Find the relaxation Time from data set

I am trying to find the relaxation time for this data set (x,y).

I tried fiting my data using this function, assuming that b is my relaxtion time, but seems this is not working.

a.*(exp(-x./b))

my data is:

x = [16702.7493630936	16751.4078531150	16801.0838041508	16850.6013020078	16899.8734422958	16948.9443455975	16997.8351928757	17046.5395954261	17095.0650194946	17143.4583950441	17191.6712810107	17239.6422806112	17287.2850789068	17334.5732726161	17381.5225220459	17428.1705992448	17474.5491609708	17520.6588114107	17566.5524137950	17612.2337078220	17657.7905844405	17703.2385658528	17748.6260969495	17793.9640960753	17839.2348906434	17884.4563031690	17929.6567579863	17974.8155082979	18019.9324942451	18065.0346013527	18110.1060663704	18155.1511195980	18200.1734470375	18245.1756011444	18290.1507004829	18335.1069159743	18380.0460122127	18424.9673052915	18469.8706090207	18514.7599107893	18559.6373842178	18604.5022579632	18649.3543472947	18694.1948255540	18739.0274105618	18783.8566673232	18828.6820348265	18873.5039048485	18918.3211629579	18963.1358307011	19007.9512692561	19052.7695430068	19097.5866920442	19142.4029424534	19187.2214655973	19232.0426453641	19276.8627908456	19321.6832380467	19366.5060884738	19411.3308371924	19456.1560275640	19500.9817830840	19545.8088202355	19590.6364403692	19635.4643657602	19680.2928981241	19725.1229629915	19769.9545476053	19814.7860919590	19859.6174318143	19904.4486119084	19949.2813137616	19994.1151213091	20038.9497219651	20083.7838602711	20128.6180599459	20173.4526732592	20218.2890940064	20263.1263274479	20307.9638465257	20352.8015224521	20397.6390839107	20442.4772879364	20487.3165698241	20532.1581748536	20577.0002872467	20621.8417664775	20666.6828225405	20711.5233541311	20756.3644216072	20801.2068114918	20846.0513857359	20890.8989301596	20935.7466244498	20980.5931128208	21025.4388689234	21070.2837781447	21115.1287278682	21159.9749892571	21204.8234060564	21249.6753093747	21294.5296529796	21339.3832732033	21384.2355817210	21429.0867789508	21473.9369011133	21518.7870175413	21563.6381401454	21608.4914649816	21653.3483318361	21698.2087993170	21743.0698537029	21787.9296871884	21832.7881188648	21877.6453025147	21922.5014124773	21967.3575979276	22012.2152359973	22057.0749931207	22101.9391088387	22146.8071376358	22191.6757315213	22236.5429812021	22281.4087275766	22326.2731739870	22371.1362441491	22415.9989637751	22460.8628283628	22505.7287758089	22550.5984728778	22595.4728321618	22640.3492691554	22685.2248637574	22730.0990896745	22774.9719070468	22819.8433319389	22864.7137450833	22909.5843500320	22954.4562192625	22999.3305780208	23044.2098186549	23089.0930932385	23133.9774657231	23178.8607175540	23223.7426328460	23268.6229305915	23313.5019235589	23358.3799401623	23403.2585637081	23448.1386549619	23493.0217510208	23537.9099887121	23582.8020601268	23627.6945055819	23672.5857098020	23717.4754713586	23762.3638114127	23807.2506539813	23852.1368011219	23897.0234648073	23941.9118913043];
y = [0.825202928094071	0.807688673381730	0.786252927159294	0.764714384319695	0.744080450754401	0.725285700600899	0.707145758962776	0.694979278221738	0.687888049472860	0.684850736070440	0.682806893749411	0.670795542273809	0.653376913760783	0.630400271197959	0.609703023623424	0.586893940685770	0.563682878477477	0.544106209431120	0.526411075782946	0.512283019941635	0.500736646787480	0.489211219027169	0.480595687420835	0.472809360333905	0.464923739678705	0.458962395241562	0.453084298423325	0.446864977239393	0.442529184067400	0.436423018016954	0.432552650938376	0.427019203614882	0.423792075963593	0.418407521908434	0.415395479275559	0.411350258553474	0.407597918669508	0.404226284563550	0.399999982159654	0.398133161349618	0.394646814687581	0.392111033460340	0.388570418884824	0.385464371729192	0.384119684825849	0.381289929230216	0.378928342179468	0.377090007072987	0.374434861330512	0.372670517960686	0.371066939598465	0.369292279664827	0.368022518474045	0.366761127838408	0.364614082097122	0.363388739522369	0.362625463397398	0.361620879708081	0.359478745300075	0.358272567050863	0.357787720634937	0.357453439636096	0.355735954402435	0.354263227197586	0.353458943048023	0.353065491418747	0.352991630163461	0.351542543972645	0.350121854608277	0.349339282376799	0.348896599572894	0.348692894990009	0.347844884276511	0.346776039775459	0.345824648247738	0.345272768480164	0.344963616656247	0.344875984204123	0.344308107604026	0.343265678404013	0.342507988033502	0.342160594662456	0.341849913300415	0.341659961087672	0.341611550747868	0.341083861844750	0.340137080650901	0.339494350647239	0.338962730267028	0.339069786421710	0.338918740876603	0.338947198743886	0.338900608649134	0.338280386209996	0.337459811476076	0.336941408959965	0.336677113754609	0.336655864432466	0.336728275737625	0.336789857219186	0.336956358307026	0.336784572249854	0.335986760984470	0.335276345610378	0.334913435678406	0.334753522628267	0.334747877501068	0.334991966112970	0.335055656693706	0.335393211860271	0.335404811280965	0.334839534592180	0.334032980036375	0.333554288722820	0.333412575631509	0.333407819168266	0.333512673422225	0.333710166501048	0.333870232273114	0.334270755120429	0.334345491764824	0.333734812733033	0.333012535707686	0.332556603367981	0.332433030563140	0.332477392294188	0.332497477821169	0.332884760554059	0.332991133592755	0.333402508157405	0.333698661877838	0.333443704777846	0.332576323356287	0.332016561638242	0.331785443171980	0.331847205581190	0.332018315288797	0.332218423905214	0.332433126318607	0.332713972256815	0.333193786924492	0.333304451452445	0.332757377898790	0.332034712640962	0.331583072571160	0.331483112096829	0.331575236767098	0.331712758306550	0.332004699402236	0.332155297554711	0.332504534122397	0.333007337862399	0.333010718964975	0.332380641128667	0.331694079591903	0.331331179422750	0.331258693346581	0.331415140544247	0.331538939655960	0.331818149307138	0.332003584473075];

