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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 r(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 r(X,Y)=
Cov(X,Y)= . However, the reverse statement is not necessarily true.
Cases of Correlation
Perfectly linearly
correlated in opposite
direction
Uncorrelated in
linear fashion
Strongly & positively
correlated in
linear fashion
Perfectly correlated in
nonlinear fashion, but
uncorrelated linearly.
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).
Day (t) Flow Q(t) Day (t) Flow Q(t) Day (t) Flow Q(t)
1 8.35 11 313.89 21 20.06
2 6.78 12 480.88 22 17.52
3 6.32 13 151.28 23 116.13
4 17.36 14 83.92 24 68.25
5 191.62 15 44.58 25 280.22
6 82.33 16 36.58 26 347.53
7 524.45 17 33.65 27 771.30
8 196.77 18 26.39 28 124.20
9 785.09 19 22.98 29 58.00
10 562.05 20 21.92 30 44.08
• 29 pairs: {(Qt, Qt+1)} = {(Q1, Q2), (Q2, Q3), …, (Q29, Q30)};
Relevant sample statistics: n=29
Qt 186.22; SQt 230.06; Qt 1 187.45; SQt 1 229.17
The 1-day auto-correlation is 0.439
Chung Mei Upper Daily Flow
800 900
700 800
Q(t+1), 1000 m^3
700
Flow (1000 cubic meters)
600
500
600
500
400
400
300
300
200
200
100
100
0
0
10 Day 20 30 0 200 400 600 800 1000
Q(t), 1000 m^3
Autocorrelation for June 2001 Daily Flows at Chung Mei Upper, HK
1.0
0.8
0.6
Autocorrelation
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
1 2 3 4 5
Time lags (Days)
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: Y β 0 X1 1 X β 2 ... X β K e.
β
2 K
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
y
x
What is a plausible mathematical model describing x & y relation?
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”?
y
^ =b +b x
y 0 1
b1
1
^
ei = yi – yi = error (residual)
^
yi
yi
b0
x
xi
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:
D D
0 and 0
b 0 b 1
n
y i b b xi 0
D n
b 2 y i b 0 b 1 xi 1 0 i 1
0 i 1
n
D 2 y b b x x 0 x y b b x 0
i i
n
b 1 i 0 1i i
i 1
i
i 1
• Expanding the above equations
n n
b b xi 0
y i n
i 1 i 1
n n n
xi y i b xi b xi2 0
i 1
i 1
i 1
• Normal equations:
n n
nb xi b yi
i 1 i 1
x b x 2 b x y
n n n
i i i i
i 1 i 1
i 1
LS Solution (2 Unknowns)
n
n
y i xi
b i 1 i 1 b y x b
ˆ ˆ
n n
n n n n
1
b xi yi n xi yi xi yi nxy
ˆ i 1 i 1 i 1 i 1
n
2
n
1 n
xi nx
2 2
x i xi
n i 1
2
i 1
i 1
Fitting a Polynomial Eq. By LS Method
y i b b xi b 2 xi2 b k xik e i , i 1,2, , n
LS criterion:
D= yi b b xi b xi2 b xik
n
2
minimize
i 1
b , , b
D
Set 0 , for j 0,1,2, , k
b j
Normal Equations are:
n n k n
bn b xi b xi y i
i 1 i 1 i 1
n n 2 n k 1 n
b xi b xi b xi y i xi
i 1 i 1 i 1 i 1
n k n k 1 n 2k n
b xi b xi b xi y i xi
k
i 1 i 1 i 1 i 1
Fitting a Linear Function of Several Variables
y b b x1 b x 2 b x k e
LS criterion :
Minimize D= yi b b xi b x1 b xk
n 2
i 1
b b , b 1 , , b k
D
Set 0 , for j 0, 1, 2, , k
b j
Normal equations:
n n n
bn b xi1 b xik y i
i 1 i 1 i 1
n n 2 n n
b xi b xi1 b xi1 xik y i xi1
i 1 i 1 i 1 i 1
n n n 2 n
b xik b xik xi1 b xik y i xik
i 1 i 1 i 1 i 1
Matrix Form of Multiple Regression by LS
y1 1 x11 x12 x1k b e 1
y 1
x 21 x22 x2 k b e 2
2
yn 1
x n1 x n 2 x nk b k e n
(Note: xij = ith observation of the jth independent variable)
or y=Xb+e in short
LS criterion is:
n
min D e i2 ε' ε y - X β ' y - X β
i 1
β ^
Set D 0 , and result in: X' ( y - X β ) 0
β
The LS solutions are: ˆ X' X 1 X' y
β
Measure of Goodness-of-Fit
R2 = Coefficient of Determination
n 2
ε
i
1 i 1
n
yi y2
i 1
= 1 - % of variation in the dependent variable, y, unexplained by
the regression equation;
= % of variation in the dependent variable, y, explained by the
regression equation.
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)
