Measuring errors on fit to hyperbolic equations
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I have an equation of the form:
Q = Qi*(1+D*b*t)^(-1/b), which is used to describe a decline in flow rates with time.
Q is the flow rate,
Qi is the hypothetical flow rate at time 0,
D is the decline rate,
b is the 'b-factor'...
When I have real data I want to estimate values for Qi, D, and b, such that my predictions of Q through time (t) are a good representation of that data.
The issue I have is that the early values are very large relative to the later values (e.g., at time steps 1,3,7,15 Q = 500, 250, 125, 62.5 when Qi = 1000, D=1, and b=1). This means my squared error at time step 1 is very important as a 10% error = 50^2, while the same 10% error in time step 15 is only 6.25^2.
This means when I use a sum of the (actual-predicted)^2 my attempt to minimize the Sum of the squared values is heavily influenced by the early data. I know this must be a common problem, but I can't figure out how to get around it.
What other methods would be better in this type of situation?
curves least-squares
asked Aug 20 at 9:49
user1563247
11
add a comment |Â
up vote
0
down vote
favorite
I have an equation of the form:
Q = Qi*(1+D*b*t)^(-1/b), which is used to describe a decline in flow rates with time.
Q is the flow rate,
Qi is the hypothetical flow rate at time 0,
D is the decline rate,
b is the 'b-factor'...
When I have real data I want to estimate values for Qi, D, and b, such that my predictions of Q through time (t) are a good representation of that data.
The issue I have is that the early values are very large relative to the later values (e.g., at time steps 1,3,7,15 Q = 500, 250, 125, 62.5 when Qi = 1000, D=1, and b=1). This means my squared error at time step 1 is very important as a 10% error = 50^2, while the same 10% error in time step 15 is only 6.25^2.
This means when I use a sum of the (actual-predicted)^2 my attempt to minimize the Sum of the squared values is heavily influenced by the early data. I know this must be a common problem, but I can't figure out how to get around it.
What other methods would be better in this type of situation?
curves least-squares
asked Aug 20 at 9:49
user1563247
11
Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
– José Carlos Santos
Aug 20 at 10:45
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I have an equation of the form:
Q = Qi*(1+D*b*t)^(-1/b), which is used to describe a decline in flow rates with time.
Q is the flow rate,
Qi is the hypothetical flow rate at time 0,
D is the decline rate,
b is the 'b-factor'...
When I have real data I want to estimate values for Qi, D, and b, such that my predictions of Q through time (t) are a good representation of that data.
The issue I have is that the early values are very large relative to the later values (e.g., at time steps 1,3,7,15 Q = 500, 250, 125, 62.5 when Qi = 1000, D=1, and b=1). This means my squared error at time step 1 is very important as a 10% error = 50^2, while the same 10% error in time step 15 is only 6.25^2.
This means when I use a sum of the (actual-predicted)^2 my attempt to minimize the Sum of the squared values is heavily influenced by the early data. I know this must be a common problem, but I can't figure out how to get around it.
What other methods would be better in this type of situation?
curves least-squares
asked Aug 20 at 9:49
user1563247
11
I have an equation of the form:
Q = Qi*(1+D*b*t)^(-1/b), which is used to describe a decline in flow rates with time.
Q is the flow rate,
Qi is the hypothetical flow rate at time 0,
D is the decline rate,
b is the 'b-factor'...
When I have real data I want to estimate values for Qi, D, and b, such that my predictions of Q through time (t) are a good representation of that data.
The issue I have is that the early values are very large relative to the later values (e.g., at time steps 1,3,7,15 Q = 500, 250, 125, 62.5 when Qi = 1000, D=1, and b=1). This means my squared error at time step 1 is very important as a 10% error = 50^2, while the same 10% error in time step 15 is only 6.25^2.
This means when I use a sum of the (actual-predicted)^2 my attempt to minimize the Sum of the squared values is heavily influenced by the early data. I know this must be a common problem, but I can't figure out how to get around it.
What other methods would be better in this type of situation?
curves least-squares
asked Aug 20 at 9:49
user1563247
11
asked Aug 20 at 9:49
user1563247
11
asked Aug 20 at 9:49
user1563247
11
asked Aug 20 at 9:49
user1563247
11
11
Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
– José Carlos Santos
Aug 20 at 10:45
add a comment |Â
Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
– José Carlos Santos
Aug 20 at 10:45
Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
– José Carlos Santos
Aug 20 at 10:45
Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
– José Carlos Santos
Aug 20 at 10:45
add a comment |Â
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Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
– José Carlos Santos
Aug 20 at 10:45