Statistical Data Set Comparison for Continuous, Dependent Data by T. C. Smith

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1 Sesson 3 Operatonal and Specal Issues Statstcal Data Set Comparson for Contnuous, Dependent Data by T. C. Smth Tmothy C. Smth Davd Taylor Model Basn (NSWC/CD) ABSTRACT Classcal statstcal methods exst for the comparson of the mean or varance of two sets of ndependent data. One of the prme requrements s that the samples be ndependent. Ths requrement s problematc for contnuous random data, such as shp moton. In ths case, samples are strongly dependent to samples near them but are ndependent only of samples far from them. The nearness or farness s determned by the autocorrelaton functon. Ths paper dscusses an approach smlar to classcal dscrete statstcal comparsons but applcable to contnuous, dependent data. Through use of the autocorrelaton functon the method determnes an equvalent ndependent number of samples wthout drastcally sub-samplng the data set. The method wll be demonstrated usng smulated roll data of the Offce of Naval Research Topsde Seres tumblehome hull form. KEYWORDS Hypothess testng; Contnuous Data INTRODUCTION Many felds, from engneerng to medcne to manufacturng, requre the statstcal comparson of data sets. The common queston asked s f two data sets are statstcally from the same populaton. Classcal statstcal methods have been developed snce the 1800 s to compare means and varances. These methods nvolve the comparson of the data sets to theoretcal probablty dstrbutons to determne f the dfference n data sets s wthn a specfed statstcal sgnfcance. The Student s t-test for mean values and F-test for varances are commonly used. In order to apply these classcal statstcal methods, a number of assumptons must be made about the data samples. The most mportant are that the data are dscrete samples and the samples are ndependent of each other. Each sample s a separate entty and s not nfluenced by other samples. Ths s often the case n materal samplng, drug trals, and manufacturng. The fll level of one bottle on an assembly lne does not depend on the fll level of other bottles on the assembly lne. However, not all data are dscrete and ndependent. Ths s the case wth shp moton data, such as roll. A roll tme hstory s a samplng of a contnuous process roll does not change n dscrete steps, but rather as a smooth contnuum. Addtonally, there s a memory effect due to waves radated from the body whch stll contnue to nfluence the body. Ths means that present moton s dependent on prevous motons. The amount and duraton of nfluence s drectly calculated from the autocorrelaton functon. The autocorrelaton functon can be used to sub-sample contnuous data to generate ndependent data samples. The data samples are randomly selected wthn data wndows that are separated by the tme of t takes the autocorrelaton functon to go to zero. Dependng on the process, for example parametrc rollng whch has a very long autocorrelaton decay tme, sub-samplng can greatly reduce the amount of data or requre a very large amount of data. Ths s both neffcent and expensve. Ths paper dscusses an approach to apply classcal statstcal methods to contnuous, dependent data usng the autocorrelaton functon to fne or penalze the data for volaton of the assumptons of dscrete, ndependent statstcs. Ths approach s further descrbed n Prestley (1981). It s applcable to any contnuous, dependent data set ncludng full- and model-scale measurements, and smulaton of shp motons. 75

2 Proceedngs of the 1 th Internatonal Shp Stablty Workshop To verfy the applcablty, we apply ths approach to the comparson of model test and smulaton data. THEORY The applcable classcal statstcal hypothess tests are: the N-test, the Student s t-tests and F-tests (Bendat and Persol 1966). The N-test uses the normal dstrbuton to compare mean values. It s approprate for large numbers of degrees of freedom. The Student s t-test compares mean values usng the t-dstrbuton, whch s approprate for relatvely small degrees of freedom, less than 30. The t-dstrbuton s a normal dstrbuton dvded by a ch squared dstrbuton. The t-dstrbuton s defned by a varance and number of degrees of freedom. For large degrees of freedom the Student s t-test approaches the N-test. The F-test uses the F-dstrbuton whch s the rato of two ch squared dstrbutons and s defned by the degrees of freedom for each ch squared dstrbuton. To compare