| M-SpatPlus44 | 0.0612 | 0.0279 | 0.0633 | 0.0281 | 0.0633 | 0.0280
| M-SpatPlus39 | 0.0571 | 0.0259 | 0.0593 | 0.0262 | 0.0599 | 0.0260
Table B.16: Estimated standard errors ($s.e._{est}$) and simulated standard errors ($s.e._{sim}$) for $\beta_{2}$ based on $300$ simulated datasets for Simulation study 2 and Scenarios 1 to 5. |  | ICAR | PCAR | BYM2
---|---|---|---|---
| Model | $s.e._{est}$ | $s.e._{sim}$ | $s.e._{est}$ | $s.e._{sim}$ | $s.e._{est}$ | $s.e._{sim}$
Scenario 1 | M-Spatial | 0.0476 | 0.0272 | 0.0519 | 0.0290 | 0.0501 | 0.0275
M-SpatPlus64 | 0.0380 | 0.0210 | 0.0404 | 0.0214 | 0.0398 | 0.0211
M-SpatPlus59 | 0.0370 | 0.0203 | 0.0393 | 0.0206 | 0.0388 | 0.0203
M-SpatPlus54 | 0.0367 | 0.0196 | 0.0390 | 0.0199 | 0.0385 | 0.0197
M-SpatPlus49 | 0.0355 | 0.0189 | 0.0378 | 0.0191 | 0.0374 | 0.0189
| M-SpatPlus44 | 0.0353 | 0.0199 | 0.0375 | 0.0201 | 0.0375 | 0.0201
| M-SpatPlus39 | 0.0341 | 0.0184 | 0.0364 | 0.0186 | 0.0367 | 0.0186
Scenario 2 | M-Spatial | 0.0381 | 0.0285 | 0.0393 | 0.0286 | 0.0399 | 0.0279
M-SpatPlus64 | 0.0371 | 0.0276 | 0.0382 | 0.0278 | 0.0392 | 0.0274
M-SpatPlus59 | 0.0380 | 0.0272 | 0.0389 | 0.0273 | 0.0401 | 0.0270
M-SpatPlus54 | 0.0374 | 0.0265 | 0.0383 | 0.0266 | 0.0395 | 0.0263
M-SpatPlus49 | 0.0368 | 0.0258 | 0.0376 | 0.0258 | 0.0389 | 0.0256
| M-SpatPlus44 | 0.0378 | 0.0261 | 0.0384 | 0.0261 | 0.0404 | 0.0261
| M-SpatPlus39 | 0.0373 | 0.0252 | 0.0379 | 0.0252 | 0.0404 | 0.0254
Scenario 3 | M-Spatial | 0.0423 | 0.0284 | 0.0446 | 0.0281 | 0.0442 | 0.0276
M-SpatPlus64 | 0.0381 | 0.0242 | 0.0425 | 0.0250 | 0.0412 | 0.0244
M-SpatPlus59 | 0.0375 | 0.0235 | 0.0415 | 0.0241 | 0.0405 | 0.0237
M-SpatPlus54 | 0.0368 | 0.0222 | 0.0407 | 0.0228 | 0.0399 | 0.0224
M-SpatPlus49 | 0.0357 | 0.0210 | 0.0394 | 0.0216 | 0.0389 | 0.0212
| M-SpatPlus44 | 0.0359 | 0.0220 | 0.0392 | 0.0223 | 0.0397 | 0.0224
| M-SpatPlus39 | 0.0350 | 0.0218 | 0.0384 | 0.0221 | 0.0389 | 0.0222
Scenario 4 | M-Spatial | 0.0384 | 0.0290 | 0.0398 | 0.0295 | 0.0399 | 0.0280
M-SpatPlus64 | 0.0378 | 0.0279 | 0.0389 | 0.0279 | 0.0395 | 0.0273
M-SpatPlus59 | 0.0391 | 0.0267 | 0.0399 | 0.0268 | 0.0409 | 0.0265
M-SpatPlus54 | 0.0385 | 0.0263 | 0.0393 | 0.0263 | 0.0404 | 0.0261
M-SpatPlus49 | 0.0380 | 0.0254 | 0.0387 | 0.0254 | 0.0399 | 0.0252
| M-SpatPlus44 | 0.0389 | 0.0262 | 0.0395 | 0.0262 | 0.0413 | 0.0262
| M-SpatPlus39 | 0.0380 | 0.0243 | 0.0386 | 0.0243 | 0.0409 | 0.0244
Scenario 5 | M-Spatial | 0.0392 | 0.0289 | 0.0404 | 0.0297 | 0.0405 | 0.0284
M-SpatPlus64 | 0.0379 | 0.0271 | 0.0388 | 0.0272 | 0.0393 | 0.0271
M-SpatPlus59 | 0.0394 | 0.0262 | 0.0402 | 0.0263 | 0.0411 | 0.0263
M-SpatPlus54 | 0.0387 | 0.0253 | 0.0394 | 0.0253 | 0.0404 | 0.0254
M-SpatPlus49 | 0.0382 | 0.0240 | 0.0389 | 0.0240 | 0.0400 | 0.0240
| M-SpatPlus44 | 0.0390 | 0.0241 | 0.0397 | 0.0241 | 0.0412 | 0.0243
| M-SpatPlus39 | 0.0384 | 0.0222 | 0.0390 | 0.0223 | 0.0411 | 0.0224
Table B.17: Empirical $95\%$ coverage probabilities of the true value of $\beta_{1}$ and $\beta_{2}$ based on $300$ simulated datasets for Simulation study 2 and Scenarios 3, 4 and 5. |  | $\beta_{1}$ | $\beta_{2}$
---|---|---|---
|  | ICAR | PCAR | BYM2 | ICAR | PCAR | BYM2
Scenario 3 | M-Spatial | 88.3333 | 85.3333 | 89.3333 | 1.3333 | 0.0000 | 0.0000
M-SpatPlus64 | 100.0000 | 100.0000 | 100.0000 | 73.6667 | 80.6667 | 78.0000
M-SpatPlus59 | 100.0000 | 100.0000 | 100.0000 | 92.6667 | 95.6667 | 95.6667
M-SpatPlus54 | 100.0000 | 100.0000 | 100.0000 | 99.0000 | 99.6667 | 99.6667
M-SpatPlus49 | 100.0000 | 100.0000 | 100.0000 | 100.0000 | 100.0000 | 100.0000
| M-SpatPlus44 | 100.0000 | 100.0000 | 100.0000 | 98.6667 | 99.6667 | 99.6667
| M-SpatPlus39 | 100.0000 | 100.0000 | 100.0000 | 98.0000 | 99.3333 | 99.0000
Scenario 4 | M-Spatial | 31.6667 | 35.0000 | 28.3333 | 0.6667 | 0.6667 | 0.3333
M-SpatPlus64 | 64.3333 | 68.3333 | 66.0000 | 8.0000 | 8.0000 | 4.0000
M-SpatPlus59 | 98.0000 | 99.0000 | 99.0000 | 70.3333 | 72.0000 | 71.0000
M-SpatPlus54 | 97.6667 | 99.0000 | 98.0000 | 88.3333 | 89.0000 | 90.6667
M-SpatPlus49 | 100.0000 | 100.0000 | 100.0000 | 94.6667 | 94.6667 | 95.6667
| M-SpatPlus44 | 100.0000 | 100.0000 | 100.0000 | 99.3333 | 99.3333 | 99.3333
| M-SpatPlus39 | 100.0000 | 100.0000 | 100.0000 | 92.6667 | 94.0000 | 95.3333
Scenario 5 | M-Spatial | 96.6667 | 98.6667 | 96.6667 | 0.3333 | 0.0000 | 0.0000
M-SpatPlus64 | 99.3333 | 99.3333 | 99.3333 | 2.3333 | 2.6667 | 2.0000
