## Green Dealers

### Lies, damn lies and statistics… or the story of how 10 households using the Green Deal financing option is inspirational

Yesterday, DECC issued a press release announcing in triumphal tones that “New research shows the Government’s Green Deal is inspiring people across the UK to install energy saving home improvements.”  The release stated that 47% of those households which had received a Green Deal assessment report had installed or were in the process of installing an energy saving measure and that a further 31% said they were likely to. This, of course, gives the impression that 78% of those having an assessment were on the cusp of installing energy saving measures. A more careful reading of the press release reveals that percentage to be taken from those that have received a report following the assessment – on which more below. The undoubted tone of the statement is that the Green Deal is working as an incentive to encourage the installation of energy efficiency measures. Whether the fact that less than half those having an assessment have actually acted on the advice is a success is debatable but not the main point of this post. My issue is with the way that the statistics have been presented to give a rather biased spin on the success of the Green Deal to date.

The press release was based upon this report. Admirably, this research (and a second study tracking progress) have been commissioned by DECC in order to provide useful information with which to evaluate the Green Deal. The raw data on which the report is based is available here.

Having read the methodology and downloaded the data, my rather quick and dirty analysis tells a rather different story to that told in the press release.

The report reveals that up to March 31st 2013, a grand total of 9000 Green Deal Assessments had been undertaken. The “official launch date” of the Green Deal is given as 28th January 2013 – this is in itself interesting, as the launch date has been somewhat difficult to pin down as shown by Guertler et al. (2013) (hat-tip to their excellent paper for the information behind the following sentence). As recently as September 2012, Energy Minister Ed Davey said “Most aspects […] will start on 1 October […] finance plans only from January 2013.” – although this was qualified by his colleague Greg Barker in October 2012 – “ [that] was not, in fact, a launch.”. One wonders whether the 9000 assessments were undertaken in the period 1 October 2012 – 31 March 2013, or between 28 January 2013 and 31 March 2013…

Of these 9000 or so, 900 were selected to be surveyed for the report – with 507 eventually responding to the survey. This seems a reasonable sample. Of these, the data sheet (Q10) shows that 64% had received their report following the assessment – with a further 13% unsure whether they had received it or had it sent to their landlord. Interestingly, though – the data sheet shows the base sample for those who had received a Green Deal Assessment Report (GDAR) to be 285 respondents (Q17 of data sheet) – which appears to include those whose assessment was arranged by a landlord. Which would be 56% of 507. The source of this discrepancy is unclear from the data sheet and worthy of further investigation.

Working with that sample of 285 – the data sheet then goes on to show which measures had been recommended to various householders, and for each of the improvements whether the respondents2:

2. In the process of doing this
3. Definitely will do this
4. Probably will do this
5. Might or might not do this
6. Probably won’t do this
7. Definitely won’t do this

Questions 19-32 in the survey relate to subgroups of the respondents who have installed (A), are about to install (B), are likely to install (C&D), Might or might not (E) or won’t install (F&G) measures. The motivation for their decision and choice of financing option are reported. These are examined below, however a caveat is needed at this point. The percentages presented in Qs 19-30 are rather difficult to interpret. They appear to be related to questions asked per measure, but presented as a single percentage figure for each category. Thus, the percentages do not sum to 100% – often summing to, for instance, 150% or more. This may be due to two factors – one respondent may respond twice if they have acted on two measures from their report. In addition some of the categories are not mutually exclusive – for instance a respondent who paid for a measure from their savings may also take advantage of the Green Deal cashback scheme – both of which are options on the financing questions (Qs 19, 25 and 29).

From the base sizes given for questions 19-21 (those who have already installed a recommended key measure 1) and questions 22-25 (those who are in the process of installation) we can determine the number of households in each of those categories. These are 91 and 43 respectively. So, in all – 134 households have installed or are in the process of installing a measure recommended following a Green Deal Assessment. And 134 of 285 is indeed 47%. However – this figure makes no mention of the 222 respondents who have had an assessment but no report. This leaves almost half those who have had an assessment out of the calculation – given considerable uncertainty to the 47% figure – although as mentioned previously the press release is carefully worded to state that the 47% is of those who have received a report

Next, I looked at the methods by which those installing measures had financed them. This, surely, is an important consideration as the Green Deal is designed as a financing scheme to encourage investments and, as former Energy Minister Chris Huhne put it, “The Green Deal is about putting energy consumers back in control of their bills and banishing Britain’s draughty homes to the history books” DECC (2011), quoted in Guertler et al (2013).