shp motons, we are most nterested n the varance, as the means are usually close to zero and the mportant part s the varaton about that mean. We wll use t to compare the varance of an ensemble of realzatons,.e., the varance of a test condton. The varance of the varance s also requred. The calculaton of the ensemble varance and varance of varance are detaled n Prestley (1981). Addtonally, we wll compare the data sets f all the realzatons n an ensemble are concatenated together nto a sngle long realzaton. Brefly restatng Prestley (1981), for an ensemble of N records, the autocorrelaton functon of each record tme hstory s determned and cut at the pont t begns to ncrease due to numercal nose. The cuttng procedure uses the envelope of the autocorrelaton functon to determne the begnnng of numercal nose and provde a smooth transton to zero. The autocorrelaton functon was transformed to a smooth spectrum as a check on the cuttng process. The ensemble autocorrelaton functon s the weghted average of the autocorrelaton functon for each record n the ensemble. The weghtng functon, W, s the number of samples n a record dvded by the total number of samples n the ensemble. The varance of the varance, VV rec, are calculated for each record usng Eqn. 1. VV rec N N rec 1 1 R rec Nrec 1 N (1) rec where N rec s the number of samples n a record and R s the autocorrelaton functon for each record or ensemble. The ensemble varance (MVa) and varance of the varance (VVa) are calculated usng Eqn. N rec MVa 0 N rec Mv Mm W Mma () VVa VV W (3) where Mv s the varance of each record, Mm s the mean value for each record, and Mma s the ensemble mean. In ths applcaton of Student s t-test, the ensemble varance takes the place of the mean and the ensemble varance of varance takes the place of the varance. The number of realzatons n the ensemble becomes the degrees of freedom. Usng the F-test to compare varances becomes more problematc. In the case of contnuous data, where there can be hundreds and thousands of data samples, some equvalent number of ndependent data samples s requred. If the number of data samples s used as the degrees of freedom, the F-test degenerates to comparng delta functons. Whle possessng a hgh degree of confdence, the result s not very helpful. Usng sub-samplng to reduce the number of degrees of freedom results n very large varances. Ths s also an unhelpful result. Hence, there s the need for an equvalent number of ndependent degrees of freedom, whch need not be an nteger. The equvalent number of ndependent degrees of freedom s the number of degrees of freedom for a ch squared dstrbuton that has the same confdence nterval as the data. The confdence nterval of a gven confdence, e.g., 95%, contans 95% of the data. For a normal dstrbuton and 95% confdence, 95% of the data are wthn 1.96 standard devatons from the 76

3 Sesson 3 Operatonal and Specal Issues mean. Usng the same concept, but wth varance, the confdence lmts for 95% are: MVa 1.96 VVa or MVa 1.69 VVa for 90% confdence. These values have already been penalzed for dependent data and are equvalent ndependent data values. The mplementaton determnes, for a gven number of degrees of freedom, the dfference between the data confdence nterval upper and lower bounds and the ch squared dstrbuton. The equvalent degree of freedom s the number that mnmzes the dfference between the data upper and lower confdence nterval bounds and a ch squared dstrbuton, as seen n Fg. 1. Numercally, ths was mplemented as a two step process. Frst, the maxmum dfference of the upper or lower bound for comparson was calculated for a range of degrees of freedom. Then the mnmum value was found. Ths process was done for both data sets to be compared as the F-test requres the degrees of freedom from both sets of data. Cumulatve Probablty x Normal 95% Normal Ch^ 95% Ch^ Fg. 1. Comparson of normal dstrbuton and the equvalent ch squared dstrbuton from matchng the confdence nterval. SHIP Smulatons were made for the Offce of Naval Research Topsde seres tumblehome hull form (Bshop 005). Ths test seres used the same underwater geometry wth three dfferent topsdes flared, wall-sded, and tumblehome. Expermental measurements of roll decay and regular transfer functons were made; hence, the lack of rregular seas model data for comparson. Stll the hull form s useful for smulaton. At ths GM, the peak of the rghtng arm curve s 0 degrees and GZ s negatve at 40.6 