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M. A Calderon.

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PROBABILITY DENSITY ANALYSIS
OF OCEAN AMBIENT AND SHIP NOISE

M. A. Calderon • Research and Development Report 1248 • 5 November 1964

U. S. NAVY ELECTRONICS LABORATORY, SAN DIEGO, CALIFORNIA 92152 . A BUREAU OF SHIPS LABORATORY







THE PROBLEM



Investigate the probability density distribution of the ampli-
tude of ocean ambient noise and ship noise; determine any
differences in the distributions which might lead to the
identification of ship noise masked by a high background-
noise level. Also, determine, by standard statistical
methods, whether the distributions are gaussian or non-
gaussian.



RESULTS



1. Ambient ocean noise was found to have a gaussian
distribution of amplitudes (in the sense that the moments of
the distribution satisfied specific tests) only when the am-
bient noise was relatively clean, i.e., the noise did not
contain high-level ship noise, biological noise, ice noise or
any of the other extraneous noises discussed in the text.

2. The group of ship-noise samples recorded at
close range contained a large number of samples that had a
non-gaussian distribution. However the other types of ex-
traneous noises were found to cause the same kind of de-
viation from a gaussian distribution, so that it was not
possible by these tests to distinguish between a sample
with ship noise and a sample with the other types of ex-
traneous noises (such as biological and ice noise).



MBL/WHOI



D 03D1 00M0503 1



RECOMMENDATIONS



1. Use the method of moments described here if
better accuracy than that given by overlays is desired to
estimate the moments and to determine whether a sample
is gaussian or non-gaussian.

2. In the probability density analysis of a noise
sample, use a range of amplitudes covering at least ±4
standard deviations; otherwise large errors in the estimates
of the moments will frequently result.

3. In future applications of the PDA, have the output
of the PDA in a digital form rather than a continuous curve,
so that the data will be available in a form more suitable
for the calculation of the moments of the distribution.



ADMINISTRATIVE INFORMATION



Work was performed under SR 004 03 01, Task 8119
(NEL L2-4) by members of the Listening Division. The
report covers work from January 1962 to June 1963 and
was approved for publication 5 November 1964.

The author wishes to express appreciation to the
members of the Listening Division who contributed their
time to perform much of the data processing; W. P. de la
Houssaye who wrote the computer program; and Elaine
Kyle who prepared the data for the computer. Thanks are
also extended to Fred Dickson, who prepared the illustra-
tions, and to G. M. Wenz, who made many helpful suggestions
during the work phase and during the writing of the
manuscript.



CONTENTS



INTRODUCTION. .. page 5
TEST PROGRAM. . . 7

Instrumentation. . . 7

Research Techniques. . . 8

Data Reduction Techniques. . .14
RESULTS... 3
CONCLUSIONS. . .3 7

PD of Ambient Ocean Noise. . .3 7

PD of Ship Noise. . .37

Comparison of Test Methods. ..37
RECOMMENDATIONS. . .38
REFERENCES. ..3 9
APPENDIX A: DESCRIPTION OF THE PDA AND ITS

OPERATION... 41-42
APPENDIX B: DETERMINATION OF NUMBER OF DATA

POINTS OF EACH SAMPLE. . .43-44



TABLES



1 Noise Samples Selected for Analysis, by Location. . . page 11

2 Number of Curves Showing Significant Values of Skewness

and Kurtosis. . . 31

3 Locations of Curves Showing Significant Values of Skewness

and Kurtosis. . . 32

4 Locations of Curves Showing Skewness and Kurtosis at 1

Per Cent Probability Level. . . 33

5 List of Curves for Which Chi-Square Was Computed. . .35

6 Curves Chosen by Normal Curve Overlay as Being Very

Closely Gaussian. . .36



ILLUSTRATIONS



1 Curve of probability density function of a gaussian random

variable. . .-page 6

2 Block diagram of Probability Density Analyzer system. . . 7
3-4 Selected probability density curves, compared with a

normal curve. . . 12-13
5-6 Examples of PD curves obtained by use of overlay method. . . 15-1 7

7 Curve obtained with Probability Density Analyzer. . . 23

8 Normalized cumulative sums of tabulated values. . . 25

9 Curves with positive and negative skewness, computed

with Edgeworth's series. . . 26

10 Curves with positive and negative kurtosis. . . 2 7

11-12 Cumulative probability of noise samples shown in figures
3 and 4. . .28-29
13 Experimental PD curve of random noise showing skewness,

calculated by Edgeworth's series. . .34
Al Theoretical and experimental PD curves of square wave

and sine wave. „ .41
Bl Number of times random noise goes into an interval

about x = for PD of 0. 4, vs. cutoff frequency of
low-pass filter. „ .4 5



INTRODUCTION



The study reported here was undertaken to investigate
the probability density distribution of the amplitudes of
ocean ambient noise and ship noise with respect to various
bandwidths in several frequency ranges. The question to
be answered was whether ambient noise, without any ship
noise or biological noises, can be considered gaussian, and
whether the presence of ship noise significantly changes the
probability of density distributions. A secondary objective
was to investigate methods of data reduction of the probability
density curves obtained with the B & K Probability Density
Analyzer, using standard statistical tests.

