Data Smoothing
`J. T. Grissom, and D. R. Koehler
`
`Citation: American Journal of Physics 39, 1314 (1971); doi: 10.1119/1.1976648
`View online: https://doi.org/10.1119/1.1976648
`View Table of Contents: http://aapt.scitation.org/toc/ajp/39/11
`Published by the American Association of Physics Teachers
`
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`
`
`PAGE 1 OF 8
`
`SONOS EXHIBIT 1015
`IPR of U.S. Pat. No. 8,942,252
`
`

`

`AJP Volume 39
`
`form x 2 = 'TJt. The classical problem of a rod of
`fixed length originally at a temperature T 0 and
`whose end faces are held at a temperature T
`cannot be solved by this technique.
`J.11 is not
`We note that
`the transformation
`completely determined due to the variability of n.
`Even with appropriate
`initial conditions for the
`transformed equation we have a one-parameter
`infinity of solutions to choose from. It is not clear
`that more than one physically meaningful solution
`should exist. This suggests the value of n that
`ultimately completely determines M should not
`be arbitrary but should reflect an important
`physical feature of the problem.
`
`ACKNOWLEDGMENT
`
`This work was supported by the Atomic Energy
`Commission.
`
`1 G. Birkhoff, Hydrodynamics (Princeton U. P. for the
`Univ. of Cincinnati, Princeton, N. J., 1950).
`2 A. G. Hansen, Similarity Analysis of Boundary Value
`Probl£ms in Engineering (Prentice-Hall, Englewood Cliffs,
`N. J., 1964).
`3 W. F. Ames, Nonlinear Partial Differential Equations
`in Engineering (Academic, New York, 1965).
`4 A.G. Hansen and T. Y. Na, J. Basic Eng. 90, 71 (1968).
`6 A. J. A. Morgan, Quart. Appl. Math. 12, 250 (1953)
`6 S. J. Kline, Similitude and Approximation Theory
`(McGraw-Hill, New York, 1965).
`
`Data Smoothing
`
`J. T. GRISSOM*
`University of Tennessee
`Knoxville, Tennessee 37916t
`D.R.KOEHLER
`Bulova Watch Company
`Flushing, New York 11370
`(Received I February 1971)
`
`A class of methods is derived for data smoothing based
`on minimizing an error function consisting of two terms.
`The first term is the weighted sum of the squares of the
`deviations between the smoothed values and the original
`data, and the second term is the weighted sum of the squares
`of the (k+ 1)st order finite differences of the smoothed
`values. The method embodies two arbitrary parameters.
`Specifying the value of k chooses one member of the class
`of methods. The other parameter is a weighting factor that
`determines the degree of smoothing achieved at each point.
`Limits on the value of the weighting factor can be imposed
`based on the statistical properties of the data.
`
`1314 / November 1971
`
`INTRODUCTION
`
`the experimental physicist will en(cid:173)
`Often
`counter situations where it is desirable to make use
`of data smoothing techniques in the analysis or
`presentation of experimental
`results. An ideal
`example is the measurement of energy spectra
`with scintillation or solid state detectors using a
`multichannel pulse height analyzer. The resulting
`data is a digital representation of an intensity
`distribution as a function of energy. In addition
`to certain types of resolution distortion,1 each data
`point can vary from the true value because of the
`It
`is the
`presence of statistical
`fluctuations.
`purpose of data smoothing to reduce the effects of
`such random variations
`in order
`that
`those
`features of the distribution
`that exceed the ex(cid:173)
`pected statistical deviations will become more
`discernible. Other obvious examples of the use of
`data smoothing techniques suggest themselves,
`but
`the underlying motivation
`is the same,
`namely,
`to
`reduce distortion
`resulting
`from
`statistical errors.
`Perhaps the most commonly used smoothing
`procedure is based on the least squares fitting of
`polynomial functions to the data points. 2 •3 Since
`the method demonstrates
`some of the basic
`features of data smoothing techniques in general,
`we will examine it in some detail. To illustrate
`
`
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`SONOS EXHIBIT 1015
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`

