theseus-growth 0.3.7
Theseus is a library of tools to use in marketing analysis
License: MIT
Requires Python: >=3.9,<4.0
Keywords: cohort analysis,analysis,marketing,media mix model

# Theseus

![alt text](https://mobiledevmemo.com/wp-content/uploads/2020/01/Version-6.png "Theseus Growth")

## Theseus provides straightforward tools for cohort analysis and general marketing performance analysis. Theseus was created by [Eric Benjamin Seufert](https://www.twitter.com/eric_seufert) of [Heracles](https://hrcls.co).

Theseus is an open source library that provides a set of common functions for use in doing analysis related to product growth: building retention profiles, projecting DAU levels, combining cohorts, segmenting cohorts by age, etc. Theseus can be used for marketing budgeting planning, scenario analysis, marketing campaign analysis, revenue projections, and in a media mix model.

Theseus is designed to be used for standalone analysis projects as well as in programmatic business intelligence environments.

Theseus is provided as open source software under the [MIT](https://choosealicense.com/licenses/mit/) license.

Note that Theseus is in a __beta__ state; bugs are to be expected.

## Documentation

The documentation for Theseus can be found in [this QuantMar thread](https://quantmar.com/529/How-can-use-the-theseus-python-library-to-do-cohort-analysis).

## Installation

Use the package manager [pip](https://pip.pypa.io/en/stable/) to install Theseus.

```bash
pip install theseus_growth
```

## Usage

Include the theseus_growth library

```python
import theseus_growth as th
```

Instantiate a Theseus object
```python
th = th.theseus()
```

Working with Theseus involves using retention profiles to build cohort projections. To get started with analysis, you'll first build a retention profile using `days` and `retention` values, where each day value corresponds to a retention value, starting from Day 1 (ie. the day after a user has entered the product). Retention values should be provided as whole numbers (not decimals), eg. 30% retention for some given day would be represented as 30 and not .30. 

The retention and day values are provided as lists, the lengths of which must match. Theseus uses the index of the values in the `days` list to associate with a value from the `retention` list, so no need to order the lists.

Here's an example:

```python
x_data = [ 1, 3, 7, 14, 30, 60, 90, 180 ]
y_data = [ 80, 70, 55, 50, 30, 22, 10, 8 ]

facebook = th.create_profile( days = x_data, retention_values = y_data )

print( facebook )
```

In this example, Day 1 retention is set to 80, Day 3 retention is set to 70, Day 7 retention is set to 55, etc. Then, these lists are supplied to the `create_profile` function to generate a retention profile (in this case, for Facebook, as per the variable name).

The curve fit to the retention data is decided by iterating over a number of different function forms to find the one that fits best with the smallest error. The functions tested are: `[ 'log', 'exp', 'linear', 'quad', 'weibull', 'power', 'interpolate' ]`. A specific function can be forced onto the data by using the `form` parameter with the `create_profile` function; when the `form` parameter is not set, `create_profile` defaults to finding the best fit function.

