Noise-injection as an Approach to Generating Random Data Sets for Online Tests and Virtual Labs

Jou nal o Labo a o y Chemical Educa ion 2021, 9(2): 26-35 DOI: 10.5923/j.jlce.20210902.02 Noise-injec ion as an App oach o Gene a ing Random Da a Se s o Online Tes s and Vi ual Labs D. González-A jona1,*, M. M. Domínguez1, G. López-Pé ez1, W. H. Mulde 2 1Depa men o Physical Chemis y, Uni e sidad de Se illa, Se illa, Spain 2Depa men o Chemis y, The Uni e si y o he Wes Indies, Mona Campus, Jamaica Abs ac A me hodology based on noise injec ion o gene a ion o andomized abula ed da a is p esen ed. The s a egy can be used bo h in online eaching and o speci ic nume ical/g aphical exe cises, when indi idualized da a se s a e equi ed simul aneously o he s uden s. Res ic ions imposed on eaching me hods du ing o he SARS-Co id-19 pandemic, especially o labo a o y sessions in chemis y o e en o p epa ing w i en exams, ha e led o a need o app oaches based on andomized da a se s based on li e a u e da a o heo e ical equa ions. Commonly a ailable sp eadshee so wa e has been used o gene a ing andom da a and o analysis and calcula ions, which acili a es easy and low cos applica ion o he me hodology p esen ed he e. Uni o m and Gaussian dis ibu ions ha e been employed o gene a e di e en ypes o noise. S a is ical analyses on linea eg ession pa ame e s o he di e en dis ibu ion and le els o injec ed noise ha e been pe o med. As examples, hese esul s a e employed o in oduce andomness in h ee ypical expe imen s pe o med in Physical Chemis y labs in ol ing he modynamics, chemical kine ics and conduc i i y o elec oly e solu ions. Li e a u e alues a e employed o he expe imen s as empla es o which di e en le els o noise a e applied. The esul s indica e ha he applica ion o noise has o be ca e ully con olled. Uni o m noise is sugges ed o da a se s ha al eady con ain na u al andom noise, whe eas Gaussian noise should be employed o da a se s c ea ed di ec ly om heo e ical o empi ical equa ions, so as o p oduce da a se s wi h a mo e na u al, ealis ic appea ance. Keywo ds Random noise-injec ion, Online lea ning, Vi ual labs 1. In oduc ion Objec i e: To build andom da a se s om well- ounded heo ies o expe imen al da a o be used in on-line lea ning wi h examples om he a ea o Physical Chemis y. Usually li e a u e da a a e employed o gene a e nume ical o g aphical exe cises in highe ( e ia y) educa ion. F om hese da a se s, by using he app op ia e heo e ical backg ound, he a ge pa ame e s a e ob ained. Some imes, i is necessa y o cons uc a plo o ob ain hese pa ame e s o an in e media e esul . Nowadays, online lea ning is commonly p ac iced and in ol es he co esponding online examina ions. This mode o cou se deli e y has now e en become he s anda d in hese imes o lockdowns as a esul o he CoVID-19 pandemic. This has made i ine i able o pe o m examina ions emo ely while a he same ime s uden s ha e ull access o In e ne and social media o ob ain in o ma ion. Unde hese condi ions, ypical es esul s do no necessa ily e lec he s uden ’s ue knowledge abou * Co esponding au ho : [email p o ec ed] (D. González-A jona) Recei ed: May 26, 2021; Accep ed: Jun. 25, 2021; Published: Jun. 30, 2021 Published online a h p://jou nal.sapub.o g/jlce he opic being examined. Thus, i seems ad isable o use some ools o s a egies o limi ins ances o chea ing. In ac , some web-based lea ning pla o ms [1,2], p o ide ools o andomize on-line es s. Fo example, ques ions can be andomly selec ed om a ques ion bank. Fu he mo e, in mul iple-choice ques ions he o de o he op ions can also be sc ambled. In he case o nume ical p oblems whe e a o mula has o be employed o pe o m calcula ions, i is possible o gene a e a numbe o di e en se s o inpu pa ame e s aken om a selec ed in e al, o example empe a u e and p essu e. Ano he s a egy ha can be employed o minimize oppo uni ies o chea ing is o change he ph asing o a nume ical p oblem, while main aining he s uc u e o a ques ion, e.g. he applica ion o a pa icula o mula, ha is, o di e si y he ex e nal appea ance o he nume ical p oblem. Addi ional nume ical p oblem se s can be easily cons uc ed, jus by changing he inpu da a uni s and/o dimensions. In many chemis y p oblems and exe cises i is equen ly equi ed o ob ain he inal o a pa ial