Week 3: Conditional Probability Implementation

-

     Implemented Conditional Probability calculations using RKHS and compared it to the existing ProbSpace method. On the bright side, the RKHS method is more accurate as we go farther from the mean, but it is also very slow in comparision. Here is a day-wise summary of my progress throughout the week:

Monday:

Studied paper on Nonparametric Conditional Density Estimation. This paper is much easier to digest compared to the previous one we were working on. Clarified some doubts about the terminology and had a discussion about the formulae with Roger sir during the weekly meeting. 

f (y | x) = f(y, x)/ f(x) =  Σ Kh2(x − Xi) * Kh1(y − Yi) /  Σ Kh2 (x − Xi)

This is the main formula we will be working with to calculate E(y|x) and later extend it to calculate E(Y|X)

Tuesday:

Implemented the equation and scaling issues aside, the results show promise to be better than the existing method in the Prob.py script. Generated dataset with Y = X2.

Shown here is the results for P(y=1|x=1) for the RKHS method vs ProbSpace method.


Wednesday:

Implemented E(Y|X) and it was clear that the RKHS method was taking far too long to produce results. For example just calculating for a range of X(-2,2) took 10 minutes. While the ProbSpace method took under 10 seconds. The reason being that, for every value of x∈(-2,2) xi , we are calculating the weighted average of E(y = yi  |  x = xi ) where y ranges from (-30,30) with a step size of 0.1. The results show that even though the RKHS method is accurate, it is still slightly worse than the ProbSpace method.



Thursday:

Fine-tuned the algorithm to run faster. This mainly involved the optimization of Y range and step size. It is a waste of time to iterate Y over negative values because for example,     P(y= -5| x = 3) is negligibly small and just impractical given that we know Y = X2 . And the real problem with ProbSpace method can only be seen when we deviate farther from the mean of the distribution, so I rescaled X to (-10,10) and Y to (0,100) with step size of 0.5 and 1 , as the error values between step sizes of 0.1 and 0.5 are not different enough justify the slow calculation. The Results are very good, as we move away from the mean ProbSpace curve just falls apart while the RKHS curve is still comparable to the ideal one.

Friday:

Tested accuracy vs different values of sigma. Here are some of the results:


Prob.py             Average Error: 15.786311330384414      Max error: 46.721149191232456

Sigma-0.1          Average Error: 1.7544741739568277     Max error: 6.819402462723623

Sigma-0.2          Average Error: 1.457341151925867       Max error: 5.0603293918499475

Sigma-0.5          Average Error: 3.6074782584714513     Max error: 6.64324125835352


Measured time taken(in seconds) in comparison to Prob.py w.r.t different step sizes


Prob.py Time: 8.134860754013062

Average Error: 15.786311330384414 Max error: 46.721149191232456


RKHS Step Size =  0.5 , Time: 719.6718242168427

Average Error: 1.3966743172847627 Max error: 5.431449933619589 


RKHS Step Size =  1 , Time: 731.5105772018433

Average Error: 1.457341151925867 Max error: 5.0603293918499475 


RKHS Step Size  =  2 , Time: 365.7068731784820

Average Error: 1.4374872842382438 Max error: 5.000025328500016 


RKHS Step Size =  5 , Time: 122.54302167892456

Average Error: 2.8136193250540837 Max error: 11.76926483287589 


RKHS Step Size =  10 , Time: 61.639594078063965

Average Error: 8.114348240870235 Max error: 30.99998027979695 


Upcoming week plans:

Explore the Dual-Tree approximation from the Fast Nonparametric Conditional Density Estimation paper. The aim is to cut the time taken for calculation while maintaining accuracy.



Comments

Post a Comment

Popular posts from this blog

Week 1: RKHS Kernel Implementation

-
Image
      My first long term goal is to improve the existing conditional probability calculation module by making use of RKHS mapping to get rid of the curse of dimensionality problem that has been haunting the traditional method. Here is a day-wise summary of my progress throughout the week: Monday: My Internship officially started on this day, I had a 1 on 1 meeting with Roger Sir regarding this week’s assignment which would be to implement an RKHS kernel for the purpose of predicting a probability distribution function, given just a set of data points ( 1000 in my case) belonging to the said distribution. This would bolster my understanding of RKHS working and will come in use when implementing more complex kernels in the future. Tuesday: Implemented the RKHS kernel and the evaluation function. Dataset for the arbitrary function X = logistic(-2,1) if choice([0,1]) else logistic(2,1) was created using the synthDataGen.py script and used as the basis for the evaluation function. Made
Read more

Week 10: Implementing UPROB

-
Image
Week 10 Report This week, I developed the UPROB module (you can check out the code here ). Since this was the main focus throughout the week and I spent all my time developing, bug-fixing , testing and documenting results, I feel that a day-wise summary would not be the most effective method to show results. Instead I have opted to explain the functionalities, working and usage of this module. This will also serve as documentation for future reference. I have listed the results and conclusion towards the end of this report. UPROB Init: Initializing a UPROB Object is very similar to that of rkhsMV except for the final argument k , which denotes the % of variables to be filtered before applying JPROB methods on the remaining ones.  R1 and R2 are two separate rkhsMV instances, one for each of the UPROB methods, condE() and condP(). The idea is to store R1 and R2 such that if we are repeatedly performing operations on the same dataset, we need not initialize a new kernel each time thus sa
Read more

Week 12: Final Week

-
  These past 12 weeks have flown by so fast! It still feels like yesterday I started working on this project. There were highs and lows but at the end of the day we overcame our obstacles to create something to be proud of. It was a great learning experience and I had fun working with the team.  Monday: Implemented the prototype of the K decrement algorithm to reduce the number of filter variables in case of shortage of data points upon filtration. Here are some results: K decrement- condP() K = 100 rf = 0.8, minmin = 20 dims, datSize, tries, K =  3 1000 1 100 Average R2: JP, UP, PS =  0.67025996194      0.726423475099       0.7434199290171855 K=100 , rf = 0.8, minmin = 5 dims, datSize, tries, K =  3 1000 5 100 Average R2: JP, UP, PS =  0.70784384556      0.87296540935         0.764876980843549 K=100 , rf = 0.8, minmin = 4 dims, datSize, tries, K =  4 1000 1 100 Average R2: JP, UP, PS =  0.70162883565      0.80123074862         0.6089856810665888  K=100 in condP() function means that w
Read more