`

`that
`the procedure, consider a set of data ( j;)
`represents experimentally measured values of a
`at a discrete set of values {x;). We
`functionj(x)
`wish to replace the set of values { f;) by another
`set of values { F;) that represent the "smoothed"
`form of the data. To generate Fk, one chooses
`from [ f;} the subset consisting of fk and the rn
`values on either side of .fk. Then an nth degree
`polynomial of the form
`
`to these 2m+ 1 points using a least
`is fitted
`squares criterion to generate the set of coefficients
`[an}. Once the coefficients have been determined
`from the least squares fit, Fk is just the value of
`In this way
`the polynomial at xk, i.e., Fk=y(xk)-
`the set (F;) is generated point by point. This
`procedure must, of course, be modified near the
`end points since for an arbitrary choice of m a
`value of J1c will not necessarily have m experi(cid:173)
`mental values on both sides of it. This situation
`can be handled by always choosing the 2rn+ 1
`points nearest the point being smoothed, with the
`additional
`requirement
`that
`the point being
`adjusted be the central value whenever the range
`of the data permits.
`the procedure
`With various modifications
`outlined above forms the basis of most polynomial
`smoothing
`techniques. The method directly
`illustrates a basic assumption of any smoothing
`technique, namely, that
`the smoothed value of
`each point should be influenced in some manner
`by the values of the data points on either side of
`the point being adjusted. This is just another way
`of saying that the function for which the data are a
`statistically distorted
`representation
`is smooth,
`or continuous, and that the interval over which
`any rapid fluctuations in the value of the function
`occurs is large with respect to the interval be(cid:173)
`tween the measured values. Because of the resolu(cid:173)
`tion distortion characteristics of the experimental
`measuring apparatus this assumption is essentially
`always satisfied even for data which should in
`principle display discontinuities or very rapid
`fluctuations. The resolution of the measuring
`apparatus causes such discontinuities
`to be dis(cid:173)
`torted into a smoothly varying function, and the
`proper design of the experiment dictates
`that
`measurements be performed in intervals that are
`
`Data Srrwothing
`
`on the order of, or smaller than, the "resolution
`width" of the measuring apparatus. 1
`Note that the polynomial smoothing technique
`is essentially a two-parameter method
`that
`is
`characteristic of any routine
`for smoothing a
`two-dimensional function for which the statistical
`uncertainty occurs in only one dimension. These
`parameters affect
`the resulting values of the
`smoothed data
`in a complex manner. First, m
`determines the number of points that exert any
`influence on the adjusted value of each data point.
`Once 1n is fixed, then n determines the degree to
`which each point is smoothed, i.e., the deviation
`between
`the smoothed value and the original
`value. For fixed rn the degree of smoothing
`increasing n. For n = 0
`decreases
`,vith
`the
`smoothing is a maximum with the smoothed value
`of each point being just the average of the nearest
`2rn+l data points. For n=2m no smoothing is
`accomplished since a polynomial of degree 2m
`will pass through all of 2rn+ 1 points. The range
`of n is thus 0~n~2m.
`type of
`There are several aspects of this
`smoothing that are unsatisfactory from a mathe(cid:173)
`matical point of view. The major difficulty is the
`absence of quantitative criteria on which to base a
`choice of m and n. It, is difficult to defend even
`a specific value for m. The only
`qualitatively
`satisfactory operational criterion is that m must
`be sufficiently small that a polynomial of reason(cid:173)
`able degree will give a good fit to the 2m+ 1
`points. This is a subjective criterion based on how
`rapidly the data fluctuate in the vicinity of the
`point being smoothed. A value of m that yields
`satisfactory
`results
`in one region of the data
`might conceivably be too large to work properly
`in another region. The choice of n is equally
`difficult to justify and in addition is related to the
`choice of m. Quite often the same degree of
`smoothing can be achieved by more than one set
`of values for m and n. What is even more difficult
`to justify a priori is the choice of polynomial
`functions to fit the data. There is no basis for
`choosing polynomials
`instead of a sum of ex(cid:173)
`ponentials or a sum of sines and cosines, for
`example. Here again such a choice is an intuitive or
`subjective one not based on any degree of rigor.
`In the following analysis we present a more
`direct approach to the problem of data smoothing
`in which the significance of the arbitrary parame-
`
`AJP Volnme 39 / 1315
`
`
`PAGE 3 OF 8
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`SONOS EXHIBIT 1015
`IPR of U.S. Pat. No. 8,942,252
`
`

