R Graphical Manual

Browse All

Last data update: 2014.03.03

R: Generating Realizations of a Hidden Markov Model

HMM_simulation

R Documentation

Generating Realizations of a Hidden Markov Model

Description

This function generates a sequence of hidden states of a Markov chain and a corresponding parallel sequence of observations.

Usage

HMM_simulation(size, m, delta = rep(1 / m, times = m), 
               gamma = 0.8 * diag(m) + rep(0.2 / m, times = m), 
               distribution_class, distribution_theta, 
               obs_range = c(NA, NA), obs_round = FALSE, 
               obs_non_neg = FALSE, plotting = 0)

Arguments

`size`	length of the time-series of hidden states and observations (also `T`).
`m`	a (finite) number of states in the hidden Markov chain.
`delta`	a vector object containing starting values for the marginal probability distribution of the `m` states of the Markov chain at the time point `t=1`. Default is `delta=rep(1/m,times=m)`.
`gamma`	a matrix (`nrow=ncol=m`) containing starting values for the transition matrix of the hidden Markov chain. Default is `gamma=0.8 * diag(m)` `+ rep(0.2 / m, times = m)`.
`distribution_class`	a single character string object with the abbreviated name of the `m` observation distributions of the Markov dependent observation process. The following distributions are supported by this algorithm: Poisson (`pois`); generalized Poisson (`genpois`); normal (`norm`, discrete log-Likelihood not applicable by this algorithm); geometric (`geom`).
`distribution_theta`	a list object containing starting values for the parameters of the `m` observation distributions that are dependent on the hidden Markov state.
`obs_range`	a vector object specifying the range for the observations to be generated. For instance, the vector `c(0,1500)` allows only observations between 0 and 1500 to be generated by the HMM. Default value is `FALSE`. See Notes for further details.
`obs_round`	a logical object. `TRUE` if all generated observations are natural. Default value is `FALSE`. See Notes for further details.
`obs_non_neg`	a logical object. `TRUE`, if non negative observations are generated. Default value is `FALSE`. See Notes for further details.
`plotting`	a numeric value between 0 and 5 (generates different outputs). NA suppresses graphical output. Default value is `0`. `0`: output 1-5 `1`: summary of all results `2`: generated time series of states of the hidden Markov chain `3`: means (of the observation distributions, which depend on the states of the Markov chain) along the time series of states of the hidden Markov chain `4`: observations along the time series of states of the hidden Markov chain `5`: simulated observations

Value

The function HMM_simulation returns a list containing the following components:

`size`	length of the generated time-series of hidden states and observations.
`m`	input number of states in the hidden Markov chain.
`delta`	a vector object containing the chosen values for the marginal probability distribution of the `m` states of the Markov chain at the time point `t=1`.
`gamma`	a matrix containing the chosen values for the transition matrix of the hidden Markov chain.
`distribution_class`	a single character string object with the abbreviated name of the chosen observation distributions of the Markov dependent observation process.
`distribution_theta`	a list object containing the chosen values for the parameters of the `m` observation distributions that are dependent on the hidden Markov state.
`markov_chain`	a vector object containing the generated sequence of states of the hidden Markov chain of the HMM.
`means_along_markov_chain`	a vector object containing the sequence of means (of the state dependent distributions) corresponding to the generated sequence of states.
`observations`	a vector object containing the generated sequence of (state dependent) observations of the HMM.

Note

Some notes regarding the default values:

gamma:
The default setting assigns higher probabilities for remaining in a state than changing into another.

obs_range:
Has to be used with caution. since it manipulates the results of the HMM. If a value for an observation at time t is generated outside the defined range, it will be regenerated as long as it falls into obs_range. It is possible just to define one boundary, e.g. obs_range=c(NA,2000) for observations lower than 2000, or obs_range=c(100,NA) for observations higher than 100.

obs_round :
Has to be used with caution! Rounds each generated observation and hence manipulates the results of the HMM (important for the normal distribution based HMM).

obs_ non_neg:
Has to be used with caution, since it manipulates the results of the HMM. If a negative value for an observation at a time t is generated, it will be re-generated as long as it is non-negative (important for the normal distribution based HMM).

Author(s)

Vitali Witowski (2013).

Examples


################################################################
### i.) Generating a HMM with Poisson-distributed data #########
################################################################

Pois_HMM_data <- 
   HMM_simulation(size = 300, 
      m = 5, 
      distribution_class = "pois", 
      distribution_theta = list( lambda=c(10,15,25,35,55)))

print(Pois_HMM_data)


################################################################
### ii.) Generating 6 physical activities with normally ########
###      distributed accelerometer counts using a HMM. #########
################################################################

## Define number of time points (1440 counts equal 6 hours of 
## activity counts assuming an epoch length of 15 seconds).
size <- 1440

## Define 6 possible physical activity ranges
m <- 6

## Start with the lowest possible state 
## (in this case with the lowest physical activity)
delta <- c(1, rep(0, times = (m - 1)))

## Define transition matrix to generate according to a 
## specific activity 
gamma <- 0.935 * diag(m) + rep(0.065 / m, times = m)

