library(tidyverse)
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library(corrr)
library(hrbrthemes)
#> NOTE: Either Arial Narrow or Roboto Condensed fonts are required to use these themes.
#> Please use hrbrthemes::import_roboto_condensed() to install Roboto Condensed and
#> if Arial Narrow is not on your system, please see http://bit.ly/arialnarrow
# -------------------------------------
# Generate Media Data
# Equation: y=a*sin(b*t)+c.unif*amp
# -------------------------------------
set.seed(1)
n <- 52 * 2 # number of data points
t <- seq(0, 4*pi, length.out = n)
b <- 8 # essentially the number of pillars in a year
c.norm1 <- rnorm(n,0,0.5)
c.norm2 <- rnorm(n,0, 0.75)
c.norm3 <- rnorm(n,0, 0.75)
amp <- 2
# generate data and calculate "y"
media_tv <- 1*sin(b*t)+c.norm1*amp # Gaussian/normal error
media_radio <- 1*sin(b*t)+c.norm2*amp # Gaussian/normal error
media_online <- 1*sin(b*t)+c.norm3*amp # Gaussian/normal error
week <- seq(1:104)
sim_df <- as.data.frame(list(week = week,
media_tv = media_tv,
media_radio = media_radio,
media_online = media_online)) %>%
mutate_at(vars(media_tv, media_radio,media_online), percent_rank)
sim_df %>%
ggplot(aes(x = week)) +
geom_line(aes(y = media_tv, color = "tv")) +
geom_line(aes(y = media_radio, color = "radio")) +
geom_line(aes(y = media_online, color = "online")) +
theme_ipsum()
# -----------------------------------------------------------------------------
# Generate Menu Price Data
# Equation: y=ARIMA(1,1,0) with AR = 0.50
# Interpretation: Price has increased by 3 dollars in the last year
# -----------------------------------------------------------------------------
price <- arima.sim(n = n, list(ar = c(0.8897, -0.4858), ma = c(0.279, 0.2488)), sd = sqrt(0.001))
mean(price); sd(price)
#> [1] -0.001195423
#> [1] 0.05821377
hist(price)
plot(price)
#price <- as.numeric(scale(price))
sim_df <- add_column(sim_df, price)
sim_df %>%
gather(key = vartype, value = value, -week) %>%
ggplot(aes(x = week, y = value)) +
geom_col()+
facet_wrap(~vartype) +
hrbrthemes::theme_ipsum()
#> Warning: attributes are not identical across measure variables;
#> they will be dropped
# --------------------------------------------------
# Scale media to [0, 1] as per paper
# media_i = x_i - min(x) / max(x) - min(x)
# --------------------------------------------------
my_normalizer <- function(x) (x - min(x)) / (max(x) - min(x))
sim_df <- sim_df %>%
mutate_at(vars(-week,-price), my_normalizer)
sim_df %>% write_csv("sim_df.csv")
sim_df %>%
gather(key = vartype, value = value, -week) %>%
ggplot(aes(x = week, y = value)) +
geom_col()+
facet_wrap(~vartype) +
hrbrthemes::theme_ipsum()
#> Warning: attributes are not identical across measure variables;
#> they will be dropped
#-----------------------------------------------------------------------------
# Generate adstock as described in the Google Paper:
# Bayesian Methods for Media Mix Modeling with Carryover and Shape Effects
#-----------------------------------------------------------------------------
library(tidyquant)
#> Loading required package: lubridate
#>
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#>
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#>
#> legend
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#> Version 0.4-0 included new data defaults. See ?getSymbols.
#> ══ Need to Learn tidyquant? ═══════════════════════════════════════════════════════════════════════════
#> Business Science offers a 1-hour course - Learning Lab #9: Performance Analysis & Portfolio Optimization with tidyquant!
#> </> Learn more at: https://university.business-science.io/p/learning-labs-pro </>
# fake data
date <- seq(from = as.Date("2018-01-01"), length.out = 104, by = "1 week")
x <- c(rep(100,5), rep(0,99)) %>%
as_tibble() %>%
add_column(date) %>%
rename(x = value)
#> Warning: Calling `as_tibble()` on a vector is discouraged, because the behavior is likely to change in the future. Use `tibble::enframe(name = NULL)` instead.
#> This warning is displayed once per session.
# -----------------------------------------------------------------------------
# Name: Carryover Effect
#
# Description: Two functions are provided for modeling the decay of ad
# effect.
#
# Geometric: This function assumes that week 1 is the most impactfull
# week of the the promo. Subsequent weeks have a slow decline
# as defined by the rate. A larger rate give a slower decline
#
# Delayed: This function assumes that a week after week 1 is the most
# impactfull week. It has a weight that is proportional to
# The normal distribution around the week of impact defined
# by theta
# -----------------------------------------------------------------------------
geom_decay <- function(rate,l,...) sum((rate^l) *...) / sum(rate^l)
delayed_decay <- function(rate,l,theta,...){
sum((rate^(l-theta)^2) *...) / sum(rate^(l-theta)^2)
}
# Examples of calculating adstock from both functions
# Since the values are calculated on a rolling window
# it is neccesary to used something like tq_mutate
# to get values for a given time-series
L = 13
x %>%
tq_mutate(select = x, mutate_fun = rollapply, width = L, align = "right",
FUN = geom_decay,
#function args
rate = 0.8,
l = seq(from = 0 , to = L-1),
#ts_mutate
col_rename = "adstock_geometric_decay"
)
#> # A tibble: 104 x 3
#> x date adstock_geometric_decay
#> <dbl> <date> <dbl>
#> 1 100 2018-01-01 NA
#> 2 100 2018-01-08 NA
#> 3 100 2018-01-15 NA
#> 4 100 2018-01-22 NA
#> 5 100 2018-01-29 NA
#> 6 0 2018-02-05 NA
#> 7 0 2018-02-12 NA
#> 8 0 2018-02-19 NA
#> 9 0 2018-02-26 NA
#> 10 0 2018-03-05 NA
#> # … with 94 more rows
x %>%
tq_mutate(select = x, mutate_fun = rollapply, width = L, align = "right",
FUN = delayed_decay,
#function args
rate = 0.8, # rate of 0.4 to 0.8 is sensible
theta = 1, # theta should be about 1 to 3
l = seq(from = 0 , to = L-1),
#ts_mutate
col_rename = "adstock_delayed_decay"
)
#> # A tibble: 104 x 3
#> x date adstock_delayed_decay
#> <dbl> <date> <dbl>
#> 1 100 2018-01-01 NA
#> 2 100 2018-01-08 NA
#> 3 100 2018-01-15 NA
#> 4 100 2018-01-22 NA
#> 5 100 2018-01-29 NA
#> 6 0 2018-02-05 NA
#> 7 0 2018-02-12 NA
#> 8 0 2018-02-19 NA
#> 9 0 2018-02-26 NA
#> 10 0 2018-03-05 NA
#> # … with 94 more rows# -----------------------------------------------------------------------------
# Name: Shape Effect
#
# Description: It is not enough to model the decay and the lag of an ad.
# The shape of its saturation is also an important funtion
# that deserves attention.
#
# Hill Function: Marketing Mix Modelers often chose between S-curves and
# C-curves when modeling media impact on sales.
# Pharmacology uses the Hill function to model receptors.
# It provides a flexible functional form that may take the
# form of both an S-curve and a C-curve which provides a
# convinient solution to parameterizing the function
# representing shape effect.
# K: Half Saturation
# S: Slope
# B: Beta
#
# Problem: It may be the case that the Slope parameter "S" may have to
# be set to 1 (S = 1). This is an issue with identifiability
# -----------------------------------------------------------------------------
# Define Function
BHill <- function(B,K,S,...) B - ((K^S * B)/(...^S + K^S))
# set up example data
x <- seq(0,1, length.out = 100) # media must be transformed to [0, 1] scale
# for ease of use
params <- tribble(
~K, ~S, ~B, ~type,
0.5, 1, 0.3, "simple_c",
0.5, 2, 0.3, "simple_s",
0.5, 0.25, 0.3, "sharp_c"
)
bhill_df <- crossing(x,params) %>%
mutate(y = BHill(B,K,S,x))
bhill_df %>%
ggplot(aes(x = x, y = y, color = type)) +
geom_line() +
labs(title = "Flexible Shape Function") +
hrbrthemes::theme_ipsum()
# =============================================================================
# Simulation
# Description: With simulated media data as "media variables" and
# simulated price as a "control variable" I applied the neccesary
# transfomations (adstock & shape) to the input variables with
# various parameters to test the ability of this model to
# discover the parameters I set
#
# Equation: Weekly sales have the following form:
#
# sales_wk = tau + BHill_tv_wk + BHill_online_wk + BHill_radio_wk
# + gamma*price_wk + e_wk
# =============================================================================
#------------------------------------------------------------
#
# Media Parameters
# ----------------
# Parameter | Media_tv | Media_radio | Media_online
# rate 0.6 0.8 0.8
# theta 5 3 4
# K 0.2 0.2 0.2
# S 1 2 2
# B 0.8 0.6 0.3
#
# Other variables
# ----------------
# Parameter | Value
# L 13
# tau 4
# gamma 0.05
# e normal(0,0.05^2)
#------------------------------------------------------------
fat_data <- sim_df %>%
add_column(date = seq(as.Date("2017-01-01"), length.out = n, by = "week")) %>%
# adstocks
tq_mutate(select = media_tv, mutate_fun = rollapply, width = L, align = "right",
FUN = delayed_decay,rate = 0.6, theta = 1,
l = seq(from = 0 , to = L-1),col_rename = "adstk_tv"
)%>%
tq_mutate(select = media_radio, mutate_fun = rollapply, width = L, align = "right",
FUN = delayed_decay,rate = 0.8, theta = 1,
l = seq(from = 0 , to = L-1),col_rename = "adstk_radio"
)%>%
tq_mutate(select = media_online, mutate_fun = rollapply, width = L, align = "right",
FUN = delayed_decay,rate = 0.8, theta = 1,
l = seq(from = 0 , to = L-1),col_rename = "adstk_online"
)%>%
# Shape
mutate(m_tv = BHill(K = 0.2, S = 1, B = 0.8,adstk_tv)) %>%
mutate(m_rd = BHill(K = 0.2, S = 1, B = 0.6,adstk_radio)) %>%
mutate(m_online = BHill(K = 0.2, S = 1, B = 0.3,adstk_online)) %>%
#and errors
mutate(e = rnorm(n = n(), mean = 0, sd = 0.25^2)) %>%
mutate(sales = 4 + m_tv + m_rd + m_online + .5 * price + e)
clean_data <- fat_data %>%
select(date, sales,m_tv,m_rd,m_online,price,e) %>%
na.omit()
clean_data
#> # A tibble: 92 x 7
#> date sales m_tv m_rd m_online price e
#> <date> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1 2017-03-26 5.08 0.597 0.471 0.227 -0.129 -0.146
#> 2 2017-04-02 5.35 0.619 0.483 0.226 -0.0789 0.0601
#> 3 2017-04-09 5.24 0.606 0.469 0.210 -0.0167 -0.0378
#> 4 2017-04-16 5.18 0.547 0.452 0.198 0.0544 -0.0471
#> 5 2017-04-23 5.07 0.470 0.435 0.200 0.122 -0.0972
#> 6 2017-04-30 5.21 0.568 0.434 0.223 0.151 -0.0909
#> 7 2017-05-07 5.36 0.633 0.430 0.233 0.125 0.00352
#> 8 2017-05-14 5.37 0.640 0.446 0.223 0.0671 0.0318
#> 9 2017-05-21 5.12 0.625 0.427 0.201 0.00162 -0.131
#> 10 2017-05-28 5.10 0.596 0.364 0.188 0.0200 -0.0628
#> # … with 82 more rowslibrary(hrbrthemes)
# some plots of the data. see if it matches the paper okay
clean_data %>%
ggplot(aes(date, sales)) + geom_line() + theme_ipsum()
clean_data %>%
select(price, m_tv, m_rd, m_online) %>%
correlate()
#>
#> Correlation method: 'pearson'
#> Missing treated using: 'pairwise.complete.obs'
#> # A tibble: 4 x 5
#> rowname price m_tv m_rd m_online
#> <chr> <dbl> <dbl> <dbl> <dbl>
#> 1 price NA -0.114 -0.0467 -0.202
#> 2 m_tv -0.114 NA 0.349 0.441
#> 3 m_rd -0.0467 0.349 NA 0.300
#> 4 m_online -0.202 0.441 0.300 NAclean_data %>%
select(sales, m_tv, m_rd, m_online, e, price) %>%
summarise_all(var) %>%
transmute(var_tv = m_tv / sales,
var_rd = m_rd / sales,
var_online = m_online / sales,
var_noise = e / sales,
price = price / sales)
#> # A tibble: 1 x 5
#> var_tv var_rd var_online var_noise price
#> <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1 0.302 0.106 0.0263 0.357 0.259clean_data %>% write_csv("clean_data.csv")media_data <- clean_data %>% select(contains("m_"))
# data Prep
N <- nrow(clean_data)
Y <- clean_data$sales
max_lag <- 13
num_media <- 3
lag_vec <- seq(0, max_lag - 1)
X_media <- array(data = media_data, dim = c(num_media))
num_ctrl <- 1
X_ctrl <- clean_data$price
stan_data <- list(N=N, Y=Y, max_lag=max_lag, num_media=num_media,
lag_vec=lag_vec,X_media=X_media,
num_ctrl=num_ctrl,X_ctrl=X_ctrl)
stan_data %>% str
#> List of 8
#> $ N : int 92
#> $ Y : num [1:92] 5.08 5.35 5.24 5.18 5.07 ...