Hope someone has a solution

Answer 1

[Star Strider]

See my latest Comment in find fitting equation and parameters to fit damped oscillations curves from set of data.

[Tesla]

Indeed I applied your script on those (x,y) but its not working.

x = [16702.7493630936	16751.4078531150	16801.0838041508	16850.6013020078	16899.8734422958	16948.9443455975	16997.8351928757	17046.5395954261	17095.0650194946	17143.4583950441	17191.6712810107	17239.6422806112	17287.2850789068	17334.5732726161	17381.5225220459	17428.1705992448	17474.5491609708	17520.6588114107	17566.5524137950	17612.2337078220	17657.7905844405	17703.2385658528	17748.6260969495	17793.9640960753	17839.2348906434	17884.4563031690	17929.6567579863	17974.8155082979	18019.9324942451	18065.0346013527	18110.1060663704	18155.1511195980	18200.1734470375	18245.1756011444	18290.1507004829	18335.1069159743	18380.0460122127	18424.9673052915	18469.8706090207	18514.7599107893	18559.6373842178	18604.5022579632	18649.3543472947	18694.1948255540	18739.0274105618	18783.8566673232	18828.6820348265	18873.5039048485	18918.3211629579	18963.1358307011	19007.9512692561	19052.7695430068	19097.5866920442	19142.4029424534	19187.2214655973	19232.0426453641	19276.8627908456	19321.6832380467	19366.5060884738	19411.3308371924	19456.1560275640	19500.9817830840	19545.8088202355	19590.6364403692	19635.4643657602	19680.2928981241	19725.1229629915	19769.9545476053	19814.7860919590	19859.6174318143	19904.4486119084	19949.2813137616	19994.1151213091	20038.9497219651	20083.7838602711	20128.6180599459	20173.4526732592	20218.2890940064	20263.1263274479	20307.9638465257	20352.8015224521	20397.6390839107	20442.4772879364	20487.3165698241	20532.1581748536	20577.0002872467	20621.8417664775	20666.6828225405	20711.5233541311	20756.3644216072	20801.2068114918	20846.0513857359	20890.8989301596	20935.7466244498	20980.5931128208	21025.4388689234	21070.2837781447	21115.1287278682	21159.9749892571	21204.8234060564	21249.6753093747	21294.5296529796	21339.3832732033	21384.2355817210	21429.0867789508	21473.9369011133	21518.7870175413	21563.6381401454	21608.4914649816	21653.3483318361	21698.2087993170	21743.0698537029	21787.9296871884	21832.7881188648	21877.6453025147	21922.5014124773	21967.3575979276	22012.2152359973	22057.0749931207	22101.9391088387	22146.8071376358	22191.6757315213	22236.5429812021	22281.4087275766	22326.2731739870	22371.1362441491	22415.9989637751	22460.8628283628	22505.7287758089	22550.5984728778	22595.4728321618	22640.3492691554	22685.2248637574	22730.0990896745	22774.9719070468	22819.8433319389	22864.7137450833	22909.5843500320	22954.4562192625	22999.3305780208	23044.2098186549	23089.0930932385	23133.9774657231	23178.8607175540	23223.7426328460	23268.6229305915	23313.5019235589	23358.3799401623	23403.2585637081	23448.1386549619	23493.0217510208	23537.9099887121	23582.8020601268	23627.6945055819	23672.5857098020	23717.4754713586	23762.3638114127	23807.2506539813	23852.1368011219	23897.0234648073	23941.9118913043];