The probability density function, as treated throughout
this report, may be defined as follows.

Lim , .

™. ' Ax. a \

p(x) = Ax-0 _t I (1)

N

N~ CO

where x is a random variable, with its range of values
divided into a large number of continuous intervals Ax.
Measure its instantaneous value a great number of times N.
Let n. be the number of measured values of x in the tth
interval {Ax . ).

The above equation can be rewritten as

Lim . , , A

At . I Ax, ...

p{x) = Ax-* v i (2)

T

T — <n

where A t . is the amount of time the signal spends in the
interval A^ and T is the total time of the sample. Equation
2 indicates more clearly how the B & K PDA measures the
probability density function. A more detailed explanation
can be found in reference 1. (See list of references at end
of report. )



The function



p(x) =



exp -Or-*) 2 /2a 3 1



where x is the mean and o~ is the standard deviation, is
illustrated in figure 1,




-3-2-10 12

AMPLITUDE IN STANDARD DEVIATION UNITS



Figure 1 . Curve of the probability density fu no-
tion of a gaussian random variable (normalized to
a unit area under the curve).



TEST PROGRAM



Instrumentation

The equipment used for the investigation is described
below and illustrated in figure 2.

An Ampex Model 350 was used as the record and play-
back recorder. This model has a good low-frequency
response to below 20 c/s.

The filters following the recorder were an Allison
Laboratories Model 2-A (used mainly as a low-pass filter)
and a B & K Band Pass Filter Set, Type 1611.

A Mcintosh amplifier, Model MC30, was used to
raise the signal level to 1 volt rms or greater.

The B & K Probability Density Analyzer, Model 160
(to be referred to as the PDA) was the main piece of equip-
ment and has been primarily designed to obtain the prob-
ability density curves of disturbances that are essentially
random in character. A brief description of the PDA and
its use in this investigation is given in Appendix A. A com-
plete and detailed description of the PDA can be obtained
from the instruction manual. 1



AMPEX 350
PLAYBACK UNIT



ALLISON LABS FILTER

MODEL 2-A

OR

B & K, TYPE 1611



Mcintosh
amplifier

MODEL MC30



B&K

PROBABILITY DENSITY

ANALYZER

MODEL 160



CRO



VAR I PLOTTER
XY RECORDER



COUNTER



Figure 2. Block diagram of Probability Density Analysis system.



An XY recorder by Electronic Associates, Inc., was
used to record the analog X and Y outputs of the PDA.

A cathode ray oscilloscope monitored the signal out-
put of the filter.

The counter used responded to frequencies of at least
10 Mc/s for use with the PDA. The counter can be used in
place of an XY recorder and, in fact, is essential if
measurements are to be made at low probability densities.



Research Techniques



Data which had been recorded for previous ambient-
noise studies were available for this study. These samples
had been recorded on 1 Of— inch reels of ^-inch tape, at 3f
inches per second, and were from three locations. Two
groups had been made in shallow water - one, about 2 miles
from the western side of an island off the coast of Southern
California, and the other in the Bering Straits. These con-
sisted of short ambient-noise samples recorded at regular
intervals throughout the day, so that one reel covered data
for one day. The third location represented was in deep
water in the North Pacific between Hawaii and Alaska; most
of these samples were of longer duration than the other two
groups, but covered only a few days.

Samples of ship noise were desired, so that their
probability density curves might be compared with those of
"clean" ambient noise. Recordings were made of ships
entering San Diego Harbor, with the sampling made at ap-
proximately the closest point of approach. These included
Navy surface ships, submarines (surfaced), and commercial
ships.

Several factors were considered in choosing the data
samples to be used in this study.

1. "Clean" ambient noise was used to determine
whether the distributions of the amplitudes were gaussian



or near-gaussian according to certain tests which will be
discussed later. Ambient noise was judged to be "clean"
when it was free from ship noise, biological noises, or any
man-made sounds when the sample was monitored. A band-
pass filter and oscilloscope were used to determine whether
60-c/s hum or any other single frequency components were
present in the noise sample.

