How to find the vector b if we know the RMSE?

If we have two vectors given by:

a=[3,6,8,20,35,45];
b=[3.0343, 6.2725, 8.5846, 18.3781, 34.2025, 44.9699];

Then its Mean Square Error MSE and Root Mean Sqaure Error RMSE are given by:

MSE = mean((a-b).^2,2);
RMSE = sqrt(MSE);

But if we know MSE and RMSE and one of the vector namely 'a', then how to find the 2nd vector b?

Answer 1

[Matt J]

That's not possible. You have, in this case, 6 unknowns but only 1 equation.

[Matt J]

Here's one simple solution for b:

a=[3,6,8,20,35,45];

RMSE=3.7;

b=a+RMSE;

rootMeanSquaredError = sqrt( mean((a-b).^2) )   %verify

rootMeanSquaredError = 3.7000

Answer 2

[John D'Errico]

There are infinitely many possible vectors b, for any given RMSE. And worse, they can have infinitely many possible shapes. This means it is flatly not possible to find a unique vector b that yields a given RMSE. Sorry.

Do you want proof?

a=[3,6,8,20,35,45];
b=[3.0343, 6.2725, 8.5846, 18.3781, 34.2025, 44.9699];

For example consider this simple vector b1:

n = length(a);
RMSEfun = @(b) sqrt(sum((a - b).^2/n));
syms x
rmsetarget = 1;

b1 = sym(a); b1(1) = x; % I will change only the first elememt of b
x1 = vpasolve(RMSEfun(b1) == rmsetarget,x)

x1 = 

0.55051025721682190180271592529411

b1 = double(subs(b1,x,x1))

b1 = 1×6

0.5505 6.0000 8.0000 20.0000 35.0000 45.0000

RMSEfun(b1)

ans = 1.0000

So by trivially changing one arbitrary element of a, I found a new vector b that yields exactly the desired RMSE. I could have perturbed ANY element and gotten the same result.

b2 = sym(a); b2(4) = x; % I will change only the first elememt of b
x2 = vpasolve(RMSEfun(b2) == rmsetarget,x)

x2 = 

17.550510257216821901802715925294

b2 = double(subs(b2,x,x2))

b2 = 1×6

3.0000 6.0000 8.0000 17.5505 35.0000 45.0000

RMSEfun(b2)

ans = 1.0000

Or, I might have chosen b in a different way.

b3 = sym(a); b3 = b3*x; % I will change EVERY element of b, proportionally
x3 = vpasolve(RMSEfun(b3) == rmsetarget,x)

x3 = 

0.96004791377243790210022857429935

b3 = double(subs(b3,x,x3))

b3 = 1×6

2.8801 5.7603 7.6804 19.2010 33.6017 43.2022

RMSEfun(b3)

ans = 1.0000

Again, there are infinitely many solutions. I chose only 3 trivial examples.

[Sadiq Akbar]

Thanks a lot dear Matt J and John D'Errico for your kind responses. Ok If we have vector 'a' as above and we have a matrix C like below:

