title: how to linearly interpolate monthly data from the 15th of each month that rolls over the 15th of the next month
body: <p>I have a monthly dataset that I want to interpolate daily. However I need to interpolate from the 15th of each month or midmonth (Jan 16th, Feb 14th, March 15th...), to the next one. </p>
  
  <pre><code>Here is my data set: 
  
  2000-01-31  0.02451
  2000-02-28  0.03392
  2000-03-31  0.15451
  2000-04-30  0.28366
  2000-05-31  0.46806
  2000-06-30  0.67766
  ...
  </code></pre>
  
  <p>First: I need these values to be set every mid-month.</p>
  
  <pre><code>2000-01-16  0.02451
  2000-02-14  0.03392
  2000-03-16  0.15451
  2000-04-15  0.28366
  2000-05-16  0.46806
  2000-06-15  0.67766
  </code></pre>
  
  <p>Second: Then daily interpolate</p>
  
  <pre><code>2000-01-16  0.02451
  2000-01-17
  2000-01-18
  ...
  2000-01-31
  2000-02-01
  2000-02-02
  ...
  
  2000-02-14  0.03392
  </code></pre>
  
  <p><strong>I'm able to interpolate each month from the 1st to the 31st using the following code:</strong> </p>
  
  <pre><code>### Linear interpolation from monthly values to daily
  import pandas as pd
  
  df.set_index(pd.date_range(start='1/1/2000', end='1/1/2010', freq='M'), inplace=True, drop=True)
  rng = pd.date_range(start='1/1/2000', end='1/1/2010', freq='D')
  df2 = df.reindex(rng, axis=0).interpolate(axis=0)
  </code></pre>
  
  <p><strong>Results</strong></p>
  
  <pre><code>RESULTS                EXPECTED RESULTS 
  2000-01-31  0.02451    2000-01-16   0.02451
  2000-02-01  0.02485    2000-02-17   0.02485
  2000-02-02  0.02518    2000-02-18   0.02518
  ...
  2000-02-26  0.03325    2000-02-12   0.03325
  2000-02-27  0.03359    2000-02-13   0.03359
  2000-02-28  0.03392    2000-02-14   0.03392
  </code></pre>
  
  <p>Any help will be greatly appreciated!</p>