Model-free Methods – a New Frontier of Science

When we make decisions or when we think our brain does not use any equations or math models. Our behaviour is fruit of certain hard-wired instincts and experience that is acquired during our lives and stored as patterns (or attractors). We sort of “feel the answer” to problems no matter how complex they may seem but without actually computing the answer. How can that be? How can a person (not to mention an animal) who has no clue of mathematics still be capable of performing fantastically complex functions? Why doesn’t a brain, with its immense memory and computational power, store some basic equations and formulae and use them when we need to make a decision? Theoretically this could be perfectly feasible. One could learn equations and techniques and store them in memory for better and more sophisticated decision-making. We all know that in reality things don’t work like that. So how do they work? What mechanisms does a brain use if it is not math models? In reality the brain uses model-free methods. In Nature there is nobody to architecture a model for you. There is no mathematics in Nature. Mathematics and math models are an invention (or discovery?) of man.  Nature doesn’t need to resort to equations or other analytical artifacts. These have been invented by man but this doesn’t mean that they really do exist. As Heisenberg put it, what we see is not Nature but Nature exposed to our way of questioning her. If we discover that “F = M * a” that doesn’t mean that Nature actually  computes this relationship each time a mass is accelerated. The relationship simply holds (until somebody disproves it).

Humans (and probably also animals) work based on  inter-related fuzzy rules which can be organised into maps, such as the one below. The so-called Fuzzy Cognitive Maps are made of nodes (bubbles) and links (arrows joining the bubbles). These links are built and consolidated  by the brain as new information linking pairs of bubbles is presented to us and becomes verifiable. Let’s take highway traffic (see map below). For example, a baby doesn’t know that “Bad weather increases traffic congestion”. However, it is a conclusion you arrive at once you’ve been there yourself a few times. The rule gets crystallised and remains in our brain for a long time (unless  sometimes alcohol dissolves it!). As time passes, new rules may be added to the picture until, after years of experience, the whole thing becomes a consolidated body of knowledge. In time, it can suffer adjustments and transformations (e.g. if new traffic rules are introduced) but the bottom line is the same. There is no math model here. Just functions (boxes) connected to each other in a  fuzzy manner, the weights being the fruit of the individual’s own experience.

Fuzzy Cognitive Maps

As a person gains experience, the rules (links) become stronger but, as new information is added, they can also become more fuzzy. This is the main difference between a teenager and an adult. For young people – who have very few data points on which to build the links – the rules are crisp (through two data point a straight line passes, while it is difficult for 1000 points to form a straight line – they will more probably form something that looks like a cigar, if at all). This is why many adults don’t see the world as black or white and why they tend to ponder their answers to questions. Again, the point is that there is no underlying math model. Just example-based learning which produces sets of inter-related Fuzzy Cognitive Maps that are stored in our memory. Clearly, one may envisage attaching a measure of complexity to each such map.

Below is a map depicting the behavior of sharks (left), dolphins (their food), and sardines (right) the food of dolphins.

10 Augmented FCM for diierent actors in a virtual world. The actors interact through linked common causal concepts such a s c hasing food and avoiding a threat. The velocity v(t) does not change at time step t. T h e F CM nds the direction and magnitude of movement. The magnitude of the velocity depends on the FCM state. If the FCM state is \run away," then the velocity i s F AST. If the FCM state is \rest," then the velocity i s S L O W. The prey choose the direction that maximizes the distance from the predator. The predator chases the prey. When a predator searches for food it swims at random 26]. Each state moves the actors through the sea. The FCM in 10 encodes limit cycles between the actors. For example, if we start with a hungry shark and We set the causal link between concept S4: FOOD SEARCH and S6: CHASE DOLPHINS equal to zero to look at shark interactions with the sh school. Then the rst state C 1 is C 1 = 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 ] This vector gives a 7-step limit cycle after four transition steps: C 1 E A = 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 ] ! C 2 = 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 ] C 2 E A = 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 ; 1 0 0 0 0 0 0 0 ] ! C 3 = 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 ] C 3 E A = 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 ] ! C 4 = 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 ]

It was this particular map that prompted, almost two decades ago, the development of OntoSpace, our flagship software product.

OntoSpace doesn’t employ math models in order to establish relationships between the parameters of a system or a process. Essentially, it emulates the functioning of the human brain.

Established originally in 2005 in the USA, Ontonix is a technology company headquartered in Como, Italy. The unusual technology and solutions developed by Ontonix focus on countering what most threatens safety, advanced products, critical infrastructures, or IT network security - the rapid growth of complexity. In 2007 the company received recognition by being selected as Gartner's Cool Vendor. What makes Ontonix different from all those companies and research centers who claim to manage complexity is that we have a complexity metric. This means that we MEASURE complexity. We detect anomalies in complex defense systems without using Machine Learning for one very good reason: our clients don’t have the luxury of multiple examples of failures necessary to teach software to recognize them. We identify anomalies without having seen them before. Sometimes, you must get it right the first and only time!

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