From ebeb45b976acea4edfa20c60c97e0e3f966aaf87 Mon Sep 17 00:00:00 2001 From: mdporter Date: Mon, 6 Nov 2023 12:51:23 -0500 Subject: [PATCH] 2024 spring semester --- homework.html | 629 ------- homework/hw0.Rmd | 255 --- homework/hw0.html | 3248 --------------------------------- homework/hw1.Rmd | 132 -- homework/hw1.html | 3052 ------------------------------- homework/hw10.Rmd | 167 -- homework/hw10.html | 1800 ------------------ homework/hw2.Rmd | 171 -- homework/hw2.html | 1810 ------------------ homework/hw3.Rmd | 144 -- homework/hw3.html | 1781 ------------------ homework/hw4.Rmd | 150 -- homework/hw4.html | 3072 ------------------------------- homework/hw5.Rmd | 132 -- homework/hw5.html | 3034 ------------------------------ homework/hw6.Rmd | 116 -- homework/hw6.html | 1756 ------------------ homework/hw7.Rmd | 138 -- homework/hw7.html | 3031 ------------------------------ homework/hw8.Rmd | 31 - homework/hw8.html | 2939 ----------------------------- homework/hw9.Rmd | 137 -- homework/hw9.html | 3040 ------------------------------ index.html | 880 --------- lectures/Rfmla.pdf | Bin 72640 -> 0 bytes lectures/Rintro.pdf | Bin 187209 -> 0 bytes lectures/boosting.pdf | Bin 961103 -> 0 bytes lectures/bootstrap.pdf | Bin 1297467 -> 0 bytes lectures/classification.pdf | Bin 3946333 -> 0 bytes lectures/clustering.pdf | Bin 3909895 -> 0 bytes lectures/crossval.pdf | Bin 1950193 -> 0 bytes lectures/density.pdf | Bin 1025898 -> 0 bytes lectures/ensembles.pdf | Bin 928037 -> 0 bytes lectures/feature-engr.pdf | Bin 468028 -> 0 bytes lectures/feature-imp.pdf | Bin 837948 -> 0 bytes lectures/forecasting.pdf | Bin 842027 -> 0 bytes lectures/forecasting2.pdf | Bin 654792 -> 0 bytes lectures/gen-classifiers.pdf | Bin 673161 -> 0 bytes lectures/intro.pdf | Bin 805450 -> 0 bytes lectures/penalized.pdf | Bin 1061406 -> 0 bytes lectures/regression.pdf | Bin 501367 -> 0 bytes lectures/review-1.pdf | Bin 471688 -> 0 bytes lectures/supervised_1.pdf | Bin 530625 -> 0 bytes lectures/supervised_2.pdf | Bin 1004004 -> 0 bytes lectures/svm.pdf | Bin 712056 -> 0 bytes lectures/taylor-expansion.pdf | Bin 221841 -> 0 bytes lectures/trees.pdf | Bin 4503864 -> 0 bytes syllabus.html | 2263 ----------------------- 48 files changed, 33908 deletions(-) delete mode 100644 homework.html delete mode 100644 homework/hw0.Rmd delete mode 100644 homework/hw0.html delete mode 100644 homework/hw1.Rmd delete mode 100644 homework/hw1.html delete mode 100644 homework/hw10.Rmd delete mode 100644 homework/hw10.html delete mode 100644 homework/hw2.Rmd delete mode 100644 homework/hw2.html delete mode 100644 homework/hw3.Rmd delete mode 100644 homework/hw3.html delete mode 100644 homework/hw4.Rmd delete mode 100644 homework/hw4.html delete mode 100644 homework/hw5.Rmd delete mode 100644 homework/hw5.html delete mode 100644 homework/hw6.Rmd delete mode 100644 homework/hw6.html delete mode 100644 homework/hw7.Rmd delete mode 100644 homework/hw7.html delete mode 100644 homework/hw8.Rmd delete mode 100644 homework/hw8.html delete mode 100644 homework/hw9.Rmd delete mode 100644 homework/hw9.html delete mode 100644 index.html delete mode 100644 lectures/Rfmla.pdf delete mode 100644 lectures/Rintro.pdf delete mode 100644 lectures/boosting.pdf delete mode 100644 lectures/bootstrap.pdf delete mode 100644 lectures/classification.pdf delete mode 100644 lectures/clustering.pdf delete mode 100644 lectures/crossval.pdf delete mode 100644 lectures/density.pdf delete mode 100644 lectures/ensembles.pdf delete mode 100644 lectures/feature-engr.pdf delete mode 100644 lectures/feature-imp.pdf delete mode 100644 lectures/forecasting.pdf delete mode 100644 lectures/forecasting2.pdf delete mode 100644 lectures/gen-classifiers.pdf delete mode 100644 lectures/intro.pdf delete mode 100644 lectures/penalized.pdf delete mode 100644 lectures/regression.pdf delete mode 100644 lectures/review-1.pdf delete mode 100644 lectures/supervised_1.pdf delete mode 100644 lectures/supervised_2.pdf delete mode 100644 lectures/svm.pdf delete mode 100644 lectures/taylor-expansion.pdf delete mode 100644 lectures/trees.pdf delete mode 100644 syllabus.html diff --git a/homework.html b/homework.html deleted file mode 100644 index 39399aa..0000000 --- a/homework.html +++ /dev/null @@ -1,629 +0,0 @@ - - - - - - - - - - - - - -Homework - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
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Homework is submitted on the course Collab -site

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Homework #0 (not graded) (Rmd)

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Due: Tue Jan 31 | 11:45am

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Homework #1 (Rmd)

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Due: Wed Feb 08 | 11:45am

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Homework #2 (Rmd)

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Due: Wed Feb 15 | 11:45am

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Homework #3 (Rmd)

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Due: Wed Feb 22 | 11:45am

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Homework #4 (Rmd)

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Due: Wed Mar 01 | 11:45am

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Homework #5 (Rmd)

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Due: Wed Mar 22 | 11:45am

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Homework #6 (Rmd)

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Due: Wed Mar 29 | 11:45am

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Homework #7 (Rmd)

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Due: Wed Apr 05 | 11:45am

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Homework #8 (Rmd)

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Due: Wed Apr 12 | 11:45am

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Homework #9 (Rmd)

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Due: Wed Apr 19 | 11:45am Fri Apr 21 | -11:45pm

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Homework #10 (Rmd)

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Due: Fri Apr 28 | 11:45am

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Project Presentations

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Due: Tue May 09 | 9am - noon

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Project Reports (Grad Students only)

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Due: Wed May 10 | 10:00pm

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- - - - - - - - - - - - - - - diff --git a/homework/hw0.Rmd b/homework/hw0.Rmd deleted file mode 100644 index ecfe7a8..0000000 --- a/homework/hw0.Rmd +++ /dev/null @@ -1,255 +0,0 @@ ---- -title: "Homework #0: Hello SYS-6018" -author: "**Your Name Here**" -date: "Due: Tue Jan 31 | 11:45am (ungraded)" -output: R6018::homework ---- - -**SYS 4582/6018 | Spring 2023 | University of Virginia ** - -******************************************* -```{r config, echo=FALSE} -source(system.file("config/hw_config.R", package="R6018")) # knitr settings -# options(dplyr.summarise.inform = FALSE) # ignore dplyr message about grouping -``` - - -# Required R packages and Directories - -::: {.solution} - -```{r packages, message=FALSE, warning=FALSE} -data.dir = 'https://mdporter.github.io/SYS6018/data/' # data directory -library(R6018) # functions for SYS-6018 -library(tidyverse) # functions for data manipulation -``` - -::: - -# Getting Help - -The purpose of this (ungraded) homework is to help get you prepared for the semester. Don't panic if you don't immediately know the answers to some of these. I expect everyone will need to look things up. Take note of the areas that are rusty and plan to spend a bit of extra time to get up to speed. If some of these items are not even remotely familiar to you, then you probably have not satisfied the pre-requisite material; review the course syllabus and speak to me about any questions. - -The teaching staff (TA and myself) are here to help! Don't wait too long before asking for help and do let us know right away if you are starting to fall behind. I will also publish the solutions right after due date. It is **highly encouraged** that you study the posted solutions. - -Use the class Teams channel to ask questions. - -# R Resources - -## a. Update R and RStudio - -- Use the latest version (R > `r with(R.version, str_c(major, ".", minor))`, RStudio > `r rstudioapi::versionInfo()$version`) - -- Help with installation and and - - -## b. Install/Update the following packages we will meet during this course: - -- Working with Data: `tidyverse` -- Utility: `remotes` -- Data: `ISLR`, `moderndive`, `MASS` -- Resampling: `boot`, `rsample` -- Modeling: `tidymodels`, `modelr`, `broom` -- Regression: `glmnet`, `FNN`, -- Classification: `e1071`, -- Trees: `rpart`, `rpart.plot`, `randomForest`, `ranger` -- Ensembles: `gbm`, `xgboost` -- Density Estimation: `fitdistrplus`, `ks` -- Clustering: `mclust`, `mixtools` -- Association Analysis: `arules` -- Networks: `igraph` - -Note: Do not call `install.packages()` in this RMarkdown document; it only needs to be done once from the console. However you will need to use `library()` in RMarkdown since it needs to be called every time the Rmarkdown is compiled. - - -## c. Install the course R package `R6018` - -The `R6018` package contains formatting for the homework and some helpful functions. This R package is not hosted on CRAN, but on github. You can install it with this command `remotes::install_github("mdporter/R6018")` on your console (make sure you have installed the `remotes` package first!). This RMarkdown file won't compile until you do this. - - -# Tidyverse Resources - -Read the following sections in [R for Data Science](https://r4ds.had.co.nz): - -## Explore -- [3 Data visualisation](https://r4ds.had.co.nz/data-visualisation.html) -- [4 Workflow: basics](https://r4ds.had.co.nz/workflow-basics.html) -- [5 Data transformation](https://r4ds.had.co.nz/transform.html) -- [6 Workflow: scripts](https://r4ds.had.co.nz/workflow-scripts.html) -- [7 Exploratory Data Analysis](https://r4ds.had.co.nz/exploratory-data-analysis.html) -- [8 Workflow: projects](https://r4ds.had.co.nz/workflow-projects.html) - -## Wrangle -- [10 Tibbles](https://r4ds.had.co.nz/tibbles.html) -- 11 Import - - [11.2](https://r4ds.had.co.nz/data-import.html#getting-started) -- 12 Tidy data - - [12.3 Pivoting](https://r4ds.had.co.nz/tidy-data.html#pivoting) - - [12.5 Missing Values](https://r4ds.had.co.nz/tidy-data.html#missing-values-3) -- [15 Factors](https://r4ds.had.co.nz/factors.html) - -## Program -- [17 Introduction](https://r4ds.had.co.nz/program-intro.html) -- [18 Pipes](https://r4ds.had.co.nz/pipes.html) -- [19 Functions](https://r4ds.had.co.nz/functions.html) -- [20 Vectors](https://r4ds.had.co.nz/vectors.html) - - 20.1-20.4 -- [21 Iteration](https://r4ds.had.co.nz/iteration.html) - - 21.1 - 21.5 - -## Model -- 23 Model basics - - [23.4 Formulas](https://r4ds.had.co.nz/model-basics.html#formulas-and-model-families) - - -## Communicate -- [27 R Markdown]((https://r4ds.had.co.nz/r-markdown.html)) -- [28 Graphics for communication](https://r4ds.had.co.nz/graphics-for-communication.html) - - *optional* -- [29 R Markdown formats](https://r4ds.had.co.nz/r-markdown-formats.html) - - *optional* - -# Save/Bookmark the following cheatsheets and resources - -## **RStudio and RMarkdown** - -- [RStudio IDE Cheatsheet](https://github.com/rstudio/cheatsheets/raw/main/rstudio-ide.pdf) - -- [R Markdown Cheatsheet](https://github.com/rstudio/cheatsheets/raw/main/rmarkdown.pdf) - -- [R Markdown Reference Guide](https://github.com/rstudio/cheatsheets/raw/main/rmarkdown-2.0.pdf) - -- [Latex Cheatsheet](https://wch.github.io/latexsheet/latexsheet-0.png) - -## **R** - -- [Base R](https://github.com/rstudio/cheatsheets/blob/main/base-r.pdf) - -- [Data Visualization Cheatsheet](https://github.com/rstudio/cheatsheets/raw/main/data-visualization.pdf) - - [`ggplot2` website](https://ggplot2.tidyverse.org/) - -- [Data Transformation Cheatsheet](https://github.com/rstudio/cheatsheets/raw/main/data-transformation.pdf) - - [`dplyr` website](https://dplyr.tidyverse.org/) - - [`tidyr` website](https://tidyr.tidyverse.org/) - -- [Factors with forcats Cheatsheet](https://github.com/rstudio/cheatsheets/raw/main/factors.pdf) - - [`forcats` website](https://forcats.tidyverse.org/) - -- [Data Import Cheatsheet](https://github.com/rstudio/cheatsheets/raw/main/data-import.pdf) - - [`readr` website](https://readr.tidyverse.org/) - - [`tidyr` website](https://tidyr.tidyverse.org/) - -- [Apply Functions Cheatsheet](https://github.com/rstudio/cheatsheets/raw/main/purrr.pdf) - - - [`purrr` website](https://purrr.tidyverse.org/) - -## **Python with RStudio/RMarkdown** (Optional) - -- [Python with R and Reticulate Cheatsheet](https://github.com/rstudio/cheatsheets/raw/main/reticulate.pdf) - - - -# Problem 1: Math Notation - -## a. What are the equations for the least squares coefficients in linear regression (in matrix notation)? Use $X$ for the design/predictor matrix, and $Y$ the vector of responses. - -::: {.solution} -Solution here -::: - -## b. Let $x_1, x_2, \ldots, x_n$ be a sample of length of time that a customer is on the phone with a call center help line. We feel comfortable modeling the data as coming from an *exponential distribution*. What is the MLE (Maximum Likelihood Estimate) of the parameter? Show your steps. - -::: {.solution} -Solution here -::: - - -# Problem 2: Coding Practice - -## a. Simulate 100 observations from the following model: - -- $X \sim N(1, 1)$ -- $Y \sim N(1 + 2X, 2)$ -- $Z = \begin{cases} 1 &\quad Y<0 \\ 2 &\quad Y \ge 0 \end{cases}$ - -::: {.solution} -Solution here -::: - -## b. Make a scatter plot of the data. Put $X$ on the x-axis and $Y$ on the y-axis. Color the points according to $Z$. - -::: {.solution} -Solution here -::: - -## c. Write a function that adds two numbers together and squares the result. - -::: {.solution} -Solution here -::: - -## d. Write a loop to calculate the mean of every column in `mtcars`. - -::: {.solution} -Solution here -::: - -# Problem 3: Statistics - -## a. Find the two [quantiles](https://en.wikipedia.org/wiki/Quantile) that capture 95% of the following data: - -```{r} -set.seed(2023) -x = runif(n=100, min=2, max=22) -``` - -::: {.solution} -Solution here -::: - -## b. Confidence Interval - -A new machine learning model, developed by UVA researchers, uses biopsy images to predict if a child has enteropathy or celiac disease. [In a study of 102 patients](https://jamanetwork.com/journals/jamanetworkopen/fullarticle/2735765), the model was able to correctly classify 95 of the images. - -Find the 90% [confidence interval](https://en.wikipedia.org/wiki/Confidence_interval) for the probability a patient is correctly classified? - -::: {.solution} -Solution here -::: - -## c. Linear Models - -- Albemarle County real estate assessment data can be found here . - -- Fit a linear regression model that predicts the `TotalValue` using the predictors: `Condition`, size (`FinSqFt`), and location (`City`). - -- What are the estimated coefficients? - -- What is the estimated `TotalValue` for home with the following characteristics? - -```{r, echo=FALSE} -tibble(FinSqFt = c(2500, 1850), - City = c('EARLYSVILLE', 'CROZET'), - Condition = c('Good', 'Fair')) -``` - -::: {.solution} -Solution here -::: - -## d. Hypothesis Testing - -Use the `movies_sample` (IMDb) data from the `moderndive` R package to performance a hypothesis test that *Action* movies are ranked lower (on average) than *Romance* movies. - -::: {.solution} -Solution here -::: - - - - - - - - - diff --git a/homework/hw0.html b/homework/hw0.html deleted file mode 100644 index 8400f1e..0000000 --- a/homework/hw0.html +++ /dev/null @@ -1,3248 +0,0 @@ - - - - - - - - - - - - - - -Homework #0: Hello SYS-6018 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
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SYS 4582/6018 | Spring 2023 | University of Virginia -

