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Three Types of Parameters
(Physical Quantities)
• Jeffreys quantities
• Cartesian quantities
• Ad-hoc quantities
1
Three Types of Parameters
(Physical Quantities)
• Jeffreys quantities
• Cartesian quantities
• Ad-hoc quantities
1
Jeffreys Parameters
• They are strictly positive
• We indifferently use the parameter r or its inverse
s =1
r
(or, more generally, any power of it, t = rk ).
• Examples:
– conductivity ρ ↔ resistivity σ = 1/ρ
– compressibility γ ↔ uncompressibility κ = 1/γ
– velocity v ↔ slowness n = 1/v
– density (of mass) ρ ↔ lightness % = 1/ρ
– frequency ν ↔ period T = 1/ν
– wavelength λ ↔ wavenumber k = 1/λ
2
Jeffreys Parameters and the Benford Effect
The first digit of a Jeffreys parameter is not evenly distributed.
Using base 10, the probability for the first digit to be n is
pn = log10
n + 1
n
digit percentage1 30.1%2 17.6%3 12.5%4 9.7%5 7.9%6 6.7%7 5.8%8 5.1%9 4.6%
Using base K,
pn = logK
n + 1
n
3
0
400
300
200
100
1 2 3 4 5 6 7 8 9
Actual statistics
Benford Model
Frequency
Digits
Surfaces and Populations of the States,
Territories and Principal Islands
(Times Atlas of the World)
0
80
60
40
20
1 2 3 4 5 6 7 8 9
Actual statistics
Benford Model
Frequency
Digits
First digit of the
Fundamental Physical Constants
(1998 CODATA least-squares adjustement)
Table DP-1. Profile of General Demographic Characteristics: 2000Geographic Area: United States
[For information on confidentiality protection, nonsampling error, and definitions, see text]
Subject Number
Total population. . . . . . . . . . . . . . . . . . . . . . . . . . 281,421,906
SEX AND AGEMale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138,053,563Female. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143,368,343
Under 5 years . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19,175,7985 to 9 years . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20,549,50510 to 14 years . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20,528,07215 to 19 years . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20,219,89020 to 24 years . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18,964,00125 to 34 years . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39,891,72435 to 44 years . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45,148,52745 to 54 years . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37,677,95255 to 59 years . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13,469,23760 to 64 years . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10,805,44765 to 74 years . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18,390,98675 to 84 years . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12,361,18085 years and over . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4,239,587
Median age (years) . . . . . . . . . . . . . . . . . . . . . . . . . . . 35.3
18 years and over . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209,128,094Male . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100,994,367Female. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108,133,727
21 years and over . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196,899,19362 years and over . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41,256,02965 years and over . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34,991,753
Male . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14,409,625Female. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20,582,128
RACEOne race . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274,595,678 97.6
White . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211,460,626 75.1Black or African American . . . . . . . . . . . . . . . . . . . 34,658,190 12.3American Indian and Alaska Native . . . . . . . . . . . 2,475,956 0.9Asian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10,242,998 3.6
Asian Indian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1,678,765 0.6Chinese . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2,432,585 0.9Filipino . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1,850,314 0.7Japanese. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 796,700 0.3Korean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1,076,872 0.4Vietnamese. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1,122,528 0.4Other Asian 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1,285,234 0.5
Native Hawaiian and Other Pacific Islander. . . . 398,835 0.1Native Hawaiian. . . . . . . . . . . . . . . . . . . . . . . . . . 140,652 -Guamanian or Chamorro . . . . . . . . . . . . . . . . . . 58,240 -Samoan. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91,029 -Other Pacific Islander 2 . . . . . . . . . . . . . . . . . . . 108,914 -
Some other race . . . . . . . . . . . . . . . . . . . . . . . . . . . 15,359,073 5.5Two or more races . . . . . . . . . . . . . . . . . . . . . . . . . . . 6,826,228 2.4
Race alone or in combination with oneor more other races: 3
White . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216,930,975 77.1Black or African American . . . . . . . . . . . . . . . . . . . . . 36,419,434 12.9American Indian and Alaska Native . . . . . . . . . . . . . 4,119,301 1.5Asian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11,898,828 4.2Native Hawaiian and Other Pacific Islander. . . . . . 874,414 0.3Some other race . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18,521,486 6.6
Subject Number Percent
HISPANIC OR LATINO AND RACETotal population. . . . . . . . . . . . . . . . . . . . . . . . . . 281,421,906 100.0
