Journal of Global Positioning Systems (2008)
Vol. 7, No. 1 : 81-90
Test Statistics in Kalman Filtering
Jian-Guo Wang
Faculty of Science and Engineering, York University
Abstract
Many estimation problems can be modeled using a
Kalman filter. One of the key requirements for Kalman
filtering is to characterize various error sources,
essentially for the quality assurance and quality co ntrol of
a system. This characterization can be evaluated by
applying the principle of multivariate statistics to the
system innovations and the measurement residuals. This
manuscript will systematically examine the test statistics
in Kalman filter on the ground of the normal, 2
χ -, t- and
F- distributions, and the strategies for global, regional and
local statistical tests as well. It is hoped that these test
statistics can generally help better understand and perform
the statistical analysis in specific applications using a
Kalman filter.
Key words: Kalman filter, test statistics, normal
distribution, 2
χ
2
χ
distribution, t-distribution, F-distribution
____________________________________________
1 Introduction
Since 1980s, Geomatics professionals both in research
and industry have increasingly shown their profound
interest in applying the Kalman filter to various
applications, such as kinematic positioning or navigation
systems, image processing, and data processing of
deformation monitoring etc. Undoubtedly, knowledge of
Kalman filtering has become essential to the Geomatics
researchers and professionals.
A Kalman filter is simply an optimal recursive data
processing algorithm that blends all available information,
including measurement outputs, prior knowledge about
the system and measuring sensors, to estimate the state
variables in such a manner that the error is statistically
minimized [Maybeck, 1979]. In practice, linear equation
system with white Gaussian noises is commonly taken as
the standard model of a Kalman filter. However, one must
generally face the following facts [Maybeck, 1979]: (1)
no mathematical model is perfect, (2) dynamic systems
are driven not only by own control inputs, but also by
disturbances which can neither be controlled nor
modelled deterministically, and (3) sensors do not provide
perfect and complete data about a system.
Hence, a Kalman filter can function properly only if the
assumptions about its model structures, dynamical
process and measurement noise are correct or realistic. It
can become divergent if any of the following situations
occurs [Schlee et al, 1967; Tarn et al, 1970; Gelb, 1974;
Stöhr, 1986; Loffeld, 1990]:
• Improper system model;
• False modeled process noise;
• False modeled measurement noise or
• Unexpected sudden changes of the state vectors
Correspondingly, one needs to study the behaviors of the
errors associated with the system model. This may be
called as system identification or system diagnostics, one
of the advanced topics in Kalman filter.
There are different ways to perform system identification.
Statistic tests belong to the essential methods of system
identification. Herewith, the system model under the Null
hypothesis is tested against one or multiple alternative
hypotheses. The statistic algorithms can be divided into
two categories. The first one is to make multiple
hypotheses abou t the stochastic ch aracteristics of a syste m
(Multiple Hypothesis Filter Detectors) [Willsky, Deyst,
Crawford, 1974, 1975 ; Willsky, 1976]. In order to reach a
statistic decision, the posteriori probabilities of the state
vectors will be calculated, for instance, through the
sequential probability ratio test- SPRT [Willsky, 1976;
Yoshimura, et al, 1979]. The second one is to perform the
system identification with the help of series of system
innovations (“Innovation-based detection”) [Mehra,
Peschon, 1971; Stöhr, 1986; Salzmann, Teunissen, 1989;
etc.]. With this method, the signal is filtered using a
normal model, until a failure is found through th e statistic
tests, for example, through GLR (Generalized likelihood
ratio) method [Willsky, 1976; Huep, 1986] or more often
through the specific test statistics based on normal, ,
t
- or
F
- distributi o n.
Since the Kalman filter was introduced, how to
characterize the error sources of a system, especially the

Wang: Test Statistics in Kalman Filtering 82
series of the system innovation, has caught certain
research attention. [Stöhr, 1986] studied the statistic tests
on the ground of Normal Distribution and -
Distribution using system innovation. [Salzmann, 1993]
summarized a three-part test procedure as Detection,
Identification and Adaptation (DIA) using system
innovation fo r Kalman filter, in wh ich the construction of
test statistics is essential. [Wang, 1997] further discussed
the test statistics no t only on th e basis of normal and -
distributions, but also on the basis of t- and F-
distributions using both of the system innovation and
measurement residuals in Kalman filter.
