklab`	A vector containing a label for each tick mark on the x axis.
`title`	Title placed on the top-center of the plot.
`ymax`	Upper boundary for y axis.
`sample_size`	Sample size for estimating the probability distribution. See `mcmc.ExpectedCumulativeTransactions`.
`covariates`	(optional) Matrix of covariates, for Pareto/NBD (Abe) model, passed to `abe.GenerateData` for simulating data.
`legend`	plot legend, defaults to 'Actual' and 'Model'.

Details

The expected transactions need to be sampled. Due to this sampling, the return result varies from one call to another. Larger values of sample_size will generate more stable results.

Value

Matrix containing actual and expected incremental repeat transactions.

Examples

## Not run: 
data("groceryElog")
cbs <- elog2cbs(groceryElog, T.cal = "2006-12-31")
inc <- elog2inc(groceryElog)
param.draws <- pnbd.mcmc.DrawParameters(cbs)
mat <- mcmc.PlotTrackingInc(param.draws,
  T.cal = cbs$T.cal,
  T.tot = max(cbs$T.cal + cbs$T.star),
  actual.inc.tracking.data = inc)

## End(Not run)
## Not run: 
data("groceryElog")
cbs <- elog2cbs(groceryElog, T.cal = "2006-12-31")
inc <- elog2inc(groceryElog)
param.draws <- pnbd.mcmc.DrawParameters(cbs)
mat <- mcmc.PlotTrackingInc(param.draws,
  T.cal = cbs$T.cal,
  T.tot = max(cbs$T.cal + cbs$T.star),
  actual.inc.tracking.data = inc)

## End(Not run)

Probability Mass Function for Pareto/GGG, Pareto/NBD (HB) and Pareto/NBD (Abe)

Description

Return the probability distribution of purchase frequencies for a random customer in a given time period, i.e. $P(X(t)=x)$ . This is estimated by generating sample_size number of random customers that follow the provided parameter draws. Due to this sampling, the return result varies from one call to another.

Usage

mcmc.pmf(draws, t, x, sample_size = 10000, covariates = NULL)
mcmc.pmf(draws, t, x, sample_size = 10000, covariates = NULL)

Arguments

`draws`	MCMC draws as returned by `*.mcmc.DrawParameters`
`t`	Length of time for which we are calculating the expected number of transactions. May also be a vector.
`x`	Number of transactions for which probability is calculated. May also be a vector.
`sample_size`	Sample size for estimating the probability distribution.
`covariates`	(optional) Matrix of covariates, for Pareto/NBD (Abe) model, passed to `abe.GenerateData` for simulating data.

Value

$P(X(t)=x)$ . If either t or x is a vector, then the output will be a vector as well. If both are vectors, the output will be a matrix.

Examples

data("groceryElog")
cbs <- elog2cbs(groceryElog)
param.draws <- pnbd.mcmc.DrawParameters(cbs,
  mcmc = 100, burnin = 50, thin = 10, chains = 1) # short MCMC to run demo fast
mcmc.pmf(param.draws, t = c(26, 52), x = 0:6)
data("groceryElog")
cbs <- elog2cbs(groceryElog)
param.draws <- pnbd.mcmc.DrawParameters(cbs,
  mcmc = 100, burnin = 50, thin = 10, chains = 1) # short MCMC to run demo fast
mcmc.pmf(param.draws, t = c(26, 52), x = 0:6)

(Re-)set burnin of MCMC chains.

Description

(Re-)set burnin of MCMC chains.

Usage

mcmc.setBurnin(draws, burnin)
mcmc.setBurnin(draws, burnin)

Arguments

`draws`	MCMC draws as returned by `*.mcmc.DrawParameters`
`burnin`	New start index.

