PHYSICS 2150
EXPERIMENTAL MODERN
PHYSICS
Lecture 5
Least Square Fits, Correlation and Covariance,
Poisson Distribution
LINE FITS
• Most
common data fitting in lab
• Example: Photoelectric
effect:
• h = Tmax +
h
• Vs =
+
e
e
• Plot
versus and fit a line to
find slope (h/e) and work
function
1.4
Stopping Potential (V)
• e Vs = Tmax
1.2
1.2
1.0
0.8
0.8
0.6
0.4
0.4
5
5.5
6.0
6.5
7.0
6
7
Frequency (1014 Hz)
7.5
8.0
8
WE ALREADY KNOW HOW
TO OBTAIN THE BEST FIT
• Measured
N data points (xi, yi) with constant uncertainty on
all y values
• Want
to find the parameters of the linear model A*x + B,
which reproduces the data best
• Probability
and B:
for obtaining (y1,...,yN) for given parameters A
p(y1 , . . . , yn |A, B)
e
1
2
(
Ax1 +B
y1
2
) · ... · e
1
2
“
AxN +B
yN
”2
LEAST SQUARES FIT
• Assume
that data y1 , . . . , yN
deviate from linear model
because of statistical
„measurement errors“
5
p( y8 )
p( y10 )
p( y9 )
4
3
2
p( y3 )
p( y6 )
p( y5 )
p( y11 )
• Assume
further that are
normally distributed,
i.e.
“
”
p( y7 )
p( y4 )
p(
y1 )
1
p( y2 2 )
4
y1 , . . . , y N
p( y)
6
8
10
e
1
2
y
y
y
2
• Employ
Maximum
Likelihood principle, i.e.
max (p( y1 ) · . . . · p( yN ))
LEAST SQUARES FIT
Minimise quadratic deviation from the model
2
=
1
N
(A xi + B
2
yi )2
i=1
( yi )
2
5
4
3
2
1
2
4
6
8
10
LEAST SQUARES FIT TO A
LINEAR MODEL
A=
N
xi yi
N x2i
xi yi
( xi )2
B=
x2i
N
N
yi
x2i
5
4
y =A·x+B
3
2
1
2
4
6
8
10
xi xi yi
( xi )2
HOW RELIABLE ARE THE
FITTED PARAMETERS?
• The
parameters A(y1,...,yN) and B(y1,...,yN) are well defined
functions of y1,...,yN.
• Thus, we
can use an error propagation calculation to find the
uncertainties of A and B:
A
B
=
=
N
y
y
N
N
x2i
(
xi )2
x2i
x2i
(
xi )2
LINEAR FITS: EXAMPLE
y = (0.53 ± 0.06) · x + (0.3 ± 0.3)
7
6
y
= 1.0
y = (0.47 ± 0.07) · x + (0.6 ± 0.4)
8
y
5
= 1.5
6
4
4
3
2
2
1
2
4
6
8
10
2
4
6
8
y = (0.4 ± 0.1)x + (1.1 ± 0.6)
8
y
= 2.0
„True Model“ ?
6
y = 0.5 · x + 0.5
4
2
2
4
6
8
10
10
LINEAR FITS: EXAMPLE
„True Model“: y = sin( 20 x)
y = (0.10 ± 0.01) · x + (0.13 ± 0.04)
1.0
1.0
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
2
4
6
8
10
2
4
6
8
10
Lesson learned: small uncertainties of the model parameters
do not imply correctness of the assumed model!
Need a measure for „how well the assumed linear
model matches the data “
LEAST SQUARE FITS TO
ARBITRARY CURVES
• How
to fit data to arbitrary function f(x|p1,...,pM) with M
parameters p1,...,pM?
• Application
of Maximum Likelihood Principle yields that the
best fit corresponds to values of p1,...,pM for which
2
=
1
N
2
i=1
(f (xi |p1 , . . . , pM )
yi )2
is minimum!
/ pi = 0 can only be solved
equation system
analytically for a small number of curves and we have to resort
to numerical methods to find the best fit parameters.
• The
2
CAVEAT:
LINEARISING NONLINEAR FUNCTIONS
• Suppose
that we want to fit data to a exponential y = Ae
B·x
to „linearise“ the data by the transformation z = ln y ,
because this gives z = ln y = ln A + B · x
• Tempting
2
7
6
1
ln(y)
y
5
4
3
2
0.5
1.0
1.5
2.0
-1
-2
1
-3
0.5
1.0
1.5
2.0
Linearised „errors“ are not any longer normally distributed!
COVARIANCE
• Take
a measurement q(x, y), which depends on pairs of data
(x1, y1),...,(xN, yN)
• Error
q=
propagation says that
q
x
x
22
+
qq
yy
yy
• Here
22
+
q q
x y
x y
we assumed that x and y follow a Gaussian
distribution and x and y are independent!
• But
what if not?
x y=0
COVARIANCE
• If
the Covariance
xy
1
=
N
N
(xi
x )(yi
y )=0
i=1
we say that the errors in x and y are correlated.
• The
xy
• This
q
covariance fulfils the Schwarz inequality
x y
can be used to show that
q
x
x
+
q
y
y
COVARIANCE AND
ERROR PROPAGATION
2
q
• In
=
2
x
q
y
+
2
2
y
+2
case of maximum positive correlation:
q
x
q
• In
q
x
2
x
q
y
+
y
case of no correlation:
q
=
q
x
2
2
x
+
q
y
2
2
y
q q
x y
x y
EXAMPLE OF CORRELATED
0
VARIABLES: THE K MASS
to measure angle θT
between pions
• Need
K+
+
+
T
K
0
p
need to measure θ±,
angles between pions and K0
• Also
θT is easy, but K0
direction can be hard if its
path is short
• Measuring
•The K0 travels a short distance and decays to π+π−
+ hits stationary neutron, producing proton and K0
Incoming
K
•
EXAMPLE OF CORRELATED
0
VARIABLES: THE K MASS
• If
+
K
+
K0 p
+
direction of the red line
is wrong, then θ+ and θ−
will be wrong by equal
amounts but ins opposite
directions.
measurements of θ+
and θ− will be correlated
(actually anti-correlated)
• our
EXAMPLE OF CORRELATED
0
VARIABLES: THE K MASS
θ+
θ-
53º
56º
56º
34º
58º
46º
48º
56º
38º
61º
27º
24º
24º
46º
22º
34º
32º
24º
42º
20º
• Covariance:
+
=
• Standard
70.6
Deviations:
= 78.5
+
• Ratio:
+
/
+
=
0.9
COVARIANCE VS. CORRELATION
covariance σxy can be normalised to create a correlation
coefficient
• The
xy
r=
or
r=
x y
• Let‘s
(xi
(xi
x )(yi
x )2
(yi
y )
y )2
assume that all points (xi, yi) lie exactly on y = A x + B:
r=
• Thus, r
A
(xi
(xi
x )2 A2
x )2
(xi
A
=
= ±1
|A|
x )2
is the wanted indicator for how well the data are
matched by a straight line!
CORRELATION COEFFICIENT
Correlation coefficients of sample data sets
COVARIANCE VS. CORRELATION
covariance σxy can be normalised to create a correlation
coefficient
• The
r=
•r
xy
x y
can vary between -1 and 1:
•r
= 0 indicates that the variables are uncorrelated
• |r|
• Sign
= 1 means the variables are completely correlated.
of r indicates direction of covariance:
• r>0
means that large x indicates y is likely large
• r<0
means that large x indicates y is likely small