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Monte Carlo Analysis Correlation Between Gaussian Input Variables Like in the univariate case, there are standard families of probability distributions for random vectors. In particular, the multivariate normal distribution with mean vector μ = (μ 1 , …, μ k ) T and variance-covariance matrix C has the density: This definition implies that each of the marginal distribution is normal and that the complete joint distribution is determined once the marginal means and the variance-covariance matrix are specified. From its definition, we see also that the density of multinormal distribution is constant on ellipsoids of the form: Figure 11-8 shows the contour ellipses of a two-dimensional normal distribution. These contour ellipses are the iso-distance curves from the mean of this normal distribution corresponding to the metric C −1 . Tip As emphasized, the relationships between the input variables are described with the concept of linear correlation. In fact “linearity” is strongly related to “gaussianity.” The two terms could be consider as synonyms in this context. It is possible, however, to describe general multivariate distributions and to represent dependence structure among the input variables. Understanding these relationships can be done with the notion of a “copula” function. For an introduction, see for example Bouye, E., Durrleman, V., Nikeghbali, A., Riboulet, G., Roncalli, T., Copulas for Finance: A Reading Guide and Some Applications, 2000, Working Paper, Groupe de Recherche Operationnelle, Credit Lyonnais. Specifying Linear Correlation with .CORREL You can define a correlation coefficient between Gaussian parameters using the .CORREL command. Refer to the .CORREL command in the Eldo Reference Manual. The following netlist is a toy example which shows the effect of linear correlation between input variables. We can indirectly visualize the joint distribution of the Gaussian input P1 and P2. 440 Eldo® User's Manual, 15.3
Monte Carlo Analysis Correlation Between Gaussian Input Variables R1 N1 0 P1 V1 N1 0 1 R2 N2 0 P2 V2 N2 0 1 .PARAM P1=10 LOT/GAUSS=10% .PARAM P2=10 LOT/GAUSS=5% .CORREL PARAM={P1,P2} CC=-0.5 .DC .MC 500 ALL .EXTRACT DC LABEL=IR1 1/I(R1) .EXTRACT DC LABEL=IR2 1/I(R2) After extraction of the simulated values one can plot the resulting scatter plot. Figure 11-9 also shows the marginal densities of the joint distribution of (IR1,IR2). Figure 11-9. Scatter Plot and Marginal Distributions Eldo® User's Manual, 15.3 441
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Table of Contents Chapter 1 Introdu
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Table of Contents Preparing and Ins
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Table of Contents Chapter 7 Running
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Table of Contents Eldo Reduction .
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Table of Contents Chapter 12 Statis
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Table of Contents Simulating a Bloc
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Table of Contents atanh . . . . . .
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Table of Contents xmin . . . . . .
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Table of Contents COS. . . . . . .
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Table of Contents One Step Relaxati
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List of Figures Figure 1-1. Cascade
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List of Figures Figure 14-1. Mixed-
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List of Tables Table 1-1. Documenta
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Chapter 1 Introduction to Eldo The
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Introduction to Eldo Getting Starte
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Introduction to Eldo Getting Starte
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Eldo Input and Output Files Introdu
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Introduction to Eldo Eldo Documenta
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Introduction to Eldo Documentation
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Chapter 2 Running Eldo This chapter
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Running Eldo Multi-Threading Eldo S
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Running Eldo Multi-Threading Eldo S
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Running Eldo Simulation Connection
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Running Eldo Simulation Connection
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Running Eldo Eldo Initialization Fi
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Running Eldo Location Maps Location
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Running Eldo Performance Bottleneck
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Running Eldo Statistics File Statis
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Running Eldo Additional Tools and U
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Chapter 3 Eldo Premier This chapter
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Eldo Premier Overview of Eldo Premi
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Eldo Premier Automatic Activation o
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Eldo Premier Handling Matrix-Like C
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Eldo Premier Eldo Premier Output El
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Eldo Premier Forcing Elaboration Ti
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Netlist reduction is activated with
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Eldo Premier Supported Commands Tab
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Eldo Premier Supported Commands Tab
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Table 3-3. Eldo Premier—Supported
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Eldo Premier Supported Models Enabl
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o Eldo Premier Other Supported Func
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Eldo Premier Miscellaneous Limitati
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Chapter 4 Eldo Netlist Setup This c
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Eldo Netlist Setup Netlist Structur
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Subcircuit Definitions and Calls El
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Eldo Netlist Setup Simulator Comman
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Eldo Netlist Setup General Aspects
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Eldo Netlist Setup Case-Sensitivity
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Eldo Netlist Setup Numerical Values
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Eldo Netlist Setup Functions, Opera
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Function PWL_LIN(xvalue, interp, x1
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Notes Eldo Netlist Setup Arithmetic
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Eldo Netlist Setup Operators Operat
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Eldo Netlist Setup Operators Bitwis
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Eldo Netlist Setup Expressions Spec
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Eldo Netlist Setup Directives Direc
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Eldo Netlist Setup Directives to Po
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Eldo Netlist Setup Working With Sub
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Eldo Netlist Setup Device Libraries
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Eldo Netlist Setup Encrypting Libra