2
χ
2
χ
A statistical test is no thing else, but a method of making
statistical decisions based on the existing system model
using experimental data. One needs statistic tests, for
example, to identify abnormal dynamic changes of the
system states, or to statistically verify the significance of
the additional parameters, such as different sensor biases,
in an integrated navigation system. Statistic tests can also
help with studying the whiteness of system innovation.
The detection of measurement outliers definitely needs
statistic tests. A lot more examples exist in practice. They
show how essential statistic tests are in Kalman filter so
that a developer has to be capable of constructing the
proper test statistics and applying them to practice.
However, there is still a lack of systematic description of
fundamentals of test statistics for Kalman filter in
textbooks. Most of the available works have missed out
this topic or, if not the case, the test statistics is mostly
based on the Normal Distribution and the-
Distribution only using system innovation. Applying of
the t- and - tests is not common.
2
χ
F
From the perspective of both research and industry, it
could be helpful to have a systematical understanding to
the fundamentals of test statistics for Kalman filter, in
order to use a Kalman filter well or develop new
algorithms. However, the existing textbooks about
Kalman filtering and applications do not normally talk
about testing statistics, although they may be found
miscellaneously in scientific publications of different
fields. This manuscript aims to fill the gaps between the
textbooks and scientific papers in the context of Kalman
filter theory and app lications.
This manuscript is organized as follows. The algorithm
of Kalman filter is summarized in Section 2. Section 3
gives the estimation of the variance factor, or more
precisely, the variance of unit weight. The statistic
characteristics of filtering solutions are described in
Section 4. Sections 5 and 6 are dedicated to building up
various test statistics for system innovation and
measurement residuals. The concluding remarks are given
in the last section.
2. ALGORITHM OF KALMAN FILTERING
The Kalman filter is a set of mathematical equations that
provide an efficient recursive means to estimate the state
of a process through minimizing its mean squared errors.
This section is to provide a brief introduction to the
discrete Kalman filter, which includes its description and
some discussion of the algorithm.
2.1. The Model
We consider a linear or linearized system with the state-
space notation and assume that the data are available over
a discrete time series },,, 10 N
ttt K{, which will often be
simplified to },,1,0 NK{. Without loss of generality, a
deterministic system input vector will be droped in all of
the expressions in this paper. Hence, at any time instant
k(k
N
≤
≤
1) the system can be written as follows:
x
)()()(),1()1( kwkkkAk
x
k
B
+
=
+
+
(1)
)1()1()1()1( +
+
=
+
+
+
kkxkCkz Δ (2)
where
x
)(kis the n-dimensional state-vector, )(kz is the
p-dimensional observation vector, )(kwis the m-
dimensional process noise vector, )(kΔis the p-
dimensional measurement noise vector, ),1( kkA + is the
nn
×
coefficient matrix of
x
)(k)(kB, is the n m
×
coefficient matrix of )(kw , )(kC is the np
×
coefficient matrix of )(kz . The random vectors )(kw and
)(k
Δ
are generally assumed to be Gaussian with zero-
mean:
))(,(~)( kQoNkw (3)
))(,(~)( kRoNk (4)
Δ
where )(kQ )(kR and are positive definite variance
matrices, respectively. Further assumptions about the
random noise are made and specified as follows (i
j
≠
):
OjwiwCov
=
(5) ))(),((
OjiCov
=
(6) ))(),((
Δ
Δ
OjiwCov
=
(7) ))(),((
Δ
Very often, we also have to assume the initial mean and
variance-covariance matrix )0(
x
and )0(
xx
D for the
system state at the time epoch 0. In addition, the initial
state
x
)0( is also assumed to be independent of )(kw
and )(k
Δ
for all k.

Wang: Test Statistics in Kalman Filtering 83
2.2. Kalman Filtering Equations
To derive the optimal estimate )(
ˆkx of )(k
x
, one may
use one of several optimality criterions to construct the
optimal filter. For example, if the least-squares method is
used, the optimality is defined in the sense of linear
unbiased minimum variance, namely,
⎪
⎭
⎪
⎬
⎫
=−−
=
min})
ˆ
)(
ˆ
{(
}
ˆ
{
T
xxxxE
xxE (8)
where x
ˆ is the unbiased minimum variance estimate of
x
.
