chapter_5_lesson_2.qmd

---
title: "Harmonic Seasonal Variables - Part 1"
subtitle: "Chapter 5: Lesson 2"
format: html
editor: source
sidebar: false
---

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

```{=html}
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    var e = document.getElementById(id);
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```{r}
#| echo: false
#| label: functions_for_sine_and_cosine_plots

# Okabe-Ito color palette
okabe_ito_colors <- c("#009E73", "#F0E442", "#0072B2", "#D55E00", "#CC79A7", "#E69F00", "#56B4E9")

plot_sine <- function(cycle_length = 12, i_values = c(1:6), amplitude = rep(1, length(i_values)), spacing = 4, title = "Sine Functions with Different Frequencies", min_t = 0, max_t = cycle_length) {
  
  create_sine_df <- function(i, cycle_length, amplitude) {
    df <- tibble(
      t = seq(from = min_t, to = max_t, length.out = 501),
      value = amplitude[which(i_values == i)] * sin(2 * pi * i * t / cycle_length),
      i = right(paste0(" ", as.character(i)), 2)
    )
  }
  
  sine_df <- tibble(t = as.integer(), value = as.numeric(), i = as.character())
  for (i in i_values) {
    sine_df <- sine_df |> bind_rows(create_sine_df(i, cycle_length, amplitude))
  }
  
  ggplot(sine_df, aes(x = t, y = value - spacing * as.numeric(i), color = i)) +
    geom_line(linewidth = 1) +
    scale_y_continuous(
      breaks = -spacing * (i_values),
      minor_breaks = NULL,
      labels = NULL
    ) +
    scale_x_continuous(
      breaks = c(-12:12),
      minor_breaks = NULL
    ) +
    scale_color_manual(values = okabe_ito_colors[1:length(i_values)], name = "i") +
    labs(x = "t", y = "Sine Value", title = title) +
    theme_minimal() +
    theme(
      plot.title = element_text(hjust = 0.5)
    )
}

plot_cosine <- function(cycle_length = 12, i_values = c(1:6), amplitude = rep(1, length(i_values)), spacing = 4, title = "Cosine Functions with Different Frequencies", min_t = 0, max_t = cycle_length) {
  
  create_cosine_df <- function(i, cycle_length, amplitude) {
    df <- tibble(
      t = seq(from = min_t, to = max_t, length.out = 501),
      value = amplitude[which(i_values == i)] * cos(2 * pi * i * t / cycle_length),
      i = right(paste0(" ", as.character(i)), 2)
    )
  }
  
  cosine_df <- tibble(t = as.integer(), value = as.numeric(), i = as.character())
  for (i in i_values) {
    cosine_df <- cosine_df |> bind_rows(create_cosine_df(i, cycle_length, amplitude))
  }
  
  ggplot(cosine_df, aes(x = t, y = value - spacing * as.numeric(i), color = i)) +
    geom_line(linewidth = 1) +
    scale_y_continuous(
      breaks = -spacing * (i_values),
      minor_breaks = NULL,
      labels = NULL
    ) +
    scale_x_continuous(
      breaks = c(-12:12),
      minor_breaks = NULL
    ) +
    scale_color_manual(values = okabe_ito_colors[1:length(i_values)], name = "i") +
    labs(x = "t", y = "Cosine Value", title = title) +
    theme_minimal() +
    theme(
      plot.title = element_text(hjust = 0.5)
    )
}
```


## Learning Outcomes

{{< include outcomes/_chapter_5_lesson_2_outcomes.qmd >}}




## Preparation

-   Read Section 5.6



## 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: Joys of Trigonometry (15 min)

In the previous lesson, we learned how to incorporate an indicator (or dummy) variable for each season in a period. If there are twelve months in a year, this requires having twelve parameters in the model. Given that many seasonal changes are gradual and can be modeled by a continuous function, we can use sines and cosines to approximate the seasonal variation. This can lead to a smaller number of parameters than is required for the indicator variable approach. 

