/**
* @brief Creates the penalty matrix D tilde of order k.
* Returns the matrix Dk premultipied by a diagonal
* matrix of weights.
*
* @param n number of observations
* @param k order of the trendfilter
* @param x locations of the responses
* @return pointer to a csparse matrix
* @see tf_calc_dktil
*/
cs * tf_calc_dktil (int n, int k, const double * x)
{
cs * delta_k;
cs * delta_k_cp;
cs * Dk;
cs * Dktil;
int i;
Dk = tf_calc_dk(n, k, x);
/* Deal with k=0 separately */
if(k == 0)
return Dk;
/* Construct diagonal matrix of differences: */
delta_k = cs_spalloc(n-k, n-k, (n-k), 1, 1);
for(i = 0; i < n - k; i++)
{
delta_k->p[i] = i;
delta_k->i[i] = i;
delta_k->x[i] = k / (x[k + i] - x[i]);
}
delta_k->nz = n-k;
delta_k_cp = cs_compress(delta_k);
Dktil = cs_multiply(delta_k_cp, Dk);
cs_spfree(Dk);
cs_spfree(delta_k);
cs_spfree(delta_k_cp);
return Dktil;
}
cs *cs_rinvwishart(const cs *A, double nu, const css *As){
int m, i, j, cnt;
cs *T, *IW, *C, *W, *tC;
csn *U;
m = A->n;
T = cs_spalloc (m, m, m*(m+1)/2, 1, 0) ;
if (!T ) return (cs_done (T, NULL, NULL, 0));
double df = nu;
cnt = 0;
for(i = 0; i<m; i++){
T->p[i] = cnt;
T->i[cnt] = i;
T->x[cnt] = sqrt(rchisq(df));
cnt++;
for(j = i+1; j<m; j++){
T->i[cnt] = j;
T->x[cnt] = rnorm(0.0,1.0);
cnt++;
}
df--;
}
T->p[m] = m*(m+1)/2;
U = cs_chol(A, As);
if(U==NULL){
PutRNGstate();
error("ill-conditioned cross-product: can't form Cholesky factor\n");
}
C = cs_multiply(U->L,T); // t(T)%*%chol(A)
tC = cs_transpose(C, TRUE); // t(CI)
W = cs_multiply(C,tC);
IW = cs_inv(W); // crossprod(t(CI))
cs_spfree(T);
cs_nfree(U);
cs_spfree(C);
cs_spfree(tC);
cs_spfree(W);
return (cs_done (IW, NULL, NULL, 1)) ; /* success; free workspace, return C */
}
cs* cs_sorted_multiply2(const cs* a, const cs* b)
{
cs* D = cs_multiply(a,b);
cs* E = cs_transpose(D,1);
cs_spfree(D);
cs* C = cs_transpose(E,1);
cs_spfree(E);
return C;
}
cs* cs_sorted_multiply(const cs* a, const cs* b)
{
cs* A = cs_transpose (a, 1) ;
cs* B = cs_transpose (b, 1) ;
cs* D = cs_multiply (B,A) ; /* D = B'*A' */
cs_spfree (A) ;
cs_spfree (B) ;
cs_dropzeros (D) ; /* drop zeros from D */
cs* C = cs_transpose (D, 1) ; /* C = D', so that C is sorted */
cs_spfree (D) ;
return C;
}
void NM_gemm(const double alpha, NumericsMatrix* A, NumericsMatrix* B,
const double beta, NumericsMatrix* C)
{
switch(A->storageType)
{
case NM_DENSE:
{
cblas_dgemm(CblasColMajor, CblasNoTrans, CblasNoTrans, A->size0, B->size1, B->size1, alpha, A->matrix0, A->size0, B->matrix0, B->size0, beta, C->matrix0, A->size0);
NM_clearSparseBlock(C);
NM_clearSparseStorage(C);
break;
}
case NM_SPARSE_BLOCK:
{
prodNumericsMatrixNumericsMatrix(alpha, A, B, beta, C);
NM_clearDense(C);
NM_clearSparseStorage(C);
break;
}
case NM_SPARSE:
{
CSparseMatrix* result = cs_add(cs_multiply(NM_csc(A), NM_csc(B)),
NM_csc(C), alpha, beta);
NM_clearDense(C);
NM_clearSparseBlock(C);
NM_clearSparseStorage(C);
NM_sparse(C)->csc = result;
C->size0 = (int)C->matrix2->csc->m;
C->size1 = (int)C->matrix2->csc->n;
break;
}
}
}
/* p = amd(A+A') if symmetric is true, or amd(A'A) otherwise */
CS_INT *cs_amd (CS_INT order, const cs *A) /* order 0:natural, 1:Chol, 2:LU, 3:QR */
{
cs *C, *A2, *AT ;
CS_INT *Cp, *Ci, *last, *W, *len, *nv, *next, *P, *head, *elen, *degree, *w,
*hhead, *ATp, *ATi, d, dk, dext, lemax = 0, e, elenk, eln, i, j, k, k1,
k2, k3, jlast, ln, dense, nzmax, mindeg = 0, nvi, nvj, nvk, mark, wnvi,
ok, cnz, nel = 0, p, p1, p2, p3, p4, pj, pk, pk1, pk2, pn, q, n, m, t ;
unsigned CS_INT h ;
/* --- Construct matrix C ----------------------------------------------- */
if (!CS_CSC (A) || order <= 0 || order > 3) return (NULL) ; /* check */
AT = cs_transpose (A, 0) ; /* compute A' */
if (!AT) return (NULL) ;
m = A->m ; n = A->n ;
dense = CS_MAX (16, 10 * sqrt ((double) n)) ; /* find dense threshold */
dense = CS_MIN (n-2, dense) ;
if (order == 1 && n == m)
{
C = cs_add (A, AT, 0, 0) ; /* C = A+A' */
}
else if (order == 2)
{
ATp = AT->p ; /* drop dense columns from AT */
ATi = AT->i ;
for (p2 = 0, j = 0 ; j < m ; j++)
{
p = ATp [j] ; /* column j of AT starts here */
ATp [j] = p2 ; /* new column j starts here */
if (ATp [j+1] - p > dense) continue ; /* skip dense col j */
for ( ; p < ATp [j+1] ; p++) ATi [p2++] = ATi [p] ;
