The Math behind Climate Change: Part 7 Trendlines from the best fitting polynomial method

Here is a typical set of data we are going to be looking at, one season in a region over a period of years, this particular graph the winters in Greenland from 1955 to 2010.

As you can see, the temperatures fluctuate a lot. Just eyeballing the situation, it looks like temperatures were dropping from the mid 1980s to the early 1990s, but from that low the trend is going to 2010, the warmest winter 







Here is a best fitting curve, this one the best fitting line. As you can see, it's fairly flat, so it would say that winters in Greenland are not showing a warming trend. Is that the end of the story?

No, we can fit any kind of curve to a set of data. If we are limited to polynomials, the next size up is the second degree, known as a parabola or quadratic equation. Now the curve hits a single lowest point known as the vertex and rises. It's also possible for a parabola to hit a single highest point. In any case, this shows a slight warming trend from the early 1980s on, but still shows little change from 1955 to 2010.

Next is the best fitting third degree polynomial, known as a cubic, which tells us a completely different story, a rising trend until the mid 1960s, a dropping trend until the early 1990s then a strong rising trend.
 









Next is the best fitting fourth degree polynomial, known as a quartic. It tells us the same story as the cubic only more so. If you enlarge the pictures, you'll see the variable R² gets larger as the degree of the polynomial rises. In layman's terms, that means the fit is better.

Next is the best fitting fifth degree polynomial, known as a quintic. The only significant difference is the little downward trend in the mid 1950s before the upward trend.

Next is the best fitting sixth degree polynomial, known as a sextic. This looks significantly different from the quintic, with a little downward trend at the end even though 2010 was obviously the warmest winter on record.

What gives?

Simply put, the polynomial approximations will get better eventually as more degrees are added, but the sequence of polynomials can't be trusted. It allows for cherry picking, which is the thing I want expressly to avoid.

Tomorrow, I will propose another way to look at trends, simpler than best fitting lines and incorporation the Oceanic Nino Intervals (ONI) that I consider vital to the study of the field.