Fitting models to data.

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Juniper Greer
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1 Fttg models to data. Prevous lectures dscussed model geerato. Start wth physcal pcture or dagram of what s happeg Make lst of assumptos (e.g., cell drug uptake s by dffuso; covecto ca be eglected) Wrte equatos to descrbe process. Ths leads to ext step: fdg parameter values for the model. How do you fd the parameter values from some data? How do you show that your model represets a mprovemet o someoe else s? Part I. Fttg models to data. Geeral dea: Choose parameters so that the model predcto s as close as possble to the data. No uque way to measure close. Several dfferet methods, some more popular tha others. Large body of theory behd each method, but ultmately t depeds o what assumptos are made about what costtutes a best model. Smple example: ft a le to some data. Method of least squares regresso You re testg a ew blood pressure treatmet. Ne patets agreed to partcpate your clcal tral. You measured ther blood pressure before (x) ad after (y) the treatmet. Fd the equato of a le that descrbes the data. The method of least squares assumes the best ft s the oe that mmzes the sum of the squared devatos betwee the predcted values o the le ad the measured values. (pass out hadout- page from Harvey Motulsky Itutve Bostatstcs book) The mathematcal procedure of least squares regresso Fttg a straght le: regresso of y o x gve data pots (x,y ), where y s depedet o x smplest approach s to ft a straght le y = a + bx

2 ftted values of y: y' = a + bx resduals: e = y - y' = y - a - bx 5 y 4 choose a ad b to make the resduals as small as possble they ca't all be zero: make sum of squares as small as possble 3 e (x,y ).e., mmze E = e = = [y a bx ] = x wth respect to a ad b the mmum must be a statoary pot = [y a bx ] = 0 so [y a bx ] = y a bx = 0 a = = E = [(y = y) b(x x) ], = [(y y) b(x x)](x x) = 0 b = (y y)(x x) = b = ad a = y bx (x x) =

3 Questo: why s best ft of y as fucto of x dfferet form best ft for x as fucto of y? 5 y 4 3 regresso of y o x regresso of x o y x Reaso: (graphcal) oe case you re mmzg the vertcal dstace betwee the pots ad the le. I the other case, you re mmzg the horzotal dstace. A mystery: why mmze the sum of squares? Easy questo: why ot mmze the sum of the devatos themselves (ot squared). Reaso: postve ad egatve errors would cacel each other out. But the f the squares are just to esure that all the terms are postve, why ot use the sum of absolute values? Reaso: least squares regresso comes from a method of parameter estmato called maxmum lkelhood. Cosder the data to be the real value plus a error. Assume the error s dstrbuted wth a ormal dstrbuto of mea zero ad varace sgma. Assume t s the same for all data pots. (Not requred. If error vares, t becomes weghted least squares.) Suppose the true model s y true (x). The the probablty of measurg y at value x s the probablty that your error was epslo_ = y - y true (x ). That s, t s the probablty f ( ε ) = σ πσ ε exp Ths s just the probablty dstrbuto for oe of your errors. What s the probablty of all the errors the data set, take together? They re assumed depedet. So t s:

4 f ( ε, ε,..., ε ) ε exp σ ε exp σ (...) ε exp σ = πσ πσ πσ Why s sgma the same all these terms? So ths s the probablty of measurg the data set y where =,,N (Note that I have gored the elephat the room: errors x. Do t worry, that ca be accouted for too. But we wo t do t here.) Now here s the key pot of maxmum lkelhood: Maxmum lkelhood turs the terpretato of ths equato aroud. Istead of seeg ths as the probablty of measurg the data set y,,yn, t s see as the probablty (ow uormalzed) of the meas ad varace beg y(x ) ad sgma, gve the data set. I other words, stead of tellg you the probablty of the data gve the parameter values, t tells you the probablty of the parameter values gve the data set. You ca t ormalze ths probablty. What you ca do s select the parameter values that maxmze t. Hece the ame maxmum lkelhood. It s easy to show that maxmum lkelhood reduces to least squares for lear models. That s, ths approach leads to tellg you that you should mmze the sum of the squared devatos stead of the sum of the absolute values. But maxmum lkelhood s a more geeral method. Beyod the le: fttg other lear models suppose x ad y are clearly correlated but the relatoshp s ot a straght le try to ft usg a more geeral relatoshp example y = a + bx + cx or y = a f (x) + a f (x) + a 3 f 3 (x) + + a m f m (x) f may be chose for coveece or from pror kowledge ('a model') of the system called a lear model because the depedece o the ukow parameters s lear the least squares approach ca be used defe E = [y + = (af(x ) +... a mfm(x))] ad set =... = = 0 a a m ths gves a lear system of equatos that ca be solved for a,, a m bult to may software packages

