prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: Who was Ralph Rokebye's brother?

Read the following text and answer briefly.

Context 1: Rokebye, Ralph of Yorks, arm  Gloucester Hall, matric  9 Nov , 1582, aged 15; student of Lincoln's Inn 1585  See Foster's Inns of Court Register 
Rokebye, Ralph of Herts (? Yorks), gent  Broadgates Hall, matric  entry 28 Feb , 1589-90, aged 14  See pedigree in Foster's Yorkshire Collection 
Rokeby, William (brother of Sir Richard, treasurer of Ireland, son of John, of Thundercliffe Grange, Yorks), fellow of King's Hall, Cambridge, D Can L ; rector of Sandal 1487, and of Halifax, Yorks, 1502, rector of Fakenham, Norfolk, 1496, chancellor of Ireland 1498, and 1515, bishop of Meath, and privy councillor 1507, archbishop of Dublin 1512, archdeacon of Surrey 1520, until his death 29 Nov , 1521  See Ath  ii  717; Cotton, i  25; & Lansdowne MS  979, ff  4, 6 
Rolfe, Augustine (Rolfus) M A  from Queen's Coll , Cambridge, 1595; incorporated 10 July, 1599 
Rolf, Richard B A  from Emanuel Coll , Cambridge, 1584-5 (incorporated 11 July, 1585); M A  1588  See Foster's Graduati Cantab 
Rolfe, William cler  fil  New Coll , matric  10 March, 1656-7, B A  1660, fellow, M A  14 Jan , 1663-4; rector of Brampton 1668, and of Stoke Bruern, Northants, 1676, until his death, buried (at Stoke Bruern) 6 Sept , 1693  See Baker's Northants, i  86 
Rolfe, William s  William, of Stoke-Bruern, Northants, cler  Brasenose Coll  7 July, 1688, aged 16; student of Inner Temple 1692, buried in the Temple church 1 March, 1692-3  See Foster's Inns of Court Reg 
Rolle, Denis youngest son John, of Steventon, Devon, equitis  Exeter Coll , matric  15 Feb , 1666-7, aged 17; brother of John same date 
Rolle, Denis s  D , of Heanton, Devon, arm  Exeter Coll  matric  24 Oct , 1687, aged 17; B A  1691, M A  1694 (as Denys), rector of Merton, Devon, 1696  See Samuel 1687, & Foster's Index Ecclesiasticus 
Rolle, (Sir) Henry of Devon, arm  fil  Broadgates Hall, matric  14 June, 1594, aged 18; student of Middle Temple 1597 (as son and heir of Henry, of Steventon, Devon, esq ), knighted 23 July, 1603, died in 1617  See Foster's Inns of Court Reg 
Rolle, Henry of Devon, arm  Exeter Coll , matric  20 March, 1606-7, aged 17; bar -at-law, Inner Temple, 1618, bencher 1633 (2s  Robert, of Heanton, Devon), M P  Callington 1621-2, 1624-5, Truro 1625-1626, 1628-9, serjeant-at-law 1640, recorder of Dorchester 1636, a judge of king's bench 1645, chief justice of upper bench 1648-55, died 30 July, 1656, buried in Shapwick church, Somerset; brother of Samuel 1605  See Ath  iii  416; & Foster's Judges and Barristers 
Rolle, Henry s  Alex , of Tavistock, Devon, gent  Christ Church, matric  23 March, 1696-7, aged 17 
Rolle, John of Devon, arm  Exeter Coll , matric  30 May, 1589, aged 15; B A  8 Feb , 1592-3, M A  25 May, 1596 
Rolle, John 1s  John, of Steventon, Devon, equitis  Exeter Coll , matric  15 Feb , 1666-7, aged 18; of Bicton, Devon; died in his father's lifetime, buried at Bicton 22 April, 1689; brother of Denis 1667, and father of Denis 1698 
Rolle, Richard s  Richard, of Cookeburye, Devon, gent  New Inn Hall, matric  26 Sept , 1634, aged 18; B A  from Jesus Coll , Cambridge, 1638, incorporated from Gloucester Hall 17 Dec , 1639, M A  2 July, 1642, rector of Sheviocke, Cornwall, 1656; father of the next-named  See Foster's Index Eccl 
Rolle, Richard s  R , of Sheviock, Cornwall, cler  St  Alban Hall, matric  3 July, 1674, aged 17; B A  1678 
Rolle, Robert (Rooles or Roales) fellow New Coll  1551-60 from Mark Lane, city of London, B A  26 June, 1555, M A  26 July, 1560, B D  22 Jan , 1572-3, D D  June, 1585, a teacher in Westminster school; perhaps canon of Combe (4) in Wells, 1574, and rector of Stoke Climsland, Devon, 1574  See O H S  i  345; & Foster's Index Eccl 
Rolle, Samuel s  Denis, of Great Torrington, Devon, arm  Exeter Coll , matric  16 July, 1687, aged 18, B A  1691; bar -at-law Middle Temple 1697; M P  Barnstaple 1705, died 1747; see Denis 1687  See Foster's Judges and Barristers 
Rolle, William B C L  14 July, 1528; perhaps vicar of Yarncombe, Devon, 1536  See Foster's Index Ecclesiasticus 
Rolles, Gabriel (Rooles) B A  from St  John's Coll , Cambridge, 1610-11, M A  1614; incorporated 13 July, 1619, rector of East Locking, Berks, 1620, as Rolle  See Foster's Graduati Cantab 
Rolles, Richard gent  Jesus Coll , matric  1 March, 1632-3, B A  next day, M A  15 Oct , 1635; perhaps created B D  20 Dec  1642, "ex regis gratia," rector of Wavendon, Bucks, and of Witham, Essex, 1646, by the Westminster assembly  See Add  MS  15,670, p  70 
Rolles, William s  Richard, of Lewknor, Oxon, gent  St  John's Coll , matrie  12 March, 1637-8, aged 17, B A  9 Nov , 1641, M A  6 July, 1644; B D  from Jesus Coll  12 Sept , 1661, rector of Wheatfield, Oxon, 1660, and of Chalfont St  Giles, Bucks, 1662  See Foster's Index Eccl 
Rolles, William created M A  from Exeter Coll  14 April, 1648 
Rolleston, Simon created M A  31 Aug , 1636 
Rolleston, Thomas of Devon, gent  Wadham Coll , matric  12 May, 1620, aged 16 
Rollinson, Francis 1584  See Rallinson 
Rollinson, William s  "Jose," of London, gent  St  John's Coll , matric  7 March, 1694-5, aged 15; perhaps brother of John Rawlinson, of New Coll  1692  See page 1236 
Rolt, Edward youngest son of Tho , of London, equitis  Merton Coll , matric  7 Nov , 1701, aged 15; of Sacomb, Herts, and Chippenham, Wilts, student of Lincoln's Inn, 1702, M P  St  Mawes 1713, Grantham 1715-22, Chippenham 1722; died 22 Dec , 1722; his father knighted 1 Oct , 1682, and died 9 Sept , 1710  See Foster's Parliamentary Dictionary 
Rolte, George s  Thomas, of St  Margarets par  Darenth, Kent, pleb  St  Alban Hall, matric  17 June, 1631, aged 18; B A  20 June, 1631, M A  29 April, 1634, incorporated at Cambridge 1639 
Romane, Edmund pleb  Balliol Coll , matric  20 Feb , 1627-8, aged 18; B A  next day, M A  3 June, 1630 
Romaine, Matthew pleb  Balliol Coll , matric  10 June, 1630, B A  same day, M A  14 May, 1633, vicar of Stoke Gaylard, Dorset, 1639; father of the next  See Foster's Index Eccl 
Romayne, Thomas s  Matth , of Stoke Gaylard, Dorset, minister  Wadham Coll , matric  17 July, 1669, aged 17; B A  from Hart Hall 1673, "the intruded" rector of Stoke Gaylard 1675  See Foster's Index Eccl 
Romayne, William (Ronayne) gent  Trinity Coll , matric  31 July, 1671, aged 16 
Rome, Harcourt s  William, of London, p p  Brasenose Coll , matric  13 Dec , 1672, aged 17 
Rome, William s  G  (? "Gul "), of Northampton (city), pleb  Brasenose Coll , matric  11 Dec , 1684, aged 16 
Romney, Joseph B A  from Emanuel Coll , Cambridge, 1610-11, M A  1614; incorporated 8 July, 1614, student of Inner Temple 1610, as of London, gent  See Foster's Inns of Court Reg 
Rone, John s  Randolph, of Hanmer, Flints, pleb  Brasenose Coll , matric  10 Oct , 1634, aged 18; D D  Trinity Coll , Dublin, 25 Jan , 1666 (as Roane), vicar of Hanmer, Flints, 1644, ejected same year, dean of Clogher 1667, bishop of Killaloe 1675, until his death 5 Sept , 1692  See Cotton's Fasti Ecc  Hib  i  467 
Rone, William of New Coll  1661  See Roane 
Roode, Edward (or Rode) B A  21 July, 1522, M A  26 Nov , 1534; perhaps canon of Southwell 1561-73 
Roode, Edward cler  fil  Merton Coll , matric  22 Nov , 1650; Eton postmaster 1649, fellow 1651, B A  2 March, 1651-2, M A  14 Dec , 1655; incorporated at Cambridge 1657, and LL D  1671; vicar of Gamlingay, co  Cambridge, rector of one moiety 1661, and of the other 1677; died at Cambridge 1689  See Burrows, 525; & O H S  iv  292 
Roode, Onesiphorus s  Edward, of Thame, Oxon, sacerd  New Inn Hall, matric  27 Oct , 1637, aged 16, B A  1 July, 1641; incorporated at Cambridge 1645; chaplain to the house of lords after the expulsion of the bishops; minister of New chapel, Tuttle-Fields, Westminster, 1648, until ejected in 1660  See Calamy, i  195 
Rood, Richard M A  from Pembroke Coll  5 Dec , 1634 
Rooke, John s  Tho , of Broadwell, co  Gloucester, pleb  Pembroke Coll , matric  1 March, 1683-4, aged 17; brother of Thomas 1693 
Rooke, John s  Tho , of Whitchurch, Wilts, gent  Balliol Coll , matric  14 Jan , 1713-14, aged 17 
Rooke, Nicholas s  Arthur, of Totnes, Devon, gent  Exeter Coll , matric  10 March, 1670-1, aged 16; B A  1674, M A  1677, rector of Dartington, Devon, 1679  See Foster's Index Eccl 
Rooke, Robert "ser " Oriel Coll , matric  1 April, 1656, B A  1659 
Rooke, Robert s  R , p p  St  Alban Hall, matric  30 March, 1677, aged 17 
Rooke, Thomas pleb  Christ Church, matric  3 May, 1659 
Rooke, William (Roock) of Dorset, pleb  Brasenose Coll , matric  entry under date 20 March, 1578-9, aged 19; B A  from St  Alban Hall 30 Jan , 1582-3, M A  9 May, 1586 
Rooke, William of Dorset, gent  New Coll , matric  12 July, 1605, aged 18; B A  21 Feb , 1608-9, chaplain, M A  16 Dec , 1611, rector of North Cheriton, Somerset, 1618  See Foster's Index Eccl 
Rooke, William s  J , of Workington, Cumberland, p p  Queen's Coll , matric  22 Oct , 1669, aged 17; B A  1674, M A  1677, B D  1690, vicar of Plumstead, Kent, 1691, and rector of Hadley, Hants, 1695  See Foster's Index Eccl 
Rookes, Christopher (Rokys or Rokkis) B A  8 July, 1522, M A  1 July, 1527, B D  supd  Oct , 1540; principal of Magdalen Hall 1529-32, vicar of Stanstead Abbots, Herts, 1534  See Foster's Index Eccl 
Rookes, Jonas B A  from Magdalen Hall 24 April, 1599, M A  11 Feb , 1601-2 (2s  William, of Roydes Hall); vicar of Penistone, Yorks, 1619, see Foster's Index Eccl ; styled fellow and bursar of University Coll  in Foster's Yorkshire Collection, possibly brother of the next-named 
Rookes, Robert of Yorks, pleb  Magdalen Hall, matric  14 May, 1602, aged 19; possibly brother of the last-named 
Ro(o)kes, William demy Magdalen Coll  1544, B A  supd  1551, fellow 1552-71, M A  27 April, 1556, B Med  supd  24 April, 1561  See Bloxam, iv  99 
Rookes, William s  William, of Rhodes Hall, Yorks, gent  University Coll , matric  30 June, 1665, aged 16; died at Oxford in 1667 
Roope, Ambrose s  A , of Dartmouth Parva, Devon, arm  Exeter Coll , matric  15 March, 1671-2, aged 16 
Roope, George s  Ant , of Bradford, Wilts, gent  Hart Hall, matric  10 Oct , 1702, aged 15 
Roope, John s  Nicholas, of Dartmouth, Devon, gent  Exeter Coll , matric  17 Nov , 1637, aged 15; student of Lincoln's Inn 1638  See Foster's Inns of Court Reg 
Roope, Nicholas of Devon, gent  Broadgates Hall, matric  6 Feb , 1606-7, aged 18; B A  6 Nov , 1610; probably father of the last-named 
Rooper, Thomas s  T , of London, gent  Trinity Coll , matric  9 July, 1699, aged 16; B A  1703, M A  19 Feb , 1705-6, as Roper 
Rooper, William of St  Alban Hall 1667  See Roper 
Roos, Brian D Can L  or doctor of decrees of the university of Valentia; incorporated 3 Feb , 1510-11; died 1529, buried in the church of Chelray  See Fasti, i  31 
Root, Isaac pleb  St  John's Coll , matric  2 July, 1658, admitted to Merchant Taylors' school 1649 (only son of Isaac, merchant taylor); born in Trinity parish 20 Aug , 1641  See Robinson, i  193 
Roots, Richard s  Tho , of Tunbridge, Kent, gent  St  John's Coll , matric  26 Dec , 1689, aged 15; demy Magdalen Coll  1690-1702, B A  1693, M A  1696, rector of Chilmarck, Wilts, 1702-27, canon of Sarum 1722, rector and vicar of Bishopstone, Wilts, 1728; brother of William 1699  See Rawl  iii  447, and xix  90; Bloxam, vi  111; & Foster's Index Eccl 
Roots, Thomas of Sussex, pleb  Magdalen Hall, matric  entry 17 Nov , 1581, aged 13; B A  supd  1 July, 1584, bar -at-law, Lincoln's Inn, 1594  See Foster's Judges and Barristers 
Rootes, Thomas s  William, of Tunbridge, Kent, pleb  St  John's Coll , matric  31 Jan , 1628-9, aged 23; B A  12 Feb , 1628-9, vicar of Long Stanton All Saints, co  Cambridge, 1630  See Add  MSS  15,669-70; & Foster's Index Eccl 
Rootes, Thomas pleb  St  John's Coll , matric  2 July, 1658; B A  1661, M A  1666; possibly father of Richard 1689, and William 1699 
Roots, William s  Tho , of Tunbridge, Kent, gent  Christ Church, matric  16 March, 1698-9, aged 18; B A  1704; clerk Magdalen Coll  1705-11, M A  1707, rector of Little Berkhampstead, Herts, 1714; brother of Richard 1689  See Bloxam, ii  85; & Foster's Index Eccl 
