WORKSHOP ON THE ANALYSIS OF NEURAL DATA 2001 MARINE BIOLOGICAL LABORATORY WOODS HOLE, MASSACHUSETTS

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1 Emery. Brown, M.D., Ph.D. WORKSHOP O THE AALYSIS OF EURAL DATA 00 MARIE BIOLOGICAL LABORATORY WOODS HOLE, MASSACHUSETTS A REVIEW OF STATISTICS PART 3: REGRESSIO AALYSIS AD THE GEERALIZED LIEAR MODEL EMERY. BROW EUROSCIECE STATISTICS RESEARCH LABORATORY DEPARTMET OF AESTHESIA AD CRITICAL CARE MASSACHUSETTS GEERAL HOSPITAL DIVISIO OF HEALTH SCIECES AD TECHOLOGY HARVARD MEDICAL SCHOOL / MIT

2 Regresson Analyss A. Smple Regresson B. Model Assumptons C. Model Fttng D. Propertes of Parameter Estmates E. Model Goodness-of-Ft F-test R Analyss of Resduals F. The Geometry of Regresson (Method of Least-Squares) Page

3 A. Smple Regresson Assume we have a data consstng of pars of two varables and we denote them as ( x, y ), ( x, y ),, ( xn, y n). For example, x and y mght be measurements of heght and weght of a set of ndvduals from a well-defned cohort. Let s assume that there s a lnear relaton between x and y. We assume that the lnear relaton may be wrtten as y = α + β x Example. For example let us consder ths example taken from Draper and Smth (98), Appled Regresson Analyss Fgure. Relaton between Monthly Steam Producton and Mean Atmospherc Temperature. The varable y s the amount of steam produced per month n a plant and varable x s the mean atmospherc temperature. There s an obvous negatve relaton. Page 3

4 B. Model Assumptons We assume ) E[ y x] = α + β x ) The x s are fxed non-random covarates ) The y s are ndependent Gaussan random varables wth mean α + β x and varance σ C. Model Fttng Our objectve s to estmate the parameters α, β and σ. Because y s assumed to have a Gaussan dstrbuton condtonal on x, a logcal approach s to use maxmum lkelhood estmaton. For these data the jont probablty densty (lkelhood) s (,,, ) f y α β σ x = ( α β, σ ) f y + x = πσ exp ( y α βx ) σ The log lkelhood s logf ( y x, α, β, σ ) log( πσ ) = ( y α βx) σ Page 4

5 Dfferentatng wth respect to α and β yelds ( y x α β σ ) log f,,, α ( y x α β σ ) log f,,, β ( y α βx ) = ( y α β ) = Settng the dervatve equal to zero yelds the normal equatons x x α α + β x = y x + βx = xy Or n matrx form x y α β = x x xy x y α β = x x xy = Page 5

6 The soluton for β and α are We may wrte any estmate as xy x / x x y y βˆ = = x x / αˆ = y β x ( )( ) ( x x) ( ) yˆ = αˆ + βˆx = y βˆx + βˆx = y+ βˆ x x If we go back and dfferentate the log lkelhood wth respect to σ lkelhood estmate Remarks ( ˆ ) σˆ = y αˆ βx /, we obtan the maxmum. Choosng α and β by the maxmum lkelhood s equvalent to the method of least square n ths case. By the method of least squares, we mnmze the sum of the squared ntal devatons of the data from the regresson lne.. The estmate of ˆα shows that every regresson lne goes through the pont ( x, y. ) 3. The resduals are y yˆ and are the components n the data whch the model does not explan. We note that ˆ ( ) y = y+ β x x ( y y ) ( y y) β ( x x) ˆ = + = 0 Page 6

7 4. The Pythagorean Relaton ( ) ( ˆ ) y yˆ = y y y y ( y yˆ ) = {( y y) ( yˆ y) } ( y y) ( y y) ( y y)( yˆ y) = +.B. ( y y) β( x x) β ( y y)( x x) = ( x x) = β ( yˆ ) y, = and ( y yˆ) = ( y y) ( yˆ y). Or ( ) ( ) y ˆ ( ˆ ) y = y y + y y Total sum of squares (TSS) = Explaned sum of squares (ESS) + Resdual sum of squares (RSS) Page 7

