Week 5: Functions

class: center, middle, inverse, title-slide

.title[
# <b>Week 5: Functions </b>
]
.subtitle[
## Data Analysis for Psychology in R 1<br><br>
]
.author[
### Patrick Sturt
]
.institute[
### Department of Psychology<br>The University of Edinburgh
]

---

# Weeks Learning Objectives
1. Understand the basic principles of functions.

2. Understand concept of data transformations.

3. Understand the calculation of z-scores.

---
# Topics for today
+ What is a function?

+ Linear and non-linear functions

+ How do we use functions in statistics?

+ An example of z-scores
  
---
# What is a function?
+ A function takes an **input**, **does something**, and provides an **output.**

+ **Input**

$$
x = 
`\begin{bmatrix}
1 \\ 
2 \\ 
3 \\ 
\end{bmatrix}`
$$

+ **Doing something**

$$
f(x) = x-2
$$

+ **An output **

$$
y = 
`\begin{bmatrix}
-1 \\ 
0 \\ 
1 \\ 
\end{bmatrix}`
$$

???
+ Functions can become as complex as we want them to be.
+ Why `$y$`?

---
# Functions and relations
+ It is important to think of the function as showing the *relationship* between input and output.

+ We can link this to the idea of relationships from week 4.

+ The function links an input (predictors, `$x$`), to an output (outcome, `$y$`)

+ So we can write

`$$y = f(x) = x-2$$`

---
# Visualising Functions
+ An important tool in understanding functions is to plot them.

+ So let's look at the following:

$$
y = f(x) = 10 + 2x
$$

???
+ This helps us both understand plots
+ And gain intuition about functions.
  
---
# Visualising Functions
+ Our input `$x$` is a vector of numbers:

$$
x = 
`\begin{bmatrix}
1 \\ 
2 \\ 
3 \\
4 \\
5 \\ 
6 \\ 
7 \\
8 \\
\end{bmatrix}`
$$

???
+ As well as the functions getting more complex, so can the inputs.
+ We are sticking with small examples to help visualize what is happening and get an intuition
    + But this could be 10,000 elements long
    + could contain values like 1.7875453
    + Computers can deal with all this for us quite easily!

---

# Visualising Simple Functions

.pull-left[
<table class="table" style="width: auto !important; margin-left: auto; margin-right: auto;">
 <thead>
  <tr>
   <th style="text-align:right;"> x </th>
   <th style="text-align:right;"> y </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 12 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 2 </td>
   <td style="text-align:right;"> 14 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 3 </td>
   <td style="text-align:right;"> 16 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 4 </td>
   <td style="text-align:right;"> 18 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 5 </td>
   <td style="text-align:right;"> 20 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 22 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 7 </td>
   <td style="text-align:right;"> 24 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 8 </td>
   <td style="text-align:right;"> 26 </td>
  </tr>
</tbody>
</table>
]

.pull-right[

```r
func_x <- tibble(
  x = c(1,2,3,4,5,6,7,8), 
* y = 10 + (2*x)
)
```

+ `tibble` is used to create a data set

+ `x` is our original data entered as a vector of numbers using `c()`

+ `y` is the output of the function `f(x) = 10+(2*x)`

]

---

# Visualising Simple Functions

.pull-right[
$$
y = f(x) = 10 + 2x
$$

+ Example row 1:

$$
10 + (2*1) = 12
$$

+ Example row 5:

$$
10 + (2*5) = 20
$$

]

---
# Visualising Functions
.pull-left[
![](data:image/png;base64,#dapR1_lec5_Functions_files/figure-html/unnamed-chunk-5-1.png)
]

.pull-right[

**Our Data**

<table class="table" style="width: auto !important; margin-left: auto; margin-right: auto;">
 <thead>
  <tr>
   <th style="text-align:right;"> x </th>
   <th style="text-align:right;"> y </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 12 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 2 </td>
   <td style="text-align:right;"> 14 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 3 </td>
   <td style="text-align:right;"> 16 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 4 </td>
   <td style="text-align:right;"> 18 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 5 </td>
   <td style="text-align:right;"> 20 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 22 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 7 </td>
   <td style="text-align:right;"> 24 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 8 </td>
   <td style="text-align:right;"> 26 </td>
  </tr>
</tbody>
</table>
]

---
# Visualising Functions

.pull-left[
![](data:image/png;base64,#dapR1_lec5_Functions_files/figure-html/unnamed-chunk-7-1.png)
]

.pull-right[

**Our Data**

---
# Visualising Functions

.pull-left[
![](data:image/png;base64,#dapR1_lec5_Functions_files/figure-html/unnamed-chunk-9-1.png)
]

.pull-right[

**Our Data**

---
# Visualising Functions

.pull-left[
![](data:image/png;base64,#dapR1_lec5_Functions_files/figure-html/unnamed-chunk-11-1.png)
]

.pull-right[

**Our Data**

---
# Visualising Functions (R-code)

.pull-left[
![](data:image/png;base64,#dapR1_lec5_Functions_files/figure-html/unnamed-chunk-13-1.png)
]

