2.1.1 Model

$$\begin{equation} \beta_j \sim \text{N}(0, \mathtt{sd}^2) \end{equation}$$

2.1.2 All defaults

sd = 1

set.seed(0)

NFix() |>
  generate(n_element = 10, n_draw = 8) |>
  plot_exch()

2.1.3 Reduce sd

sd = 0.01

set.seed(0)

NFix(sd = 0.01) |>
  generate(n_element = 10, n_draw = 8) |>
  plot_exch()

2.2.1 Model

$$\begin{align} \beta_j & \sim \text{N}(0, \tau^2) \\ \tau & \sim \text{N}^+(0, \mathtt{s}) \end{align}$$

2.2.2 All defaults

s = 1

set.seed(0)

N() |>
  generate(n_element = 10, n_draw = 8) |>
  plot_exch()

2.2.3 Reduce s

s = 0.01

set.seed(0)

N(s = 0.01) |>
  generate(n_element = 10, n_draw = 8) |>
  plot_exch()

3.1.1 Model

$$\begin{align} \beta_1 & \sim \text{N}(0, \mathtt{sd}^2) \\ \beta_j & \sim \text{N}(\beta_{j-1}, \tau^2), \quad j = 2, \cdots, J \\ \tau & \sim \text{N}^+(0, \mathtt{s}^2) \end{align}$$

3.1.2 All defaults

s = 1, sd = 1

set.seed(0)

RW() |>
  generate(n_draw = 8) |>
  plot_cor_one()

3.1.3 Reduce s

s = 0.01, sd = 1

set.seed(0)

RW(s = 0.01) |>
  generate(n_draw = 8) |>
  plot_cor_one()

3.1.4 Reduce s and sd

s = 0.01, sd = 0

set.seed(0)

RW(s = 0.01, sd = 0) |>
  generate(n_draw = 8) |>
  plot_cor_one()

3.2.1 Model

$$\begin{align} \beta_1 & \sim \text{N}(0, \mathtt{sd}^2) \\ \beta_2 & \sim \text{N}(\beta_1, \mathtt{sd\_slope}^2) \\ \beta_j & \sim \text{N}(2\beta_{j-1} - \beta_{j-2}, \tau^2), \quad j = 3, \cdots, J \\ \tau & \sim \text{N}^+(0, \mathtt{s}^2) \end{align}$$

3.2.2 All defaults

s = 1, sd = 1, sd_slope = 1

set.seed(0)

RW2() |>
  generate(n_draw = 8) |>
  plot_cor_one()

3.2.3 Reduce s

s = 0.01, sd = 1, sd_slope = 1

set.seed(0)

RW2(s = 0.01) |>
  generate(n_draw = 8) |>
  plot_cor_one()

3.2.4 Reduce s and sd_slope

s = 0.01, sd = 1, sd_slope = 0.01

set.seed(0)

RW2(s = 0.01, sd_slope = 0.01) |>
  generate(n_draw = 8) |>
  plot_cor_one()

3.2.5 Reduce s, sd, and sd_slope

s = 0.01, sd = 0, sd_slope = 0.01

set.seed(0)

RW2(s = 0.01, sd = 0, sd_slope = 0.01) |>
  generate(n_draw = 8) |>
  plot_cor_one()

3.3.1 Model

$$\begin{equation} \beta_j \sim \text{N}\left(\phi_1 \beta_{j-1} + \cdots + \phi_{\mathtt{n\_coef}} \beta_{j-\mathtt{n\_coef}}, \omega^2\right) \end{equation}$$ TMB derives a value of $\omega$ that gives each $\beta_j$ variance $\tau^2$. The prior for $\tau$ is $$\begin{equation} \tau \sim \text{N}^+(0, \mathtt{s}^2). \end{equation}$$ The prior for each $\phi_k$ is $$\begin{equation} \frac{\phi_k + 1}{2} \sim \text{Beta}(\mathtt{shape1}, \mathtt{shape2}). \end{equation}$$

3.3.2 All defaults

n_coef = 2, s = 1

set.seed(0)

AR() |>
  generate(n_draw = 8) |>
  plot_cor_one()