}
ATp [m] = p2 ; /* finalize AT */
A2 = cs_transpose (AT, 0) ; /* A2 = AT' */
C = A2 ? cs_multiply (AT, A2) : NULL ; /* C=A'*A with no dense rows */
cs_spfree (A2) ;
}
else
{
C = cs_multiply (AT, A) ; /* C=A'*A */
}
cs_spfree (AT) ;
if (!C) return (NULL) ;
cs_fkeep (C, &cs_diag, NULL) ; /* drop diagonal entries */
Cp = C->p ;
cnz = Cp [n] ;
P = cs_malloc (n+1, sizeof (CS_INT)) ; /* allocate result */
W = cs_malloc (8*(n+1), sizeof (CS_INT)) ; /* get workspace */
t = cnz + cnz/5 + 2*n ; /* add elbow room to C */
if (!P || !W || !cs_sprealloc (C, t)) return (cs_idone (P, C, W, 0)) ;
len = W ; nv = W + (n+1) ; next = W + 2*(n+1) ;
head = W + 3*(n+1) ; elen = W + 4*(n+1) ; degree = W + 5*(n+1) ;
w = W + 6*(n+1) ; hhead = W + 7*(n+1) ;
last = P ; /* use P as workspace for last */
/* --- Initialize quotient graph ---------------------------------------- */
for (k = 0 ; k < n ; k++) len [k] = Cp [k+1] - Cp [k] ;
len [n] = 0 ;
nzmax = C->nzmax ;
Ci = C->i ;
for (i = 0 ; i <= n ; i++)
{
head [i] = -1 ; /* degree list i is empty */
last [i] = -1 ;
next [i] = -1 ;
hhead [i] = -1 ; /* hash list i is empty */
nv [i] = 1 ; /* node i is just one node */
w [i] = 1 ; /* node i is alive */
elen [i] = 0 ; /* Ek of node i is empty */
degree [i] = len [i] ; /* degree of node i */
}
mark = cs_wclear (0, 0, w, n) ; /* clear w */
elen [n] = -2 ; /* n is a dead element */
Cp [n] = -1 ; /* n is a root of assembly tree */
w [n] = 0 ; /* n is a dead element */
/* --- Initialize degree lists ------------------------------------------ */
for (i = 0 ; i < n ; i++)
{
d = degree [i] ;
if (d == 0) /* node i is empty */
{
elen [i] = -2 ; /* element i is dead */
nel++ ;
Cp [i] = -1 ; /* i is a root of assembly tree */
w [i] = 0 ;
}
else if (d > dense) /* node i is dense */
{
nv [i] = 0 ; /* absorb i into element n */
elen [i] = -1 ; /* node i is dead */
nel++ ;
Cp [i] = CS_FLIP (n) ;
nv [n]++ ;
}
else
{
if (head [d] != -1) last [head [d]] = i ;
next [i] = head [d] ; /* put node i in degree list d */
head [d] = i ;
}
}
while (nel < n) /* while (selecting pivots) do */
{
/* --- Select node of minimum approximate degree -------------------- */
for (k = -1 ; mindeg < n && (k = head [mindeg]) == -1 ; mindeg++) ;
if (next [k] != -1) last [next [k]] = -1 ;
head [mindeg] = next [k] ; /* remove k from degree list */
elenk = elen [k] ; /* elenk = |Ek| */
nvk = nv [k] ; /* # of nodes k represents */
nel += nvk ; /* nv[k] nodes of A eliminated */
/* --- Garbage collection ------------------------------------------- */
if (elenk > 0 && cnz + mindeg >= nzmax)
{
for (j = 0 ; j < n ; j++)
{
if ((p = Cp [j]) >= 0) /* j is a live node or element */
{
Cp [j] = Ci [p] ; /* save first entry of object */
Ci [p] = CS_FLIP (j) ; /* first entry is now CS_FLIP(j) */
}
}
for (q = 0, p = 0 ; p < cnz ; ) /* scan all of memory */
{
if ((j = CS_FLIP (Ci [p++])) >= 0) /* found object j */
{
Ci [q] = Cp [j] ; /* restore first entry of object */
Cp [j] = q++ ; /* new pointer to object j */
for (k3 = 0 ; k3 < len [j]-1 ; k3++) Ci [q++] = Ci [p++] ;
}
}
cnz = q ; /* Ci [cnz...nzmax-1] now free */
}
/* --- Construct new element ---------------------------------------- */
dk = 0 ;
nv [k] = -nvk ; /* flag k as in Lk */
p = Cp [k] ;
pk1 = (elenk == 0) ? p : cnz ; /* do in place if elen[k] == 0 */
pk2 = pk1 ;
for (k1 = 1 ; k1 <= elenk + 1 ; k1++)
{
if (k1 > elenk)
{
e = k ; /* search the nodes in k */
pj = p ; /* list of nodes starts at Ci[pj]*/
ln = len [k] - elenk ; /* length of list of nodes in k */
}
else
{
e = Ci [p++] ; /* search the nodes in e */
pj = Cp [e] ;
ln = len [e] ; /* length of list of nodes in e */
}
for (k2 = 1 ; k2 <= ln ; k2++)
{
i = Ci [pj++] ;
if ((nvi = nv [i]) <= 0) continue ; /* node i dead, or seen */
dk += nvi ; /* degree[Lk] += size of node i */
nv [i] = -nvi ; /* negate nv[i] to denote i in Lk*/
Ci [pk2++] = i ; /* place i in Lk */
if (next [i] != -1) last [next [i]] = last [i] ;
if (last [i] != -1) /* remove i from degree list */
{
next [last [i]] = next [i] ;
}
else
{
head [degree [i]] = next [i] ;
}
}
if (e != k)
{
Cp [e] = CS_FLIP (k) ; /* absorb e into k */
w [e] = 0 ; /* e is now a dead element */
}
}
if (elenk != 0) cnz = pk2 ; /* Ci [cnz...nzmax] is free */