If you `print` the `facebook` variable, the output will reveal a number of pieces of information about the retention profile:

```python
{'x': [1, 3, 7, 14, 30, 60, 90, 180], 'y': [80, 70, 55, 50, 30, 22, 10, 8], 'y_collapsed': [80.0, 70.0, 55.0, 50.0, 30.0, 22.0, 10.0, 8.0], 'x_collapsed': [1, 3, 7, 14, 30, 60, 90, 180], 'interpolation_f': <scipy.interpolate.interpolate.interp1d object at 0x10c6234f8>, 'interpolation_s': <scipy.interpolate.fitpack2.InterpolatedUnivariateSpline object at 0x10c638588>, 'params': {'log': array([11.69432981,  0.85932489, 91.18858849]), 'exp': array([6.81055507e+01, 4.01937193e-02, 1.00786302e+01]), 'linear': array([-0.36314103, 58.10116222]), 'quad': array([ 4.23356783e-03, -1.09641452e+00,  6.94411850e+01]), 'weibull': array([136.70664663,   0.99893803]), 'power': array([88.3002565,  0.3123284]), 'interpolate': None}, 'errors': {'log': 61.1068291195336, 'exp': 101.38898207577283, 'linear': 1412.367783723572, 'quad': 364.49321231183075, 'weibull': 12824.82253493541, 'power': 440.6176923037875}, 'best_fit': 'log', 'retention_profile': 'best_fit', 'retention_projection': (array([  1,   2,   3,   4,   5,   6,   7,   8,   9,  10,  11,  12,  13,
        14,  15,  16,  17,  18,  19,  20,  21,  22,  23,  24,  25,  26,
        27,  28,  29,  30,  31,  32,  33,  34,  35,  36,  37,  38,  39,
        40,  41,  42,  43,  44,  45,  46,  47,  48,  49,  50,  51,  52,
        53,  54,  55,  56,  57,  58,  59,  60,  61,  62,  63,  64,  65,
        66,  67,  68,  69,  70,  71,  72,  73,  74,  75,  76,  77,  78,
        79,  80,  81,  82,  83,  84,  85,  86,  87,  88,  89,  90,  91,
        92,  93,  94,  95,  96,  97,  98,  99, 100, 101, 102, 103, 104,
       105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117,
       118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130,
       131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143,
       144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156,
       157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169,
       170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180]), [100.0, 80.72474913359267, 73.46379035522745, 68.40395854720839, 64.51667685670971, 61.35945860038563, 58.70096154455828, 56.40491084756263, 54.384231538266555, 52.579888860453295, 50.95000745540398, 49.46380467615453, 48.097988591666066, 46.834508880262675, 45.65909322498185, 44.56026139229167, 43.52864131907821, 42.55648257398489, 41.63730255356898, 40.76562419809498, 39.936778210530946, 39.14675163202042, 38.392070317383634, 37.669706592412716, 36.97700588336105, 36.3116278251625, 35.67149854953705, 35.054771699032194, 34.459796319341784, 33.88509022316152, 33.32931774346239, 32.79127103579817, 32.269854271186304, 31.764070199363758, 31.273008668265803, 30.79583676761319, 30.331790328468763, 29.88016656089222, 29.440317651599635, 29.01164517522338, 28.593595198171947, 28.185653974577036, 27.78734415043222, 27.398221405576763, 27.017871474283382, 26.645907494353807, 26.281967642193266, 25.92571301762287, 25.576825747436132, 25.235007281103094, 24.899976855723068, 24.57147011044914, 24.24923783325221, 23.933044825140172, 23.622668868865617, 23.317899790794513, 23.01853860601659, 22.724396737987817, 22.43529530504138, 22.15106446700662, 21.87154282596005, 21.59657687581418, 21.326020496044592, 21.059734485374676, 20.79758613169244, 20.539448814872614, 20.285201639528808, 20.034729095029036, 19.787920740381438, 19.544670911838594, 19.30487845128266, 19.06844645364413, 18.835282031775264, 18.605296097351072, 18.378403156503964, 18.154521119018952, 17.933571120024055, 17.715477353206225, 17.500166914670615, 17.287569656638155, 17.07761805024684, 16.870247056785416, 16.66539400674492, 16.462998486125443, 16.263002229482055, 16.065349019235967, 15.86998459081596, 15.676856543229135, 15.485914254692815, 15.297108802987182, 15.11039289021575, 14.925720771683615, 14.743048188626403, 14.562332304542082, 14.383531644896777, 14.206606039992167, 14.031516570797578, 13.858225517564094, 13.686696311050966, 13.516893486206541, 13.34878263815699, 13.18233038036621, 13.01750430483952, 12.854272944252656, 12.692605735895398, 12.53247298732623, 12.373845843641718, 12.216696256270339, 12.060996953206299, 11.90672141060432, 11.753843825661477, 11.602339090716569, 11.452182768502269, 11.30335106848878, 11.15582082426181, 11.009569471881221, 10.86457502916953, 10.720816075882823, 10.578271734719507, 10.436921653124443, 10.296745985849213, 10.157725378230722, 10.019840950153323, 9.883074280660807, 9.747407393187231, 9.612822741376846, 9.47930319546505, 9.346832029194132, 9.215392907238623, 9.084969873116748, 8.955547337565534, 8.827110067358433, 8.69964317454533, 8.573132106096097, 8.447562633929394, 8.322920845309952, 8.199193133597916, 8.076366189334905, 7.954426991652241, 7.833362799987498, 7.713161146095985, 7.5938098263449945, 7.475296894278529, 7.357610653441469, 7.240739650452227, 7.124672668313735, 7.009398719952898, 6.8949070419793514, 6.781187088654434, 6.668228526062208, 6.556021226474144, 6.444555262900209, 6.33382090381852, 6.223808608077022, 6.114509019960167, 6.005912964414506, 5.898011442426906, 5.790795626549496, 5.684256856566179, 5.578386635294848, 5.473176624520619, 5.368618641055164, 5.264704652917132, 5.16142677562982, 5.058777268631218, 4.95674853179267, 4.8553331020422945, 4.754523650089098, 4.65431297724453, 4.554694012337748, 4.455659808721521, 4.357203541365507, 4.259318504033715, 4.161998106543535, 4.065235872103301, 3.969025434725765, 3.873360536714898, 3.778235026223541, 3.6836428548796363, 3.5895780754783857])}
```

