6-1 . Introduction
fig 6-1,1 : Program Execution Display Shot
6-2 . Aim
6-3 .Preparation and Mathematical Definitions of the Model
fig 6-3,1: The Whole Grasshopper Definition with parts
6-3-1. Static Tensegrity Structure (part1)
fig 6-3-1,1: One Tensegrity Component
Fig 6-3-1,2: Arrayed Tensegrity Components
fig 6-3-2, 1 :Hooke's law (fig from Wikipedia)
Fig 6-3-2,2: Cables and Their Shortest Length
Fig 6-3-2,3: Core parts of the Spring
Fig 6-3-2,4: Physics Engine
Fig 6-3-2,5: Colour of the Strings
6-3-3. Membranes on the tensegrity structure (Part3)
Fig 6-3-3,1: Membranes
6-3-4. The Sun (Part4)
fig 6-3-4,1: Grasshopper Sun Definition
6-3-5. The Evaluative and Recording function (Part5)
fig 6-3-5,1: Grasshopper Evaluation Definition
6-4 . The Unique Feature / Limitation of ‘Galapagos’
fig 6-4,1 : Genome-Map and Mating Method (Drawn by K.Hotta)
The ‘Genome-Map’ is useful in adjusting the breeding factor and other aspects of the mating process. The Genome Map is an approximate 2 dimensional map made by using projections from higher dimensions. The number of genes (an N-dimensional value) is compressed to 2 dimensions. (It is assumed that all the genomes in a species have the same number of genes. This is not technically a limitation of Evolutionary Algorithms, even though it is currently a limitation of Galapagos). All the individuals in a certain population are demonstrated as dots on a 2 dimensional grid (this grid axis does not have a meaning because it is a projection). The distance between two dots on this grid is practically analogous with the distance between the individual’s genomes in higher dimensional gene-space. The only information a Genome Map conveys is which genomes are more similar (close together) and which genomes are more different (far apart).
In contrast ‘Zoophilic Mating’ is the method of mating by excluding everyone near the one who is mating. Those genomes are totally different and often incompatible as the genome distance is huge. This method is also recognized as harmful especially when the fitness landscape has multiple mountains ( fitness landscape is defined in Chapter3-3-3), or a population has more than a single group of genomes. Here, in iteration, candidates are climbing their own little fitness peak. When two different individuals, who are climbing different mountains, make offspring generally, they tend to fall down into a fitness valley somewhere in the middle of the fitness landscape, in other words they tend to be non-optimal and not fit for purpose.
fig 6-4,2 Genome Graph and Mutation (Drawn by K.Hotta)
As the ‘genome’ or gene sequence is the key driver for GA, the Genome Graph is one useful way to visualize gene sequences as a 2 dimensional graph. Thanks to this way of representation, subspecies or lone species can be identified at a glance. A common method for displaying multi-dimensional points on a two- dimensional medium is to draw them as a series of lines that connect different values on a set of vertical bars. Each bar represents a single dimension. This enables people to quite easily display not just points with any number of dimensions, but even points with a different number of dimensions in the same graph. In the fig 5-4-1, the example genome, which consists of 5 genomes, is shown. It is common in actual GA procedures as well as in Galapagos, for each value or gene to be given a decimal value between 0.0 and 1.0, such as G0=0.25, G1=0.5, G2=1.0, G3=0.5,G4=0.5 on the graph. In this way, every unique genome has a unique graph line.
This figure illustrates a Point Mutation, where a single gene value has changed. The original value of G2 was 1.0 but it became 0.75 after this operation. This is currently the only mutation type that is possible in Galapagos. An alternative method, Inversion Mutation is available where two adjacent gene values are swapped. This operation has a significant benefit only when neighbouring genes have a very explicit relationship. Otherwise, it will probably have a detrimental effect on the procedure as this operation produces too huge a modification. In the fig 5-4-1, G0 and G1’s values have been swapped horizontally. Davit Rutten also mentions Addition Mutations and Deletion Mutations, which affect the number of genes. At present Galapagos only works on fixed-size genomes, but this is not a logical or practical limitation and this may be overcome in future releases.
6-5 . Four Candidates
6-6 . Graph Approximation and Visualization
6-7. Comparison of the 4 Candidates
fig 6-7,1: Comparison in 4 candidates (drawn by K.Hotta)
The solid lines have had the curve-fitting method applied to them while the dotted lines are the actual scores. The vertical axis shows the score, which is the number of shadows on the evaluation plane. The horizontal axis indicates the sun's location in degrees. The very beginning and the very end of the solid lines leap up but those can be ignored as a side effect of the curve fitting methodology.
6-8 . The Comparing the Computing Time for the Kinetic Candidates
fig 6-8,1: The comparison of different computing times, using the Kinetic candidate.
In this graph, similar to the previous graph, the solid line and dotted lines indicate the curve-fitting methodology and the actual scores. The vertical axis and horizontal axis show the shadow score, and sun’s location in degrees respectively. In this graph all the candidates are kinetically controlled roofs or ‘Kinetic’ candidates but different time periods are used in conducting the experiment. For example, one of the candidates, 'Kinetic 100s' is evaluated over a period of 100 seconds; ‘Kinetic300’ is evaluated over a period of 300 seconds. The other conditions are kept constant. The high score of the ‘Kinetic ultimate’ candidate is included in red.
Again when the sun moves to -70 degrees, the GA calculation resets. In this way every 10 degrees, the computing procedure resets. So this candidate is chopped and its calculation reset 18 times in total over the total sun period of 180 degrees.
Fig 6-9,1: The Comparison between different chopped calculations
fig 6-9,2: Total sum of the score in six candidates
6-10 . Discussion and Conclusion
fig 6-10,1: A diagram of ever changing fitness landscape
The left hand diagram shows the abstract fitness landscape at a certain time (let’s say 10:00AM in here as the initial point). There might be several hills, and one of them is the highest achieved value. In that time frame it may or may not reach the top of the hill, it is not certain but GA can usually achieve a good value. However, in the next time frame (for example 11:00AM: the right hand figure), the previous optimized answer has already become old. Because, the fitness landscape has changed, with the time. This
When only the sun’s position is concerned, this ever changing fitness landscape is a continuous function. Because the sun's locus is one curved, continuous line, the sun rising from the east side then going down in the west along a circular path. However, when the actual environmental situation is considered, it is impossible to ignore the disturbances such as clouds which block the sun’s rays or temporary light sources such as light pollution from neighbours. These are unpredictable noise factors that make this fitness landscape a discontinuous function.
A normal GA solver has no idea about the shape of the fitness landscape when it starts calculating.