## Define parameters 
## (here: means and standard deviations for m=6 normal 
##  distributions that define the distribution in 
##  a phsycial acitivity level)
distribution_theta <- list(mean = c(0,100,400,600,900,1200), 
   sd = rep(x = 200, times = 6))

### Assume for each count an upper boundary of 2000
obs_range <-c(NA,2000)

### Accelerometer counts shall not be negative
obs_non_neg <-TRUE

### Start simulation
accelerometer_data <- 
   HMM_simulation(size = size, 
     m = m, 
     delta = delta, 
     gamma = gamma, 
     distribution_class = "norm", 
     distribution_theta = distribution_theta, 
     obs_range = obs_range, 
     obs_non_neg= obs_non_neg, plotting=0)

print(accelerometer_data)

Results


R version 3.3.1 (2016-06-21) -- "Bug in Your Hair"
Copyright (C) 2016 The R Foundation for Statistical Computing
Platform: x86_64-pc-linux-gnu (64-bit)

R is free software and comes with ABSOLUTELY NO WARRANTY.
You are welcome to redistribute it under certain conditions.
Type 'license()' or 'licence()' for distribution details.

R is a collaborative project with many contributors.
Type 'contributors()' for more information and
'citation()' on how to cite R or R packages in publications.

Type 'demo()' for some demos, 'help()' for on-line help, or
'help.start()' for an HTML browser interface to help.
Type 'q()' to quit R.

> library(HMMpa)
> png(filename="/home/ddbj/snapshot/RGM3/R_CC/result/HMMpa/HMM_simulation.Rd_%03d_medium.png", width=480, height=480)
> ### Name: HMM_simulation
> ### Title: Generating Realizations of a Hidden Markov Model
> ### Aliases: HMM_simulation
> ### Keywords: ~kwd1 ~kwd2
> 
> ### ** Examples
> 
> 
> ################################################################
> ### i.) Generating a HMM with Poisson-distributed data #########
> ################################################################
> 
> Pois_HMM_data <- 
+    HMM_simulation(size = 300, 
+       m = 5, 
+       distribution_class = "pois", 
+       distribution_theta = list( lambda=c(10,15,25,35,55)))
> 
> print(Pois_HMM_data)
$size
[1] 300

$m
[1] 5

$delta
[1] 0.2 0.2 0.2 0.2 0.2

$gamma
     [,1] [,2] [,3] [,4] [,5]
[1,] 0.84 0.04 0.04 0.04 0.04
[2,] 0.04 0.84 0.04 0.04 0.04
[3,] 0.04 0.04 0.84 0.04 0.04
[4,] 0.04 0.04 0.04 0.84 0.04
[5,] 0.04 0.04 0.04 0.04 0.84

$distribution_class
[1] "pois"

$distribution_theta
$distribution_theta$lambda
[1] 10 15 25 35 55


$markov_chain
  [1] 1 1 1 1 1 5 5 5 5 4 4 4 4 4 4 4 4 4 1 1 1 1 1 1 1 1 1 2 2 1 1 1 1 4 4 4 4
 [38] 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 1 4 5 5 5 5 5 5 5
 [75] 1 5 5 5 5 5 5 5 5 5 5 5 5 5 4 4 4 5 2 2 2 2 2 2 3 3 1 1 1 3 3 3 3 3 3 1 1
[112] 1 1 1 1 1 3 3 1 1 1 1 5 5 5 5 5 5 5 5 5 5 2 2 2 4 4 4 4 4 1 1 1 1 1 1 1 1
[149] 1 1 1 1 1 1 1 1 1 1 1 1 1 4 4 4 4 4 4 4 4 4 4 1 1 5 5 5 5 5 5 5 5 5 5 5 4
[186] 4 4 3 3 5 5 4 4 4 4 4 4 4 4 4 4 4 2 4 4 4 4 4 4 4 4 4 4 1 1 1 1 1 1 1 1 1
[223] 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
[260] 2 2 2 2 2 2 2 2 2 2 3 5 5 1 1 1 1 1 1 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
[297] 3 3 3 3

$means_along_markov_chain
  [1] 10 10 10 10 10 55 55 55 55 35 35 35 35 35 35 35 35 35 10 10 10 10 10 10 10
 [26] 10 10 15 15 10 10 10 10 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35
 [51] 35 35 35 35 35 35 35 35 35 35 35 35 25 25 25 10 35 55 55 55 55 55 55 55 10
 [76] 55 55 55 55 55 55 55 55 55 55 55 55 55 35 35 35 55 15 15 15 15 15 15 25 25
[101] 10 10 10 25 25 25 25 25 25 10 10 10 10 10 10 10 25 25 10 10 10 10 55 55 55
[126] 55 55 55 55 55 55 55 15 15 15 35 35 35 35 35 10 10 10 10 10 10 10 10 10 10
[151] 10 10 10 10 10 10 10 10 10 10 10 35 35 35 35 35 35 35 35 35 35 10 10 55 55
[176] 55 55 55 55 55 55 55 55 55 35 35 35 25 25 55 55 35 35 35 35 35 35 35 35 35
[201] 35 35 15 35 35 35 35 35 35 35 35 35 35 10 10 10 10 10 10 10 10 10 10 10 10
[226] 10 10 10 10 15 15 15 15 15 15 15 15 35 10 10 10 10 15 15 15 15 15 15 15 15
[251] 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 15 25 55 55 10 10 10
[276] 10 10 10 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25

