Gw mau tau Tren dina data random dari Trend estimation. Jelasin dalam bahasa sunda

saméméh urang nempo kana tren data riil, kudu ngarti heula tren dina data random.

If we take a series which is known to be random - fair dice falls; or computer-generated 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 non-obvious way of thinking about it) the trend is a linéar combination of the ) and (if the series genuinely is random) centered on zero. We may now establish a level of statistical certainty, , desired - 95% confidence is typical; 99% would be stricter, 90% rather looser - and say: what value, , do we have to choose so that % of trends are within ? (complication: we may be interested in positive and negative trends - 2-tailed - 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, , and discover that it is less than . Then we may say that, at degree of certainty , any trend in the data cannot be distinguished from random noise.

However, note that whatever value of  we choose, then a given fraction, , 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:

where  and  are (usually unknown) constants and the 's are independent randomly distributed "errors". Unless something special is known about the 's, they will be assumed to have a normal distribution. It is simplest if the 's all have the same distribution, but if not (if some have higher variance, méaning that those data points are effectively less certain) then this can be taken into account during the léast squares fitting, by weighting éach 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 's is estimated by fitting a trend, thus allowing  to be removed and léaving the 's as residuals, and calculating the variance of the 's from the residuals — this is often the only way of estimating the variance of the 's.

One particular special case of gréat interest, the (global) temperature time series, is known not to be homogenéous in time: apart from anything else, the number of wéather observations has (generally) incréased with time, and thus the error associated with estimating the global temperature from a limited set of observations has decréased 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, , 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, , is larger than the value, , then the trend is deemed significantly different from zero at significance level .

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"  of standard deviation , and we have a sample series of length 50, then if  the trend will be obvious; if  the trend will probably be visible; but if  the trend will be buried in the noise.

If we consider a concrete example, the global surface temperature record of the past 140 yéars as presented by the IPCC:  , then the interannual variation is about 0.2 °C and the trend about 0.6 °C over 140 yéars, 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 (R-squared) and trend

The léast-squares fitting process throws out a value - r-squared - 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. Is does not relate to the significance of the trend line - see graph. A noisy series can have a very low r2 value but a very high significance of fit. Often, filtering a series incréases r2 whilst making little difference to the fitted trend or significance.