esul om he pa ame e s ex ac ed om a, usually linea , plo . This kind o scena io is no commonly implemen ed in web-based lea ning pla o ms. Mo eo e , hese kind o exe cises, cons uc ion o g aphs, a e e y common in ace- o- ace Jou nal o Labo a o y Chemical Educa ion 2021, 9(2): 26-35 27 eaching labs, whe e a linea eg ession analysis is usually o be pe o med. Un o una ely, du ing las yea ’s lock-downs, no ace- o- ace labs we e held. The e a e some so wa e pla o ms [3-6, and e e ences he ein], ha p o ide simula ed lab expe imen s o i ual labs, ha can be used as p epa a o y aining p io o he eal lab expe ience. This pa ial solu ion necessi a es on he one hand o modi y some opics and secondly, in ol es wo ex a cos s, an economic one and a cos in e ms o ime in es ed by he ins uc o in implemen ing he new scena io. Ob iously, he ace- o- ace lea ning mode is essen ial o a lab-based p og am. Video u o ials can help, bu can ne e subs i u e o he hands-on expe ience. In o de o educe he loss incu ed as a esul o hese lea ning objec i es being comp omised, a di e en app oach has been adop ed las yea . A new amewo k was designed o each lab expe imen , adap ed o he new lea ning en i onmen . This included ques ions abou he s ep by s ep p ocedu e o he expe imen as well as p oblems a ising om some unusual ou comes ha could be ob ained on an ac ual lab day. Da a se s we e p o ided o indi idual s uden s o wo k wi h, wi h andomized o iginal lab da a. Thus, some noise was supe imposed on he o iginal da a, while main aining he signi icance and a iabili y o he esul s ha can be ob ained om hese new andomized da a. In his pape , he use o con olled noise injec ion o gene a e a clus e o andomized abula ed da a, o use in online eaching, is discussed. The p ocedu e will be analyzed s ep-by-s ep and illus a ed wi h some examples. Fi s , a b ie in oduc ion abou he di e en kinds o noise o be employed, o he abo e men ioned pu pose, and i s ela ionship wi h he basic concep s o accu acy and p ecision, will be p esen ed. Nex , he use o sp eadshee so wa e o cons uc di e en noise dis ibu ions will be explained and e alua ed. And inally, he p ocedu e will be explained and applied o i ualize some lab expe imen s and g aphical exe cises in he ield o Physical Chemis y. The s a egy in oduced in his pape has been success ully employed since he las academic yea o online es s and lab i ualiza ion. The me hodology p o ides a as and inexpensi e way o de eloping andomized da a se s wi h ealis ic and eliable esul s o use in online es s. 2. Gene a ing Random Digi al Noise The undamen al limi o he esolu ion and accu acy and model analysis om da a is hei le el o noise. The noise can be ca ego ized acco ding o he ocus o in e es . Thus in ins umen a ion he noise is ca ego ized by he in e e ing sou ce: powe lines, empe a u e e ec s on he ins umen s and senso s and andom noise om he ins umen i sel . This las ype o noise can be analyzed using s a is ical and p obabilis ic p inciples. Thus, he goal in Digi al Signal P ocessing is o elimina e he in e e ing noise wi hou al e ing he signal o in e es . The cha ac e iza ion o noise and i s le el is a undamen al ask. S a is ical analysis can be applied o he signal. Thus, he mean alue and s anda d de ia ion p o ide in o ma ion abou he signal a e age le el and sp ead. These wo pa ame e s can u nish in o ma ion abou he accu acy and p ecision o he signal. La ge de ia ions o he mean om he ue alue and la ge s anda d de ia ions indica e lowe signal quali y o high noise le el. I he sou ce o noise is iden i ied, he quali y o he signal can be imp o ed by emo ing he sou ce. Bu andom noise can only be pa ially il e ed ou o minimized and special ca e has o be aken when il e ing o a oid dis o ion o he signal unde s udy. Using hose basics concep s om Digi al Signal P ocessing (DSP), he in e se me hodology has been employed o noise injec ion in o a p e iously gene a ed da a se [7]. The amoun and ype o andomized noise ha e o be con olled o a oid comp omising he da a analysis, keeping in mind he a ge o o e ing ealis ic indi idualized da a o each s uden . The e a e many s anda d p obabili y dis ibu ions (Binomial, Poisson, Uni o m, Chi-Squa ed, Gaussian, Be noulli, Logno mal…) and mos o hem can be gene a ed om andom numbe s [8]. He