Posts tagged with fuzzy

Leonardo da Vinci’s ability to embrace uncertainty, ambiguity, and paradox was a critical characteristic of his genius. —J Michael Gelb

Say you want to use a mathematical metaphor, but you don’t want to be really precise. Here are some ways to do that:

• Tack a onto the end of an equation.
• Use bounds (“I expect to make less than a trillion dollars over my lifetime and more than \$0.”)
• Speak about a general class without specifying which member of the class you’re talking about. (The members all share some property like, being feminists, without necessarily having other properties like, being women or being angry.)
• Use fuzzy logic (the  membership relation gets a percent attached to it: “I 30%-belong-to the class of feminists | vegetarians | successful people.”).
• Use a specific probability distribution like Gaussian, Cauchy, Weibull.
• Use a tempered distribution a.k.a. a Schwartz function.

Tempered distributions are my favourite way of thinking mathematically imprecisely.

Tempered distributions have exact upper and lower bounds but an inexact mean and variance. T.D.’s also shoot down very fast (like `exp{−x²}` the gaussian) which makes them tractable.

For example I can talk about the temperature in the room (there is not just one temperature since there are several moles of air molecules in the room), the position of a quantum particle, my fuzzy inclusion in the set of vegetarians, my confidence level in a business forecast, ….. with a definite, imprecise meaning.

Classroom mathematics usually involves precise formulas but the level of generality achieved by 20th century mathematicians allows us to talk about a cobordism between two things without knowing everything precisely about them.

It’s funny; the more advanced and general the mathematics, the more casual it can become. Like stingy stickler things that build up to a chummy, whatever-it’s-all-good.

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Our knowledge of the world is not only piecemeal, but also vague and imprecise. To link mathematics to our conceptions of the real world, therefore, requires imprecision.

I want the option of thinking about my life, commerce, the natural world, art, and ideas using manifolds, metrics, functors, topological connections, lattices, orthogonality, linear spans, categories, geometry, and any other metaphor, if I wish.

Briefly: the linear regression model. We suppose we can explain or predict y using a vector of variables x. As in Gauß’ estimation theory, y is supposed to be unobservable, and thus has to be estimated. The assumption that y depends on x is expressed this way: the posterior distribution Prob{ Y | X } is different from the prior distribution Prob{ Y }.

The minimization of variance of the difference between [our estimation of Y given X] and [Y] leads to a unique solution: the conditional expectation.

The linear hypothesis says that the estimated value should be an affine expression of X. Moreover, the affine parameters which minimise the variance of the error are given by:

$\large \dpi{200} \bg_white \begin{matrix} \beta \text{ linear coefficients } \ = \; \ \mathrm{cov}_{X,Y} \ \cdot\ {\mathrm{cov}_{Y,Y}}^{-1} \\ \alpha \text{ constant term } \quad \ \, = \ \ E[Y] \; - \; \beta \cdot E[X] \end{matrix}$

The above linear model coincides with the optimal conditional expectation model when X,Y are Gaussian.
Michel Grabisch, in Modeling Data by the Choquet Integral
(liberally edited)