M-SpatPlus59 | 100.0000 | 100.0000 | 100.0000 | 59.0000 | 61.0000 | 62.6667
M-SpatPlus54 | 100.0000 | 100.0000 | 100.0000 | 82.0000 | 83.6667 | 85.6667
M-SpatPlus49 | 100.0000 | 100.0000 | 100.0000 | 93.3333 | 94.0000 | 94.6667
| M-Spatplus44 | 100.0000 | 100.0000 | 100.0000 | 99.3333 | 99.3333 | 99.3333
| M-SpatPlus39 | 100.0000 | 100.0000 | 100.0000 | 96.6667 | 97.3333 | 98.3333
Table B.18: DIC and WAIC based on 300 simulated data sets for Simulation Study 2 and Scenarios 3, 4 and 5. |  | ICAR | PCAR | BYM2
---|---|---|---|---
| Model | DIC | WAIC | DIC | WAIC | DIC | WAIC
Scenario 3 | M-Spatial | 967.9454 | 954.5795 | 967.5495 | 952.1199 | 967.9909 | 951.7634
M-SpatPlus64 | 969.1749 | 951.7115 | 970.9422 | 951.6440 | 970.9431 | 950.8905
M-SpatPlus59 | 970.2056 | 951.4510 | 972.0560 | 951.7777 | 972.2680 | 951.2902
M-SpatPlus54 | 972.2309 | 953.1453 | 973.6655 | 952.8703 | 974.4395 | 953.3604
M-SpatPlus49 | 973.2094 | 954.5434 | 974.3598 | 953.7926 | 975.3738 | 954.5668
| M-SpatPlus44 | 974.3461 | 955.1660 | 975.2471 | 954.0045 | 976.5248 | 955.1149
| M-SpatPlus39 | 974.7621 | 955.7078 | 975.6728 | 954.5366 | 976.9174 | 955.6734
Scenario 4 | M-Spatial | 966.8835 | 950.9785 | 967.8758 | 948.4934 | 967.5878 | 947.7641
M-SpatPlus64 | 967.1693 | 949.2779 | 968.0115 | 946.6388 | 968.0098 | 946.4287
M-SpatPlus59 | 969.9224 | 948.3949 | 970.8501 | 946.1590 | 971.2409 | 947.0300
M-SpatPlus54 | 970.9355 | 949.3026 | 971.8493 | 947.1336 | 972.3405 | 948.2459
M-SpatPlus49 | 972.1451 | 950.7576 | 973.0600 | 948.5624 | 973.5596 | 949.7205
| M-SpatPlus44 | 974.2859 | 952.6960 | 975.0717 | 950.2682 | 975.6985 | 951.5074
| M-SpatPlus39 | 978.3585 | 957.7842 | 979.0163 | 955.1304 | 979.6206 | 956.2045
Scenario 5 | M-Spatial | 965.3720 | 950.3049 | 966.0718 | 947.7260 | 965.6266 | 946.7197
M-SpatPlus64 | 965.5671 | 947.2383 | 966.5097 | 945.1004 | 966.4342 | 944.4985
M-SpatPlus59 | 968.6013 | 946.5191 | 969.6242 | 944.7155 | 969.9139 | 945.2128
M-SpatPlus54 | 969.9959 | 947.6624 | 971.0162 | 945.9452 | 971.4843 | 946.7808
M-SpatPlus49 | 971.3763 | 949.1848 | 972.3350 | 947.4099 | 972.8820 | 948.4109
| M-SpatPlus44 | 973.5362 | 951.3029 | 974.4013 | 949.3644 | 975.0159 | 950.4543
| M-SpatPlus39 | 978.1803 | 956.8477 | 978.9062 | 954.5679 | 979.5386 | 955.6113
Table B.19: Average value of mean absolute relative bias (MARB) and mean relative root mean prediction error (MRRMSE) of the relative risks based on $300$ simulated datasets for Simulation study 2 and Scenarios 1 to 5. |  | ICAR | PCAR | BYM2
---|---|---|---|---
| Model | MARB | MRRMSE | MARB | MRRMSE | MARB | MRRMSE
Scenario 1 | M-Spatial | 0.0675 | 0.1550 | 0.0649 | 0.1567 | 0.0639 | 0.1557
M-SpatPlus64 | 0.0652 | 0.1567 | 0.0632 | 0.1585 | 0.0626 | 0.1579
M-SpatPlus59 | 0.0660 | 0.1579 | 0.0643 | 0.1597 | 0.0642 | 0.1592
M-SpatPlus54 | 0.0657 | 0.1583 | 0.0638 | 0.1601 | 0.0639 | 0.1598
M-SpatPlus49 | 0.0694 | 0.1608 | 0.0671 | 0.1623 | 0.0676 | 0.1622
| M-SpatPlus44 | 0.0695 | 0.1612 | 0.0671 | 0.1626 | 0.0677 | 0.1626
| M-SpatPlus39 | 0.0737 | 0.1655 | 0.0705 | 0.1666 | 0.0715 | 0.1669
Scenario 2 | M-Spatial | 0.0634 | 0.1580 | 0.0597 | 0.1597 | 0.0602 | 0.1594
M-SpatPlus64 | 0.0623 | 0.1585 | 0.0584 | 0.1601 | 0.0589 | 0.1598
M-SpatPlus59 | 0.0608 | 0.1606 | 0.0571 | 0.1621 | 0.0583 | 0.1621
M-SpatPlus54 | 0.0616 | 0.1615 | 0.0580 | 0.1630 | 0.0593 | 0.1630
M-SpatPlus49 | 0.0622 | 0.1632 | 0.0593 | 0.1646 | 0.0601 | 0.1647
| M-SpatPlus44 | 0.0616 | 0.1649 | 0.0588 | 0.1663 | 0.0595 | 0.1664
| M-SpatPlus39 | 0.0685 | 0.1705 | 0.0651 | 0.1718 | 0.0657 | 0.1718
Scenario 3 | M-Spatial | 0.0668 | 0.1545 | 0.0647 | 0.1555 | 0.0650 | 0.1555
M-SpatPlus64 | 0.0625 | 0.1571 | 0.0614 | 0.1598 | 0.0608 | 0.1592
M-SpatPlus59 | 0.0618 | 0.1584 | 0.0617 | 0.1612 | 0.0609 | 0.1608
M-SpatPlus54 | 0.0633 | 0.1612 | 0.0624 | 0.1634 | 0.0627 | 0.1637
M-SpatPlus49 | 0.0669 | 0.1641 | 0.0650 | 0.1657 | 0.0659 | 0.1665
| M-SpatPlus44 | 0.0652 | 0.1648 | 0.0637 | 0.1663 | 0.0644 | 0.1673
| M-SpatPlus39 | 0.0674 | 0.1661 | 0.0654 | 0.1674 | 0.0665 | 0.1684
Scenario 4 | M-Spatial | 0.0621 | 0.1540 | 0.0588 | 0.1557 | 0.0596 | 0.1554
M-SpatPlus64 | 0.0600 | 0.1552 | 0.0564 | 0.1569 | 0.0572 | 0.1566
M-SpatPlus59 | 0.0586 | 0.1596 | 0.0550 | 0.1612 | 0.0569 | 0.1612
M-SpatPlus54 | 0.0595 | 0.1608 | 0.0559 | 0.1623 | 0.0578 | 0.1625
M-SpatPlus49 | 0.0608 | 0.1627 | 0.0578 | 0.1642 | 0.0594 | 0.1643
