Curve Fitting: Fitting to the Curve y = a*e^(b*x^2)
Introduction
This blog is to fit data to the equation y =
a*e^(b*x^2). A famous example is the
equation:
y = 2/√(π) * e^(x^2)
which the indefinite integral of 0 to real number t is the
basis of the error function.
Turning y = a*e^(b*x^2) into a Linear Form
y = a*e^(b*x^2)
ln y = ln (a*e^(b*x^2))
ln y = ln a + (b)*x^2
Y’ = A’ + B’ * x^2
Where: Y’ = ln
y, A’ = ln a, B’ = b
Results: e^A, B
Inputs: square each
of the data points of X, take the natural logarithm for Y.
Example
I use the HP Prime for this example.
Fit the data below to the curve y = a*e^(b*x^2).
x

y

0.75

0.642931

0.35

0.998284

0

1.128379

0.35

0.998284

0.62

0.768267

0.94

0.466951

Adjust data:
x

C1 = x^2

y

C2 = ln y
(to 6 decimal places)

0.75

0.5625

0.642931

0.441718

0.35

0.1225

0.998284

0.001717

0

0

1.128379

0.120782

0.35

0.1225

0.998284

0.001717

0.62

0.3844

0.768267

0.263618

0.94

0.8836

0.466951

0.761531

Running the linear_regression(C1^2, LN(C2)) returns the
matrix [B’, A’]:
[0.998758, 0.120567]
The results must be adjusted: e^(.120567) = 1.128136
Final adjusted curve:
y ≈ 1.128136 * e^(0.998758*x^2)
Using the Exponential Regression Model
We can use exponential regression function as an alternative
to the linear regression.
For the input, square all of the x data.
x

C1 = x^2

C2 = y

0.75

0.5625

0.642931

0.35

0.1225

0.998284

0

0

1.128379

0.35

0.1225

0.998284

0.62

0.3844

0.768267

0.94

0.8836

0.466951

The function exponential_regression from the HP Prime that
fits data to the model y = B * A^x.
To fit this model asked, y = a*e^(b*x^2), we will need to
take the natural logarithm of B.
Hence:
exponential_regression(C1^2, C2) returns
[0.368337, 1.128136]
With ln(0.368337) ≈ 0.998757, the required equation is
y = 1.128136*e^(0.998757*x^2)
Summary
To fit data to the curve y = a*e^(b*x^2):
Linear Regression, Y’ = A’ + B’*X’

Exponential Regression, Y’ = B’*A’^X’

Input: X’ = x^2, Y’
= ln y
Output: a = e^A’, b
= B’

Input: X’ = x^2, Y’ = y
Output: a = A’, b = ln(B’)

Eddie
This blog is property of Edward Shore, 2018.
The Correlation Calculator displays correlations for major, exotic and cross currency pairs. It is a useful analytical tool that helps in making a sound trading decision.
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