Under the given stochastic conditions in Section 2.1, one
can derive the Kalman filtering for the state vector at
1+k:
)/1()1()/1(
ˆ
)1(
ˆkkdkGkkxkx ++++=+ (9)
and its variance-covariance matrix
)1()1()1()1()1({
)/1()}1()1({)1(
++++++−
+++−=+
kGkRkGkCkGE
kkDkCkGEkD
T
xxxx (10)
where )(
ˆ
),1()/1(
ˆkxkkAkkx +=+ (11)
)()()(
),1()(),1()/1(
kBkQkB
kkAkDkkAkkD
T
T
xxxx
+
++=+ (12)
)/1(
ˆ
)1()1()/1( kkxkCkzkkd++−+=+ (13)
)1(
)1()/1()1()/1(
++
+++=+
kR
kCkkDkCkkDT
xxdd (14)
)/1()1()/1()1( 1kkDkCkkDkG dd
T
xx +++=+ − (15)
Here )/1(
ˆkkx + is the one-step prediction of the state
vector from the past epoch k with its variance matrix
)/1( kkDxx+, )/1( kkd + is the system innovation
vector with its variance matrix )/1( kkDdd +, and
)1( +kG is the Kalman gain matrix.
An essential characteristic of the sequence )0/1(d, …,
)/1(iid +, …, )/1( kkd+ is that they are independent
from each other epochwise [Stöhr, 1986; Chui, Chen,
1987]:
OjjdiidCov =++ )}/1(),/1({ for (
j
i≠) (16)
The stochastic characteristics of )/1( kk +d are
obviously the mixture of the stochastic information from
the real observation noise {}),2(),1(KΔ
Δ
and the system
noise }),1(),0({ Kww . Traditionally, the system
innovation sequences are analyzed and used to build up
the test statistics.
2.3. An alternate Derivation of Kalman Filtering
Let us analyze the error sources in Kalman filter in a
different way. The optimal estimate )1(
ˆk
+
x of
)1(kx
+
at the instant k is always associated with the
stochastic information, which may be divided into three
independent grou ps:
a. The real observation noise )1( +kΔ,
b. The system noise )(kw ,
c. The noise from the predicted )/1( kkx+ through
ˆ
)(
ˆkx , on which the stochastic characteristics of
)}(, k),2(),1({
Δ
Δ
Δ
K, )}1(,),1(),0({
−
kw are
propagated thro ugh the system state model.
ww K
If these different error resources could be studied
separately, it could be very helpful to evaluate the
performance of a system in Kalman filter. Along with this
line of thinking, the system model as in 2.1 can be
reformulated through the three groups of the observation
or residual equations as follows:
=+
=+
=+
)1(
)1(
)1(
kv
kv
k
z
w
x
l
l
l
v
)1( +kC
)(
ˆ
)(
ˆ
kw
kw
)1(
ˆ
)()1(
ˆ
+
−
+
kx
kBkx
)1(
)1(
)1(
+−
+−
+
−
kl
kl
kl
z
w
x
)19(
)18
)17(
(
where the independent (pseudo-)observation groups are
simply listed by
)1()1(
)()1(
)(
ˆ
),1()1(
0
+=+
=+
+=+
kzkl
kwkl
kxkkAkl
z
w
x
)22(
)21(
)20(
with their variance-covariance matrices by
),1()(),1()1( kkAkDkkAkD T
xxll xx ++=+ (23)
)()1( kQkD wwll
+
=
(24)
)1()1(
+
=
kRkD zzll
+
(25)
)1(klx
+
, )1(klw
+
and )1()1( +=+ kzklz are the n-,
m- and p-dimentional measurement or pseudo-
measurement vectors, respectively. Usually okw
=
)(
0.

Wang: Test Statistics in Kalman Filtering 84
Again, by applying the least squares method, the identical
estimate )1(
ˆ+kx of )1( +kx as in the section 2.2 can be
obtained. For more details on this alternate derivation of
Kalman filter and its advantages, the reader is referred to
[Wang, 1997; Caspary and Wang, 1998].
})(),(),(diag{)( kDkDkDkD zzwwxx llllllll
=
This alternate derivation of Kalman filtering will directly
make the measurement residual vectors available for error
analysis and possibly to build up the test statistics in
Kalman filter, since it is based on the measurement
residual vectors. One can now analyse any of three
measurement vectors through their own residual vectors.
