ࡱ;   !"#$%&'()*+,-./01234578A:;<=>?@6BCDRoot EntryF0ЧBook `CompObj^SummaryInformation(9 FMicrosoft Excel 5.0 WorksheetBiff5Excel.Sheet.5;  Oh+'Oh+'0H`hpG#0&     [ ``iݜ*o?H6A^BbzbO/(fvEE>j(9jJ2 """)))000___UUUMMMBBB999 ```( +9%I;/]E:IS+!!lYQGj2ga1aS {Cg..&YQFh.IRj#SjJu3lJA7e ,N, QdoV YC6r3w_GqC-}zn#&s[ Rp$L 6n{DuqVJ4H2Aph6B"}3%6Z\H"BMXR ӕ$sVo<gX .` @ !)!$F6XL.nZ@A!uw_?X!!$LA!ӏ\GW? !!$ LA~ "|@"R)L@"l(w@!!"&u@!!"wđ?!!"ucz?!!~ #@#n@^@#LWy@"!#6v{@"!#A=?"!#2ta>ſ"!~ $@$~@$P@#!$@#!$|?#!$3[Qֿ#!'&Dة$e $e $A;'&I,}X$e $e $A;'')(S:@&e $e $A7''6HZ@&e $e $A7!)ln y = -0.01225x+26.32772$)ln(y-4) = -0.05341x+107.7501!*y=exp(-0.01225x+26.32772)%*y-4 = exp(-0.05341x+107.7501)#+y=exp(-0.05341x+107.7501)+4R-JThe value of y=4 as a different lower bound was suggested by the WRD data./Graph: VVMN'VABCD FHIJKJL:M N:P R T  : U :V JX :Z )@[ l \ sandy B=hxy,8X@"1xArial1xArial1xArial1xArial1xArial1xArial"$"#,##0_);\("$"#,##0\)"$"#,##0_);[Red]\("$"#,##0\) "$"#,##0.00_);\("$"#,##0.00\)%""$"#,##0.00_);[Red]\("$"#,##0.00\)5*2_("$"* #,##0_);_("$"* \(#,##0\);_("$"* "-"_);_(@_),))_(* #,##0_);_(* \(#,##0\);_(* "-"_);_(@_)=,:_("$"* #,##0.00_);_("$"* \(#,##0.00\);_("$"* "-"??_);_(@_)4+1_(* #,##0.00_);_(* \(#,##0.00\);_(* "-"??_);_(@_)                + ) , *   ! !  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The x-axis need not be the horizontal asymptote of this curve, however.ogConsider the following example and fit two exponentials--one with horizontal asymptote y=0 and one with.& a different horizontal asymptote.)!The general form of the curve is: y=Ce^(ax) + bSKwhere a < 0 and y=b is the lower bound of the exponential; C is a constant.TLThe added term shows how much the curve is lifted above or below the x-axis.IAExponential curve fit to projected crude birth data, 2005 to 2025)!Source: WRD data for Bangladesh. y=0 y=4 YearWRD proj exp. proj. exp. proj.WRD proj y-4=Ce^(ax)T@T@LN(y-0)LN(y-4) y=0 y=4."2*jRons2-WXM-{< ! " # $ & ' ) * J+ :-  / :~ T@ Hg+@1 Lۑf@$J +D >:uS:@A5 n;iG@ F6XD .nZ@A p'Ǘ?'D A! I?' D A~ !h@!7!@!2@ !%!$J +L>:uS:@A!\ :] ^ _ OAGIt appears that the WRD projection, while exponential in general shape,TBLmay have been made using lines of different slopes joined at 2015 (3 above).ZCRThe exponential that has y=4 as a horizontal asymptote appears to be closer to theeD]criteria used to make forecasts than does the exponential with y=0 as a horizontal asymptote..F& B. The logistic curve (variant).kHc One variant of the logistic curve, in which the S-shape appears flatter is the Gompertz curve;cI[it is used to model growth of various kinds, from financial to population. The reason the jJbcurve is flatter becomes evident when the logistic equation is written as a differential equation,WKOdP/dt = P(a-b*P), and the Gompertz is also written in an equivalent manner, as LdP/dt = P(a-b*ln P)nMf the logarithmic factor tends to flatten out the curve and make the S-shape less curved than wouldN a logistic fit.,P$General form for the Gompertz curve:Ry=q*e^((-ce)^(-bx))hT`where q is selected prior to making any analysis and is the value of the upper bound selected bykUcthe user on carrying capapcity or other bases, and b and c are constants to be determined dependinguVmon the values selected for q and the beginning and ending times chosen. There are numerous equivalent forms.]XUExample. Gompertz curve fit to WRD data from 1955-2025, Bangladesh total population. ZYearZ Pop. mil. 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