degree [k] = dk ; /* external degree of k - |Lk\i| */
Cp [k] = pk1 ; /* element k is in Ci[pk1..pk2-1] */
len [k] = pk2 - pk1 ;
elen [k] = -2 ; /* k is now an element */
/* --- Find set differences ----------------------------------------- */
mark = cs_wclear (mark, lemax, w, n) ; /* clear w if necessary */
for (pk = pk1 ; pk < pk2 ; pk++) /* scan 1: find |Le\Lk| */
{
i = Ci [pk] ;
if ((eln = elen [i]) <= 0) continue ;/* skip if elen[i] empty */
nvi = -nv [i] ; /* nv [i] was negated */
wnvi = mark - nvi ;
for (p = Cp [i] ; p <= Cp [i] + eln - 1 ; p++) /* scan Ei */
{
e = Ci [p] ;
if (w [e] >= mark)
{
w [e] -= nvi ; /* decrement |Le\Lk| */
}
else if (w [e] != 0) /* ensure e is a live element */
{
w [e] = degree [e] + wnvi ; /* 1st time e seen in scan 1 */
}
}
}
/* --- Degree update ------------------------------------------------ */
for (pk = pk1 ; pk < pk2 ; pk++) /* scan2: degree update */
{
i = Ci [pk] ; /* consider node i in Lk */
p1 = Cp [i] ;
p2 = p1 + elen [i] - 1 ;
pn = p1 ;
for (h = 0, d = 0, p = p1 ; p <= p2 ; p++) /* scan Ei */
{
e = Ci [p] ;
if (w [e] != 0) /* e is an unabsorbed element */
{
dext = w [e] - mark ; /* dext = |Le\Lk| */
if (dext > 0)
{
d += dext ; /* sum up the set differences */
Ci [pn++] = e ; /* keep e in Ei */
h += e ; /* compute the hash of node i */
}
else
{
Cp [e] = CS_FLIP (k) ; /* aggressive absorb. e->k */
w [e] = 0 ; /* e is a dead element */
}
}
}
elen [i] = pn - p1 + 1 ; /* elen[i] = |Ei| */
p3 = pn ;
p4 = p1 + len [i] ;
for (p = p2 + 1 ; p < p4 ; p++) /* prune edges in Ai */
{
j = Ci [p] ;
if ((nvj = nv [j]) <= 0) continue ; /* node j dead or in Lk */
d += nvj ; /* degree(i) += |j| */
Ci [pn++] = j ; /* place j in node list of i */
h += j ; /* compute hash for node i */
}
if (d == 0) /* check for mass elimination */
{
Cp [i] = CS_FLIP (k) ; /* absorb i into k */
nvi = -nv [i] ;
dk -= nvi ; /* |Lk| -= |i| */
nvk += nvi ; /* |k| += nv[i] */
nel += nvi ;
nv [i] = 0 ;
elen [i] = -1 ; /* node i is dead */
}
else
{
degree [i] = CS_MIN (degree [i], d) ; /* update degree(i) */
Ci [pn] = Ci [p3] ; /* move first node to end */
Ci [p3] = Ci [p1] ; /* move 1st el. to end of Ei */
Ci [p1] = k ; /* add k as 1st element in of Ei */
len [i] = pn - p1 + 1 ; /* new len of adj. list of node i */
h %= n ; /* finalize hash of i */
next [i] = hhead [h] ; /* place i in hash bucket */
hhead [h] = i ;
last [i] = h ; /* save hash of i in last[i] */
}
} /* scan2 is done */
degree [k] = dk ; /* finalize |Lk| */
lemax = CS_MAX (lemax, dk) ;
mark = cs_wclear (mark+lemax, lemax, w, n) ; /* clear w */
/* --- Supernode detection ------------------------------------------ */
for (pk = pk1 ; pk < pk2 ; pk++)
{
i = Ci [pk] ;
if (nv [i] >= 0) continue ; /* skip if i is dead */
h = last [i] ; /* scan hash bucket of node i */
i = hhead [h] ;
hhead [h] = -1 ; /* hash bucket will be empty */
for ( ; i != -1 && next [i] != -1 ; i = next [i], mark++)
{
ln = len [i] ;
eln = elen [i] ;
for (p = Cp [i]+1 ; p <= Cp [i] + ln-1 ; p++) w [Ci [p]] = mark;
jlast = i ;
for (j = next [i] ; j != -1 ; ) /* compare i with all j */
{
ok = (len [j] == ln) && (elen [j] == eln) ;
for (p = Cp [j] + 1 ; ok && p <= Cp [j] + ln - 1 ; p++)
{
if (w [Ci [p]] != mark) ok = 0 ; /* compare i and j*/
}
if (ok) /* i and j are identical */
{
Cp [j] = CS_FLIP (i) ; /* absorb j into i */
nv [i] += nv [j] ;
nv [j] = 0 ;
elen [j] = -1 ; /* node j is dead */
j = next [j] ; /* delete j from hash bucket */
next [jlast] = j ;
}
else
{
jlast = j ; /* j and i are different */
j = next [j] ;
}
}
}
}
/* --- Finalize new element------------------------------------------ */
for (p = pk1, pk = pk1 ; pk < pk2 ; pk++) /* finalize Lk */
{
i = Ci [pk] ;
if ((nvi = -nv [i]) <= 0) continue ;/* skip if i is dead */
nv [i] = nvi ; /* restore nv[i] */
d = degree [i] + dk - nvi ; /* compute external degree(i) */
d = CS_MIN (d, n - nel - nvi) ;
if (head [d] != -1) last [head [d]] = i ;
next [i] = head [d] ; /* put i back in degree list */
last [i] = -1 ;
head [d] = i ;
mindeg = CS_MIN (mindeg, d) ; /* find new minimum degree */
degree [i] = d ;
Ci [p++] = i ; /* place i in Lk */
}
nv [k] = nvk ; /* # nodes absorbed into k */
if ((len [k] = p-pk1) == 0) /* length of adj list of element k*/
{
