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Adamu Mustapha PhD
There are number of multivariate Geostatistical analyses used 
in environmental studies to identify the spatial and temporal 
variation of the datasets. 
1. Hierarchical Agglomerative Cluster Analysis (HACA) 
2. Principal Component Analysis (PCA) 
3. Multiple Linear Regression (MLR) 
4. Pearson’s Product Moment Correlation Analysis 
5. Discriminant Analysis
1. Hierarchical Agglomerative Cluster Analysis (HACA) 
HACA is a multivariate Geostatistical technique whose primary 
purpose is to assemble similar objects based on characteristic 
they possess (Shrestha and Kazama, 2007) 
The level of similarities at which observation are merged are 
used to construct a dendrogram of clusters (Singh et al., 2004; 
Chen et al., 2007; Juahir et al. 2011, Mustapha et al. 2012). 
The resultant clusters exhibit high internal (within clusters) 
homogeneity and high external (between groups) heterogeneity.
Jakara Dam 
0 3 Km 
Sources apportionment of Jakara 
Basin (Upstream) 
Domestic sources 
Industrial sources 
Agricultural sources 
Nigeria 
S1 S2 
S3 
S4 
S5 
S6 
S7 
S15 
S11 S12 
S9 
S8 
S10 
S13 S14 
S16 
S17 
S18 
S19 
S20 
S21 
S22 
S23 
S25 
S26 
S30 
S29 
S28 
S27 
S1, S2, . .. Sampling sites 
S24 
Sampling Points
2. Principal component analysis and or factor analysis (PCA) 
PCA is a multivariate Geostatistical statistical technique that 
examine the underlying pattern or relationship of a large number 
of variables. It is use to get information about inter-relationships 
among a set of variables 
PCA group the variables into smaller and more meaningful set 
of factors
How do we determine the number of factors to be retained? 
We use the Kaiser’s-one- Criterion also known as the eigen-value 
rule of >1 
We equally use the Catell’s scree plot 
It produce plot of the eigenvalues, looking at the plot where it 
becomes horizontal, then Cartell;s recommends retaining all the 
factors above this points. 
These factors with eigenvalues 1 and >1 contribute the most 
variance in the data sets.
The Important parameters in the factor have factor high factor 
loading. Liu et al. 2003 suggest the following loading on 
parameters 
0 – 0.4 Low loading 
0.5 -0.7 Moderate loading 
> 0.7 High loading
Parameters Unit PC1 PC2 PC3 PC4 PC5 
Pb mg L-1 0.960 -0.111 0.105 0.060 -0.098 
Cd mg L-1 0.953 -0.101 0.145 -0.049 -0.040 
Cr mg L-1 0.940 -0.037 0.148 -0.049 -0.060 
Hg mg L-1 0.854 -0.070 0.020 -0.086 -0.299 
Fe mg L-1 0.706 0.172 0.327 -0.353 -0.078 
EC μS/cm -0.659 -0.139 -0.181 0.094 -0.144 
Ni mg L-1 0.620 -0.014 -0.331 0.602 -0.234 
BOD5 mg L-1 0.537 0.594 -0.114 -0.399 0.399 
DS mg L-1 -0.156 0.835 0.557 0.074 -0.167 
TS mg L-1 0.042 0.670 0.514 0.133 -0.036 
pH -0.418 0.260 0.633 0.217 -0.073 
DO mg L-1 0.166 -0.583 -0.617 -0.017 0.240 
COD mg L-1 0.332 0.527 0.565 0.187 0.090 
Turbidity NTU 0.342 -0.009 0.126 0.788 0.191 
Hardness mg L-1 0.178 -0.257 0.367 0.162 0.809 
Eigen value 5.53 2.21 1.96 1.42 1.13 
% Variance 36.91 14.79 13.06 9.49 7.57
3. Multiple Linear Regression (MLR) 
 MLR is used to fit a model to our data and use it to predict the 
value of the Y (DVs) from one or more IV’s. 
 Predicting out come from one or several predictors. 
 Mathematical techniques LSM is used to establish the line that 
best describes the data. 
