How do you recover the original time series $q_k$ from modeled differences $Z_t$?

Conditions

• You have forecasts for the differenced series $Z_t$.
• You know the last observed value of the original series $q_l$.
• The relationship $Z_t = q_{t+1} - q_t$ holds.

Reasoning, step by step

1. Start with the definition $Z_{k-1} = q_k - q_{k-1}$.
2. Rearrange to solve for $q_k$: $q_k = Z_{k-1} + q_{k-1}$.
3. Recursively substitute for $q_{k-1}, q_{k-2}, \dots$ until reaching the last known value $q_l$.
4. Sum all the intermediate $Z$ terms.
5. Add the last known actual value $q_l$ to the sum.
6. Resulting formula: $q_k = \sum_{i=1}^{k-l} Z_{k-i} + q_l$.

Example

The board shows the derivation: $q_k = Z_{k-1} + q_{k-1} = Z_{k-1} + Z_{k-2} + q_{k-2} = \dots = \sum_{i=1}^{k-1} Z_{k-i} + q_1$. The speaker explains this as doing the reverse transformation.

Common misconceptions

• Forgetting to add the initial or last known value $q_l$.
• Summing the wrong range of $Z$ terms.
• Thinking the model predicts $q_k$ directly without inversion.

Watch the explanation

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