$observations
  [1] 19  7 14 12  8 40 58 58 63 41 32 32 37 28 32 33 34 34  8 13  8 14 11  8  7
 [26]  7  9 14 13  9 13  7 10 33 31 32 39 44 35 32 28 29 34 29 40 34 31 31 28 42
 [51] 33 38 36 43 25 33 43 41 41 27 29 28 23 17 22 11 35 65 53 53 47 43 63 52 11
 [76] 55 58 58 51 47 65 64 53 54 50 49 63 56 31 38 25 49 19 14 16 10 17 11 20 21
[101]  7 13  9 27 26 24 23 23 28 14 13 10 14 10  9 17 22 24 10  7  7 12 56 47 66
[126] 54 68 56 57 54 39 45  8 18 16 42 48 37 31 25 12  8 20 19  6  9 10  5 15 13
[151]  8  8 14 14 12 10 14 10 13 10  8 39 26 36 33 30 34 38 31 23 32 16  8 51 53
[176] 60 46 42 59 44 48 61 65 48 33 38 31 32 22 47 57 36 45 27 31 34 41 36 32 39
[201] 37 31 14 50 38 35 27 35 39 37 38 40 33  7 10 13  6 12 11 14 10 11 11 12 10
[226] 12 10  5 12 14 11 10 13 14 11 11 12 37 11  5 13 14 14 16 21 22 15 18 17 11
[251] 11 16 17  7 16 10 13 11 19 22 20  9 14 15 13 19 17 13 10 18 64 45 10  8 11
[276] 14 14  8 33 34 34 26 21 16 23 34 21 23 22 28 30 27 29 27 22 16 30 26 33 28

> 
> 
> ################################################################
> ### ii.) Generating 6 physical activities with normally ########
> ###      distributed accelerometer counts using a HMM. #########
> ################################################################
> 
> ## Define number of time points (1440 counts equal 6 hours of 
> ## activity counts assuming an epoch length of 15 seconds).
> size <- 1440
> 
> ## Define 6 possible physical activity ranges
> m <- 6
> 
> ## Start with the lowest possible state 
> ## (in this case with the lowest physical activity)
> delta <- c(1, rep(0, times = (m - 1)))
> 
> ## Define transition matrix to generate according to a 
> ## specific activity 
> gamma <- 0.935 * diag(m) + rep(0.065 / m, times = m)
> 
> ## Define parameters 
> ## (here: means and standard deviations for m=6 normal 
> ##  distributions that define the distribution in 
> ##  a phsycial acitivity level)
> distribution_theta <- list(mean = c(0,100,400,600,900,1200), 
+    sd = rep(x = 200, times = 6))
> 
> ### Assume for each count an upper boundary of 2000
> obs_range <-c(NA,2000)
> 
> ### Accelerometer counts shall not be negative
> obs_non_neg <-TRUE
> 
> ### Start simulation
> accelerometer_data <- 
+    HMM_simulation(size = size, 
+      m = m, 
+      delta = delta, 
+      gamma = gamma, 
+      distribution_class = "norm", 
+      distribution_theta = distribution_theta, 
+      obs_range = obs_range, 
+      obs_non_neg= obs_non_neg, plotting=0)
> 
> print(accelerometer_data)
$size
[1] 1440

$m
[1] 6

$delta
[1] 1 0 0 0 0 0

$gamma
           [,1]       [,2]       [,3]       [,4]       [,5]       [,6]
[1,] 0.94583333 0.01083333 0.01083333 0.01083333 0.01083333 0.01083333
[2,] 0.01083333 0.94583333 0.01083333 0.01083333 0.01083333 0.01083333
[3,] 0.01083333 0.01083333 0.94583333 0.01083333 0.01083333 0.01083333
[4,] 0.01083333 0.01083333 0.01083333 0.94583333 0.01083333 0.01083333
[5,] 0.01083333 0.01083333 0.01083333 0.01083333 0.94583333 0.01083333
[6,] 0.01083333 0.01083333 0.01083333 0.01083333 0.01083333 0.94583333

$distribution_class
[1] "norm"

