Demetris Koutsoyiannis contributed the following excellent piece as a comment on a previous post. I have made it into a post to ensure it gets the widest distribution.

### Hurst, Joseph, colours and noises: The importance of names in an important natural behaviour

â€œWhatâ€™s in a name? That which we call a rose

By any other name would smell as sweet.

William Shakespeare, â€œRomeo and Juliet, Act 2 scene 2

Is the name given to a physical phenomenon or in a scientific concept (e.g. a mathematical object) really unimportant? Let us start with a characteristic example, the term â€œregression”. The term was coined by Frances Galton who studied biological data and noticed that the offspring population were closer to the overall mean size than the parent population. For example, sons of unusually short fathers have heights typically closer to the mean height than their fathers. Today we know that this does not manifest a peculiar biological phenomenon but a normal and global statistical behaviour. The slope of the least squares straight line of two variables x and y is r_xy * s_y / s_x, where s_x and s_y are the standard deviations of the variables and r_xy is the correlation coefficient. In the example of the height of fathers and sons, s_x = s_y, so the slope is precisely r_xy, which (by definition) is not greater than one; hence the â€œregression” towards the mean. Today no one has any problem with this generally accepted term, even though clearly it is not a good name. No one has problem to understand the statistical (rather than biological or physical) origin of the â€œregression” and its irrelevance with time: For example the fathers of exceptionally short people also tend to be closer to the mean than their sons. Just interchange y and x (and the axes in the graph) and you will have again another line whose slope (in the new graph) will be again r_xy, that is, not greater than unity. However, until people understood these simple truths, the improper term must have caused several fallacies (see Regression fallacies in the Wikipedia article â€œRegression toward the mean”, http://en.wikipedia.org/wiki/Regression_toward_the_mean).

Thus, it could be maintained that, at least at the initial stages of the study of a scientific concept and before its establishment and wide dissemination, the names used are closely related to understanding and explanation. Vit Klemes (1974), in a pioneering and famous paper notes how important the explanation and understanding is and how a model can hinder them: â€œIndeed it is the very success of an operational model that by diverting further attention from the problem, often delays satisfactory explanation and understanding. I think that a bad name can, too, hinder explanation and understanding, and that poor explanation and understanding puts bounds to progress in modelling. Klemes uses the example of the Ptolemaic planetary model saying â€œIt was exactly because it â€˜workedâ€™ so well (its predictions of position of stars were more than accurate enough for the contemporary needs) that it hampered progress in astronomy for centuries. Here I could add that the geocentric model of Ptolemy (90-168 AD), despite its successful predictions was in fact a regression (with the literal rather than the statistical meaning of the word â€˜regressionâ€™): four centuries earlier Aristarchus of Samos (310-230 BC) had formulated the heliocentric model of the solar system (1800 years before Copernicus, who admits this in a note), and figured out how to measure the distances to and sizes of the Sun and the Moon. And at about the same time, Eratosthenes (276-194 BC) measured with an error of only 3% the circumference of the earth, based on the angle of the sunâ€™s rays at different places at noon; this happened 1700 years before Columbus, who must have used Ptolemyâ€™s estimate (underestimated by about 30%) of the circumference of the earth, thus giving the incorrect name â€˜Indiansâ€™ to the people of the new continent.

Coming back to the geophysical bevaviour that Hurst discovered, it is interesting to quote again Vit Klemes: â€œFortunately the success of fractional noises does not seem to be so universal that it could pose a similar danger to progress in hydrology and related sciences. My interpretation of this sentence is that Klemes (a) makes a clear distinction of the natural Hurst phenomenon and its mathematical modeling; (b) he disapproves the â€˜fractional noiseâ€™ as a model for this behaviour; but (c) simultaneously accepts the natural behaviour and seeks for an explanation of it (as seen from the entire context). I concur with Vit Klemes that it is important to make the distinction of the natural behaviour and the mathematical model; the fact that some fail to do this distinction always creates confusion. Besides, any mathematical model is only an approximation of reality; so one has the right not to like even a successful model and seek for a better one. In my opinion, â€˜fractional Gaussian noiseâ€™ is a good model (if demystified somewhat) but its name is not good. Nevertheless, this is not the only name associated with the Hurst phenomenon; both the natural behaviour and the models devised have been given a plethora of names, which alone creates confusion. Besides, several of these names are not good enough. In my opinion, the inappropriate names is one of the reasons (obviously there are additional ones that are not discussed here) that this natural behaviour has been regarded as a puzzle or a mystery, perhaps metaphysical, and that its consequences were not understood or were neglected, more than half a century after its discovery by Hurst.

Here are the lists of names (perhaps not complete) separated into two categories, names for the natural behaviour and names for models, along with my comments on the names:

#### A. Natural behaviour

A1. Hurst phenomenon: This is the best in my opinion; it respectfully attributes the behaviour to the engineer E. H. Hurst who discovered and studied it in geophysics. Here I wish to point out that the some people have identified the behaviour with the properties of a statistic called â€˜rescaled rangeâ€™ that Hurst used to report the behaviour. This, in my opinion, is not an ideal statistic (see Koutsoyiannis, 2002, 2003, 2006) and there is no reason to continue using it today and to identify the phenomenon with properties of this statistic.

A2. Joseph effect: this was coined by Benoit Mandelbrot (1977) who associated it to the biblical story of the seven fat and the seven thin cows. I have used it in talks addressed to general audience and I found that it helps people to approach the concept. However, the periodicity it implies and its association with the â€˜magicalâ€™ number seven (which some in the audiences have tried to point out) do not make it a good scientific term.

A3. Long memory: This is the worst name in my opinion. It stimulates people to imagine a mechanism inducing long memory (e.g. hundreds of years) and of course it is difficult to conceptualize such a mechanism. On the contrary, the mechanism dominating in this behaviour could be better characterized as absence of memory, as I tried to explain elsewhere (Koutsoyiannis, 2002).

A4. Long-range dependence: It is better than â€˜long memoryâ€™ as it is free of the metaphorical meaning of â€˜memoryâ€™. It is mathematically precise, so it is good to be used to describe a property of a model (that is, a stochastic process). However, it may be misleading in describing a natural behaviour and it does not point to any physical mechanism.

A5. Long-term persistence: â€˜Persistenceâ€™ is a term more understandable, in physical terms, than â€˜dependenceâ€™, whose conceptualization lies in stochastic processes. The term however implies some mystery, as in â€˜Joseph effectâ€™.

