TES TING A NONLINEAR REGRESS ION SPEe IFICAT ION:
by
A
0
RONALD GALIANT
Institute of Statistics
Mimeograph Series No. 1017
Raleigh - June 1975
A NONREGTJ1AR CASE
TESTING A NONLINEAR REGRESSION SPECIFICATION:
A NONREGULAR CASE
by
A. RONALD GALIANT*
*1\
Associate Professor of Statistics and Economics, Department of Statistics,
North Carolina State University, Raleigh, North Carolina 27607. Presently
on leave and with the Graduate School of Business, University of Chicago,
Chicago, Illinois 60637.
ABSTRACT
The article considers a statistical test of whether or not a nonlinear regression specification should contain an additional additive nonlinear term in the response function.
The regularity conditions used to
obtain the asymptotic distributions of the usual test statistics for
parameters of nonlinear regression models are violated when the null
hypothesis - that this additional term is not present - is true.
Moreover,
standard iterative algorithms are likely to perform poorly when, in fact,
the data support the null hypothesis.
Methods designed to circumvent these
mathematical and computational difficulties are described in the article
and are illustrated with examples.
-1-
1.
INTRODUCTION
In nonlinear regression analysis it is helpful to be able to choose
between two model specifications:
and
on the basis of sample data:
puts
x
observed responses
(t = 1, 2, •.• , n) •
t
dime nS iona 1,
g(x,'f)
fact, vary with
til.
and
to k-dimensional in-
The unknown parameters are
.,., univa ria te, and
functional forms
Yt
til
h(x,w)
The errors
e
which is
'f,
v-d imens iona 1 •
are known and
h(x,w)
uThe
does, in
are assumed to be normally and
t
independently distributed with mean zero and unknown variance (;]2
Parametrically, the situation described is equivalent to testing:
H:
.,.
=a
against
.,.:f=
A:
0
w , and a 2 as nuisance parameters.
regarding
A natural impulse is to employ one of the nonlinear regression
analogues of the tests used in linear regression, either the Likelihood
Ratio Test or the test based on the asymptotic normality of the least
squares estimator
[8J.
Both of these tests - they are not equivalent in
nonlinear regression - depend on the unconstrained least squares estimator
A
e
A
A
A
= ('f,W,.,.)
•
When the null hypotheses is true, this dependence causes two
difficulties:
1.
Likely, the attempt to fit the full model
Yt
= g(xt,'f) + .,.h(xt,w) +
et
using one of the standard first derivative iteritive alogirthms [la, 141
will failor, at best, converge very slowly.
-22.
The regularity conditions [12,13,4J used to obtain the asymptotic
properties of the unconstrained least squares estimator
e
are violated;
these asymptotic properties are needed to derive the asymptotic null
distribution of these two test statistics [8l.
The first difficulty can be illustrated using the model
ConSider an attempt to determine from the data whether the exponential term
should be included; the correspondences with the notation above are:
·x
h(x,w)
=e
w3
Table 1 illustrates how the performance of the modified Gauss-Newton method
deteriorates as
H becomes more nearly true.
Table 1 was constructed as follows.
A sample of thirty normally
distributed errors with mean zero and variance
.001
was generated.
To
this sample (fixed throughout) was added the response function
using the parameter choices shown in Table 1 and the design points,
given in the Appendix of [7l.
determined by grid search.
(0)8i
= 9i
... 1
The least squares estimator
From the start value
(i
= 1,
2, 3, 4) ,
"'-
e
was
xt '
1.
PERFORMANCE OF THE MODIFIED GAUSS-NEWTON METHOD
True Parameter Values
2
"Ok
Least Squares Estimate
,.
"9
63
a2
4
Number of Modified Gauss-Newton
2
. a1
a2
a3
0
1
-1
-.5
.001
-.0259
1.02
- 1.12
-.505
.00117
4
0
1
-1'
-.3
.001
-.0260
1.02
- 1.20
-.305
.00117
5
0
'I
-1
-.1
.001
-.0265
1.02
- 1.71
-.108
.00118
6
0
1
-1
-.05
.001
-.0272
1.02
- 3.16
-.0641
.00117
7
0
.1
-1
-.01
.001
-.0275
1.02
-28.7
-.163
.00117
20
0
1
-1
-.005
.001
-.0268
1.01
- .•0971
.0106
.00119
a
0
1
-1
-.001
.001
-.0266
1.01
.134
.0132
.00119
202
0
1
-1
0
.001
-.0266
1.01
.142
.0139
.00119
69
a
a4
a
-
Algorithm had failed to converge after 500 iterations
:;
s
Iterations From A Start of
9i
- .1
-3-
an attempt was made to recompute the least squares estimator using the
modified Gauss-Newton method and the stopping rule:
successive iterations,
and
Stop when two
(i+1)8 , do not differ in the fifth
significant digit (properly rounded) of any component.