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Required R packages and Directories

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data.dir = 'https://mdporter.github.io/SYS6018/data/' # data directory
-library(R6018)     # functions for SYS-6018
-library(tidyverse) # functions for data manipulation    
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-
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Getting Help

-

The purpose of this (ungraded) homework is to help get you prepared -for the semester. Don’t panic if you don’t immediately know the answers -to some of these. I expect everyone will need to look things up. Take -note of the areas that are rusty and plan to spend a bit of extra time -to get up to speed. If some of these items are not even remotely -familiar to you, then you probably have not satisfied the pre-requisite -material; review the course syllabus and speak to me about any -questions.

-

The teaching staff (TA and myself) are here to help! Don’t wait too -long before asking for help and do let us know right away if you are -starting to fall behind. I will also publish the solutions right after -due date. It is highly encouraged that you study the -posted solutions.

-

Use the class Teams channel to ask questions.

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-
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R Resources

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a. Update R and RStudio

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b. Install/Update the following packages we will meet during this -course:

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    -
  • Working with Data: tidyverse
  • -
  • Utility: remotes
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  • Data: ISLR, moderndive, -MASS
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  • Resampling: boot, rsample
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  • Modeling: tidymodels, modelr, -broom
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  • Regression: glmnet, FNN,
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  • Classification: e1071,
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  • Trees: rpart, rpart.plot, -randomForest, ranger
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  • Ensembles: gbm, xgboost
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  • Density Estimation: fitdistrplus, ks
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  • Clustering: mclust, mixtools
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  • Association Analysis: arules
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  • Networks: igraph
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Note: Do not call install.packages() in this RMarkdown -document; it only needs to be done once from the console. However you -will need to use library() in RMarkdown since it needs to -be called every time the Rmarkdown is compiled.

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c. Install the course R package R6018

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The R6018 package contains formatting for the homework -and some helpful functions. This R package is not hosted on CRAN, but on -github. You can install it with this command -remotes::install_github("mdporter/R6018") on your console -(make sure you have installed the remotes package first!). -This RMarkdown file won’t compile until you do this.

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Tidyverse Resources

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Read the following sections in R for -Data Science:

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Wrangle

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Program

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Model

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Communicate

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Problem 1: Math Notation

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a. What are the equations for the least squares coefficients in -linear regression (in matrix notation)? Use \(X\) for the design/predictor matrix, and -\(Y\) the vector of responses.

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Solution here

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b. Let \(x_1, x_2, \ldots, x_n\) be -a sample of length of time that a customer is on the phone with a call -center help line. We feel comfortable modeling the data as coming from -an exponential distribution. What is the MLE (Maximum -Likelihood Estimate) of the parameter? Show your steps.

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Solution here

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Problem 2: Coding Practice

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a. Simulate 100 observations from the following model:

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  • \(X \sim N(1, 1)\)
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  • \(Y \sim N(1 + 2X, 2)\)
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  • \(Z = \begin{cases} 1 &\quad Y<0 \\ -2 &\quad Y \ge 0 \end{cases}\)
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Solution here

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b. Make a scatter plot of the data. Put \(X\) on the x-axis and \(Y\) on the y-axis. Color the points -according to \(Z\).

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Solution here

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c. Write a function that adds two numbers together and squares the -result.

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Solution here

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d. Write a loop to calculate the mean of every column in -mtcars.

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Solution here

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Problem 3: Statistics

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a. Find the two quantiles that capture -95% of the following data:

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set.seed(2023)
-x = runif(n=100, min=2, max=22)
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Solution here

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b. Confidence Interval

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A new machine learning model, developed by UVA researchers, uses -biopsy images to predict if a child has enteropathy or celiac disease. -In -a study of 102 patients, the model was able to correctly classify 95 -of the images.

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Find the 90% confidence -interval for the probability a patient is correctly classified?

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Solution here

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c. Linear Models

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Solution here

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d. Hypothesis Testing

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Use the movies_sample (IMDb) data from the -moderndive R package to performance a hypothesis test that -Action movies are ranked lower (on average) than -Romance movies.

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Solution here

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- - - - - - - - - - - - - - - - diff --git a/homework/hw1.Rmd b/homework/hw1.Rmd deleted file mode 100644 index 65e4c2c..0000000 --- a/homework/hw1.Rmd +++ /dev/null @@ -1,132 +0,0 @@ ---- -title: "Homework #1: Supervised Learning" -author: "**Your Name Here**" -date: "Due: Wed Feb 08 | 11:45am" -output: R6018::homework ---- - -**SYS 6018 | Sp 2023 | University of Virginia** - ------------------------------------------------------------------------- - -```{r config, echo=FALSE} -source(system.file("config/hw_config.R", package="R6018")) # knitr settings -``` - - -# Required R packages and Directories - -::: {.solution} -```{r packages, message=FALSE, warning=FALSE} -data.dir = 'https://mdporter.github.io/SYS6018/data/' # data directory -library(R6018) # functions for SYS-6018 -library(tidyverse) # functions for data manipulation -``` -::: - - -# Problem 1: Evaluating a Regression Model - -## a. Create a set of functions to generate data from the following distributions: -\begin{align*} -X &\sim \mathcal{N}(0, 1) \\ -Y &= -1 + .5X + .2X^2 + \epsilon \\ -\epsilon &\sim \mathcal{N}(0,\, \sigma) -\end{align*} - - -::: {.solution} -Add solution here -::: - - -## b. Simulate $n=100$ realizations from these distributions using $\sigma=3$. Produce a scatterplot and draw the true regression line $f(x) = E[Y \mid X=x]$. -- Use `set.seed(611)` prior to generating the data. - - -::: {.solution} -Add solution here -::: - -## c. Fit three polynomial regression models using least squares: linear, quadratic, and cubic. Produce another scatterplot, add the fitted lines and true population line $f(x)$ using different colors, and add a legend that maps the line color to a model. -- Note: The true model is quadratic, but we are also fitting linear (less complex) and cubic (more complex) models. - - -::: {.solution} -Add solution here -::: - -## d. Simulate a *test data* set of 10,000 observations from the same distributions. Use `set.seed(612)` prior to generating the test data. -- Calculate the estimated mean squared error (MSE) for each model. -- Are the results as expected? - -::: {.solution} -Add solution here -::: - -## e. What is the best achievable MSE? That is, what is the MSE if the true $f(x)$ was used to evaluate the test set? How close does the best method come to achieving the optimum? - - -::: {.solution} -Add solution here -::: - - -## f. The MSE scores obtained in part *d* came from one realization of training data. Here will we explore how much variation there is in the MSE scores by replicating the simulation many times. - -- Re-run parts b. and c. (i.e., generate training data and fit models) 100 times. - - Do not generate new testing data -- Calculate the MSE for all simulations. -- Create kernel density or histogram plots of the resulting MSE values for each model. -- Use `set.seed(613)` prior to running the simulation and do not set the seed in any other places. -- Use the same test data from part d. (This question is only about the variability that comes from the training data). - - -::: {.solution} -Add solution here -::: - -## g. Show a count of how many times each model was the best. That is, out of the 100 simulations, count how many times each model had the lowest MSE. - -::: {.solution} -Add solution here -::: - -## h. Write a function that implements the simulation in part *f*. The function should have arguments for i) the size of the training data $n$, ii) the standard deviation of the random error $\sigma$, and iii) the test data. Use the same `set.seed(613)`. - -::: {.solution} -Add solution here -::: - - -## i. Use your function to repeat the simulation in part *f*, but use $\sigma=2$. Report the number of times each model was best (you do not need to produce any plots). - -- First generate new test data with ($n = 10000$, $\sigma = 2$, using `seed = 612`). - - -::: {.solution} -Add solution here -::: - -## j. Repeat *i*, but now use $\sigma=4$ and $n=300$. - - -- First generate new test data with ($n = 10000$, $\sigma = 4$, using `seed = 612`). - -::: {.solution} -Add solution here -::: - - -## k. Describe the effects $\sigma$ and $n$ has on selection of the best model? Why is the *true* model form (i.e., quadratic) not always the *best* model to use when prediction is the goal? - -::: {.solution} -Add solution here -::: - - - - - - - diff --git a/homework/hw1.html b/homework/hw1.html deleted file mode 100644 index 42e2933..0000000 --- a/homework/hw1.html +++ /dev/null @@ -1,3052 +0,0 @@ - - - - - - - - - - - - - - -Homework #1: Supervised Learning - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
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SYS 6018 | Sp 2023 | University of Virginia

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Required R packages and Directories

-
-
data.dir = 'https://mdporter.github.io/SYS6018/data/' # data directory
-library(R6018)     # functions for SYS-6018
-library(tidyverse) # functions for data manipulation    
-
-
-
-

Problem 1: Evaluating a Regression Model

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a. Create a set of functions to generate data from the following -distributions:

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\[\begin{align*} -X &\sim \mathcal{N}(0, 1) \\ -Y &= -1 + .5X + .2X^2 + \epsilon \\ -\epsilon &\sim \mathcal{N}(0,\, \sigma) -\end{align*}\]

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Add solution here

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b. Simulate \(n=100\) realizations -from these distributions using \(\sigma=3\). Produce a scatterplot and draw -the true regression line \(f(x) = E[Y \mid -X=x]\).

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c. Fit three polynomial regression models using least squares: -linear, quadratic, and cubic. Produce another scatterplot, add the -fitted lines and true population line \(f(x)\) using different colors, and add a -legend that maps the line color to a model.

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  • Note: The true model is quadratic, but we are also fitting linear -(less complex) and cubic (more complex) models.
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d. Simulate a test data set of 10,000 observations from the -same distributions. Use set.seed(612) prior to generating -the test data.

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  • Calculate the estimated mean squared error (MSE) for each -model.
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e. What is the best achievable MSE? That is, what is the MSE if the -true \(f(x)\) was used to evaluate the -test set? How close does the best method come to achieving the -optimum?