#> $ max_lag : num 13
#> $ num_media: num 3
#> $ lag_vec : int [1:13] 0 1 2 3 4 5 6 7 8 9 ...
#> $ X_media :Classes 'tbl_df', 'tbl' and 'data.frame': 92 obs. of 3 variables:
#> ..$ : num [1:92] 0.597 0.619 0.606 0.547 0.47 ...
#> ..$ : num [1:92] 0.471 0.483 0.469 0.452 0.435 ...
#> ..$ : num [1:92] 0.227 0.226 0.21 0.198 0.2 ...
#> ..- attr(*, "na.action")= 'omit' Named int [1:12] 1 2 3 4 5 6 7 8 9 10 ...
#> .. ..- attr(*, "names")= chr [1:12] "1" "2" "3" "4" ...
#> ..- attr(*, "dim")= int 3
#> $ num_ctrl : num 1
#> $ X_ctrl : num [1:92] -0.1289 -0.0789 -0.0167 0.0544 0.1218 ...library(rstan)
#> Loading required package: StanHeaders
#> rstan (Version 2.19.2, GitRev: 2e1f913d3ca3)
#> For execution on a local, multicore CPU with excess RAM we recommend calling
#> options(mc.cores = parallel::detectCores()).
#> To avoid recompilation of unchanged Stan programs, we recommend calling
#> rstan_options(auto_write = TRUE)
#>
#> Attaching package: 'rstan'
#> The following object is masked from 'package:tidyr':
#>
#> extract
clean_data <- read_csv("clean_data.csv")
#> Parsed with column specification:
#> cols(
#> date = col_date(format = ""),
#> sales = col_double(),
#> m_tv = col_double(),
#> m_rd = col_double(),
#> m_online = col_double(),
#> price = col_double(),
#> e = col_double()
#> )
media_data <- clean_data %>% select(contains("m_"))
long_media_array <- c(clean_data$m_tv,clean_data$m_rd,clean_data$m_online)
# data Prep
N <- nrow(clean_data)
Y <- clean_data$sales
max_lag <- 13
num_media <- 3
lag_vec <- seq(0, max_lag - 1)
X_media <- array(data = media_data, dim = c(3,13))
X_media <- array(data = long_media_array, dim = c(92,3,13))
num_ctrl <- 1
X_ctrl <- clean_data %>% select(price) %>% as.vector()
stan_data <- list(N=N, Y=Y, max_lag=max_lag, num_media=num_media,
lag_vec=lag_vec,X_media=X_media,
num_ctrl=num_ctrl,X_ctrl=X_ctrl)
m.stan <- stan(file = "model.stan",data = stan_data, iter = 3000, chains = 1, control = list(max_treedepth = 15))
#> Warning in readLines(file, warn = TRUE): incomplete final line found on '/Users/
#> alexhallam/r/media_mix_sim/model.stan'
#>
#> SAMPLING FOR MODEL 'model' NOW (CHAIN 1).
#> Chain 1:
#> Chain 1: Gradient evaluation took 0.001078 seconds
#> Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 10.78 seconds.
#> Chain 1: Adjust your expectations accordingly!
#> Chain 1:
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#> Chain 1:
#> Chain 1: Elapsed Time: 321.411 seconds (Warm-up)
#> Chain 1: 238.065 seconds (Sampling)
#> Chain 1: 559.476 seconds (Total)
#> Chain 1:
#> Warning: There were 18 divergent transitions after warmup. Increasing adapt_delta above 0.8 may help. See
#> http://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmup
#> Warning: Examine the pairs() plot to diagnose sampling problems
#summary(m.stan)m.stan
#> Inference for Stan model: model.
#> 1 chains, each with iter=3000; warmup=1500; thin=1;
#> post-warmup draws per chain=1500, total post-warmup draws=1500.
#>
#> mean se_mean sd 2.5% 25% 50% 75% 97.5%
#> noise_var 0.01 0.00 0.00 0.00 0.00 0.01 0.01 0.01
#> tau 4.02 0.03 0.47 2.87 3.78 4.11 4.36 4.71
#> beta_medias[1] 1.06 0.02 0.44 0.43 0.75 0.99 1.29 2.09
#> beta_medias[2] 0.95 0.02 0.41 0.32 0.65 0.91 1.18 1.88
#> beta_medias[3] 0.98 0.02 0.51 0.22 0.58 0.89 1.30 2.13
#> gamma_ctrl[1] 0.39 0.00 0.13 0.13 0.30 0.39 0.49 0.67
#> retain_rate[1] 0.51 0.00 0.19 0.16 0.36 0.51 0.65 0.85
#> retain_rate[2] 0.49 0.00 0.18 0.15 0.36 0.49 0.62 0.84
#> retain_rate[3] 0.49 0.00 0.18 0.16 0.36 0.49 0.62 0.84
#> delay[1] 6.10 0.08 3.40 0.41 3.22 6.15 9.01 11.71
#> delay[2] 6.17 0.09 3.53 0.30 2.97 6.26 9.19 11.74
#> delay[3] 6.03 0.09 3.54 0.34 2.91 6.09 9.16 11.72
#> ec[1] 0.59 0.01 0.18 0.26 0.46 0.59 0.74 0.91
#> ec[2] 0.54 0.01 0.19 0.19 0.40 0.54 0.67 0.89
#> ec[3] 0.40 0.01 0.21 0.09 0.23 0.39 0.55 0.83
#> slope[1] 2.68 0.05 1.26 0.99 1.76 2.45 3.33 5.82
#> slope[2] 2.63 0.04 1.27 0.89 1.68 2.38 3.29 5.84
#> slope[3] 2.22 0.04 1.25 0.66 1.34 1.99 2.77 5.50
#> mu[1] 5.24 0.00 0.02 5.20 5.22 5.24 5.25 5.28
#> mu[2] 5.29 0.00 0.02 5.25 5.27 5.29 5.30 5.32
#> mu[3] 5.27 0.00 0.01 5.24 5.26 5.27 5.28 5.30
#> mu[4] 5.21 0.00 0.02 5.18 5.20 5.21 5.23 5.24
#> mu[5] 5.16 0.00 0.02 5.11 5.14 5.15 5.17 5.20
#> mu[6] 5.28 0.00 0.02 5.24 5.27 5.28 5.30 5.33
#> mu[7] 5.34 0.00 0.02 5.29 5.32 5.34 5.35 5.39
#> mu[8] 5.32 0.00 0.02 5.29 5.31 5.32 5.33 5.36
#> mu[9] 5.24 0.00 0.01 5.21 5.23 5.24 5.25 5.27
#> mu[10] 5.15 0.00 0.02 5.12 5.14 5.15 5.16 5.19
#> mu[11] 5.08 0.00 0.02 5.04 5.06 5.08 5.09 5.11
#> mu[12] 4.93 0.00 0.04 4.85 4.90 4.93 4.95 5.00
#> mu[13] 5.11 0.00 0.02 5.08 5.10 5.11 5.12 5.14
#> mu[14] 5.22 0.00 0.02 5.19 5.21 5.22 5.24 5.26
#> mu[15] 5.23 0.00 0.02 5.20 5.22 5.24 5.25 5.27
#> mu[16] 5.17 0.00 0.02 5.13 5.16 5.17 5.18 5.21
#> mu[17] 5.17 0.00 0.01 5.14 5.16 5.17 5.17 5.19
#> mu[18] 5.20 0.00 0.01 5.18 5.19 5.20 5.20 5.22
#> mu[19] 5.22 0.00 0.02 5.19 5.21 5.22 5.23 5.25
#> mu[20] 5.29 0.00 0.02 5.26 5.28 5.29 5.30 5.32
#> mu[21] 5.30 0.00 0.02 5.27 5.29 5.30 5.31 5.33
#> mu[22] 5.21 0.00 0.02 5.18 5.20 5.21 5.23 5.25
#> mu[23] 5.01 0.00 0.03 4.94 4.99 5.01 5.03 5.07
#> mu[24] 4.97 0.00 0.02 4.92 4.95 4.97 4.98 5.01
#> mu[25] 5.08 0.00 0.01 5.05 5.07 5.08 5.09 5.10
#> mu[26] 5.15 0.00 0.02 5.11 5.14 5.15 5.16 5.18
#> mu[27] 5.20 0.00 0.01 5.17 5.19 5.20 5.21 5.23
#> mu[28] 5.21 0.00 0.01 5.19 5.21 5.21 5.22 5.23
#> mu[29] 5.19 0.00 0.01 5.17 5.19 5.19 5.20 5.22
#> mu[30] 5.09 0.00 0.02 5.04 5.08 5.09 5.11 5.14
#> mu[31] 5.12 0.00 0.02 5.08 5.10 5.11 5.13 5.15
#> mu[32] 5.17 0.00 0.02 5.14 5.16 5.17 5.18 5.20
#> mu[33] 5.24 0.00 0.01 5.21 5.23 5.24 5.25 5.27
#> mu[34] 5.21 0.00 0.01 5.19 5.20 5.21 5.22 5.24
#> mu[35] 5.15 0.00 0.01 5.12 5.14 5.15 5.16 5.18
#> mu[36] 5.08 0.00 0.02 5.04 5.06 5.08 5.09 5.11
#> mu[37] 5.09 0.00 0.02 5.05 5.08 5.09 5.10 5.13
#> mu[38] 5.17 0.00 0.02 5.13 5.15 5.17 5.18 5.20
#> mu[39] 5.20 0.00 0.02 5.17 5.19 5.20 5.22 5.24
#> mu[40] 5.26 0.00 0.01 5.23 5.25 5.26 5.27 5.29
#> mu[41] 5.27 0.00 0.01 5.24 5.26 5.27 5.28 5.29
#> mu[42] 5.21 0.00 0.01 5.19 5.20 5.21 5.22 5.23
#> mu[43] 5.09 0.00 0.01 5.06 5.08 5.09 5.10 5.12
#> mu[44] 4.93 0.00 0.03 4.87 4.91 4.93 4.95 4.99
#> mu[45] 5.09 0.00 0.02 5.05 5.07 5.09 5.10 5.12
#> mu[46] 5.26 0.00 0.01 5.24 5.25 5.26 5.27 5.29
#> mu[47] 5.32 0.00 0.02 5.28 5.30 5.31 5.33 5.35
#> mu[48] 5.24 0.00 0.02 5.20 5.23 5.24 5.26 5.29
#> mu[49] 5.09 0.00 0.03 5.04 5.07 5.09 5.11 5.15
#> mu[50] 5.03 0.00 0.03 4.97 5.01 5.03 5.05 5.09
#> mu[51] 5.09 0.00 0.02 5.06 5.08 5.09 5.10 5.12
#> mu[52] 5.23 0.00 0.01 5.21 5.22 5.23 5.23 5.25
#> mu[53] 5.29 0.00 0.01 5.27 5.28 5.29 5.30 5.32
#> mu[54] 5.28 0.00 0.01 5.26 5.28 5.28 5.29 5.31
#> mu[55] 5.28 0.00 0.01 5.25 5.27 5.28 5.29 5.31
#> mu[56] 5.19 0.00 0.02 5.16 5.18 5.19 5.20 5.23
#> mu[57] 5.15 0.00 0.02 5.11 5.14 5.15 5.17 5.20
#> mu[58] 5.26 0.00 0.02 5.22 5.25 5.26 5.27 5.30
#> mu[59] 5.29 0.00 0.02 5.25 5.28 5.29 5.30 5.33
#> mu[60] 5.25 0.00 0.02 5.21 5.24 5.25 5.27 5.29
#> mu[61] 5.19 0.00 0.02 5.15 5.17 5.19 5.20 5.23
#> mu[62] 5.09 0.00 0.02 5.05 5.07 5.09 5.10 5.13
#> mu[63] 5.01 0.00 0.02 4.97 4.99 5.01 5.02 5.04
#> mu[64] 5.14 0.00 0.02 5.10 5.12 5.14 5.15 5.17
#> mu[65] 5.26 0.00 0.02 5.23 5.25 5.26 5.28 5.30
#> mu[66] 5.30 0.00 0.02 5.26 5.28 5.30 5.31 5.33
#> mu[67] 5.28 0.00 0.01 5.26 5.27 5.28 5.29 5.30
#> mu[68] 5.24 0.00 0.01 5.22 5.23 5.24 5.25 5.27
#> mu[69] 5.18 0.00 0.02 5.15 5.17 5.18 5.19 5.21
#> mu[70] 5.19 0.00 0.01 5.18 5.19 5.19 5.20 5.21
#> mu[71] 5.23 0.00 0.01 5.21 5.23 5.23 5.24 5.26
#> mu[72] 5.31 0.00 0.02 5.28 5.30 5.31 5.32 5.35
#> mu[73] 5.28 0.00 0.02 5.24 5.27 5.28 5.29 5.32
#> mu[74] 5.12 0.00 0.02 5.07 5.10 5.12 5.13 5.16
#> mu[75] 5.03 0.00 0.03 4.97 5.01 5.03 5.05 5.09
#> mu[76] 5.03 0.00 0.04 4.96 5.01 5.03 5.06 5.10
#> mu[77] 5.17 0.00 0.02 5.14 5.16 5.17 5.18 5.20
#> mu[78] 5.27 0.00 0.01 5.25 5.26 5.27 5.28 5.29
#> mu[79] 5.23 0.00 0.01 5.21 5.22 5.23 5.24 5.26
#> mu[80] 5.19 0.00 0.02 5.16 5.18 5.19 5.20 5.22