Cp [k] = -1 ; /* k is a root of the tree */
w [k] = 0 ; /* k is now a dead element */
}
if (elenk != 0) cnz = p ; /* free unused space in Lk */
}
/* --- Postordering ----------------------------------------------------- */
for (i = 0 ; i < n ; i++) Cp [i] = CS_FLIP (Cp [i]) ;/* fix assembly tree */
for (j = 0 ; j <= n ; j++) head [j] = -1 ;
for (j = n ; j >= 0 ; j--) /* place unordered nodes in lists */
{
if (nv [j] > 0) continue ; /* skip if j is an element */
next [j] = head [Cp [j]] ; /* place j in list of its parent */
head [Cp [j]] = j ;
}
for (e = n ; e >= 0 ; e--) /* place elements in lists */
{
if (nv [e] <= 0) continue ; /* skip unless e is an element */
if (Cp [e] != -1)
{
next [e] = head [Cp [e]] ; /* place e in list of its parent */
head [Cp [e]] = e ;
}
}
for (k = 0, i = 0 ; i <= n ; i++) /* postorder the assembly tree */
{
if (Cp [i] == -1) k = cs_tdfs (i, k, head, next, P, w) ;
}
return (cs_idone (P, C, W, 1)) ;
}
/* Cholesky update/downdate */
int demo3 (problem *Prob)
{
cs *A, *C, *W = NULL, *WW, *WT, *E = NULL, *W2 ;
int n, k, *Li, *Lp, *Wi, *Wp, p1, p2, *p = NULL, ok ;
double *b, *x, *resid, *y = NULL, *Lx, *Wx, s, t, t1 ;
css *S = NULL ;
csn *N = NULL ;
if (!Prob || !Prob->sym || Prob->A->n == 0) return (0) ;
A = Prob->A ; C = Prob->C ; b = Prob->b ; x = Prob->x ; resid = Prob->resid;
n = A->n ;
if (!Prob->sym || n == 0) return (1) ;
rhs (x, b, n) ; /* compute right-hand side */
printf ("\nchol then update/downdate ") ;
print_order (1) ;
y = cs_malloc (n, sizeof (double)) ;
t = tic () ;
S = cs_schol (1, C) ; /* symbolic Chol, amd(A+A') */
printf ("\nsymbolic chol time %8.2f\n", toc (t)) ;
t = tic () ;
N = cs_chol (C, S) ; /* numeric Cholesky */
printf ("numeric chol time %8.2f\n", toc (t)) ;
if (!S || !N || !y) return (done3 (0, S, N, y, W, E, p)) ;
t = tic () ;
cs_ipvec (S->pinv, b, y, n) ; /* y = P*b */
cs_lsolve (N->L, y) ; /* y = L\y */
cs_ltsolve (N->L, y) ; /* y = L'\y */
cs_pvec (S->pinv, y, x, n) ; /* x = P'*y */
printf ("solve chol time %8.2f\n", toc (t)) ;
printf ("original: ") ;
print_resid (1, C, x, b, resid) ; /* print residual */
k = n/2 ; /* construct W */
W = cs_spalloc (n, 1, n, 1, 0) ;
if (!W) return (done3 (0, S, N, y, W, E, p)) ;
Lp = N->L->p ; Li = N->L->i ; Lx = N->L->x ;
Wp = W->p ; Wi = W->i ; Wx = W->x ;
Wp [0] = 0 ;
p1 = Lp [k] ;
Wp [1] = Lp [k+1] - p1 ;
s = Lx [p1] ;
srand (1) ;
for ( ; p1 < Lp [k+1] ; p1++)
{
p2 = p1 - Lp [k] ;
Wi [p2] = Li [p1] ;
Wx [p2] = s * rand () / ((double) RAND_MAX) ;
}
t = tic () ;
ok = cs_updown (N->L, +1, W, S->parent) ; /* update: L*L'+W*W' */
t1 = toc (t) ;
printf ("update: time: %8.2f\n", t1) ;
if (!ok) return (done3 (0, S, N, y, W, E, p)) ;
t = tic () ;
cs_ipvec (S->pinv, b, y, n) ; /* y = P*b */
cs_lsolve (N->L, y) ; /* y = L\y */
cs_ltsolve (N->L, y) ; /* y = L'\y */
cs_pvec (S->pinv, y, x, n) ; /* x = P'*y */
t = toc (t) ;
p = cs_pinv (S->pinv, n) ;
W2 = cs_permute (W, p, NULL, 1) ; /* E = C + (P'W)*(P'W)' */
WT = cs_transpose (W2,1) ;
WW = cs_multiply (W2, WT) ;
cs_spfree (WT) ;
cs_spfree (W2) ;
E = cs_add (C, WW, 1, 1) ;
cs_spfree (WW) ;
if (!E || !p) return (done3 (0, S, N, y, W, E, p)) ;
printf ("update: time: %8.2f (incl solve) ", t1+t) ;
print_resid (1, E, x, b, resid) ; /* print residual */
cs_nfree (N) ; /* clear N */
t = tic () ;
N = cs_chol (E, S) ; /* numeric Cholesky */
if (!N) return (done3 (0, S, N, y, W, E, p)) ;
cs_ipvec (S->pinv, b, y, n) ; /* y = P*b */
cs_lsolve (N->L, y) ; /* y = L\y */
cs_ltsolve (N->L, y) ; /* y = L'\y */
cs_pvec (S->pinv, y, x, n) ; /* x = P'*y */
t = toc (t) ;
printf ("rechol: time: %8.2f (incl solve) ", t) ;
print_resid (1, E, x, b, resid) ; /* print residual */
t = tic () ;
ok = cs_updown (N->L, -1, W, S->parent) ; /* downdate: L*L'-W*W' */
t1 = toc (t) ;
if (!ok) return (done3 (0, S, N, y, W, E, p)) ;
printf ("downdate: time: %8.2f\n", t1) ;
t = tic () ;
cs_ipvec (S->pinv, b, y, n) ; /* y = P*b */
cs_lsolve (N->L, y) ; /* y = L\y */
cs_ltsolve (N->L, y) ; /* y = L'\y */
cs_pvec (S->pinv, y, x, n) ; /* x = P'*y */
t = toc (t) ;
printf ("downdate: time: %8.2f (incl solve) ", t1+t) ;
print_resid (1, C, x, b, resid) ; /* print residual */
return (done3 (1, S, N, y, W, E, p)) ;
}
/**
* @brief Creates the penalty matrix of order k.