In each case considered in Table 1, all factors are held fixed save
~*
, the true value of
1/ , and the computational task is the same.
performance of the algorithm is seen to deteriorate as
becomes smaller.
The
~-
'~I
The inference to be drawn from Table 1 is:
(=
le 4 \)
If a non-
linear regression specification is not supported by the data one can
expect problems with iterative nonlinear least squares algorithms.
This
inference is also supported by Tables 1 and 2 of [51.
The mathematical difficulties are caused by the violation of two
standard [6,4,12J regularity conditions; the first, that the (almost sure)
limit
has a unique minimum at the true value
(almost sure) limit
is non-singular.
e*
of
e
and, the second, that the
matrix with typical element
The consequence of these violations is that it does not
appear possible to derive the .asymptotic properties of
W when
H
is true
under assumptions having sufficient generality to be useful in applications.
Specifically,
n
W will not converge in probability to a constant so that as
becomes large the responses are not constrained to lie in the linear space
defined by the gradient of the response function evaluated at
result, the methods of proof employed in,
deduce that, under
~.&.,
As a
[4,6,8,12J cannot be used to
H, test statistics depending on
behave as their linear regression analogs.
e*
e
will eventually
-4While it may be possible to discover the asymptotic properties of
"
e
usi.ng other methods of proof, it would seem a fruitless task in view of the
computational problems noted previously.
For this reason, this article
does not consider tests for
A which depend on the computation
of
e•
A
H against
The approach, rather, shall be to extend Hartley's method [7, Sec. 4J
to .this problem.
It is instructive to consider when the situation of testing
against
A using data which support
H
is likely to arise.
H
This is the
situation which causes computational difficulties and is the situation where
the methods suggested here will be the most useful.
Computational problems of the nature discussed above are unlikely to
arise, in the author's experience, when plots of the data visually suggest
the nonlinear model.
For example, in the cases considered in
plots of the observed response
Yt
against the input
x
3t
any visual impression of exponential growth for values of
.1.
Table 1,
fail to reveal
ITI
smaller than
It is improbable, therefore, that one would attempt to fit a nonlinear
model which is not supported visually if one is merely attempting to
represent data parametrically without reference to the substantive
discipline(s);involved.
Consequently, substantive considerations will
likely have suggested
A
rather than data analytic considerations; moreover,
a statistical test of
H against
A will likely have substantive relavance.
As we shall see, it will be helpful if these same substantive considerations
also imply
probab1~
2.
values for
w
AN EXTENSION OF HARTLEY'S TEST
Hartley's test [7, Sec. 4l would serve for testing
the model were linear in the remaining parameters once
H against
T
A
if
was specified.
-5Cne linearity requirement may be eli.minated, as is
sef-~n
below, when
satisfies regularity conditions sufficient to obtain, asymptotically,
g(x,')
the linear regression results on which Hartley's test is based.
results in hand, Hartley's idea may be applied:
blem, testing
Aift
where
H against
With these
Replace the original pro-
A , with a new problem, testing
H against
A#
is chosen so that the test statistic for the new problem will be
analytically and computationally tractable when
power when
A
H
is true yet
n~tain
good
is true.
Consider, then, the nonlinear regression model
where the additional regressors
parameters.
Zt
g(x,~)
The function
(w x 1)
is assumed to be such that, and the
regressors
Zt
[6,
are satisfied by the model A#; see
2J
Sec.
do not depend on any unknown
are chosen to be such that, the regularity conditi.ons of
formal statement of the regularity conditions.
[7,
Sec.
2J
for a less
Notice that satisfaction
of the regularity conditions is not affected by whether or not
0
=0
since the additional regressors do not depend on unknown parameters.
