Apply a Box–Cox Transformation to a Time Series 

TimeSeriesMap can be used to transform values of time series.

The data contains the civilian unemployment rate in the United States from 1948 through 1993.

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X timeFrame = Interval[{DateObject[{1948, 1, 1}], DateObject[{1993, 12, 1}]}];

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X data = EntityValue[Entity["Country", "UnitedStates"], EntityProperty["Country", "CivilianUnemploymentRate", {"Date" -> timeFrame, "Frequency" -> "Monthly"}]]

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X DateListPlot[data, ImageSize -> Medium]

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Define a Box–Cox transformation.

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X BoxCox[x_, \[Lambda]_, \[Alpha]_, \[Sigma]_] := With[{y = QuantityMagnitude[x]}, Piecewise[{{(((y - \[Alpha])/\[Sigma])^\[Lambda] - 1)/\[Lambda], \[Lambda] > 0 || \[Lambda] < 0}}, Log[(y - \[Alpha])/\[Sigma]]]]

Use TimeSeriesMap to apply a Box–Cox transformation to the data with few choices of λ.

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X lambdas = {1/2, 0, -1/2}; newdata = Table[TimeSeriesMap[BoxCox[#, \[Lambda], .02, .016] &, data], {\[Lambda], lambdas}];

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X Grid[{MapThread[ DateListPlot[#1, ImageSize -> 180, PlotRange -> All, FrameTicks -> {{Table[{x, NumberForm[N[x], {5, 2}]}, {x, Min[#1], Max[#1], (Max[#1] - Min[#1])/5}], Automatic}, {Automatic, Automatic}}, PlotLabel -> Row[{"\[Lambda] = ", #2}]] &, {newdata, lambdas}]}]

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