You won't ever actually interact directly with a retention profile variable, but you can see that it contains:
+ The original X and Y (the `days` and `retention` lists) data provided;
+ A projection (in the `retention_projection` variable);
+ Two `_collapsed` variables that contain the average values for each of the `days` and `retention` lists (in this example, only one value was provided for each day, so the `y_collapsed` list is the same as the `y` list, which was provided);
+ A `params` dict that contains coefficients for a number of different shape functions;
+ Some other miscellaneous data, like interpolation models;

With the Facebook retention profile created, cohort projections can be generated from it. First, the profile can be visualized with the `plot_retention` function:

```python
th.plot_retention( facebook )
```

Which should output a graph that looks like this:

![alt text](https://mobiledevmemo.com/wp-content/uploads/2020/01/fb_retention.png "Facebook retention profile graph")

Now a DAU projection based on cohorts can be generated -- in the Theseus library, this is called a **`forward DAU projection`**. First, we'll create a list of cohorts, meaning a list containing the numbers of new users that joined the product on a daily basis, with each number representing a sequential day.

Then, the `project_cohorted_DAU` function can be used to create a Pandas DataFrame containing the number of DAU present in the product, given the new users that joined via the cohorts, on the basis of the `facebook` retention profile. In this example, the function will take 4 inputs:

+ `profile`: the retention profile to use;
+ `periods`: the number of periods to project forward
+ `cohorts`: a list of new user values 
+ `start_date`: the date at which the cohorts are added and from which the projection is made

```python
#cohorts are daily new user values, eg. the number of new users
#joining the product on a given day
cohorts = [1000, 1000, 1000, 1000, 1000 ]

facebook_DAU = th.project_cohorted_DAU( profile = facebook, periods = 50, 
    cohorts = cohorts, start_date = 1 )

print( facebook_DAU )
```

The output of this should look like:

```python
                1     2     3     4     5    6    7    8    9   10  ...   41  \
cohort_date                                                         ...        
1            1000   807   734   684   645  613  587  564  543  525  ...  285   
2               0  1000   807   734   684  645  613  587  564  543  ...  290   
3               0     0  1000   807   734  684  645  613  587  564  ...  294   
4               0     0     0  1000   807  734  684  645  613  587  ...  298   
5               0     0     0     0  1000  807  734  684  645  613  ...  303   

              42   43   44   45   46   47   48   49   50  
cohort_date                                               
1            281  277  273  270  266  262  259  255  252  
2            285  281  277  273  270  266  262  259  255  
3            290  285  281  277  273  270  266  262  259  
4            294  290  285  281  277  273  270  266  262  
5            298  294  290  285  281  277  273  270  266  

[5 rows x 50 columns]
```

This DataFrame table shows how many of the original cohorts are present on any given day; the cohort numbers run down the Y axis and the days run across the X axis.

To see this as a total, the `DAU_total` function can be used:

```python
facebook_total = th.DAU_total( facebook_DAU )

print( facebook_total )
```

The output of which should look like:

```python
          1    2     3     4     5     6     7     8     9     10  ...    41  \
Value                                                              ...         
DAU    1000  808  1734  2491  3186  4548  5911  7274  8637  10000  ...  4312   

         42    43    44    45    46    47    48    49    50  
Value                                                        
DAU    4246  4182  4119  4058  3999  3941  3888  3831  3779  

[1 rows x 50 columns]
```

This table represents the total number of DAU present in the product from those five cohorts over the course of a 50-period timeline.


