04_D_ClassicalDecomposition_fromScratch.qmd

---
title: "04_D_Classical_Decomposition_fromScratch"
author: "Juan Garbayo - TS Analysis"
format: html
editor: source
params:
  print_sol: true
  print_sol_int: true
toc: true
toc-location: left
toc-depth: 6
self-contained: true
---

```{r, include=FALSE, eval=FALSE}
TODOs:
1. Justificar mejor por qué se ajusta el seasonal component a los valores 0 y m.
Hacer esta justificación en un apartado al final de notebook, para los alumnos
que estén interesados en entenderlo, ya que no es necesario, lo que me intere
* Additive: se ajusta a 0 porque, al quitar la trend de manera aditiva te va a
quedar algo que se mueve por encima y por debajo de 0. El valor medio del detrended
no tiene por qué ser 0, 

2. Dejar claro cuáles son los objetivos:
  - Que practiquen implementar un algoritmo
  - Que practiquen la comparación de resultados y que practiquen incluir comprobaciones
    intermedias
  - Que entiendad el significado de todos los términos de la descomposición, así
    como el orden en el que se obtienen:
      1. Trend
      2. Detrended
      3. Seasonal
      4. Remainder
```

# References

NOTE: this material is not original, but rather an extension of the following reference, which is publicly available.

1. Hyndman, R.J., & Athanasopoulos, G., 2021. *Forecasting: principles and practice*. 3rd edition. [Link](https://otexts.com/fpp3/)

# Packages

```{r, error=FALSE, warning=FALSE, message = FALSE}
library(fpp3)
```

# Classical decomposition

It is the starting point for most other methods of time series decomposition. In addition, it is simple enough that it can be easily implemented from scratch, which is something we are going to do in class.

Essentially are 2 variants of classical decomposition:

  * Classical decomposition of **aditive schemes**
  * Classical decomposition of **multiplicative schemes**
  
The algorithm depends on **whether the length seasonal period is even or uneven.** Examples:

* If we have daily data we may consider, for example, a period of one week. That seasonal period would be of length 7 (7 days).
    * NOTE: For daily data we could also consider other seasonal periods: one month, one quarter... even one year.
* If we have monthly data, we could consider a yearly seasonal period. That seasonal period would be of length 12 (12 months).
* If we have quarterly data, we could consider a yearly seasonal period. That seasonal period would be of length 4 (4 quarters).
* ...

Because the algorithm **approximates the trend with a centered moving average**, the even or uneven nature of the seasonal period is important. See the separate notebook on moving averages, which has an associated video to understand this:

* **Even seasonal period ($m$ even):**  trend approximated by a 2 x $m$ moving average.
* **Uneven seasonal period $m$:** trend approximated by an $m$ moving average.
  
# Additive schemes - classical decomposition

## Step 1 - Estimate trend

$m$: length of the seasonal period.

* If $m$ is an even number, compute the trend-cycle component $T_t$ using 2 x $m$-MA. 
* If $m$ is an odd number compute the trend-cycle component $T_t$  using an $m$-MA.

**Output**: the **trend component $T_t$**

## Step 2 - Detrended series

Calculate the de-trended series: $D_t = y_t - T_t$

**Output**: the **detrended series $D_t$**

## Step 3 - Compute seasonal component

**Main assumption to compute the seasonal component**: the seasonal component remains constant over time. This is one of the main weaknesses of the classical decomposition.

### **Step 3.1.** 

Average the de-trended values for each component of the season.

* Example: with monthly data the seasonal component for march is the average of all the detrended March values in the data
  
* **Output**: a vector of length $m$, with $m$ being the length of the seasonal compoent.
    * Let us call this vector **$S_{unadj}$** for *unadjusted seasonal component*
        
<br>
    
### **Step 3.2.** 

Adjust $S_{unadj}$, the output of step 3.1 to ensure that the sum of its **values adds to 0**.

