1. Statistical Forecasting
Jan Verkade
November 3, 2016

2. Statistical Forecasting = forecasting from data
What does that mean?
What other types of forecasting do you know?

3. Regression analysis
Regression analysis: predicting future values of a variable using information about other variables
Predictor: the variable that you want to forecast
Predictand: the variable that you use as input
what we hope to find is that the different variables do not vary independently (in a statistical sense), but that they tend to vary together.
we assume that the future will behave like the past

4. Regression models
A predictand may depend on predictor(s) in varying ways:
y ~ x
y ~ a + bx
y ~ x2 …

5. The linear (regression) model
𝑌 𝑡 = 𝑏 0 + 𝑏 1 𝑋 1𝑡 + 𝑏 2 𝑋 2𝑡 + …+ 𝑏 𝑘 𝑋 𝑘𝑡
prediction for Y is a straight-line function of each of the X-variables
contributions of different X variables to predictions are additive
slopes b1, b2, etc: coefficients of the variables
intercept b0

6. Justification of linear model for regression assumptions
Why should we assume that relationships between variables are linear?
Because linear relationships are the simplest non-trivial relationships that can be imagined (hence the easiest to work with), and.....
Because the "true" relationships between our variables are often at least approximately linear over the range of values that are of interest to us, and...
Even if they're not, we can often transform the variables in such a way as to linearize the relationships.

7. Fitting a linear model
We fit a linear model through an objective function: minimise the mean squared error (MSE)
Steps:
Standardize variables: convert them to units of standard-deviations-from-the-mean
Calculate average product of standardized values
Minimize mean squared error
Subsitute, re-arrange and solve for b0 and b1

8. Fitting a linear model
Standardize variables: convert them to units of standard-deviations-from-the-mean
𝑋 𝑡 ∗ = 𝑋 𝑡 −𝑚𝑒𝑎𝑛(𝑋) 𝑠𝑡𝑑𝑒𝑣(𝑋)
𝑌 𝑡 ∗ = 𝑌 𝑡 −𝑚𝑒𝑎𝑛(𝑌) 𝑠𝑡𝑑𝑒𝑣(𝑌)

9. Fitting a linear model
Standardize variables: convert them to units of standard-deviations-from-the-mean
Calculate average product of standardized values
𝑟 𝑋𝑌 = 1 𝑛 𝑋 1 ∗ 𝑌 1 ∗ + 𝑋 2 ∗ 𝑌 2 ∗ +…+ 𝑋 𝑛 ∗ 𝑌 𝑛 ∗

10. Fitting a linear model
Standardize variables: convert them to units of standard-deviations-from-the-mean
Calculate average product of standardized values
Minimize mean squared error
𝑌 𝑡 ∗ = 𝑟 𝑋𝑌 𝑋 𝑡 ∗

11. Fitting a linear model
Standardize variables: convert them to units of standard-deviations-from-the-mean
Calculate average product of standardized values
Minimize mean squared error
Subsitute, re-arrange and solve for b0 and b1
𝑌 𝑡 −𝑚𝑒𝑎𝑛(𝑌) 𝑠𝑡𝑑𝑒𝑣(𝑌) = 𝑟 𝑋𝑌 𝑋 𝑡 −𝑚𝑒𝑎𝑛(𝑋) 𝑠𝑡𝑑𝑒𝑣(𝑋)
𝑌 𝑡 ∗ = 𝑟 𝑋𝑌 𝑋 𝑡 ∗
𝑌 𝑡 = 𝑏 0 + 𝑏 1 𝑋 1𝑡
𝑏 1 = 𝑟 𝑋𝑌 𝑠𝑡𝑑𝑒𝑣(𝑌) 𝑠𝑡𝑑𝑒𝑣(𝑋)
𝑏 0 =𝑚𝑒𝑎𝑛 𝑌 − 𝑏 1 𝑚𝑒𝑎𝑛(𝑋)

12. Exercise: piezometric head within a levee

13. Exercise: piezometric head within a levee
river water level
water pressure sensor

14. Exercise: piezometric head within a levee
Use voorhavendijk.xls
Explore the data by building a scatter (x,y) plot
Determine mean and standard deviations
Determine standardized values; then explore…
marginal distributions (ecdf of either variable)
joint distribution (scatter plot)
Determine the coefficient of correlation
Determine the coefficients of the regression equation
Verify by using Excel’s built-in function to show regression line

15. Exercise: piezometric head within a levee

16. Exercise: piezometric head within a levee
Discuss: is the linear model a good model?

17. Exercise: piezometric head within a levee
How to use / interpret the regression line?

18. Exercise: piezometric head within a levee
Use voorhavendijk.xls
Explore the data by building a scatter (x,y) plot
Determine mean and standard deviations
Determine standardizes values; then explore…
marginal distributions (ecdf of either variable)
joint distribution (scatter plot)
Determine the coefficient of correlation
Determine the coefficients of the regression equation
Verify by using Excel’s built-in function to show regression line
Explore the residuals by plotting an empirical cumulative density function.
What is the mean value?
How are the residuals distributed?

19. LM-model: residuals

20. LM-model: residuals
mean: e-18
stdev:

21. Exercise: piezometric head within a levee
How to use / interpret the regression line?

22. Forecasting errors Intrinsic risk: signal v noise
Parameter risk: uncertain parameter values
Model risk: the risk of choosing the wrong model (linear model v quadratic model, for example)

23. Confidence Intervals v Prediction Intervals

24. An alternative statistical technique: Quantile Regression
Principles:
QR is a method for describing conditional quantiles
Rather than minimising the mean squared error (MSE)
QR is based on minimising the mean absolute error (MAE)
This yields not the sample mean but the sample median
Other quantiles may be derived by adding weights to errors
E.g. weight = .1 for positive errors and .9 for negative errors
Fitting models may be done in transformed space to account for heteroscedasticity

27. Application in real-time hydrologic forecasting: post-processing
Ensemble techniques
Post-processing techniques

28. Application in real-time hydrologic forecasting: post-processing
Once a record of forecasts is in place
This record can be analysed for ‘forecast errors’
And these records can be assumed to occur in future forecasts also

29. 1: Find a relationship between forecast and obs
5 december 2017
1: Find a relationship between forecast and obs

30. 2. Apply that relation to new forecasts

31. And here’s your forecast
5 december 2017
And here’s your forecast

32. Famous forecasting quotes
"I have seen the future and it is very much like the present, only longer."
--Kehlog Albran, The Profit
 Pretty concise description of statistical forecasting:
We search for statistical properties of a time series that are constant in time (levels, trends, seasonal patterns, correlations and autocorrelations, etc.)
We then predict that those properties will describe the future as well as the present

33. Famous forecasting quotes
"Prediction is very difficult, especially if it's about the future."
--Nils Bohr, Nobel laureate in Physics
warning of the importance of validating a forecasting model out-of-sample.
It's often easy to find a model that fits the past data well--perhaps too well!—
but quite another matter to find a model that correctly identifies those patterns in the past data that will continue to hold in the future.