data analysis view markdown
pqrs
 Goal: inference  conclusion or opinion formed from evidence
 PQRS
 P  population
 Q  question  2 types
 hypothesis driven  does a new drug work
 discovery driven  find a drug that works
 R  representative data colleciton
 simple random sampling = SRS
 w/ replacement: $var(\bar{X}) = \sigma^2 / n$
 w/out replacement: $var(\bar{X}) = (1  \frac{n}{N}) \sigma^2 / n$
 simple random sampling = SRS
 S  scrutinizing answers
visualization
First 5 parts here are based on the book storytelling with data by cole nussbaumer knaflic
 difference between showing data + storytelling with data
understand the context (1)
 who is your audience? what do you need them to know/do?
 exploratory vs explanatory analysis
 slides (need little details) vs email (needs lots of detail)  usually need to make both in slideument
 should know how much nonsupporting data to show
 distill things down into a 3minute story or a 1sentence Big Idea
 easiest to start things on paper/postit notes
choose an effective visual (2)
 generally avoid pie/donut charts, 3D charts, 2nd yaxes
 tables
 best for when people will actually read off numbers
 minimalist is best
 bar charts should basically always start at 0
 horizontal charts typically easy to read
 on axes, retain things like dollar signs, percent, etc.
eliminate clutter (3)
 gestalt principles of vision
 proximity  close things are grouped
 similarity  similar things are grouped
 connection  connected things are grouped
 enclosure
 closure
 continuity
 generally good to have titles and such at topleft!
 diagonal lines / text should be avoided
 centeraligned text should be avoided
 label lines directly
focus attention (4)
 visual hierarchy  outlines what is important
tell a story / think like a designer (5)
 affordances  aspects that make it obvious how something will be used (e.g. a button affords pushing)
 “You know you’ve achieved perfection, not when you have nothing more to add, but when you have nothing to take away” (Saint‐Exupery, 1943)
 stories have different parts, which include conflict + tension
 beginning  introduce a problem / promise
 middle  what could be
 end  call to action
 horizontal logic  people can just read title slides and get out what they need
 can either convince ppl through conventional rhetoric or through a story
visual summaries
 numerical summaries
 mean vs. median
 sd vs. iq range
 visual summaries
 histogram
 kernel density plot  Gaussian kernels
 with bandwidth h $K_h(t) = 1/h K(t/h)$
 plots
 box plot / piechart
 scatter plot / qq plot
 qq plot = probability plot  easily check normality
 plot percentiles of a data set against percentiles of a theoretical distr.
 should be straight line if they match
 transformations = feature engineering
 log/sqrt make longtail data more centered and more normal
 deltamethod  sets comparable bw (wrt variance) after log or sqrt transform: $Var(g(X)) \approx [g’(\mu_X)]^2 Var(X)$ where $\mu_X = E(X)$
 if assumptions don’t work, sometimes we can transform data so they work
 transform x  if residuals generally normal and have constant variance
 corrects nonlinearity  transform y  if relationship generally linear, but nonconstant error variance
 stabilizes variance  if both problems, try y first  BoxCox: Y’ = $Y^l : if : l \neq 0$, else log(Y)
 least squares
 inversion of pxp matrix ~O(p^3)
 regression effect  things tend to the mean (ex. bball children are shorter)
 in high dims, l2 worked best
 kernel smoothing + lowess
 can find optimal bandwidth
 nadarayawatson kernel smoother  locally weighted scatter plot smoothing
 \(g_h(x) = \frac{\sum K_h(x_i  x) y_i}{\sum K_h (x_i  x)}\) where h is bandwidth  loess  multiple predictors / lowess  only 1 predictor
 also called local polynomial smoother  locally weighted polynomial
 take a window (span) around a point and fit weighted least squares line to that point
 replace the point with the prediction of the windowed line
 can use local polynomial fits rather than local linear fits

