chapter_7_lesson_1.qmd

---
title: "Introduction to Non-stationary Models and Differencing"
subtitle: "Chapter 7: Lesson 1"
format: html
editor: source
sidebar: false
---



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```{r}
#| include: false
source("common_functions.R")
```

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## Learning Outcomes

{{< include outcomes/_chapter_7_lesson_1_outcomes.qmd >}}




## Preparation

-   Read Sections 7.1-7.2
-   Read [Prof. Frenzel's Blog Post](https://prof-frenzel.medium.com/kb-time-series-analysis-part-4-autoregressive-models-ed824838bd4c)


## Learning Journal Exchange (10 min)

-   Review another student's journal

-   What would you add to your learning journal after reading another student's?

-   What would you recommend the other student add to their learning journal?

-   Sign the Learning Journal review sheet for your peer


## Class Activity: Non-seasonal ARIMA Models (15 min)

### Effect of Differencing


In [Chapter 4 Lesson 2](https://byuistats.github.io/timeseries/chapter_4_lesson_2.html#McDonalds), 
we found that if we compute the first difference of the price of McDonald's stock from July 2020 through December 2023, the differences can be modeled as white noise.
Sometimes differencing can remove trends.

Consider the case of a random walk and a linear trend with white noise errors

#### Random Walk

<!-- Check Your Understanding -->

::: {.callout-tip icon=false title="Check Your Understanding"}

Consider the random walk

$$
  x_t = x_{t-1} + w_t
$$

where $\{w_t\}$ is a white noise process. 

-   What is the model for the first differences of this time series?

<!-- White noise -->

:::


#### Linear Trend with White Noise Errors

<!-- Check Your Understanding -->

::: {.callout-tip icon=false title="Check Your Understanding"}

Consider a time series with a linear trend and white noise errors.

$$
  x_t = a + bt + w_t
$$

where $\{w_t\}$ is a white noise process. 

-   What is the model for the first differences of this time series?
<!-- $\nabla x_t = x_t - x_{t-1} = w_t$, which is white noise. -->
-   What is the model obtained by subtracting $a+bt$ from this series?
<!-- $\nabla x_t = x_t - x_{t-1} = b + w_t - w_{t-1}$, which is an $MA(1)$ process -->
-   What are some potential concerns of using differencing to eliminate a deterministic trend?
<!-- It can lead to an unnecessarily complicated model. -->

:::


### Fitting an ARIMA Model when the Difference Model has a Non-Zero Mean

(See the last sentence in the first paragraph on page 138.)

### Differencing a Time Series or the Logarithm of a Time Series

If the difference of a time series demonstrates an increasing trend, taking the logarithm before differencing can eliminate the increasing variation in the differences. As an example, consider the Australian electricity production series given in the book.

```{r}
#| code-fold: true
#| code-summary: "Show the code"
#| output: false

pacman::p_load("tsibble", "fable", "feasts",
    "tsibbledata", "fable.prophet",
    "tidyverse", "patchwork")
cbe <- read_table("https://byuistats.github.io/timeseries/data/cbe.dat") |>
  select(elec) |>
  mutate(
    date = seq(
      ymd("1958-01-01"),
      by = "1 months",
      length.out = n()),
      year_month = tsibble::yearmonth(date)) |>
  as_tsibble(index = year_month)

cbe |>
  mutate(
    `Diff series` = elec - lag(elec),
    `Diff log-series` = log(elec) - lag(log(elec))) |>
  pivot_longer(
    cols = all_of(c("elec", "Diff series", "Diff log-series"))) |>
  mutate(name = factor(name, levels =c("elec","Diff series", "Diff log-series"))) |>
  ggplot(aes(x = date, y = value)) +
  geom_line() +
  facet_wrap(~name, ncol = 1, scales = "free", strip.position = "left") +
  labs(x = "Time", y = "") +
  scale_x_date(breaks = "5 years", date_labels = "%Y") +
  theme_bw()
```

