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Gaussian adaptation

Gaussian adaptation

Gaussian adaptation (GA) is also an evolutionary algorithm designed for the maximization of manufacturing efficiency in the statistical processing of signal processing systems. In short, GA is a stochastic adaptive process where a number of samples of an n- dimensional vector x [ T = ( 1 , 2 , …, n )] are taken from a multivariate Gaussian distribution , N ( m , M ), having meanm and moment matrix M . The samples are tested for fail or pass. The first- and second-order moments of the Gaussian restricted to the pass are m * and M * samples .

The outcome of x as a sample is determined by a function s ( x ), 0 < s ( x ) < q ≤ 1, such that s ( x ) is the probability that x will be selected as a pass sample. The average probability of finding pass samples (yield) is

{\ displaystyle P (m) = \ int s (x) N (xm) \, dx}

Then the theorem of GA states:

For any s ( x ) and for any value of  < q , there always exists a Gaussian pdf [ probability density function ] that is adapted for maximum dispersion. The necessary conditions for a local optimum are m = m * and Mproportional to M *. The dual problem is also solved: P is maximized while keeping the dispersion constant (Kjellström, 1991).

Proofs of the theorem can be found in the papers by Kjellström, 1970, and Kjellström & Taxén, 1981.

Since dispersion is defined as the exponential of entropy / disorder / average information it time immediately follows que la theorem is valid également For Those concepts. Altogether, this means that Gaussian adaptation may carry out a simultaneous maximization of yield and average information (without any need for the yield or the average information to be defined as criteria functions).

The theorem is valid for all regions of acceptability and all Gaussian distributions . It can be used by cyclic repetition of random variation and selection (like the natural evolution). In every cycle, the number of Gaussian distributed points is sampled and tested for membership in the region of acceptability. The center of gravity of the Gaussian, m , is then moved to the center of gravity of the approved (selected) points, m *. Thus, the process converges to a state of equilibrium fulfilling the theorem. A solution is always approximate because of the center of gravity is always determined for a limited number of points.

It was used for the first time in 1969 as a pure optimization algorithm making the regions of acceptability smaller and smaller (in analogy to simulated annealing , Kirkpatrick 1983). Since 1970 it has been used for both ordinary optimization and yield maximization.

Natural evolution and Gaussian adaptation

It has also been compared to the natural evolution of populations of living organisms. In this case s ( x ) is the probability que le individual Having an array x of phenotypes will survive by giving offspring to the next generation; Definition of individual fitness given by Hartl 1981. The yield, P , is replaced by the average fitness determined by a large population.

Phenotypes are often Gaussian distributed in a large population and a necessary condition for the natural evolution to be able to fulfill the theorem of Gaussian adaptation, with respect to all Gaussian quantitative characters, is that it can push the center of gravity of the Gaussian to the center of gravity of the selected individuals. This may be done by the Hardy-Weinberg law . This is possible because the theorem of Gaussian adaptation is valid for any region of independent acceptance of structure (Kjellström, 1996).

In this case the rules of genetic variation such as crossover, inversion, transposition etcetera can be seen as random number generators for the phenotypes. So, in this sense, Gaussian adaptation can be seen as a genetic algorithm.

How to climb a mountain

Mean fitness may be calculated that the distribution of parameters and the structure of the landscape is known. The real landscape is a profile of a landscape along a line in a room spanned by such parameters. The red curve is the mean based on the curve at the bottom of the curve. It is achieved by letting the curve curve along the x -axis, calculating the mean at every location. Can be seen, small peaks and pits are smoothed out. Thus, if evolution is started with a relatively small variance (the red bell curve), then it will take place on the red curve. The process can get stuck for millions of years at B or C, as long as the hollows to the right of these points remain, and the mutation rate is too small.