On face value (taking account of the caveats above regarding presented percentages), I conclude that only 7 households have taken advantage of the Green Deal Financing scheme and got to the point where an energy efficiency measure has been installed. A further 3 households intend to use Green Deal financing to finance measures being installed currently. So, in total, it appears that 10 households are using the Green Deal financing option of the 507 who have had the initial assessment. Those who have not started any work, but are “definitely” or “probably” going to install measures are more positive about the financing option – with 31% of the 86 involved stating that they intend to use Green Deal financing – some 27 respondents. This is not quite the message that would be gleaned from the press release.

Understandably, the Green Deal Cashback scheme (which is effectively a capital subsidy for works undertaken) is more popular. It appears that ~15 respondents that have had work completed have used the cashback voucher scheme – with a further 9 of those currently having work carried out intending to use the cashback.

Tellingly – when asked why they did not or will not use Green Deal financing 18% of the 150 in this category said they “were not aware of it”, with a further 23% saying that they “Don’t like borrowing/taking out finance/prefer to pay up-front”. These seem like major issues for a government “flagship policy”. This should all be set against a backdrop where orders for energy efficiency measures have fallen through the floor (non-paywalled summary here), whilst the Energy Minister Ed Davey announces in this press release that ““This is great news for the energy efficiency industry as well, because this shows a genuine appetite among householders for more energy efficient homes.”

The government should be lauded for making data available to those who want to delve behind the headlines. However, the presentation of those data in a way that rather over-eggs the success of policies, whilst unsurprising, is less laudable.

The first official statistics on the Green Deal are due tomorrow. I await them, and how they are reported, with bated breath.

As always – comments are welcome.

References:

Pedro Guertler, David Robson and Sarah Royston (2013) Somewhere between a ‘Comedy of errors’ and ‘As you like it’? A brief history of Britain’s ‘Green Deal’ so far in proceeding of ECEEE Summer Study 2013 – available from http://www.ukace.org/wp-content/uploads/2013/06/1-306-13_Guertler.pdf
DECC 2011 “Homes and economy to benefit from energy and climate policies – Huhne” Department of Energy & Climate Change – available from https://www.gov.uk/government/news/homes-and-economy-to-benefit-from-energy-and-climate-policies-huhne

Footnotes:

1. Note that this does not mean that the respondent has installed everything that was recommended on the GDAR, but that at least one measure was installed.
2. It should be noted here that this information is recorded per measure recommended to the respondent. It is to be expected that some households had few (if any) measures recommended, whilst others had multiple measures recommended.

Posted in Uncategorized | 26 Comments

## Heronian triangles

A non Heronian triangular heron

Despite the picture, Heronian triangles are not named for the Heron, but actually for Hero of Alexandria (c. 10–70 AD), an ancient Greek mathematician and engineer (links are to Wikipedia).

Such triangles are interesting in that both their side lengths and areas are integer.  It should be fairly easy to see that all pythagorean triangles fulfil this property (which, by the way, shows that my heron picture is in fact not made of heronian triangles at all – as all the right angle triangles are isoceles which means they cannot have all integer sides…)

I’ve been having a little look at a problem asking for all triangles with integer sides which have an integer ratio of area to perimeter.  The problem is delimited by giving a maximum value of that ratio – for the original see Problem 283 at projecteuler.net.  A little thought shows that to have an integral ratio, the ratio must first be rational.  For a triangle to have integer sides and a rational area:perimeter ratio, the area must be integral – therefore the triangles in question must be Heronian.

What makes the puzzle a bit tricky is that not all Heronian triangles are right angled (pythagorean).  If they were – it is relatively easy to generate all pythagorean triples with the required property.  However, the non-pythagorean Heronian triangles are harder to generate and it is rather harder to convert the limit on the ratio to the largest Heronian triangle which might fall within the constraint.

This post simply records my line of thinking on a first inspection of the problem:

Consider pythagorean triples (3,4,5) and (5,12,13).  The triangle with sides 3,4,5 has area 6 (=(3*4)/2) and perimeter 12.  Thus, it has an area:perimeter ratio 0.5.  For ease of notation – let’s call the area:perimeter ratio r.  If we double the side lengths (i.e. triangle with sides 6,8,10) the triangle has area 24 and perimeter 24 => r=1.  So, we have  a triangle fulfilling our criterion.  Similarly with 5,12,13: area=30, perimeter=30 => r=1.