Value

2-element list with MCMC draws

Examples

data("groceryElog")
cbs <- elog2cbs(groceryElog)
param.draws <- pnbd.mcmc.DrawParameters(cbs,
  mcmc = 100, burnin = 50, thin = 10, chains = 1) # short MCMC to run demo fast
param.draws.stable <- mcmc.setBurnin(param.draws, burnin = 80)
data("groceryElog")
cbs <- elog2cbs(groceryElog)
param.draws <- pnbd.mcmc.DrawParameters(cbs,
  mcmc = 100, burnin = 50, thin = 10, chains = 1) # short MCMC to run demo fast
param.draws.stable <- mcmc.setBurnin(param.draws, burnin = 80)

Calculate the log-likelihood of the NBD model

Description

Calculate the log-likelihood of the NBD model

Usage

nbd.cbs.LL(params, cal.cbs)
nbd.cbs.LL(params, cal.cbs)

Arguments

`params`	NBD parameters - a vector with r and alpha, in that order.
`cal.cbs`	Calibration period CBS. It must contain columns for frequency `x` and total time observed `T.cal`.

Value

The total log-likelihood for the provided data.

Examples

data("groceryElog")
cbs <- elog2cbs(groceryElog)
params <- nbd.EstimateParameters(cbs)
nbd.cbs.LL(params, cbs)
data("groceryElog")
cbs <- elog2cbs(groceryElog)
params <- nbd.EstimateParameters(cbs)
nbd.cbs.LL(params, cbs)

NBD Conditional Expected Transactions

Description

Uses NBD model parameters and a customer's past transaction behavior to return the number of transactions they are expected to make in a given time period.

Usage

nbd.ConditionalExpectedTransactions(params, T.star, x, T.cal)
nbd.ConditionalExpectedTransactions(params, T.star, x, T.cal)

Arguments

`params`	NBD parameters - a vector with `r` and `alpha`, in that order.
`T.star`	Length of time for which we are calculating the expected number of transactions.
`x`	Number of repeat transactions in the calibration period `T.cal`, or a vector of calibration period frequencies.
`T.cal`	Length of calibration period, or a vector of calibration period lengths.

Value

Examples

data("groceryElog")
cbs <- elog2cbs(groceryElog, T.cal = "2006-12-31")
params <- nbd.EstimateParameters(cbs)
xstar.est <- nbd.ConditionalExpectedTransactions(params, cbs$T.star, cbs$x, cbs$T.cal)
sum(xstar.est) # expected total number of transactions during holdout
data("groceryElog")
cbs <- elog2cbs(groceryElog, T.cal = "2006-12-31")
params <- nbd.EstimateParameters(cbs)
xstar.est <- nbd.ConditionalExpectedTransactions(params, cbs$T.star, cbs$x, cbs$T.cal)
sum(xstar.est) # expected total number of transactions during holdout

Parameter Estimation for the NBD model

Description

Estimates parameters for the NBD model via Maximum Likelihood Estimation.

Usage

nbd.EstimateParameters(cal.cbs, par.start = c(1, 1), max.param.value = 10000)
nbd.EstimateParameters(cal.cbs, par.start = c(1, 1), max.param.value = 10000)

Arguments

`cal.cbs`	Calibration period CBS. It must contain columns for frequency `x` and total time observed `T.cal`.
`par.start`	Initial NBD parameters - a vector with `r` and `alpha` in that order.
`max.param.value`	Upper bound on parameters.

Value

List of estimated parameters.

References

Ehrenberg, A. S. (1959). The pattern of consumer purchases. Journal of the Royal Statistical Society: Series C (Applied Statistics), 8(1), 26-41. doi:10.2307/2985810

Examples

data("groceryElog")
cbs <- elog2cbs(groceryElog)
nbd.EstimateParameters(cbs)
data("groceryElog")
cbs <- elog2cbs(groceryElog)
nbd.EstimateParameters(cbs)

Simulate data according to NBD model assumptions

Description

Simulate data according to NBD model assumptions

Usage

nbd.GenerateData(n, T.cal, T.star, params, date.zero = "2000-01-01")
nbd.GenerateData(n, T.cal, T.star, params, date.zero = "2000-01-01")

Arguments

`n`	Number of customers.
`T.cal`	Length of calibration period.
`T.star`	Length of holdout period. This may be a vector.
`params`	NBD parameters - a vector with `r` and `alpha` in that order.
`date.zero`	Initial date for cohort start. Can be of class character, Date or POSIXt.