#> mu[81] 5.18 0.00 0.01 5.15 5.17 5.18 5.19 5.20
#> mu[82] 5.13 0.00 0.01 5.11 5.12 5.13 5.14 5.15
#> mu[83] 5.13 0.00 0.01 5.11 5.13 5.14 5.14 5.16
#> mu[84] 5.18 0.00 0.01 5.16 5.17 5.18 5.19 5.21
#> mu[85] 5.25 0.00 0.02 5.22 5.24 5.25 5.26 5.28
#> mu[86] 5.23 0.00 0.02 5.19 5.22 5.23 5.24 5.27
#> mu[87] 5.19 0.00 0.01 5.17 5.19 5.20 5.20 5.22
#> mu[88] 5.08 0.00 0.02 5.05 5.07 5.08 5.09 5.12
#> mu[89] 5.08 0.00 0.02 5.04 5.06 5.08 5.09 5.11
#> mu[90] 5.19 0.00 0.01 5.17 5.18 5.19 5.20 5.21
#> mu[91] 5.27 0.00 0.02 5.23 5.25 5.27 5.28 5.30
#> mu[92] 5.32 0.00 0.02 5.28 5.31 5.32 5.33 5.36
#> cum_effect 0.23 0.00 0.00 0.23 0.23 0.23 0.23 0.23
#> cum_effects_hill[1,1] 0.52 0.01 0.19 0.19 0.38 0.50 0.65 0.90
#> cum_effects_hill[1,2] 0.46 0.01 0.21 0.12 0.30 0.43 0.60 0.93
#> cum_effects_hill[1,3] 0.36 0.01 0.25 0.04 0.16 0.30 0.49 0.92
#> cum_effects_hill[2,1] 0.54 0.01 0.18 0.22 0.40 0.52 0.67 0.91
#> cum_effects_hill[2,2] 0.47 0.01 0.21 0.13 0.31 0.44 0.61 0.93
#> cum_effects_hill[2,3] 0.35 0.01 0.25 0.04 0.16 0.29 0.49 0.92
#> cum_effects_hill[3,1] 0.53 0.01 0.19 0.20 0.39 0.51 0.65 0.90
#> cum_effects_hill[3,2] 0.46 0.01 0.21 0.12 0.30 0.43 0.59 0.93
#> cum_effects_hill[3,3] 0.33 0.01 0.24 0.03 0.14 0.27 0.46 0.90
#> cum_effects_hill[4,1] 0.47 0.01 0.19 0.14 0.33 0.46 0.60 0.87
#> cum_effects_hill[4,2] 0.44 0.01 0.21 0.11 0.28 0.41 0.57 0.92
#> cum_effects_hill[4,3] 0.31 0.01 0.23 0.03 0.12 0.25 0.44 0.88
#> cum_effects_hill[5,1] 0.39 0.01 0.19 0.07 0.24 0.37 0.51 0.81
#> cum_effects_hill[5,2] 0.42 0.01 0.21 0.09 0.25 0.39 0.55 0.91
#> cum_effects_hill[5,3] 0.31 0.01 0.24 0.03 0.12 0.25 0.44 0.88
#> cum_effects_hill[6,1] 0.49 0.01 0.19 0.16 0.35 0.48 0.62 0.88
#> cum_effects_hill[6,2] 0.42 0.01 0.21 0.09 0.25 0.39 0.55 0.91
#> cum_effects_hill[6,3] 0.35 0.01 0.24 0.04 0.16 0.29 0.48 0.91
#> cum_effects_hill[7,1] 0.55 0.01 0.18 0.23 0.42 0.53 0.68 0.91
#> cum_effects_hill[7,2] 0.41 0.01 0.21 0.09 0.25 0.39 0.55 0.90
#> cum_effects_hill[7,3] 0.36 0.01 0.25 0.05 0.17 0.30 0.50 0.92
#> cum_effects_hill[8,1] 0.56 0.01 0.18 0.24 0.42 0.54 0.69 0.92
#> cum_effects_hill[8,2] 0.43 0.01 0.21 0.10 0.27 0.40 0.57 0.91
#> cum_effects_hill[8,3] 0.35 0.01 0.24 0.04 0.16 0.29 0.48 0.91
#> cum_effects_hill[9,1] 0.54 0.01 0.18 0.22 0.41 0.52 0.67 0.91
#> cum_effects_hill[9,2] 0.41 0.01 0.21 0.09 0.25 0.38 0.54 0.90
#> cum_effects_hill[9,3] 0.32 0.01 0.24 0.03 0.13 0.26 0.44 0.88
#> cum_effects_hill[10,1] 0.52 0.01 0.19 0.19 0.38 0.50 0.65 0.89
#> cum_effects_hill[10,2] 0.34 0.01 0.21 0.05 0.17 0.31 0.45 0.85
#> cum_effects_hill[10,3] 0.29 0.01 0.23 0.02 0.11 0.23 0.43 0.86
#> cum_effects_hill[11,1] 0.44 0.01 0.19 0.11 0.29 0.42 0.56 0.85
#> cum_effects_hill[11,2] 0.33 0.01 0.21 0.04 0.16 0.29 0.44 0.84
#> cum_effects_hill[11,3] 0.29 0.01 0.23 0.02 0.11 0.23 0.42 0.86
#> cum_effects_hill[12,1] 0.28 0.01 0.18 0.03 0.13 0.25 0.39 0.69
#> cum_effects_hill[12,2] 0.36 0.01 0.21 0.06 0.20 0.34 0.49 0.87
#> cum_effects_hill[12,3] 0.27 0.01 0.22 0.02 0.09 0.21 0.39 0.82
#> cum_effects_hill[13,1] 0.42 0.01 0.19 0.09 0.27 0.40 0.54 0.83
#> cum_effects_hill[13,2] 0.41 0.01 0.21 0.09 0.25 0.39 0.55 0.91
#> cum_effects_hill[13,3] 0.30 0.01 0.23 0.02 0.11 0.24 0.43 0.86
#> cum_effects_hill[14,1] 0.53 0.01 0.19 0.20 0.39 0.51 0.66 0.90
#> cum_effects_hill[14,2] 0.44 0.01 0.21 0.11 0.28 0.41 0.58 0.92
#> cum_effects_hill[14,3] 0.32 0.01 0.24 0.03 0.13 0.26 0.45 0.89
#> cum_effects_hill[15,1] 0.55 0.01 0.18 0.23 0.41 0.53 0.68 0.91
#> cum_effects_hill[15,2] 0.44 0.01 0.21 0.11 0.28 0.41 0.58 0.92
#> cum_effects_hill[15,3] 0.32 0.01 0.24 0.03 0.13 0.26 0.45 0.89
#> cum_effects_hill[16,1] 0.52 0.01 0.19 0.18 0.38 0.50 0.64 0.89
#> cum_effects_hill[16,2] 0.42 0.01 0.21 0.09 0.25 0.39 0.55 0.91
#> cum_effects_hill[16,3] 0.32 0.01 0.24 0.03 0.13 0.26 0.45 0.89
#> cum_effects_hill[17,1] 0.50 0.01 0.19 0.16 0.36 0.48 0.62 0.88
#> cum_effects_hill[17,2] 0.40 0.01 0.21 0.08 0.24 0.37 0.53 0.90
#> cum_effects_hill[17,3] 0.31 0.01 0.24 0.03 0.12 0.25 0.44 0.88
#> cum_effects_hill[18,1] 0.50 0.01 0.19 0.17 0.36 0.49 0.63 0.89
#> cum_effects_hill[18,2] 0.40 0.01 0.21 0.08 0.24 0.38 0.54 0.90
#> cum_effects_hill[18,3] 0.32 0.01 0.24 0.03 0.13 0.26 0.45 0.88
#> cum_effects_hill[19,1] 0.53 0.01 0.18 0.21 0.39 0.51 0.66 0.90
#> cum_effects_hill[19,2] 0.40 0.01 0.21 0.08 0.24 0.38 0.53 0.90
#> cum_effects_hill[19,3] 0.30 0.01 0.23 0.02 0.11 0.24 0.43 0.86
#> cum_effects_hill[20,1] 0.56 0.01 0.18 0.24 0.42 0.54 0.69 0.92
#> cum_effects_hill[20,2] 0.44 0.01 0.21 0.11 0.28 0.41 0.58 0.92
#> cum_effects_hill[20,3] 0.33 0.01 0.24 0.03 0.14 0.27 0.46 0.90
#> cum_effects_hill[21,1] 0.56 0.01 0.18 0.24 0.42 0.54 0.69 0.92
#> cum_effects_hill[21,2] 0.45 0.01 0.21 0.11 0.28 0.42 0.58 0.92
#> cum_effects_hill[21,3] 0.34 0.01 0.24 0.03 0.14 0.28 0.47 0.90
#> cum_effects_hill[22,1] 0.51 0.01 0.19 0.18 0.37 0.49 0.64 0.89
#> cum_effects_hill[22,2] 0.42 0.01 0.21 0.10 0.26 0.40 0.56 0.91
#> cum_effects_hill[22,3] 0.29 0.01 0.23 0.02 0.11 0.23 0.42 0.85
#> cum_effects_hill[23,1] 0.39 0.01 0.19 0.07 0.24 0.37 0.51 0.80
#> cum_effects_hill[23,2] 0.37 0.01 0.21 0.06 0.20 0.34 0.49 0.87
#> cum_effects_hill[23,3] 0.22 0.01 0.19 0.01 0.06 0.16 0.31 0.75
#> cum_effects_hill[24,1] 0.35 0.01 0.19 0.05 0.20 0.33 0.47 0.77
#> cum_effects_hill[24,2] 0.33 0.01 0.21 0.04 0.17 0.30 0.45 0.85
#> cum_effects_hill[24,3] 0.26 0.01 0.22 0.01 0.09 0.20 0.37 0.81
#> cum_effects_hill[25,1] 0.42 0.01 0.19 0.09 0.28 0.41 0.54 0.83
#> cum_effects_hill[25,2] 0.36 0.01 0.21 0.06 0.20 0.34 0.49 0.87
#> cum_effects_hill[25,3] 0.32 0.01 0.24 0.03 0.13 0.26 0.45 0.89
#> cum_effects_hill[26,1] 0.45 0.01 0.19 0.12 0.31 0.44 0.57 0.85
#> cum_effects_hill[26,2] 0.41 0.01 0.21 0.09 0.25 0.39 0.55 0.90
#> cum_effects_hill[26,3] 0.35 0.01 0.24 0.04 0.16 0.29 0.48 0.91
#> cum_effects_hill[27,1] 0.48 0.01 0.19 0.14 0.33 0.46 0.60 0.87
#> cum_effects_hill[27,2] 0.43 0.01 0.21 0.10 0.26 0.40 0.56 0.91
#> cum_effects_hill[27,3] 0.35 0.01 0.25 0.04 0.16 0.29 0.48 0.91
#> cum_effects_hill[28,1] 0.50 0.01 0.19 0.17 0.37 0.49 0.63 0.89
#> cum_effects_hill[28,2] 0.40 0.01 0.21 0.08 0.24 0.38 0.53 0.90
#> cum_effects_hill[28,3] 0.35 0.01 0.24 0.04 0.15 0.29 0.48 0.91
#> cum_effects_hill[29,1] 0.52 0.01 0.19 0.19 0.38 0.50 0.65 0.90
#> cum_effects_hill[29,2] 0.36 0.01 0.21 0.06 0.20 0.34 0.49 0.87
#> cum_effects_hill[29,3] 0.33 0.01 0.24 0.03 0.14 0.27 0.46 0.90
#> cum_effects_hill[30,1] 0.50 0.01 0.19 0.17 0.36 0.49 0.63 0.88
#> cum_effects_hill[30,2] 0.28 0.01 0.20 0.02 0.12 0.25 0.39 0.80
#> cum_effects_hill[30,3] 0.32 0.01 0.24 0.03 0.13 0.26 0.45 0.89
#> cum_effects_hill[31,1] 0.46 0.01 0.19 0.13 0.32 0.45 0.59 0.86
#> cum_effects_hill[31,2] 0.33 0.01 0.21 0.04 0.16 0.30 0.44 0.84
#> cum_effects_hill[31,3] 0.35 0.01 0.25 0.04 0.16 0.29 0.48 0.91
#> cum_effects_hill[32,1] 0.49 0.01 0.19 0.15 0.35 0.48 0.61 0.88
#> cum_effects_hill[32,2] 0.35 0.01 0.21 0.05 0.18 0.32 0.47 0.86
#> cum_effects_hill[32,3] 0.36 0.01 0.25 0.04 0.16 0.30 0.49 0.92
#> cum_effects_hill[33,1] 0.51 0.01 0.19 0.18 0.37 0.50 0.64 0.89
#> cum_effects_hill[33,2] 0.40 0.01 0.21 0.08 0.24 0.37 0.53 0.90
#> cum_effects_hill[33,3] 0.37 0.01 0.25 0.05 0.17 0.31 0.50 0.93
#> cum_effects_hill[34,1] 0.48 0.01 0.19 0.14 0.34 0.47 0.60 0.87
#> cum_effects_hill[34,2] 0.40 0.01 0.21 0.08 0.24 0.38 0.53 0.90
#> cum_effects_hill[34,3] 0.36 0.01 0.25 0.04 0.16 0.30 0.49 0.92
#> cum_effects_hill[35,1] 0.43 0.01 0.19 0.10 0.28 0.41 0.55 0.84
#> cum_effects_hill[35,2] 0.38 0.01 0.21 0.07 0.22 0.36 0.51 0.89
#> cum_effects_hill[35,3] 0.33 0.01 0.24 0.03 0.14 0.27 0.46 0.90
#> cum_effects_hill[36,1] 0.38 0.01 0.19 0.07 0.23 0.36 0.50 0.80
#> cum_effects_hill[36,2] 0.36 0.01 0.21 0.06 0.19 0.33 0.48 0.86
#> cum_effects_hill[36,3] 0.30 0.01 0.23 0.02 0.12 0.24 0.43 0.87
#> cum_effects_hill[37,1] 0.41 0.01 0.19 0.09 0.27 0.40 0.54 0.83
#> cum_effects_hill[37,2] 0.33 0.01 0.21 0.05 0.17 0.31 0.45 0.85
#> cum_effects_hill[37,3] 0.29 0.01 0.23 0.02 0.11 0.23 0.42 0.86
#> cum_effects_hill[38,1] 0.51 0.01 0.19 0.18 0.38 0.50 0.64 0.89
