I'm trying to estimate an Okun's law equation with a dlm using the dlm package in R. I can estimate the non-time varying model using nls as follows:
const_coef <- nls(formula = dur~ b1*dur_lag1 + b2*(d2lgdp-b0) + b3*d2lrulc_lag2 ,
start = list(b0 =0.1, b1=0.1, b2=0.1, b3=0.1),
data = mod_data)
the dlm model I want to be able to estimate allows for b1 and b0 in the above to follow random walks. I can do this in Eviews by declaring the measurement equation and appending the states (below is some code provided by the authors of the original paper which I can replicate:
'==========================
' SPECIFY THE KALMAN FILTER
'==========================
'Priors on state variables
vector(2) mprior
mprior(1) = 4 'Prior on starting value for trend GDP growth (annual average GDP growth over 1950s)
mprior(2) = 0 'Prior on starting value for lagged dependent variable
sym(2) vprior
vprior(1,1) = 5 'Prior on variance of trend GDP growth (variance of annual GDP growth over 1950s)
vprior(2,2) = 1 'Prior on variance of lagged dependent variable
'Specify coefficient vector
coef(8) ckf
'Declare state space
sspace ss1
ss1.append dur = lag*dur(-1) + ckf(2)*(d2lgdp-trend)+ckf(3)*D2LRULC(-2)+[var=exp(ckf(4))] 'Measurement equation
ss1.append #state trend = 1*trend(-1) + [var = exp(ckf(5))] 'State equation for trend GDP growth (random walk)
ss1.append #state lag = 1*lag(-1) + [var = exp(ckf(6))] 'State equation for lagged dependent variable (random walk)
'Apply priors to state space
ss1.append #mprior mprior
ss1.append #vprior vprior
'Set parameter starting values
param ckf(2) -0.0495 ckf(3) 0.01942 ckf(4) -2.8913 ckf(5) -4.1757 ckf(6) -6.2466 'starting values for parameters
'=====================
' ESTIMATE THE MODEL
'=====================
'Estimate state space
smpl %estsd %ested 'Estimation sample
ss1.ml(m=500,showopts) 'Estimate Kalman filter by maximum likelihood
freeze(mytab) ss1.stats
I'm really not sure how to do this with the dlm package. I've tried the following:
buildSS <- function(v){
dV <- exp(v[1]) # Variance of the measurment equation (ckf4)
dW <- c(exp(v[2]), # variance of the lagged dep (ckf6)
0, # variance of the coef on d2lgdp ckf(2) set to 0
0, # variance of the coef on d2lrulc ckf(3) set to 0
exp(v[3]) # variance of the random walk intercept (ckf5)
)
beta.vec <- c(1,v[4],v[5],1) # Params ckf(2) ckf3(3)
okuns <- dlmModReg(mod_data.tvp[,-1], addInt = TRUE, dV =dV, dW = dW, m0 = beta.vec)
}
#'Set parameter starting values
ckf4Guess <- -2.8913
ckf2guess <- -0.0495
ckf3guess <- 0.01942
ckf5guess <- -4.1757
ckf6guess <- -6.2466
params <- c(ckf4Guess,
ckf5guess,
ckf6guess,
ckf2guess,
ckf3guess)
tvp_mod.mle <- dlmMLE(mod_data.tvp[,"dur"] , parm = params, build = buildSS)
tvp_mod <- buildSS(tvp_mod.mle$par)
tvp_filter <- dlmFilter(mod_data$dur,tvp_mod)
The above code runs, but the outputs are not correct. I am not specifying the the states properly. Does anyone have any experience in building dlms with mutlvirate regression in R?
I think I have gotten to a solution - I've managed to recreate the estimates in the paper which estimates this model using Eviews (also checked this using Eviews).
