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Descriptive Statistics quantitatively describe the main features of a collection of data. My salary is $45,000. It’s a middle salary in my company Staff. Jones Benefits are highly related to working age What should I make of all this???!!! How do salaries vary across the company? HR manager employee

Mean> mean(x); > mean(x,trim=a) Median> median(x) Mode> sort(table(x)) Standard deviation> sd(x) Variance > var(x) the median absolute deviation > mad(c(x)) interquartile range> IQR(x) Range> range(x) Descriptive Statistics in R

Data Dimensions > length(x) [1] > nrow(X) [1] 2030 > ncol(X) [1] > dim(X) [1] Matrix X ….

Vectorization in R Matrix X > apply( X, MARGIN=1, FUN= mean) > apply( X, MARGIN=2, FUN= mean)

boxplot(X) Good for small data sets Easy to compar e groups side b y side 1.5*IQR defines outlier

The Big Six Minimum, 1 st Q, Median, Mean, 3 rd Q, Maximu m > summary(X)

R tries to understand you > summary(X)

Histograms: > hist(X)

Correlation > cor(wt,mpg) [1] > plot(x=wt,y=mpg)

Scatterplot Matrix Iris dataset 150 flowers 5 variables Goingslo, flickrflickr

Scatterplot Matrix > pairs(data)

> coplot(lat ~ long | depth)

Linear Regression Why? What?  Prediction of future or unknown observations  Assessment of relationship between variables  General description of data structure

Variable Selection Why?  Simplification  Elimination of multicollinearity and noise  Time and money saving How?  Testing-based Variable Selection Methods - Backward, Forward, Stepwise  Criterion-based Procedures What?  AIC = n ln(RSS/n) + 2(p)

Example: U.S. State Fact and Figures Life Expectancy  Population, Income, Illiteracy, Murder, HS Grad, Frost, Area > g <- lm(Life.Exp ~ Population + Income + Illiteracy + Murder + HS.Grad + Frost + Area, data = statedata) > summary(g) Selected R code  Linear Regression  AIC > step(g) > anova(g) Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 7.094e e < 2e-16 *** Population 5.180e e Income e e Illiteracy 3.382e e Murder e e e-08 *** HS.Grad 4.893e e * Frost e e Area e e Analysis of Variance Table Response: Life.Exp Df Sum Sq Mean Sq F value Pr(>F) Population Income e-05 *** Illiteracy e-07 *** Murder e-08 *** HS.Grad ** Frost Area Residuals AIC = n ln(RSS/n) + 2(p)

Continued: U.S. State Fact and Figures Start: AIC= Life.Exp ~ Population + Income + Illiteracy + Murder + HS.Grad + Frost + Area Df Sum of Sq RSS AIC - Area Income Illiteracy <none> Population Frost HS.Grad Murder Step: AIC= Life.Exp ~ Population + Income + Illiteracy + Murder + HS.Grad + Frost Df Sum of Sq RSS AIC - Illiteracy Income <none> Population Frost HS.Grad Murder

Step: AIC= Life.Exp ~ Population + Murder + HS.Grad + Frost Df Sum of Sq RSS AIC <none> Population Frost HS.Grad Murder Coefficients: (Intercept) Population Murder HS.Grad Frost 7.103e e e e e-03 Effect on Response Variable of One Unit Change of Predict Variable

What is Principal Component Analysis (PCA)? Two general approaches of reducing variables : feature selection and feature extraction  Feature Selection : “Akaike Information Criterion”(AIC), BIC or Back-Substitution  Feature extraction : “Principal Component Analysis”(PCA) is most widely used  Create several artificial variables  Built-in functions in R = Convenient!

Actual Pima Data pregnantglucosediastolictricepsinsulinbmidiabetesagetest …. ( Imagine a data set with many more (~1000) columns ) (Imagine a Linear Regression: Which variables affect diabetes in what ways?)

PCA Example: Pima Indians The National Institute of Diabetes and Digestive and Kidney Diseases conducte d a study on 768 adult female Pima Indians living near Phoenix. 9 Variables (8 continuous, 1 categorical)  pregnant: Number of times pregnant  Glucose : Plasma glucose concentration at 2 hours in an oral glucose tolerance test  Diastolic : Diastolic blood pressure (mm Hg)  Triceps : Triceps skin fold thickness (mm)  Insulin : 2-Hour serum insulin (mu U/ml)  Bmi : Body mass index (weight in kg/(height in metres squared))  Diabetes : Diabetes pedigree function  Age : Age (years)  Test : diabetes (coded 0 if negative, 1 if positive) Next Slide: PCA Implementation

What principal components might look like: PC1 : 1*Insulin *Glucose +.. PC2 : 1*Glucose *Age *DiastolicBP +.. PC3 : 0.92 * DiastolicBP *Triceps  Principal components : What are they composed of? (less important)  Difference with Linear Regression

- Goal: obtain summary about data in lower dimensions - - How many dimensions? - R code in the next slide:

Brief : R-Code > data.pca <- prcomp(data[,-9]); summary(data.pca); Importance of components: PC1 PC2 PC3 PC4 PC5 PC6 PC7 Standard deviation Proportion of Variance Cumulative Proportion > data.pca Rotation: PC1 PC2 PC3 PC4 PC5 PC6 PC7 pregnant e e+00 glucose e e-04 Diastolic e e-03 triceps e e-04 insulin e e-03 bmi e e-03 age e e-01 > barplot(totalrep, main="Representation of Principal Components", xlab="Principal Component", ylab="% of Total Variance") > biplot(data.pca, xlabs=rep('+',768), xlim = c(-0.05,0.3), ylim = c(-0.15,0.12)); abline(h=0,v=0);