* Returns the matrix Dk as a suite sparse style matrix.
*
* @param n number of observations
* @param k order of the trendfilter
* @param x locations of the responses
* @return pointer to a csparse matrix
* @see tf_calc_dktil
*/
cs * tf_calc_dk (int n, int k, const double * x)
{
long int i;
int tk = 1; /* "this k" - will iterate until ts = k */
cs * D1;
cs * D1_cp;
cs * Dk;
cs * Dk_cp;
cs * delta_k;
cs * delta_k_cp;
cs * D1_x_delta;
cs * Dk_next;
cs * T;
cs * eye;
/* Deal with k=0 separately */
if(k == 0)
{
T = cs_spalloc (n, n, n, 1, 1) ;
for (i = 0 ; i < n; i++) cs_entry (T, i, i, 1);
eye = cs_compress (T);
cs_spfree (T);
return eye;
}
/* Contruct one 'full D1', which persists throughout
and another copy as Dk */
D1 = cs_spalloc(n-tk, n, (n-tk)*2, 1, 1);
Dk = cs_spalloc(n-tk, n, (n-tk)*2, 1, 1);
D1->nz = (n-tk)*2;
Dk->nz = (n-tk)*2;
for (i = 0; i < (n-tk)*2; i++)
{
D1->p[i] = (i+1) / 2;
Dk->p[i] = D1->p[i];
D1->i[i] = i / 2;
Dk->i[i] = D1->i[i];
D1->x[i] = -1 + 2*(i % 2);
Dk->x[i] = D1->x[i];
}
/* Create a column compressed version of Dk, and
delete the old copy */
Dk_cp = cs_compress(Dk);
cs_spfree(Dk);
for (tk = 1; tk < k; tk++)
{
/* 'reduce' the virtual size of D1 to: (n-tk-1) x (n-tk),
compress into compressed column, saving as D1_cp */
D1->nz = (n-tk-1)*2;
D1->m = n-tk-1;
D1->n = n-tk;
D1_cp = cs_compress(D1);
/* Construct diagonal matrix of differences: */
delta_k = cs_spalloc(n-tk, n-tk, (n-tk), 1, 1);
for(i = 0; i < n - tk; i++)
{
delta_k->p[i] = i;
delta_k->i[i] = i;
delta_k->x[i] = tk / (x[tk + i] - x[i]);
}
delta_k->nz = n-tk;
delta_k_cp = cs_compress(delta_k);
D1_x_delta = cs_multiply(D1_cp, delta_k_cp);
/* Execute the matrix multiplication */
Dk_next = cs_multiply(D1_x_delta, Dk_cp);
/* Free temporary cs matricies created in each loop */
cs_spfree(D1_cp);
cs_spfree(delta_k);
cs_spfree(delta_k_cp);
cs_spfree(D1_x_delta);
cs_spfree(Dk_cp);
Dk_cp = Dk_next;
}
cs_spfree(D1);
return Dk_cp;
}
/**
* @brief Main wrapper for fitting a trendfilter model.
* Takes as input either a sequence of lambda tuning parameters, or the number
* of desired lambda values. In the latter case the function will also calculate
* a lambda sequence. The user must supply allocated memory to store the output,
* with the function itself returning only @c void. For default values, and an
* example of how to call the function, see the function tf_admm_default.
*
* @param y a vector of responses
* @param x a vector of response locations; must be in increasing order
* @param w a vector of sample weights
* @param n the length of y, x, and w
* @param k degree of the trendfilter; i.e., k=1 linear
* @param family family code for the type of fit; family=0 for OLS
* @param max_iter maximum number of ADMM interations; ignored for k=0
* @param lam_flag 0/1 flag for whether lambda sequence needs to be estimated
* @param lambda either a sequence of lambda when lam_flag=0, or empty
* allocated space if lam_flag=1
* @param nlambda number of lambda values; need for both lam_flag=0 and 1
* @param lambda_min_ratio minimum ratio between min and max lambda; ignored for lam_flag=0
* @param beta allocated space of size n*nlambda to store the output coefficents
* @param obj allocated space of size max_iter*nlambda to store the objective
* @param iter allocated space of size nlambda to store the number of iterations
* @param status allocated space of size nlambda to store the status of each run
* @param rho tuning parameter for the ADMM algorithm
* @param obj_tol stopping criteria tolerance
* @param alpha_ls for family != 0, line search tuning parameter
* @param gamma_ls for family != 0, line search tuning parameter
* @param max_iter_ls for family != 0, max number of iterations in line search
* @param max_iter_newton for family != 0, max number of iterations in inner ADMM
* @param verbose 0/1 flag for printing progress
* @return void
* @see tf_admm_default
*/
void tf_admm (double * y, double * x, double * w, int n, int k, int family,
int max_iter, int lam_flag, double * lambda,
int nlambda, double lambda_min_ratio, double * beta,
double * obj, int * iter, int * status, double rho,
double obj_tol, double alpha_ls, double gamma_ls,
int max_iter_ls, int max_iter_newton, int verbose)
{
int i;
int j;
double max_lam;
double min_lam;
double * temp_n;
double * beta_max;
double * alpha;
double * u;
cs * D;
cs * Dt;
cs * Dk;
cs * Dkt;
cs * DktDk;
gqr * Dt_qr;
gqr * Dkt_qr;
beta_max = (double *) malloc(n * sizeof(double));
temp_n = (double *) malloc(n * sizeof(double));
alpha = (double *) malloc(n * sizeof(double)); /* we use extra buffer (n vs n-k) */
u = (double *) malloc(n * sizeof(double)); /* we use extra buffer (n vs n-k) */
/* Assume w does not have zeros */
for(i = 0; i < n; i++) temp_n[i] = 1/sqrt(w[i]);
D = tf_calc_dk(n, k+1, x);
Dk = tf_calc_dktil(n, k, x);
Dt = cs_transpose(D, 1);
diag_times_sparse(Dt, temp_n); /* Dt = W^{-1/2} Dt */
Dkt = cs_transpose(Dk, 1);
Dt_qr = glmgen_qr(Dt);
Dkt_qr = glmgen_qr(Dkt);
DktDk = cs_multiply(Dkt,Dk);
/* Determine the maximum lambda in the path, and initiate the path if needed
* using the input lambda_min_ratio and equally spaced log points.