```{r}
#| label: fig-BooksFigure7Dot1
#| fig-cap: "Plot of Australian electricity production, first differences, and first differences of the logarithm of the series"
#| echo: false

cbe |>
  mutate(
    `Diff series` = elec - lag(elec),
    `Diff log-series` = log(elec) - lag(log(elec))) |>
  pivot_longer(
    cols = all_of(c("elec", "Diff series", "Diff log-series"))) |>
  mutate(name = factor(name, levels =c("elec","Diff series", "Diff log-series"))) |>
  ggplot(aes(x = date, y = value)) +
  geom_line() +
  facet_wrap(~name, ncol = 1, scales = "free", strip.position = "left") +
  labs(x = "Time", y = "") +
  scale_x_date(breaks = "5 years", date_labels = "%Y") +
  theme_bw()
```



### Integrated Time Series of Order d

::: {.callout-note icon=false title="Definition of an Integrated Series of Order $d$, $I(d)$"}

We say that a time series is **integrated of order d** if the $d^{th}$ difference of $\{x_t\}$ is a white noise process $\{w_t\}$. Expressed differently, we write this as ${\nabla^d x_t = w_t}$. We denote an integrated time series of order $d$ as $I(d)$.

:::

Recall that $\nabla^d \equiv \left( 1 - \mathbf{B} \right)^d$.
So, either of the following can be used to indicate an integrated time series of order $d$:

\begin{align*}
  \nabla^d x_t &= w_t \\
  ~\\
  \left( 1 - \mathbf{B} \right)^d x_t &= w_t 
\end{align*}


#### Special Case: $I(1)$

<!-- Check Your Understanding -->

::: {.callout-tip icon=false title="Check Your Understanding"}

-   What model is given by the special case $I(1)$?
<!-- Random walk -->

:::


### Second-Order Differencing and Lagged Differences

A linear trend can be removed by first-order differencing. A curved trend can sometimes be eliminated by second order differencing. 

In some cases, a lagged difference is more appropriate. For example, if you have monthly data and need to remove additive seasonal effects, you may want to take a difference with a lag of 12. This subtracts sequential January observations from each other. This models the year-over-year growth. 

Notice that taking a lag 12 difference 

$$  
  \left( 1 - \mathbf{B}^{12} \right) x_t  = x_t - x_{t-12}
$$

is very different from taking the twelfth differences 

\begin{align*}
  \nabla^{12} x_t 
    &= \left( 1 - \mathbf{B} \right)^{12} x_t \\
    &= x_t - 12 x_{t-1} + 66 x_{t-2} - 220 x_{t-3} + 495 x_{t-4} - 792 x_{t-5} + 924 x_{t-6} \\
    & ~~~~~~~~~~~~~~~~~~~ - 792 x_{t-7} + 495 x_{t-8} - 220 x_{t-9} + 66 x_{t-10} - 12 x_{t-11} + x_{t-12}
\end{align*}


### ARIMA Process

#### ARIMA 

::: {.callout-note icon=false title="Definition of an ARIMA Process"}

A time series is said to follow an **$ARIMA(p,d,q)$ process** if the $d^{th}$ differences of the time series follow an $ARMA(p,q)$ process.

:::

Suppose we let $y_t = \left( 1 - \mathbf{B} \right)^d x_t$. The series $\{y_t\}$ follows an $ARMA(p,q)$ process if
$\theta_p \left(\mathbf{B} \right) y_t = \phi_q \left(\mathbf{B} \right)w_t$.

Substituting, we find that $\{x_t\}$ follows an $ARIMA(p,d,q)$ process if

$$
    \theta_p \left(\mathbf{B} \right) \left( 1 - \mathbf{B} \right)^d x_t = \phi_q \left(\mathbf{B} \right) w_t
$$

where $\theta_p \left(\mathbf{B} \right)$ and $\phi_q \left(\mathbf{B} \right)$ are polynomials of orders $p$ and $q$, respectively.


#### Special Case: $IMA(d,q)$ Process

::: {.callout-note icon=false title="Definition of an IMA Process"}

A time series $\{x_t\}$ follows an **$IMA(d,q)$ process** if it can be expressed as:

$$
    \left( 1 - \mathbf{B} \right)^d x_t = \phi_q \left(\mathbf{B} \right) w_t
$$

Note that $IMA(d,q) \equiv ARIMA(0,d,q)$.

:::

<!-- Check Your Understanding -->

::: {.callout-tip icon=false title="Check Your Understanding"}

-   Solve for $x_t$ in an $IMA(1,1)$ process.
<!-- $x_t = x_{t-1} + w_t + \beta w_{t-1}$ -->

:::

#### Special Case: $ARI(p,d)$ Process

::: {.callout-note icon=false title="Definition of an ARI Process"}

A time series $\{x_t\}$ follows an **$ARI(p,d)$ process** if it can be expressed as:

$$
    \theta_p \left(\mathbf{B} \right) \left( 1 - \mathbf{B} \right)^d x_t = w_t
$$

Note that $ARI(p,d) \equiv ARIMA(p,d,0)$.

:::

<!-- Check Your Understanding -->

::: {.callout-tip icon=false title="Check Your Understanding"}

-   Solve for $x_t$ in an $ARI(1,1)$ process.
<!-- $x_t = \alpha x_{t-1} + x_{t-1} - \alpha x_{t-2} + w_t $ -->

:::


### Simulating an ARIMA Process

We can simulate data from the ARIMA process

$$
  x_t = 0.5 x_{t-1} + x_{t-1} - 0.5 x_{t-2} + w_t + 0.3 w_{t-1}
$$

using the following R code.

