Another 'm' estimate

We continue exploring the new definition of health = v*m. Particularly the role of the m-factor in health. Previously we used the correlation coefficient to estimate m, which was somewhat crude and will be replaced with a better estimate.

The  experiment is controlled by two functions:
delivery activation: [state[1, set point = p[1] - 0.1, state[0,k]];{k,1,46}
change state: [state[1, i+1] = state[0, k], state[1,k]]; {k,1, 46}
delivery: [2, 1, While[p[1] > set point], 2];

The following parameters were estimated:

1. Tolerance velocity was  estimated with regression.
2. Min[velocity] and  Max[velocity].

Max[velocity] and Min[velocity]

Movement of a cell in the CA is controlled by three probabilities: p[inward], p[forward], and p[outward]. Max[velocity], and Min[velocity], express the maximal movement outward, and respectively inward. Both are controlled by m.


The CA  consists of two structures, a core of constant width, and an oscillating rim . Resources are accumulated only in the core. Resources produced in the rim are delivered into the environment. The  wider the core the more resources are accumulated. Core and rim  velocities are defined respectively as Min[velocity] and Max[velocity] - Min[velocity].
The m-factor  is defined:  m = Min[velocity]/[Max velocity], and health = v * m.

The parameters were estimated in 46 CA. Each was triggered by a different CA-0 state, using the change state function. Some CA have a distinct core (CA-1) while other, like CA-2 lack it. When the core vanishes its velocity v = 0, health = 0 whereupon the CA-2 dies and  is replaced with a new zygote.

The figure depicts health in CA triggered by the 46 CA-0 states. Most of the CA remain isolated  and  like CA-1 occupy the central line.  Their health is about 2.4.

CA product

The main task of a CA is to accumulate tolerance (resources). Resources generated at one instant (step) are called product (input). One part of the product becomes CA structure (CA core). The other part, or output (CA rim), is supplied to their environment. Core and rim width are controlled by the m-factor.

CA-3 is the healthiest since its core is the widest. CA-4 whose core is narrower is less healthy than CA-3 and CA-1. Nevertheless CA-4 is more active than CA-1. It oscillates more vigorously, and has a larger output. It may be less healthy  but is more productive.  We shall define output velocity = (Max[velocity] - Min[velocity])/Max[velocity].

Output[CA-1] = 0.708
Output[CA-2] = 1.000
Output[CA-3] =  0.703
Output[CA-4] =  0.854

In CA-1 output is moderate. The entire CA-2 product becomes an output. Its health is low  and when it dies it is replaced with a new zygote. In most CA clustering in the center, health and output are balanced. They occupy a spectrum which spans between CA which accumulate their entire product, and those who deliver their product into the environment. In between CA exhibit different proportions of both properties.

delivery activation: [state[j , set point = p[j] - 0.1, state[0, k]]
First argument: the CA which is activated
Second argument: set point definition
Third argument: the state k of CA-0 which activates delivery.
p[j]:  daily production.

delivery: [j, j-1, While[p[j-1] > set point], 2]
First argument: CA receiving the delivery. 
Second argument: Delivering CA.
Third argument: Delivery condition.
Fourth argument: Delivery amount.
p[j]:  daily production.

change state: [state[j, i+1] = state[0. i], state[0, k]]
First argument: state transfer.
Second argument: the state k of  CA-0 which activates the change.

augment state: [state[j, i+1] += state[j-1, i], While[p[j-1] > p[ j ]]]
First argument: state augmentation.
Second argument: Delivery condition.
p[j]:  daily production.