a=[3,6,8,20,35,45];
C=[3.06181363232819	5.96396501049181	8.25079270073043	19.3505885706877	34.7265808000449	44.8913140030355
3.64001925634431	5.98364764723370	7.12130211241516	22.8057608404000	36.1740001634801	45.3970372885559
3.17011616342810	6.05427732183907	7.91500668708759	19.9901769028977	35.1392373306947	45.0414427941172
3.49623311262451	5.85707368306556	7.22998790626857	23.3316283132859	36.0831540000424	45.2112712565938
2.92047890619507	5.98073992423870	7.34629031683134	21.2274196556183	35.6133968156586	45.2249296263882
3.28122649019689	6.02197261480197	7.41012412411829	21.8214811093497	35.8433664638944	45.1754763409129
3.05055366989385	5.84500765946933	7.61619649499705	20.3444163443428	35.1600477907806	45.0203965490118
3.35137473414291	6.10591183779609	7.32913457862805	21.9676479850177	36.2969052709137	45.4852890551514
2.99792332258418	6.19555775159305	8.23992097434602	18.1929809703303	34.4571498128698	44.9129222625995
2.70762350515748	5.71793601618055	7.99634975738420	20.9545663754584	34.9282939099770	44.8266237805697
2.72993580293976	5.97668202364062	7.90347847954929	19.8450798182115	34.8470433236845	44.7384449308857
3.52252967239948	6.25064919405331	7.30104417950632	22.7736601823865	36.2583066749613	45.4735956835323
3.58799168878030	6.16090800852346	7.79338617886263	18.6363451925863	34.6056000650003	45.0579200938259
2.89980201196849	6.17299107034898	7.96656134906137	19.1264729797469	34.7318506628196	44.9088362540612
2.80641348052595	5.85937240368011	8.36046805043236	17.9523361529969	34.3210666838908	44.5816784232150
3.35108056904191	6.02600494921349	7.10138149096324	23.3582711432715	36.9285417417436	45.6235211401919
3.40930601706609	5.46074415567056	7.35724139097203	22.2846204622025	36.5023931695037	45.5367023417295
2.82041760258753	5.74391909232112	8.02198530637453	19.6817967543074	35.0389171515998	44.6992654809265
3.25958459443168	5.99361694362547	7.35255548288549	21.7677612028838	36.1524948175466	45.4770017095515
3.51114112386012	6.21148662476894	7.24964397380967	22.1982330695283	36.3063556903631	45.5239357081812
2.29584903911312	5.83549400284958	8.54054844745186	18.5537438829656	34.0853573877348	44.5350320841707
3.61218359357534	5.85147244384986	7.99202838603309	21.0812059027521	35.0824577747278	44.9500404294298
3.55377981739408	6.08975103162044	6.48702470271272	25.2885798683930	37.8306434788573	46.1055286331553
2.24245142507239	6.11915900360233	8.49006459096238	15.4481575399741	33.1797482761758	44.3782537528582
3.26564292723495	6.09052787524414	7.95176634970064	21.7575437650067	35.5791040128730	45.1802246755359
2.48626946032580	6.01819868762952	8.23437709281147	18.6786644149176	34.3924265721998	44.7174243468455
3.04474387497122	5.99632070648553	7.72168979585755	21.2822602414133	35.0917061049533	45.0999309780041
2.97132191442379	5.77809404551583	7.53511830859875	22.7227532111206	35.9170757823308	45.0668787918347
3.43571877641989	5.90022479161200	7.86354828844215	21.1146979172011	35.4986177116211	45.1800685051214
3.36342216734938	5.91413292832505	7.66211483090569	19.9467835547251	35.1325939287918	44.9803752275789
2.99114623645280	5.77436749773500	7.73875951025142	20.8110178555897	34.9166197510954	44.8521266732482
3.74788699378821	6.09077875356616	6.65878127532021	23.4578457136708	37.0161347867600	45.8588086413667
3.53960487171837	6.08873356596946	6.73644916679078	22.8308974217458	36.8948813771955	45.6902746612877
2.64095118721226	5.93482635110647	7.87164798158551	19.1998896662842	34.3033821630704	44.5493763520004
3.20620519654466	5.99037026523446	7.56775240647675	21.6637252749642	35.8933215291068	45.4240766541374
2.90757183520566	5.94984883453035	7.79431793181772	19.8310792707498	34.8809789520528	44.9724156686921
3.14127233980399	6.14035723192385	7.42691470227650	23.0647333785142	35.6336203979671	45.2455577737031
3.37231538168801	6.00604735034748	8.13531894178816	20.1381803643340	35.2739743173277	45.1642479318647
3.05409695118852	6.13175079188580	8.05311990854475	19.1457183327093	34.7266641695383	44.9376513790470
2.83923126708206	6.01802157985948	8.01630562859383	19.7023750712290	34.7288880986983	44.7248446241133
3.21650812888087	5.84012971332292	7.97756081678195	20.4116492893986	35.2246819834515	45.0960279732789
2.47436621845811	5.64663754798863	7.98277511873552	20.8657490985375	34.9490184013146	45.0052042302480
3.33077916258149	6.17202310810380	7.71752912195908	21.2369438540115	35.4844036732228	45.2401484923104
3.51836980671015	6.26907798512552	7.47611827970451	21.8629128188055	35.6340883270279	45.3898335266407
3.11787502230456	5.53497495094766	7.37478973469679	21.5908786697722	36.2190579579977	45.2699751973316
3.41687682208616	5.98773658182777	8.37484252814834	21.2092966255901	35.6716459124982	45.2699527417330
2.96466922968465	6.16393956479268	8.04159422356011	20.6684303274655	35.4278420310650	45.2084435902796
3.32474182746288	6.04741718104917	6.37747247657930	26.1190233305269	37.6566544026611	45.8879752141604
2.75600773839185	6.06846441388402	7.64308244981665	21.6352718011813	35.7491634975853	45.3197138478112
2.90776150033578	5.95585351321632	7.59244820802842	22.6069994030822	36.2554360950468	45.4642258642193