Friday, November 28, 2014 11
Regression analysis is to derived a prediction equation 
Y=bo +b1x1+b2x2+b3x3+……bpxp 
Where: 
Y = dependent variable 
Xs = independent variables 
bo = Y-intercept 
b1 = regression coefficient
 Before interpreting the result of MLR, there is need to check 
for assumptions of regression analysis. i.e. Normality, linearity 
and multicolinearity (Berry, 1993). 
Friday, November 28, 2014 13
The normal p-p plot of regression standardized residuals revealed all 
observed Values fall roughly along the straight line. This indicates 
residuals are from normally Distributed population 
Friday, November 28, 2014 14
Assumption sof linear regression model 
Colinearity/Multicolinearity 
Problem with correlation between Ivs that occurs when 
Ivs are highly correlated which make it difficult to 
determine the contribution of Ivs. 
Tolerance value 
Variance Inflation Factor (VIF) 
Condition index
a. Tolerance 
This is the amount of variability not explain by other Ivs, 
small tolerance value indicates high Multicolinearity smaller 
than 0.10 
b. Variance Inflation factor (VIF) 
This is the inverse of the tolerance. The cutoff threshold of 
VIF must be >1.0
c. Condition index statistics 
Condition Index (CI) is a measure of the relative amount of 
variance associated with an eigen value. A large CI indicates a 
high degree of collinearity 
A value of CI greater than 15 indicates a possible problem and an 
index greater than 30 suggests a serious problem with collinearity 
(Kutner et al. 2004).
R = 0.986 
R2 = 0.971 
Model R 
R 
Square 
Adjusted 
R Square 
SE of the 
Estimate 
R Square 
Change 
Change Statistics 
F 
Change 
df1 df2 
Sig. F 
Change 
Durbin- 
Watson 
1 0.986 0.971 0.840 2.331 0.971 7.382 15 5 0.018 2.651 
Friday, November 28, 2014 18
Model 
Estimates of coefficient for the model 
Unstandadized 
BETA Std. Error 
Standardized 
Coefficients 
BETA t Sig. Tolerance VIF 
1 (Constant) 102.748 39.602 2.594 0.018 
Iron mg/l 0.438 0.127 0.778 3.449 0.000 0.250 15.897 
Mercury 
mg/l 2.442 1.906 3.500 1.281 0.000 0.333 1304.69 
Chromium 
mg/l -0.852 0.672 -3.188 -1.267 0.000 0.290 1105.85 
Cadmium 
mg/l -5.695 2.019 -11.900 -2.821 0.000 0.540 3110.806 
Lead mg/l 3.719 1.317 12.358 2.823 0.001 0.889 3350.478 
From the table the largest beta coefficient is 3.719 (lead), the 
variable make a unique contribution in explaining DV. 
Friday, November 28, 2014 19
4. Pearson’s Product Moment Correlation Analysis 
Identify the significant relationship between bivariate 
   
n xy x y 
( )( ) 
 
2 2 2 2 
    
( ) ( ) 
r 
n x n y y 
 
      
Table 2 Guildford rule of thumb for interpreting correlation analysis (r) 
r value Interpretation 
0.0 to 0.29 Negligible or little correlation 
0.3 to 0.49 Low correlation 
0.5 to 0.69 Moderate or marked correlation 
0.7 to 0.89 High correlation 
0.9 to 1.00 Very high correlation
Headache Fever Backpain JointPain StickInjuries Scabies Rashes Catarh Cough Breathprob Diarrhoea EyeProblem StomachPain 
Headache 1 0.505 0.547 .788** 0.575 .679* 0.308 0.191 -0.419 0.268 0.043 0.184 -0.049 
Fever 1 0.624 0.525 0.571 .786** 0.183 0.498 0.085 .686* 0.204 .762* 0.619 
Backpain 1 0.537 .862** .849** .701* 0.543 0.156 .823** 0.056 0.344 0.352 
JointPain 1 0.551 .672* 0.246 0.181 -0.246 0.422 -0.443 0.344 -0.171 
StickInjuries 1 .827** 0.452 0.369 0.301 .778** -0.083 0.207 0.064 
Scabies 1 0.58 0.389 -0.087 .816** 0.023 0.57 0.367 
Rashes 1 0.23 -0.058 0.528 -0.047 0.134 0.324 
Catarh 1 0.308 0.441 0.18 .656* 0.437 
Cough 1 0.314 -0.221 -0.012 0.054 
Breathprob 1 -0.141 0.534 0.346 
Diarrhoea 1 0.058 0.624 
EyeProblem 1 0.602 
StomachPain 1
Thank you

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Kano GIS Day 2014 - The Application of Multivariate Geostatistical analyses in Environmental Data

  • 2. There are number of multivariate Geostatistical analyses used in environmental studies to identify the spatial and temporal variation of the datasets. 1. Hierarchical Agglomerative Cluster Analysis (HACA) 2. Principal Component Analysis (PCA) 3. Multiple Linear Regression (MLR) 4. Pearson’s Product Moment Correlation Analysis 5. Discriminant Analysis
  • 3. 1. Hierarchical Agglomerative Cluster Analysis (HACA) HACA is a multivariate Geostatistical technique whose primary purpose is to assemble similar objects based on characteristic they possess (Shrestha and Kazama, 2007) The level of similarities at which observation are merged are used to construct a dendrogram of clusters (Singh et al., 2004; Chen et al., 2007; Juahir et al. 2011, Mustapha et al. 2012). The resultant clusters exhibit high internal (within clusters) homogeneity and high external (between groups) heterogeneity.