silhouette plots  good clusters members are close to each other and far from other clustersf
 popular graphic method for K selection
 measure of separation between clusters $s(i) = \frac{b(i)  a(i)}{max(a(i), b(i))}$
 a(i)  ave dissimilarity of data point i with other points within same cluster
 b(i)  lowest average dissimilarity of point i to any other cluster
 good values of k maximize the average silhouette score
 lackoffit test  based on repeated Y values at same X values
imbalanced data
 randomly oversample minority class
 randomly undersample majority class
 weighting classes in the loss function  more efficient, but requires modifying model code
 generate synthetic minority class samples
 smote (chawla et al. 2002)  interpolate betwen points and their nearest neighbors (for minority class)  some heuristics for picking which points to interpolate
 adasyn (he et al. 2008)  smote, generate more synthetic data for minority examples which are harder to learn (number of samples is proportional to number of nearby samples in a different class)
 smrt  generate with vae
 smote (chawla et al. 2002)  interpolate betwen points and their nearest neighbors (for minority class)  some heuristics for picking which points to interpolate
 selectively removing majority class samples
 tomek links (tomek 1976)  selectively remove majority examples until al lminimally distanced nearestneighbor pairs are of the same class
 nearmiss (zhang & mani 2003)  select samples from the majority class which are close to the minority class. Example: select samples from the majority class for which the average distance of the N closest samples of a minority class is smallest
 edited nearest neighbors (wilson 1972)  “edit” the dataset by removing samples that don’t agree “enough” with their neighborhood
 feature selection and extraction
 minority class samples can be discarded as noise  removing irrelevant features can reduce this risk
 feature selection  select a subset of features and classify in this space
 feature extraction  extract new features and classify in this space
 ideas
 use majority class to find different low dimensions to investigate
 in this dim, do density estimation
 residuals  iteratively reweight these (like in boosting) to improve performance
 incorporate sampling / classweighting into ensemble method (e.g. treat different trees differently)
 ex. undersampling + ensemble learning (e.g. IFME, Becca’s work)
 algorithmic classifier modifications
 misc papers
 enrichment (jegierski & saganowski 2019)  add samples from an external dataset
 ref
 imblancedlearn package with several methods for dealing with imbalanced data
 good blog post
 Learning from classimbalanced data: Review of methods and applications (Haixiang et al. 2017)
 sample majority class w/ density (to get best samples)
 logspline  doesn’t scale
missingdata imputation
 Missing value imputation: a review and analysis of the literature (lin & tsai 2019)
 Causal Inference: A Missing Data Perspective (ding & li, 2018)
 different missingness mechanisms (little & rubin, 1987)
 MCAR = missing completely at random  no relationship between the missingness of the data and any values, observed or missing
 MAR = missing at random  propensity of missing values depends on observed data, but not the missing data
 can easily test for this vs MCAR
 MNAR = missing not at random  propensity of missing values depends both on observed and unobserved data
 connections to causal: MCAR is much like randomization, MAR like ignorability (although slightly more general), and MNAR like unmeasured unconfounding
 imputation problem: propensity of missing values depends on the unobserved values themselves (not ignorable)
 simplest approach: drop rows with missing vals
 mean/median imputation
 probabilistic approach
 EM approach, MCMC, GMM, sampling
 matrix completion: lowrank, PCA, SVD
 nearestneighbor / matching: hotdeck
 (weighted) prediction approaches
 linear regr, LDA, naive bayes, regr. trees
 can do weighting using something similar to inverse propensities, although less common to check things like covariate balance
 multiple imputation: impute multiple times to get better estimates
 MICE (passes / imputes data multiple times sequentially)
 can perform sensitivity analysis to evaluate the assumption that things are not MNAR
 two standard models for nonignorable missing data are the selection models and the patternmixture models (Little and Rubin, 2002, Chapter 15)
 performance evaluation
 acc at finding missing vals
 acc in downstream task
 applications
 Purposeful Variable Selection and Stratification to Impute Missing FAST Data in Trauma Research (fuchs et al. 2014)
matrix completion

A Survey on Matrix Completion: Perspective of Signal Processing (li…zhao, 2019)
 formulations
 Exact matrix completion via convex optimization (candes & recht, 2012)
 $\min {\boldsymbol{M}} \operatorname{rank}(\boldsymbol{M})$, s.t. $\left\boldsymbol{M}{\Omega}\boldsymbol{X}_{\Omega}\right_F \leq \delta$:  this is NPhard
 nuclear norm approxmiation

$\min {\boldsymbol{M}}\boldsymbol{M}*$, s.t. $ \boldsymbol{M}{\Omega}\boldsymbol{X}{\Omega} _F \leq \delta$  this has be formulated as semidefinite programming, nuclear norm relaxation, or robust PCA

 minimum rank approximation helps with the assumption that the data are corrupted by noise (e.g. ADMiRA (lee & bresler, 2010))
 $\min {\boldsymbol{M}}\left(\boldsymbol{M}){\Omega}\boldsymbol{X}_{\Omega}\right_F^2$, s.t. $\operatorname{rank}(\boldsymbol{M}) \leq r$
 matrix factorization is a faster but nonconvex approximation (e.g. LMaFit (wen, yin, & zhang, 2012))
 $\min {\boldsymbol{U}, \boldsymbol{V}, \boldsymbol{Z}}\left\boldsymbol{U} \boldsymbol{V}^T\boldsymbol{Z}\right_F^2$, s.t. $\boldsymbol{Z}{\Omega}=\boldsymbol{X}_{\Omega}$
 $\ell_p$Norm minimization  use a different norm than Frobenius to handle specific types of noise
 Adaptive outlier pruning (yan, yang, & osher, 2013)  better handles outliers
 Exact matrix completion via convex optimization (candes & recht, 2012)
 algorithms
 gradientbased
 gradient descent
 accelerated proximal descent
 bregman iteration
 nongradient
 block coordinate descent
 ADMM: alterntating direction method of multipliers
 gradientbased
 formulations