#--------------------------------------------------------------------------------------------------------------------------
# tvp model full model - dur = alpha*dur(-1)+ beta(dgdp-potential) + gamma*wages
#--------------------------------------------------------------------------------------------------------------------------
# Construct DLM
OkunsDLMfm <- dlm(
FF = matrix(c(1,1,1,1),ncol = 4, byrow = TRUE),
V = matrix(1),
GG = matrix(c(1,0,0,0,
0,1,0,0,
0,0,1,0,
0,0,0,1), ncol = 4, byrow = TRUE),
W = matrix(c(1,0,0,0,
0,1,0,0,
0,0,1,0,
0,0,0,1), ncol = 4, byrow = TRUE),
JFF = matrix(c(1,2,3,0),ncol = 4, byrow = TRUE),
X = cbind(mod_data$dur_lag1,mod_data$d2lgdp, mod_data$d2lrulc_lag2), # lagged dep var, dgdp, wages.
m0 = c(0,0,0,0),
C0 = matrix(c(1e+07,0,0,0,
0,1e+07,0,0,
0,0,1e+07,0,
0,0,0,1e+07), ncol = 4, byrow = TRUE)
)
buildOkunsFM <- function(p){
V(OkunsDLMfm) <- exp(p[2])
GG(OkunsDLMfm)[1,1] <- 1
GG(OkunsDLMfm)[2,2] <- 1
GG(OkunsDLMfm)[3,3] <- 1
GG(OkunsDLMfm)[4,4] <- 1
W(OkunsDLMfm)[1,1] <- exp(p[3])
W(OkunsDLMfm)[2,2] <- 0
W(OkunsDLMfm)[3,3] <- 0
W(OkunsDLMfm)[4,4] <- exp(p[4])
m0(OkunsDLMfm) <- c(0,0,0,p[1]*4)
C0(OkunsDLMfm)[1,1] <- 1
C0(OkunsDLMfm)[4,4] <- 5
return(OkunsDLMfm)
}
okuns.estfm <- dlmMLE(y = mod_data$dur, parm = c(-0.049,-1.4,-6,-5), build = buildOkunsFM)
OkunsDLM1fm <- buildOkunsFM(okuns.estfm$par)
The time varying level, the estimate of potential output, is derived by dividing the 4 element of the state vector by the second * by negative 1.
Not sure if this is best way to specify the DLM, but the results from the model are very close to what is reported (within 0.01) of the results from using Eviews. That being said, very open to any other specifications.
Related
I'm trying to model an estimation of the price elasticity of demand for each customer using GAM model, a model like this:
\ln D = \ln P + \ln P \cdot \sum_{i=1}^{20} f(X_i)
PED = \frac{\partial \ln D {\partial \ln P} = 1 + \sum_{i=1}^{20} f(X_i)
https://latex.codecogs.com/svg.image?$$&space;\ln&space;D&space;=\ln&space;P&space;+&space;\ln&space;P&space;\cdot&space;\sum_{i=1}^{20}&space;f(X_i)\\PED&space;=&space;\frac{\partial&space;\ln&space;D}{\partial&space;\ln&space;P}&space;=&space;1&space;+&space;&space;\sum_{i=1}^{20}&space;f(X_i)
where $D$ is Demand, $P$ is rate, PED is price elasticity of demand and $X_i$ is a set of customer's variable.
Since $PED$ is not observable, i want to estimate PED from the model created for log demand using gam model, but I have trying some difficulty in how to estimate that way.
I tried to get the each splines to calculate PED, but i failed. I know there is a package called gratia with derivatives function, but i dont understand how to use it to calculate ped.
Once the model to estimate demand is created, I will need to estimate the price elasticity of demand for each customer, but for these customers I don't have the rate variable, only the 20 personal variables.
I read some links:
https://stats.stackexchange.com/questions/495775/first-derivative-of-fitted-gam-changes-according-to-specified-model-distribution
https://stats.stackexchange.com/questions/590167/how-can-i-calculate-a-derivative-of-a-global-smooth-and-group-level-smooths-with
https://stats.stackexchange.com/questions/32013/what-is-the-mathematical-model-formula-corresponding-to-this-gam-model-fit-in-r
Really appreciate for any explanation, advices or other way to model my data.
Thanks
EDIT
What i've tried:
#create the dataset
A <- sample(x = 0:1000, size = 5000, replace = TRUE)
B <- sample(x = 0:1000, size = 5000, replace = TRUE)
C <- sample(x = 0:1000, size = 5000, replace = TRUE)
D <- sample(x = 0:1000, size = 5000, replace = TRUE)
log.R <- log(rbeta(5000, 5,10)*10) #log rate
log.Y <- log(rgamma(5000, 10, 20)*10000) #log demand
mydata <- data.frame(A, B, C, D, log.R, log.Y)
#the model
model <- gam(log.Y ~ s(A, by=log.R) + s(B, by=log.R) + s(C, by=log.R) + s(D, by=log.R), data = mydata, method = "REML")
mfx <- marginaleffects(model, variables = "log.R", eps = 10^-5)
head(mfx)
mfx returns a 'dydx' column, is it the elasticity of my data used to model?