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Chapter 5 Simulator Compatibility E
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Simulator Compatibility Hybrid Comp
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Simulator Compatibility Devices in
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Simulator Compatibility Devices in
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Simulator Compatibility Commands in
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Simulator Compatibility Options in
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Netlist in Compat Mode Some aspects
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Simulator Compatibility Arithmetic
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Simulator Compatibility Spectre Com
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Eldo can be launched on the Spectre
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Simulator Compatibility Working wit
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Error Code Related Topics Spectre C
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Simulator Compatibility Spectre to
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spect2el Command Reference Simulato
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Simulator Compatibility spect2el Co
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Chapter 6 Setting Up An Analysis Th
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DC and Operating Point Analyses The
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Setting Up An Analysis Performing a
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Setting Up An Analysis Operating Po
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Setting Up An Analysis DC Convergen
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Setting Up An Analysis Saving and R
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Setting Up An Analysis Transient An
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.SAVE END in Combination with .REST
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AC Analysis Results Setting Up An A
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Loop Stability Analysis Setting Up
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Setting Up An Analysis Pole Zero An
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Setting Up An Analysis Pole Zero An
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Hand Selection Setting Up An Analys
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Setting Up An Analysis Transfer Fun
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Setting Up An Analysis Noise Analys
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Setting Up An Analysis Transient No
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Transient Noise Analysis Principles
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Figure 6-10. PSD of a Flicker Noise
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o Setting Up An Analysis Transient
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Setting Up An Analysis Statistical
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AC Sensitivity Analysis Results Set
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Design of Experiments Analysis Resu
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Setting Up An Analysis Aging and Re
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High Impedance Node Check Setting U
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Results Setting Up An Analysis View
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Setting Up An Analysis RF Analyses
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Chapter 7 Running Parametric Analys
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Running Parametric Analyses Tempera
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Running Parametric Analyses .DATA P
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Running Parametric Analyses Removin
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Running Parametric Analyses .ALTER
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Running Parametric Analyses Using O
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Chapter 8 Specifying Simulation Out
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Specifying Simulation Output Output
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Specifying Simulation Output Types
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Examples of Wildcard Usage in Subci
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See Also • .PLOT command in the E
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Compound Waveforms Specifying Simul
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Specifying Simulation Output Tabula
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Specifying Simulation Output Dynami
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Specifying Simulation Output Extrac
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Specifying Simulation Output Plot a
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3. Right-click on the sweep paramet
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Specifying Simulation Output Exampl
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Specifying Simulation Output Exampl
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Safe Operating Area (SOA) Checks Sp
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Specifying Simulation Output Plotti
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See Also • .CHECKSOA in the Eldo
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Specifying Simulation Output Safe O
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Specifying Simulation Output High I
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Specifying Simulation Output High I
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• .OPTION PRINT_OPTION=0 (in the
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Specifying Simulation Output Increm
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Chapter 9 Analyzing Simulation Resu
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File Extension Table 9-1. Eldo Stan
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File Extension _csdf.trX _csdf.acX
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Interpreting the Output Log (.chi)
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Analyzing Simulation Results Simula
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Analyzing Simulation Results Simula
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Analyzing Simulation Results Error
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Analyzing Simulation Results EZwave
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Related Topics EZwave Joint Wavefor
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Analyzing Simulation Results Marchi
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Analyzing Simulation Results Diagno
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Analyzing Simulation Results Non-Co
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Nodes/Devices Impacting Time-Step A
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Analyzing Simulation Results Perfor
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Analyzing Simulation Results Using
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Analyzing Simulation Results Statis
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Monte Carlo Analysis Importance Sam
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Monte Carlo Analysis Further Exampl
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Monte Carlo Analysis Parameter Nami
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Five independent random variables w