If we call this adjusted seasonal component $S$, in classical decomposition it will be a vector of length $m$, just like **$S_{unadj}$**. We want the components of that vector to add up to 0:

$$
\text{Condition we want to attain:  }\sum_{i=1}^{m} S_i= 0
$$

We want this because, if this is the case, the seasonal component does not contribute to the average of the time series over one period:

$$
\frac{\sum_{i=1}^my_i}{m} = \frac{\sum_{i=1}^m(T_i + S_i + R_i)}{m} \underset{\underset{\sum_{i=1}^mS_i = 0}{\uparrow}}{=} \frac{\sum_{i=1}^m(T_i + R_i)}{m}
$$

The vector resulting from step 3.1, $S_{unadj \ i}$ does not satisfy this condition. We are going to correct each of the components of $S_{unadj \ i}$ by the same amount $x$ in order to fulfill the condition above:

$$
S_i = S_{Unadj \ i} + x
$$

To determine x:

#### Step 3.2.1 

Sum all the components of the vector $S_{unadj}$. Lets call this output $a$. This result is the total deviation from 0 we want to correct.

$$
\sum_{i=1}^{m}{S_{Unadj \ i}} = a
$$

#### Step 3.2.2 

Obtain $x$ from the equation of the condition we want to attain:

$$
\sum_{i=1}^{m} S_i = 0 \rightarrow \sum_{i=1}^{m} (S_{Unadj \ i} + x) = a + \sum_{i=1}^{m}x = a + m \cdot x \underset{\begin{array}{c} \uparrow \\ \text{condition} \\ \text{imposed} \end{array}}{=} 0 \rightarrow \text{    Solving for x } \rightarrow x = -\frac{a}{m}
$$

By adding this amount to each $S_{unadj}$, we will obtain the seasonal component $S$

* **Output**: the adjusted seasonal component $S$.

## Step 4 - Compute the remainder component

$R_t = D_t - S_t = y_t - T_t - S_t$ 

## Example: computation using the function `classical_decomposition()`:

```{r}
# Filter the series and select relevant columns
us_retail_employment <- 
  us_employment %>%
  filter(year(Month) >= 1990, Title == "Retail Trade") %>%
  select(-Series_ID)

# Let us compute the classical decomposition using the function in the feasts library:
classical_dec <- us_retail_employment %>%
  
  # Fit the classical decomposition model specifying an additive scheme
  model(
    clas_deic = classical_decomposition(Employed, type = "additive")
  ) %>%
  
  # Extract components from the fitted model
  components()

select(classical_dec, Month, Employed, trend, seasonal, random)
```

```{r}
classical_dec %>% 
  autoplot()
```


## Example: manual computation implementing the described algorithm.

### Step 1: compute the trend using moving averages

```{r}
# Compute moving averages to estimate the trend
# Because the seasonal period is 12, we need a 2x12-MA
manual_decomposition <- us_retail_employment %>%
  mutate(
    
    `12-MA` = slider::slide_dbl(Employed, mean,
                .before = 5, .after = 6, .complete = TRUE),
    `2x12-MA` = slider::slide_dbl(`12-MA`, mean,
                .before = 1, .after = 0, .complete = TRUE)
    
  )

# Plot the computed trend
manual_decomposition %>%
  autoplot(Employed, colour = "gray") +
  geom_line(aes(y = `2x12-MA`), colour = "#D55E00") +
  labs(y = "Persons (thousands)",
       title = "Total employment in US retail")
```

### Step 2 - detrend the time series

```{r}
# Compute the detrended component:
manual_decomposition <- 
  manual_decomposition %>%
  mutate(
      detrended = Employed - `2x12-MA`,
    )
manual_decomposition %>% autoplot(detrended)
```
```{r}

```

This is the resulting detrended component, which contains the seasonal + remainder component: $D_t = y_t - T_t = S_t + R_t$

### Step 3: compute the seasonal component

#### Step 3.1. Compute $S_{unadj}$ averaging the de-trended values for each element of the season:

We will store the result in a separate dataframe called `df_seasonal`. This dataframe will contain 2 columns:

1. An identifier for the number of the month. This identifier will be based solely on the month, disregarding the year. More generally, we need to compute an identifier for each element of the seasonal component.
    * We will use this identifier to group the data and compute the mean value
2. `s_unadj`: we compute it averaging the detrended series grouping based on the previously created identifier. For example, with monthly data, the seasonal component for March is the average of all the detrended March values in the data.