2. All noise samples should be stationary for their
entire length. When the sample is ambient ocean noise,
this condition will not in general be true. For a noise
sample to be stationary it is necessary for the sample
parameters, the means and the variances, to remain un-
changed as measured from samples taken at different times.
It is possible that no significant difference in the sample
parameters will be found if the time between samples is
short enough. In a previous study 3 it was concluded that
ocean noise is a slowly varying, not a stationary, process.
This conclusion was based on a comparison of samples that
were 3 or more minutes apart. However, no significant
difference was found among the values of some other samples
which were only 3 minutes or less apart. Thus it appears
reasonable to assume that ocean noise is stationary during

a short interval of time (less than 3 minutes).

3. The PDA requires a noise sample of about 30
minutes duration for a complete automatic analysis of the
amplitudes from -3. 00 to +3. 00 standard deviations.

The need for a long noise sample that is stationary
can be satisfied by recording a short noise sample on mag-
netic tape and then making a loop of the tape. A loop length
was selected according to the following requirements.

a. The loop should be short enough so that the noise
could be considered stationary and so that the entire loop
could be analyzed for each amplitude interval. The PDA
(in the particular position used) requires 30 seconds to
sweep a range of amplitudes equal to the window width,
which is 0. 1 times the rms value of the input signal. A
sample length of 7 seconds met all the above requirements



and this gives a loop size of 52. 5 inches, which was
conveniently handled.

b. The recorded noise on the loop should be con-
tinuous, i.e., there should be no blank intervals on the
loop, since a blank interval would change the average rms
value of the recorded noise.

A typical analysis procedure was as follows. A portion
of data was selected for analysis from the recorded data
available. The noise was re-recorded on a loop. The loop
was played back at l\ ips and the analysis proceeded as
indicated by the diagram in figure 2. The filter was set to
the desired bandwidth, and the noise was amplified to 1 volt
rms or greater. The PDA was carefully calibrated and
adjusted just before each analysis. Its input level of noise
was adjusted to 1 volt rms by its potentiometer, thus
normalizing its output.

Probability density of the amplitudes was recorded on
the Y scale of the XY recorder and the amplitude around
which the probability density was measured was on the X
scale. Scale factors were selected to give a deflection of
4 inches on the Y scale for a probability density range of
to 0. 4, and a deflection of 1 inch per standard deviation of
amplitude on the X scale. The automatic sweep time of the
PDA was set at X = -3. 00 standard deviations, and would
automatically sweep through to X = +3. 00 standard deviations,
based on a 1-volt rms input. Total running time was about
30 minutes. This procedure was repeated for each band-
width on every loop analyzed.

Table 1 lists the number of samples analyzed from
each location, the total number of probability density
curves obtained from the samples, and the filter used to
analyze these curves. When the Allison Laboratories filter
was used, the system cutoff frequency at the low end was
about 20 c/s and the upper cutoff frequency was determined
by the filter which was set at 2500, 1500, 1200, 6 00, 300,
or 150 c/s. The B & K filter was used in both the octave



10



and third-octave positions for center band frequencies of
100, 200, 400, 800, and 1600 c/s.



TABLE 1. NOISE SAMPLES SELECTED FOR ANALYSES, BY LOCATION.
(FOR THE BANDWIDTHS USED, SEE ABOVE)



LOCATION


NUMBER OF
NOISE SAMPLES


NUMBER OF P D
CURVES OBTAINED


FILTER USED FOR ANALYSIS OF DATA


SOUTHERN
CALIFORNIA


9


29


8 SAMPLES WITH ALLISON LABS FILTER;
1 SAMPLE WITH ALLI SON LABS AND B & K


BERING STRAITS


9


24


ALLISON LABS


NORTH PACIFIC


8


65


B&K


SAN DIEGO
(SHIP NOISE
IN HARBOR)


9


36


ALLISON LABS



Actual probability density curves of ambient noise
are shown in figures 3 and 4. The large fluctuations in
some of the traces are caused by substantial variations in
the level of the noise sample. Since some of the curves
appeared to be closely gaussian, the methods used to
measure the parameters of the distribution included over-
lays, calculated moments, and cumulative probability
graphs. Tests of significance and the chi-square "good-
ness of fit" tests were used to determine what values of
skewness and kurtosis were improbable at a 5 or 1 per cent
probability level.



11



0.5



0.4



0.3



0.2



0.1



0.5



0.4



0.3



0.2



0.1



DATA TAKEN IN SHALLOW WATER (So. Calif.




i i 1 r

DATA TAKEN IN BERING STRAITS




-2 -1

AMPLITUDE



1 2

IN STANDARD DEVIATION UNITS



Figure 3. Exampl es of some PD curves taken in
shallow water, compared with a normal curve.



