# Number of the Month

## Yes, it’s boring old mathematics, but …

… you need to understand it if you are dabbling in areas such as signal theory.

I see that friends of Willis Eschenbach have pointed out to him that he has rediscovered the Discrete Fourier Transform. In my remarks last month I was only concerned with the misinterpretation of one of my laws. I was reluctant to embarrass an entertaining author or to join in one of those interminable and opaque blog-comments debates. Now that the cat is out of the bag it might be useful to add a few thoughts that might be relevant.

The sinusoids are one of a family of functions that can be used to form an orthogonal basis. This means that they are amenable to treatment by powerful methods based on an analogy with algebraic geometry, in which those functions behave like Cartesian coordinates. Every point or path in the function space can be expressed in terms of those coordinates. To physicists and engineers the most important example of this sort of treatment is the Fourier transform, but in pure mathematics it is just one of many.

There is a symmetry in the Fourier transform, which means that the phenomenon once known as Gibb’s oscillation can occur in either the frequency or time domain, depending on whether there is a sharp cut-off in the other. This means, among other important things, that the Uncertainty Principle has to be dealt with by the application of Window Functions. This means that two workers might produce slightly differing results for the spectrum of a block of discrete data: that is uncertainty. In these pages I have tried to devise explanations (see our FAQs) that avoid higher mathematics, but inevitably they are (to greater or lesser extent) inadequate.

While on the subject, a remark on the nature of “Red Noise” might be appropriate. It appears to be the current jargon for 1/f noise. The trouble with jargon is that it is inconstant. At one time it was called Brown Noise, after the first observer of Brownian motion, to which it applies. A colleague of mine used to call it Bastard Noise, which will be recognised as appropriate by anyone involved, as we were, in trying to develop new semiconductor devices (microsensors). It can be modelled by time-integrating White Noise (since the Fourier transformation of the operation of integration is multiplication by the reciprocal of frequency). In practice and in general it means big trouble.

A further relevant reminder is that discrete transformations and operations (such as convolution) are circular (In my beginning is my end –  East Coker). The effects of this are not entirely negative: I once devised a measuring instrument that was based on circular deconvolution, realised by a research student (who was later Professor Chowdry in Pakistan). It is, however, a potential hazard to those who ignore it. Also note that the Eschenbach modelling in question includes a non-linear process (a variation on peak-picking), which is not amenable to linear algebraic analysis.

Mathematics is not boring and it can be exciting. In our dominant culture (of the political and media classes) it is considered perfectly respectable to boast of total ignorance of it. Once, at a climate conference, I offered my skate-boarding theory of learning. Children know that learning to skate-board is hard and painful, but they rightly anticipate the rewards of mastering it: so it is with mathematics, but they are not given this message; quite the contrary. Early in my research career I came up against the problems of applied statistics and digital signal processing. I came to realise that I was just dabbling, so I acquired a few basic books and gave myself the mathematical background. The best book I have found for a general mathematical background to digital signal processing is  Mirsky’s An Introduction to Linear Algebra (Oxford. 1955), but it is demanding. For a rigorous statistical background, no one has bettered the great Cramér and his Mathematical Methods of Statistics. Meanwhile, beware dabblers who propose new methods of their own. Stick to the known and trusted ones.

04/05/14

Footnote: and there's more.

Footnote 2: The relationship between frequency and period is reciprocal (a rectangular hyperbola). So, if you plot data against period rather than frequency, regular phenomena (such as Gibb’s oscillations) are squashed up to the left and stretched out to the right.

Footnote 3: Oh dear! Reading through the comments on those WUWT pieces, reminds me to mention the great Papoulis, whose books on probability and signal theory were such an inspiration to me.