Consider the sine wave with the following parameters:

-   $A$:$~~$    the amplitude, 
-   $f$:$~~~$    the frequency or the number of cycles per sampling interval, and
-   $\phi$:$~~~$ the phase shift.

$$
  A \sin ( 2 \pi f t + \phi )
$$

Here is an interactive plot of this function. Adjust the values of $A$, $f$, and $\phi$ to see their effect on the function.

<iframe src="https://www.desmos.com/calculator/sgbxo2c0hx" width="750" height="500" style="border: 1px solid #ccc" frameborder=0></iframe>

Notice that this sine function is not linear in the parameters $A$ and $\phi$. 

$$
  A \sin ( 2 \pi f t + \phi )
$$

One of the trigonometric sum and difference identities is:

$$
  \sin(\theta + \phi) 
    = \cos(\phi) \sin(\theta) + \sin(\phi) \cos(\theta) 
$$

We apply this to our sine function.

\begin{align*}
  A \sin ( 2 \pi f t + \phi )
    &= \underbrace{A \cos( \phi )}_{\alpha_s} \cdot \sin ( 2 \pi f t ) + \underbrace{A \sin( \phi )}_{\alpha_c} \cdot \cos ( 2 \pi f t ) \\
    &= \alpha_s \cdot \sin ( 2 \pi f t ) + \alpha_c \cdot \cos ( 2 \pi f t ) 
\end{align*}

We have transformed this from something that is not linear in the parameters $A$ and $\phi$ to an expression that is linear in the parameters $\alpha_s$ and $\alpha_c$.

We can denote the frequency of a sine function as $f = \frac{i}{s}$, where $s$ is the number of seasons in a cycle and $i$ is some integer. This leads the following representation:

\begin{align*}
  A \sin \left( \frac{2 \pi i t}{s} + \phi \right)
    &= \underbrace{A \cos( \phi )}_{\alpha_s} \cdot \sin \left( \frac{2 \pi i t}{s} \right) + \underbrace{A \sin( \phi )}_{\alpha_c} \cdot \cos \left( \frac{2 \pi i t}{s} \right) \\
    &= \alpha_s \cdot \sin \left( \frac{2 \pi i t}{s} \right) + \alpha_c \cdot \cos \left( \frac{2 \pi i t}{s} \right) 
\end{align*}

@fig-sineFrequencies and @fig-cosineFrequencies illustrate these sine and cosine functions with various values of $i$, where $s = 12$.

<!-- Beginning of two columns -->
::: columns
::: {.column width="45%"}

```{r}
#| code-fold: true
#| code-summary: "Show the code"
#| label: fig-sineFrequencies
#| fig-cap: "Sine Functions with Various Frequencies"

plot_sine()
```

:::

::: {.column width="10%"}
<!-- empty column to create gap -->
:::

::: {.column width="45%"}

```{r}
#| code-fold: true
#| code-summary: "Show the code"
#| label: fig-cosineFrequencies
#| fig-cap: "Cosine Functions with Various Frequencies"

plot_cosine()
```

:::
:::
<!-- End of two columns -->


One key objective of this lesson is to use a linear combination of functions like those above to model the seasonal component of a time series.

## Class Activity: Fourier Series (15 min)

### What is a Fourier Series?

We now explore an important mathematical concept that allows us to approximate any periodic function. If we have an infinite number of terms, the Fourier Series described below gives an exact representation of the function.