Hispanic or Latino (of any race) . . . . . . . . . . . . . . . . 35,305,818 12.5Mexican . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20,640,711 7.3Puerto Rican. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3,406,178 1.2Cuban . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1,241,685 0.4Other Hispanic or Latino . . . . . . . . . . . . . . . . . . . . 10,017,244 3.6
Not Hispanic or Latino . . . . . . . . . . . . . . . . . . . . . . . . 246,116,088 87.5White alone. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194,552,774 69.1
RELATIONSHIPTotal population. . . . . . . . . . . . . . . . . . . . . . . . . . 281,421,906 100.0
In households. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273,643,273 97.2Householder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105,480,101 37.5Spouse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54,493,232 19.4Child. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83,393,392 29.6
Own child under 18 years . . . . . . . . . . . . . . . . 64,494,637 22.9Other relatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15,684,318 5.6
Under 18 years . . . . . . . . . . . . . . . . . . . . . . . . . 6,042,435 2.1Nonrelatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14,592,230 5.2
Unmarried partner . . . . . . . . . . . . . . . . . . . . . . . 5,475,768 1.9In group quarters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7,778,633 2.8
Institutionalized population. . . . . . . . . . . . . . . . . . . 4,059,039 1.4Noninstitutionalized population . . . . . . . . . . . . . . . 3,719,594 1.3
HOUSEHOLD BY TYPETotal households. . . . . . . . . . . . . . . . . . . . . . . . . 105,480,101 100.0
Family households (families). . . . . . . . . . . . . . . . . . . 71,787,347 68.1With own children under 18 years . . . . . . . . . . 34,588,368 32.8
Married-couple family . . . . . . . . . . . . . . . . . . . . . . . 54,493,232 51.7With own children under 18 years . . . . . . . . . . 24,835,505 23.5
Female householder, no husband present . . . . . 12,900,103 12.2With own children under 18 years . . . . . . . . . . 7,561,874 7.2
Nonfamily households . . . . . . . . . . . . . . . . . . . . . . . . 33,692,754 31.9Householder living alone . . . . . . . . . . . . . . . . . . . . 27,230,075 25.8
Householder 65 years and over . . . . . . . . . . . . 9,722,857 9.2
Households with individuals under 18 years . . . . . 38,022,115 36.0Households with individuals 65 years and over . . 24,672,708 23.4
Average household size. . . . . . . . . . . . . . . . . . . . . . . 2.59 (X)Average family size. . . . . . . . . . . . . . . . . . . . . . . . . . . 3.14 (X)
HOUSING OCCUPANCYTotal housing units. . . . . . . . . . . . . . . . . . . . . . . 115,904,641 100.0
Occupied housing units . . . . . . . . . . . . . . . . . . . . . . . 105,480,101 91.0Vacant housing units. . . . . . . . . . . . . . . . . . . . . . . . . . 10,424,540 9.0
For seasonal, recreational, oroccasional use . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3,578,718 3.1
Homeowner vacancy rate (percent). . . . . . . . . . . . . 1.7 (X)Rental vacancy rate (percent). . . . . . . . . . . . . . . . . . 6.8 (X)
HOUSING TENUREOccupied housing units . . . . . . . . . . . . . . . . . . 105,480,101 100.0
Owner-occupied housing units . . . . . . . . . . . . . . . . . 69,815,753 66.2Renter-occupied housing units . . . . . . . . . . . . . . . . . 35,664,348 33.8
Average household size of owner-occupied units. 2.69 (X)Average household size of renter-occupied units . 2.40 (X)
- Represents zero or rounds to zero. (X) Not applicable.1 Other Asian alone, or two or more Asian categories.2 Other Pacific Islander alone, or two or more Native Hawaiian and Other Pacific Islander categories.3 In combination with one or more of the other races listed. The six numbers may add to more than the total population and the six percentages
may add to more than 100 percent because individuals may report more than one race.
Source: U.S. Census Bureau, Census 2000.
U.S. Census Bureau, Census 2000
2
X =
10
x
0.1
1
10
100
0.2
0.5
2
5
20
50
x =
log
10 X
−1
0
1
2
−0.5
0.5
1.5
Distance for a Jeffreys Parameter
The distance between two values r1 and r2 of a Jeffreys quantity is(base of logarithm arbitrary, we may use base e for mathematicaldevelopments, base 10 for numerical developments, or base 2 formusic):
D = | logr2
r1|
Example: the distance between two musical notes, as characterizedby their period T their frequency ν = 1/T or their pulsationω = 2 π ν (logarithm in base 2 to have the musician’s octave asdistance unit)
D = | logT2
T1| = | log
ν2
ν1| = | log
ω2
ω1|
This expression is invariant (i) when we change the parameter byits inverse, (ii) when we change units, (iii) when we multiply thevalues of the parameter by a constant. It is the only expressionthat has these invariances and that is additive
D(r1, r2) + D(r2, r3) = D(r2, r3)
4
Remark: with this definition of distance, the center of a se-quence of values (of a Jeffreys parameter), {r1, r2, . . . , rp} is
center =( p∏
i=1
ri
)1/p
In particular, for two values,
center =√
r1 r2
5
Homogeneous Probability Density for aJeffreys Parameter
With this definition of distance for a Jeffreys quantity, it is easyto show that the homogeneous probability density (that associatesequal probabilities to intervals of equal length) is
f(r) =k
r; (k arbitrary constant)
(in fact it is not normalizable, so we should rather call it a ‘measuredensity’)
It is invariant when we change one Jeffreys parameter by its inverse:the Jacobian rule shows that the probability density for the inverseparameter s = 1/r is
g(s) =k
s
In fact, it is invariant when we change a Jeffreys parameter by anypower of a Jeffreys parameter.