```{r}
set.seed(1)
n <- 10000
x <- rnorm(n)
w <- rnorm(n)
for (i in 3:n) {
  x[i] <- 0.5 * x[i - 1] + x[i - 1] - 0.5 * x[i - 2] + w[i] + 0.3 * w[i - 1]
}
arima(x, order = c(1, 1, 1))
```

This is an ARIMA(1,1,1) process with parameters $\alpha = 0.5$ and $\beta = 0.3$.


<!-- Check Your Understanding -->

::: {.callout-tip icon=false title="Check Your Understanding"}

-   Modify the code above to simulate from an $ARIMA(2,1,2)$ process with parameters $\alpha_1 = 0.5$, $\alpha_2 = 0.2$, $\beta_1 = 0.4$, and $\beta_2 = 0.1$.

<!-- $$(1 - \alpha_1 B - \alpha_2 B^2) (1-B) x_t = (1 + \beta_1 B + \beta_2 B^2) w_t$$ -->

:::



## Class Activity: Fitting an ARIMA Process - Exchange Rates (10 min)

The data file [exchange_rates.parquet](https://byuistats.github.io/timeseries/data/exchange_rates.parquet) gives the exchange rates for foreign currencies. The daily-observed values in the time series are the amount in the foreign currency equivalent to one U. S. dollar. We will consider the exchange rates to convert one dollar into Euros.

```{r}
#| label: tbl-exchangeRatesTS
#| tbl-cap: "Select values of the time series representing the exchange rate to convert US$1 into Euros"
#| code-fold: true
#| code-summary: "Show the code"

exchange_ts <- rio::import("data/exchange_rates.parquet") |>
  filter(currency == "USD.EUR") |>
  as_tsibble(index = date) |>
  na.omit()

exchange_ts |>
  display_partial_table(6,3)
```


```{r}
#| label: fig-exchangeRateTimePlot
#| fig-cap: "Time plot of the exchange rate to convert US$1 into Euros"
#| code-fold: true
#| code-summary: "Show the code"
 
exchange_ts |>
  autoplot(.vars = rate) + labs(title = exchange_ts$currency[1])
```


```{r}
#| label: fig-exchangeRateACFpacf
#| fig-cap: "Correlogram and partial correlogram for the time series representing the exchange rate to convert US$1 into Euros"
#| code-fold: true
#| warning: false
#| code-summary: "Show the code"

acf_plot <- exchange_ts |> select(rate) |> ACF() |> autoplot(var = .resid)

pacf_plot <- exchange_ts |> select(rate) |> PACF() |> autoplot(var = .resid)

acf_plot | pacf_plot

```



```{r}
# Fit the ARIMA Model

exchange_model <- exchange_ts |>
  model(
    auto = ARIMA(rate ~ 1 + pdq(0:2,0:1,0:2) + PDQ(0, 0, 0)),
    
