Hello!

I don't like much black boxes too, but I think the origin of the problem lies in the way training and test sets were built. Defining

train = RandomSample[tuples, 7];
test = Complement[tuples, train];

and plotting the prevision

ListPlot[{Map[Flatten@Apply[List, #] &, train], 
  Map[Flatten@Apply[List, #] &, test], 
  Transpose[{{0.27, 0.9, 1, 1.1, 1.2, 1.3, 1.4, 1.5}, yPredictions}]},
  PlotStyle -> {Black, Green, Red}, 
 PlotLegends -> {"Training data", "Test data", "Predictions"}]

you should obtain something correct.

Hi Mohammad,

Using a simple neural network will generate a much better fit than Predict with Method -> "NeuralNetwork".

y = {0.44, 0.33, 0.3, 0.3, 0.32, 0.34, 0.4, 0.53, 0.68, 0.83};
x = {0.04, 0.08, 0.12, 0.16, 0.2, 0.24, 0.32, 0.48, 0.64, 0.8};

points = Transpose[{x, y}];
trainingData = Rule @@@ points;

net = NetChain[{32, Tanh, 1}]
trained = NetTrain[net, trainingData]

Show[
 ListPlot[points, PlotLegends -> LineLegend[{"Data"}]],
 Plot[trained[x], {x, 0, .8}, PlotLegends -> LineLegend[{"Trained"}]],
 Plot[trained[x], {x, 0.8, 1.2}, PlotStyle -> Red, PlotLegends -> LineLegend[{"Extrapolated"}]],
 PlotRange -> All]

But, as @Jim Baldwin said "Don't extrapolate with black box functions". If you have a model that you expect the data to follow, try fitting.

With such a small dataset, excluding 30% from training is likely to produce worse results. You can evaluate the deviations at each x value and then compute StandardDeviation or whatever metric you want.

trained = NetTrain[net, trainingData]
deviations = y - trained[x];
StandardDeviation@deviations

To train on a subset

{train, test} = ResourceFunction["TrainTestSplit"][trainingData, "TrainingSetSize" -> Scaled[.7]];
trained = NetTrain[net, train]

Use the test subset to evaluate the quality of the model

measurements = 
 NetMeasurements[trained, test, 
   {"StandardDeviation", "MeanDeviation", "RSquared", "MeanSquare"}]

Hi Alex,

Unless you have a much larger dataset I would not split it into train / test. The results are much worse compared to using all of the data, and will be very sensitive to the random selection. I did not call SeedRandom before the random split, so your results will probably be very different.

predictOnTrained = MapAt[trained[#] &, train, {All, 2}]
predictOnTest = MapAt[trained[#] &, test, {All, 2}]

ListPlot[{points, List @@@ predictOnTrained, List @@@ predictOnTest},
 PlotLegends -> {"Data", "Train Predictions", "Test Predictions"},
 PlotStyle -> ColorData[112]]

You might not like the values but to get values just do what you did for predicting the test and training values:

yPredictions = Map[cfunc, {{0.27}, {0.9}, {1}, {1.1}, {1.2}, {1.3}, {1.4}, {1.5}}]
ListPlot[{Transpose[{x[[1 ;; 7]], y[[1 ;; 7]]}], 
  Transpose[{x[[8 ;; 10]], y[[8 ;; 10]]}], 
  Transpose[{{0.27, 0.9, 1, 1.1, 1.2, 1.3, 1.4, 1.5}, yPredictions}]},
  PlotStyle -> {Black, Green, Red}, PlotLegends -> {"Training data", "Test data", "Predictions"}]

This is non-Mathematica advice: Don't extrapolate with black box functions.