`

`n
`
`f·2
`
`= I:~,
`
`i=l Ui
`
`(1)
`
`Consider the quantity
`
`El=
`-
`
`.;., (F;-f;)2
`,L.,---
`a?
`i=l
`
`J. T. Grissom and D.R. Koehler
`
`ters can be interpreted directly and for which
`quantitative criteria aid in choosing the parame(cid:173)
`ters based on the statistical properties of the data.
`
`I. A DIRECT APPROACH
`
`Consider a set of n data points { f;} that repre(cid:173)
`sent the measured values of the function f(x)
`with f; = f (x;). Due to the presence of statistical
`variations
`in the measuring procedure each of
`the values f; differs from the corresponding true
`value F; with Fi-Ji=
`f;. For the sake of simplicity
`we limit our attention to data where the error in
`the measurement of f; greatly exceeds the ex(cid:173)
`pected error in x;. Let u; denote the standard
`deviation in f;. The value of u; is either experi(cid:173)
`mentally determined or statistically
`estimated
`from the data.
`Any smoothing technique must be based on a
`few fundamental assumptions. For a two-dimen(cid:173)
`sional function f (x), where the major statistical
`error is confined to only one dimension with
`relatively negligible error
`in the
`independent
`variable, the smoothing procedure should contain
`two arbitrary parameters as discussed above. One
`of these can be related to the number of neigh(cid:173)
`boring points that influence each of the smoothed
`values, and the other parameter can determine
`the degree of smoothing at each point. These are
`the two degrees of freedom in the smoothing
`process. Note that the former degree of freedom is
`related
`to the independent variable while the
`latter one affects only the dependent variable.
`Consequently,
`the two-parameter nature of the
`smoothing process derives from the two-dimen(cid:173)
`sional character of the data.
`two
`The smoothed data
`itself must satisfy
`essential requirements. First, it must be a sta(cid:173)
`tistically valid representation of the original data,
`and second, it must represent a continuous func(cid:173)
`tion.
`· A smoothing procedure should embody
`quantitative measures of the degree to which
`each of these requirements
`is satisfied. Quite
`obviously the two requirements cannot be totally
`independent but are in fact opposite conditions.
`The more the data are smoothed the less the
`agreement between
`the
`initial and smoothed
`values. For the sake of discussion we will refer to
`these two conditions as, respectively, the require-
`ments of "reproducibility" and "smoothness."
`
`1316 / November 1971
`
`where E 1 is a positive quantity whose magnitude
`measures the agreement between the smoothed
`data and the original values. Each of the squared
`deviations in Eq. (1) is weighted by the inverse
`of the corresponding variance a}. Thus E 1 is a
`quantitative measure of reproducibility. Mini(cid:173)
`mizing the value of E1 with respect to the choice of
`( F;} in the least squares sense results in the trivial
`solution
`
`i=l,2,
`
`... ,n,
`
`which satisfies the reproducibility criterion exactly
`but violates the smoothness requirement.
`To measure the smoothness of the solution we
`will make use of the finite difference Taylor series
`representation
`
`Fi+s =F;+soF;+[s(s+l)
`
`/2!]o 2F;+ · · ·
`
`+[s(s+l)
`
`· · · (s+k-1)/k!JokF;+Rki,
`
`where s is an integer, okF; represents
`the kth
`finite difference of F;, and Rki is the remainder
`after k+ 1 terms. We will limit ourselves to the
`case s = I for which
`
`The finite differences are of the form
`
`oF;=Fi-F;-1,
`o2F; = F;-2F;-1+F;-2,
`o3F;=F;-3F;-1+3F;-2-F;_3,
`
`(3)
`
`where
`
`kJ_)=-k!
`(k-J)!j!'
`
`(
`
`are the binomial coefficients. From Eqs. (2) and
`
`
`PAGE 4 OF 8
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`SONOS EXHIBIT 1015
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`