e, he Uni o m and Gaussian dis ibu ions will be employed o gene a e he andom alues o be injec ed in o da a se s. The Uni o m dis ibu ion applies o a ini e numbe in e al, usually any alue be ween 0 and 1. The main cha ac e is ic is ha selec ing any numbe in his ange has he same p obabili y, in his case wi h a mean alue o 0.5 and a a iance o 1 12 . This kind o dis ibu ion is also known as ‘whi e noise’, and has been employed in C yp og aphy and in Mon e Ca lo simula ions. The o he well-known dis ibu ion is he Gaussian o no mal dis ibu ion. In his case he alues a e dis ibu ed so as o c ea e a bell-shape a ound a mean alue ha coincides wi h he mode, and p obabili ies alling o exponen ially o alues away om he mean. This ype o dis ibu ion is commonly ound in many na u al p ocesses, e.g. i s exponen ial cha ac e is encoun e ed in he Maxwell-Bol zmann dis ibu ion, applicable o physicochemical p ocesses. These wo kinds o dis ibu ion can be easily gene a ed by using sp ead-shee p og ams. The well-known Excel o Lib eO ice p og ams can be used in e changeably. The andom numbe algo i hm employed he e has been imp o ed o he la es e sion o Excel wo kshee [9]. Despi e he imp o emen o he algo i hm, his is no ye ecommended o use in p o essional c yp og aphy no o Mon e Ca lo simula ions, bu i is pe ec ly adequa e o mee ing he objec i es o his pape . Excel uses he RAND() unc ion o gene a e a eal numbe be ween 0 and 1. E e y ime he wo k shee is modi ied, a new andom numbe is gene a ed o all he cells which con ain ha unc ion. The numbe gene a ed has a uni o m dis ibu ion wi h a mean alue o 0.5 and a s anda d de ia ion o 1 12 [10]. 28 D. González-A jona e al.: Noise-injec ion as an App oach o Gene a ing Random Da a Se s o Online Tes s and Vi ual Labs Ano he popula way o gene a ing uni o m dis ibu ions is by using he modulus ope a ion (MOD) [11]. This me hod has some weaknesses when i comes o gene a ing uly andom sequences, bu o he objec i es o his pape i is s ill use ul. The Cen al Limi Theo em can be in oked o gene a e he no mal dis ibu ion wi hou using any new Excel unc ion. Simply adding wel e andom numbe s be ween 0-1, an excellen app oxima ion o a no mal dis ibu ion is ob ained wi h mean alue o six (sum o each indi idual uni o m mean) and one as s anda d de ia ion (squa e oo o sum o he a iances) [12]. This dis ibu ion can be easily modi ied o p oduce selec ed mean and s anda d de ia ion alues. The e a e some o he algo i hms, based in he Box-Mulle T ans o ma ion, capable o gene a ing s anda d no mal dis ibu ions om jus wo uni o m dis ibu ions: 2ln(RAND()) cos(2 RAND())x   [13]. This algo i hm has he ad an age o a sho e de ini ion o he cons uc ion unc ion ha simpli ies w i ing i in sp eadshee , hence minimizing he isk o ypos. Figu e 1. Sp eadshee algo i hms employed o he ype o da a se dis ibu ion gene a ion wi h a selec ed mean and s anda d de ia ion Da a se s, ha ing mo e han 5000 poin s, ollowing bo h dis ibu ions, Uni o m (U) and No mal (N), ha e been gene a ed by he di e en equa ions displayed in Figu e 1. Al hough he mean and s anda d de ia ion can be selec ed a bi a ily, in his wo k ze o and one, espec i ely, we e always chosen. Table 1. Selec ed s a is ics pa ame e s o he di e en dis ibu ions gene a ed Table 1 collec s some s a is ical pa ame e s (mean, s anda d de ia ion and max and min alues) o he di e en da a se s gene a ed wi h he algo i hms desc ibed abo e. I can be seen ha bo h algo i hms o uni o m dis ibu ion pe o m iden ically and p oduce he expec ed esul s, especially he s anda d de ia ions ob ained. Bo h algo i hms using he no mal dis ibu ion also p o ide he expec ed esul s. The numbe o RAND imes employed in he unc ion is also shown in Table 1, showing hei in luence on he s a is ical pa ame e s o he dis ibu ions. Thus, an inc ease in he use o he RAND unc ion implies a highe alue o he s anda d de ia ion and b oade ange o he ex eme alues max and min. Figu e 2 shows ha i is ha d o dis inguish be ween he wo ypes o dis ibu ions, uni o m o no mal, jus by isual inspec ion o he sequence o alues. The cons uc ion o a his og am c ea ing a se ies o bins (class in e als) wi h an inc emen o 0.02 each and so ing alues acco ding o he in e als (bins) in which hey occu , leads o he dis ibu ions shown in Figu