$distribution_theta
$distribution_theta$mean
[1]    0  100  400  600  900 1200

$distribution_theta$sd
[1] 200 200 200 200 200 200


$markov_chain
   [1] 1 1 4 4 4 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
  [38] 1 1 1 1 1 1 1 1 1 1 1 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 2 2 2 2 2
  [75] 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 5 5 5 5 5 5 5 1 6 6 6 6 5 5 5 5 5 5 5 4 4 4
 [112] 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
 [149] 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
 [186] 4 4 4 4 4 4 4 4 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3
 [223] 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
 [260] 1 1 1 3 3 3 3 3 3 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 1 1 1 1
 [297] 1 1 1 1 1 1 1 1 1 5 5 5 5 5 5 5 5 5 5 5 5 2 2 6 6 6 5 5 5 5 5 5 5 5 5 5 2
 [334] 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 5 5
 [371] 5 5 5 5 5 5 5 5 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1
 [408] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 5 5 5 5 2 2 6 6 6 6 6 6 6
 [445] 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 5 5 5 5 5 5 5 5 5 5
 [482] 5 5 5 5 5 5 6 6 6 6 6 6 6 6 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 6 6 6 6 6 6 6
 [519] 6 6 6 6 6 6 6 6 6 5 5 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
 [556] 6 6 6 6 6 6 6 6 6 6 6 6 6 6 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
 [593] 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 6 6 6 6 6 6 6 6 6 6 6 6
 [630] 6 1 1 1 1 1 1 1 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
 [667] 4 4 4 4 4 4 4 4 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 3 3 3 3 3 3 3 3 3 3
 [704] 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 6 6 6 6 6 6 6 6 6 6 6 6
 [741] 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5
 [778] 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6
 [815] 6 6 6 6 6 6 6 6 6 6 6 6 6 1 1 1 1 5 5 5 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4 4 4
 [852] 4 4 4 4 4 4 4 4 4 4 4 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
 [889] 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3
 [926] 3 3 3 3 3 3 3 3 3 3 5 5 5 5 5 5 3 3 3 3 3 6 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5
 [963] 5 5 5 5 5 5 5 5 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
[1000] 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6
[1037] 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 5 5 2 2 2 2 2 2 2 2 2 2 5 5 5 5 5 5
[1074] 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
[1111] 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 5 5 5 5 5 4 4 4 5 5 5 5 5 3 3 3 3 3 3 3 3
[1148] 3 3 3 3 3 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5
[1185] 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 1 1
[1222] 1 1 1 1 1 1 1 1 1 1 1 5 5 1 1 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
[1259] 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
[1296] 3 3 3 3 3 3 6 6 6 6 6 6 6 6 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
[1333] 2 2 2 2 6 6 6 6 6 1 6 6 6 6 6 6 6 6 6 6 6 6 6 3 3 3 3 3 3 3 3 3 3 3 3 3 3
[1370] 3 3 3 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 1 1 1
[1407] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2

