MEA 593 Climate Risk Analysis for Adaptation Instructor – Fredrick Semazzi Lecture-3 Mean

1. MEA 593 Climate Risk Analysis for Adaptation Instructor – Fredrick Semazzi Lecture-3 Mean Correlation Standard Deviation Empirical Orthogonal Functions (EOF) Diagnostic Methods

2. Common descriptive statistics • Count (frequencies): The number of occurrences of a repeating event in a given time interval • Percentage: a percentage is a number or ratio expressed as a fraction of 100 • Mean: Sum of the values divided by the number of values • Ranking: Refers to the relative position in terms value • Mode: The most common value among a group of numbers • Median: The numerical value separating the higher half of a data sample from the lower half • Range: The difference between the highest and the lowest values in a set of numbers • Variance: Measures how far a set of numbers is spread out; A small variance indicates that the data points tend to be very close to the mean (expected value) and hence to each other; (A variance of zero indicates that all the values are identical) • Standard deviation: Square root of variance is called the standard deviation • p-value: This is a measure of statistical significance (confidence in the results). The p-value is the probability of an observed result arising by chance. A researcher will often declare the outcome statistically significant if the p-value turns out to be less than a certain significance level (i.e. very remote possibility), often 0.05or 0.01

3. Main Topics • Calculation and interpretation of the mean • Calculation and interpretation of the variance and standard deviation • Calculation and interpretation of the correlation • Interpretation of EOF modes (calculation of EOFs not required)

4. References • Physics of climate. José P. Peixoto and Abraham Oort. • Applied Multivariate Statistical Analysis. Richard Johson and Dean W. Wichern. • North et. al (1982). Mon. Wea. Rev; 110, 699-706 • Richman, M.B., 1986: Rotation of principal components. J. Climatol., 6, 293-335.

5. Primary Data Set Data Matrix/Mean Matrix/ Variance Matrix time space Data Mean Variance

6. Standardized Matrix for input data Standardized data matrix Mean Variance D= (nxp) D= (nxp)

7. Sample Variance / Correlation Matrix Defining: Covariance, variance & correlation at a particular location(s):

8. Sample Variance/Correlation Matrix (continued) R= R= *R is a square symmetric matrix The elements of R are standardized variances i.e. correlations.

9. Principal Component Analysis (PCA) or Empirical Orthogonal Functions Analysis (EOF) For the analysis model or observational multi-dimensional climate data is difficult to detect or isolate the dominant mode of climate variability by simple inspection of the time series of maps. PCA or EOF techniques are concerned with explaining the variance-covariance (i.e. information) structure through a few modes of variability that account for large % of the original variance. The resulting transformation simplifies interpretation.

10. Construction of E0F Timeseries Correlation = Matrix =Data data map at t=kth Column . E0Fi , amp 1 Var=1 amp (E0Fi , t=k) t:=n E0Fi , ampi Var=i t:=k t:=1 t:=n E0Fi , ampp Var=p t:=1 t:=n

11. Main Steps In Construction of Eigen Functions • Construction of standardized data matrix • Construction of covariance or correlation matrix (R) • Solve characteristic equation for the covariance/correlation matrix to obtain eigen value/eigen vector pairs • Determine cutoff for “noise” & signal E0Fs. A rule of thumb is to retain only those components with variance () greater than one or that explain at least a proportion 1/p of the total variance. This rule doesn’t always work & more sophisticated criteria exist.

12. Semazzi & Sud (1986)

13. Main Steps (continued) • Plot (i) Histogram for eigen values & separation between ‘noise’ & ‘signal’ modes can be detected (ii) E0F patterns for dominant modes (iii) E0F time series for dominant modes 6. If needed reconstruct data matrix by combining contribution of a subset of eigen modes. This is one way of filtering the original data set by ignoring the ‘noise’ modes

14. Example of the application of EOF analysis to the Atlantic SSTs

15. Semazzi & Sud (1986)

16. Atlantic EOFs (regular) Largest loading Over Northern tropical Atlantic Largest loading Over Southern tropical Atlantic ENSO signal (largest loading over the Pacific) Semazzi & Sud (1986)

17. Atlantic EOFs (rotated) Semazzi & Sud (1986)

18. Example of the application of EOF analysis to the Pacific SSTs

20. Examples of EOFs for NC Climate EPA Report An Urban Scale AQ Planning Tool For Global Change Adaption NARR EOF Analysis An Urban Scale AQ Planning Tool For Global Change Adaption EOF Analysis for North America Region http://climlab.meas.ncsu.edu/EPA-Project1/epa-project1.html

21. Geopotential EOF-1 (North America)

22. Geopotential EOF-1 (Southeast US)

23. Geopotential EOF-1 (NC)

24. Geopotential EOF-2 (North America)

25. Geopotential EOF-2 (Southeast US)

26. Geopotential EOF-2 (NC)

27. Geopotential EOF-3 (North America)

28. Geopotential EOF-3 (Southeast US)

29. Geopotential EOF-3 (NC)

30. Wind Composite with Precipitation EOF1 (Northeast US)

31. Wind Composite with Precipitation EOF2 (Northeast US)

32. Wind Composite with Precipitation EOF3 (Northeast US)

33. Useful Link for NC Data North Carolina Climate Office (NCCO) website [http://www.nc-climate.ncsu.edu/climate/climdiv.php]

34. Summary • Calculation and interpretation of the mean • Calculation and interpretation of the variance and standard deviation • Calculation and interpretation of the correlation • Interpretation of EOF modes (calculation of EOFs not required)