Posts Tagged ‘unit root’

The dietician responds

April 16, 2010

Bart claimed that his weight in insensitive to his food intake, because a stochastic model of his body weight can not be rejected.

As a dietician, I beg to differ with this conclusion. The fact that one can construct a stochastic model that envelops the observed evolution of his bodyweight is by itself not very informative. Many finite datasets of a quantity that is known to be physically driven can probably mathematically be described by a stochastic model.

If a few years from now Bart’s bodyweight increases to outside the 95% confidence interval of this stochastic model, the current model would be falsified. No problem. It’s probably possible to construct another stochastic model, tested not on the period 1978 -1992 (1880-1935), but e.g. on 1978 – 1994, or whatever period works, to obtain an even broader envelope of potential outcomes. (After all, if the model is based on a period where Bart’s personal energy balance already led to an even bigger increase in his bodyweight, it will probably widen even more in potential outcome). So even if Bart wanted to try to falsify this hypothesis by e.g. eating as many brownies as he can over next years, chances are the hypothesis could be easily amended to still encompass his new weight.

Conversely, if the model was tested on 1978 – 1988 (1880 – 1920), what would it have looked like? Would the upswing in Bart’s bodyweight in recent years still be within the 95% confidence interval? Perhaps it would, but I wouldn’t bank on it.

In any case, the stochastic model predicts equal chances of Bart’s body weight to increase or decrease, irrespective of Bart’s eating, sporting, and other relevant habits or state of health. That runs counter to the accumulated knowledge of the human body, not to mention to conservation of energy. How much did you weight as a baby? According to the stochastic model, Bart’s body weight could as easily have turned out to be 60 kg rather than his current 100 kg, despite his eating habits. That’s preposterous.

I don’t claim to be able to predict Bart’s bodyweight to the gram, but I do claim to be able to make a more skillful prediction than this stochastic model. Namely, if I’d have access to data on his relevant habits (eating, drinking, sporting, sickness, state of metabolism, etc), I could explain within certain boundaries (much tighter than the stochastic model boundaries) how his bodyweight changed over the past decades, and why.

What’s worthwhile in this context is to have an explanatory model/framework for your bodyweight. If you change your eating habits to such and such, how is your bodyweight likely to respond? That’s an important question. An answer that your bodyweight is insensitive to what you eat, and that it could vary anywhere between -25 kg and +25 kg of your current weight is uniformative to the extreme.

According to the stochastic model there are no explanatory, deterministic variables for your body weight; it just varies within very wide bounds. As such, it is an essentially meaningless prediction. Choosing to believe this model gives you the benefit of eating to your heart’s content, presumably without it influencing your body weight. Actually, your body weight may tend to go down as it approaches the less likely boundaries of the prediction interval. Even when eating all those mars bars! I don’t blame you for wanting to believe in it.

However, I would urge you to take my thoughts into consideration when deciding about your eating habits. But in the end, it’s your choice; it’s your body after all. You chose how to deal with your own body.

At this crucial point the analogy breaks down.

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A ‘rooty’ solution to my weight gain problem

April 1, 2010

I just love brownies, chocolate fudge cake and the like. As a result of eating too many of those –so my dietician told me- I have gained weight over the past years. According to my dietician, somebody’s body weight depends on the ratio of their caloric input and output (i.e. someone’s personal ‘energy balance’). I also  believed that. Until recently.

Here’s a graph of my body weight over the past 32 years:

As you can see from this graph, I’ve been on quite a few diets. But often, as soon as I had lost a few pounds, they came back when I lost my appetite in carrots and hunted down the chocolate aisle again. In the nineties, I did quite a bit of sports, which prevented my weight from increasing too much. I’ve stopped since; it just makes me tired.

Over the past 8 years, despite the yo-yo effect of sometimes losing as much as 5 kg over the course of a few months, my weight has increased. My dietician told me that unless I change my eating and sporting habits in a sustainable way, my weight will probably keep yo-yo-ing up.