::: {.callout-note title="Fourier Series"}

The **Fourier Series** is an infinite series representation of a smooth function $f(t)$ with period $s$:

$$
  f(t) = \frac{A_{s_0}}{2} + \sum_{i=1}^{\infty} \left\{ \alpha_{s_i} \sin \left( \frac{2\pi i t}{s} \right) + \alpha_{c_i} \cos \left( \frac{2\pi i t}{s} \right) \right\}
$$

The coefficients $\alpha_{s_i}$ and $\alpha_{c_i}$ are defined by the integrals:

$$
  \alpha_{s_i} = \frac{2}{s} \int_0^s f(t) \cos \left( \frac{2\pi i t}{s} \right) \; dt
  ~~~~~~~~~~~~~~~~~~
  \alpha_{c_i} = \frac{2}{s} \int_0^s f(t) \sin \left( \frac{2\pi i t}{s} \right) \; dt
$$

(You will not need to compute any of these integrals.)
:::

<!-- Sample Fourier Series -->

As an example, we will approximate the periodic function illustrated here with a Fourier series.

```{r}
#| echo: false
#| label: use_this_chunk_to_generate_fourier_data

step_size <- 0.001

fourier_df <- tibble(x = seq(-1.75, 1.75, step_size)) |>
  mutate(
    terms_1 = sin(pi * x),
    terms_2 = terms_1 + sin(3 * pi * x) / 3,
    terms_3 = terms_2 + sin(5 * pi * x) / 5,
    terms_4 = terms_3 + sin(7 * pi * x) / 7,
    terms_5 = terms_4 + sin(9 * pi * x) / 9,
    terms_6 = terms_5 + sin(11 * pi * x) / 11,
    terms_7 = terms_6 + sin(13 * pi * x) / 13,
    terms_8 = terms_7 + sin(15 * pi * x) / 15,
    terms_9 = terms_8 + sin(17 * pi * x) / 17,
    terms_10 = terms_9 + sin(19 * pi * x) / 19,
    terms_15 = terms_10 + sin(21 * pi * x) / 21 + sin(23 * pi * x) / 23 
                        + sin(25 * pi * x) / 25 + sin(27 * pi * x) / 27 
                        + sin(29 * pi * x) / 29,
    terms_25 = terms_15 + sin(31 * pi * x) / 31 + sin(33 * pi * x) / 33 
                        + sin(35 * pi * x) / 35 + sin(37 * pi * x) / 37 
                        + sin(39 * pi * x) / 39 + sin(41 * pi * x) / 41 
                        + sin(43 * pi * x) / 43 + sin(45 * pi * x) / 45
                        + sin(47 * pi * x) / 47 + sin(49 * pi * x) / 49,
    terms_50 = terms_25 + sin(51 * pi * x) / 51 + sin(53 * pi * x) / 53 
      + sin(55 * pi * x) / 55 + sin(57 * pi * x) / 57 + sin(59 * pi * x) / 59 
      + sin(61 * pi * x) / 61 + sin(63 * pi * x) / 63 + sin(65 * pi * x) / 65 
      + sin(67 * pi * x) / 67 + sin(69 * pi * x) / 69 + sin(71 * pi * x) / 71 
      + sin(73 * pi * x) / 73 + sin(75 * pi * x) / 75 + sin(77 * pi * x) / 77 
      + sin(79 * pi * x) / 79 + sin(81 * pi * x) / 81 + sin(83 * pi * x) / 83 
      + sin(85 * pi * x) / 85 + sin(87 * pi * x) / 87 + sin(89 * pi * x) / 89 
      + sin(91 * pi * x) / 91 + sin(93 * pi * x) / 93 + sin(95 * pi * x) / 95 
      + sin(97 * pi * x) / 97 + sin(99 * pi * x) / 99,
    terms_100 = terms_50 + sin(101 * pi * x) / 101 + sin(103 * pi * x) / 103 + sin(105 * pi * x) / 105 + sin(107 * pi * x) / 107 + sin(109 * pi * x) / 109 + sin(111 * pi * x) / 111 + sin(113 * pi * x) / 113 + sin(115 * pi * x) / 115 + sin(117 * pi * x) / 117 + sin(119 * pi * x) / 119 + sin(121 * pi * x) / 121 + sin(123 * pi * x) / 123 + sin(125 * pi * x) / 125 + sin(127 * pi * x) / 127 + sin(129 * pi * x) / 129 + sin(131 * pi * x) / 131 + sin(133 * pi * x) / 133 + sin(135 * pi * x) / 135 + sin(137 * pi * x) / 137 + sin(139 * pi * x) / 139 + sin(141 * pi * x) / 141 + sin(143 * pi * x) / 143 + sin(145 * pi * x) / 145 + sin(147 * pi * x) / 147 + sin(149 * pi * x) / 149 + sin(151 * pi * x) / 151 + sin(153 * pi * x) / 153 + sin(155 * pi * x) / 155 + sin(157 * pi * x) / 157 + sin(159 * pi * x) / 159 + sin(161 * pi * x) / 161 + sin(163 * pi * x) / 163 + sin(165 * pi * x) / 165 + sin(167 * pi * x) / 167 + sin(169 * pi * x) / 169 + sin(171 * pi * x) / 171 + sin(173 * pi * x) / 173 + sin(175 * pi * x) / 175 + sin(177 * pi * x) / 177 + sin(179 * pi * x) / 179 + sin(181 * pi * x) / 181 + sin(183 * pi * x) / 183 + sin(185 * pi * x) / 185 + sin(187 * pi * x) / 187 + sin(189 * pi * x) / 189 + sin(191 * pi * x) / 191 + sin(193 * pi * x) / 193 + sin(195 * pi * x) / 195 + sin(197 * pi * x) / 197 + sin(199 * pi * x) / 199
  ) |>
  pivot_longer(names_to = "terms", values_to = "Value", cols = starts_with("terms")) |>
  mutate(terms = str_remove(terms, "terms_"))
```