$means_along_markov_chain
   [1]    0    0  600  600  600    0    0    0    0    0    0    0    0    0
  [15]    0    0    0    0    0    0    0    0    0    0    0    0    0    0
  [29]    0    0    0    0    0    0    0    0    0    0    0    0    0    0
  [43]    0    0    0    0    0    0 1200 1200 1200 1200 1200 1200 1200 1200
  [57] 1200 1200 1200 1200 1200 1200 1200 1200 1200 1200 1200 1200 1200  100
  [71]  100  100  100  100  100  100  100  100  100  100  100  100  100  100
  [85]  100  100  100  100  100  900  900  900  900  900  900  900    0 1200
  [99] 1200 1200 1200  900  900  900  900  900  900  900  600  600  600  600
 [113]  600  600  600  600  600  600  600  600  600  600  600  600  600  600
 [127]  600  600  600  600  600  600  600  600  600  600  600  600  600  600
 [141]  600  600  600  600  600  600  600  600  600  600  600  600  600  600
 [155]  600  600  600  600  600  600  600  600  600  600  600  600  600  600
 [169]  600  600  600  600  600  600  600  600  600  600  600  600  600  600
 [183]  600  600  600  600  600  600  600  600  600  600  600  100  100  100
 [197]  100  100  100  100  100  100  100  100  100  100  100  100  100  100
 [211]  100  100  100  100  100  100  100  100  100  100  400  400  400  400
 [225]  400  400  400  400  400  400  400  400  400  400  400  400  400  400
 [239]  400  400  400  400  400  400  400  400  400  400  400  400  400  400
 [253]  400  400  400  400  400  400  400    0    0    0  400  400  400  400
 [267]  400  400  100  100  100  100  100  600  600  600  600  600  600  600
 [281]  600  600  600  400  400  400  400  400  400  400  400  400    0    0
 [295]    0    0    0    0    0    0    0    0    0    0    0  900  900  900
 [309]  900  900  900  900  900  900  900  900  900  100  100 1200 1200 1200
 [323]  900  900  900  900  900  900  900  900  900  900  100  100  100  100
 [337]  100  100  100  100  100  100  100  100  100  100  100  100  100  100
 [351]  100  100  100  100  100  100  100  100  100  100  100  100  100  400
 [365]  400  400  400  400  900  900  900  900  900  900  900  900  900  900
 [379]  100  100  100  100  100  100  100  100  100  100  100  100  100  100
 [393]  100  100  100    0    0    0    0    0    0    0    0    0    0    0
 [407]    0    0    0    0    0    0    0    0    0    0    0    0    0    0
 [421]    0    0    0    0    0  100  100  100  100  100  100  900  900  900
 [435]  900  100  100 1200 1200 1200 1200 1200 1200 1200 1200 1200 1200 1200
 [449] 1200 1200 1200 1200 1200 1200 1200 1200 1200 1200 1200 1200 1200 1200
 [463] 1200 1200 1200 1200 1200 1200 1200 1200 1200  900  900  900  900  900
 [477]  900  900  900  900  900  900  900  900  900  900  900 1200 1200 1200
 [491] 1200 1200 1200 1200 1200  600  600  600  600  600  600  600  600  600
 [505]  600  600  600  600  600  600  600 1200 1200 1200 1200 1200 1200 1200
 [519] 1200 1200 1200 1200 1200 1200 1200 1200 1200  900  900  900  900  900
 [533]  900  900  900  900  900  900  900  900 1200 1200 1200 1200 1200 1200
 [547] 1200 1200 1200 1200 1200 1200 1200 1200 1200 1200 1200 1200 1200 1200
 [561] 1200 1200 1200 1200 1200 1200 1200 1200 1200    0    0    0    0    0
 [575]    0    0    0    0    0    0    0    0    0    0    0    0    0    0
 [589]    0    0    0    0    0    0    0    0    0    0    0    0  100  100
 [603]  100  100  100  100  100  100  100  100  100  100  100  100  100  100
 [617]  100 1200 1200 1200 1200 1200 1200 1200 1200 1200 1200 1200 1200 1200
 [631]    0    0    0    0    0    0    0  600  600  600  600  600  600  600
 [645]  600  600  600  600  600  600  600  600  600  600  600  600  600  600
 [659]  600  600  600  600  600  600  600  600  600  600  600  600  600  600
 [673]  600  600 1200 1200 1200 1200 1200 1200 1200 1200 1200 1200 1200 1200
 [687] 1200 1200 1200 1200 1200 1200  600  400  400  400  400  400  400  400
 [701]  400  400  400  400  400  400  400  400  400  400  400  400  400  400