8 5. Correlaton and Regresson ote that the correlaton coeffcent for x and y s r xy = ( x x)( y y) ( x x) ( y y) and recall that β Hence, β = ( x x)( y y) ( x x) ( y y) = ( x x ) r xy The regresson coeffcent s a scale verson of the correlaton coeffcent. D. Propertes of the Parameter Estmates The varances of the parameter estmates Var ( βˆ ) = σ ( x x) Page 8

9 Var ( αˆ ) = = σ x ( x x) If we estmate σ by σˆ then ρ confdence ntervals for the parameters based on the t- dstrbuton are t βˆ ± n, ρ / σˆ ( x x) x αˆ ± tn, ρ / ( x x ) σˆ We also nvert the above statstcs to test hypotheses about the regresson coeffcents by constructng a t-test. To construct a confdence nterval for a predcted y k value at a gven x value, x k, we note that the var ( y ) k σ = + ( x x) k and hence an approxmate ρ confdence nterval s ( x x) σ Example. (contnued) yˆ ( x x) k k ± tn, ρ / + ˆ σ ( x x) Page 9

10 Fgure. Ft and Confdence Intervals for a Smple Lnear Regresson Model E. Model Goodness-of-Ft A crucal (f not the most crucal) step n statstcal analyses of data s measurng goodness of ft. That s, how well does the model agree wth the data. We dscuss three graphcal and two statstcal measures of goodness-of-ft.. F-test Gven the null hypothess H0 : β = 0,.e. that there s no lnear relaton n x and y, we can test ths explctly usng an F-test defned as F /( p ) ( ) ESS RSS / p y yˆ ( y y) ( p ) p, p = = ( ) ( p ) Page 0

11 where ESS s the explaned sum of squares; RSS s the resdual sum. We reject ths null hypothess for large values of the F statstc. Ths suggests that f the amount of the varance n the data that the regresson explans s large relatve to the amount whch s unexplaned then we reject the null hypothess.. R-Squared Another measure of goodness-of-ft s the R. It s defned as R ESS TSS = = ( yˆ y) ( y y) The R measures the fracton of varance n the data explaned by the regresson equaton. The R-Squared and the F-statstc are related as ( ) ( ) ( ) ( ) F p / p R = + F p / p Clearly, as the F-statstc ncreases, so does the R. Example. Data Analyss Summary (Contnued) Parameter Estmate Standard Error t-statstc ˆα = ˆβ = TSS = 63.8 ESS = RSS = 7.3 F-statstc (, 3) = 57.5, R = Factod: The square of the t-statstc (n-p degrees of freedom) s the F-statstc (, n-p). 3. Graphcal Measures of Goodness-of-Ft. Plot the Raw Data Is the relaton lnear?. Plot the Ft vs. the ResdualsIs there a relaton between the explaned and the unexplaned structure n the data? 3. Plot the Resduals Lack of ft of the model Page

12 Fgure 3. Plot of Resduals versus x. Page

13 Fgure 4. Plot of the Resduals aganst the Predcted y values. F. The Geometry of Regresson Analyss (Method of Least-Squares) Regresson analyss has an ntutvely appealng geometrc nterpretaton. z 3 y ε z z ŷ x x Page 3

14 R = ŷ y F statstc ŷ ε I. Dervaton and Propertes of the Least Squares Estmates The procedure used to estmate the parameters n the lnear regresson models n ths study s least squares. In ths secton we frst derve the least squares estmates, and then by makng some dstrbutal assumptons about our model, we next derve the statstcal propertes of the estmates. We begn by assumng a model of the form Y = Xβ + ε () where Y s an n l vector of dependent observatons. X s an n p matrx of explanatory observatons,.e. the matrx contans p n l dmensonal vectors. β s an n l vector of parameters. ε s an n l vector of errors. In order to estmate our vector of parameters β usng least squares, we assume that X s of rank p or, n other words, that the p column vectors n X are lnearly ndependent. Therefore, we want to fnd the estmates of β, ˆβ, whch mnmzes ˆ ε ' ˆ ε = ( Y Xβˆ)'( Y Xβˆ) = YY ' βˆ' XY ' Y' Xβˆ + βˆ' X' Xβˆ (.) Because ˆ ' β X ' Y s a scalar t equals ts transpose Y ' Xβˆ. Consequently (.) smplfes to ˆ ε' ˆ ε = YY ' βˆ' XY ' + βˆ' Xβˆ To fnd the least squares estmates of β we dfferentate (.3) wth respect to ˆβ and set the dervatve equal to zero. (.3) Page 4

