- #1
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I write math related tutorials and would appreciate comments/advice about terminology that may be potentially ambiguous or confusing. As an example, I use ##f(x;\beta)## to define a generic continuous function with independent variable ##x## and function parameters ##\beta = (\beta_1, \beta_2, \dots, \beta_n)## that are also considered variables (to be determined by an iterative optimization process). The partial derivative of ##f(x;\beta)## with respect to the ##k##th component of ##\beta##, or ##\beta_k## would be:
$$\frac{\partial f(x;\beta)}{\partial \beta_k} = .....$$
However, later on, ##\beta_k## is referred to as the ##k##th iterate of the values of the vector ##\beta##, for example: ##\beta_{k+1} = \beta_k + \Delta \beta##. Thus, ##\beta_k## has a different interpretation based on context. I've seen use of the following to denote iterates, which I don't like (seems too complicated):
$$\beta^{(k+1)} = \beta^{(k)} + \Delta \beta \quad\text{or}\quad \beta_{(k+1)} = \beta_{(k)} + \Delta \beta$$
I've also thought about using a different symbol for the vector, so that:
$$\phi = (\beta_1, \beta_2, \dots, \beta_n)
\qquad
\frac{\partial f(x;\phi)}{\partial \beta_k} = .....
\qquad
\phi_{k+1} = \phi_k + \Delta \phi$$
which seems confusing to me because there appears to be no connection between ##f## and ##\beta_k##. Lastly, I've thought about using a capital greek letter to refer to the vector, something like:
$$\Phi = (\phi_1, \phi_2, \dots, \phi_n)
\qquad
\frac{\partial f(x;\Phi)}{\partial \phi_k} = .....
\qquad
\Phi_{k+1} = \Phi_k + \Delta \Phi$$
I think the last choice is probably the most clear, but I don't like the bold symbols sprinkled around in the tutorial. What I really want to do is have two different meanings of ##\beta_k##, one as vector component and the other as a vector iterate. Is this bad form?
$$\frac{\partial f(x;\beta)}{\partial \beta_k} = .....$$
However, later on, ##\beta_k## is referred to as the ##k##th iterate of the values of the vector ##\beta##, for example: ##\beta_{k+1} = \beta_k + \Delta \beta##. Thus, ##\beta_k## has a different interpretation based on context. I've seen use of the following to denote iterates, which I don't like (seems too complicated):
$$\beta^{(k+1)} = \beta^{(k)} + \Delta \beta \quad\text{or}\quad \beta_{(k+1)} = \beta_{(k)} + \Delta \beta$$
I've also thought about using a different symbol for the vector, so that:
$$\phi = (\beta_1, \beta_2, \dots, \beta_n)
\qquad
\frac{\partial f(x;\phi)}{\partial \beta_k} = .....
\qquad
\phi_{k+1} = \phi_k + \Delta \phi$$
which seems confusing to me because there appears to be no connection between ##f## and ##\beta_k##. Lastly, I've thought about using a capital greek letter to refer to the vector, something like:
$$\Phi = (\phi_1, \phi_2, \dots, \phi_n)
\qquad
\frac{\partial f(x;\Phi)}{\partial \phi_k} = .....
\qquad
\Phi_{k+1} = \Phi_k + \Delta \Phi$$
I think the last choice is probably the most clear, but I don't like the bold symbols sprinkled around in the tutorial. What I really want to do is have two different meanings of ##\beta_k##, one as vector component and the other as a vector iterate. Is this bad form?