Tuesday, January 31, 2006

The Stochastic Bass Model

Background

The Bass model is used in marketing in order to forecast the diffusion of sales of a new product and takes the following form [1]:
dN(t)/dt= (P + Q / M x N(t)) x (M − N(t))
where the parameters have the following meaning:
N(t): number of adopters of new product at time t
M: total market size
Q: internal influence, e.g. word of mouth
Solving the above differential equation gives the Ricatti function:
N(t)= M x (1 - exp(-(P + Q) x t)) / (1 + Q / P x exp(-(P + Q) x t))
with the following derivative (equivalent to the oil production):
dN(t)/dt= M x (P + Q) ^2 / P x exp(-(P + Q) x t)) / (1 + Q / P x exp(-(P + Q) x t))^2
New sales can be seen as a combination of two set of buyers:
1. innovators which are not influenced in the timing of their purchase by the number of people who have already bought the product;
2. imitators which are inlfuenced by others
Intermediate quantities can be computed:
Residual_Market(t)= M - N(t)
Innovators(t)= P x Residual_Market(t)
Word_of_Mouth(t)= N(t) / M
Accessibility_of_Imitators(t)= q x Word_of_Mouth(t) x Residual_Market(t)
dN(t)/dt= Innovators(t) + Accessibility_of_Imitators(t)
Note that if you set P= 0 the above equa. diff. is exactly the logistic function:
dN(t)/dt= Q / M x N(t) x (M − N(t))
Application to Oil Depletion

In the case of oil depletion, N(t) will be the cumulative oil production, Q will be the growth rate (usually noted K) and M the Ultimate Recoverable Ressource (URR). The Generalized Bass Model has been investigated by Guseo et al. [2,3] for the modeling of oil depletion (see thread on peakoil.com: Peak prediction based on the Riccati equation: 2007!). The diffusion by internal influence controled by P plays a minor role in the oil production and generally has a very small value. We can interpret that small value by the fact the oil demand is driven by the economic growth and imposed by the maturity of the oil infrastucture without any innovations. The figure below gives an example of the Bass model applied to the US production (EIA data). We can see that the Bass model curve is very similar to the logistic model because of the small value of P. The fitting was performed using the Marquardt's method for non-linear least squares which smoothly switches from a simple downhill search to Newton's method as it approaches the minimum.

The Stochastic Bass Model

Recently, some have proposed a time-discrete stochastic interpretation of the Bass model which is seen as a pure birth random process [1]. At time t, for each M - N(t) individual susceptible buyer, the probability of an acquisition at time t+1 is the following:
P(Adopter(t+1) | Susceptible(t))= p + q / M x N(t)
You can demonstrate that realizations of the Stochastic Bass Model (SBM) will converge in mean toward the Ricatti function. Applied to the oil production (or consumption), the above transition probability is the probability of a produced barrel to be consumed at time t. The figure below gives an example applied to the US production where three different SBM are shown with different values of p. Each curve is the average value of 2,000 runs. We can see that the parameter p controls in fact the time shift of the curve.

That's it for now, in a next post we will explore the use of this stochastic Bass model for the stochastic "tracking" of an oil production curve.

[1] Georg Holtz, An individual level diffusion model, carefully derived from the Bass-Model. Working Paper. September 2004.

[2]GUSEO, R., DALLA VALLE, A. and GUIDOLIN, M. (2005). World Oil Depletion Models: Price Effects Compared with Strategic or Technological Interventions

[3] GUSEO, R. e DALLA VALLE, A. (2004). Oil and Gas Depletion: Diffusion Models and Forecasting under Strategic Intervention, Atti della XLII Riunione Scientifica della Società Italiana di Statistica, Bari 9-11/6/2004, Vol. Sessioni Spontanee, 733-36, CLEUP, Padova.

Friday, January 27, 2006

The 4 Biggest Oil Exporters

This first blog is a follow up on discussions around the future production of the world four top oil exporters (Iran, Norway, Saudi Arabia and Russia) that have been posted recently on theoildrum.com: Hubbert Linearization Analysis of the Top Three Net Oil Exporters.

The Hubbert linearization technique is applied on Saudi Arabia, Norway, Iran and Russia from which I derive the URR (Qinf) and the growth rate (K). The resulting logistic curve is then shifted in order to match the last production level (2004). I note Qt the cumulative production in 2004. The production data comes mainly from the BP statistical review which gives data from 1965 to 2004. Cumulative production values (Q) are corrected for missing pre-1965 production data.

Saudi Arabia:

Fitting a logistic curve on Saudi Arabia production is quite difficult because production was constrained by quotas and political interventions (embargos). The circled data points are those used for the logistic fit.

The parameters of the logistic fit are the following:

Qt(2004): 110.6 Gb (62.9% of Qinf)
k: 0.0777
Qinf: 175.8 Gb
thalf: 2003.5
Russia:

Russia is also quite a difficult case because of the production collapse in the 80s and the strong rebound in the late 90s.

The parameters of the logistic fit are the following:

Qt(2004): 138.8 Gb (91.6% of Qinf)
k: 0.1083
Qinf: 151.6 Gb
thalf: 1995.5

Norway:

We know that Norway production has peaked and will probably not return near its peak production level. The linear fit in the P/Q vs Q representation is quite clean:

The parameters of the logistic fit are the following:

Qt(2004): 19.5 Gb (63.9% of Qinf)
k: 0.1647
Qinf: 30.5 Gb
thalf: 2000

Iran

The parameters of the logistic fit are the following:

Qt(2004): 19.5 Gb (47.0% of Qinf)
k: 0.0473
Qinf: 121.1 Gb
thalf: 2009

The summation of the 4 production profiles is given below and is obviously pessimistic. The main reason is that Russia is predicted to collapse in 2005 which is obviously not the case. Production forecasts for Norway and Iran seem realistic and their URR are near the ASPO estimates. A more careful modeling of the Russian multi-peak production is necessary... more to come.