07-sculpting-data.Rmd

# Sculpting data using models


## Parametric regression

* **Parametric** means that the researcher or analyst assumes in advance that the data fits some type of distribution (e.g. the normal distribution). E.g. one may assume that
$$\color{blue}{y_i} = \color{red}{\beta_0} + \color{red}{\beta_1} \color{blue}{x_i} + \color{red}{\beta_2} \color{blue}{x_i^2} + \epsilon_i,$$
where $\epsilon_i \sim N(0, \color{red}{\sigma^2})$.

  * $\color{red}{red} = \text{estimated}$
  * $\color{blue}{blue} = \text{observed}$
* Because some type of distribution is assumed in advance, parametric fitting can lead to fitting a smooth curve that misrepresents the data.

## Example

```{r quad-good-fit, fig.height = 3, fig.width = 6}
set.seed(1)
tibble(id = 1:200) %>% 
  mutate(x = runif(n(), -10, 10),
         y = x^2 + rnorm(n(), 0, 5)) %>% 
  ggplot(aes(x, y)) +
  geom_point() +
  geom_smooth(method = "lm", se = FALSE,
              formula = y ~ poly(x, 2),
              size = 2, color = "red")
```

Still assuming a quadratic fit:

```{r quad-bad-fit, fig.height = 3, fig.width = 6}
tibble(id = 1:200) %>% 
  mutate(x = runif(n(), -10, 10),
         y = x^3 + rnorm(n(), 0, 5)) %>% 
  ggplot(aes(x, y)) +
  geom_point() +
  geom_smooth(method = "lm", se = FALSE,
              formula = y ~ poly(x, 2),
              size = 2, color = "red")
```


## Simulating data from parametric models

* If a model is say:
$$y = x^2 + e, \qquad e \sim N(0, 2^2)$$
we can simulate say $200$ observations from this model for $x\in(-10,10)$ by code as shown on the right. 
```{r sim-quad, include = FALSE}
set.seed(1)
df <- tibble(id = 1:200) %>% 
        mutate(x = runif(n(), -10, 10),
               y = x^2 + rnorm(n(), 0, 2))
df
```
```{r sim-plot, fig.height = 3, fig.width = 4}
ggplot(df, aes(x, y)) +
  geom_point()
```

## Logistic regression

* Not all parametric models assume Normally distributed errors.
* Logistic regression models the relationship between a set of explanatory variables $(x_{i1}, ..., x_{ik})$ and a set of .monash-blue[**binary outcomes**] $Y_i$ for $i = 1, ..., r$.
* We assume that $Y_i \sim B(n_i, p_i)$ and the model is given by 

$$\text{logit}(p_i) = \text{ln}\left(\dfrac{p_i}{1 - p_i}\right) = \beta_0 + \beta_1x_{i1} + ... + \beta_k x_{ik}.$$
* The function $f(p) = \text{ln}\left(\dfrac{p}{1 - p}\right)$ is called the .monash-blue[**logit**] function, continuous with range $(-\infty, \infty)$, and if $p$ is the probablity of an event, $f(p)$ is the log of the odds.

## Menarche

In 1965, the average age of 25 homogeneous groups of girls was
recorded along with the number of girls who have reached
menarche out of the total in each group.

```{r menarche-data, include = FALSE}
data(menarche, package = "MASS")
skimr::skim(menarche)
```
```{r menarche-plot, fig.height = 4, fig.width = 5}
ggplot(menarche, aes(Age, Menarche/Total)) + 
  geom_point() +
  geom_smooth(method = "glm",
              formula = y ~ x,
              se = FALSE,
              method.args = list(family = "binomial"))
```

## Simulating data from logistic regression 

```{r logit, echo = TRUE}
fit1 <- glm(Menarche/Total ~ Age, 
            family = "binomial", 
            data = menarche)
(beta <- coef(fit1))
```

* The fitted regression model is given as:
$$\text{logit}(\hat{p}_i) = \hat{\beta}_0  + \hat{\beta}_1 x_{i1}.$$
* Taking the exponential of both sides and rearranging we get
$$\hat{p}_i = \dfrac{1}{1 + e^{-(\hat{\beta}_0  + \hat{\beta}_1 x_{i1})}}.$$

```{r sim-logistic, echo = TRUE}
menarche %>% 
  rowwise() %>% # simulating from first principles
  mutate(
    phat = 1/(1 + exp(-(beta[1] + beta[2] * Age))),
    simMenarche = rbinom(1, Total, phat)) #<<
```

 If simulating data from a model object, `simulate` function usually can do this for you!