We will start by creating the mentioned identifier for each month (disregarding the year) on our dataframe:

```{r}
manual_decomposition <- 
  
  manual_decomposition %>% 
  
  # Compute an identifier for each month, regardless
  # of the year. This will define the subsequent groups
  mutate(
    n_month = month(Month)
  )

manual_decomposition
```

Now we will use this newly created identifier to compute the averages of the detrended time series for each month regardless of the year. Expressed in more general terms, we are computing the averages of the detrended time series for each element of the seasonal component.


```{r}
df_seasonal <- 
  
  manual_decomposition %>%
  
  # group by the created identifier
  index_by(n_month) %>%
  
  # compute the average seasonal component for every month
  summarise(
    s_unadj = mean(detrended, na.rm = TRUE)
  ) 

df_seasonal

#Check the aspect of the obtained unadjusted seasonal component
df_seasonal %>%
  
  autoplot(s_unadj) +
  
  scale_x_continuous(breaks = seq(1, 12),
                     minor_breaks = seq(1, 12)) +
  
  geom_point(color = "red")
```
```{r}

```

This is the resulting unadjusted seasonal component $S_{unadj}$:

* The first point is the unadjusted value of the seasonal component for January.
* The second point is the unadjusted value of the seasonal component for February.
* The third point is the unadjusted value of the seasonal component for March.
* ...

#### Step 3.2 Adjust the seasonal component

Let us check if the values of the component add up to 0 as we wanted to impose:

```{r}
# Seasonal period
m = 12

#The sum does not add up to 0
a = sum(df_seasonal$s_unadj)
a
```

We can see that the sum is not 0. Let us compute the correction $x$ using the equation above:

```{r}
# Correction x (see formula above)
x = - a / m
```

Now we apply that correction:

```{r}
df_seasonal <- 
  
  df_seasonal %>%
  
  # Correction so that the sum of the seasoan components is 0
  mutate(
        seasonal_c = s_unadj  + x
    ) %>% 
  
  # Drop s_unadj since we do not need it anymore.
  select(n_month, seasonal_c)

# Check that the sum is indeed 0
sum(df_seasonal$seasonal_c)
```

As you can see, the seasonal component is now 0. Great. Let us bring the seasonal component back into our original dataset. For this we will use a `left join` operation. Remember that we created previously a column called `n_month` that now will come in handy when doing this join:

```{r}
# Bring seasonal component into the dataset (I chose to use a "join" statement, very convenient)
manual_decomposition <- 
  left_join(manual_decomposition, df_seasonal, by = "n_month")

manual_decomposition
```

### Step 4: compute the remainder component removing the seasonal component from the detrended series

Now that we have the seasonal component, we obtain the remainder component by subtracting it from the de-trended series.

```{r}
# Compute remainder component:
manual_decomposition <- 
  manual_decomposition %>%
  mutate(
        remainder = detrended - seasonal_c
      )

select(manual_decomposition, Month, `2x12-MA`, seasonal_c, remainder)
```

### Test result:

We have all the components. Using the function `all.equal()` we can check that our computation matches exactly the output of the function `classical_decomposition`

```{r}
all.equal(classical_dec$trend, manual_decomposition$`2x12-MA`)
all.equal(classical_dec$seasonal, manual_decomposition$seasonal_c)
all.equal(classical_dec$random, manual_decomposition$remainder)
```

### Extra: generate a plot of the components we just created

```{r}
# As an additional ggplot visualization exercise, below the code to visualize
# the decomposition we have computed manually

# Define levels for the factor variable to ensure proper ordering
# in the facet grid figure.
factor_levels = c("Employed", "trend", "seasonal_c", "remainder")

manual_decomposition_long <- 
  
  manual_decomposition %>% 

    # Rename so that the labels are consistent
    rename(trend = `2x12-MA`) %>% 
  
    # Select the columns of interest
    select(Month, Employed, trend, seasonal_c, remainder) %>% 
  
    # Generate long format based on a single variable so that we can
    # facet_grid following this variable
    pivot_longer(cols = Employed:remainder, names_to = "component") %>% 
    
    # Convert to factor to ensure proper ordering in the figure
    mutate(
      component = factor(component, levels=factor_levels)
    )

# Now generate the graph
manual_decomposition_long %>% 
  
  ggplot(aes(x = Month, y = value)) +
  
  geom_line() +
  
  # scales = "free_y" so that each component has an adapted y-axis range
  facet_grid(component ~ ., scales = "free_y")
```

We can see that it looks just like the decomposition returned by the fable library, which is logical, since all the components are the same

# Multiplicative scheme - classical decomposition

In this case, the algorithm is just the same. In fact the estimation of the trend is identical. However, now subtractions are replaced by divisions.