degrees. The prncpal dmensons are lsted n Table 1. The smulaton realzatons were made for longcrested and short-crested Sea State 8 seas. The sgnfcant wave heght was 11.5m wth a 13.5 sec average perod for both seaways. The spectral shape was Bretschneder. The short-crested seas used fve headng components wth cos n spreadng over ±45 degrees. Sx realzatons were made for each seaway. The shp speed was 16 knots and the relatve wave headng was 0 degrees (40 degrees off starboard bow). Table 1. Prncpal dmensons for full-scale model ONR tumblehome as smulated. Parameter Unts Length between m perpendculars Beam m Draft fwd m 5.50 Draft aft m 5.50 Dsplacement tonnes SW 8,487.8 GM m Blge keel span m 1.50 SIMULATION PROGRAM The shp moton smulaton program used for comparson was FREDYN. FREDYN s a tme doman, quas-nonlnear, dynamc stablty model that can smulate the motons of a free-runnng shp n a seaway under control of an autoplot. The archtecture of FREDYN s based on the de Kat and Paullng model (1989) whch n essence adds together the relevant force contrbutons n the equatons of moton. The theoretcal model conssts of a blended non-lnear strp theory approach, where lnear radaton and dffracton potental flow, and non-lnear Froude-Krylov are added to maneuverng and vscous drag forces. The radaton and dffracton forces are calculated over the calm water wetted surface. The Froude- Krylov forces have a hydrostatc component and a hydrodynamc component from the ncdent wave and are ntegrated over the nstantaneous wetted surface. The maneuverng and vscous drag forces nclude: hull resstance; vscous dampng, e.g., roll dampng; rudder; skeg; propeller; and wnd forces. The wave feld s modeled as a summaton of sne waves. 77

4 Proceedngs of the 1 th Internatonal Shp Stablty Workshop COMPARISON A Student s t and F-test were done for the same test condton bow seas, 16 knots, and Sea State 8. Sx FREDYN realzatons were made for each seaway. Each realzaton had the same wave spectrum but a dfferent wave tme hstory. The porton of the realzaton where the wave force ramped up was excluded from analyss. The total tme of all the realzatons equaled 34 mnutes of full-scale tme. Usng Prestley (1981), the wave elevaton and roll angle ensemble varance and varance of the varance are gven n Table for the two seaways. Table 3 presents the varance and equvalent ndependent degrees of freedom for concatenated data. As a check, the ensemble and concatenated varances and varance of varances are very close to each other. Table. and roll statstcs by record and ensemble. Long-crested (m ) Short-crested (m ) Record Avg Ensemble Var Var of Var # records Equv Indep DoF Avg DoF Also, the sum of the ndvdual record equvalent ndependent degrees of freedom s comparable to the concatenated equvalent ndependent degrees of freedom. Wth these data, we now proceed wth the Student s t and F-tests takng our null hypothess that the varances are equal for 95% confdence (=0.05). Wthout lookng at the results, we can expect to accept the waves as the same sgnfcant wave heghts that were specfed, but not the roll as the spread waves wll excte the shp from dfferent headngs. Or we can expect to reject both as spreadng could affect both the waves and response. From smply lookng at the results, we expect the latter due to the large dfferences between them. Table 3. and roll statstcs for concatenaton of all records. Long-crested (m ) Short-crested (m ) Var Var of Var Equv Indep DoF Tme (sec) Student s t-test As mentoned earler, we are substtutng the varance for the mean and the varance of varance for the varance n the standard formulaton. The number of degrees of freedom equals the number of records n the ensemble. Table 4 has the Student s t-test parameters calculated assumng unequal varances and unequal samples (degrees of freedom). Assumng equal number of samples s a specal case of the unequal samples assumpton and no error s ntroduced by usng equal number of samples. And the test statstc s: x1 x (4) S1 S n 1 n 1 1 Where x, S, and n are the mean, varance, and number of degrees of freedom, respectvely, for data set n the standard formulaton. We are 78