## Diagnostics for logistic regression models

* One diagnostic is to compare the observed and expected proportions under the logistic regression fit.

```{r fit-logistic, echo = TRUE}
df1 <- menarche %>% 
  mutate(
    pexp = 1/(1 + exp(-(beta[1] + beta[2] * Age))),
    pobs = Menarche / Total)
```
```{r plot-logistic, fig.height = 4, fig.width = 4}
ggplot(df1, aes(pobs, pexp)) +
  geom_point() + 
  geom_abline(slope = 1, intercept = 0,
              color = "red") +
  labs(x = "Observed proportion",
       y = "Expected proportion")
```

* Goodness-of-fit type test is used commonly to assess the fit as well.

* E.g. Hosmer–Lemeshow test, where test statistic is given as 


$$H = \sum_{i = 1}^r \left(\dfrac{(O_{1i} - E_{1g})^2}{E_{1i}} + \dfrac{(O_{0i} - E_{0g})^2}{E_{0i}}\right)$$
where $O_{1i}$ $(E_{1i})$ and $O_{0i}$ $(E_{0i})$ are observed (expected) frequencies for successful and non-successful events for group $i$, respectively.

```{r HLtest, echo = TRUE}
vcdExtra::HLtest(fit1)
```

## Diagnostics for linear models

Assumptions for linear models

For $i \in \{1, ..., n\}$,

$$Y_i = \beta_0 + \beta_1x_{i1} + ... + \beta_{k}x_{ik} + \epsilon_i,$$
where $\epsilon_i \sim NID(0, \sigma^2)$ or in matrix format,

$$\boldsymbol{Y} = \mathbf{X}\boldsymbol{\beta} + \boldsymbol{\epsilon}, \quad \boldsymbol{\epsilon} \sim N(\boldsymbol{0}, \sigma^2 \mathbf{I}_n)$$

* $\boldsymbol{Y} = (Y_1, ..., Y_n)^\top$,
* $\boldsymbol{\beta} = (\beta_0, ..., \beta_k)^\top$,
* $\boldsymbol{\epsilon} = (\epsilon_1, ..., \epsilon_n)^\top$, and
* $\mathbf{X} = \begin{bmatrix}\boldsymbol{1}_n & \boldsymbol{x}_1 & ... & \boldsymbol{x}_k \end{bmatrix}$, where $\boldsymbol{x}_j =(x_{1j}, ..., x_{nj})^\top$ for $j \in \{1, ..., k\}$

For $i \in \{1, ..., n\}$,

$$Y_i = \beta_0 + \beta_1x_{i1} + ... + \beta_{k}x_{ik} + \epsilon_i,$$
where $\color{red}{\epsilon_i \sim NID(0, \sigma^2)}$ or in matrix format,

$$\boldsymbol{Y} = \mathbf{X}\boldsymbol{\beta} + \boldsymbol{\epsilon}, \quad \color{red}{\boldsymbol{\epsilon} \sim N(\boldsymbol{0}, \sigma^2 \mathbf{I}_n)}$$

* $\boldsymbol{Y} = (Y_1, ..., Y_n)^\top$,
* $\boldsymbol{\beta} = (\beta_0, ..., \beta_k)^\top$, 
* $\boldsymbol{\epsilon} = (\epsilon_1, ..., \epsilon_n)^\top$, and 
* $\mathbf{X} = \begin{bmatrix}\boldsymbol{1}_n & \boldsymbol{x}_1 & ... & \boldsymbol{x}_k \end{bmatrix}$, where 
* $\boldsymbol{x}_j =(x_{1j}, ..., x_{nj})^\top$ for $j \in \{1, ..., k\}$

This means that we assume

1. $E(\epsilon_i) = 0$ for $i \in \{1, ..., n\}.$
2. $\epsilon_1, ..., \epsilon_n$ are independent.
3. $Var(\epsilon_i) = \sigma^2$ for $i \in \{1, ..., n\}$ (i.e. homogeneity).
4. $\epsilon_1, ..., \epsilon_n$ are normally distributed.

So how do we check it?

