GLOSSARY

Page tree

Versions Compared

Old Version 34

changes.mady.by.user Arthur Aslanyan (Nafta College)

Saved on

compared with

New Version Current

changes.mady.by.user Arthur Aslanyan (Nafta College)

Saved on

Key

  • This line was added.
  • This line was removed.
  • Formatting was changed.

...


The constrained matching is used to one may wish to ensure the smooth transition from the training dataset to future model predictions.

Anchor
Unconstrained
Unconstrained
Unconstrained matching

...


The value of the model rate at the initial time moment is set to training dataset: 

LaTeX Math Inline
body--uriencoded--q%5e*(t=0) = q%5e*_0 = q_0
 and the two other model properties
LaTeX Math Inline
body--uriencoded--\%7B D\tau_0, \, b \%7D
 are being varied to achieve the best fit to the training dataset.

Exponential Production DeclineHyperbolic Production DeclineHarmonic Production Decline

LaTeX Math Inline
bodyb=0

LaTeX Math Inline
body0<b<1

LaTeX Math Inline
bodyb=1

LaTeX Math Block
anchor1
alignmentleft
q(t)=q_0 \exp \left( -Dt/\tau_0 \, t \right)
LaTeX Math Block
anchor003NF
alignmentleft
q(t) = \frac{q_0}{ \left( 1+b \cdot Dt/\tau_0 \cdot t \right)^{1/b} }
LaTeX Math Block
anchor1
alignmentleft
q(t)=\frac{q_0}{1+Dt/\tau_0 \, t}


The best-fit model may not match: 

...

Anchor
Current_rate
Current_rate
Match the value of the current rate 
LaTeX Math Inline
body--uriencoded--q%5e*(t=t_N) = q_N

...


To ensure the smooth transition from historical data The value of the model rate at the current time moment is set to training dataset: 

LaTeX Math Inline
body
[(t_1,q_1)... (
--uriencoded--q%5e*(t=t_N
,
) = q_N
)]
 to the production forecasts in future time moments  and the two other model properties
LaTeX Math Inline
body--uriencoded--
[(t_%7BN+1%7D,q_%7BN+1%7D), ...]
\%7B \tau_0, \, b \%7D
 are being varied to achieve the best fit to the training dataset. one may wish to constrain the model by firm matching the production at the last historical moment
LaTeX Math Inline
body(t_N, q_N)
 which leads to the following form of Arp's model:


Q(t) - Q_N = q_N \, (\tau_0 + t_N) \cdot \ln \frac{q_N}{q(t)}
Exponential Production DeclineHyperbolic Production DeclineHarmonic Production Decline

LaTeX Math Inline
bodyb=0

LaTeX Math Inline
body0<b<1

LaTeX Math Inline
bodyb=1

LaTeX Math Block
anchor1
alignmentleft
q(t)=q_N \cdot \exp \big[ -D_0 \cdot (t-t_N)/\tau_0 \big]
LaTeX Math Block
anchor003NF
alignmentleft
q(t) = q_N \cdot \left[ \frac{1+b \cdot Dt_0 \cdot t_NN/\tau_0 }
{ 1+b \cdot Dt/\tau_0 \cdot t   } \right]^{1/b}
LaTeX Math Block
anchor1
alignmentleft
q(t) =  q_N \cdot  \left[ \frac{1+Dt_0 \cdot t_NN/\tau_0 }
{ 1+ Dt/\tau_0 \cdot t   } \right]
LaTeX Math Block
anchor1
alignmentleft
Q(t) - Q_N = [ q_N - q(t)] \, \tau_0
LaTeX Math Block
anchor1
alignmentleft
Q(t) - Q_N = \frac{q_N^b \, (\tau_0 + b \, t_N)}{1-b} \left[ q_N^{1-b} - q^{1-b}(t) \right]
LaTeX Math Block
anchor1
alignmentleft



This ensures the smooth transition from historical data

LaTeX Math Inline
body[(t_1,q_1)... (t_N, q_N)]
 to the production forecasts in future time moments
LaTeX Math Inline
body--uriencoded--[(t_%7BN+1%7D,q_%7BN+1%7D), ...]
.

The best-fit model may not match: 

...

Anchor
Current_cumulative
Current_cumulative
Match the value of the current cumulative 
LaTeX Math Inline
body--uriencoded--Q%5e*(t=t_N) = Q_N

...