 [715]  400  400  400  400  400  400  400  400  400  400  400  400  400  400
 [729] 1200 1200 1200 1200 1200 1200 1200 1200 1200 1200 1200 1200 1200 1200
 [743] 1200 1200 1200 1200 1200 1200 1200 1200 1200 1200 1200 1200 1200 1200
 [757] 1200 1200  900  900  900  900  900  900  900  900  900  900  900  900
 [771]  900  900  900  900  900  900  900  900  900  900  900  900  900  900
 [785]  900  900  900  900  900  900  900  900  900  900  900  900  900  900
 [799]  900  900  900  900  900  900  900  900  900  900  900 1200 1200 1200
 [813] 1200 1200 1200 1200 1200 1200 1200 1200 1200 1200 1200 1200 1200 1200
 [827] 1200    0    0    0    0  900  900  900    0    0    0    0    0    0
 [841]    0    0    0    0    0    0    0    0  600  600  600  600  600  600
 [855]  600  600  600  600  600  600  600  600  100  100  100  100  100  100
 [869]  100  100  100  100  100  100  100  100  100  100  100  100  100  100
 [883]  100  100  100  100  100  100  100  100  100  100  100  100  100  100
 [897]  100  100  100  100  100  100  100  100  100  100  100  100  100  100
 [911]  100  100  100  100  100  400  400  400  400  400  400  400  400  400
 [925]  400  400  400  400  400  400  400  400  400  400  400  900  900  900
 [939]  900  900  900  400  400  400  400  400 1200  900  900  900  900  900
 [953]  900  900  900  900  900  900  900  900  900  900  900  900  900  900
 [967]  900  900  900  900  400  400  400  400  400  400  400  400  400  400
 [981]  400  400  400  400  400  400  400  400  400  400  400  400  400  400
 [995]  400  400  400  400  400  400  400  400  400  400  400  400  400  400
[1009]  400  400  400  400  400  400  400  400  400  900  900  900  900  900
[1023]  900  900  900  900 1200 1200 1200 1200 1200 1200 1200 1200 1200 1200
[1037] 1200 1200 1200 1200 1200 1200 1200 1200 1200 1200 1200 1200 1200 1200
[1051] 1200 1200 1200 1200 1200  900  900  100  100  100  100  100  100  100
[1065]  100  100  100  900  900  900  900  900  900  900  900  900  900  900
[1079]  900  900  900  900  900  900  900  900  900  900  900  900  900  900
[1093]  900  400  400  400  400  400  400  400  400  400  400  400  400  400
[1107]  400  400  400  400  400  400  400  400  400  400  400  400  400  400
[1121]  400  400  400  400  400  400  900  900  900  900  900  600  600  600
[1135]  900  900  900  900  900  400  400  400  400  400  400  400  400  400
[1149]  400  400  400  400  900  900  900  900  900  900  900  900  900  900
[1163]  900  900  900  900  900  900  900  900  900  900  900  900  900  900
[1177]  900  900  900  900  900  900  900  900  900  900  900  900  900  900
[1191]  900  900  900  900  900  900  900  900  900  900  900  900  900  900
[1205]  900  900  900  900  900  900  900  900  900  900  900  900  900  900
[1219]  900    0    0    0    0    0    0    0    0    0    0    0    0    0
[1233]  900  900    0    0  400  400  400  600  600  600  600  600  600  600
[1247]  600  600  600  600  600  600  600  600  600  600  600  600  600  600
[1261]  600  600  600  600  600  600  600  600  600  600  600  600  600  600
[1275]  600  600  600  400  400  400  400  400  400  400  400  400  400  400
[1289]  400  400  400  400  400  400  400  400  400  400  400  400  400 1200
[1303] 1200 1200 1200 1200 1200 1200 1200    0    0    0    0    0    0    0
[1317]  100  100  100  100  100  100  100  100  100  100  100  100  100  100
[1331]  100  100  100  100  100  100 1200 1200 1200 1200 1200    0 1200 1200
[1345] 1200 1200 1200 1200 1200 1200 1200 1200 1200 1200 1200  400  400  400
[1359]  400  400  400  400  400  400  400  400  400  400  400  400  400  400
[1373] 1200 1200 1200 1200 1200 1200 1200 1200 1200 1200 1200 1200 1200 1200
[1387] 1200 1200 1200 1200 1200 1200 1200 1200 1200 1200 1200 1200 1200 1200
[1401] 1200 1200 1200    0    0    0    0    0    0    0    0    0    0    0
[1415]    0    0    0    0    0    0    0    0    0    0    0    0    0    0
[1429]    0    0    0    0    0    0    0    0    0  100  100  100

