Tutorial to evaluate measurement uncertainty as applied in 21NRM05 STASIS

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Tu o ial o e alua e measu emen unce ain y as applied in
21NRM05 STASIS
S anda isa ion o sa e implan scanning in MRI
Theo e ical in oduc ion
When e alua ing measu emen unce ain y, he e y basic sou ce o ely on is he documen JCGM
100:2008 E alua ion o measu emen da a — Guide o he exp ession o unce ain y in measu emen
(GUM 1995 wi h mino co ec ions). This Guide es ablishes gene al ules o e alua ing and exp essing
unce ain y in measu emen ha a e in ended o be applicable o a b oad spec um o measu emen s.
I p o ides gene al ules o e alua ing and exp essing unce ain y in measu emen .
Following a e e y ew exce p s om he Guide, used as a basic example o e alua ing he
measu emen da a.
1. Es ima e o he ou pu quan i y
In mos cases, a measu and Y is no measu ed di ec ly, bu is de e mined om N inpu quan i ies X1,
X2, ....XN h ough a unc ional ela ionship
Y = (X1, X2, ..., XN ) (1)
An es ima e y o he measu and Y, is ob ained om Equa ion (1) using inpu es ima es x1, x2, ..., xN o
he alues o he N inpu quan i ies X1, X2, ..., XN. Thus he ou pu es ima e y, which is he esul o he
measu emen , is gi en by
𝑦=𝑌=1𝑛∑𝑌𝑘
𝑛𝑘=1 =1𝑛∑𝑓(𝑋1,𝑘,𝑋2,𝑘,…,𝑋𝑁,𝑘,)
𝑛𝑘=1 (2)
Tha is, y is aken as he a i hme ic mean o a e age o n independen de e mina ions Yk o Y, each
de e mina ion ha ing he same unce ain y and each being based on a comple e se o obse ed alues
o he N inpu quan i ies Xi ob ained a he same ime.
2. Measu emen unce ain y o he inpu quan i ies
E alua ion o he measu emen unce ain y o inpu quan i i es can be ca ied ou in wo ways:
a. Type A e alua ion o measu emen unce ain y,
b. Type B e alua ion o measu emen unce ain y.
Type A e alua ion o measu emen unce ain y (VIM 2.28)
I is e alua ion o a componen o measu emen unce ain y by a s a is ical analysis o measu ed quan i y
alues ob ained unde de ined measu emen condi ions
Fea u es:
1. Employed o inpu quan i ies which a e based on eplica e measu emen s (i.e. measu emen s unde
epea abili y condi ions)
2. A leas 4 eplica e measu emen s a e necessa y o ype A e alua ion
3. The unce ain y is ela ed o he es ima e o inpu quan i y, i.e. o he a i hme ic mean in his case
𝑢A=√1
𝑛(𝑛−1)∑ (𝑥𝑖−𝑥)2
𝑛𝑖=1 (3)
whe e
n numbe o eplica e measu emen s
21NRM05 STASIS
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xi he i h measu emen esul o eplica e measu emen s
𝑥 es ima e o he o e all measu emen esul – a i hme ic mean
Type B e alua ion o measu emen unce ain y (VIM 2.29)
I is e alua ion o a componen o measu emen unce ain y de e mined by means o he han a Type A
e alua ion o measu emen unce ain y.
Fea u es:
1. employed o inpu quan i ies ob ained om o he sou ces
2. i bounda ies a (some imes designa ed as zmax) o he app oxima ed alues a e gi en, he unce ain y
e alua ed by he ype B me hod can be calcula ed as
𝑢𝐵=𝑎𝑘 (4)
whe e k is he alue belonging o he selec ed app oxima ion o he p obabili y dis ibu ion:
EXAMPLES. Type B e alua ion o measu emen unce ain y is based on in o ma ion:
- associa ed wi h au ho i a i e published quan i y alues,
- associa ed wi h he quan i y alue o a ce i ied e e ence ma e ial,
- ob ained om a calib a ion ce i ica e,
- abou d i ,
- ob ained om he accu acy class o a e i ied measu ing ins umen ,
- ob ained om limi s deduced h ough pe sonal expe ience.
