LIST OF LECTURES:

* ADDITIONAL MATERIAL:

MATLAB Tutorials: MathWorks Tutorial , MIT tutorial , more links here
MATH Reviews on Calculus, Linear Algebra and ODEs by Paul Dawkins
Additional Copyright Material - [ web ]

* BACKGROUND REVIEW: Matrix and Vector Algebra, Fundamental Statistical
Measures, Multivariable Probability Densities, Sample Estimates, Correlation and
Covariance, Function and Sums of Random Variables, Central Limit Theorem.

Topic 1:
Why is statistical analysis useful.
Homework 1: 01-hw.pdf
References: Davis-lect-1.pdf , 01-notes.pdf , 01-figues.pdf

MATLAB Codes: EXAMPLE_lorenz.m , bs_lorenz.m, bs_lorenz_rhs.m, bs_compute_sample_pdf1.m , bs_compute_sample_pdf2.m , utl_shapiro2.m, utl_contourfill.m

Topic 2:
An overview of the statistical methods. How does it all fit together ?
References: 02-notes.pdf, 02-figues.pdf

Topic 3 :
Fundamental Statistical Measures. Univariate Statistics and PDFs.
References: Davis-lect-2.pdf , Wunsch Chap. 2 (pp. 27-41), 03-notes.pdf

Topic 4 :
Fundamental Statstical Measures. Multivariate Statistics and JPDFs.
References: Davis-lect-2.pdf , Wunsch Chap. 2 (pp. 27-41), 04-notes.pdf , CentralLimitTheorem.pdf (S. Gille)

Topic 5 :
Statistically Optimal Linear Estimators: relationship between least squares and conditional joint PDF estimates. (MATLAB examples in class)
References: Davis-lect-3.pdf

MATLAB Codes: EXAMPLE_lsq_jpdf.m , bs_generate_XY.m , bs_rand.m , bs_estimate_pdf2_condx.m , bs_compute_sample_pdf1.m , bs_compute_sample_pdf2.m , utl_shapiro2.m

* COMBINING MODELS AND OBSERVATIONS: Interpolation and Function Fitting, Least
Square modeling and Singular Vector Expansion, Uncertainties in Estimates, Inverse
Methods, Statistical vs. Dynamical Constraints.

Topic 6 :
Testing a model against observations: Introduction to Least Squares (LSQ)
References: Wunsch Chap. 1 , Wunsch Chap. 2 ( 41-57) , 06-lsq-review.pdf
Linear Algebra Review: 06-linalg-notes.pdf from Wuncsh Chap. 2 (pp. 1-27) , Paul Dawkins

Topic 7 :
Interpolation and function fitting with LSQ: The CO2 curve and SST spatial maps
References: Wunsch Chap. 2 ( 41-57) , CO2.pdf , LSQ_SST.pdf
MATLAB Codes: EXAMPLE_CO2_curve.m , CO2.mat , Feb98_SST.mat , EXAMPLE_lsq_2D_function_fit.m
MATLAB Codes: EXAMPLE_lsq_fourier.m, utl_sincos.m, utl_sincos_2d.m

Topic 8 :
LSQ and Inverse Modeling: Reconstructing the source of a pollutant with an advection diffusion model
References: Wunsch Chap. 1 , Wunsch Chap. 2 ( 41-57) , LSQ_dispersion.pdf
MATLAB Codes: EXAMPLE_lsq_dispersion.m

Topic 9 :
Lagrange Multiplyers and Adjoints
References: Wunsch Chap. 2 ( 58-68) , 09-adjoint.pdf

* TIME SERIES ANALYSIS : Time and Frequency Domain Models, Stationarity, Auto-
Regression Models, Spectral Analysis and Coherence, Trend Analysis and
Significance, Estimating errors in time series reconstruction.

Material is taken from the following references:

Hartmann Web notes Chapter 6
Time Series pdfbook Chapter 1-4
Wilks Chapter 8

Topic 10 :
Frequency domain, Spectrum and Autocovariance function
References: Hartmann Chapter 6, Time Series pdfbook Chapter 1 or Wilks Chapter 8, 10-timeseries-intro.pdf

Topic 11-12 :
Review Convolution and Cross-correlation, Aliasing, DFT and Tapering
References: C. Hoyos Powerpoint Slide [ ppt1 | ppt2 ], TimeSeriesCodes.zip
Homework 3: 03-hw.pdf

Topic 13 :
Time domain models
References: Hartmann Chapter 6, Time Series pdfbook Chapter 4

Practice Exam Questions : prac_exam_questions.pdf ,
Exam : exam.pdf

Topic 14 :
Analysis of two or more signals, Cross-Spectra and Coherence
References: Hartmann Chapter 6c, Time Series pdfbook Chapter 4,
class notes Coherece.pdf

* FORECASTING AND EXTRAPOLATION : Mulitvariate Statistically Optimal Linear Estimators, Regression models, space and time models, objective mapping (multivariate regression), covariance modeling.

Topic 15 :
Covariance Modeling, Basic Theory
References: Hartmann from Chapter 3 and 5 , CovModel_Theory.pdf

Examples in the time and Yule-Walker Equations : CovModel_TimeEX.pdf

Example in the space domain and the multivariate optimal interpolation: CovModel_SpaceEX.pdf and CovModel_SpaceEX_fig.pdf

Example of Objective Mapping: EXAMPLE_ObjMap.m, SST.txt, DecorrelationLength.m

* SIGNAL DECOMPOSITION : Multivariate eigenfunction analysis, EOFs, PCA, CCA, and
Wavelet analysis

Topic 16 :
Empirical Orthogonal Functions (EOFs) / Principal Component Analysis (PCA), Maximum Covariance Analysis (MCA), Combined EOFs (SVD) and Canonical Correlation Analysis (CCA)
References: Hartmann

Topic 17 :
Space/Time filters (e.g. high-pass, low-pass, band-pass) and Wavelet analysis
References: Hartmann, WaveletClass.ppt, Wavelet_Torrence_compo1998.pdf (MATLAB programs)

* STUDENT PRESENTATION - Presentation Guidelines