| M-SpatPlus44 | 0.0602 | 0.1643 | 0.0570 | 0.1657 | 0.0584 | 0.1659
| M-SpatPlus39 | 0.0691 | 0.1706 | 0.0654 | 0.1718 | 0.0665 | 0.1719
Scenario 5 | M-Spatial | 0.0594 | 0.1511 | 0.0571 | 0.1528 | 0.0574 | 0.1524
M-SpatPlus64 | 0.0559 | 0.1527 | 0.0534 | 0.1545 | 0.0536 | 0.1543
M-SpatPlus59 | 0.0547 | 0.1573 | 0.0519 | 0.1589 | 0.0532 | 0.1590
M-SpatPlus54 | 0.0559 | 0.1590 | 0.0533 | 0.1606 | 0.0548 | 0.1608
M-SpatPlus49 | 0.0575 | 0.1606 | 0.0550 | 0.1622 | 0.0565 | 0.1625
| M-SpatPlus44 | 0.0564 | 0.1625 | 0.0539 | 0.1640 | 0.0554 | 0.1643
| M-SpatPlus39 | 0.0642 | 0.1688 | 0.0611 | 0.1701 | 0.0624 | 0.1703
Table B.20: Length of the $95\%$ credible intervals of $\beta_{1}$ and $\beta_{2}$ based on $300$ simulated datasets for Simulation study 2 and Scenarios 1 to 5. |  | $\beta_{1}$ | $\beta_{2}$
---|---|---|---
| Model | ICAR | PCAR | BYM2 | ICAR | PCAR | BYM2
Scenario 1 | M-Spatial | 0.3208 | 0.3293 | 0.3328 | 0.1871 | 0.2039 | 0.1972
M-SpatPlus64 | 0.2407 | 0.2435 | 0.2478 | 0.1495 | 0.1587 | 0.1565
M-SpatPlus59 | 0.2304 | 0.2331 | 0.2382 | 0.1455 | 0.1546 | 0.1527
M-SpatPlus54 | 0.2280 | 0.2309 | 0.2358 | 0.1443 | 0.1534 | 0.1516
M-SpatPlus49 | 0.2215 | 0.2245 | 0.2298 | 0.1395 | 0.1485 | 0.1469
| M-SpatPlus44 | 0.2154 | 0.2184 | 0.2259 | 0.1389 | 0.1474 | 0.1476
| M-SpatPlus39 | 0.2036 | 0.2067 | 0.2161 | 0.1341 | 0.1429 | 0.1442
Scenario 2 | M-Spatial | 0.2694 | 0.2800 | 0.2785 | 0.1499 | 0.1543 | 0.1569
M-SpatPlus64 | 0.2579 | 0.2678 | 0.2655 | 0.1460 | 0.1502 | 0.1542
M-SpatPlus59 | 0.2515 | 0.2615 | 0.2582 | 0.1495 | 0.1530 | 0.1575
M-SpatPlus54 | 0.2457 | 0.2555 | 0.2524 | 0.1472 | 0.1505 | 0.1552
M-SpatPlus49 | 0.2408 | 0.2503 | 0.2481 | 0.1448 | 0.1478 | 0.1531
| M-SpatPlus44 | 0.2355 | 0.2453 | 0.2455 | 0.1485 | 0.1509 | 0.1588
| M-SpatPlus39 | 0.2232 | 0.2331 | 0.2358 | 0.1467 | 0.1488 | 0.1588
Scenario 3 | M-Spatial | 0.3175 | 0.3197 | 0.3234 | 0.1664 | 0.1753 | 0.1736
M-SpatPlus64 | 0.2584 | 0.2567 | 0.2635 | 0.1499 | 0.1671 | 0.1619
M-SpatPlus59 | 0.2464 | 0.2452 | 0.2522 | 0.1474 | 0.1632 | 0.1591
M-SpatPlus54 | 0.2341 | 0.2330 | 0.2404 | 0.1445 | 0.1598 | 0.1570
M-SpatPlus49 | 0.2266 | 0.2261 | 0.2340 | 0.1402 | 0.1547 | 0.1531
| M-SpatPlus44 | 0.2179 | 0.2178 | 0.2274 | 0.1410 | 0.1540 | 0.1560
| M-SpatPlus39 | 0.2118 | 0.2118 | 0.2215 | 0.1377 | 0.1509 | 0.1532
Scenario 4 | M-Spatial | 0.2855 | 0.2953 | 0.2940 | 0.1509 | 0.1566 | 0.1570
M-SpatPlus64 | 0.2699 | 0.2788 | 0.2764 | 0.1486 | 0.1530 | 0.1554
M-SpatPlus59 | 0.2547 | 0.2641 | 0.2601 | 0.1535 | 0.1568 | 0.1608
M-SpatPlus54 | 0.2486 | 0.2579 | 0.2541 | 0.1514 | 0.1546 | 0.1589
M-SpatPlus49 | 0.2434 | 0.2525 | 0.2494 | 0.1492 | 0.1522 | 0.1570
| M-SpatPlus44 | 0.2385 | 0.2477 | 0.2471 | 0.1529 | 0.1554 | 0.1625
| M-SpatPlus39 | 0.2234 | 0.2327 | 0.2346 | 0.1494 | 0.1517 | 0.1610
Scenario 5 | M-Spatial | 0.3051 | 0.3141 | 0.3113 | 0.1540 | 0.1585 | 0.1592
M-SpatPlus64 | 0.2764 | 0.2849 | 0.2825 | 0.1488 | 0.1526 | 0.1544
M-SpatPlus59 | 0.2595 | 0.2683 | 0.2650 | 0.1548 | 0.1579 | 0.1614
M-SpatPlus54 | 0.2515 | 0.2602 | 0.2576 | 0.1520 | 0.1550 | 0.1589
M-SpatPlus49 | 0.2456 | 0.2539 | 0.2521 | 0.1501 | 0.1531 | 0.1572
| M-SpatPlus44 | 0.2405 | 0.2489 | 0.2490 | 0.1534 | 0.1560 | 0.1621
| M-SpatPlus39 | 0.2246 | 0.2332 | 0.2356 | 0.1507 | 0.1534 | 0.1618
Table B.21: Average value of mean absolute relative bias (MARB) and mean relative root mean prediction error (MRRMSE) of $\beta_{1}$ based on $300$ simulated datasets for Simulation study 2 and Scenarios 1 to 5. |  | ICAR | PCAR | BYM2
---|---|---|---|---
| Model | MARB | MRRMSE | MARB | MRRMSE | MARB | MRRMSE
Scenario 1 | M-Spatial | 0.8006 | 0.8317 | 0.8439 | 0.8738 | 0.8399 | 0.8696
M-SpatPlus64 | 0.2523 | 0.3038 | 0.2473 | 0.2997 | 0.2635 | 0.3130
M-SpatPlus59 | 0.2762 | 0.3223 | 0.2766 | 0.3227 | 0.3076 | 0.3500
M-SpatPlus54 | 0.1391 | 0.2129 | 0.1365 | 0.2114 | 0.1541 | 0.2233
M-SpatPlus49 | 0.0534 | 0.1595 | 0.0527 | 0.1594 | 0.0663 | 0.1642
| M-SpatPlus44 | 0.0632 | 0.1645 | 0.0618 | 0.1642 | 0.0499 | 0.1611
| M-SpatPlus39 | 0.1601 | 0.2101 | 0.1598 | 0.2100 | 0.1407 | 0.1961
Scenario 2 | M-Spatial | 0.7923 | 0.8164 | 0.8174 | 0.8414 | 0.8222 | 0.8448
M-SpatPlus64 | 0.6038 | 0.6326 | 0.5998 | 0.6295 | 0.6044 | 0.6326
M-SpatPlus59 | 0.4398 | 0.4764 | 0.4267 | 0.4652 | 0.4265 | 0.4635
M-SpatPlus54 | 0.4486 | 0.4828 | 0.4380 | 0.4738 | 0.4424 | 0.4765