```{r}
#| echo: false
#| label: fourier_plot_function

plot_fourier <- function(df = fourier_df, num_terms) {
  plot_df <- df |>
  filter(terms == num_terms)
  
  my_plot <- plot_df |>
    ggplot(aes(x = x, y = Value)) +
    geom_segment(aes(x = -1.75, y = 0, xend = 1.75, yend = 0), linewidth = 1, colour = "black") +
    geom_segment(aes(x = -1.75, y = 0.785, xend = -1, yend = 0.785, colour = "Target Function"), linewidth = 1) +
    geom_segment(aes(x = -1, y = 0.785, xend = -1, yend = -0.785, colour = "Target Function"), linewidth = 1) +
    geom_segment(aes(x = -1, y = -0.785, xend = 0, yend = -0.785, colour = "Target Function"), linewidth = 1) +
    geom_segment(aes(x = 0, y = -0.785, xend = 0, yend = 0.785, colour = "Target Function"), linewidth = 1) +
    geom_segment(aes(x = 0, y = 0.785, xend = 1, yend = 0.785, colour = "Target Function"), linewidth = 1) +
    geom_segment(aes(x = 1, y = 0.785, xend = 1, yend = -0.785, colour = "Target Function"), linewidth = 1) +
    geom_segment(aes(x = 1, y = -0.785, xend = 1.75, yend = -0.785, colour = "Target Function"), linewidth = 1) +
    geom_line(aes(colour = "Approximation"), linewidth = 1, alpha = 0.75) +
    scale_color_manual(values = c("Target Function" = "#E69F00", "Approximation" = "#56B4E9")) +
    theme_minimal() +
    coord_cartesian(ylim = c(-1,1)) +
    labs(title = paste0("Fourier Series Approximation with ", plot_df$terms |> head(1), " Terms")) +
    theme(
      panel.grid.major = element_blank(),
      panel.grid.minor = element_blank(),
      axis.text.y = element_blank(),
      legend.position = c(0.9, 0.8),
      legend.title = element_blank(),
      plot.title = element_text(hjust = 0.5)
    )
  