The `project_cohorted_DAU` can be used to project DAU out given some set of cohorts and a retention profile, but it can also be used to generate the number of new users needed to reach a DAU target over a timeline, given some existing set of cohorts.

In this example, the `cohorts` list contains five cohorts of 1000 new users each. If a marketing analyst wanted to know how many _additional_ cohorts, and of what size, would be needed in order to get the user base to 10,000 DAU, then they could use `project_cohorted_DAU` to do that by adding two parameters: `DAU_target` and `DAU_target_timeline`. `DAU_target` is the targeted number of DAU, and `DAU_target_timeline` is the number of days over which the additional new users will be added.

In action:

```python
facebook_DAU = th.project_cohorted_DAU( profile = facebook, periods = 50, cohorts = cohorts, 
    DAU_target = 10000, DAU_target_timeline = 10, start_date = 1 )

print( facebook_DAU )
```

Should produce the following output:

```python
                1     2     3     4     5     6     7     8     9    10  ...  \
cohort_date                                                              ...   
1            1000   807   734   684   645   613   587   564   543   525  ...   
2               0  1000   807   734   684   645   613   587   564   543  ...   
3               0     0  1000   807   734   684   645   613   587   564  ...   
4               0     0     0  1000   807   734   684   645   613   587  ...   
5               0     0     0     0  1000   807   734   684   645   613  ...   
6               0     0     0     0     0  1613  1302  1184  1103  1040  ...   
7               0     0     0     0     0     0  1757  1418  1290  1201  ...   
8               0     0     0     0     0     0     0  1853  1495  1361  ...   
9               0     0     0     0     0     0     0     0  1934  1561  ...   
10              0     0     0     0     0     0     0     0     0  2005  ...   

              41   42   43   44   45   46   47   48   49   50  
cohort_date                                                    
1            285  281  277  273  270  266  262  259  255  252  
2            290  285  281  277  273  270  266  262  259  255  
3            294  290  285  281  277  273  270  266  262  259  
4            298  294  290  285  281  277  273  270  266  262  
5            303  298  294  290  285  281  277  273  270  266  
6            496  489  481  474  467  461  454  448  441  435  
7            549  541  532  524  517  509  502  495  488  481  
8            588  579  570  562  553  545  537  529  522  514  
9            624  614  604  595  586  577  569  561  553  545  
10           657  647  636  627  617  608  599  590  581  573  

[10 rows x 50 columns]
```

This table reveals that the additional DNU needed to get to 10,000 overall DAU within the 10-period timeframe is: 1613, 1757, 1853, 1934, 2005. _Note that this approach seeks to minimize the number of total DNU added on any given day within the timeline_.

To get only the DNU (new users) values from a forward DAU projection, the `get_DNU` function can be used:

```python
#get DNU from a DAU projection
facebook_DNU = th.get_DNU( facebook_DAU )
print( facebook_DNU )
```

The output of which should look like:

```python
       cohort_date    1       2       3       4       5       6       7  \
Value                                                                     
DNU           1000  1.0  1000.0  1000.0  1000.0  1710.0  1881.0  1994.0   

            8       9  ...   41   42   43   44   45   46   47   48   49   50  
Value                  ...                                                    
DNU    2090.0  2171.0  ...  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  

[1 rows x 51 columns]
```

And to reduce `facebook_DAU` to only the total DAU values over the projection timeline, the `DAU_total` function can be used again:

```python
        1     2     3     4     5     6     7     8     9     10  ...    41  \
DAU                                                               ...         
0    1000  1807  2541  3225  3870  5096  6322  7548  8774  10000  ...  4384   

       42    43    44    45    46    47    48    49    50  
DAU                                                        
0    4318  4250  4188  4126  4067  4009  3953  3897  3842  

[1 rows x 50 columns]
```

Note that this shows DAU reaching 10,000 by Day 10.