`

`(3) and the properties of the binomial coefficients
`one can show that
`
`Rki=F,-+1- kf (-).i-
`
`.1-1
`
`1 (k~l)Fi-(j-1J
`.7
`
`(4)
`
`the. magnitudes of
`From Eq. (,t:i) we see that
`the Rki directly indicate the relative smoothness
`of the solution { F, /. Consider
`
`n-1
`B2= ,I: R1:/
`i-k+I
`
`(6)
`
`defined so that E2 is a positive quantity whose
`{Fi j .
`value measures
`the smoothness of the
`Minimizing E 2 with respect to the choice of the
`l Fi)
`in the
`least squares sense results
`in the
`trivial solution
`
`Fi=const;
`
`1'.=l, 2, .. ·, 11,
`
`which satisfies the smoothness criterion exactly
`but in general violates the reproducibility require(cid:173)
`ment.
`the error functions E1 and
`We can combine
`E 2 that measure, respectively, the degree to which
`the reproducibility
`and smoothness criteria are
`individually satisfied into a single error function
`E such that
`
`(7)
`
`where a is a constant. The value of E measures
`both the smoothness of the curve and the extent
`to which
`the smoothed values
`reproduce
`the
`initial data. The value of a determines the relative
`weight of E 2 with respect
`to E 1. Smoothing
`is
`accomplished by minimizing the value of E with
`respect to the choice of the {Fi). The smoothed
`solution { Fi) is thus determined by solving the
`set of n simultaneous equations
`
`Data Smoothing
`
`smoothing routine based on the arbitrary pararne(cid:173)
`ters k and a. The value of k specifies the number
`of terms in the finite difference series representa(cid:173)
`tion of Eq.
`(2), which is analogous at least to
`determining
`the number of neighboring points
`that are sampled to adjust the value of each of the
`initial data points. This analogy is not rigorous
`the value of each F; resulting
`since in actuality
`from the solution of Eqs. (8) depends on all of the
`data values. The parameter a fixes the relative
`weights of E 1 and E2 in the minimization of E,
`which is analogous
`to determining
`the degree
`of smoothing at each point. Later we will discuss
`the dependence of the solution on k and a.
`:1finimizing E by Eq. (8) automatically mini(cid:173)
`mizes
`the deviations
`between
`the
`smoothed
`solution and
`the original values as far as is
`consistent with the degree of smoothing achieved.
`This in turn is determined by the value of k and a
`or more specifically, by the weighting factor a
`once k is specified. Thus neither E 1 nor E2 is
`minimized absolutely, but their weighted sum is
`minimized
`to satisfy both
`the requirements of
`reproducibility and smoothness.
`
`II. MATRIX FORMALISM
`
`We will rewrite the smoothing procedure using a
`matrix formalism to make it easier to appreciate
`the mathematical
`structure of the technique and
`to
`interpret
`the
`significance of
`the various
`quantities
`that comprise the method.
`Let f represent an nX I column matrix whose
`elements consist of the n values { fd. Similarly F
`represents an nX 1 matrix composed of the n
`smoothed values f F,-l. Then t is the matrix given
`by
`
`t= F-f.
`
`(9)
`
`If S is an n X n matrix whose elements s;; are
`given by
`
`(10)
`
`the usual Kronecker delta,
`where Oii represents
`then Eq. ( 1) can be written in matrix form as
`
`E 1 = (F-f)TS-
`
`1 (F-f)
`
`aE/aF;=O;
`
`j=l,
`
`2, .. ·, 11,
`
`(8)
`
`or
`
`with le and a fixed.
`We have now
`
`formulated
`
`a
`
`least
`
`squares
`
`in terms of the matrix t and its transpose.
`
`(lla)
`
`(llb)
`
`AJP Volume 89 / 1817
`
`
`PAGE 5 OF 8
`
`SONOS EXHIBIT 1015
`IPR of U.S. Pat. No. 8,942,252
`
`