Value

List of length 2:

`cbs`	A data.frame with a row for each customer and the summary statistic as columns.
`elog`	A data.frame with a row for each transaction, and columns `cust`, `date` and `t`.

Examples

n <- 200  # no. of customers
T.cal <- 32  # length of calibration period
T.star <- 32  # length of hold-out period
params <- c(r = 0.85, alpha = 4.45)  # purchase frequency lambda_i ~ Gamma(r, alpha)
data <- nbd.GenerateData(n, T.cal, T.star, params)
cbs <- data$cbs  # customer by sufficient summary statistic - one row per customer
elog <- data$elog  # Event log - one row per event/purchase
n <- 200  # no. of customers
T.cal <- 32  # length of calibration period
T.star <- 32  # length of hold-out period
params <- c(r = 0.85, alpha = 4.45)  # purchase frequency lambda_i ~ Gamma(r, alpha)
data <- nbd.GenerateData(n, T.cal, T.star, params)
cbs <- data$cbs  # customer by sufficient summary statistic - one row per customer
elog <- data$elog  # Event log - one row per event/purchase

Calculate the log-likelihood of the NBD model

Description

Calculate the log-likelihood of the NBD model

Usage

nbd.LL(params, x, T.cal)
nbd.LL(params, x, T.cal)

Arguments

`params`	NBD parameters - a vector with `r` and `alpha`, in that order.
`x`	Frequency, i.e. number of re-purchases.
`T.cal`	Total time of observation period.

Value

A numeric vector of log-likelihoods.

Simulate data according to Pareto/GGG model assumptions

Description

Simulate data according to Pareto/GGG model assumptions

Usage

pggg.GenerateData(n, T.cal, T.star, params, date.zero = "2000-01-01")
pggg.GenerateData(n, T.cal, T.star, params, date.zero = "2000-01-01")

Arguments

`n`	Number of customers.
`T.cal`	Length of calibration period. If a vector is provided, then it is assumed that customers have different 'birth' dates, i.e. $max(T.cal)-T.cal$ .
`T.star`	Length of holdout period. This may be a vector.
`params`	A list of model parameters `r`, `alpha`, `s`, `beta`, `t` and `gamma`.
`date.zero`	Initial date for cohort start. Can be of class character, Date or POSIXt.

Value

List of length 2:

`cbs`	A data.frame with a row for each customer and the summary statistic as columns.
`elog`	A data.frame with a row for each transaction, and columns `cust`, `date` and `t`.

References

Platzer, M., & Reutterer, T. (2016). Ticking away the moments: Timing regularity helps to better predict customer activity. Marketing Science, 35(5), 779-799. doi:10.1287/mksc.2015.0963

Examples

params <- list(t = 4.5, gamma = 1.5, r = 5, alpha = 10, s = 0.8, beta = 12)
data <- pggg.GenerateData(n = 200, T.cal = 32, T.star = 32, params)
cbs <- data$cbs  # customer by sufficient summary statistic - one row per customer
elog <- data$elog  # Event log - one row per event/purchase
params <- list(t = 4.5, gamma = 1.5, r = 5, alpha = 10, s = 0.8, beta = 12)
data <- pggg.GenerateData(n = 200, T.cal = 32, T.star = 32, params)
cbs <- data$cbs  # customer by sufficient summary statistic - one row per customer
elog <- data$elog  # Event log - one row per event/purchase

Pareto/GGG Parameter Draws

Description

Returns draws from the posterior distributions of the Pareto/GGG parameters, on cohort as well as on customer level.

