::Free Statistics and Forecasting Software::

All rights reserved. The non-commercial (academic) use of this software is free of charge. The only thing that is asked in return is to cite this software when results are used in publications.

This free online software (calculator) reproduces Example 6.6 (Newton-Raphson for Ex. 6.3) in Shumway R.H., Stoffer D.S., Time Series Analysis and Its Applications (2nd ed. with R examples). The module includes all necessary functions to reproduce their results (Kfilter.R, Ksmooth.R, and ex66.txt). In addition, some hard-coded numbers have been replaced by user-defined parameters.
Note: many combinations of parameter values will not yield satisfactory results.

Click here to edit the underlying code of this R Module.

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Number of values (?)

Seed (?)

phi (?)

SD of state equation (?)

SD of obs. equation (?)

Click here to edit the underlying code of this R Module.

---

Source code of R module
1	par1 <- as.numeric(par1)
2	par2 <- as.numeric(par2)
3	par3 <- as.numeric(par3)
4	par4 <- as.numeric(par4)
5	par5 <- as.numeric(par5)
6	Kfilter = function(num,y,A,mu0,Sigma0,Phi,cQ,cR){
7	Q=t(cQ)%*%cQ
8	R=t(cR)%*%cR
9	N=num+1
10	nseries=ncol(as.matrix(y))
11	yobs=matrix(0,N,nseries)
12	yobs[2:N,]=y # yobs is y with first row=zeros
13	xp=vector("list",N) # xp=x_t^{t-1}
14	Pp=vector("list",N) # Pp=P_t^{t-1}
15	xf=vector("list",N) # xf=x_t^t
16	Pf=vector("list",N) # Pf=x_t^t
17	innov=vector("list",N) # innovations
18	sig=vector("list",N) # innov var-cov matrix
19	like=0 # -log(likelihood)
20	xf[[1]]=mu0
21	Pf[[1]]=Sigma0
22	for (i in 2:N){
23	xp[[i]]=Phi%*%xf[[i-1]]
24	Pp[[i]]=Phi%*%Pf[[i-1]]%*%t(Phi)+Q
25	siginv=A[[i]]%*%Pp[[i]]%*%t(A[[i]])+R
26	sig[[i]]=(t(siginv)+siginv)/2 # make sure sig is symmetric
27	siginv=solve(sig[[i]]) # now siginv is sig[[i]]^{-1}
28	K=Pp[[i]]%*%t(A[[i]])%*%siginv
29	innov[[i]]=as.matrix(yobs[i,])-A[[i]]%*%xp[[i]]
30	xf[[i]]=xp[[i]]+K%*%innov[[i]]
31	Pf[[i]]=Pp[[i]]-K%*%A[[i]]%*%Pp[[i]]
32	like= like + log(det(sig[[i]])) + t(innov[[i]])%*%siginv%*%innov[[i]]
33	}
34	like=0.5*like
35	list(xp=xp,Pp=Pp,xf=xf,Pf=Pf,like=like,innov=innov,sig=sig,Kn=K)
36	}
37	Ksmooth = function(num,y,A,mu0,Sigma0,Phi,cQ,cR){
38	kf=Kfilter(num,y,A,mu0,Sigma0,Phi,cQ,cR)
39	N=num+1
40	xs=vector("list",N) # xs=x_t^n
41	Ps=vector("list",N) # Ps=P_t^n
42	J=vector("list",N) # J=J_t
43	xs[[N]]=kf$xf[[N]]
44	Ps[[N]]=kf$Pf[[N]]
45	for(k in N:2) {
46	J[[k-1]]=(kf$Pf[[k-1]]%*%t(Phi))%*%solve(kf$Pp[[k]])
47	xs[[k-1]]=kf$xf[[k-1]]+J[[k-1]]%*%(xs[[k]]-kf$xp[[k]])
48	Ps[[k-1]]=kf$Pf[[k-1]]+J[[k-1]]%*%(Ps[[k]]-kf$Pp[[k]])%*%t(J[[k-1]])
49	}
50	list(xs=xs,Ps=Ps,J=J,xp=kf$xp,Pp=kf$Pp,xf=kf$xf,Pf=kf$Pf,like=kf$like,Kn=kf$K)
51	}
52	Linn=function(para){
53	phi=para[1]
54	sigw=para[2] # this is the standard dev
55	sigv=para[3] # this is the standard dev
56	mu0=0
57	Sigma0=(sigw^2)/(1-phi^2)
58	Sigma0[Sigma0<0] =0 # this makes sure Sigma0 is never negative
59	kf = Kfilter(num,y,A,mu0,Sigma0,phi,sigw,sigv)
60	return(kf$like)
61	}
62	set.seed(par2)
63	num=par1 #100
64	N=num+1
65	x =arima.sim(n=N, list(ar = par3, sd=par4)) # x[1]=x_0, x[2]=x_1,...
66	v = rnorm(num,0,par5)
67	y=x[2:N]+v
68	data=ts(y)
69	u=ts.intersect(data,lag(data,-1),lag(data,-2));
70	varu=var(u);
71	coru=cor(u);
72	phi=coru[1,3]/coru[1,2];
73	q = (1-phi^2)*varu[1,2]/phi;
74	r = varu[1,1] - q/(1-phi^2);
75	warning = ""
76	if (q < 0)
77	{
78	warning = "Warning: initial estimate for SD(w) could not be computed. Initial value is set to 1. "
79	q=1
80	}
81	if (r < 0)
82	{
83	warning = paste(warning,"Warning: initial estimate for SD(v) could not be computed. Initial value is set to 1. ")
84	r=1
85	}
86	initpar=c(phi,sqrt(q),sqrt(r))
87	initpar # view the initial estimates
88	A=matrix(1,N,1)
89	A=as.list(A,N) # A has to be a list, A[[i+1]] for y[i]. A[[1]] not used.
90	est=optim(initpar, Linn, NULL, method = "BFGS", hessian = TRUE, control=list(trace=1,REPORT=1))
91	est # for a summary
92	stderr=sqrt(diag(solve(est$hessian)))
93	cbind(est$par,stderr) # list estimates and SEs
94	load(file="createtable")
95	a<-table.start()
96	a<-table.row.start(a)
97	a<-table.element(a,"Parameter",1,TRUE)
98	a<-table.element(a,"Value",1,TRUE)
99	a<-table.element(a,"S.D.",1,TRUE)
100	a<-table.row.end(a)
101	a<-table.row.start(a)
102	a<-table.element(a,"#simulated values",header=TRUE)
103	a<-table.element(a,par1)
104	a<-table.element(a,"-")
105	a<-table.row.end(a)
106	a<-table.row.start(a)
107	a<-table.element(a,"random seed",header=TRUE)
108	a<-table.element(a,par2)
109	a<-table.element(a,"-")
110	a<-table.row.end(a)
111	a<-table.row.start(a)
112	a<-table.element(a,"true phi",header=TRUE)
113	a<-table.element(a,par3)
114	a<-table.element(a,"-")
115	a<-table.row.end(a)
116	a<-table.row.start(a)
117	a<-table.element(a,"true SD(w)",header=TRUE)
118	a<-table.element(a,par4)
119	a<-table.element(a,"-")
120	a<-table.row.end(a)
121	a<-table.row.start(a)
122	a<-table.element(a,"true SD(v)",header=TRUE)
123	a<-table.element(a,par5)
124	a<-table.element(a,"-")
125	a<-table.row.end(a)
126	a<-table.row.start(a)
127	a<-table.element(a,"estimated phi",header=TRUE)
128	a<-table.element(a,est$par[1])
129	a<-table.element(a,stderr[1])
130	a<-table.row.end(a)
131	a<-table.row.start(a)
132	a<-table.element(a,"SD(w)",header=TRUE)
133	a<-table.element(a,est$par[2])
134	a<-table.element(a,stderr[2])
135	a<-table.row.end(a)
136	a<-table.row.start(a)
137	a<-table.element(a,"SD(v)",header=TRUE)
138	a<-table.element(a,est$par[3])
139	a<-table.element(a,stderr[3])
140	a<-table.row.end(a)
141	a<-table.end(a)
142	table.save(a,file="mytable1.tab")
143	if (warning != "")
144	{
145	load(file="createtable")
146	a<-table.start()
147	a<-table.row.start(a)
148	a<-table.element(a,"Warnings about Parameter Estimation",1,TRUE)
149	a<-table.row.end(a)
150	a<-table.row.start(a)
151	a<-table.element(a,warning)
152	a<-table.row.end(a)
153	a<-table.end(a)
154	table.save(a,file="mytable2.tab")
155	}