15 ( ˆ ε' ˆ ε) = X ' Y + ( X ' X) βˆ βˆ (.4) Solvng for ˆβ gves That βˆ = ( X ' X ) X ' Y ( βˆ' X ' Xβˆ) βˆ = ( X ' X) βˆ (.5) follows from the fact that βˆ' ( X ' X) βˆ s a quadratc form. Furthermore, we know that ( X ' X) has an nverse snce we ntally assumed X had rank p. Hence, ( X ' X) s postve defnte of rank p and has nverse ( X ' X). The second order condtons for the mnmzaton of ε' ε follows drectly snce ( X ' X) s postve defnte. In order to test hypotheses about our data we need to make some dstrbutonal assumptons about our model. The standard assumptons for the lnear regresson model are: I. ε ~ ( 0, Iσ ) common varance σ.. The ε s are ndependent, normally dstrbuted random errors wth mean 0 and II. The pn-dmensonal vectors comprsng X are taken as fxed constants. From these two assumptons we see that Y beng a lnear functon of ε, s also normally dstrbuted wth mean X β and varance Iσ. Usng these dstrbutonal propertes we now derve the dstrbutons of the least squares estmates. Proposton. The least squares estmate of varance σ ( X ' X). β, βˆ, s normally dstrbuted wth means β and Proof: To show that β s normally dstrbuted we have only to notce that β s a lnear functon of Y, snce lnear functons of ndependent normally dstrbuted random varables are also normally dstrbuted. To fnd the mean of the dstrbuton of β consder Page 5

16 { } ( ) ( ) ( ) ( ) ( ) E βˆ = E X ' X X ' Y = X ' X X ' Xβ + X ' X X ' E ε (.6) But from Assumpton I, we know that E( ε ) = 0, and snce ( X ' X) X ' X = I, t follows that E ( βˆ ) = β. To determne the varance of ˆβ, we consder { " } Var( βˆ ) = Var ( X ' X ) X ' Y = ( X ' X) X" Var( YY ') X( X ' X) But Var( YY ') = σ I, so we can rewrte (7) as βˆ = σ Var( ) ( X ' X ) ( X ' X )( X ' X ) I = ( X ' X) σ I Proposton. The least squares estmate of Y, Yˆ, where Yˆ = Xβˆ s ( Xβσ, X( X ' X) X ') Proof: the fact that Ŷ has a normal dstrbuton follows drectly from the fact that t s a lnear combnaton of Y. We fnd the mean of Ŷ by consderng { βˆ } { β ε } E( Yˆ ) = E X = E X( X ' X) X ' X + X( X ' X) X ' E( ) Because ( X ' X) ( X ' X) = I and E( ε) = 0, we get EY ( ˆ ) = Xβˆ. (.7) (.8). (.9). To fnd the varance of Ŷ we wrte Var( Yˆ ) = Var{ X ( X ' X ) X ' Y} = X( X ' X) X ' Var( YY ') X( X ' X) X ' (.0) Snce we know that Var( YY ') = σ I, substtutng ths nto (.0) and rearrangng terms gves Page 6

17 Var( Yˆ ) = X ( X ' X ) ( X ' X )( X ' X ) X ' σ I = X( X ' X) X ' σ I (.) Proposton.3 The least squares estmate of ε, ˆ ε, where ˆ ε = Y Yˆ s (0,( I H) σ ), where H = X( X ' X) X '. To complete the proof of ths Proposton, we need the fact that the matrx I H s symmetrc and dempotent. A matrx A s sad to be dempotent s A = A and A s symmetrc f A' = A. Proof: The fact that ˆε has a normal dstrbuton follows drectly snce t s a lnear functon of Y. In order to fnd the mean of ˆε to consder { ˆ } { } E( ˆ ε ) = E Y Y = E( I X( X ' X) X ' Y) (.) We recall that EY ( ) = Xβ. Therefore, (.) smplfes to E( ˆ ε) = X β X( X ' X" ) X ' Xβ = Xβ Xβ = 0 (.3) We fnd the varance of ˆε by frst rewrtng ˆε as Hence, But I H ˆ ε = Y Yˆ = ( I H) Y Var( ˆ ε ) = ( I H ) Var( YY ')( I H )' s symmetrc, and Var( YY ') = σ I, so from (.5) we get (.4) (.5) Var( ε) = ( I H)( I H ) σ I. (.6) We recall that I H s dempotent and fnally we have Var( ε) = ( I H ) σ We notce that all the estmates just derved are unbased n that the expected values of ther dstrbutons are the parameters they estmate. ow we can fnd an unbased estmate of σ. In the next Proposton we demonstrate that such an estmate s σ, where Page 7

