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by: Louie Hansen


Louie Hansen
GPA 3.87

Christopher Monsere

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Christopher Monsere
Class Notes
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This 22 page Class Notes was uploaded by Louie Hansen on Tuesday September 1, 2015. The Class Notes belongs to CE 507 at Portland State University taught by Christopher Monsere in Fall. Since its upload, it has received 28 views. For similar materials see /class/168210/ce-507-portland-state-university in Civil Engineering at Portland State University.

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Date Created: 09/01/15
Probit Kristie Gladhill CE507 Statistical Method Presentation 1222008 7 r Portland State 39 U N I V E R S l TV History Probit or probability unit term developed in 1934 by Charles Bliss Finney further refined very similar to logit before computing power readily available look up tables for this probitipj M EEIf1I2p I function available in statistics packages ex MATLAB function erfinv 7 a o O I 39 Portland State 0 U N l V E RSITV Defining Probit Probit or probability unit inverse cumulative distribution function CDF associated with the normal distribution a transformation to make a sigmoid relationship linear can calculate confidence intervals for doseresponse quantities 39 O gt O I 39 Portland State 0 gt U NlVERSITV Defining Probit quot Pry1x ltDxb 5 where D is the standard cumulative normal probability distribution and xb is called the probit score Since xb has a normal distribution interpreting probit coefficients requires thinking in the Z normal quantile metric The interpretation of a probit coefficient b is that a oneunit increase in the predictor leads to increasing the probit score by b standard deviations U from httpWWW gseis ucla educoursesed231 cnoz es3probiz 1 html 0 Portland State DNIVERSITV Defining Probit The probit function is the inverse cumulative distribution function CDF or quantile function associated with the standard normal distribution It has applications in exploratory statistical graphics and specialized regression modeling of binary response variables For the standard normal distribution often denoted NO1 the CDF is commonly denoted Clgtz Clgtz is a continuous monotone increasing sigmoid function whose domain is the real line and range is 01 As an example consider the familiar fact that the NO1 distribution places 95 of probability between 196 and 196 and islsymmetric around zero It follows pr 196 0025 1 bl1ilfi The probit function gives the 39inverse39 computation generating a value of an NO1 random variable associated with specified cumulative probability Formally the probit function is the inverse of Clgtz denoted D 1p I Continuing the examrprobit H025 1Qfi prohitl95 In gentpmbit133 I 39 quot l i v a probttipj VIE erf l3 ll from httpen wikipedia orgwikiProbiz 7 7 7 o O 39 F P Portland State 39 DNIVERSITV Defining Probit 171 P where Y39 is the probit transformed value 5 used to be added to avoid negative values in hand calculation p is the proportion p responderstotal number and inverse F p is the 100p quantile from the standard normal distribution from httpWWW statsdirecz comhelpregressionand correlationprobit mm 7 7 A 7 o O 39 F 39 Portland State 39 DNIVERSITV Use of PROBIT Probit is used typically with dosage data Bliss needed a way to treat data of of pest killed by a pesticide 39 F Portland State U N l V E l S l TV Resources and References httpenwikipediaorqwikiProbi39t httpwwqueis uclaeducoursesed231 cnotesBprobit1 html httgzllwwwstatsdi rectcomhelpreqression and correlationprobithtm all accessed 12208 x gt r r Portland State 39 U N I V E R S l TV Introduction to Nonlinear Regression Presented by Wei Feng CE 507 Transportation Research Method Portland State University 12022008 Why Choose a Nonslinear Regression With known theoretical shape but unknown parameters 170 180 Weight Kg 110 120 130 140 150 160 Weight lb With unknown E theoretical shape I 0 50 100 150 200 250 Days Differences between Linear and Nonalinear Regresson At least one parameter is related to the explanatory variables in a nonlinear fashion There is usually no closed form expression for the least squares parameter gt ChiZDoF R2 Correlation CODR2 Least squares estimates of the parameters are therefore derived from an iterative process using numerical methods Steps of Solving Nonalinear Regression 1 Linearization 2 Numerical iteration methods gt Minimize RSS Residual Sum of Square Linearization Examples Iny Ina blnX or y a bX yaXb lt yzaebx Znylnabx 0ry a bX Numerical Iteration Method Choose a model Estimate initial parameter values Constrain a parameter Handle replicate values if any Choose stopping rules Choose a model Known theoretical shape with unknown parameters gt Example BPR model V0 i1aiQiCi 8i VqiivOJ Ci H Unknown model gt Quadratic polynomial gt Cubic polynomial gt Power gt Exponential Estimate Initial Parameter Values BPR model 170 Hi1aiqiCi i VCIiI v0 Ci Constrain parameter gt VO60mph gt Ci 1200 vehhane Parameterinitialization gt ai O4 gt bi 015 Handle Replicate Values If variables are independent gt Treat them separately If variables are dependent gt Use average of the replicate values Stopping Rules Set maximal times of iteration gt Eg 10000 Set error difference gt Eg R2aik1 bik1 R2aik bik lt 005 Example Weight loss model y 30 31239t e Dependent variable y Independent variable t Parameters BO 31 6 Weightkg Result 170 180 160 400 350 8 396 o g E m dgom m l 5390 130 130 2IDO 21130 Days SSR B0 B1 9 6754349 9O 95 120 4018081 8272629 10130457 13871374 3924489 8139868 10265836 14185859 392447 8137375 10268417 14191052 Weight loss model y 81374 10268 2DavS14191 Softwares of Nonalinear Regression Excel Solver SPSS Origin Lab provide over 200 models in different academic fields and support selfdefined models MATLAB QUESTIONS


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