.pull-right[

**R-code**

```r
*fun1 <- function(x) 10+(2*x)

ggplot(func_x, aes(x, y)) + 
* geom_point(colour = "red", size = 2) +
* stat_function(fun = fun1) +
  scale_x_continuous(name = "x", breaks = c(0:10), 
                     labels = c(0:10), limits = c(0,10)) +
  scale_y_continuous(name = "y", breaks = c(seq(0,30,5)), 
                     labels = c(seq(0,30,5)), limits = c(0,30)) 
```
]

---
# Multiple arguments
+ Functions can take multiple arguments. Consider:

$$
y = f(x,z) = 10 + (x*z)
$$

+ Where:

$$
x = 
`\begin{bmatrix}
1 \\ 
2 \\ 
3 \\ 
\end{bmatrix}`
$$

$$
z = 
`\begin{bmatrix}
1 \\ 
2 \\ 
3 \\ 
\end{bmatrix}`
$$

---

# Multiple arguments

.pull-left[
<table class="table table-striped" style="width: auto !important; margin-left: auto; margin-right: auto;">
 <thead>
  <tr>
   <th style="text-align:right;"> x </th>
   <th style="text-align:right;"> z </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 2 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 3 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 2 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 2 </td>
   <td style="text-align:right;"> 2 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 2 </td>
   <td style="text-align:right;"> 3 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 3 </td>
   <td style="text-align:right;"> 1 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 3 </td>
   <td style="text-align:right;"> 2 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 3 </td>
   <td style="text-align:right;"> 3 </td>
  </tr>
</tbody>
</table>
]

.pull-right[

+ Notice that when we have multiple inputs, our rows correspond to pairs of inputs.

+ So `$x$` = 1, pairs with:
  + `$z$` = 1
  + `$z$` = 2
  + `$z$` = 3
  
+ and so on for all values of `$x$`

]

---

# Multiple arguments

.pull-left[
<table class="table table-striped" style="width: auto !important; margin-left: auto; margin-right: auto;">
 <thead>
  <tr>
   <th style="text-align:right;"> x </th>
   <th style="text-align:right;"> z </th>
   <th style="text-align:right;"> f(x,z) </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 11 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 2 </td>
   <td style="text-align:right;"> 12 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 3 </td>
   <td style="text-align:right;"> 13 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 2 </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 12 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 2 </td>
   <td style="text-align:right;"> 2 </td>
   <td style="text-align:right;"> 14 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 2 </td>
   <td style="text-align:right;"> 3 </td>
   <td style="text-align:right;"> 16 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 3 </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 13 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 3 </td>
   <td style="text-align:right;"> 2 </td>
   <td style="text-align:right;"> 16 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 3 </td>
   <td style="text-align:right;"> 3 </td>
   <td style="text-align:right;"> 19 </td>
  </tr>
</tbody>
</table>
]

.pull-right[

$$
y = f(x,z) = 10 + (x*z)
$$

+ Example 1, row 2

$$
10 + (1*2) = 12
$$

+ Example, row 8

$$
10 + (3*2) = 16
$$

]

---
# Linear vs non-linear functions
+ Each of the examples so far have been linear functions.
    + If we plot them, we get a straight line (or flat surface)
    
+ Can also have non-linear functions:
    + A non-linear function would contain powers or roots

---
# Non-linear functions

.pull-left[
![](data:image/png;base64,#dapR1_lec5_Functions_files/figure-html/unnamed-chunk-17-1.png)

]

.pull-right[

**Example of non-linear function**

$$
y = f(x) = 5 + x^2
$$

]

---
# Why are functions important?
+ There are going to be lots of examples of functions in action.

+ Two primary examples are:
    + **Data transformations**
    + **Describing formal models**

+ We will start with transformations, and come back to models at the end of the course.

---
# z-scores
+ One of the most common transformations in data analysis is standardizing variables.

+ What is standardizing?

+ It is putting all variables onto the same scale so they can be compared.

+ We refer to standardized variables as `$z$`-scores (the reason we will explain later)

+ `$z$`-score:

`$$z = \frac{x - \mu}{\sigma}$$`

---
# Z-score for measured variable
+ `$z$`-score for `$x$`:

`$$z_{x_i} = \frac{x_i - \bar{x}}{s_x}$$`

+ Where
  + `$x_i$` = individual score on `$x$`
  + `$\bar{x}$` = mean of `$x$`
  + `$s_x$` = standard deviation of `$x$`

---
# z-scores
+ A `$z$`-score will have a mean = 0, and a SD = 1.

+ What this means is there is a standard way to interpret `$z$`-scores.

+ `$z$`-score = 1.5, means a respondent is 1.5 SD above the mean.

+ `$z$`-score = -2, means a respondent is 2 SD below the mean.

---
# Summary of today
+ Functions take input, do something, and produce an output.

+ Functions can have multiple arguments, be linear or non-linear

+ Typically we will visualize functions

+ We use functions frequently in statistics.

+ In fact almost everything we are going to see involves functions.

---
# Next tasks

+ This week:
  + Complete your lab
  + Come to office hours
  + No weekly quiz - submit group Formative Report A by Friday 12 noon.

+ Next week, there will be no new lecture material:
  + No lectures on Monday and Tuesday next week
  + Use next week to review and catch up on weeks 1-5.
  + In week 7 we will begin looking at probability.

+ Labs will still take place next week:
  + Make sure you still go to labs next week
  + You will receive in-person feedback on Formative Report A