Usage

pggg.mcmc.DrawParameters(
  cal.cbs,
  mcmc = 2500,
  burnin = 500,
  thin = 50,
  chains = 2,
  mc.cores = NULL,
  param_init = NULL,
  trace = 100
)
pggg.mcmc.DrawParameters(
  cal.cbs,
  mcmc = 2500,
  burnin = 500,
  thin = 50,
  chains = 2,
  mc.cores = NULL,
  param_init = NULL,
  trace = 100
)

Arguments

`cal.cbs`	Calibration period customer-by-sufficient-statistic (CBS) data.frame. It must contain a row for each customer, and columns `x` for frequency, `t.x` for recency , `T.cal` for the total time observed, as well as the sum over logarithmic intertransaction times `litt`. A correct format can be easily generated based on the complete event log of a customer cohort with `elog2cbs`.
`mcmc`	Number of MCMC steps.
`burnin`	Number of initial MCMC steps which are discarded.
`thin`	Only every `thin`-th MCMC step will be returned.
`chains`	Number of MCMC chains to be run.
`mc.cores`	Number of cores to use in parallel (Unix only). Defaults to `min(chains, detectCores())`.
`param_init`	List of start values for cohort-level parameters.
`trace`	Print logging statement every `trace`-th iteration. Not available for `mc.cores > 1`.

Details

See demo('pareto-ggg') for how to apply this model.

Value

List of length 2:

`level_1`	list of `mcmc.list`s, one for each customer, with draws for customer-level parameters `k`, `lambda`, `tau`, `z`, `mu`
`level_2`	`mcmc.list`, with draws for cohort-level parameters `r`, `alpha`, `s`, `beta`, `t`, `gamma`

References

Platzer, M., & Reutterer, T. (2016). Ticking away the moments: Timing regularity helps to better predict customer activity. Marketing Science, 35(5), 779-799. doi:10.1287/mksc.2015.0963

Examples

data("groceryElog")
cbs <- elog2cbs(groceryElog, T.cal = "2006-12-31")
param.draws <- pggg.mcmc.DrawParameters(cbs,
  mcmc = 20, burnin = 10, thin = 2, chains = 1) # short MCMC to run demo fast

# cohort-level parameter draws
as.matrix(param.draws$level_2)
# customer-level parameter draws for customer with ID '4'
as.matrix(param.draws$level_1[["4"]])

# estimate future transactions
xstar.draws <- mcmc.DrawFutureTransactions(cbs, param.draws, cbs$T.star)
xstar.est <- apply(xstar.draws, 2, mean)
head(xstar.est)
data("groceryElog")
cbs <- elog2cbs(groceryElog, T.cal = "2006-12-31")
param.draws <- pggg.mcmc.DrawParameters(cbs,
  mcmc = 20, burnin = 10, thin = 2, chains = 1) # short MCMC to run demo fast

# cohort-level parameter draws
as.matrix(param.draws$level_2)
# customer-level parameter draws for customer with ID '4'
as.matrix(param.draws$level_1[["4"]])

# estimate future transactions
xstar.draws <- mcmc.DrawFutureTransactions(cbs, param.draws, cbs$T.star)
xstar.est <- apply(xstar.draws, 2, mean)
head(xstar.est)

Pareto/GGG Plot Regularity Rate Heterogeneity

Description

Plots and returns the estimated gamma distribution of k (customers' regularity in interpurchase times).

Usage

pggg.plotRegularityRateHeterogeneity(
  draws,
  xmax = NULL,
  fn = NULL,
  title = "Distribution of Regularity Rate k"
)
pggg.plotRegularityRateHeterogeneity(
  draws,
  xmax = NULL,
  fn = NULL,
  title = "Distribution of Regularity Rate k"
)

Arguments

`draws`	MCMC draws as returned by `pggg.mcmc.DrawParameters`.
`xmax`	Upper bound for x-scale.
`fn`	Optional function to summarize individual-level draws for k, e.g. 'mean'.
`title`	Plot title.