3.3.3 Increase n_coef

set.seed(0)

AR(n_coef = 3) |>
  generate(n_draw = 8) |>
  plot_cor_one()

3.3.4 Reduce s

set.seed(0)

AR(s = 0.01) |>
  generate(n_draw = 8) |>
  plot_cor_one()

3.3.5 Specify ‘along’ and ‘by’ dimensions

set.seed(0)

AR() |>
  generate(n_along = 20, n_by = 3, n_draw = 8) |>
  plot_cor_many()

3.3.6 Specify ‘along’ and ‘by’ dimensions, set con = "by"

set.seed(0)

AR(con = "by") |>
  generate(n_along = 20, n_by = 3, n_draw = 8) |>
  plot_cor_many()

3.4.1 Model

$$\begin{equation} \beta_j \sim \text{N}\left(\phi \beta_{j-1}, \omega^2\right) \end{equation}$$ TMB derives a value of $\omega$ that gives each $\beta_j$ variance $\tau^2$. The prior for $\tau$ is $$\begin{equation} \tau \sim \text{N}^+(0, \mathtt{s}^2). \end{equation}$$ The prior for $\phi$ is $$\begin{equation} \frac{\phi - \mathtt{min}}{\mathtt{max} - \mathtt{min}} \sim \text{Beta}(\mathtt{shape1}, \mathtt{shape2}). \end{equation}$$

3.4.2 All defaults

s = 1, min = 0.8, max = 0.98

set.seed(0)

AR1() |>
  generate(n_draw = 8) |>
  plot_cor_one()

3.4.3 Reduce s

set.seed(0)

AR1(s = 0.01) |>
  generate(n_draw = 8) |>
  plot_cor_one()

3.4.4 Reduce s and modify min, max

set.seed(0)

AR1(s = 0.01, min = -1, max = 1) |>
  generate(n_draw = 8) |>
  plot_cor_one()

3.5.1 Model

$$\begin{align} \beta_j & = \alpha_j + \lambda_j \\ \alpha_1 & \sim \text{N}(0, \mathtt{sd}^2) \\ \alpha_j & \sim \text{N}(\alpha_{j-1}, \tau^2), \quad j = 2, \cdots, J \\ \tau & \sim \text{N}^+\left(0, \mathtt{s}^2\right) \\ \lambda_j & \sim \text{N}(0, \mathtt{sd\_seas}^2), \quad j = 1, \cdots, \mathtt{n\_seas} - 1 \\ \lambda_j & = -\sum_{s=1}^{j-1} \lambda_{j-s}, \quad j = \mathtt{n\_seas},\; 2 \mathtt{n\_seas}, \cdots \\ \lambda_j & \sim \text{N}(\lambda_{j-\mathtt{n\_seas}}, \omega^2), \quad \text{otherwise} \\ \omega & \sim \text{N}^+\left(0, \mathtt{s\_seas}^2\right) \end{align}$$

3.5.2 All defaults

s = 1, sd = 1, s_seas = 0, sd_seas = 1

set.seed(0)

RW_Seas(n_seas = 4) |>
  generate(n_draw = 8) |>
  plot_cor_one()

3.5.3 Reduce s, sd

s = 0.01, sd = 0, s_seas = 0, sd_seas = 1

set.seed(0)

RW_Seas(n_seas = 4, s = 0.01, sd = 0) |>
  generate(n_draw = 8) |>
  plot_cor_one()

3.5.4 Increase s_seas

s = 0.01, sd = 0, s_seas = 1, sd_seas = 1

set.seed(0)

RW_Seas(n_seas = 4, s = 0.01, sd = 0, s_seas = 1) |>
  generate(n_draw = 8) |>
  plot_cor_one()

3.5.5 Reduce sd_seas

s = 0.01, sd = 0, s_seas = 1, s_seas = 0, sd_seas = 0.01

set.seed(0)

RW_Seas(n_seas = 4, s = 0.01, sd = 0, sd_seas = 0.01) |>
  generate(n_draw = 8) |>
  plot_cor_one()