18 Proposton.4 σˆ YY ' βˆ XY ' = ( n p) s an unbased estmate of σ. In order to complete the proof of ths Proposton we need the followng two results from the theory of matrx algebra, and the dstrbuton theory of quadratc forms: For two conformng matrces A and B trace ( AB ) = trace ( BA). If A s a symmetrc matrx and e s a vector of uncorrelated random varables wth common varance σ and mean 0, Proof: We consder frst EeAe ( ' ) = σ trace ( A ). E ( σˆ ) { ' βˆ ' X ' Y ' } E Y Y = ( n p) (.7) Substtutng for β ˆ ' gves E ( σˆ ) { ' ' ( ' ) } E Y Y Y X X X XY = (.8) ( n p) If we now substtute for Y ' and Y, (.8) becomes E( σˆ {( β ' ' + ε ')( β + ε) ( β ' ' + ε ') ( ' ) ( β + ε} E X X X X X X X X 0 = n p (.9) Expandng (.9), takng expectatons and smplfyng yelds { ε' ε ε' ( ' " ) ' ε} E I X X X X E( σ ) = ( n p) (.0) Page 8

19 In equaton (.0) we have the expectaton of two quadratc forms. Applyng the frst of two results stated ntally, we get E( σˆ ) = { σ trace( I) σ trace ( X( X ' X) X ')} (.) ( n p) Snce I s an nxn dentty matrx ts trace s n. Applyng the second of the two results, we stated, we fnd that trace ( X( X ' X) X ') = trace ( X ' X( X ' X) ). (.) Because ( X ' X) and ( X ' X) are pxp, trace ( X ' X( X ' X) ) Substtutng back nto (.) gves = p. σ n σ p ˆ E( σ ) = = σ n p (.3) Havng found the least squares estmates for our lnear regresson model and derved ther statstcal propertes, we now would lke to demonstrate an mportant optmalty property assocated wth our estmate ˆβ. Specfcally, we want to show that ˆβ s the best lnear unbased estmate of β. That s, n the class of lnear unbased estmates of β, βˆ has the smallest varance-covarance matrx. Ths well-known property of the least squares estmates s known as the Gauss-Markov Proposton. It s stated and proved below as Proposton.5. Proposton.5 In the class of lnear unbased estmates of β, the least squares estmate ˆβ the smallest varance. has Proof: Consder an alternatve lnear estmate of β, and call t β *, where β * = AY Let A= ( XX) X + C. We now want to determne C such that β * s unbased wth the smallest varances of all lnear estmates. Therefore, we consder E( β ) = AE( Y) = ( X ' X) ( X ' X) β + CXC' = β CXC ' (.4) Page 9

20 In order for β * to be unbased we requre that CX = 0. { } Var( β*) = E ( β * β)( β * β ') ow we want to fnd the varance of β * (.5) Substtutng for β * n (.5) gves { } ( ) { } Var( β ) = E ( X ' X ) X C ( X β ε ) β X ' X X C X β ε β ' (.6) Expandng terms, takng expectatons and smplfyng yelds Var( β ) = ( X ' X) X ' E( εε ') X( X ' X) + Cεε ' C' = σ ( X ' X) + CC' (.7) We now notce that Var( β*) Var( β) 0 (.8) The equalty n (.8) holds f and only f we choose CC ' = 0. Ths mples that the elements of C must all be zero. We have demonstrated that the least squares estmate of β, has the ˆβ smallest varance-covarance matrx among the class of all lnear unbased estmates. We note however that ˆβ s not the only estmate n ths class wth the smallest varance. Moreover, t s possble to construct based lnear estmates or unbased nonlnear estmates whose varance s smaller than ˆβ. By makng smlar arguments for Ŷ and ˆε, we can also prove that they beng lnear functons of ˆβ, they are the b.l.u.e. of Y and ε, respectvely. We now llustrate one fnal mathematcal property of our least squares estmates, whch s helpful for understandng the theory behnd ths method of estmaton. Proposton.6 ˆ ε '( Yˆ ) = 0 To prove ths Proposton, we need the same result for the matrx X( X ' X) X that we stated n the proof of Proposton.3 for the matrx I X( X ' X) X '. That s X( X ' X) X ' s also symmetrc and dempotent. Proof: Agan let X( X ' X) X ' = H and recall from (.4) that ˆ ( ) ε = I H Y. Hence, ( ˆ) ( ) ( ) ˆ ε ' Y = I H Y ' HY = Y ' I H ' HY (.9) Page 0