Roper, Francis s  Robert, of Trimdon, co  Durham, gent  Corpus Christi Coll , matric  16 Dec , 1661, aged 18; probably identical with Francis, son of Robert, of Kelloe, co  Durham, farmer, was admitted sizar of St  John's Coll , Cambridge, 21 Sept , 1658, aged 16; fellow, B A  1662-3, M A  1666, B D  1673, vicar of Waterbeach, co  Cambridge, 1678, canon of Ely 1686-90, rector of Northwold, Norfolk, 1687, died 13 April, 1719  See Mayor, 138; Surtees' Durham, i  107; & Foster's Index Eccl 
Roper, John (or Rooper) demy Magdalen Coll , from Berks, M A  fellow, 1483, D D  disp  27 June, 1506, (first) Margaret professor of divinity, 1500, vice-chancellor of the university 1505, and 1511, principal of Salesurry and George Hall, rector of Witney, Oxon, 1493, vicar of St  Mary's church, Oxford, canon of Cardinal Coll  1532; died May, 1534  See Ath  i  76; & Landsowne MS  979, f  118 
Roper, John B A  disp  4 July, 1512 
Roper, Thomas of Trinity Coll  1699  See Rooper 
Roper, Philip of Kent, arm  Gloucester Hall, matric  7 Sept , 1588, aged 15 (subscribes Rooper) 
Roper, William (subscribes Rooper) of co  Hereford, militis fil  St  Alban Hall, matric  entry dated 5 June, 1607, aged 13; probably of Malmains, Kent, 2nd son of Sir Christopher Roper, afterwards 2nd baron Teynham  See Foster's Peerage 
Roscarrock, Henry of Cornwall, arm  Hart Hall, matric  entry under date 17 Dec , 1576, aged 21; probably son of Thomas, of Roscarrock, and brother of the next, and of Richard 1581 
Roscarrock, John B A  11 Feb , 1576-7; perhaps from Exeter Coll  (and 1s  Thomas, of Roscarrock, Cornwall); died 24 Nov , 1608; brother of Henry and Richard  See O H S  xii  65 
Roscarrock, Nicolas (Roiscariot) B A  supd  3 May, 1568, student Inner Temple 1571, as of Roscarrock, Cornwall  See Foster's Inns of Court Reg 
Roscarrock, Richard of Cornwall, arm  Broadgates Hall, matric  entry under date circa 1581, aged 19; student of Middle Temple 1583 (as 3s  Thomas, of Roscarrock, Cornwall, esq ), brother of Henry and John  See Foster's Inns of Court Reg 
Rosdell, Christopher of Yorks, pleb  St  Edmund Hall, matric  entry under date 22 Dec , 1576, aged 22, B A  4 July, 1576; rector of St  Bennet Sherehog, London, 1579, and vicar of Somerton, Somerset, 1582  See Foster's Index Eccl 
Rose, Christopher s  John, of Marlow, Bucks, gent  Christ Church, matric  13 Feb , 1622-3, aged 21, B A  same day; rector of Hutton, Essex, 1642  See Foster's Index Ecclesiasticus 
Rose, Christopher s  Giles, of Lynn Regis, Norfolk, gent  Lincoln Coll , matric  8 July, 1670, aged 15; student of Gray's Inn, 1673  See Foster's Gray's Inn Register 
Rose, Gilbert Augustinian Canon, B D  supd  22 May, 1512, and supd  12 Dec , 1519, for incorporation as D D 
Rose, Henry "ser " Lincoln Coll , matric  22 July, 1658, B A  16 Jan , 1660-1, fellow 1662 from Pirton, Oxon, M A  1663 (incorporated at Cambridge 1688), B D  1672; minister of All Saints, Oxford, but running much into debt, and marrying beneath himself, left his fellowship and church about 1674, retired to London, and at length to Ireland  See Ath  iv  561 
Rose, Hugh s  "Dav  Ni " (Nigg 4to ), of Ross, Scotland, p p  (subs  pleb )  Balliol Coll , matric  3 April, 1707, aged 20; B A  1709 
Rose, John B A  8 June, 1519, fellow Merton Coll  1523, M A  31 March, 1525; one of these names vicar of Shoreham, Kent, 1536  See Foster's Index Ecclesiasticus 
Rose, John of co  Leicester, pleb  Merton Coll , matric  24 Nov , 1581, aged 21 
Rose, John s  Jeremy, of Swell, co  Gloucester, pleb  Corpus Christi Coll , matric  12 Dec , 1623, aged 15; B A  4 July, 1626 
Rose, John s  Rich , of Halberton, Devon, gent  Exeter Coll , matric  14 May, 1688, aged 17 
Rose, John s  J , of West Derby, co  Lancaster, pleb  University Coll , matric  7 March, 1712-13, aged 18, B A  1716; rector of Bilborough, Notts, 1722  See Foster's Index Eccl 
Rose, Jonathan s  Th , of Mickleton, co  Gloucester, pleb  St  Alban Hall, matric  16 May, 1677, aged 18; B A  9 Feb , 1680-1 
Rose, Joseph s  Thomas, of Sturminster Newton, Dorset, pleb  Oriel Coll , matric  12 Dec , 1623, aged 19 
Rose, Richard B A  from Exeter Coll  14 June, 1621; perhaps student of Middle Temple 1622 (as son and heir of John, of Lyme, Dorset, gent ), and M P  Lyme Regis April-May, 1640, 1640 (l p ), till his death after 1648  See Foster's Inns of Court Reg  & Foster's Parliamentary Dictionary 
Rose, Richard arm  Exeter Coll , matric  29 March, 1656; student of Lincoln's Inn 1659, as 4s  Richard, of Wootton Fitzwarren, Dorset, esq  See Foster's Inns of Court Reg 
Rose, Richard s  Richard, of Monks Kirby, co  Warwick, pleb  Magdalen Coll , matric  3 May, 1672, aged 16 (as Rosse); chorister 1670-6  See Bloxam, i  95 
Rose, Richard s  R(ichard), of Wyng, Bucks, gent  Trinity Coll , matric  7 May, 1680, aged 16; bar -at-law, Inner Temple, 1699  See Foster's Judges and Barristers 
Rose, Stephen of co  Gloucester, pleb  Corpus Christi Coll , matric  21 Jan , 1619-20, aged 16; B A  13 Nov , 1621, M A  2 July, 1625, vicar of Aldermaston 1627, and rector of Barkham 1633, and of Arborfield, Berks, 1640, and perhaps of Hartley Mawditt, Hants, 1652  See Foster's Index Ecclesiasticus 
Rose, Stephen "ser " Lincoln Coll , matric  19 Nov , 1650 
Rose, Stephen "servi  fil " Magdalen Coll , matric  19 Nov , 1650 (subscribes "serv ") 
Rose, Stephen "ser " Magdalen Coll , subscribed 23 Nov , 1655; B A  from Wadham Coll  1659, vicar of Cold Overton, co  Leicester, 1662-3, and rector of Woolhampton, Berks, 1667-95, father of Temple  See Foster's Index Eccl 
Rose, Temple s  Step , of Woolhampton, Berks, cler  Trinity Coll , matric  29 March, 1693, aged 17, B A  1696 
Rose, Thomas Minorite, B D  22 June, 1509 
Rose, Thomas of Herts, pleb  Magdalen Hall, matric  10 Oct , 1589, aged 15 
Rose, Thomas s  Seth, of Telscombe, Sussex, sacerd  Oriel Coll , matric  5 June, 1640, aged 18; his father rector of Telscombe 1604, etc  See Foster's Index Eccl 
Rose, Thomas s  Edw , 
Context 2: UT College of Liberal Arts: College of Liberal Arts University of Texas at Austin Departments Graduate Resources Undergraduate Resources Courses Online Courses Dean's Office Alumni & Giving Faculty by Department Search the College of Liberal Arts
next profile Jeffrey Tulis Associate Professor — Ph D ,
E-mail: tulis@austin utexas edu
Office: MEZ 3 152
Political Theory and American Politics
Professor Tulis's interests bridge the fields of political theory and American politics, including more specifically, American political development, constitutional theory, political philosophy and the American presidency  His publications include The Presidency in the Constitutional Order (LSU, 1981; Transaction, 2010), The Rhetorical Presidency (Princeton, 1987), The Constitutional Presidency (Johns Hopkins 2009), The Limits of Constitutional Democracy (Princeton, 2010) and recent journal articles and chapters on constitutional interpretation, the logic of political change, and the meaning of political success  Four collections of essays on The Rhetorical Presidency with responses by Tulis have been published, including a special double issue of Critical Review: An Interdisciplinary Journal of Politics and Society, (2007), where his book is described as "one of the two or three most important and perceptive works written by a political scientist in the twentieth century "
He has served as President of the Politics and History Section of the American Political Science Association  He received the President's Associates Teaching Excellence Award at the University of Texas  He has held research fellowships from NEH, ACLS, Olin Foundation, Harvard Law School, and the Mellon Preceptorship at Princeton University, where he taught before moving to Texas  He has held visiting positions at Notre Dame and Harvard  He has served as associate chair of the Department of Government from 1989-2001 and was acting chair during 1992-93  and for part of each year between 1989 and 2001  During the academic year 2008-09, he was a Laurance S  Rockefeller Visiting Fellow at the University Center for Human Values at Princeton  During Spring 2016, he was a Dahrendorf Visiting Fellow at the London School of Economics and Political Science 
His forthcoming books include: Legacies of Losing in American Politics, with Nicole Mellow (University of Chicago Press, Fall 2017), and an expanded edition of The Rhetorical Presidency in the Princeton Classics series (Princeton, Fall 2017)  For two decades he served as co-editor of the Johns Hopkins Series in Constitutional Thought, and he currently co-edits (with Sanford Levinson) Constitutional Thinking, a Series at the University Press of Kansas 
GOV 370L • Pres In Constitutional Ord 38840 • Spring 2017 Meets MW 2:30PM-4:00PM CAL 221 show description
GOV 370 Seminar: The Presidency in the Constitutional Order
Spring 2017 Unique # 38840
MW 2:30 to 4pm GDC 2 402
Jeffrey K  Tulis
In this Seminar we will discuss a series of constitutional problems including: the problem of executive energy in the American Constitution; presidential selection and the problem of political legitimacy; separation of powers; delegation of powers, the constitutional status of war and foreign affairs, administration and bureaucracy and the meaning of leadership in the constitutional order 
Seminar will meet twice a week and regular attendance and thorough preparation for discussion is expected  Unexcused absence from more than three classes will result in failure of the participation component of the course  There will also be pop quizzes on the reading that will count as part of your participation grade  In addition to class participation, course requirements include four short analytic essays, and one in-class test  The course grade will be calculated as follows:
Seminar participation: 20%
In-class test: 20%
Three analytic essays 60% (20% each)
Class participation is especially important  Preparation for seminar and for your in-class test will be enhanced by careful note taking on the readings  If students appear to be unprepared, pop quizzes will be given and the grades on them will affect the participation component of your course grade 
Texts: (tentative)
Joseph M  Bessette and Jeffrey K  Tulis, The Constitutional Presidency
Michael Nelson, The Presidency in the Political System (tenth edition)
Richard Ellis and Michael Nelson, Debating the Presidency (third edition)
The Federalist (any edition, or online) GOV 310L • American Government-Honors 38335 • Fall 2016 Meets TTH 3:30PM-5:00PM BEN 1 106 show description
GOV 310 (Honors) (38335) Fall 2016
TTH 3:30-5:00pm, BEN 1 106
This honors seminar offers an introduction to American politics that emphasizes the confluence of ideas, mores, institutions, and interests, in the constitutional system  This course covers more theory, and the readings are more demanding, than other versions of GOV 310  One of the main objectives of the course is to deepen your understanding of the practical aspects of contemporary public affairs by developing your ability to understand the theoretical foundations of American politics  Although we cover the nuts and bolts of politics there is much more theory in this version of GOV 310  If you have registered for this section mainly because 310 is a legislative requirement that you need to fulfill, this is not the right version for you  There is a substantial workload in this class 
Regular attendance, thorough and timely preparation, and active participation are all necessary to do well 
Four essays (approximately 1000 words each)  Three of these will be assigned analytic essay topics  The last will be a book review of a title chosen by the student from a long list of provided possibilities  (15% each essay, 60% of total course grade)
Two in-class tests  These will count 15% each, 30% of total course grade 
Class participation  (10% of course grade)  Both informed participation and occasional leadership of the seminar will be graded 
No make-up exams or late papers, except for documented medical or other emergencies 
Mark Landy and Sidney M  Milkis, American Government: Enduring Principles, Critical Choices, Third Edition
Mary Nichols and David Nichols, Readings in American Government, Ninth Edition
Thomas Mann and Norman Ornstein, Its Even Worse Than It Looks: How the American Constitutional System Collided With the New Politics of Extremism
