But as commenter Frank noted, this is not the only significant difference that might be of interest. So in this post, I'll show some plots that help to bound the range of signficance.

The first plot shows, for each start and end month, the lowest trend that the observed trend would be significantly less than. That is useful for testing predictions that might be failing on the cool side. For example, if you think that there is a claim that the trend over a period should have been 2°C/century, you can see where the actual trend was

*significantly*(at 95%) below that.

The second plot, below the jump, shows the converse. What is the highest trend that was significantly (95%) below the observed. This is probably mainly of interest in establishing whether a trend was significantly above zero, but you might be interested in other values - if you think a theory significantly under-predicts.

The final plot is of the t-statistic - the trend normalized by its standard error. This lets you look at other degrees of significance - mainly relative to zero, but in conjunction with the other plots, you can work out other trend comparisons too.

I've retained the apparatus whereby you can check each point (by clicking) against its plot, echoing the numerical trend value (and period). I've changed from earlier plots of this kind by allowing trend periods down to 1 year.

#### Purpose

I should at this stage say that my purpose here is to show how the much invoked arithmetic of trend fitting works out. I'm not saying that it is always a good thing to do, and there are cautions about what significance means.Saying that a trend is significantly different from a base trend is saying that the on the null hypothesis that the data is formed from the base trend plus random noise the observed result is improbable. Cautions:

- Lack of significance does not mean the base trend is right. It just means it is consistent with this data. Many other possibilities would also be consistent.
- Significance does not mean that any physics, say AGW, is disproved. It just means that there may be something more than random variation plus trend.
- And it may not even mean that. It says the result, on the null hypothesis, is improbable. But improbable things happen. There are about half a million dots on the longest plots. If they were independent, and each had a 5% chance of being in a certain range, then that means 25000 significant dots. Even with correlation, you might still expect something like 5% of the area to show as significant.

#### Plots

So here is the first plot, showing which trends the observed trend would be significantly less than. To use it, pick a trend (color) you want to test from the legend. That andLand and Ocean Hadcrut GISSLO NOAA UAH MSU.RSS TempLS | Land Only BEST GissTs CRUTEM NOAAland Sea Surface HADSST2 |

And here is the second plot, showing trends where the observed would be significantly greater. The brown marks the edge of the area where the trend is significantly greater than zero.

Land and Ocean Hadcrut GISSLO NOAA UAH MSU.RSS TempLS | Land Only BEST GissTs CRUTEM NOAAland Sea Surface HADSST2 |

And the third plot, which shows the t-statistic, or ratio of the trend to its standard error. For the number of degrees of freedom here, this is distributed normally, and 1.96, marked in brown, is the level of 95% significance. 1.64 is 90%, and 2.58 is 99%. One observation here is that there is only a small fringe region where a choice of a different test level would alter the result..

Land and Ocean Hadcrut GISSLO NOAA UAH MSU.RSS TempLS | Land Only BEST GissTs CRUTEM NOAAland Sea Surface HADSST2 |

Hey Nick,

ReplyDeleteCould you state what regression model is used for linear trends? AR(1), AR(2) ARMA(1,1)...?

Barry,

ReplyDeleteI'm using Ar(1). I've written about the issues here, here, and here. In the last, I'm specifically looking at Ar(1) and alternatives, and giving the case for Ar(1).

ReplyDelete"Could you state what regression model is used for linear trends? AR(1), AR(2) ARMA(1,1)...?"Why would you use any of these when the global trend is deterministic and not stochastic?

That said, I do agree that if any physics-based stochastic model is used it should be the Ornstein-Uhlenbeck process, which statisticians refer to as AR(1)

How the Ornstein–Uhlenbeck process can be considered as the continuous-time analogue of the discrete-time AR(1) process?

My question is exactly what is the random walk element that you are trying to extract? The only thing I see that is close to random is volcanic activity.

The trend may be deterministic, but we have to calculate it in the presence of variation that we can't predict, and that affects the outcome. So we'd like to know, if that variation had worked out differently, how would that affect our answer for trend. Ar(1) is our model for how it might have worked out differently. It isn't a perfect model; it's a matter of finding the best you can.

DeleteYet the Ornstein-Uhlenbeck process has a strong reversion to the mean, which means that the only way you will see any kind of large low-frequency fluctuations is if the diffusion and potential parameter is somehow set to give that. Otherwise it will just bounce around the mean without any kind of trend appearing. So to get the low-frequency movements means that you probably won't be able to pick up the faster ENSO fluctuations.

DeleteBottom-line is that essentially what you are saying is that you keep the AR(1) model around so you can laugh at how bad it works to explain anything like ENSO. That's the only way I will use it for these climate time series.

Taking the Auckland tidal gauge SLH data to extract SOI

http://contextearth.com/2016/04/13/seasonal-aliasing-of-tidal-forcing-in-mean-sea-level-height/#comment-177421

Delete"Otherwise it will just bounce around the mean without any kind of trend appearing"AR(1) isn't modelling the trend; it is modelling the residuals. And the only use made of it here is to see how much it increases uncertainty of trend.

In this post I show how it interacts with oscillations that do appear in the acf, which I think are related to ENSO.

How can you be modeling the residuals if you don't even have a model for ENSO?

DeleteIts a good thing that I am asking these pointed questions. Like I mentioned, you ought to look closely at how I characterize mean SLH from tidal gauge readings at Sydney and Aukland here -- http://contextearth.com/2016/04/13/seasonal-aliasing-of-tidal-forcing-in-mean-sea-level-height/

The goal is to decompose the time series until the residual is white noise.