1
What is covariance?
1.1
Formal explanation
1.1.1
Probability distribution and measurements
Let X and Y be random variables described by a probability density ρ(X, Y ), which
RR
should be normalized to unity (
ρ(X, Y )dXdY = 1). The probability density can be thought
as a underlying law of the physical behavior, while the X or Y are just the observable
symptoms. By measuring the X, Y we through the dice, and the probability of observing
result (X, Y ) is p(X, Y ) = ρ(X, Y )dXdY .
For every variable F = F (X, Y ) dependent on (X, Y ) we can calculate its expectation
values:
Z Z
hF i =
F (X, Y )ρ(X, Y )dXdY .
In practice, the probability distribution ρ(X, Y ) is usually unknown and in the experiment we observe = measure the random realizations of the random variables (X, Y ). If we
do N measurements, we will get a set of results
{(Xi , Yi )}N
i=1 = {(X1 , Y1 ), (X2 , Y2 ), . . . , (XN , YN )} .
The probability distribution will be “encoded” inside the measurements, therefore we can
calculate any variable’s F expectation value as simple as:
N
1 X
F (Xi , Yi ) .
hF i ≈
N i=1
There are a lot of useful stuff that can be calculated, but a we will focus only on the few
chosen ones.
1.1.2
Moments
Moments describe how one of the variables (X or Y ) are distributed, thus using those
we can characterize the unknown “projections” of ρ(X, Y ) on X or Y :