#> cum_effects_hill[38,2] 0.36 0.01 0.21 0.06 0.20 0.34 0.49 0.87
#> cum_effects_hill[38,3] 0.28 0.01 0.22 0.02 0.10 0.22 0.40 0.84
#> cum_effects_hill[39,1] 0.55 0.01 0.18 0.23 0.41 0.53 0.68 0.91
#> cum_effects_hill[39,2] 0.34 0.01 0.21 0.05 0.18 0.32 0.47 0.85
#> cum_effects_hill[39,3] 0.33 0.01 0.24 0.03 0.14 0.27 0.46 0.90
#> cum_effects_hill[40,1] 0.55 0.01 0.18 0.23 0.41 0.53 0.68 0.91
#> cum_effects_hill[40,2] 0.39 0.01 0.21 0.08 0.23 0.37 0.52 0.89
#> cum_effects_hill[40,3] 0.35 0.01 0.25 0.04 0.16 0.29 0.49 0.92
#> cum_effects_hill[41,1] 0.53 0.01 0.18 0.21 0.40 0.51 0.66 0.91
#> cum_effects_hill[41,2] 0.41 0.01 0.21 0.08 0.24 0.38 0.54 0.90
#> cum_effects_hill[41,3] 0.36 0.01 0.25 0.04 0.16 0.30 0.49 0.92
#> cum_effects_hill[42,1] 0.51 0.01 0.19 0.18 0.37 0.50 0.64 0.89
#> cum_effects_hill[42,2] 0.41 0.01 0.21 0.08 0.24 0.38 0.54 0.90
#> cum_effects_hill[42,3] 0.34 0.01 0.24 0.04 0.15 0.28 0.47 0.90
#> cum_effects_hill[43,1] 0.44 0.01 0.19 0.11 0.30 0.43 0.56 0.85
#> cum_effects_hill[43,2] 0.37 0.01 0.21 0.07 0.21 0.35 0.50 0.88
#> cum_effects_hill[43,3] 0.31 0.01 0.24 0.03 0.12 0.25 0.44 0.88
#> cum_effects_hill[44,1] 0.34 0.01 0.19 0.05 0.19 0.32 0.46 0.76
#> cum_effects_hill[44,2] 0.30 0.01 0.20 0.03 0.14 0.26 0.41 0.81
#> cum_effects_hill[44,3] 0.32 0.01 0.24 0.03 0.13 0.26 0.45 0.89
#> cum_effects_hill[45,1] 0.44 0.01 0.19 0.11 0.30 0.43 0.56 0.85
#> cum_effects_hill[45,2] 0.36 0.01 0.21 0.06 0.19 0.33 0.48 0.86
#> cum_effects_hill[45,3] 0.34 0.01 0.24 0.04 0.15 0.28 0.47 0.90
#> cum_effects_hill[46,1] 0.54 0.01 0.18 0.21 0.40 0.52 0.66 0.91
#> cum_effects_hill[46,2] 0.41 0.01 0.21 0.09 0.25 0.39 0.55 0.91
#> cum_effects_hill[46,3] 0.36 0.01 0.25 0.04 0.16 0.30 0.49 0.92
#> cum_effects_hill[47,1] 0.55 0.01 0.18 0.24 0.42 0.53 0.68 0.91
#> cum_effects_hill[47,2] 0.43 0.01 0.21 0.10 0.26 0.40 0.56 0.91
#> cum_effects_hill[47,3] 0.35 0.01 0.25 0.04 0.16 0.29 0.49 0.92
#> cum_effects_hill[48,1] 0.53 0.01 0.19 0.20 0.39 0.51 0.66 0.90
#> cum_effects_hill[48,2] 0.37 0.01 0.21 0.06 0.20 0.34 0.49 0.87
#> cum_effects_hill[48,3] 0.33 0.01 0.24 0.03 0.13 0.26 0.45 0.89
#> cum_effects_hill[49,1] 0.48 0.01 0.19 0.15 0.34 0.47 0.61 0.88
#> cum_effects_hill[49,2] 0.27 0.01 0.19 0.02 0.11 0.23 0.38 0.77
#> cum_effects_hill[49,3] 0.29 0.01 0.23 0.02 0.11 0.23 0.42 0.86
#> cum_effects_hill[50,1] 0.43 0.01 0.19 0.10 0.28 0.41 0.55 0.84
#> cum_effects_hill[50,2] 0.25 0.01 0.19 0.02 0.10 0.21 0.36 0.75
#> cum_effects_hill[50,3] 0.32 0.01 0.24 0.03 0.13 0.26 0.45 0.89
#> cum_effects_hill[51,1] 0.43 0.01 0.19 0.10 0.29 0.42 0.55 0.84
#> cum_effects_hill[51,2] 0.35 0.01 0.21 0.05 0.18 0.32 0.47 0.86
#> cum_effects_hill[51,3] 0.33 0.01 0.24 0.03 0.14 0.27 0.46 0.90
#> cum_effects_hill[52,1] 0.50 0.01 0.19 0.17 0.36 0.49 0.63 0.89
#> cum_effects_hill[52,2] 0.41 0.01 0.21 0.09 0.25 0.39 0.54 0.90
#> cum_effects_hill[52,3] 0.35 0.01 0.25 0.04 0.16 0.29 0.48 0.91
#> cum_effects_hill[53,1] 0.54 0.01 0.18 0.22 0.40 0.52 0.67 0.91
#> cum_effects_hill[53,2] 0.43 0.01 0.21 0.10 0.27 0.40 0.57 0.91
#> cum_effects_hill[53,3] 0.36 0.01 0.25 0.04 0.17 0.30 0.50 0.92
#> cum_effects_hill[54,1] 0.55 0.01 0.18 0.24 0.42 0.53 0.68 0.91
#> cum_effects_hill[54,2] 0.41 0.01 0.21 0.09 0.25 0.39 0.54 0.90
#> cum_effects_hill[54,3] 0.35 0.01 0.25 0.04 0.16 0.29 0.49 0.92
#> cum_effects_hill[55,1] 0.53 0.01 0.18 0.21 0.40 0.51 0.66 0.91
#> cum_effects_hill[55,2] 0.42 0.01 0.21 0.09 0.25 0.39 0.55 0.91
#> cum_effects_hill[55,3] 0.36 0.01 0.25 0.04 0.16 0.30 0.49 0.92
#> cum_effects_hill[56,1] 0.41 0.01 0.19 0.09 0.27 0.40 0.54 0.83
#> cum_effects_hill[56,2] 0.43 0.01 0.21 0.10 0.27 0.41 0.57 0.92
#> cum_effects_hill[56,3] 0.35 0.01 0.25 0.04 0.16 0.29 0.48 0.91
#> cum_effects_hill[57,1] 0.37 0.01 0.19 0.06 0.23 0.35 0.49 0.79
#> cum_effects_hill[57,2] 0.42 0.01 0.21 0.10 0.26 0.40 0.56 0.91
#> cum_effects_hill[57,3] 0.33 0.01 0.24 0.03 0.14 0.27 0.46 0.90
#> cum_effects_hill[58,1] 0.51 0.01 0.19 0.17 0.37 0.49 0.63 0.89
#> cum_effects_hill[58,2] 0.45 0.01 0.21 0.11 0.29 0.42 0.58 0.92
#> cum_effects_hill[58,3] 0.30 0.01 0.23 0.02 0.12 0.24 0.43 0.87
#> cum_effects_hill[59,1] 0.55 0.01 0.18 0.24 0.42 0.53 0.68 0.91
#> cum_effects_hill[59,2] 0.46 0.01 0.21 0.13 0.30 0.43 0.60 0.93
#> cum_effects_hill[59,3] 0.30 0.01 0.23 0.02 0.12 0.24 0.43 0.87
#> cum_effects_hill[60,1] 0.55 0.01 0.18 0.24 0.42 0.53 0.68 0.91
#> cum_effects_hill[60,2] 0.45 0.01 0.21 0.11 0.29 0.42 0.59 0.92
#> cum_effects_hill[60,3] 0.30 0.01 0.23 0.02 0.12 0.24 0.43 0.87
#> cum_effects_hill[61,1] 0.52 0.01 0.19 0.19 0.38 0.50 0.65 0.90
#> cum_effects_hill[61,2] 0.42 0.01 0.21 0.09 0.25 0.39 0.55 0.91
#> cum_effects_hill[61,3] 0.29 0.01 0.23 0.02 0.11 0.23 0.42 0.86
#> cum_effects_hill[62,1] 0.45 0.01 0.19 0.12 0.31 0.44 0.57 0.86
#> cum_effects_hill[62,2] 0.39 0.01 0.21 0.07 0.22 0.36 0.51 0.89
#> cum_effects_hill[62,3] 0.28 0.01 0.22 0.02 0.10 0.22 0.40 0.84
#> cum_effects_hill[63,1] 0.37 0.01 0.19 0.06 0.22 0.35 0.49 0.79
#> cum_effects_hill[63,2] 0.34 0.01 0.21 0.05 0.17 0.31 0.46 0.85
#> cum_effects_hill[63,3] 0.31 0.01 0.23 0.03 0.12 0.25 0.44 0.87
#> cum_effects_hill[64,1] 0.40 0.01 0.19 0.08 0.25 0.38 0.52 0.81
#> cum_effects_hill[64,2] 0.40 0.01 0.21 0.08 0.24 0.38 0.53 0.90
#> cum_effects_hill[64,3] 0.34 0.01 0.24 0.04 0.15 0.28 0.47 0.90
#> cum_effects_hill[65,1] 0.48 0.01 0.19 0.14 0.34 0.47 0.60 0.87
#> cum_effects_hill[65,2] 0.45 0.01 0.21 0.12 0.29 0.42 0.59 0.92
#> cum_effects_hill[65,3] 0.36 0.01 0.25 0.04 0.17 0.30 0.50 0.92
#> cum_effects_hill[66,1] 0.51 0.01 0.19 0.17 0.37 0.49 0.63 0.89
#> cum_effects_hill[66,2] 0.46 0.01 0.21 0.13 0.30 0.43 0.60 0.93
#> cum_effects_hill[66,3] 0.37 0.01 0.25 0.05 0.18 0.31 0.51 0.93
#> cum_effects_hill[67,1] 0.52 0.01 0.19 0.19 0.39 0.51 0.65 0.90
#> cum_effects_hill[67,2] 0.45 0.01 0.21 0.11 0.29 0.42 0.58 0.92
#> cum_effects_hill[67,3] 0.35 0.01 0.24 0.04 0.15 0.29 0.48 0.91
#> cum_effects_hill[68,1] 0.53 0.01 0.18 0.21 0.40 0.51 0.66 0.90
#> cum_effects_hill[68,2] 0.41 0.01 0.21 0.08 0.24 0.38 0.54 0.90
#> cum_effects_hill[68,3] 0.31 0.01 0.24 0.03 0.13 0.25 0.44 0.88
#> cum_effects_hill[69,1] 0.52 0.01 0.19 0.19 0.38 0.50 0.65 0.90
#> cum_effects_hill[69,2] 0.38 0.01 0.21 0.07 0.22 0.36 0.51 0.88
#> cum_effects_hill[69,3] 0.29 0.01 0.23 0.02 0.11 0.23 0.42 0.85
#> cum_effects_hill[70,1] 0.50 0.01 0.19 0.16 0.36 0.48 0.62 0.88
#> cum_effects_hill[70,2] 0.39 0.01 0.21 0.08 0.23 0.37 0.52 0.89
#> cum_effects_hill[70,3] 0.33 0.01 0.24 0.03 0.14 0.27 0.46 0.90
#> cum_effects_hill[71,1] 0.49 0.01 0.19 0.15 0.35 0.47 0.61 0.88
#> cum_effects_hill[71,2] 0.44 0.01 0.21 0.11 0.28 0.41 0.58 0.92
#> cum_effects_hill[71,3] 0.35 0.01 0.25 0.04 0.16 0.29 0.48 0.91
#> cum_effects_hill[72,1] 0.51 0.01 0.19 0.18 0.38 0.50 0.64 0.89
#> cum_effects_hill[72,2] 0.47 0.01 0.21 0.13 0.31 0.44 0.61 0.93
#> cum_effects_hill[72,3] 0.37 0.01 0.25 0.05 0.17 0.30 0.50 0.93
#> cum_effects_hill[73,1] 0.48 0.01 0.19 0.15 0.34 0.47 0.61 0.87
#> cum_effects_hill[73,2] 0.47 0.01 0.21 0.14 0.32 0.45 0.61 0.93
#> cum_effects_hill[73,3] 0.35 0.01 0.25 0.04 0.16 0.29 0.49 0.91
#> cum_effects_hill[74,1] 0.36 0.01 0.19 0.06 0.21 0.34 0.48 0.78
#> cum_effects_hill[74,2] 0.46 0.01 0.21 0.12 0.30 0.43 0.60 0.93
#> cum_effects_hill[74,3] 0.33 0.01 0.24 0.03 0.14 0.27 0.46 0.89
#> cum_effects_hill[75,1] 0.31 0.01 0.18 0.04 0.17 0.29 0.43 0.74
#> cum_effects_hill[75,2] 0.43 0.01 0.21 0.10 0.27 0.41 0.57 0.92
#> cum_effects_hill[75,3] 0.29 0.01 0.23 0.02 0.11 0.23 0.42 0.85
#> cum_effects_hill[76,1] 0.36 0.01 0.19 0.06 0.22 0.35 0.49 0.79
#> cum_effects_hill[76,2] 0.41 0.01 0.21 0.09 0.25 0.39 0.55 0.91
#> cum_effects_hill[76,3] 0.22 0.01 0.20 0.01 0.06 0.16 0.32 0.75
#> cum_effects_hill[77,1] 0.46 0.01 0.19 0.13 0.32 0.45 0.58 0.86
#> cum_effects_hill[77,2] 0.41 0.01 0.21 0.09 0.25 0.39 0.55 0.90
#> cum_effects_hill[77,3] 0.29 0.01 0.23 0.02 0.11 0.23 0.42 0.86
#> cum_effects_hill[78,1] 0.52 0.01 0.19 0.19 0.38 0.50 0.65 0.90
#> cum_effects_hill[78,2] 0.43 0.01 0.21 0.11 0.27 0.41 0.57 0.92
