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Author(s): Alex Ocampo, Chong Zhang, Sangwon Hyun, Yiqun Hu, 2012
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Descriptive Statisticsquantitatively 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
manageremployee
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] 1000-------------------------> nrow(X)
[1] 2030> ncol(X)
[1] 100000> dim(X)
[1] 2034 100000
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 compare groups side by side
• 1.5*IQR defines outlier
The Big Six
Minimum, 1st Q, Median, Mean, 3rd Q, Maximum
> summary(X)
R tries to understand you
> summary(X)
Histograms: > hist(X)
Correlation
> cor(wt,mpg) [1] -0.8676594> plot(x=wt,y=mpg)
Scatterplot Matrix
• Iris dataset• 150 flowers• 5 variables Goingslo, flickr
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+01 1.748e+00 40.586 < 2e-16 ***Population 5.180e-05 2.919e-05 1.775 0.0832 . Income -2.180e-05 2.444e-04 -0.089 0.9293 Illiteracy 3.382e-02 3.663e-01 0.092 0.9269 Murder -3.011e-01 4.662e-02 -6.459 8.68e-08 ***HS.Grad 4.893e-02 2.332e-02 2.098 0.0420 * Frost -5.735e-03 3.143e-03 -1.825 0.0752 . Area -7.383e-08 1.668e-06 -0.044 0.9649
Analysis of Variance TableResponse: Life.Exp Df Sum Sq Mean Sq F value Pr(>F) Population 1 0.4089 0.4089 0.7372 0.395434 Income 1 11.5946 11.5946 20.9028 4.218e-05 ***Illiteracy 1 19.4207 19.4207 35.0116 5.228e-07 ***Murder 1 27.4288 27.4288 49.4486 1.308e-08 ***HS.Grad 1 4.0989 4.0989 7.3895 0.009494 ** Frost 1 2.0488 2.0488 3.6935 0.061426 . Area 1 0.0011 0.0011 0.0020 0.964908 Residuals 42 23.2971 0.5547AIC = n ln(RSS/n) + 2(p)
Continued: U.S. State Fact and Figures
Start: AIC=-22.18Life.Exp ~ Population + Income + Illiteracy + Murder + HS.Grad + Frost + Area
Df Sum of Sq RSS AIC- Area 1 0.0011 23.298 -24.182- Income 1 0.0044 23.302 -24.175- Illiteracy 1 0.0047 23.302 -24.174<none> 23.297 -22.185- Population 1 1.7472 25.044 -20.569- Frost 1 1.8466 25.144 -20.371- HS.Grad 1 2.4413 25.738 -19.202- Murder 1 23.1411 46.438 10.305
Step: AIC=-24.18Life.Exp ~ Population + Income + Illiteracy + Murder + HS.Grad + Frost
Df Sum of Sq RSS AIC- Illiteracy 1 0.0038 23.302 -26.174- Income 1 0.0059 23.304 -26.170<none> 23.298 -24.182- Population 1 1.7599 25.058 -22.541- Frost 1 2.0488 25.347 -21.968- HS.Grad 1 2.9804 26.279 -20.163- Murder 1 26.2721 49.570 11.569
Continued: U.S. State Fact and Figures
Step: AIC=-28.16Life.Exp ~ Population + Murder + HS.Grad + Frost
Df Sum of Sq RSS AIC<none> 23.308 -28.161- Population 1 2.064 25.372 -25.920- Frost 1 3.122 26.430 -23.877- HS.Grad 1 5.112 28.420 -20.246- Murder 1 34.816 58.124 15.528
Coefficients:(Intercept) Population Murder HS.Grad Frost 7.103e+01 5.014e-05 -3.001e-01 4.658e-02 -5.943e-03
Effect on Response Variable ofOne 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
pregnant
glucosediastoli
ctriceps insulin bmi
diabetes
age test
1 6 148 72 35 0 33.6 0.627 50 12 1 85 66 29 0 26.6 0.351 31 03 8 183 64 0 0 23.3 0.672 32 14 1 89 66 23 94 28.1 0.167 21 05 0 137 40 35 168 43.1 2.288 33 16 5 116 74 0 0 25.6 0.201 30 0
….
( 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 conducted 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 + 0.01*Glucose + ..PC2 : 1*Glucose + 0.12*Age +
0.12*DiastolicBP + ..PC3 : 0.92 * DiastolicBP + 0.31*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:
-0.30 -0.25 -0.20 -0.15 -0.10 -0.05 0.00
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-4000 -3000 -2000 -1000 0
-15
00
-10
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-50
00
50
01
00
0
pregnant
glucose
diastolictricepsinsulin
bmiage
Brief : R-Code
> data.pca <- prcomp(data[,-9]); summary(data.pca);Importance of components: PC1 PC2 PC3 PC4 PC5 PC6
PC7Standard deviation 116.002 30.5411 19.7630 14.0777 10.6155 6.76973 2.78575Proportion of Variance 0.889 0.0616 0.0258 0.0131 0.00744 0.00303 0.00051Cumulative Proportion 0.889 0.950 0.976 0.9890 0.996 0.999 1.00000 > data.pca
Rotation: PC1 PC2 PC3 PC4 PC5 PC6 PC7pregnant 0.002 -0.02 0.02 0.05 2e-01 -0.005 -1e+00glucose -0.098 -0.97 -0.14 -0.12 -9e-02 0.051 -9e-04Diastolic -0.016 -0.14 0.92 0.26 -2e-01 0.076 1e-03triceps -0.061 0.06 0.31 -0.88 3e-01 0.221 4e-04insulin -0.993 0.09 -0.02 0.07 -2e-04 -0.006 -1e-03bmi -0.014 -0.05 0.13 -0.19 2e-02 -0.971 3e-03age 0.004 -0.14 0.13 0.30 9e-01 -0.015 2e-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);
Representation of Principal Components
Principal Component
% o
f To
tal V
ari
an
ce
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