  • 4. Jakara Dam 0 3 Km Sources apportionment of Jakara Basin (Upstream) Domestic sources Industrial sources Agricultural sources Nigeria S1 S2 S3 S4 S5 S6 S7 S15 S11 S12 S9 S8 S10 S13 S14 S16 S17 S18 S19 S20 S21 S22 S23 S25 S26 S30 S29 S28 S27 S1, S2, . .. Sampling sites S24 Sampling Points
  • 5.
  • 6. 2. Principal component analysis and or factor analysis (PCA) PCA is a multivariate Geostatistical statistical technique that examine the underlying pattern or relationship of a large number of variables. It is use to get information about inter-relationships among a set of variables PCA group the variables into smaller and more meaningful set of factors
  • 7. How do we determine the number of factors to be retained? We use the Kaiser’s-one- Criterion also known as the eigen-value rule of >1 We equally use the Catell’s scree plot It produce plot of the eigenvalues, looking at the plot where it becomes horizontal, then Cartell;s recommends retaining all the factors above this points. These factors with eigenvalues 1 and >1 contribute the most variance in the data sets.
  • 8.
  • 9. The Important parameters in the factor have factor high factor loading. Liu et al. 2003 suggest the following loading on parameters 0 – 0.4 Low loading 0.5 -0.7 Moderate loading > 0.7 High loading
  • 10. Parameters Unit PC1 PC2 PC3 PC4 PC5 Pb mg L-1 0.960 -0.111 0.105 0.060 -0.098 Cd mg L-1 0.953 -0.101 0.145 -0.049 -0.040 Cr mg L-1 0.940 -0.037 0.148 -0.049 -0.060 Hg mg L-1 0.854 -0.070 0.020 -0.086 -0.299 Fe mg L-1 0.706 0.172 0.327 -0.353 -0.078 EC μS/cm -0.659 -0.139 -0.181 0.094 -0.144 Ni mg L-1 0.620 -0.014 -0.331 0.602 -0.234 BOD5 mg L-1 0.537 0.594 -0.114 -0.399 0.399 DS mg L-1 -0.156 0.835 0.557 0.074 -0.167 TS mg L-1 0.042 0.670 0.514 0.133 -0.036 pH -0.418 0.260 0.633 0.217 -0.073 DO mg L-1 0.166 -0.583 -0.617 -0.017 0.240 COD mg L-1 0.332 0.527 0.565 0.187 0.090 Turbidity NTU 0.342 -0.009 0.126 0.788 0.191 Hardness mg L-1 0.178 -0.257 0.367 0.162 0.809 Eigen value 5.53 2.21 1.96 1.42 1.13 % Variance 36.91 14.79 13.06 9.49 7.57
  • 11. 3. Multiple Linear Regression (MLR)  MLR is used to fit a model to our data and use it to predict the value of the Y (DVs) from one or more IV’s.  Predicting out come from one or several predictors.  Mathematical techniques LSM is used to establish the line that best describes the data. Friday, November 28, 2014 11
  • 12. Regression analysis is to derived a prediction equation Y=bo +b1x1+b2x2+b3x3+……bpxp Where: Y = dependent variable Xs = independent variables bo = Y-intercept b1 = regression coefficient
  • 13.  Before interpreting the result of MLR, there is need to check for assumptions of regression analysis. i.e. Normality, linearity and multicolinearity (Berry, 1993). Friday, November 28, 2014 13
  • 14. The normal p-p plot of regression standardized residuals revealed all observed Values fall roughly along the straight line. This indicates residuals are from normally Distributed population Friday, November 28, 2014 14