12




£ o



^°-5

CO

<

CO

o
en



0.4



0.3



0.2



0.1



i r



n 1 r~

-DATA TAKEN IN
SAN DIEGO HARBOR (Ship Noise)




-2-10 12

AMPLITUDE IN STANDARD DEVIATION UNITS



Figure 4. Examples of some PD curves taken
both shallow and deep water, compared with a
no rmal curve.



13



Data Reduction Techniques



OVERLAY METHOD

Since it was expected that the probability density-
curves obtained with the PDA would have a gaussian or
nearly gaussian distribution, an overlay with a gaussian
curve was used. The curve had parameters of a mean
equal to zero and a standard deviation equal to one. Figure
5 illustrates the use of this method with two curves, one
judged to be gaussian and the other non-gaussian. Some
probability density curves obtained with the PDA were
judged to be very nearly gaussian.

One disadvantage of the overlay method is that de-
cisions about how well a particular curve compares with
the overlay are purely subjective. Skewness and kurtosis
can be detected, but the magnitudes of these moments cannot
be estimated with accuracy. An extension of the overlay
method which will allow estimates of skewness and kurtosis
is described here.

The extension is an overlay with several curves in-
stead of just one. Each curve has a different set of values
for skewness and kurtosis. The curves are positioned
over the actual probability density curve and the parameters
are estimated by interpolation between the two closest
curves. The curves of the overlay can be computed with
the use of Edge worth's series approximation for nearly
gaussian distributions. 3 The first four terms of this series
are



fix) = h(x)-—h 3 (x)+-^hUx)+— h e (x) (3)

where h(x) is the normalized gaussian distribution, h (x)
is the nth derivative of h(x), g is the standardized skew-
ness, and g is the standardized kurtosis.



14




-2-10123
AMPLITUDE IN STANDARD DEVIATION UNITS



Figure 5. Examples of two PD curves which
were det ermined to he gauss i an or non- gaus s i an,
using a normal curve as an overlay.



15



Estimates of the skewness and kurtosis can be found
with the above method; but it does not give any indication of
whether these estimates are significantly different from the
expected values, if the sample is taken from a gaussian
distribution. Using the previous overlay, a method can be
developed so that a sample can be accepted or rejected at
any desired level of probability. Basically the method is
to have two of the curves on the overlay plotted so that they
will represent the maximum deviations allowed in the par-
ticular parameter of a sample with (N) points. The method
will be developed for kurtosis, but a similar method can be
used for skewness.



The variance of kurtosis is given by 4

var(o ) = 24/N (4)



for large TV. This holds for a sample taken from a normal
parent population. The standard deviation of kurtosis is
(24/710 2; if the kurtosis is distributed normally, then from
the ratio of a particular value of kurtosis (g s ') and the
standard deviation we can obtain the probability of getting a
value of kurtosis as large or larger than g ' . The ratio is



(24/70 2



The probability of getting a value of kurtosis as large as or
larger than g ' is given by the amount of area under a
normal curve outside the -7? and +7? standard deviations.
A value of 7? = 1. 96 corresponds to a probability level of



(5)



16



5 per cent, or l/20th the total area. A ratio as large as
1.96 may be considered sufficiently improbable and hence
g ' can be assumed to result from a non-gaussian distribution.
The sample would therefore be rejected as coming from a
gaussian distribution. The value of g s ' therefore depends
onf, g ' - 1.96(24///) s. Edgeworth's series would then
be used to compute two curves, one with -g 3 ' (for negative
kurtosis) and one with +g 3 ' (for positive kurtosis). These
curves would represent the limits, at a 5 per cent
probability level, within which a sample of N points would be
considered as coming from a gaussian distribution.

Figure 6 shows two curves as they would appear in
the overlay. These two curves are the limits for a sample



0.5



g' ■ +0.50




-3



g' =-0.50



-2-10 12

AMPLITUDE IN STANDARD DEVIATION UNITS



Figure 6. Overlay indicating g 3 ' of +0.50 and of -0.50,
A curve having a value of kurtosis as large or larger
than these values will be non- gauss i an at a 5 per cent
level for a sample of 3 70 points or, e qui val ent 1 y , a
bandwidth of about 55 c/s.



17



of bandwidth about 55 c/s, with N given by the equation N =

6.7/, where / is the bandwidth. The equation is obtained

o o

from Appendix B, using a time constant T = 2.3 seconds.

The overlay method was not used extensively because
of the complexity that comes from considering different
values of N and also different combinations of skewness and
kurtosis in the same sample. A method using computed
moments of the curves is described next; it was felt that
this method would yield accurate values of the mean,
standard deviation, skewness, and kurtosis.