A6: Scaling behaviour: This is a concise and fashionable term, expressing the equivalence of (time) scales in this behaviour. I have used it a lot. The problem is that scaling is not a physical mechanism but a result of one or more other physical mechanisms or principles (perhaps the maximum entropy, as I tried to show in Koutsoyiannis, 2005a,b). Thus, it does not help understanding the physical concept.

A7. Multi-scale fluctuation: I have coined this term and I believe it demystifies the behaviour and makes it easily understandable. We are familiar with daily, seasonal and annual irregular fluctuations of weather and hydrologic quantities. If we expand these fluctuations at larger scales, say tens and hundreds of years (and there is no reason why we should not), then we obtain the Hurst behaviour (Koutsoyiannis, 2002).

#### B. Mathematical models

Firstly I should clarify that the names of models listed below do not refer to a single stochastic process but to two closely related processes; the fist (with names B1-B4) is a cumulative, continuous time, non-stationary process (such as in the cumulated rainfall depth at a site, which increases in time ever). The second (with names B5-B10) is the discrete-time stationary process that is obtained by taking the differences of the first process at equidistant times (such as in the annual rainfall depth); it could be also a continuous time process if derivative is used instead of difference. The second process more directly corresponds to what we study in geophysics and therefore is the most commonly used in branches of geophysics such as in climatology and hydrology.

A general observation is that several of the names of models contain the term â€˜noiseâ€™. I do not find this a good term for geophysics. Generally, â€˜noiseâ€™ is used (e.g. in electronics, information and communication) in contrast to â€™signalâ€™ and the distinction implies that there is some signal that contains information, which is contaminated by a (random) noise. Noise should be identified and removed from the signal to recover the maximum of information. Such a distinction may not have a meaning in geophysics. The evolution in time of temperature or rainfall, as we measure it at a site, has the characteristics of â€˜noiseâ€™ rather than those of a typical signal of anthropic origin, i.e. it is irregular or random. Yet it has some structure and certainly it is the â€™signalâ€™ of nature, so I do not think we could classify it as â€˜noiseâ€™. In recent studies, some attempted to find the signal-to-noise ratio in hydrological time series. They successfully applied algorithms (e.g. from the chaotic literature) to obtain a certain value of signal-to-noise ratio, but they failed to explain what signal and what noise represents. This failure is expected in my opinion, because natureâ€™s signs are â€™signalsâ€™ in their entirety even though look like â€˜noiseâ€™.

B1. Self-similar process: This name is the most widely used in the mathematical literature today â€“ but not so much in geophysics. It is a precise and concise name.

B2. Wiener spiral (or Wiener helix): This name (honouring the mathematician Norbert Wiener) was given to the process by the Russian mathematician Andrei Nikolaevich Kolmogorov (1940), who introduced and was the first to study it, to model turbulence. It is amazing that Kolmogorov introduced the process ten years before Hurstâ€™s celebrated paper and simultaneously that this contribution is so very little known to geophysical community (including myself). Thus, the name given by Kolmogorov is not at all used today, even though we still use the name â€˜Wiener processâ€™ for the limiting form of the random walk process, which is a special (non-interesting, i.e. without Hurst behaviour) case of a self-similar process.

B3. Semi-stable process: This name was given to the process by the American mathematician John Lamperti (1962). Again this name (and perhaps Lampertiâ€™s significant contribution to its study) has been forgotten today.

B4. Fractional brownian noise. This is due to Benoit Mandelbrot (1965) and it is the most widespread. It may be a name mathematically rigorous, but I do not like it as a whole and each of the three terms separately. The first term, fractional, is not easily understandable, unless combined with fractals, which is not necessary. The second term, brownian, points to brownian motion (the movement of a particle in a liquid subjected to collisions and other forces), which again is not necessary as there not direct connection of the process with the brownian motion. The third term, noise, is unsuccessful as I described above.

B5. Stationary intervals of a self-similar process. This is a mathematically rigorous name of the discrete time stationary process but I think that it is too wordy and it is difficult to understand the meaning it communicates.

B6. Fractional gaussian noise. This corresponds to fractional brownian noise and again is due to Benoit Mandelbrot (1965). I do not like this name too for the reasons explained in B4 and the additional reason that it restricts our view to processes that are gaussian. The gaussian distribution may be not the case for several geophysical processes that are asymmetric (non-gaussian).

B7. Fractional ARIMA process (abbreviated as FARIMA or ARFIMA). This is based on Hoskingâ€™s (1984) work on fractional differencing (in fact meaning taking a weighted sum of infinite terms) of a Box-Jenkins ARMA process, which results in a process with long-range dependence. I do not think that this term communicates any information that helps in understanding.

B8. Red noise. I do not have enough information about the prevailing of this term and the message it contains (for example, the articles in Wikipedia â€œColors of noiseï¿½? http://en.wikipedia.org/wiki/Colors_of_noise and â€œRed noise http://en.wikipedia.org/wiki/Red_noise are not very informative). Perhaps the â€˜redâ€™ colour points to the fact that the multi-scale fluctuation, when studied in the frequency domain, results in high values of power spectrum for low frequencies. Besides, in the visible light, red is the colour with the lowest frequency. However, the scaling behaviour certainly could be not modelled as â€œmonochromatic red, because its power spectrum extends to the entire frequency domain. Besides, I do not like the term â€˜noiseâ€™ as I explained above. Therefore, I do not think that this name is successful.

B9. Brown noise. This has been used as synonymous to â€˜red noiseâ€™. Perhaps it is better than â€˜red noiseâ€™, as brown is not one of the colours of the visible light spectrum (it is a mixture of colours) and also reminds â€˜brownianâ€™, as discussed above.

B10. Simple scaling signal or Simple scaling stochastic process (abbreviated as an SSS process). I have proposed these terms and the abbreviation (Koutsoyiannis, 2003) thinking that they are less misleading than other terms described above, more understandable and more helpful in understanding of the process described by this name. In fact, the definition of the process is a simple scaling relationship involving a power-law of time scale. The abbreviation SSS could be derived in other ways, too (observe that most of the terms above start with an â€™sâ€™); for example â€™stationarized self-smimilarâ€™ with â€™stationarizedâ€™ standing either for taking the â€™stationary intervalsâ€™ (or taking the difference at equidistant times as explained above) or for taking the â€™stationary derivativeâ€™ (at any time instant) in the continuous time version.

### References

Hosking, J. R. M. (1984), Modeling persistence in hydrological time series using fractional differencing, Water Resources Research, 20(12) 1898-1908.

H. E. Hurst (1950), Long-Term Storage Capacity of Reservoirs, Proceedings of the American Society of Civil Engineers, 76(11).