$observations
   [1]    0.000000   90.717841  605.103831  405.960897  483.496713   60.446817
   [7]   28.192710    0.000000  240.730155  286.358221    0.000000    0.000000
  [13]    0.000000    0.000000   49.270735    0.000000    0.000000  399.597957
  [19]   51.712193  165.277121  265.474603    0.000000   95.914739    0.000000
  [25]    0.000000  182.707438  134.779271    0.000000    0.000000  248.481644
  [31]   44.328458  249.115919   45.003066    0.000000  178.860428    0.000000
  [37]    0.000000    0.000000   30.832351    0.000000  125.368705   61.736540
  [43]    0.000000    0.000000    0.000000  144.212924  334.489141   22.850444
  [49] 1438.417290 1234.770773 1151.753310 1241.148666 1344.310593 1203.296657
  [55] 1173.523896  806.270976 1087.698778 1486.049933 1192.783144 1178.423934
  [61] 1234.778941 1127.799555 1242.899646 1264.507343 1038.940104 1200.402153
  [67] 1472.097886 1571.167396  902.803218    0.000000  283.285604  172.807631
  [73]   53.934797    0.000000    0.000000    0.000000  320.621018  172.681509
  [79]  167.764588    0.000000   56.777903    0.000000  154.118930    4.615203
  [85]  248.892674  623.458823   69.570790    0.000000   83.932081  523.335595
  [91]  729.710625  737.634848  683.154908  695.979053  990.795476  630.043309
  [97]    0.000000  981.818772 1281.571445 1007.787244 1024.126003 1070.084847
 [103] 1171.394249  598.255301  713.196264 1166.200992  984.691996  870.594570
 [109]  982.841936  457.736820  321.839128  714.273429  528.088082 1030.645533
 [115]  811.981710  418.927215  687.733061  553.652087  614.520267  577.398813
 [121]  789.946660  960.560583  526.570698  529.833065  627.876311  752.758749
 [127]  644.416096  431.139184  456.693202  526.994170  589.459933  868.538030
 [133]  568.798916    0.000000  576.644882  635.526475  536.854556  578.277216
 [139]  549.541038  548.036187  432.362219  767.502742  289.844047  235.943563
 [145]  527.581169  235.318494  523.598507  620.167587  633.749133  541.220615
 [151]  483.454992  683.354829  523.327379  544.426204  670.846803  796.904181
 [157]  425.885110  517.829495  653.158291  651.715712  522.652973  549.219083
 [163]  479.654096  755.831486  789.879469  623.578288  850.178980  542.948471
 [169]  402.520950  444.322354  309.873452  516.382264  446.879949  525.873352
 [175]  344.006882  782.493852  404.419225  588.814409  430.475361  705.999181
 [181]  817.318241  645.599455  782.545558  602.461962  556.337181  458.130847
 [187]  895.293464  752.810315  733.551575  518.131607  553.222847  708.570624
 [193]  760.169878  366.963629    0.000000    0.000000    0.000000    0.000000
 [199]   61.573107    5.736829  261.852834    0.000000  219.689442    0.000000
 [205]  193.340701  450.580104   72.558655   66.129911   56.875147    0.000000
 [211]    0.000000    0.000000  276.677863   44.772692    0.000000    0.000000
 [217]  298.916697   51.441970  436.215778  219.128145  516.758119  547.964148
 [223]  432.891677  208.301098  657.059719  301.762363  215.679145  569.410872
 [229]  593.183836  773.888950   96.506299  622.900096  781.225918  647.820972
 [235]  397.615947  373.867206  450.804619  322.797827  255.822949  566.091883
 [241]  650.983540  596.254099  358.753242  325.541032  499.680784  611.301378
 [247]  653.061433  260.363708  320.302695  271.588688  632.334378  301.706587
 [253]  250.999054  747.052352  406.953073  598.015105  423.799963  129.373496
 [259]  302.915716  198.378116    0.000000    0.000000  428.344790  464.412247
 [265]  295.232401  651.618018  586.805932  252.748369   61.986258    0.000000
 [271]    5.907867  515.108576  379.327344  731.796957  855.108002  791.988292
 [277]  673.344454  428.014088  841.396134  540.938938  770.449331  444.591475
 [283]  901.605891  329.294925  417.187899  440.644016  410.177745  502.895179
 [289]  157.190237  498.217228  242.881829  273.892714   50.997790    0.000000
 [295]    0.000000    0.000000    1.130098    0.000000  162.744537  317.403876
 [301]    0.000000   20.074348  137.685598   81.009359  115.575236  811.842600
 [307]  972.359713  745.208627  843.875151 1309.946693 1275.143711 1136.555177
 [313] 1004.620654 1057.223311  874.265276 1143.743161  413.184196  408.983314
 [319]  594.804981  962.981436 1427.388534 1249.858873  642.359814 1085.043580
 [325]  809.293372 1036.835304  707.406169 1297.723795  994.628394 1091.957965
 [331] 1011.516827  880.841720  199.556304  191.376312  180.734162  311.391463
 [337]    0.000000  363.341576   81.912928  653.460310    0.000000   29.329984
 [343]   81.971690  375.850606   59.107101   44.475052    0.000000  165.440376
 [349]    0.000000   28.873384    0.000000    0.000000   45.119736    9.908418
 [355]   39.846203    0.000000  214.982430   84.625831    0.000000  114.443439
 [361]  118.449732  352.181497   96.430577  268.117547  342.774151  607.695380
 [367]  316.709579  207.384939 1269.674838  959.514524  724.853712  866.521381
 [373]  844.718853  905.858766  814.284512  997.214818 1069.279066  730.441283
 [379]  223.356972  190.215478  247.149594   44.536336    0.000000    0.000000
 [385]    0.000000  353.809909  345.773261    0.000000  149.177615    0.000000
 [391]    0.000000  381.829732  336.813210  121.601043    0.000000    0.000000
 [397]  152.639160  194.401052    0.000000  140.583682    0.000000    0.000000
 [403]    0.000000  297.279899    0.000000    0.000000   79.544806    0.000000
 [409]  266.825817    0.000000   18.895397    0.000000  312.729556  166.078708
 [415]  183.805296  110.429405    0.000000  127.277473  112.491527    0.000000
 [421]    0.000000    0.000000    0.000000    0.000000  194.536987   91.640296
 [427]    0.000000  441.420257  187.341909  272.303073    0.000000 1046.783744
 [433] 1047.723584 1211.735177  884.127621  268.914098    9.426682 1023.998656
 [439] 1270.590837 1138.190742 1209.797165 1371.510985 1275.395564 1235.521506
 [445] 1117.506175 1104.669803 1375.044361 1260.155704 1034.757717 1214.157099
 [451] 1078.679507 1061.241841 1259.855329 1477.859105  959.152695 1242.352958
 [457] 1404.914449  977.971367 1195.467587 1167.543719 1193.947306 1215.146257
 [463] 1494.137977 1168.617923  989.571890 1010.897479 1074.138888 1024.235033