The measurement residual vectors are the functions of the
system innovation vector epochwise
)1/()()1/()()( 1−−= −kkdkKkkDkDkv xxllll xxxx (26)
)1/()()1/()1()1()( 1−−−−= −kkdkKkkDkBkQkv xx
T
llww
......(27)
)1/(})()({)(−−= kkdEkKkCkvzzll (28)
Similar to (16), we can readily prove the following results
of independence:
OjvivCov =)}(),({ for (ji ≠) (29)
3. VARIANCE OF WEIGHT UNIT
The posteriori estimation of the variance of weight unit
2
0
σ is essential in Geomatics. Some confusion has been
out there in applications of Kalman filer, because the
variance-covariance matrices are directly used in Kalman
filter. Surely the variance of weight unit, also called as
variance factor, should be close to unity for a perfect
model of system. However, this barely happens in
practice.
An algorithm for the estimation of unknown variance
factor 2
0
σ was constructed on the ground of the normal-
Gamma distribution in [Koch, 1990]. It allows estimating
the variance factor together with the state vector in
Kalman filter. Alternatively, 2
0
σ
can also be estimated by
taking advantages of the sequences of the system
innovation or the measurement residual vectors in [Wang,
1997 etc]. The single epoch estimate of 2
0
σ
, also called as
the local variance of unit weight, is given by
)(
)()()(
)(
ˆ1
2
0kr
kvkDkv
kll
T
l
−
= σ
(30)
where T
])(),(),([)( kvkvkvkv T
l
T
l
T
lzwx
= (31)
(32)
and )(kr is the number of the redundant measurements at
epoch k (tNk tt
<
≤
0). An alternate expression exists:
)(
)1/()1/()1/(
)(
ˆ1
2
0kr
kkdkkDkkd
kdd
T
l−−−
=
−
σ
(33)
The proof of equivalence between (30) and (33) can be
referred to [Pelzer, 1987; Tao, 1992; Wang, 1997]. One
can also estimate the variance factor 2
0
σ over a specific
time interval as the regional estimate of variance of unit
weight. For example, over a certain specified time interval
from epoch (
−
sk
+
1) to epoch k, one can order the
system innovation for these s epochs as follows
T
])1/(),...,3/2(
),2/1([(s)
−−+−+
−+−+=
kkdskskd
skskdd
TT
T
r (34)
with its variance matrix
})1/(),...,3/2(
),2/1(diag{
(s)(s)
−−+−+
−+−+=
kkDskskD
skskDD
dddd
dddd rr (35)
The regional estimate of 2
0
σ is then equal to
(s)
(s)(s)
)(
ˆ1)()(
20r
sdsd
T
rf
dDd
krr
−
= σ
(36)
(s)
r
f where is the total number of the redundant
measurements of s epochs:
∑
=
+−=
s
j
rjskrf
1
)((s) (37)
Furthermore, the global estimate of 2
0
σ for all of the past
k epochs can be calculated by
)(
)()(
)(
ˆ
1)()(
20kf
kdDkd
kg
kdkd
T
g
ggg
−
= σ (38)
where T
])1/(,),1/2(),0/1([)( −= kkdddkdTTT
gK (39)
})1/(),...,1/2(),0/1(diag{
)()(
=
kkDDDD ddddddkdkd gg
−
… …(40)
∑
=
=
k
j
gjrkf
1
)()( (41)

Wang: Test Statistics in Kalman Filtering 85
4. STATISTIC CHARACTERISTICS OF
FILTERING SOLUTIONS
In order to evaluate the quality of the solutions and
construct different test statistics, the statistic distributions
of various random vectors are used in Kalman filter on the
ground of the hypothesis: the normal distributed process
and measurement noise as given in 3.1 will be discussed.
At an arbitrary epoch k, all of the derived random
variables or vectors are the functions of the measurement
vector TT
z
T
w
T
xklklklk )](),(),([) =l( (see (20) ~ (22)).
Among them, )1−/(kkd , )(
ˆkx, )(kv , )( ˆ2
0k
σ are
essential for quality control of a system. By applying the
law of error propagation, their distributions are easily
known as:
))1/(,0(~)1/ −− kkDNkk dd
(d (42)
))(,(~) kDNk xxx (
ˆ
x
μ
(43)
))(,0(~) kDNk vv (v (44)
))((~)()()(
)1/()1/()1/(
21
1
krkvkDkv
kkdkkDkk
ll
T
dd
χ
−
−
=
−−−dT (45)
where ),(ba
N
represents a normal distribution with a
and b as its expectation and variance, respectively.