#> cum_effects_hill[78,3] 0.34 0.01 0.24 0.03 0.15 0.28 0.47 0.90
#> cum_effects_hill[79,1] 0.52 0.01 0.19 0.19 0.38 0.50 0.65 0.90
#> cum_effects_hill[79,2] 0.42 0.01 0.21 0.09 0.25 0.39 0.55 0.91
#> cum_effects_hill[79,3] 0.36 0.01 0.25 0.04 0.17 0.30 0.50 0.92
#> cum_effects_hill[80,1] 0.47 0.01 0.19 0.14 0.33 0.46 0.60 0.87
#> cum_effects_hill[80,2] 0.41 0.01 0.21 0.08 0.24 0.38 0.54 0.90
#> cum_effects_hill[80,3] 0.36 0.01 0.25 0.05 0.17 0.30 0.50 0.92
#> cum_effects_hill[81,1] 0.44 0.01 0.19 0.11 0.30 0.43 0.56 0.85
#> cum_effects_hill[81,2] 0.42 0.01 0.21 0.09 0.25 0.39 0.55 0.91
#> cum_effects_hill[81,3] 0.34 0.01 0.24 0.04 0.15 0.28 0.47 0.90
#> cum_effects_hill[82,1] 0.43 0.01 0.19 0.10 0.29 0.42 0.55 0.85
#> cum_effects_hill[82,2] 0.39 0.01 0.21 0.07 0.22 0.36 0.52 0.89
#> cum_effects_hill[82,3] 0.32 0.01 0.24 0.03 0.13 0.26 0.45 0.89
#> cum_effects_hill[83,1] 0.46 0.01 0.19 0.12 0.31 0.45 0.58 0.86
#> cum_effects_hill[83,2] 0.39 0.01 0.21 0.07 0.22 0.36 0.52 0.89
#> cum_effects_hill[83,3] 0.31 0.01 0.24 0.03 0.12 0.25 0.44 0.88
#> cum_effects_hill[84,1] 0.52 0.01 0.19 0.19 0.38 0.51 0.65 0.90
#> cum_effects_hill[84,2] 0.37 0.01 0.21 0.06 0.21 0.34 0.50 0.87
#> cum_effects_hill[84,3] 0.34 0.01 0.24 0.03 0.15 0.28 0.47 0.90
#> cum_effects_hill[85,1] 0.56 0.01 0.18 0.25 0.43 0.54 0.69 0.92
#> cum_effects_hill[85,2] 0.41 0.01 0.21 0.09 0.25 0.38 0.54 0.90
#> cum_effects_hill[85,3] 0.35 0.01 0.24 0.04 0.15 0.29 0.48 0.91
#> cum_effects_hill[86,1] 0.55 0.01 0.18 0.23 0.41 0.52 0.67 0.91
#> cum_effects_hill[86,2] 0.41 0.01 0.21 0.09 0.25 0.39 0.55 0.91
#> cum_effects_hill[86,3] 0.35 0.01 0.25 0.04 0.16 0.29 0.48 0.91
#> cum_effects_hill[87,1] 0.49 0.01 0.19 0.15 0.35 0.48 0.61 0.88
#> cum_effects_hill[87,2] 0.41 0.01 0.21 0.09 0.25 0.39 0.54 0.90
#> cum_effects_hill[87,3] 0.36 0.01 0.25 0.04 0.16 0.30 0.49 0.92
#> cum_effects_hill[88,1] 0.40 0.01 0.19 0.08 0.26 0.39 0.52 0.82
#> cum_effects_hill[88,2] 0.38 0.01 0.21 0.07 0.21 0.35 0.51 0.88
#> cum_effects_hill[88,3] 0.34 0.01 0.24 0.03 0.15 0.28 0.47 0.90
#> cum_effects_hill[89,1] 0.40 0.01 0.19 0.08 0.26 0.39 0.52 0.82
#> cum_effects_hill[89,2] 0.35 0.01 0.21 0.06 0.19 0.33 0.48 0.86
#> cum_effects_hill[89,3] 0.34 0.01 0.24 0.04 0.15 0.28 0.48 0.91
#> cum_effects_hill[90,1] 0.48 0.01 0.19 0.15 0.34 0.47 0.61 0.87
#> cum_effects_hill[90,2] 0.38 0.01 0.21 0.07 0.22 0.36 0.51 0.89
#> cum_effects_hill[90,3] 0.35 0.01 0.24 0.04 0.15 0.29 0.48 0.91
#> cum_effects_hill[91,1] 0.55 0.01 0.18 0.24 0.42 0.53 0.68 0.91
#> cum_effects_hill[91,2] 0.39 0.01 0.21 0.07 0.22 0.36 0.51 0.89
#> cum_effects_hill[91,3] 0.35 0.01 0.25 0.04 0.16 0.29 0.48 0.91
#> cum_effects_hill[92,1] 0.57 0.01 0.18 0.26 0.44 0.55 0.70 0.92
#> cum_effects_hill[92,2] 0.41 0.01 0.21 0.09 0.25 0.39 0.55 0.91
#> cum_effects_hill[92,3] 0.36 0.01 0.25 0.04 0.16 0.30 0.49 0.92
#> lag_weights[1] 0.10 0.01 0.25 0.00 0.00 0.00 0.01 0.94
#> lag_weights[2] 0.17 0.01 0.32 0.00 0.00 0.00 0.12 0.99
#> lag_weights[3] 0.19 0.01 0.32 0.00 0.00 0.00 0.26 0.98
#> lag_weights[4] 0.19 0.01 0.32 0.00 0.00 0.00 0.24 0.99
#> lag_weights[5] 0.18 0.01 0.31 0.00 0.00 0.00 0.22 0.98
#> lag_weights[6] 0.17 0.01 0.30 0.00 0.00 0.00 0.20 0.98
#> lag_weights[7] 0.17 0.01 0.30 0.00 0.00 0.00 0.21 0.99
#> lag_weights[8] 0.18 0.01 0.31 0.00 0.00 0.00 0.21 0.98
#> lag_weights[9] 0.19 0.01 0.32 0.00 0.00 0.00 0.25 0.99
#> lag_weights[10] 0.19 0.01 0.31 0.00 0.00 0.00 0.27 0.98
#> lag_weights[11] 0.19 0.01 0.32 0.00 0.00 0.00 0.28 0.98
#> lag_weights[12] 0.17 0.01 0.32 0.00 0.00 0.00 0.13 0.99
#> lag_weights[13] 0.10 0.01 0.25 0.00 0.00 0.00 0.01 0.95
#> lp__ 170.51 0.14 3.19 163.24 168.56 170.84 172.86 175.64
#> n_eff Rhat
#> noise_var 1376 1.00
#> tau 274 1.00
#> beta_medias[1] 470 1.00
#> beta_medias[2] 656 1.00
#> beta_medias[3] 681 1.00
#> gamma_ctrl[1] 1491 1.00
#> retain_rate[1] 1952 1.00
#> retain_rate[2] 1623 1.00
#> retain_rate[3] 1556 1.00
#> delay[1] 1775 1.00
#> delay[2] 1644 1.00
#> delay[3] 1682 1.00
#> ec[1] 510 1.00
#> ec[2] 673 1.00
#> ec[3] 627 1.01
#> slope[1] 770 1.00
#> slope[2] 1012 1.00
#> slope[3] 900 1.00
#> mu[1] 1464 1.00
#> mu[2] 1342 1.00
#> mu[3] 1381 1.00
#> mu[4] 1118 1.00
#> mu[5] 1278 1.00
#> mu[6] 1290 1.00
#> mu[7] 1124 1.00
#> mu[8] 1142 1.00
#> mu[9] 1553 1.00
#> mu[10] 1463 1.00
#> mu[11] 1161 1.00
#> mu[12] 831 1.00
#> mu[13] 1100 1.00
#> mu[14] 1536 1.00
#> mu[15] 1577 1.00
#> mu[16] 1585 1.00
#> mu[17] 1419 1.00
#> mu[18] 1350 1.00
#> mu[19] 1605 1.00
#> mu[20] 1385 1.00
#> mu[21] 1321 1.00
#> mu[22] 1368 1.00
#> mu[23] 993 1.01
#> mu[24] 1009 1.00
#> mu[25] 1440 1.00
#> mu[26] 1581 1.00
#> mu[27] 1432 1.00
#> mu[28] 1410 1.00
#> mu[29] 1282 1.00
#> mu[30] 1161 1.00
#> mu[31] 1353 1.00
#> mu[32] 1297 1.00
#> mu[33] 1166 1.00
#> mu[34] 1369 1.00
#> mu[35] 1287 1.00
#> mu[36] 1332 1.00
#> mu[37] 1339 1.00
#> mu[38] 1460 1.00
#> mu[39] 1149 1.00
#> mu[40] 990 1.00
#> mu[41] 1082 1.00
#> mu[42] 1429 1.00
#> mu[43] 1246 1.00
#> mu[44] 1190 1.00
#> mu[45] 1659 1.00
#> mu[46] 1087 1.00
#> mu[47] 1092 1.00
#> mu[48] 1312 1.00
#> mu[49] 1377 1.00
#> mu[50] 1159 1.00
#> mu[51] 1497 1.00
#> mu[52] 1324 1.00
#> mu[53] 1086 1.00
#> mu[54] 932 1.00
#> mu[55] 1087 1.00
#> mu[56] 1350 1.00
#> mu[57] 1330 1.00
#> mu[58] 1393 1.00
#> mu[59] 1603 1.00
#> mu[60] 1605 1.00
#> mu[61] 1504 1.00
#> mu[62] 1153 1.00
#> mu[63] 1388 1.00
#> mu[64] 1420 1.00
#> mu[65] 1272 1.00
#> mu[66] 1254 1.00
#> mu[67] 1287 1.00
#> mu[68] 1482 1.00
#> mu[69] 1487 1.00
#> mu[70] 1311 1.00
#> mu[71] 1278 1.00
#> mu[72] 1257 1.00
#> mu[73] 1278 1.00
#> mu[74] 1367 1.00
#> mu[75] 1270 1.00
#> mu[76] 1075 1.01
#> mu[77] 1065 1.00
#> mu[78] 1261 1.00
#> mu[79] 1490 1.00
#> mu[80] 1662 1.00
#> mu[81] 1271 1.00
#> mu[82] 1020 1.00
#> mu[83] 1091 1.00
#> mu[84] 1450 1.00
#> mu[85] 1167 1.00
#> mu[86] 1545 1.00
#> mu[87] 1584 1.00
#> mu[88] 1784 1.00
#> mu[89] 1616 1.00
#> mu[90] 1338 1.00
#> mu[91] 1051 1.00
#> mu[92] 1015 1.00
#> cum_effect 1 1.00
#> cum_effects_hill[1,1] 544 1.00
#> cum_effects_hill[1,2] 697 1.00
#> cum_effects_hill[1,3] 591 1.01
#> cum_effects_hill[2,1] 558 1.00
#> cum_effects_hill[2,2] 698 1.00
#> cum_effects_hill[2,3] 591 1.01
#> cum_effects_hill[3,1] 550 1.00
#> cum_effects_hill[3,2] 697 1.00
#> cum_effects_hill[3,3] 596 1.01
#> cum_effects_hill[4,1] 510 1.00
#> cum_effects_hill[4,2] 694 1.00
#> cum_effects_hill[4,3] 598 1.01
#> cum_effects_hill[5,1] 451 1.00
#> cum_effects_hill[5,2] 690 1.00
#> cum_effects_hill[5,3] 598 1.01
#> cum_effects_hill[6,1] 524 1.00
#> cum_effects_hill[6,2] 690 1.00
#> cum_effects_hill[6,3] 592 1.01
#> cum_effects_hill[7,1] 567 1.00
#> cum_effects_hill[7,2] 688 1.00
#> cum_effects_hill[7,3] 589 1.01
#> cum_effects_hill[8,1] 572 1.00
#> cum_effects_hill[8,2] 693 1.00
#> cum_effects_hill[8,3] 592 1.01
#> cum_effects_hill[9,1] 562 1.00
#> cum_effects_hill[9,2] 687 1.00
#> cum_effects_hill[9,3] 598 1.01
#> cum_effects_hill[10,1] 544 1.00
#> cum_effects_hill[10,2] 657 1.00
#> cum_effects_hill[10,3] 599 1.01
#> cum_effects_hill[11,1] 485 1.00
#> cum_effects_hill[11,2] 651 1.00
#> cum_effects_hill[11,3] 600 1.01
#> cum_effects_hill[12,1] 374 1.00
#> cum_effects_hill[12,2] 670 1.00
#> cum_effects_hill[12,3] 599 1.01
#> cum_effects_hill[13,1] 471 1.00
#> cum_effects_hill[13,2] 689 1.00
#> cum_effects_hill[13,3] 599 1.01
#> cum_effects_hill[14,1] 553 1.00
#> cum_effects_hill[14,2] 694 1.00
#> cum_effects_hill[14,3] 596 1.01
#> cum_effects_hill[15,1] 565 1.00
#> cum_effects_hill[15,2] 694 1.00
#> cum_effects_hill[15,3] 597 1.01
#> cum_effects_hill[16,1] 542 1.00
#> cum_effects_hill[16,2] 689 1.00
#> cum_effects_hill[16,3] 597 1.01
#> cum_effects_hill[17,1] 529 1.00
#> cum_effects_hill[17,2] 684 1.00
#> cum_effects_hill[17,3] 598 1.01
#> cum_effects_hill[18,1] 534 1.00
#> cum_effects_hill[18,2] 686 1.00
#> cum_effects_hill[18,3] 597 1.01
#> cum_effects_hill[19,1] 554 1.00
#> cum_effects_hill[19,2] 685 1.00
#> cum_effects_hill[19,3] 599 1.01
#> cum_effects_hill[20,1] 573 1.00
#> cum_effects_hill[20,2] 694 1.00
#> cum_effects_hill[20,3] 596 1.01
