# pnbd.PlotDropoutRateHeterogeneity: Pareto/NBD Plot Dropout Rate Heterogeneity In BTYD: Implementing BTYD Models with the Log Sum Exp Patch

## Description

Plots and returns the estimated gamma distribution of mu (customers' propensities to drop out).

## Usage

 `1` ```pnbd.PlotDropoutRateHeterogeneity(params, lim = NULL) ```

## Arguments

 `params` Pareto/NBD parameters - a vector with r, alpha, s, and beta, in that order. r and alpha are unobserved parameters for the NBD transaction process. s and beta are unobserved parameters for the Pareto (exponential gamma) dropout process. `lim` The upper-bound of the x-axis. A number is chosen by the function if none is provided.

## Details

This returns the distribution of each customer's exponential parameter that determines their lifetime (using the Pareto/NBD assumption that a customer's lifetime can be modeled with an exponential distribution).

## Value

Distribution of customers' propensities to drop out.

## Examples

 ```1 2 3 4``` ```params <- c(0.55, 10.56, 0.61, 11.64) pnbd.PlotDropoutRateHeterogeneity(params) params <- c(0.55, 10.56, 3, 11.64) pnbd.PlotDropoutRateHeterogeneity(params) ```

### Example output

```Loading required package: hypergeo
[,1]         [,2]         [,3]         [,4]        [,5]
x.axis.ticks     0  0.003153792  0.006307583  0.009461375  0.01261517
heterogeneity  Inf 27.759092104 20.420226327 16.805132923 14.48014252
[,6]        [,7]        [,8]       [,9]      [,10]
x.axis.ticks   0.01576896  0.01892275  0.02207654 0.02523033 0.02838412
heterogeneity 12.79484187 11.48712187 10.42699037 9.54111444 8.78428895
[,11]      [,12]     [,13]      [,14]      [,15]      [,16]
x.axis.ticks  0.03153792 0.03469171 0.0378455 0.04099929 0.04415308 0.04730687
heterogeneity 8.12677401 7.54800237 7.0331759 6.57129934 6.15398192 5.77467505
[,17]      [,18]      [,19]      [,20]      [,21]      [,22]
x.axis.ticks  0.05046066 0.05361446 0.05676825 0.05992204 0.06307583 0.06622962
heterogeneity 5.42816875 5.11024881 4.81745692 4.54691886 4.29621891 4.06330635
[,23]     [,24]    [,25]      [,26]      [,27]      [,28]
x.axis.ticks  0.06938341 0.0725372 0.075691 0.07884479 0.08199858 0.08515237
heterogeneity 3.84642473 3.6440575 3.454886 3.27775475 3.11164787 2.95566476
[,29]      [,30]      [,31]      [,32]     [,33]     [,34]
x.axis.ticks  0.08830616 0.09145995 0.09461375 0.09776754 0.1009213 0.1040751
heterogeneity 2.80900428 2.67095030 2.54085995 2.41815396 2.3023085 2.1928481
[,35]     [,36]     [,37]     [,38]     [,39]     [,40]
x.axis.ticks  0.1072289 0.1103827 0.1135365 0.1166903 0.1198441 0.1229979
heterogeneity 2.0893404 1.9913905 1.8986372 1.8107491 1.7274216 1.6483740
[,41]     [,42]     [,43]    [,44]     [,45]     [,46]
x.axis.ticks  0.1261517 0.1293055 0.1324592 0.135613 0.1387668 0.1419206
heterogeneity 1.5733471 1.5021013 1.4344146 1.370081 1.3089087 1.2507198
[,47]     [,48]    [,49]     [,50]     [,51]     [,52]
x.axis.ticks  0.1450744 0.1482282 0.151382 0.1545358 0.1576896 0.1608434
heterogeneity 1.1953479 1.1426381 1.092446 1.0446346 0.9990784 0.9556579
[,53]     [,54]     [,55]     [,56]     [,57]     [,58]
x.axis.ticks  0.1639972 0.1671510 0.1703047 0.1734585 0.1766123 0.1797661
heterogeneity 0.9142617 0.8747848 0.8371287 0.8012008 0.7669137 0.7341852
[,59]     [,60]     [,61]     [,62]     [,63]     [,64]
x.axis.ticks  0.1829199 0.1860737 0.1892275 0.1923813 0.1955351 0.1986889
heterogeneity 0.7029378 0.6730983 0.6445977 0.6173709 0.5913560 0.5664948
[,65]     [,66]     [,67]     [,68]     [,69]     [,70]
x.axis.ticks  0.2018427 0.2049964 0.2081502 0.2113040 0.2144578 0.2176116
heterogeneity 0.5427321 0.5200158 0.4982962 0.4775266 0.4576624 0.4386615
[,71]     [,72]     [,73]     [,74]     [,75]     [,76]
x.axis.ticks  0.2207654 0.2239192 0.2270730 0.2302268 0.2333806 0.2365344
heterogeneity 0.4204840 0.4030918 0.3864488 0.3705209 0.3552755 0.3406817
[,77]     [,78]     [,79]     [,80]     [,81]     [,82]
x.axis.ticks  0.2396882 0.2428419 0.2459957 0.2491495 0.2523033 0.2554571
heterogeneity 0.3267099 0.3133324 0.3005223 0.2882545 0.2765047 0.2652500