Klemes, V. (1974), The Hurst phenomenon: A puzzle?, Water Resources Research, 10(4) 675-688.

Kolmogorov, A. N. (1940), Wienersche Spiralen und einige andere interessante Kurven in Hilbertschen Raum, Comptes Rendus (Doklady) Acad. Sci. USSR (N.S.) 26, 115â€“118.

Koutsoyiannis, D. (2002), The Hurst phenomenon and fractional Gaussian noise made easy, Hydrological Sciences Journal, 47(4), 573-595.

Koutsoyiannis, D. (2003), Climate change, the Hurst phenomenon, and hydrological statistics, Hydrological Sciences Journal, 48(1), 3-24.

Koutsoyiannis, D. (2005a), Uncertainty, entropy, scaling and hydrological stochastics, 1, Marginal distributional properties of hydrological processes and state scaling, Hydrological Sciences Journal, 50(3), 381-404.

Koutsoyiannis, D. (2005b), Uncertainty, entropy, scaling and hydrological stochastics, 2, Time dependence of hydrological processes and time scaling, Hydrological Sciences Journal, 50(3), 405-426.

Koutsoyiannis, D. (2006), Nonstationarity versus scaling in hydrology, Journal of Hydrology, (in press).

Lamperti, J. W. (1962), Semi-stable stochastic processes, Transactions of the American Mathematical Society, 104, 62-78.

Mandelbrot, B. B. (1965), Une classe de processus stochastiques homothetiques a soi: Application a la loi climatologique de H. E. Hurst, Compte Rendus Academie Science, 260, 3284-3277.

Mandelbrot, B. B. (1977), The Fractal Geometry of Nature, Freeman, New York.

From the online dictionary the word signal has an clear origin [Middle English, from Old French, from Medieval Latin signle, from neuter of Late Latin signlis, of a sign, from Latin signum, sign. See sign.] with a primary meaning invoking primative language: An indicator, such as a gesture or colored light, that serves as a means of communication.

Noise OTOH has a more obscure origin [Middle English, from Old French, perhaps from Vulgar Latin *nausea, discomfort, from Latin nausea, seasickness. See nausea.] that shows the negative meaning: Sound or a sound that is loud, unpleasant, unexpected, or undesired.

This reminds me of a recent comment by Marty Ringo where he pointed out an essay by Henk Tenekes on Roger Pielke’s (Sr.) site (Climate Science). It seems to me that the emphasis on one or the other has practical implications for a climate policy . Here he quoted:

“The constraints imposed by the planetary ecosystem require continuous adjustment and permanent adaptation. Predictive skills are of secondary importance.”

This is a lot to think about, particularly the implications. Is there still a simple noise/signal dichotomy? Or is it the case that simple binaries such a signal/noise, trend/residual, prediction/chaos, amelioration/adaptation look a bit shaky under radical models of noise you propose?

I think one question people would ask is how do you incorporate driving factors in these models. I notice you removed periodic signals in your analysis but not recent trends? Is that inconsistent?

From the online dictionary the word signal has an clear origin [Middle English, from Old French, from Medieval Latin signle, from neuter of Late Latin signlis, of a sign, from Latin signum, sign. See sign.] with a primary meaning invoking primative language: An indicator, such as a gesture or colored light, that serves as a means of communication.

Noise OTOH has a more obscure origin [Middle English, from Old French, perhaps from Vulgar Latin *nausea, discomfort, from Latin nausea, seasickness. See nausea.] that shows the negative meaning: Sound or a sound that is loud, unpleasant, unexpected, or undesired.

This reminds me of a recent comment by Marty Ringo where he pointed out an essay by Henk Tenekes on Roger Pielkeâ€™s (Sr.) site (Climate Science). It seems to me that the emphasis on one or the other has practical implications for a climate policy . Here he quoted:

â€œThe constraints imposed by the planetary ecosystem require continuous adjustment and permanent adaptation. Predictive skills are of secondary importance.â€?

This is a lot to think about, particularly the implications.Â Is there still a simple noise/signal dichotomy?Â Or is it the case that simple binaries such a signal/noise, trend/residual, prediction/chaos, amelioration/adaptation look a bit shaky under radical models of noise you propose?

I think one question people would ask is how do you incorporate driving factors in these models.Â I notice you removed periodic signals in your analysis but not recent trends?Â Is that inconsistent?

David,

Regarding your last question, I would give the following answer:

1. It is difficult, if not impossible, to remove anything from a time series of observations. We sometimes think that we can remove “periodic signals” by standardizing a time series on a sub-annual scale. For example, in a monthly time series we think we can remove periodicity by subtracting the mean and dividing by standard deviation of each month separately. This may be a widespread simple approximate technique but it cannot remove periodicity. Take, for instance the skewness coefficient for each moth of the standardized time series. This will be exactly the same as in the original time series, before standardization. Take the lag-one autocorrelation. Again, this will be exactly the same as in the original time series. (The proof is simple for both). As both skewness and autocorrelation vary with month in the original time series, they will vary in the standardized time series, too. Thus, we have not removed periodicity.

2. Things are even worse with trends. As I have tried to show in Koutsoyiannis (2006) a ‘trend’ is an essential part of the time series and if removed it distorts the time series largely and gives a false impression for the natural behaviour. In addition, trend-looking patterns of natural time series are not trends but parts of large scale fluctuations. Besides, I do not think that anyone could define a trend in a time series objectively.

3. These comments do not mean that we cannot do anything to model natural phenomena with stochastic processes. Modelling does not presuppose decomposing of time series into parts such as ‘trendy’, periodic and random components. This is a bad modelling practice, in my opinion. There exist consistent ‘holistic’ modelling techniques.

4. Specifically, at annual and overannual time scales, we could use an SSS process as a model, which will reproduce the trend-looking behaviour, as I have tried to show in Koutsoyiannis (2006). If one wishes to reproduce exactly an observed ‘trend’ in the past, there are two solutions: (a) One can use an explicit conditional simulation technique as described in Koutsoyiannis (2000). (b) One can use a Monte Carlo conditional simulation technique as you did in your article “A new temperature reconstruction” (http://landshape.org/enm/?p=15).