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f. The MSE scores obtained in part d came from one -realization of training data. Here will we explore how much variation -there is in the MSE scores by replicating the simulation many -times.

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  • Re-run parts b. and c. (i.e., generate training data and fit models) -100 times. -
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  • Calculate the MSE for all simulations.
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  • Create kernel density or histogram plots of the resulting MSE values -for each model.
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g. Show a count of how many times each model was the best. That is, -out of the 100 simulations, count how many times each model had the -lowest MSE.

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h. Write a function that implements the simulation in part -f. The function should have arguments for i) the size of the -training data \(n\), ii) the standard -deviation of the random error \(\sigma\), and iii) the test data. Use the -same set.seed(613).

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i. Use your function to repeat the simulation in part f, -but use \(\sigma=2\). Report the number -of times each model was best (you do not need to produce any -plots).

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j. Repeat i, but now use \(\sigma=4\) and \(n=300\).

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k. Describe the effects \(\sigma\) -and \(n\) has on selection of the best -model? Why is the true model form (i.e., quadratic) not always -the best model to use when prediction is the goal?

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- - - - - - - - - - - - - - - - diff --git a/homework/hw10.Rmd b/homework/hw10.Rmd deleted file mode 100644 index 54d1876..0000000 --- a/homework/hw10.Rmd +++ /dev/null @@ -1,167 +0,0 @@ ---- -title: "Homework #10: Clustering" -author: "**Your Name Here**" -date: "Due: Fri Apr 28 | 11:45am" -output: R6018::homework ---- - -**SYS 4582/6018 | Spring 2023 | University of Virginia ** - -******************************************* -```{r config, echo=FALSE} -source(system.file("config/hw_config.R", package="R6018")) # knitr settings -# options(dplyr.summarise.inform = FALSE) # ignore dplyr message about grouping -``` - - - -# Required R packages and Directories {-} - -::: {.solution} -```{r packages, message=FALSE, warning=FALSE} -data.dir = 'https://mdporter.github.io/SYS6018/data/' # data directory -library(R6018) # functions for SYS-6018 -library(tidyverse) # functions for data manipulation -library(mclust) # functions for mixture models -library(mixtools) # poisregmixEM() function -``` -::: - - -# Problem 1: Customer Segmentation with RFM (Recency, Frequency, and Monetary Value) - -RFM analysis is an approach that some businesses use to understand their customers' activities. At any point in time, a company can measure how recently a customer purchased a product (Recency), how many times they purchased a product (Frequency), and how much they have spent (Monetary Value). There are many ad-hoc attempts to segment/cluster customers based on the RFM scores (e.g., here is one based on using the customers' rank of each dimension independently: ). In this problem you will use the clustering methods we covered in class to segment the customers. - - -The data for this problem can be found here: <`r file.path(data.dir, "RFM.csv")`>. Cluster based on the Recency, Frequency, and Monetary value columns. - - -::: {.solution} -Load data here -::: - - -## a. Implement hierarchical clustering. - -- Describe any pre-processing steps you took (e.g., scaling, distance metric) -- State the linkage method you used with justification. -- Show the resulting dendrogram -- State the number of segments/clusters you used with justification. -- Using your segmentation, are customers 1 and 99 in the same cluster? - -::: {.solution} -Add solution here -::: - - - -## b. Implement k-means. - -- Describe any pre-processing steps you took (e.g., scaling) -- State the number of segments/clusters you used with justification. -- Using your segmentation, are customers 1 and 99 in the same cluster? - -::: {.solution} -Add solution here -::: - -## c. Implement model-based clustering - -- Describe any pre-processing steps you took (e.g., scaling) -- State the number of segments/clusters you used with justification. -- Describe the best model. What restrictions are on the shape of the components? -- Using your segmentation, are customers 1 and 99 in the same cluster? - -::: {.solution} -Add solution here -::: - -## d. Discussion - -Discuss how you would cluster the customers if you had to do this for your job. Do you think one model would do better than the others? - -::: {.solution} -Add solution here -::: - - - -# Problem 2: Poisson Mixture Model - -The pmf of a Poisson random variable is: -\begin{align*} -f_k(x; \lambda_k) = \frac{\lambda_k^x e^{-\lambda_k}}{x!} -\end{align*} - -A two-component Poisson mixture model can be written: -\begin{align*} -f(x; \theta) = \pi \frac{\lambda_1^x e^{-\lambda_1}}{x!} + (1-\pi) \frac{\lambda_2^x e^{-\lambda_2}}{x!} -\end{align*} - - - -## a. Parameters - -What are the parameters of the model? - -::: {.solution} -Add solution here -::: - -## b. Log-likelihood - -Write down the log-likelihood for $n$ independent observations ($x_1, x_2, \ldots, x_n$). - -::: {.solution} -Add solution here -::: - -## c. Responsibilities - -Suppose we have initial values of the parameters. Write down the equation for updating the *responsibilities* using the notation given in the problem statement. - -::: {.solution} -Add solution here -::: - - -## d. Updating model parameters - -Suppose we have responsibilities, $r_{ik}$ for all $i=1, 2, \ldots, n$ and $k=1,2$. Write down the equations for updating the parameters. - -::: {.solution} -Add solution here -::: - - -## e. Two-component Poisson mixture model - -Fit a two-component Poisson mixture model, report the estimated parameter values, and show a plot of the estimated mixture pmf for the following data: - -```{r, echo=TRUE} -#-- Run this code to generate the data -set.seed(123) # set seed for reproducibility -n = 200 # sample size -z = sample(1:2, size=n, replace=TRUE, prob=c(.25, .75)) # sample the latent class -theta = c(8, 16) # true parameters -y = ifelse(z==1, rpois(n, lambda=theta[1]), rpois(n, lambda=theta[2])) -``` - -- Note: The function `poisregmixEM()` in the R package `mixtools` is designed to estimate a mixture of *Poisson regression* models. We can still use this function for our problem of pmf estimation if it is recast as an intercept-only regression. To do so, set the $x$ argument (predictors) to `x = rep(1, length(y))` and `addintercept = FALSE`. - - Look carefully at the output from this model. The `beta` values (regression coefficients) are on the log scale. The model output of `lambda` are the prior probabilities. - -::: {.solution} -Add solution here -::: - - -## f. Extra Credit - -**2 pts Extra Credit**: Write a function that estimates this two-component Poisson mixture model using the EM approach. Show that it gives the same result as part *e*. - -- Note: you are not permitted to copy code. Write everything from scratch and use comments to indicate how the code works (e.g., the E-step, M-step, initialization strategy, and convergence should be clear). -- Cite any resources you consulted to help with the coding. - -::: {.solution} -Add solution here -::: diff --git a/homework/hw10.html b/homework/hw10.html deleted file mode 100644 index f5e657a..0000000 --- a/homework/hw10.html +++ /dev/null @@ -1,1800 +0,0 @@ - - - - - - - - - - - - - - -Homework #10: Clustering - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
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SYS 4582/6018 | Spring 2023 | University of Virginia -

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Required R packages and Directories

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data.dir = 'https://mdporter.github.io/SYS6018/data/' # data directory
-library(R6018)     # functions for SYS-6018
-library(tidyverse) # functions for data manipulation   
-library(mclust)    # functions for mixture models
-library(mixtools)  # poisregmixEM() function
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Problem 1: Customer Segmentation with RFM (Recency, Frequency, and -Monetary Value)

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RFM analysis is an approach that some businesses use to understand -their customers’ activities. At any point in time, a company can measure -how recently a customer purchased a product (Recency), how many times -they purchased a product (Frequency), and how much they have spent -(Monetary Value). There are many ad-hoc attempts to segment/cluster -customers based on the RFM scores (e.g., here is one based on using the -customers’ rank of each dimension independently: https://joaocorreia.io/blog/rfm-analysis-increase-sales-by-segmenting-your-customers.html). -In this problem you will use the clustering methods we covered in class -to segment the customers.

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The data for this problem can be found here: https://mdporter.github.io/SYS6018/data//RFM.csv. -Cluster based on the Recency, Frequency, and Monetary value columns.

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Load data here

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a. Implement hierarchical clustering.

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  • Describe any pre-processing steps you took (e.g., scaling, distance -metric)
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  • State the linkage method you used with justification.
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  • Show the resulting dendrogram
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  • State the number of segments/clusters you used with -justification.
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  • Using your segmentation, are customers 1 and 99 in the same -cluster?
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Add solution here

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b. Implement k-means.

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  • Describe any pre-processing steps you took (e.g., scaling)
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  • State the number of segments/clusters you used with -justification.
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  • Using your segmentation, are customers 1 and 99 in the same -cluster?
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Add solution here

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c. Implement model-based clustering

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  • Describe any pre-processing steps you took (e.g., scaling)
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  • State the number of segments/clusters you used with -justification.
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  • Describe the best model. What restrictions are on the shape of the -components?
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  • Using your segmentation, are customers 1 and 99 in the same -cluster?
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d. Discussion

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Discuss how you would cluster the customers if you had to do this for -your job. Do you think one model would do better than the others?

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Problem 2: Poisson Mixture Model

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The pmf of a Poisson random variable is: \[\begin{align*} -f_k(x; \lambda_k) = \frac{\lambda_k^x e^{-\lambda_k}}{x!} -\end{align*}\]

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A two-component Poisson mixture model can be written: \[\begin{align*} -f(x; \theta) = \pi \frac{\lambda_1^x e^{-\lambda_1}}{x!} + (1-\pi) -\frac{\lambda_2^x e^{-\lambda_2}}{x!} -\end{align*}\]

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a. Parameters

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What are the parameters of the model?

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b. Log-likelihood

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Write down the log-likelihood for \(n\) independent observations (\(x_1, x_2, \ldots, x_n\)).

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c. Responsibilities

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Suppose we have initial values of the parameters. Write down the -equation for updating the responsibilities using the notation -given in the problem statement.

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d. Updating model parameters

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Suppose we have responsibilities, \(r_{ik}\) for all \(i=1, 2, \ldots, n\) and \(k=1,2\). Write down the equations for -updating the parameters.

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e. Two-component Poisson mixture model

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Fit a two-component Poisson mixture model, report the estimated -parameter values, and show a plot of the estimated mixture pmf for the -following data:

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#-- Run this code to generate the data
-set.seed(123)             # set seed for reproducibility
-n = 200                   # sample size
-z = sample(1:2, size=n, replace=TRUE, prob=c(.25, .75)) # sample the latent class
-theta = c(8, 16)          # true parameters
-y = ifelse(z==1, rpois(n, lambda=theta[1]), rpois(n, lambda=theta[2]))
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  • Note: The function poisregmixEM() in the R package -mixtools is designed to estimate a mixture of Poisson -regression models. We can still use this function for our problem -of pmf estimation if it is recast as an intercept-only regression. To do -so, set the \(x\) argument (predictors) -to x = rep(1, length(y)) and -addintercept = FALSE. -
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f. Extra Credit

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2 pts Extra Credit: Write a function that estimates -this two-component Poisson mixture model using the EM approach. Show -that it gives the same result as part e.