*/
max_lam = tf_maxlam(n, y, Dt_qr, w);
if (!lam_flag)
{
min_lam = max_lam * lambda_min_ratio;
lambda[0] = max_lam;
for (i = 1; i < nlambda; i++)
lambda[i] = exp((log(max_lam) * (nlambda - i -1) + log(min_lam) * i) / (nlambda-1));
}
rho = rho * pow( (x[n-1] - x[0])/n, (double)k);
/* Initiate alpha and u for a warm start */
if (lambda[0] < max_lam * 1e-5)
{
for (i = 0; i < n - k; i++)
{
alpha[i] = 0;
u[i] = 0;
}
} else {
/* beta_max */
for (i = 0; i < n; i++) temp_n[i] = -sqrt(w[i]) * y[i];
glmgen_qrsol (Dt_qr, temp_n);
for (i = 0; i < n; i++) beta_max[i] = 0;
cs_gaxpy(Dt, temp_n, beta_max);
/* Dt has a W^{-1/2}, so in the next step divide by sqrt(w) instead of w. */
for (i = 0; i < n; i++) beta_max[i] = y[i] - beta_max[i]/sqrt(w[i]);
/* alpha_max */
tf_dxtil(x, n, k, beta_max, alpha);
/* u_max */
switch (family)
{
case FAMILY_GAUSSIAN:
for (i = 0; i < n; i++) u[i] = w[i] * (beta_max[i] - y[i]) / (rho * lambda[0]);
break;
case FAMILY_LOGISTIC:
for (i = 0; i < n; i++) {
u[i] = logi_b2(beta_max[i]) * w[i] * (beta_max[i] - y[i]) / (rho * lambda[0]);
}
break;
case FAMILY_POISSON:
for (i = 0; i < n; i++) {
u[i] = pois_b2(beta_max[i]) * w[i] *(beta_max[i] - y[i]) / (rho * lambda[0]);
}
break;
default:
for (i = 0; i < nlambda; i++) status[i] = 2;
return;
}
glmgen_qrsol (Dkt_qr, u);
}
/* Iterate lower level functions over all lambda values;
* the alpha and u vectors get used each time of subsequent
* warm starts
*/
for (i = 0; i < nlambda; i++)
{
/* warm start */
double * beta_init = (i == 0) ? beta_max : beta + (i-1)*n;
for(j = 0; j < n; j++) beta[i*n + j] = beta_init[j];
switch (family)
{
case FAMILY_GAUSSIAN:
tf_admm_gauss(y, x, w, n, k, max_iter, lambda[i], beta+i*n, alpha,
u, obj+i*max_iter, iter+i, rho * lambda[i], obj_tol,
DktDk, verbose);
break;
case FAMILY_LOGISTIC:
tf_admm_glm(y, x, w, n, k, max_iter, lambda[i], beta+i*n, alpha, u, obj+i*max_iter, iter+i,
rho * lambda[i], obj_tol, alpha_ls, gamma_ls, max_iter_ls, max_iter_newton,
DktDk, &logi_b, &logi_b1, &logi_b2, verbose);
break;
case FAMILY_POISSON:
tf_admm_glm(y, x, w, n, k, max_iter, lambda[i], beta+i*n, alpha, u, obj+i*max_iter, iter+i,
rho * lambda[i], obj_tol, alpha_ls, gamma_ls, max_iter_ls, max_iter_newton,
DktDk, &pois_b, &pois_b1, &pois_b2, verbose);
break;
}
/* If there any NaNs in beta: reset beta, alpha, u */
if(has_nan(beta + i * n, n))
{
for(j = 0; j < n; j++) beta[i*n + j] = 0;
for(j = 0; j < n-k; j++) { alpha[j] = 0; u[j] = 0; }
status[i] = 1;
printf("Numerical error in lambda[%d]=%f",i,lambda[i]);
}
}
cs_spfree(D);
cs_spfree(Dt);
cs_spfree(Dk);
cs_spfree(Dkt);
cs_spfree(DktDk);
glmgen_gqr_free(Dt_qr);
glmgen_gqr_free(Dkt_qr);
free(temp_n);
free(beta_max);
free(alpha);
free(u);
}
/**
* @brief Main wrapper for fitting a trendfilter model.
* Takes as input either a sequence of lambda tuning parameters, or the number
* of desired lambda values. In the latter case the function will also calculate
* a lambda sequence. The user must supply allocated memory to store the output,
* with the function itself returning only @c void. For default values, and an
* example of how to call the function, see the function tf_admm_default.
