analytic_jacobian.rmd

---
output: html_document
editor_options: 
  chunk_output_type: console
---


```{r functions}
# functions to get delta and weights -- see the TPC paper
get_delta <- function(w_h, beta_sk, x_hk){
  beta_x <- exp(beta_sk %*% t(x_hk))
  log(w_h / colSums(beta_sk))
}

get_weights <- function(beta_sk, delta_h, x_hk){
  # get all weights
  beta_x <- beta_sk %*% t(x_hk)
  # add delta to every row of beta_x and transpose
  beta_xd <- apply(beta_x, 1 , function(m) m + delta_h) 
  exp(beta_xd)
}

vi_to_msk <- function(vi){
  # function to convert a vector of length i to a matrix of dimension s by k
  matrix(vi, nrow=nrows, byrow=FALSE)
}

msk_to_vi <- function(msk){
  # function to convert a vector of length i to a matrix of dimension s by k
  as.vector(msk)
}

```



```{r create_data}
# create the poisson problem ----
h <- 6 # number households
k <- 3 # number characteristics
s <- 4 # number states

# x_hk is an h x k matrix of characteristics of households
x_hk <- matrix(seq(2, by=2, length.out=(h * k)), nrow=h, byrow=TRUE)
x_hk

# beta is an s x k matrix of coefficients
beta_sk <- matrix(seq(2, 4, length.out=s * k), nrow=s, byrow = TRUE)
beta_sk

# w_h is a vector of national weights for each household
w_h <- seq(10, by=2, length.out=h)
w_h

# delta is an h-length vector of individual constants in the formula for state weights
delta_h <- get_delta(w_h, beta_sk, x_hk)
delta_h

# w_h is an h-length vector of individual national weights

w_hs <- get_weights(beta_sk, delta_h, x_hk)
w_hs

targets <- 


```