---
title: "sigmoid"
author: "Bastiaan Quast"
date: "`r Sys.Date()`"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{sigmoid}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

The `sigmoid()` function returns the sigmoid value of the input(s), by default this is done using the **standard logistic function**.

```{r}
library(sigmoid)
sigmoid(3)
```

Inputs can also be tensors, such as vectors, matrices, or arrays.

```{r}
sigmoid(-5:5)
sigmoid( matrix(-3:5,nrow=3) ) # etc.
```

The `sigmoid()` function is a wrapper, which by default uses the `logistic()` function, it can also use other methods.

## Gompertz

```{r}
sigmoid( -5:5, method='Gompertz' )
```

These functions can also be accessed directly.

```{r}
Gompertz(-1:-5)
```

## Rectified Linear Unit (ReLU)

Rectified Linear Unit (ReLU)

```{r}
sigmoid( -5:5, method='ReLU')
```

Leaky Rectified Linear Unit

```{r}
sigmoid( -5:5, method="leakyReLU")
```

## Mappings

These mappings are similar but not identical.

```{r sigmoid-shape}
library(ggplot2)

input = -5:5

df = data.frame(input, logistic(input), Gompertz(input))

ggplot(df, aes(input, logistic(input))) + geom_line() +
  geom_line(aes(input,Gompertz(input)), colour='red')
```

## Inverse

The wrapper can also apply the inverse of the method, returning the original values.

```{r}
sigmoid( sigmoid(-5:5), inverse=TRUE )
```

Which also works for other methods.

```{r}
sigmoid( sigmoid(-5:5, method='Gompertz'), method='Gompertz', inverse=TRUE )
```

In addition to this, the SoftMax algorithm can be pre applied.

```{r, results='hold'}
sigmoid( -3:5 )
sigmoid( -3:5, SoftMax = TRUE )
```

Several parameters can be specified (for details see `help(logistic)`, etc.). This can for instance be used to preserve greater entropy.

```{r, message=FALSE}
x = seq(1,5, by=0.05)
qplot(sigmoid(x))
qplot( sigmoid(x, k=sd(x), x0=mean(x) ) )
```