Newer approaches
 Simple, Fast, and Flexible Framework for Matrix Completion with Infinite Width Neural Networks (radhakrishnan…belkin, uhler, 2022)  use NTK for FCN/CNN to do matrix completion
 Graph Convolutional Matrix Completion (van den berg, kipf, & welling, 2017)
 Transformer methods for image inpainting (e.g. maskaware tranfsormer, 2022)
preprocessing
 often good to discretize/binarize features
 e.g. from genomics

whitening

get decorrelated features $Z$ from inputs $X$

$W=$ whitening matrix , selected based on problem goals:
 PCA: Maximal compression of $\mathbf{X}$ in $\mathbf{Z}$
 ZCA: Maximal similarity between $\mathbf{X}$ and $\mathbf{Z}$
 Cholesky: Inducing structure: $\operatorname{Cov}(X, Z)$ is lowertriangular with positive diagonal elements
 $W$ is constrained as to enforce $\Sigma_{Z}=I$
 PCA: Maximal compression of $\mathbf{X}$ in $\mathbf{Z}$

principles
breiman
 conversation
 moved sf > la > caltech (physics) > columbia (math) > berkeley (math)
 info theory + gambling
 CART, ace, and prob book, bagging
 ucla prof., then consultant, then founded stat computing at berkeley
 lots of cool outside activities
 ex. selling ice in mexico
 2 cultures paper
 generative  data are generated by a given stochastic model
 stat does this too much and needs to move to 2
 ex. assume y = f(x, noise, parameters)
 validation: goodnessoffit and residuals
 predictive  use algorithmic model and data mechanism unknown
 assume nothing about x and y
 ex. generate P(x, y) with neural net
 validation: prediction accuracy
 axioms
 Occam
 Rashomon  lots of different good models, which explains best?  ex. rf is not robust at all
 Bellman  curse of dimensionality
 might actually want to increase dimensionality (ex. svms embedded in higher dimension)
 industry was problemsolving, academia had too much culture
 generative  data are generated by a given stochastic model
box + tukey
 questions
 what points are relevant and irrelevant today in both papers?
 relevant
 box
 thoughts on scientific method
 solns should be simple
 necessity for developing experimental design
 flaws (cookbookery, mathematistry)
 tukey
 separating data analysis and stats
 all models have flaws
 no best models
 lots of goold old techniques (e.g. LSR)  irrelevant
 some of the data techniques (I think)
 tukey multipleresponse data has been better attacked (graphical models)
 how do you think the personal traits of Tukey and Box relate to the scientific opinions expressed in their papers?
 probably both pretty critical of the science at the time
 box  great respect for Fisher
 both very curious in different fields of science
 what is the most valuable msg that you get from each paper?
 box  data analysis is a science
 tukey  models must be useful
 no best models
 find data that is useful
 no best models
 what points are relevant and irrelevant today in both papers?
 box_79 “science and statistics”
 scientific method  iteration between theory and practice
 learning  discrepancy between theory and practice
 solns should be simple
 fisher  founder of statistics (early 1900s)
 couples math with applications
 data analysis  subiteration between tentative model and tentative analysis
 develops experimental design
 flaws
 cookbookery  forcing all problems into 1 or 2 routine techniques
 mathematistry  development of theory for theory’s sake
 scientific method  iteration between theory and practice
 tukey_62 “the future of data analysis”
 general considerations
 data analysis  different from statistics, is a science
 lots of techniques are very old (LS  Gauss, 1803)
 all models have flaws
 no best models
 must teach multiple data analysis methods
 spotty data  lots of irregularly nonconstant variability
 could just trim highest and lowest values
 winzorizing  replace suspect values with closest values that aren’t
 must decide when to use new techniques, even when not fully understood
 want some automation
 FUNOP  fulll normal plot
 can be visualized in table
 could just trim highest and lowest values

spotty data in more complex situations
 FUNORFUNOM
 multipleresponse data
 understudied except for factor analysis
 multipleresponse procedures have been modeled upon how early singleresponse procedures were supposed to have been used, rather than upon how they were in fact used
 factor analysis
 reduce dimensionality with new coordinates
 rotate to find meaningful coordinates
 can use multiple regression factors as one factor if they are very correlated
 regression techniques always offer hopes of learning more from less data than do variancecomponent techniques

flexibility of attack
 ex. what unit to measure in
 general considerations
models
 normative  fully interpretable + modelled
 idealized
 probablistic
 ~mechanistic  somewhere in between
 descriptive  based on reality
 empirical
exaggerated claims
 video by Rob Kass
 concepts are ambiguous and have many mathematical instantiations
 e.g. “central tendency” can be mean or median
 e.g. “information” can be mutual info (reduction in entropy) or squared correlation (reduction in variance)
 e.g. measuring socioeconomic status and controlling for it
 regression “when controlling for another variable” makes causal assumptions
 must make sure that everything that could confound is controlled for
 Idan Segev: “modeling is the lie that reveals the truth”
 picasso: “art is the lie that reveals the truth”
 box: “all models are wrong but some are useful”  statistical pragmatism
 moves from true to useful  less emphasis on truth
 “truth” is contingent on the purposes to which it will be put
 the scientific method aims to provide explanatory models (theories) by collecting and analyzing data, according to protocols, so that
 the data provide info about models
 replication is possible
 the models become increasingly accurate
 scientific knowledge is always uncertain  depends on scientific method