Cite this software as:

Wessa P., (2008), Reproduction of Example 6.6 in Time Series Analysis and Its Applications (v1.0.4) in Free Statistics Software (v1.1.23-r2), Office for Research Development and Education, URL http://www.wessa.net/rwasp_shumwaystofferex66.wasp/

The R code is based on :

Shumway R.H., Stoffer D.S., Time Series Analysis and Its Applications (2nd ed. with R examples), Springer, 2006

Shumway R.H., Stoffer D.S., Time Series Analysis and Its Applications (2nd ed. with R examples), URL http://www.stat.pitt.edu/stoffer/tsa2/#Rchapter6

Belisle, C. J. P. (1992) Convergence theorems for a class of simulated annealing algorithms on Rd., J Applied Probability, vol. 29, 885-895.

Byrd, R. H., Lu, P., Nocedal, J. and Zhu, C. (1995) A limited memory algorithm for bound constrained optimization., SIAM J. Scientific Computing, vol. 16, 1190-1208.

Fletcher, R. and Reeves, C. M. (1964) Function minimization by conjugate gradients., Computer Journal, vol. 7, 148-154.

Nash, J. C. (1990), Compact Numerical Methods for Computers., Linear Algebra and Function Minimisation., Adam Hilger.

Nelder, J. A. and Mead, R. (1965), A simplex algorithm for function minimization., Computer Journal, vol. 7, 308-313.

Nocedal, J. and Wright, S. J. (1999), Numerical Optimization., Springer.

To cite Wessa.net in publications use:
Wessa, P. (2008), Free Statistics Software, Office for Research Development and Education,
version 1.1.23-r2, URL http://www.wessa.net/

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