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Monte Carlo Analysis Problems in St
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Figure 11-23. Four Random Samples f
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Monte Carlo Analysis Obsolete Featu
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Chapter 12 Statistical Experimental
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Statistical Experimental Design and
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Statistical Modeling for Discrete C
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Chapter 13 Optimization The purpose
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Optimization Optimization in Eldo O
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Table 13-1. Eldo Optimization Metho
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Figure 13-2. Discretization of Desi
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Optimization Specifying Tracking wi
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Scalar Design Objectives Optimizati
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Optimization Specifying Design Obje
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Optimization Eldo Optimizer/Search
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Conducting an Eldo Optimization Opt
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Optimization of Sweep Simulations O
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o o Goal Value — Goal value of th
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Chapter 14 Working with S, Y, Z Par
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Working with S, Y, Z Parameters S,
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Working with S, Y, Z Parameters Out
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Working with S, Y, Z Parameters Tec
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• Simulation of passive networks
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Working with S, Y, Z Parameters Ins
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VAR_LIST_BEGIN / VAR_LIST_END SEG_L
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Electrothermal Simulation Reporting
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Chapter 17 IBIS Models Support in E
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Chapter 18 Eldo Control Language Th
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Eldo Control Language Examples Eldo
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Example Dir 22-plot_mos_ operating_
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Temporary Files Eldo Control Langua
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Debug Mode Eldo Control Language Ad
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Eldo Control Language Advanced Simu
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Eldo Control Language Advanced Simu
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Eldo Control Language Examples Usin
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Eldo Control Language .EXTRACT Alte
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Eldo Control Language .PRINTFILE Al
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Eldo Control Language .PRINTFILE Al
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Eldo Control Language .OPTIMIZE Alt
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Eldo Control Language .MPRUN with M
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Chapter 19 Post-Processing Library
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Post-Processing Library Post-Proces
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Post-Processing Library General Usa
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Post-Processing Library Macro-Like
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Post-Processing Library Working wit
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Post-Processing Library evalExpr Re
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Post-Processing Library defineVec N
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Post-Processing Library EZwave Wave
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Post-Processing Library ABS ABS Eld
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Post-Processing Library ACOSH ACOSH
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Post-Processing Library ACOTH ACOTH
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Post-Processing Library ASINH ASINH
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Post-Processing Library ATANH ATANH
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Post-Processing Library COMPRESS CO
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Post-Processing Library COSH COSH E
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Post-Processing Library COTH COTH E
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Post-Processing Library DEG DEG Eld
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Post-Processing Library DERIVE DERI
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Post-Processing Library EXP EXP Eld
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Post-Processing Library GMARGIN GMA
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Post-Processing Library IMAG IMAG E
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Post-Processing Library INTEGRAL IN
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Post-Processing Library INVDB INVDB
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Post-Processing Library LOG10 LOG10
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Post-Processing Library MAX MAX Eld
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Post-Processing Library PHASE PHASE
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Post-Processing Library RAD RAD Eld
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Post-Processing Library RELATION RE
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Post-Processing Library RMS RMS Eld
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Post-Processing Library SETTLINGTIM
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Post-Processing Library SINH SINH E
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Post-Processing Library SQRT SQRT E
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Post-Processing Library TAN TAN Eld
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Post-Processing Library TPD TPD Eld
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Post-Processing Library TPDDU TPDDU
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Post-Processing Library TPDUU TPDUU
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Post-Processing Library VMIN VMIN E
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Post-Processing Library XCOMPRESS X
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Post-Processing Library XWAVE XWAVE
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Post-Processing Library IIPX IIPX E
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Post-Processing Library Built-In DS
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Post-Processing Library CONV CONV E
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Post-Processing Library CORRELO •
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Post-Processing Library CT • fmin
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Post-Processing Library FFT • nor
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Post-Processing Library HDIST HDIST
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Post-Processing Library PERIODO PER
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Post-Processing Library PSD PSD Eld
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Post-Processing Library SAMPLER 10
