'symlog'-like Plot with a mixed log-linear-log scale

Question

Python's matplotlib has the possibility to use a symmetric log (symlog) axis, as demonstrated in the documentation. You can set a negative threshold and positive threshold, between which the plot is linear (so it can go through 0), and then beyond those thresholds the scale is logarithmic (decades go in both the positive and negative directions).

Then, you can show sign-indefinite functions that grow large in absolute value, even if in some places they grow very negative.

Is there any easy way to achieve this effect or a utility to make such a plot in Mathematica?

Answer 1

You may use the ScalingFunctions option which is available for most plotting functions.

We need a logarithmic function that passes through zero that we can flip and mirror in the negative domain. With such a function we will not need to specify a linear portion and thus have a smooth function over the domain. A simple solution is Log[x + 1].

Plot[Log[x + 1], {x, 0, 10}]

Using Abs and Sign this equation and its inverse can be setup as scaling functions.

symlog =
  {
   Function[x, Sign[x] (Log[Abs[x] + 1])],
   Function[y, Sign[y] (Exp[Abs[y]] - 1)]
   };

Then with Plot

Plot[x, {x, -10, 10},
 ScalingFunctions -> symlog]

and ListPlot

ListPlot[
 Transpose@ConstantArray[Range[-10, 10, .25], 2],
 ScalingFunctions -> symlog]

Hope this helps.

Answer 2

Creating this plot we need the function that maps from and to these new axes. Just like we have with logarithmic axes we need the function Log and its opposite Exp in the ScalingFunction option.

So let's define these two functions:

ClearAll[ConvertPoint, UnConvertPoint]
ConvertPoint[n_?NumericQ, {down_, up_}] := Module[{},
  If[n < 0,
   -ConvertPoint[-n, {-up, -down}]
   ,
   If[n < up,
    n
    ,
    Log[n/up] + up
    ]
   ]
  ]
UnConvertPoint[n_?NumericQ, {down_, up_}] := Module[{},
  If[n < 0,
   -UnConvertPoint[-n, {-up, -down}]
   ,
   If[n < up,
    n
    ,
    Exp[n - up] up
    ]
   ]
  ]

Given an input these will convert these back and from these new coordinates.

Now we modify the built-in ListPlot function:

ClearAll[ListSymmetricLogPlot];
ListSymmetricLogPlot[data_List, threshold_?NumericQ, opts : OptionsPattern[]] := 
 ListSymmetricLogPlot[data, {-threshold, threshold}, opts]
ListSymmetricLogPlot[data_List, thresholds : {downthres_, upthres_}, opts : OptionsPattern[]] := 
 Module[{xmin, xmax, ymin, ymax, vticks1, vticks2, vticks3, vticks, vticksright, tmp},
  {{xmin, xmax}, {ymin, ymax}} = CoordinateBounds[data];
  vticks1 = If[ymin < downthres,
    tmp = Charting`ScaledTicks[{Log, Exp}][Log[-downthres], Log[-ymin]];
    tmp[[All, 1]] = Minus@*Exp /@ tmp[[All, 1]];
    tmp[[All, 2]] = Replace[tmp[[All, 2]], {x_?NumericQ :> -x, _Superscript[a_, b_] :> Superscript[-a, b]}, {1}];
    tmp
    ,
    {}
    ];
  vticks2 = Charting`ScaledTicks["Linear"][downthres, upthres, 4];
  vticks3 = If[ymax > upthres,
    tmp = Charting`ScaledTicks[{Log, Exp}][Log@upthres, Log@ymax];
    tmp[[All, 1]] = Exp /@ tmp[[All, 1]];
    tmp
    ,
    {}
    ];
  vticks = vticksright = DeleteDuplicatesBy[SortBy[Join[vticks1, vticks2, vticks3], First],
      First];
  vticksright[[All, 2]] = "";
  ListPlot[data, opts,
   ScalingFunctions -> {None, {ConvertPoint[#, thresholds] &, UnConvertPoint[#, thresholds] &}},
   PlotRange -> All,
   FrameTicks -> {{vticks, vticksright}, Automatic},
   Ticks -> {Automatic, vticks}
   ]
  ]

We can now test it out:

ListSymmetricLogPlot[{#,#}&/@Range[-10,10,0.2],0.5,ImageSize->600]
ListSymmetricLogPlot[{#,Tan[#]}&/@Range[-0.4995Pi,0.4995Pi,0.001Pi],{-1,1},Joined->True,Frame->True,ImageSize->600]

Giving:

Another test:

ListSymmetricLogPlot[Join[Table[{-x,-Exp[x-5]},{x,0,10,0.01}],Table[{x,Exp[x-5]},{x,0,10,0.01}]],{-3,3}]

Giving:

One limitation is now that you need to input {x,y} pairs as data just the y values {y1,y2,y3,…} does not work.

Answer 3

Edmund's Answer works, but there is a small error in it: it is missing parentheses to correctly define the inverse function. So the correct symlog function definition should be:

symlog = {
  Function[x, Sign[x] * Log[Abs[x] + 1]], 
  Function[y, Sign[y] * (Exp[Abs[y]] - 1)]};
}

The Error only becomes visible when you try to plot values in the range [-1,1]. Without this fix, mathematica shows a vertical axis without ticks. The fix resolves this.

Moreover, there has been made the suggestion to apply a scale to the functions, as shown here. This allows to observe a proper log-scale even if the function scale is at very small orders of magnitude. In Mathematica, I use this scaled version of symlog:

ssymlog[mag_] := {
  Function[x, Sign[x]*Log[1 + Abs[x]*10^mag]], 
  Function[y, Sign[y]*10^-mag*(Exp[Abs[y]] - 1)]
}

Here is an image that shows the difference on a function of mine, where the ordinary symlog scale does not really add much of a stretch to the function: