1. VII. Observation-Parameter Statistics
Composite scaled sensitivities for estimated parameters
Parameter variances and covariances
Leverage and Influence
Parameter correlation coefficients
Parameter uniqueness
Parameter standard deviations, confidence intervals, and coefficients of variation
Parameter confidence intervals
Reasonable ranges
Measures of model nonlinearity
For preliminary evaluation, use fit-independent statistics (all but the influence statistics.
VII. Parameter Statistics

2. 1. Composite Scaled Sensitivities (Book, p. 124-125)
CSS were used initially to help decide which parameters to estimate.
Recall that they indicate the total amount of information provided by the observations for the estimation of each individual parameter.
Can be also be used as an initial gauge of the reliability with which the parameters are estimated. Parameters with larger values of CSS are generally estimated more reliably than those with smaller values of CSS.
After the regression has converged, it is important to recalculate CSS for all parameters, to check whether the conclusions made for the initial CSS about which parameters to estimate still apply. Should additional parameters be estimated? Should changes be made in the use of prior information?
DO EXERCISE 7.1a (p. 145)
What is the difference between initial and final CSS values?
Effect of model nonlinearity and scaling?
Use CSS to explain small weighted residuals for prior information.
Because the regression is nonlinear, sensitivities, can change as parameters change, and therefore CSS’s can change. For CSS, the statistic is scaled by the parameter value, so if there is a large change in the parameter value, there could be a large change in the CSS. Therefore, it is important to reevaluate the CSS (and other measures based on sensitivity).
In the case presented, there is little change in the CSS for the parameters with prior, and the CSS is still much smaller than the other parameters. Therefore, the results indicate that the regression should not be attempted w/o prior. The fact that the parameter values are very near the prior values (as seen by the small weighted residuals) indicates that this is the dominating information for these parameters.
VII. Parameter Statistics

3. 1. Composite Scaled Sensitivities
from the starting steady-state model
An alternative:
The relative parameter coefficients of variation can also be plotted.
- They equal the parameter standard deviation divided by the parameter value.
Not fit-independent, but all are multiplied by the same value of the standard error of the regression so relative values are useful.
Account for parameter correlation
Composite scaled sensitivities from the final steady-state model
(Book, Fig. 7-5a, p. 146)

4. 2. Parameter Variances and Covariances
The variance-covariance matrix for the parameters is:. (Hill and Tiedeman, 2007, p. 125, eq 7.1)
V(b)=s2(XTw X) -1
b is a vector of parameter values
s2 is he calculated error variance (measure of model fit)
X is the matrix of sensitivities of the simulated equivalents
to the observations, calculated at b
w is the weight matrix

5. 2. Parameter variance-covariance matrix
V(b)=s2(XTw X)-1
Five versions
With optimized parameter values and only optimized parameters.
With optimized parameter values and all defined parameters
With nonoptimal parameter values
Alternate observation sets – observations omitted or added
With predictions
For versions a and b, s2 is calculated using optimal parameter values and statistics affected by s2 can be meaningful
For versions c, d, and e, use statistics for which s2 divides out
For now we will work with version (a)

6. 2. Parameter Variances and Covariances (book p. 126)
The diagonal elements of the matrix are the parameter variances; the off-diagonal elements are the parameter covariances. For a 3 parameter problem, the matrix is:
Var (1) Cov(1,2) Cov(1,3)
Cov (2,1) Var (2) Cov (2,3)
Cov (3,1) Cov (3,2) Var (3)

7. 2. Parameter Variances and Covariances (Book, p. 126)
The parameter variance-covariance matrix is used to evaluate parameter uncertainty and parameter correlation.
Generally, we use statistics computed from the parameter variance-covariance matrix, rather than elements of the matrix itself.

8. 2. Regression Parameter Variances and Covariances
In regression, the parameter values are estimated indirectly using observations.
This can be accomplished because the simulation model is based on equations that relate observations to parameter values.
Because of this indirect way of estimating parameter values, parameter variances and covariances are calculated using sensitivities.
Interpretation of the variance and correlation of parameters estimated by regression :
The variance indicates the range over which a parameter value could extend without affecting model fit too adversely.
The parameter correlation coefficients indicate whether coordinated changes in the parameter values could produce the same simulated values and, therefore, the same model fit.