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  • Note: you are not permitted to copy code. Write everything from -scratch and use comments to indicate how the code works (e.g., the -E-step, M-step, initialization strategy, and convergence should be -clear).
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- - - - - - - - - - - - - - - - diff --git a/homework/hw2.Rmd b/homework/hw2.Rmd deleted file mode 100644 index d82ec63..0000000 --- a/homework/hw2.Rmd +++ /dev/null @@ -1,171 +0,0 @@ ---- -title: "Homework #2: Resampling" -author: "**Your Name Here**" -date: "Due: Wed Feb 15 | 11:45am" -output: R6018::homework ---- - -**SYS 6018 | Sp 2023 | University of Virginia** - ------------------------------------------------------------------------- - -```{r config, echo=FALSE} -source(system.file("config/hw_config.R", package="R6018")) # knitr settings -# options(dplyr.summarise.inform = FALSE) # ignore dplyr message about grouping -``` - - -# Required R packages and Directories - - -::: {.solution} -```{r packages, message=FALSE, warning=FALSE} -data.dir = 'https://mdporter.github.io/SYS6018/data/' # data directory -library(R6018) # functions for SYS-6018 -library(tidyverse) # functions for data manipulation -library(tidymodels)# optional functions for tidymodels framework -``` -::: - - - -# Problem 1: Bootstrapping - -Bootstrap resampling can be used to quantify the uncertainty in a fitted curve. - -## a. Generate Data - -Create a set of functions to generate data from the following distributions: -\begin{align*} -X &\sim \mathcal{U}(0, 2) \qquad \text{Uniform in $[0,2]$}\\ -Y &= 1 + 2x + 5\sin(5x) + \epsilon \\ -\epsilon &\sim \mathcal{N}(0,\, \sigma=2.5) -\end{align*} - -::: {.solution} -Add solution here -::: - -## b. Simulate $n=100$ realizations from these distributions. - -- Produce a scatterplot and draw the true regression line $f(x) = E[Y \mid X=x]$. -- Use `set.seed(211)` prior to generating the data. - -::: {.solution} -Add solution here -::: - - -## c. Fit a 5th degree polynomial. - -- Produce a scatterplot and draw the *estimated* regression curve. - - -::: {.solution} -Add solution here -::: - -## d. Bootstrap sampling - -Make 200 bootstrap samples. For each bootstrap sample, fit a 5th degree polynomial and make predictions at `eval_pts = seq(0, 2, length=100)` - -- Set the seed (use `set.seed(212)`) so your results are reproducible. -- Produce a scatterplot with the original data and add the 200 bootstrap curves - -::: {.solution} -Add solution here -::: - - -## e. Calculate the pointwise 95% confidence intervals from the bootstrap samples. - -That is, for each $x \in {\rm eval\_pts}$, calculate the upper and lower limits such that only 5% of the curves fall outside the interval at $x$. - -- Remake the plot from part *c*, but add the upper and lower boundaries from the 95% confidence intervals. - -::: {.solution} -Add solution here -::: - - -# Problem 2: V-Fold cross-validation with $k$ nearest neighbors - -Run 10-fold cross-validation on the data generated in part 1b to select the optimal $k$ in a k-nearest neighbor (kNN) model. Then evaluate how well cross-validation performed by evaluating the performance on a large test set. The steps below will guide you. - - -## a. Select $k$ using $10$-fold cross-validation - -Use $10$-fold cross-validation to find the value of $k$ (i.e., neighborhood size) that provides the smallest cross-validated MSE using a kNN model. - -- Search over $k=3,4,\ldots, 40$. -- Use `set.seed(221)` prior to generating the folds to ensure the results are replicable. -- Show the following: - - the optimal $k$ (as determined by cross-validation) - - the corresponding estimated MSE - - produce a plot with $k$ on the x-axis and the estimated MSE on the y-axis (optional: add 1-standard error bars). -- Notation: The $k$ is the tuning paramter for the kNN model. The $v=10$ is the number of folds in V-fold cross-validation. Don't get yourself confused. - -::: {.solution} -Add solution here -::: - - - -## b. What is the optimal *edf* for the kNN model? - -The $k$ (number of neighbors) in a kNN model determines the effective degrees of freedom *edf*. Be sure to use the correct sample size when making this calculation. - -- Produce a plot similar to that from part *a*, but use *edf* (effective degrees of freedom) on the x-axis. - -::: {.solution} -Add solution here -::: - - - -## c. Optimal $k$ for final model - -After running cross-validation, a final model fit from *all* of the training data needs to be produced to make predictions. What value of $k$ would you choose? Why? - - -::: {.solution} -Add solution here -::: - -## d. Cross-validation performance - -Simulate a test data set of $50000$ observations from the same distributions. Use `set.seed(223)` prior to generating the test data. - -- Fit a set of kNN models, using the full training data, and calculate the mean squared error (MSE) on the test data for each model. Use the same $k$ values in *a*. -- Report the optimal $k$, the corresponding *edf*, and MSE based on the **test set**. - -::: {.solution} -Add solution here -::: - -## e. Performance Plots - -Plot both the cross-validation estimated and (true) error calculated from the test data on the same plot. See Figure 5.6 in ISL (pg 182) as a guide. - -- Produce two plots: one with $k$ on the x-axis and one with *edf* on the x-axis. -- Each plot should have two lines: one from part *a* and one from part *d* - -::: {.solution} -Add solution here -::: - -## f. Did cross-validation work? - -Based on the plots from *e*, does it appear that cross-validation worked as intended? How sensitive is the choice of $k$ on the resulting test MSE? - -::: {.solution} -Add solution here -::: - - - - - - - - diff --git a/homework/hw2.html b/homework/hw2.html deleted file mode 100644 index cbdcaf6..0000000 --- a/homework/hw2.html +++ /dev/null @@ -1,1810 +0,0 @@ - - - - - - - - - - - - - - -Homework #2: Resampling - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
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SYS 6018 | Sp 2023 | University of Virginia

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Required R packages and Directories

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data.dir = 'https://mdporter.github.io/SYS6018/data/' # data directory
-library(R6018)     # functions for SYS-6018
-library(tidyverse) # functions for data manipulation 
-library(tidymodels)# optional functions for tidymodels framework
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Problem 1: Bootstrapping

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Bootstrap resampling can be used to quantify the uncertainty in a -fitted curve.

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a. Generate Data

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Create a set of functions to generate data from the following -distributions: \[\begin{align*} -X &\sim \mathcal{U}(0, 2) \qquad \text{Uniform in $[0,2]$}\\ -Y &= 1 + 2x + 5\sin(5x) + \epsilon \\ -\epsilon &\sim \mathcal{N}(0,\, \sigma=2.5) -\end{align*}\]

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b. Simulate \(n=100\) realizations -from these distributions.

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c. Fit a 5th degree polynomial.

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d. Bootstrap sampling

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Make 200 bootstrap samples. For each bootstrap sample, fit a 5th -degree polynomial and make predictions at -eval_pts = seq(0, 2, length=100)

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  • Set the seed (use set.seed(212)) so your results are -reproducible.
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e. Calculate the pointwise 95% confidence intervals from the -bootstrap samples.

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That is, for each \(x \in {\rm -eval\_pts}\), calculate the upper and lower limits such that only -5% of the curves fall outside the interval at \(x\).

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Problem 2: V-Fold cross-validation with \(k\) nearest neighbors

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Run 10-fold cross-validation on the data generated in part 1b to -select the optimal \(k\) in a k-nearest -neighbor (kNN) model. Then evaluate how well cross-validation performed -by evaluating the performance on a large test set. The steps below will -guide you.

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a. Select \(k\) using \(10\)-fold cross-validation

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Use \(10\)-fold cross-validation to -find the value of \(k\) (i.e., -neighborhood size) that provides the smallest cross-validated MSE using -a kNN model.

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  • Notation: The \(k\) is the tuning -paramter for the kNN model. The \(v=10\) is the number of folds in V-fold -cross-validation. Don’t get yourself confused.
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b. What is the optimal edf for the kNN model?

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The \(k\) (number of neighbors) in a -kNN model determines the effective degrees of freedom edf. Be -sure to use the correct sample size when making this calculation.

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c. Optimal \(k\) for final -model

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After running cross-validation, a final model fit from all -of the training data needs to be produced to make predictions. What -value of \(k\) would you choose? -Why?

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d. Cross-validation performance

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Simulate a test data set of \(50000\) observations from the same -distributions. Use set.seed(223) prior to generating the -test data.

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  • Fit a set of kNN models, using the full training data, and calculate -the mean squared error (MSE) on the test data for each model. Use the -same \(k\) values in a.
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e. Performance Plots

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Plot both the cross-validation estimated and (true) error calculated -from the test data on the same plot. See Figure 5.6 in ISL (pg 182) as a -guide.

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f. Did cross-validation work?

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Based on the plots from e, does it appear that -cross-validation worked as intended? How sensitive is the choice of -\(k\) on the resulting test MSE?

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Add solution here

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- - - - - - - - - - - - - - - - diff --git a/homework/hw3.Rmd b/homework/hw3.Rmd deleted file mode 100644 index 53d4609..0000000 --- a/homework/hw3.Rmd +++ /dev/null @@ -1,144 +0,0 @@ ---- -title: "Homework #3: Penalized Regression" -author: "**Your Name Here**" -date: "Wed Feb 22 | 11:45am" -output: R6018::homework ---- - -**SYS 6018 | Sp 2023 | University of Virginia** ------------------------------------------------------------------------- - -```{r config, echo=FALSE} -source(system.file("config/hw_config.R", package="R6018")) # knitr settings -# options(dplyr.summarise.inform = FALSE) # ignore dplyr message about grouping -``` - - - -# Required R packages and Directories - -::: {.solution} -```{r packages, message=FALSE, warning=FALSE} -data.dir = 'https://mdporter.github.io/SYS6018/data' # data directory -library(mlbench) -library(glmnet) -library(R6018) # functions for SYS-6018 -library(tidyverse) # functions for data manipulation -``` -::: - -# Problem 1: Optimal Tuning Parameters - -In cross-validation, we discussed choosing the tuning parameter values that minimized the cross-validation error. Another approach, called the "one-standard error" rule [ISL pg 214, ESL pg 61], uses the values corresponding to the least complex model whose cv error is within one standard error of the best model. The goal of this assignment is to compare these two rules. - -Use simulated data from `mlbench.friedman1(n, sd=2)` in the `mlbench` R package to fit *lasso models*. The tuning parameter $\lambda$ (corresponding to the penalty on the coefficient magnitude) is the one we will focus one. Generate training data, use k-fold cross-validation to get $\lambda_{\rm min}$ and $\lambda_{\rm 1SE}$, generate test data, make predictions for the test data, and compare performance of the two rules under a squared error loss using a hypothesis test. - - -Choose reasonable values for: - -- Number of cv folds ($K$) - - Note: you are free to use repeated CV, repeated hold-outs, or bootstrapping instead of plain cross-validation; just be sure to describe what do did so it will be easier to follow. -- Number of training and test observations -- Number of simulations -- If everyone uses different values, we will be able to see how the results change over the different settings. -- Don't forget to make your results reproducible (e.g., set seed) - -This pseudo code will get you started: -```yaml -library(mlbench) -library(glmnet) - -#-- Settings -n.train = # number of training obs -n.test = # number of test obs -K = # number of CV folds -alpha = # glmnet tuning alpha (1 = lasso, 0 = ridge) -M = # number of simulations - -#-- Data Generating Function -getData <- function(n) mlbench.friedman1(n, sd=2) # data generating function - -#-- Simulations -# Set Seed Here - -for(m in 1:M) { - -# 1. Generate Training Data -# 2. Build Training Models using cross-validation, e.g., cv.glmnet() -# 3. get lambda that minimizes cv error and 1 SE rule -# 4. Generate Test Data -# 5. Predict y values for test data (for each model: min, 1SE) -# 6. Evaluate predictions - -} - -#-- Compare -# compare performance of the approaches / Statistical Test -``` - -## a. Code for the simulation and performance results - -::: {.solution} -Add solution here -::: - - -## b. Hypothesis Test - -Describe and state the results of a hypothesis test comparing $\lambda_{\rm min}$ and $\lambda_{\rm 1SE}$. - -::: {.solution} -Add solution here -::: - - - -# Problem 2 Prediction Contest: Real Estate Pricing - -This problem uses the [realestate-train](`r file.path(data.dir, 'realestate-train.csv')`) and [realestate-test](`r file.path(data.dir, 'realestate-test.csv')`) (click on links for data). - -The goal of this contest is to predict sale price (in thousands) (`price` column) using an *elastic net* model. Evaluation of the test data will be based on the root mean squared error ${\rm RMSE}= \sqrt{\frac{1}{m}\sum_i (y_i - \hat{y}_i)^2}$ for the $m$ test set observations. - - -## a. Data - -Load the data and create necessary data structures for running *elastic net*. - -- You are free to use any data transformation or feature engineering -- Note: there are some categorical predictors so at the least you will have to convert those to something numeric (e.g., one-hot or dummy coding). - -::: {.solution} -Add solution here -::: - - -## b. Prediction - -Use an *elastic net* model to predict the `price` of the test data. - -- You are free to use any data transformation or feature engineering -- You are free to use any tuning parameters -- Report the $\alpha$ and $\lambda$ parameters you used to make your final predictions. -- Describe how you choose those tuning parameters - -::: {.solution} -Add solution here -::: - -## c. Submit predictions - -Submit a .csv file (ensure comma separated format) named `lastname_firstname.csv` that includes your predictions in a column named *yhat*. We will use automated evaluation, so the format must be exact. - -- You will receive credit for a proper submission; the top five scores will receive 2 bonus points. - ::: {.solution} -Add solution here -::: - -## d. Anticipated Performance - -Report the anticipated performance of your method in terms of RMSE. We will see how close your performance assessment matches the actual value. - -::: {.solution} -Add solution here -::: - diff --git a/homework/hw3.html b/homework/hw3.html deleted file mode 100644 index f157738..0000000 --- a/homework/hw3.html +++ /dev/null @@ -1,1781 +0,0 @@ - - - - - - - - - - - - - - -Homework #3: Penalized Regression - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
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SYS 6018 | Sp 2023 | University of Virginia

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Required R packages and Directories

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-
data.dir = 'https://mdporter.github.io/SYS6018/data' # data directory
-library(mlbench)
-library(glmnet)
-library(R6018)     # functions for SYS-6018
-library(tidyverse) # functions for data manipulation
-
-
-
-

Problem 1: Optimal Tuning Parameters

-

In cross-validation, we discussed choosing the tuning parameter -values that minimized the cross-validation error. Another approach, -called the “one-standard error” rule [ISL pg 214, ESL pg 61], uses the -values corresponding to the least complex model whose cv error is within -one standard error of the best model. The goal of this assignment is to -compare these two rules.

-

Use simulated data from mlbench.friedman1(n, sd=2) in -the mlbench R package to fit lasso models. The -tuning parameter \(\lambda\) -(corresponding to the penalty on the coefficient magnitude) is the one -we will focus one. Generate training data, use k-fold cross-validation -to get \(\lambda_{\rm min}\) and \(\lambda_{\rm 1SE}\), generate test data, -make predictions for the test data, and compare performance of the two -rules under a squared error loss using a hypothesis test.