## Step 1 - Estimate trend

$m$: length of the seasonal period.

* If $m$ is an even number, compute the trend-cycle component $T_t$ using 2 x $m$-MA. 
* If $m$ is an odd number compute the trend-cycle component $T_t$  using an $m$-MA.

## Step 2 - Detrended series

Calculate the de-trended series: $y_t / \hat{T_t}$

## Step 3 - Compute seasonal component

**Main assumption to compute the seasonal component**: the seasonal component remains constant over time. This is one of the main weaknesses of the classical decomposition.

### **Step 3.1.** 

Average the de-trended values for each component of the season.

* Example: with monthly data the seasonal component for march is the average of all the detrended March values in the data
  
* **Output**: a vector of length $m$, with $m$ being the length of the seasonal compoent.
    * Let us call this vector **$S_{unadj}$** for *unadjusted seasonal component*
        
<br>
    
### **Step 3.2.** 

Adjust each $S_{unadj\text{ i}}$ (output of step 3.1) by a specific amount to obtain the adjusted seasonal component $S_i$, forcing it to add up to $m$:

$$
\text{Condition to attain: } \sum_{i=1}^{m}S_i = \sum_{i=1}^{m}(S_{unadj\text{ i}} + x) = m
$$

This condition is interesting because, thanks to it, the sum of the time series over a seasonal period becomes a weighted average of $T_iR_i$ in which $S_i/m$ is the percent weight of each $T_iR_i$:

$$
\frac{\sum_{i=1}^m(T_i S_i R_i)}{m} = \underbrace{\sum_{i=1}^m\frac{S_i}{m}(T_i R_i) }_{\text{weighted avg.} \\ \text{of } R_iT_i \text{ since } \sum_{i=1}^mS_i/m = 1}
$$

To attain this condition we will follow the steps below:

#### Step 3.2.1

Sum all the components of the vector $Sunadj_i$. Lets call this output $a$. In general, $a$ will be different from $m$, although it can be very close.

$$
\sum_{i=1}^mS_{unadj\text{ i}} = a
$$
    
#### Step 3.2.2 

Compute the necessary correction $x$ to be added to each $S_{unadj\text{ i}}$ so that the resulting $S$ adds to $m$.

If we assume that we are going to correct each $S_{unadj\text{ i}}$ by the same amount $x$, the resulting correction would be:

$$
\begin{align*}
\text{Condition we want to attain}: && \sum_{i=1}^{m}S_i = \sum_{i=1}^{m}(S_{unadj\text{ i}} + x) \underset{\begin{array}{c} \uparrow \\ \text{condition} \\ \text{imposed} \end{array}}{=} m \\
\text{Developing the sums: } && \sum_{i=1}^m{S_i} = \sum_{i=1}^mS_{unadj\text{ i}} + \sum_{i=1}^mx = a + m \cdot x \underset{\begin{array}{c} \uparrow \\ \text{condition} \\ \text{imposed} \end{array}}{=} m \\
\text{Hence, solving for x: } && x = \frac{m - a}{m} = 1-\frac{a}{m}
\end{align*}
$$

If we made this correction, we would already fulfill the condition we wanted to attain.

$$
\sum_{i=1}^{m}S_i = \sum_{i=1}^{m}(S_{unadj\text{ i}} + x) = a + m \cdot x = a + m \cdot (1-\frac{a}{m}) = a + m - a = m
$$

However, **we are going to make a different correction for each $S_{unadj\text{ i}}$ instead of the same correction for each of them**. Specifically, the correction $x_i$ for each $S_{unadj\text{ i}}$ results from scaling the previous constant correction $x$ by a factor proportional to $S_{unadj\text{ i}}$ as shown below:

$$
x_i = m \cdot \frac{S_{unadj\text{ i}}}{a} \cdot x = m \cdot \frac{S_{unadj\text{ i}}}{a} \cdot (1-\frac{a}{m})
$$