ρ (X) = R ρ(X, Y )dY ,
X
ρY (Y ) = R ρ(X, Y )dX .
Distributions ρX (X) and ρY (Y ) are called marginal densities.
There are quite a few of different types of moments, but we will remember only the
two main types.
• The raw moments are calculated as
Z Z
Z
k
k
νk (X) = hX i =
X ρ(X, Y )dXdY = X k ρX (X)dX .
1
ρ(X)
FWHM≈2.355σ
μX
X
Figure 1 Gaussian distribution described by Eq. 1
The most important raw moment is the variable expectation value:
Z
ν1 (X) = hXi =
XρX (X)dX ≈
N
1 X
Xi .
N i=1
It characterize the “center-of-mass” of the distribution, which however is not necessarily the most probable outcome of the measurement. Usually the ν1 (X) is denoted
as µX , and due to the lack of letters in Latin+Greek alphabets we will be very confused
by the next item in the list.
• The central moments are the second important parameters:
Z
k
µk (X) = h(X − hXi) i = X k ρX (X)dX .
The µ1 (X) = hX − hXii = 0 is really boring, however1
µ2 (X) = h (X − hXi)2 i = hX 2 i − 2 hXihXi +hXi2 = hX 2 i − hXi2 = var(X)
|
{z
}
| {z }
X 2 −2XhXi+hXi2
hXi2
is really cool and is known as variance of X and thus denoted as var(X). This value
is closely related to the standard deviation which is defined as
p
p
σX = var(X) = hX 2 i − hXi2 .
Both variance and standard deviation show how the variable is “spread” around the
expectation value.
The best illustration of these concepts is the Gaussian distribution (see Fig. 1)
1
(X − µX )2
ρX (X) = √
exp −
.
2
2σX
2πσ
1
h1i =
R
1ρX (X)dX =
RR
ρ(X, Y )dXdY = 1 by definition.
2
(1)
2
The expectation value of X is hXi = µX and the variance is var(X) = σX
(i.e. σX is the
standard deviation).
But please always remember the following: FWHM (full width
for
at half 2 maximum)
(X−µX )
∈ (0; 1], the
Gaussian distribution is NOT the standard deviation σ. Since exp − 2σ2
X
2
X)
condition for FWHM is 12 = exp − (X−µ
. Solutions of this equations are
2σ 2
X
X̃± = (X − µX )± = ±
p
2 ln(2)σX ,
thus taking the difference between the edges of Gaussian distribution at half-way-to-maximum
we will get FWHM:
p
FWHM = X̃+ − X̃− = 2 2 ln(2)σX ≈ 2.354820045 · σX .
1.1.3
Covariance
If we want to know something about how the variables X and Y are distributed jointly,
we can use a quantity called covariance. It is calculated as
cov(X, Y ) = h(X − hXi) · (Y − hY i)i = hXY i − hXihY i .
It is fairly simple to compute from the measurements:
!
!
N
N
N
1 X
1 X
1 X
Xi Yi −
Xi ·
Yi ,
cov(X, Y ) ≈
N i=1
N i=1
N i=1
|
{z
} |
{z
} |
{z
}
≈hXY i
≈hXi
≈hY i
and such calculation is usually done by special functions in high-level languages (in MATLAB
it is cov, and in Python one can use either numpy.cov or scipy.stats.cov).
To understand the meaning of this relation we need a concept of independent events.
T
The events A and B are independent, if their probability is p(A B) = p(A) · p(B). In case
of continuous distribution ρ(X, Y ) this means factorizability of probability density function:
ρ(X, Y ) = ρX (X) · ρY (Y ) .
This means that
Z Z
n m
hX Y i =
X n Y m ρX (X) · ρY (Y )dXdY =
|
{z
}
ρ(X,Y )dXdY
Z
=
|
Z
m
X ρX (X)dX ·
Y ρY (Y )dY = hX n ihY m i
{z
} |
{z
}
n
hX n i
hY m i
therefore hXY i = hXihY i and thus for independent X and Y cov(X, Y ) = 0.
3
De
tec
tor
"A"
te
Source
De
↑A
r"
o
t
c
B"
↓B
(|↑A↓B⟩–|↓A↑B⟩ )/√2
Figure 2 Prototypical scheme to measure entanglement of two electrons in singlet state.
The covariance normed at the standard deviations is called Pearson correlation coefficient:
ρX,Y =
cov(X, Y )
hXY i − hXihY i
cov(X, Y )
=p
.
=p
σX · σY
var(X) · var(Y )
(hX 2 i − hXi2 ) · (hY 2 i − hY i2 )
Statisticians love this one more then the pure covariance, however, in the physics it’s completely opposite.
1.2
Simple example of how covariance works
Let us consider a prototypical quantum cryptography scheme (see Fig. 2): (1)
• a source that produces two electrons flying in opposite directions,
• two detectors (“A” and “B”) to catch electrons and measure their spin on the vertical
axis.
In this system there are four possible measurement outcomes:
1. ↑A and ↑B with spins sA = + 21 and sB = + 12 ,
2. ↑A and ↓B with sA = + 21 and sB = − 12 ,
3. ↓A and ↑B with sA = − 21 and sB = + 21 ,
4. ↓A and ↓B with sA = − 21 and sB = − 21 ,
where lower index denotes result of the observation on the detector A or B. There can be
a three basic situations, that can be happening.
4
1.2.1
Anticorrelated electrons
The source can produce entangled pair of electrons created from a singlet state |ψi =
↑A ↓B i+| ↓A ↑B i) In this case probabilities of the outcomes will be p(↑A ↑B ) = p(↓A ↓B ) = 0
and p(↑A ↓B ) = p(↓A ↑B ) = 21 . The average spin orientations hsA i and hsB i measured at the
detectors will be