3.30661696996288	5.85137325215910	7.49047516990492	21.4550750391531	35.6636829506993	45.1592258431526
3.29681093918724	5.97381195653750	7.84010834147814	20.3548509211370	35.2551081094494	45.1385757677216
2.77059240289126	5.87701489831775	8.01023522712290	17.8686443734179	34.1147673132632	44.6224480898093
3.19087794514061	6.03406058821330	8.21016596305990	20.0598994023779	34.9031485269876	45.0100576935752
2.77419370832795	5.73578361735032	7.93565309751826	19.3270418326096	34.6891553019825	44.9661943488721
3.35371293975863	6.55495527031325	7.57550437758558	20.9725407438628	35.6462049864952	45.3380349598383
3.03024988571427	6.12954274986963	8.12933028160170	22.6853072854205	36.2019355410636	45.4525592454471
2.82880230213902	6.08628346540213	8.14937526855258	20.4898792850174	34.9823877851767	44.8734336101725
3.59809358151616	5.93095732502467	7.30550902656001	22.9657557804347	36.4822975437894	45.5349864431456
3.27893548966474	5.95686576174613	7.19490078050851	23.6727651860279	37.0651562572697	45.8623307523661
3.74970191843468	6.33374448808416	6.71895893016287	23.5300187531225	37.3100472455620	45.9953672890887
2.89820634763269	6.11084839071748	7.93358074224729	21.4127856322212	35.5521107686301	45.1974030181117
3.48302053562852	6.21468796903211	7.65730070858571	22.1177465597439	35.9568194563380	45.6122581225029
3.65060217406274	5.80741627134660	7.86741030155943	20.9124338714913	35.3228734375037	45.0070429607647
3.11560272645808	5.82692849886746	8.41831361085076	19.8196219409120	34.8352250855341	44.9505352973710
3.83051511617102	6.28347059234528	6.62587680015654	24.4769298597694	37.4119999202422	45.8228972232651
2.65079093356038	5.48112592826721	7.73991838133814	21.9886077847821	35.6838413286826	45.0149434093148
2.73731810473870	5.74214668513713	7.58743663290870	22.9103735101729	35.8672347836476	45.2277615671829
3.40201605544853	6.27790095056006	7.12842444790291	22.9429694365717	36.5653856735330	45.6774176349821
3.94018199667914	5.84248455577022	6.65154139649531	25.1242425768365	37.9365809615722	45.8298359769128
3.32575201997402	5.93365733110741	7.81554398750806	21.8481000230686	35.3637232308201	45.1020930797667
2.90019786899456	5.93724497780101	7.78052654122994	19.1000347354423	34.6715666102677	44.7428208044880
3.35233919868015	5.32627874068234	6.29351713003076	24.8600653190066	37.7033727189708	45.8048963414889
2.85688555580955	5.75494649211542	8.53246809462648	17.8499224906123	33.9037870754662	44.4794449790230
2.88075164671379	6.02070698097154	7.70249566232490	21.7394861208954	35.4146575894307	45.1574646680885
3.29130780260455	6.24847008395789	7.42804606437577	22.2079506918747	36.1756417771051	45.5191043478324
3.31809736987560	6.00599610964350	7.56390483648941	22.2081720643857	36.1566319347143	45.4570835652416
3.35596434219596	5.81652093061360	7.49187380460815	22.2835815236262	35.8735114736686	45.3990479773432
3.18656223581782	5.86683957165756	7.43720102783068	22.5506214040870	36.7937131916687	45.6545462901012
2.64077813257503	5.81236262802082	7.93777718359068	19.5541952751087	34.5580691765605	44.9496374844442
3.65006744031758	6.20382268497635	6.78672927679406	23.8490663381923	37.3651312569556	45.9991247783639
3.74767050746826	6.61504878435988	6.87884618799552	24.4472761030029	37.5485776203323	46.0897335861142
2.68844165995351	6.03931939958729	8.37430013673645	17.0870537982334	33.6334282742943	44.3833025020339
2.62054651418076	5.93185797914460	7.77104714502923	20.8081986276793	35.2269290104115	45.1161732889870
3.30841995840698	5.97158072463165	7.33788252641810	22.1187999635024	36.2200278272868	45.5914832289491
3.10103975930973	5.72982416665390	7.51541039393971	21.8839767155586	35.8271676838862	45.2842238428012
3.84303016803870	6.22819772020755	6.79423198126980	24.0807943918709	37.4054910123664	45.8415416034950
2.80907324554871	6.13462234042477	7.90490687076948	20.3184791571846	35.0721327941573	45.0470761625650
3.41281719706610	5.36715775148355	7.27158565256160	21.1216322280325	35.4091106480877	44.8146754767184
2.95711451039460	5.89220164854351	8.26564524551711	19.2674759701783	34.4654553239045	44.7055987388237
3.75968413330627	6.26327143837108	7.09137709788582	23.4058562655476	37.2198439372000	46.0346427131310
3.39184594385267	6.00194106724290	7.64519315736743	21.5050985397539	35.4255264508545	45.2005055769687
3.26187135300294	6.23065919912977	8.35742072579332	20.4605881241588	35.3677080112872	45.2518807723672
2.86569975049581	5.97536308994709	8.48798555737800	18.8170057482218	34.8226122662561	44.9462972651760
3.40866176193584	5.70664874630864	7.28150595256083	22.8938572547888	36.2303217310112	45.3446098712551
2.93466564723072	5.97652046822375	7.57545088710525	21.1613590586160	35.7222655608491	45.2220795373392
3.77584809038320	6.48500805420988	6.77399342073342	24.0615839749439	37.7513968709761	46.3510729154031
2.74849459020479	5.69521532591511	8.03237221095755	18.0705773461830	34.1941845422416	44.8989749639214
3.07422286585787	6.23748232957458	8.05274981119599	20.1044920269881	34.4540600765602	44.8930137875219
3.03431689503796	6.27247141873906	8.58457086719847	18.3780622908586	34.2025151061166	44.9699143790029];