5 Retur to example of treatmet teded to decrease blood pressure. Whch of the followg two graphs should we ft a le to: - Fal blood pressure vs. tal (both are depedet expermetal measuremets) - Chage blood pressure vs. tal (chage s calculated by subtractg the two expermetal measured values) (Pass out hadout showg the two graphs.) Moral: careful about adjustg data before fttg t. Fttg olear models ofte the depedece o ukow parameters s olear examples y = ax b, P = P 0 e kt, VmaxC V =, C = Ae -at + Be -bt K + C m The basc prcple of turg t to a fucto mmzato problem stays the same: a geeral olear problem ca be wrtte y = f(x, a, a,, a m ) as before, defe E = [y y ] where y = f (x,a,a,...,a m) = Key dffereces that make olear optmzato/mmzato dfferet from lear: olear fucto may have may local mma choce of tal guess s mportat mpossble to be sure global mmum s foud dffculty creases sgfcatly as umber of parameters creases HANDOUT showg fuctos of varables vsually Idetfable parameter: deftve, arrow bowl No-detfable parameter: log valley How do you choose a tal guess? By kowg order of magtude, pror formato, etc. Example: Say parameter represets average blood pressure of patets populato who were o kdey dalyss. Take the average blood pressure of ormal patets as tal guess. Or, perhaps fd value lterature measured t aother patet populato. Most mportatly, you pck a value that s physcally reasoable. May umercal methods developed. I practce, usually ot ecessary to wrte your ow code. Packages are used. Mathematca: NMImze, FdMmum Matlab has routes as well.

6 Part II. How to show your model s better tha other models. IMPORTANT POINT: No statstcal tests ca compesate for bad caddate models. No substtute for physcal tuto. Two dfferet cases of comparg models: Nested. New model adds o terms or factors volvg addtoal parameters to a exstg model No-ested. Caot make the two models the same by settg some parameters to specal values (such as zero or oe). NESTED MODELS Curret model descrbes cell uptake of a drug. d dt C = k( C e C ) Should you add saturato? d dt C C k + s C e = ece F-test s method to decde f addto of saturato factor/parameter s warrated from the data. Addg extra parameters wll always gve a better ft to the data, but may ot be worth t. Model has p parameters Model has p > p parameters There are data pots. The F statstc s gve by where RSS m s the resdual sum of squares of model m. RSS = E = = [y y ]

7 Compare the computed F value wth tables of the F dstrbuto for (p p, p) degrees of freedom, ad the desred level of certaty (usually chose as 95% cofdece that s, 0.05). If calculated F s greater tha value gve table for F dstrbuto, the the model wth more parameters s better. Note that f the data set used s lmted, t may ot be possble to dscover the true model. For example, f the data cover oly a small rage of cocetrato, they may ot show saturato, eve though saturato s a real effect. So, drawg coclusos about whch model s better, remember that the data have lmtatos! NON-NESTED MODELS F-test caot be used. Methods here le at the tersecto of formato theory ad probablty theory. Oe example s the Akake Iformato Crtero. Why coecto to formato theory? Model fttg ca be regarded as the same problem as data compresso. N data pots (bts of formato) are reduced to p parameters (bts of formato). Example of data compresso: Rppg compact dsk to mp3 (lossy). If you rp to wav, t s lossless. Corrected Akake (AIC) for smaller data sets: ( p + )( p + ) AIC c = N l( RMS ) ( p + ) N p N = umber of data pots p = umber of parameters RMS = E / = [y y ] = Why s there P+ ths formula? Because you had to estmate the varace of the error too! Hgher AIC meas better model. AIC caot tell you how much better oe model s tha aother. It just tells you how models rak. Notce how models are rewarded for gvg less devato from the data, but pealzed for havg more parameters. Ths s as t should be.

ESS Line Fitting

ESS Line Fitting ESS 5 014 17. Le Fttg A very commo problem data aalyss s lookg for relatoshpetwee dfferet parameters ad fttg les or surfaces to data. The smplest example s fttg a straght le ad we wll dscuss that here