y = [0.825202928094071	0.807688673381730	0.786252927159294	0.764714384319695	0.744080450754401	0.725285700600899	0.707145758962776	0.694979278221738	0.687888049472860	0.684850736070440	0.682806893749411	0.670795542273809	0.653376913760783	0.630400271197959	0.609703023623424	0.586893940685770	0.563682878477477	0.544106209431120	0.526411075782946	0.512283019941635	0.500736646787480	0.489211219027169	0.480595687420835	0.472809360333905	0.464923739678705	0.458962395241562	0.453084298423325	0.446864977239393	0.442529184067400	0.436423018016954	0.432552650938376	0.427019203614882	0.423792075963593	0.418407521908434	0.415395479275559	0.411350258553474	0.407597918669508	0.404226284563550	0.399999982159654	0.398133161349618	0.394646814687581	0.392111033460340	0.388570418884824	0.385464371729192	0.384119684825849	0.381289929230216	0.378928342179468	0.377090007072987	0.374434861330512	0.372670517960686	0.371066939598465	0.369292279664827	0.368022518474045	0.366761127838408	0.364614082097122	0.363388739522369	0.362625463397398	0.361620879708081	0.359478745300075	0.358272567050863	0.357787720634937	0.357453439636096	0.355735954402435	0.354263227197586	0.353458943048023	0.353065491418747	0.352991630163461	0.351542543972645	0.350121854608277	0.349339282376799	0.348896599572894	0.348692894990009	0.347844884276511	0.346776039775459	0.345824648247738	0.345272768480164	0.344963616656247	0.344875984204123	0.344308107604026	0.343265678404013	0.342507988033502	0.342160594662456	0.341849913300415	0.341659961087672	0.341611550747868	0.341083861844750	0.340137080650901	0.339494350647239	0.338962730267028	0.339069786421710	0.338918740876603	0.338947198743886	0.338900608649134	0.338280386209996	0.337459811476076	0.336941408959965	0.336677113754609	0.336655864432466	0.336728275737625	0.336789857219186	0.336956358307026	0.336784572249854	0.335986760984470	0.335276345610378	0.334913435678406	0.334753522628267	0.334747877501068	0.334991966112970	0.335055656693706	0.335393211860271	0.335404811280965	0.334839534592180	0.334032980036375	0.333554288722820	0.333412575631509	0.333407819168266	0.333512673422225	0.333710166501048	0.333870232273114	0.334270755120429	0.334345491764824	0.333734812733033	0.333012535707686	0.332556603367981	0.332433030563140	0.332477392294188	0.332497477821169	0.332884760554059	0.332991133592755	0.333402508157405	0.333698661877838	0.333443704777846	0.332576323356287	0.332016561638242	0.331785443171980	0.331847205581190	0.332018315288797	0.332218423905214	0.332433126318607	0.332713972256815	0.333193786924492	0.333304451452445	0.332757377898790	0.332034712640962	0.331583072571160	0.331483112096829	0.331575236767098	0.331712758306550	0.332004699402236	0.332155297554711	0.332504534122397	0.333007337862399	0.333010718964975	0.332380641128667	0.331694079591903	0.331331179422750	0.331258693346581	0.331415140544247	0.331538939655960	0.331818149307138	0.332003584473075];