e 3. This ope a ion can be easily pe o med in Excel by using he Da a Analysis ool. Figu e 2. Sca e ed alues a ound ze o ob ained by using he uni o m dis ibu ion unc ion scaled be ween -0.5 and 0.5, U, and a no mal dis ibu ion algo i hm, N. Bo h ha e he same  = 1/√12 Jou nal o Labo a o y Chemical Educa ion 2021, 9(2): 26-35 29 Figu e 3. Compa a i e his og ams o di e en dis ibu ions, wi h ze o as mean alue. Red: Uni o m,  = 1/√12; Blue: No mal,  = 1/√12; Viole : No mal,  = 1/(2√12) and G een: No mal log based:  = 1/(2√12) Figu e 3 shows he his og ams o di e en ypes o dis ibu ions. The alues o he uni o m dis ibu ion ( ed ba s) a e andom dis ibu ed showing no pa e n, wi h hei alues pe ec ly con ined be ween -0.5 and +0.5. Ne e heless he no mal dis ibu ion (blue ba s) wi h he same s anda d de ia ion,  = 1/√12, p oduces he bell-shaped pa e n, bu he e a e alues g ea e han 0.5 and below -0.5. These la e alues accoun o app oxima ely 5% o he en i e se . The iole and g een his og am a e ob ained o a no mal dis ibu ion using RAND unc ion and he Box-Mulle ans o ma ion, bo h scaled o a s anda d de ia ion 1/(2√12), hal o he blue his og am. Bo h his og ams a e independen o he algo i hm employed. Mo eo e , wi h s anda d de ia ion alue selec ed, less han 0.5% o he alues lays ou side o he -0.5/+0.5 in e al. The selec ion o he dis ibu ion ype and i s scaling is ou nex ask. 3. Analysis o he In luence o Noise Type on Linea Reg ession A common ask pe o med by s uden s is o c ea e a plo om he da a o ex ac in o ma ion abou a physical phenomenon. Among he di e en kinds o g aphical analyses, linea eg ession is he mos equen ly encoun e ed p ocedu e. In o de o analyze he in luence o noise on he linea pa ame e s, a con olled le el o andom noise has been in oduced in o he da a along bo h coo dina e axes. The amoun o noise in oduced a each da a poin is se o a selec ed pe cen age o he ac ual alue, hence he new alue luc ua es andomly a ound he o iginal alue. The uni o m dis ibu ion p o ides easy con ol o he ange o ou pu alues, and has he e been se o a y be ween -0.5 and 0.5, so wi h a mean alue o ze o. Thus o each da a poin , a andom pe cen age o luc ua ion a ound i s ac ual alue can be added, employing he equa ion:   New da a Ac ual da a 1 (RAND (-0.5 0.5)) % noise/100     whe e, RAND(-0.5↔0.5), ep esen s he algo i hm o ob ain a eal numbe be ween -0.5 o 0.5 wi h a selec ed ype o dis ibu ion. Thus, he pe cen age o noise is dis ibu ed andomly (unde a uni o m o no mal dis ibu ion), a ound he ac ual alue, adding o sub ac ing hal he pe cen age o he noise selec ed. Howe e , his p ocedu e implies ha he noise- ee alue ze o is singula , i.e. i has no noise added using he abo e equa ion. Ne e heless, his can be emedied i an ex a pe cen age o andom noise is included o each da a poin as an o se , posi i e o nega i e, educing his singula i y. The amoun o noise used o p oduce he o se is chosen a a le el ha is less by a ac o o en compa ed o he le el o noise selec ed. As s a ed be o e, a no mal dis ibu ion wi h he same s anda d de ia ion as he uni o m one, p oduces alues ou side o hose selec ed o he uni o m dis ibu ion. Thus, o main ain app oxima ely he same in e al o alues, he s anda d de ia ion o he no mal dis ibu ion should be scaled, di iding by 1/(2√12) ≈1/7, as can be seen in igu e 3, p oducing alues ha a e mo e concen a ed a ound he mean alue. In his sense, when noise is injec ed using scaling o he no mal dis ibu ion he ac ual alues a e less biased. All hese di e en app oxima ions o injec ing noise in he da a se ha e been es ed and analyzed: uni o m dis ibu ion, no mal dis ibu ions wi h di e en s anda d de ia ion scales, and o each case, wi h and wi hou addi ional o se . The analyses ha e been pe o med o e a da a se con aining en poin s ha we e gene a ed using a simple linea ela ionship o he o m y = x + 1. Fo each ype o noise dis ibu ion, i e di e en pe cen ages o noise le els we e injec ed: 0.5%, 1%, 2%, 5% and 10% o he o dina e and 0.2%, 0.5%, 1%, 2% and 5% o he abscissa. The sum o e squa ed de ia ions a ies app oxima ely om 2·10-4, o 0.2% o 0.1 o 5% in he case o he uni o m dis ibu ion applied o he abscissa and om 1·10-3 o 0.5% o 0.5 o 10% in he o dina e. When he co ec ion o he singula noise- ee alue as men ioned abo e is applied, he