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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "First we input some librar
ies:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "with(stats): readli
b(randomize):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 104 "We can reset th
e random number seed with the function \"randomize.\"  This can be don
e at any point below." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "ra
ndomize(12345);   # this sets the seed to 12345" }{TEXT -1 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 95 "Now we give a function to make a c
ertain number of plots of an AR(1) with Gaussian white noise:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "ar1_gaussian := proc(num_plo
ts, time_ser_length, c, phi, sigma)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
16 "  local i,t,k,y;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "  y := arra
y(0..time_ser_length);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "  y[0] :=
 0;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "  for i from 1 to num_plots \+
do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "    for t from 1 to time_ser_
length do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "      y[t] := c + phi*
y[t-1] + sigma*stats[random, normald](1)" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 7 "    od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "    prin
t(plot(y[k],k=0..time_ser_length,numpoints=10*time_ser_length+1,adapti
ve=false));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "  od;" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 181 "S
o to get 2 plots of a time series of length 500 based on the AR(1) pro
cess Y_t = c + phi * Y_\{t-1\} + eps_t with eps_t a Gaussian white noi
se, with c=0, phi=1, and sigma=1, we type:" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 26 "ar1_gaussian(2,500,0,1,1);" }{TEXT -1 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 100 "Now we give a similar program, ex
cept that it uses noise that is +1 or -1 each with probability 1/2." }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "ar1_disc_uniform := proc(n
um_plots, time_ser_length, c, phi, sigma)" }{TEXT -1 0 "" }}{PARA 0 ">
 " 0 "" {MPLTEXT 1 0 16 "  local i,t,k,y;" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 33 "  y := array(0..time_ser_length);" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 12 "  y[0] := 0;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "
  for i from 1 to num_plots do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 38 " \+
   for t from 1 to time_ser_length do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 81 "      y[t] := c + phi*y[t-1] + sigma*(2*stats[random, discreteun
iform[0,1]](1)-1)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "    od;" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "    print(plot(y[k],k=0..time_ser_l
ength,numpoints=10*time_ser_length+1,adaptive=false));" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 5 "  od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "Now for a few tests:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "randomize(123);" }{TEXT -1 
0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "ar1_disc_uniform(1,6
0,0,1,1);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 
"ar1_disc_uniform(1,600,0,1,1);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 80 "Now we use a Cauchy distribution as our noise.  This
 should look very different:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 61 "ar1_cauchy := proc(num_plots, time_ser_length, c, phi, sigma)" }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "  local i,t,k,y;" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 33 "  y := array(0..time_ser_length);" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 12 "  y[0] := 0;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 30 "  for i from 1 to num_plots do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 38 "    for t from 1 to time_ser_length do" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 65 "      y[t] := c + phi*y[t-1] + sigma*stats[random,cau
chy[0,1]](1)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "    od;" }}{PARA 0 "
> " 0 "" {MPLTEXT 1 0 89 "    print(plot(y[k],k=0..time_ser_length,num
points=10*time_ser_length+1,adaptive=false));" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 5 "  od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "Now for a test run:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "ar1_cauchy(2,500,0,1,1);" }{TEXT 
-1 0 "" }}}}{MARK "3 0 1" 0 }{VIEWOPTS 1 1 0 1 1 1803 }