#3.6a)

library(astsa)
## Warning: package 'astsa' was built under R version 4.1.3
n<-length(varve)
var(varve)
## [1] 412.6488
var(varve[1:(n/2)])
## [1] 133.4574
var(varve[((n+2)/2):n])
## [1] 594.4904

When we look at the varance of the varve dataset, the first half of the data has a variance of 133.5 while the second half of the dataset has a variance of 594.5. This suggests that there is heteroscedasticity in the varve data set and that the variance of the data is increasing over time.

If we look at the graph and qqplot of the varve data as well as the log transformed varve data, we see that the log transformed data produces a much more normal fit and periods of increased variability are less obvious in the plot. For the orginal data, there is a noticeable period of increased variability from about 350 to 500. The qqplot also suggests that the orginal varve data is skewed right. The log transformed data produces a much more normal qqplot. These observations are supported by the histograms of the varve and log(varve) data; the varve histogram is strongly skewed to the right while the log(varve) histogram shows a mostly symmetrical and Normal distribution of the data.

#3.6b)

The plot of the varve data overall looks very similar to the plot of the global temperature records; there appears to be an underlying trend overlaid with white noise. The varve data is also increasing from about 100 to 400 years, which matches with the increasing trend in the global records data. The log(varve) data almost appears as though it is part of a larger cyclic pattern, but we would need more data to check this theory.

#3.6c)

There appears to be a high level of correlation between varve values for several intervals. From about lag(0) to lag(100), there is a significant positive correlation, while from about lag(150) to lag(320) there is a significant negative correlation. Overall, the graph reveals strong evidence of correlation between varve values that are about 300 years or less apart from each other.

#3.6d)

The difference of the log transformed varve data appears to create a reasonably stationary series. When we look at the plot of the data, the mean appears to be constant and centered at zero so the mean is not dependent on the time. The variability also appears to be relatively constant. In constrast, the orginal and log data both had underlying trends that made the mean nonconstant and dependent on time. When we look at the sample ACF for the differenced log data, it supports the conclusion that the values are not significantly correlated - less than 5% of lag values have a significant ACF value. We also do not see any patterns in the ACF plot that would suggest that the series is nonstationary.

A practical interpretation of the differenced log data is as the percentage change. That is, it gives the predicted % increase in varve for a 1% increase in the time t.

#3.7)

Using a moving average smoother of 10 years, we can more clearly see the trends in both graphs - much of the year-to-year fluctuation has been smoothed out and we con see more long-term temperature trends. Using kernel smoothing also set to a span of 10 years, we get much of the same effect and see an even smoother curve with fewer year-to-year fluctuations. Finally, the loess curve and confidence interval show the smoothest fit to the data with all year-to-year fluctations smoothed away. It is very clear to see the decade-by-decade trends in global temperature using the loess curve. For the land temperature we see an increase from about 1180 to 1930, no change for about 1930 to 1960, and then a steeper increase from about 1960 to 2018. For the ocean temperature we see a decrease from about 1880 to 1910, and then a steady increase from about 1910 to 2018.

Part II

#IIa)

Yes, the two plots are similar - both appear to have a mean of zero and a relatively constant variance. The main difference is that the UIN series appears to have a smaller variance than the 1.2 model, which is expected since the the \(\psi\) values are smaller in magnitude for the UIN series.

#IIb)

AR1=c(1,-1.5,.75)
polyroot(AR1)
## [1] 1+0.57735i 1-0.57735i
AR2=c(1,-2,2)
polyroot(AR2)
## [1] 0.5+0.5i 0.5-0.5i
AR3=c(1,-.5,.5)
polyroot(AR3)
## [1] 0.5+1.322876i 0.5-1.322876i
1^2+(.57735)^2
## [1] 1.333333
.5^2+.5^2
## [1] 0.5
.5^2+1.322876^2
## [1] 2.000001

The roots of the origial model 1.2 and the inverse UIN model are ouside of the unit circle, while the original UIN model has both roots inside the unit circle.

#IIc) Both the original model and the inverse UIN model are causal and stationary, while the original UIN model is neither causal nor stationary. We can see this confirmed in the plots of all three functions, where the the original and inverse UIN series both have a mean of zero and a constant variance, as well as roots that are outside of the unit cirlce. In contrast the original UIN series has a clearly nonconstant variance that is dependent on time which shows that this series is nonstationary, and has roots that are inside of the unit circle.