Linear Regression Analysis for STARDEX


 Ginger Sparks
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1 Lnear Regresson Analss for STARDEX Malcolm Halock, Clmatc Research Unt The followng document s an overvew of lnear regresson methods for reference b members of STARDEX. Whle t ams to cover the most common and relevant methods for calculatng trends and ther levels of statstcal sgnfcance, there wll nevtabl be omssons. Please send an correctons or comments to Malcolm Halock Trend Calculaton Least squares (used n dagnostc tool) Least squares lnear regresson s a maxmum lkelhood estmate.e. gven a lnear model, what s the lkelhood that ths data set could have occurred? The method attempts to fnd the lnear model that maxmses ths lkelhood. Suppose each data pont has a measurement error that s ndependentl random and normall dstrbuted around the lnear model wth a standard devaton σ. The probablt that our data (± some fxed probabltes at each pont: N ( ) a + bx P α exp σ Maxmsng ths s equvalent to mnmsng: ( a + bx ) σ at each pont) occurred s the product of the If the standard devaton σ at each pont s the same, then ths s equvalent to mnmsng: ( ( a + bx )) Solvng ths b fndng a and b for whch partal dervatves wth respect to a and b are zero, gves the best ft parameters for the regresson constant and coeffcent (α and β): S xxs S xs x α NS x S xs β where NS xx ( S x ) and S x x, S, S x x, S xx x For further nformaton, see Wlks (995) or Press et al. (986). Mnmum Absolute Devaton Least squares lnear regresson, lke man statstcal technques, assumes that the departures from the lnear model (errors) are normall dstrbuted. Technques that do not rel on such assumptons are termed robust.
2 Least squares regresson s also senstve to outlers. Although most of the errors ma be normall dstrbuted, a few ponts wth large errors can have a large affect on the estmated parameters. Technques that are not so senstve to outlers are termed resstant. A more resstant method for lnear trend analss s to assume that the errors are dstrbuted as a twosded exponental. Ths dstrbuton, wth ts larger tals, allows a hgher probablt of outlers: Pr { ( a + )} ~ exp ( a + ) bx ( ) bx Smlarl to the process of least squares, ths requres that we mnmse: N ( a + ) bx The soluton to ths needs to be found numercall. Example code can be found n Press et al. (986). Threegroup resstant lne Ths method derves ts resstance from the fact that one of the smplest resstant measures of a sample s the medan. Data are dvded nto three groups dependng on the rank of the x values. The left group contans the ponts wth the lowest thrd of x values. In a tme seres ths s equvalent to the frst thrd of the seres. Smlarl, the mddle and rght groups contan ponts wth the mddle and hghest thrd of ranked x values respectvel. Next the x and medan values are determned for the three groups to gve the ponts x,. ( x, L L ), ( ) M M x, and ( ) R R The slope of the lne s taken as the gradent of the lne through the medans of the left and rght groups: b 0 x R R x L L The ntercept of the lne s calculated b fndng the three lnes wth slope b 0 that pass x,, then averagng ther ntercept: through each of the ponts ( ) L L 3 x,, ( x, ) and ( ) M M [( b x ) + ( b x ) + ( b x )] a0 L 0 L M 0 M R 0 R R R The threegroup resstant lne method usuall requres teraton. After the frst pass to fnd a0 and b 0, the process can be repeated on the resduals to fnd a and b. The teratons are contnued untl the adjustment to the slope s suffcentl small n magntude (at most %). The fnal slope and ntercept s the sum of those from each teraton. Further nformaton can be found n Hoagln et al. (983). Logstc Regresson Lnear regresson has been generalsed under the feld of generalsed lnear modellng, of whch logstc regresson s a specal case. Ths method utlses the bnomal dstrbuton and can therefore be used to model counts of extreme events.
3 Often n a seres, the varance of the resduals (from the lnear model) vares wth the magntude of the data. Ths goes aganst the assumptons of least squares regresson, whch assumes resduals to have constant varance, but s a natural element of the bnomal dstrbuton and logstc regresson. Therefore data do not need to be normalsed. The logstc regresson model expresses the probablt π of a success (e.g. an event above a partcular threshold) as a functon of tme: ( ) α + β t η π Snce the probablt of a success s n the range [0,], t needs to be transformed to the range, usng a lnk functon: ( ) x η( x) log x Solvng for π gves: α + β t e π ( t; α, β ) α + t + e β We are not fttng a straght lne to the counts and therefore can not refer to a sngle trend value. The odds rato s used to express the relatve change n the rato of events to nonevents t : over the perod (,t ) π ( t ) π ( t ) Θ π ( t ) π ( t ) e β ( t t ) Model fttng can be done usng a maxmum lkelhood method. Further nformaton about logstc regresson, together wth an example usng extreme precptaton n Swtzerland, can be found n Fre and Schär (00). Sgnfcance Testng Confdence ntervals for least squares Often the standard devatons σ for the observatons are not known. If we assume that the lnear model does ft well and that all observatons have the same standard devaton σ, the assumpton that the resduals are normall dstrbuted around the lnear model mples that: ( ( a + bx )) σ N estmated., wth N appearng n the denomnator because two parameters are From the above, t can be shown that the regresson coeffcent b wll be normall dstrbuted wth varance: Var[ b] σ ( x x) Snce the varance of b s estmated, Student s tdstrbuton s used to defne the multpler t for the confdence lmts for the regresson coeffcent: b β± t Var(b)