The i-th component )1/( −kkdi in )1/(
−
kkd is
normally distributed as follows:
),0(~)1/( 2)1/( − −kkdi Nkk σ
i
d
(i = 1, 2, …, p; k = 1, 2, … N) (46)
Any arbitrary subvector of )1/( −kkd is also normally
distributed. Based on the independency of the innovation
vectors between two arbitrary epochs shown in (16), the
vectors )(sdr and )(kdg as in (34) and (39) belong to
the following normal distributions:
),0(~)( )()( sdsdr rr
DNsd (47)
),0(~)( )()( kdkdg gg DNkd (48)
wherein )()( sdsd rr D and )()( kdkdgg D are as in (35) and
(40). Analog to (46), any arbitrary components )(kdgi
and )(sdri for the i-th type of observations also belong to
the normal distribution:
),0(~)( )()( kdkdgi gigi
DNkd (i = 1, 2, …, p) (49)
),0(~)( )()( sdsdririri DNsd (i = 1, 2, …, p) (50)
where, for i = 1, 2, …, n+m +p,
(
)
T
iiigi kkdddkd )1/(,),1/2(),0/1()( −= L (51)
(
),1/2(),/1()(
=
−
+
−
+−
−
skskdskskdsd iiri
+
)
T
ikkd )1/(,−L (52)
with their corresponding variance-covariance matrices
)()( kdkdgigi
D
=
(
)
2)1/( 2)1/2( 2)0/1(,,, −kkdddiii
diag σσσ L (53)
(
,, 2)1/2(
2)/1()()( +−+−−+−
=skskdskskdsdsd iiriri diagD σσ
)
2)1/(
,−kkdi σ L (54)
The i-th component )(k
i
v of )(kv is of the normal
distribution, too:
),0(~)( 2)(kvi i Nkv σ
(i = 1, 2, …, n+m+p; k = 1, 2, …, N) (55)
For any arbitrary subvector of )(kv, e.g. )(k
x
l
v, )(kv w
l
or )(kv z
l, the normal distributions apply. The global
cumulative measurement residual vector for all of the past
epochs can be defined as
(
)
T
TTT
gkvvvkv )(,),2(),1()( L= (56)
and the regional cumulative from the past s epochs as
(
)
T
TTT
rkvskvskvsv)(,),2(),1()( L+−+−=
… … (57)
with the following corresponding variance-covariance
matrices
)()( kvkv gg
D
=
(
)
)()()2()2()1()1( ,,,kvkvvvvv DDDdiag L (58)
(
D,, )2()2()1()1()()( +−+−+−+−
=
skvskvskvskvsvsv DDdiag
rr
)
)()( kvkv
,DL (59)
So )(kvg and )(k
r
v are obviously normally distributed:
),0(~)( )()( kvkvg gg
DNkv (60)
),0(~)()()( svsvr rr DNsv (61)
Similar to (49) and (50), the individual components of
)(kvg and )(k
r
v:

Wang: Test Statistics in Kalman Filtering 86
(
T
iiigi kvvvkv )(,),2(),1()( L=
) (62)
( ),2(),1()( +
−
+−= skvskvsv iiri
)
T
ikv)(,L (63)
belong to the following normal distributions
),0(~)( )()( kvkvgigigi DNkv (64)
),0(~)( )()( svsvri riri DNsv (65)
with i = 1, 2, …, n+m+p, where
(
)
2)( 2)2( 2)1()()( ,,, kvvvkvkv iiigigi diagD
σσσ L= (66)
(
)
2)( 2)2( 2)1()()( ,,, kvskvskvsvsv iiiriri diagD
σσσ L
+−+−
=
… … (67)
are the variance matrices of )(kvgi and )(svri .
For multiple components in )(kdg, )(sdr, )(kvg or
)(kvr, the same rule applies.
5. TEST STATISTICS FOR SYSTEM
INNOVATION
This section will construct test statistics using system
innovation. Under the assumption that no outliers exist in
measurements, one could diagnose the possible failure
caused by inappropriate state equations. Contrarily, one
can identify the possible outliers under the assumption if
the system model is assumed to be correct. The cause of a
failure may be ambiguous and need to be analyzed in
more details.
In this and next sections, we turn to perform statistic tests
the epoch k = 1, 2, … from the very beginning to an
arbitrary epoch. The statistic tests will be introduced in
three different levels, namely, global for all of the past k
epochs, regional for an arbitrary continuous epoch group,
e.g., the s epochs in the past, and local for a single epoch
(often the current epoch). The first two tests are very
meaningful for the identification of systematic errors and
the local one aims directly at the potential outliers or the
unexpected sudden state changes.