[Ux30, ix1, ix2] = unique(x, 'stable');
USlope30 = y(ix1);
Ux30 = x(ix1);

[minv,idx1] = min(USlope30);                                                % Minimum & First Index
idx2 = find(USlope30(idx1:end) < -1E-4, 1, 'last') + idx1;                  % Second Index
rlxv = exp(-1)                                                              % Relaxation Constant

rlxv = 0.3679

rlxt = interp1(USlope30(idx1:idx2), Ux30(idx1:idx2), rlxv*minv, 'pchip')    % Relaxation Time

Error using matlab.internal.math.interp1
Interpolation requires at least two sample points for each grid dimension.

Error in interp1 (line 188)
VqLite = matlab.internal.math.interp1(X,V,method,method,Xqcol);

figure
plot(Ux30, USlope30, '.-b', 'DisplayName','Unique Data')
hold on
plot(rlxt, rlxv*minv, 'sr', 'DisplayName','Relaxation Time')
hold off
grid
legend('Location','best')

[Star Strider]

These are different data, so this is a different problem.

Try this —

x = [16702.7493630936	16751.4078531150	16801.0838041508	16850.6013020078	16899.8734422958	16948.9443455975	16997.8351928757	17046.5395954261	17095.0650194946	17143.4583950441	17191.6712810107	17239.6422806112	17287.2850789068	17334.5732726161	17381.5225220459	17428.1705992448	17474.5491609708	17520.6588114107	17566.5524137950	17612.2337078220	17657.7905844405	17703.2385658528	17748.6260969495	17793.9640960753	17839.2348906434	17884.4563031690	17929.6567579863	17974.8155082979	18019.9324942451	18065.0346013527	18110.1060663704	18155.1511195980	18200.1734470375	18245.1756011444	18290.1507004829	18335.1069159743	18380.0460122127	18424.9673052915	18469.8706090207	18514.7599107893	18559.6373842178	18604.5022579632	18649.3543472947	18694.1948255540	18739.0274105618	18783.8566673232	18828.6820348265	18873.5039048485	18918.3211629579	18963.1358307011	19007.9512692561	19052.7695430068	19097.5866920442	19142.4029424534	19187.2214655973	19232.0426453641	19276.8627908456	19321.6832380467	19366.5060884738	19411.3308371924	19456.1560275640	19500.9817830840	19545.8088202355	19590.6364403692	19635.4643657602	19680.2928981241	19725.1229629915	19769.9545476053	19814.7860919590	19859.6174318143	19904.4486119084	19949.2813137616	19994.1151213091	20038.9497219651	20083.7838602711	20128.6180599459	20173.4526732592	20218.2890940064	20263.1263274479	20307.9638465257	20352.8015224521	20397.6390839107	20442.4772879364	20487.3165698241	20532.1581748536	20577.0002872467	20621.8417664775	20666.6828225405	20711.5233541311	20756.3644216072	20801.2068114918	20846.0513857359	20890.8989301596	20935.7466244498	20980.5931128208	21025.4388689234	21070.2837781447	21115.1287278682	21159.9749892571	21204.8234060564	21249.6753093747	21294.5296529796	21339.3832732033	21384.2355817210	21429.0867789508	21473.9369011133	21518.7870175413	21563.6381401454	21608.4914649816	21653.3483318361	21698.2087993170	21743.0698537029	21787.9296871884	21832.7881188648	21877.6453025147	21922.5014124773	21967.3575979276	22012.2152359973	22057.0749931207	22101.9391088387	22146.8071376358	22191.6757315213	22236.5429812021	22281.4087275766	22326.2731739870	22371.1362441491	22415.9989637751	22460.8628283628	22505.7287758089	22550.5984728778	22595.4728321618	22640.3492691554	22685.2248637574	22730.0990896745	22774.9719070468	22819.8433319389	22864.7137450833	22909.5843500320	22954.4562192625	22999.3305780208	23044.2098186549	23089.0930932385	23133.9774657231	23178.8607175540	23223.7426328460	23268.6229305915	23313.5019235589	23358.3799401623	23403.2585637081	23448.1386549619	23493.0217510208	23537.9099887121	23582.8020601268	23627.6945055819	23672.5857098020	23717.4754713586	23762.3638114127	23807.2506539813	23852.1368011219	23897.0234648073	23941.9118913043];