#> cum_effects_hill[21,1] 572 1.00
#> cum_effects_hill[21,2] 695 1.00
#> cum_effects_hill[21,3] 595 1.01
#> cum_effects_hill[22,1] 538 1.00
#> cum_effects_hill[22,2] 691 1.00
#> cum_effects_hill[22,3] 600 1.01
#> cum_effects_hill[23,1] 449 1.00
#> cum_effects_hill[23,2] 672 1.00
#> cum_effects_hill[23,3] 595 1.00
#> cum_effects_hill[24,1] 420 1.00
#> cum_effects_hill[24,2] 655 1.00
#> cum_effects_hill[24,3] 599 1.01
#> cum_effects_hill[25,1] 473 1.00
#> cum_effects_hill[25,2] 671 1.00
#> cum_effects_hill[25,3] 597 1.01
#> cum_effects_hill[26,1] 493 1.00
#> cum_effects_hill[26,2] 688 1.00
#> cum_effects_hill[26,3] 593 1.01
#> cum_effects_hill[27,1] 514 1.00
#> cum_effects_hill[27,2] 692 1.00
#> cum_effects_hill[27,3] 592 1.01
#> cum_effects_hill[28,1] 535 1.00
#> cum_effects_hill[28,2] 686 1.00
#> cum_effects_hill[28,3] 593 1.01
#> cum_effects_hill[29,1] 548 1.00
#> cum_effects_hill[29,2] 670 1.00
#> cum_effects_hill[29,3] 595 1.01
#> cum_effects_hill[30,1] 531 1.00
#> cum_effects_hill[30,2] 626 1.00
#> cum_effects_hill[30,3] 597 1.01
#> cum_effects_hill[31,1] 506 1.00
#> cum_effects_hill[31,2] 652 1.00
#> cum_effects_hill[31,3] 592 1.01
#> cum_effects_hill[32,1] 523 1.00
#> cum_effects_hill[32,2] 664 1.00
#> cum_effects_hill[32,3] 591 1.01
#> cum_effects_hill[33,1] 539 1.00
#> cum_effects_hill[33,2] 684 1.00
#> cum_effects_hill[33,3] 588 1.01
#> cum_effects_hill[34,1] 516 1.00
#> cum_effects_hill[34,2] 685 1.00
#> cum_effects_hill[34,3] 590 1.01
#> cum_effects_hill[35,1] 478 1.00
#> cum_effects_hill[35,2] 679 1.00
#> cum_effects_hill[35,3] 595 1.01
#> cum_effects_hill[36,1] 442 1.00
#> cum_effects_hill[36,2] 669 1.00
#> cum_effects_hill[36,3] 599 1.01
#> cum_effects_hill[37,1] 469 1.00
#> cum_effects_hill[37,2] 656 1.00
#> cum_effects_hill[37,3] 600 1.01
#> cum_effects_hill[38,1] 542 1.00
#> cum_effects_hill[38,2] 671 1.00
#> cum_effects_hill[38,3] 600 1.01
#> cum_effects_hill[39,1] 566 1.00
#> cum_effects_hill[39,2] 661 1.00
#> cum_effects_hill[39,3] 595 1.01
#> cum_effects_hill[40,1] 565 1.00
#> cum_effects_hill[40,2] 683 1.00
#> cum_effects_hill[40,3] 591 1.01
#> cum_effects_hill[41,1] 556 1.00
#> cum_effects_hill[41,2] 687 1.00
#> cum_effects_hill[41,3] 591 1.01
#> cum_effects_hill[42,1] 539 1.00
#> cum_effects_hill[42,2] 687 1.00
#> cum_effects_hill[42,3] 594 1.01
#> cum_effects_hill[43,1] 487 1.00
#> cum_effects_hill[43,2] 676 1.00
#> cum_effects_hill[43,3] 598 1.01
#> cum_effects_hill[44,1] 415 1.00
#> cum_effects_hill[44,2] 635 1.00
#> cum_effects_hill[44,3] 597 1.01
#> cum_effects_hill[45,1] 489 1.00
#> cum_effects_hill[45,2] 668 1.00
#> cum_effects_hill[45,3] 594 1.01
#> cum_effects_hill[46,1] 557 1.00
#> cum_effects_hill[46,2] 689 1.00
#> cum_effects_hill[46,3] 591 1.01
#> cum_effects_hill[47,1] 569 1.00
#> cum_effects_hill[47,2] 692 1.00
#> cum_effects_hill[47,3] 591 1.01
#> cum_effects_hill[48,1] 551 1.00
#> cum_effects_hill[48,2] 672 1.00
#> cum_effects_hill[48,3] 596 1.01
#> cum_effects_hill[49,1] 520 1.00
#> cum_effects_hill[49,2] 616 1.00
#> cum_effects_hill[49,3] 599 1.01
#> cum_effects_hill[50,1] 478 1.00
#> cum_effects_hill[50,2] 607 1.00
#> cum_effects_hill[50,3] 597 1.01
#> cum_effects_hill[51,1] 479 1.00
#> cum_effects_hill[51,2] 665 1.00
#> cum_effects_hill[51,3] 595 1.01
#> cum_effects_hill[52,1] 533 1.00
#> cum_effects_hill[52,2] 688 1.00
#> cum_effects_hill[52,3] 592 1.01
#> cum_effects_hill[53,1] 558 1.00
#> cum_effects_hill[53,2] 692 1.00
#> cum_effects_hill[53,3] 590 1.01
#> cum_effects_hill[54,1] 568 1.00
#> cum_effects_hill[54,2] 688 1.00
#> cum_effects_hill[54,3] 591 1.01
#> cum_effects_hill[55,1] 556 1.00
#> cum_effects_hill[55,2] 689 1.00
#> cum_effects_hill[55,3] 591 1.01
#> cum_effects_hill[56,1] 470 1.00
#> cum_effects_hill[56,2] 693 1.00
#> cum_effects_hill[56,3] 592 1.01
#> cum_effects_hill[57,1] 437 1.00
#> cum_effects_hill[57,2] 691 1.00
#> cum_effects_hill[57,3] 596 1.01
#> cum_effects_hill[58,1] 537 1.00
#> cum_effects_hill[58,2] 696 1.00
#> cum_effects_hill[58,3] 599 1.01
#> cum_effects_hill[59,1] 569 1.00
#> cum_effects_hill[59,2] 697 1.00
#> cum_effects_hill[59,3] 599 1.01
#> cum_effects_hill[60,1] 569 1.00
#> cum_effects_hill[60,2] 696 1.00
#> cum_effects_hill[60,3] 599 1.01
#> cum_effects_hill[61,1] 548 1.00
#> cum_effects_hill[61,2] 690 1.00
#> cum_effects_hill[61,3] 599 1.01
#> cum_effects_hill[62,1] 494 1.00
#> cum_effects_hill[62,2] 680 1.00
#> cum_effects_hill[62,3] 600 1.01
#> cum_effects_hill[63,1] 435 1.00
#> cum_effects_hill[63,2] 657 1.00
#> cum_effects_hill[63,3] 598 1.01
#> cum_effects_hill[64,1] 456 1.00
#> cum_effects_hill[64,2] 685 1.00
#> cum_effects_hill[64,3] 594 1.01
#> cum_effects_hill[65,1] 515 1.00
#> cum_effects_hill[65,2] 696 1.00
#> cum_effects_hill[65,3] 590 1.01
#> cum_effects_hill[66,1] 536 1.00
#> cum_effects_hill[66,2] 697 1.00
#> cum_effects_hill[66,3] 586 1.01
#> cum_effects_hill[67,1] 548 1.00
#> cum_effects_hill[67,2] 696 1.00
#> cum_effects_hill[67,3] 593 1.01
#> cum_effects_hill[68,1] 555 1.00
#> cum_effects_hill[68,2] 687 1.00
#> cum_effects_hill[68,3] 598 1.01
#> cum_effects_hill[69,1] 544 1.00
#> cum_effects_hill[69,2] 678 1.00
#> cum_effects_hill[69,3] 600 1.01
#> cum_effects_hill[70,1] 530 1.00
#> cum_effects_hill[70,2] 682 1.00
#> cum_effects_hill[70,3] 595 1.01
#> cum_effects_hill[71,1] 521 1.00
#> cum_effects_hill[71,2] 694 1.00
#> cum_effects_hill[71,3] 592 1.01
#> cum_effects_hill[72,1] 541 1.00
#> cum_effects_hill[72,2] 698 1.00
#> cum_effects_hill[72,3] 589 1.01
#> cum_effects_hill[73,1] 518 1.00
#> cum_effects_hill[73,2] 698 1.00
#> cum_effects_hill[73,3] 591 1.01
#> cum_effects_hill[74,1] 428 1.00
#> cum_effects_hill[74,2] 697 1.00
#> cum_effects_hill[74,3] 596 1.01
#> cum_effects_hill[75,1] 397 1.00
#> cum_effects_hill[75,2] 693 1.00
#> cum_effects_hill[75,3] 600 1.01
#> cum_effects_hill[76,1] 431 1.00
#> cum_effects_hill[76,2] 689 1.00
#> cum_effects_hill[76,3] 595 1.00
#> cum_effects_hill[77,1] 502 1.00
#> cum_effects_hill[77,2] 688 1.00
#> cum_effects_hill[77,3] 600 1.01
#> cum_effects_hill[78,1] 546 1.00
#> cum_effects_hill[78,2] 693 1.00
#> cum_effects_hill[78,3] 594 1.01
#> cum_effects_hill[79,1] 545 1.00
#> cum_effects_hill[79,2] 689 1.00
#> cum_effects_hill[79,3] 590 1.01
#> cum_effects_hill[80,1] 514 1.00
#> cum_effects_hill[80,2] 686 1.00
#> cum_effects_hill[80,3] 589 1.01
#> cum_effects_hill[81,1] 486 1.00
#> cum_effects_hill[81,2] 689 1.00
#> cum_effects_hill[81,3] 594 1.01
#> cum_effects_hill[82,1] 481 1.00
#> cum_effects_hill[82,2] 681 1.00
#> cum_effects_hill[82,3] 597 1.01
#> cum_effects_hill[83,1] 500 1.00
#> cum_effects_hill[83,2] 680 1.00
#> cum_effects_hill[83,3] 598 1.01
#> cum_effects_hill[84,1] 548 1.00
#> cum_effects_hill[84,2] 674 1.00
#> cum_effects_hill[84,3] 594 1.01
#> cum_effects_hill[85,1] 575 1.00
#> cum_effects_hill[85,2] 687 1.00
#> cum_effects_hill[85,3] 593 1.01
#> cum_effects_hill[86,1] 564 1.00
#> cum_effects_hill[86,2] 689 1.00
#> cum_effects_hill[86,3] 592 1.01
#> cum_effects_hill[87,1] 523 1.00
#> cum_effects_hill[87,2] 687 1.00
#> cum_effects_hill[87,3] 591 1.01
#> cum_effects_hill[88,1] 460 1.00
#> cum_effects_hill[88,2] 677 1.00
#> cum_effects_hill[88,3] 594 1.01
#> cum_effects_hill[89,1] 460 1.00
#> cum_effects_hill[89,2] 666 1.00
#> cum_effects_hill[89,3] 593 1.01
#> cum_effects_hill[90,1] 518 1.00
#> cum_effects_hill[90,2] 679 1.00
#> cum_effects_hill[90,3] 593 1.01
#> cum_effects_hill[91,1] 568 1.00
#> cum_effects_hill[91,2] 680 1.00
#> cum_effects_hill[91,3] 592 1.01
#> cum_effects_hill[92,1] 582 1.00
#> cum_effects_hill[92,2] 689 1.00
#> cum_effects_hill[92,3] 591 1.01
#> lag_weights[1] 917 1.00
#> lag_weights[2] 1135 1.00
#> lag_weights[3] 1618 1.00
#> lag_weights[4] 1829 1.00
#> lag_weights[5] 1335 1.00
#> lag_weights[6] 921 1.00
#> lag_weights[7] 898 1.00
#> lag_weights[8] 1024 1.00
#> lag_weights[9] 1469 1.00
#> lag_weights[10] 1478 1.00
#> lag_weights[11] 1288 1.00
#> lag_weights[12] 980 1.00
#> lag_weights[13] 688 1.00
#> lp__ 555 1.00
#>
#> Samples were drawn using NUTS(diag_e) at Sat Feb 15 23:16:20 2020.