References

Platzer, M., & Reutterer, T. (2016). Ticking away the moments: Timing regularity helps to better predict customer activity. Marketing Science, 35(5), 779-799. doi:10.1287/mksc.2015.0963

Examples

data("groceryElog")
cbs <- elog2cbs(groceryElog, T.cal = "2006-12-31")
param.draws <- pggg.mcmc.DrawParameters(cbs,
  mcmc = 20, burnin = 10, thin = 2, chains = 1) # short MCMC to run demo fast
pggg.plotRegularityRateHeterogeneity(param.draws)
data("groceryElog")
cbs <- elog2cbs(groceryElog, T.cal = "2006-12-31")
param.draws <- pggg.mcmc.DrawParameters(cbs,
  mcmc = 20, burnin = 10, thin = 2, chains = 1) # short MCMC to run demo fast
pggg.plotRegularityRateHeterogeneity(param.draws)

Plot timing patterns of sampled customers

Description

Plot timing patterns of sampled customers

Usage

plotTimingPatterns(
  elog,
  n = 40,
  T.cal = NULL,
  T.tot = NULL,
  title = "Sampled Timing Patterns",
  headers = NULL
)
plotTimingPatterns(
  elog,
  n = 40,
  T.cal = NULL,
  T.tot = NULL,
  title = "Sampled Timing Patterns",
  headers = NULL
)

Arguments

`elog`	Event log, a `data.frame` with columns `cust` and transaction time `t` or `date`.
`n`	Number of sampled customers.
`T.cal`	End of calibration period, which is visualized as a vertical line.
`T.tot`	End of observation period
`title`	Plot title.
`headers`	Vector of length 2 for adding headers to the plot, e.g. `c("Calibration", "Holdout")`.

Examples

data("groceryElog")
plotTimingPatterns(groceryElog, T.tot = "2008-12-31")
plotTimingPatterns(groceryElog, T.cal = "2006-12-31", headers = c("Calibration", "Holdout"))
data("groceryElog")
plotTimingPatterns(groceryElog, T.tot = "2008-12-31")
plotTimingPatterns(groceryElog, T.cal = "2006-12-31", headers = c("Calibration", "Holdout"))

Simulate data according to Pareto/NBD model assumptions

Description

Simulate data according to Pareto/NBD model assumptions

Usage

pnbd.GenerateData(n, T.cal, T.star, params, date.zero = "2000-01-01")
pnbd.GenerateData(n, T.cal, T.star, params, date.zero = "2000-01-01")

Arguments

`n`	Number of customers.
`T.cal`	Length of calibration period. If a vector is provided, then it is assumed that customers have different 'birth' dates, i.e. $max(T.cal)-T.cal$ .
`T.star`	Length of holdout period. This may be a vector.
`params`	A list of model parameters `r`, `alpha`, `s`, `beta`.
`date.zero`	Initial date for cohort start. Can be of class character, Date or POSIXt.

Value

List of length 2:

`cbs`	A data.frame with a row for each customer and the summary statistic as columns.
`elog`	A data.frame with a row for each transaction, and columns `cust`, `date` and `t`.

Examples

params <- list(r = 5, alpha = 10, s = 0.8, beta = 12)
data <- pnbd.GenerateData(n = 200, T.cal = 32, T.star = 32, params)
cbs <- data$cbs  # customer by sufficient summary statistic - one row per customer
elog <- data$elog  # Event log - one row per event/purchase
params <- list(r = 5, alpha = 10, s = 0.8, beta = 12)
data <- pnbd.GenerateData(n = 200, T.cal = 32, T.star = 32, params)
cbs <- data$cbs  # customer by sufficient summary statistic - one row per customer
elog <- data$elog  # Event log - one row per event/purchase

Pareto/NBD (HB) Parameter Draws

Description

Returns draws from the posterior distributions of the Pareto/NBD (HB) parameters, on cohort as well as on customer level.

Usage

pnbd.mcmc.DrawParameters(
  cal.cbs,
  mcmc = 2500,
  burnin = 500,
  thin = 50,
  chains = 2,
  mc.cores = NULL,
  use_data_augmentation = TRUE,
  param_init = NULL,
  trace = 100
)
pnbd.mcmc.DrawParameters(
  cal.cbs,
  mcmc = 2500,
  burnin = 500,
  thin = 50,
  chains = 2,
  mc.cores = NULL,
  use_data_augmentation = TRUE,
  param_init = NULL,
  trace = 100
)