    a000 = ARIMA(rate ~ 1 + pdq(0,0,0) + PDQ(0, 0, 0)),
    a001 = ARIMA(rate ~ 1 + pdq(0,0,1) + PDQ(0, 0, 0)),
    a002 = ARIMA(rate ~ 1 + pdq(0,0,2) + PDQ(0, 0, 0)),
    a100 = ARIMA(rate ~ 1 + pdq(1,0,0) + PDQ(0, 0, 0)),
    a101 = ARIMA(rate ~ 1 + pdq(1,0,1) + PDQ(0, 0, 0)),
    a102 = ARIMA(rate ~ 1 + pdq(1,0,2) + PDQ(0, 0, 0)),
    a200 = ARIMA(rate ~ 1 + pdq(2,0,0) + PDQ(0, 0, 0)),
    a201 = ARIMA(rate ~ 1 + pdq(2,0,1) + PDQ(0, 0, 0)),
    a202 = ARIMA(rate ~ 1 + pdq(2,0,2) + PDQ(0, 0, 0)),
    
    a011 = ARIMA(rate ~ 1 + pdq(0,1,1) + PDQ(0, 0, 0)),
    a012 = ARIMA(rate ~ 1 + pdq(0,1,2) + PDQ(0, 0, 0)),
    a110 = ARIMA(rate ~ 1 + pdq(1,1,0) + PDQ(0, 0, 0)),
    a111 = ARIMA(rate ~ 1 + pdq(1,1,1) + PDQ(0, 0, 0)),
    a112 = ARIMA(rate ~ 1 + pdq(1,1,2) + PDQ(0, 0, 0)),
    a210 = ARIMA(rate ~ 1 + pdq(2,1,0) + PDQ(0, 0, 0)),
    a211 = ARIMA(rate ~ 1 + pdq(2,1,1) + PDQ(0, 0, 0)),
    a212 = ARIMA(rate ~ 1 + pdq(2,1,2) + PDQ(0, 0, 0))
    )
```

Here is one way to determine which model is selected by the "auto" process.

```{r}
#| code-fold: true
#| code-summary: "Show the code"

exchange_model |>
  select(auto)
```

We now examine all the fitted models to determine the value of the residual mean squared error (`sigma2`), log-likelihood, AIC, AICc, and BIC. For the log-likelihood, larger values are preferable. For all other measures, smaller values are preferred.

```{r}
#| code-fold: true
#| code-summary: "Show the code"
#| output: false

exchange_model |>
  glance()
```


```{r}
#| label: tbl-exchangeRateSeveralFittedModels
#| tbl-cap: "Values used in the model selection process for the time series representing the exchange rate to convert US$1 into Euros"
#| echo: false

exchange_model |>
  glance() |>
  display_arima_models()
```

Suppose we choose to apply the "auto" model, which is $ARIMA(1,1,1)$.
The model parameters are summarized here:

```{r}
#| code-fold: true
#| code-summary: "Show the code"
#| output: false

exchange_model |>
  select(auto) |>
  coefficients()
```

```{r}
#| echo: false

exchange_model |>
  select(auto) |>
  coefficients() |>
  display_table()
```


The following plots give the acf and pacf of the residuals from this model.

```{r}
#| label: fig-exchangeRateResidACFpacf
#| fig-cap: "Correlogram and Partial Correlogram for the residuals from the ARIMA(1,1,1) model for the daily exchange rates to convert US$1 into Euros"
#| code-fold: true
#| code-summary: "Show the code"
#| warning: false

model_resid <- exchange_model |>
  select(auto) |>
  residuals()

acf_plot <- model_resid |> ACF() |> autoplot(var = .resid)

pacf_plot <- model_resid |> PACF() |> autoplot(var = .resid)

acf_plot | pacf_plot
```

Here is a histogram of the residuals from our model.