#> For each parameter, n_eff is a crude measure of effective sample size,
#> and Rhat is the potential scale reduction factor on split chains (at
#> convergence, Rhat=1).rstan::get_posterior_mean(m.stan)
#> mean-chain:1
#> noise_var 5.331325e-03
#> tau 4.022695e+00
#> beta_medias[1] 1.061030e+00
#> beta_medias[2] 9.516126e-01
#> beta_medias[3] 9.766150e-01
#> gamma_ctrl[1] 3.928343e-01
#> retain_rate[1] 5.050962e-01
#> retain_rate[2] 4.920488e-01
#> retain_rate[3] 4.932218e-01
#> delay[1] 6.102818e+00
#> delay[2] 6.165042e+00
#> delay[3] 6.033674e+00
#> ec[1] 5.929361e-01
#> ec[2] 5.353357e-01
#> ec[3] 4.036159e-01
#> slope[1] 2.677443e+00
#> slope[2] 2.631627e+00
#> slope[3] 2.222469e+00
#> mu[1] 5.238545e+00
#> mu[2] 5.285312e+00
#> mu[3] 5.267987e+00
#> mu[4] 5.214458e+00
#> mu[5] 5.155157e+00
#> mu[6] 5.284885e+00
#> mu[7] 5.337371e+00
#> mu[8] 5.322957e+00
#> mu[9] 5.241752e+00
#> mu[10] 5.150476e+00
#> mu[11] 5.076132e+00
#> mu[12] 4.928041e+00
#> mu[13] 5.107885e+00
#> mu[14] 5.223832e+00
#> mu[15] 5.234774e+00
#> mu[16] 5.170774e+00
#> mu[17] 5.165796e+00
#> mu[18] 5.197543e+00
#> mu[19] 5.215900e+00
#> mu[20] 5.288624e+00
#> mu[21] 5.302331e+00
#> mu[22] 5.213929e+00
#> mu[23] 5.010438e+00
#> mu[24] 4.967476e+00
#> mu[25] 5.076095e+00
#> mu[26] 5.147449e+00
#> mu[27] 5.198569e+00
#> mu[28] 5.211650e+00
#> mu[29] 5.194351e+00
#> mu[30] 5.092464e+00
#> mu[31] 5.115228e+00
#> mu[32] 5.168391e+00
#> mu[33] 5.242663e+00
#> mu[34] 5.210742e+00
#> mu[35] 5.151674e+00
#> mu[36] 5.075467e+00
#> mu[37] 5.088358e+00
#> mu[38] 5.165898e+00
#> mu[39] 5.203421e+00
#> mu[40] 5.263580e+00
#> mu[41] 5.269046e+00
#> mu[42] 5.211705e+00
#> mu[43] 5.093970e+00
#> mu[44] 4.929134e+00
#> mu[45] 5.085432e+00
#> mu[46] 5.261347e+00
#> mu[47] 5.315077e+00
#> mu[48] 5.243238e+00
#> mu[49] 5.094495e+00
#> mu[50] 5.031395e+00
#> mu[51] 5.088192e+00
#> mu[52] 5.226122e+00
#> mu[53] 5.293281e+00
#> mu[54] 5.284696e+00
#> mu[55] 5.279985e+00
#> mu[56] 5.191200e+00
#> mu[57] 5.151516e+00
#> mu[58] 5.259518e+00
#> mu[59] 5.289290e+00
#> mu[60] 5.252831e+00
#> mu[61] 5.187039e+00
#> mu[62] 5.086212e+00
#> mu[63] 5.006292e+00
#> mu[64] 5.135806e+00
#> mu[65] 5.264354e+00
#> mu[66] 5.296168e+00
#> mu[67] 5.278050e+00
#> mu[68] 5.243855e+00
#> mu[69] 5.182439e+00
#> mu[70] 5.194619e+00
#> mu[71] 5.234416e+00
#> mu[72] 5.312213e+00
#> mu[73] 5.280617e+00
#> mu[74] 5.115383e+00
#> mu[75] 5.030901e+00
#> mu[76] 5.031898e+00
#> mu[77] 5.170058e+00
#> mu[78] 5.268340e+00
#> mu[79] 5.233995e+00
#> mu[80] 5.186275e+00
#> mu[81] 5.176022e+00
#> mu[82] 5.131971e+00
#> mu[83] 5.134740e+00
#> mu[84] 5.183810e+00
#> mu[85] 5.249235e+00
#> mu[86] 5.229523e+00
#> mu[87] 5.194602e+00
#> mu[88] 5.080393e+00
#> mu[89] 5.075647e+00
#> mu[90] 5.189475e+00
#> mu[91] 5.265386e+00
#> mu[92] 5.320813e+00
#> cum_effect 2.274121e-01
#> cum_effects_hill[1,1] 5.177459e-01
#> cum_effects_hill[1,2] 4.584694e-01
#> cum_effects_hill[1,3] 3.555236e-01
#> cum_effects_hill[2,1] 5.378850e-01
#> cum_effects_hill[2,2] 4.709962e-01
#> cum_effects_hill[2,3] 3.537306e-01
#> cum_effects_hill[3,1] 5.266422e-01
#> cum_effects_hill[3,2] 4.557501e-01
#> cum_effects_hill[3,3] 3.296193e-01
#> cum_effects_hill[4,1] 4.698817e-01
#> cum_effects_hill[4,2] 4.378280e-01
#> cum_effects_hill[4,3] 3.096318e-01
#> cum_effects_hill[5,1] 3.895034e-01
#> cum_effects_hill[5,2] 4.180688e-01
#> cum_effects_hill[5,3] 3.140342e-01
#> cum_effects_hill[6,1] 4.899499e-01
#> cum_effects_hill[6,2] 4.172740e-01
#> cum_effects_hill[6,3] 3.490374e-01
#> cum_effects_hill[7,1] 5.501015e-01
#> cum_effects_hill[7,2] 4.129622e-01
#> cum_effects_hill[7,3] 3.638127e-01
#> cum_effects_hill[8,1] 5.569502e-01
#> cum_effects_hill[8,2] 4.306705e-01
#> cum_effects_hill[8,3] 3.485061e-01
#> cum_effects_hill[9,1] 5.436667e-01
#> cum_effects_hill[9,2] 4.090017e-01
#> cum_effects_hill[9,3] 3.154680e-01
#> cum_effects_hill[10,1] 5.173451e-01
#> cum_effects_hill[10,2] 3.354841e-01
#> cum_effects_hill[10,3] 2.949147e-01
#> cum_effects_hill[11,1] 4.353048e-01
#> cum_effects_hill[11,2] 3.251075e-01
#> cum_effects_hill[11,3] 2.911221e-01
#> cum_effects_hill[12,1] 2.781495e-01
#> cum_effects_hill[12,2] 3.619316e-01
#> cum_effects_hill[12,3] 2.690854e-01
#> cum_effects_hill[13,1] 4.161076e-01
#> cum_effects_hill[13,2] 4.148500e-01
#> cum_effects_hill[13,3] 2.975082e-01
#> cum_effects_hill[14,1] 5.307641e-01
#> cum_effects_hill[14,2] 4.403161e-01
#> cum_effects_hill[14,3] 3.249908e-01
#> cum_effects_hill[15,1] 5.470521e-01
#> cum_effects_hill[15,2] 4.388754e-01
#> cum_effects_hill[15,3] 3.230341e-01
#> cum_effects_hill[16,1] 5.150564e-01
#> cum_effects_hill[16,2] 4.156203e-01
#> cum_effects_hill[16,3] 3.227322e-01
#> cum_effects_hill[17,1] 4.969516e-01
#> cum_effects_hill[17,2] 3.993994e-01
#> cum_effects_hill[17,3] 3.129899e-01
#> cum_effects_hill[18,1] 5.038489e-01
#> cum_effects_hill[18,2] 4.041953e-01
#> cum_effects_hill[18,3] 3.176104e-01
#> cum_effects_hill[19,1] 5.317967e-01
#> cum_effects_hill[19,2] 4.023790e-01
#> cum_effects_hill[19,3] 2.994409e-01
#> cum_effects_hill[20,1] 5.578491e-01
#> cum_effects_hill[20,2] 4.382098e-01
#> cum_effects_hill[20,3] 3.298174e-01
#> cum_effects_hill[21,1] 5.564550e-01
#> cum_effects_hill[21,2] 4.455809e-01
#> cum_effects_hill[21,3] 3.359775e-01
#> cum_effects_hill[22,1] 5.085675e-01
#> cum_effects_hill[22,2] 4.232014e-01
#> cum_effects_hill[22,3] 2.887202e-01
#> cum_effects_hill[23,1] 3.865633e-01
#> cum_effects_hill[23,2] 3.662665e-01
#> cum_effects_hill[23,3] 2.154642e-01
#> cum_effects_hill[24,1] 3.477011e-01
#> cum_effects_hill[24,2] 3.324469e-01
#> cum_effects_hill[24,3] 2.587131e-01
#> cum_effects_hill[25,1] 4.187524e-01
#> cum_effects_hill[25,2] 3.638346e-01
#> cum_effects_hill[25,3] 3.201074e-01
#> cum_effects_hill[26,1] 4.464447e-01
#> cum_effects_hill[26,2] 4.119064e-01
#> cum_effects_hill[26,3] 3.474715e-01
#> cum_effects_hill[27,1] 4.752340e-01
#> cum_effects_hill[27,2] 4.264632e-01
#> cum_effects_hill[27,3] 3.505701e-01
#> cum_effects_hill[28,1] 5.045614e-01
#> cum_effects_hill[28,2] 4.037669e-01
#> cum_effects_hill[28,3] 3.466008e-01
#> cum_effects_hill[29,1] 5.227207e-01
#> cum_effects_hill[29,2] 3.624296e-01
#> cum_effects_hill[29,3] 3.337851e-01
#> cum_effects_hill[30,1] 4.985603e-01
#> cum_effects_hill[30,2] 2.839807e-01
#> cum_effects_hill[30,3] 3.247995e-01
#> cum_effects_hill[31,1] 4.641591e-01
#> cum_effects_hill[31,2] 3.259869e-01
#> cum_effects_hill[31,3] 3.507667e-01
#> cum_effects_hill[32,1] 4.884865e-01
#> cum_effects_hill[32,2] 3.497756e-01
#> cum_effects_hill[32,3] 3.575387e-01
#> cum_effects_hill[33,1] 5.112219e-01
#> cum_effects_hill[33,2] 3.987652e-01
#> cum_effects_hill[33,3] 3.670250e-01
#> cum_effects_hill[34,1] 4.781039e-01
#> cum_effects_hill[34,2] 4.024923e-01
#> cum_effects_hill[34,3] 3.582962e-01
#> cum_effects_hill[35,1] 4.258158e-01
#> cum_effects_hill[35,2] 3.847624e-01
#> cum_effects_hill[35,3] 3.343121e-01
#> cum_effects_hill[36,1] 3.771745e-01
#> cum_effects_hill[36,2] 3.600635e-01
#> cum_effects_hill[36,3] 3.030257e-01
#> cum_effects_hill[37,1] 4.138420e-01
#> cum_effects_hill[37,2] 3.347791e-01
#> cum_effects_hill[37,3] 2.926297e-01
#> cum_effects_hill[38,1] 5.145510e-01
#> cum_effects_hill[38,2] 3.642221e-01
#> cum_effects_hill[38,3] 2.785457e-01
#> cum_effects_hill[39,1] 5.493880e-01
#> cum_effects_hill[39,2] 3.442303e-01
#> cum_effects_hill[39,3] 3.327978e-01
#> cum_effects_hill[40,1] 5.467892e-01
#> cum_effects_hill[40,2] 3.943931e-01
#> cum_effects_hill[40,3] 3.545520e-01
#> cum_effects_hill[41,1] 5.343942e-01
#> cum_effects_hill[41,2] 4.063385e-01
#> cum_effects_hill[41,3] 3.560805e-01
#> cum_effects_hill[42,1] 5.099368e-01
#> cum_effects_hill[42,2] 4.072849e-01
#> cum_effects_hill[42,3] 3.412557e-01