5.1. Global Test Statistics
Global tests can be introduced in two different ways to
investigate the system behaviors. Right after the first k
epochs are completed, one can perform the statistic tests
with all of the system innovation information from the
past and with their individual components by constructing
the corresponding – test statistics.
2
χ
With all of the past k epochs together (k = 1, 2, …, N), the
null hypothesis about )(kdg
0)(
0:
=
kdg H or
~
)(: 2
0
2)(0 ==
σχ
kdg
EH 0.1 (68)
and its alternative
0)(
1:
≠
kdg H or
~
)(: 2
0
2)(1=≠
σχ
kdg
EH 0.1 (69)
can be performed according to (48) by using the test
statistic [Salzmann, Teunissen, 1989]
))(,(~)()( 21 )()(
2)( kfkdDkd gggkdkd
T
gkd ggg
αχχ
−
=
g
(70)
at a significance level with the Type I error
α
.
),(
2f
αχ
is the )1(
α
−
- critical value from –
Distribution with the degrees of freedom of f after (41).
The null hypothesis (68) will be rejected if
2
χ
))(,(
22 )( kfggkdg
αχ > χ (71)
The test can easily be extended to the i-th component
)(kdgi in )(kdg for i = 1, 2, …, p and k = 1, 2, …, N.
Under the null hypothesis
0)(:
0kdH gi
=
(72)
against its alternative
(73) 0)(:
1kdH gi
≠
Based on (49), the test statistic can be given by
)(~)()( 21 )()(
2)( kkdDkd gikdkd T
gikdgigigi
χχ
−
= (74)
If
),(
22 )( k
gikdgi
αχχ >
gi
(i = 1, …, p) (75)
under the given significance level
α
, the null
hypothesis will be rejected.
5.2. Regional Tests
For the regional system diagnose, the processed k epochs
can be grouped at the user’s wish. Without loss of the
generality, the discussion here will be limited to two
groups. We consider having the first group for the first k –
s epochs (from 1 to epoch k – s) and the second group for

Wang: Test Statistics in Kalman Filtering 87
the rest of the epochs (from epoch k – s + 1 to epoch k) as
)( sk−dr (equivalent to )s(k
g
d
−
) and )(sdr.
The null hypothesis about )(sdr is
0.1)(:
0=sdr
H (76)
against the alternative
0)(:
1≠sdr
H (77)
On the ground of the test statistic [Willsky, 1976; Stöhr,
1986; Salzmann, Teunissen, 1989], the following test is
performed
))((~)()( 21 )()()( sfsdDsd rrsdsd
T
rs rrr
χ
−
=
2
d
χ
(78)
at the significance level of r
α
, where the number )(sfr
is the degrees of freedom as in (37) . For the second group
)( skdr−2
χ
also has the - distribution as
)()(1)()()( skdDskd gskdskd
T
gsk ggg −−= −−−−
2
d
χ
))s((~ 2kf g− χ
(79)
An additional F–test statistic can be constructed to test the
variance homogeneity between (78) and (79) or (70)
because )(sdr is independent from )( skdr−. This F–
Test is given by
))(),((~
)(
ˆ
)(
ˆ
20
20
)( skfsfF
sk
sgr
g
r
s
r−
−
=
σ
σ
Fd (80)
with )(
)()(
)(
ˆ1)()(
20sf
sdDsd
sr
rsdsd
T
r
rrr
−
= σ
(81)
)(
)()(
)(
ˆ
1)()(
20skf
skdDskd
sk g
gskdskd
T
g
grg
−
−−
=−
−−−
σ (82)
),(baF in (80) is the critical value of the Fisher
distribution with the 1st degrees of freedom a for the
numerator and the 2nd one b for the denominator. This test
is always one-sided under a user- specified Type I error
α
as the significance level. An exchange between the
numerator and the denominator may need in case
)(
ˆ20
g
σ sk − greater than)(
ˆ20s
r
σ . This test is commonly
employed to diagnose the significant difference between
the first k – s epochs and the rest of s epochs.