```{r}
#| label: fig-exchangeRateResidHistogram
#| fig-cap: "Histogram of the residuals from the ARIMA(1,1,1) model for the daily exchange rates to convert US$1 into Euros"
#| code-fold: true
#| code-summary: "Show the code"

model_resid |>
  mutate(density = dnorm(.resid, mean(model_resid$.resid), sd(model_resid$.resid))) |>
  ggplot(aes(x = .resid)) +
    geom_histogram(aes(y = after_stat(density)),
        color = "white", fill = "#56B4E9", binwidth = 0.001) +
    geom_line(aes(x = .resid, y = density)) +
    theme_bw() +
    labs(
      x = "Values",
      y = "Frequency",
      title = "Histogram of Residuals"
    ) +
    theme(
      plot.title = element_text(hjust = 0.5)
    )
```



<!-- Check Your Understanding -->

::: {.callout-tip icon=false title="Check Your Understanding"}

-   Write the fitted model:
$$
  x_t = ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$$
-   Does the model provide an appropriate fit for the data?

:::

Here is a forecast for the next 7 days based on our model.

```{r}
final_model <- exchange_model |>
  select(auto)

temps_forecast <- final_model |>
  select(auto) |>
  forecast(h = "7 days")

temps_forecast |>
  autoplot(exchange_ts, level = 95) +
  geom_line(aes(y = .fitted, color = "Fitted"),
    data = augment(final_model)) +
  scale_color_discrete(name = "") +
  labs(
    x = paste0(
                "Date (",
                format(ymd(min(exchange_ts$date)), "%d %b %Y"),
                  " - ",
                format(ymd(max(exchange_ts$date)), "%d %b %Y"),
                ")"
              ),
    y = "Exchange Rate",
    title = "Exchange Rate for Converting US$1 to Euros",
    subtitle = "7-Day Forecast Based on our AR(1,1,1) Model"
  ) +
  theme_minimal() +
  theme(
    plot.title = element_text(hjust = 0.5),
    plot.subtitle = element_text(hjust = 0.5)
  )
```










## Small-Group Activity: Fitting an ARIMA Process - Microsoft Stock Prices (20 min)

A time series given the daily closing price for Microsoft (MSFT) stock is given below. To handle the gaps in the data, we define a new variable, `t`, which gives the observation number.

```{r}
#| code-fold: true
#| code-summary: "Show the code"

# Set symbol and date range
symbol <- "MSFT"                # Abercrombie & Fitch stock trading symbol
date_start <- "2020-01-01"
date_end <- "2024-03-28"

# Fetch stock prices
df_stock <- tq_get(symbol, from = date_start, to = date_end, get = "stock.prices")

# Transform data into tsibble
stock_ts <- df_stock |>
  mutate(
    dates = date, 
    value = close
  ) |>
  dplyr::select(dates, value) |>
  as_tibble() |> 
  arrange(dates) |>
  mutate(t = 1:n()) |>
  as_tsibble(index = t, key = NULL)
```


<!-- Check Your Understanding -->

::: {.callout-tip icon=false title="Check Your Understanding"}

Using the daily closing prices of Microsoft stock, do the following.

-   Make a time plot of the data.
-   Create a correlogram and partial correlogram of the stock prices.
-   Fit candidate $ARIMA(p,d,q)$ models to the data.
-   Choose the "best" model, and justify your selection.
-   Generate a correlagram and partial correlogram of the residuals from your chosen model.
-   Make a histogram of the residuals from your model.
-   Did your your model account for the  the time series?
-   Predict the value 60 trading days in the future. 

*Note:* The "time" index is just an integer sequence in the `stock_ts` tsibble. So, apply the `forecast()` function as `forecast(h = 60)`, rather than `forecast(h = "60 days")`.

:::


```{r}
#| include: false
#| echo: false
#| eval: false

# Generate time series plot using plot_ly
plot_ly(stock_ts, x = ~dates, y = ~value, type = 'scatter', mode = 'lines') |>
  layout(
    xaxis = list(title = "Date"),
    yaxis = list(title = "Value"),
    title = paste0("Time Plot of ", symbol, " Daily Closing Price (", format(ymd(date_start), "%d %b %Y"), " - ", format(ymd(date_end), "%d %b %Y"),")")
  )

stock_model <- stock_ts |>
  model(
    auto = ARIMA(value ~ 1 + pdq(0:2,0:1,0:2) + PDQ(0, 0, 0)),
    
