https://www.researchgate.net/post/What_is_the_difference_between_Prediction_and_Forecast

To my knowledge, "projection" has nothing to do with prediction from statistical models. It has a physical and a mathematical meaning (mapping a higher-dimensional to a lower-dimensional space) but its use related to preditcions seems to be wrong. I know that people say thinkgs like "if we project this into future..." but this is a colloquialism lacking a clear scientific fundament.

And "prophecy" clearly does not relate to statistical models at all. If you wish you may take "prophecy" and "forecast" as synonyms, with the difference that the latter may not be a result of a traceable data analysis.

Forecast is out-of-sample

Suggest you read about it here: 

http://en.wikipedia.org/wiki/Prediction

http://en.wikipedia.org/wiki/Forecasting

Interesting!

Forecasting is the use of past data/information to determine the future outcome. This may be done through technical skills, statistics models and software, designed formulas and so on.

Prediction is slightly linked to forecasting, but however it has a gambling aspect. prediction may involve using inborn or accumulated skills to determine the future outcome. For example predicting the winner between Real Madrid vs Manchester United soccer match. Prediction is not quantitative in nature but uses approximations and determine sides.

Projection is mainly used by governments and companies in setting targets, Projection involves setting a standard or a level at which the economy or firm should be at the end of the year. For example projecting that company sales will increase at least by 10%, Government expenditure to reduce by 5%. Projection is used to strengthen policies. Mostly projection magnitudes are not met, hence they are reviewed quarterly or mid yearly. Howevever, just like forecasting, it uses past data to determine levels of operation.

Prophecy. This involves, the use of special skills to tell the future. It is related to spiritual gift. it may not be reliable, since it is difficult to prove and hence subject to abuse and manipulation.

There is apparently some room for interpretation here. When you are thinking in terms of statistical science, regression is the common element here: Forecasting involves a time series for individual members of a population; prediction involves the relationship between data sets; but projection and anticipated may have other futuristic implications (see footnote at bottom of first page of the paper at the link below). All of those terms involve regression for which you can statistically estimate variance. However, when you get to prophecy, you have left this realm.   

(Note that the "variance of the prediction error" is a term found in econometrics and model-based estimation for survey sampling statistics; perhaps elsewhere.  It's square root, by the way, is STDI in SAS PROC REG, a regression procedure not involving time series.)   

Article Projected Variance for the Model-Based Classical Ratio Estimator