09/05/14

A dirge for the dead children

Ghost:
Murder most foul, as in the best it is;
But this most foul, strange and unnatural.
Hamlet

Daily Telegraph main headline; Emphasis is given to the tragic numbers of children in the statistics. If you keep sweeping the dirt under the carpet, the hump will eventually give the game away. So it has proved in the case of the murderous neglect of the astonishing and continual rise in childhood asthma. If you kill a child by neglecting to feed it you are rightly accused of vile murder. These children, however, were not only murdered by neglect. They have been trampled underfoot by the March of the Zealots.

Some seem to think that statistical frauds carried out by corrupt organisations, such as the EPA and its epidemiological allies, are victimless crimes. They are not. A consequence is that funds are not only officially purloined from the enslaved masses, but they are diverted from combating the real threats that those people face.

In the middle of the last century, British children walked to school in thick sulphurous fog, caused by the burning of low quality coal for heating, which penetrated their homes and classrooms. In public buildings, such as cinemas, they were immersed in clouds of tobacco smoke. Yet childhood asthma was almost unknown: the few victims were shut away like valetudinarian characters in Victorian fiction. Since that time, childhood asthma has continually risen and it is now casually accepted as an unfortunate fact of life. This was brought home to me by the sight of children descending from the school bus, a significant number clutching their blue salbutamol inhalers.

Note that the current story only deals with the treatment of the illness. The prevention of disease has become a playground for fanatics, fantasists and money-grubbers (see, for example the case of cholesterol and statins).

Public money pours by the trillions into imaginary disease and climate catastrophes, yet relatively almost nothing goes to a cause of juvenile agony and death that has grown inexorably for decades. The real tragedy is not just the death but the horrific way of dying. A brush with death from pneumonia taught me all about the terrible torment of being unable to breathe. Only a week ago we published a reminder that the particle wars began as a gleam in the eyes of the bureaucratic enemy within, just a quarter of a century before. That is now another sink for substantial public funds that could have been spent on preventing genuine suffering

As a correspondent (a medic) wrote to the Telegraph the very next day:

The national review into asthma deaths shows a truly worrying problem. Although I accept that some of these deaths may result from sufferers underestimating their disease, I fear a significant number may be attributed to a reluctance to admit people to hospital.

There are risks associated with the NHS policies that offer incentives to GPs to reduce acute hospital admissions, and that encourages hospitals to reduce the number of beds available. It is sadly inevitable that putting pressure on doctors to think twice about admissions will lead to disaster.

This is especially a risk with a disease such as asthma, where deterioration can be unpredictable and rapid.

It has become a cliché of modern zealotry to ignore the effects of costly state campaigns and ascribe any timely changes of statistics to your own hate-object (see, for example, the salties only last month) and the zealots always keep a watch out for statistics that appear to justify their bans and interventions, by publicising those that drift in the “right” direction at the right time and ignoring those that don’t. As in this case, however, a related technique is used to divert blame from the generals at the rear to the unfortunates on the front line.

Unlike the 7,000 imaginary dead of our most recent month-number, these young people were real human beings with real names, real post-mortems, real death certificates and real mourning parents.

07/05/14

## The tribulations of science

A little learning is a dangerous thing.
Drink deep, or taste not the Pierian Spring;
There shallow draughts intoxicate the brain,
and drinking largely sobers us again.
Alexander Pope,
An Essay on Criticism

If quantum mechanics hasn't profoundly shocked you, you haven't understood it yet.
Niels Bohr

If I have seen a little further it is by standing on the shoulders of Giants.
Isaac Newton

The death-throes of the global warming fallacy are going to be of long duration, painful and ugly.
Number Watch, March 2013

In the cases covered in the two pieces above there is a common thread, even though the criticism applies to cases emanating from opposite “sides” of the arguments about the conduct of science. Science is built on the foundations of the achievements of our predecessors, but it depends on precious concepts such as openness and reproducibility. Traditional working scientists expect to experience forthright criticism of theories and experimental evidence that they offer and they are trained to deal with this in a courteous and impersonal manner. If necessary, they hold a conference to clear the air. All this has changed. The big difference now is that there is one new, united establishment and it has swallowed up not only political institutions, but also scientific ones. Even the once pre-eminent institution, the Royal Society, finds itself promoting censorship and intolerance. So, for example, the new protagonists proclaim that “denial” is not about the science. Everything to them has to be ideological. Even the words have changed connotations: denial has a particularly nasty political association, while The Science now means something quite different from Science tout court.