And when i will apply this model to newdata, i got an error:
newdat = data.frame(A = 750, B = 500, C = 398, D = 740)
marginaleffects(model, variables = "log.R", eps = 10^-5, newdata= newdat, slope = 'dydx')
Error: There is no valid predictor variable. Please change the `variables` argument or supply a new data frame to the `newdata` argument.
What should I do?
I'm working with the train() function from the caret package to fit multiple regression and ML models to test their fit. I'd like to write a function that iterates through all model types and enters the best fit into a dataframe. Biggest issue is that caret doesn't provide all the model fit statistics that I'd like so they need to be derived from the raw output. Based on my exploration there doesn't seem to be a standardized way caret outputs each models fit.
Another post (sorry don't have a link) created this function which pulls from fit$results and fit$bestTune to get pre calculated RMSE, R^2, etc.
get_best_result <- function(caret_fit) {
best = which(rownames(caret_fit$results) == rownames(caret_fit$bestTune))
best_result = caret_fit$results[best, ]
rownames(best_result) = NULL
best_result
}
One example of another fit statistic I need to calculate using raw output is BIC. The two functions below do that. The residuals (y_actual - y_predicted) are needed along with the number of x variables (k) and the number of rows used in the prediction (n). k and n must be derived from the output not the original dataset due to the models dropping x variables (feature selection) or rows (omitting NAs) based on its algorithm.
calculate_MSE <- function(residuals){
# residuals can be replaced with y_actual-y_predicted
mse <- mean(residuals^2)
return(mse)
}
calculate_BIC <- function(n, mse, k){
BIC <- n*log(mse)+k*log(n)
return(BIC)
}
The real question is is there a standardized output of caret::train() for x variables or either y_actual, y_predicted, or residuals?
I tried fit$finalModel$model and other methods but to no avail.
Here is a reproducible example along with the function I'm using. Please consider the functions above a part of this reproducible example.
library(rlist)
library(data.table)
# data
df <- data.frame(y1 = rnorm(50, 0, 1),
y2 = rnorm(50, .25, 1.5),
x1 = rnorm(50, .4, .9),
x2 = rnorm(50, 0, 1.1),
x3 = rnorm(50, 1, .75))
missing_index <- sample(1:50, 7, replace = F)
df[missing_index,] <- NA
# function to fit models and pull results
fitModels <- function(df, Ys, Xs, models){
# empty list
results <- list()
# number of for loops
loops_counter <- 0
# for every y
for(y in 1:length(Ys)){
# for every model
for(m in 1:length(models)){
# track loops
loops_counter <- loops_counter + 1
# fit the model
set.seed(1) # seed for reproducability
fit <- tryCatch(train(as.formula(paste(Ys[y], paste(Xs, collapse = ' + '),
sep = ' ~ ')),
data = df,
method = models[m],
na.action = na.omit,
tuneLength = 10),
error = function(e) {return(NA)})
# pull results
results[[loops_counter]] <- c(Y = Ys[y],
model = models[m],
sample_size = nrow(fit$finalModel$model),
RMSE = get_best_result(fit)[[2]],
R2 = get_best_result(fit)[[3]],
MAE = get_best_result(fit)[[4]],
BIC = calculate_BIC(n = length(fit$finalModel),
mse = calculate_MSE(fit$finalModel$residuals),
k = length(fit$finalModel$xNames)))
}
}
# list bind
results_df <- list.rbind(results)
return(results_df)
}
linear_models <- c('lm', 'glmnet', 'ridge', 'lars', 'enet')
fits <- fitModels(df, c(y1, y2), c(x1,x2,x3), linear_models)
A hidden Markov model (HMM) is one in which you observe a sequence of observations, but do not know the sequence of states the model went through to generate the observations. Analyses of hidden Markov models seek to recover the sequence of hidden states from the observed data.
I have data with both observations and hidden states (observations are of continuous values) where the hidden states were tagged by an expert. I would like to train a HMM that would be able - based on a (previously unseen) sequence of observations - to recover the corresponding hidden states.
Is there any R package to do that? Studying the existing packages (depmixS4, HMM, seqHMM - for categorical data only) allows you to specify a number of hidden states only.