3. E alua ion o measu emen unce ain y uy o he ou pu quan i y y
1. Fo non-co ela ed inpu quan i ies (no common in luence on pai s o inpu quan i ies):
𝑢𝑦2=∑𝐴𝑖2𝑢𝑥𝑖
2
𝑚
𝑖=1 (5)
whe e Ai (Aj espec i ely) a e sensi i i y coe icien s, which can be calcula ed as
𝐴𝑖=𝜕𝑓(𝑋1,𝑋2,...𝑋𝑚)
𝜕𝑋𝑖|𝑋1=𝑥1,...𝑋𝑚=𝑥𝑚 (6)
2. Fo co ela ed inpu quan i ies (common in luence on pai s o inpu quan i ies exis s):
𝑢𝑦2=∑𝐴𝑖2𝑢𝑥𝑖
2
𝑚
𝑖=1 +2∑ ∑ 𝐴𝑖𝐴𝑗𝑢𝑥𝑖,𝑗
𝑚−1
𝑗<𝑖
𝑚
𝑖=2 (7)
whe e 𝑢𝑥𝑖,𝑗 is a co a iance among es ima es x1, x2,... xm o co ela ed inpu quan i ies X1, X2,... Xm
3. I ce ain co ela ion be ween he wo inpu quan i ies Xi and Xj exis s, i.e. i one quan i y somehow
depends on he o he one, hei co a iance mus be conside ed as a pa o he o e all unce ain y o
measu emen . Co a iance can inc ease o dec ease he o e all unce ain y o measu emen .
3.a E alua ion o co a iance by he ype A me hod
I wo inpu quan i ies Xi and Xj wi h es ima es xi and xj a e co ela ed, co a iance e alua ed by he ype
A me hod is
𝑢A 𝑥𝑖,𝑗=1
𝑛(𝑛−1)∑ (𝑥𝑖𝑘−𝑥𝑖)
𝑛𝑘=1 (𝑥𝑗𝑘−𝑥𝑗) (8)
3.b E alua ion o co a iance by he ype B me hod
21NRM05 STASIS
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I wo inpu quan i ies Xi and Xj wi h es ima es xi and xj a e co ela ed, he co a iance e alua ed by he
ype B me hod is
𝑢B 𝑥𝑖,𝑗=𝑟𝑥𝑖,𝑗𝑢𝑥𝑖𝑢𝑥𝑗 (9)
whe e
𝑟𝑥𝑖,𝑗is a co ela ion coe icien be ween es ima es xi and xj.
𝑢𝑥𝑖 esp. 𝑢𝑥𝑗 a e unce ain ies o es ima es xi and xj.
Finding he co ela ion be ween he powe deposi ed in o he implan
and he ollowing empe a u e inc ease
1. Scope
De e mina ion o he ela ionship be ween he induced powe and he empe a u e inc ease o he
implan .
2. Technical speci ica ion ISO/TS 10974
In he chap e 8, he poin 8.4.4.4 s a es ha he local empe a u e ise T o SAR a a poin loca ion in
a ho spo p oduced by he AIMD can be ela ed o he o al powe deposi ion using a calib a ed RF
powe injec ion me hod. Fo each AIMD ho spo , a con e sion ac o m be ween T o SAR a he ho
spo and injec ed powe is expe imen ally de e mined (i.e. T = mPinjec o SAR = mPinjec ).
3. The ask
Fo each implan , de ine a coe icien , wi h associa ed unce ain y/con idence le el, ha allows
es ima ing he empe a u e inc ease a e a speci ied ime ins an gi en he powe deposi ed in o he
implan . The inpu da a a e ep esen ed by a pai – deposi ed powe and a co esponding empe a u e
ise.