M-SpatPlus49 | 0.3310 | 0.3712 | 0.3216 | 0.3637 | 0.3233 | 0.3635
| M-SpatPlus44 | 0.1332 | 0.2217 | 0.1235 | 0.2174 | 0.1222 | 0.2156
| M-SpatPlus39 | 0.0304 | 0.1623 | 0.0368 | 0.1651 | 0.0395 | 0.1639
Scenario 3 | M-Spatial | 0.8103 | 0.8347 | 0.8355 | 0.8602 | 0.8133 | 0.8378
M-SpatPlus64 | 0.2987 | 0.3408 | 0.2809 | 0.3250 | 0.2833 | 0.3270
M-SpatPlus59 | 0.2849 | 0.3251 | 0.2730 | 0.3144 | 0.2871 | 0.3267
M-SpatPlus54 | 0.1723 | 0.2271 | 0.1634 | 0.2199 | 0.1762 | 0.2295
M-SpatPlus49 | 0.0178 | 0.1415 | 0.0074 | 0.1406 | 0.0129 | 0.1400
| M-SpatPlus44 | 0.0739 | 0.1570 | 0.0810 | 0.1604 | 0.0761 | 0.1578
| M-SpatPlus39 | 0.0235 | 0.1376 | 0.0299 | 0.1387 | 0.0164 | 0.1360
Scenario 4 | M-Spatial | 1.0548 | 1.0756 | 1.0739 | 1.0947 | 1.0995 | 1.1188
M-SpatPlus64 | 0.8148 | 0.8390 | 0.8169 | 0.8414 | 0.8285 | 0.8518
M-SpatPlus59 | 0.4753 | 0.5099 | 0.4623 | 0.4986 | 0.4622 | 0.4973
M-SpatPlus54 | 0.4700 | 0.5032 | 0.4598 | 0.4944 | 0.4627 | 0.4960
M-SpatPlus49 | 0.3457 | 0.3855 | 0.3359 | 0.3778 | 0.3368 | 0.3774
| M-SpatPlus44 | 0.1719 | 0.2464 | 0.1623 | 0.2411 | 0.1619 | 0.2401
| M-SpatPlus39 | 0.0240 | 0.1585 | 0.0178 | 0.1596 | 0.0193 | 0.1579
Scenario 5 | M-Spatial | 0.5787 | 0.6219 | 0.5447 | 0.5910 | 0.5826 | 0.6238
M-SpatPlus64 | 0.3976 | 0.4488 | 0.4029 | 0.4542 | 0.4070 | 0.4557
M-SpatPlus59 | 0.0902 | 0.2156 | 0.0787 | 0.2125 | 0.0678 | 0.2060
M-SpatPlus54 | 0.1071 | 0.2178 | 0.0984 | 0.2151 | 0.0926 | 0.2102
M-SpatPlus49 | 0.0482 | 0.1852 | 0.0392 | 0.1846 | 0.0369 | 0.1814
| M-SpatPlus44 | 0.1255 | 0.2244 | 0.1332 | 0.2299 | 0.1413 | 0.2339
| M-SpatPlus39 | 0.1989 | 0.2636 | 0.2041 | 0.2685 | 0.2077 | 0.2704
Table B.22: Average value of mean absolute relative bias (MARB) and mean relative root mean prediction error (MRRMSE) of $\beta_{2}$ based on $300$ simulated datasets for Simulation study 2 and Scenarios 1 to 5. |  | ICAR | PCAR | BYM2
---|---|---|---|---
| Model | MARB | MRRMSE | MARB | MRRMSE | MARB | MRRMSE
Scenario 1 | M-Spatial | 0.5088 | 0.5267 | 0.6421 | 0.6583 | 0.5893 | 0.6051
M-SpatPlus64 | 0.0274 | 0.1087 | 0.0255 | 0.1098 | 0.0345 | 0.1109
M-SpatPlus59 | 0.0309 | 0.1061 | 0.0327 | 0.1081 | 0.0263 | 0.1050
M-SpatPlus54 | 0.0854 | 0.1301 | 0.0856 | 0.1314 | 0.0850 | 0.1300
M-SpatPlus49 | 0.1297 | 0.1604 | 0.1303 | 0.1616 | 0.1277 | 0.1587
| M-SpatPlus44 | 0.1976 | 0.2213 | 0.2003 | 0.2241 | 0.1936 | 0.2181
| M-SpatPlus39 | 0.3479 | 0.3599 | 0.3492 | 0.3614 | 0.3440 | 0.3564
Scenario 2 | M-Spatial | 0.4378 | 0.4603 | 0.4570 | 0.4789 | 0.4984 | 0.5175
M-SpatPlus64 | 0.3371 | 0.3643 | 0.3471 | 0.3738 | 0.3855 | 0.4091
M-SpatPlus59 | 0.1968 | 0.2392 | 0.2017 | 0.2436 | 0.2116 | 0.2509
M-SpatPlus54 | 0.1568 | 0.2053 | 0.1616 | 0.2092 | 0.1683 | 0.2134
M-SpatPlus49 | 0.0945 | 0.1599 | 0.0949 | 0.1601 | 0.1065 | 0.1666
| M-SpatPlus44 | 0.0619 | 0.1445 | 0.0601 | 0.1436 | 0.0474 | 0.1391
| M-SpatPlus39 | 0.2732 | 0.3009 | 0.2709 | 0.2988 | 0.2641 | 0.2931
Scenario 3 | M-Spatial | 0.7698 | 0.7827 | 0.8939 | 0.9049 | 0.8795 | 0.8902
M-SpatPlus64 | 0.3011 | 0.3245 | 0.3167 | 0.3406 | 0.3150 | 0.3377
M-SpatPlus59 | 0.2010 | 0.2327 | 0.2063 | 0.2390 | 0.2004 | 0.2329
M-SpatPlus54 | 0.0659 | 0.1289 | 0.0721 | 0.1349 | 0.0556 | 0.1250
M-SpatPlus49 | 0.0115 | 0.1057 | 0.0009 | 0.1080 | 0.0188 | 0.1079
| M-SpatPlus44 | 0.1098 | 0.1553 | 0.1034 | 0.1522 | 0.1133 | 0.1592
| M-SpatPlus39 | 0.1181 | 0.1606 | 0.1122 | 0.1574 | 0.1193 | 0.1630
Scenario 4 | M-Spatial | 0.7372 | 0.7513 | 0.7891 | 0.8029 | 0.8242 | 0.8360
M-SpatPlus64 | 0.5784 | 0.5950 | 0.5887 | 0.6051 | 0.6296 | 0.6442
M-SpatPlus59 | 0.3016 | 0.3298 | 0.3060 | 0.3340 | 0.3115 | 0.3385
M-SpatPlus54 | 0.2395 | 0.2732 | 0.2430 | 0.2763 | 0.2454 | 0.2779
M-SpatPlus49 | 0.1667 | 0.2094 | 0.1664 | 0.2093 | 0.1722 | 0.2133
| M-SpatPlus44 | 0.0244 | 0.1333 | 0.0257 | 0.1335 | 0.0329 | 0.1348
| M-SpatPlus39 | 0.1900 | 0.2256 | 0.1884 | 0.2242 | 0.1827 | 0.2198
Scenario 5 | M-Spatial | 0.8536 | 0.8657 | 0.9415 | 0.9531 | 0.9496 | 0.9602
M-SpatPlus64 | 0.6253 | 0.6399 | 0.6285 | 0.6431 | 0.6637 | 0.6774
M-SpatPlus59 | 0.3510 | 0.3747 | 0.3518 | 0.3756 | 0.3568 | 0.3804
M-SpatPlus54 | 0.2712 | 0.2992 | 0.2725 | 0.3005 | 0.2739 | 0.3019
M-SpatPlus49 | 0.1982 | 0.2316 | 0.1973 | 0.2309 | 0.2004 | 0.2336
| M-SpatPlus44 | 0.0699 | 0.1394 | 0.0696 | 0.1393 | 0.0749 | 0.1426
| M-SpatPlus39 | 0.1601 | 0.1950 | 0.1600 | 0.1949 | 0.1555 | 0.1917