All human societies have establishments and each is eventually succeeded by another, but the first instinct of any establishment is to defend itself and its power, for that is the prime interest of its members.

It is a remarkable concurrence that the shocking development at GWPF, while still reverberating, was almost immediately followed by outrageous bullying behaviour by the establishment at the University of Queensland. Both the and Shollenberger  cases are gross examples of the renewed militancy of the establishment on the defensive. In the UK there is a comparable political example in the hysterical media smear campaign against a small party, UKIP, which dares to say what most Britons think about the EU and its works.

Training as a scientist imparts the ability to approach a problem in a scientific manner. First there is defining and understanding the problem, but the vital second stage is the literature search, including appropriate text books. This is why the recent stuff from WUWT was so dispiriting. Even if you are a genius, you are not, in one lifetime, going to retrace all the steps taken by your illustrious predecessors. The ease of implementing a sophisticated procedure by computer program is a trap that can lure the unwary into misguided dabbling, even if it does produce pretty pictures. It is particularly unfortunate that an important resource such as WUWT should be damaged by such activity. Risking the hostility of serious physicists, signal processors and other scholars is a game not worth the candle. Likewise, seemingly endless blog discussions, mainly from the ill-informed, add nothing to the sum of human knowledge: quite the reverse. This situation has seriously deteriorated with the introduction of hand-held computer devices and social networks. There is a new body of ignoramuses who simply muddy the waters, the thumb-typists. You can often recognise them by their shared misspellings (to for too, your for you’re, loose for lose etc.) to say nothing of their irritating acronyms and initialisms. If they engaged their brains before activating their opposed digits, the world would be a better place. It is a dangerous illusion of modern society that anyone is qualified to comment on anything. In their own time, the contributions of the great scientists, such as Newton, were understood by a negligible proportion of the population. That is not to denigrate the progress due to amateurs, who were significant drivers of scientific advance; but who also did the necessary homework to achieve an adequate level of scholarship (a word that has all but lost its meaning, even in the groves of academe).

Apart from all this there is the problem of perverse official forms of science. No wonder that medical writers are confused! The adoption of low standards of statistical significance by epidemiologists made it inevitable that contradictory studies follow each other with increasing frequency; which was all confidently predicted. Epidemiology is so corrupted by this insouciance that it is worse than useless: it is destructive. It grants political power to autonomous bodies, such as the EPA and the Brussels bureaucracy, to interfere with human liberty and economic progress; though these have now reached a stage of confidence in their authority whereat they simply make unsupported statements, without even resorting to phoney calculations or statistical fraud.

As for the climate scare, it is self-evident that the thinking portion of the public are terminally bored with it. The posture of the adherents is reminiscent of Ceausescu on that balcony, but they are not going to give up their privileges without a fight and that will make it all the more unpleasant for everybody. Although it is a political phenomenon rather than a scientific one, the political activists have seized control over the funding of science.

Whither science?

21/05/14

## Down with the establishment!

Voters across the EU have delivered a blow against the Establishment in the elections for the Powerless Pretend Parliament in Brussels. In the UK, the Eurosceptic UKIP has topped the poll. Ordinary people have been obliged to stand aside watching their institutions being colonised by the invaders and their vital industries being trashed in the name of the carbophobic religion. Now they have had a chance to speak with their crosses on ballot papers, even if it is only in a parody election based on the anti-democratic party-list system. People, unlike the media, have not forgotten the dirty tricks by which the loathed Lisbon Constitution was foisted on them.

I have not been a member of a political party since that photograph of that hug-a-husky stunt by Dave appeared, but I do have a soft spot for UKIP (and not just because they are the only party that ever invited me to address them). They are justifiably obsessed with the appalling unelected supreme power in Brussels and its allies in national parliaments, but they do approach other political problems with an open mind.