4 The assumpton that the resduals are normall dstrbuted can be tested wth a quantlequantle (QQ) plot of the resduals aganst the quantles from a Gaussan dstrbuton. For further nformaton, see Wlks (995) or Press et al. (986). Lnear Correlaton The lnear correlaton coeffcent (Pearson productmoment coeffcent of lnear correlaton) s used wdel to assess relatonshps between varables and has a close relatonshp to least squares regresson. The correlaton coeffcent s defned b: ( x, ) cov rx.e. the rato of the covarance of x and to the product of ther standard σ xσ devatons. In a least squares lnear model, the varance of the predctand can be proportoned nto the varance of the regresson lne and the varance of the predctand around the lne: SSTSSR+SSE Sum of Squares Total Sum of Squares Regresson + Sum of Squares Error In a good lnear relatonshp between the predctor and predctand, SSE wll be much smaller than SSR.e. the spread of ponts around the lne wll be much smaller than the varance of the lne. Ths goodness of ft can be descrbed b the coeffcent of determnaton: R SSR SST varance of predctand explaned b the predctor It can be shown that the coeffcent of determnaton s the same as the square of the correlaton coeffcent. The correlaton coeffcent can therefore be used to assess how well the lnear model fts the data. Assessng the sgnfcance of a sample correlaton s dffcult, however, as there s no wa to calculate ts dstrbuton for the null hpothess (that the varables are not correlated). Most tables of sgnfcance use the approxmaton that, for a small number of ponts and normall dstrbuted data, the followng statstc s dstrbuted for the null hpothess lke Student s tdstrbuton: N t r () r The common bass of the correlaton coeffcent and least squares lnear regresson means that the share the same shortcomngs such as lmted resstance to outlers. See Wlks (995) or Press et al. (986) for further nformaton. Spearman rankorder correlaton coeffcent Non parametrc correlaton statstcs are an attempt to overcome the lmted resstance and robustness of the lnear correlaton coeffcent, as well as the uncertant n determnng ts sgnfcance. If x and data values are replaced b ther rank, we are left wth the set of ponts (,j),,j,n whch are drawn from an accuratel known dstrbuton. Although we are gnorng some nformaton n the data, ths s far outweghed b the benefts of greater robustness and resstance.
5 The Spearman rankorder correlaton coeffcent s just the correlaton coeffcent of these ranked data. Sgnfcance s tested as for the lnear correlaton coeffcent usng (), but n ths case the approxmaton does not depend on the dstrbutons of the data. See Press et al. (986) for further nformaton. KendallTau (used n dagnostc tool) Kendall s Tau dffers from the Spearman rankorder correlaton n that t onl uses the relatve orderng of ranks when comparng ponts. It s calculated over all possble pars of data ponts usng the followng: τ concordant dscordant concordant + dscordant + samex concordant + dscordant + samey where concordant s the number of pars where the relatve orderng of x and are the same, dscordant where the are the opposte, samex where the x values are the same and samey where the values are the same. τ s approxmatel normall dstrbuted wth zero mean and varance: 4N + 0 Var( τ ) 9N( N ) One advantage of Kendall s tau over the Spearman coeffcent s the problem of assgnng ranks when data are ted. Kendall s tau s onl concerned whether a rank s hgher or lower than another, and can therefore be calculated b comparng the data themselves rather than ther rank. When data are lmted to onl a few dscrete values, Kendall s tau s a more sutable statstc. See Press et al. (986) for further nformaton. Resamplng Resamplng procedures are used extensvel b clmatologsts and could be used to assess the sgnfcance of a lnear trend. The bootstrap method nvolves randoml resamplng data (wth replacement) to create new samples, from whch the dstrbuton of the null hpothess can be estmated. Therefore no assumpton needs be made about the sample dstrbuton. If enough random samples are generated, the sgnfcance of an observed lnear trend can be assessed b where t appears n the dstrbuton of trends from the random samples. A problem, however, s that the maxmum lkelhood dervaton of the least squares estmate for the lnear trend assumed that data resduals about the lne were normall dstrbuted. Therefore f the dstrbuton of the resduals s not Gaussan, then the least squares estmate s not vald. Stll, bootstrappng could be used to test the sgnfcance of a least squares lnear trend, gven that ths ma not be the best trend estmate. An mportant assumpton n resamplng s that observatons are ndependent. Zwers (990) showed that, for the case of assessng the sgnfcance of the dfference n two sample means, the presence of seral correlaton greatl affected the results. A method has been proposed b Ebsuzak (997) whereb random samples are taken n the frequenc doman (wth random phase) to retan the seral correlaton of the data n each sample.
6 References Ebsuzak, W., 997: A method to estmate the statstcal sgnfcance of a correlaton when the data are serall correlated. J. Clm., 0, Fre, C. and C. Schär, 00. Detecton probablt of trends n rare events: theor and applcaton to heav precptaton n the alpne regon. J. Clm., 4, Hoagln, D.C., F. Mosteller and J.W. Tuke, 983. Understandng robust and explorator data analss. Wle Press, W.H., B.P. Flanner, S.A. Teukolsk and W.T. Vetterlng, 986. Numercal recpes: The art of scentfc computng. Cambrdge Unv. Press, Wlks, D.S., 995. Statstcal Methods n the Atmospherc Scences. Academc Press Zwers, F. W., 990. The effect of seral correlaton on statstcal nferences made wth resamplng procedures. J. Clm., 3,
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