# MBIR Training for a 2.5D DL network in X-ray CT

Obaidullah Rahman  Department of Electrical Engineering, University of Notre
Dame, USA  Madhuri Nagare  School of Electrical Electrical and Computer
Engineering, Purdue University, West Lafayette, USA  Ken D. Sauer  Department
of Electrical Engineering, University of Notre Dame, USA  Charles A. Bouman
School of Electrical Electrical and Computer Engineering, Purdue University,
West Lafayette, USA  Roman Melnyk  General Electric Healthcare, Waukesha, USA
Brian Nett  General Electric Healthcare, Waukesha, USA  Jie Tang  General
Electric Healthcare, Waukesha, USA

Abstract In computed tomographic imaging, model based iterative reconstruction
methods have generally shown better image quality than the more traditional,
faster filtered backprojection technique. The cost we have to pay is that MBIR
is computationally expensive. In this work we train a 2.5D deep learning (DL)
network to mimic MBIR quality image. The network is realized by a modified
Unet, and trained using clinical FBP and MBIR image pairs. We achieve the
quality of MBIR images faster and with a much smaller computation cost.
Visually and in terms of noise power spectrum (NPS), DL-MBIR images have
texture similar to that of MBIR, with reduced noise power. Image profile
plots, NPS plots, standard deviation, etc. suggest that the DL-MBIR images
result from a successful emulation of an MBIR operator.
---

##  1 Introduction

X-ray computed tomography has become a important tool in applications such as
healthcare diagnostics, security inspection, and non-destructive testing. The
industry preferred method of reconstruction is filtered backprojection (FBP)
and its popularity is owed to its speed and low computational cost. Iterative
methods such as model-based iterative reconstruction (MBIR) generally have
better image quality than FBP and do better in limiting image artifacts [1,
2].

MBIR is a computationally expensive and potentially slow reconstruction method
since it entails repeated forward projection of the estimated image and back
projection of the sinogram residual error. Even with fast GPUs becoming the
norm, MBIR may take minutes compared to an FBP reconstruction that can be
performed in seconds. The computational cost and reconstruction time have been
deterents in wide adoption of MBIR.

In recent years, deep learning has made serious inroads in CT applications. It
is applied in sinogram and image domains and sometimes in both. It has been
applied in low signal correction [3], image denoising [4, 5], and metal
artifact reduction [6].

Ziabari, et al [7] showed that a 2.5D deep neural network, with proper
training, can effectively learn a mapping from an FBP image to MBIR. In this
paper we expand on their work and study the characteristics of output from
networks trained to simulate MBIR with a highly efficient neural network
implementation.

##  2 Methods

We will first train a deep neural network, which we will, similarly to [7],
entitle DL-MBIR. Our aim is to train the network to closely approximate MBIR
images from FBP images. The training input is FBP images and the target is
MBIR images from the same data. Let $X_{FBP}$ be the input to the network,
$X_{MBIR}$ be the target, and $\sigma$ represents a hypothetical mapping such
that $\sigma:X_{MBIR}\rightarrow X_{FBP}$. Let $f_{DL\text{-}MBIR_{Z}}$ be the
DL neural network with $Z$ number of input channels. During the training
phase:

$\displaystyle\hat{f}_{DL\text{-}MBIR_{Z}}=\operatorname*{argmin}_{f_{DL\text{-}MBIR_{Z}}}{\big{|}}{\big{|}}f_{DL\text{-}MBIR_{Z}}(X_{FBP})-X_{MBIR}{\big{|}}{\big{|}}_{2}$
(1)

$f_{DL\text{-}MBIR_{Z}}$ can be thought of as the inverse of $\sigma$, i.e.
$f_{DL\text{-}MBIR_{Z}}=\sigma^{-1}$. During the training phase, the weights
of $f_{DL\text{-}MBIR_{Z}}$ are randomly initialized and then adjusted in
several iterations using error backpropagation. Once the number of iterations
is exhausted or the convergence criteria is met, the training stops.

For training, 4 pairs of clinical exams were selected. Each pair had one FBP
image volume and the corresponding MBIR volume. Each image volume had about
200 slices, resulting in about 800 training image pairs. A modified version of
Unet [8] was chosen as the network architecture. The learning rate was set to
$0.0001$ and 2 GPUs were used. Training and inferencing were done on
Tensorflow/Keras. Training was run for 300 epochs. 3 versions of DL-MBIR were
trained: $DL\text{-}MBIR_{1}$ was trained with inputs with 1 channel i.e. 1
axial slice, $DL\text{-}MBIR_{3}$ was trained with 3-channel inputs and
$DL\text{-}MBIR_{5}$ was trained with 5-channel inputs. Having adjacent slices
in the input provides additional information to the DL network [7] and helps
train it better. Figure 1 shows the DL architecture and the training setup.

Figure 1: DL architecture and training setup. This is a modified form of Unet.
Layers on the left denote the contracting path where features are compressed
from image towards latent space but the number of features increases. Layers
on the right denote the expanding path where feature are decompressed from
latent space towards corrected image but the number of features decreases.