The Facebook `forward DAU projection` can be visualized with the `plot_forward_DAU_stacked`, which takes three required parameters:
+ `forward_DAU`: the forward DAU projection being visualized (in this case, the `facebook_DAU` variable);
+ `forward_DAU_labels`: a list of the cohort names as labels for the stacked bars. The length of this list needs to match the number of cohorts in the forward DAU projection;
+ `forward_DAU_dates`: a list of dates as labels for the X axis. The length of this list needs to match the number of periods in the forward DAU projection;

To visualize the Facebook forward DAU projection that reaches the DAU target of 10,000:

```python
th.plot_forward_DAU_stacked( forward_DAU = facebook_DAU, 
    forward_DAU_labels = list( facebook_DAU.index ), 
    forward_DAU_dates = list( facebook_DAU.columns ), 
)
```

This should produce a graph that looks like this:

![alt text](https://mobiledevmemo.com/wp-content/uploads/2020/01/facebook_forward_DAU_projection.png "Facebook forward DAU projection")

Note that anything can be provided in the `forward_DAU_labels` and `forward_DAU_dates` parameters. For instance, to give the X axis actual date values (starting from January 1, 2020) and to make the legend more readable, the following can be done:

```python
from datetime import date, timedelta
th.plot_forward_DAU_stacked( forward_DAU = facebook_DAU, 
    forward_DAU_labels = [ 'Cohort ' + str( x ) for x in list( facebook_DAU.index ) ], 
    forward_DAU_dates = [ date(2020, 1, 1) + timedelta(days=int( x ) - 1 ) for x in list( facebook_DAU.columns ) ]
)
```

This should produce a graph that looks like this:

![alt text](https://mobiledevmemo.com/wp-content/uploads/2020/01/facebook_forward_DAU_readable.png.png "Facebook forward DAU projection")

To create a second retention profile -- this time, for Google -- the `create_profile` profile can be used again. This time, the `profile_max` parameter will be supplied: when `profile_max` is provided, the retention profile is projected out to that day (when it is not provided, the retention profile is only projected out to the maximum value provided in the `days` parameter). Also, with the Google retention profile, a much larger dataset of days and retention values will be supplied, so the curve fit is done against many more (arbitrarily produced) data points:

```python
import numpy as np
import random

x_data = [ 1, 14, 60 ]
y_data = [ 40, 22, 10 ]

new_x = []
for i, x in enumerate( x_data ):
    this_x = x
    for z in np.arange( 1, 100 ):
        this_y = float( y_data[ i ] * ( 1 + ( random.randint( -20, 20 ) / 100 ) ) )
        y_data.append( this_y )
        new_x.append( this_x )
        
x_data.extend( new_x )

google = th.create_profile( days = x_data, retention_values = y_data, profile_max = 180 )

th.plot_retention( google )
```

The output of this should look something like this (the red dots are the actual values from `retention_values`):

![alt text](https://mobiledevmemo.com/wp-content/uploads/2020/01/google_retention_profile.png "Google retention profile")

To build a forward DAU projection for Google, the following can be run. Note that the start_date is set to 10:

```python
cohorts = [ 2000, 4000, 1200, 2200, 1700, 1300, 4200, 9200 ]
google_DAU = th.project_cohorted_DAU( profile = google, periods = 40, cohorts = cohorts, 
    DAU_target = 20000, DAU_target_timeline = 20, start_date = 10 )

from datetime import date, timedelta
th.plot_forward_DAU_stacked( forward_DAU = google_DAU, 
    forward_DAU_labels = [ 'Cohort ' + str( x ) for x in list( google_DAU.index ) ], 
    forward_DAU_dates = [ date(2020, 1, 1) + timedelta(days=int( x ) - 1 ) for x in list( google_DAU.columns ) ]
)
```

![alt text](https://mobiledevmemo.com/wp-content/uploads/2020/01/google_forward_DAU-1.png "Google forward DAU")

Note the lumpiness of the first few days of DAU -- this is a result of 1) the volatile number of DAU in the initial cohorts and 2) the relatively low Google retention ( 40% on Day 1). Also note the dates on the X axis: the chart starts on January 10th, 2020 since the `start_date` variable is set to 10.