For the i-th component )(sdri in )(sdr, one can also
construct a test based on (50) and another F–test
analogue to (80). It runs
2
χ
0)(:
0sdHri
=
(83)
against the alternative
0)(:
1
≠
sdHri
2
χ
(84)
by using the test statistics
)(~)()( 21 )()(
2)( ssdDsd risdsd T
risd ririri
χχ
−
= (85)
An F-test runs for their variance homogeneity between
(85) and (79) or (70) as follows
))(,(~
)(
ˆ
/)()(
20
1)()(
)( skfsF
sk
ssdDsd
Fg
g
risdsd
T
ri
sd riri
gi −
−
=
−
σ
ri
… …(86)
for i = 1, …, p at the significant level of
α
.
5.3. Local Tests
Through the local system diagnose, the tests can be
introduced for the innovation vector as a whole and for its
components, re spectively .
At an arbitrary epoch k, the null hypothesis for
)1/( kkd
−
0)1/(:
0kkdH
−
=
(86)
against the alternative
0)1/(:
0kkdH
−
≠
(87)
can be given. Its test statistic runs
)1/()1/()1/( 12 )1/( −−−=−
−kkdkkDkkd dd
T
kkd
χ
))((~2kr
χ
l
(88)
at the significance level of
α
with the degrees of
freedom r(k).
A further F–test statistic can be introduced as
))(),((~
)(
ˆ
)(
ˆ
20
2
0
)1/( sfkrF
s
k
Fr
r
kkd
σ
σ
=
−(k = 2, 3, …, N) (89)
for the variance homogeneity between (81) and (33) or
(88). s means arbitrary specific epochs between epoch 1
and epoch k – 1.
The quadratic form in (45) contains the entire information
from the system innovation for an arbitrary epoch.
Therefore, the causes of a system failure must be
localized after the rejection of a 2)1/( −kkd χ
)1/( −kkd
F
or
test. It should orient to the individual error sources, e.g.

Wang: Test Statistics in Kalman Filtering 88
the individual measurements or the individual process
noise factors etc. in kinematic positioning or navigation.
One should perform the further statistic tests for the
individual measurements.
In order to perform the statistic tests for multiple
components in )1/(−kkd , the method for detection of
position displacements in deformation analysis can be
employed. More on this can be found in [Chrzanowski,
Chen, 1986].
The test statistic for single component of )1/(
−
kkd can
directly be constructed on the ground of the normal
distribution or the t–distribution. The null hypothesis is
0))1/((:
0=−kkdEi
H (90)
with its alternative
0))1/((:
1≠−kkdEi
H (91)
for i = 1,2, …, p and k = 1, 2, …, N. According to (46) the
test is performed
)1,0(~
)1/(
)1/(
)1/( N
kkd
kkd
i
kkd
i
i−
−
−
=
σ
N
li
(92)
at the significant level of
α
. The null hypothesis will be
accepted if the two-sided test satisfies
2
1
)1/(
2
1
)1/(
li
i
li u
kkd
ukkd
i
αα
σ
−
−
−≤
−
≤−
( i = 1, 2, …, p; k = 1, 2, …, N) (93)
where
2
1li
α
−
u is a )
2
1(li
α
−-critical value from the
standard normal distribution. Furthermore, based on the
past system information, (93) can be extended to the
following t–test
))((~
)(
ˆ
/)1/(
0
)1/(
)1/( sft
s
kkd
r
r
kkdi
kki
σ
σ
−
−
−
=Tdi
(i = 1, 2, …, p; k = 2, 3, …, N) (94)
The most common case is to test the current epoch k vs.
the past k – 1 e po chs.
The differences between (93)-(94) and (88)-(89) are
obvious. However, which one is preferable will absolutely
depend on the user. A t-test or an F-test may deliver the
more reliable results of fit to the real data, while a normal
or a 2 χ
)(k
x
l
)(k
w
l
)(k
z
l
2)(kdg
χ
2)(sdr
χ
2)1/(−kkd
χ
)(k
)(k
gi )(kvg
0)(:
0
test is introduced with respect to the a-priori
assumption.
6. TEST STATISTICS FOR MEASUREMENT
RESIDUALS
As it can be seen, the system innovation mixes up
different types of information. But it is transferred to the
individual measurement residuals epoch by epoch through
(26) ~ (28), i.e., the residual vector v for the
predicted state vector, the residual vector v for the
process noise and the residual vectorv for the direct
measurements. In this way, these different types of
random information can separately be studied. On the
basis of the fact that (30) and (33) are equivalent, the test
statistics , and in (70), (78) and
(88) can also be derived using the measurement residual
vector v. But it is not necessary to be repeated here.
Therefore, only the test statistics for the individual
components will be discussed in this section.
