Interpolate daily values from monthly averages

Question

I have a list of monthly production guarantees and I want to estimate daily values. Dividing monthly totals by days/month works, but when graphed, leads to a chunky piece-wise plot. I could use a spline interpolation, but this would not guarantee that the monthly totals would be correct.

What is the best way to interpolate daily values to provide a smooth curve while maintaining monthly totals?

Jan 50
Feb 60
Mar 85
Apr 98
May 111
Jun 113
Jul 113
Aug 105
Sep 91
Oct 75
Nov 54
Dec 45

Answer 1

A possible idea : define a table of $X_i$ and $Y_i$ such that $X_i$ represents the number of days from January 1st and $Y_i$ the cumulated values from January 1st.

So $X_0=0$, $Y_0=0$, then $X_1=30$, $Y_1=50$, then $X_2=30+28=58$, $Y_2=50+60=110$, then $X_3=58+31=89$, $Y_3=110+85=195$ and so on up to the end of the year. This should give you something like

   0       0
  30      30 
  58     110
  89     195
 119     293
 150     404
 180     517
 211     630
 242     735
 272     826
 303     901
 333     955
 364    1000

Using your data shows a very nice and smooth curve for function $Y(X)$

Using now cubic splines : cumulated values (the $Y$'s) will be respected. Otherwise, use any other standard interpolation technique.

Now, as an example : you want to know the value for day $251$; interpolating for $X=250$ gives $Y=760.606$; interpolating for $X=251$ gives $Y=763.747$; so $763.747-760.606=3.141$. Isn't funny to find, as a result of a random calculation something lookig like $\pi$ ?