If the mutation rate is high enough, the disorder or variance may increase and the parameter (s) may become distributed like the green bell curve. Then the climbing will take place on the green curve, which is even more smoothed out. Because the hollows to the right of B and C have now disappeared, the process may continue to the peaks at D. But of course the landscape puts a limit on the disorder or variability. Besides – dependent on the landscape – the process can be very fast, and if the ratio between the time and the process is a peak and the time of transition to the next peak is very high, it may well look like a punctuated equilibrium as suggested by Gould (see Ridley).

Computer simulation of Gaussian adaptation

Thus far the theory only considers mean values ​​of an infinite number of individuals. In reality however, the number of individuals is always limited, which gives rise to an uncertainty in the estimation of m and M (the moment matrix of the Gaussian). And this can also affect the efficiency of the process. Unfortunately, this is at least theoretically.

The implementation of normal adaptation is a fairly simple task. The adaptation of m by the sample (individual) at a time, for example

m ( i + 1) = (1 – a ) m ( i ) + ax

where x is a pass sample, and a <a suitable constant so that the inverse of a represents the number of individuals in the population.

M May in principle be updated after-every step are leading to a feasible spot

x = m + y according to:
M ( i + 1) = (1 – 2 b ) M ( i ) + 2 byy T ,

where T is the transpose of y and b << 1 is another suitable constant. In order to guarantee a suitable Increase of average information , there shoulds be Normally distributed with time matrix μ million , Where the scalar μ > 1 is used pour augmenter average information ( information entropy , disorder, diversity) at a suitable rate. But M will never be used in the calculations. Instead we use the matrix W defined by WW T = M .

Thus, we have y = Wg , where g is normally distributed with the moment matrix uU , and U is the unit matrix. W and T may be updated by the formulas

W = (1 – b ) W + byg T and T = (1 – b ) T + bgy T

because multiplication gives

M = (1 – 2 b ) M + 2 byy T ,

where terms including 2 have been neglected. THUS, M Will Be Indirectly adapté with good approximation. In practice it will suffice to update W only

W ( i + 1) = (1 – b ) W ( i ) + BYG T .

This is the formula used in a simple 2-dimensional model of a brain satisfying the Hebbian rule of associative learning; see the next section (Kjellström, 1996 and 1999).

The Figure below illustrates the effect of Increased average information in a Gaussian pdf used to climb a mountain Crest (the two lines Represent the Contour line). Both the red and green cluster have equal mean fitness, about 65%, but the green cluster has a much higher average information . The effect of this adaptation is a very high-dimensional case, a high-dimensional case, the efficiency of the process may be increased by many orders of magnitude.

The evolution in the brain

In the brain the evolution of DNA-messages is supposed to be replaced by an evolution of signal patterns and the phenotypic landscape is replaced by a mental landscape, the complexity of which will hardly be second to the trainer. The metaphor with the mental landscape is based on the assumption that some signal patterns give rise to a better well-being or performance. For instance, the control of a group of muscles leads to a pronunciation of a word or performance of a piece of music.

In this simple model it is assumed that the brain consists of interconnected components that may add, multiply and delay signal values.

  • A nerve cell kernel may add signal values,
  • can synapse multiply with a constant
  • An axon may delay values.

This paper is based on the theory of digital filters and neural networks.

Gaussian distributed signal patterns. This may be possible since some neurons fire at random (Kandel et al.). The stem is also a disordered structure surrounded by more ordered shells (Bergström, 1969), and according to the central limit theorem the sum of signals from many neurons may be Gaussian distributed. The triangular boxes represent synapses and the boxes with the kernels.

In the cortex signals are supposed to be tested for feasibility. Where a signal is accepted in the synapses in the synapses with the Hebbian theory. The figure shows a 2-dimensional computer simulation of Gaussian adaptation according to the last formula in the preceding section.

m and W are updated selon:

1 = 0.9 1 + 0.1 x 1; 2 = 0.9 2 + 0.1 2 ;
11 = 0.9 11 + 0.1 1 ; 12 = 0.9 12 + 0.1 2 ;
21 = 0.9 21 + 0.1 1 ; 22 = 0.9 22 + 0.1 2 ;

As Kjellström, 1996, 1999 and 2002), he is very much like a small brain ruled by the theory of Hebbian learning .