To ensure the smooth transition from historical data

LaTeX Math Inline
body[(t_1,q_1)... (t_N, q_N)]
 to the production forecasts in future time moments The value of the model rate at the initial time moment
LaTeX Math Inline
body--uriencoded--
[
q%5e*(t
_%7BN+1%7D,q_%7BN+1%7D), ...]
=0) = q%5e*_0
is set to achieve the match between the values of current cumulative from model prediction and training dataset   one may wish to constrain the model by firm matching the production at the last historical moment
LaTeX Math Inline
body--uriencoded--Q%5e*(t=t_N
, q
) = Q_N
)
 which leads to the following form of Arp's model:

Exponential Production DeclineHyperbolic Production DeclineHarmonic Production Decline

LaTeX Math Inline
bodyb=0

LaTeX Math Inline
body0<b<1

LaTeX Math Inline
bodyb=1

LaTeX Math Block
anchor1
alignmentleft
q(t) =q \frac{Q_N \cdot \exp \big[ -D_0/\tau_0}{1-\exp(-t_N/\tau_0)}  \cdot \exp(t-t/\tau_N0) \big]
LaTeX Math Block
anchor003NF
alignmentleft
q(t) = q_N \cdot \left[ \frac{(1+-b) \cdot DQ_0 \cdot t_N }
N/\tau_0}{ 1+b \cdot D_0 \cdot t  } \right]^{1/b}
LaTeX Math Block
anchor1
alignmentleft
q(t) =  q_N \cdot  \left[ \frac{1+D_0 \cdot t_N }
{ 1+ D_0 \cdot t  } \right]
LaTeX Math Block
anchor1
alignmentleft
Q(t) - Q_N = [ q_N - q(t)] \, \tau_0
LaTeX Math Block
anchor1
alignmentleft
Q(t) - Q_N = \frac{q_N^b \, (\tau_0 + b \, t_N)}{1-b} \left[ q_N^{1-b} - q^{1-b}(t) \right] - \left( 1 + b \, t_N/\tau_0 \right) ^{-b/(1-b)}} \cdot \frac{1}{\left(1 + b\, t/\tau_0 \right)^{1/b}}
LaTeX Math Block
anchor1
alignmentleft
Qq(t) - =\frac{Q_N = q_N \, (/\tau_0}{\ln (1+ t_N/\tau_0)} \cdot \ln \frac{q_N1}{q(t)1+t/\tau_0}


The best-fit model may not match: 

...

Anchor
Current_rate_cumulative
Current_rate_cumulative
Match the value of the current rate and cumulative 
LaTeX Math Inline
body--uriencoded--q%5e*(t=t_N) = q_N
, 
LaTeX Math Inline
body--uriencoded--Q%5e*(t=t_N) = Q_N

...


The value of the model rate at the current time moment and decline pace are set to match both current rate To ensure the smooth transition from historical data

LaTeX Math Inline
body
[(t_1,q_1)... (
--uriencoded--q%5e*(t=t_N
,
) = q_N
)] to the production forecasts in future time moments
 and current cumulative 
LaTeX Math Inline
body--uriencoded--
[
Q%5e*(t
_%7BN+1%7D,q_%7BN+1%7D), ...] one may wish to constrain the model by firm matching the production at the last historical moment
LaTeX Math Inline
body(t_N, q_N)
 which leads to the following form of Arp's model:
=t_N) = Q_N
.

This makes Exponential Production Decline and Harmonic Production Decline are fully set while Hyperbolic Production Decline has opportunity to vary one model parameter

LaTeX Math Inline
body--uriencoded--\%7B b \%7D
 to achieve the best fit to the training dataset.

Exponential Production DeclineHyperbolic Production DeclineHarmonic Production Decline

LaTeX Math Inline
bodyb=0

LaTeX Math Inline
body0<b<1

LaTeX Math Inline
bodyb=1

LaTeX Math Block
anchor1
alignmentleft
q(t) =q \frac{Q_N \cdot \exp \big[ -D_0/\tau_0}{1-\exp(-t_N/\tau_0)}  \cdot \exp(t-t/\tau_N0) \big]
LaTeX Math Block
anchor003NF
alignmentleft
q(t) = q_N \cdot \left[ \frac{(1+-b) \cdot DQ_0 \cdot t_N }
N/\tau_0}{ 1+b \cdot D_0 \cdot t  } \right]^{1/b}
LaTeX Math Block
anchor1
alignmentleft
q(t) =  q_N \cdot  \left[ \frac{1+D_0 \cdot t_N }
{ 1+ D_0 \cdot t  } \right]
LaTeX Math Block
anchor1
alignmentleft
Q(t) - Q_N = [ q_N - q(t)] \, \tau_0
LaTeX Math Block
anchor1
alignmentleft
Q(t) - Q_N = \frac{q_N^b \, (\tau_0 + b \, t_N)}{1-b} \left[ q_N^{1-b} - q^{1-b}(t) \right]- \left( 1 + b \, t_N/\tau_0 \right) ^{-b/(1-b)}} \cdot \frac{1}{\left(1 + b\, t/\tau_0 \right)^{1/b}}
LaTeX Math Block
anchor1
alignmentleft
Qq(t) - =\frac{Q_N = q_N \, (/\tau_0}{\ln (1+ t_N/\tau_0)} \cdot \ln \frac{q_N1}{q(t)1+t/\tau_0}


The best-fit model may not match: 

...