alues o he sum o squa ed de ia ions a e e y simila . Applying he scaled no mal dis ibu ion, he alues o he sum o e squa ed de ia ions a e app oxima ely om 3·10-5, o 0.2% o 0.04 o 5% o he abscissa and om 2·10-4 o 0.5% o 0.1 o 10% o he o dina e. No signi ican changes a e obse ed when a small o se is added o minimize he singula ze o alue. These alues o he sum o e squa ed de ia ions ag ee wi h he expec ed beha io , namely ha noise gene a ed by he uni o m dis ibu ion will p oduce s onge luc ua ions wi h poo e eg ession s a is ics han he scaled no mal dis ibu ion. In o de o analyze he in luence o he kind o noise dis ibu ion and i s le el in bo h a iables, linea eg ession s a is ics ha e been pe o med by means o he LINTEST ou ine in Excel. 30 D. González-A jona e al.: Noise-injec ion as an App oach o Gene a ing Random Da a Se s o Online Tes s and Vi ual Labs Table 2. Linea eg ession s a is ics by using he LINTEST unc ion in Excel o di e en uni o m noise le els added o he o dina e only and o bo h coo dina es (x,y), espec i ely Uni o m noise a o dina e y% Reg . Pa am. 2% 5% 10% slpe/in cp 1.00195512 0.9936106 0.9995866 0.9946863 1.0102083 0.967860847 s de slpe/in cp 1.8104E-03 1.2279E-02 4.6607E-03 3.1611E-02 9.2095E-03 6.2462E-02 2/ s de y 9.9997E-01 1.8988E-02 9.9980E-01 4.8882E-02 9.9925E-01 9.6590E-02 F/ d 3.0630E+05 9 4.5997E+04 9 1.2032E+04 9 ss eg/ss esid 1.1043E+02 3.2448E-03 1.0991E+02 2.1505E-02 1.1226E+02 8.3966E-02 Uni o m noise a bo h coo dina es x%/y% Reg . Pa am. 1%/2% 2%/5% 5%/10% slpe/in cp 1.00184749 0.9968192 1.0005897 0.9899704 1.0050629 0.993968265 s de slpe/in cp 2.4575E-03 1.6662E-02 5.9783E-03 4.0531E-02 9.8773E-03 6.7106E-02 2/ s de y 9.9995E-01 2.5777E-02 9.9968E-01 6.2635E-02 9.9913E-01 1.0412E-01 F/ d 1.6619E+05 9 2.8012E+04 9 1.0354E+04 9 ss eg/ss esid 1.1043E+02 5.9801E-03 1.0990E+02 3.5308E-02 1.1224E+02 9.7565E-02 Legend: slpe: slope; in cp: in e cep ; s de slpe/in cp: s anda d e o ; 2: coe icien o de e mina ion; s de y: s anda d e o y es ima e; F: F s a is ic; d : deg ees o eedom; ss eg: eg ession sum o squa es; ss esid: esidual sum o squa es. Table 2 summa izes, as an example, he esul s o a eg ession s a is ical analysis o he injec ion o di e en le els o uni o m noise. The sub- able a he op epo s esul s ob ained when noise is added only o he o dina e alues, while he lowe sub- able gi es LINTEST ou pu when noise is added o bo h coo dina es. The % le el o noise injec ed is indica ed in he column heade s. Clea ly, om he da a in Table 2, lowe noise added implies be e s a is ics o he linea eg ession pa ame e s ( 2, s d e , esidual, and % ela i e e o ) indica i e o a g ea e eliabili y o he model equa ion. Figu e 4. Rela i e e o associa ed wi h di e en pe cen ages o uni o m noise le els injec ed along he o dina e Figu es 4 and 5 show he ela i e e o in he o dina e, and a close-up iew o one poin o he da a se , (2, 3), espec i ely. Di e en le els o uni o m dis ibu ion noise ha e been employed, in igu e 4 only in o dina e and in igu e 5 bo h coo dina es. The shaded a ea in igu e 5 app oxima ely delinea es he limi s o he (x, y) luc ua ion ange. An analysis o he % ela i e e o in slopes, in e cep s and eg ession esiduals o he di e en le els o noise and o he di e en kinds o dis ibu ions has been pe o med. The esul s a e summa ized in 3D plo o ma o ease o compa ison. Many plo s o his ype ha e been gene a ed, and he ones shown he e can be conside ed as ypical o he gene al pa e n. Figu e 5. Close-up iew o he app oxima e a ea o luc ua ion when di e en pe cen ages o uni o m noise le els a e injec ed along bo h coo dina es x, y Figu es 6, 7 and 8, show column diag ams o he ela i e e o s in he slope, in e cep and he squa ed sum o he eg ession esiduals o Uni o m ( ed) and No mal scaled (blue) dis ibu ions a di e en le els o noise o he o dina e only and o bo h coo dina es. The % o noise is indica ed in he plo s. In gene al, he ela i e e o in he in e cep is app oxima ely i e imes highe han ha in he slope, no ing he di e en scales in igu es 6 and 7. Addi ionally, he e o in he in e cep inc eases when he ex apola ion is made a om he da a se in e al. Mo e o en han no , when hese calcula ions a e epea ed many imes, i is mo e likely o ob ain highe ela i e e o s and eg ession esiduals o he uni