5. At a sub-annual scale, the answer is to use a cyclostationary, rather than a stationary model, without any removal of periodicity. A simple model of this category is the periodic autoregressive model of order 1 (PAR(1)). This, however, will not reproduce the Hurst behaviour and the trend-looking patterns at overannual scales. Here there are at least two solutions: (a) One can couple PAR(1) or another simple model of this type that describes short scale cyclostationary properties, with an SSS process that takes account of large scale properties. This is described in Koutsoyiannis (2001). (b) One can use a cyclostationary SSS modelling strategy as described in Langousis and Koutsoyiannis (2006). Both cases are more difficult that typical stochastic modelling. However, the first model in Langousis and Koutsoyiannis (2006) is simple enough and can be implemented conveniently; also the single-site case of Koutsoyiannis (2001) associated with a PAR(1) model is simple enough and could be made even simpler (Koutsoyiannis and Manetas, 1996).

References

Koutsoyiannis, D. (2000), A generalized mathematical framework for stochastic simulation and forecast of hydrologic time series, Water Resources Research, 36(6), 1519-1533,.

Koutsoyiannis, D. (2001), Coupling stochastic models of different time scales, Water Resources Research, 37(2), 379-392.

Koutsoyiannis, D. (2006), Nonstationarity versus scaling in hydrology, Journal of Hydrology (in press).

Koutsoyiannis, D., and A. Manetas (1996), Simple disaggregation by accurate adjusting procedures, Water Resources Research, 32(7), 2105-2117.

Langousis, A., and D. Koutsoyiannis (2006), A stochastic methodology for generation of seasonal time series reproducing overyear scaling, Journal of Hydrology (in press).

David,

Regarding your last question, I would give the following answer:

1. It is difficult, if not impossible, to remove anything from a time series of observations. We sometimes think that we can remove “periodic signals” by standardizing a time series on a sub-annual scale. For example, in a monthly time series we think we can remove periodicity by subtracting the mean and dividing by standard deviation of each month separately. This may be a widespread simple approximate technique but it cannot remove periodicity. Take, for instance the skewness coefficient for each moth of the standardized time series. This will be exactly the same as in the original time series, before standardization. Take the lag-one autocorrelation. Again, this will be exactly the same as in the original time series. (The proof is simple for both). As both skewness and autocorrelation vary with month in the original time series, they will vary in the standardized time series, too. Thus, we have not removed periodicity.

2. Things are even worse with trends. As I have tried to show in Koutsoyiannis (2006) a ‘trend’ is an essential part of the time series and if removed it distorts the time series largely and gives a false impression for the natural behaviour. In addition, trend-looking patterns of natural time series are not trends but parts of large scale fluctuations. Besides, I do not think that anyone could define a trend in a time series objectively.

3. These comments do not mean that we cannot do anything to model natural phenomena with stochastic processes. Modelling does not presuppose decomposing of time series into parts such as ‘trendy’, periodic and random components. This is a bad modelling practice, in my opinion. There exist consistent ‘holistic’ modelling techniques.

4. Specifically, at annual and overannual time scales, we could use an SSS process as a model, which will reproduce the trend-looking behaviour, as I have tried to show in Koutsoyiannis (2006). If one wishes to reproduce exactly an observed ‘trend’ in the past, there are two solutions: (a) One can use an explicit conditional simulation technique as described in Koutsoyiannis (2000). (b) One can use a Monte Carlo conditional simulation technique as you did in your article “A new temperature reconstruction” (http://landshape.org/enm/?p=15).

5. At a sub-annual scale, the answer is to use a cyclostationary, rather than a stationary model, without any removal of periodicity. A simple model of this category is the periodic autoregressive model of order 1 (PAR(1)). This, however, will not reproduce the Hurst behaviour and the trend-looking patterns at overannual scales. Here there are at least two solutions: (a) One can couple PAR(1) or another simple model of this type that describes short scale cyclostationary properties, with an SSS process that takes account of large scale properties. This is described in Koutsoyiannis (2001). (b) One can use a cyclostationary SSS modelling strategy as described in Langousis and Koutsoyiannis (2006). Both cases are more difficult that typical stochastic modelling. However, the first model in Langousis and Koutsoyiannis (2006) is simple enough and can be implemented conveniently; also the single-site case of Koutsoyiannis (2001) associated with a PAR(1) model is simple enough and could be made even simpler (Koutsoyiannis and Manetas, 1996).

References

Koutsoyiannis, D. (2000), A generalized mathematical framework for stochastic simulation and forecast of hydrologic time series, Water Resources Research, 36(6), 1519-1533,.

Koutsoyiannis, D. (2001), Coupling stochastic models of different time scales, Water Resources Research, 37(2), 379-392.

Koutsoyiannis, D. (2006), Nonstationarity versus scaling in hydrology, Journal of Hydrology (in press).

Koutsoyiannis, D., and A. Manetas (1996), Simple disaggregation by accurate adjusting procedures, Water Resources Research, 32(7), 2105-2117.

Langousis, A., and D. Koutsoyiannis (2006), A stochastic methodology for generation of seasonal time series reproducing overyear scaling, Journal of Hydrology (in press).

Demetris, under natural models perhaps do you think something should be added about “Entropy maximizing at all scales” and some discussion of intensive and extensive entropy?

Demetris, under natural models perhaps do you think something should be added about “Entropy maximizing at all scales” and some discussion of intensive and extensive entropy?

Demetri,

I think that your article is unique. To be honest although I am doing research on the field for more than 5 years, I once more find myself being a student of your philosophical questions. Indeed a rose smells as nice even if one uses different names for it, as William Shakespeare said. But there is an issue here. Everybody knows what a rose is. It happens that it satisfies 3 of our 5 senses. A blind can touch it, a person without hands can smell it… So only a person that cannot smell, see and touch would not enjoy a rose… This probability is really small (i.e. 0.0125% if we assume that the probability that one looses one of his/her 5 senses is 5%). I do not know a lot of people that have lost 3 of their five senses. That is why terminology is not important in such cases when the expected conclusion is so obvious for almost everybody. However Hurst phenomenon is something that we still trace in data trying to understand its sources. So an accurate description of it is more than important, and I agree with you. Your article took me back to my highschool years and reminded me a conversation that I had with my Professor in Mathematics. I was sick with the symbols currently used in mathematics, and I asked him what do we need them since we can describe them using words. His answer was exactly what you are talking about in your article: “If I was asking you to walk on a relatively wide ramp placed 1 meter over the ground you wouldn’t need a support. How about asking you to walk on this ramp when it is placed 100m over the ground…. I am sure that you would ask for a handgrip….” His words become clearer to me now. He was mentioning to me the need of right terminology when you try to describe something that you cannot see, or even better, when what you see out of it is just a part of the truth and not the truth itself…