  return(my_plot)
}
```

::: panel-tabset
#### Target

```{r}
#| echo: false
#| label: plot_target_function

fourier_df |>
  filter(terms == 1) |>
  filter(x == floor(x)) |> # To speed up the processing of this chunk
  ggplot(aes(x = x, y = Value)) +
    geom_segment(aes(x = -1.75, y = 0, xend = 1.75, yend = 0), linewidth = 1, colour = "black") +
    geom_segment(aes(x = -1.75, y = 0.785, xend = -1, yend = 0.785, colour = "Target Function"), linewidth = 1) +
    geom_segment(aes(x = -1, y = 0.785, xend = -1, yend = -0.785, colour = "Target Function"), linewidth = 1) +
    geom_segment(aes(x = -1, y = -0.785, xend = 0, yend = -0.785, colour = "Target Function"), linewidth = 1) +
    geom_segment(aes(x = 0, y = -0.785, xend = 0, yend = 0.785, colour = "Target Function"), linewidth = 1) +
    geom_segment(aes(x = 0, y = 0.785, xend = 1, yend = 0.785, colour = "Target Function"), linewidth = 1) +
    geom_segment(aes(x = 1, y = 0.785, xend = 1, yend = -0.785, colour = "Target Function"), linewidth = 1) +
    geom_segment(aes(x = 1, y = -0.785, xend = 1.75, yend = -0.785, colour = "Target Function"), linewidth = 1) +
    scale_color_manual(values = c("Target Function" = "#E69F00")) +
    theme_minimal() +
    coord_cartesian(ylim = c(-1,1)) +
    labs(
      title = "Periodic Target Function",
      x = "x",
      y = "Values"
    ) +
    theme(
      panel.grid.major = element_blank(),
      panel.grid.minor = element_blank(),
      axis.text.y = element_blank(),
      legend.position = "none",
      legend.title = element_blank(),
      plot.title = element_text(hjust = 0.5)
    )

```

#### 1 term

```{r}
#| echo: false
#| label: plot_fourier_approx_1

plot_fourier(num_terms = 1)
```
$$
  f(x) \approx sin(\pi x)
$$

```{r}
#| echo: false

plot_sine(cycle_length = 2, i_values = c(1), amplitude = c(1), spacing = 0, title = "Functions Summed for this Approximation", min_t = -1.75, max_t = 1.75) +
  theme(legend.position = c(0.95, 0.85))
```


#### 2 terms

```{r}
#| echo: false
#| label: plot_fourier_approx_2

plot_fourier(num_terms = 2)
```
$$
  f(x) \approx \sin(\pi x) + \frac{1}{3} \sin(3 \pi x)
$$

```{r}
#| echo: false

plot_sine(cycle_length = 2, i_values = c(1,3), amplitude = c(1, 1/3), spacing = 0, title = "Functions Summed for this Approximation", min_t = -1.75, max_t = 1.75) +
  theme(legend.position = c(0.95, 0.85))
```

#### 3 terms

```{r}
#| echo: false
#| label: plot_fourier_approx_3

plot_fourier(num_terms = 3)
```
$$
  f(x) \approx \sin(\pi x) + \frac{1}{3} \sin(3 \pi x) + \frac{1}{5} \sin(5 \pi x)
$$

```{r}
#| echo: false

plot_sine(cycle_length = 2, i_values = c(1,3,5), amplitude = c(1, 1/3, 1/5), spacing = 0, title = "Functions Summed for this Approximation", min_t = -1.75, max_t = 1.75) +
  theme(legend.position = c(0.95, 0.85))
```



#### 4 terms

```{r}
#| echo: false
#| label: plot_fourier_approx_4

plot_fourier(num_terms = 4)
```
$$
  f(x) \approx \sin(\pi x) + \frac{1}{3} \sin(3 \pi x) + \frac{1}{5} \sin(5 \pi x) + \frac{1}{7} \sin(7 \pi x)
$$

