GPS-X Technical Reference

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Optimizer 398 Detailed Statistical Report The detailed statistical report provides additional information that is helpful when doing parameter estimation. It allows the user to assess how well their model fits the measured data. The detailed statistical report is activated by setting the detailed statistical report option to ON in the Static sub-section of the Optimizer form. The statistics included in the detailed statistical report are discussed below. Variance-Covariance and Correlation Matrices The variance-covariance matrix is defined by Equation 14.22 for the sum of squares objective function and by Equation 14.24 for the maximum likelihood objective function. This matrix contains the variances of the individual parameter estimates (diagonal elements) and the covariances between the different parameter estimates (offdiagonal elements). Parameters with large variances may be unnecessary. The elements of the correlation matrix are calculated as follows: Equation 14.28 where ( V ˆ) k , l is the (k,l) element of the variance-covariance matrix. By definition, the elements of the correlation matrix are always between zero and one. Unless some parameters are on their bounds the diagonal elements are always equal to one. The off-diagonal elements indicate the degree of correlation between pairs of parameters. A large off-diagonal element indicates a high degree of correlation between a pair of parameters. This provides evidence that some of the parameters in the model are unnecessary and the model is over parameterized. This may indicate that the model is not adequate for the task at hand or that the data do not provide enough information to allow estimation of all of the model parameters (Draper and Smith, 1981). The variance-covariance and correlation matrices are only reported when the printing of confidence limits switch in the Confidence Limits sub-section in the Optimizer form is set to ON. GPS-X Technical Reference
399 Optimizer Percentage Variation Explained by Regression This is a statistic often reported by linear least squares packages. In GPS-X it has been modified to handle nonlinear maximum likelihood problems with heteroscedasticity. The statistic, adapted from Steiner et al. (1990), is calculated as follows for each target variable: Equation 14.29 It is multiplied by 100 to get a percentage. The variable SS j is the weighted residual sum of squares for response j and is defined by: Equation 14.30 The variable SSmean j is the total weighted sum of squares corrected for the mean and is defined below: Equation 14.31 z where j as: Equation 14.32 is a weighted average of the measured values for response j and is calculated GPS-X Technical Reference
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GPS-X Technical Reference GPS-X Ver
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Table of Contents iii Table of Cont
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v Table of Contents Sedimentation M
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vii Table of Contents Real Time Clo
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ix Table of Contents Figure 6-6 - P
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xi Table of Contents Figure 10-8 -
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xiii Table of Contents Figure 15-6
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xv Table of Contents Table 6-2 - Mo
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17 Modelling Fundamentals Experimen
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19 Modelling Fundamentals The proce
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21 Modelling Fundamentals Overview
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23 Modelling Fundamentals Primary a
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25 Modelling Fundamentals Vesilin
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27 GPS-X Objects CHAPTER 2 GPS-X Ob
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29 GPS- X Objects The mass balance
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31 GPS- X Objects Combining Equatio
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33 GPS-X State Variable Libraries S
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35 GPS-X State Variable Libraries C
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37 GPS-X State Variable Libraries C
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39 GPS-X Composite Variable Librari
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41 GPS-X Composite Variable Calcula
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55 Composite Variable Calculations
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61 Influent Models CHAPTER 5 Influe
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63 Influent Models Figure 5-3 - Inf
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65 Influent Models The three influe
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67 Influent Models Total runoff (Q
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69 Influent Models These inputs are
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71 Influent Models CODFractions COD
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73 Influent Models TSSCOD Figure 5-
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75 Influent Models Figure 5-13 - CN
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77 Influent Models CN and CNIP Libr
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83 Influent Models INFLUENT OBJECTS
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85 Influent Models Figure 5-22 - In
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87 Influent Models Chemical Dosage
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89 Influent Models Figure 5-29 - Ac
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91 Influent Models Figure 5-33 - Se
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93 Influent Models Figure 5-36 - Se
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95 Suspended Growth Models where: V
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97 Suspended Growth Models The baro
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99 Suspended Growth Models Oxygen M
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101 Suspended Growth Models In the
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103 Suspended Growth Models The SOT
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105 Suspended Growth Models Standar
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107 Suspended Growth Models The reg
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109 Suspended Growth Models SUMMARY
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111 Suspended Growth Models Figure
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113 Suspended Growth Models General
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115 Suspended Growth Models Number
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117 Suspended Growth Models Elevati
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121 Suspended Growth Models AERATIO
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123 Suspended Growth Models The ent
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125 Suspended Growth Models The org
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127 Suspended Growth Models Process
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129 Suspended Growth Models It i
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131 Suspended Growth Models Nitroge
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133 Suspended Growth Models Include
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135 Suspended Growth Models
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137 Suspended Growth Models Althoug
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139 Suspended Growth Models Particu
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141 Suspended Growth Models 9. Grow
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143 Suspended Growth Models 21. Sto
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145 Suspended Growth Models 33. Gro
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147 Suspended Growth Models To mode
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149 Suspended Growth Models Algebra
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151 Suspended Growth Models The sto
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153 Suspended Growth Models Decay o
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155 Suspended Growth Models SUGGEST
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157 Suspended Growth Models Special
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159 Suspended Growth Models Table 6
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161 Suspended Growth Models The GPS
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163 Suspended Growth Models where:
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165 Suspended Growth Models The max
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167 Suspended Growth Models If yo
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169 Suspended Growth Models ANAEROB
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171 Suspended Growth Models Simple
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173 Suspended Growth Models With th
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175 Suspended Growth Models Manual
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177 Suspended Growth Models Equatio
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179 Suspended Growth Models The met
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181 Suspended Growth Models Pond Mo
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183 Suspended Growth Models Special
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185 Suspended Growth Models Se
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187 Suspended Growth Models aerated
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193 Suspended Growth Models Calcula
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195 Suspended Growth Models The vap
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197 Suspended Growth Models Oxygen
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199 Suspended Growth Models Estimat
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203 Suspended Growth Models Enterin
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209 Suspended Growth Models MODELLI
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211 Suspended Growth Models TEMPERA
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213 Suspended Growth Models Table 6
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215 Suspended Growth Models The PAC
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217 Suspended Growth Models CHAPTER
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219 Attached-Growth Models Each lay
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221 Attached-Growth Models Equation
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223 Attached-Growth Models The mode
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225 Attached-Growth Models The phys
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227 Attached-Growth Models Liquid F
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229 Attached-Growth Models This con
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231 Attached-Growth Models The next
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233 Attached-Growth Models The SBC
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235 Attached-Growth Models Based on
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237 Attached-Growth Models Operatio
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239 Attached-Growth Models Hydrauli
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241 Attached-Growth Models ADVANCED
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243 Attached-Growth Models Filtrati
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245 Attached-Growth Models Operatio
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247 Attached-Growth Models This for
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249 Attached-Growth Models HYBRID S
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251 Attached-Growth Models Figure 7
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253 Sedimentation and Flotation Mod
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267 Sand Filtration Models Floatati
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269 Sand Filtration Models The back
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271 Sand Filtration Models The mass
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273 Sand Filtration Models Figure 9
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275 Digestion Models Figure 10-1 -
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277 Digestion Models Seven yield co
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279 Digestion Models Based on the a
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281 Digestion Models Combining the
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283 Digestion Models The Operationa
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285 Digestion Models Kinetic and St
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287 Digestion Models The rest of th
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291 Digestion Models ADM1 State Var
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293 Digestion Models AEROBIC DIGEST
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297 Other Models CHAPTER 11 Other M
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299 Others Models High volume for p
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301 Others Models The sludge pretre
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303 Others Models With a large over
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305 Others Models Dosage Controller
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307 Others Models The set of above
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309 Others Models There are three p
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311 Others Models DISINFECTION UNIT
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313 Others Models DEWATERING Four m
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315 Others Models The optimal polym
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317 Others Models DISC AND BELT MIC
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319 Others Models Figure 11-12 - Pl
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321 Others Models Physical Setup Fo
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323 Others Models Exponential const
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325 Others Models BLACK BOX The Bla
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327 Tools Object Figure 11-19 - Spe
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329 Tools Object Figure 11-21 - Inp
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331 Tools Object Typical Model Outp
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333 Tools Object The variable speed
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335 Tools Object SLUDGE EFFLUENT BU
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337 Tools Object PH TOOL Take a sam
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339 Tools Object These concentratio
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341 Tools Object LOW-PASS FILTERING
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343 Tools Object Control Variable P
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345 Tools Object The individual mod
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Page 361 and 362: 361 Operating Cost Models CHAPTER 1
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Page 369 and 370: 369 Operating Cost Models CALIBRATI
Page 371 and 372: 371 Operating Cost Models Category
Page 373 and 374: 373 Optimizer If after the reflecti
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Page 381 and 382: 381 Optimizer SUMMARY OF THE OPTIMI
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Page 387 and 388: 387 Optimizer APPENDIX A: MAXIMUM L
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Page 405 and 406: 405 Optimizer To check for violatio
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Page 409 and 410: 409 Optimizer APPENDIX C: NOMENCLAT
Page 411 and 412: 411 Optimizer APPENDIX D: REFERENCE
Page 413 and 414: 413 Miscellaneous CHAPTER 15 Miscel
Page 415 and 416: 415 Miscellaneous As compared to th
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Page 419 and 420: 419 Miscellaneous RESIDUAL PLOTS Fo
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Page 423 and 424: 423 Miscellaneous REAL TIME CLOCK T
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Page 429 and 430: 429 Miscellaneous The smoothing fun
Page 431 and 432: 431 Miscellaneous The Gear's Stiff
Page 433 and 434: Appendix A - Petersen Matrices A-1
Page 437 and 438: ASM2d in CNPLIB Appendix A - Peters
Page 439 and 440: 15 Lysis of XBP 16 Lysis of XPP 17
Page 443 and 444: NewGeneral in CNPLIB Appendix A - P
Page 445 and 446: NewGeneral in CNPLIB Appendix A - P
Page 447 and 448: Component i Process Rates Appendix
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Appendix A - Petersen Matrices A-17
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2 Appendix A - Prefermenter Peterse
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D-2 Appendix D - References Brockwe
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D-4 Appendix D - References Hur, D.
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D-6 Appendix D - References Reilly,
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D-8 Appendix D - References Wuhrman
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