We may check that this correction also fulfills the desired condition:

```{r, eval=FALSE, include=FALSE}
# TODO: COMPROBAR SI ESTA CORRECTION ES CORRECTA
df_seasonal <- df_seasonal %>%
  mutate(
    seasonal_c = s_unadj / mean(s_unadj, na.rm = TRUE)  # Normalize seasonal component
  ) %>%
  select(n_month, seasonal_c) 

#
```


$$
\begin{align*}
\sum_{i=1}^{m}S_i = \sum_{i=1}^{m}(S_{unadj\text{ i}} + x_i) = \sum_{i=1}^{m}(S_{unadj\text{ i}}) + \sum_{i=1}^{m}(m \cdot \frac{S_{unadj\text{ i}}}{a} \cdot (1-\frac{a}{m})) = \\ a  + \sum_{i=1}^{m}(m \cdot \frac{S_{unadj\text{ i}}}{a} \cdot (1-\frac{a}{m}))
\end{align*}
$$

Considering that $a$, $m$ and $(1-\frac{a}{m})$ are constants, we may take them out of the sum. Continuing where we left off:

$$
\begin{align*}
\sum_{i=1}^{m}S_i = a  + \sum_{i=1}^{m}(m \cdot \frac{S_{unadj\text{ i}}}{a} \cdot (1-\frac{a}{m})) = a + \frac{m}{a}(1-\frac{a}{m})\cdot \sum_{i=1}^{m}S_{unadj\text{ i}} = a + \frac{m}{a}(1-\frac{a}{m})\cdot a = \\ a + \frac{m}{a} \cdot a - \frac{m}{a} \cdot \frac{a}{m} \cdot a = a + m - a = m
\end{align*}
$$

In conclusion, the correction $x_i$ to be added to each $S_{unadj\text{ i}}$ to obtain the corrected $S_i$ is:

$$
x_i = m \cdot \frac{S_{unadj\text{ i}}}{a} \cdot x = m \cdot \frac{S_{unadj\text{ i}}}{a} \cdot (1-\frac{a}{m})
$$

And the adjusted seasonal component $S$ will be computed out of $S_{unadj}$ as follows

$$
S_i = S_{unadj\text{ i}} + x_i
$$

* **Output**: the adjusted seasonal component $S$. **Check at this point that indeed $\sum_{i=1}^{m}S_i=m$**

## Step 4 - Compute the remainder component

$R_t = \frac{y_t}{T_tS_t} = \frac{D_t}{S_t}$

# Exercise 1

Consider the last five years of the Gas data from `aus_production`

```{r}
gas <- tail(aus_production, 5*4) %>% select(Gas)
gas
```

a. Use `classical_decomposition` with `type=multiplicative` to calculate the trend-cycle and seasonal components

```{r, include = params$print_sol}
# Let us compute the classical decomposition using the function in the feasts library:
classical_dec <- 
  gas %>%
  model(
    c_mult = classical_decomposition(Gas, type = "multiplicative")
  ) %>%
  components()

classical_dec
```

b. Compute and plot the seasonally adjusted data.

```{r, include=params$print_sol}
classical_dec %>%
  
  as_tsibble() %>%
  
  autoplot(Gas, colour = "gray") +
  geom_line(aes(y=season_adjust), colour = "#0072B2") +
  labs(y = "Gas production (Petajoules)",
       title = "Australian gas production")
```