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Choose reasonable values for:

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    -
  • Number of cv folds (\(K\)) -
      -
    • Note: you are free to use repeated CV, repeated hold-outs, or -bootstrapping instead of plain cross-validation; just be sure to -describe what do did so it will be easier to follow.
    • -
  • -
  • Number of training and test observations
  • -
  • Number of simulations
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  • If everyone uses different values, we will be able to see how the -results change over the different settings.
  • -
  • Don’t forget to make your results reproducible (e.g., set seed)
  • -
-

This pseudo code will get you started:

-
library(mlbench)
-library(glmnet)
-
-#-- Settings
-n.train =        # number of training obs
-n.test =         # number of test obs
-K =              # number of CV folds
-alpha =          # glmnet tuning alpha (1 = lasso, 0 = ridge)
-M =              # number of simulations
-
-#-- Data Generating Function
-getData <- function(n) mlbench.friedman1(n, sd=2) # data generating function
-
-#-- Simulations
-# Set Seed Here
-
-for(m in 1:M) {
-
-# 1. Generate Training Data
-# 2. Build Training Models using cross-validation, e.g., cv.glmnet()
-# 3. get lambda that minimizes cv error and 1 SE rule
-# 4. Generate Test Data
-# 5. Predict y values for test data (for each model: min, 1SE)
-# 6. Evaluate predictions
-
-}
-
-#-- Compare
-# compare performance of the approaches / Statistical Test
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a. Code for the simulation and performance results

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Add solution here

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b. Hypothesis Test

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Describe and state the results of a hypothesis test comparing \(\lambda_{\rm min}\) and \(\lambda_{\rm 1SE}\).

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Add solution here

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Problem 2 Prediction Contest: Real Estate Pricing

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This problem uses the realestate-train -and realestate-test -(click on links for data).

-

The goal of this contest is to predict sale price (in thousands) -(price column) using an elastic net model. -Evaluation of the test data will be based on the root mean squared error -\({\rm RMSE}= \sqrt{\frac{1}{m}\sum_i (y_i - -\hat{y}_i)^2}\) for the \(m\) -test set observations.

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a. Data

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Load the data and create necessary data structures for running -elastic net.

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    -
  • You are free to use any data transformation or feature -engineering
  • -
  • Note: there are some categorical predictors so at the least you will -have to convert those to something numeric (e.g., one-hot or dummy -coding).
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Add solution here

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b. Prediction

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Use an elastic net model to predict the price -of the test data.

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  • You are free to use any data transformation or feature -engineering
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  • You are free to use any tuning parameters
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  • Report the \(\alpha\) and \(\lambda\) parameters you used to make your -final predictions.
  • -
  • Describe how you choose those tuning parameters
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Add solution here

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c. Submit predictions

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Submit a .csv file (ensure comma separated format) named -lastname_firstname.csv that includes your predictions in a -column named yhat. We will use automated evaluation, so the -format must be exact.

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  • You will receive credit for a proper submission; the top five scores -will receive 2 bonus points. ::: {.solution} Add solution here :::
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d. Anticipated Performance

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Report the anticipated performance of your method in terms of RMSE. -We will see how close your performance assessment matches the actual -value.

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Add solution here

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- - - - - - - - - - - - - - - - diff --git a/homework/hw4.Rmd b/homework/hw4.Rmd deleted file mode 100644 index 3bf2f06..0000000 --- a/homework/hw4.Rmd +++ /dev/null @@ -1,150 +0,0 @@ ---- -title: "Homework #4: Classification" -author: "**Your Name Here**" -date: "Due: Wed Mar 01 | 11:45am" -output: R6018::homework ---- - -**SYS 4582/6018 | Spring 2023 | University of Virginia ** - -******************************************* -```{r config, echo=FALSE} -source(system.file("config/hw_config.R", package="R6018")) # knitr settings -# options(dplyr.summarise.inform = FALSE) # ignore dplyr message about grouping -``` - -::: {style="background-color:yellow; color:red; display: block; border-color: black; padding:1em"} -This is an **independent assignment**. Do not discuss or work with classmates. -::: - - -# Required R packages and Directories - -::: {.solution} -```{r packages, message=FALSE, warning=FALSE} -data.dir = 'https://mdporter.github.io/SYS6018/data/' # data directory -library(glmnet) # for glmnet() functions -library(yardstick) # for evaluation metrics -library(R6018) # functions for SYS-6018 -library(tidyverse) # functions for data manipulation -``` -::: - - -# Crime Linkage - -Crime linkage attempts to determine if two or more unsolved crimes share a common offender. *Pairwise* crime linkage is the more simple task of deciding if two crimes share a common offender; it can be considered a binary classification problem. The linkage training data has 8 evidence variables that measure the similarity between a pair of crimes: - -- `spatial` is the spatial distance between the crimes -- `temporal` is the fractional time (in days) between the crimes -- `tod` and `dow` are the differences in time of day and day of week between the crimes -- `LOC`, `POA,` and `MOA` are binary with a 1 corresponding to a match (type of property, point of entry, method of entry) -- `TIMERANGE` is the time between the earliest and latest possible times the crime could have occurred (because the victim was away from the house during the crime). -- The response variable indicates if the crimes are linked ($y=1$) or unlinked ($y=0$). - - -These problems use the [linkage-train](https://mdporter.github.io/SYS6018/data/linkage_train.csv) and [linkage-test](https://mdporter.github.io/SYS6018/data/linkage_test.csv) datasets (click on links for data). Load the data here: - - - -::: {.solution} -Add Solution Here -::: - - - - -# Problem 1: Penalized Regression for Crime Linkage - -## a. Penalized linear regression - -Fit a penalized *linear* regression model to predict linkage. Use a lasso, ridge, or elasticnet penalty (your choice). - -- Report the value of $\alpha$ used (if elasticnet) -- Report the value of $\lambda$ used -- Report the estimated coefficients - -::: {.solution} -Add Solution Here -::: - - -## b. Penalized logistic regression - -Fit a penalized *logistic* regression model to predict linkage. Use a lasso, ridge, or elasticnet penalty (your choice). - -- Report the value of $\alpha$ used (if elasticnet) -- Report the value of $\lambda$ used -- Report the estimated coefficients - -::: {.solution} -Add Solution Here -::: - - -## c. ROC Curves (training data) - -Produce one plot that has the ROC curves, using the *training data*, for both models (from part a and b). Use color and/or linetype to distinguish between models and include a legend. - -::: {.solution} -Add Solution Here -::: - - -## d. ROC Curve (hold-out data) - -Recreate the ROC curve from the penalized logistic regression model using repeated hold-out data. The following steps will guide you: - -- Fix $\alpha=.75$ -- Run the following steps 25 times: -i. Hold out 500 observations -ii. Use the remaining observations to estimate $\lambda$ using 10-fold CV -iii. Predict the probability of linkage for the 500 hold-out observations -iv. Store the predictions and hold-out labels -- Combine the results and produce the hold-out based ROC curve -- Note: by estimating $\lambda$ each iteration, we are incorporating the uncertainty present in estimating that tuning parameter. - -::: {.solution} -Add Solution Here -::: - - -## e. Contest Part 1: Probability - -Predict the estimated *probability* of linkage for the test data (using any model). - -- Submit a .csv file (ensure comma separated format) named `lastname_firstname_1.csv` that includes the column named **p** that is your estimated posterior probability. We will use automated evaluation, so the format must be exact. -- You are free to use any tuning parameters -- You are free to use any data transformation or feature engineering -- You will receive credit for a proper submission; the top five scores will receive 2 bonus points. -- Your probabilities will be evaluated with respect to the mean negative Bernoulli log-likelihood (known as the average *log-loss* metric) -$$ -L = - \frac{1}{M} \sum_{i=1}^m [y_i \log \, \hat{p}_i + (1 - y_i) \log \, (1 - \hat{p}_i)] -$$ -where $M$ is the number of test observations, $\hat{p}_i$ is the prediction for the $i$th test observation, and $y_i \in \{0,1\}$ are the true test set labels. - -::: {.solution} -Add Solution Here -::: - - -## f. Contest Part 2: Linkages - -Predict the linkages for the test data (using any model). - -- Submit a .csv file (ensure comma separated format) named `lastname_firstname_2.csv` that includes the column named **linkage** that takes the value of 1 for linkages and 0 for unlinked pairs. We will use automated evaluation, so the format must be exact. -- You are free to use any tuning parameters. -- You are free to use any data transformation or feature engineering. -- Your labels will be evaluated based on total cost, where cost is equal to `1*FP + 8*FN`. This implies that False Negatives (FN) are 8 times as costly as False Positives (FP) -- You will receive credit for a proper submission; the top five scores will receive 2 bonus points. Note: you only will get bonus credit for one of the two contests. - - -::: {.solution} -Add Solution Here -::: - - - - - - diff --git a/homework/hw4.html b/homework/hw4.html deleted file mode 100644 index bb9eb2f..0000000 --- a/homework/hw4.html +++ /dev/null @@ -1,3072 +0,0 @@ - - - - - - - - - - - - - - -Homework #4: Classification - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
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SYS 4582/6018 | Spring 2023 | University of Virginia -

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This is an independent assignment. Do not discuss or -work with classmates.

-
-
-

Required R packages and Directories

-
-
data.dir = 'https://mdporter.github.io/SYS6018/data/' # data directory
-library(glmnet)    # for glmnet() functions
-library(yardstick) # for evaluation metrics
-library(R6018)     # functions for SYS-6018
-library(tidyverse) # functions for data manipulation   
-
-
-
-

Crime Linkage

-

Crime linkage attempts to determine if two or more unsolved crimes -share a common offender. Pairwise crime linkage is the more -simple task of deciding if two crimes share a common offender; it can be -considered a binary classification problem. The linkage training data -has 8 evidence variables that measure the similarity between a pair of -crimes:

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    -
  • spatial is the spatial distance between the crimes
  • -
  • temporal is the fractional time (in days) between the -crimes
  • -
  • tod and dow are the differences in time of -day and day of week between the crimes
  • -
  • LOC, POA, and MOA are binary -with a 1 corresponding to a match (type of property, point of entry, -method of entry)
  • -
  • TIMERANGE is the time between the earliest and latest -possible times the crime could have occurred (because the victim was -away from the house during the crime).
  • -
  • The response variable indicates if the crimes are linked (\(y=1\)) or unlinked (\(y=0\)).
  • -
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These problems use the linkage-train -and linkage-test -datasets (click on links for data). Load the data here:

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Add Solution Here

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-

Problem 1: Penalized Regression for Crime Linkage

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a. Penalized linear regression

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Fit a penalized linear regression model to predict linkage. -Use a lasso, ridge, or elasticnet penalty (your choice).

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    -
  • Report the value of \(\alpha\) used -(if elasticnet)
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  • Report the value of \(\lambda\) -used
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  • Report the estimated coefficients
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Add Solution Here

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b. Penalized logistic regression

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Fit a penalized logistic regression model to predict -linkage. Use a lasso, ridge, or elasticnet penalty (your choice).

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    -
  • Report the value of \(\alpha\) used -(if elasticnet)
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  • Report the value of \(\lambda\) -used
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  • Report the estimated coefficients
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Add Solution Here

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c. ROC Curves (training data)

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Produce one plot that has the ROC curves, using the training -data, for both models (from part a and b). Use color and/or -linetype to distinguish between models and include a legend.

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Add Solution Here

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d. ROC Curve (hold-out data)

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Recreate the ROC curve from the penalized logistic regression model -using repeated hold-out data. The following steps will guide you:

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    -
  • Fix \(\alpha=.75\)
  • -
  • Run the following steps 25 times:
  • -
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  1. Hold out 500 observations
  2. -
  3. Use the remaining observations to estimate \(\lambda\) using 10-fold CV
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  5. Predict the probability of linkage for the 500 hold-out -observations
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  7. Store the predictions and hold-out labels
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  • Combine the results and produce the hold-out based ROC curve
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  • Note: by estimating \(\lambda\) -each iteration, we are incorporating the uncertainty present in -estimating that tuning parameter.
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Add Solution Here

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e. Contest Part 1: Probability

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Predict the estimated probability of linkage for the test -data (using any model).

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    -
  • Submit a .csv file (ensure comma separated format) named -lastname_firstname_1.csv that includes the column named -p that is your estimated posterior probability. We will -use automated evaluation, so the format must be exact.
  • -
  • You are free to use any tuning parameters
  • -
  • You are free to use any data transformation or feature -engineering
  • -
  • You will receive credit for a proper submission; the top five scores -will receive 2 bonus points.
    -
  • -
  • Your probabilities will be evaluated with respect to the mean -negative Bernoulli log-likelihood (known as the average -log-loss metric) \[ -L = - \frac{1}{M} \sum_{i=1}^m [y_i \log \, \hat{p}_i + (1 - y_i) \log -\, (1 - \hat{p}_i)] -\] where \(M\) is the number of -test observations, \(\hat{p}_i\) is the -prediction for the \(i\)th test -observation, and \(y_i \in \{0,1\}\) -are the true test set labels.
  • -
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Add Solution Here

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f. Contest Part 2: Linkages

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Predict the linkages for the test data (using any model).