The Likelihood Ratio Test statistic
H:
against
Yt
A#
= g(Xt'~) +
TIt for testing
et
may be derived as follows.
Define
(~t,o#)
to be the least
squares estimator obtained by minimizing
n
~t=l
[Y t
- g(Xt'~) - z~ol
2
and set
2
(cr )#
Define
~
= (lin)
n
~t~l [Yt - g(Xt'~#) - z~o#J
2
•
to be the least squares estimator obtained by minimizing
n
~t=l [Yt - g(Xt,~)J
2
-6and set
...... 2
cr
n
....
= (lIn) ~t=l [Yt - g(Xt,~)J
2
•
The test statistic is
and
n~
H is rejected when
is larger than the appropriate size
~
critical point.
The asymptotic critical point is
= 1 + wF
c#
where
~
I(n - u - w)
denotes the upper
F
~
variable with
a.lOO
percentage point of an
w numerator degrees freedom and
degrees freedom.
A#
[6, Sec. 2] are
then the additional assumption of
will automatically be satisfied when H
random
n - u - w denominator
If the regularity conditions of
satisfied by the model
F
[8, Sec. 2J
is true with the consequence that
lim
P[T-I~ > c1~ IH] = a •
n.... oo
test of
H
n~
H when
Thus, the test - reject
> c# - is, asymptotically, a size
against A •
should be
It remains to consider how the additional regressors
chosen.
1.
Relevant are two restrictions implied by the regularity conditions:
which is a random variable
Every component of the w-vector
must be distributed independently of the errors e
2.
Let
G(~)
be the
("ota~j)g(Xt'~)
where
let
n
Z
~
be the
by
t
n
by
u
matrix with
is the row index and
w matrix with
t
j
row
•
~ypical
element
is the column index;
z~.
The (almost sure)
limit matrix
must have full rank.
Thus, the columns of Z
combinations of the 'columns of
G(~)
should not be exact linear
for admissible values of
~ J:./ •
-7The objective governing the choice of the additional regressors i.stl:1".d those which will maximize the power of the test when
should attempt to find those
Zt
which best approximate
A
is true.
h(xt'W~''')
One
- in the
sense of maximizing the ratio
h'(W*)Z(Z'Z)-IZ'h(w*)/h'(W*)h(W*)
where
(n x 1)
- while attempting, simultaneously, to keep the number of columns in
small as possible; see Hartley [9J.
Z
as
We consider, next, how this might be
done in applications.
In a situation where substantive considerations or previous
experimental evidence suggest a single point estimate
natural choice is
Zt
= h(Xt,~i)
~i
for
w
the
•
If, instead of a point estimate, ranges of plausible values for the
components of
ware available then a representative selection of values for
w
(W'lE1. :
i = I, 2, ••• , K}
whose components fall within these ranges can be chosen - either dete,ministically or by random sampling from a distribution defined on the
plausible values - and the vectors
h(~t)
made the columns of
Z.
If,
1.
following this procedure, the number of columns of
Z would be un-
reasonably large, their number may be reduced as follows.
Decompose the
matrix
H
= [h(~); ..• ;
h(~)l
into its principal component vectors and choose the first few of these to
make up
Z ; equivalently, obtain the singular value decomposition [1]
-8-
H - USV' and choose the first few columns of
U
to make up
Z •
A portion of the sample data may be used to refine this procedure,
i f des ired.
x
Select, by inspecting the inputs
t
'
a representative subset
(j = 1, 2, ... , n*)
+
I
of the sample - u + vi observations will suffice.
If the data consist of
replicates of a designed experiment one of the replicates may be chosen
randomly and used as the subsamp1e.
y
= g(x t. ,~) +
t.
J
Th(x
J
t
j
Attempt to fit the model
,w) + e t.
J
by least squares to obtain a point estimate
The average
W of the
~t
(i
~
=
~F
1, 2, .0., K)
for use as described above.
may be used as the start
value for the iterations; convergence may be checked by trying a few of the
w#
~
as start values to see if convergence to the same solution obtains.
As
seen previously, the least squares estimator probably can be found without
difficulty when the data support
A ; if the computations prove troublesome,
find the minima
(~ ,'r)
with respect to
associated with the smallest
whose associated
M.
~
= 1,
2, •.• , K}
(y t ,x
j
t
M.
(w4t:
~
i
= 1,
•• 0' Ki'1
M.