y = [0.825202928094071	0.807688673381730	0.786252927159294	0.764714384319695	0.744080450754401	0.725285700600899	0.707145758962776	0.694979278221738	0.687888049472860	0.684850736070440	0.682806893749411	0.670795542273809	0.653376913760783	0.630400271197959	0.609703023623424	0.586893940685770	0.563682878477477	0.544106209431120	0.526411075782946	0.512283019941635	0.500736646787480	0.489211219027169	0.480595687420835	0.472809360333905	0.464923739678705	0.458962395241562	0.453084298423325	0.446864977239393	0.442529184067400	0.436423018016954	0.432552650938376	0.427019203614882	0.423792075963593	0.418407521908434	0.415395479275559	0.411350258553474	0.407597918669508	0.404226284563550	0.399999982159654	0.398133161349618	0.394646814687581	0.392111033460340	0.388570418884824	0.385464371729192	0.384119684825849	0.381289929230216	0.378928342179468	0.377090007072987	0.374434861330512	0.372670517960686	0.371066939598465	0.369292279664827	0.368022518474045	0.366761127838408	0.364614082097122	0.363388739522369	0.362625463397398	0.361620879708081	0.359478745300075	0.358272567050863	0.357787720634937	0.357453439636096	0.355735954402435	0.354263227197586	0.353458943048023	0.353065491418747	0.352991630163461	0.351542543972645	0.350121854608277	0.349339282376799	0.348896599572894	0.348692894990009	0.347844884276511	0.346776039775459	0.345824648247738	0.345272768480164	0.344963616656247	0.344875984204123	0.344308107604026	0.343265678404013	0.342507988033502	0.342160594662456	0.341849913300415	0.341659961087672	0.341611550747868	0.341083861844750	0.340137080650901	0.339494350647239	0.338962730267028	0.339069786421710	0.338918740876603	0.338947198743886	0.338900608649134	0.338280386209996	0.337459811476076	0.336941408959965	0.336677113754609	0.336655864432466	0.336728275737625	0.336789857219186	0.336956358307026	0.336784572249854	0.335986760984470	0.335276345610378	0.334913435678406	0.334753522628267	0.334747877501068	0.334991966112970	0.335055656693706	0.335393211860271	0.335404811280965	0.334839534592180	0.334032980036375	0.333554288722820	0.333412575631509	0.333407819168266	0.333512673422225	0.333710166501048	0.333870232273114	0.334270755120429	0.334345491764824	0.333734812733033	0.333012535707686	0.332556603367981	0.332433030563140	0.332477392294188	0.332497477821169	0.332884760554059	0.332991133592755	0.333402508157405	0.333698661877838	0.333443704777846	0.332576323356287	0.332016561638242	0.331785443171980	0.331847205581190	0.332018315288797	0.332218423905214	0.332433126318607	0.332713972256815	0.333193786924492	0.333304451452445	0.332757377898790	0.332034712640962	0.331583072571160	0.331483112096829	0.331575236767098	0.331712758306550	0.332004699402236	0.332155297554711	0.332504534122397	0.333007337862399	0.333010718964975	0.332380641128667	0.331694079591903	0.331331179422750	0.331258693346581	0.331415140544247	0.331538939655960	0.331818149307138	0.332003584473075];