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    -
  • Submit a .csv file (ensure comma separated format) named -lastname_firstname_2.csv that includes the column named -linkage that takes the value of 1 for linkages and 0 -for unlinked pairs. We will use automated evaluation, so the format must -be exact.
  • -
  • You are free to use any tuning parameters.
  • -
  • You are free to use any data transformation or feature -engineering.
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  • Your labels will be evaluated based on total cost, where cost is -equal to 1*FP + 8*FN. This implies that False Negatives -(FN) are 8 times as costly as False Positives (FP)
    -
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  • You will receive credit for a proper submission; the top five scores -will receive 2 bonus points. Note: you only will get bonus credit for -one of the two contests.
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Add Solution Here

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- - - - - - - - - - - - - - - - diff --git a/homework/hw5.Rmd b/homework/hw5.Rmd deleted file mode 100644 index e15e562..0000000 --- a/homework/hw5.Rmd +++ /dev/null @@ -1,132 +0,0 @@ ---- -title: "Homework #5: Density Estimation" -author: "**Your Name Here**" -date: "Due: Wed Mar 22 | 11:45am" -output: R6018::homework ---- - -**SYS 4582/6018 | Spring 2023 | University of Virginia ** - -******************************************* -```{r config, echo=FALSE} -source(system.file("config/hw_config.R", package="R6018")) # knitr settings -options(dplyr.summarise.inform = FALSE) # ignore dplyr message about grouping -``` - - -# Required R packages and Directories - -::: {.solution} -```{r packages, message=FALSE, warning=FALSE} -data.dir = 'https://mdporter.github.io/SYS6018/data/' # data directory -library(R6018) # functions for SYS 6018 -library(ks) # functions for KDE -library(tidyverse) # functions for data manipulation -``` -::: - - - -# Problem 1 Geographic Profiling - -Geographic profiling, a method developed in criminology, can be used to estimate the [home location (roost) of animals](https://www.sciencedirect.com/science/article/pii/S0022519305004157) based on a collection of sightings. The approach requires an estimate of the distribution the animal will travel from their roost to forage for food. - -A sample of $283$ distances that pipistrelle bats traveled (in meters) from their roost can be found at: - -- **Bat Data**: <`r file.path(data.dir, 'geo_profile.csv')`> - -One probability model for the distance these bats will travel is: -\begin{align*} -f(x; \theta) = \frac{x}{\theta} \exp \left( - \frac{x^2}{2 \theta} \right) -\end{align*} -where the parameter $\theta > 0$ controls how far they are willing to travel. - - -## a. Derive the MLE for $\theta$ (i.e., show the math). - -::: {.solution} -Add solution here -::: - - -## b. What is the MLE of $\theta$ for the bat data? (Use results from a, or use computational methods.) - -::: {.solution} -Add solution here -::: - -## c. Using the MLE value of $\theta$ from part b, compute the estimated density at a set of evaluation points between 0 and 8 meters. Plot the estimated density. - -::: {.solution} -Add solution here -::: - -## d. Estimate the density using KDE. Report the bandwidth you chose and produce a plot of the estimated density. - -::: {.solution} -Add solution here -::: - - - - -## e. Which model do you prefer, the parametric or KDE? - -::: {.solution} -Add solution here -::: - - - - -# Problem 2: Interstate Crash Density - -Interstate 64 (I-64) is a major east-west road that passes just south of Charlottesville. Where and when are the most dangerous places/times to be on I-64? The crash data (link below) gives the mile marker and fractional time-of-week for crashes that occurred on I-64 between mile marker 87 and 136 in 2016. The time-of-week data takes a numeric value of *\.\*, where the dow starts at 0 for Sunday (6 for Sat) and the decimal gives the time of day information. Thus `time=0.0417` corresponds to Sun at 1am and `time=6.5` corresponds to Sat at noon. - -- **Crash Data**: <`r file.path(data.dir, 'crashes16.csv')`> - - -## a. Extract the crashes and make a scatter plot with mile marker on x-axis and time on y-axis. - -::: {.solution} -Add solution here -::: - - - -## b. Use KDE to estimate the *mile marker* density. - -- Report the bandwidth. -- Plot the density estimate. - -::: {.solution} -Add solution here -::: - -## c. Use KDE to estimate the temporal *time-of-week* density. - -- Report the bandwidth. -- Plot the density estimate. - -::: {.solution} -Add solution here -::: - -## d. Use KDE to estimate the bivariate mile-time density. - -- Report the bandwidth parameters. -- Plot the bivariate density estimate. - -::: {.solution} -Add solution here -::: - - -## e. Based on the estimated density, approximate the most dangerous place and time to drive on this strech of road. Identify the mile marker and time-of-week pair. - -::: {.solution} -Add solution here -::: - - - \ No newline at end of file diff --git a/homework/hw5.html b/homework/hw5.html deleted file mode 100644 index edd01cd..0000000 --- a/homework/hw5.html +++ /dev/null @@ -1,3034 +0,0 @@ - - - - - - - - - - - - - - -Homework #5: Density Estimation - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
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SYS 4582/6018 | Spring 2023 | University of Virginia -

-
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Required R packages and Directories

-
-
data.dir = 'https://mdporter.github.io/SYS6018/data/' # data directory
-library(R6018)     # functions for SYS 6018
-library(ks)        # functions for KDE
-library(tidyverse) # functions for data manipulation   
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Problem 1 Geographic Profiling

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Geographic profiling, a method developed in criminology, can be used -to estimate the home -location (roost) of animals based on a collection of sightings. The -approach requires an estimate of the distribution the animal will travel -from their roost to forage for food.

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A sample of \(283\) distances that -pipistrelle bats traveled (in meters) from their roost can be found -at:

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One probability model for the distance these bats will travel is: -\[\begin{align*} -f(x; \theta) = \frac{x}{\theta} \exp \left( - \frac{x^2}{2 \theta} -\right) -\end{align*}\] where the parameter \(\theta > 0\) controls how far they are -willing to travel.

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a. Derive the MLE for \(\theta\) -(i.e., show the math).

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Add solution here

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b. What is the MLE of \(\theta\) -for the bat data? (Use results from a, or use computational -methods.)

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Add solution here

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c. Using the MLE value of \(\theta\) from part b, compute the estimated -density at a set of evaluation points between 0 and 8 meters. Plot the -estimated density.

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Add solution here

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d. Estimate the density using KDE. Report the bandwidth you chose -and produce a plot of the estimated density.

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Add solution here

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e. Which model do you prefer, the parametric or KDE?

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Problem 2: Interstate Crash Density

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Interstate 64 (I-64) is a major east-west road that passes just south -of Charlottesville. Where and when are the most dangerous places/times -to be on I-64? The crash data (link below) gives the mile marker and -fractional time-of-week for crashes that occurred on I-64 between mile -marker 87 and 136 in 2016. The time-of-week data takes a numeric value -of <dow>.<hour/24>, where the dow starts at 0 for -Sunday (6 for Sat) and the decimal gives the time of day information. -Thus time=0.0417 corresponds to Sun at 1am and -time=6.5 corresponds to Sat at noon.

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a. Extract the crashes and make a scatter plot with mile marker on -x-axis and time on y-axis.

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Add solution here

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b. Use KDE to estimate the mile marker density.

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  • Report the bandwidth.
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Add solution here

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c. Use KDE to estimate the temporal time-of-week -density.

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Add solution here

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d. Use KDE to estimate the bivariate mile-time density.

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e. Based on the estimated density, approximate the most dangerous -place and time to drive on this strech of road. Identify the mile marker -and time-of-week pair.

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Add solution here

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- - - - - - - - - - - - - - - - diff --git a/homework/hw6.Rmd b/homework/hw6.Rmd deleted file mode 100644 index fff8932..0000000 --- a/homework/hw6.Rmd +++ /dev/null @@ -1,116 +0,0 @@ ---- -title: "Homework #6: SVM and Generative Classifiers" -author: "**Your Name Here**" -date: "Due: Wed Mar 29 | 11:45am" -output: R6018::homework ---- - -**SYS 4582/6018 | Spring 2023 | University of Virginia ** - -******************************************* -```{r config, echo=FALSE} -source(system.file("config/hw_config.R", package="R6018")) # knitr settings -# options(dplyr.summarise.inform = FALSE) # ignore dplyr message about grouping -``` - -# Required R packages and Directories - -::: {.solution} -```{r packages, message=FALSE, warning=FALSE} -data_dir = 'https://mdporter.github.io/SYS6018/data/' # data directory -library(R6018) # functions for SYS-6018 -library(tidyverse) # functions for data manipulation -library(e1071) # svm functions -# Add other libraries here -``` -::: - - - -# Problem 1: Handwritten Digit Recognition - -The MNIST database of handwritten digits () was an early driver of machine learning innovation. The data are greyscale values (0-255) of handwritten single digits (0-9) from a 28 x 28 image (784 dimensional vector). I did some preprocessing, include principal component analysis (PCA), and reduced dimensionality down to 30 predictors (`X1`, `X2`, ..., `X30`). There are ten possible labels $\{0, 1, \ldots, 9\}$ corresponding to each digit. - - -## a. Load the MNIST training and testing data. -The data are `.rds` format. Training data has 1000 samples from each class. The test data has only one sample from each class. -[Training Data](`r file.path(data_dir, "mnist_train.rds")`) -[Testing Data](`r file.path(data_dir, "mnist_test.rds")`) - - -::: {.solution} - -Add Solution Here - -::: - - - -## b. Quadratic Discriminant Analysis (QDA) - -Implement quadratic discriminant analysis (QDA) step-by-step. Do the following: - -i. For each digit, estimate the mean and variance-covariance of the predictors. -ii. Estimate the prior class probabilities. -iii. Use i. and ii. to estimate $\Pr(Y = k \mid X=x)$, with the help of Bayes theorem, for $k = 0, 1, \ldots, 9$, for the ten observations in the test set. This should produce a 10 x 10 object where the rows correspond to the test observations, and the 10 columns the probability of each digit. You can use existing functions for obtaining the MV Gaussian density (e.g., `mvtnorm::dmvnorm()`). -iv. Indicate how well the model predicts the test data. You choose the metric(s) for evaluation. - - -::: {.solution} - -Add Solution Here - -::: - - -# Problem 2: One-vs-Rest - -In KNN and the generative models (LDA/QDA, MDA, KDA, naive bayes), it is straightforward to fit a model with more than two classes. Other methods, like Logistic Regression and Support Vector Machines, are designed to deal with outcome variables that take only two values. However we can still use binary classifiers for a multi-class problems. One approach, called *one-vs-rest* is the easiest to implement (, and see ISL 9.4.2). - -For outcome variables that take K values, K models will be fit. Model 1 will be fit to discriminant class $1$ from all the other classes ($\{2,\ldots, K\}$). Model 2 will be fit to discriminate class $2$ from all the other classes ($\{1, 3, 4, \ldots, K\}$), etc. The estimated class for observation $Y_i$ is the one receiving the highest probability score (this assumes equal costs of mis-classification). - -Details: To fit model $k$ on the training data, code $Y=1$ if the label is $k$ and $Y=0$ if the label is not $k$ (thus comparing class $k$ vs all the rest). Then on the test data, calculate $\hat{p}_k(x_i)$, the estimated probability that $Y = 1$ according to model $k$. The estimated class label for test observation $i$ is $\arg\max_{1\leq k \leq K} \hat{p}_k(x_i)$. - -We will use the MNIST digit data for this problem. - -## a. Support Vector Machines (SVM) for 2-class problem - -To get warmed up we will fit an SVM model for a 2-class problem. In this problem, treat the digit 0 as the class of interest (`+1`) and all other digits as the negative class (`-1`). -Fit a SVM model, using the *radial basis kernel*. - -- Normally, you will need to estimate the tuning parameters for each model. But to simplify, you can use the default value of `gamma` and set `cost = 100`. - -Make predictions on the 10 test observations and show the results. - -- If you use the `e1071::svm()` function, set `probability = TRUE`. Then with `predict.svm(..., probability=TRUE) %>% attr("probabilities")` you can extract a probability matrix. The probabilities are estimated with [Platt scaling](https://en.wikipedia.org/wiki/Platt_scaling). - -- The model is intended to distinguish between 0 and everything else, so hopefully it correctly identifies which observation is 0. - -::: {.solution} - -Add Solution Here - -::: - -## b. Game time. Implement one-vs-rest for the MNIST data. - -- Hint: this may take a few minutes to run. Use chunk option `cache=TRUE` on the code chunk that takes a long time to run. After you knit once (and don't change anything in the chunk), then subsequent knits will fetch the saved data and you won't have to wait. - -::: {.solution} - -Add Solution Here - -::: - - - - - - - - - - - - - diff --git a/homework/hw6.html b/homework/hw6.html deleted file mode 100644 index a21a11e..0000000 --- a/homework/hw6.html +++ /dev/null @@ -1,1756 +0,0 @@ - - - - - - - - - - - - - - -Homework #6: SVM and Generative Classifiers - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
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SYS 4582/6018 | Spring 2023 | University of Virginia -

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Required R packages and Directories

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-
data_dir = 'https://mdporter.github.io/SYS6018/data/' # data directory
-library(R6018)     # functions for SYS-6018
-library(tidyverse) # functions for data manipulation   
-library(e1071)     # svm functions
-# Add other libraries here
-
-
-
-

Problem 1: Handwritten Digit Recognition

-

The MNIST database of handwritten digits (http://yann.lecun.com/exdb/mnist/) was an early driver -of machine learning innovation. The data are greyscale values (0-255) of -handwritten single digits (0-9) from a 28 x 28 image (784 dimensional -vector). I did some preprocessing, include principal component analysis -(PCA), and reduced dimensionality down to 30 predictors -(X1, X2, …, X30). There are ten -possible labels \(\{0, 1, \ldots, 9\}\) -corresponding to each digit.

