1. ESTIMATION METHODS We know how to calculate confidence intervals for estimates of  and  2 Now, we need procedures to calculate  and  2, themselves Several methods to do this, we’ll look at only one: MAXIMUM LIKELIHOOD First, define Likelihood: L(y 1, y 2, …., y N ) is the joint probability density evaluated at the observations y i where y 1, y 2, …., y N are sample observations of random variables Y 1, Y 2, …., Y N PDF of random variables Y 1, Y 2, …., Y N

2. MAXIMUM LIKELIHOOD METHOD Choose the parameter values that maximize Example: Apply method to estimates of  and  2 for a normal population. Let y 1, y 2, …., y N be a random sample of the normal population Find Maximum Likelihood

3. Simplify by taking the log N (L): Taking derivative with respect to  and  2 Making them equal to zero to get the maximum, the maximum likelihood:

4. Making them equal to zero to get maximum likelihood estimators of mean and variance: are the Maximum Likelihood estimators of  and  2 is an unbiased estimator of , but is not unbiased for  2 substituting  hat into

5. can be adjusted to the unbiased estimator: So, for a normally distributed oceanographic data set, we can readily obtain Maximum Likelihood estimates of  and  2 This technique (ML) is really useful for variables that are not normally distributed. Spectral energy values from current velocities or sea level, show  2 rather than normal distribution Following the ML procedure, we find that the mean of the spectral values is and the variance is 2

6. So, with the ML approach you can calculate the best parameters that fit certain models. For instance, you can apply it to a pulse of current velocity data to obtain the best dissipation value  and fitting coefficient  in the inertial subrange, on the basis of Kolmogorov’s law for turbulence:

7. As another example, you can apply it to a segment of temperature gradient in a profile to obtain the best Batchelor length scale (or wave number  B ) and dissipation of temperature variance  T, to get dissipation values on the basis of Batchelor spectrum for turbulence: Steinbuck et al., 2009 So in general, to apply the ML method to a sample: - Determine appropriate PDF for sample values - Find joint likelihood function - Take natural logs - Differentiate wrt parameter of interest - Set derivative = 0 to find max - Obtain value of parameter

8. LINEAR ESTIMATION (REGRESSION) Consider the values y of a random variable Y called dependent variable. The values y are a function of one or more non-random variables x 1, x 2, …, x N called independent variables. The random variable can be modeled (represented) as: The random variable  (not to be confused with dissipation used before) gives the departure from linearity and has a specific PDF with mean of zero. Simple linear regression:

9. If N independent variables are involved then we have a multiple linear regression: A powerful method to fit the independent variables x 1, x 2, …, x N to the dependent variable y is the method of least squares The simplest case is to fit a straight line to a set of points using the “best” coefficients b 0, b 1 The method of least squares does what we do by eye, i.e., minimize deviations (residuals) between data points and fitted line. x y

10. Let: where: is the deterministic portion of the data is the residual or error To find b 0, b 1 minimize the sum of the squared errors (SSE) Sum of Squares Total (data variance) Sum of Squares Regression (variance explained by regression)

11. To minimize the sum of the squared errors (SSE) Two equations, two unknowns; solve for the parameters

12. Regression line splits the scatter of observations such that the positive residuals cancel out with negative residuals x y Regression line always goes through

13. Percent explained variance R 2 : Sum of Squares Total (data variance) Sum of Squares Regression (variance explained by regression) Goodness of Fit (Correlation of Determination) Least squares can be used to fit any curve – we’ll see it in harmonic analysis Least squares can be considered a Maximum Likelihood Estimator

14. x y x’ y’  x cos  -x sin  (x, 0) (0, y) y cos  y sin  Rotation of axes  can be obtained from linear regression of scatter diagram

15. Rotation of axes

16. CORRELATION Concept linked to time series analysis Correlation coefficient: determines how well two variables co-vary in time or space. For two random variables, x and y the correlation coefficient can be: C xy is the covariance of x and y, and s x and s y are the stdev

17. AUTOCORRELATION x are the measurements L represents a lag N is the total number of measurements overbar represents mean over the N measurements r x is the autocorrelation coefficient for x r x oscillates between -1 and 1 r x equals1 at L = 0