6.1. Global Tests
For the i-th component v in , the null
hypothesis is
(
i = 1, 2, …, n+m+p) (95)
=
kvH gi
0)(:
1
against the alternative
≠
kvH gi
)()( 1)()(
2)( kvDkv gikvkv T
gikv gigigi
−
= χ )(~ 2k χ
χVk
gi ()
2)( 2k
χ >
(96)
The test statistic is given by
(97)
with the degrees of freedom of k. The null hypothesis will
be rejected if
( i = 1, 2, …, n+m+p) (98)
at the significant level of gi α
)(s
ri )(s r
0)(:
0
.
6.2. Regional Tests
For the i-th component v from v, a χ2–test and
a F–test can be constructed. The null hypothesis is
=
svH ri
0)(:
1
(99)
with the alternative
≠
svH ri
)(~)()( 21 )()(
2)( ssvDsv risvsv T
risv ririri
χχ
−
=
(100)
The corresponding test statistic is given by
( i = 1, 2, …, n+m+p) (101)

Wang: Test Statistics in Kalman Filtering 89
with the degrees of freedom of s. For the variance
homogeneity between the independent and
similar to (80), the F–test statistic can be
introduced as
2)(svri
χ
2)( skd g−
χ
))(,(~
)(
ˆ
/)()(
20
1)()(
)( skfsF
sk
ssvDsv
g
g
risvsv
T
ri
sv riri
gi −
−
=
−
σ
F
H
H
… … (102)
6.3. Local Tests
The single outlier detection at a single epoch can be
modeled through the null hypothesis
0)(:
0=kvi (103)
against the alternative
(104)
0)(:
1≠kvi
for i = 1, 2, …, m+n+p and k = 1, 2, … after the test of its
standardized residual
)1,0(~
)(
)(
)( N
kv
kv
i
kv
i
i
σ
=N (105)
at the significant level of li
α
. The null hypothesis (102)
will be accepted if
2
1
)(
2
1
)(
li
i
li u
kv
ukv
i
αασ
−− ≤≤−
(
i = 1, 2, …, m+n+p; k = 1, 2, …, N) (106)
Besides a t-test between (105) and (78) or (70) can be
introduced as follows
))((~
)(
ˆ
/)(
0
)(
)( sft
s
kv
r
r
kvi
ki
σ
σ
=Tvi
x
l)(k)(k
z
l
(k
)k)(klz
2
χ
(i = 1, 2, …, n+m+p; k = 2, …, N) (107)
For any epo ch k, one can use v, and v,
along with v to investigate the statistic characteristics
of measurement vectors , and ,
especially two latter ones. Multiple components are
possibly diagnosed together in the same way as
mentioned in 5.3.
)(kvw
l
) (lw
)
(klx
7. CONCLUDING REMARKS
Based on the standard model of Kalman filter, different
test statistics have been elaborated on the basis of the
normal, -,
t
- and
F
-distributions in this manuscript.
This work can be conducive to better understanding of the
statistic fundamentals in Kalman filter, provides some
insights into statistic testing methods and applications.
In particular, the system innovation vector is transformed
to the residual vectors of three measurement and pseudo-
measurement groups by the aid of an alternative
derivation of Kalman filter algorithm. This makes
possible to construct test statistics directly using the
measurement residual vectors so that the system diagnosis
can directly aim at different error sources of interests to
users. The given posteriori estimate of variance of weight
unit in Section 3 can be used either to scale the variance
and covariance matrices, or to reveal the difference
between the model and the processed data set. On the
ground of statistic characteristics of filter solutions
summarized in the section 4, the test statistics using the
series of system innovation are constructed in the section
5 globally with - test, regionally either with - test
or
2
χ
2
χ
F
- test, and locally either with
t
- test or the normal
test according to the normal distribution. Analogous to the
section 5, the section 6 constructs the corresponding test
statistics using the measurement residuals. Fortunately,
- test,
2
χ
F
- test,
t
- test and the normal test are four
most commonly used statistic tests. The choice between
a- test and a
2
χ
F
- test, or between a
t
- test and a
normal test, wherever two parallel tests are available, is
left to the user.
How to construct a test statistics is more or less a
theoretical task. But how to efficiently design the
procedures to introduce the statistic tests in practice
mostly depends on the understanding about the theory and
the application. Practical experience plays an essential
role in helping deliver a realistic and reliable test scheme.
This manuscript however has limited to the testing
statistics for general purposes instead of a specific
application.
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