4. Theo e ical backg ound
I is assumed ha he ela ionship be ween he deposi ed powe (deno ed as X), and he empe a u e
di e ence a e a gi en ime (deno ed as Y), can be app oxima ed by linea eg ession. The wo o ms
o linea eg ession can be employed:
a. he linea eg ession (line) which c osses he ze o poin (in e sec ion o x and y axes), i.e.
𝑌=𝛽 𝑋
b. he linea eg ession (line) which is shi ed om he ze o poin (in e sec ion o x and y axes), i.e.
𝑌=𝛼+ 𝛽 𝑋
Fo he linea eg ession in he o m o 𝒀=𝜷 𝑿
1. Employing he leas squa es me hod, he unknown coe icien b can be de e mined as ollows:
𝑏= 1
∑𝑥𝑖2∑𝑥𝑖𝑦𝑖
whe e
xi is he measu ed powe inpu alue ( om he supplied da a),
yi is he measu ed empe a u e ise ( om he supplied da a).
2. The unce ain y ub o he coe icien b can be de e mined as ollows:
𝑢𝑏=1
√∑𝑥𝑖2𝑠,
21NRM05 STASIS
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whe e s can be es ima ed by a o mula o sample esidual a iance
𝑠=√1
𝑛−1∑[𝑦𝑖−(𝑏𝑥𝑖)]2
𝑛
𝑖=1
3. The unce ain y uy o he calcula ed alue y = bx can be de e mined as ollows:
𝑢𝑦=𝑥 𝑢𝑏=𝑥 1
√∑𝑥𝑖2𝑠
4. Coe icien o de e mina ion R2 is a s a is ical measu e o how well he eg ession p edic ions
app oxima e he eal da a poin s
𝑅2=1−∑ (𝑦𝑖−𝑏𝑥𝑖)2
𝑛𝑖=1
∑ (𝑦𝑖−𝑦)2
𝑛𝑖=1
Fo he linea eg ession in he o m o 𝒀=𝜶 + 𝜷 𝑿
1. Employing he leas squa es me hod, he unknown coe icien s a, and b can be de e mined as
ollows:
𝑎=∑𝑥𝑖2∑𝑦𝑖−∑𝑥𝑖∑𝑥𝑖𝑦𝑖
𝑛∑𝑥𝑖2−(∑𝑥𝑖)2
𝑏=𝑛∑𝑥𝑖𝑦𝑖−∑𝑥𝑖∑𝑦𝑖
𝑛∑𝑥𝑖2−(∑𝑥𝑖)2
2. The unce ain ies ua, ub o bo h coe icien s a, and b can be de e mined as ollows:
𝑢𝑎2=∑𝑥𝑖2
𝑛∑𝑥𝑖2−(∑𝑥𝑖)2𝑠2
𝑢𝑏2=𝑛
𝑛∑𝑥𝑖2−(∑𝑥𝑖)2𝑠2
whe e
𝑠2=1
𝑛−2∑(𝑦𝑖−𝑦)2=1
𝑛−2∑(𝑦𝑖−𝑎−𝑏𝑥𝑖)2
3. The co a iance ua,b be ween he wo coe icien s a, and b can be de e mined as ollows:
𝑢𝑎,𝑏=−∑𝑥𝑖
𝑛∑𝑥𝑖2−(∑𝑥𝑖)2𝑠2
4. The unce ain y uy o he calcula ed alue y can be de e mined as ollows ( he o mula o uy2 is s a ed
he e):
uy2 = ua2 + x2ub2 + 2xua,b
5. Coe icien o de e mina ion R2 is a s a is ical measu e o how well he eg ession p edic ions
app oxima e he eal da a poin s
𝑅2=1−∑ (𝑦𝑖−𝑎−𝑏𝑥𝑖)2
𝑛𝑖=1
∑ (𝑦𝑖−𝑦)2
𝑛𝑖=1
21NRM05 STASIS
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5. G aphic example
Ankle pla e - linea eg ession in he o m o 𝒀=𝜷 𝑿
Ankle pla e - linea eg ession in he o m o Y = a + b X