Plot $Y_i$ vs $x_i$ to see if there is $\approx$ a linear relationship between $Y$ and $x$.

```{r, echo=F, fig.height=3, fig.width=3, fig.align="center"}
dat <- read_csv(here::here("data/sleep.csv"))
dat %>%
  ggplot(aes(log(BodyWt), log(BrainWt))) + geom_point() +
  geom_smooth(method="lm", se=F) +
  labs(x="log(Body) (kg)", y="log(Brain) (g)") 
```

A boxplot of the residuals $R_i$ to check for symmetry.
```{r, echo=F, fig.height=2.5, fig.width=2.5, fig.align="center", dev='svg', dev.args=list(bg = "transparent")}
lm(log(BrainWt) ~ log(BodyWt), data=dat) %>%
  augment() %>% 
  ggplot(aes(1,.std.resid)) + 
  geom_boxplot() + 
  labs(x="", y="Residual") + 
  theme(axis.text.x = element_blank())
```

To check the homoscedasticity assumption, plot $R_i$ vs $x_i$. There should be no obvious patterns.

```{r, echo=F, fig.height=2.2, fig.width=2.2, fig.align="center", dev='svg', dev.args=list(bg = "transparent")}
lm(log(BrainWt) ~ log(BodyWt), data=dat) %>%
  augment() %>% 
  ggplot(aes(`log(BodyWt)`, .std.resid)) + 
  geom_point() + 
  labs(x="log(Body) (kg)", y="Residual") +
  geom_hline(yintercept=0, color="blue") 
```
A normal Q-Q plot, i.e. a plot of the ordered residuals vs $\Phi^{-1}(\frac{i}{n+1})$.

```{r, echo=F, fig.height=2.2, fig.width=2.2, fig.align="center", dev='svg', dev.args=list(bg = "transparent")}
lm(log(BrainWt) ~ log(BodyWt), data=dat) %>%
  augment() %>% 
  ggplot(aes(sample=.std.resid)) + 
  stat_qq() + stat_qq_line(color="blue")
```

## Assessing (A1) $E(\epsilon_i) = 0$ for $i=1,\ldots,n$

* It is a property of the least squares method that $$\sum_{i=1}^n R_i = 0,\quad
\text{so}\quad \bar R_i  = 0$$ for $R_i = Y_i - \hat{Y}_i$, hence (A1)  will always appear valid "overall". 
* Trend in residual versus fitted values or covariate can indicate "local"
failure of (A1). 
* What do you conclude from the following plots?

```{r sim-plots, echo=F, fig.width=12, fig.height=3, dev='svg', dev.args=list(bg = "transparent")}
set.seed(2019)
n <- 100
x <- seq(0, 1, length.out = n)
y1 <- x + rnorm(n) / 3                 #  Linear
y2 <- 3 * (x - 0.5) ^ 2 + 
  c(rnorm(n / 2)/3, rnorm(n / 2)/6)  #  Quadratic
y3 <- - 0.25 * sin(20 * x - 0.2) + x + rnorm(n) / 3    #  Non-linear
g1 <- lm(y1 ~ x) %>% augment() %>%
  ggplot(aes(.fitted, .resid)) + 
  geom_point() + 
  labs(x="Fitted Values", y="Residual", tag="(1)") +
  geom_hline(yintercept=0, color="blue") 
g2 <- lm(y2 ~ x) %>% augment() %>%
  ggplot(aes(.fitted, .resid)) + 
  geom_point() + 
  labs(x="Fitted Values", y="Residual", tag="(2)") +
  geom_hline(yintercept=0, color="blue") 
g3 <- lm(y3 ~ x) %>% augment() %>%
  ggplot(aes(.fitted, .resid)) + 
  geom_point() +
  labs(x="Fitted Values", y="Residual", tag="(3)") +
  geom_hline(yintercept=0, color="blue") 

g1 + g2 + g3
```

## Assessing (A2)-(A3) 

(A2) $\epsilon_1, \ldots ,\epsilon_n$ are independent

* If (A2) is correct, then residuals should appear randomly scattered
about zero if plotted against fitted values or covariate.
* Long sequences of positive residuals followed by sequences of negative residuals in $R_i$ vs $x_i$ plot suggests that the error terms are not independent.

(A3) $Var(\epsilon_i) = \sigma^2$ for $i=1,\ldots,n$

* If (A3) holds then the spread of the residuals should be roughly the same across the fitted values or covariate. 