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a. Load the MNIST training and testing data.

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The data are .rds format. Training data has 1000 samples -from each class. The test data has only one sample from each class. Training -Data Testing -Data

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Add Solution Here

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b. Quadratic Discriminant Analysis (QDA)

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Implement quadratic discriminant analysis (QDA) step-by-step. Do the -following:

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  1. For each digit, estimate the mean and variance-covariance of the -predictors.
  2. -
  3. Estimate the prior class probabilities.
  4. -
  5. Use i. and ii. to estimate \(\Pr(Y = k -\mid X=x)\), with the help of Bayes theorem, for \(k = 0, 1, \ldots, 9\), for the ten -observations in the test set. This should produce a 10 x 10 object where -the rows correspond to the test observations, and the 10 columns the -probability of each digit. You can use existing functions for obtaining -the MV Gaussian density (e.g., mvtnorm::dmvnorm()).
  6. -
  7. Indicate how well the model predicts the test data. You choose the -metric(s) for evaluation.
  8. -
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Add Solution Here

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Problem 2: One-vs-Rest

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In KNN and the generative models (LDA/QDA, MDA, KDA, naive bayes), it -is straightforward to fit a model with more than two classes. Other -methods, like Logistic Regression and Support Vector Machines, are -designed to deal with outcome variables that take only two values. -However we can still use binary classifiers for a multi-class problems. -One approach, called one-vs-rest is the easiest to implement -(https://en.wikipedia.org/wiki/Multiclass_classification, -and see ISL 9.4.2).

-

For outcome variables that take K values, K models will be fit. Model -1 will be fit to discriminant class \(1\) from all the other classes (\(\{2,\ldots, K\}\)). Model 2 will be fit to -discriminate class \(2\) from all the -other classes (\(\{1, 3, 4, \ldots, -K\}\)), etc. The estimated class for observation \(Y_i\) is the one receiving the highest -probability score (this assumes equal costs of mis-classification).

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Details: To fit model \(k\) on the -training data, code \(Y=1\) if the -label is \(k\) and \(Y=0\) if the label is not \(k\) (thus comparing class \(k\) vs all the rest). Then on the test -data, calculate \(\hat{p}_k(x_i)\), the -estimated probability that \(Y = 1\) -according to model \(k\). The estimated -class label for test observation \(i\) -is \(\arg\max_{1\leq k \leq K} -\hat{p}_k(x_i)\).

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We will use the MNIST digit data for this problem.

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a. Support Vector Machines (SVM) for 2-class problem

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To get warmed up we will fit an SVM model for a 2-class problem. In -this problem, treat the digit 0 as the class of interest -(+1) and all other digits as the negative class -(-1). Fit a SVM model, using the radial basis -kernel.

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  • Normally, you will need to estimate the tuning parameters for each -model. But to simplify, you can use the default value of -gamma and set cost = 100.
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Make predictions on the 10 test observations and show the -results.

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    -
  • If you use the e1071::svm() function, set -probability = TRUE. Then with -predict.svm(..., probability=TRUE) %>% attr("probabilities") -you can extract a probability matrix. The probabilities are estimated -with Platt -scaling.

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  • The model is intended to distinguish between 0 and everything -else, so hopefully it correctly identifies which observation is -0.

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Add Solution Here

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b. Game time. Implement one-vs-rest for the MNIST data.

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  • Hint: this may take a few minutes to run. Use chunk option -cache=TRUE on the code chunk that takes a long time to run. -After you knit once (and don’t change anything in the chunk), then -subsequent knits will fetch the saved data and you won’t have to -wait.
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Add Solution Here

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- - - - - - - - - - - - - - - - diff --git a/homework/hw7.Rmd b/homework/hw7.Rmd deleted file mode 100644 index 417e655..0000000 --- a/homework/hw7.Rmd +++ /dev/null @@ -1,138 +0,0 @@ ---- -title: "Homework #7: Tree Ensembles" -author: "**Your Name Here**" -date: "Due: Wed Apr 05 | 11:45am" -output: R6018::homework ---- - -**SYS 4582/6018 | Spring 2023 | University of Virginia ** - -******************************************* -```{r config, echo=FALSE} -source(system.file("config/hw_config.R", package="R6018")) # knitr settings -# options(dplyr.summarise.inform = FALSE) # ignore dplyr message about grouping -``` - - -::: {style="background-color:yellow; color:red; display: block; border-color: black; padding:1em"} - -This is an **independent assignment**. Do not discuss or work with classmates. - -::: - - -# Required R packages and Directories {-} - -::: {.solution} -```{r packages, message=FALSE, warning=FALSE} -data_dir = 'https://mdporter.github.io/SYS6018/data/' # data directory -library(R6018) # functions for SYS-6018 -library(tidyverse) # functions for data manipulation -library(randomForest) -# Add other libraries here -``` -::: - -# Problem 1: Tree Splitting for classification - -Consider the Gini index, classification error, and entropy impurity measures in a simple classification setting with two classes. - -Create a single plot that displays each of these quantities as a function of $p_m$, the estimated probability of an observation in node $m$ being from class 1. The x-axis should display $p_m$, ranging from 0 to 1, and the y-axis should display the value of the Gini index, classification error, and entropy. - -::: {.solution} - -Add solution here - -::: - - -# Problem 2: Combining bootstrap estimates - -```{r, echo=FALSE} -p_red = c(0.2, 0.25, 0.3, 0.4, 0.4, 0.45, 0.7, 0.85, 0.9, 0.9) -``` - -Suppose we produce ten bootstrapped samples from a data set containing red and green classes. We then apply a classification tree to each bootstrapped sample and, for a specific value of $X$, produce the following 10 estimates of $\Pr(\text{Class is Red} \mid X)$: $\{`r stringr::str_c(p_red, sep=", ")`\}$. - -## a. Majority Voting - -ISLR 8.2 describes the *majority vote* approach for making a hard classification from a set of bagged classifiers. What is the final classification for this example using majority voting? - -::: {.solution} - -Add solution here - -::: - - -## b. Average Probability - -An alternative is to base the final classification on the average probability. What is the final classification for this example using average probability? - -::: {.solution} - -Add solution here - -::: - - - -## c. Unequal Cost of Misclassification - -Suppose the cost of mis-classifying a Red observation (as Green) is twice as costly as mis-classifying a Green observation (as Red). How would you modify both approaches to make better final classifications under these unequal costs? Report the final classifications. - -::: {.solution} - -Add solution here - -::: - - - -# Problem 3: Random Forest Tuning - -Random forest has several tuning parameters that you will explore in this problem. We will use the `Boston` housing data from the `MASS` R package (See the ISLR Lab in section 8.3.3 for example code). - -- Note: remember that `MASS` can mask the `dplyr::select()` function. - -## a. Tuning Parameters - -List all of the random forest tuning parameters in the `randomForest::randomForest()` function. Note any tuning parameters that are specific to classification or regression problems. Indicate the tuning parameters you think will be most important to optimize? - -::: {.solution} - -Add solution here - -::: - - - -## b. RF CV performance - -Use a random forest model to predict `medv`, the median value of owner-occupied homes (in $1000s). Use the default parameters and report the 10-fold cross-validation MSE. - -::: {.solution} - -Add solution here - -::: - - - -## c. Effects of `mtry` and `ntree` - -Now we will vary the tuning parameters of `mtry` and `ntree` to see what effect they have on performance. - -- Use a range of reasonable `mtry` and `ntree` values. -- Use 5 times repeated out-of-bag (OOB) to assess performance. That is, run random forest 5 times for each tuning set, calculate the OOB MSE each time and use the average for the MSE associated with the tuning parameters. -- Use a plot to show the average MSE as a function of `mtry` and `ntree`. -- Report the best tuning parameter combination. -- Note: random forest is a stochastic model; it will be different every time it runs. Set the random seed to control the uncertainty associated with the stochasticity. -- Hint: If you use the `randomForest` package, the `mse` element in the output is a vector of OOB MSE values for `1:ntree` trees in the forest. This means that you can set `ntree` to some maximum value and get the MSE for any number of trees up to `ntree`. - - -::: {.solution} - -Add solution here - -::: diff --git a/homework/hw7.html b/homework/hw7.html deleted file mode 100644 index 03e23bd..0000000 --- a/homework/hw7.html +++ /dev/null @@ -1,3031 +0,0 @@ - - - - - - - - - - - - - - -Homework #7: Tree Ensembles - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
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SYS 4582/6018 | Spring 2023 | University of Virginia -

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This is an independent assignment. Do not discuss or -work with classmates.

-
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Required R packages and Directories

-
-
data_dir = 'https://mdporter.github.io/SYS6018/data/' # data directory
-library(R6018)     # functions for SYS-6018
-library(tidyverse) # functions for data manipulation  
-library(randomForest)
-# Add other libraries here
-
-
-
-

Problem 1: Tree Splitting for classification

-

Consider the Gini index, classification error, and entropy impurity -measures in a simple classification setting with two classes.

-

Create a single plot that displays each of these quantities as a -function of \(p_m\), the estimated -probability of an observation in node \(m\) being from class 1. The x-axis should -display \(p_m\), ranging from 0 to 1, -and the y-axis should display the value of the Gini index, -classification error, and entropy.

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Add solution here

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Problem 2: Combining bootstrap estimates

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Suppose we produce ten bootstrapped samples from a data set -containing red and green classes. We then apply a classification tree to -each bootstrapped sample and, for a specific value of \(X\), produce the following 10 estimates of -\(\Pr(\text{Class is Red} \mid X)\): -\(\{0.2, 0.25, 0.3, 0.4, 0.4, 0.45, 0.7, 0.85, -0.9, 0.9\}\).

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a. Majority Voting

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ISLR 8.2 describes the majority vote approach for making a -hard classification from a set of bagged classifiers. What is the final -classification for this example using majority voting?

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Add solution here

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b. Average Probability

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An alternative is to base the final classification on the average -probability. What is the final classification for this example using -average probability?

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Add solution here

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c. Unequal Cost of Misclassification

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Suppose the cost of mis-classifying a Red observation (as Green) is -twice as costly as mis-classifying a Green observation (as Red). How -would you modify both approaches to make better final classifications -under these unequal costs? Report the final classifications.

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Add solution here

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Problem 3: Random Forest Tuning

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Random forest has several tuning parameters that you will explore in -this problem. We will use the Boston housing data from the -MASS R package (See the ISLR Lab in section 8.3.3 for -example code).

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  • Note: remember that MASS can mask the -dplyr::select() function.
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a. Tuning Parameters

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List all of the random forest tuning parameters in the -randomForest::randomForest() function. Note any tuning -parameters that are specific to classification or regression problems. -Indicate the tuning parameters you think will be most important to -optimize?

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Add solution here

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b. RF CV performance

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Use a random forest model to predict medv, the median -value of owner-occupied homes (in $1000s). Use the default parameters -and report the 10-fold cross-validation MSE.

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Add solution here

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c. Effects of mtry and ntree

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Now we will vary the tuning parameters of mtry and -ntree to see what effect they have on performance.

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  • Use a range of reasonable mtry and ntree -values.
  • -
  • Use 5 times repeated out-of-bag (OOB) to assess performance. That -is, run random forest 5 times for each tuning set, calculate the OOB MSE -each time and use the average for the MSE associated with the tuning -parameters.
  • -
  • Use a plot to show the average MSE as a function of -mtry and ntree.
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  • Report the best tuning parameter combination.
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  • Note: random forest is a stochastic model; it will be different -every time it runs. Set the random seed to control the uncertainty -associated with the stochasticity.
  • -
  • Hint: If you use the randomForest package, the -mse element in the output is a vector of OOB MSE values for -1:ntree trees in the forest. This means that you can set -ntree to some maximum value and get the MSE for any number -of trees up to ntree.
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Add solution here

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- - - - - - - - - - - - - - - - diff --git a/homework/hw8.Rmd b/homework/hw8.Rmd deleted file mode 100644 index 7ae4f3b..0000000 --- a/homework/hw8.Rmd +++ /dev/null @@ -1,31 +0,0 @@ ---- -title: "Homework #8: Kaggle and Stacking" -author: "**Your Name Here**" -date: "Due: Wed Apr 12 | 11:45am" -output: R6018::homework ---- - -**SYS 4582/6018 | Spring 2023 | University of Virginia ** - -******************************************* -```{r config, echo=FALSE} -source(system.file("config/hw_config.R", package="R6018")) # knitr settings -# options(dplyr.summarise.inform = FALSE) # ignore dplyr message about grouping -``` - - -# Problem 1: Kaggle - -You are to make at least one official entry in the [House Prices: Advanced Regression Techniques](https://www.kaggle.com/c/house-prices-advanced-regression-techniques/overview) Kaggle contest. - -- You will need to register in Kaggle (its free) -- Read the details of the contest. Understand the data and evaluation function. -- Make at least one submission -- If you get a score on the public leaderboard of $\text{RMSE}<0.50$ (note RMSE is calculated on the log scale), you receive full credit, otherwise, you'll lose 10 points. - - I'll allow [teaming](https://www.kaggle.com/c/house-prices-advanced-regression-techniques/team) to achieve the score, but only if everyone on the team produces a model that is used to generate the final submission (e.g., stacking or model averaging) -- Submit the following in Collab: - - Code - - kaggle name (or team name) so we can ensure you had a valid submission. - - your score and current ranking on the kaggle leaderboard -- Top 5 scores get 2 bonus points - - Teams will split their bonus points among team members diff --git a/homework/hw8.html b/homework/hw8.html deleted file mode 100644 index 68049a8..0000000 --- a/homework/hw8.html +++ /dev/null @@ -1,2939 +0,0 @@ - - - - - - - - - - - - - - -Homework #8: Kaggle and Stacking - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
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SYS 4582/6018 | Spring 2023 | University of Virginia -

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Problem 1: Kaggle

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You are to make at least one official entry in the House -Prices: Advanced Regression Techniques Kaggle contest.