*
* @param x a vector of data locations; must be in increasing order
* @param y a vector of responses
* @param w a vector of sample weights
* @param n the length of x, y, and w
* @param k polynomial degree of the fitted trend; i.e., k=1 for linear
* @param family family code for the type of fit; family=0 for OLS
* @param max_iter maximum number of ADMM interations; ignored for k=0
* @param beta0 initialization value of beta for first lambda; ignored if NULL
* @param lam_flag 0/1 flag for whether lambda sequence needs to be estimated
* @param lambda either a sequence of lambda when lam_flag=0, or empty
* allocated space if lam_flag=1
* @param nlambda number of lambda values; need for both lam_flag=0 and 1
* @param lambda_min_ratio minimum ratio between min and max lambda; ignored for lam_flag=0
* @param df allocated space of nlambda to store the output df values
* @param beta allocated space of size n*nlambda to store the output coefficents
* @param obj allocated space of size max_iter*nlambda to store the objective
* @param iter allocated space of size nlambda to store the number of iterations
* @param status allocated space of size nlambda to store the status of each run
* @param rho tuning parameter for the ADMM algorithm
* @param obj_tol stopping criteria tolerance
* @param obj_tol_newton for family != 0, stopping criteria tolerance for prox Newton
* @param alpha_ls for family != 0, line search tuning parameter
* @param gamma_ls for family != 0, line search tuning parameter
* @param max_iter_ls for family != 0, max number of iterations in line search
* @param max_iter_newton for family != 0, max number of iterations in inner ADMM
* @param verbose 0/1 flag for printing progress
* @return void
* @see tf_admm_default
*/
void tf_admm ( double * x, double * y, double * w, int n, int k, int family,
int max_iter, double * beta0, int lam_flag, double * lambda,
int nlambda, double lambda_min_ratio, int tridiag, int * df,
double * beta, double * obj, int * iter, int * status,
double rho, double obj_tol, double obj_tol_newton, double alpha_ls, double gamma_ls,
int max_iter_ls, int max_iter_newton, int verbose)
{
int i;
int j;
int numDualVars;
double max_lam;
double min_lam;
double * temp_n;
double * beta_max;
double * alpha;
double * u;
double * A0;
double * A1;
double * v;
cs * D;
cs * Dt;
cs * Dk;
cs * Dkt;
cs * DktDk;
gqr * Dt_qr;
gqr * Dkt_qr;
beta_max = (double *) malloc(n * sizeof(double));
temp_n = (double *) malloc(n * sizeof(double));
v = (double *) malloc(n * sizeof(double));
numDualVars = tridiag ? k : 1;
/* we use extra buffer below (n vs n-k) */
alpha = (double *) malloc(n * numDualVars * sizeof(double));
u = (double *) malloc(n * numDualVars * sizeof(double));
/* Assume w does not have zeros */
for (i = 0; i < n; i++) temp_n[i] = 1/sqrt(w[i]);
D = tf_calc_dk(n, k+1, x);
Dk = tf_calc_dktil(n, k, x);
Dt = cs_transpose(D, 1);
diag_times_sparse(Dt, temp_n); /* Dt = W^{-1/2} Dt */
Dkt = cs_transpose(Dk, 1);
Dt_qr = glmgen_qr(Dt);
Dkt_qr = glmgen_qr(Dkt);
DktDk = cs_multiply(Dkt,Dk);
/* Determine the maximum lambda in the path */
max_lam = tf_maxlam(n, y, Dt_qr, w);
/* and if it is too small, return a trivial solution for Gaussian case */
if (family == FAMILY_GAUSSIAN) {
if (max_lam < 1e-12) {
for (i=0; i<nlambda; i++) {
for (j=0; j<n; j++) beta[i*n+j] = y[j];
obj[i*(max_iter+1)] = 0;
df[i] = n;
}
cs_spfree(D);
cs_spfree(Dt);
cs_spfree(Dk);
cs_spfree(Dkt);
cs_spfree(DktDk);
glmgen_gqr_free(Dt_qr);
glmgen_gqr_free(Dkt_qr);
free(temp_n);
free(beta_max);
free(alpha);
free(u);
return;
}
}
else {
max_lam += 1;
}
/* Initiate the path if needed using the input lambda_min_ratio and
* equally spaced points in log space. */
if (!lam_flag) seq_logspace(max_lam,lambda_min_ratio,nlambda,lambda);
/* Augmented Lagrangian parameter */
rho = rho * pow((x[n-1] - x[0])/(double)(n-1), (double)k);
/* Initiate alpha and u for a warm start */
if (lambda[0] < max_lam * 1e-5)
for (i = 0; i < n - k; i++) alpha[i] = u[i] = 0;
else {
/* beta_max */
if (beta0 == NULL)
calc_beta_max(y,w,n,Dt_qr,Dt,temp_n,beta_max);
else
memcpy(beta_max, beta0, n*sizeof(double));
/* Check if beta = weighted mean(y) is better than beta */
double yc = weighted_mean(y,w,n);
for (i = 0; i < n; i++) temp_n[i] = yc;
double obj1 = tf_obj(x,y,w,n,k,max_lam,family,beta_max,v);
double obj2 = tf_obj(x,y,w,n,k,max_lam,family,temp_n,v);
