CAP 5625: Programming Assignment 31 CAP 5625: Programming Assignment 3 Due on Canvas by...

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CAP 5625: Programming Assignment 31    CAP 5625: Programming Assignment 3    Due on Canvas by Friday, December 2, 2022 at 11:59pm    Preliminary instructions  You may consult with other students currently taking CAP 5625 in your section (section 001 for  in-person, section 003 for fully online) at FAU on this programming assignment. If you do  consult with others, then you must indicate this by providing their names with your submitted  assignment. However, all analyses must be performed independently, all source code must be  written independently, and all students must turn in their own independent assignment. Note  that for this assignment, you may choose to pair up with one other student in your section of  CAP 5625 and submit a joint assignment. If you choose to do this, then both your names must  be associated with the assignment and you will each receive the same grade.    Though it should be unnecessary to state in a graduate class, I am reminding you  that you may not turn in code (partial or complete) that is written or inspired by  others, including code from other students, websites, past code that I release  from prior assignments in this class or from past semesters in other classes I  teach, or any other source that would constitute an academic integrity violation.  All instances of academic integrity violations will receive a zero on the  assignment and will be referred to the Department Chair and College Dean for  further administrative action.    You may choose to use whatever programming language you want. However, you must provide  clear instructions on how to compile and/or run your source code. I recommend using a  modern language, such as Python, R, or Matlab as learning these languages can help you if you  were to enter the machine learning or artificial intelligence field in the future.    All analyses performed and algorithms run must be written from scratch. That is, you may not  use a library that can perform gradient descent, cross validation, ridge regression, logistic  (multinomial) regression, optimization, etc. to successfully complete this assignment (though  you may reuse your relevant code from Programming Assignments 1 and 2). The goal of this  assignment is not to learn how to use particular libraries of a language, but it is to instead  understand how key methods in statistical machine learning are implemented. With that  stated, I will provide 10% extra credit if you additionally implement the assignment using built- in statistical or machine learning libraries (see Deliverable 7 at end of the document).    Note, credit for deliverables that request graphs, discussion of results, or specific values will not  be given if the instructor must run your code to obtain these graphs, results, or specific values.    Brief overview of assignment  In this assignment you will still be analyzing human genetic data from ? = 183 training  observations (individuals) sampled across the world. The goal is to fit a model that can predict  2    (classify) an individual’s ancestry from their genetic data that has been projected along ? = 10  top principal components (proportion of variance explained is 0.2416) that we use as features  rather than the raw genetic data, as the training would take too long to complete for this  assignment with the raw data. I recognize that we have not covered precisely what principal  components are, but we will get to this in Module 11, and for now you should just treat them as  highly informative features as input to your classifier. Specifically, you will perform a penalized  (regularized) logistic (multinomial) regression fit using ridge regression, with the model  parameters obtained by batch gradient descent. Your predictions will be based on ? = 5  continental ancestries (African, European, East Asian, Oceanian, or Native American). Ridge  regression will permit you to provide parameter shrinkage (tuning parameter ? ≥ 0) to mitigate  overfitting. The tuning parameter ? will be chosen using five-fold cross validation, and the best- fit model parameters will be inferred on the training dataset conditional on an optimal tuning  parameter. This trained model will be used to make predictions on new test data points.    Training data  Training data for these observations are given in the attached  TrainingData_N183_p10.csv comma-separated file, with individuals labeled on each  row (rows 2 through 184), and input features (PC1, PC2, …, PC10) and ancestry label given on  the columns (with the first row representing a header for each column).     Test data  Test data are given in the attached TestData_N111_p10.csv comma-separated file, with  individuals labeled on each row (rows 2 through 112), and input features (PC1, PC2, …, PC10),  and ancestry label given on the columns (with the first row representing a header for each  column). There are five individuals with Unknown ancestry, 54 individuals with Mexican  ancestry, and 52 individuals with African American ancestry. Each of the five Unknown  individuals belong to one of the five ancestries represented in the training set, and each  ancestry is represented once in the five Unknown individuals. The Mexican and African  American individuals have a range of ancestry proportions based on historical mixing of  ancestors of diverse ancestry. You will use the class probabilities from logistic (multinomial)  regression to predict the ancestry proportion of each of these putatively mixed samples.    Detailed description of the task  Recall that the task of performing a ridge-penalized logistic regression fit to training data  {(?1, ?1), (?2, ?2), … , (?? , ??)} is to minimize the cost function  ?(?, ?) =  − log ℒ(?) + ?∑∑??? 2 ? ?=1 ? ?=1   where   3    ? = [ ?01 ?02 ⋯ ?0? ?11 ?12 ⋯ ?1? ⋮ ⋮ ⋱ ⋮ ??1 ??2 ⋯ ??? ]  is the (? + 1) × ? matrix of model parameters, where  log ℒ(?) =  ∑[∑??? (?0? +∑?????? ? ?=1 ) ? ?=1 − log(∑exp(?0ℓ +∑?????ℓ ? ?=1 ) ? ℓ=1 )] ? ?=1   is the likelihood function, where ???, ? = 1,2, … ,? and ? = 1,2, … , ?, is an indicator variable  defined as  ??? = { 1 if ?? = ? 0 if ?? ≠ ?   and where the input ? features are standardized (i.e., centered and divided by their standard  deviation). Moreover, recall that batch gradient descent updates each parameter ???, ? = 0,1, … , ? and ? = 1,2, … , ?, as follows:  ??? ≔ ??? − ? ? ???? ?(?, ?)  where ? is the learning rate and where the partial derivative of the cost function with respect  to parameter ??? is  ? ???? ?(?, ?) = {        −∑[??? − ??(??; ?)] ? ?=1 if ? = 0 −∑???[??? − ??(??; ?)] ? ?=1 + 2???? if ? > 0   where  ??(??; ?) = exp(?0? + ∑ ?????? ? ?=1 ) ∑ exp (?0ℓ + ∑ ?????ℓ ? ?=1 ) ? ℓ=1   To implement this algorithm, depending on whether your chosen language can quickly compute  vectorized operations, you may implement batch gradient descent using either Algorithm 1 or  Algorithm 2 below (choose whichever you are more comfortable implementing). Note that in  languages like R, Python, or Matlab, Algorithm 2 (which would be implemented by several  nested loops) may be much slower than Algorithm 1. Also note that if you are implementing  Algorithm 1 using Python, use numpy arrays instead of Pandas data frames for computational  speed. For this assignment, assume that we will reach the minimum of the cost function within  a fixed number of steps, with the number of iterations being 10,000.    4    You may need to play with different learning rate values in order to identify a learning rate that  is not too large and not too small, such that it is likely for the algorithm to converge in a  reasonable period of time. I would consider values of the learning rate ? such that 2??

1 CAP 5625: Programming Assignment 3 Due on Canvas by Friday, December 2, 2022 at 11:59pm Preliminary instructions You may consult with other students currently taking CAP 5625 in...

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