When we define another subset of models suitable to the specific thing being modelled, then we will just as easily be able to come up with a set of explicit symbolic formulae.
Not necessarily. Closed-form solutions are not guaranteed to exist for your particular subset of models and, in fact, often do not, forcing you to use numeric methods with all the associated problems.
Sorry, hadn’t seen this (note to self: mail alerts).
Is this really true, even if we pick a similarly restricted set of models? I mean, consider a set of equations which can only contain products of a number of variables : like (x_1)^a (x_2)^b = const1 ,(x_1)^d (x_2)^e = const2 .
Is this nonlinear? Yes. Can it be solved easily? Of course. In fact it is easily transformable to a set of linear equations through logarithms.
That’s what I’m kinda getting at : I think there is usually some transform that can convert your problem into a linear, or, in general, easy problem. Am I more correct now?
I think there is usually some transform that can convert your problem into a linear, or, in general, easy problem.
I don’t think this is true. The model must reflect the underlying reality and the underlying reality just isn’t reliably linear, even after transforms.
Now, historically people used to greatly prefer linear models. Why? Because they were tractable. And for something that you couldn’t convert into linear, well, you just weren’t able to build a good model. However in our computer age this no longer holds.
For an example consider what nowadays is called “machine learning”. They are still building models, but these tend to be highly non-linear models with no viable linear transformations.
Not necessarily. Closed-form solutions are not guaranteed to exist for your particular subset of models and, in fact, often do not, forcing you to use numeric methods with all the associated problems.
Sorry, hadn’t seen this (note to self: mail alerts).
Is this really true, even if we pick a similarly restricted set of models? I mean, consider a set of equations which can only contain products of a number of variables : like (x_1)^a (x_2)^b = const1 ,(x_1)^d (x_2)^e = const2 .
Is this nonlinear? Yes. Can it be solved easily? Of course. In fact it is easily transformable to a set of linear equations through logarithms.
That’s what I’m kinda getting at : I think there is usually some transform that can convert your problem into a linear, or, in general, easy problem. Am I more correct now?
I don’t think this is true. The model must reflect the underlying reality and the underlying reality just isn’t reliably linear, even after transforms.
Now, historically people used to greatly prefer linear models. Why? Because they were tractable. And for something that you couldn’t convert into linear, well, you just weren’t able to build a good model. However in our computer age this no longer holds.
For an example consider what nowadays is called “machine learning”. They are still building models, but these tend to be highly non-linear models with no viable linear transformations.