Demetri,

I think that your article is unique. To be honest although I am doing research on the field for more than 5 years, I once more find myself being a student of your philosophical questions. Indeed a rose smells as nice even if one uses different names for it, as William Shakespeare said. But there is an issue here. Everybody knows what a rose is. It happens that it satisfies 3 of our 5 senses. A blind can touch it, a person without hands can smell it… So only a person that cannot smell, see and touch would not enjoy a rose… This probability is really small (i.e. 0.0125% if we assume that the probability that one looses one of his/her 5 senses is 5%). I do not know a lot of people that have lost 3 of their five senses. That is why terminology is not important in such cases when the expected conclusion is so obvious for almost everybody. However Hurst phenomenon is something that we still trace in data trying to understand its sources. So an accurate description of it is more than important, and I agree with you. Your article took me back to my highschool years and reminded me a conversation that I had with my Professor in Mathematics. I was sick with the symbols currently used in mathematics, and I asked him what do we need them since we can describe them using words. His answer was exactly what you are talking about in your article: “If I was asking you to walk on a relatively wide ramp placed 1 meter over the ground you wouldn’t need a support. How about asking you to walk on this ramp when it is placed 100m over the ground…. I am sure that you would ask for a handgrip….” His words become clearer to me now. He was mentioning to me the need of right terminology when you try to describe something that you cannot see, or even better, when what you see out of it is just a part of the truth and not the truth itself…

Re #3: Yes, but I have to think it a little.

Re #3: Yes, but I have to think it a little.

With Vit Klemes’s permission, I am posting here his following comment sent to me by email.

—–Original Message—–

From: Vit Klemes

Sent: Monday, March 20, 2006 1:11 AM

To: Demetris Koutsoyiannis

Subject: Re: From cartoons to Hurst (again)

Very good, Demetris!

Well, I also am somewhat allergic to ‘names’ which distort the meaning of the thing being named but, having lived in the “English language environment” for 40 years, I have got used to the fact that ‘names’ often mean just ‘labels’ and have sometimes used this practice myself (remember my use of ‘nonstationarity’?). But when one does this intentionally (say, for the sake of semantic economy), one should clearly define the change in the meaning of the ‘name’ used (but, as you know, this is risky because some readers may not read his definition).

I remember that, when I complained about some such misleading names, one late Canadian colleague of mine used to rebuff it jokingly as “typical European logic chopping”. But I still think that English is far too ‘tolerant’ in this regard. Or could it have something to do with Winston Churchill’s observation that “The English never draw a line without blurring it”?

I don’t mind true labels like ‘Hurst effect’ or ‘Madelbrot set’, but don’t like labels which pretend to be names. For example ‘standard error’ where one is not concerned with measurement errors but with variability of observed phenomena. My most favourite example comes from meteorological terminology: they call the real observed values ‘anomalies’ and their computed averages ‘normals’. I claim that it is exactly the other way around: their ‘normals’ are anomalies since they are seldom observed, while their ‘anomalies’ are the real normal observations.

By the way, I also had some thoughts about ‘names’ in section 4.1 of my paper “Statistics and Probability: Wrong Remedies for a Confused Hydrologic Modeller” (In: Statistics for the Environment, Vol.2, V.Barnett and K.F.Turkman, eds., Ch. 19, pp.345 – 366, John Wiley, Chichester, 1994) from which I am reprinting its first few paragraphs below.

Cordially,

Vit

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4.1 What’s in a name!

The suggestive power of names is well known. Had, for instance, Hamburg rather than Berlin been a divided city, President Kennedy would, no doubt, have thought twice before proudly proclaiming “Ich bin ein Hamburger!”

Ten years ago, in an extremely stimulating interview televised on the NOVA program, the world-renowned American physicist, the late Richard Feynman, told a story how, as a boy, he learned about inertia from his father. When he once asked him why it was that the ball in his toy wagon rolled to the back of the wagon when he pulled it forward, and to its front when he stopped it,

“… he says nobody knows. He said, ‘The general principle is that things that are moving try to keep moving and things that are standing still tend to stand still unless you push on them hard.’ And he says this tendency is called inertia, but nobody knows why it’s true. Now that’s a deep understanding. He doesn’t give me a name. He knew the difference between KNOWING THE NAME OF SOMETHING and KNOWING SOMETHING, which I learned very early” [Feynman, 1983; emphasis added].

I believe that this early knowledge was one of the most important keys to Feynman’s later accomplishments as a scientist and a teacher. I also believe that the lack of this knowledge is one of the main roots of confusions in statistical and stochastic hydrologic modelling. This is why I am taking the liberty of elaborating on the following point which may seem trivial.

……

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With Vit Klemes’s permission, I am posting here his following comment sent to me by email.

—–Original Message—–

From: Vit Klemes

Sent: Monday, March 20, 2006 1:11 AM

To: Demetris Koutsoyiannis

Subject: Re: From cartoons to Hurst (again)

Very good, Demetris!

Well, I also am somewhat allergic to ‘names’ which distort the meaning of the thing being named but, having lived in the “English language environment” for 40 years, I have got used to the fact that ‘names’ often mean just ‘labels’ and have sometimes used this practice myself (remember my use of ‘nonstationarity’?). But when one does this intentionally (say, for the sake of semantic economy), one should clearly define the change in the meaning of the ‘name’ used (but, as you know, this is risky because some readers may not read his definition).

I remember that, when I complained about some such misleading names, one late Canadian colleague of mine used to rebuff it jokingly as “typical European logic chopping”. But I still think that English is far too ‘tolerant’ in this regard. Or could it have something to do with Winston Churchill’s observation that “The English never draw a line without blurring it”?

I don’t mind true labels like ‘Hurst effect’ or ‘Madelbrot set’, but don’t like labels which pretend to be names. For example ‘standard error’ where one is not concerned with measurement errors but with variability of observed phenomena. My most favourite example comes from meteorological terminology: they call the real observed values ‘anomalies’ and their computed averages ‘normals’. I claim that it is exactly the other way around: their ‘normals’ are anomalies since they are seldom observed, while their ‘anomalies’ are the real normal observations.

By the way, I also had some thoughts about ‘names’ in section 4.1 of my paper “Statistics and Probability: Wrong Remedies for a Confused Hydrologic Modeller” (In: Statistics for the Environment, Vol.2, V.Barnett and K.F.Turkman, eds., Ch. 19, pp.345 – 366, John Wiley, Chichester, 1994) from which I am reprinting its first few paragraphs below.

Cordially,

Vit

——————————————————————-

4.1 What’s in a name!

The suggestive power of names is well known. Had, for instance, Hamburg rather than Berlin been a divided city, President Kennedy would, no doubt, have thought twice before proudly proclaiming “Ich bin ein Hamburger!”