```{r}
#| echo: false

plot_sine(cycle_length = 2, i_values = c(1,3,5,7), amplitude = c(1, 1/3, 1/5, 1/7), spacing = 0, title = "Functions Summed for this Approximation", min_t = -1.75, max_t = 1.75) +
  theme(legend.position = c(0.95, 0.85))
```

#### 5 terms

```{r}
#| echo: false
#| label: plot_fourier_approx_5

plot_fourier(num_terms = 5)
```
$$
  f(x) \approx sin(\pi x) + \frac{1}{3} \sin(3 \pi x) + \frac{1}{5} \sin(5 \pi x) + \frac{1}{7} \sin(7 \pi x) + \frac{1}{9} \sin(9 \pi x)
$$

```{r}
#| echo: false

plot_sine(cycle_length = 2, i_values = c(1,3,5,7,9), amplitude = c(1, 1/3, 1/5, 1/7, 1/9), spacing = 0, title = "Functions Summed for this Approximation", min_t = -1.75, max_t = 1.75) +
  theme(legend.position = c(0.95, 0.85))
```

#### 10 terms

```{r}
#| echo: false
#| label: plot_fourier_approx_10

plot_fourier(num_terms = 10)
```
$$
  f(x) \approx sin(\pi x) + \frac{1}{3} \sin(3 \pi x) + \frac{1}{5} \sin(5 \pi x) + \cdots + \frac{1}{17} \sin(17 \pi x) + \frac{1}{19} \sin(19 \pi x)
$$

#### 15 terms

```{r}
#| echo: false
#| label: plot_fourier_approx_15

plot_fourier(num_terms = 15)
```
$$
  f(x) \approx sin(\pi x) + \frac{1}{3} \sin(3 \pi x) + \frac{1}{5} \sin(5 \pi x) + \cdots + \frac{1}{27} \sin(27 \pi x) + \frac{1}{29} \sin(29 \pi x)
$$

#### 25 terms

```{r}
#| echo: false
#| label: plot_fourier_approx_25

plot_fourier(num_terms = 25)
```
$$
  f(x) \approx sin(\pi x) + \frac{1}{3} \sin(3 \pi x) + \frac{1}{5} \sin(5 \pi x) + \cdots + \frac{1}{47} \sin(47 \pi x) + \frac{1}{49} \sin(49 \pi x)
$$

#### 50 terms

```{r}
#| echo: false
#| label: plot_fourier_approx_50

plot_fourier(num_terms = 50)
```
$$
  f(x) \approx sin(\pi x) + \frac{1}{3} \sin(3 \pi x) + \frac{1}{5} \sin(5 \pi x) + \cdots + \frac{1}{97} \sin(97 \pi x) + \frac{1}{99} \sin(99 \pi x)
$$

#### 100 terms