`

`J. T. Grissom and D. R. Koehler
`
`In matrix form Eqs. (4) and (5) become
`
`R=OF,
`
`(12)
`
`where R is an n X 1 matrix whose elements are the
`quantities Rki, and O is an nXn matrix operator
`whose elements depend on the value of k through
`Eq.
`(5). From Eq.
`(4) we choose to rewrite
`Eq. (12) in the form
`
`where A denotes the variation of E with respect
`to the elements of F. Writing E explicitly gives
`1 (F-f) + (OF)TW- 1OF,
`
`E= (F-f)TS-
`
`(20)
`
`and applying the A operator results in the matrix
`equation
`
`cs-1+orw-
`
`1o)F-S-
`
`1f=0,
`
`R= (1-T)F,
`
`or
`
`(13)
`
`identity matrix, and T is a
`where I is the nXn
`transformation matrix whose elements depend
`again on the specific value chosen for k. Finally,
`from Eq. (13)
`
`R=F-TF,
`
`(14)
`
`and R has the same form as t: in Eq. (9). We will
`see that R and t: play similar roles in the mathe(cid:173)
`matical form of the smoothed solution.
`From Eq. ( 6) , E 2 can be written in terms of the
`matrix R and its transpose as
`
`(15)
`
`(11) we can choose to
`By analogy with Eqs.
`modify the definition of E 2 to include a weighting
`matrix w- 1 such that
`
`(16)
`where w- 1 is an nXn diagonal matrix whose
`elements are the weighting factors
`
`F= (I+sorw-
`
`1O)- 1£.
`
`(21)
`
`(21) represents a formal solution for
`Equation
`the smoothed values F in terms of the original
`data f, the matrix operator O defined in Eq. (12),
`the variance matrix S of the experimental data,
`and the weight matrix W. The form of Eq. (21)
`illustrates that the smoothed solution is generated
`from f by applying the transformation operator
`
`(I+sorw-
`
`1O)- 1.
`
`(22)
`
`The first term in this operator essentially arises
`from the requirement of reproducibility. The
`second term results from the smoothing require(cid:173)
`ment itself and also contains the information for
`fixing the relative weighting of the smoothing with
`respect
`to
`the
`reproducibility. The
`rationale
`underlying these observations becomes apparent
`identically
`if we let the weighting factors a;
`vanish so that w- 1 becomes the zero matrix.
`in the trivial solution F = f from
`This results
`Eq. (21).
`
`(W- 1) ;j= a;o;j,
`
`(17)
`
`III. SMOOTHING CRITERIA
`
`The set of constants a; are as yet unspecified and
`are completely arbitrary. Examination of Eqs.
`(11) and '(16) displays the similar form of E 1 and
`E 2 from which it can be seen that R and t: are
`analogous quantities, and the weighting matrix W
`is analogous to the variance matrix S. The total
`error function now becomes
`
`The solution again results from the set of equa(cid:173)
`tions obtained by minimizing the value of E. Thus
`
`.:iE=O,
`
`(19)
`
`1318 / November 1971
`
`To complete the solution we must discuss some
`criteria for choosing the values of k and a;.
`In Eq. ( 2-1) , 0 depends directly on the value of k
`through Eqs.
`(5) and
`(12), which define the
`elements of O. In addition we must specify the
`elements of the weighting matrix w- 1 in Eq. (21).
`In the matrix formulation w- 1 was taken to be a
`completely general diagonal matrix in analogy to
`the matrix s-1. This choice gives the terms E1
`form and empha(cid:173)
`and E 2 the same mathematical
`sizes the similar roles played by the smoothness
`and reproducibility
`criteria
`in specifying
`the
`solution. However, since w- 1 is an nXn matrix,
`we now have to specify the n values of the a;
`
`
`PAGE 6 OF 8
`
`SONOS EXHIBIT 1015
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`
`