  • 15. Assumption sof linear regression model Colinearity/Multicolinearity Problem with correlation between Ivs that occurs when Ivs are highly correlated which make it difficult to determine the contribution of Ivs. Tolerance value Variance Inflation Factor (VIF) Condition index
  • 16. a. Tolerance This is the amount of variability not explain by other Ivs, small tolerance value indicates high Multicolinearity smaller than 0.10 b. Variance Inflation factor (VIF) This is the inverse of the tolerance. The cutoff threshold of VIF must be >1.0
  • 17. c. Condition index statistics Condition Index (CI) is a measure of the relative amount of variance associated with an eigen value. A large CI indicates a high degree of collinearity A value of CI greater than 15 indicates a possible problem and an index greater than 30 suggests a serious problem with collinearity (Kutner et al. 2004).
  • 18. R = 0.986 R2 = 0.971 Model R R Square Adjusted R Square SE of the Estimate R Square Change Change Statistics F Change df1 df2 Sig. F Change Durbin- Watson 1 0.986 0.971 0.840 2.331 0.971 7.382 15 5 0.018 2.651 Friday, November 28, 2014 18
  • 19. Model Estimates of coefficient for the model Unstandadized BETA Std. Error Standardized Coefficients BETA t Sig. Tolerance VIF 1 (Constant) 102.748 39.602 2.594 0.018 Iron mg/l 0.438 0.127 0.778 3.449 0.000 0.250 15.897 Mercury mg/l 2.442 1.906 3.500 1.281 0.000 0.333 1304.69 Chromium mg/l -0.852 0.672 -3.188 -1.267 0.000 0.290 1105.85 Cadmium mg/l -5.695 2.019 -11.900 -2.821 0.000 0.540 3110.806 Lead mg/l 3.719 1.317 12.358 2.823 0.001 0.889 3350.478 From the table the largest beta coefficient is 3.719 (lead), the variable make a unique contribution in explaining DV. Friday, November 28, 2014 19
  • 20. 4. Pearson’s Product Moment Correlation Analysis Identify the significant relationship between bivariate    n xy x y ( )( )  2 2 2 2     ( ) ( ) r n x n y y        Table 2 Guildford rule of thumb for interpreting correlation analysis (r) r value Interpretation 0.0 to 0.29 Negligible or little correlation 0.3 to 0.49 Low correlation 0.5 to 0.69 Moderate or marked correlation 0.7 to 0.89 High correlation 0.9 to 1.00 Very high correlation
  • 21. Headache Fever Backpain JointPain StickInjuries Scabies Rashes Catarh Cough Breathprob Diarrhoea EyeProblem StomachPain Headache 1 0.505 0.547 .788** 0.575 .679* 0.308 0.191 -0.419 0.268 0.043 0.184 -0.049 Fever 1 0.624 0.525 0.571 .786** 0.183 0.498 0.085 .686* 0.204 .762* 0.619 Backpain 1 0.537 .862** .849** .701* 0.543 0.156 .823** 0.056 0.344 0.352 JointPain 1 0.551 .672* 0.246 0.181 -0.246 0.422 -0.443 0.344 -0.171 StickInjuries 1 .827** 0.452 0.369 0.301 .778** -0.083 0.207 0.064 Scabies 1 0.58 0.389 -0.087 .816** 0.023 0.57 0.367 Rashes 1 0.23 -0.058 0.528 -0.047 0.134 0.324 Catarh 1 0.308 0.441 0.18 .656* 0.437 Cough 1 0.314 -0.221 -0.012 0.054 Breathprob 1 -0.141 0.534 0.346 Diarrhoea 1 0.058 0.624 EyeProblem 1 0.602 StomachPain 1