```{r}
#| echo: false
#| label: plot_fourier_approx_100

plot_fourier(num_terms = 100)
```
$$
  f(x) \approx sin(\pi x) + \frac{1}{3} \sin(3 \pi x) + \frac{1}{5} \sin(5 \pi x) + \cdots + \frac{1}{197} \sin(197 \pi x) + \frac{1}{199} \sin(199 \pi x)
$$

:::
<!-- End of Fourier series example -->

We can continue to get more and more precise estimates of the function by just adding more terms to the sum.
So, the function $f(t)$ can be approximated to any desired level of precision by truncating the series after a sufficient number of terms. For the purpose of this discussion, we will define "one term" as an expression of the form 
$$
  \left\{ \alpha_{s_i} \sin \left( \frac{2\pi i t}{s} \right) + \alpha_{c_i} \cos \left( \frac{2\pi i t}{s} \right) \right\}
$$

### Fitting a Seasonal Component

The Fourier series exists for any smooth (continuously differentiable) function. Note theoretically, this allows us to obtain the value of the function at any real value by using this series. 
However, for a discrete time series with $s$ seasons, we only need to evaluate the function at a finite number of points: 
$t = 1, 2, 3, \ldots, s-1, s$. 
For example, in the previous lesson, we used twelve indicator variables to pass through twelve points in a monthly seasonal component to a time series with annual cycles.

It turns out, that we only need six terms (which involves twelve coefficients) to fit monthly data with annual cycles.
In general, we only need to obtain $\left\lfloor \frac{s}{2} \right\rfloor$ terms of this sum to fit the seasonal values perfectly.

| Pattern            | Number of Seasons, $s$ | Maximum terms in the sum, $\left\lfloor \frac{s}{2} \right\rfloor$ |
|--------------------|------------------------|------------------------|
| Days in a Week     | 7                      | 3                      |
| Quarters in a Year | 4                      | 2                      |
| Months in a Year   | 12                     | 6                      |

: A few examples of seasonal patterns and the corresponding values of $s$ and $\left\lfloor \frac{s}{2} \right\rfloor$ {#tbl-sAndFloorFunction}

Note that if $s$ is even and $i=\frac{s}{2}$, 
$$
  \sin \left( \frac{2\pi i t}{s} \right) = \sin \left( \frac{2\pi \cdot \frac{s}{2} \cdot t}{s} \right) = \sin \left( \pi t \right) = 0
$$
for all integer values of $t$. So, this term must be omitted from the model. If we try to include it in the model, the coefficient will be rediculously large, as R trys to make the product of something very close to 0 (the value from the sine function) and the coefficient multiply to some reasonably small number.

The method for fitting seaonal components using indicator variables does not assume any relationship between successive seasons. However, values observed in January are often highly correlated with values observed in February, etc. Fitting a seasonal component using terms in the Fourier Series can often yield a good approximation for the periodic cycles with only a few terms.

For a time series with $s$ seasons per cycle, our additive model can be written as:

\begin{align*}
  x_t
    &= m_t + s_t + z_t \\
    &= m_t +
        \sum_{i=1}^{\left\lfloor \frac{s}{2} \right\rfloor} \left\{ \alpha_{s_i} \sin \left( \frac{2\pi i t}{s} \right) + \alpha_{c_i} \cos \left( \frac{2\pi i t}{s} \right) \right\}
      + z_t
\end{align*}

The term $m_t$ can take a variety of forms, including:

-   Linear:       $~~~~~~~~~~~$ $m_t = \alpha_0 + \alpha_1 t$
-   Quadratic:    $~~~~~~$ $m_t = \alpha_0 + \alpha_1 t + \alpha_2 t^2$
-   Exponential:  $~~~$ $m_t = \alpha_0 e^{\alpha_1 t}$
-   Any other functional form

