Correlation and Regression Analysis Many engineering design and analysis problems involve factors that are interrelated and dependent. E.g., (1) runoff volume, rainfall; (2) evaporation, temperature, wind speed; (3) peak discharge, drainage area, rainfall intensity; (4) crop yield, irrigated water, fertilizer. Due to inherent complexity of system behaviors and lack of full understanding of the procedure involved, the relationship among the various relevant factors or variables are established empirically or semi-empirically. Regression analysis is a useful and widely used statistical tool dealing with investigation of the relationship between two or more variables related in a non-deterministic fashion. If a variable Y is related to several variables X1, X2, …, XK and their relationships can be expressed, in general, as Y = g(X1, X2, …, XK) where g(.) = general expression for a function; Y = Dependent (or response) variable; X1, X2,…, XK = Independent (or explanatory) variables.

Correlation When a problem involves two dependent random variables, the degree of linear dependence between the two can be measured by the correlation coefficient r(X,Y), which is defined as where Cov(X,Y) is the covariance between random variables X and Y defined as   where <Cov(X,Y)< and  (X,Y)  . Various correlation coefficients are developed in statistics for measuring the degree of association between random variables. The one defined above is called the Pearson product moment correlation coefficient or correlation coefficient. If the two random variables X and Y are independent, then (X,Y)= Cov(X,Y)= . However, the reverse statement is not necessarily true.

Cases of Correlation Perfectly linearly correlated in opposite direction Strongly & positively correlated in linear fashion Perfectly correlated in nonlinear fashion, but uncorrelated linearly. Uncorrelated in

Calculation of Correlation Coefficient Given a set of n paired sample observations of two random variables (xi, yi), the sample correlation coefficient ( r) can be calculated as

Auto-correlation Consider following daily stream flows (in 1000 m3) in June 2001 at Chung Mei Upper Station (610 ha) located upstream of a river feeding to Plover Cove Reservoir. Determine its 1-day auto-correlation coefficient, i.e., r(Qt, Qt+1). 29 pairs: {(Qt, Qt+1)} = {(Q1, Q2), (Q2, Q3), …, (Q29, Q30)}; Relevant sample statistics: n=29 The 1-day auto-correlation is 0.439

Chung Mei Upper Daily Flow

Regression Models due to the presence of uncertainties a deterministic functional relationship generally is not very appropriate or realistic. The deterministic model form can be modified to account for uncertainties in the model as Y = g(X1, X2, …, XK) + e where e = model error term with E(e)=0, Var(e)=s2. In engineering applications, functional forms commonly used for establishing empirical relationships are  Additive: Y = b0 + b1X1 + b2X2 + … + bKXK +e Multiplicative: +e.

Least Square Method Suppose that there are n pairs of data, {(xi, yi)}, i=1, 2,.. , n and a plot of these data appears as What is a plausible mathematical model describing x & y relation? x y

Least Square Method Considering an arbitrary straight line, y =b0+b1 x, is to be fitted through these data points. The question is “Which line is the most representative”?

Least Square Criterion What are the values of b0 and b1 such that the resulting line “best” fits the data points? But, wait !!! What goodness-of-fit criterion to use to determine among all possible combinations of b0 and b1 ? The least squares (LS) criterion states that the sum of the squares of errors (or residuals, deviations) is minimum. Mathematically, the LS criterion can be written as:   Any other criteria that can be used?

Normal Equations for LS Criterion The necessary conditions for the minimum values of D are: and Expanding the above equations Normal equations:

LS Solution (2 Unknowns)

Fitting a Polynomial Eq. By LS Method

Fitting a Linear Function of Several Variables

Matrix Form of Multiple Regression by LS or y = X b + e in short LS criterion is: The LS solutions are:

Measure of Goodness-of-Fit

Example 1 (LS Method)

Example 1 (LS Method)

LS Example

LS Example (Matrix Approach)

LS Example (by Minitab w/ b0)

LS Example (by Minitab w/o b0)

LS Example (Output Plots)