The media section of the establishment are now publishing articles trying to explain the phenomenon, but they carefully avoid mention of their own contribution, with that hysterical but orchestrated smear campaign against a small party with few resources. The older Britons, in particular, still maintain their traditional regard for fair play. Unfortunately, in national elections people are obliged to vote tactically against the bunch they fear most; so this is not likely to be repeated. But at least the people have spoken at last.

26/05/14

We have long abandoned any attempt to keep up with the flood of junk science that makes up the contents of those space fillers in the popular newspapers, but every now and then there is one that cries out for comment. This one made the media headlines; in the Telegraph front page headline - Cynics three times more likely to suffer from dementia. That’s one in the eye for the nasty old cynics who refuse to accept that all is for the best in the best of all possible worlds. There are, however, one of two points that the recidivists among them may wish to consider:

 The Trojan Number of 1,500 people shrinks to a mere 14 who fill the criteria for both alleged cause and effect (though the Trojan Number is 622 in The Times and the original article abstract). According to that abstract – “Cynical distrust was not associated with dementia in the crude analyses, but those with the highest level of cynical distrust had higher risk of dementia after adjusting for confounders.” Thus the claimed result is entirely due to the adjusting. Just like global warming! Researchers adjusted for other factors that “could” affect dementia risk, “such as” high blood pressure, high cholesterol and smoking (all from the fashionable list of targets for every disease, but naturally excluding any they did not think of). There were obviously a lot of data and anecdotes collected, which points to the study being something of a data dredge. The undemanding 95% confidence interval for the main conclusion was 1.15–8.55, a suggestively wide range. The last sentence in the abstract is “We acknowledge the need for larger replication studies.” A nice euphemism! It all fits in neatly with the current campaign to brand scepticism as a psychiatric disease.

Of course, the inclusion of smoking was compulsory under the rules of the game as now played. It would be interesting to know in which direction that particular adjustment went. I have been fortunate in later life to have received care from two outstanding consultants (a surgeon and a physician). They independently observed that on their visits to old people’s homes it was startlingly evident that the smokers exhibited a higher mental acuity than the non-smokers. Naturally, they would never have been allowed to publish such an observation under The Censorship.

29/05/14

PS: it even spawned a follow up article. Also, oddly enough, 14 is exactly the expected number of victims in our imaginary example on Trojan Numbers.

30/05/14

## Number of the month - 65

Prinny has spoken! He has now turned his attack on capitalism. Admittedly capitalism is a pretty poor way of running things but, like democracy, it survives because every other system that has been tried has proved worse. Likewise, the saving grace of constitutional monarchy is that the head of state is apolitical. You only have to look at what goes on in the world to see the chaos that ensues from elevating a mere politician to that role. Unfortunately, the more he gradually takes on the reins from his elderly and gracious mother, the less he emulates her admirable public disinterest in things political. His inadequacies have been sufficiently exposed in mellifluous Monktonian prose to allow us to skip the detail, but perhaps a word or two on his thought processes would not be amiss.

I do not accuse him of the modern sin of simply growing old (after all, I was at junior school when his birth was so joyfully announced) but I do condemn his failure to use all those years to develop a capacity for reason. There are two things you need to know about his belief system:

 He has zero understanding of the concept of evidence He has absolute faith in absolute faith

That is why his arguments are circular and self-defeating and his pronouncements tend to be redolent of insular arrogance. He cocoons himself in the company of sycophantic courtiers, so never experiences the stimulus and correction of debate.

So 65 is the age of the heir to the throne; an age at which you might expect the development of some humility and understanding. Only four months ago he was the origin of our number of the month, after a gratuitous and intemperate onslaught of insult against some greater men than himself. There a few things more contemptible than using a position of privilege to hurl abuse. So we dedicate yet another month number to His Royal Haughtiness.

31/05/14