if(obj2 < obj1) memcpy(beta_max, temp_n, n*sizeof(double));
/* alpha_max */
if (tridiag && k>0)
{
tf_dx1(x, n, 1, beta_max, alpha + (n*k-n));
for (j=k-1; j >= 1; j--)
tf_dx1(x, n, k-j+1, alpha + (n*j), alpha + (n*j-n));
}
else if (k>0)
tf_dxtil(x, n, k, beta_max, alpha);
/* u_max */
if (tridiag)
for (j=0; j<k; j++) memset(u + (n*j), 0, (n-k+j) * sizeof(double));
else {
for (i = 0; i < n; i++)
u[i] = w[i] * (beta_max[i] - y[i]) / (rho * lambda[0]);
if(family == FAMILY_LOGISTIC)
for (i = 0; i < n; i++) u[i] *= logi_b2(beta_max[i]);
else if(family == FAMILY_POISSON)
for (i = 0; i < n; i++) u[i] *= pois_b2(beta_max[i]);
glmgen_qrsol (Dkt_qr, u);
// for (i = 0; i < n-k; i++) u[i] = 0;
}
}
if (tridiag && k>0)
{
/* Setup tridiagonal systems */
A0 = (double*) malloc(n*k*sizeof(double));
A1 = (double*) malloc(n*k*sizeof(double));
for (j=2; j <= k; j++)
{
form_tridiag(x, n, k-j+2, 1, 1, A0+(n*j-n), A1+(n*j-n));
}
}
/* Iterate lower level functions over all lambda values;
* the alpha and u vectors get used each time of subsequent
* warm starts */
for (i = 0; i < nlambda; i++)
{
/* warm start */
double *beta_init = (i == 0) ? beta_max : beta + (i-1)*n;
for(j = 0; j < n; j++) beta[i*n + j] = beta_init[j];
if (tridiag)
{
form_tridiag(x, n, 1, rho * lambda[i], 0, A0, A1);
for (j=0; j < n; j++) A0[j] = A0[j] + w[j];
}
switch (family) {
case FAMILY_GAUSSIAN:
if (tridiag)
tf_admm_gauss_tri(x, y, w, n, k, max_iter, lambda[i], df+i, beta+i*n,
alpha, u, obj+i*(1+max_iter), iter+i, rho * lambda[i],
obj_tol, A0, A1, verbose);
else
tf_admm_gauss(x, y, w, n, k, max_iter, lambda[i], df+i, beta+i*n,
alpha, u, obj+i*(1+max_iter), iter+i, rho * lambda[i],
obj_tol, DktDk, verbose);
break;
case FAMILY_LOGISTIC:
tf_admm_glm(x, y, w, n, k, max_iter, lambda[i], tridiag, df+i, beta+i*n,
alpha, u, obj+i*(1+max_iter_newton), iter+i, rho * lambda[i], obj_tol,
obj_tol_newton, alpha_ls, gamma_ls, max_iter_ls, max_iter_newton,
DktDk, A0, A1, &logi_b, &logi_b1, &logi_b2, verbose);
break;
case FAMILY_POISSON:
tf_admm_glm(x, y, w, n, k, max_iter, lambda[i], tridiag, df+i, beta+i*n,
alpha, u, obj+i*(1+max_iter_newton), iter+i, rho * lambda[i], obj_tol,
obj_tol_newton, alpha_ls, gamma_ls, max_iter_ls, max_iter_newton,
DktDk, A0, A1, &pois_b, &pois_b1, &pois_b2, verbose);
break;
default:
printf("Unknown family, stopping calculation.\n");
status[i] = 2;
}
/* If there any NaNs in beta: reset beta, alpha, u */
if (has_nan(beta + i*n, n))
{
double yc = weighted_mean(y,w,n);
switch(family) {
case FAMILY_POISSON:
yc = (yc > 0) ? log(yc) : -DBL_MAX;
break;
case FAMILY_LOGISTIC:
yc = (yc > 0) ? ( yc < 1 ? log(yc/(1-yc)) : DBL_MAX) : -DBL_MAX;
break;
default: break;
}
for (j = 0; j < n; j++) beta[i*n + j] = yc;
for (j = 0; j < n-k; j++) alpha[j] = 0;
for (j = 0; j < n; j++) u[j] = w[j] * (beta[i*n+j] - y[j]) / (rho * lambda[i]);
glmgen_qrsol (Dkt_qr, u);
if (tridiag) for (j = 0; j < n*k; j++) alpha[j] = u[j] = 0;
status[i] = 1;
}
}
cs_spfree(D);
cs_spfree(Dt);
cs_spfree(Dk);
cs_spfree(Dkt);
cs_spfree(DktDk);
glmgen_gqr_free(Dt_qr);
glmgen_gqr_free(Dkt_qr);
free(beta_max);
free(temp_n);
free(alpha);
free(u);
free(v);
if (tridiag && k>0)
{
free(A0);
free(A1);
}
}
int main ( void )
{
cs *A;
cs *AT;
cs *C;
cs *D;
cs *Eye;
int i;
int m;
cs *T;
printf ( "\n" );
printf ( "CS_DEMO1:\n" );
printf ( " Demonstration of the CSPARSE package.\n" );
/*
Load the triplet matrix T from standard input.
*/
T = cs_load ( stdin );
/*
Print T.
*/
printf ( "T:\n" );
cs_print ( T, 0 );
/*
A = compressed-column form of T.
*/
A = cs_triplet ( T );
printf ( "A:\n" );
cs_print ( A, 0 );
/*
Clear T.
*/
cs_spfree ( T );
/*
AT = A'.
*/
AT = cs_transpose ( A, 1 );
printf ( "AT:\n" );
cs_print ( AT, 0 );
/*
M = number of rows of A.
*/
m = A->m;
/*
Create triplet identity matrix.
*/
T = cs_spalloc ( m, m, m, 1, 1 );
for ( i = 0; i < m; i++ )
{
cs_entry ( T, i, i, 1 );
}
/*
Eye = speye ( m )
*/
Eye = cs_triplet ( T );
cs_spfree ( T );
/*
Compute C = A * A'.
*/
C = cs_multiply ( A, AT );
/*
Compute D = C + Eye * norm (C,1).
*/
D = cs_add ( C, Eye, 1, cs_norm ( C ) );
printf ( "D:\n" );
cs_print ( D, 0 );
/*
Clear A, AT, C, D, Eye,
*/
cs_spfree ( A );
cs_spfree ( AT );
cs_spfree ( C );
cs_spfree ( D );
cs_spfree ( Eye );
/*
Terminate.
*/
printf ( "\n" );
printf ( "CS_DEMO1:\n" );
printf ( " Normal end of execution.\n" );
return ( 0 );
}