I thought I would continue my thoughts on 3 Dimensional Fuzzy Sets,
One of the problems that the model presented in my last blog post faces, as it was presented to me by Robert Cox this morning, is that the system would have trouble handing invalid or incorrect data. When the training is not a series perfect solutions, then our system could potentially build up a less than optimal solution…
How do we handle this?
You will recall that I was considering the benefits of having each point handled as a object rather than simply just a piece of data. This gives us a great flexibility as to how we deal with the diagrams. We can actually add an extra modifier to our objects – that being an accuracy or uncertainty factor.
Let’s consider our 3 Dimensional model we were using earlier. If we were to add a whole heap of points in a very localised area, they would provide affirmation… but if these points are less than optimal, we would not want our system to be trained to produce the solution. This is where our accuracy or uncertainty factor comes in. This means that if we have a small space of very mixed data then we could request the system to assess whether the uncertainty of that region, using the uncertainty we could use our existing areas of influence to create a map of the certainty of certain areas. A lower certainty, the lower that area will output to our result sets. If a area is sufficiently uncertain, then if we are in the testing stage, we could request the system to perform it’s test, to compare the different potential of each situation.
Of course, when changes can be counteracted by others later on, we may be unable to get a clear assessment of what was actually a successful or unsuccessful change. In this case, we can instead ask the system to firstly seek more suitable alternatives, and if none can be found then to take the modified potential of our fuzzy set. Taking both the standard mapping of our point to the fuzzy set, but also applying the uncertainties of the area.
The picture I have in my mind about these situations is somewhat like a series of topological maps. You can download maps which show different aspects, rainfall patterns, ground conditions, area boundaries, heights etc. In the same way, I think we can adapt our fuzzy system to be amazingly intuitive as to how it deals with different scenarios.
Also, during our testing stage, we could ask our system to create a test case. To have two situations which are identical. Every input stays the same no matter what happens – a control case. As we provide more of these to the system, it can test the different possibilities and the subsequent results. As we apply these test cases we would need to allow them to accumulate data, but also to assess the scenario and look at the input… what if we change one input slightly? The system needs to develop fuzzy concepts of what is happening, so that it knows when to respond and at what stage it will need to respond.
In fact, perhaps mixing it with different genetic algorithms during the training stage would be beneficial – particularly since human beings rarely have the optimal solution themselves – particularly in the realm of computer games and strategies. Such adaptation by the AI would be particularly useful in scenarios when there a variety of different prospective strategies that could provide equally useful results.
Hmm…
Hi Owen,My research is with fuzzy sets. I have a very simple q to you, by a 3d fuzzy set do you mean two dimensions with two variable as in x,z and the third dimension for its membership degree?
Also in a 3d fuzzy set do you think refering the memebership function we need to choose words like 2d membership function because of 2 variables in the 3d fuzzy set?
Thanks
Thanks
Hi Shohan,
Thanks for the questions!
By a 3 Dimensional Fuzzy Set, I mean a Fuzzy Set with Dimensions (x, y, z (aka. Membership)). It is 3 Dimensional in that it is graphed to a 3D plane denoting memberships.
When we fuzzify an input, what we pass through the set would be two-dimensions of input, but what we receive out is the 3rd dimension.
No, I do not think that we need to use the term “2 Dimensional”. We are not looking at points in a 2D plane, our Fuzzy Set is graphed in a 3D plane because it does have 3 Dimensions of information.
When we add points to our Fuzzy Set, each point (x,y) is given a third value (z) (the membership) which is an extra dimension in and of itself.
Does that sound reasonable enough?
I agree totally with you. The reason as I mentioned was reviewers in water resources research journal declined my paper stating “my 3D (3 dimensional) fuzzy set with 2 variables and a memebership degree is actually a 2D fuzzy set because of it having two variables”. They totally got that wrong. I greatly appreciete your response. My supervisor and I would resubmit our paper based on our “right” definition of the 3D fuzzy set.You have really cleared my confusion. Thanks for the help.
Glad to hear it,
I would note that if you were to get a 3 Dimensional landscape, you could take the X and Y coordinates of the landscape and get the output of the height of terrain at that particular point.
Just because the function to transform (X, Y) -> (Z) only takes two inputs, the initial system has been prepped with a 3 Dimensional model.
if it was a simple equation
x + y = z
It has two inputs and a singular output, but this would be two dimensional, as it is not referencing an extra plane.
However, ours is
f(x + y) = z
and within the function of f, we actually have a static reference to the 3 Dimensional plane (our fuzzy set)
Hi Owen,
I agree that with x,y and the membership grade we get the 3d fuzzy set. But i am a bit unsure whether from a 3d fuzzy set we can refer a 3d fuzzy membership function. I think we can but literatures term the traditional fuzzy set as 1d membership function and therefore it makes our 3d fuzzy set having 2d memebrship function.
The following paper mentions 3d f.m.f for a 3d fuzzy set.
A Three-Dimensional Fuzzy Control Methodology for a Class of Distributed Parameter Systems, Han-Xiong Li; Xian-Xia Zhang; Shao-Yuan Li; Fuzzy Systems, IEEE Transactions on Volume: 15 , Issue: 3
However if you follow this link, you will see that they are reffering the traditional fuzzy set having 1d fuzzy membership function. So that woul make our 3d fuzzy set having 2d Fuzzy MF.
http://www.bluerockresearch.com/papers/ieee_kelly.pdf
Both are contradictory. I would appreciete your help on this because i am submiting my paper and I would like to know whether my 3d fuzzy set is developing a 3d or 2d fuzzy membership function.
Hmm…
Well one thing you should consider is in which domain you are developing for, depending on the field/area there may be an entirely different way of noting the dimensions of a set.
Eg. the Complex Number Plane
It could also be a lame excuse from someone who doesn’t want to accept Fuzzy Theory as real mathematics.
Believe it or not, though Zadeh spent years proving the validity of Fuzzy Logic, there are still people who would prefer to be done with the whole thing. (Generally protesting loudly because there is no excluded middle)
Out of curiosity, what was the topic of the thesis? What type of journal were you submitting to?
my research is in risk hazard management and i am using fuzzy set to represent uncertainty of two variables. want to submit in water resources research journal.
my 3d fuzzy set looks exactly like a pyramid when using triangular MF. Its a very simple 3d Fuzzy set with x dimension for one variable, y dimention for second variable and z dimension is membership degree.