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Question: How can I tell Mathematica to replace a sub-expression (in a simplified expression) with a symbol if the symbol is defined to be that sub-expression?

Say you have defined:

f[x_, y_] := some_complicated_expr_using_x_and_y_1 g[x_, y_] := some_complicated_expr_using_x_and_y_2 h := some_complicated_expr_using_f_and_g

then you do D[h,x] // Simplify, you would get an expression that has sub-expressions some_complicated_expr_using_x_and_y_1 and some_complicated_expr_using_x_and_y_2.

How can I tell Mathematica to replace some_complicated_expr_using_x_and_y_1 with f[x,y]?

For a concrete example, consider:

h[μ_, r_] := Sum[x[μ, s]*w[s, r], {s, 1, n}] yhat[μ_, r_] := σ[h[μ, r]] e := (1/(2 m)) Sum[Sum[(y[μ, r] - yhat[μ, r])^2, {r, 1, p}], {μ, 1, m}] h[μ, r] yhat[μ, r] e D[e, w[i, j]]

I you evaluate the above you will get

mse

How can I get it to produce:

enter image description here

(which equals to Out[97])

I have tried ReplaceRepeated but got no luck:

enter image description here

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  • $\begingroup$ to get halfway there you need to have the unevaluated forms of the expressions on the rhs of ruleh and ruleyhat. That is, use HoldForm[h[μ, r]] and HoldForm[yhat[μ, r]]. $\endgroup$ Commented Jan 20, 2018 at 5:55

1 Answer 1

Reset to default
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Here is the result of the operation D[e, w[i, j]]:

Clear[yhat]; expr=(1/(2 m) (\!\( \*UnderoverscriptBox[\(\[Sum]\), \(\[Mu] = 1\), \(m\)]\( \*UnderoverscriptBox[\(\[Sum]\), \(r = 1\), \(p\)]\((\(-2\)\ \(( \*UnderoverscriptBox[\(\[Sum]\), \(s = 1\), \(n\)]\* TemplateBox[{RowBox[{"i", ",", "s"}]}, "KroneckerDeltaSeq"]\ \* TemplateBox[{RowBox[{"j", ",", "r"}]}, "KroneckerDeltaSeq"]\ x[\[Mu], s])\)\ y[\[Mu], r]\ \* SuperscriptBox["\[Sigma]", "\[Prime]", MultilineFunction->None][ \*UnderoverscriptBox[\(\[Sum]\), \(s = 1\), \(n\)]w[s, r]\ x[\[Mu], s]] + 2\ \(( \*UnderoverscriptBox[\(\[Sum]\), \(s = 1\), \(n\)]\* TemplateBox[{RowBox[{"i", ",", "s"}]}, "KroneckerDeltaSeq"]\ \* TemplateBox[{RowBox[{"j", ",", "r"}]}, "KroneckerDeltaSeq"]\ x[\[Mu], s])\)\ \[Sigma][ \*UnderoverscriptBox[\(\[Sum]\), \(s = 1\), \(n\)]w[s, r]\ x[\[Mu], s]]\ \* SuperscriptBox["\[Sigma]", "\[Prime]", MultilineFunction->None][ \*UnderoverscriptBox[\(\[Sum]\), \(s = 1\), \(n\)]w[s, r]\ x[\[Mu], s]])\)\)\))) /. \[Sigma][a_] -> yhat[\[Mu], r] /. Derivative[1][\[Sigma]][a_] -> yhat'[\[Mu], r]

Try this:

expr /. \[Sigma][a_] -> yhat[\[Mu], r] /.Derivative[1][\[Sigma]][a_] -> yhat'[\[Mu], r]

yielding thew following:

enter image description here

Have fun!

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