Trend estimation
From Academic Kids

A series of measurements of a process may be treated as a time series, and then trend estimation is the application of statistical techniques to make and justify statements about trends in the data. Assuming the underlying process is a physical system that is incompletely understood, one may thereby construct a model, independent of anything known about the physics of the process, to explain the behaviour of the measurement. In particular, one may wish to know if the measurements exhibit an increasing or decreasing trend, that can be statistically distinguished from random behaviour. For example, take daily average temperatures at a given location, from winter to summer; or the global temperature series over the last 100 years.
Particularly in that latter case, issues of homogeneity (is the series equally reliable throughout its length?) are important. For the moment we shall simplify the discussion and neglect those points. This page does not attempt a full mathematical treatment, merely an exposition.
Contents 
Fitting a trend: leastsquares
Given a set of data, and the desire to produce some kind of "model" of that data (model, in this case, meaning a function fitted through the data) there are a variety of functions that can be chosen for the fit. But if there is no prior understanding of the data, the simplest function to fit is a straight line and thus this is the "default".
Continuing, once it has been decided to fit a straight line, there are various ways to do so, but the overwhelming default is the leastsquares fit, equivalent to minimisation of the L^{2} norm. See least squares.
Thus, given a set of data points <math>x_i<math>, and data values <math>y_i<math>, one choses <math>a<math> and <math>b<math> so that
 <math>\sum \{[(ax_i + b)  y_i]^2\}<math>
is minimised. This can always be done, in closed form [1] (http://mathworld.wolfram.com/LeastSquaresFitting.html).
For the rest of this article, "trend" will mean the least squares trend. It's what it means in 99% of cases everywhere else.
Now, we have a trend. But is it significant? And what do we mean by significant?
Trends in random data
Before we can consider trends in real data, we need to understand trends in random data.
If we take a series which is known to be random  fair dice falls; or computergenerated random numbers  and fit a trend line through the data, the chances of a truly zero trend are negligible. But we would probably expect the trend to be "small". If we take a series with a given degree of noise, and a given length (say, 100 points), and generate a large number of such series (say, 100,000 series), we can then calculate the trends from these 100,000 series, and empirically establish a distribution of trends that are to be expected from such random data  see diagram. Such a distribution will be normal (Central limit theorem except in pathological cases, since (in a slightly nonobvious way of thinking about it) the trend is a linear combination of the <math>y_i<math>) and (if the series genuinely is random) centered on zero. We may now establish a level of statistical certainty, <math>S<math>, desired  95% confidence is typical; 99% would be stricter, 90% rather looser  and say: what value, <math>V<math>, do we have to choose so that <math>S<math>% of trends are within <math>V<math>? (complication: we may be interested in positive and negative trends  2tailed  or may have prior knowledge that only positive, or only negative, trends are of interest).
In the above discussion the distribution of trends was calculated empirically, from a large number of trials. In simple cases (normally distributed random noise being a classic) the distribution of trends can be calculated exactly.
Suppose we then take another series with approximately the same variance properties as our random series. We do not know in advance whether it "really" has a trend in it, so we calculate the trend, <math>T<math>, and discover that it is less than <math>V<math>. Then we may say that, at degree of certainty <math>S<math>, any trend in the data cannot be distinguished from random noise.
However, note that whatever value of <math>S<math> we choose, then a given fraction, <math>1S<math>, of truly random series will be declared (falsely, by construction) to have a significant trend. Conversely, a certain fraction of series that "really" have a trend will not be declared to have a trend.
Data as trend plus noise
To analyse a (time) series of data, we assume that it may be represented as trend plus noise:
<math>x_i = at_i + b + e_i<math>
where <math>a<math> and <math>b<math> are (usually unknown) constants and the <math>e<math>'s are independent randomly distributed "errors". Unless something special is known about the <math>e<math>'s, they will be assumed to have a normal distribution. It is simplest if the <math>e<math>'s all have the same distribution, but if not (if some have higher variance, meaning that those data points are effectively less certain) then this can be taken into account during the least squares fitting, by weighting each point by the inverse of the variance of that point.
In most cases, where only a single time series exists to be analysed, the variance of the <math>e<math>'s is estimated by fitting a trend, thus allowing <math>at+b<math> to be removed and leaving the <math>e<math>'s as residuals, and calculating the variance of the <math>e<math>'s from the residuals — this is often the only way of estimating the variance of the <math>e<math>'s.
One particular special case of great interest, the (global) temperature time series, is known not to be homogeneous in time: apart from anything else, the number of weather observations has (generally) increased with time, and thus the error associated with estimating the global temperature from a limited set of observations has decreased with time. In fitting a trend to this data, this can be taken into account, as described above.
Once we know the "noise" of the series, we can then assess the significance of the trend by making the null hypothesis that the trend, <math>a<math>, is not significantly different from 0. From the above discussion of trends in random data with known variance, we know the distribution of trends to be expected from random (trendless) data. If the calculated trend, <math>a<math>, is larger than the value, <math>V<math>, then the trend is deemed significantly different from zero at significance level <math>S<math>.
Noisy time series, and an example
It is harder to see a trend in a noisy time series. For example, if the true series is 0, 1, 2, 3 all plus some independent normally distributed "noise" <math>e<math> of standard deviation <math>E<math>, and we have a sample series of length 50, then if <math>E=0.1<math> the trend will be obvious; if <math>E=100<math> the trend will probably be visible; but if <math>E=10000<math> the trend will be buried in the noise.
If we consider a concrete example, the global surface temperature record of the past 140 years as presented by the IPCC: [2] (http://www.grida.no/climate/ipcc_tar/wg1/figspm1.htm), then the interannual variation is about 0.2°C and the trend about 0.6°C over 140 years, with 95% confidence limits of 0.2°C (by coincidence, about the same value as the interannual variation). Hence the trend is statistically different from 0. This alone, however, tells us nothing about the physical causes of the temperature change.
Goodness of fit (Rsquared) and trend
Randomdataplustrendr2.png
The leastsquares fitting process throws out a value  rsquared (<math>r^2<math>)  which is the square of the residuals of the data after the fit. It says what fraction of the variance of the data is explained by the fitted trend line. It does not relate to the significance of the trend line  see graph. A noisy series can have a very low <math>r^2<math> value but a very high significance of fit. Often, filtering a series increases <math>r^2<math> whilst making little difference to the fitted trend or significance.
Real data is autocorrelated
The above discussion assumed the data could be represented as trend + noise, with the noise at each data point being independent (markov, white noise). The assumption that the noise is markov arrises from the principle of minimal information (One simply does not make any assumptions about the noise.). This is important, as it makes an enormous difference to the ease with which the statistics can be analysed. Real (for example climate data, data from any physical system) data only fulfills this criterion if analysed on a time scale which is long enough (markov approximation).
[This section in need of expansion: examples.]