# How do you derive a function that describes a series?

It's been a really long time since I've done calculus or any other kind of math beyond tip calculation. I was given a spreadsheet that calculates and plots a growth curve over time based on a handful of inputs. It goes out for one year. I'd like to be able to make educated (though not necessarily precise) predictions of values along the curve for any amount of time, but if I can avoid it, I'd rather not just extend the length of the spreadsheet. The series is nonlinear, the first values look like this:

540 545 561 599 672 780 904 1,016 1,098 1,151 1,193 1,238 1,297 1,367 1,444

THat's the first ten days of the year, given a 'seed' value of 500 (which of course is one of the variables).

Like I said, I don't need it to be super accurate, just generally describe the slope over time, I just don't know where to even begin figuring out the equation.

Thanks, Paul

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Point of vocabulary: you probably mean "sequence", not "series". Also, the set of data you gave has fifteen elements for ten days, i.e. more than one data point per day: is that accurate? If so, is the data generated on a regular time interval (that interval being, I guess, one data point every 16 hours)? – Alex Basson Jan 2 '11 at 17:31
Problem is "time series" is frequently used to refer to a sequence indexed by time. – Raskolnikov Jan 2 '11 at 19:37
No, sorry, headspace and timing error. It's one data point per day. – Paul Jan 3 '11 at 2:23

You are describing a "regression". Depending on how the data looks plotted on a graph, you should choose a type of regression that looks like it will fit the data (quadratic, cubic, linear, exponential, logarithmic, logistic, etc.). However, I would suggest you go to the stackexchange for statistics, since this doesn't really qualify as mathematics. Thanks! (Please note that I mean no disrespect. I am merely pointing out that statisticians will be able to tell you how to avoid common mistakes, like trying to get "too good a fit". There are infinitely many functions that go through a finite number of points, and there is not a unique function that gives a "best fit" without restricting the class of functions (to some of the classes I mentioned above)).

Point of order (Poo): You do not need to do this by hand. Most spreadsheet software supports these types of computations (excel, for instance)!

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Maybe I missed it; the Excel examples I found were all using linear regression, and since this is modeling a growth curve (potentially viral), it seemed a poor fit for linear regression (the only of the types you mentioned thatI recognize by sight. How do you know "by the look" what regression method to use? – Paul Jan 3 '11 at 2:27
OK, based on your advice, I did some more googling, decided from examples that my data is exponential and did a best fit curve equation that works pretty well after going through all the recommended analysis. Thanks! – Paul Jan 3 '11 at 3:42
@Paul: No problem! When I say "by the look", I mean that there is a bit of art to it. If the growth looks exponential and you expect it to be exponential, you should try an exponential curve. If it looks polynomial, try quadratic or cubic regressions. Remember, the more data you have, the more accurately you can predict what type of curve you should have. – deeeez Jan 3 '11 at 4:21
Well, the sample I have is a year (365 data points) and is pretty clearly exponential along those points. But it's modeling online community population growth, which clearly can't be expected to be exponential for all time, I just dont' have any data outside that year. – Paul Jan 4 '11 at 5:52

I'm not clear on the boundary between mathematics and statistics in this area. Any good book on numerical methods will have routines for extrapolation. My favorite is chapter 3 of Numerical Recipes. Excel has routines for doing this, but a text like NR will have useful cautionary notes.

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