1.
and proceed as
replacing
To Jr eserve independence, the da. ta points
0
( J. = 1 "
)
2
••• , n*)
j
used to acquire information concerning W
computation of
~
One may, if desired, retain those
0
1.
are very near the smallest of the
above with this smaller set
i
wit
and retain as the point estimate that
T#
0
should be deleted prior to the
-9('¥4ft, oiF)
The reason that i t is eas ier to compute
NF:
Yt
= g(x t ,,¥) +
+ ct
z~S
(~,~,~)
than it is to compute
for
for
from data which support
is that
S
enters the model
unknown parameters.
MF
linearly and
l1sll
The length
of
8
Zt
does not depend on
aD.)'
can move toward zero in the
iterations without causing other parameters to become indeterminate or the
matrix
[G('¥):Z]'[G('¥):Z]
to become nearly singular.
The Likelihood Ratio Test statistic
against
A4ft
normality of
T4ft
rather than a test statistic
S4ft
was selected to test
H
based on the asymptotic
84ft
because the MOnte-Carlo evidence in Tables land 2 of [8]
support this choice; we are acting on the presumption that, for well chosen
A4ft, the more powerful test of
test of
H against
A.
H against
will be the more powerful
A4ft
If deSired, a test using
may be substituted
84ft
and the considerations above remain valid.
One proceeds as follows.
Let
(,¥4ft,8#)
be the least squares
estimat~
for
and let
2
n
2
(s )# = ~ t=l [Y t - g(x t ,'¥#) - z'84ftl
t . I(n - u - w) •
Evaluate the matrix
G('¥)
at
'¥
= '¥#
.
C# = ([G('¥4ft):Z]'[G('¥4ft):Zll-
.
Let
of
C:~2
and put
l
•
be the matrix formed by deleting the first
C ; then
u
rows and columns
~lO-
.H
an
is rejected when
F
exceeds the upper
S#
random variable with
~.lOO
percentage point
w numerator degrees freedom and
F
of
~
n - u - w
denominator degrees freedom.
3.
EXAMPLES
The first example is a reconsideration of the testing problem described
in [5, See. 5l.
The data shown in Figure A are preschool boy's weight/
height ratios plotted against age and were obtained from [2]; the tabular
values are given in the Appendix.
The question to be decided is whether
the data support the choice of a three segment quadratic-quadratic-linear
polynomial response function as opposed to a two segment quadratic-linear
response function.
continuous in
In both cases, the response function is required to be
(= age)
x
and to have a continuous first derivative in
Formally,
and
where
when
z
~
when
z
< 0
0
see [5, Sec. 2].
The correspondences with the notation of Section 1 are:
g(x,~) = ~l
h(x,w)
~
+
~2x
= T2 (W -
= (81'
+
~3T2(~4 - x)
x) ,
8 2 , 83 , 84) ,
x •
POUNDS PER
A. PRESCHOOL BOYS
WEIGHT HEIGHT RATIO
INCH
1.1
o
_o~
o
1.0
0
o~~vn 0
co.
. "o--v. , 0
'0
o
0.9
o
0
0
0
o~vvo
o
o
0.8
0.7
0.6
0.5
0.4
0&3
I
o
I
,
4
8
'"iF
12
,&
,
•
I
16 20 24 28
I
•
,
,
I
,
I
,
,
'''Tn'''~---'·'·:-''!
32 36 40 44 48 52 56 60 64 68
MONTHS
'2
-11-
w the points wit
Choosing as plausible values for
the matrix
=4
,
w~
... 8 , w ~
:=
H of Section 2 has typical row
The first principal component vector of
Zt
= [(2.08)T 2 (4
x 10
H, with elements
- x t ) + (14.07)T 2 (8 - x t ) + (39.9)T/12 - x t )
l
-4 ,
was chosen as the additional regressor to obtain:
.....2
cr = .0005264 ,
2
(cr )4F
= .0005235
,
llF = 1.006 ,
P[llF > 1.006\H]
= .485
These data give little support to
•
A •
The model
Mt:
Yt
= ~l + ~2Xt
I
+ ~3T2(~4 - x t ) + 5Z t + e t
does not satisfy the regularity conditions of [6, Sec. 2J because the
second derivative
~4
is not continuous in
; we are relying on the asymptotic theory in [4l to justify the use of
TIF in this instance.