The term $z_t$ is a (possibly autocorrelated) time series with mean zero.

We will now focus on the seasonal term, $s_t$.
The full seasonal term when considering 12 months in a year is:

\begin{align*}
  s_t 
    &= \sum_{i=1}^{\left\lfloor \frac{s}{2} \right\rfloor} \left\{ \alpha_{s_i} \sin \left( \frac{2\pi i t}{s} \right) + \alpha_{c_i} \cos \left( \frac{2\pi i t}{s} \right) \right\} \\
    &= \sum_{i=1}^{6} \left\{ \alpha_{s_i} \sin \left( \frac{2\pi i t}{12} \right) + \alpha_{c_i} \cos \left( \frac{2\pi i t}{12} \right) \right\} \\
    &=~~~~ \left\{ \alpha_{s_1} \sin \left( \frac{2\pi \cdot 1 t}{12} \right) + \alpha_{c_1} \cos \left( \frac{2\pi \cdot 1 t}{12} \right) \right\} & \leftarrow i = 1 \\
    &~~~~~+ \left\{ \alpha_{s_2} \sin \left( \frac{2\pi \cdot 2 t}{12} \right) + \alpha_{c_2} \cos \left( \frac{2\pi \cdot 2 t}{12} \right) \right\} & \leftarrow i = 2 \\
    &~~~~~+ \left\{ \alpha_{s_3} \sin \left( \frac{2\pi \cdot 3 t}{12} \right) + \alpha_{c_3} \cos \left( \frac{2\pi \cdot 3 t}{12} \right) \right\} & \leftarrow i = 3 \\
    &~~~~~+ \left\{ \alpha_{s_4} \sin \left( \frac{2\pi \cdot 4 t}{12} \right) + \alpha_{c_4} \cos \left( \frac{2\pi \cdot 4 t}{12} \right) \right\} & \leftarrow i = 4 \\
    &~~~~~+ \left\{ \alpha_{s_5} \sin \left( \frac{2\pi \cdot 5 t}{12} \right) + \alpha_{c_5} \cos \left( \frac{2\pi \cdot 5 t}{12} \right) \right\} & \leftarrow i = 5 \\
    &~~~~~+ \left\{ \alpha_{s_6} \sin \left( \frac{2\pi \cdot 6 t}{12} \right) + \alpha_{c_6} \cos \left( \frac{2\pi \cdot 6 t}{12} \right) \right\} & \leftarrow i = 6 \\
    &=~~~~ \left\{ \alpha_{s_1} \sin \left( \frac{2\pi \cdot 1 t}{12} \right) + \alpha_{c_1} \cos \left( \frac{2\pi \cdot 1 t}{12} \right) \right\}  \\
    &~~~~~+ \left\{ \alpha_{s_2} \sin \left( \frac{2\pi \cdot 2 t}{12} \right) + \alpha_{c_2} \cos \left( \frac{2\pi \cdot 2 t}{12} \right) \right\} \\
    &~~~~~+ \left\{ \alpha_{s_3} \sin \left( \frac{2\pi \cdot 3 t}{12} \right) + \alpha_{c_3} \cos \left( \frac{2\pi \cdot 3 t}{12} \right) \right\} \\
    &~~~~~+ \left\{ \alpha_{s_4} \sin \left( \frac{2\pi \cdot 4 t}{12} \right) + \alpha_{c_4} \cos \left( \frac{2\pi \cdot 4 t}{12} \right) \right\} \\
    &~~~~~+ \left\{ \alpha_{s_5} \sin \left( \frac{2\pi \cdot 5 t}{12} \right) + \alpha_{c_5} \cos \left( \frac{2\pi \cdot 5 t}{12} \right) \right\} \\
    &~~~~~+ \left\{ \phantom{\alpha_{s_6} \sin \left( \frac{2\pi \cdot 6 t}{12} \right) +}~~ \alpha_{c_6} \cos \left( \frac{2\pi \cdot 6 t}{12} \right) \right\} \\
\end{align*}

Note that $\sin \left( \frac{2\pi \cdot 6 t}{12} \right) = 0$ for all integer values of $t$, so we can omit the term $\alpha_{s_6} \sin \left( \frac{2\pi \cdot 6 t}{12} \right)$.

As noted above, we can often use a relatively small subset of these terms to get a good approximation of the seasonal component.


### Simulation

The following simulation illustrates harmonic seasonal terms. 
The values S1, S2, S3, $\ldots$ represent the coefficients on the sine functions: $\alpha_{s_1}, ~ \alpha_{s_2}, ~ \alpha_{s_3}, ~ \ldots, ~ \alpha_{s_6}$. Similarly, the values C1, C2, C3, $\ldots$ represent the coefficients on the corresponding cosine terms: $\alpha_{c_1}, ~ \alpha_{c_2}, ~ \alpha_{c_3}, ~ \ldots, ~ \alpha_{c_6}$.

Adjust the values of the parameters to create different seasonal patterns. Note that this is just a sum of sine and cosine function with various frequencies and amplitudes.

```{=html}
 <iframe id="harmonicSeasons" src="https://posit.byui.edu/content/3fcf5813-76fe-44ee-ab7f-b4a99884c855" style="border: none; width: 100%; height: 950px" frameborder="0"></iframe>
```








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