Now consider right angled triangles 12,16,20 (3,4,5 * 4) and 5,12,13.  The former has r=2 and the latter r=1 – both within the set we are looking for.  If we then sit them together with the edges of length 12 touching (adjoin them), we form another triangle with side lengths 13,20 and 21.  As the right angled triangles have integral ares, the triangle formed by their adjunction must also have integral area and therefore this must be Heronian.  In this case, the triangle so formed has area = 126 and perimeter = 54.  r = 7/3.  It doesn’t fit our criterion, but all is not lost.  If we multiply the whole thing by 3, it does!  Thus – triangle with sides 39,60,63 is a Heronian triangle with r=1 again.  This is the adjunction of right angled triangles 36,48,60 and 15,36,39 with r=6 and r=3 respectively (non-primitive pythagorean triangles = (3,4,5)*12 and (5,12,13)*3) to give a Heronian with r=1.

So, we can see that adjoining two right angled triangles with (relatively) large r gives a Heronian with fairly large side lengths and a perimeter larger than either right angled triangle, but with a much smaller r (=1 in the example case).  This means we might have to find some very large Heronian triangles to ensure I’ve caught all of them with ratio <= 1000.  So, thoughts continue.  Parameterisations should help with both generation of the triangles and turning the limits on r into limits on the size of the triangle (either in terms of perimeter or longest side).  In fact turning the limit on r into a limit on size of right-angled triangles is fairly easy – we need to find triangles with a maximum non-hypotenuse side length of 2*maximum_r + 1.  I’ll sign off now – proof of that in a future post, maybe.

## In which I think about relative importance

I have been thinking a lot recently about the relative importance, or weighting, that people give to various activities in their own lives and the lives of others. It’s an area in which I am not particularly well versed and would welcome comments from philosophers, psychologists, sociologists or, in fact, anyone with an opinion to share. This post represents little more than a documentation of musings so far. It is not a researched piece, but is recorded here to be picked up later – by me and (hopefully) others.

My thinking in this direction started to form from a combination of observations in my own life, bolstered by some reading I did as part of my ‘real’ research (about which more in a future post) and some vicarious observations I made. It struck me that people find certain activities in their lives in some way non-negotiable, unquestionable, imperative or, maybe, essential. I’m aware that all those words are somewhat loaded and part of the reason for writing this post is to tease out some language to describe what I’m getting at. I’ll start from a few examples and work out from there – one from my research and an observation of behaviour to illustrate.

1. Personal – my partner and I requested some assistance from a relative. The help would have allowed us to conduct leisure activities separately on the same evening. Unfortunately the relative couldn’t help. Later, my work required that I take a trip away which encompassed the same evening. The relative then revised the decision and helped my partner to attend another event. It is clear that the relative viewed my work commitment as imperative, whereas my social commitment was not.
2. Research – my research focuses on people’s willingness to alter (in quantity or time) their use of electricity. In a large number of papers, presentations and other material on this subject, a number of aspects of electricity use are taken as ‘givens’. For example, “People will watch television when they want to watch television” could be quoted as a premise in a number of papers I read, even without the usual standards of academic evidence, without much question. It is taken as unquestionable that people will watch television whenever they like – irrespective of (reasonable) influence.
3. Observed – an acquaintance from one city took a job in another city. She felt that she could not move her family as they had many ties in their original city. However, the job was a career progression. Rather than not take the job, she commutes each week, spending a number of days each week away from home. She is not unique in my peer group – it appears that a number of people find career progression an imperative and continue with an arrangement that is likely to be inconvenient at least.

Having considered a number of such examples for a while, it seemed to me that there is possibly interest beyond anecdote if we can draw out some general principles. Analysing a number of situations from this point of view has given me some interesting insight into what is happening. My thoughts have begun to crystallise into something more general. If there is a discrepancy between what people involved in a negotiation (or conversation) view as imperative, common understanding becomes very difficult. In turn, debate and resolution become virtually unattainable. This appears to be observable in situations from the minutiae of a single point in a business meeting to stand-offs between large institutions. It seems also that non-negotiables, or even perception thereof, may ‘lock-in’ a particular behaviour or norm in a situation – either for an individual or group.

I think this mode of understanding can be powerful at a number of scales. In daily conversations, say between cohabiting partners, a difference in opinion on what is imperative can lead to oft repeated arguments, whereas as collective imperatives reinforced by repeated discussion will determine the lifestyle of that partnership. On a much larger scale, collective acceptance of the non-negotiable primacy of rational economic evaluation of any action determines not only what institutions do, but also how actions are described and debated (in this instance, in terms of investment, payback period). Where people or institutions having different ideas on the non-negotiables in such evaluation engage, the result is often incomprehension between debating parties – quite often descending into mud-slinging. An example of this is often seen when environmentalists engage with rational economists with regard to climate change. They fundamentally find different elements of the debate important and therefore, usually, each fails to even see the others’ position as other than ill thought out.