o m han o he scaled no mal dis ibu ion. Gene ally, hese conclusions a e s ill alid when he compa ison is made using he no mal dis ibu ion wi hou Jou nal o Labo a o y Chemical Educa ion 2021, 9(2): 26-35 31 scaling wi h he uni o m dis ibu ion. Howe e , he e o le el using he no mal dis ibu ion wi hou scaling is highe and close o ha ob ained using he uni o m dis ibu ion. This beha io can be expec ed as ex eme alues unde he no mal dis ibu ion ha e ela i ely low p obabili y. In conclusion, he injec ion o andom noise in o a da a se can be use ul o gene a ing di e en da a se s o use in online es s and i ual labs. Figu e 6. Rela i e % e o in he slope pa ame e o di e en noise le els and o bo h dis ibu ions, uni o m (U) and no mal (N) Figu e 7. Rela i e % o e o in he in e cep pa ame e o di e en noise le els and o bo h dis ibu ions, uni o m (U) and no mal (N) Figu e 8. Squa ed sum o he eg ession esiduals o di e en noise le els and o bo h dis ibu ions, uni o m (U) and no mal (N) 4. Examples o Injec ion o Con olled Random Noise In his sec ion, h ee kinds o expe imen s om he unde g adua e Physical Chemis y lab will be employed as examples o he applica ion o he s a egy p esen ed abo e. 4.1. Chemical Kine ics o he Fading o Phenolph halein in S ong Alkaline Media unde Pseudo-Fi s O de Condi ions This is a classical expe imen used o eaching chemical kine ics, in ol ing he analysis o he change wi h ime o he abso bance o a phenolph halein solu ion a di e en high alkaline concen a ions. The ading is second o de o e all, pa ial o de s equal o one o each componen , OH- and he Phenolph halein anion. The goal is he de e mina ion o pa ial o de s and he second o de a e cons an . This kine ic expe imen has been pa o ou lab p og am o decades. I is ad isable o pe o m he expe imen a cons an ionic s eng h by adding di e en concen a ions o an ine sal . The gene al expe imen al condi ions ha e been ecen ly desc ibed [14], and epo ed esul s will be employed as ini ial da a se . Table 3. Excel able con aining he ini ial expe imen al condi ions and li e a u e da a o he kine ics o phenolph halein ading. The ionic s eng h is main ained a 0.435M A new able is c ea ed in Excel om he da a in able 3, by andomizing OH- concen a ions, ex inc ion coe icien and concen a ion o phenolph halein and he second o de a e cons an . The kind and he % le el o noise can be selec ed indi idually o each pa ame e . The goal is o add jus enough noise o andomize he ini ial da a se , while keeping he esul s ob ained close o li e a u e alues, wi h a ole ance le el o 20% being ad isable. Taking in o accoun ha Excel andomizes he cell con en s each ime a cell is modi ied a any posi ion on he wo kshee , i is equi ed o copy hese gene a ed alues elsewhe e on he wo kshee . In his way he new da a emain ixed o he es o he calcula ions. This inal able can be conside ed o be he da a se co esponding o he esul s o a lab expe imen . A new wo kshee is gene a ed con aining he abso bance alues o he ou alkaline concen a ions o di e en sampling imes, by using he equa ions depic ed in igu e 9. These abso bance alues, in ou case, a e andomized wi h a new pe cen age o noise, o mimic he possible luc ua ions o he abso bance du ing he ac ual measu emen p ocess. In he o iginal lab expe imen , he abso bance is au oma ically ead e e y second, and he eac ion is moni o ed o 5 min. Li e a u e Exp. Se C OH /M C Phen. /M Ex _Coe (M·cm)-1 ksd (Ms)-1 O iginal 1 0.4350 1.25E-05 30000 2.00E-02 Da a 2 0.3125 Sou ce 3 0.1875 Re . [14] 4 0.0325 32 D. González-A jona e al.: Noise-injec ion as an App oach o Gene a ing Random Da a Se s o Online Tes s and Vi ual Labs Figu e 9. Equa ions employed in he kine ic s udy o he ading o phenolph halein. The symbols ha e hei usual meanings This wo kshee , oge he wi h he li e a u e da a o he ini ial concen a ions and mola ex inc ion coe icien , a e he andomized da a o be supplied online o s uden s indi idually as an Excel ile. Al e na i ely, i he examina ion is ca ied ou ace- o- ace, as in a semina , hose abso bance alues can be sampled, ia Excel Da a Analysis, o ob ain a andomized small se o abso bance da a ha can be eadily p o ided o he s uden s. Figu e 10. Sc een sho displaying he kine ics analysis esul s. 20% noise e o added abso bance da a Figu e 10 shows