[Ux30, ix1, ix2] = unique(x, 'stable');
USlope30 = y(ix1);
Ux30 = x(ix1);

[maxv,idx1] = max(USlope30)                                                % Maximum & First Index

maxv = 0.8252

idx1 = 1

[minv,idx2] = min(USlope30)                                                % Minimum & second Index

minv = 0.3313

idx2 = 157

rlxv = exp(-1)                                                              % Relaxation Constant

rlxv = 0.3679

rlxd = rlxv*(maxv-minv) + minv

rlxd = 0.5130

rlxt = interp1(USlope30, Ux30, rlxd, 'linear')    % Relaxation Time

rlxt = 1.7610e+04

figure
plot(Ux30, USlope30, '-b', 'DisplayName','Unique Data')
hold on
plot(rlxt, rlxd, 'sr', 'DisplayName','Relaxation Time')
hold off
grid
legend('Location','best')

It is necessary to restrict the calculations and interpolation to the range of the data (between ‘minv’ and ‘maxv’) in order to calculate the relaxation time correctly. The ‘rlxd’ value is the relaxation value for these data, and ‘rlxt’ the interpolated relaxation time.

.

[Tesla]

I see now. But do you think 17610 is a relaxation time, because I expect it would be around 21000, where the courbe is stabilized

[Star Strider]

I am using the data provided, and am calculating the relaxation time based on those data.

How do you define ‘relaxation time’? When I looked it up (to be certain that I remembered it correctly) it was defined as or about 0.368 of the mximum value. The curve appears to be asymptotic to about 3.2, so I used that range to calculate it.