My point here is not, I think, one of platonic idealism or some form of essentialism. I am not arguing that there are some objective characteristics which define a group or are in some way virtuous. Nor am I arguing that the importance of certain actions is an entirely individual matter – rather that the importance or non-negotiability is determined as a product of individual experience, social consensus (or influence) and vicarious observation. What I am trying to argue is that these perceptions of what form the essential elements of a persons life have a very deep influence on the path that they choose and, by implication, the paths followed by groups of various sizes.

I’m trying to place this idea within what I know of theories of structure and agency. In some ways, carried to its conclusion the idea would tend toward an ontology which minimises the effect of cognition on individual agency. Actions would be pre-determined if they were to be seen as purely due to satisfying a number of imperatives. However, that would be to neglect both the possibility of the imperatives changing over time and the possibility that imperatives may contradict each other in some situations.

The above is rather a stream of conciousness on an idea that has been intriguing me. I think this idea bears some more thought and maybe generalises to a scale of relative importance rather than the simple binary treatment implied by ‘non-negotiable’ or ‘imperative’ and their opposites. I am absolutely sure that there is a lot of thought out there about this very subject – probably hiding in the literature of disciplines with which I am not so familiar having originally trained as an engineer. I shall no doubt return to this theme as I think more about it and try to develop it into something more coherent. As I said at the top, comments most welcome.

## Identifying a polynomial

A tweet has got me musing again. @ColinTheMathmo sent out a tweet making a seemingly audacious claim that from just two data points he could deduce all the coefficients and powers in a polynomial that you had defined as long as the coefficients and powers were zero or positive integers

i.e. you pick a polynomial of the form:

$$y = \sum {a_i \cdot x^i} \qquad for \qquad i \in Z^*, a_i \in Z^* \qquad (1)$$

He then supplies an $$x$$ value, you return the corresponding $$y$$ value, this procedure is repeated once only and then he can tell you all the $$a_i$$ values in your polynomial.

I looked at this tweet on the way to work one day and starting thinking about it. I wondered whether you cleverly supplied some complex numbers for the $$x$$ values which could indicate the powers of x in use, but Colin assured me that the $$x$$ values could be in $$Z$$ too. In my head, I started drawing graphs. Let’s call the first number asked for $$x_1$$. I could see that, given $$y_1$$ for that value, there are a finite number of graphs for polynomials satisfying $$(1)$$ that go through that point and that for the constraints given, these graphs were all monotonically increasing in $$x$$.

I could see all this in my mind’s eye, but couldn’t quite work out how that would let me uniquely identify each one – i.e. at what point all those lines no longer cross. So, when I had a minute, I wrote a little python program to spit some graphs out. Below is one graph for a particular sample pair ($$x_1=2$$,$$y_1=41$$) showing all the possible polynomials satisfying $$(1)$$ that go through that point. For those who would like to look at the graphs in a bit more detail – there’s a gallery at the end of the post with various $$x,y$$ pairs and zoom levels. Scrappy Python code to generate these is also available on request.

Click the graph to see a full size, better resolution, image

From the picture above and a bit of reasoning that any higher powers of $$x$$ will cross all curves at lower values of $$x$$, I’m pretty convinced that for any point you ask for, if you work out the shallowest polynomial of order 2 through the point and the steepest straight line and then find their intersection, all the graphs beyond that point are distinct, will never cross and are monotonically increasing in $$x$$.

With this insight, if I define
\begin{aligned}
b & ={\lfloor}{\frac{y_1}{x_1}}{\rfloor} \\
c & = y_1-{x_1}^2 – (y_1 mod x_1)
\end{aligned}
The intersection is at
$$x = \frac{b + \sqrt{b^2 – 4c}}{2}$$
So, we ask for $$x_2$$ greater than that value and the polynomial is uniquely identified. For instance, in the example above, this works out to $$x_2 \geq 18$$ How to then get back to each co-efficient is interesting, but basically a search problem.

Now, Colin has reliably informed me that this is way too complicated and also challenged me as to whether I can prove it will always work – which I’m not sure I can, but I think the above sort of shows it.