Q-Q Plots

* The function `qqnorm(x)` produces a Q-Q plot of the ordered vector `x` against the quantiles of the normal distribution.
* The $n$ chosen normal quantiles $\Phi^{-1}(\frac{i}{n+1})$ are easy to calculate but more sophisticated ways exist:
   * $\frac{i}{n+1} \mapsto \frac{i-3/8}{n+1/4}$, default in `qqnorm`. 
   * $\frac{i}{n+1} \mapsto \frac{i-1/3}{n+1/3}$, recommended by Hyndman and Fan (1996).

### In R

```{r, echo = T, eval = F}
fit <- lm(y ~ x)
```


By "hand"
```{r, eval=F, echo = T}
plot(qnorm((1:n) / (n + 1)), sort(resid(fit)))
```

By `base`

```{r, eval=F, echo = T}
qqnorm(resid(fit))
qqline(resid(fit))
```

By `ggplot2`

```{r, eval=F, echo = T}
data.frame(residual = resid(fit)) %>% 
  ggplot(aes(sample = residual)) + 
  stat_qq() + stat_qq_line(color="blue")
```

##  Examining the simulated data further

```{r, echo=F, fig.height=4, fig.width=8, fig.align="center"}
dat3sim <- data.frame(y=c(y1, y2, y3), x=rep(x, times=3), Simulation=rep(1:3, each=length(x)))
dat3sim %>%
  ggplot(aes(x,y)) + geom_point() +
  geom_smooth(method="lm", se=F) + facet_grid( . ~ Simulation) +
  theme(axis.text = element_blank())
```

```{r, echo=F, fig.height=4, fig.width=8, fig.align="center",y2dev='svg', dev.args=list(bg = "transparent")}
M1 <- lm(y1 ~ x); M2 <- lm(y2 ~ x); M3 <- lm(y3 ~ x)
data.frame(residual=c(resid(M1), resid(M2), resid(M3)),
           Simulation=rep(1:3, each=length(x))) %>%
  ggplot(aes(x=factor(Simulation), y=residual)) + geom_boxplot() + labs(x="", y="Residual")
```


```{r, echo=F, fig.height=4, fig.width=8, fig.align="center",y2dev='svg', dev.args=list(bg = "transparent")}
data.frame(residual=c(resid(M1), resid(M2), resid(M3)),
           Simulation=rep(1:3, each=length(x))) %>%
  ggplot(aes(sample=residual)) + stat_qq() + 
  stat_qq_line(color="blue") +
  facet_grid( . ~ Simulation) 
```
Simulation scheme
```{r, echo = TRUE}
n <- 100
x <- seq(0, 1, length.out = n)
y1 <- x + rnorm(n) / 3                  #  Linear
y2 <- 3 * (x - 0.5) ^ 2 + 
  c(rnorm(n / 2)/3, rnorm(n / 2)/6)     #  Quadratic
y3 <- -0.25 * sin(20 * x - 0.2) + 
  x + rnorm(n) / 3                      #  Non-linear

M1 <- lm(y1 ~ x); M2 <- lm(y2 ~ x); M3 <- lm(y3 ~ x)

```

## Revisiting outliers

* We defined [outliers in week 4](https://eda.numbat.space/lectures/lecture-04a#9) as "observations that are significantly different from the majority" when studying univariate variables.
* There is actually no hard and fast definition. <br><br>

We can also define an outlier as a data point that emanates from a different model than do the rest of the data.

* Notice that this makes this definition *dependent on the model* in question.

## Pop Quiz

Would you consider the yellow points below as outliers?

```{r, echo=F, fig.height=5, fig.width=10, fig.align="center"}
n <- 20
set.seed(1)
shifty <- rep(0, n); shifty[5] <- 10
g1 <- data.frame(x=seq(1, 20, 1)) %>% 
    mutate(y=x + rnorm(n, 0, 1)) %>% 
  ggplot(aes(x, y + shifty, color=factor(shifty))) + geom_point(size=6) + 
  labs(x="x", y="y", tag="(A)") +
  guides(color=F) +
  scale_color_manual(values = c("black", "yellow"))

n <- 20
set.seed(2)
g2 <- data.frame(x=c(seq(1, 19, 1), 30)) %>% 
    mutate(y=x + rnorm(n, 0, 1)) %>% 
  ggplot(aes(x, y, color=factor(c(rep(0,19), 1)))) + geom_point(size=6) + 
  labs(x="x", y="y", tag="(B)") +
  guides(color=F)+
  scale_color_manual(values = c("black", "yellow"))

g1 + g2 
```