Doing an exponential regression —

x = [16702.7493630936	16751.4078531150	16801.0838041508	16850.6013020078	16899.8734422958	16948.9443455975	16997.8351928757	17046.5395954261	17095.0650194946	17143.4583950441	17191.6712810107	17239.6422806112	17287.2850789068	17334.5732726161	17381.5225220459	17428.1705992448	17474.5491609708	17520.6588114107	17566.5524137950	17612.2337078220	17657.7905844405	17703.2385658528	17748.6260969495	17793.9640960753	17839.2348906434	17884.4563031690	17929.6567579863	17974.8155082979	18019.9324942451	18065.0346013527	18110.1060663704	18155.1511195980	18200.1734470375	18245.1756011444	18290.1507004829	18335.1069159743	18380.0460122127	18424.9673052915	18469.8706090207	18514.7599107893	18559.6373842178	18604.5022579632	18649.3543472947	18694.1948255540	18739.0274105618	18783.8566673232	18828.6820348265	18873.5039048485	18918.3211629579	18963.1358307011	19007.9512692561	19052.7695430068	19097.5866920442	19142.4029424534	19187.2214655973	19232.0426453641	19276.8627908456	19321.6832380467	19366.5060884738	19411.3308371924	19456.1560275640	19500.9817830840	19545.8088202355	19590.6364403692	19635.4643657602	19680.2928981241	19725.1229629915	19769.9545476053	19814.7860919590	19859.6174318143	19904.4486119084	19949.2813137616	19994.1151213091	20038.9497219651	20083.7838602711	20128.6180599459	20173.4526732592	20218.2890940064	20263.1263274479	20307.9638465257	20352.8015224521	20397.6390839107	20442.4772879364	20487.3165698241	20532.1581748536	20577.0002872467	20621.8417664775	20666.6828225405	20711.5233541311	20756.3644216072	20801.2068114918	20846.0513857359	20890.8989301596	20935.7466244498	20980.5931128208	21025.4388689234	21070.2837781447	21115.1287278682	21159.9749892571	21204.8234060564	21249.6753093747	21294.5296529796	21339.3832732033	21384.2355817210	21429.0867789508	21473.9369011133	21518.7870175413	21563.6381401454	21608.4914649816	21653.3483318361	21698.2087993170	21743.0698537029	21787.9296871884	21832.7881188648	21877.6453025147	21922.5014124773	21967.3575979276	22012.2152359973	22057.0749931207	22101.9391088387	22146.8071376358	22191.6757315213	22236.5429812021	22281.4087275766	22326.2731739870	22371.1362441491	22415.9989637751	22460.8628283628	22505.7287758089	22550.5984728778	22595.4728321618	22640.3492691554	22685.2248637574	22730.0990896745	22774.9719070468	22819.8433319389	22864.7137450833	22909.5843500320	22954.4562192625	22999.3305780208	23044.2098186549	23089.0930932385	23133.9774657231	23178.8607175540	23223.7426328460	23268.6229305915	23313.5019235589	23358.3799401623	23403.2585637081	23448.1386549619	23493.0217510208	23537.9099887121	23582.8020601268	23627.6945055819	23672.5857098020	23717.4754713586	23762.3638114127	23807.2506539813	23852.1368011219	23897.0234648073	23941.9118913043];
y = [0.825202928094071	0.807688673381730	0.786252927159294	0.764714384319695	0.744080450754401	0.725285700600899	0.707145758962776	0.694979278221738	0.687888049472860	0.684850736070440	0.682806893749411	0.670795542273809	0.653376913760783	0.630400271197959	0.609703023623424	0.586893940685770	0.563682878477477	0.544106209431120	0.526411075782946	0.512283019941635	0.500736646787480	0.489211219027169	0.480595687420835	0.472809360333905	0.464923739678705	0.458962395241562	0.453084298423325	0.446864977239393	0.442529184067400	0.436423018016954	0.432552650938376	0.427019203614882	0.423792075963593	0.418407521908434	0.415395479275559	0.411350258553474	0.407597918669508	0.404226284563550	0.399999982159654	0.398133161349618	0.394646814687581	0.392111033460340	0.388570418884824	0.385464371729192	0.384119684825849	0.381289929230216	0.378928342179468	0.377090007072987	0.374434861330512	0.372670517960686	0.371066939598465	0.369292279664827	0.368022518474045	0.366761127838408	0.364614082097122	0.363388739522369	0.362625463397398	0.361620879708081	0.359478745300075	0.358272567050863	0.357787720634937	0.357453439636096	0.355735954402435	0.354263227197586	0.353458943048023	0.353065491418747	0.352991630163461	0.351542543972645	0.350121854608277	0.349339282376799	0.348896599572894	0.348692894990009	0.347844884276511	0.346776039775459	0.345824648247738	0.345272768480164	0.344963616656247	0.344875984204123	0.344308107604026	0.343265678404013	0.342507988033502	0.342160594662456	0.341849913300415	0.341659961087672	0.341611550747868	0.341083861844750	0.340137080650901	0.339494350647239	0.338962730267028	0.339069786421710	0.338918740876603	0.338947198743886	0.338900608649134	0.338280386209996	0.337459811476076	0.336941408959965	0.336677113754609	0.336655864432466	0.336728275737625	0.336789857219186	0.336956358307026	0.336784572249854	0.335986760984470	0.335276345610378	0.334913435678406	0.334753522628267	0.334747877501068	0.334991966112970	0.335055656693706	0.335393211860271	0.335404811280965	0.334839534592180	0.334032980036375	0.333554288722820	0.333412575631509	0.333407819168266	0.333512673422225	0.333710166501048	0.333870232273114	0.334270755120429	0.334345491764824	0.333734812733033	0.333012535707686	0.332556603367981	0.332433030563140	0.332477392294188	0.332497477821169	0.332884760554059	0.332991133592755	0.333402508157405	0.333698661877838	0.333443704777846	0.332576323356287	0.332016561638242	0.331785443171980	0.331847205581190	0.332018315288797	0.332218423905214	0.332433126318607	0.332713972256815	0.333193786924492	0.333304451452445	0.332757377898790	0.332034712640962	0.331583072571160	0.331483112096829	0.331575236767098	0.331712758306550	0.332004699402236	0.332155297554711	0.332504534122397	0.333007337862399	0.333010718964975	0.332380641128667	0.331694079591903	0.331331179422750	0.331258693346581	0.331415140544247	0.331538939655960	0.331818149307138	0.332003584473075];