Bruce Ackerman,Before the Next Attack: Preserving Civil Liberties in an Age of Terrorism GOV 381L • Constitutional Conflict 38660 • Fall 2016 Meets W 3:30PM-6:30PM BAT 5 102 show description
GOV 381L Fall 2016
Constitutional Conflict
W 3:30-6:30pm, BAT 5 102
Many of the most important debates regarding the nature and character of contemporary American politics are essentially arguments regarding the structure of separation of powers  In this seminar we will consider such questions as whether the American system is prone to deadlock of stalemate in the construction of national policy; whether conflict is a hindrance to institutional responsibility or an essential attribute of responsibility; whether there are “political questions” especially suitable to resolution between President and Congress; how one can distinguish salutary from pathological conflict, and whether it is truly possible to harness the ambition of office holders to the duties of their office 
More specifically, we will review literature and arguments regarding constitutional reform; divided government; separation of powers theory; and case studies of Supreme Court appointments; the budget process; and war powers and foreign affairs  In these contexts we will also discuss current controversies surrounding war authorization, intelligence and secrecy, sequestration, government shut downs and budget resolutions, and debt ceiling politics 
The course is designed to accommodate two different student needs: it will provide a good overview of important literature relevant to the comprehensive examination in American politics and it will provide opportunities for research  This subject area is a treasure trove of “hot” topics, publication possibilities, subjects for MA theses and Ph D  dissertations  I will tailor the written requirements to the objectives of individual students 
1  All students will prepare a short analytic essay early in the semester, and an annotated bibliography at mid-semester  These assignments will count (30%) of the grade 
2  Students interested primarily in exam preparation will complete an examination near the end of the semester based on study questions assigned in advance  OR
Students interested in research will write a 20-25 page paper  (60%)
3  A basic requirement of the course is that students prepare for each seminar by carefully reading the material assigned for that week  Class discussion is an essential component of the course  (10%)
Tentative Texts:
Jones, Separate But Equal Branches
Silverstein, Imbalance of Powers
Wilson & Schram, Separation of Powers and Good Government
Burgess, Contest for Constitutional Authority
Farrier, Passing the Buck: Congress, the Budget and Deficits
Weissman, A Culture of Deference
Zeisberg, War Powers: The Politics of Constitutional Authority
Fisher, Congressional Abdication on War and Spending
Lowi, The End of Liberalism GOV 379S • Regime Persp Amer Poltc-Honors 38105 • Spring 2016 Meets TH 3:30PM-6:30PM GAR 1 134 (also listed as CTI 335, LAH 350) show description
GOV 379S Regime Perspectives on American Politics
This is a seminar on American politics and culture  Two purposes govern the selection of texts for the course and guide our discussion of them  All of our texts attempt to look at American politics as a whole  Most books and courses on America look at only a part, such as the Presidency, or elections, or popular culture  Here we attempt to think about how the parts of America fit together  Even when these texts speak about a part, for example an institution such as the presidency or the Congress, they present the topic from a vantage point on the whole polity  To see the polity as a whole also means that we will have to revisit and rethink aspects of our political life that we take for granted – that we don’t examine because those parts have become so natural or familiar to us  Seeing the polity whole enables us to render the familiar unfamiliar, to make what we take for granted strange and new 
To see the polity as a whole requires that we get some distance from our subject, much as to see the planet earth as a whole requires one to look at it from outer space  Just as it is difficult to get visual perspective on a place living within it, it is difficult to understand the promise or pathologies of a regime from within  To get critical distance from our politics, we will closely study three sets of texts that look at American politics from a distance  The first part of the course will recover the perspective of the founding debate between Federalists and Anti-federalists  This fundamental debate reveals what is a stake in the basic architecture of the American regime  The second part of the course is a close study of Tocqueville’s Democracy in America  Regarded by many as the best book ever written on democracy and the best book written on America, Tocqueville sees our polity whole because he looks at it from the vantage point of Europe, in general, and France, in particular  In the third part of the seminar we think about American politics from the perspective of thoughtful commentators who feel only nominally included in the polity  Half in and half out, these extraordinary black American writers reveal fissures and fault lines in the American regime  We end the class with a discussion of America’s place in the world today – examining a speech by a writer who articulately raises challenges to our self-understanding that are inarticulately expressed today in rage and ranting from enemies of the United States 
Three take home analytic essays, chosen from a list of topics I provide, each weighted 25% of the course grade  Late essays will not be accepted, except with a doctor’s excuse or a Dean’s excuse for family emergency 
OR as an option: you may write the two short essays (both together weighted 25%) and do a longer 15 page paper on a topic of your choice in consultation with me (weighted 50% of your course grade)  Government honors students who are thinking of doing an honors thesis next year may prefer this option to begin to develop research and writing skills for longer work  Students who prefer this option will need to designate their preferred third short essay and have discussed with me a topic for their long paper by March 30  Texts:
Selected Anti-Federalist writings
Tocqueville, Democracy in America
Essays, speeches and articles by Frederick Douglass, W E B  Dubois, Booker T  Washington, James Baldwin and Ralph Ellison GOV 382M • Democratic Theory 38120 • Spring 2016 Meets M 3:30PM-6:30PM BAT 1 104 show description
GOV 382M (38120)
Democratic Theory Spring 2016
This is a graduate seminar on contemporary topics in democratic theory  Topics to be covered include: democratic epistemology; deliberative democracy; the meaning of the people; oracular democracy; agonistic democracy; and possibly new theories of republicanism, representation and partisanship 
Texts (tentative)
Helene Landemore, Democratic Reason
Jeffrey Edward Green, The Eyes of the People
Amy Gutmann and Dennis Thompson, Why Deliberative Democracy?
Alan Keenan, Democracy in Question
Jason Frank, Constituent Moments
Jason Frank, Publius and Political Imagination
Nadia Urbanati, Democracy Disfigured
Russell Muirhead, Partisanship in a Polarized Age
Bryan Garsten, manuscript
Active seminar participation; an annotated bibliography or review essay; a research/analytic paper  GOV 310L • American Government-Honors 37615 • Fall 2015 Meets TTH 2:00PM-3:30PM BEN 1 106 show description
TTH 2-3:30/BEN 1 106
Bruce Ackerman,Before the Next Attack: Preserving Civil Liberties in an Age of Terrorism GOV 370L • Presidency In Constitutl Order 37845 • Fall 2015 Meets TTH 5:00PM-6:30PM PAR 310 show description
GOV 370L (37845)
TTH 5-6:30 PAR 310
The Presidency in the Constitutional Order
A study of the place of the presidency in the American political order that stresses tension between power and accountability inherent in the office and the system  Topics include: separation of powers, presidential selection, impeachment, relations with Congress and bureaucracy, emergency powers, presidential character, and leadership 
This is a very demanding writing flag class  If you are enrolling in this class just in order to satisfy the writing flag, you are in the wrong class  Interest in political theory and willingness to work very hard are necessary for success in this class 
Joseph M  Bessette, The Constitutional Presidency
Andrew Rudalevige, The New Imperial Presidency
Bruce Ackerman, The Rise and Decline of the American Republic
Michael Nelson, ed , The Presidency in the Political System
Michael Nelson, ed , The Evolving Presidency
Louis Fisher, Constitutional Conflicts Between Congress and the President
Active and prepared class participation
Regular quizzes on the reading
Four analytic essays (approximately 1200 words) 
One term paper, (approximately 5000 words)  GOV 379S • Regime Persp On Amer Politics 38100 • Spring 2015 Meets T 3:30PM-6:30PM MEZ 1 104 (also listed as LAH 350) show description
Essays, speeches and articles by Frederick Douglass, W E B  Dubois, Booker T  Washington, James Baldwin and Ralph Ellison GOV 382M • Tocqueville 38135 • Spring 2015 Meets M 3:30PM-6:30PM BAT 5 102 show description
This graduate seminar will be devoted to close readings of two principal writings of Tocqueville: Democracy in America and The Ancien Regime and the Revolution  We will also assess some of the best secondary studies of Tocqueville, including work by Sheldon Wolin, Harvey Mansfield, Delba Winthrop, Jon Elster, Francois Furet, and a book by Pierre Manent 
Course requirements will include two very short analytic essays and one seminar paper (20-25 pages)  GOV 310L • American Government-Honors 38722 • Fall 2014 Meets TTH 2:00PM-3:30PM GAR 2 112 show description
Joseph M  Bessette and John J  Pitney, American Government and Politics: Deliberation, Democracy and Citizenship
Mary Nichols and David Nichols, Readings in American Government
Bruce Ackerman,Before the Next Attack: Preserving Civil Liberties in an Age of Terrorism GOV 370L • Presidency In Constitutl Order 38977 • Fall 2014 Meets TTH 9:30AM-11:00AM CBA 4 332 show description
A study of the place of the presidency in the American political order that stresses
tension between power and accountability inherent in the office and the system 
Topics include: separation of powers, presidential selection, impeachment,
relations with Congress and bureaucracy, emergency powers, presidential
character, and leadership 
This is a very demanding writing flag class  If you are enrolling in this class just in order
to satisfy the writing flag, you are in the wrong class  Interest in political theory and willingness
to work very hard are necessary for success in this class 
One term paper, (approximately 5000 words)  GOV 379S • Regime Persp On Amer Politics 39395 • Spring 2014 Meets T 3:30PM-6:30PM MEZ 1 104 (also listed as CTI 335, LAH 350) show description
Essays, speeches and articles by Frederick Douglass, W E B  Dubois, Booker T  Washington, James Baldwin and Ralph Ellison GOV 381L • Constitutional Conflict 39415 • Spring 2014 Meets M 3:30PM-6:30PM BAT 1 104 show description
Lowi, The End of Liberalism GOV 330K • The American President 39140 • Fall 2013 Meets MW 3:00PM-4:30PM MEZ B0 306 show description
This course offers an over view of the place of the presidency in the American political order  Topics covered include: constitutional design of the office; nominations and elections; legislative leadership; leadership of the bureaucracy; staffing and organizing the White House; the presidency and the judiciary; war and emergencies  We will spend extra time this fall on the presidential campaign and election of 2012 
Two in-class examinations (50% of the final grade)
One short (1000 word) take-home essay (30% of the final grade)
Class participation and quizzes (20% of the final grade)
Richard J  Ellis, The Development of the American Presidency (Routledge, 2012)
Richard J  Ellis and Michael Nelson, eds, Debating the American Presidency, (2nd edition, CQ Press, 2009)
Packet of selected primary texts (to be linked or posted on Blackboard)  GOV 330K • The American President 39145 • Fall 2013 Meets MW 5:00PM-6:30PM MEZ B0
Context 3: Paper Info