c. Compute manually the classical decomposition of the series (as we did in class, but keep in mind that this scheme is multiplicative and that now the seasonal period is 4 instead of 12).

```{r, include=params$print_sol}
### 1. Compute the trend with a 2x4 MA

# Compute moving averages
# Because the seasonal period is 12, we need a 2x12-MA
manual_decomposition <- gas %>%
  mutate(
    `4-MA` = slider::slide_dbl(Gas, mean,
                     .before = 1, .after = 2, .complete = TRUE),
    `2x4-MA` = slider::slide_dbl(`4-MA`, mean,
                      .before = 1, .after = 0, .complete = TRUE)
  )

# Plot the computed trend
manual_decomposition %>%
  autoplot(Gas, colour = "gray") +
  geom_line(aes(y = `2x4-MA`), colour = "#D55E00") +
  labs(y = "Gas production (Petajoules)",
       title = "Australian gas production")

# Check
all.equal(classical_dec$trend, manual_decomposition$`2x4-MA`)
```

```{r, include=params$print_sol}
### 2. Compute the de-trended component:

# Compute the detrended component:
manual_decomposition <- manual_decomposition %>%
  mutate(detrended = Gas / `2x4-MA`,
         n_quarter = quarter(Quarter)) # Add an identifier for the month (used later for a join statement)

manual_decomposition %>% autoplot(detrended)
```

```{r, include=params$print_sol}
### 3. Compute the unadjustedseasonal component grouping and 
# averaging the detrended component
res <- manual_decomposition %>%
  
  index_by(n_quarter) %>%
  
  #Compute seasonal component
  summarise(
    S_unadj = mean(detrended, na.rm = TRUE)
  ) 

res
```

```{r, include=params$print_sol}
### 4. Correct the seasonal components so that they add to m = 4
m = 4
a = sum(res$S_unadj)
x_i = res$S_unadj * (m/a) * (1 - a/m) # Vector of corrections

res <- res %>%
  # Correction so that the sum of the adjusted seasonal adds up to 4
  mutate(seasonal_c = S_unadj + x_i)

# Check that the corrected seasonal component adds up to 4
sum(res$seasonal_c)
```

```{r, include=params$print_sol}
#Check the aspect of the obtained seasonal component
res %>%
  autoplot(seasonal_c) +
  geom_point(color="red")
```

```{r, include=params$print_sol}
# Bring seasonal component into the dataset 
# (I chose to use a "join" statement, very convenient)
manual_decomposition <- left_join(manual_decomposition, res, by = "n_quarter")
manual_decomposition
```

```{r, include=params$print_sol}
# Compute remainder component:
manual_decomposition <- 
  manual_decomposition %>%
  mutate(remainder = detrended / seasonal_c)

select(manual_decomposition, Quarter, Gas, `2x4-MA`, seasonal_c, remainder)
```

```{r, include = params$print_sol}
# Comparison of manual computations with computations from the library:
all.equal(classical_dec$trend, manual_decomposition$`2x4-MA`)
all.equal(classical_dec$seasonal, manual_decomposition$seasonal_c)
all.equal(classical_dec$random, manual_decomposition$remainder)
```

d. Change one observations to be an outlier (e.g., add 300 to one observation), and recompute the seasonally adjusted data. What is the effect of the outlier?

```{r, include=params$print_sol}
gas %>%
  mutate(Gas = if_else(Quarter == yearquarter("2007Q4"), Gas + 300, Gas)) %>%
  model(decomp = classical_decomposition(Gas, type = "multiplicative")) %>%
  components() %>%
  as_tsibble() %>%
  autoplot(season_adjust) +
  labs(title = "Seasonally adjusted data", y = "Petajoules")
```

```{r, include=params$print_sol, eval=FALSE}
* The outlier is more or less in the middle of the series.
* The trend is approximated using a 2x4-MA. Since the outlier is in the middle 
of the series, this information is in the region where the moving average can 
be computed and the information of the outlier is captured by the trend component.
* The detrended component also inherits the information of the outlier, 
since it is computed as $y_t/T_t$. 
* The seasonal component is computed averaging this detrended component and 
therefore is also affected by the outlier. It will not accurately approximate 
the seasonal component.
```

e. Does it make any difference if the outlier is near the end rather than in the middle of the series?

```{r, include=params$print_sol, eval=FALSE}
As a result the seasonally adjusted data still has seasonality, because we have 
not adequately captured the seasonal component.
```

e. Does it make any difference if the outlier is near the end rather than in the middle of the series?

```{r, include=params$print_sol, eval=FALSE}
gas %>%
  mutate(Gas = if_else(Quarter == yearquarter("2010Q2"), Gas + 300, Gas)) %>%
  model(decomp = classical_decomposition(Gas, type = "multiplicative")) %>%
  components() %>%
  as_tsibble() %>%
  autoplot(season_adjust) +
  labs(title = "Seasonally adjusted data", y = "Petajoules")
```

```{r, include=params$print_sol, eval=FALSE}
* The outlier is located at the last point of the series.
* The trend is approximated using a 2x4-MA. Since the outlier is at the very 
end of the series, this information is in the region where the 2x4 moving 
average cannot be computed and the information of the outlier is not captured 
by the trend component.
* The detrended component is defined in exactly the same region as the trend 
approximation, since it is computed as $y_t/T_t$. Therefore the detrended 
component does not capture the information about the outlier.
* The seasonal component is computed averaging this detrended component. 
Therefore it does not inherit information from the outlier. The seasonal 
component is not affected by the presence of this outlier.
```