And we find its RMSE given below:

RMSE=[0.309925349536485
1.32854625508834
0.100258360121278
1.48195811952683
0.628150579881412
0.864809907233644
0.230411719324083
1.03068337172397
0.781387212199642
0.430398162933991
0.181934907865268
1.31181940341704
0.636759421360166
0.383915798346207
0.914335796849431
1.64911521639988
1.19973554075008
0.220423729686980
0.928023339651249
1.13154234351589
0.812353212365206
0.512379120908612
2.57577285073583
2.05131650936839
0.767885671103765
0.647315621057416
0.538835110529774
1.19199171447276
0.538829093097871
0.213858787094643
0.366470498459897
1.78538007836936
1.52625044501880
0.496345397898337
0.813971482965366
0.127033024692744
1.30527448552268
0.215290366969390
0.372268601458720
0.209853114694237
0.224386681935183
0.438482287814006
0.583258792765323
0.879397959864566
0.886088271283586
0.619250170277863
0.342332966622459
2.82928319318677
0.766940511780096
1.20862364183965
0.702233428268855
0.232576316093124
0.960617605349140
0.125720290771078
0.335554080457334
0.590334473888534
1.21756292335773
0.229266007817692
1.42149712083561
1.79004372052030
1.87539379361390
0.628068245521539
1.01423670298790
0.485589699807721
0.216193503346578
2.20590608472883
0.901964791696282
1.26358907561752
1.44734733529171
2.52643000385019
0.785571026877701
0.417535300857816
2.41752091045457
1.03758926040220
0.744502718006506
1.08015261982739
1.05783958815740
1.04527297632092
1.32418448220170
0.306784642351348
1.97245891688670
2.22318517660875
1.35226999687439
0.391413192446921
1.06942068971933
0.878672192860379
2.05605255803595
0.169542119628818
0.653359315068986
0.406822773422638
1.78279167608156
0.678973706925456
0.331743970571321
0.530816859521561
1.34015859920727
0.592284213820656
2.16919003141260
0.869809655010314
0.253345767225850
0.783657094970932];

Then how can we find the whole matrix again if we bring some changes in the given RMSE? like if I change the RMSE as below:

RMSE1=RMSE-0.000000000000012;

Now how can we find the new matrix C1 of the same size as is C?

[Sadiq Akbar]

@Matt J and @John D'Errico

I think now we have more than 6 equations.

[John D'Errico]

I'm sorry, but NO.