Title: Efficient nonparametric estimation of Toeplitz covariance matrices
Publish Date: March 20, 2023
Author List: Karolina Klockmann (from Department of Statistics and Operations Research, Universität Wien), Tatyana Krivobokova (from Department of Statistics and Operations Research, Universität Wien)

Figure

Figure 1: Spectral density functions (first row) and autocovariance functions (second row) for examples 1, 2, 3 
Figure 2: Distance between the first atom and the first center of mass of aquaporin (left) and the opening diameter y t over time t (right) 
black line in the left plot) confirms that the covariance matrix estimated with our VST-DCT method almost completely decorrelates the channel diameter Y on the training data set Next, we estimated the regression coefficients β with the usual PLS algorithm, ignoring the dependence in the data Finally, we estimated β with PLS that takes into account dependence using our covariance estimator Σ Based on these regression coefficient estimators, the prediction on the test set was calculated The plot on the right side of Figure 2 shows the Pearson correlation between the true channel diameter on the test set and the prediction on the same test set based on raw (grey) and decorrelated data (black) 
Figure 3: On the left, the auto-correlation function of Y (grey) and of Σ−1/2 Y (black), where Σ is estimated with the VST-DCT method; On the right, correlation between the true values on the test data set and prediction based on partial least squares (in grey) and corrected partial least squares (black) 
Uniform distributionThe observations follow a uniform distribution with covariance matrices Σ 1 , Σ 2 , Σ 3 of examples 1, 2, 3, i e , Y i = Σ 1/2 j X i , j = 1, 3, with X 1 ,    the parameter innov of the R function arima sim is used to pass the innovations X 1 ,    , X n i i d Table4, 5 and 6 show respectively the results for (A) p = 5000, n = 1, (B) p = 1000, n = 50 and (C) p = 5000, n = 10 
(A) p = 5000, n = 1: Errors of the Toeplitz covariance matrix and the spectral density estimators with respect to the spectral and L 2 norm, respectively, as well as the average computation time of the covariance estimators in seconds for one Monte Carlo sample (last column) 
(C) p = 5000, n = 10: Errors of the Toeplitz covariance matrix and the spectral density estimators with respect to the spectral and L 2 norm, respectively, as well as the average computation time of the covariance estimators in seconds for one Monte Carlo sample (last column) 
(A) p = 5000, n = 1: Errors of the Toeplitz covariance matrix and the spectral density estimators with respect to the spectral norm and the L 2 norm, respectively Average computation time of the covariance estimators in seconds for one Monte Carlo sample (last column) 
(B) p = 1000, n = 50: Errors of the Toeplitz covariance matrix and the spectral density estimators with respect to the spectral norm and the L 2 norm, respectively Average computation time of the covariance estimators in seconds for one Monte Carlo sample (last column) 
(C) p = 5000, n = 10: Errors of the Toeplitz covariance matrix and the spectral density estimators with respect to the spectral norm and the L 2 norm, respectively Average computation time of the covariance estimators in seconds for one Monte Carlo sample (last column) 

abstract

A new nonparametric estimator for Toeplitz covariance matrices is proposed  This estimator is based on a data transformation that translates the problem of Toeplitz covariance matrix estimation to the problem of mean estimation in an approximate Gaussian regression  The resulting Toeplitz covariance matrix estimator is positive definite by construction, fully data-driven and computationally very fast 
Moreover, this estimator is shown to be minimax optimal under the spectral norm for a large class of Toeplitz matrices  These results are readily extended to estimation of inverses of Toeplitz covariance matrices  Also, an alternative version of the Whittle likelihood for the spectral density based on the Discrete Cosine Transform (DCT) is proposed 
The method is implemented in the R package vstdct that accompanies the paper 

Introduction

Estimation of covariance and precision matrices is a fundamental problem in statistical data analysis with countless applications in the natural and social sciences  Covariance matrices with a Toeplitz structure arise in the study of stationary stochastic n = 1, to the best of our knowledge, there is no fully data-driven approach for selecting the banding/tapering/thresholding parameter 
suggested first to split the time series into non-overlapping subseries and then apply the cross-validation criterion of   However, it turns out that the right choice of the subseries length is crucial for this approach, but there is no data-based method available for this  In this work, an alternative way to estimate a Toeplitz covariance matrix and its inverse is chosen 
Our approach exploits the one-to-one correspondence between Toeplitz covariance matrices and their spectral densities  First, the given data are transformed into approximate Gaussian random variables whose mean equals to the logarithm of the spectral density  Then, the log-spectral density is estimated by a periodic smoothing spline with a data-driven smoothing parameter 
Finally, the resulting spectral density estimator is transformed into an estimator for Σ or its inverse  It is shown that this procedure leads to an estimator that is fully data-driven, automatically positive definite and achieves the minimax optimal convergence rate under the spectral norm over a large class of Toeplitz covariance matrices 
In particular, this class includes Toeplitz covariance matrices that correspond to long-memory processes with bounded spectral densities  Moreover, the computation is very efficient, does not require iterative or resampling schemes and allows to apply any inference and adaptive estimation procedures developed in the context of nonparametric Gaussian regression 
Estimation of the spectral density from a stationary time series is a research topic with a long history  Earlier nonparametric methods are based on smoothing of the (log-)periodogram, which itself is not a consistent estimator   Another line of nonparametric methods for estimating the spectral density is based on the Whittle likelihood, which is an ap-proximation to the exact likelihood of the time series in the frequency domain 
For example, estimated the spectral density from a penalized Whittle likelihood, while used polynomial splines to estimate the log-spectral density function maximizing the Whittle likelihood  Recently, Bayesian methods for spectral density estimation have been proposed (see , but these may become very computationally intensive in large samples due to posterior sampling 
The minimax optimal convergence rate for nonparametric estimators of Hölder continuous spectral densities from Gaussian stationary time series was obtained by under the L p , 1 ≤ p ≤ ∞, norm  Only few works on spectral density estimation show the optimality of the corresponding estimators  In particular, and derived convergence rates of their estimators for the log-spectral density under the L 2 norm, while neglecting the Whittle likelihood approximation error 
In general, most works on spectral density estimation do not exploit further the close connection to the corresponding Toeplitz covariance matrix estimation  In particular, an upper bound for the L ∞ risk of a spectral density estimator automatically provides an upper bound for the risk of the corresponding Toeplitz covariance matrix estimator under the spectral norm 
This fact is used to establish the minimax optimality of our nonparametric estimator for the Toeplitz covariance matrices  The main contribution of this work is to show that our proposed spectral density estimator is not only numerically very efficient, but also achieves the minimax optimal rate in the L ∞ norm, which in turn ensures the minimax optimality of the corresponding Toeplitz covariance matrix estimator 
The paper is structured as follows  In Section 2, the model is introduced and ap-proximate diagonalization of Toeplitz covariance matrices with the discrete cosine transform is discussed  Moreover, an alternative version of the Whittle's likelihood is proposed  In Section 3, new estimators for the Toeplitz covariance matrix and the precision matrix are derived, while in Section 4 their theoretical properties are presented 
Section 5 contains simulation results, Section 6 presents a real data example, and Section 7 closes the paper with a discussion  The proofs are given in the appendix to the paper 

