Test Time Series Second Order Curvature Significance

test.ts.2nd.order.sig <- function(x, graph=F)
{ 
    ## Purpose: Test Time Series 2nd Order Significance. 
    ## Arguments:
    ##    x: a numeric vector (time series)
    ## Return:  P-value for testing the 2nd order significance. 
    ## Author: Feiming Chen, Date: 10 Apr 2017, 14:57
    ## ________________________________________________

    d <- data.frame(Response = x, Time = seq(x))
    m1 <- lm(Response ~ Time, d)
    m2 <- lm(Response ~ Time + I(Time^2), d)
    a <- anova(m1, m2)
    model.compare.P.value <- a[["Pr(>F)"]][2]
    
    if (graph) {
        plot(d$Time, d$Response, xlab="Time", ylab="Response", type="b")
        lines(d$Time, fitted(m1), col="red")
        lines(d$Time, fitted(m2), col="blue")
        title(main=paste("Model Comparison P-value =", round(model.compare.P.value, 3)))
    }

    model.compare.P.value
}
if (F) {                                # Unit Test
    x <- rnorm(100)
    test.ts.2nd.order.sig(x, graph = T)
    r <- replicate(10000, test.ts.2nd.order.sig(rnorm(20))) # should follow a uniform distribution. 
    ## summary(r)
    ##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
    ## 0.00011 0.25800 0.50600 0.50500 0.75500 1.00000 
}