EDIT:
Example:
data.tagged.by.expert = data.frame(
hidden.state = c("Wake", "REM", "REM", "NonREM1", "NonREM2", "REM", "REM", "Wake"),
sensor1 = c(1,1.2,1.2,1.3,4,2,1.78,0.65),
sensor2 = c(7.2,5.3,5.1,1.2,2.3,7.5,7.8,2.1),
sensor3 = c(0.01,0.02,0.08,0.8,0.03,0.01,0.15,0.45)
)
data.newly.measured = data.frame(
sensor1 = c(2,3,4,5,2,1,2,4,5,8,4,6,1,2,5,3,2,1,4),
sensor2 = c(2.1,2.3,2.2,4.2,4.2,2.2,2.2,5.3,2.4,1.0,2.5,2.4,1.2,8.4,5.2,5.5,5.2,4.3,7.8),
sensor3 = c(0.23,0.25,0.23,0.54,0.36,0.85,0.01,0.52,0.09,0.12,0.85,0.45,0.26,0.08,0.01,0.55,0.67,0.82,0.35)
)
I would like to create a HMM with discrete time t whrere random variable x(t) represents the hidden state at time t, x(t) {"Wake", "REM", "NonREM1", "NonREM2"}, and 3 continuous random variables sensor1(t), sensor2(t), sensor3(t) representing the observations at time t.
model.hmm = learn.model(data.tagged.by.user)
Then I would like to use the created model to estimate hidden states responsible for newly measured observations
hidden.states = estimate.hidden.states(model.hmm, data.newly.measured)
Data (training/testing)
To be able to run learning methods for Naive Bayes classifier, we need longer data set
states = c("NonREM1", "NonREM2", "NonREM3", "REM", "Wake")
artificial.hypnogram = rep(c(5,4,1,2,3,4,5), times = c(40,150,200,300,50,90,30))
data.tagged.by.expert = data.frame(
hidden.state = states[artificial.hypnogram],
sensor1 = log(artificial.hypnogram) + runif(n = length(artificial.hypnogram), min = 0.2, max = 0.5),
sensor2 = 10*artificial.hypnogram + sample(c(-8:8), size = length(artificial.hypnogram), replace = T),
sensor3 = sample(1:100, size = length(artificial.hypnogram), replace = T)
)
hidden.hypnogram = rep(c(5,4,1,2,4,5), times = c(10,10,15,10,10,3))
data.newly.measured = data.frame(
sensor1 = log(hidden.hypnogram) + runif(n = length(hidden.hypnogram), min = 0.2, max = 0.5),
sensor2 = 10*hidden.hypnogram + sample(c(-8:8), size = length(hidden.hypnogram), replace = T),
sensor3 = sample(1:100, size = length(hidden.hypnogram), replace = T)
)
Solution
In the solution, we used Viterbi algorithm - combined with Naive Bayes classifier.
At each clock time t, a Hidden Markov Model consist of
an unobserved state (denoted as hidden.state in this case) taking a finite number of states
states = c("NonREM1", "NonREM2", "NonREM3", "REM", "Wake")
a set of observed variables (sensor1, sensor2, sensor3 in this case)
Transition matrix
A new state is entered based upon a transition probability distribution
(transition matrix). This can be easily computed from data.tagged.by.expert e.g. using
library(markovchain)
emit_p <- markovchainFit(data.tagged.by.expert$hidden.state)$estimate
Emission matrix
After each transition is made, an observation (sensor_i) is produced according to a conditional probability distribution (emission matrix) which depends on the current state H of hidden.state only. We will replace emmision matrices by Naive Bayes classifier.