a sc eensho o he Excel ou pu when 20% Gaussian scaled noise is injec ed in o he abso bance alues. Ini ially he abso bance is gene a ed using p ima y da a wi h 10% noise le el injec ed. As can be seen, e en wi h a high le el o noise injec ed, i is s ill possible o ob ain alues close o hose epo ed in li e a u e. Thus, his expe imen al design is e y obus and use ul o eaching chemical kine ics. 4.2. Es ima ion o he Limi ing Mola Conduc i i y o an Elec oly e Based on Kohl ausch’s Law Ob aining he in ini e dilu ion mola conduc i i y o an elec oly e is ano he classical expe imen in he Physical Chemis y eaching lab. The expe imen is easily ca ied ou wi h a cheap po able conduc i i y me e . In his case he s uden s ha e o ob ain a linea Kohl ausch’s ela ionship be ween he mola conduc i i y and he squa e oo o he elec oly e concen a ion and om he in e cep a ze o concen a ion, he limi ing mola conduc i i y o he elec oly e is ob ained. The compa ison wi h li e a u e da a o e s in o ma ion abou s uden ’s skills in p epa ing solu ions and hei glasswa e-cleaning p o ocols. The expe imen al da a a e he speci ic conduc i i ies o he elec oly e solu ion. The speci ic conduc i i ies o he mos dilu e solu ions ha e o be co ec ed o he con ibu ion om he sol en , usually wa e . The ini ial da a se o di e en ue elec oly es a e culled om he li e a u e, [15]. Ano he possible app oach is o gene a e he da a om he Debye-Hückel-Onsage equa ion, knowing he limi ing mola concen a ion and he cons an s A and B, see igu e 11. Figu e 11. Debye-Hückel-Onsage equa ion and pa ame e s A, B, o wa e a 25°C A se o ini ial alues o mola conduc i i ies a di e en concen a ion is selec ed, in ou case, KNO3, K2SO4, KIO3 and sodium oxala e. These concen a ions can be andomized o ob ain he di e en mola conduc i i ies. F om hem, a se o andomized speci ic conduc i i ies can be gene a ed. Fo his case, uni o m dis ibu ion- ype noise has been selec ed. The expe imen al concen a ion ange is usually om 10-4 M o 10-2 M, bu can be modi ied. The speci ic conduc i i y o low elec oly e concen a ions has o be co ec ed o he wa e con ibu ion, and his can be ano he sou ce o andomiza ion. The andomiza ion p ocess is analogous o ha used in he kine ics expe imen . A wo kshee con ains he ini ial da a, and om hese he andomized se is gene a ed. Figu es 12 and 13 show he li e a u e da a o Kohl ausch’s plo s o he selec ed sal s and hose ecalcula ed a e andomiza ion, espec i ely. Figu e 12. Li e a u e-gene a ed Kohl ausch plo s o di e en selec ed sal s Jou nal o Labo a o y Chemical Educa ion 2021, 9(2): 26-35 33 Figu e 13. Es ima ed Kohl ausch plo s o di e en sal s a e andomiza ion p oduced by adding 5% o uni o m noise o he li e a u e da a In his case, a low le el o noise should be injec ed because he pa ame e o in e es is ob ained om he in e cep , he one mos sensi i e o highe noise le els, as was illus a ed ea lie . Ne e heless, wi h his noise le el in combina ion wi h li e a u e in o ma ion, many andomized da a se s can be cons uc ed. This s a egy can be applied also o ob ain he limi ing conduc i i y o a weak elec oly e and i s dissocia ion equilib ium cons an . In his case, he pa ame e s a e ob ained by ex apola ing a om he da a se in e al. Consequen ly, special ca e should be aken in choosing he le el and ype o noise o be injec ed. 4.3. Es ima ion o he En halpy o Vapo iza ion Based on he Clausius-Clapey on Equa ion The de e mina ion o he en halpy o apo iza ion o wa e cons i u es ano he s anda d lab expe imen in he Physical Chemis y lab. The ypical expe imen is based on he measu emen o he change o he wa e apo p essu e wi h empe a u e when he apo and he liquid wa e phases a e in equilib ium. The esul s a e analyzed by using he app oxima e Clausius-Clapey on (CC) exp ession, whe e he mola olume o wa e is neglec ed wi h espec o ha o he liquid, he en halpy o apo iza ion is independen o empe a u e and ideal gas beha io is assumed o he wa e apo [16]. The CC equa ion can be o mula ed o he liquid/ apo equilib ium: 1 Ln A Vap ap H PRT     whe e P ap is he wa e apo p essu e, A is a cons an , ΔH ap is he mola en halpy o apo iza ion o wa e , assumed o be cons an in he empe a u e ange conside ed, R he gas cons an and T he empe a u e in K. In ou lab, a dis illa ion glasswa e sys