Set up and diagonalization of Toeplitz matrices

Let Y 1 ,       , Y n i i d  ∼ N p (0 p , Σ), where Σ is a (p × p)-dimensional positive definite covariance matrix with a Toeplitz structure, that is, Σ = {σ |i−j| } p i,j=1 0  The sample size n may tend to infinity or to be a constant  The case n = 1 corresponds to a single observation of a stationary time series, and in this case the data are simply denoted by Y ∼N p (0 p , Σ) 
The dimension p is assumed to grow  The spectral density function f , corresponding to a Toeplitz covariance matrix Σ, is given by so that for f ∈ L 2 (−π, π) the inverse Fourier transform implies Hence, Σ is completely characterized by f , and the non-negativity of the spectral density function implies the positive definiteness of the covariance matrix 
Moreover, the decay of the autocovariance σ k is directly connected to the smoothness of f   Finally, the convergence rate of a Toeplitz covariance estimator and that of the corresponding spectral density estimator are directly related via Σ ≤ f ∞ := sup x∈ |f (x)|, where • denotes the spectral norm (see  
As in , we introduce a class of positive definite Toeplitz covariance matrices with Hölder continuous spectral densities  For β = γ + α > 0, where The optimal convergence rate for estimating Toeplitz covariance matrices over P β (M 0 , M 1 ) depends crucially on β  It is well known that the k-th Fourier coefficient of a function whose γ-th derivative is α-Hölder continuous decays at least with order O(k −β ) (see  
Hence, β determines the decay rate of the autocovariances σ k , which are the Fourier coefficients of the spectral density f , as k → ∞  In particular, this implies that for β ∈ (0, 1], the class P β (M 0 , M 1 ) includes Toeplitz covariance matrices corresponding to long-memory processes with bounded spectral densities, since the sequence of corresponding autocovariances is not summable 
A connection between Toeplitz covariance matrices and their spectral densities is further exploited in the following lemma  Lemma 1  Let Σ ∈ P β (M 0 , M 1 ) and let x j = (j − 1)/(p − 1), j = 1,    , p, then where δ i,j is the Kroneker delta, O(•) terms are uniform over i, j = 1,       , p and divided by √ 2 when i, j ∈ {1, p} is the Discrete Cosine Transform I (DCT-I) matrix 
The proof can be found in Appendix A 1  This result shows that the DCT-I matrix approximately diagonalizes Toeplitz covariance matrices and that the diagonalization error depends to some extent on the smoothness of the corresponding spectral density  In the spectral density literature the discrete Fourier transform (DFT) matrix
, where i is the imaginary unit, is typically employed to approximately diagonalize Toeplitz covariance matrices  Using the fact that introduced an approximation for the likelihood of a single Gaussian stationary time series (case n = 1), the so-called Whittle likelihood (1) The quantity , where F j denotes the j-th column of F , is known as the periodogram at the j-th Fourier frequency 
Note that due to periodogram symmetry, only p/2 data points I 1 ,    , I p/2 are available for estimating the mean f (2πj/p), j = 1,       , p/2 , where x denotes the largest integer strictly smaller than x  The Whittle likelihood has become a popular tool for parameter estimation of stationary time series, e g , for nonparametric and parametric spectral density estimation or for estimation of the Hurst exponent, see e g , ;  
Lemma 1 yields the following alternative version of the Whittle likelihood where W j = (D t j Y ) 2   Note that this likelihood approximation is based on twice as many data points W j as the standard Whittle likelihood  Thus, it allows for a more efficient use of the data Y to estimate the parameter of interest, such as the spectral density or the Hurst parameter 
Equations ( ) or (2) invite for the estimation of f by maximizing the (penalized) likelihood over certain linear spaces (e g , spline spaces), as suggested e g , in or   However, such an approach requires well-designed numerical methods to solve the corresponding optimization problem, since the spectral density in the second term of (1) or ( ) is in the denominator, which does not allow to obtain a closed-form expression for the estimator and often leads to numerical instabilities 
Also, the choice of the smoothing parameter becomes challenging  Therefore, we suggest an alternative approach that allows the spectral density to be estimated as a mean in an approximate Gaussian regression  Such estimators have a closed-form expression, do not require an iterative optimization algorithm and a smoothing parameter can be easily obtained with any conventional criterion 
First Hence, for W j = (D t j Y ) 2 , j = 1,       , p it follows with Lemma 1 that where Γ(a, b) denotes a gamma distribution with a shape parameter a and a scale parameter b  Note that the random variables W 1 ,       , W p are only asymptotically independent  Obviously, E(W j ) = f (πx j ) + O(1), j = 1,      
, p  To estimate f from W 1 ,       , W p , one could use a generalized nonparametric regression framework with a gamma distributed response, see e g , the classical monograph by   However, this approach requires an iterative procedure for estimation, e g , a Newton-Raphson algorithm, with a suitable choice for the smoothing parameter at each iteration step 
Deriving the L ∞ rate for the resulting estimator is also not a trivial task  Instead, we suggest to employ a variance stabilizing transform of that converts the gamma regression into an approximate Gaussian regression  In the next section we present the methodology in more detail for a general setting with n ≥ 1 

Methodology

For Y i ∼ N p (0 p , Σ), i = 1,       , n, it was shown in the previous section that with Lemma 1 the data can be transformed into gamma distributed random variables     , n, j = 1,       , p, where for each fixed i the random variable W i,j has the same distribution as W j given in (3)  Now the approach of Cai et al  ( ) is adapted to the setting n ≥ 1 
First, the transformed data points W i,j are binned, that is, fewer new variables     , T   Note that the number of observations in a bin is m = np/T   In Theorem 1 in Section 4, we show that setting T = p υ for any υ ∈ ((4 − 2 min{β, 1})/3, 1) leads to the minimax optimal rate for the spectral density estimator 
To simplify the notation, m is handled as an integer (otherwise, one can discard several observations in the last bin)  Next, applying the variance stabilizing transform (VST) ∼ where H(y) = {φ(m/2) + log (2y/m)} / √ 2 and φ is the digamma function (see   Now, the scaled and shifted log-spectral density H(f ) can be estimated with a periodic smoothing spline
where h > 0 denotes a smoothing parameter, q ∈ N is the penalty order and S per (2q − 1) a space of periodic splines of degree 2q − 1  The smoothing parameter h can be chosen either with generalized cross-validation (GCV) as derived in or with the restricted maximum likelihood, see   Once an estimator H(f ) is obtained, application of the inverse transform function H −1 (y) = m exp √ 2y − φ (m/2) /2 yields the spectral density estimator f = H −1 H(f )  
Finally, using the inverse Fourier transform leads to the fol- The precision matrix Ω is estimated by the inverse Fourier transform of the reciprocal of the spectral density estimator, i e , Ω = (ω |i−j| ) p i,j=1 with ωk = The estimation procedure for Σ and Ω can be summarised as follows  1  Data Transformation:
where D is the (p × p)-dimensional DCT-I matrix as given in Lemma 1 and D j is its j-th column  2  Binning: Set T = p υ for any υ ∈ ((4 − 2 min{β, 1})/3, 1) and calculate W i,j , k = 1,     



ground_truth: Sir Richard.

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: What is Professor Tulis's forthcoming book?

Read the following text and answer briefly.