In order to get a fuller picture of product DAU, the Facebook and Google forward DAU projections can be combined with the `combine_DAU` function. The totals for each forward DAU projection should be used, otherwise the graph would be too busy to read:

```python
google_total = th.DAU_total( google_DAU )

combined_DAU = th.combine_DAU( DAU_totals = [ facebook_total, google_total ], labels = [ "Facebook", "Google" ] )

th.plot_forward_DAU_stacked( forward_DAU = combined_DAU, 
    forward_DAU_labels = list( combined_DAU.index ), 
    forward_DAU_dates = [ date(2020, 1, 1) + timedelta(days=int( x ) - 1 ) for x in list( combined_DAU.columns ) ]
)
```

The output of the above should look like:

![alt text](https://mobiledevmemo.com/wp-content/uploads/2020/01/download.png "Combined Facebook and Google forward DAU")

One important aspect of cohort analysis is age segmentation: breaking the user base out into segments based on user age. Theseus comes with two functions to do this: `project_aged_DAU` and `project_exact_aged_DAU`. 

`project_aged_DAU` presents the DAU projection in terms of _minimum_ user ages: it can display the number of users that are _at least_ X days old on a given date.

`project_exact_aged_DAU` presents the DAU projection in terms of _absolute_ user ages: it can display the number of users that are _exactly_ X days old on a given date.

Both functions take five parameters:
+ `profile`: the retention profile being used for the projection;
+ `periods`: the number of periods for which the forward DAU projection is being made;
+ `cohorts`: the cohorts that are being projected forward;
+ `start_date`: the start date of the projection;
+ `ages`: a list of ages that the projection should be broken down for. For `project_aged_DAU`, the forward DAU projection will be broken out to show the number of users per day that are _at least_ as old as every age in the list. For `project_exact_aged_DAU`, the forward DAU projection will be broken out to show the number of users per day that are _exactly_ as old as every age in the list.

An example of `project_aged_DAU`:

```python
x_data = [ 1, 14, 30, 90 ]
y_data = [ 25, 18, 12, 8 ]

#form options: 'log', 'exp', 'linear', 'quad', 'weibull', 'power'
snapchat = th.create_profile( days = x_data, retention_values = y_data, profile_max = 120 )

snapchat_aged_DAU = th.project_aged_DAU( snapchat, 20, [ 100, 200, 300, 400, 500 ], 
    start_date = 1, ages = [ 3, 7, 14 ] )

print( snapchat_aged_DAU )
```

This should produce output that looks like the following:

```python
     1  2   3   4    5    6    7    8    9   10   11   12   13   14   15   16  \
age                                                                             
3    0  0  24  71  143  236  352  343  336  327  320  312  305  296  290  282   
7    0  0   0   0    0    0   22   65  130  214  320  312  305  296  290  282   
14   0  0   0   0    0    0    0    0    0    0    0    0    0   18   55  108   

      17   18   19   20  
age                      
3    275  268  260  254  
7    275  268  260  254  
14   180  268  260  254 
```

Taking column 10 as an example: 869 users are _at least_ 3 days old, 563 users are _at least_ 7 days old, and 0 users are _at least_ 14 days old. 

An example of `project_exact_aged_DAU`:

```python
snapchat_exact_aged_DAU = th.project_exact_aged_DAU( snapchat, 20, [ 100, 200, 300, 400, 500 ], 
    start_date = 1, ages = [ 3, 7, 14 ] )

print( snapchat_exact_aged_DAU )
```

This should produce output that looks like the following:

```python
     1  2   3   4   5   6    7   8   9  10   11 12 13  14  15  16  17  18 19  \
age                                                                            
3    0  0  24  48  73  97  121   0   0   0    0  0  0   0   0   0   0   0  0   
7    0  0   0   0   0   0   22  44  66  88  110  0  0   0   0   0   0   0  0   
14   0  0   0   0   0   0    0   0   0   0    0  0  0  18  37  55  74  93  0   

    20  
age     
3    0  
7    0  
14   0
```

Note that each "age" only has five values; this is because there are only five cohorts provided in the example (each cohort will be an exact age only once).

