Rate of Inflation of Gas

Posted on June 27, 2012 by Siddharth Jain

Raw Data: http://www.randomuseless.info/gasprice/gasprice.txt

Model:
p(t) = p_0(1+r)^t where r is the rate of inflation. This can also be written as:
p(t) = p_0e^{\gamma t}
Given series of data points (p_i,t_i) construct cost fn.
e = \sum_i (p_i - p_0e^{\gamma t_i})^2
Need to find p_0 and \gamma that minimize e.
\frac{de}{dp_0} = \sum_i 2(p_i - p_0e^{\gamma t_i})(-e^{\gamma t_i}) = 0
\frac{de}{d\gamma} = \sum_i 2(p_i - p_0e^{\gamma t_i})(-p_0t_ie^{\gamma t_i}) = 0
Simplify above two equations to
\sum_i \alpha_i \beta_i = 0 ….(1)
and
\sum_i \alpha_i \beta_i t_i = 0 ….(2)
where
\alpha_i = p_i - p_0 \beta_i
\beta_i = e^{\gamma t_i}
Simplify (1) to give
p_0 = \frac{\sum_i p_i \beta_i}{\sum_i \beta_i^2} ….(3)
substitute value of p_0 from (3) into (2) to give
f(\gamma) = \sum_i (p_i - p_0(\gamma)\beta_i(\gamma))\beta_i(\gamma) t_i = 0
Now use Newton’s method to find roots of f(\gamma)
\gamma_{n+1} = \gamma_n - \frac{f(\gamma_n)}{f^{'}(\gamma_n)}
After lots of algebra, verify
f^{'}(\gamma) = -2p_0 \beta_i^2 t_i^2 - p_0^{'} \beta_i^2 t_i + p_i \beta_i t_i^2
p_0^{'}(\gamma) = \frac{bc-ad}{c^2}
where
a = 2\sum_i \beta_i p_i
b = \sum_i p_i \beta_i t_i
c = \sum_i \beta_i^2
d = \sum_i \beta_i^2 t_i

Code: http://sdrv.ms/MWaGWo

Output of program (Newton’s method):
\gamma = 0.0584642367944379, residual = 8264.22684488131
\gamma = 0.0394587424266077, residual = 755.065800336963
\gamma = 0.0373150889455502, residual = 11.93657393052
\gamma = 0.0372800836836129, residual = 0.00330268916627574
Rate of inflation: 3.798%
Residual = 0.00330268916627574
Iterations = 4

Graph:

http://db.tt/PeU0ZBmK

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