I was gonna go back to drinking carrot juice again, but then somebody pointed out that my weight increase had nothing to do with my eating too much chocolate or anything like that. Huh? 

He pointed out to me that the timeseries of my weight versus time (as shown in the graph above) contains a unit root! No, that’s not a consequence of eating too much carrots; it’s a characteristic of the time series. So what, you my ask? Well,

a deterministic trend is inconsistent with a unit root

Though admittedly,

it can contain a drift parameter, which indeed predicts a ‘deterministic’ rise in a certain period

According to this theory, my body weight just varies stochastically, e.g. between the blue lines in the graph below:

As you can see, the theory is valid: My weight has indeed remained between the blue lines. And for the next few years, my weight will be between 55 and 105 kg, irrespective of what I eat and how much I sport! After all, that would be deterministic, wouldn’t it? (i.e. my eating and other habits determining my weight)

Wow, if that’s the case, then I’ll stop my carrot juice diet right now and run to the corner store for a box of mars bars!! And I’ll cancel further consultations with my dietician. Energy balance… such nonsense. Never thought I’d be so happy with a root!

 

PS: This post is not meant to ridicule the arguments made in favor of a unit root. It is meant to draw attention to the fact that the physical (or biological in this case) context of the quantity we’re investigating is very important. If someone is riding a bike downhill, I could wonder if the bike could have gotten to where it is all by itself, and conclude that I cannot possibly predict when the bike will reach the valley. But that ignores the (deterministic) effect of the guy who is riding the bike. Share your favorite analogy in the comments!

[Some typos edited]

The relevance of rooting for a unit root

March 18, 2010

So what if the global average temperature series contained a unit root? It would mean that ordinary least squares regression may lead to spurious results in terms of inflated trend significance. It would *not* mean that phsyics-based climate models are suddenly invalid or that AGW is suddenly falsified (just as gravity is not falsified by observing a bird in the sky).

On a previous post, ‘VS’ commented that

“(…) global temperature contains a stochastic rather than deterministic trend, and is statistically speaking, a random walk.”

He later clarified (updated):

I agree with you that temperatures are not ‘in essence’ a random walk, just like many (if not all) economic variables observed as random walks are in fact not random walks.

And later still:

“I’m not ‘disproving’ AGWH here.
I’m not claiming that temperatures are a random walk.
I’m not ‘denying’ the laws of physics.”

However, many commenters started chiming in with a sense of “Yeah, somebody is taking on climate science and seems to have refuted it all!” Uhm, no.

Basically, a random walk towards warmer air temperatures would cause either a negative radiative imbalance at the top of the atmosphere, or the energy would have to come from other parts of the earth’s system. Neither is the case. It’s actually opposite: There is a positive radiation imbalance and other reservoirs (e.g. oceans, cryosphere) are also gaining more energy. Which makes sense, in the face of a radiative forcing.

Explaining the increase in global average temperatures by a mere ‘random walk’ would violate conservation of energy.

Ramanathan and Feng describe the earth’s radiation balance as follows:

So the process of the net incoming (downward solar energy minus the reflected) solar energy warming the system and the outgoing heat radiation from the warmer planet escaping to space goes on, until the two components of the energy are in balance. On an average sense, it is this radiation energy balance that provides a powerful constraint for the global average temperature of the planet.

I.e. The global average temperature only changes over climatic timescales (multiple decades or longer) if there is an imbalance in the radiation budget. As is now indeed the case. Climate is to a certain extent deterministic, irrespective of unit roots.

The presence/absence of a unit root (dependent on the nature of the assumed trend amongst other choices) does not disprove/prove that the extra greenhouse gases we put in the atmosphere are warming the planet.

Update: This discussion has focussed on global average air temperatures, but changes have been observed in many other parts of the earth system that point to a changing (warming) climate: Sea level rise, ocean heat content, ice sheets , sea ice, glaciers, ecosystems, radiation budget. A statement along the lines of ‘nothing anomalous is happening’ should take all these changes into account.


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