[,83]     [,84]     [,85]     [,86]     [,87]     [,88]
x.axis.ticks  0.2586109 0.2617647 0.2649185 0.2680723 0.2712261 0.2743799
heterogeneity 0.2544686 0.2441395 0.2342430 0.2247601 0.2156727 0.2069636
[,89]     [,90]     [,91]     [,92]     [,93]     [,94]
x.axis.ticks  0.2775337 0.2806874 0.2838412 0.2869950 0.2901488 0.2933026
heterogeneity 0.1986165 0.1906156 0.1829460 0.1755935 0.1685444 0.1617857
[,95]     [,96]    [,97]     [,98]     [,99]    [,100]
x.axis.ticks  0.2964564 0.2996102 0.302764 0.3059178 0.3090716 0.3122254
heterogeneity 0.1553051 0.1490906 0.143131 0.1374154 0.1319336 0.1266755
[,1]        [,2]      [,3]       [,4]      [,5]       [,6]
x.axis.ticks     0 0.007294549 0.0145891 0.02188365 0.0291782 0.03647275
heterogeneity    0 0.038543448 0.1416235 0.29271329 0.4780184 0.68610298
[,7]       [,8]       [,9]      [,10]      [,11]      [,12]
x.axis.ticks  0.0437673 0.05106185 0.05835639 0.06565094 0.07294549 0.08024004
heterogeneity 0.9075624 1.13473577 1.36145509 1.58282554 1.79503414 1.99518315
[,13]      [,14]     [,15]     [,16]     [,17]     [,18]
x.axis.ticks  0.08753459 0.09482914 0.1021237 0.1094182 0.1167128 0.1240073
heterogeneity 2.18114517 2.35143765 2.5051141 2.6416705 2.7609642 2.8631450
[,19]     [,20]    [,21]     [,22]     [,23]     [,24]
x.axis.ticks  0.1313019 0.1385964 0.145891 0.1531855 0.1604801 0.1677746
heterogeneity 2.9485954 3.0178807 3.071705 3.1108768 3.1362760 3.1488304
[,25]     [,26]     [,27]     [,28]     [,29]     [,30]
x.axis.ticks  0.1750692 0.1823637 0.1896583 0.1969528 0.2042474 0.2115419
heterogeneity 3.1494938 3.1392289 3.1189926 3.0897252 3.0523412 3.0077220
[,31]    [,32]     [,33]     [,34]     [,35]     [,36]
x.axis.ticks  0.2188365 0.226131 0.2334256 0.2407201 0.2480147 0.2553092
heterogeneity 2.9567111 2.900110 2.8386752 2.7731177 2.7041012 2.6322424
[,37]     [,38]     [,39]     [,40]    [,41]     [,42]
x.axis.ticks  0.2626038 0.2698983 0.2771929 0.2844874 0.291782 0.2990765
heterogeneity 2.5581116 2.4822334 2.4050884 2.3271144 2.248709 2.1702292
[,43]     [,44]     [,45]     [,46]     [,47]     [,48]
x.axis.ticks  0.3063711 0.3136656 0.3209602 0.3282547 0.3355493 0.3428438
heterogeneity 2.0919975 2.0143001 1.9373911 1.8614941 1.7868042 1.7134905
[,49]     [,50]     [,51]    [,52]     [,53]     [,54]
x.axis.ticks  0.3501384 0.3574329 0.3647275 0.372022 0.3793166 0.3866111
heterogeneity 1.6416976 1.5715475 1.5031421 1.436564 1.3718799 1.3091392
[,55]     [,56]     [,57]     [,58]     [,59]     [,60]
x.axis.ticks  0.3939057 0.4012002 0.4084948 0.4157893 0.4230839 0.4303784
heterogeneity 1.2483785 1.1896216 1.1328807 1.0781583 1.0254477 0.9747342
[,61]     [,62]     [,63]     [,64]     [,65]     [,66]
x.axis.ticks  0.4376730 0.4449675 0.4522621 0.4595566 0.4668512 0.4741457
heterogeneity 0.9259966 0.8792072 0.8343334 0.7913380 0.7501801 0.7108157
[,67]     [,68]     [,69]     [,70]     [,71]     [,72]
x.axis.ticks  0.4814403 0.4887348 0.4960294 0.5033239 0.5106185 0.5179130
heterogeneity 0.6731980 0.6372785 0.6030067 0.5703313 0.5391999 0.5095597
[,73]     [,74]     [,75]     [,76]     [,77]     [,78]
x.axis.ticks  0.5252076 0.5325021 0.5397967 0.5470912 0.5543858 0.5616803
heterogeneity 0.4813579 0.4545414 0.4290579 0.4048551 0.3818818 0.3600874
[,79]     [,80]     [,81]     [,82]     [,83]     [,84]
x.axis.ticks  0.5689749 0.5762694 0.5835639 0.5908585 0.5981530 0.6054476
heterogeneity 0.3394223 0.3198380 0.3012871 0.2837235 0.2671023 0.2513800
[,85]     [,86]     [,87]     [,88]     [,89]     [,90]
x.axis.ticks  0.6127421 0.6200367 0.6273312 0.6346258 0.6419203 0.6492149
heterogeneity 0.2365145 0.2224651 0.2091922 0.1966581 0.1848261 0.1736612
[,91]     [,92]     [,93]     [,94]     [,95]     [,96]
x.axis.ticks  0.6565094 0.6638040 0.6710985 0.6783931 0.6856876 0.6929822
heterogeneity 0.1631295 0.1531986 0.1438376 0.1350166 0.1267074 0.1188825
[,97]     [,98]      [,99]     [,100]
x.axis.ticks  0.7002767 0.7075713 0.71486584 0.72216039
heterogeneity 0.1115162 0.1045836 0.09806115 0.09192633
```

BTYD documentation built on Nov. 18, 2021, 1:10 a.m.