The second example illustrates how the ideas in Section 2 may be
extended in a natural way to a situation studied by Feder [3J.
The data
shown in Figure B are from [llJ and were obtained to study the specific
retention volume of the organic liquid methylene chloride in the polymer
polyethylene terephthalate at selected temperatures; the tabular values
are given in the Appendix.
The question is whether the data support a
single quadratic polynomial model
12
In (cc./gm.) .
1.50
B. SPECIFIC RETENTION VOLUME OF METHYLENE
CHLORIDE IN POLYETHYLENE TEREPHTHALATE
1.26
1.02
0.78
0.54
0.30 '
.
2.540
I
I
2.672
I
Y
I
2.804
2.936
(l/oK) x 10 3
I
3.068
.J
3.200
-12H:
Yt
2
= 81
+ 8 2X t + 8 3X t + e t
or a segmented quadratic-quadratic polynomial model
A:
where the latter response function is restricted to be continuous in
but is E£! restricted to have continuous first derivitive in
x
= 86
•
x - there
18 4 1
will be a jump discontinuity in the first derivative of gap
x
at
The answer to the question is visllally obvious from Figure B but,
nonetheless, we shall confirm it by means of a statistical test.
Choosing as plausible values for
~
= 2.85
, and
Ht
w~
= 2.9
= [T 1 (2.8
"W
we construct
q=
= 86
H
the points w
(n x 6)
2.8,
with typical row
- x t ), T (2.85 - x ), T (2.9 - x ) ,
l
t
1
t
T2 (2.8 - x t ), T2 (2.85 - x t ), T (2.9 - x t )] •
2
The first two principal component vectors of
Zlt
= Ht ·(·52,
Z2t
= Ht ·(-6.4,
H, with elements
.68, .84, .11, .17, .25)' ,
-.94, 6.6, -2.7, -3.1, -2.8)'
were chosen as additional regressors to obtain:
,..,,2
a
=
.02381 ,
2
(a ):fF = .003001 ,
TIF
= 7.935
,
P[TIF > 7.93S\H]
= .0000895
These data strongly support
•
A which, as was noted, is visually obvious.
Observe that the model
AiF:
is linear in its parameters so that
P[T# > 7.93SIH]
=
.0000895
exactly, not asymptotically - granted normally distributed errors.
-13For this problem Feder [3J has shown that the Likelihood Ratio Test
statistic
T
= ,..,.21'::.2
a fa
(normalized as
nT)
random variable when
of
is not asymptotically distributed as a Chi-square
H holds.
Moreover, he argues that the distribution
T will depend on the particular arrangement of design points
x
t
•
Thus, it would appear impossible to obtain a single set of tables for the use
of
T
in applications; it would be necessary to find critical points by
Monte-Carlo simulations at each instance.
substitutes the statistic
The methodology suggested here
TIF allowing the use of tables of
F.
In
fact, as was noted, for this particular example the model chosen for
A#
is linear in the parameters; hence, exact size, not asymptotic size, is
achieved by the test - granted the normality assumption.
.14FOOTNOTES
1../ The asterisk is used in connection with parameters • 'f~\', ,.'1",
W '1'. ,
and
e'l'< -
to emphasize that it is the true, but unknown, value which is meant.
The omission of the asterisk does not necessarily denote the contrary.
~/ThiS is technically correct if one wishes to apply the results of [6,8J
straightforwardly.
It appears that the nonlinear theory could be
strengthened to allow
Z
to have rank less than
w.
Such a
generalization would have no practical utility in the present context.
-15REFERENCES
[1]
Bussinger, P. A. and Golub, G. H., "Singular Value Decomposition of a
Complex Matrix", Communications of the ACM, 12 (October 1969),
564..565.
[2]
Eppright, E. S., Fox, H. M., Fryer, B. A., Lamkin, G. H., Vivian, V. M.,
and Fuller, E. S., "Nutrition of Infants and Preschool Children in
the North Central Region of the United States of America", World
Review of Nutrition and Dietetics, 14 (1972), 269-332.
[3J
Feder, P. I., "The Log Likelihood Ratio in Segmented Regression", The
Annals of Statistics, 3 (January 1975) 84-97.
[4J
Gallant, A. R., "Inference for Nonlinear Models", Institute of Statistics
Mimeograph Series, No. 875.