## Outlying values 

* As with simple linear regression the fitted model should not be used to predict $Y$ values for $\boldsymbol{x}$ combinations that are well away from the set of observed $\boldsymbol{x}_i$ values. 
* This is not always easy to detect!

```{r, fig.height = 4}
tibble(id = 1:20) %>% 
  mutate(x1 = runif(n()),
         x2 = 1 - 4 * x1 + x1^2 + rnorm(n(), 0, 0.1)) %>% 
  add_row(x1 = 0.6, x2 = 0.6) %>% 
  ggplot(aes(x1, x2)) +
  geom_point(size = 4) + 
  annotate("text", x = 0.55, y = 0.55, label = "P", 
           size = 10)
```

* Here, a point labelled P has $x_1$ and $x_2$ coordinates well within their respective ranges but P is not close to the observed sample values in 2-dimensional space. 

* In higher dimensions this type of behaviour is even harder to detect but we need to be on guard against extrapolating to extreme values. 

## Leverage 

* The matrix $\mathbf{H} = \mathbf{X}(\mathbf{X}^\top\mathbf{X})^{-1}\mathbf{X}^\top$ is referred to as the .monash-blue[**hat matrix**].
* The $i$-th diagonal element of $\mathbf{H}$, $h_{ii}$, is called the .monash-blue[**leverage**] of the $i$-th observation.
* Leverages are always between zero and one,
$$0 \leq h_{ii} \leq 1.$$
* Notice that leverages are not dependent on the response!
* Points with high leverage can exert a lot of influence on the parameter estimates

## Studentized residuals

In order to obtain residuals with equal variance, many texts recommend using the .monash-blue[**studentised residuals**]
$$R_i^* = \dfrac{R_i} {\hat{\sigma} \sqrt{1 - h_{ii}}}$$
for diagnostic checks.

## Cook's distance

* Cook's distance, $D$, is another measure of influence: 
\begin{eqnarray*}
D_i &=& \dfrac{(\hat{\boldsymbol{\beta}}- \hat{\boldsymbol{\beta}}_{[-i]})^\top Var(\hat{\boldsymbol{\beta}})^{-1}(\hat{\boldsymbol{\beta}}- \hat{\boldsymbol{\beta}}_{[-i]})}{p}\\
&=&\frac{R_i^2 h_{ii}}{(1-h_{ii})^2p\hat\sigma^2},
\end{eqnarray*}
where $p$ is the number of elements in $\boldsymbol{\beta}$, $\hat{\boldsymbol{\beta}}_{[-i]}$ and $\hat Y_{j[-i]}$ are least squares estimates and the fitted value obtained by fitting the model ignoring the $i$-th data point $(\boldsymbol{x}_i,Y_i)$, respectively.

## Social media marketing

Data collected from advertising experiment to study the impact of three advertising medias (youtube, facebook and newspaper) on sales.

```{r marketing-data, include = FALSE}
data(marketing, package="datarium")
skimr::skim(marketing)
```
```{r marketing-plot, fig.height = 6, fig.width = 7}
GGally::ggpairs(marketing, progress=F)
```

## Extracting values from models in R

* The leverage value, studentised residual and Cook's distance can be easily extracted from a model object using `broom::augment`.
  * `.hat` is the leverage value
  * `.std.resid` is the studentised residual
  * `.cooksd` is the Cook's distance

```{r, echo = TRUE}
fit <- lm(sales ~ youtube * facebook, data = marketing)
broom::augment(fit)

```

## Non-parametric regression

### LOESS

* LOESS (LOcal regrESSion) and LOWESS (LOcally WEighted
Scatterplot Smoothing) are .monash-blue[**non-parametric regression**] methods (LOESS is a generalisation of LOWESS)
* **LOESS fits a low order polynomial to a subset of neighbouring data** and can be fitted using `loess` function in `R`
* a user specified "bandwidth" or "smoothing parameter" $\color{blue}{\alpha}$ determines how much of the data is used to fit each local polynomial.

```{r dummy, fig.height = 3.2, fig.width = 6}
df2 <- tibble(id = 1:200) %>% 
  mutate(x = runif(n(), -10, 10),
         y = 0.5 * x + 3 * sin(x) + rnorm(n(), 0, 2))

