Four Year Plan
Recently, I have been doing some business planning as part of Limerick’s LEAP programme. One of the things they teach you is the importance of realistic financial planning. Realistic means that the numbers reflect known facts and uncertainties. Projections on a timeframe of 3 or 4 years often turn out wrong, but an unrealistic plan is worse than useless.
With the arrival of the IMF in late 2010, the then Irish government published a Four Year Plan for National Economic Recovery. A key assumption is Nominal GDP growth rates for the years 2011-2014. (“Nominal” means not adjusted for inflation.) The plan contains pessimistic and optimistic growth scenarios. Are these numbers realistic?
The history of Irish 4-year Nominal GDP  growth is shown below (uncompounded  e.g. the rate for 1961 is the sum of  rates for 1961,1962,1963 & 1964). The two scenarios from the Four Year Plan are grafted on the end.
How likely is it that the outcome will lie within the expected forecast range? One way to answer this question is to treat historical growth rates as random numbers drawn from a probability distribution. The empirical 4-year growth probability distribution (along with the goverment’s forecast range) is shown below.
The shaded area gives the probability that the 2011-2014 outcome is between the governments pessimistic and optimistic projections. The probability is about 40%. A less than 50% chance that growth falls between the two limiting cases is poor. The Four Year Plan is not realistic in the sense that the range of uncertainty in the forecasts should be increased.
There is some good news here. A more realistic plan will include more optimistic growth scenarios.
February 21, 2011
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Bee vs MacBook Pro
vs
Bees are efficient pollinators, and play a critical role in agriculture and natural ecosystems. Recently, researchers at Queen Mary College London reported experiments suggesting that bees are even more efficient than previously thought. When faced with a new arrangement of flowers, a bumblebee is able to determine the shortest path linking the flowers. In effect, tiny bee-brains solve a complex Traveling Salesman Problem (TSP) in a matter of seconds on the fly. This is surpising because TSP is a notorious “hard” problem in optimization. A TSP with N nodes means finding the shortest out of O(N!) possible paths. For 30 flowers, thats about 1032 paths, most of which are very inefficient and a waste of bee energy.
Suppose the bumblebee arrives in the North-East corner of a flower bed which consists of 300 randomly distributed pink, orange and yellow flowers.
The shortest tour solution on the right was obtained using the fast and exact Concorde TSP solver. This took 5 seconds on a MacBook Pro 2.4GHz Intel Core 2 Duo. The optimal tour length ≈13 which is less than 10% the length of a typical randomly chosen tour.
The processor in this MacBook Pro weighs about 20g. A bumblebee weighs about 200mg, with a brain not more than 1% of that, say 2mg. If the bumblebee is really finding the optimal tour, then gramme for gramme, the humble bumblebee brain appears to outperform the MacBook by a factor of order 20,000!
To save the MacBook’s blushes, let’s hope the bee is doing something simpler. For example, using the nearest neighbour algorithm (visit the nearest flower not already visited). In the above example, the nearest-neighbour tour length is ≈16, which gets the bee most of the way there in terms of efficiency. This algorithm, while not exact, takes only 0.02 seconds on the MacBook Pro.
By the way, Charles Darwin was aware of the surprising power of small insect brains. “It is certain that there may be extraordinary activity with an extremely small absolute mass of nervous matter.. the brain of an ant is one of the most marvellous atoms of matter in the world, perhaps more so than the brain of man.“ – Charles Darwin, Origin of Species, 1859
Here is the R code used above.
library("TSP")
concorde_path("/usr/local/bin")
Nflowers <- 300
garden <- matrix(runif(Nflowers*2),Nflowers,2)
garden[1, ]<-c(1,1)
garden.dist <- dist(garden)
garden.tsp <- TSP(garden.dist)
garden.path <- solve_TSP(garden.tsp,method = "concorde")
garden.tour <- as.vector(TOUR(garden.path))
ex=par(mfrow=c(1,3))
plot(garden,xlab="flower garden",ylab="",pch=16,col=c("orange","pink","yellow"),xaxt="n",yaxt="n",cex=1.5)
points(garden, pch=1,cex=1.5)
plot(garden,xlab="concorde",ylab="",pch=16,col=c("orange","pink","yellow"),xaxt="n",yaxt="n",cex=1.5)
lines(garden[garden.tour,],col="black",lwd=2)
points(garden, pch=1,cex=1.5)
par(ex)
A handy script to help  with the external installation of concorde TSP on the Mac is available here. The TSP package also needs to be installed.
October 28, 2010
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