I’m going to have a think about a few of my other ideas, which are (in no particular order)

• ask for 1 as the first x ($$x_1 = 1$$), thus giving you at least the sum of all co-efficients. I have a feeling this might be useful
• put in the value that you get back from the first answer as the second value you ask for (not sure why I think that might be good – I doubt it really)
• find a value of $$x_2$$ which is $$f(x_1)$$ but lower than the value outlined above which can similarly always have the graphs distinct. I’m not hopeful.
Gallery of graphs
Click on any of the images below for the full size picture
Posted in Maths, Puzzles | 2 Comments

## Sums of consecutive integers

This post is inspired by a tweet from @standupmaths who sends out puzzles tagged #mathspuzzle irregularly but fairly frequently. I won’t put out any spoilers on the blog, but once the solution is out on twitter, I may occasionally muse on these here. It’s my first go at both maths blogging and embedding equations into a webpage, so any comments welcome. I’m pretty sure I’ve gone round the houses on this one, so shortened method / notation welcome (as well as pointing out any errors that I’ve made, obviously).

The problem was stated thus: “#MathsPuzzle: What is the only number 10 →20 that is not a sum of consecutive numbers? (eg. Not 12 because 3 + 4 + 5 = 12)”. Tweeters fairly quickly answered 16 and the follow up question (abbreviated) was – how does this generalise? To which the answer is the only positive integers that cannot be represented as the sum of consecutive integers are the set $$x \in \{2^n\}$$ where n is an integer.

After finding this result, I tried to express why $$\{2^n\}$$ and only $$\{2^n\}$$ exhibit this property in 140 characters and failed miserably. So, I decided to blog about it and write the method I followed in my head down – if only to clarify to myself how I came to the result (and whether it was valid).

I began by using the result:
\begin{aligned} \sum_{i=1}^{i=j}{i} = \frac {j \cdot (j+1)}{2} \end{aligned}

therefore, the sum of integers between j and k is:
\begin{aligned} \sum_{i=1}^{i=k}{i} \quad – \quad \sum_{i=1}^{i=j}{i} \qquad & = \qquad \frac {k \cdot (k+1)}{2} – \frac {j \cdot (j+1)}{2} \\ & = \qquad \frac {k^2 – j^2 + k – j}{2} \\ & = \qquad \frac {(k+j) \cdot (k – j) + (k – j)}{2} \\ & = \qquad \frac {(k+j+1) \cdot (k-j)}{2} \end{aligned}

therefore, if x is to be expressed as the sum of some set of consecutive integers, the following must hold:

\begin{aligned} x \qquad &= \qquad \frac {(k+j+1) \cdot (k-j)}{2} \\ 2x \qquad &= \qquad{(k+j+1) \cdot (k-j)} \qquad (1)\end{aligned}

We notice that if both $$k$$ and $$j$$ are odd, or both $$k$$ and $$j$$ are even, then both $$k + j$$ and $$k – j$$ are even, therefore $$(k+j+1)$$ is odd and $$(k-j)$$ is even. Conversely, if one of $$k$$ or $$j$$ is even and the other odd, both $$(k + j)$$ and $$(k – j)$$ are odd $$(k+j+1)$$ is even and $$(k-j)$$ is odd.

So we may state that if $$x$$ is the sum of consecutive integers then we must be able to express $$2x$$ as the product of one even and one odd integer

If we next consider any integer as a product of primes, we can say that for any integer
$$\label{primefactors} i = 2^a \cdot 3^b \cdot 5^c \cdot 7^d \cdots \qquad (2)$$ and so on for all primes.

If we set $$i = 2x$$ then we can combine (1) and (2) to see that for x to be the sum of consecutive integers,

\begin{aligned} 2x \qquad &= \qquad {(k+j+1) \cdot (k-j)} \\&= \qquad 2^a \cdot 3^b \cdot 5^c \cdot 7^d \cdots \\ \\ \text{therefore} \\ \\{(k+j+1) \cdot (k-j)}&= \qquad 2^a \cdot 3^b \cdot 5^c \cdot 7^d \cdots \qquad \qquad (3) \end{aligned}

As all primes are odd except 2, and $$(\text{odd}\times\text{odd})$$ is always odd, we can say that the right hand side of (3) will always be $$(\text{even}\times\text{odd})$$ except in the case where $$b = c = d = \cdots = 0$$ where it will be $$2^a$$. However, for $$x$$ to be the sum of consecutive digits, this expression is required to be $$(\text{even}\times\text{odd})$$, which holds in all cases except where $$2x = 2^a \rightarrow x = 2^{a-1} \rightarrow x \in \{2^n\}$$