##  3 Results

A cardiac FBP image was inferenced on the trained DL-MBIR network. Inference
time for every network was between 4 and 6 seconds, and it goes up with the
increase in the number of input channels. The MBIR version of the same exam
was also available. Figure 3 shows a comparison, for 4 slices –
LABEL:sub@fig:Image_10, LABEL:sub@fig:Image_50, LABEL:sub@fig:Image_90, and
LABEL:sub@fig:Image_170 in the image volume, among MBIR image, FBP image, and
the outputs of $DL\text{-}MBIR_{Z}$, where $Z=1,\ 3,\ 5$. Figure 4 shows a
comparison, for the same slices in the image volume, among difference between
images and the MBIR images. Figure 5 has a profile plot to show the comparison
of $DL\text{-}MBIR_{Z}$ and FBP images w.r.t the MBIR images.

Peak signal to noise ratio (PSNR) is another measure of similarity between
images and is closely related to mean squared error. A higher value would mean
that the image is closer to the reference image. Figure 2 is a plot of PSNR of
all slices within the images with MBIR as the reference image. Table 1 has
some other metrics of comparison among the images, such as averaged (across
slices) PSNR, standard deviation (std) within regions of interest (RoIs), and
average CT number within those RoIs.

The noise power spectrum (NPS) is a reliable tool for demonstrating similarity
in the image texture. To measure NPS, uniform region patches from one of the
cardiac chambers were extracted from each image. Then NPS was measured for all
patches and averaged. Then 1D radial profile was measured from the 2D NPS.
Figure 6 shows NPS in the uniform region within one of the cardiac chambers.

Figure 2: PSNR (with MBIR image as reference). PSNR values are in dB. X axis
represents the axial slices in the image volume.

(a)

(b)

(c)

(d)

Figure 3: Reconstructed image. (left to right): $MBIR$, $FBP$,
$DL\text{-}MBIR_{1}$, $DL\text{-}MBIR_{3}$, $DL\text{-}MBIR_{5}$.
LABEL:sub@fig:Image_10, LABEL:sub@fig:Image_50, LABEL:sub@fig:Image_90 and
LABEL:sub@fig:Image_170 represent different slices in the image volume. WW/WL
450/0 HU.

(a)

(b)

(c)

(d)

Figure 4: Difference image w.r.t. MBIR. (left to right): $FBP$,
$DL\text{-}MBIR_{1}$, $DL\text{-}MBIR_{3}$, $DL\text{-}MBIR_{5}$.
LABEL:sub@fig:Error_10, LABEL:sub@fig:Error_50, LABEL:sub@fig:Error_90 and
LABEL:sub@fig:Error_170 represent different slices in the image volume. WW/WL
150/0 HU.

(a)

(b)

Figure 5: Image profile. LABEL:sub@fig:Image_160_yline One axial slice, from left to right, of: $MBIR$, $FBP$, $DL\text{-}MBIR_{1}$, $DL\text{-}MBIR_{3}$, $DL\text{-}MBIR_{5}$ LABEL:sub@fig:Profile_160 Profile plot of the images along the yellow line. | $MBIR$ | $FBP$ | $DL\text{-}MBIR_{1}$ | $DL\text{-}MBIR_{3}$ | $DL\text{-}MBIR_{5}$
---|---|---|---|---|---
PSNR | - | 64.98 | 68.99 | 70.63 | 71.08
std | 25.07 | 46.65 | 26.39 | 27.24 | 25.85
average | 74.68 | 75.20 | 77.0 | 71.52 | 74.43
Table 1: Performance metrics. PSNR (dB) is calculated w.r.t the MBIR images
and averaged for all axial slices. Standard deviation (std) and the average
value are in HU, with water being 0 and air -1000. Figure 6: NPS plot
demonstrating similar behavior of DL-MBIR with MBIR for noise power.

##  4 Discussion

Visually, all DL-MBIR images bear close resemblance to the MBIR images in
figure 3. It is confirmed by the difference images in figure 4. In the profile
plot of Figure 5, the DL-MBIR profiles closely follow that of MBIR.

All DL-MBIR images have higher PSNR than that of FBP, with
$DL\text{-}MBIR_{5}$ having the best. Ziabari, et al [7] achieved a PSNR gain
over FBP of 3.4 dB for $DL\text{-}MBIR_{1}$, compared to 4.1 dB with the
current implementation. For $DL\text{-}MBIR_{5}$, we have improved the result
from 4.25 dB to 6.1 dB over FBP. DL-MBIR images have standard deviations
nearly the same as MBIR, with $DL\text{-}MBIR_{5}$ outperforming the rest. The
average within the chosen RoI is more or less preserved in all images.

In figure 6, noise power spectrum (NPS) plots of DL-MBIR images are quite
close to that of MBIR, indicating that the DL-MBIR image texture is also
similar to that of MBIR, and it appears this attribute is learned well by the
network. Due to its type of adaptive regularization, MBIR may create
distinctive texture in the surviving image noise. The attenuation of noise is
a clear gain; however this texture and its effect on low-contrast
detectability may be of concern to some users.

##  5 Conclusion

We trained a U-net, 2.5D DL network that effectively estimates MBIR results
from FBP input images. The computation cost is also signifcantly less than
that of MBIR. All metrics – NPS, PSNR, standard deviation, profile plots
demonstrate that DL-MBIR images have all the features of MBIR including noise
reduction and noise texture.

## References

  * [1] Jean-Baptiste Thibault, Ken D Sauer, Charles A Bouman and Jiang Hsieh  “A three-dimensional statistical approach to improved image quality for multislice helical CT”  In _Medical physics_ 34.11 Wiley Online Library, 2007, pp. 4526–4544
  * [2] Zhou Yu et al.  “Fast model-based X-ray CT reconstruction using spatially nonhomogeneous ICD optimization”  In _IEEE Transactions on image processing_ 20.1 IEEE, 2010, pp. 161–175
  * [3] Mingrui Geng et al.  “Unsupervised/semi-supervised deep learning for low-dose CT enhancement”  In _arXiv preprint arXiv:1808.02603_ , 2018
  * [4] Qingsong Yang et al.  “Low-dose CT image denoising using a generative adversarial network with Wasserstein distance and perceptual loss”  In _IEEE transactions on medical imaging_ 37.6 IEEE, 2018, pp. 1348–1357
  * [5] Masakazu Matsuura, Jian Zhou, Naruomi Akino and Zhou Yu  “Feature-Aware Deep-Learning Reconstruction for Context-Sensitive X-ray Computed Tomography”  In _IEEE Transactions on Radiation and Plasma Medical Sciences_ 5.1 IEEE, 2020, pp. 99–107
  * [6] Muhammad Usman Ghani and W Clem Karl  “Fast enhanced CT metal artifact reduction using data domain deep learning”  In _IEEE Transactions on Computational Imaging_ 6 IEEE, 2019, pp. 181–193
  * [7] Amirkoushyar Ziabari et al.  “2.5 D deep learning for CT image reconstruction using a multi-GPU implementation”  In _2018 52nd Asilomar Conference on Signals, Systems, and Computers_ , 2018, pp. 2044–2049  IEEE
  * [8] Olaf Ronneberger, Philipp Fischer and Thomas Brox  “U-net: Convolutional networks for biomedical image segmentation”  In _International Conference on Medical image computing and computer-assisted intervention_ , 2015, pp. 234–241  Springer


