Apologies in advance, time series is not my strength.

Say I want to predict f(T+1) using f(T-1, T-2, ..., T-N) -- for example using a multi-level preceptron.

If I want to enhance this using some other signals say using weather data on times T-1, T2, ..., T-N

Would this still be considered Time Series Analysis?



Yes, it's still generally considered to be time series.

For example, models of this kind - ones depending on past $x$ and past $y$ - include transfer function models (see, for example, Box and Jenkins (1972) Time series analysis: Forecasting and control), and ARIMA models with regressors (also see ARIMAX models). On the second, there are some questions here on CrossValidated, such as:

Time Series Forecasting with Daily Data: ARIMA with regressor

ARIMA and external regressors in SAS and R

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