Context 2: 306 show description
Packet of selected primary texts (to be linked or posted on Blackboard)  GOV 381L • American Founding 39040 • Spring 2013 Meets T 6:30PM-9:30PM BAT 1 104 show description
NOTE WELL: Course meets Tuesdays, 6:30 to 9:30pm
Batts Hall 1 104
This is a seminar on American political thought and constitutional design  It is designed for students of American politics and political theory  The principal themes include: 1) the nature of founding and its constitutive significance; 2) the relation of structure and power in American politics; 3) the meaning and significance of the Federalist/Anti-Federalist debate; 4) the philosophic background of the American founding; and 5) the relevance of the founding to debate to prospects for, and pathologies of, American politics today 
We will conduct a close reading of the Madison’s Notes, of The Federalist, and selected Anti-Federalist writings  We will also study a larger and growing body of secondary literature on the constitutional convention, ratification and early American political thought 
James Madison, Notes of the Debates: In the Federal Convention of 1787
The Federalist (Rossiter, ed )
The Anti-Federalist (Storing, ed )
David Brian Robertson, The Constitution and America’s Destiny (2005)
Pauline Maier, Ratification (2012)
Gordon Wood, The Idea of America (2011)
Jack Rakove, Original Meanings: Politics & Ideas in the Making of the Constitution
Herbert Storing, What the Anti-Federalists Were For (1981)
Numerous essays and articles (to be posted on line or gathered in packet)
Grading: Active seminar participation, including three short papers and presentations (40%) and one article-length seminar paper (60%) T C 357 • Amer Founding/Probs Const Des 43095 • Spring 2013 Meets M 3:30PM-6:30PM CRD 007B show description
The American Founding and Problems of Constitutional Design
Jeffrey Tulis, Associate Professor, Department of Government
Sanford Levinson, Professor, School of Law
This Plan II seminar will be built around a close reading of the debates that informed the drafting and ratification of the U S  Constitution  We aim to recover the perspective of these founding thinkers -- their way of thinking -- as much as their concrete ideas, in order to raise fundamental questions about the American political order today  Are some of the most important pathologies of American politics today rooted in design features of our original political architecture? Are the original answers to basic founding questions (such as "how democratic is our Constitution?) still adequate for contemporary circumstances? What features of the Constitution should we preserve and what features should we amend, if possible? Would it be good for the polity as a whole to reconsider these questions in a new constitutional convention today, or would such an event be a political nightmare? Our reading will include notes from the founding conventions, writings by Federalists and Anti-Federalists, and present-day critiques of the American political order  Our aim will be to generate a dialogue between the thought of the founders and some of the best present day critics and supporters of the Constitution 
James Madison, Notes of the Debates in the Federal Convention
The Federalist, ed  Clinton Rossiter
The Anti-Federalist, ed  Herbert Storing
Pauline Maier, Ratification: The People Debate the Constitution, 1787-1788
Sanford Levinson, Framed: America’s 51 Constitutions and the Crisis of Governance
Bruce Ackerman, The Decline and Fall of the American Republic
Robert Goldwin, ed  How Democratic is the Constitution?
a course packet of selected articles, essays, and additional primary materials 
Class participation, including at least one presentation of a short discussion paper 25%
One take-home analytic essay 25%
One term paper 50%
About the Professors:
Professor Tulis's interests bridge the fields of political theory and American politics, including more specifically, American political development, constitutional theory, political philosophy and the American presidency  He received the President's Associates Teaching Excellence Award at the University of Texas  He has held research fellowships from NEH, ACLS, Olin Foundation, Harvard Law School, and the Mellon Preceptorship at Princeton University, where he taught before moving to Texas  He has held visiting positions at Notre Dame and Harvard  During the academic year 2008-09, he was a Laurance S  Rockefeller Visiting Fellow at the University Center for Human Values at Princeton 
Proefessor Levinson holds the W  St  John Garwood and W  St  John Garwood, Jr  Centennial Chair in Law, he joined the University of Texas Law School in 1980  Previously a member of the Department of Politics at Princeton University, he is also a Professor in the Department of Government at the University of Texas  The author of over 350 articles and book reviews in professional and popular journals--and a regular contributor to the popular blog Balkinization  He received the Lifetime Achievement Award from the Law and Courts Section of the American Political Science Association in 2010  He has been a visiting faculty member of the Boston University, Georgetown, Harvard, New York University, and Yale law schools in the United States and has taught abroad in programs of law in London; Paris; Jerusalem; Auckland, New Zealand; and Melbourne, Australia 
GOV 330K • The American President 38675 • Fall 2012 Meets MW 3:00PM-4:30PM MEZ B0 306 show description
Packet of selected primary texts (to be linked or posted on Blackboard)  GOV 330K • The American President 38675 • Fall 2011 Meets MW 3:30PM-5:00PM WAG 420 show description
see syllabus GOV 330K • The American President 38680 • Fall 2011 Meets MW 5:30PM-7:00PM UTC 1 146 show description
see syllabus GOV 379S • Regime Persp On Amer Polit-Hon 39110 • Spring 2011 Meets W 3:30PM-6:30PM BAT 5 102 (also listed as CTI 326, LAH 350) show description
To see the polity as a whole requires that we get some distance from our subject, much as to see the planet earth as a whole requires one to look at it from outer space  Just as it is difficult to get visual perspective on a place living within it, it is difficult to understand the promise or pathologies of a regime from within it  To get critical distance from our politics, we will closely study three sets of texts that look at American politics from a distance  The first part of the course will recover the perspective of the founding debate between Federalists and Anti-federalists  This fundamental debate reveals what is a stake in the basic architecture of the American regime  The second part of the course is a close study of Tocqueville’s Democracy in America  Regarded by many as the best book ever written on democracy and the best book written on America, Tocqueville sees our polity whole because he looks at it from the vantage point of Europe, in general, and France, in particular  In the third part of the seminar we think about American politics from the perspective of thoughtful commentators who feel only nominally included in the polity  Half in and half out, these extraordinary black American writers reveal fissures and fault lines in the American regime  We end the class with a discussion of America’s place in the world today – examining a speech by a writer who articulately raises challenges to our self-understanding that are inarticulately expressed today in rage and ranting from enemies of the United States 
Four take home writing assignments  Analytic essays, each 1000-1500 words  (Grades weighted: 10%, 25%, 25%, and 25%) Late essays will not be accepted, except with a doctor’s excuse or a Dean’s excuse for family emergency  Regular preparation and class participation: 15% 
OR as an option: By prior arrangement with me by the due date of the second analytic essay, students may substitute one longer research paper (15 – 20 pages) for two of the last three analytic papers This paper will be on a topic of the students choosing , if I approve, and the due date will be the same as the last assigned analytic essay  This project would count 50% of the students course grade 
Selected writings by Frederick Douglass, W E B  Dubois, Ralph Ellison, James Baldwin
Solzhenitsyn, “A World Split Apart”
Tocqueville, Democracy in America GOV 382M • Tocqueville 39150 • Spring 2011 Meets T 6:30PM-9:30PM BAT 5 102 show description
See syllabus GOV 370L • President, Congress, And Court 38695 • Fall 2010 Meets TTH 8:00AM-9:30AM UTC 3 112 show description
Course Description: A Study of the political relationship of the President, Congress and Court in the American constitutional order  Has this relationship changed over the course of American history? Is American national politics prone to stalemate or deadlock between the branches regarding major issues of public policy? Do we have a new “imperial presidency?” Should the Court arbitrate disputes between the President and Congress over custody of their respective powers? Has Congress abdicated its constitutional responsibilities? We will examine questions like these in light of practical problems such as executive privilege and secrecy, the war on terror, budget politics and controversies regarding appointments to the Supreme Court  Grading:Three in class essay tests, for which study questions will be distributed in advance  The exam questions will be chosen from the list of study questions  (25% each) One short take home essay (10% each)  Class participation and attendance (15%)  Tentative Texts: The FederalistFisher, Congressional Abdication on War and SpendingRudalevige, The New Imperial PresidencyBessette and Tulis, The Constitutional PresidencySkowronek, Presidency in Political TimeGoldsmith, The Terror PresidencyA course packet of articles and essays GOV 370L • President, Congress, And Court 38700 • Fall 2010 Meets TTH 5:00PM-6:30PM UTC 3 122 show description
Course Description: A Study of the political relationship of the President, Congress and Court in the American constitutional order  Has this relationship changed over the course of American history? Is American national politics prone to stalemate or deadlock between the branches regarding major issues of public policy? Do we have a new “imperial presidency?” Should the Court arbitrate disputes between the President and Congress over custody of their respective powers? Has Congress abdicated its constitutional responsibilities? We will examine questions like these in light of practical problems such as executive privilege and secrecy, the war on terror, budget politics and controversies regarding appointments to the Supreme Court  Grading:Three in class essay tests, for which study questions will be distributed in advance  The exam questions will be chosen from the list of study questions  (25% each) One short take home essay (10% each)  Class participation and attendance (15%)  Tentative Texts: The FederalistFisher, Congressional Abdication on War and SpendingRudalevige, The New Imperial PresidencyBessette and Tulis, The Constitutional PresidencySkowronek, Presidency in Political TimeGoldsmith, The Terror PresidencyA course packet of articles and essays GOV 312L • Iss & Policies In Amer Gov-Hon 38698 • Spring 2010 Meets MW 3:30PM-5:00PM UTC 3 104 show description
Government 312L satisfies the second half of the mandated six hours of government that every UT student must take  Course covers analysis of varying topics concerned with American political institutions and policies, including the United States Constitution, and assumes basic knowledge of government from GOV 310L, which is a prerequiste  May be taken for credit only once  GOV 370L • President, Congress, And Court 38966 • Spring 2010 Meets MW 5:00PM-6:30PM MEZ B0 306 show description
Prerequisite: Six semester hours of lower-division coursework in government 
GOV 370L • President, Congress, And Court 39295 • Fall 2009 Meets TTH 2:00PM-3:30PM UTC 3 112 show description
GOV 370L • President, Congress, And Court 39435 • Spring 2008 Meets MW 3:00PM-4:30PM PAR 203 show description
GOV 312L • Iss & Policies In Am Gov-Hon-W 38615-38620 • Spring 2007 Meets MW 11:00AM-12:00PM MEZ B0 306 show description
Government 312L satisfies the second half of the mandated six hours of government that every UT student must take  Course covers analysis of varying topics concerned with American political institutions and policies, including the United States Constitution, and assumes basic knowledge of government from GOV 310L, which is a prerequiste  May be taken for credit only once  GOV 312L • Iss & Policies In Am Gov-Hon-W 37600-37605 • Spring 2006 Meets MW 11:00AM-12:00PM MEZ B0 306 show description
Government 312L satisfies the second half of the mandated six hours of government that every UT student must take  Course covers analysis of varying topics concerned with American political institutions and policies, including the United States Constitution, and assumes basic knowledge of government from GOV 310L, which is a prerequiste  May be taken for credit only once  GOV 312L • Iss & Policies In Am Gov-Hon-W 34900-34905 • Spring 2004 Meets MW 11:00AM-12:00PM BUR 134 show description
Government 312L satisfies the second half of the mandated six hours of government that every UT student must take  Course covers analysis of varying topics concerned with American political institutions and policies, including the United States Constitution, and assumes basic knowledge of government from GOV 310L, which is a prerequiste  May be taken for credit only once  GOV 312L • Iss & Policies In Am Gov-Hon-W 34495-34500 • Spring 2003 Meets MW 11:00AM-12:00PM UTC 1 130 show description
Government 312L satisfies the second half of the mandated six hours of government that every UT student must take  Course covers analysis of varying topics concerned with American political institutions and policies, including the United States Constitution, and assumes basic knowledge of government from GOV 310L, which is a prerequiste  May be taken for credit only once  Publications
Tulis, JK (2011), "Plausible Futures," in Dunn, Charles W  (ed ) The Presidency in the Twenty-First Century, University Press of Kentucky Tulis, J K  and Macedo, S  (2010) The Limits of Constitutional Democracy, Princeton University Press Tulis, J K  and Macedo, S  (2010) "Constitutional Boundaries," in The Limits of Constitutional Democracy, Princeton University Press Tulis, JK (2010), "The Possibility of Constitutional Statesmanship," in Tulis, JK and Macedo, S (eds ) The Limits of Constitutional Democracy, Princeton University Press Tulis, J  (2009) The Constitutional Presidency  Johns Hopkins University Press Tulis, J  (2009) Impeachment in the Constitutional Order  In J  Tulis & J M  Bessette (Eds ), The Constitutional Presidency  Johns Hopkins University Press Tulis, J  & Bessette, J M  (2009) On the Constitution, Politics, and the Presidency  In J  Tulis & J M  Bessette (Eds ), The Constitutional Presidency  Johns Hopkins University Press Tulis, J (and Bessette, J M) (2010) The Presidency in the Constitutional Order: Historical Perspectives, Reissued Classics Series, Transaction Publishers,Tulis, J and Bessette, J M  (2010, "Introduction to the Transaction Edition," The Presidency in the Constitutional Order: Historical Perspectives, Transaction Publishers 
Tulis, JK, (2009) "The Two Constitutional Presidencies," in Nelson, Michael (ed ) The Presidency in the Political System, Congressional Quarterly Press Tulis, J  & Mellow, N  (2007) Andrew Johnson and the Politics of Failure  In S  Skowronek & M  Glassman (Eds ), Formative Acts: Reckoning with Agency in American Politics  Philadelphia: University of Pennsylvania Press Tulis, J  (2007, September) The Rhetorical Presidency in Retrospect  Critical Review: An Interdisciplinary Journal of Politics and Society, 19(2&3)  Curriculum Vitae
Context 3:  , T 

VST:

where k are asymptotically i i d  Gaussian variables  Inverse VST: Estimate the spectral density f with f = H −1 H(f ) , where Note that Σ and Ω are positive definite matrices by construction, since their spectral density functions f and f −1 are non-negative, respectively  Unlike the banding and tapering estimators, the autocovariance estimators σk are controlled by a single smoothing parameter h, which can be estimated fully data-driven with several available automatic methods, which are numerically efficient and well-studied 
In addition, one can also use methods for adaptive mean estimation, see e g , , which in turn leads to adaptive Toeplitz covariance matrix estimation  All inferential procedures developed in the Gaussian regression context can also be adopted accordingly 

Theoretical Properties

In this section, we study the asymptotic properties of the estimators f , Σ and Ω  The results are established under the asymptotic scenario where p → ∞ and p/n → c ∈ (0, ∞], that is, the dimension p grows, while the sample size n either remains fixed or also grows but not faster than p  This corresponds to the asymptotic scenario when the sample covariance matrix is inconsistent 
Let f be the spectral density estimator defined in Section 3, i e , f = m exp{ √ 2 H(f ) − φ(m/2)}/2, where H(f ) is given in (4), m = np/T and φ is the digamma function  Furthermore, let Σ be the Toeplitz covariance matrix estimator and Ω the corresponding precision matrix defined in equations ( ) and (6), respectively 
The following theorem shows that both Σ and Ω attain the minimax optimal rate of convergence over the class and hT → ∞, then with T = p υ for any υ ∈ ((4 − 2 min{β, 1})/3, 1) and q = max{1, γ}, the spectral density estimator f , the corresponding covariance matrix estimator Σ and the precision matrix estimator Ω satisfy sup
For h {log(np)/(np)}   The proof of Theorem 1 can be found in the Appendix A 3 and is the main result of our work  The most important part of this proof is the derivation of the convergence rate for the spectral density estimator f under the L ∞ norm  In the original work, established an L 2 rate for a wavelet nonparametric mean estimator in a gamma regression where the data are assumed to be independent 
In our work, the spectral density estimator f is based on the gamma distributed data W i,1 ,       , W i,p , which are only asymptotically independent  Moreover, the mean of these data is not exactly f (πx 1 ),       , f (πx p ), but is corrupted by the diagonalization error given in Lemma 1  This error adds to the error that arises via binning and VST and that describes the deviation from a Gaussian distribution, as derived in  
Finally, we need to obtain an L ∞ rather than an L 2 rate for our spectral density estimator  Overall, the proof requires different tools than those used in   To get the L ∞ rate for f , we first derive that for the periodic smoothing spline estimator H(f ) of the log-spectral density  To do so, we use a closed-form expression of its effective kernel obtained in , thereby carefully treating various (dependent) errors that describe deviations from a Gaussian nonparametric regression with independent errors and mean f (πx i ) 
Note also that although the periodic smoothing spline estimator is obtained on T binned points, the rate is given in terms of the vector dimension p  Then, using the Cauchy-Schwarz inequality and a mean value argument, this rate is translated into the L ∞ rate for the spectral density estimator f   To obtain the rate for the Toeplitz covariance matrix estimator is enough to note that

