07-model.Rmd

---
title: "Model"
date: "2019-03-25"
output: html_document
---

```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE)
```

# Models

There are two parts to a model:

1. First, you define a *family of models* where you specify what is the generic relationship between dependent and independent variables. 

y = a + b * x, where y and x are variables in your data and a and b are model coeficients that are unknown.

2. Next, you generate a *fitted model* by finding the model from the family that is closest to your data. 

Fitted model can look something like y = 30 + 3 * x.

## Linear model

```{r}
library(tidyverse)
library(broom)
library(modelr)
```


First, lets have 1:1 relationship between dependent (y) and independent variable (x).

Simplest model y = x where Intercept is always set to 0 and slope = 1.

Generate data:

```{r}
x <- 0:100
y <- x
```


Plot:

```{r}
p <- tibble(x, y) %>% 
  ggplot() +
  geom_line(aes(x, y)) +
  geom_abline(linetype = "dashed")
p
```

Let's add variable Intercept to the model.

y = a + x

Lets assign random value 30 to a:

```{r}
x <- 0:100 
a <- 30
y <- a + x
```



```{r}
tibble(x, y) %>% 
  ggplot() +
  geom_line(aes(x, y)) +
  geom_abline(linetype = "dashed")
```

Setting a to 30 moved line upwards and y = 30 when line crosses x axis.

Intercept determines y value when x = 0.

What happens when we multiply x with constant instead of addition:

y =  b * x

Let's assign random value 3 to b.

```{r}
x <- 0:200
b <- 3
y <- b * x
```


```{r}
tibble(x, y) %>% 
  ggplot() +
  geom_line(aes(x, y)) +
  geom_abline(linetype = "dashed")
```

Now we changed slope of the line, which specifies how much y changes when x changes 1 unit.
When b = 3, then x increases by 1 then y increases by 3 units.

Full simple linear formula with tunable Intercept and slope 

y = a + b * x

```{r}
a <- 30
b <- 3
x <-  0:100 
y <-  a + b * x
```


```{r}
tibble(x, y) %>% 
  ggplot() +
  geom_line(aes(x, y)) +
  geom_abline(linetype = "dashed")
```


Fitting Sepal.length vs Petal.length using iris dataset:
```{r}
model <- lm(Sepal.Length ~ Petal.Length, data = iris)
summary(model)
```

### Visualising a model

augment function from broom package adds predicted values from model to original data frame under variable .fitted:

```{r}
augment(model, iris) %>% 
  ggplot(aes(Petal.Length, Sepal.Length)) +
  geom_point(aes(color = Species)) +
  geom_line(aes(y = .fitted))
```

Predictions can be added to data also with add_predictions() function from modelr package:
```{r}
library(modelr)
add_predictions(iris, model)
```


To predict Sepal.length from our model we need model coeficients -- Intercept and slope.

Model coeficients can be extracted with coef() function:
```{r}
coef(model)
```

Here a = (Intercept) ja b (slope) = Petal.Length ehk 0.41.

## Predict from linear model

Let's extract model coeficients and plug into a linear model formula:
```{r }
a <- coef(model)[1] # Intercept
b <- coef(model)[2] # slope
petal_length <- iris$Petal.Length
sepal_length <-  a + b * petal_length
```

We can estimate goodness of fit from our predictions by calculating the root-mean-square error (RMSE). 

RMSE is a measure of the differences between values predicted by a model and the values observed.

RMSE is always non-negative, and a value of 0 (almost never achieved in practice) would indicate a perfect fit to the data. In general, a lower RMSD is better than a higher one.

Here we calculate RSME "manually" to illustrate how it's obtained:
```{r}
(sepal_length - iris$Sepal.Length) %>%
  (function(x) x^2) %>% 
  sum() %>% 
  (function(x) x / length(sepal_length)) %>% 
  sqrt()
```

But there are also functions around to calculate RMSE. 
This one comes from modelr package.
```{r}
rmse(model, iris)
```

Residuals, difference between predicted and original values, can be obtained also by resid() function:
```{r}
resid(model)
```

And residuals can be added to original dataframe with add_residuals function from modelr:
```{r}
iris %>% add_residuals(model)
```

How does fit look like, are residuals ~normally distributed:
```{r}
iris %>% 
  add_residuals(model) %>% 
  ggplot() +
  geom_histogram(aes(resid), bins = 30)
```