21 snce Y ˆ = HY. But both I H and H are symmetrc and dempotent, so that ˆ ε '( Y) = Y '( H H) Y = 0 (.30) What ths result says s that the vector of estmated errors s orthogonal to the vector of predcted values of the dependent varable. In other words, regresson decomposes Y nto two orthogonal components Ŷ and ˆε. The predcted values of Y, Ŷ, s the orthogonal projecton of Y on to the subspace determned by X. We llustrate ths geometrcally n Fgure A. for the case when n=3 and p=. We return to ths geometrc nterpretaton of regresson n ths next secton, where we dscuss the F-test. I. The F-Test Our prmary method for testng hypotheses about our regresson models s the F-test. In ths secton, we outlne ts dervaton and gve ts geometrc nterpretaton. Let s wrte β as β β = β where β s a k l column vector (.) β s a ( p k) l column vector If we wsh to test the hypothess that β = 0 we can do so by usng the followng steps to construct the F-test. ) Compute the Regresson Sum of Squares (SSR 0 ) under H 0 that β = 0. ) Subtract (SSR 0 ) from the Regresson Sum of Squares (SSR A ) obtaned under the alternatve hypothess H A β 0. 3) Dvde ths dfference by k, the number of parameters n β. 4) Dvde ths new statstc by the Mean-Squared Error under H A (MSE A ). If the normalty assumptons regardng ε stated n Secton I hold, under H 0 the rato ( ) SSR SSR k A MSE A 0 / (.) has an F-dstrbuton wth k and n-p degrees of freedom. We reject H 0 at the α level of sgnfcance f our observed F-statstc exceeds the ( α ) quantle of the F-dstrbuton wth k, n- Page

22 p degrees of freedom. Although we do not prove here that ths statstc has an F-dstrbuton, we show geometrcally why ths rato s reasonable for the test of ths hypothess. For the geometrc argument, we agan consder the case where n = 3 p =, shown n Fgure A.. Here the null hypothess s that the coeffcent of X s zero, and therefore K =. From Fgure A. we see that ####$ SSR = OA ny A #####$ SSR = OB ny 0 (.3) MSE A #####$ = AC To test the hypothess that β = 0, we fnd that the approprate F statstc s #####$ #####$ F = AB AC, (.4) ###$ We notce that AB s the dfference between the projectons of Y under H A and Y under H 0. Ths geometrc descrpton of the F-test shows that H 0 s rejected f Y s sgnfcantly closer to the subspace spanned by X and X than the one spanned by X alone. G. Generalzed Lnear Model (GLM) Lnear Model Y = Xβ + ε ε (0, σ I) y = µ + ε where µ s the mean of y n X n p β p Components of the Generalzed Lnear Model. Random Component for y s chosen from the exponental famly. Systematc Component are covarates x, x,, xp produce a lnear predctor η gven by p η = x β. j= j j 3. The lnk between the random and systematc components η = g( µ ) where g s monotonc dfferentable functon. Page

23 Revst the Exponental Famly of Dstrbuton Gven y a data vector from a probablty dstrbuton belongng to the exponental famly. We may express the probablty densty or mass functon of y as { } f ( y, θ, φ) = exp yθ b( θ))/ a( φ) + c( y, φ). y a(), b () and c () functons to be specfed f φ s known then we have an exponental famly model wth a canoncal parameter. Example : Gaussan Model { } fy ( y, θφ, ) = exp ( y µ ) /σ πσ = exp ( yµ µ / ) / σ ( y / σ + log( πσ )), where θ µ, φ σ, b( θ) θ /, c( y, φ) { y / σ log(πσ } = = = = +. Example 3: Bnomal Model n Pr( Y = y) = p ( p) ( n y) y y n y p n = exp ylog + nlog( p) + log p ( n y) y n = log( /( )), =, ( ) = log( ), (, ) = log ( n y ) y where θ p p φ b θ n p c y φ Example 4: Posson Model Pr( Y = y) = λ exp( λ) / y y ( y λ λ y ) = exp log log where θ = log λ, φ =, b( θ) = λ, c( y, φ) = log y Mean and Varance of y (We wll skp ths stuff n the lecture; we can revst t n a tutoral). Let '( θ, φ, y) = log f ( y, θ, φ). Two standard propertes of lkelhood functons y Page 3