```
```{r df2-plot, fig.height = 3}
ggplot(df2, aes(x, y)) +
  geom_point() +
  geom_smooth(se = FALSE, color = "red",
              # note "loess" doesn't take method.args! 
              # looks like a BUG in ggplot2
              method = stats::loess, size = 1,
              method.args = list(span = 0.4)) 
```

* $\alpha \in \left(\frac{\lambda + 1}{n}, 1\right)$ (default `span=0.75`) where $\lambda$ is the degree of the local polynomial (default `degree=2`) and $n$ is the number of observations.
* Large $\alpha$ produce a smoother fit.
* Small $\alpha$ overfits the data with the fitted regression capturing the random error in the data.

## How `span` changes the loess fit

```{r loess-span, fig.height = 5, fig.width = 8, out.width = "90%"}
library(gganimate)
fits <- tibble(span = seq(.1, 1, .05)) %>%
  rowwise() %>%
  do(mutate(augment(loess(y ~ x, df2, span = .$span)),
            span = .$span))

p <- ggplot(fits, aes(x, y)) +
  geom_point() +
  geom_line(aes(y = .fitted), color = "red", size = 1.2) +
  labs(title = 'span = {closest_state}')

p + transition_states(
    span,
    transition_length = 2,
    state_length = 1
  )
```

Code inspired by http://varianceexplained.org/files/loess.html


## How `loess` works

```{r animate-loess, gganimate = list(nframes = 500), fig.height = 5, fig.width = 8, out.width = "90%"}
dat <- df2 %>%
  crossing(center = unique(df2$x)) %>%
  mutate(dist = abs(x - center)) %>%
  filter(rank(dist) / n() <= .4) %>%
  mutate(weight = (1 - (dist / max(dist)) ^ 3) ^ 3) %>% 
  arrange(x, center)
p <- ggplot(dat, aes(x, y)) +
  geom_point(data = df2, color = "gray") +
  geom_point(aes(alpha = weight), color = "#6600cc") +
  geom_smooth(data = df2,
              se = FALSE, color = "red",
              method = stats::loess, size = 1,
              method.args = list(span = 0.4)) +
  geom_smooth(aes(group = center, weight = weight), method = stats::lm, se = FALSE, formula = y ~ poly(x, 2)) +
  geom_vline(aes(xintercept = center), lty = 2) +
  labs("span = 0.4") +
  guides(alpha = FALSE)#+
  #ggforce::facet_wrap_paginate(~center, nrow = 3, ncol = 3)
p + transition_states(center, 
                      transition_length = 1, 
                      state_length = 1)
```

## US economic time series

This dataset was produced from US economic time series data available from http://research.stlouisfed.org/fred2. 


```{r economics-data, include = FALSE}
data(economics, package = "ggplot2")
skimr::skim(economics)
```
```{r economics-plot, fig.height = 6, fig.width = 7}
ggplot(economics, aes(date, uempmed)) + 
  geom_point() +
  geom_smooth(method = loess, se = FALSE,
              method.args = list(span = 0.1)) +
  labs(x = "Date", y = "Median unemployment duration")
```

## How to fit LOESS curves in R?

### Model fitting

The model can be fitted using the `loess` function where 

* the default  span is 0.75 and 
* the default local polynomial degree is 2.

```{r loess, echo = TRUE}
fit <- economics %>% 
          mutate(index = 1:n()) %>% 
          loess(uempmed ~ index, #<<
                data = ., #<<
                span = 0.75, #<<
                degree = 2) #<<
```

### Showing it on the plot

In `ggplot`, you can add the loess using `geom_smooth` with `method = loess` and method arguments passed as list:
```{r loess-ggplot, echo = TRUE, fig.height = 3}
ggplot(economics, aes(date, uempmed)) +
  geom_point() + 
  geom_smooth(method = loess,  #<<
              method.args = list(span = 0.75, #<<
                                 degree = 2)) #<<
```

## Why non-parametric regression?

* Fitting a line to a scatter plot where noisy data values, sparse data points or weak inter-relationships interfere with your ability to see a line of best fit.
* Linear regression where least squares fitting doesn't create a line of good fit or is too labour intensive to use.
* Data exploration and analysis.
* Recall: In a parametric regression, some type of distribution is assumed in advance; therefore fitted model can lead to fitting a smooth curve that
misrepresents the data.
* In those cases, non-parametric regression may be a better choice. 
* *Can you think of where it might be useful?*


## Bluegills

Data were collected on length (in mm) and the age (in years) of 78 bluegills captured from Lake Mary, Minnesota in 1981.