Simulation Study

In this section, we compare the performance of the proposed Toeplitz covariance estimator, denoted as VST-DCT, with the tapering estimator of and with the sample covariance matrix  We consider Gaussian vectors Y 1 ,    , (3) such that the corresponding spectral density is Lipschitz continuous but not differentiable: f (x) = 1 44{| sin(x + 0 5π)| 1 7 + 0 45} 
In particular, var(Y i ) = 1 44 in all three examples  Figure shows the spectral densities and the corresponding autocorrelation functions for the three examples  A Monte Carlo simulation with 100 iterations is performed using R (version 4 1 2, seed 42)  For our VST-DCT estimator, we use a cubic periodic spline, i e , q = 2 is set in (4) 
The binning parameters are set to T = 500 bins with m = 10 points for (A) and T = 500 bins with m = 100 points for both (B) and (C)  To select the regularisation parameter for our estimator, we implemented the restricted maximum likelihood (ML) method, generalized cross validation (GCV) and the corresponding oracle versions, i e , as if Σ were known The tapering parameter
where T ap k (Σ ν 2 ) is the tapering estimator of with parameter k  If n = 1, that is, under scenario (A), suggest to split the time series Y into l non-overlapping subseries of length p/l and then proceed as before to select the tuning parameter k  To the best of our knowledge, there is no data-driven method for selecting this parameter l 
Using the true covariance matrix Σ, we selected l = 30 subseries for the example 1 and l = 15 subseries for the exam-ples 2 and 3  The parameter k can then be chosen by cross-validation as above  We employ this approach under scenario (A) instead of an unavailable fully data-driven criterion and name it semi-oracle 
Finally, for all three scenarios (A), (B) and (C), the oracle tapering parameter is computed using grid search for each Monte Carlo sample as kor = arg min k=2,3,   ,p/2 T ap k ( Σ) − Σ , where Σ is the sample covariance matrix  To speed up the computation, one can replace the spectral norm by the 1 norm, as suggested by  
In Tables , the errors of the Toeplitz covariance estimators with respect to the spectral norm and the computation time for one Monte Carlo iteration are given for scenarios (A), (B) and (C), respectively  To illustrate the goodness-of-fit of the spectral density, the L 2 norm f − f 2 is also computed 
The results show that the tapering and VST-DCT estimator perform overall similar in terms of the spectral norm risk  This is not surprising as both estimators are proved to be rate-optimal  Moreover, both the tapering and VST-DCT estimators are clearly superior to the inconsistent sample Toeplitz covariance matrix 
A closer look at the numbers shows that the VST-DCT method has better constants, i e , VST-DCT estimators have somewhat smaller errors in the spectral norm than the tapering estimators across all examples, but especially under scenario (C)  The oracle estimators show similar behaviour, but are slightly less variable compared to the data-driven estimators 
In general, both the tapering and VST-DCT estimators perform best for example 1, second best for example 3 and worst for example 2, which traces back to functions complexity  In terms of computational time, both methods are similarly fast for scenarios (A) and (B)  For scenario (C), the tapering method is much slower due to the multiple high-dimensional matrix multiplications in the cross-validation method 
It is expected that for larger p the tapering estimator is much more computationally intensive than the corresponding VST-DCT estimator  (1) polynomial σ ( (  To test how robust our approach is to deviations from the Gaussian assumption, we simulated the data from gamma and uniform distributions and conducted a simulation study for the same scenarios and examples 
The results are very similar to those of the Gaussian distribution, see supplementary materials for the details 

Application to Protein Dynamics

We revisit the data analysis of protein dynamics performed in Krivobokova et al  (2012) and   We consider data generated by the molecular dynamics (MD) simulations for the yeast aquaporin (Aqy1) -the gated water channel of the yeast Pichi pastoris  MD simulations are an established tool for studying biological systems at the atomic level on timescales of nano-to microseconds 
The data are given as Euclidean coordinates of all 783 atoms of Aqy1 observed in a 100 nanosecond time frame, split into 20 000 equidistant observations  Additionally, the diameter of the channel y t at time t is given, measured by the distance between two centers of mass of certain residues of the protein 
The aim of the analysis is to identify the collective motions of the atoms responsible for the channel opening  In order to model the response variable y t , which is a distance, based on the motions of the protein atoms, we chose to represent the protein structure by distances between atoms and certain fixed base points instead of Euclidean coordinates 
That is, we calculated where A t,i ∈ R 3 , i = 1,       , 783 denotes the i-th atom of the protein at time t, B j ∈ R 3 , j = 1, 2, 3, 4, is the j-th base point and d(•, •) is the Euclidean distance  Figure shows the diameter y t and the distance between the first atom and the first center of mass  It can therefore be concluded that a linear model Y = Xβ + holds, where
  This linear model has two specific features which are intrinsic to the problem: first, the observations are not independent over time and second, X t is high-dimensional at each t and only few columns of X are relevant for Y   have shown that the partial least squares (PLS) algorithm performs exceptionally well on this type of data, leading to a small-dimensional and robust representation of proteins, which is able to identify the atomic dynamics relevant for Y  
Singer et al  ( ) studied the convergence rates of the PLS algorithm for dependent observations and showed that decorrelating the data before running the PLS algorithm improves its performance  Since Y is a linear combination of columns of X, it can be assumed that Y and all columns of X have the same correlation structure 
Hence, it is sufficient to estimate Σ = cov(Y ) to decorrelate the data for the PLS algorithm, i e , Σ −1/2 Y = Σ −1/2 Xβ + Σ −1/2 results in a standard linear regression with independent errors  Our goal now is to estimate Σ and compare the performance of the PLS algorithm on original and decorrelated data 
For this purpose, we divided the data set into a training and a test set (each with p = 10 000 observations)  First, we tested whether the data are stationary  The augmented Dickey-Fuller test confirmed stationarity for Y with a p-value< 0 01  The Hurst exponent of Y is 0 85, indicating moderate long-range dependence supported by a rather slow decay of the sample autocovariances (see grey line in the left plot of Figure ) 
Therefore, we set q = 1 for the VST-DCT estimator to match the low smoothness of the corresponding spectral density  Moreover, the smoothing parameter is selected with the restricted maximum likelihood method and T = 550 bins are used  Obviously, the performance of the PLS algorithm on the decorrelated data is significantly better for the small number of components 
In particular, with just one PLS component, the correlation between the true opening diameter on the test set and its prediction that takes into account the dependence in the data is already 0 54, while it is close to zero for PLS that ignores the dependence in the data  showed that the estimator of β based on one PLS component is exactly the ensemble-weighted maximally correlated mode (ewMCM), which is defined as the collective mode of atoms that has the highest probability to achieve a specific alteration of the response Y  
Therefore, an accurate estimator of this quantity is crucial for the interpretation of the results and can only be achieved if the dependence in the data is taken into account  Estimating Σ with a tapered covariance estimator has two practical problems  First, since we only have a single realization of a time series Y (n = 1), there is no datadriven method for selecting the tapering parameter 
Second, the tapering estimator turned out not to be positive definite for the data at hand  To solve the second problem, we truncated the corresponding spectral density estimator ftap to a small positive value, i e , f + tap = max{ ftap , 1/ log(p)} (see   To select the tapering parameter with cross-validation, we experimented with different subseries lengths and found that the tapering estimator is very sensitive to this choice 
For example, estimating the tapered covariance matrix based on subseries of length 8/15/30 yields a correlation of 0 42/0 53/0 34 between the true diameter and the first PLS component, respectively  Altogether, our proposed estimator is fully data-driven, fast even for large sample sizes, automatically positive definite and can handle certain long-memory processes 
In contrast, the tapering estimator is not data-driven and must be manipulated to become positive definite  Our method is implemented in the R package vstdct 

Discussion

In this paper, we proposed a simple, fast, fully data-driven, automatically positive definite and minimax optimal estimator of Toeplitz covariance matrices from a large class that also includes covariance matrices of certain long-memory processes  Our estimator is derived under the assumption that the data are Gaussian 
However, simulations show that the suggested approach yields robust estimators even when the data are not normally distributed  In the context of spectral density estima- , for mixing processes (see Theorem 5 3 of Rosenblatt, 2012), as well as for non-linear processes (see   Since DFT and DCT matrices are closely related, we expect that equation (3) also holds asymptotically for these non-Gaussian time series, but consider a rigorous analysis to be beyond the scope of this paper 
In fact, our numerical experiments have even shown that if the spectral density is estimated from W j = f (πx j ) + j , that is, as if W j were Gaussian instead of gamma distributed, then the resulting spectral density estimator has almost the same L ∞ risk (and hence the corresponding covariance matrix has almost the same spectral norm) 
Of course, such an estimator would lead to a wrong inference about f (πx j ), since the growing variance of W j would be ignored  Since our approach translates Toeplitz covariance matrix estimation into a mean estimation in an approximate Gaussian nonparametric regression, all approaches developed in the context of Gaussian nonparametric regression, such as (locally)
adaptive estimation, as well as the corresponding (simultaneous) inference, can be directly applied  Bayesian tools for adaptive estimation and inference in Gaussian nonparametric regression as proposed in can also be employed 

Appendix

Throughout the appendix, we denote by c, c 1 , C, C 1 ,       etc  generic constants, that are independent of n and p  To simplify the notation, the constants are sometimes skipped and we write for less than or equal to up to constants  We embed the p-dimensional Toeplitz matrix Σ = toep(σ 0 ,       , σ p−1 ) in a (2p − 2)dimensional circulant matrix Σ = toep(σ 0 ,      
, σ p−1 , σ p−2 ,       , σ 1 )  Then, Σ = with the conjugate transpose U * , and Λ is a diagonal matrix with the k-th diagonal value for k = 1,    , p given by Furthermore, Σ = V * ΛV , where V ∈ C (2p−2)×p contains the first p columns of U   In particular, b(j, r) = b(j, 2p−r) and c(j, r) = −c(j, 2p−r) for r = p+1,      
, 2p−2  Together, we have (A 1) Some calculations show that for r = 1,       p Using the Taylor expansion of cot(x) for 0 < |x| < π one obtains for r = 1,   



ground_truth: Legacies of Losing in American Politics and an expanded edition of The Rhetorical Presidency in the Princeton Classics series.

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: What is the advantage of decorrelating the data before running the PLS algorithm?

Read the following text and answer briefly.

