P8111FinalProject
rk
May 13, 2015
The original accelerometer dataset consists of 4551 observations over 960 variables (961 if age is included). In building a regression model using the principle components as coefficients I started by first coverting the indepentent variables (activity) into principle components using the stats package in R and the function prcomp. Using PCA, I was able to reduce the dimension of the original dataset to one in which a couple components explained >90% of the variability in the dataset.
## loading the data
data = load("~/Desktop/P8111_FinalProjectData.RDA")
data = data.frame(age,activity)
pca1 = prcomp(activity,scale=TRUE)
#pca2 = princomp(activity,scores=TRUE,cor=TRUE)
#pca2$loadings[,1] ##loadings
#screeplot(pca2,type="line",main="Scree Plot")
#summary(pca1)
screeplot(pca1,type="line")
From the screeplot, we see that the first component explains roughly 60% of the variation in the data. Additionally, given the elbow in the plot at component 6 we can see that using the first 6 principle components explains >90% of the variation in the activity dataset. Thus in building a regression model with principal components I will chose the use the first 6 components in the model as it does a good job at explaining most of the variation in the data
library(pls)##
## Attaching package: 'pls'
##
## The following object is masked from 'package:stats':
##
## loadingspcrmod = pcr(age~activity,ncomp=7)
summary(pcrmod)## Data: X dimension: 4551 960
## Y dimension: 4551 1
## Fit method: svdpc
## Number of components considered: 7
## TRAINING: % variance explained
## 1 comps 2 comps 3 comps 4 comps 5 comps 6 comps 7 comps
## X 10.448 16.769 21.95 26.33 30.05 33.44 36.04
## age 4.266 6.876 17.67 21.84 29.09 29.58 29.58predact = predict(pca1,activity)
data2 = data.frame(age,predact)
## linmod1 = lm(age~predact,data=data2)
## summary(linmod1)
## AIC(linmod1)
## linmodx = lm(age~predact[,1],data=data2)
linmod2 = lm(age~PC1 + PC2 + PC3 + PC4 + PC5 + PC6, data=data2)
summary(linmod2)##
## Call:
## lm(formula = age ~ PC1 + PC2 + PC3 + PC4 + PC5 + PC6, data = data2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -20.107 -3.238 0.386 3.980 31.981
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 68.48275 0.08197 835.443 < 2e-16 ***
## PC1 0.41915 0.01054 39.773 < 2e-16 ***
## PC2 -0.28798 0.01208 -23.847 < 2e-16 ***
## PC3 -0.21365 0.01296 -16.486 < 2e-16 ***
## PC4 -0.22560 0.01360 -16.589 < 2e-16 ***
## PC5 -0.07005 0.01423 -4.924 8.78e-07 ***
## PC6 -0.04425 0.01446 -3.061 0.00222 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5.53 on 4544 degrees of freedom
## Multiple R-squared: 0.3754, Adjusted R-squared: 0.3746
## F-statistic: 455.2 on 6 and 4544 DF, p-value: < 2.2e-16AIC(linmod2)## [1] 28490.15PCA is a non parametric statistical technique that examines the interrelations amongst sets of variables. It chooses a variable system for the data set such that the greatest variance of the data lies on teh first first principal component (or eigenvector), the second greatest variance on the second principal component, and so on. The principal components are uncorrelated and thus when using it is possible to run a linear regression analysis using the components as predictors without having to deal with the issue of multicolinearity of the explanatory dataset. The correlations between the component and the original variables are known as the component loadings. The component loadings tell us how much of the variation in the variable is explained by the component. The greater the respective variable’s component loading, the more important the variable is to that principal component.
After running the PCA, I was able to reduce the 960 dimension dataset to just 6 based on the largest eigenvectors/values (the first 6 principal components). Based on the principal component regression model using the 6 components, we see that the highest loadings are for variables minute 168,171,175,176,and 178 to 192. Using the principal component regression model: Age ~ PC1 + PC2 + PC3 + PC4 + PC5 + PC6 resulted a statistically significant model predicting age on principal components 1 to 6 at the alpha = <0.05 level of significance. The principal component regression model explains 37.5% of the variance in age. The coefficient of principal component 1 is 0.4195, which is to suggest for every unit change in each independent variable (activity/ minute_xx) that went into component 1 there is a change of 0.4195 in age.
library(glmnet)## Loading required package: Matrix
## Loaded glmnet 1.9-8library(ggplot2)
library(reshape2)
library(dplyr)##
## Attaching package: 'dplyr'
##
## The following object is masked from 'package:stats':
##
## filter
##
## The following objects are masked from 'package:base':
##
## intersect, setdiff, setequal, unionlibrary(gridExtra)## Loading required package: gridlibrary(reshape2)
## building a lasso regresison model
## choosing lambda using cross validation
mod1 = lm(age~.,data=data)
y = data$age
X.des = model.matrix(mod1)[,-1]
set.seed(12)
grid.lam = seq(-7, 5, by = .5)
lam = 2^(grid.lam)
GROUP = sample(c(1:5), prob = rep(.2, 5), dim(data)[1], replace = TRUE)
MSE = matrix(NA, nrow = dim(data)[1], ncol = length(lam))
COEFS = matrix(NA, nrow = dim(X.des)[2], ncol = length(lam))
for(l in 1:length(lam)){
for(i in 1:5){
X.train = X.des[which(GROUP != i),]
X.valid = X.des[which(GROUP == i),]
y.train = y[which(GROUP != i)]
y.valid = y[which(GROUP == i)]
model.cur = glmnet(X.train, y.train, lambda = lam[l])
predictions = predict(model.cur, newx = X.valid)
MSE[which(GROUP == i),l] = (y.valid - predictions)^2
}
COEFS[,l] = coef(glmnet(X.des, y, lambda = lam[l]))[-1]
}
plot.dat = data.frame(grid = grid.lam, MSE = apply(MSE, 2, mean))p1 =ggplot(plot.dat, aes(x = grid, y = MSE)) + geom_path() +
labs(x = expression(log[2](lambda)), ylab = "CV MSE")
## coefficient plot for ridge regression
rownames(COEFS) = colnames(X.des)
colnames(COEFS) = grid.lam
plot.dat = melt(COEFS)
p2 =ggplot(plot.dat, aes(x = Var2, y = value, group = Var1, color = Var1)) + geom_path() +
labs(x = expression(log[2](lambda)), ylab = "Coefficient")
grid.arrange(p1, p2, nrow = 1, ncol = 2)
The above graphs show the lambda and coefficent variable selection when applying the lambda penalization for lasso regression based on cross validation for selection of lambda.
Lam.Final = lam[which(apply(MSE, 2, mean) == min(apply(MSE, 2, mean)))]
model.lasso = glmnet(X.des, y, lambda = Lam.Final)
(coef(model.lasso))## 961 x 1 sparse Matrix of class "dgCMatrix"
## s0
## (Intercept) 7.327327e+01
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## minute_.960 .I ran a lasso regression model using cross validation to choose a lambda for penalization. The lambda chosen from cross validation was 0.177. In the final lasso model there is 42.01% deviation explained by the model on the outcome age. The signifcant predictors based on the lasso model are shown above under coef(model.lasso). The lasso penalization causes several of the variables to be shrunke to 0 and thus allowing a method of variable selection from the activity dataset. Based on the lasso regression coefficient printout above, the non 0 values for activity variables seem to be predictive of age.
Overall, the principal component model seems to be a better fit given that it takes into account overall variation of each predictor for mutiple components that when taken in account for >90% of the total variation in activity.