 [469] 1245.464409  812.702802 1544.493382 1207.585120  869.240546  996.600371
 [475] 1131.581919  907.495753 1180.209370 1236.013379 1233.110800 1450.318062
 [481]  742.675854  923.019641  957.335324  782.023179  607.155246  731.457186
 [487]  854.075067 1375.427875 1356.605144 1219.744426 1202.672500  979.619548
 [493]  978.645121 1005.060149  953.389293  676.514536  454.128027  432.002193
 [499]  728.052888  770.583306  873.600530  738.419855  748.983215  529.494140
 [505]  721.532999  508.214742  719.574356  419.230639  759.938381  471.081282
 [511]  684.975711 1389.495518 1189.544608  936.503272 1088.248040 1268.720655
 [517] 1554.347192 1303.821346 1259.055360 1331.731310 1412.544709 1306.832691
 [523] 1202.349933 1015.805253 1141.697715 1422.615451  998.142136  410.928248
 [529]  767.369254 1248.947543 1086.151749  812.445122 1288.327251  836.369822
 [535]  692.321842 1220.749609  838.892585  722.542223  869.492966 1050.701291
 [541] 1141.003383 1152.519141 1647.138699 1163.755368 1500.640581 1023.190875
 [547] 1187.522670 1011.638108 1569.345990 1489.157679 1083.643041 1300.260734
 [553] 1462.155416  803.340445 1361.928053 1217.649099 1465.410695 1212.813773
 [559] 1137.044203 1393.887011  940.721663 1140.728437 1291.623398  759.089524
 [565] 1531.516272 1148.997191 1271.154972 1336.327467 1067.742689    0.000000
 [571]    0.000000    0.000000    0.000000    0.000000    0.000000  110.515607
 [577]   93.445659    0.000000    0.000000    0.000000  360.763668   82.197253
 [583]    0.000000    0.000000  134.020713    0.000000    0.000000  123.469171
 [589]  269.762762  100.634180  163.034951  147.801344  106.129107    0.000000
 [595]  167.577796    0.000000    0.000000    0.000000    0.000000    0.000000
 [601]    0.000000    0.000000    0.000000  145.495392  228.387279    0.000000
 [607]  127.854451    0.000000    0.000000  274.215150    0.000000  320.584110
 [613]   65.580183  248.130033  121.993801  644.065345    0.000000  932.946188
 [619] 1107.104711 1498.236679 1639.096896 1250.445467 1252.707388 1100.573176
 [625] 1268.065734 1684.139374 1228.322188 1474.511510 1187.208671 1249.152834
 [631]    0.000000    0.000000  127.259198    0.000000   34.118691  292.558620
 [637]  131.545132  516.035865  597.011624  553.440139  263.850555  511.912169
 [643]  638.787237  897.872645  330.597184  844.863795  382.222405  458.731949
 [649]  706.804326  749.756987  559.520144  989.561292  613.383185  692.915231
 [655]  934.151407  538.858988  627.740031  753.424946  701.013488  768.022058
 [661]   98.774389  491.864333  582.344168  681.006272  758.056666  640.400613
 [667]  943.879750  606.893483  412.265330  591.078434  353.939604  842.621161
 [673]  673.249376  476.948846 1137.721854 1230.892511 1189.779115 1469.785793
 [679] 1018.522704 1187.728300 1532.491775 1204.319744 1168.658809 1257.405313
 [685] 1418.998438 1159.685168 1093.633981 1246.739762 1548.682150 1466.435720
 [691]  949.286291 1271.481924  678.368010  320.346559  251.464916  505.717206
 [697]  167.815192  531.561536  631.687679  299.285951  448.060339  540.450191
 [703]  769.922264  355.677382  290.578409   21.716572  495.511669  263.938649
 [709]  172.129947  701.327498  489.461375  516.581733  268.143658  413.248150
 [715]  383.390789  416.777445  400.613110   54.999328  817.347675  381.873305
 [721]  396.371473  247.680511  211.901500  684.527653  550.658382  447.220830
 [727]  674.941792  346.990211 1351.308698 1031.941004 1195.117230 1002.148697
 [733] 1105.070024 1235.535182  853.419201 1327.893101 1109.970873 1273.186015
 [739] 1238.832002 1327.607974 1056.935250 1041.578220  922.923340  976.819509
 [745] 1033.988757 1249.443312 1464.495640 1241.888302 1299.588896 1335.501473
 [751] 1387.024193 1081.848676  953.946050 1404.330735 1207.602419 1494.851291
 [757] 1180.223206 1082.567372 1202.810126  748.096487  927.293478  807.622222
 [763] 1050.729869  928.096826  801.120001  915.378912  618.675073  978.437239
 [769] 1137.296575 1039.076455  892.126737  962.018655 1142.607639  730.947995
 [775]  808.997055  768.089858 1166.471602  771.051892  765.079316  850.379114
 [781]  775.131292 1142.451426  881.080512  956.119220  947.953014  835.009712
 [787]  829.340463 1318.612503 1203.511966  981.533763 1116.478716  975.115195
 [793] 1241.660232  825.989387  796.989495  870.800657 1042.949267  976.191290
 [799] 1165.698867  994.385113  910.014898  882.832720 1113.129824  594.558411
 [805]  991.556849 1062.936762 1044.057291  794.175223 1259.244362 1564.151226
 [811] 1498.163022 1340.267817 1628.493481 1218.764305 1499.675743  946.579094
 [817] 1396.355545 1050.716673 1187.644962 1225.477412 1194.060443 1104.252708
 [823] 1227.505764 1044.244935 1099.774243 1420.061025 1392.689530    0.000000
 [829]  159.964378    0.000000  156.406517 1114.652313  987.777877  864.291928
 [835]   91.171661  402.028048    0.000000   41.959141   52.648483    0.000000
 [841]    0.000000    0.000000   46.996769    0.000000  160.607541    0.000000
 [847]    0.000000  140.291983  784.580674  688.234696  865.298487  477.623153
 [853]  656.270332  555.187451  428.896609  684.308421  725.907471  322.536576
 [859]  747.105239  429.718903  395.373890  460.958719    0.000000  344.092607
 [865]   53.930955    0.000000    0.000000    0.000000    0.000000  107.055666
 [871]  237.369740  149.200496    7.866825  504.713637  195.687421  212.301658
 [877]    9.881056  208.391833  106.585135  291.624387  111.979735    0.000000
 [883]  294.018311  297.269705   54.229837  113.607241  200.398829  337.466872
 [889]    0.000000    0.000000  254.752324  138.432670    0.000000    0.000000
 [895]   86.807526  315.439422    0.000000  371.646329  401.661022  289.834504
 [901]   69.253244   61.783147   81.570489    0.000000   89.348830    0.000000
 [907]   76.561678  291.736844  210.895018  433.380796   14.022948   50.371545
 [913]  216.891709    0.000000  213.085443  326.743127  248.171600  655.647724
 [919]  328.005845  225.263972  498.518188  324.798433  779.256501  748.722952
 [925]  883.916315  622.087507  215.129623  469.484090  613.851363  534.124633
 [931]  284.391446  111.802536  738.65947