Hello sympy people!
I couldn't find any information on the topic of "Error Analysis" in sympy
(possibly because terms like "standard", "error", and "propagation" don't
really catch this top well in search engines ;)
Anyway, I just went ahead and wrote a basic converter from any expression
to the expression for the error; I just documented it in the docstring if
you are curious.
It's a little rudimentary and could be better -- namely the error of (2*a)
is reported as 2*sqrt(ae**2/a**2)*Abs(a) instead of just 2*ae -- but it
seems to at least produce correct results.
Please let me know if there is already a better way to do this or if you
find a bug with this code (or if you find the code useful)!
Thanks!
-David Mashburn
def
GetCombinedErrorExpression(sympyExpression,variables,errorVariables,verbose=False,top=True):
'''Takes any symbolc expression with variables involving basic +-*/^
and converts
it to the symbolic expression for the standard errors using error
propagation rules
"variables" is the list of variables to use; anything else is
assumed to be constant
"errorVariables" is the corresponding list of variables for the
errors that will be used in the output
errorVariables are simple symbols used to generate the return
expression
verbose prints the expression and all sub-expressions generated by
recursion
For cases not easily handlable, this simply returns None
WARNING! No error handling is done in the case that there are
variables IN exponents;
you should only use error-free constants in exponents (but
bases can have errors)
Simple examples:
a,b,c,ae,be,ce = sympy.symbols('a,b,c,ae,be,ce')
GetCombinedErrorExpression(a**2,[a],[ae])
--> 2*a*ae
GetCombinedErrorExpression(a+b+c,[a,b,c],[ae,be,ce]) )
--> sqrt(ae**2 + be**2 + ce**2)
GetCombinedErrorExpression(a*b*c,[a,b,c],[ae,be,ce]) )
--> sqrt(ce**2/c**2 + be**2/b**2 + ae**2/a**2)*Abs(a*b*c)
GetCombinedErrorExpression(a/b,[a,b],[ae,be])
--> sqrt(be**2/b**2 + ae**2/a**2)*Abs(a/b)
GetCombinedErrorExpression(a+2*b+c**2,[a,b,c],[ae,be,ce])
(pretty form) -->
_____________________________
/ 2 2
/ 2 2 2 4*be *|b|
/ ae + 4*c *ce + ----------
/ 2
\/ b
'''
if verbose:
print sympyExpression
if top: # Check that input variables are correct, but only do it once!
assert hasattr(variables,__len__)
assert hasattr(errorVariables.__len__)
assert len(variables)==len(errorVariables)
for v in variables:
assert v.__class__==sympy.Symbol
for v in errorVariables:
assert v.__class__==sympy.Symbol
# Handle cases for sympyExpression (can be any of Add, Mul, Symbol,
Integer, Float, Number)
if sympyExpression.__class__==sympy.Add:
subExpressions = sympyExpression.as_coeff_add()[1]
subExpressionsError = [
GetCombinedErrorExpression(s,variables,errorVariables,verbose,top=False)
for s in subExpressions ]
if None in subExpressionsError:
return None
else:
return sympy.sqrt(sum([s**2 for s in subExpressionsError]))
elif sympyExpression.__class__==sympy.Mul:
subExpressions = sympyExpression.as_coeff_mul()[1]
subExpressionsError = [
GetCombinedErrorExpression(s,variables,errorVariables,verbose,top=False)
for s in subExpressions ]
if None in subExpressionsError:
return None
else:
errorsDivVars = [ subExpressionsError[i]/subExpressions[i]
for i in range(len(subExpressions)) ]
return sympy.Abs(sympyExpression)*sympy.sqrt(sum([s**2 for s in
errorsDivVars]))
elif sympyExpression.__class__==sympy.Pow:
subExpression,exponent = sympyExpression.as_base_exp()
subExpressionError =
GetCombinedErrorExpression(subExpression,variables,errorVariables,verbose,top=False)
if subExpressionError==None:
return None
return
sympyExpression*sympy.Abs(exponent)*subExpressionError/subExpression
elif sympyExpression.__class__ in (sympy.Symbol,
sympy.Integer,
sympy.Float,
sympy.Number):
if sympyExpression in variables:
return errorVariables[variables.index(sympyExpression)] #
return the appropriate error var
else:
return 0 # assume it's a constant
return sympyExpression
else: # kick any other cases as exceptions...
return None
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