5 Sesson 3 Operatonal and Specal Issues substtutng varance for the mean and varance of varance for the varance. For 95% confdence, the crtcal t value s.8 for a two-taled test. If the test statstc s greater than the crtcal value, the null hypothess s rejected and the varances cannot be consdered equal. Thus, nether the waves nor roll can be consdered to have the same varance. The Student s t-test was appled to the concatenated data. Those results are n Table 5. In ths case, the large number of degrees of freedom results n a normal dstrbuton as seen by the crtcal t value equalng nearly 1.961, the normal dstrbuton value for 95% confdence. Agan, both the waves and roll cannot be consdered to have the same varance. Table 4 Student's t-test results for ensemble data. Elevaton Angle Mean () ( VoV) Number of DoF Pooled alpha Number of DoF T-Crtcal.8.8 Test Statstc Reject Reject Table 5. Student's t-test results for concatenated data. Elevaton Angle Mean () ( VoV) Number of DoF Pooled alpha Number of DoF T-Crt Test Statstc Reject Reject F-test The F-test statstc s the rato of the two varances. As general practce, the rato s formed to be greater than one, though ths s not necessary. Agan, f the test statstc s greater than the crtcal value, the null hypothess s rejected and the varances cannot be consdered equal. If the rato s formed as less than one, then the probablty s taken as one mnus the probablty, and now the test statstc needs to be greater than the crtcal value to accept the null hypothess. The F-test results for the ensemble data are n Table 6 and the concatenated data are n Table 7. The ensemble average number of equvalent ndependent degrees of freedom s used for the ensemble data. Usng ensemble data, both the waves and roll angle can be accepted as from the same data set. Ths s opposte the Student s t-test results. Usng concatenated data, ths case accepted the waves as havng the same varance and rejected roll. Ths s the frst of our expected results. Table 6. F-test results for ensemble data. Elevaton Angle Ensemble Avg Equv Indep DoF alpha F-Crt Test Statstc Accept Accept Comparng the maxmum and mnmum records n each ensemble usng the F-test also produced mxed results. In ths case, we know the records come from the same data set and we should accept them as so. However, long-crested roll was rejected wth a test statstc of 1.93 vce a crtcal value of Ths reflects the large varance of varance seen n the long-crested roll ensemble. Ths s a functon of non-lnearty n the system ncreasng the uncertanty. 79

6 Proceedngs of the 1 th Internatonal Shp Stablty Workshop Table7. F-test results for concatenated data. Elevaton Angle Concatenated Equv Indep DoF alpha F-Crt Test Statstc Accept Reject CONCLUSION The statstcal comparson of data sets s a common problem. Statstcal formulatons snce the 1800 s assume dscrete ndependent data samples. As a result, the typcal statstcal comparsons such as Student s t- and F-test are not strctly applcable to contnuous, dependent data, e.g., shp roll moton. Ths paper dscussed an approach usng the autocorrelaton functon to fne dependent data to calculate the varance of varance. The paper also demonstrated a confdence nterval matchng technque to determne the equvalent ndependent values for a tme hstory of dependent data. Interestngly, the equvalent number of ndependent values was more than smply dvdng the total tme by the tme for the autocorrelaton to go to zero. Ths mples more nformaton was retaned and greater accuracy attaned than smply sub-samplng the tme hstory. Also, both the ensemble and concatenated data approach produced comparable varance, varance of the varance, and equvalent ndependent degrees of freedom. Ths approach was appled to both the Student s t- and F-tests to compare long-crested to short-crested smulaton data. Both hypothess test results met an expected outcome, albet dfferent ones. The Student s t-test s somewhat preferred as t produced consstent results wth both ensemble and concatenated data. The results do show the danger of blndly followng the hypothess testng results. ACKNOWLEDGEMENT The author s most grateful to Dr. Vadm Belenky for helpful dscussons concernng ths work. REFERENCES Bendat, J. S. and Persol, A. G., (1966) Measurement and Analyss of Random Data, John Wley & Sons Bshop, R. C., Belknap, W. F., Turner, C., Smon, B. S., and Km, J. H., (005) Parametrc Investgaton on the Influence of GM, Dampng, and Above- Water Form on the Response of Model 5613, NSWCCD-TR /7. de Kat, J. O. and Paullng, J. R., (1989) The Smulaton of Shp Motons and Capszng n Severe Seas, SNAME Trans,, vol. 97. Prestley, M. B. (1981) Spectral Analyss and Tme Seres Vol. 1 Unvarate Seres, Academc Press 80