Which fit looks better?

```{r bluegills-data, include = FALSE}
bg_df <- read.table(here::here("data/bluegills.txt"),
           header = TRUE)
skimr::skim(bg_df)
```
```{r bluegills-plot1, fig.height = 4, fig.width = 6}
ggplot(bg_df, aes(age, length)) + 
  geom_point() +
  geom_smooth(method = lm, se = FALSE) +
  labs(tag = "(A)", title = "Linear regression", x = "Age (in years)", y = "Length (in mm)")
```
```{r bluegills-plot2, fig.height = 4, fig.width = 6}
ggplot(bg_df, aes(age, length)) + 
  geom_point() +
  geom_smooth(method = lm, se = FALSE,
              formula = y ~ poly(x, 2)) +
  labs(tag = "(B)", title = "Quadratic regression",
       x = "Age (in years)", y = "Length (in mm)")
```

* Let's have a look at the residual plots.
* Do you see any patterns on either residual plot?

```{r bluegills-fit, include = FALSE}
fit1 <- lm(length ~ age, data = bg_df)
fit2 <- lm(length ~ poly(age, 2), data = bg_df)
df1 <- augment(fit1)
df2 <- mutate(augment(fit2), age = bg_df$age)
summary(fit1)
summary(fit2)
```
```{r bluegills-resplot1, fig.height = 4, fig.width = 6}
ggplot(df1, aes(age, .std.resid)) + 
  geom_point() +
  geom_hline(yintercept = 0) +
  labs(x = "Age", y = "Residual",
       tag = "(A)", title = "Linear regression")
```

```{r bluegills-resplot2, fig.height = 4, fig.width = 6}
ggplot(df2, aes(age, .std.resid)) + 
  geom_point() +
  geom_hline(yintercept = 0) +
  labs(x = "Age", y = "Residual",
       tag = "(B)", title = "Quadratic regression")
```

The structure is easily visible with the LOESS curve:

```{r bluegills-lresplot1, fig.height = 4, fig.width = 6}
ggplot(df1, aes(age, .std.resid)) + 
  geom_point() +
  geom_hline(yintercept = 0) +
  labs(x = "Age", y = "Residual",
       tag = "(A)", title = "Linear regression") +
  geom_smooth(method = loess, color = "red",
              se = FALSE)
```

```{r bluegills-lresplot2, fig.height = 4, fig.width = 6}
ggplot(df2, aes(age, .std.resid)) + 
  geom_point() +
  geom_hline(yintercept = 0) +
  labs(x = "Age", y = "Residual",
       tag = "(B)", title = "Quadratic regression") +
  geom_smooth(method = loess, color = "red",
              se = FALSE)
```

## Soil resistivity in a field

This data contains measurement of soil resistivity of an agricultural field.

```{r cleveland-data, include = FALSE}
data(cleveland.soil, package = "agridat")
skimr::skim(cleveland.soil)
```

```{r cleveland-plot1, fig.height = 5, fig.width = 4}
ggplot(cleveland.soil, aes(easting, northing)) +
  geom_point()
```

```{r cleveland-plot2, fig.height = 5, fig.width = 7}
library(lattice)
cloud(resistivity ~ easting * northing, pch = ".", data = cleveland.soil)
```

## Conditioning plots (Coplots)

```{r coplots, echo = TRUE, fig.height = 5.7, fig.width = 10}
library(lattice)
xyplot(resistivity ~ northing | equal.count(easting, 12),
       data = cleveland.soil, cex = 0.2,  
       type = c("p", "smooth"), col.line = "red", 
       col = "gray", lwd = 2)
```

## Coplots via `ggplot2`

* Coplots with `ggplot2` where the panels have overlapping observations is tricky.
* Below creates a plot for non-overlapping intervals of `easting`:

```{r ggcoplots, echo = TRUE, fig.height = 4, fig.width = 10}
ggplot(cleveland.soil, aes(northing, resistivity)) +
  geom_point(color = "gray") + 
  geom_smooth(method = "loess", color = "red", se = FALSE) +
  facet_wrap(~ cut_number(easting, 12))
```