`

`to k, and the method becomes a
`in addition
`2n+ 1 parameter
`technique. Although one could
`possibly formulate certain criteria for choosing
`2n+ 1 parameters,
`the difficulty involved quickly
`becomes considerably greater. As discussed pre(cid:173)
`viously,
`the
`two-dimensional character of the
`function being smoothed imposes the requirement
`of a minimum of two degrees of freedom without
`loss of generality. In the interest of practicality
`we therefore choose the simplest form possible
`for w- 1, namely
`
`w-1 =al,
`
`the problem reduces again to one of
`so that
`specifying only two parameters.
`It is necessary to first investigate some of the
`general effects of varying k and a. Once k is fixed
`the form of O is specified and the solution depends
`on a through the equation
`
`F= (I+aSOTQ)-
`
`1£.
`
`Recalling that F results from minimizing E, we
`can best understand
`the effects of varying a by
`examining E 1 and E 2• Increasing the value of a for
`fixed k emphasizes smoothness at the expense of
`reproducibility,
`resulting
`in a solution
`that
`is
`increasingly smooth but deviates further from the
`experimental data. Stated differently,
`the value
`of Ji} is minimized independently of a, but as a
`increases Er increases while E2 decreases, within
`limits. The value of B2 goes from some finite upper
`limit to zero as a increases without
`limit. The
`maximum value of E2 is given by the sum of the
`squares of the (le+l)st
`finite differences of the
`experimental data according to Eqs. (i)) and (6).
`The value of Er is zero when a= 0 and increases
`to some finite upper limit as a increases without
`limit. Changing the value of le will necessarily
`alter the upper limits on Er and E2 and will yield a
`new solution corresponding
`to a different mini(cid:173)
`mum value of B, but the general dependence of
`Er and E2 on the value of a will remain the same.
`Increasing le for fixed a will also result in a solution
`that is less Rmooth but that more accurately re(cid:173)
`produces the initial data.
`The number of data points will set some limits
`on the value of le. Examination of Eqs. (3) shows
`that le can only assume integer values from le= 0 to
`k=n-2, where n is the total number of experi-
`
`Data Snwolhing
`
`mental values in the original data. Generally, one
`deals with data for which n»2 so that the values
`of k used in practice are far below the upper limit.
`In reality, the smoothing procedure formulated
`here is not one method but is a whole class of
`similarly structured
`smoothing
`techniques, one
`k.
`for each allowed value of the parameter
`Thus k can be thought of as an index that specifies
`the type of smoothing procedure to be used from
`the class of least squares leth-order finite-difference
`methods. In practice it becomes unnecessary
`to
`consider other than the firnt few values of k, e.g.,
`k=O, 1, 2, :3. For values greater than k=3
`the
`reRults become redundant.
`To determine a we can use the error function
`Er. If the reproducibility criterion is satisfied so
`that the Rmoothed data form a valid representa(cid:173)
`tion of the initial values then we would expect the
`deviations
`to be on the order of the corre(cid:173)
`t;
`sponding Rtandard deviationR <Ti, i.e.,
`
`fi=f?i-fi~<r;.
`
`(24)
`
`If Ei is much greater than ui we would suspect that
`the smoothing process has gone beyond just the
`removal of statistical variations and has over(cid:173)
`smoothed
`in thiR region. Combining Eqs. (24)
`and ( 1) gives
`
`(2.5)
`
`as an estimate of the maximum allowed value of
`Er. For Er= n the average deviation between the
`experimental values and the smoothed solution is
`one standard deviation of the experimental data.
`For Er»n
`the average deviation exceeds the
`value for which the smoothed data can be accepted
`as a valid representation of the initial quantities.
`If Er«n
`the induced deviations will be smaller
`than the standard deviations on the average, but
`very little smoothing will be achieved.
`
`IV. SUMMARY
`
`The practical use of the methods derived above
`requires the services of a digital computer. Equa(cid:173)
`tion (23) represents a set of n equations
`in n
`If n is very large it becomes quite
`unknowns.
`tedious to perform the calculations by hand, and
`the method works best on data for which n»l.
`
`AJP Vol1tme 39 I 1319
`
`
`PAGE 7 OF 8
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`SONOS EXHIBIT 1015
`IPR of U.S. Pat. No. 8,942,252
`
`