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Post-Processing Library Accessing W
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Post-Processing Library DISP_FILE D
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Post-Processing Library Additional
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Post-Processing Library GETSIZE GET
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Post-Processing Library SETPOINT SE
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Chapter 20 Speed and Accuracy Eldo
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Speed and Accuracy Speed and Accura
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Speed and Accuracy Integration Meth
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Speed and Accuracy Time Step Contro
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Speed and Accuracy Time Step Contro
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Speed and Accuracy Global Tuning of
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Speed and Accuracy Global Tuning of
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Speed and Accuracy Simulation of La
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Speed and Accuracy Tips for Improvi
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Speed and Accuracy IEM Overview Whe
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Speed and Accuracy IEM Tolerance Pa
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IEM Alone (.OPTION IEM) Speed and A
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Integration Method Speed and Accura
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Chapter 21 Examples The most produc
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Examples Table 21-1. Eldo Examples
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Netlist Explanation Examples Exampl
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Examples Example 2—Transient and
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Examples Example 3—AC and Noise A
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Examples Example 3—AC and Noise A
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Figure 21-8. Low Pass Filter Exampl
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Examples Example 5—Transient Anal
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Simulation Results Examples Example
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Figure 21-15. Non-inverting Amplifi
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Examples Example 8—DC Sensitivity
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Examples Example 9—Transient and
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Examples Example 11—Loop Stabilit
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Examples Example 12—Overshoot Ext
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Summary of Eldo commands used in th
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Examples Example 16—Transient Ana
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Examples Example 17—Monte Carlo S
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Examples Example 17—Monte Carlo S
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Examples Example 18—Numerical Int
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Examples Example 19—DC Mismatch C
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Examples Example 21—Post-Layout S
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Examples Example 22—Extract Gain
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Examples Example 22—Extract Gain
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Examples Example 23—Parametric Se
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Examples Example 23—Parametric Se
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Examples Example 24—Cell Characte
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Examples Example 25—Setting up an
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Examples Example 26—Spectre Compa
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Examples Example 27—Monte Carlo A
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Examples Tutorial—Using Power Ana
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Examples Tutorial—High Impedance
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Examples Transient Noise Analysis E
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Appendix A Error and Warning Messag
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Error Number Description Table A-1.
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Error and Warning Messages Global E
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Error and Warning Messages Errors R
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Error Number Table A-4. Errors Rela
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Error and Warning Messages Miscella
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Error and Warning Messages Warning
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Error and Warning Messages Global W
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Error and Warning Messages Warnings
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Warning Number Table A-11. Warnings
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Warning Number Table A-14. Warnings
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Warnings Related to Subcircuits The
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Warning Number Table A-16. Miscella
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Warning Number Warning 1021092 Warn
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Appendix B Troubleshooting Most use
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Troubleshooting File Troubleshootin
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Appendix C Eldo Interactive Mode El
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Eldo Interactive Mode Reading Infor
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Command [.]LSMODEV LSMODPAR [.]LS
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Eldo Interactive Mode Change Featur
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Eldo Interactive Mode Change Featur
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STOP IF Eldo Interactive Mode Cont
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Appendix D Eldo Conversion Utilitie
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Eldo Conversion Utilities Utility t
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Appendix E STMicroelectronics Model
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STMicroelectronics Models What Does
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Css Css Csb Csb Csb+CJSB Cbd Cdbd C
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Index — Symbols — ! comment del
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PWL_LIN(), 111 PWR(VAL1, VAL2), 110
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32-bit and 64-bit modes, 42 An Intr
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LIMIT function, 110 Limiting Output
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DSPF backannotation, 392 Examples,
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Noise Model, 1373 Syntax Errors, 36
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Third-Party Information This sectio
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3. The name of the author may not b
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USE, DATA, OR PROFITS; OR BUSINESS
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End-User License Agreement The late
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7.1. Mentor Graphics warrants that
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