- Working with Gradient Boosting Machine (GMB)
- Looking closer in the data set
- Cleaning up the data set
- Results after clean up
- Next steps
August 23, 2019
Index
Working with GMB
What does GBM do?
- Each tree is grown using information from previous one
- After evaluating the first tree
- increase weights of observations that are hard to classify
- and lower the weights for those that are easy to classify
- In other words, given the current model
- fit a tree using the current residuals
- rather than the outcome
- This method slowly converts weak learners into strong ones
Let's try to predict the 5th tracking hit
- Sample of 10K tracks
- Training phase
- Inputs for GBM:
- x_1, x_2, x_3, x_4, y_1, y_2, y_3, y_4, z_1, z_2, z_3, z_4
- 7K tracks
- Number of trees = 200
- Trees shrinkage = 0.1 (10%)
- Trees interaction depth = 10
- GBM distribution = gaussian
- Inputs for GBM:
- Testing phase
- 3K tracks
- 39 trees between 10 and 200 (in steps of 5)
Relative influence from each input coordinate
Best number of trees for prediction (I)
- Trying to predict 5th tracking hit
- Compute Minimum Mean Squared Error for each number of tree \(n_{tree}\)
- Considering
- 39 trees between 10 and 200 (in steps of 5)
- N = 3K tracks
- Considering
\[MSE_{n_{tree}} = \frac{\sum_{i=1}^{N} (x^{Test}_{i} - x^{Predicted}_{i})^2}{N}\]
Best number of trees for prediction (II)
- Minimum MSE = 3576 \(mm^2\) for 45 trees
Checking predicted results for 5th hit
- We observe a considerable fluctuation
- A dream case would be a straight diagonal line
Looking closer in the data
Plotting all hits in X-Y plane (I)
- Hits from 10K tracks with |z| < 300 mm
- We observe an overlay of the plates
Plotting all hits in X-Y plane (II)
- Adding a track
- Some layers have more than 1 hit
Plotting all hits in X-Y plane (III)
- If we try to predict the 5th hit using previous 4
- It could bring difficult to GBM
Plotting all hits in X-Y plane (IV)
- It seems we may have up to 4 hits in the same layer
Plotting all hits in Z-Y plane
- And the same happens in Z-Y (or Z-X) plane
- So let's try to remove duplicated events
Let's clean up the data set
Method to clean up the data set (I)
- Compute 3D distance between each 2 subsequent hits
- in the same track
Method to clean up the data set (II)
- Looking closer
- A pick for 3D distance < 20 mm seems to be due duplicated hits in the same layer
Method to clean up the data set (III)
- Looking even closer
- Shoud we consider a tighter cut?
A cut of 20 mm is reasonable?
- Looking the central region of the detector
- Distance between plates seems to be higher than 20 mm
- So a cut in 20 mm looks fine
Comparing tracks after clean up (X-Y)
Comparing tracks after clean up (Z-Y)
Predicted results after clean up (I)
- What about the horizontal points in a straight line?
- Does it seem reasonable? Actually it is pure Physics!!!
Predicted results after clean up (II)
- It seems GBM has learned Physics
- finding a 5th hit even when there is no hit
Predicted results after clean up (III)
- Now for Y and Z predictions
- Results are not satisfied for Z, needing more studies
Next Steps
To-Do-List
- Tune GBM parameters
- Try to predict all the other hits in the track
- Find nearest neighbor point of a predicted point
Moving ahead
Tune GBM parameters
- Training sample (7K tracks) to find 5th hit (from previous 4)
- Shrinkage = 9, Interaction deepth = 10
Clustering heat map cells
- Unfortunately, Minimum MSE = 4454 mm\(^2\)
- which corresponds to 6.6 cm (error too big!!!)
New idea: cutting out parameters (I)
- Let's try to predict a 4th point from previous 3 ones
- Minimum MSE = 2122 mm\(^2\) (4.6 cm)
New idea: cutting out parameters (II)
Next steps (finally)
- Counting out more parameters
- Try to predict all the other hits in the track
- Find nearest neighbor point of a predicted point