Ten years ago, in an extremely stimulating interview televised on the NOVA program, the world-renowned American physicist, the late Richard Feynman, told a story how, as a boy, he learned about inertia from his father. When he once asked him why it was that the ball in his toy wagon rolled to the back of the wagon when he pulled it forward, and to its front when he stopped it,

“… he says nobody knows. He said, ‘The general principle is that things that are moving try to keep moving and things that are standing still tend to stand still unless you push on them hard.’ And he says this tendency is called inertia, but nobody knows why it’s true. Now that’s a deep understanding. He doesn’t give me a name. He knew the difference between KNOWING THE NAME OF SOMETHING and KNOWING SOMETHING, which I learned very early” [Feynman, 1983; emphasis added].

I believe that this early knowledge was one of the most important keys to Feynman’s later accomplishments as a scientist and a teacher. I also believe that the lack of this knowledge is one of the main roots of confusions in statistical and stochastic hydrologic modelling. This is why I am taking the liberty of elaborating on the following point which may seem trivial.

……

——————————————————————

Demetris, very interesting remakrs and summary. I’ve been struck for some time by the difference in metaphors between climate science and econometrics for dealing with data sets of autocorrelated series. Climate scientists used the metaphor of “signal” and “noise” – terms which would simply not occur to a business statistician, and, in the Hockey Stick field, are quick to do so without previously establishing that the “proxy” series is actually a proxy for (say) temperature.

Demetris, very interesting remakrs and summary. I’ve been struck for some time by the difference in metaphors between climate science and econometrics for dealing with data sets of autocorrelated series. Climate scientists used the metaphor of “signal” and “noise” – terms which would simply not occur to a business statistician, and, in the Hockey Stick field, are quick to do so without previously establishing that the “proxy” series is actually a proxy for (say) temperature.

Dear fellows, when you are discussing extending entropy, I find, in a Shannonese sense, that you are doing just that to mine. :-)

More seriously, I have a question regarding #2.1 and 2.2. In a pure simulation world, if I make a model with a trend, some periodicity and some (pseudo) random component, I can remove. (There is a computational issue here if the random variables are treated as unknown once generated. This is — I think — analogous to the public key cipher deciphering problem, but I assume the computational issue can be put aside.) The removal is just algorithmic processing. Now if you make the model and hand me only the data, I might be able to remove the trend/periodicity depending on how devious you are and how clever I am. But after the fact of my attempt, you can judge whether or not I did correctly remove the trend/periodicity.

In the physical world we are usually trying to find the model (or probability distribution or whatever one wishes to call it). Thus, we are not in the position of the modeler who can reverse the process or the student who can have the teacher tell him if he is correct. However, we proceed as if nature were created by some hidden process which if we got it “right,” we can proceed like our modeler in the pure simulation world and in that sense we go about removing trends and periodicities (presuming there is some reason for doing so in the first place).

Now why can’t I look at the process in the manner just described? This (approach) is not to say that I will get the removal correct, only that it is not damned from the start.

PS: Vogelsang, “Trend Function Hypothesis Testing in the Presence of Serial Correlation,” Econometrica 66, Jan 1998, and Fomby and Vogelsang, “Tests of Common Deterministic Trend Slopes Applied to Quarterly Global Temperature Data” working paper somewhere on the Web [and easier to pull the computational details from if you are like me and think there is too much asymptotics in the econometric literature] give an interesting approach to the test of trend over a fairly broad set of serial correlation models (although I do not know if that included SSS- FGN-Hurst phenomenon-etc. — I find myself in “nominal” quandary -– models).

Dear fellows, when you are discussing extending entropy, I find, in a Shannonese sense, that you are doing just that to mine. :-)

More seriously, I have a question regarding #2.1 and 2.2. In a pure simulation world, if I make a model with a trend, some periodicity and some (pseudo) random component, I can remove. (There is a computational issue here if the random variables are treated as unknown once generated. This is — I think — analogous to the public key cipher deciphering problem, but I assume the computational issue can be put aside.) The removal is just algorithmic processing. Now if you make the model and hand me only the data, I might be able to remove the trend/periodicity depending on how devious you are and how clever I am. But after the fact of my attempt, you can judge whether or not I did correctly remove the trend/periodicity.

In the physical world we are usually trying to find the model (or probability distribution or whatever one wishes to call it). Thus, we are not in the position of the modeler who can reverse the process or the student who can have the teacher tell him if he is correct. However, we proceed as if nature were created by some hidden process which if we got it â€œright,â€? we can proceed like our modeler in the pure simulation world and in that sense we go about removing trends and periodicities (presuming there is some reason for doing so in the first place).

Now why canâ€™t I look at the process in the manner just described? This (approach) is not to say that I will get the removal correct, only that it is not damned from the start.

PS: Vogelsang, â€œTrend Function Hypothesis Testing in the Presence of Serial Correlation,â€? Econometrica 66, Jan 1998, and Fomby and Vogelsang, â€œTests of Common Deterministic Trend Slopes Applied to Quarterly Global Temperature Dataâ€? working paper somewhere on the Web [and easier to pull the computational details from if you are like me and think there is too much asymptotics in the econometric literature] give an interesting approach to the test of trend over a fairly broad set of serial correlation models (although I do not know if that included SSS- FGN-Hurst phenomenon-etc. — I find myself in â€œnominalâ€? quandary -â€“ models).

Demetris, is the reference in 4a “A generalized mathematical framework for stochastic simulation and forecast of hydrologic time series”? If so, this article appears not to be freely available. I think its important to know what approach(s) you are advocating, and if there is any chance of explaining or otherwise making this accessible, that would be very helpful.

Demetris, is the reference in 4a “A generalized mathematical framework for stochastic simulation and forecast of hydrologic time series”? If so, this article appears not to be freely available. I think its important to know what approach(s) you are advocating, and if there is any chance of explaining or otherwise making this accessible, that would be very helpful.

Thanks for dropping by Vit. This phenomena is starting to be known as ‘trendiness’. I wonder if that will get things moving in the field. Regards.

Thanks for dropping by Vit. This phenomena is starting to be known as ‘trendiness’. I wonder if that will get things moving in the field. Regards.

Thinking of the questions put by Marty Ringo in #8, I would put some additional ones.

1. The following numbers are synthetic, generated by a mathematical model. Can anybody decompose it into components such as trends, periodicities or whatever, and can one infer the type of the generating model?

0.057 0.204 0.469 0.108 0.422 0.046 0.437 0.175 0.371 0.085

0.487 0.602 0.633 0.854 0.529 0.579 0.260 0.695 0.564 0.181

0.991 0.679 0.657 0.648 0.392 0.543 0.293 0.769 0.183 0.932

0.538 0.339 0.335 0.978 0.732 0.325 0.760 0.821 0.651 0.554

0.374 0.692 0.982 0.922 0.604 0.815 0.969 0.986 0.859 0.940

2. In the post “A new temperature reconstruction” (http://landshape.org/enm/?p=15) David has plotted some synthetic “trendy-looking” time series. Can anybody remove trends?

3. Can anybody give a rigorous definition of “trend” for an observed time series? (I cannot give a definition as I explain in my paper “Nonstationarity versus scaling in hydrology”, Journal of Hydrology, 2006, http://www.itia.ntua.gr/e/docinfo/673/).

Notification 1: Fuzzy or cyclical definitions are not acceptable.

Notification 2: Definitions implying that the underlying process is a sequence of IID variables are not acceptable.

Notification 3: Definitions referring to time series generated by a stochastic process or a mathematical model of any type are not acceptable.

4. If something is not defined, can it be found?

Thinking of the questions put by Marty Ringo in #8, I would put some additional ones.

1. The following numbers are synthetic, generated by a mathematical model. Can anybody decompose it into components such as trends, periodicities or whatever, and can one infer the type of the generating model?

0.057 0.204 0.469 0.108 0.422 0.046 0.437 0.175 0.371 0.085

0.487 0.602 0.633 0.854 0.529 0.579 0.260 0.695 0.564 0.181

0.991 0.679 0.657 0.648 0.392 0.543 0.293 0.769 0.183 0.932

0.538 0.339 0.335 0.978 0.732 0.325 0.760 0.821 0.651 0.554

0.374 0.692 0.982 0.922 0.604 0.815 0.969 0.986 0.859 0.940

2. In the post “A new temperature reconstruction” (http://landshape.org/enm/?p=15) David has plotted some synthetic “trendy-looking” time series. Can anybody remove trends?

3. Can anybody give a rigorous definition of “trend” for an observed time series? (I cannot give a definition as I explain in my paper “Nonstationarity versus scaling in hydrology”, Journal of Hydrology, 2006, http://www.itia.ntua.gr/e/docinfo/673/).

Notification 1: Fuzzy or cyclical definitions are not acceptable.

Notification 2: Definitions implying that the underlying process is a sequence of IID variables are not acceptable.

Notification 3: Definitions referring to time series generated by a stochastic process or a mathematical model of any type are not acceptable.

4. If something is not defined, can it be found?

Re #9: Preprint made available on line on itia.

Re #9: Preprint made available on line on itia.

RE #12. Thanks! The paper is here.

RE #12. Thanks! The paper is here.

Re #12 & 13 Thanks.

Re #11.1 1) as a practical matter of deciphering, the set of 50 numbers to 2 significant figures doesn’t contain enough information normally to determine the algorithm. But if we had, say, 1 million to standard double precision and we sent it to NSA (the US National Security Agency — the code breakers), would you want to bet that if the data is not random (as in from nuclear decay or some such process), they could not determine the algorithm given sufficient time? To paraphrase G.B. Shaw, at this point “we are haggling over the price of computation.”

I presume I am deliberately missing the point: that is can I determine the generating model if said data were measurements of physical phenomena. I will answer that with an evasive “some times” and hope I don’t have give citations to the examples were statistical analysis has found the model (as opposed to confirmed the model).

Re #11 I can’t give a serious answer now. I’m on my “writer’s block” break: I have to explain a trend to my client. Well, not exactly but close enough to offer a somewhat analogous — and hopefully amusing — tangent and limerick. In 1964 the US Supreme Court heard the Jacobellis v. Ohio case on obscenity. The Court wrestled with terms like “prurient interests” and “redeeming social significance” with less than satisfactory results and which led Justice Potter Stewart to given his oft cited quote as the lack of definition of obscenity, “I know it when I see it.”

A trend is a trend is a trend.

But the question is, will it bend?

Will it alter its course

through some unforeseen force

and come to a premature end?

[Sir Alexander Cairncross]

PS: Dear Prof. Koutsoyiannis, my levity here does not mean I do not take your questions seriously. Merely, that I am taking a break from serious thinking (and haven’t read your papers yet). Will read; will think; may offer comments.

Re #12 & 13 Thanks.

Re #11.1 1) as a practical matter of deciphering, the set of 50 numbers to 2 significant figures doesn’t contain enough information normally to determine the algorithm. But if we had, say, 1 million to standard double precision and we sent it to NSA (the US National Security Agency — the code breakers), would you want to bet that if the data is not random (as in from nuclear decay or some such process), they could not determine the algorithm given sufficient time? To paraphrase G.B. Shaw, at this point “we are haggling over the price of computation.”

I presume I am deliberately missing the point: that is can I determine the generating model if said data were measurements of physical phenomena. I will answer that with an evasive “some times” and hope I don’t have give citations to the examples were statistical analysis has found the model (as opposed to confirmed the model).

Re #11 I can’t give a serious answer now. I’m on my “writer’s block” break: I have to explain a trend to my client. Well, not exactly but close enough to offer a somewhat analogous — and hopefully amusing — tangent and limerick. In 1964 the US Supreme Court heard the Jacobellis v. Ohio case on obscenity. The Court wrestled with terms like “prurient interests” and “redeeming social significance” with less than satisfactory results and which led Justice Potter Stewart to given his oft cited quote as the lack of definition of obscenity, “I know it when I see it.”

A trend is a trend is a trend.

But the question is, will it bend?

Will it alter its course

through some unforeseen force

and come to a premature end?

[Sir Alexander Cairncross]

PS: Dear Prof. Koutsoyiannis, my levity here does not mean I do not take your questions seriously. Merely, that I am taking a break from serious thinking (and haven’t read your papers yet). Will read; will think; may offer comments.

#11.1 The puzzle you post is interesting and thought provoking. Because you give so little to go on except that it was generated by a synthetic model there is little to start with. I think I will post it as a quiz. Perhaps there are some students who are interested in it, and you could tell us the answer in 2 weeks say?

#11.3 As to three, I would have thought that a non-zero average slope would be be sufficient. Surely there are tests of significance of that slope that do not assume IID, but anyway, wouldn’t average slope capture the basic concept of generally going up or down?

#11.1 The puzzle you post is interesting and thought provoking. Because you give so little to go on except that it was generated by a synthetic model there is little to start with. I think I will post it as a quiz. Perhaps there are some students who are interested in it, and you could tell us the answer in 2 weeks say?