3.6.1 Model

$$\begin{align} \beta_j & = \alpha_j + \lambda_j \\ \alpha_1 & \sim \text{N}(0, \mathtt{sd}^2) \\ \alpha_2 & \sim \text{N}(\alpha_1, \mathtt{sd\_slope}^2) \\ \alpha_j & \sim \text{N}(2\alpha_{j-1} - \alpha_{j-2}, \tau^2), \quad j = 3, \cdots, J \\ \tau & \sim \text{N}^+\left(0, \mathtt{s}^2\right) \\ \lambda_j & \sim \text{N}(0, \mathtt{sd\_seas}^2), \quad j = 1, \cdots, \mathtt{n\_seas} - 1 \\ \lambda_j & = -\sum_{s=1}^{j-1} \lambda_{j-s}, \quad j = \mathtt{n\_seas},\; 2 \mathtt{n\_seas}, \cdots \\ \lambda_j & \sim \text{N}(\lambda_{j-\mathtt{n\_seas}}, \omega^2), \quad \text{otherwise} \\ \omega & \sim \text{N}^+\left(0, \mathtt{s\_seas}^2\right) \end{align}$$

3.6.2 All defaults

s = 1, sd = 1, sd_slope = 1, s_seas = 0, sd_seas = 1

set.seed(0)

RW2_Seas(n_seas = 4) |>
  generate(n_draw = 8) |>
  plot_cor_one()

3.6.3 Reduce s, sd, sd_slope, sd_seas

s = 0.01, sd = 0, sd_slope = 0.01, s_seas = 0, sd_seas = 0.01

set.seed(0)

RW2_Seas(n_seas = 4, s = 0.01, sd = 0, sd_slope = 0.01, sd_seas = 0.01) |>
  generate(n_draw = 8) |>
  plot_cor_one()

3.7.1 Model

$$\begin{align} \beta_j & = \alpha_j + \epsilon_j \\ \alpha_j & = \left(j - \frac{J + 1}{2}\right) \eta \\ \eta & \sim \text{N}\left(\mathtt{mean\_slope}, \mathtt{sd\_slope}^2 \right) \\ \epsilon & \sim \text{N}(0, \tau^2) \\ \tau & \sim \text{N}^+\left(0, \mathtt{s}^2\right) \end{align}$$

3.7.2 All defaults

s = 1, mean_slope = 0, sd_slope = 1

set.seed(0)

Lin() |>
  generate(n_draw = 8) |>
  plot_cor_one()

3.7.3 Reduce s

s = 0, mean_slope = 0, sd_slope = 1

set.seed(0)

Lin(s = 0) |>
  generate(n_draw = 8) |>
  plot_cor_one()

3.7.4 Modify mean_slope, sd_slope

s = 1, mean_slope = 0.2, sd_slope = 0.1

set.seed(0)

Lin(mean_slope = 0.2, sd_slope = 0.1) |>
  generate(n_draw = 8) |>
  plot_cor_one()

3.7.5 Specify ‘along’ and ‘by’ dimensions

set.seed(0)

Lin() |>
  generate(n_along = 20, n_by = 3, n_draw = 8) |>
  plot_cor_many()

3.7.6 Specify ‘along’ and ‘by’ dimensions, set con = "by"

set.seed(0)

Lin(con = "by") |>
  generate(n_along = 20, n_by = 3, n_draw = 8) |>
  plot_cor_many()

3.8.1 Model

$$\begin{align} \beta_j & = \alpha_j + \epsilon_j \\ \alpha_j & = \left(j - \frac{J + 1}{2}\right) \eta \\ \eta & \sim \text{N}\left(\mathtt{mean\_slope}, \mathtt{sd\_slope}^2 \right) \\ \epsilon_j & \sim \text{N}\left(\phi_1 \epsilon_{j-1} + \cdots + \phi_{\mathtt{n\_coef}} \epsilon_{j-\mathtt{n\_coef}}, \omega^2\right) \tau & \sim \text{N}^+\left(0, \mathtt{s}^2\right) \\ \frac{\phi_k + 1}{2} & \sim \text{Beta}(\mathtt{shape1}, \mathtt{shape2}) \end{align}$$