fcn = @(b,x) b(1) .* exp(-b(2).*(x+b(3))) + b(4);

format longE
[B,fv] = fminsearch(@(b)norm(y-fcn(b,x)), [y(1)-y(end); 1E-5; x(1); y(end)])

 
Exiting: Maximum number of function evaluations has been exceeded
         - increase MaxFunEvals option.
         Current function value: 0.125979 

B = 4×1

1.0e+00 * 1.913972897204954e+01 1.098162124832478e-03 -1.342996988512001e+04 3.316783006640941e-01

fv = 
     1.259793255107448e-01

rlxv = exp(-1)                                                              % Relaxation Constant

rlxv = 
     3.678794411714423e-01

rlxd = rlxv*(y(1)-y(end))+y(end)

rlxd = 
     5.134414833904892e-01

rlxt = fzero(@(x)fcn(B,x)-rlxd, 1E+3)

rlxt = 
     1.767052485722764e+04

figure
plot(x, y, '.', 'DisplayName','Data')
hold on
plot(x, fcn(B,x), '-r', 'DisplayName','Exponential Regression')
plot(rlxt, fcn(B,rlxt), 'sr', 'DisplayName','Relaxation Time')
hold off
grid
legend('Location','best')

Solving the exponential regression for the relaxation value (‘rlxd’) produces a similar result to the interpolated value.

EDIT — (27 Sep 2022 at 12:18)

If you want to extrapolate the data (although I do not recommend that) to get a relaxation time that is quite literally ‘off the chart’, choose an appropriate method and see with the results are —

lrlxt = interp1(y, x, rlxv*max(y), 'linear', 'extrap')

lrlxt = 
     4.090588276293874e+04

mrlxt = interp1(y, x, rlxv*max(y), 'makima', 'extrap')

mrlxt = 
     5.581972955859375e+08

prlxt = interp1(y, x, rlxv*max(y), 'pchip', 'extrap')

prlxt = 
     6.684953091031247e+08

Other method options are possible as well.

.

[Tesla]

I see your point, that was so helpful. Thank you very much

[Star Strider]

As always, my pleasure!