    a000 = ARIMA(value ~ 1 + pdq(0,0,0) + PDQ(0, 0, 0)),
    a001 = ARIMA(value ~ 1 + pdq(0,0,1) + PDQ(0, 0, 0)),
    a002 = ARIMA(value ~ 1 + pdq(0,0,2) + PDQ(0, 0, 0)),
    a100 = ARIMA(value ~ 1 + pdq(1,0,0) + PDQ(0, 0, 0)),
    a101 = ARIMA(value ~ 1 + pdq(1,0,1) + PDQ(0, 0, 0)),
    a102 = ARIMA(value ~ 1 + pdq(1,0,2) + PDQ(0, 0, 0)),
    a200 = ARIMA(value ~ 1 + pdq(2,0,0) + PDQ(0, 0, 0)),
    a201 = ARIMA(value ~ 1 + pdq(2,0,1) + PDQ(0, 0, 0)),
    a202 = ARIMA(value ~ 1 + pdq(2,0,2) + PDQ(0, 0, 0)),
    
    a011 = ARIMA(value ~ 1 + pdq(0,1,1) + PDQ(0, 0, 0)),
    a012 = ARIMA(value ~ 1 + pdq(0,1,2) + PDQ(0, 0, 0)),
    a110 = ARIMA(value ~ 1 + pdq(1,1,0) + PDQ(0, 0, 0)),
    a111 = ARIMA(value ~ 1 + pdq(1,1,1) + PDQ(0, 0, 0)),
    a112 = ARIMA(value ~ 1 + pdq(1,1,2) + PDQ(0, 0, 0)),
    a210 = ARIMA(value ~ 1 + pdq(2,1,0) + PDQ(0, 0, 0)),
    a211 = ARIMA(value ~ 1 + pdq(2,1,1) + PDQ(0, 0, 0)),
    a212 = ARIMA(value ~ 1 + pdq(2,1,2) + PDQ(0, 0, 0))
    ) 

stock_model |>
  glance() 

stock_model |>
  select(auto)

#########

stock_model |>
  glance() |>
  display_arima_models()

####

model_resid <- stock_model |>
  select(auto) |>
  residuals()

acf(model_resid$.resid, main = "ACF of Residuals from ARIMA Model")

pacf(model_resid$.resid, main = "ACF of Residuals from ARIMA Model")


model_resid |>
  mutate(density = dnorm(.resid, mean(model_resid$.resid), sd(model_resid$.resid))) |>
  ggplot(aes(x = .resid)) +
    geom_histogram(aes(y = after_stat(density)),
        color = "white", fill = "#56B4E9", binwidth = 2) +
    geom_line(aes(x = .resid, y = density)) +
    theme_bw() +
    labs(
      x = "Values",
      y = "Frequency",
      title = "Histogram of Residuals"
    ) +
    theme(
      plot.title = element_text(hjust = 0.5)
    )

######

stock_ts$diff = stock_ts$value - lag(stock_ts$value) 
stock_ts |>
  na.omit() |>
  autoplot(.vars = diff) + 
    labs(
      title = paste("Time Plot of Differences in Daily", symbol, "Stock Prices"),
      subtitle = 
        paste0(
          format(ymd(date_start), "%d %b %Y"),
            " - ",
          format(ymd(date_end), "%d %b %Y")
          ),
      x = "Date",
      y = "Difference",
    ) +
    theme_minimal() +
    theme(
      plot.title = element_text(hjust = 0.5),
      plot.subtitle = element_text(hjust = 0.5)
    )
```









## Homework Preview (5 min)

-   Review upcoming homework assignment
-   Clarify questions




::: {.callout-note icon=false}

## Download Homework

<a href="https://byuistats.github.io/timeseries/homework/homework_7_1.qmd" download="homework_7_1.qmd"> homework_7_1.qmd </a>

:::





<a href="javascript:showhide('Solutions1')"
style="font-size:.8em;">Class Activity</a>
  
::: {#Solutions1 style="display:none;"}
    

:::




<a href="javascript:showhide('Solutions2')"
style="font-size:.8em;">Class Activity</a>
  
::: {#Solutions2 style="display:none;"}
    

:::




<a href="javascript:showhide('Solutions3')"
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