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I am working with regression tree. I have four predictors. There is a exponential relationship between predictor and dependent variable. But after building predictive model I cannot understand whether my model is a right one or not. Which tests can assure that my model is a good one?. For regression tree, is it necessary to test all the assumptions which is applied to linear regression.

I have checked and found that residuals are homoscedastic but in q-q plot residuals does not lies along the diagonal line. Here I attached the q-q plot please check it and suggests me whether residuals are normally distributed or not.enter image description here

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  • $\begingroup$ Welcome to the site! To see if it's a "good" model you would need at least one other model to compare it to. I have not so far seen QQ plots or heteroskedasticiy tests with tree-based methods. These models are fairly robust and do not per se require extra characteristics of the data such as homoskedasticity or no autocorrelation. Parameter tuning is an important aspect for trees (tree depth, number of leaves, etc.). The best way of comparing two models is via a test sample data (i.e. computing the test error). $\endgroup$ – PaulG Apr 4 at 16:05
  • $\begingroup$ If you use linear regression as a pure prediction machine as well, a correctly specified model equation (aka "linearity") is the only relevant assumption of linear regression. $\endgroup$ – Michael M Apr 4 at 19:17
  • $\begingroup$ Thank you for your answer. I prune the tree using cost-complexity approach with the help of cross-validation and best model is chosen based on MSE value and the model performs well on the test data. So for tree based method it is not necessary to test assumption like heteroskedasticity, autocorrelation. Am i right or not? $\endgroup$ – tarequzzaman shaheen Apr 4 at 21:13
  • $\begingroup$ There are better ways to estimate the tree than ordinary squared error loss under heteroscedasticity and non-normality. $\endgroup$ – BigBendRegion Apr 5 at 0:45
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With tree regression, you can be a little more relaxed about assumptions. In particular, you simply give up on the "linearity" (or more precisely, the correct functional specification) assumption, because the natural process obviously does not follow the piecewise flat segmentation that is assumed by the tree model. Instead, you simply acknowledge that the model is wrong with regard to the "linearity" assumption, but proceed anyway, hoping that the flexibility and ease of interpretation of the model overcome this glaringly incorrect assumption.

The normality and constant variance assumptions have at least two useful aspects as regards tree regression. (1) The within-node prediction bounds $\hat y \pm 2 rmse$ make sense under these conditions; (2) The least squares estimation criteria also makes sense under these conditions. Absent these conditions, alternate estimation procedures and prediction bounds are better.

In your case, it is clear from the outlier-prone character of the q-q plot that you could do better my using something other than least squares. Perhaps mean absolute deviations.

It is also clear that your model will miss observations occasionally by many more standard deviations (eg, 5) than you might expect had the distributions been normal. This is not necessarily a problem; it's just good to know how your model works.

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  • $\begingroup$ Thank you so much for your answer. Can you please explain how do you understand that my model miss observation occasionally ? Form this q-q plot can we say that it is a normal distribution ? $\endgroup$ – tarequzzaman shaheen Apr 4 at 22:34
  • $\begingroup$ No it is heavy-tailed. The large vertical slopes show that there are occasional observations that are much farther from the mean than you would expect, had the distributions been normal. $\endgroup$ – BigBendRegion Apr 5 at 0:47
  • $\begingroup$ Thank you so much for your answer and suggestion. $\endgroup$ – tarequzzaman shaheen Apr 5 at 11:12

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