interpolation
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
You can actually get entire functions this way. See Ahlfors 5.2.3 (canonical products) Theorem 7 (Weierstrass) and Exercise 1 (with hint). You get series expansions at least some of which (many) are convergent.
Seems to have finally worked, scan from pages 194 and 196 of Complex Analysis by Lars V. Ahlfors, second edition 1966.
Trying jpeg, Lineart setting, 600 dpi, evidently about 5.1 MB. This is the highest degree of difficulty, the judges have been coming down pretty hard on gymnasts who overreach, but after a disappointing rotation on the vault it is time to throw it all out there for the floor exercise. No, after three minutes attempting to download, she is going with the more conservative 300 dpi. Not a popular choice, there are boos from around the arena. But at 1.4 MB it seems she has just about avoided disqualification, just nine people-seconds to upload on imgur.
I have come up with a simpler equation while formulating my question.
Idea is to interpolate between X and Y axis proportionally to distance to them.
Let me guide you through with an example:
// Input
X = 0.5; Y = 0.1
// This is base interpolation that prefers closer edge
x = A * (1-Y) + B * X + C * Y + D * (1-X); // Sum = 2
// Now calculate proportion between distances to edges
distX = 0.5 - Abs(X-0.5); // Distance to edge 0..0.5 range
distY = 0.5 - Abs(Y-0.5); // Distance to edge 0..0.5 range
if distX + distY > 0.001 then //0.001 is to deal with FP precision loss
begin
distSum = distX + distY;
coefX := distX / distSum; // 0..1 range
coefY := distY / distSum; // 0..1 range
end
else
begin
coefX := 0.5;
coefY := 0.5;
end;
// Our example values
coefX = 0.5 / 0.6 = 0.83
coefY = 0.1 / 0.6 = 0.17
// Add X/Y weights into equation, to get Sum = 1
x = A * (1-Y) * coefX + B * X * coefY + C * Y * coefX + D * (1-X) * coefY;
// Our example values
x = A * 0.747 + B * 0.085 + C * 0.083 + D * 0.085; // Sum = 1
Sorry I'm not familiar with Excel formulas close enough to build a diagram, but it looks much alike this:
Benefits are: no need for iterations, knowing ABCD values and X Y position of x is enough.
Best Answer
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
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 $\pi$ ?