# The extraordinary mid-infrared spectral properties of FeLoBAL Quasars

D. Farrah11affiliation: Astronomy Centre, University of Sussex, Brighton, UK
T. Urrutia22affiliation: Spitzer Science Center, California Institute of
Technology, Pasadena, CA 91125, USA M. Lacy33affiliation: National Radio
Astronomy Observatory, Charlottesville, Virginia, USA V.
Lebouteiller44affiliation: Department of Astronomy, Cornell University,
Ithaca, NY, USA H. W. W. Spoon44affiliation: Department of Astronomy, Cornell
University, Ithaca, NY, USA J. Bernard-Salas44affiliation: Department of
Astronomy, Cornell University, Ithaca, NY, USA N. Connolly55affiliation:
Physics Department, Hamilton College, Clinton, NY 13323, USA J.
Afonso66affiliation: Observatório Astronómico de Lisboa, Faculdade de
Ciências, Universidade de Lisboa, Tapada da Ajuda, 1349-018 Lisbon, Portugal
77affiliation: Centro de Astronomia e Astrofísica da Universidade de Lisboa,
Lisbon, Portugal B. Connolly88affiliation: Department of Physics and
Astronomy, University of Pennsylvania, Philadelphia, PA 19104-6396, USA J.
Houck44affiliation: Department of Astronomy, Cornell University, Ithaca, NY,
USA

###### Abstract

We present mid-infrared spectra of six FeLoBAL QSOs at $1<z<1.8$, taken with
the Spitzer space telescope. The spectra span a range of shapes, from hot dust
dominated AGN with silicate emission at 9.7$\mu$m, to moderately obscured
starbursts with strong Polycyclic Aromatic Hydrocarbon (PAH) emission. The
spectrum of one object, SDSS 1214-0001, shows the most prominent PAHs yet seen
in any QSO at any redshift, implying that the starburst dominates the mid-IR
emission with an associated star formation rate of order 2700 M⊙ yr-1. With
the caveats that our sample is small and not robustly selected, we combine our
mid-IR spectral diagnostics with previous observations to propose that FeLoBAL
QSOs are at least largely comprised of systems in which (a) a merger driven
starburst is ending, (b) a luminous AGN is in the last stages of burning
through its surrounding dust, and (c) which we may be viewing over a
restricted line of sight range.

###### Subject headings:

galaxies: active – quasars: absorption lines – infrared: galaxies – galaxies:
evolution

##  1\. Introduction

It is now well-established that there exist intimate links between the growth
of supermassive black holes and the production of stars in galaxies. Indirect
evidence for such links comes from, for example, the tight relationship
between central black hole and stellar bulge masses (e.g. Magorrian et al.
1998; Gebhardt et al. 2000; Gadotti & Kauffmann 2009), and the coeval stellar
populations in many massive elliptical galaxies (Dunlop et al., 1996; Ellis et
al., 1997; Rakos et al., 2007), suggesting their stars formed within less than
a Gyr of each other. Direct evidence comes from the existence of galaxies that
simultaneously harbor both high rates of star formation (of order tens to
thousands M⊙yr-1) and rapid accretion onto a central black hole (Sanders &
Mirabel, 1996; Genzel et al., 1998; Farrah et al., 2003; Lonsdale et al.,
2006), all enveloped in large quantities of dust, making them extremely
luminous in the infrared. The importance of these links for understanding the
assembly history of galaxies over at least a substantial fraction of the age
of the Universe is demonstrated by the strong evolution of the luminosity
function of IR-luminous galaxies with redshift (Saunders et al., 1990; Le
Floc’h et al., 2005), and the existence of a profusion of IR-luminous sources
in the high-redshift Universe (e.g. Barger et al. 1998; Eales et al. 1999;
Coppin et al. 2006; Austermann et al. 2009).

An insightful way to study these links is to identify active galaxies at ‘key’
points, where they are either rapidly building up stellar or central black
hole mass, or shifting from one evolutionary phase to the next. One such point
may be the ”youthful” QSOs; if the starburst precedes or is coeval with the
QSO111The (optical) QSO phase is not necessarily the period in which the SMBH
gains the bulk of its mass; indeed, there is evidence that the major periods
of BH growth are coeval with or precede the starburst, see e.g. Martínez-
Sansigre et al. (2005); Merloni et al. (2010); Treister et al. (2010), then
young QSOs should be those in which the starburst is drawing to a close, and a
QSO is starting to emerge from its surrounding dust. Such objects would
provide an excellent laboratory for testing feedback mechanisms between the
starburst and AGN (e.g. Silk & Rees 1998; Ciotti & Ostriker 2007; Lagos et al.
2008; Moe et al. 2009; Ceverino & Klypin 2009).

Finding the youthful QSO population is a challenging and ongoing problem (e.g.
Sanders et al. 1988; Canalizo & Stockton 2001; Lacy 2006; Coppin et al. 2008;
Georgakakis et al. 2009; Lípari et al. 2009). Significant attention has
focused on the Broad Absorption Line (BAL) QSOs (Lynds, 1967; Weymann et al.,
1991; Brotherton et al., 1998; Schmidt & Hines, 1999; Arav et al., 2001; Green
et al., 2001; Hall et al., 2002; Reichard et al., 2003; Priddey et al., 2007;
Gibson et al., 2009), whose properties222BAL QSOs come in 3 subtypes. High
Ionization BAL QSOs (HiBALs) show absorption in CIV $\lambda$1549, NV
$\lambda$1240, SiIV $\lambda$1394 and Ly$\alpha$, and comprise about 10% of
all QSOs (Trump et al., 2006). Low Ionization BAL QSOs (LoBALs) additionally
show absorption in MgII $\lambda$2799 and other lower ionization species, and
comprise $\sim 1.3$% of all QSOs. Finally, FeLoBAL QSOs, in addition to
showing all the absorption lines seen in LoBALs, also show weak iron
absorption features (Hazard et al., 1987; Becker et al., 1997). They account
for around $0.33\%$ of all QSOs, though the exact fraction is unclear. have
been explained either as arising from being observed at a particular
orientation, or because they are young objects still partially surrounded by
dust. The properties of HiBALs are now thought to be primarily an orientation
effect (Surdej & Hutsemekers, 1987; Murray et al., 1995; Schmidt & Hines,
1999; Gallagher et al., 2007; Doi et al., 2009), while the LoBALs remain
controversial, with evidence favoring both orientation and evolution (e.g.
Voit et al. 1993; Ogle et al. 1999; Gallagher et al. 2007; Ghosh & Punsly
2007; Montenegro-Montes et al. 2008; Urrutia et al. 2009; Zhang et al. 2010).
For FeLoBALs there is also controversy, but the evidence favoring the
evolution scenario is (arguably) stronger than for the LoBALs. Based on rest-
frame UV and optical spectra, Hall et al (2002) suggest that FeLoBALs are
young objects still enveloped in dust. Similar conclusions are reached by
Gregg et al (2002), who also postulate that FeLoBALs may be associated with
mergers. Further evidence comes from the discovery of iron absorption lines in
the UV spectra of two low-redshift ULIRGs which harbor obscured AGN (Farrah et
al., 2005). Finally, mid-IR photometry observations (Farrah et al. 2007a,
hereafter F07) found that many FeLoBAL QSOs were IR-luminous, and may harbor
high rates of star formation.