Arguments

`cal.cbs`	Calibration period customer-by-sufficient-statistic (CBS) data.frame. It must contain a row for each customer, and columns `x` for frequency, `t.x` for recency and `T.cal` for the total time observed. A correct format can be easily generated based on the complete event log of a customer cohort with `elog2cbs`.
`mcmc`	Number of MCMC steps.
`burnin`	Number of initial MCMC steps which are discarded.
`thin`	Only every `thin`-th MCMC step will be returned.
`chains`	Number of MCMC chains to be run.
`mc.cores`	Number of cores to use in parallel (Unix only). Defaults to `min(chains, detectCores())`.
`use_data_augmentation`	deprecated
`param_init`	List of start values for cohort-level parameters.
`trace`	Print logging statement every `trace`-th iteration. Not available for `mc.cores > 1`.

Details

See demo('pareto-ggg') for how to apply this model.

Value

2-element list:

level_1 list of mcmc.lists, one for each customer, with draws for customer-level parameters lambda, tau, z, mu
level_2 mcmc.list, with draws for cohort-level parameters r, alpha, s, beta

References

Ma, S. H., & Liu, J. L. (2007, August). The MCMC approach for solving the Pareto/NBD model and possible extensions. In Third international conference on natural computation (ICNC 2007) (Vol. 2, pp. 505-512). IEEE. doi:10.1109/ICNC.2007.728

Abe, M. (2009). "Counting your customers" one by one: A hierarchical Bayes extension to the Pareto/NBD model. Marketing Science, 28(3), 541-553. doi:10.1287/mksc.1090.0502

Examples

data("groceryElog")
cbs <- elog2cbs(groceryElog, T.cal = "2006-12-31")
param.draws <- pnbd.mcmc.DrawParameters(cbs,
  mcmc = 100, burnin = 50, thin = 10, chains = 1) # short MCMC to run demo fast

# cohort-level parameter draws
as.matrix(param.draws$level_2)
# customer-level parameter draws for customer with ID '4'
as.matrix(param.draws$level_1[["4"]])

# estimate future transactions
xstar.draws <- mcmc.DrawFutureTransactions(cbs, param.draws, cbs$T.star)
xstar.est <- apply(xstar.draws, 2, mean)
head(xstar.est)
data("groceryElog")
cbs <- elog2cbs(groceryElog, T.cal = "2006-12-31")
param.draws <- pnbd.mcmc.DrawParameters(cbs,
  mcmc = 100, burnin = 50, thin = 10, chains = 1) # short MCMC to run demo fast

# cohort-level parameter draws
as.matrix(param.draws$level_2)
# customer-level parameter draws for customer with ID '4'
as.matrix(param.draws$level_1[["4"]])

# estimate future transactions
xstar.draws <- mcmc.DrawFutureTransactions(cbs, param.draws, cbs$T.star)
xstar.est <- apply(xstar.draws, 2, mean)
head(xstar.est)

Package 'BTYDplus'

Help Index

Simulate data according to Pareto/NBD (Abe) model assumptions

Description

Usage

Arguments

Value

Examples

Pareto/NBD (Abe) Parameter Draws

Description

Usage

Arguments

Details

Value

References

See Also

Examples

Convert Event Log to customer-level summary statistic

Description

Usage

Arguments

Details

Value

Examples

Convert Event Log to Transaction Counts

Description

Usage

Arguments

Details

Value

Examples

Estimate Regularity in Intertransaction Timings

Description

Usage

Arguments

Details

Value

References

Examples

Event log for customers of an online grocery store.

Description

Usage

Format

Source

References

(M)BG/CNBD-k Log-Likelihood

Description

Usage

Arguments

Value

References

(M)BG/CNBD-k Conditional Expected Transactions

Description

Usage

Arguments

Value

References

Examples

(M)BG/CNBD-k Parameter Estimation

Description

Usage

Arguments

Value

References

See Also

Examples

(M)BG/CNBD-k Expectation

Description

Usage

Arguments

Details

Value

References

Examples

(M)BG/CNBD-k Expected Cumulative Transactions

Description

Usage

Arguments

Details

Value