em con aining wa e which can be hea ed and is connec ed o a acuum pump is employed. A simple h ee-way manually ope a ed al e allows easy con ol o he p essu e in he sys em. The a angemen pe mi s he measu emen o apo p essu e (P ap) and he empe a u e (T) da a a equilib ium, om oom empe a u e o he no mal boiling poin . Figu e 14 shows he linea plo s (ln P ap s. 1/T) acco ding o he CC equa ion ob ained by h ee s uden s wi h di e en le els o labo a o y skills. As can be no ed, he deg ee o da a sca e is highe o lowe empe a u es, whe e equilib ium condi ions a e less easy o a ain. Figu e 14. Expe imen al Clausius-Clapey on plo s based on da a, independen ly ob ained by h ee unde g adua e s uden s Li e a u e da a [15], ob ained as P ap s /°C, o he phase equilib ium be ween wa e apo /liquid a he condi ions p e ailing in he lab (25°C-100°C) a e plo ed as a CC linea plo in Fig. 15. Figu e 15. Wa e apo /liquid equilib ium Clausius-Clapey on plo om li e a u e da a [15] o he empe a u e ange employed in he lab No wi hs anding he good linea ela ionship, sligh di e ences be ween he da a poin s and he s aigh line can be no ed specially o he wo empe a u es limi s. This is ano he sign o he accu acy o he assump ions used in applying he CC equa ion. Taking in o accoun all hese conside a ions, he con olled injec ion o noise has been applied o gene a e 34 D. González-A jona e al.: Noise-injec ion as an App oach o Gene a ing Random Da a Se s o Online Tes s and Vi ual Labs andomized P ap/T da a om li e a u e sou ces. Bo h ypes o noise ha e been injec ed, he Uni o m and he No mal (log based, scaled). Di e en le els o noise ha e been applied o he expe imen al a iables: apo p essu e in kPa and empe a u e in Celsius scale, anging om 10% o 40% o P ap and 1% o 5% o empe a u e. The loga i hmic cha ac e o he o dina e a iable in he CC equa ion, in addi ion o he ac ha he pa ame e o in e es is ob ained om he slope, pe mi high le els o noise o he apo p essu e. This ac ensu es he success ul pe o mance o hese expe imen s in he unde g adua e lab. The hea o apo iza ion ob ained by a mode a ely skilled s uden is always close o he li e a u e alue, boos ing s uden ’s sel -con idence. None heless, he le el o noise applied o empe a u e alues has o be con olled ca e ully o ob ain signi ican esul s, and a maximum luc ua ion ange o 5% is ecommended. Figu e 16. Clausius-Clapey on plo s ob ained by injec ing o li e a u e da a Uni o m- and No mal- (log based, scaled) ype noise, wi h a 40% ange o P ap in kPa and 5% o empe a u e in °C Figu e 16, shows he in luence o noise injec ion in he sampled li e a u e da a om Fig. 15. The same conclusions eached in sec ion 3 apply he e. Injec ion o uni o m noise p oduces poo e linea eg ession pa ame e s han he no mal scaled noise a he same pe cen age le el. E en hough andomized Gaussian noise p oduces an ou lie , he slope ob ained is close o he li e a u e da a. The occu ence o hese ou lie alues in a da a se is no e y likely. The slopes ob ained om plo s wi h injec ed Gaussian noise a e likely o end up close o he li e a u e alue han hose ob ained wi h uni o m noise. 5. Conclusions In conclusion, some ecommenda ions can be implemen ed o andomized da a se s by using injec ion o noise. Fi s , he applica ion o noise has o be ca e ully con olled, ha is, he esul s ob ained om he analysis o he noisy da a se ha e o be ealis ic and eliable. Fo each speci ic si ua ion and se o condi ions, a check should be pe o med p io o selec ing he app op ia e amoun o noise. Secondly, i is ad isable o use uni o m (whi e) noise o da a se s ha al eady con ain na u al andom noise, such as he da a se s culled om li e a u e. And inally, no mal dis ibu ion (Gaussian) noise should be employed o da a se s gene a ed di ec ly om heo e ical o simula ed equa ions, p oducing noisy da a se s ha ha e a mo e na u al appea ance. ACKNOWLEDGEMENTS The au ho s a e indeb ed o M. A. Gue a Aguila -Galindo o his con inuous, deep in ol emen wi h and dedica ion o he Labs o he Physical Chemis y Dp . o he Uni e si y o Se ille. REFERENCES [1] Blackboa d online EdTech, Lea ning Pla o m homepage. [Online]. A ailable: h ps://www.blackboa d.com/. Accessed Ap il 2021. [2] Collabo a i e Lea ning Managemen Sys em. 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