METHOD OF MOMENTS

The method of moments is basically a general method
of forming estimates of the parameters of a distribution by
means of a set of measured sample values. The first few
moments of the actual distribution are calculated and these
are used as estimates of the moments of the parent population.
On the basis of these moments a suitable theoretical dis-
tribution curve is selected. For any particular distribution
curve the moments are functions of the parameters of that
curve. The parameters are determined and tests of sig-
nificance are made on the skewness and kurtosis.

The moments about the origin are defined as 5



m ' = £ p.(x)x. (6)

r v i>

i
where p^(x) is the probability that a value selected at ran-
dom from the population will lie in the i-th class. The
variate x with which we are concerned may be discrete or
continuous.



The moment



m 1 ' - E P t ^ x t (7)



18



is defined as the mean value of x, m ' - x.

Another more important set of moments is obtained
by changing the origin to the arithmetic mean. Equation 8
defines the moments about the mean.

m = £ p.(x) (x. - x\ (8)



For computing purposes, the relations between the

m and the Tn ' are convenient. Expressing the m in terms

of the rn ' we have the relations
r

m 1 = m 1 ' (9a)

m 2 = m 3 ' - (ot 1 'f (9b)

m 3 = m 3 ' - 3m a 'm 1 ' + 2(% ') 3 (9c)

ff2 4 = ot 4 ' - 4m 3 , m 1 ' + Qm 3 '{m x 'f - 3(m ± ') 4 (9d)

Grouping errors are negligible, so Sheppard's
corrections are not applied.

- These moments can be expressed in standard units
by the use of a standardized variable z, by dividing the

variable x by s , the standard deviation.

J x



(x-x)
z -

s

X



(10)



The standardized moments are defined by the equations

772

a = — — , for r = 1, 2, 3, and 4 (11)

r r

s
x



19



a 2


= 1




m a


a 3






s ;




«r


a 4


m 4




s




X



The first four standardized moments are

a-L = (12a)

(12b)
(12c)



(I2d)



The third moment, a 3 , is a measure of the skewness of the
distribution. A positive value indicates a distribution with
a longer positive tail than a negative tail.

The fourth standardized moment, a 4 , is a measure
of the kurtosis of the distribution. In some cases it. is a
measure of the "peakedness" of the distribution, though it
is now understood that the length and size of the tails are
very important in this measurement.

For a normal curve the values of a 3 and a 4 will be
and 3, respectively. We redefine the skewness and kur-
tosis as

Q, r « 3 (13a)

g 2 = « 4 - s (i3b)

so that g is for a normal curve.

It is not very likely that the third and fourth moments
of a random sample will be zero. Depending on the distri-
bution and on the actual sample values, the third and fourth
moments will have some value different from zero. To de-
termine whether this difference is significant, it is neces-
sary to use the variances of the third and fourth moments. 4

varfo, ) = 6N(N-l)(N-2)- 1 (N-l)- 1 (N-3)- 1 (14a)



20



var(£ 3 ) - 24A"(^-l) 2 (^-3)- 1 (^-2)- 1 (^-3)- 1 (#-5)- 1 (14b)

For large N use,

var(^) = 6/N (15a)

var(p a ) = 24/27 . (15b)

The hypothesis to be tested is that the data sample is
taken from a gaussian distribution. To test the hypothesis
compare g to (6/N)z and g 3 to (24/#)s (see ref. 5), then



if



> 1. 96 reject the hypothesis at the 5 per cent level



(6 /N)s

> 2. 57 reject the hypothesis at the 1 per cent level.



Similarly, for g r



if > 1.96 reject the hypothesis at the 5 per cent level

(24/tf)*

> 2. 57 reject the hypothesis at the 1 per cent level.



CHI-SQUARE "GOODNESS OF FIT" TEST

The x 3 test will be applied to the hypothesis that a
sample of N individuals forms a random sample from a
population with a given probability distribution. The param-
eters of a distribution are known and are not estimated
from the sample itself. Later a modification will be given
for the situation where the parameters are estimated from
the sample.



21



The quantity*



(F.-Np. f

L -V- (16)



s



is a measure of the deviation of the sample from the ex-
pectation, where F . is the number of observed frequencies
in the tth interval, and Np . is the number of expected fre-
quencies in the ith interval as predicted by the theoretical
distribution. Karl Pearson proved that the above quantity,
in the limit, is the ordinary \ 2 distribution which is now
tabulated in most statistics books.


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Online LibraryM. A CalderonProbability density analysis of ocean ambient and ship noise → online text (page 1 of 3)