library(caret)
library(klaR)
library(e1071)
model = train(hidden.state ~ .,
data = data.tagged.by.expert,
method = 'nb',
trControl=trainControl(method='cv',number=10)
)
Viterbi algorithm
To solve the problem, we use Viterbi algorithm with the initial probability of 1 for "Wake" state and 0 otherwise. (We expect the patient to be awake in the beginning of the experiment)
# we expect the patient to be awake in the beginning
start_p = c(NonREM1 = 0,NonREM2 = 0,NonREM3 = 0, REM = 0, Wake = 1)
# Naive Bayes model
model_nb = model$finalModel
# the observations
observations = data.newly.measured
nObs <- nrow(observations) # number of observations
nStates <- length(states) # number of states
# T1, T2 initialization
T1 <- matrix(0, nrow = nStates, ncol = nObs) #define two 2-dimensional tables
row.names(T1) <- states
T2 <- T1
Byj <- predict(model_nb, newdata = observations[1,])$posterior
# init first column of T1
for(s in states)
T1[s,1] = start_p[s] * Byj[1,s]
# fill T1 and T2 tables
for(j in 2:nObs) {
Byj <- predict(model_nb, newdata = observations[j,])$posterior
for(s in states) {
res <- (T1[,j-1] * emit_p[,s]) * Byj[1,s]
T2[s,j] <- states[which.max(res)]
T1[s,j] <- max(res)
}
}
# backtract best path
result <- rep("", times = nObs)
result[nObs] <- names(which.max(T1[,nObs]))
for (j in nObs:2) {
result[j-1] <- T2[result[j], j]
}
# show the result
result
# show the original artificial data
states[hidden.hypnogram]
References
To read more about the problem, see Vomlel Jiří, Kratochvíl Václav : Dynamic Bayesian Networks for the Classification of Sleep Stages , Proceedings of the 11th Workshop on Uncertainty Processing (WUPES’18), p. 205-215 , Eds: Kratochvíl Václav, Vejnarová Jiřina, Workshop on Uncertainty Processing (WUPES’18), (Třeboň, CZ, 2018/06/06) [2018] Download
I am using 'KFAS' package from R to estimate a state-space model with the Kalman filter. My measurement and transition equations are:
y_t = Z_t * x_t + \eps_t (measurement)
x_t = T_t * x_{t-1} + R_t * \eta_t (transition),
with \eps_t ~ N(0,H_t) and \eta_t ~ N(0,Q_t).
So, I want to estimate the variances H_t and Q_t, but also T_t, the AR(1) coefficient. My code is as follows:
library(KFAS)
set.seed(100)
eps <- rt(200, 4, 1)
meas <- as.matrix((arima.sim(n=200, list(ar=0.6), innov = rnorm(200)*sqrt(0.5)) + eps),
ncol=1)
Zt <- 1
Ht <- matrix(NA)
Tt <- matrix(NA)
Rt <- 1
Qt <- matrix(NA)
ss_model <- SSModel(meas ~ -1 + SSMcustom(Z = Zt, T = Tt, R = Rt,
Q = Qt), H = Ht)
fit <- fitSSM(ss_model, inits = c(0,0.6,0), method = 'L-BFGS-B')
But it returns: "Error in is.SSModel(do.call(updatefn, args = c(list(inits, model), update_args)),: System matrices (excluding Z) contain NA or infinite values, covariance matrices contain values larger than 1e+07"
The NA definitions for the variances works well, as documented in the package's paper. However, it seems this cannot be done for the AR coefficients. Does anyone know how can I do this?
Note that I am aware of the SSMarima function, which eases the definition of the transition equation as ARIMA models. Although I am able to estimate the AR(1) coef. and Q_t this way, I still cannot estimate the \eps_t variance (H_t). Moreover, I am migrating my Kalman filter codes from EViews to R, so I need to learn SSMcustom for other models that are more complicated.
Thanks!
It seems that you are missing something in your example, as your error message comes from the function fitSSM. If you want to use fitSSM for estimating general state space models, you need to provide your own model updating function. The default behaviour can only handle NA's in covariance matrices H and Q. The main goal of fitSSM is just to get started with simple stuff. For complex models and/or large data, I would recommend using your self-written objective function (with help of logLik method) and your favourite numerical optimization routines manually for maximum performance. Something like this:
library(KFAS)
set.seed(100)
eps <- rt(200, 4, 1)
meas <- as.matrix((arima.sim(n=200, list(ar=0.6), innov = rnorm(200)*sqrt(0.5)) + eps),
ncol=1)
Zt <- 1
Ht <- matrix(NA)
Tt <- matrix(NA)
Rt <- 1
Qt <- matrix(NA)
ss_model <- SSModel(meas ~ -1 + SSMcustom(Z = Zt, T = Tt, R = Rt,
Q = Qt), H = Ht)
objf <- function(pars, model, estimate = TRUE) {
model$H[1] <- pars[1]
model$T[1] <- pars[2]
model$Q[1] <- pars[3]
if (estimate) {
-logLik(model)
} else {
model
}
}
opt <- optim(c(1, 0.5, 1), objf, method = "L-BFGS-B",
lower = c(0, -0.99, 0), upper = c(100, 0.99, 100), model = ss_model)
ss_model_opt <- objf(opt$par, ss_model, estimate = FALSE)
Same with fitSSM:
updatefn <- function(pars, model) {
model$H[1] <- pars[1]
model$T[1] <- pars[2]
model$Q[1] <- pars[3]
model
}
fit <- fitSSM(ss_model, c(1, 0.5, 1), updatefn, method = "L-BFGS-B",
lower = c(0, -0.99, 0), upper = c(100, 0.99, 100))
identical(ss_model_opt, fit$model)
I am trying to use XGBoost to model claims frequency of data generated from unequal length exposure periods, but have been unable to get the model to treat the exposure correctly. I would normally do this by setting log(exposure) as an offset - are you able to do this in XGBoost?