Also note that if 1 is passed in the `ages` list for `project_exact_aged_DAU`, it produces a list of DNU:

```python
snapchat_exact_aged_DAU = th.project_exact_aged_DAU( snapchat, 20, [ 100, 200, 300, 400, 500 ], 
    start_date = 1, ages = [ 1 ] )

print( snapchat_exact_aged_DAU )
```

```python
       1    2    3    4    5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20
age                                                                      
1    100  200  300  400  500  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0
```

One interesting use case for user base segments is calculating percentages of the overall user base that are at least X days old -- often, certain monetization moments are only available to users after some amount of time, so being able to break a user base out into age groups to know what percentage of the user base is capable of monetizing is helpful. This can be done with Theseus by creating a Total forward DAU projection and combining it with the aged projection:

```python
snapchat_DAU = th.project_cohorted_DAU( profile = snapchat, periods = 20, cohorts = [ 100, 200, 300, 400, 500 ], 
    start_date = 1 )

snapchat_total = th.DAU_total( snapchat_DAU )

combined_DAU = th.combine_DAU( DAU_totals = [ snapchat_aged_DAU, snapchat_total ], 
    labels = [ [ "Age " + str( x ) for x in list( snapchat_aged_DAU.index ) ], "Total" ] 
)

for x in list( snapchat_aged_DAU.index ):
    combined_DAU.loc[ 'Age ' + str( x ) + ' Pct' ] = combined_DAU.apply( lambda z: ( z[ 'Age ' + str( x )] / z[ 'Total' ] ) )

print( combined_DAU )
```

This would output something that looks like:

```python
                1      2           3           4           5           6  \
profile                                                                    
Age 3         0.0    0.0   24.000000   71.000000  143.000000  236.000000   
Age 7         0.0    0.0    0.000000    0.000000    0.000000    0.000000   
Age 14        0.0    0.0    0.000000    0.000000    0.000000    0.000000   
Total       100.0  224.0  373.000000  545.000000  742.000000  360.000000   
Age 3 Pct     0.0    0.0    0.064343    0.130275    0.192722    0.655556   
Age 7 Pct     0.0    0.0    0.000000    0.000000    0.000000    0.000000   
Age 14 Pct    0.0    0.0    0.000000    0.000000    0.000000    0.000000   

                   7           8           9          10     11     12     13  \
profile                                                                         
Age 3       352.0000  343.000000  336.000000  327.000000  320.0  312.0  305.0   
Age 7        22.0000   65.000000  130.000000  214.000000  320.0  312.0  305.0   
Age 14        0.0000    0.000000    0.000000    0.000000    0.0    0.0    0.0   
Total       352.0000  343.000000  336.000000  327.000000  320.0  312.0  305.0   
Age 3 Pct     1.0000    1.000000    1.000000    1.000000    1.0    1.0    1.0   
Age 7 Pct     0.0625    0.189504    0.386905    0.654434    1.0    1.0    1.0   
Age 14 Pct    0.0000    0.000000    0.000000    0.000000    0.0    0.0    0.0   

                    14          15          16          17     18     19  \
profile                                                                    
Age 3       296.000000  290.000000  282.000000  275.000000  268.0  260.0   
Age 7       296.000000  290.000000  282.000000  275.000000  268.0  260.0   
Age 14       18.000000   55.000000  108.000000  180.000000  268.0  260.0   
Total       296.000000  290.000000  282.000000  275.000000  268.0  260.0   
Age 3 Pct     1.000000    1.000000    1.000000    1.000000    1.0    1.0   
Age 7 Pct     1.000000    1.000000    1.000000    1.000000    1.0    1.0   
Age 14 Pct    0.060811    0.189655    0.382979    0.654545    1.0    1.0   

               20  
profile            
Age 3       254.0  
Age 7       254.0  
Age 14      254.0  
Total       254.0  
Age 3 Pct     1.0  
Age 7 Pct     1.0  
Age 14 Pct    1.0 
```

In order to actually work with these projections, Theseus comes with two file output functions: `to_excel` and `to_json`. 

`to_excel` can take three parameters:
+ `df`: the forward DAU projection dataframe being output;
+ `file_name`: the name of the file that will be output (optional)
+ `sheet_name`: the name of the sheet that the data will be written to (optional)

`to_excel` will save a .xlsx file in the directory from which the Theseus object is being executed.

`to_json` can take two parameters:
+ `df`: the forward DAU projection dataframe being output;
+ `file_name`: the name of the file that will be output (optional)

`to_json` will save a .json file in the directory from which the Theseus object is being executed.




## Contributing
Pull requests are welcome. For major changes, please open an issue first to discuss what you would like to change.

Please make sure to update tests as appropriate.

## License
[MIT](https://choosealicense.com/licenses/mit/)