07-temporal.Rmd

# Temporal and Spatial Effects
```{r, echo=FALSE, message=FALSE,warning=FALSE}
rm(list=ls())
library(dplyr)
library(flair)
library(lme4)
library(MCMCglmm)
library(scales)
library(squidSim)
library(knitr)
set.seed(25)
```
## Simple Temporal Effects

We might have measured a variable over the course of a certain time period (e.g. 20 years). We might expect that there is stochastic year-to-year variation, which we can simulate already. However we might also want to simulate patterns in that temporal data. We can treat the levels associated with a particular grouping factor (e.g. year) as both a factor and continuous.

To treat a grouping factor as continuous, we use `covariate=TRUE` in the parameter list. In this way we can simulate a linear effect of year:
```{r}
squid_data <- simulate_population( 
  data_structure= make_structure(structure = "year(20) + sex(2)/individual(50)",repeat_obs=20),
  parameters=list(
    year_cont = list(
      group="year",
      names= "year_cont",
      covariate=TRUE,
      beta=0.3
    ),
    year = list(
      vcov = 0.8
    ),
    residual=list(
      vcov = 1
      )
    )
)
```

note we have specified `group` in the parameter list. This enables us to link a set of parameters to the grouping factor in the data structure. This doesn't have to be specified and defaults to the name of the list item.


```{r,fig.width=6,fig.height=6}
data <- get_population_data(squid_data)
head(data)
plot(y ~ year_cont, data)
```


Here we can see there is within year variation, year to year variation, as well as a linear directional year effect.
```{r}
lmer(y ~ year_cont + (1|year), data)
```

In a similar way we can also simulate a quadratic effect of time.

```{r,fig.width=6,fig.height=6}
squid_data <- simulate_population(
  data_structure = make_structure(structure = "year(20) + sex(2)/individual(50)",repeat_obs=20),
  parameters=list(
    year_cont = list(
      group="year",
      names= c("year_cont"),
      covariate=TRUE,
      beta=c(0.3)
    ),
    interactions=list(
      names= c("year_cont:year_cont"),
      beta=c(-0.05)
    ),
    year = list(
      vcov = 1
    ),
    residual=list(
      vcov = 0.8
    )
  )
)

data <- get_population_data(squid_data)

plot(y~year_cont,data)
```

<br>

## Cyclical Temporal Effects
The `squidR` function in the {squid} R package uses the sinusoidal equation to implement cyclical temporal effects:

<div class="alert alert-info">

$$
y = A sin(B(x - C)) + D
$$
</div>

 where A is the amplitude, $B/2\pi$ is the period $C/B$ is the horizontal shift and D is the vertical shift. We can visualise this

```{r}
 
time <- 1:20

amplitude <- 10       # |A| = the amplitude
period <- 10
h_shift <- 3
v_shift <- 5

B <- (2*pi) / abs(period) # 2pi/|B| = the period 
cyclic_effect <- amplitude*sin(B*time - B^2*h_shift ) + v_shift

plot(cyclic_effect~time)

```

We can simulate this using the model part of the `simulate_population()`, adding the extra parameters for the cyclical effects into the year_cont part of the list.

```{r}
squid_data <- simulate_population(
  data_structure= make_structure(structure = "year(20) + sex(2)/individual(50)",repeat_obs=1),

  parameters=list(
    year_cont = list(
      group="year",
      names= "linear_effect",
      covariate=TRUE,
      beta=0.3,
      amplitude = 2,       # |A| = the amplitude
      period = 10,
      h_shift = 3,
      v_shift = 5
    ),
    year = list(
      vcov = 1.2
    ),
    residual=list(
      vcov = 1
      )
    ), 

  model=" B =(2*pi) / abs(period);
          cyclic_effect = amplitude*sin(B*I(linear_effect) - B^2*h_shift ) + v_shift;
          y = linear_effect + cyclic_effect + year_effect + residual"
)

data <- get_population_data(squid_data)

plot(y~year,data)


```

## Temporal Autocorrelation {#temporalauto}



## Spatial Autocorrelation {#spatialauto}