#11.3 As to three, I would have thought that a non-zero average slope would be be sufficient. Surely there are tests of significance of that slope that do not assume IID, but anyway, wouldn’t average slope capture the basic concept of generally going up or down?

David,

Re significant trends, see the “PS” is comment 8. Will send you the more accessable article.

David,

Re significant trends, see the “PS” is comment 8. Will send you the more accessable article.

Hi Marty, thanks for the references. As DK asked for a definition of ‘trends’ I was suggesting that the word trends really just means average slope. This is of course relative – e.g global temperature has been declining since 1998, increasing since 1900, declining since the holocene optimum 10000 years ago, but increasing since the peak of the last glacial. Like the stockmarket, whether a stock or index is trending up or down depends entirely on the scale you are looking at.

As to a more rigorous taxonomy of trends, I like the idea of categorizing whether the moments are finite or infinite. When you look at trends like this, ‘normal’ least squares analysis assumes that the series are non-stationary, or that is, the first moment, or the mean is infinite. A ‘stationary’ series has a finite mean, and may have infinite variance, and so on. I would have thought it important to try to establish these properties first.

Whether something ‘has a trend’ or ‘is trendy’ or ‘trends’ are clearly different things that are easily confused. My first paragraph is about the third meaning, the second paragraph is about the second. The first meaning is what I think DK is concerned with and implies a ‘something’ that may or may not be ‘removed’. That is something I find confusing and profound about DK’s paper that he refers to above. He claims that the notion of ‘having a trend’ is self-contradictory: that we are simultaneously asserting non-stationarity (of the trend) and stationarity (of the errors). It doesn’t seem problematic, but that may be because I am so used to doing it. (like ten times a day when I worked in trace metal analysis). But a scatter plot (for an instrument calibration) is not the same as a time series, so there may not be the problem with scatter plots, and a problem peculiar to time series.

Hi Marty, thanks for the references. As DK asked for a definition of ‘trends’ I was suggesting that the word trends really just means average slope. This is of course relative – e.g global temperature has been declining since 1998, increasing since 1900, declining since the holocene optimum 10000 years ago, but increasing since the peak of the last glacial. Like the stockmarket, whether a stock or index is trending up or down depends entirely on the scale you are looking at.

As to a more rigorous taxonomy of trends, I like the idea of categorizing whether the moments are finite or infinite. When you look at trends like this, ‘normal’ least squares analysis assumes that the series are non-stationary, or that is, the first moment, or the mean is infinite. A ‘stationary’ series has a finite mean, and may have infinite variance, and so on. I would have thought it important to try to establish these properties first.

Whether something ‘has a trend’ or ‘is trendy’ or ‘trends’ are clearly different things that are easily confused. My first paragraph is about the third meaning, the second paragraph is about the second. The first meaning is what I think DK is concerned with and implies a ‘something’ that may or may not be ‘removed’. That is something I find confusing and profound about DK’s paper that he refers to above. He claims that the notion of ‘having a trend’ is self-contradictory: that we are simultaneously asserting non-stationarity (of the trend) and stationarity (of the errors). It doesn’t seem problematic, but that may be because I am so used to doing it. (like ten times a day when I worked in trace metal analysis). But a scatter plot (for an instrument calibration) is not the same as a time series, so there may not be the problem with scatter plots, and a problem peculiar to time series.

David,

If you identify tend with slope, I can agree. In this case it is correct to say that now the temperature has a negative trend (because at this very moment that I am writing this note, here in Bologna it is afternoon so the temperature decreases with time). Also, we can speak of a “zero trend” – if the derivative is zero.

I can also agree if you identify trend with “average slope” but I would ask two questions: “average at which time scale?” and “slope at what time(s) exactly?” So, if you reply these questions, I think that there is not any problem, but on the other hand we do not add any information about the process at hand, introducing the term “trend”.

The problem starts when we try to “dress up” the term “trend” with either of two properties:

(a) A property of “globality”, implying that we have a phenomenon lasting for a long time or even forever, so that we feel obliged to describe it as a “nonstationarity”.

(b) A property of “statistical significance”, implying that this “trend” is something extraordinary, so that it really is a “nonstationarity”. But statistical significance depends on the model we use in the null hypothesis; usually this is IID, which is good when we play dice but not good when we study natural or human-related phenomena.

In this respect, I do not favour the use of the term “trend” thinking that it creates problems rather than helps in understanding. In addition, I think that we should always distinguish models from reality and in case that we use models we should always do it consciously and admit it. For example, if we had determined a “statistically significant trend” using, say, Kendall’s test, we must be conscious that we used a model, specifically the IID model, and that the result is specific to this model; perhaps it would be opposite if we had used another model, say the SSS model.

Having said that, I also think that it is not accurate to say “a series may have infinite variance”. The infinite variance is for the process (i.e. model), not for the series (i.e. reality). A series of observations with finite length has always finite variance.

David,

If you identify tend with slope, I can agree. In this case it is correct to say that now the temperature has a negative trend (because at this very moment that I am writing this note, here in Bologna it is afternoon so the temperature decreases with time). Also, we can speak of a “zero trend” â€“ if the derivative is zero.

I can also agree if you identify trend with “average slope” but I would ask two questions: “average at which time scale?” and “slope at what time(s) exactly?” So, if you reply these questions, I think that there is not any problem, but on the other hand we do not add any information about the process at hand, introducing the term “trend”.

The problem starts when we try to “dress up” the term “trend” with either of two properties:

(a) A property of “globality”, implying that we have a phenomenon lasting for a long time or even forever, so that we feel obliged to describe it as a “nonstationarity”.

(b) A property of “statistical significance”, implying that this “trend” is something extraordinary, so that it really is a “nonstationarity”. But statistical significance depends on the model we use in the null hypothesis; usually this is IID, which is good when we play dice but not good when we study natural or human-related phenomena.

In this respect, I do not favour the use of the term “trend” thinking that it creates problems rather than helps in understanding. In addition, I think that we should always distinguish models from reality and in case that we use models we should always do it consciously and admit it. For example, if we had determined a “statistically significant trend” using, say, Kendallâ€™s test, we must be conscious that we used a model, specifically the IID model, and that the result is specific to this model; perhaps it would be opposite if we had used another model, say the SSS model.

Having said that, I also think that it is not accurate to say “a series may have infinite variance”. The infinite variance is for the process (i.e. model), not for the series (i.e. reality). A series of observations with finite length has always finite variance.