9. 2. Parameter Variances and Covariancesb2
Linear objective function:
No correlation, b1 less sensitive
minimum
~Var(b2)
~Var(b1)
Can change b1 and have little change in the objective function. Objective function changes more quickly with b2 b1

10. 2. Parameter Variances and Covariancesb2
Linear objective function
Strong, negative correlation
Can change b1 and b2 together and have little change in the objective function
minimum b1

11. 2. Parameter Variances and Covariances
Obtaining the common expression of the variance-covariance matrix requires that the weight matrix be defined in a certain way. If the weight matrix is diagonal, then each of the weights need to be defined as:
Defining the weight matrix in this manner produces the smallest calculated parameter variances (smallest uncertainty in the parameter estimates).
See Appendix C of H&T (p. 396) for a mathematical proof of why the weights need to be defined in this manner.

12. 3. Leverage and Influence
Based on V(b)
Important to understand the role that observations play in the regression
Estimates can be largely affected by very few observations
Two characteristics are important
Leverage – depends only on the type, location, and time of the observation
Influence – depends on the observed value as well

13. 3. Leverage (p. 134)
Leverage Statistics identify observations for which observed values potentially have a big effect on regression results
If the observation is inconsistent with other observations, the observation will dominate the estimated parameter values if it has high leverage.
Measures a potential effect.
May or may not be an actual effect. ( )
parameter or the transpose of the ith row of the X matrix
is a vector composed of the scaled sensitivities of the ith
leverage of ith parameter is
where 1 x h X i T - = w

14. 3. Leverage for linear regression
Leverage = (1/n) + [(xi-(Sx/n))2]/SSx
Leverage is large when the X for an observation is far from the mean of the X’s.
In linear regression, the X values are equivalent to our sensitivities.
The idea that high leverage parameters are those for which the sensitivities are somehow different carries over to multiple, nonlinear regression.

15. 3. Influence: Cook’s D and DFBETAS
Influence Statistics incorporate calculated residuals to determine the actual effect of the observation in the regression
Cook’s D is a measure of how a set of parameter estimates would change with omission of an observation, relative to how well the parameters are estimated given the entire set of observations (in file with extension ._rc)
DFBETAS measures the importance of one observation to one parameter. Specifically, the influence of observation i on parameter j, scaled by the variance of parameter j when estimated using all observations (in file with ._rb)
residual_analysis.exe calculates these measures and prints them in UCODE_2005 data-exchange files

16. 3. Leverage vs. Influence
High leverage
Low influence 19
Exclude obs 18
All obs
High leverage
High influence
Exclude obs 19 18
Regression line when the more influential of the points considered is omitted from the regression

17. Exercises EXERCISE 7.1b: Evaluate leverage statistics (p. 146)
Statistics needed are in ex5.2c_ucode._so
EXERCISE 7.1c: Evaluate importance of using influence statistics (p. 146)
DFBETAS are in file ex5.2c_ucode._rb
Cook’s D are in file ex5.2c_ucode._rc
Produced in exercise 6.2e by running the residual_analysis computer program
Use the equations in Hill and Tiedeman (2007) to calculate the critical values. Compare them to the graphs to identify influential observations.
Cood
Cook’s D: 4/(ND+NPR) DFBETAS: 2/(ND+NPR)1/2

18. 4. Parameter Correlation Coefficients (Book, p. 127)
Parameter correlation coefficients (from the _pcc file) are used primarily to assess parameter uniqueness
Computed from parameter variance-covariance matrix values
Correlation coefficients are typically presented as a matrix.
Independent of model fit (s2 cancels out in calculation). Correlation coefficients depend only on sensitivities of the simulated equivalents to the parameters, and weights.
Because of model nonlinearity, parameter correlation coefficients are a function of parameter values.
Variances are always positive, covariances (and therefore, correlation coefficients) can be positive or negative.

19. 4. Parameter Correlation Coefficients
If the absolute value of a correlation for a parameter pair is greater than about 0.95, then it may not be possible to estimate the 2 parameters uniquely using the available regression data. Changing the parameter values in a coordinated manner may produce very similar model results.
In this case, it is essential to restart the regression with different initial parameter values and check whether the regression converges to the same estimates. If not, then the parameter estimates are not unique
Absolute values close to 1.0 may also cause failure of the regression to converge to a set of ‘optimal’ values.
DO EXERCISE 7.1d (p. 148)

20. 4. Parameter Correlation Coefficients
If correlations are large in absolute value and negative, increasing one parameter and decreasing another will produce similar conditions
Absolute values close to 1.0 may cause these problems:
No single set of parameter values will produce the smallest value of the sum of squared, weighted residuals and regression may have trouble converging. OR
Solution may be non-unique: different solutions would result from using different initial parameter values.

21. 4. Correlation Coefficients and Inaccurate Sensitivities
If using a perturbation method to calculate sensitivities using, for example, UCODE_2005 or PEST, the sensitivities generally are less accurate than those computed using the sensitivity equation method using, for example, MODFLOW-2000.
Inaccurate sensitivities produce parameter correlation coefficients close to 1.0 that are reliable, but correlation coefficients far from 1.0 that may be inaccurate. The problem becomes worse as one or both of the parameters involved in the correlation become less sensitive.
Correlation coefficients far from 1.0 may mean either there is a lack of correlation or the correlations are not accurate enough.
It is important to test for nonuniqueness even if the calculated correlations are not close to 1.0
FINISH EXERCISE 7.1d

22. 4. Parameter Correlation Coefficients
EXERCISE 7.2: Consider all the different correlation coefficients presented (p. 155)
R = (6.11a)
= (6.18)
(7.5)

23. 5. Further tests for nonunique parameters
While useful, for a variety for reasons parameter correlation coefficients can fail to detect nonunique solutions.
Furthe investigation methods include starting the regression from alternative starting values of the parameters and using global sensitivity analysis and regression methods.
In this class we consider alternative starting values of the parameters.

24. Exercise 7.1e. Detecting non-unique parameter estimates – instructions for MFI2005
First, perform a regression run without flow observation and prior information:
In MFI2K, save the ex5.2c dataset as dataset ex7.1e and use the ex7.1e dataset in this exercise
To omit the flow observation, click on “Deactivate” under the Options menu in MFI, and deactivate RVOB
In “Observations>Single-Head Observations”, make the name of the output file ex7.1e._os
From the UCODE menu, remove the prior information equations
Perform regression
Do part 1 of the Problem on p. 150 of Hill and Tiedeman
Run UCODE_2005 with SensitivityAnalysis=yes to obtain a parameter correlation matrix
Next, include the flow observation and prior information, and start the regression from different initial values:
Include the flow observation by activating RVOB
In the PES file, include the prior information equations, and change MAX-ITER to 10
In the SEN file, change the starting parameter values to the values in set 1 of table 10, perform nonlinear regression, and rename the _ot file from this run. Then, change the values to those in set 2 of table 10, and perform nonlinear regression
Do part 2 of the Problem on p. 151 of Hill and Tiedeman

25. 6. Parameter Standard Deviation and Coefficient of Variation (Book, p
The parameter standard deviation (σ) for parameter bj is the square root of the parameter variance (on the diagonal of the parameter variance-covariance matrix):
The parameter standard deviation is easier to interpret than the parameter variance, because it is in the same units as the parameter value.
The parameter coefficient of variation is:
The coefficient of variation is dimensionless, and can be used to compare the relative precision of different parameter estimates

26. 7. Individual Linear Confidence Intervals (Book, p. 138)
The linear individual confidence interval (CI) on a parameter is calculated as:
A 95 percent individual linear CI on a parameter has a 95 percent probability of containing the true parameter value (when the model is sufficiently linear with respect to the parameters)

27. 7. Individual Linear Confidence Intervals
Student-t probability distribution: Similar to normal distribution except adjusts for smaller sample sizes. As n becomes large, the distribution approaches the normal distribution
n t(n,0.05) N(0.05)
Infinity

28. 7. Individual Linear Confidence Intervals
Student-t probability distribution: Similar to normal distribution except adjusts for smaller sample sizes. As n becomes large, the distribution approaches the normal distribution
Width of a confidence interval can be thought of as precision of estimate
Three important assumptions are made in deriving the linear confidence intervals:
The model is correct
The parameter values are normally distributed
The model is linear near the optimal values

29. 7. Individual Linear Confidence Intervals
For a linear model, the parameter estimates are normally distributed if the true observation errors are normally distributed.
However, because the true errors are unknown, we analyze the weighted residuals instead, by constructing a normal probability plot and evaluating RN2.
Model linearity can be evaluated using the modified Beale’s measure (exercise 7.3)
DO EXERCISE 7.1f (p. 151): Evaluate precision by examining standard deviations, linear confidence intervals, and coefficients of variation