`

`AJP Volume 39
`
`On the other hand, the equations can be handled
`well on a digital computer for n on the order of
`102 or greater. At this point the computational
`procedures may be summarized.
`The method begins by specifying k. In practice
`it is sufficient to limit consideration
`to a single
`value of k, for example k = 3. Once k is fixed the
`matrix operator O can be computed from Eqs. (5)
`and (12). From the experimental data or from
`auxiliary measurements
`the elements of
`the
`variance matrix S are determined by using
`Eq. (10). Finally, a first guess of the value of a is
`made, and the solution F is found from Eq. (23).
`This solution
`is tested
`for reproducibility by
`computing E 1 from Eq. (11) and comparing E1
`with n. If necessary the value of a is reduced and
`
`the calculations repeated until a value of E 1«n is
`obtained. Then a is increased in steps until E 1~n
`or some other predetermined upper limit that is
`deemed acceptable. The degree of smoothing
`increases and the statistical reliability decreases
`with increasing a. If desired the value of k can be
`changed and the entire procedure repeated.
`
`*U.S. Radiological Health Physics Fellow.
`t Present Address: Sandia Laboratories, Albuquerque,
`N. M. 87115.
`1 J. T. Grissom and D. R. Koehler, Amer. J. Phys. 35,
`753 (1967).
`2 K. L. Nielsen, Methods in Numerical Analysis (Mac(cid:173)
`millan, New York, 1967), p. 295.
`3 J. Singer, Elements of Numerical Analysis (Academic,
`New York, 1964), p. 351.
`
`The Response to Crisis-A Contemporary Case Study
`
`T. P. SWETMAN
`Cavendish Laboratory
`Cambridge University
`Cambridge, England
`(Received 28 December 1970; revised 1 April 1971)
`
`to a
`The initial response of the scientific community
`recent crisis situation is documented. The example chosen
`is the 1964 discovery of the two-body decay of the K2°
`meson: The more immediate reactions to this discovery,
`and to its implications for time reversal symmetry, are
`considered.
`
`INTRODUCTION
`
`The reaction of the scientific community to the
`changes in paradigm, necessitated by new dis(cid:173)
`coveries that do not fit the currently accepted
`theoretical matrix, has long been a subject of
`study for historians of science. This paper docu(cid:173)
`ments a fairly recent crisis in the field of sub(cid:173)
`nuclear physics and attempts to indicate the initial
`theoretical and experimental
`responses to this
`crisis; the discovery concerned is the t\vo-body
`decay of the long-lived neutral K meson, and the
`preliminary experimental work was performed in
`1964.1 I shall endeavor here to delineate
`the
`reactions to this discovery during the year fol(cid:173)
`lowing its publication in the scientific literature.
`The historian of science T. S. Kuhn has de(cid:173)
`scribed the period of crisis as leading to "The
`proliferation
`of competing
`articulations,
`the
`willingness to try anything,
`the expression of
`explicit discontent,
`the recourse to philosophy
`and to debate over fundamentals.2''
`I hope that
`the present paper will illuminate this process in a
`contemporary context. I shall begin by sketching
`the relevant
`theoretical framework prior to the
`discovery, in order to clarify the significance of
`the 1964 experiment.
`
`1320 / November 1971
`
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`PAGE 8 OF 8
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`SONOS EXHIBIT 1015
`IPR of U.S. Pat. No. 8,942,252
`
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