Page History

...

• Unconstrained matching 
• Constrained matching:
  • Match the value of the initial rate 

    LaTeX Math Inline
    body --uriencoded--q%5e*(t=0) = q%5e*_0 = q_0

  • Match the value of the current rate

    LaTeX Math Inline
    body --uriencoded--q%5e*(t=t_N) = q_N

  • Match the value of the current cumulative

    LaTeX Math Inline
    body --uriencoded--Q%5e*(t=t_N) = Q_N

  • 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 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

...

q(t) = \frac{Q_N/\tau_0}{1-\exp(-t_N/\tau_0)}  \cdot \exp(-t/\tau_0) 

q(t) = \frac{(1-b) \cdot Q_N/\tau_0}{ 1 - \left( 1 + b \, t_N/\tau_0 \right) ^{-b/(1-b)}} \cdot \frac{1}{\left(1 + b\, t/\tau_0 \right)^{1/b}}

q(t)=\frac{Q_N/\tau_0}{\ln (1+ t_N/\tau_0)} \cdot \frac{1}{1+t/\tau_0}

Q(t) - Q_N = [ q^*_N - q(t)] \, \tau_0

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]

Q(t) - Q_N = q^*_N \, (\tau_0 + t_N) \cdot \ln \frac{q^*_N}{q(t)}

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

...

...

...

...

q(t)=q_N = \frac{Q_N/\tau_0}{1-\exp(-t_N/\tau_0)}  \cdot \exp \big[ -(t-t_N)/\tau_0) \big]

q(t) = q_N \cdot \left[ \frac{(1+-b) \cdot tQ_N/\tau_0 }
{ 1 - \left( 1 + b \cdot, t_N/\tau_0  } \right]) ^{-b/(1/-b}

q(t) =  q_N \cdot  \left[)}} \cdot \frac{1+t_N/\tau_0 }
{\left(1 1+ b\, t/\tau_0  } \right])^{1/b}}

Q(t) - Q_N = [ q_N - q(t)] \, \tau_0

Q(t) - Q_N = \frac{qQ_N^b \, (\N/\tau_0 + b \, t_N)}{1-b} \left[ q_N^{1-b} - q^{1-b}(t) \right]

Q(t) - Q_N = q_N \, (\tau_0 + t_N) \cdot \ln \frac{q_N}{q(t)\ln (1+ t_N/\tau_0)} \cdot \frac{1}{1+t/\tau_0}

The best-fit model may not match: 

...