05-models-for-nonstationary-time-series.Rmd

# Models for Nonstationary Time Series

## 5.1

## 5.2

## 5.3

## 5.4

## 5.5

## 5.6

## 5.7

## 5.8

## 5.9

## 5.10

## Winnebago

### a {-}

The plot in \@ref(fig:winnebago2) has a trend that seems almost exponential,
as well as a seasonal pattern by which sales seem to slump later in the year
and surge in the spring months.

```{r winnebago2, fig.cap = "Monthly unit sales of recreational vehicles."}
data(winnebago)
winnebago <- as.xts(winnebago)
xyplot(winnebago, ylab = "Sales")
```

### b {-}

We take the log of sales and it looks like we have made the trend linear 
in time \@ref(fig:winnebago-log)

```{r winnebago-log, fig.cap = "Logged monthly sales."}
winnebago_log <- log(winnebago)
xyplot(winnebago_log, ylab = expression(log(sales)))
```

## c {-}

We produce the result in figure \@ref(fig:winnebago-comp). The patterns are
similar but it seems like the fractional relative changes are greater in
magnitude for the larger values of sales

```{r winnebago-comp, fig.cap = "Comparison between differences of logs and fractional relative changes."}
winnebago_frac <- diff(winnebago) / lag(winnebago, 1)
winnebago_logdiff <- diff(log(winnebago))
winnebago_comp <- merge.xts(frac = winnebago_frac, logdiff = winnebago_logdiff)
xyplot(winnebago_comp, screens = c(1, 1), auto.key = TRUE,
       ylab = "Sales",
       col = c("darkorange2", "navy"))
```

## Standard & Poor

### a {-}

There is an exponential trend in the time series (Figure \@ref(fig:sp)) that,
however, seems to perhaps level off after 1970 or so.

```{r sp, fig.cap = "Quarterly values of the Standard and Poor index."}
data(SP)
SP <- as.xts(SP)
xyplot(SP, grid = TRUE)
```

### b {-}

In Figure \@ref(fig:sp-log) we've transformed the time series of the S&P index
by taking the log. The series is "more" linear but there is still an
exponential pattern.

```{r sp-log, fig.cap = "Logged values of the Standard and Poor's index."}
sp_log <- log(SP)
xyplot(sp_log, ylab = expression(log(value)), grid = TRUE)
```

### c {-}

Next, we compute the fractional relative changes and the differences of 
natural logartihms (Figure \@ref(fig:sp-comp)). We see that there is little
difference between the series and only really so for the higher numbers of
sales.

```{r sp-comp, fig.cap = "Differences in logs and fractional relative differences."}
SP_frac <- diff(SP) / lag(SP, 1)
SP_logdiff <- diff(log(SP))
SP_comp <- merge.xts(frac = SP_frac, logdiff = SP_logdiff)
xyplot(SP_comp, screens = c(1, 1), auto.key = TRUE,
       ylab = "Sales",
       col = c("darkorange2", "navy"))
```

## Air passengers

### a {-}

We plot the monthly intervational airline passenger counts in
Figure\@ref(fig:airpass) and note that we have a strong seasonal trend and
perhaps a slight exponential increase. Variance seems to increase as well.

```{r airpass, fig.cap = "Monthly airline passenger totals."}
data(airpass)
airpass <- as.xts(airpass)
xyplot(airpass, ylab = "Passengers")
```

### b {-}

As in previous exercises, we take the natural log of the dependent variable,
in this case passenger totals, and note that we have made the trend linear
(or maybe exponentially decreasing?) but notably have stabilized the variance of
the series.

```{r airpass-log, fig.cap = "Logged monthly airline passenger totals."}
airpass_log <- log(airpass)
xyplot(airpass_log, ylab = expression(log(airpass)))
```

### c {-}

Next, we compute the fractional relative changes and the differences of 
natural logartihms (Figure \@ref(fig:sp-comp)). We see that there is little
difference between the series and only really so for the higher numbers of
sales.

```{r airpass-comp, fig.cap = "Differences in logs and fractional relative differences."}
airpass_frac <- diff(airpass) / lag(airpass, 1)
airpass_logdiff <- diff(log(airpass))
airpass_comp <- merge.xts(frac = airpass_frac, logdiff = airpass_logdiff)
xyplot(airpass_comp, screens = c(1, 1), auto.key = TRUE,
       ylab = "Sales",
       col = c("darkorange2", "navy"))
```

## Rainfall

### a {-}

In this exercise we consider annual rainfall data for Los Angeles. We use `TSA::BoxCox.ar()`
to train a power model to the time series (Figure \@ref(fig:larain)), optimizing
via lok-likelihood maximization. 

```{r larain, fig.cap = "Box-Cox training on the annual rainfall data."}
data(larain)
larain <- as.xts(larain)
obj <- BoxCox.ar(larain)
lambda <- obj$lambda[which.max(obj$loglike)]
```

The best value of $\lambda$ is `r lambda`.

### b {-}

We apply the power transformation and see the results of Q-Q plots for both
the untransformed and the transformed time series in Figure \@ref(fig:larain-qq)

```{r larain-qq, fig.cap = "Q-Q plots of untransformed and transformed time series data.", warning = FALSE}
larain_trans <- larain ^ lambda
xyplot(list(Untransformed = larain, Transformed = larain_trans),
       FUN = qqmath, y.same = FALSE)
```

### d {-}

Next we plot $Y_t$ versus $Y_{t-1}$ in Figure \@ref(fig:lahrain-laglag) --
no correlation is evident from this picture, nor would we expect that
our transformatin would induce any given that it is not at all based on the
previous values.

```{r lahrain-laglag, fig.cap = "Lag 0 against lag 1."}
xyplot(zlag(larain_trans) ~ larain_trans,
       ylab = "Lag 1",
       xlab = "Lag 0")
```