Extended Probability & Statistics

Social Networks Modeling

The shifted Gompertz distribution is the distribution of the maximum of independent exponential and extreme-value distributed random variables. The distribution can be used to model the growth and decline of interest in social networks. The CDF of the shifted Gompertz distribution takes the following form.

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CDF[ShiftedGompertzDistribution[\[Lambda], \[Xi]], x]

Weekly counts for interest on Facebook from Google Trends.

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ts = TemporalData[TimeSeries, {CompressedData[" 1:eJyFz2tPwjAUBmDA+7wRNCpeQBTRiAGvXLoNYYytPW03QHSybiYav/tX/Ume GRO/mPg0p+l72pykpbeP8VsmlUqlsT5xS/8n82smM/Nt9sdcYv4vC2gxoWlL mqYto5WVVbS2tp7N5nK5jY3N7Z2tnXx+d3dvbx/XQbFwUCgUi4eHpdLRcbl8 UjmtnJ1XqxeoVqvV65dX9cvrZvP2pnHXaLUbjWaLIJMYuk4MwzTNTue+1+1a qO9a9qA/cPq2PRg4ILnLgHPGgLkuABfSdRyHUs5AAghvyD3peb4U+IhLJoCB NxLUk0JwgXcCRNIT2AYpPS4YByo5cGBsFL+oOFaRCqP4/VWpOFQqCKbhVEXh 9BmFKopjTFEQBs9PweRxNHkYj/yh7/ueEHI8xDGAg7AAqMsodRw3OTP2HWxK XfyFbVl2z+p2kGn1jATRSZsQXW+RNkpSmxjkC/7xXxc= "], { TemporalData`DateSpecification[{2006, 8, 26, 0, 0, 0.}, { 2015, 7, 11, 0, 0, 0.}, {1, "Week"}]}, 1, {"Continuous", 1}, { "Discrete", 1}, 1, { ValueDimensions -> 1, DateFunction -> Automatic, ResamplingMethod -> {"Interpolation", InterpolationOrder -> 1}}}, True, 314.1];
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DateListPlot[ts, ImageSize -> Medium, PlotTheme -> "Detailed", Filling -> Axis]

Fitting the data to a truncated shifted Gompertz distribution.

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rawcounts = ts["Values"]; length = Length[rawcounts]; x = Range[length] - 0.5; wdata = WeightedData[x, rawcounts];
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edist = EstimatedDistribution[wdata, TruncatedDistribution[{0, length}, ShiftedGompertzDistribution[\[Lambda], \[Xi]]], {{\[Lambda], 1}, {\[Xi], 6}}]

Compare the predictions from the model to the data.

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counts = Total[rawcounts] PDF[edist , x]; DateListPlot[{rawcounts, counts}, {ts["FirstDate"], Automatic, "Week"}, Filling -> Axis, PlotLegends -> {"data", "model"}, ImageSize -> Medium, PlotTheme -> "Detailed"]

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