3.8.2 All defaults

n_coef = 2, s = 1, mean_slope = 0, sd_slope = 1

set.seed(0)

Lin_AR() |>
  generate(n_draw = 8) |>
  plot_cor_one()

3.8.3 Reduce s, sd_slope

s = 0.1, mean_slope = 0, sd_slope = 0.01

set.seed(0)

Lin_AR(s = 0.1, sd_slope = 0.01) |>
  generate(n_draw = 8) |>
  plot_cor_one()

3.9.1 Model

$$\begin{align} \beta_j & = \alpha_j + \epsilon_j \\ \alpha_j & = \left(j - \frac{J + 1}{2}\right) \eta \\ \eta & \sim \text{N}\left(\mathtt{mean\_slope}, \mathtt{sd\_slope}^2 \right) \\ \epsilon_j & \sim \text{N}\left(\phi \epsilon_{j-1}, \omega^2\right) \tau & \sim \text{N}^+\left(0, \mathtt{s}^2\right) \\ \frac{\phi - \mathtt{min}}{\mathtt{max} - \mathtt{min}} \sim \text{Beta}(\mathtt{shape1}, \mathtt{shape2}) \end{align}$$

3.9.2 All defaults

s = 1, min = 0.8, max = 0.98, mean_slope = 0, sd_slope = 1

set.seed(0)

Lin_AR1() |>
  generate(n_draw = 8) |>
  plot_cor_one()

3.9.3 Modify min, max

s = 1, min = -1, max = 1, mean_slope = 0, sd_slope = 1

set.seed(0)

Lin_AR1(min = -1, max = 1) |>
  generate(n_draw = 8) |>
  plot_cor_one()

3.9.4 Reduce s, sd_slope

s = 0.1, min = 0.8, max = 0.98, mean_slope = 0, sd_slope = 0.02

set.seed(0)

Lin_AR1(s = 0.1, sd_slope = 0.02) |>
  generate(n_draw = 8) |>
  plot_cor_one()

3.10.1 Model

$$\begin{align} \pmb{\beta} & = \bm{X} \pmb{\alpha} \\ \alpha_1 & \sim \text{N}(0, \mathtt{sd}^2) \\ \alpha_2 & \sim \text{N}(\alpha_1, \mathtt{sd\_slope}^2) \\ \alpha_j & \sim \text{N}(2\alpha_{j-1} - \alpha_{j-2}, \tau^2), \quad j = 3, \cdots, J \\ \tau & \sim \text{N}^+\left(0, \mathtt{s}^2\right) \\ \end{align}$$

3.10.2 All defaults

n_comp = NULL, s = 1, sd = 1, sd_slope = 1

set.seed(0)

Sp() |>
  generate(n_draw = 8) |>
  plot_cor_one()

3.10.3 Specify n_comp

n_comp = 15, s = 1, sd = 1, sd_slope = 1

set.seed(0)

Sp(n_comp = 5) |>
  generate(n_draw = 8) |>
  plot_cor_one()

3.10.4 Reduce s, sd, sd_slope

n_comp = NULL, s = 0.01, sd = 0.01, sd_slope = 0.01

set.seed(0)

Sp(s = 0.01, sd = 0.01, sd_slope = 0.01) |>
  generate(n_draw = 8) |>
  plot_cor_one()

4.1.1 Model

$$\begin{equation} \pmb{\beta} = \pmb{F} \pmb{\alpha} + \pmb{g} \end{equation}$$

4.1.2 All defaults

n_comp = NULL, indep = TRUE

set.seed(0)

SVD(HMD) |>
  generate(n_draw = 8) |>
  plot_svd_one()

4.1.3 Increase n_comp

n_comp = 5, indep = TRUE

SVD(HMD, n_comp = 5) |>
  generate(n_draw = 8) |>
  plot_svd_one()

4.1.4 Set indep to FALSE

SVD(HMD, indep = FALSE) |>
  generate(n_draw = 8) |>
  plot_svd_one()

4.1.5 Multiple units

SVD(HMD, indep = FALSE) |>
  generate(n_draw = 8, n_element = 3) |>
  plot_svd_many()