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You recover the original series $q_k$ by summing the predicted differences $Z_t$ and adding them to the last known actual value $q_l$ (or $q_1$ depending on the starting index). The formula is $q_k = \sum_{i=1}^{k-l} Z_{k-i} + q_l$.

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.

You recover the original series $q_k$ by summing the predicted differences $Z_t$ and adding them to the last known actual value $q_l$ (or $q_1$ depending on the starting index). The formula is $q_k = \sum_{i=1}^{k-l} Z_{k-i} + q_l$.

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.

You would use a second difference if the first differenced series is still not stationary. The goal is to choose the lowest differencing order $d$ that makes the series stationary.

Conditions: The original series is non-stationary.; First differencing ($d=1$) does not result in a stationary series.; The modeler can inspect the stationarity of the differenced plots.

You would use a second difference if the first differenced series is still not stationary. The goal is to choose the lowest differencing order $d$ that makes the series stationary.

Conditions: The original series is non-stationary.; First differencing ($d=1$) does not result in a stationary series.; The modeler can inspect the stationarity of the differenced plots.

The equation for ARIMA(1,1,1) is $Z_t = \phi_1 Z_{t-1} + \theta_1 \epsilon_{t-1} + \epsilon_t$, where $Z_t$ is the differenced series, $\phi_1$ is the autoregressive coefficient, $\theta_1$ is the moving-average coefficient, and $\epsilon_t$ is the error term.

Conditions: The model is ARIMA(1,1,1).; $Z_t$ is the first difference of the original series $q_t$.; $p=1, d=1, q=1$.

The equation for ARIMA(1,1,1) is $Z_t = \phi_1 Z_{t-1} + \theta_1 \epsilon_{t-1} + \epsilon_t$, where $Z_t$ is the differenced series, $\phi_1$ is the autoregressive coefficient, $\theta_1$ is the moving-average coefficient, and $\epsilon_t$ is the error term.

Conditions: The model is ARIMA(1,1,1).; $Z_t$ is the first difference of the original series $q_t$.; $p=1, d=1, q=1$.

Differencing helps because for a linear function, the change from one point to the next is constant. By taking successive differences, the linear trend is removed, leaving a series that fluctuates around a constant mean, which satisfies the stationarity requirement for ARMA models.

Conditions: The original series has an approximately linear trend with noise.; The goal is to achieve stationarity for modeling.

Differencing helps because for a linear function, the change from one point to the next is constant. By taking successive differences, the linear trend is removed, leaving a series that fluctuates around a constant mean, which satisfies the stationarity requirement for ARMA models.

Conditions: The original series has an approximately linear trend with noise.; The goal is to achieve stationarity for modeling.

The differenced series $z_t$ is defined as the difference between the next value of the original series and the current value. Mathematically, it is expressed as $z_t = q_{t+1} - q_t$.

Conditions: The original series is indexed by time $t$.; The values represent observations at discrete time points (e.g., months).

The differenced series $z_t$ is defined as the difference between the next value of the original series and the current value. Mathematically, it is expressed as $z_t = q_{t+1} - q_t$.

Conditions: The original series is indexed by time $t$.; The values represent observations at discrete time points (e.g., months).