30. 7. Linear 95% Confidence Intervals on Estimated Parameter Values
If the model and weighting are correct and the mean of the true errors is zero,
there is a 95% chance that the true parameter values vall within the calculated ranges.
Fig. 7.7, p. 153

31. 8. Comparing Estimated Parameter Values with Reasonable Ranges (Book, p. 140)
It is important to compare the regression estimates of the parameter and the calculated confidence intervals with the reasonable ranges of parameter values. (But you set the reasonable ranges, so be careful.)
3 common situations:
Parameter
Value 1 2 3
Situation
Range of
Reasonable
Values
1: Parameter estimate and most of confidence intervals lie within reasonable range
2: Parameter estimate and confidence intervals lie outside reasonable range
3: Parameter estimate lies outside the range but part of confidence intervals lies within the range

32. 8. Comparing Estimated Parameter Values with Reasonable Ranges1 2 3
Range of
Reasonable
Values
Situation 1: Desirable – enough information to estimate the parameter precisely, and the estimate is consistent with independent information (we have learned something: the probable range is smaller than the original range of reasonable values)
Situation 2: Problematic – enough information to estimate the parameter precisely, but the estimate is inconsistent with independent information (we need to reconsider model, choice of parameters, measurements …: why does the outcome go against our expectations)
Situation 3: Inconclusive - there is enough information to estimate the parameter, but not with much precision (we did not learn anything new)

33. 8. Comparing Estimated Parameter Values with Reasonable Ranges1 2 3
Situation
Range of
Reasonable
Values
Situation 2 Example: Perhaps the reasonable range is determined from aquifer test data in only one part of a large K zone, and there are head data throughout the K zone. The head data in the part of the zone away from the aquifer test might be influencing the K estimate that is unreasonable. This may mean that K needs to be represented by 2 separate zones. X O X O
X: aquifer test data O O X X X O X O O O
O: head data X X O O O O O O K1 K1 K2

34. 8. Comparing Estimated Parameter Values with Reasonable Ranges1 2 3
Range of
Reasonable
Values
DO EXERCISE 7.1g (p. 153) Compare estimated values with reasonable ranges

35. 8. Comparing Estimated Parameter Values with Reasonable Ranges1 2 3
Situation
Range of
Reasonable
Values
Situation 3: Modeler needs to think about (1) what additional data could provide more information towards estimating the parameter, and (2) improvements to the conceptual model of the system.
Consider adding more observation data, such as flows or concentrations
Regression estimate of parameter may become more reasonable and CI may become smaller
Situations 2 and 3: Adding prior does not address the fundamental problem. There is enough information for the regression to converge to a particular parameter value. The modeler needs to think about what is causing the unreasonable estimates.
DO EXERCISE 7.1g (p. 153) Compare estimated values with reasonable ranges

36. Confidence intervals, starting, final, and true values, compared to reasonable ranges
(Book, Figure 7-7, p.153)
GW_Chart can make graphs like these.
_pc

37. 9. Test linearity of model
Linear confidence intervals are only valid when the model is sufficiently linear.
Model linearity is evaluated by using the modified Beale’s measure and total and intrinsic model linearity measures

38. Modified Beale’s Measure. ex7.3_ucode.#modlin
USING FSTAT = , BEALES MEASURE =
IF BEALES MEASURE IS GREATER THAN , THE MODEL IS NONLINEAR.
IF BEALES MEASURE IS LESS THAN E-01, THE MODEL IS EFFECTIVELY LINEAR,
AND LINEAR CONFIDENCE INTERVALS ARE FAIRLY ACCURATE IF THE RESIDUALS ARE
NORMALLY DISTRIBUTED.

39. From output file ex7.3_ucode.#modlinadv_conf
##########################################################
###########
########### TOTAL NONLINEARITY (BNT) =
########### INTRINSIC NONLINEARITY (BNI) =
###########CRITICAL VALUES FOR BOTH MEASURES:
########### >1.0 highly nonlinear
########### to 1.0 non-linear
########### to 0.09 moderately nonlinear
########### <0.01 effectively linear

40. End of VII. Parameter Statistics
Composite scaled sensitivities for estimated parameters
Parameter variances and covariances
Leverage and Influence
Parameter correlation coefficients
Parameter uniqueness
Parameter standard deviations, confidence intervals, and coefficients of variation
Parameter confidence intervals
Reasonable ranges
Measures of model nonlinearity
End of VII. Parameter Statistics
VII. Parameter Statistics