(A similar question was posted here: xgboost, offset exposure?)
To illustrate the issue, the R code below generates some data with the fields:
x1, x2 - factors (either 0 or 1)
exposure - length of policy period on observed data
frequency - mean number of claims per unit exposure
claims - number of observed claims ~Poisson(frequency*exposure)
The goal is to predict frequency using x1 and x2 - the true model is: frequency = 2 if x1 = x2 = 1, frequency = 1 otherwise.
Exposure can't be used to predict the frequency as it is not known at the outset of a policy. The only way we can use it is to say: expected number of claims = frequency * exposure.
The code tries to predict this using XGBoost by:
Setting exposure as a weight in the model matrix
Setting log(exposure) as an offset
Below these, I've shown how I would handle the situation for a tree (rpart) or gbm.
set.seed(1)
size<-10000
d <- data.frame(
x1 = sample(c(0,1),size,replace=T,prob=c(0.5,0.5)),
x2 = sample(c(0,1),size,replace=T,prob=c(0.5,0.5)),
exposure = runif(size, 1, 10)*0.3
)
d$frequency <- 2^(d$x1==1 & d$x2==1)
d$claims <- rpois(size, lambda = d$frequency * d$exposure)
#### Try to fit using XGBoost
require(xgboost)
param0 <- list(
"objective" = "count:poisson"
, "eval_metric" = "logloss"
, "eta" = 1
, "subsample" = 1
, "colsample_bytree" = 1
, "min_child_weight" = 1
, "max_depth" = 2
)
## 1 - set weight in xgb.Matrix
xgtrain = xgb.DMatrix(as.matrix(d[,c("x1","x2")]), label = d$claims, weight = d$exposure)
xgb = xgb.train(
nrounds = 1
, params = param0
, data = xgtrain
)
d$XGB_P_1 <- predict(xgb, xgtrain)
## 2 - set as offset in xgb.Matrix
xgtrain.mf <- model.frame(as.formula("claims~x1+x2+offset(log(exposure))"),d)
xgtrain.m <- model.matrix(attr(xgtrain.mf,"terms"),data = d)
xgtrain <- xgb.DMatrix(xgtrain.m,label = d$claims)
xgb = xgb.train(
nrounds = 1
, params = param0
, data = xgtrain
)
d$XGB_P_2 <- predict(model, xgtrain)
#### Fit a tree
require(rpart)
d[,"tree_response"] <- cbind(d$exposure,d$claims)
tree <- rpart(tree_response ~ x1 + x2,
data = d,
method = "poisson")
d$Tree_F <- predict(tree, newdata = d)
#### Fit a GBM
gbm <- gbm(claims~x1+x2+offset(log(exposure)),
data = d,
distribution = "poisson",
n.trees = 1,
shrinkage=1,
interaction.depth=2,
bag.fraction = 0.5)
d$GBM_F <- predict(gbm, newdata = d, n.trees = 1, type="response")
At least with the glm function in R, modeling count ~ x1 + x2 + offset(log(exposure)) with family=poisson(link='log') is equivalent to modeling I(count/exposure) ~ x1 + x2 with family=poisson(link='log') and weight=exposure. That is, normalize your count by exposure to get frequency, and model frequency with exposure as the weight. Your estimated coefficients should be the same in both cases when using glm for Poisson regression. Try it for yourself using a sample data set
I'm not exactly sure what objective='count:poisson' corresponds to, but I would expect setting your target variable as frequency (count/exposure) and using exposure as the weight in xgboost would be the way to go when exposures are varying.
I have now worked out how to do this using setinfo to change the base_margin attribute to be the offset (as a linear predictor), ie:
setinfo(xgtrain, "base_margin", log(d$exposure))