---
title: "Using Tidy FFT"
subtitle: "Analyze Time Series data with tidy_fft"
author: "Steven P. Sanderson II, MPH"
date: "`r Sys.Date()`"
output: 
  rmarkdown::html_vignette:
    toc: true
    toc_depth: 2
vignette: >
  %\VignetteIndexEntry{Using Tidy FFT}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>",
  fig.width = 8,
  fig.height = 5
)
```

```{r setup}
library(healthyR.ts)
```

# Introduction

In this vignette we will discuss how to use the `tidy_fft` function, what it does,
and what it produces.

# The Function

The `tidy_fft` function has only a few parameters, six to be exact. There are some
sensible defaults made. It is important that when you use this function, that you
supply it with a full time-series data set, one that has no missing data in it as
this will affect your results.

## Funcation and Parameters

The function and its full parameters are as follows:

```r
tidy_fft(
  .data,
  .date_col,
  .value_col,
  .frequency = 12L,
  .harmonics = 1L,
  .upsampling = 10L
)
```
The `.data` argument is the actual formatted data that will get passed to the function,
the time series data. The `.date_col` argument is the column that holds the datetime
of interest. The `.value` column is the column that holds the value that is being
analyzed by the function, this can be counts, averages, any type of value that is
in the time series. The `.frequency` argument details the cyclical nature of the data, 
is it 12 for monthly, 7 for weekly, etc. The `.harmonics` argument will tell the 
function how many times the `fft` should be run internally and how many filters should
be made. Finally the `.upsampling` argument will tell the function how much the
function should up sample the time parameter.

Let us now work through a simple example.

# Example

## Data

Lets get started with some data.

```{r example_data}
suppressPackageStartupMessages(library(dplyr))
suppressPackageStartupMessages(library(ggplot2))
suppressPackageStartupMessages(library(timetk))

data_tbl <- AirPassengers %>%
    ts_to_tbl() %>%
    select(-index)
```

Now that we have our sample data, let's check it out.

```{r data_glimpse}
glimpse(data_tbl)
```

## Plot Data

Lets take a look at a time series plot of the data.

```{r ts_plt}
suppressPackageStartupMessages(library(timetk))

data_tbl %>%
  plot_time_series(
    .date_var = date_col,
    .value    = value
  )
```

Now that we know what our data looks like, lets go ahead and run it through the function
and assign it to a variable called `output`

## Run Function

```{r run_func, message=FALSE, warning=FALSE}
output <- tidy_fft(
  .data = data_tbl,
  .date_col = date_col,
  .value_col = value,
  .harmonics = 8,
  .frequency = 12,
  .upsampling = 5
)
```

Now that we have run the function, let's take a look at the output.

# Output

The function invisibly returns a list object, hence the need to assign it to a variable.
There are a total of 4 different sections of data in the list that are returned. These 
are:

  * data
  * plots
  * parameters
  * model

## Output Data

In this section we will go over all of the data components that are returned. We
can access all of the data in the usual format `output$data`, which in of itself 
will return another list of objects, 7 to be specific. Lets go through them all.

### data

The data element accessed by `output$data$data` is the original data with a few
elements added to it. Let's take a look:

```{r data_data}
output$data$data %>%
  glimpse()
```

### error_data

The error_data element accessed by `output$data$error_data` is a `tibble` that
has the original data, plus a few other elements and an error term that is the
actual value minus the harmonic output. This is done for each harmonic level.

```{r error_data}
output$data$error_data %>%
  glimpse()
```

### input_vector

The input_vector is just the value column that was passed to the function.
```{r input_vector}
output$data$input_vector
```

### maximum_harmonic_tbl

The maximum_harmonic_tbl is a `tibble` that has data regarding the maximum harmonic
entered into the function, this will be the most flexible data returned.

```{r max_har_tbl}
output$data$maximum_harmonic_tbl %>%
  glimpse()
```

### differenced_value_tbl

The `differenced_value_tbl` is a `tibble` that has a lag 1 difference of the value
column supplied.

```{r diff_val_tbl}
output$data$differenced_value_tbl %>%
  glimpse()
```

### dff_tbl

The `dff_tbl` is a `tibble` that is returned that has the fft values, the complex,
real and imaginary parts.

```{r dff_tbl}
output$data$dff_tbl %>%
  glimpse()
```

### ts_obj

The last data piece of the data section is the `ts_obj`. This is a `ts` version of
the `input_vector`

```{r ts_obj}
output$data$ts_obj
```

## Output Plots

There are a total of five plots that are returned in the list. Three of them are `ggplot` plots and two of them are `plotly::ggplotly` plots.

### harmonic_plt

The `harmonic_plot` is a `ggplot` plot that shows all of the harmonic waves on the same graph if you set `.harmonics` greater than 1.

```{r har_plt, message=FALSE, warning=FALSE}
output$plots$harmonic_plot
```

### diff_plot

The `diff_plot` is a `ggplot` plot of the lag 1 `differenced_value_tbl`

```{r diff_val_plt, message=FALSE, warning=FALSE}
output$plots$diff_plot
```

### max_har_plot

The `max_har_plot` is a `ggplot` plot of the maximum harmonic wave entered into `.harmonics`

```{r max_har_plt, message=FALSE, warning=FALSE}
output$plots$max_har_plot
```

### harmonic_plotly

The `harmonic_plotly` is a `plotly::ggplotly` plot of the `harmonic_plot`

```{r har_pltly, message=FALSE, warning=FALSE}
output$plots$harmonic_plotly
```

### max_har_plotly

The `max_har_plotly` is a `plotly::ggplotly` plot of the `max_har_plot`

```{r max_har_pltly, message=FALSE, warning=FALSE}
output$plots$max_har_plotly
```

## Output Parameters

### parameters

The `parameters` element is a list of input parameters and internal parameters.

```{r parameters}
output$parameters
```

## Output Model

The `model` portion has four pieces to it which we will look at below.

### m

The parameter `m` is an internal parameter that is equal to `.harmonics` / 2. This is fed into `TSA::harmonic` along with the `ts_obj`

The parameter `harmonic_obj` is the object returned from `TSA::harmonic`

The parameter `harmonic_model` is the harmonic model from the `TSA::harmonic`

The parameter `model_summary` is a summary of the harmonic model.

```{r m}
output$model$m
```

```{r harmonic_obj}
output$model$harmonic_obj %>% head()
```

```{r har_model}
output$model$harmonic_model
```

```{r har_summary}
output$model$model_summary
```