→ Any power of a Jeffreys parameter is a Jeffreys parameter
6
Acceptable Probability Density fora Jeffreys Parameter
• It must vanish for negative values of the parameter
• Typically, it is invariant of form when change fromthe parrameter to its inverse
• Typically, it has the ‘one-over-the-parameter’ limitwhen the ‘dispersion’ tends to ∞
Example: log-normal probability density
f(r) =1
(2π)1/2 σ
1
rexp
(− 1
2 σ2
(log
r
r0
)2 )≈ k
r
s =1
r
g(s) =1
(2π)1/2 σ
1
sexp
(− 1
2 σ2
(log
s
s0
)2 )≈ k
s
7
0 0.5 1 1.5 2
0
1
2
3
4
0.1
0.20.4
0.81.6
3.2
Log-normal probability density
Cartesian Parameters
A parameter x is said Cartesian is the natural definitionof distance between two values x1 and x2 is
D = |x2 − x1|(like for a Cartesian coordinate in an Euclidean space).
In fact, the only Cartesian parameters are the loga-rithms of a Jeffreys parameter: if r is Jeffreys, then
x = logr
R(where R is some fixed value)
is Cartesian.Cartesian parameters are easy to handle. I strongly
suggest to ‘take the logarithm’ of all Jeffreys parameters,to handle Cartesian parameters only.
Typical model: the Gaussian (or normal)
f(r) =1
(2π)1/2 σexp
(− 1
2
(x− x0)2
σ2
)≈ k
8
Ad-hoc Parameters
They usually correspond to old definitions with historical interestonly, and could (should) be replaced by Jeffreys parameters.
Example: Poisson’s ratio of an elastic solid
−1 < σ <1
2
One should better use the uncompressibility (bulk modulus) κ andthe shear modulus µ
κ > 0 ; µ > 0
(these are eigenvalues of the positive definite stiffness tensor cijk! ),together with the mass density ρ
ρ > 0
→When choosing to use an ad-hoc parameter (like Poisson’s ratio)in an inverse problem, it is very easy to make some mistakes
9
f( ) =1
(1 + )(1 2 )−σσ σ
-1 -0.5 0 0.50
σ
Hom
ogen
eous
Pro
babi
lity
Den
sity
Ad-hoc Parameters
They usually correspond to old definitions with historical interestonly, and could (should) be replaced by Jeffreys parameters.
Example: Poisson’s ratio of an elastic solid
−1 < σ <1
2
One should better use the uncompressibility (bulk modulus) κ andthe shear modulus µ
κ > 0 ; µ > 0
(these are eigenvalues of the positive definite stiffness tensor cijk! ),together with the mass density ρ
ρ > 0
→When choosing to use an ad-hoc parameter (like Poisson’s ratio)in an inverse problem, it is very easy to make some mistakes
9
Ad-hoc Parameters (2)
Sometimes, because of its widespread use, one may choose to workwith some parameters that are not Jeffreys. For instance, whenchoosing {ρ, Vp, Vs} for a elastic solid (mass density and veloci-ties of compressional and shear waves), instead of {ρ, κ, µ} (massdensity, bulk and shear modulus).
Homogeneous Probability Density
• Start with the logarithmic parameters {ρ∗, κ∗, µ∗} :
f(ρ∗, κ∗, µ∗) = k
• Use the Jacobian rule to pass to the Jeffreys parameters {ρ, κ, µ}
g(ρ, κ, µ) =k
ρ κµ
• Use the Jacobian rule again to pass to {ρ, Vp, Vs} . One finds(good exercise!)
h(ρ, Vp, Vs) =k
ρ Vp Vs
(34 − V 2
sV 2
p
)
10
0 1 2 3 4 50
1
2
3
4
5
f(v, w) =1
v w
34
w2
v2
( )−
velocity of longitudinal waves (v)
velo
city
of
shea
r w
aves
(w)
Warning! Using a Gaussian model to represent someinformation on the three parameters {ρ, Vp, Vs} wouldbe inconsistent. The Gaussian model could be used forthe logarithmic parameters {ρ∗, κ∗, µ∗} . Then, the Ja-cobian rule allows to represent the same information inany other equivalent set of three parameters.
11
u
v
v=v(u)
In order for this definition to accept the interpretation of "the conditional probabi-lity density for u , given the constraint v = v(u) ," two conditions have to be satisfied: i) v must be a Cartesian variable (i.e., |v2-v1| is a distance), and ii) the relation v= v(u) must only be weakly non- linear.