1/2
1/2
0
0


z }| {
z }| {
z }| {
z }| {


hsA i = + 1 p(↑A ↑B ) + + 1 p(↑A ↓B ) + − 1 p(↓A ↑B ) + − 1 p(↓A ↓B ) = 0
2
2
2
2
.
1
1
1

hsB i = + 2 p(↑A ↑B ) + − 2 p(↑A ↓B ) + + 2 p(↓A ↑B ) + − 12 p(↓A ↓B ) = 0


| {z }
| {z }
| {z }
| {z }


0
0
√1 (|
2
1/2
1/2
However, the cross-term for the entangled state will be:
1/2
0
z }|
{ 1 1 z }| {
1
1
hsA sB i = +
+
p(↑A ↑B ) + +
−
p(↑A ↓B ) +
2
2
2
2
0
z 1/2
}| { 1 1 z }| {
1
1
1
+
−
+ −
p(↓A ↑B ) + −
p(↓A ↓B ) = − ,
2
2
2
2
4
and thus
1
.
4
Negative covariance means that if one value of sA positive, then the value of sB is probably
negative.
cov(sA , sB ) = hsA sB i − hsA ihsB i = −
1.2.2
Non-correlated electrons
If we break the entanglement we will get independent electrons, which will give equal
probabilities for all four outcomes: p(↑A ↑B ) = p(↓A ↓B ) = p(↑A ↓B ) = p(↓A ↑B ) = 41 . This will
not change the expectation values of the single detector measurements:

1/4
1/4
1/4
1/4


z }| {
z }| {
z }| {
z }| {


hsA i = + 1 p(↑A ↑B ) + + 1 p(↑A ↓B ) + − 1 p(↓A ↑B ) + − 1 p(↓A ↓B ) = 0
2
2
2
2
,
1
1
1

hsB i = + 2 p(↑A ↑B ) + − 2 p(↑A ↓B ) + + 2 p(↓A ↑B ) + − 12 p(↓A ↓B ) = 0


| {z }
| {z }
| {z }
| {z }


1/4
1/4
1/4
1/4
but the cross-term will now change:
1/4
z 1/4
}| { 1 1 z }| {
1
1
hsA sB i = +
+
p(↑A ↑B ) + +
−
p(↑A ↓B ) +
2
2
2
2
1/4
z 1/4
}| { 1 1 z }| {
1
1
+ −
+
p(↓A ↑B ) + −
−
p(↓A ↓B ) = 0 ,
2
2
2
2
and we will see the independence of the electrons as a
cov(sA , sB ) = hsA sB i − hsA ihsB i = 0 .
5
1.2.3
Positively correlated electrons
However, we can set the source to produce entangled electrons from a triple state, that
can be one of three possibilities:




| ↑A ↑B i ,
|ψi =
√1 (| ↑A ↓B i + | ↓A ↑B i) ,
2



| ↓A ↓B i .
This will give a set of probabilities p(↑A ↑B ) = p(↓A ↓B ) =
hsA i and hsB i will still be zero, but the cross-term
1
2
and p(↑A ↓B ) = p(↓A ↑B ) = 0. The
0
z 1/2
}| { 1 1 z }| {
1
1
hsA sB i = +
+
p(↑A ↑B ) + +
−
p(↑A ↓B ) +
2
2
2
2
1/2
0
z }|
{ 1 1 z }| {
1
1
1
p(↓A ↑B ) + −
p(↓A ↓B ) = + ,
+ −
+
−
2
2
2
2
4
will be positive, and thus the covariance
cov(sA , sB ) = hsA sB i − hsA ihsB i = +
1
.
4
This positive correlation means that when sA positive, the sB is also positive.
1.2.4
All together now!
To summarize:
• negative covariance = anti-relation between variables (when one is increased, the other
is decreased, and vice versa),
• zero covariance = independent variables,
• positive covariance = when one value is increased, the second is increased as well (and
vice versa).
2
TOF-TOF covariance maps
2.1
What is TOF covariance map?
First, we need to bring back the memory of what is time-of-flight (TOF) mass spectroscopy (MS). The simplest setup (see Fig. 3) consists of the two plates charged positively and
negatively. This capacitor-like construction forms an uniform electric field. When the ions are
produced at distance L from the detector (being close to the negative plate) they are accelerated by Lorentz force F = qE towards negative plate. Depending on the mass-to-charge
6
"A": mA, qA
+
vA
E
vB
vA
vB
"B": mB, qB
L
–
x
Figure 3 Simplest time-of-flight (TOF) spectrometer scheme and the ion trajectories in the
Coulomb explosion reaction. L is the distance to the detector, E is the electric field strength
along perpendicular to the detector, v is the particle velocity and v its projection on the electric
field direction, m and q are the mass and the charge of the ion.
ratio, the time from ion production to registration will be different: the heavier ions will
come later then the lighter ones. This simple scheme provides a very fine mass-to-charge
resolution.
In the end, the MS signal is the I(TOF), ion current as a function of TOF. When it is
digitized, the continuous signal turns into a discrete vector-like quantity:
I = ( I1 , I2 , . . . , IM ) ,
|{z} |{z}
|{z}
I(TOF1 ) I(TOF2 )
I(TOFM )
where TOFα = TOF1 + dt · (α − 1), dt is the discretization time step and M is the number
of MS points taken. We can perform N measurements of the TOF MS, which will give us a
N
set of independent results {Ii = (I1i , I2i , . . . , IM i ) = {Iαi }M
α=1 }i=1 . Based on this ensamble of
points we can calculate M × M size covariance map (4; 5) consisting of elements
!
!
N
N
N
1 X
1 X
1 X
Cαβ = cov( Iα , Iβ ) =
Iαi Iβi −
Iαi ·
Iβi .
(2)
|{z}
N i=1
N i=1
N i=1
|{z}
I(TOFα ) I(TOFβ )
From the meaning of the covariance we can expect to see the underlying reactions.
• If fragments A and B are formed independently, in the corresponding part of the map
will be cov(I(A), I(B)) = 0.
• If the fragment A is formed from parent ion B, the increase of the A will deplete
amount = intensity of the parent, and vice versa, less fragmentation of B will yield
in smaller amounts of observed A. Therefore we expect from this part of the map
cov(I(A), I(B)) < 0.
7
• If the A and B are produced in the same reaction sequence from the same parent
species, the increase/decrease of one ion can be noticed in the connected fragment,
therefore cov(I(A), I(B)) > 0.
However, life is a little bit more interesting than that.
2.2
Coulomb explosions
The Coulomb explosion is a reaction of the kind
ABq+ → AqA + + BqB + .
We have then two ions flying away from each other (Fig. 3). According to the momentum
conservation law, the final velocities of the fragments (vA for AqA + and vB for BqB + ) should
be related to the initial velocity of ABq+ (vAB ) as
(mA + mB ) vAB = mA vA + mB vB ,
{z
}
|
mAB
where m are the masses of the parent and the fragments. Let us consider only the projection
of the velocities on the electric field direction (v, see Fig. 3):
mAB vAB = mA vA + mB vB .
(3)
Each of the fragments will move towards the detector according to the second Newton’s
law mẍ = F = qE, where q is the charge of the ion and E is the electric field strength (we
consider only the electric field direction). This we can solve taking the initial position of the
ion x = 0 and velocity v:
qE t2
x(t) = v · t +
· ,
m
|{z} 2
a
where a is the ion acceleration in the constant electric field. We need to know the time when
the ion will reach the detector located at distance L from the reaction place. It can be found
according tox(t) = L, which is a quadratic equation for TOF t
a 2
t +v·t−L=0 .
2
The solution is
√
2aL − v 2 − v
t=
.
a
√
√
Since the applied field is strong enough, we can write 2aL − v 2 ≈ 2aL, thus
r
2L v t=
+ −
,
a
a
| {z } | {z }
δt
t0
8
slow
op
e
=
q
2/
q
1
fast
t0B
time-of-flight
sl
fast
slow
t0A
time-of-flight
Figure 4 Formation of the slope in Coulomb explosion on the covariance maps
where t0 =
q
2L
E
·
m
q
is the time when the velocity-less ion will reach the detector,2 and
δt = −
mv
q E
(4)
is the TOF drift. Obviously, the ions that are already flying towards the detector v > 0 will
reach the final destination earlier (t < t0 ), and those, that initially run away from detector
v < 0 will come later (t > t0 ).
We want to know how the arrival times of AqA + and BqB + are connected = correlated.
From Eq. 3 we can write
mAB vAB
mA
vB =
−
vA .
mB
mB
The time drift (Eq. 4) for AqA + is δtA = − mqAA vEA , while for BqB + it is
δtB = −
mB vB
mAB vAB qA mA vA
qA
=−
+
= δtAB − δtA .
qB E
q
E
q q E
qB
| B{z } B | A{z }
−δtA
δtAB
2
It should be the average arrival time for this sort of ions.
9
This gives the TOFs for ions AqA + and BqB + :