Raleigh:
North Carolina State
University, 1973.
[5J
Gallant, A. R. and Fuller, W. A., "Fitting Segmented Polynomial
Regression Models Whose Join Points Have to Be Estimated",
Journal
of the American Statistical Association, 68 (March 1973), 144-147.
[6J
Gallant, A. R., "The Power of the Likelihood Ratio Test of Location in
Nonlinear Regression Models", Journal of the American Statisti.cal
Association, 70 (March 1975), 198-203.
[7l
Gallant, A. R., "Nonlinear Regression", The American Statistician, 29
(May 1975), 73-81.
[8l
Gallant, A. R., "Testing a Subset of the Parameters of a Nonlinear
Regression Model", Journal of the American Statistical A.ssociation,
70 (In press, December 1975).
[9J
Hartley, H. Q., "Exact Confidence Regions for the Parameters in NonLinear Regression Laws", Biometri!<:a, 51 (December
1964), 347-35.3.
-16[10] Hartley, H. 0., "The Modified Gauss-Newton Method for the Fitting of
Non-Linear Regression Functions by Least Squares",
Technometri~s.
3 (May 1961), 269-280.
[llJ Hsiung, P.O., Study of the Interaction of Organic Liquids With
i
Polymers by Means of Gas Chromatography.
Raleigh:
Ph.D. Dissertation,
North Carolina State University.
[12J Jennrich, R. I., "Asymptotic Properties of Non-Linear Least Squares
Estimators", The i\.nnals of Mathematical Statistics, 40 (April 1969),
633-643.
[131 Malinvaud, E., "The Consistency of Nonlinear Regressions", The Annals
of Mathematical Statistics, 41 (June 1970), 956-969.
[14J Marquardt, D. W., "An Algorithm for Least-Squares Estimation of NonLinear Parameters", Journal of the Society for Industrial and
Applied Mathematics, 11 (June 1963), 431-441.
2.
BOYS' WEIGHT/HEIGHT VS. AGE
W/H
AGE
W/H
AGE
W/H
AGE
0.46
0.47
0.56
0.5
1.5
2.5
3.5
4.5
5.5
6.5
7.5
8.5
9.5
10.5
11.5
12.5
13.5
14.5
15.5
16.5
17 .5
18.5
19.5
20.5
21.5
22.5
23.5
0.88
0.81
0.83
0.82
0.82
0.86
0.82
0.85
0.88
0.86
0.91
0.87
0.87
0.87
0.85
.0.90
0.87
0.91
0.90
0.93
0.89
0.89
0.92
0.89
24.5
25.5
26.5
27.5
28.5
29.5
30.5
31.5
32.5
33.5
34.5
35.5
36.5
37.5
38.5
39.5
40.5
41.5
42.5
43.5
44.5
45.5
46.5
47.5
0.92
0.96
0.92
0.91
0.95
0.93
0.93
0.98
0.95
0.97
0.97
0.96
0.97
0.94
0.96
1.03
0.99
1.01
0.99
0.99
0.97
1.01
0.99
1.04
48.5
49.5
50.5
51.5
52.5
53.5
54.5
55.5
56.5
57.5
58.5
59.5
60.5
61.5
62.5
63.5
64.5
65.5
66.5
67.5
68.5
69.5
70.5
71.5
0~61
0.61
0.67
0.68
0.78
0.69
0.74
0.77
0.78
0.75
0.80
0.78
0.82
0.77
0.80
0.81.
0.18
0.87
0.80
0.83
0.81
Source;
Epprf.ght .!S.
.!!
(21
3.
SPECIFIC· RETENTION VOLUME OF METHYLENE CHLORIDE IN POLYETHYLENE TEREPHTHAIATE
3
RECIPROCAL X 10 OF
TEMPERATURE IN DEGREES KELVIN
2.54323
2.60960
2.67952
2.75330
2.79173
2.82965
2.87026
2.91120
2.94637
3.00030
3.04228
3.09214
3.13971
3.19081
Source:
Hsiung [11]
NATURAL LOGARITHM OF
SPECIFIC VOLUME rn CC. PER GM.
1.16323
1.10458
0.98832
0.87471
0.62060
0.51175
0.35371
0.66954
0.85555
1.07086
1.22272
1.29113
1.38480
1.46728