environmental modeling basic testing methods - statistics iii
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Environmental Modeling Basic Testing Methods - Statistics III. 1. Covariance. Joint variation of two variables about their common mean Covariance. 2. Simple Regression. Regression: models relationships between variables - PowerPoint PPT PresentationTRANSCRIPT
Environmental ModelingEnvironmental Modeling
Basic Testing Methods - Statistics IIIBasic Testing Methods - Statistics III
1. Covariance1. Covariance
► Joint variation of two variables about their common mean
► Covariance
2. Simple Regression2. Simple Regression
► Regression: models relationships between variables
Yi = 0 + 1Xi + ei, 0 - intercept, i – slope
Y is the dependent variableX is the independent variableSimple regression has one independent variable Multiple regression has more than one indep var
2. Simple Regression..2. Simple Regression..
► We can fit a line through the cloud of dots► Only one position is the best fit Y = b0 + bX
Y
X
2. Simple Regression..2. Simple Regression..
► Least square methods can help identify the best fit, following two conditions
n
(Yi - Yi)2 = minimum; ei = 0 (Yi - Yi = ei)
1 ► Parameters estimated in the process: b0, b
Y = b0 + bX
b0 – intercept, b – regression coefficient
3. Goodness of Fit3. Goodness of Fit 1
►Total Sum of Squares: SSt = (Yi - Y)2 n
1
►Sum Squares of Regression: SSr = (Yi - Y)2
n
1
►Sum Squares of residuals: SSe = (Yi - Yi)2
n
SSt = SSr + SSe
CoefficientsCoefficients
► Coefficient of determination (goodness of fit):
R2 = SSr/SSt
► Coefficient of correlation: R = R2 = SSr/SSt
r = Cov(x,y)/sxsy
Adjusted R2
(k-1)(1 - R2) ►Adjusted R2: R2
a = R2 - -----------------
N - k N - sample size k - number of independent variables
4. Test of Regression Model4. Test of Regression Model
► General F test: equality of two variances
► Null hypothesis: S12 = S2
2
S12
F = ---------- S2
2
Test of Regression ModelTest of Regression Model
► Compare the computed F value to the critical F value for specified degrees of freedom for both variances and level of significance
► If the computed F>critical F, reject the null, accept otherwise
► Check the p value, if p<, reject the null hypothesis
F Test for Regression ModelF Test for Regression Model
► F test for regression model: ► Null hypothesis: SSr = SSe
SSr/k
F = --------------, SSe/N-k-1
k - number of parameters excluding b0
N - sample size
t Test for bt Test for b
► t test for individual parameters b ► Null hypothesis: bi = 0
bi
t = ------, Sbi - standard error of bi
Sbi
5. Multiple Regression5. Multiple Regression
► Yi = b0 + b1X1 + b2X2 + b3X3 + ... + bmXm + ei
► Y = b0 + b1X1 + b2X2 + b3X3 + ... + bmXm
Regression ResultsRegression Results
► Analysis of varianceAnalysis of varianceDF DF Sum of Squares Sum of Squares Mean SquareMean Square
Regression Regression 3 3 97747.0918497747.09184 32583.0306132583.03061ResidualResidual 36 36 7061.68316 7061.68316 196.15787 196.15787
F = 166.10616F = 166.10616 Signif F = 0.0000Signif F = 0.0000
Multiple rMultiple r 0.87328 0.87328R SquareR Square 0.76262 0.76262Adjusted R Square Adjusted R Square 0.75701 0.75701Standard ErrorStandard Error 14.00564 14.00564
Regression ResultsRegression Results
► Variables in the EquationVariables in the EquationVariableVariablebb Se b Se b Beta Beta t t Sig t Sig t
XX11 0.1917 0.1917 0.0017150.001715 0.7259980.725998 6.262 6.262 0.00000.0000
XX22 -0.0829-0.0829 0.0012190.001219 -0.994050-0.994050 -16.161 -16.161 0.00000.0000
XX33 -4.9594-4.9594 11.07978511.079785 -0.052423-0.052423 -0.4841 -0.4841 0.03100.0310
XX44 5.3639 5.3639 7.39087.3908 7.92737.9273 -0.932-0.932 0.09260.0926
Y = 0.1917XY = 0.1917X11 - 0.0829X - 0.0829X22 - 4.9594X - 4.9594X33 + 5.3639X + 5.3639X44