forecastingregressionterminologytime series
I was wondering what difference and relation are between forecast and prediction? Especially in time series and regression?
For example, am I correct that:
In time series, forecasting seems to mean to estimate a future values given past values of a time series.
In regression, prediction seems to mean to estimate a value whether it is future, current or past with respect to the given data.
No, that method is not valid in general.
Here's a simple, illustrative counterexample. Assume that you have a random walk without drift:
$$Y_t = Y_{t-1} + \varepsilon_t$$ $$\varepsilon_t \sim \mathcal{N}(0,1)$$
This process falls in the TBATS class (it is just an "ANN"-type ETS model with $\alpha=1$, without any complex seasonality, or Box-Cox transform, or ARMA errors).
Here's what it looks like if you use your method on simulated data:
The "simulated path" is flat and has a small variance, whereas the original data will stray from its mean level quite a bit. It does not "look" like the original data at all.
If we repeat the procedure many times and compute the empirical quantiles for the middle 95% of the distribution at each horizon, you will see that they are wrong compared to the prediction intervals reported by forecast.tbats (if the method worked, they should match the outer, grey intervals):
Many time series models can be viewed as a transformation of a sequence of uncorrelated random variables; the exact transformation depends on the model. Given a specific transformation, you can generally take the residuals (call them $\hat{\varepsilon_t}$), resample them, and then apply this transformation to simulate from the same process.
For example, the random walk transforms a sequence of uncorrelated variables $\varepsilon_t$ by the recursion stated above (the cumulative sum). If your original series ends at $T$, you can sample $\varepsilon^*_{T+1}$, from $\{ \hat{\varepsilon_1}, \ldots, \hat{\varepsilon_T} \}$, and apply the same recursion to obtain a simulated value for $Y_{T+1}$, like this:
$$Y_{T+1}^* = Y_T + \varepsilon^*_{T+1}$$
If you compute the quantiles as before, you should come close to the grey area.
In general, therefore, this kind of model-based bootstrap requires slightly different code for different models, to perform different transformations on the resampled $\varepsilon^*_t$. The function simulate.ets handles this for you for the ETS class, but there still does not seem to be an equivalent for TBATS in the package, as far as I can tell.
In statistics, imputation is the process of replacing missing data with substituted values. When substituting for a data point, it is known as "unit imputation"; when substituting for a component of a data point, it is known as "item imputation". Because missing data can create problems for analyzing data, imputation is seen as a way to avoid pitfalls involved with listwise deletion of cases that have missing values. That is to say, when one or more values are missing for a case, most statistical packages default to discarding any case that has a missing value, which may introduce bias or affect the representativeness of the results. Imputation preserves all cases by replacing missing data with a probable value based on other available information. Once all missing values have been imputed, the data set can then be analysed using standard techniques for complete data.
On the other hand,
Forecasting is the process of making statements about events whose actual outcomes (typically) have not yet been observed. A commonplace example might be estimation of some variable of interest at some specified future date. Prediction is a similar, but more general term. Both might refer to formal statistical methods employing time series, cross-sectional or longitudinal data, or alternatively to less formal judgmental methods. Usage can differ between areas of application: for example, in hydrology, the terms "forecast" and "forecasting" are sometimes reserved for estimates of values at certain specific future times, while the term "prediction" is used for more general estimates, such as the number of times floods will occur over a long period.
Best Answer
Your distinction sounds reasonable. There was a similar discussion at the analyticbridge website, where several people make various distinctions but none of them seem to agree.
The closest one was, "Forecasting would be a subset of prediction. Any time you predict into the future it is a forecast. All forecasts are predictions, but not all predictions are forecasts, as when you would use regression to explain the relationship between two variables."
So as you say, "forecast" implies time series and future, while "prediction" does not.
Note that there is also a term "projection" which is distinct from forecast or prediction, in some disciplines.