t = t + δt ,
A
0A
A
tB = t0B + δtAB −
qA
δt
qB A
.
The times tA and tB are connected through the variable δtA that shows initial velocity of
fragment AqA + . Thus the positive correlation = covariance in coordinates (tA , tB ) (i.e. in the
coordinates of the covariance map, (TOFi , TOFj )) for ions AqA + and BqB + will look like a
straight line with a slope − qqBA (see Fig. 4). Those straight lines with negative slopes are the
clear signatures of the Coulomb explosions. If there are some intermediate or later reactions,
the shape of covariances will change in very wierd ways (see (2) and (3) for more information).
2.3
Partial covariance maps
The free electron lasers (FELs) are known to have large fluctuations of the pulse energies.
Let us consider ion yields of two ions: A and B. Assuming that the formation of all the ions
happens as a result of one-photon process, we can assume that the ion yields of ions will
be proportional to the power of the FEL (W ):

A = a · W ,
B = b · W ,
where a and b are the true random ion yields, that depend only on the intermolecular
random variables, i.e. a and b are independent of W . This means that han W m i = han ihW m i.
Let us calculate the covariance of A and B that we will get from the covariance maps:
cov(A, B) = hABi − hAi hBi = habihW 2 i − haihbihW i2 .
| {z }
|{z} |{z}
habihW 2 i
haihW i hbihW i
But in reality we want to have
cov(a, b) = habi − haihbi ,
so let’s clean it up in cov(A, B). To do that we need to add and subtract missing haihbihW 2 i,
and by doing that we’ll get
cov(A, B) = habihW 2 i − haihbihW 2 i +haihbi hW 2 i − hW i2 = hW 2 i·cov(a, b)+haihbi var(W ) .
{z
}
|
{z
}
|
hW 2 i·cov(a,b)
var(W )
The first term is what we want: it represents the true intramolecular relation between ions.
The second term is called false covariance: it will give a nonzero covariance even if the
ions in reality are unrelated (cov(a, b) = 0). To clean out the first term we need to express
the second addend in the terms of observables A, B, W .
10
Let us calculate covariance of A and B with W :