Context 3:    p where the O term does not depend on j and the hidden constant does not depend on r, p  If i = j, equations (A 1) -(A 3) imply where the O terms do not depend on j 
Since the complex exponential function is Lipschitz continuous with constant L = 1, it holds λ r = λ j + L r,j |r − j|p −1 where −1 ≤ L r,j ≤ 1 is a constant depending on r, j  Then, , it is sufficient to consider j = 1,    , p − 1  We begin with first sum  For a shorter notation, we use k := r − 1 and l := j − 1 in the following 
Then, summing the squares of the first term in (A 4) for l = 0,    , p−2 on sums of reciprocal powers  If p is even, then the residual terms are given by where φ and φ (1) denote the digamma function and its derivative  If p is odd, similar remainder terms can be derived  To see that R i (l, p) = O(p −1 ) for i = 1, 2, 3 and uniformly in l we use that asymptotically φ(x)∼ log(x)−1/(2x) and
The mixed term are both of the order p −1   Furthermore, since the harmonic sum diverges at a rate of log(p)  Finally, λ j = f (x j )+O{log(p)p −β } by the uniform approximation properties of the discrete Fourier series for Hölder continuous functions (see   All together, we have shown that (DΣD) j,j = where the O terms are uniform over j = 1,    , p 
Case i = j and |i − j| is even In this case, (DΣD) i,j = a i a j uniformly in i, j  To show that a i a j 2p−2 r=1 λ r c(i, r)c(j, r) = O(p −1 ), we proceed similarly as before  Setting k=r−1, l=j−1, m=i−1 and using that l =m and |l−m| is even, one obtains where for even p the residual terms are given by If p is odd, analogous residual terms can be derived 
Using similar techniques as before, one can show that the two residual terms and the remaining mixed and square terms vanish at a rate of the order O(p −1 ) and uniformly in i, j  Case i = j and |i − j| is odd |r − i| and |r − j| are either odd and even, or even and odd  Without loss of generality, assume that |r − i| is even 
Then, (DΣD) i,j = a i a j 2p−2 r=1 λ r b(i, r)c(j, r)  Since b(i, •) is an even function, c(j, •) is an odd function and λ r = λ 2p−r , it follows (DΣD) i,j = 0  The structure of the proof is as follows  First, we derive the L ∞ rate of the periodic smoothing spline estimator H(f )  Then, using the Cauchy-Schwarz inequality and a mean value argument, the convergence rate of the spectral density estimator f is
∞ the first claim of the theorem follows  Finally, we prove the second statement on the precision matrices  For the sake of clarity, some technical lemmas used in the proof are listed separately in A 4  hT → ∞, then with T = p υ for any υ ∈ ((4 − 2 min{1, β})/3, 1), the estimator H(f ) described in Section 3 with q = max{1, γ} satisfies
Proof : Application of the triangle inequality yields a bias-variance decomposition Set T = 2T − 2 and x k = (k − 1)/ T for k = 1,    , T   Using Lemma 4, we can write where Mirroring and renumerating ζ k , η k , k is similar as for Y * k , k = 1,    , T   Using the above representation, one can write First we reduce the supremum to a maximum over a finite number of points 
If q > 1, then W (•, x k ) is Lipschitz continuous with constant L > 0  In this case, it holds almost surely that sup ) is a piecewise linear function with knots at x j = j/ T   The factor (ζ k + ξ k ) can be considered as stochastic weights that do not affect the piecewise linear property  Thus, the supremum is attained at one of the knots x j = j/ T , j = 1,    , T , and (A 7) is also valid for q = 1 
Again with (a + b) 2 ≤ 2a 2 + 2b 2 we obtain We start with bounding   This requires a bound on 1 • ψ 2 denotes the sub-Gaussian norm  In case of a Gaussian random variable the norm equals to the variance  Thus with Lemma 2 and Lemma 4, we obtain Lemma 1 6 of ) then yields Recall that T = p υ for some fixed υ ∈ ((4 − 2 min{1, β})/3, 1) 
Using the inequality log(x) ≤ x a /a one can find constants x υ , C υ > 0 depending on υ but not on n, p such that log(2 T ) log(p) Next, we derive a bound for the second term The exponential decay property of the kernel K stated in Lemma 2 yields The first term in (A 9) can be bounded again with Lemma 1 6 of  
We use the fact that for not necessarily independent random variables X 1 ,    , X N R and R > 0 are constants  This is a consequence of Lemma 1 of which yields , it follows that N i=1 a i X i has a subGaussain distribution and the subGaussian norm is bounded by 2R( N i=1 a 2 i ) 1/2   See for further details on the subgaussian distribution 
T h   For the second inequality Lemma 2(ii) is used  Applying Lemma 1 6 of then yields To bound the second term in (A 9), we use the moment bounds for ξ k derived in Lemma 4  Then, for all integers > 1 Combining the error bounds (A 10) and (A 11) and choosing R=m −1/2 gives By assumption T = p υ and m = np (1−υ) for some fixed υ ∈ ((4 − 2 min{1, β})/3, 1) 
If is an integer such that ≥ 1/(1 − υ), then where we used log(x) ≤ x a /a with a = 1/(4 )  Consider 1/2 < β ≤ 1 and let 0 < χ < 1 be a constant  Applying log(x) ≤ x a /a twice with a = χ/(2 ) yields For any fixed υ∈((4 − 2 min{1, β})/3, 1) one can find an integer which is independent of n, p such that the right side of (A 12) holds 
Since p/n → c ∈ (0, ∞] and thus n/p = O(1) and p −1 = O(n −1 ), it follows for satisfying (A 12) that In total, choosing an integer Using the representation in Lemma 4 once more gives for each x ∈ [0, 1] The bounds on k in Lemma 4 imply Consider the case that β ≥ 1  In particular, q = γ and f (q) is α-Hölder continuous 
Since f is a periodic function with f (x) ∈ [δ, M 0 ] and H(y) ∝ φ(m/2)+ log (2y/m), it follows that {H(f )} (q) is also α-Hölder continuous  Extending g := H(f ) to the entire real line, we get Expanding g(t) in a Taylor series around x and using that h −1 K h is a kernel of order 2q, see Lemma 2(iii), it follows that for any x ∈ [0, 1]
where ξ x,t is a point between x and t  Using the fact that the kernel K h decays exponentially and that g (q) is α-Hölder continuous on [δ, M 0 ] with some constant L, the logarithm is Lipschitz continuous on a compact interval, it follows g = H(f ) is β-Hölder continuous  Expanding g to the entire line and using Lemma 2(iii) with
In a similar way as before, one obtains Note that T −β =o(h β ) as β > 1/2, T h → ∞ and h → 0 by assumption  Since the derived bounds are uniform for x ∈ [0, 1] it holds Putting the bounds A 13 and A 14 together gives If h > 0 such that h → 0 and hT → ∞, then with T = p υ for any υ ∈ ((4 − 2 min{1, β})/3, 1), the estimator f described in Section 3 with q = max{1, γ} satisfies
Proof : By the mean value theorem, it holds for some function g between H(f ) and To show that the second term on the right hand side of (A 15) is negligible we use the moment generating function of H(f ) ∞   In the next paragraph, we derive the asymptotic order of E[exp{λ H(f ) ∞ }] for n, p → ∞, where λ > 0 may depend on n, p or not 
By the exponential decay property of the kernel K stated in Lemma 2 holds First, H(f ) ∞ is bounded with the maximum over a finite number of points  Calculating the derivative of s : Since δ δx s(x) > 0 almost surely for x = x k , the extrema occur at x k , k = 1,    , T   Thus, for λ > 0 the moment generating function of H(f ) ∞ is bounded by
Let M j = ( T h) −1 T k=1 γ h (x j , x k ), which by Lemma 2 is bounded uniformly in j by some global constant M > 0  By the convexity of the exponential function we obtain √ 2 and by assumption 0 ≤ δ ≤ f ≤ M 0   Using Lemma 3, Q k can be written as a sum of m = np/T independent gamma random variables, i e 
The moment generating function of | log(X)| when X follows a Γ(a, b)-distribution is given by where Γ(a) is the gamma function and γ(a, b) is the lower incomplete gamma function  In particular, To derive the asymptotic order of E[exp{λ H(f ) ∞ }] for n, p → ∞ we first establish the asymptotic order of the ratio Γ(a + t)/Γ(a) for a → ∞ 
We distinguish the two cases where t is independent of a and where t linearly depends on a  Thus, for 0 < t < a and t independent of a, equation (A 17) implies for a → ∞ that Γ(a + t)/Γ(a) = O(a t )  Similarly, it can be seen that Γ(a − t)/Γ(a) = O(a −t )  If 0 < t < a and t linearly depends on a, i e  t = ca for some constant c ∈ (0, 1), then we get Γ(a ± t)/Γ(a) = O(a ±t exp{a}) for a → ∞ 
Hence, for a fixed λ not depending on n, p and such that 0 < λ < m/( √ 2M j ) we get for sufficiently large n, p If λ = cm such that 0 < λ < m/( √ 2M j ), then for sufficiently large n, p b∈{cδ/m,cM 0 /m} (bm/2) for some constant L > 1  Set K = min j=1,   , T 1/( √ 2M j ) which is a constant independent of n, p  Altogether, we showed that for 0 < λ < Km and n, p → ∞
Bounding the right hand side of (A 15) for some constants c 0 , c 1 > 0 and n, p → ∞ Since g lies between H(f ) and H(f ), and f almost surely pointwise  Thus, for C > f ∞ = M 0 it holds where c 1 := H(C − M 0 )  Applying Markov inequality for t = cm with c ∈ (0, K) and C = 2L 4/c + M 0 where c, K, L are the constants in gives
Together with Proposition 1 follows Using the fact that the spectral norm of a Toeplitz matrix is upper bounded by the sup norm of its spectral density we get sup According to the mean value theorem, for a function g between H(f ) and H(f ), it holds that some constant c 1 > 0 not depending on n, p  Chosing the same constant C as in section A 3 2 it follows
Noting that 1/f ∞ ≤ 1/δ and 2/m exp {φ(m/2)} ∈ [0 25, 1] for m ≥ 1, (A 18) implies for some constants c 2 , c 3 > 0 and n, p → ∞ Since the derived bounds hold for each Σ(f ) ∈ F β , we get all together sup This section states some technical lemmata needed for the proof of Theorem 1  The proofs can be found in the supplementary material 
The first lemma lists some properties of the kernel K h and its extension K h on the real line  The proof is based on   Lemma 2  Let h > 0 be the bandwith parameter depending on N   (i) There are constants 0 < C < ∞ and 0 < γ < 1 such that for all for x, t ∈ [0, 1] Lemma 3 states that the sum of the correlated gamma random variables in each bin can be rewritten as a sum of independent gamma random variables 
for i = 1,    , n and j = (k − 1)m + 1,    , km, and x j = (j − 1)/(2p − 2)  Finally, Lemma 4 gives explicit bounds for the stochastic and deterministic errors of the variance stabilizing transform  Thus, it quantifies the difference to an exact Gaussian regression setting  This result is a generalization of Theorem 1 of Cai et al 
(2010) adapted to our setting with n ≥ 1 observations and correlated observations  √ 2 can be written as where for the proof of the first statement  Furthermore, for x, t ∈ [0, 1] holds In particular, for some constants C 1 , C 2 > 0 depending on γ ∈ (0, 1) but not on h and x, it holds h (iii) See Lemma 15 of with p = 2q − 1 
It is sufficient to show the statement for n = 1 by independence of the Y i   Then, the number of points per bin is m = p/T   For simplicity, the index i is skipped in the following  First, we write Q k as a matrix-vector product and refactor it so that it corresponds to a sum of independent scaled χ 2 random variables 
In the second step, we calculate the scaling factors  Let E (km) be a diagonal matrix with ones on the (k − 1)m + 1,    , km-th entries and otherwise zero diagonal elements  Then, By Theorem 1 of for the gamma distribution it follows where Wi,j iid  ∼ Γ(1/2, 2 f (x * k )) and such that Cov( Wi,j , Wi,h ) = Cov(W i,j , W i,h ) for j = (k − 1)p/T + 1,    , kp/T and h ∈ {1,    , p} \ {(k − 1)p/T + 1,    , kp/T } 
Let θ be the maximum difference of the observations' means in each bin  Then, θ = max are defined via quantile coupling, it holds Z k = Φ −1 {F Q( Qk )} (see   Furthermore, define the uniform random variables Let ρ = Cov(Z k , Z l )  Then, the identity implies F Z,Z (x, y) − Φ(x)Φ(y) ≥ 0 for all x, y ∈ R ⇐⇒ ρ ≥ 0, (see  
Since Cov( Qk , Ql ) ≥ 0 and the ratio of two densities is non-negative, x = − 2/m, it follows that f Q(x) is monotone decreasing for x ≥ − 2/m  Furthermore, F Q(− m/2) ≤ 0 5 for all m ∈ N as f Q(x) is right-skewed  In particular, − m/2 ≤ F −1 Q (1/2) for all m ∈ N  Finally, since f Q(− 2/m) → φ(0) for m → ∞ there is a constant c > 0 not depending on m such that
The simulation study in Section 5 is performed in the same way, but with the uniform and the gamma distribution instead of the Gaussian distribution 



ground_truth: Decorrelating the data before running the PLS algorithm improves the performance of the algorithm.