#> cum_effects_hill[43,1] 4.386029e-01
#> cum_effects_hill[43,2] 3.749885e-01
#> cum_effects_hill[43,3] 3.120519e-01
#> cum_effects_hill[44,1] 3.404171e-01
#> cum_effects_hill[44,2] 2.975977e-01
#> cum_effects_hill[44,3] 3.196449e-01
#> cum_effects_hill[45,1] 4.410700e-01
#> cum_effects_hill[45,2] 3.573434e-01
#> cum_effects_hill[45,3] 3.401487e-01
#> cum_effects_hill[46,1] 5.357314e-01
#> cum_effects_hill[46,2] 4.149270e-01
#> cum_effects_hill[46,3] 3.556343e-01
#> cum_effects_hill[47,1] 5.530193e-01
#> cum_effects_hill[47,2] 4.262096e-01
#> cum_effects_hill[47,3] 3.536702e-01
#> cum_effects_hill[48,1] 5.280604e-01
#> cum_effects_hill[48,2] 3.658539e-01
#> cum_effects_hill[48,3] 3.252756e-01
#> cum_effects_hill[49,1] 4.836446e-01
#> cum_effects_hill[49,2] 2.682770e-01
#> cum_effects_hill[49,3] 2.938741e-01
#> cum_effects_hill[50,1] 4.254780e-01
#> cum_effects_hill[50,2] 2.538760e-01
#> cum_effects_hill[50,3] 3.245571e-01
#> cum_effects_hill[51,1] 4.275607e-01
#> cum_effects_hill[51,2] 3.507211e-01
#> cum_effects_hill[51,3] 3.339378e-01
#> cum_effects_hill[52,1] 5.015422e-01
#> cum_effects_hill[52,2] 4.100663e-01
#> cum_effects_hill[52,3] 3.503474e-01
#> cum_effects_hill[53,1] 5.378121e-01
#> cum_effects_hill[53,2] 4.287123e-01
#> cum_effects_hill[53,3] 3.602849e-01
#> cum_effects_hill[54,1] 5.520135e-01
#> cum_effects_hill[54,2] 4.102978e-01
#> cum_effects_hill[54,3] 3.545811e-01
#> cum_effects_hill[55,1] 5.348224e-01
#> cum_effects_hill[55,2] 4.161330e-01
#> cum_effects_hill[55,3] 3.562721e-01
#> cum_effects_hill[56,1] 4.149311e-01
#> cum_effects_hill[56,2] 4.320249e-01
#> cum_effects_hill[56,3] 3.511945e-01
#> cum_effects_hill[57,1] 3.713373e-01
#> cum_effects_hill[57,2] 4.243088e-01
#> cum_effects_hill[57,3] 3.300979e-01
#> cum_effects_hill[58,1] 5.076783e-01
#> cum_effects_hill[58,2] 4.475300e-01
#> cum_effects_hill[58,3] 3.006004e-01
#> cum_effects_hill[59,1] 5.528391e-01
#> cum_effects_hill[59,2] 4.625261e-01
#> cum_effects_hill[59,3] 3.034067e-01
#> cum_effects_hill[60,1] 5.526545e-01
#> cum_effects_hill[60,2] 4.484084e-01
#> cum_effects_hill[60,3] 3.027052e-01
#> cum_effects_hill[61,1] 5.227739e-01
#> cum_effects_hill[61,2] 4.184623e-01
#> cum_effects_hill[61,3] 2.943918e-01
#> cum_effects_hill[62,1] 4.479670e-01
#> cum_effects_hill[62,2] 3.869540e-01
#> cum_effects_hill[62,3] 2.776620e-01
#> cum_effects_hill[63,1] 3.687498e-01
#> cum_effects_hill[63,2] 3.363803e-01
#> cum_effects_hill[63,3] 3.090642e-01
#> cum_effects_hill[64,1] 3.961559e-01
#> cum_effects_hill[64,2] 4.009364e-01
#> cum_effects_hill[64,3] 3.409719e-01
#> cum_effects_hill[65,1] 4.764295e-01
#> cum_effects_hill[65,2] 4.506118e-01
#> cum_effects_hill[65,3] 3.604872e-01
#> cum_effects_hill[66,1] 5.068033e-01
#> cum_effects_hill[66,2] 4.620743e-01
#> cum_effects_hill[66,3] 3.688586e-01
#> cum_effects_hill[67,1] 5.237806e-01
#> cum_effects_hill[67,2] 4.479190e-01
#> cum_effects_hill[67,3] 3.453100e-01
#> cum_effects_hill[68,1] 5.336182e-01
#> cum_effects_hill[68,2] 4.075199e-01
#> cum_effects_hill[68,3] 3.147490e-01
#> cum_effects_hill[69,1] 5.177384e-01
#> cum_effects_hill[69,2] 3.816043e-01
#> cum_effects_hill[69,3] 2.872861e-01
#> cum_effects_hill[70,1] 4.972445e-01
#> cum_effects_hill[70,2] 3.933854e-01
#> cum_effects_hill[70,3] 3.319862e-01
#> cum_effects_hill[71,1] 4.858347e-01
#> cum_effects_hill[71,2] 4.399085e-01
#> cum_effects_hill[71,3] 3.494394e-01
#> cum_effects_hill[72,1] 5.139956e-01
#> cum_effects_hill[72,2] 4.689021e-01
#> cum_effects_hill[72,3] 3.653859e-01
#> cum_effects_hill[73,1] 4.813359e-01
#> cum_effects_hill[73,2] 4.733841e-01
#> cum_effects_hill[73,3] 3.535383e-01
#> cum_effects_hill[74,1] 3.588387e-01
#> cum_effects_hill[74,2] 4.594600e-01
#> cum_effects_hill[74,3] 3.281733e-01
#> cum_effects_hill[75,1] 3.144272e-01
#> cum_effects_hill[75,2] 4.322541e-01
#> cum_effects_hill[75,3] 2.890026e-01
#> cum_effects_hill[76,1] 3.629243e-01
#> cum_effects_hill[76,2] 4.143426e-01
#> cum_effects_hill[76,3] 2.176162e-01
#> cum_effects_hill[77,1] 4.584654e-01
#> cum_effects_hill[77,2] 4.131309e-01
#> cum_effects_hill[77,3] 2.894964e-01
#> cum_effects_hill[78,1] 5.203030e-01
#> cum_effects_hill[78,2] 4.346735e-01
#> cum_effects_hill[78,3] 3.390936e-01
#> cum_effects_hill[79,1] 5.184211e-01
#> cum_effects_hill[79,2] 4.162129e-01
#> cum_effects_hill[79,3] 3.624077e-01
#> cum_effects_hill[80,1] 4.749758e-01
#> cum_effects_hill[80,2] 4.055395e-01
#> cum_effects_hill[80,3] 3.635701e-01
#> cum_effects_hill[81,1] 4.376459e-01
#> cum_effects_hill[81,2] 4.162180e-01
#> cum_effects_hill[81,3] 3.412771e-01
#> cum_effects_hill[82,1] 4.308284e-01
#> cum_effects_hill[82,2] 3.889113e-01
#> cum_effects_hill[82,3] 3.209998e-01
#> cum_effects_hill[83,1] 4.556869e-01
#> cum_effects_hill[83,2] 3.878532e-01
#> cum_effects_hill[83,3] 3.137771e-01
#> cum_effects_hill[84,1] 5.228835e-01
#> cum_effects_hill[84,2] 3.714605e-01
#> cum_effects_hill[84,3] 3.376435e-01
#> cum_effects_hill[85,1] 5.606345e-01
#> cum_effects_hill[85,2] 4.084856e-01
#> cum_effects_hill[85,3] 3.472042e-01
#> cum_effects_hill[86,1] 5.459495e-01
#> cum_effects_hill[86,2] 4.149246e-01
#> cum_effects_hill[86,3] 3.499224e-01
#> cum_effects_hill[87,1] 4.875487e-01
#> cum_effects_hill[87,2] 4.094277e-01
#> cum_effects_hill[87,3] 3.560995e-01
#> cum_effects_hill[88,1] 4.012663e-01
#> cum_effects_hill[88,2] 3.789817e-01
#> cum_effects_hill[88,3] 3.396898e-01
#> cum_effects_hill[89,1] 4.016994e-01
#> cum_effects_hill[89,2] 3.537649e-01
#> cum_effects_hill[89,3] 3.430115e-01
#> cum_effects_hill[90,1] 4.809929e-01
#> cum_effects_hill[90,2] 3.827208e-01
#> cum_effects_hill[90,3] 3.452765e-01
#> cum_effects_hill[91,1] 5.516625e-01
#> cum_effects_hill[91,2] 3.861167e-01
#> cum_effects_hill[91,3] 3.492242e-01
#> cum_effects_hill[92,1] 5.711076e-01
#> cum_effects_hill[92,2] 4.144362e-01
#> cum_effects_hill[92,3] 3.555888e-01
#> lag_weights[1] 1.000924e-01
#> lag_weights[2] 1.702420e-01
#> lag_weights[3] 1.878956e-01
#> lag_weights[4] 1.895120e-01
#> lag_weights[5] 1.784495e-01
#> lag_weights[6] 1.712769e-01
#> lag_weights[7] 1.710582e-01
#> lag_weights[8] 1.811478e-01
#> lag_weights[9] 1.907943e-01
#> lag_weights[10] 1.901259e-01
#> lag_weights[11] 1.892111e-01
#> lag_weights[12] 1.692939e-01
#> lag_weights[13] 9.967880e-02
#> lp__ 1.705108e+02
list_of_draws <- extract(m.stan)
predicted_sales <- summary(m.stan, pars = "mu", probs = NULL)$summary %>%
as_tibble() %>%
select(mean) %>%
rename(pred_sales = mean)
pred_and_sales <- predicted_sales %>%
add_column(sales = clean_data$sales) %>%
mutate(index = row_number())
pred_and_sales %>%
ggplot(aes(x = index)) +
geom_line(aes(y = sales), color = "black") +
geom_line(aes(y = pred_sales), color = "red")
#look at functions learned from model
x <- seq(0,1, length.out = 100)
tv_pred <- BHill(B = 1.20, K = 0.50, S = 2.23,x)
rd_pred <- BHill(B = 0.95, K = 0.50, S = 2.45,x)
online_pred <- BHill(B = 0.90, K = 0.50, S = 1.59,x)
m_tv <- BHill(K = 0.2, S = 1, B = 0.8,x)
m_rd <- BHill(K = 0.2, S = 1, B = 0.6,x)
m_online <- BHill(K = 0.2, S = 1, B = 0.3,x)
as_tibble(list(tv_actual = m_tv, tv_pred = tv_pred)) %>%
mutate(index = row_number()) %>%
ggplot(aes(x = index)) +
geom_line(aes(y = tv_actual, color = "actual")) +
geom_line(aes(y = tv_pred, color = "pred"))
as_tibble(list(rd_actual = m_rd, rd_pred = rd_pred)) %>%
mutate(index = row_number()) %>%
ggplot(aes(x = index)) +
geom_line(aes(y = rd_actual, color = "actual")) +
geom_line(aes(y = rd_pred, color = "pred"))
as_tibble(list(online_actual = m_online, online_pred = online_pred)) %>%
mutate(index = row_number()) %>%
ggplot(aes(x = index)) +
geom_line(aes(y = online_actual, color = "actual")) +
geom_line(aes(y = online_pred, color = "pred"))
rgamma(n = 100, shape = 2, scale = .25) %>% hist()