24 The expected value of the score functon s zero ' E = 0 θ. () ' ' E + E = 0 θ θ. () The expected Fsher Informaton s the negatve of the varance of the score functon '( θ, y) = { y( θ) b( θ) }/ a( φ) + c( y, φ), (3) whence ' = { y b ( θ )}/ a( φ) θ (4) θ ' = b ( θ )/ a( φ). (5) Eqs. () and (4) mply ' 0 = E = { µ b ( θ )}/ a( φ) θ or Ey [ ] = µ = b ( θ ). Usng Eqs. 4 and 5 n Eq., we obtan b ( θ ) Var( y) 0 = +, a φ ( ) a ( φ) Hence Var( y) = b ( θ ) a( φ). The varance depends on the mean and a functon, whch depends on the dsperson parameter; b ( θ ) s the varance functon. Page 4

25 Model Summary Table Gaussan Posson Bnomal Gamma Inverse Gaussan Canoncal µ log λ p log lnk: g ( µ ) p λ λ MAI POIT. WE MAY WRITE EXPLICITLY THE LIK OF THE SYSTEMATIC AD RADOM µ = β. I THIS WAY, WE EXTED THE SIMPLE REGRESSIO MODEL WITH STRUCTURE AS g( ) X GAUSSIA ERRORS TO REGRESSIO MODELS FOR THE ETIRE EXPOETIAL FAMILY OF PROBABILITY DISTRIBUTIOS. Devance (Goodness of Ft Crteron) Let '( µ ; y) = log f( y; θ) and defne the devance as D*( y; µ ) = '( y; y) '( µ ; y) ote that the second expresson s the log lkelhood of the data. Hence mnmzng D*( y ; µ ) s equvalent to maxmzng the log lkelhood. otce that n the Gaussan case, the devance s D*( y; µ ˆ) = ( y µ ˆ ) / σ whch s just the resdual sum of squares (RSS). Analyss of Devance. Instead of analyss of varance we use analyss of devance, whch s addtve for nested sets of models ft by maxmum lkelhood. D*( y ; µ ) s analyzed as an approxmately ch-squared quantty. Devance Resduals. The devance resduals are defned as D*( y; µ ˆ) d = where we defne r d, = sgn( d) d. Plots of these quanttes can be analyzed n the same way the regresson resduals are analyzed. For generalzed lnear models n whch there s a natural tme dependence n the observatons we wll show that a global measure of model goodness of ft may be constructed usng the tmerescalng Proposton (Berman, 983; Ogata, 988; Brown et al., 00). Page 5

26 Remarks. GLM allows generalzaton (extenson) of lnear models to non-gaussan settng.. A unfed computatonal approach. 3. Can use non-canoncal lnk and varance functons. 4. Wdely dstrbuton Splus and non Matlab. 5. ow there s Bayesan GLM. 6. Assess goodness-of-ft wth an analyss of devance. euroscence Data Analyss Examples Usng GLM. Random Threshold Model (Brllnger (988, 99)) Smplfed verson J k x µ jxt j kyt k pt α β x j= k= pt log = + + xt = 0 spke from neuron otherwse xat t yt = 0 spke from neuron otherwse y at t. pt x probablty of a spke at t from neuron x.. Prospectve and Retrospectve Encodng by Hppocampal Place Cells (Frank, Brown and Wlson, 000) A rat runnng on a W-maze, multple sngle unt actvty s recorded from CA regon of the hppocampus and the entorhnal cortex. t ndexes tral P t probablty of gong rght on an outbound journey from the center arm of the W-maze. pt log = µ + α jλj( t) pt J, where λ j () t s the frng rate of the neuron at locaton j pror to the choce pont on pass t. 3. Cockroach Cercal System Response to Wnd Stmul (Davdowtz and Rmberg, 000). n = number of spkes at tme t (dscrete tme bns of 0msec) t j= Page 6

27 vx () t x drecton wnd velocty vy () t y drecton wnd velocty a t x drecton wnd acceleraton a t y drecton wnd acceleraton x () y () Posson lnk functon log λ ( t) = µ + α v ( t) + α v ( t) + β a ( t) + β a ( t), x y x y λ () t spke rate at tme t. References Draper R, Smth H. (98). Appled Regresson Analyss, nd ed., Wley: ew York. McCullagh, P. elder JA (989). Generalzed Lnear Models, nd ed, Chapman and Hall: London. Page 7