Testing pairwise orthognoality

[danielpower1859]

What are the properties we like?

• multiple attractors and their complement
• evo works
• ...?

What is is about certain random attractor pairs that creates the properties we like?

• is it pairwise orthogonality?

What does this property look like? How can we define it in a general way?

[danielpower1859]

new function 
def correspondence_patterns_between_binary_vectors(v1, v2):
    '''
    for two vectors of the same length
    returns the proportion of entries which are:
        1 in both
        1 in v1, 0 in v2
        0 in v1, 1 in v2
        0 in both
        '''
        assert len(v1) == len(v2)
# to do :
# check inputs are binary

HH = np.sum( v1 * v2) 
HL = np.sum(v1 * (1-v2))
LH = np.sum((1-v1) * v2)
LL = np.sum((1-v1) * (1-v2))

assert HH+HL+LH+LL == len(v1)
return HH, HL, LH, LL

Results: for 50:50 vectors (ecah vector has 50 ones and 50 0s) The number of HL is always == number of LH and # HH == # LL

[danielpower1859]

HH:Hl = 4 attractor states, each with 50% high /low?? 30:20 = fail 24:26 = fail 27:23 = fail 23:27 = fail 22:28 = fail 23:27 = fail 26:24 = fail 25:25 = Success!!!

Looks highly likely that, with the current parameter set, pairwise orthogonality is essential for 50:50 species density split

[danielpower1859]

idea:

• function that measures the species density at end.
• Want a good 50 high, 50 low split
• then compare this to HH:HL proportion

[danielpower1859]

Looks very much like its only variations of 1 1 1 1 1 0 0 0 0 0 1 0 1 0 1 0 1 0 1 0

That work. I.e. shuffled versions of the paired pattern. Damn

[danielpower1859]

The system currently only works with random compositions such that each vector has the same proportion (25%) of each of:

• species that are favoured in both patterns
• Species that are favoured in v1 but not v2
• Species that are favoured in v2 but not v1
• species that are not favoured in either

`def pair_wise_orthogonal_patterns(N): ''' returns two binary np.vectors that are pairwise orthogonal. i.e. each has 25%: HH, HL, LH, LL '''

assert N%4 ==0

v = []
for i in range(int(N/4)):
    v.extend([[1,1], [1,0], [0,1], [0,0]])
shuffle(v)
v = np.array(v)
return  v.transpose()[0], v.transpose()[1]`

[danielpower1859]

HOWEVER!! I haven't checked what happens to evolution on the patterns created by assymetric patterns. Although given how sensitive the "balanced" pattern is.. i don't hold much hope.

[danielpower1859]

Have been testing system with HH: HL 26:24 Reducing m_values looks promising... playing in the range nm_values - 3 - 4 K = 3. # base carrying capacity 3. M = 0.5 # growth rate 0.5 is default

M_values: must be high enough to divide HL from LH (or vice versa) low enogh that HH and HL are v. similar

0.34 seems v good

[danielpower1859]

Useful to have a function that compares density of HH with density of HL, LH and LL species Returns proportions.

[danielpower1859]

Next step here: test 26:24 patterns with 0.0034 parameter Will low learning rate work?

YES :-)

[danielpower1859]

Patterns: 23:27, 24,:26, 25:25, 26:24, 27:23 All can be parameterised for to produce systems with 2 attractor pairs, which Hebbian evolution can act upon to enlarge the BoA for the larger (lower MCC) attractor.

These 5 patterns account for about 68% of patterns produced by the shuffle 50 1s and 50 zeros method.

Pairwise orthogonality (equal proportions of HH:HL:LH:LL) adds a lot of stability