Gaussian adaptation and free will

Gaussian adaptation as an evolutionary model of the brain obeying the Hebbian theory of associative learning offers an alternative view of free will to the ability of the process to maximize the mental fitness of signal patterns in the brain by climbing a mental landscape in analogy with phenotypic evolution.

Such a random process gives us lots of freedom of choice, but hardly any will. An illusion of will may, however, emanate from the ability of the process to maximize mean fitness, making the process goal seeking. I. e., It prefers higher peaks in the landscape, or better alternatives prior to worse. In this way an illusive will may appear. A similar view by Zohar 1990. See also Kjellström 1999.

A theorem of efficiency for random search

The efficiency of Gaussian adaptation to the theory of information by Claude E. Shannon (see information content ). When an event occurs with probability P , then the information -log ( P ) may be achieved. For instance, if the mean fitness is P , the Information Gained For Each individual selected for survival will be -log ( P ) – on the average – and the work / time needed to get the information is proportional to 1 / P . Thus, if efficiency, E is defined as information by the work / time needed to get it we have:

E = – P log ( P ).

This function attains its maximum when P = 1 / e = 0.37. The same result was obtained by Gaines with a different method.

E = 0 if P = 0, for a process with infinite mutation rate, and if P = 1, for a process with mutation rate = 0 (provided that the process is alive). This measure of efficiency is valid for a large class of random search processes that certain conditions are at hand.

The search should be statistically independent and efficient in different parameter directions. This requirement Approximately May be Fulfilled When the time matrix of the Gaussian has-been adapté for maximum average informationto Some area of acceptability, Because linear transformations of the whole process do not affect efficiency.

2 All individuals have equal cost and the derivative at P = 1 is <0.

Then, the following theorem can be proved:

All levels of efficiency, which satisfy the above conditions, are asymptotically proportional to -P log ( P / q ) when the number of dimensions increases, and are maximized by P = q exp (-1) (Kjellström, 1996 and 1999).

The figure above shows a possible effect of Gaussian adaptation. To the left is the most chaotic when P = 0, while there is perfect order where P = 1.

In an example by Rechenberg, 1971, 1973, a random walk is pushed a corridor maximizing the parameter 1 . In this case the region of acceptability is defined as a ( n -1) -dimensional interval in the parameters 2 , 3 , …, n , but a 1 -value below the last accepted will not be accepted. Since P can never exceed 0.5 in this case, the maximum speed towards higher 1 -values ​​is reached for P = 0.5 / e = 0.18, in agreement with the findings of Rechenberg.

It is also possible that the definition of information is more important than that it is necessary for the proof of the theorem. Then, because the formula can be interpreted as information by the work needed to get information, this is also an indication that -log ( P ) is a good candidate for being a measure of information.

The Stauffer and Grimson algorithm

Gaussian adaptation has been used for other purposes as shadow removal by “The Stauffer-Grimson algorithm” which is equivalent to Gaussian adaptation as used in the section “Computer simulation of Gaussian adaptation” above. In both cases, the maximum likelihood method is used for estimation of mean values ​​by adaptation at one sample at a time.

But there are differences. In the Stauffer-Grimson case the information is not used for the control of a random number generator for centering, maximization of mean fitness, average information or manufacturing yield. The adaptation of the moment has also been compared to “the evolution in the brain” above.

See also

  • Entropy in thermodynamics and information theory
  • Fisher’s fundamental theorem of natural selection
  • Free will
  • Genetic algorithm
  • Hebbian learning
  • Content information
  • Simulated annealing
  • Stochastic optimization
  • Covariance Matrix Adaptation Evolution Strategy (CMA-ES)
  • Unit of selection


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