You have a matrix C, and a vector a.

size(C)
ans =
100     6

size(a)
ans =
1     6

For every row of C, you subtract the vector a. And then compute an RMSE against each row of C. That allowed you to compute an RMSE for each row.

Now you tell us that you want to change the RMSE by some constant amount.

The matrix C has 100*6 = 600 elements. Now you want to infer what the =value of all 600 unknown elements, having specified only 100 numbers?

You did not think about my answer, did you? Now you have 600 unknowns. And you have 100 equations. Is 600 SIGNIFICANTLY greater than 100?

Even if you have only 1 more unknown than you have equations, you cannot solve the problem uniquely. But with 500 more unknowns than you have equations, you somehow think it is possible to solve this problem?

Let me say it again. YOU CANNOT INFER THE VALUE OF 600 UNKNOWNS FROM ONLY 100 PIECES OF INFORMATION.

Just wanting a problem to have a solution is not sufficient for it to have a solution. Just wanting a solution very badly does not make the solution suddenly possible.

[Sadiq Akbar]

Thank you very much dear John D'Errico for your kind response. Yes, mathematically such a system in which unknowns are more than no. of equations cannot be solved uniquely. But actually I was thinking about the formula of RMSE. Since formula is:

RMSE=sqrt((a-b)^2/N), where N is the number of elements in a or b. Now if we sqaure its both sides mathematiclly, then RMSE^2 = (a-b)^2/N. This implies that

(a-b)^2=N*RMSE^2 or a-b=sqrt(N*RMSE^2) or b=a-sqrt(N*RMSE^2). So isn't it possible then?

[Torsten]

I think you have to look up the formula for RMSE. Where is the sum in your expression RMSE=sqrt((a-b)^2/N) ?

https://en.wikipedia.org/wiki/Root-mean-square_deviation

[Sadiq Akbar]

Thanks a lot dear Torsten for your kind response. Yes you are right. But here I have taken a and b as vectors e.g

a=[x1 x2 x3 x4] and b=[y1 y2 y3 y4], Now since the last step above is:

b=a-sqrt(N*RMSE^2), so it will be

[y1 y2 y3 y4]=[x1 x2 x3 x4]-sqrt(N*RMSE^2), Now since in given post, the value of 'a' is known i.e., a=[3,6,8,20,35,45]; and value of sqrt(N*RMSE^2) is also known because RMSE is known, so [y1 y2 y3 y4] i.e., b can be found easily. Isn't that?

[Torsten]

Isn't that?

No.

N * RMSE^2 = (x1-y1)^2 + (x2-y2)^2 + (x3-y3)^2 + (x4-y4)^2

Now show me how you want to solve uniquely for y1, y2, y3 and y4 given N, RMSE and x1, x2, x3 and x4.

As noted several times by other participants, you have only 1 equation (the one above), but 4 unknowns.

[Matt J]

As noted several times by other participants, you have only 1 equation (the one above), but 4 unknowns.

Yes. In fact the equation is that of a hypersphere of radius sqrt(N)*RMSE and centered at a. Any point b on the surface of this hypersphere is a solution.

[John D'Errico]

@Sadiq Akbar

I will keep on saying it. You do not have sufficient information to solve for the unknowns. It is one equation only. And you have multiple unknowns. In your last example, we have:

a = [3,6,8,20];

With N and RMSE given, we have

N = 4;
RMSE = 1; % I'll just pick a number for RMSE
syms y [1,4]

ans = 

N * RMSE^2 == sum((a - y).^2)

Do you recognize this as the equation of a sphere in 4 dimensions? If not, you should. The center of the sphere is the point a=[3 6 8 20], and the square of the radius is given here as 4=2^2.

But ANY point on the surface of that sphere is a solution to your problem. ANY point. Need I repeat that? ANY POINT. How many points lie on the surface of a sphere? (Infinitely many.)

There is NO solution to your question. You cannot solve for the unknown vector (here y or b as you prefer.) You can keep on insisting there is a solution, but the mathematics says you are completely wrong. There are infinitely many solutions and there is no way to choose any specific solution, beyond saying the solution lies SOMEWHERE on the surface of that sphere.

[Sadiq Akbar]

Thanks a lot for your kind response. Amazing explanation. Yes, I am convinced with your explanation. But is there any trick that we can use because most of the time tricks also work though trick may be against mathematics rules? Isn't that?