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  • You will need to register in Kaggle (its free)
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  • Read the details of the contest. Understand the data and evaluation -function.
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  • Make at least one submission
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  • If you get a score on the public leaderboard of \(\text{RMSE}<0.50\) (note RMSE is -calculated on the log scale), you receive full credit, otherwise, you’ll -lose 10 points. -
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    • I’ll allow teaming -to achieve the score, but only if everyone on the team produces a model -that is used to generate the final submission (e.g., stacking or model -averaging)
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  • Submit the following in Collab: -
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    • Code
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    • kaggle name (or team name) so we can ensure you had a valid -submission.
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    • your score and current ranking on the kaggle leaderboard
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- - - - - - - - - - - - - - - - diff --git a/homework/hw9.Rmd b/homework/hw9.Rmd deleted file mode 100644 index cf93f43..0000000 --- a/homework/hw9.Rmd +++ /dev/null @@ -1,137 +0,0 @@ ---- -title: "Homework #9: Feature Importance" -author: "**Your Name Here**" -date: "Due: Fri Apr 21 | 11:45pm" -output: R6018::homework ---- - -**SYS 4582/6018 | Spring 2023 | University of Virginia ** - -******************************************* -```{r config, echo=FALSE} -source(system.file("config/hw_config.R", package="R6018")) # knitr settings -# options(dplyr.summarise.inform = FALSE) # ignore dplyr message about grouping -``` - -::: {style="background-color:yellow; color:red; display: block; border-color: black; padding:1em"} -This is an **independent assignment**. Do not discuss or work with classmates. -::: - -# Required R packages and Directories - -::: {.solution} -```{r packages, message=FALSE, warning=FALSE} -data.dir = 'https://mdporter.github.io/SYS6018/data/' # data directory -library(R6018) # functions for SYS-6018 -library(tidyverse) # functions for data manipulation -``` -::: - - - -# Problem 1: Permutation Feature Importance - -The [R package `titanic`](https://cran.r-project.org/web/packages/titanic/index.html) contains data on who survived the Titanic sinking. The passengers have been split into a training (`titanic_train`) and testing (`titanic_test`) set. We are going to use this data to investigate feature importance. -Use `Pclass`, `Sex`, `Age`, `Fare`, `SibSp`, `Parch`, and `Embarked` for the predictor variables (features) and `Survived` as the outcome variable. - -::: {style="background-color:lightblue; display: block;"} - -NOTE: Use the training data `titanic_train` as the test data. We would not usually do this, but after posting the homework I was notified that the test data does not contain the labels which prohibits analysis of the permutation feature importance scores. -In summary, set `test = titanic_*train*`. - -::: - -## a. Method 1: Built-in importance scores - -Fit a tree ensemble model (e.g., Random Forest, boosted tree) on the training data. You are free to use any method to select the tuning parameters. - -Report the built-in feature importance scores and produce a barplot with feature on the x-axis and importance on the y-axis. - -::: {.solution} -Add solution here -::: - -## b. Performance - -Report the performance of the model from (a.) on the test data. Use the log-loss (where $M$ is the size of the test data): -$$ -\text{log-loss}(\hat{p}) = - \frac{1}{M} \sum_{i=1}^m [y_i \log \, \hat{p}_i + (1 - y_i) \log \, (1 - \hat{p}_i)] -$$ - -::: {.solution} -Add solution here -::: - -## c. Method 2: Permute *after* fitting - -Use the fitted model from question (a.) to perform permutation feature importance. Shuffle/permute each variable individually on the *test set* before making predictions. Record the loss. Repeat $M=10$ times and produce a boxplot of the change in loss (change from reported loss from part b.). - -::: {.solution} -Add solution here -::: - -## d. Method 3: Permute *before* fitting - -For this approach, shuffle/permute the *training data* and re-fit the ensemble model. Evaluate the predictions on the (unaltered) test data. Repeat $M=10$ times (for each predictor variable) and produce a boxplot of the change in loss. - -::: {.solution} -Add solution here -::: - - -## e. Understanding - -Describe the benefits of each of the three approaches to measure feature importance. - -::: {.solution} -Add solution here -::: - -# Problem 2: Effects of correlated predictors - -This problem will illustrate what happens to the importance scores when there are highly associated predictors. - -## a. Create an almost duplicate feature - -Create a new feature `Sex2` that is 95% the same as `Sex`. Do this by selecting 5% of training ($n=45$) and testing ($n=21$) data and flip the `Sex` value. - -::: {.solution} -Add solution here -::: - -## b. Method 1: Built-in importance - -Fit the same model as in Problem 1a, but use the new data that includes `Sex2` (i.e., use both `Sex` and `Sex2` in the model). Calculate the built-in feature importance score and produce a boxplot. - -::: {.solution} -Add solution here -::: - - - -## c. Method 2: Permute *after* fitting - -Redo Method 2 (problem 1c) on the new data/model and produce a boxplot of importance scores. - -::: {.solution} -Add solution here -::: - -## d. Method 3: Permute *before* fitting - -Redo Method 3 (problem 1d) on the new data and produce a boxplot of importance scores. - -::: {.solution} -Add solution here -::: - - -## e. Understanding - -Describe how the addition of the almost duplicated predictor impacted the feature importance results. - -::: {.solution} -Add solution here -::: - - diff --git a/homework/hw9.html b/homework/hw9.html deleted file mode 100644 index 3baeeb8..0000000 --- a/homework/hw9.html +++ /dev/null @@ -1,3040 +0,0 @@ - - - - - - - - - - - - - - -Homework #9: Feature Importance - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
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SYS 4582/6018 | Spring 2023 | University of Virginia -

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This is an independent assignment. Do not discuss or -work with classmates.

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Required R packages and Directories

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data.dir = 'https://mdporter.github.io/SYS6018/data/' # data directory
-library(R6018)     # functions for SYS-6018
-library(tidyverse) # functions for data manipulation   
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Problem 1: Permutation Feature Importance

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The R -package titanic contains data on who survived the -Titanic sinking. The passengers have been split into a training -(titanic_train) and testing (titanic_test) -set. We are going to use this data to investigate feature importance. -Use Pclass, Sex, Age, -Fare, SibSp, Parch, and -Embarked for the predictor variables (features) and -Survived as the outcome variable.

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NOTE: Use the training data titanic_train as the test -data. We would not usually do this, but after posting the homework I was -notified that the test data does not contain the labels which prohibits -analysis of the permutation feature importance scores. In summary, set -test = titanic_*train*.

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a. Method 1: Built-in importance scores

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Fit a tree ensemble model (e.g., Random Forest, boosted tree) on the -training data. You are free to use any method to select the tuning -parameters.

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Report the built-in feature importance scores and produce a barplot -with feature on the x-axis and importance on the y-axis.

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Add solution here

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b. Performance

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Report the performance of the model from (a.) on the test data. Use -the log-loss (where \(M\) is the size -of the test data): \[ -\text{log-loss}(\hat{p}) = - \frac{1}{M} \sum_{i=1}^m [y_i \log \, -\hat{p}_i + (1 - y_i) \log \, (1 - \hat{p}_i)] -\]

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Add solution here

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c. Method 2: Permute after fitting

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Use the fitted model from question (a.) to perform permutation -feature importance. Shuffle/permute each variable individually on the -test set before making predictions. Record the loss. Repeat -\(M=10\) times and produce a boxplot of -the change in loss (change from reported loss from part b.).

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Add solution here

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d. Method 3: Permute before fitting

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For this approach, shuffle/permute the training data and -re-fit the ensemble model. Evaluate the predictions on the (unaltered) -test data. Repeat \(M=10\) times (for -each predictor variable) and produce a boxplot of the change in -loss.

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Add solution here

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e. Understanding

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Describe the benefits of each of the three approaches to measure -feature importance.

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Add solution here

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Problem 2: Effects of correlated predictors

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This problem will illustrate what happens to the importance scores -when there are highly associated predictors.

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a. Create an almost duplicate feature

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Create a new feature Sex2 that is 95% the same as -Sex. Do this by selecting 5% of training (\(n=45\)) and testing (\(n=21\)) data and flip the Sex -value.

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Add solution here

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b. Method 1: Built-in importance

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Fit the same model as in Problem 1a, but use the new data that -includes Sex2 (i.e., use both Sex and -Sex2 in the model). Calculate the built-in feature -importance score and produce a boxplot.

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Add solution here

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c. Method 2: Permute after fitting

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Redo Method 2 (problem 1c) on the new data/model and produce a -boxplot of importance scores.

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Add solution here

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d. Method 3: Permute before fitting

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Redo Method 3 (problem 1d) on the new data and produce a boxplot of -importance scores.

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Add solution here

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e. Understanding

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Describe how the addition of the almost duplicated predictor impacted -the feature importance results.

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Add solution here

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Course Schedule

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ClassDateDayTopicReading (Pre-Class)Notes
1Jan 19ThuIntroductionPre-Req material; R and Rstudio
Modern Data Science with -R sections: 2,3,4,6,7,9
HW 0 assigned; Get textbook(s); update software; review -pre-req material
2Jan 24TueR and TidyverseR Tidyverse
-3-8, 10-11, 12.3, 12.5, 15, 17-19, 20.1-20.4, 21.1-21.5, 23.4, -27-29
3Jan 26ThuLinear Regression -with RISL 3.1-3.6R formula -interface
4Jan 31TueSupervised Learning -IISL 1, 2.1, 7.1
ESL 1, 2.1-2.4
HW 0 Completed
5Feb 02ThuSupervised Learning -IIISL 2.2-2.3
ESL 2.5-2.9
6Feb 07TueResampling: Bootstrap -and SplinesISL 5.2, 7.2-7.4
ESL 7.11, 5.1-5.3
(optional) -Bootstrapping -Regression Models
HW 1 Due
7Feb 09ThuResampling: -Cross-Validation and Model SelectionISL 5.1
ESL 7.10
ISL Chap 3 (review)
8Feb 14TuePenalized -RegressionISL 6.1-6.2
ESL 3.3-3.4
HW 2 Due
9Feb 16ThuPenalized RegressionISL 6.4-6.5 (optional 6.3)
ESL 3.5-3.6
10Feb 21TueRisk Modeling and -ClassificationISL 4.1-4.3
ESL 4.1-4.4
HW 3 Due
11Feb 23ThuRisk Modeling and ClassificationESL 9.1
12Feb 28TueDensity -EstimationIPSUR: -pg. 209-218 (MLE intro)
Maximum -Likelihood Estimation: pg. 221-223
IPSUR -Chap. 5-7 (Random Variable Review, if necessary)
HW 4 Due
13Mar 02ThuReview #1
Mar 07TueNo Class
Mar 09ThuNo Class
14Mar 14TueDensity EstimationSilverman -2.1-2.6
15Mar 16ThuGenerative -ClassifiersISL 4.4-4.5; ESL 6.6
16Mar 21TueSupport Vector Machines -(SVM)ISL 9.1-9.6
ESL 12.1-12.3 or MMDS 12.3
HW 5 Due
17Mar 23ThuTree-Based -MethodsISL 8.1, 8.3.1-8.3.2
ESL 9.2
18Mar 28TueTree-Based MethodsISL 8.2, 8.3.3-8.3.4
ESL 15
HW 6 Due
19Mar 30ThuEnsemblesISL 8.2
20Apr 04TueBoostingISL 8.2.3; ESL 10.1-10.9
LogitBoost -paper
HW 7 Due
21Apr 06ThuBoostingESL 10.10-10.14
XGBoost paper, CatBoost -paper, LightGBM
Taylor -Expansion
22Apr 11TueFeature -EngineeringISLR 6.3HW 8 Due
23Apr 13ThuFeature -ImportanceInterpretable -Machine Learning: Permutation Feature Importance
24Apr 18TueReviewHW 9 Due
25Apr 20ThuClusteringPrimary: ISL 12.4-12.5; ITDM 7.1; 7.5
Secondary: -ESL 14.3.1-14.3.8; 14.3.12; ITDM 7.2-7.3
26Apr 25TueClusteringPrimary: ESL 8.5.1
Secondary: Gaussian Mixture Models: -11.1-11.3
27Apr 27ThuTime Series and -ForecastingForecasting -Principles and Practice Chapters 2-3
Facebook Prophet
HW 10 Due
28May 02TueTime Series and -Forecasting (Part 2)Forecasting -Principles and Practice Chapters 5.10, 8
May 09Tue(9am-12) Final Presentations
May 10Wed(10pm) Final Project Due
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