haihW i
haihW 2 i


z }| { z}|{


cov(A, W ) = hAW i − hAi hW i = hai · var(W ) ,

cov(B, W ) = hBW i − hBi hW i = hbi · var(W ) ,


| {z } |{z}


2
hbihW i
hbihW i
therefore
haihbi var(W ) =
cov(A, W ) · cov(W, B)
.
var(W )
Now we can calculate hW 2 i · cov(a, b) which is called partial covariance (pcov) as
pcov(A, B) = cov(A, B) −
cov(A, W ) · cov(W, B)
.
var(W )
In principle, this value represents the “true” covariance of ionic signals.
N
If we have the same measured data {Ii = (I1i , I2i , . . . , IM i ) = {Iαi }M
α=1 }i=1 as in section
2.1 we would have to calculate the following matrix elements:
Pαβ = Cαβ −
1
N
PN
2
i=1 Wi
·
−
1
P
N
1
N
i=1
Wi
N
1 X
Iαi Wi −
N i=1
2 ·
N
1 X
Iαi
N i=1
!
·
N
1 X
Wi
N i=1
N
1 X
Iβi Wi −
N i=1
·
!!
·
N
1 X
Iβi
N i=1
!
·
N
1 X
Wi
N i=1
!!
where Cαβ is the covariance map defined by Eq. 2.
If we denote the M × M TOF-TOF covariance matrix as cov(I, I) and a define a vector
of length M for ion intensity covariance with FEL power W :
cov(I, W ) = (cov(I1 , W ), cov(I2 , W ), . . . , cov(IM , W )) ,
then we can write partial covariance matrix pcov(I, I) as
pcov(I, I) = cov(I, I) −
cov(I, W ) ⊗ cov(W, I)
,
var(W )
where ⊗ is an outer product. When using high-level languages (such as MATLAB and Python)
please use this or equivalent expressions, otherwise the calculation of the partial covariance
will take forever.
In reality, the FEL power can be not the only random variable that creates the false covariances. The molecular beam can fluctuate, the other lasers can also have power variations,
the detection devices can do some stuff, so a lot of possible things can happen. If there are
11
multiple random variables {ξk }K
k=1 = ξ that are independent from each other (cov(ξk , ξl ) = 0
for k 6= l), then the generalized covariance can be calculated:
K
X
cov(I, ξk ) ⊗ cov(ξk , I)
pcov(I, I|ξ) = cov(I, I) −
k=1
var(ξk )
.
However, other variables except for FEL power are usually unknown, therefore people take
the assumption that the FEL power correction factor looks exactly like the other ones, i.e.
cov(I, W ) ⊗ cov(W, I)
cov(I, ξ) ⊗ cov(ξ, I)
∝
,
var(W )
var(ξ)
so to get rid of all the false covariances we can just take a partial covariance modified by a
scale factor s > 1:
pcov(I, I) = cov(I, I) − s ·
2.4
cov(I, W ) ⊗ cov(W, I)
.
var(W )
Real life example
In FLASH beamtime we did the XUV-IR destructions of different PAHs, in particular of
fluorene (C13 H10 ). It can go the following reaction pathway:

C H + XUV → C H2+ + 2e− ,
13 10
13 10
2+
C13 H → Cn H+ + C13−n H+ ,
10
x
10−x
and the second Coulomb explosion can be registered in the TOF-TOF covariance maps. Fig.
5 shows Coulomb explosions for n = 2, 3, 4.
• We can calculate the the covariance map (top picture in Fig. 5), but the Coulomb
explosion lines are just slightly above the false covariances.
• If we compute partial covariance correcting for the FEL (XUV) intensity (middle picture
in Fig. 5) the background noise and decreases, and the lines appear a little more.
• But if we scale XUV correction matrix to the optimum (bottom picture in Fig. 5), we
will get rid of most of the noise and the lines will appear more vividly.
12
cov
pcov
scaled
pcov
Figure 5 Fluorene (C13 H10 ) TOF-TOF covariance, partial covariance and partial covariance with
optimized scale coefficient (from top to bottom).
13
References
[1] Ekert, A. K. Quantum cryptography based on bell’s theorem. Phys. Rev. Lett. 67 (Aug
1991), 661–663.
[2] Eland, J. The dynamics of three-body dissociations of dications studied by the triple
coincidence technique pepipico. Molecular Physics 61, 3 (1987), 725–745.
[3] Eland, J. H. D. Dynamics of fragmentation reactions from peak shapes in multiparticle
coincidence experiments. Laser Chemistry 11 (1991), 259–263.
[4] Frasinski, L. J. Covariance mapping techniques. Journal of Physics B: Atomic, Molecular
and Optical Physics 49, 15 (jul 2016), 152004.
[5] Zhaunerchyk, V., Frasinski, L. J., Eland, J. H. D., and Feifel, R. Theory and simulations of
covariance mapping in multiple dimensions for data analysis in high-event-rate experiments. Phys. Rev. A 89 (May 2014), 053418.
14