Excellent distinguishing evidence of the ‘youth vs. orientation’ debate for
FeLoBAL QSOs would be observations that show directly that high rates of star
formation accompany the AGN. Given their reddened nature, a good place to look
for such evidence is in the md-infrared, in which the spectral shapes of
reddened AGN differ markedly from those of star-forming regions. The
outstanding capabilities of the Infrared Spectrograph (IRS, Houck et al. 2004)
on-board the Spitzer space telescope (Werner et al., 2004) provided a huge
step forward in available mid-IR spectroscopic capabilities, with the
opportunity to shed light on the FeLoBAL phenomenon. In this paper, we use
Spitzer-IRS observations of six FeLoBAL QSOs to examine the idea that they are
young objects. We assume a spatially flat cosmology, with $H_{0}=70$ km s-1
Mpc-1, $\Omega=1$, and $\Omega_{\Lambda}=0.7$.

##  2\. Sample Selection

We selected our sample in early 2006, with the requirements that candidates be
confirmed as FeLoBAL QSOs via rest-frame UV spectroscopy, and lie in a
redshift range where we can observe useful diagnostics with the IRS.
Accordingly, we chose six objects (Table 1) at random from F07. The F07 sample
is drawn randomly from the FeLoBAL QSO population known at the time, lie in
the redshift range $1.0<z<1.8$, thus placing important mid-IR spectral
features in the IRS bandpass, and have photometry observations at 24$\mu$m,
70$\mu$m and 160$\mu$m, which provide good constraints on IR luminosities.

A downside of this selection though is that our sample is heterogeneous. When
we selected our sample, the few known FeLoBAL QSOs had been found in several
different surveys. Since then, larger, homogeneous samples of FeLoBAL QSOs
have been published (Trump et al., 2006; Scaringi et al., 2009), but these
samples were not available to us. Accordingly, some level of bias is
inevitable in our sample, though it is difficult to quantify what effect this
may have. We therefore simply list the origins of our sample; three objects
were found via the Sloan Digital Sky Survey (Hall et al., 2002; Adelman-
McCarthy et al., 2008), one (ISO 0056-2738) was discovered serendipitously
from followup of distant clusters (Duc et al., 2002), and two (SDSS 1427+2709
& SDSS 1556+3517) were discovered during spectroscopic followup of quasars
from the FIRST survey (Becker et al., 1997; Najita et al., 2000). SDSS
1556+3517 is radio-loud, while the other five are radio quiet.

Furthermore, at least three of our sample appear to have atypical333We include
the qualifier as the rest-frame UV absorption line properties of FeLoBAL QSOs
have not been studied exhaustively iron absorption features (Hall et al.,
2002). SDSS 1154+0300 has troughs where the absorption remains significant at
velocities comparable to the spacing between absorption features (so
$\gtrsim$12,000 km s-1), causing them to overlap each other. SDSS 2215-0045
and possibly SDSS 1214-0001 have stronger FeIII absorption than FeII
absorption, suggesting that the gas in which the BALs occur is dense, hot, and
moderately highly ionized. We therefore do not correlate mid-IR spectral
properties with those of the iron absorption features.

##  3\. Methods

We observed with the IRS using the first order of the short-low module, and
both orders of the long-low module, giving observed-frame wavelength coverage
of 7.5$\mu$m to 35$\mu$m. As the 24$\mu$m fluxes of our sample span a range of
values, we used different observation times for each object so as to to give a
signal-to-noise of at least 15 in the continuum at observed-frame 24$\mu$m.
Observations were performed in staring mode, using ‘high’ accuracy peak up
observations performed with the blue array from a nearby Two Micron All-Sky
Survey (2MASS, Skrutskie et al. 2006) star.

The data were processed through the Spitzer Science Center’s pipeline software
(version 18.7), which performs standard tasks such as ramp fitting and dark
current subtraction, and produces Basic Calibrated Data (BCD) frames. Starting
with these frames, we produced reduced spectra using the SMART v8.0 software
package, following the methods described in Lebouteiller et al. (2010), which
we summarize here. Individual frames were cleaned of ‘bad’ pixels using the
IRSCLEAN task. The first and last five pixels, corresponding to regions of
reduced sensitivity on the detector, were then removed. The individual frames
at each nod position were then median combined with equal weighting on each
resolution element. Sky background was removed from each image by subtracting
the image for the same order taken with the other nod position (i.e. ‘nod-nod’
sky subtraction). One-dimensional spectra were then extracted using ‘optimal’
extraction with default parameters, and defringed using the internal SMART
algorithm. We found that, in all cases, the sources were point-like, with a
FWHM that was never wider than the PSF. We also checked that the sources were
centered in the IRS slit by extracting spectra using ‘simple’ (i.e. not PSF
weighted) extraction. We found that the resulting spectra were of slightly
lower signal-to-noise but consistent with the optimally extracted spectra,
confirming that any slit offset is insignificant.

This procedure results in separate spectra for each nod and for each order.
The spectra for each nod were inspected; features present in only one nod were
treated as artifacts and removed. The two nod positions were then combined. As
the two nod positions have slightly offset wavelength grids, we combined the
nods by first interleaving them, and then interpolating the fully sampled
spectrum onto a reference wavelength grid. The nod-combined spectra in the
three orders were then merged to give the final spectrum for each object.
Overall, we obtained excellent continuum matches between different orders.
Only in one case (SDSS 1556+3517) was there a significant order mismatch,
between the SL1 and LL2 orders. For this object, we scaled the SL1 spectrum by
a factor of 1.10 to match the blue end of the LL2 continuum

There remain however several uncertainties over error propagation in the IRS
reduction process. For example, defringing is still not completely understood,
so some residuals likely remain that are not included in the ‘formal’ errors.
Another example is the order mismatch between SL1 and LL2 for SDSS 1556+3517 -
a PSF weighted ’optimal’ extraction should in principle not produce this
effect, and while the scaling is small, we do not understand why it is
necessary. Overall therefore, we regard the resulting errors on each
resolution element to be somewhat smaller than they should be, though the
degree of this underestimate is likely insignificant.

##  4\. Results

The IRS spectra are presented in Figure 1. Spectral measurements are presented
in Table 2. We found one further object serendipitously in the slit of ISO
0056-2738, which appears to be a PAH dominated system at $z\simeq 1.42$, but
do not consider it further in this paper.

###  4.1. Spectral Features

The spectra show a variety of spectral features. Four objects show one or more
broad emission features at 6.2$\mu$m, 7.7$\mu$m, 11.2$\mu$m and 12.7$\mu$m,
attributed to bending and stretching modes in Polycyclic Aromatic Hydrocarbons
(PAHs, the 12.7$\mu$m feature also contains a contribution from
[NeII]$\lambda$12.81). The redshifts determined from the PAHs were in all
cases consistent with the optical emission line redshifts, rather than the
redshifts at which the UV absorption features peak, placing the source of the
PAH emission in the host galaxy rather than in the outflow. The PAH fluxes and
equivalent widths (EWs) were computed by integrating the flux after
subtracting off a spline interpolated local continuum. An example of the
spline fits is shown in Figure 2.

At least three objects contain a broad feature centered approximately at
9.7$\mu$m, seen in both emission and absorption, that arises from an Si-O
stretching mode in Silicate dust (Knacke & Thomson, 1973). We measured the
strengths of these features via:

$S_{sil}=ln\left(\frac{F_{obs}(9.7\mu m)}{F_{cont}(9.7\mu m)}\right)$ (1)

where $F_{obs}$ is the observed flux density at rest-frame 9.7$\mu$m, and
$F_{cont}$ is the flux at the same wavelength deduced from a spline fit to the
continuum on either side (Spoon et al., 2007; Levenson et al., 2007; Sirocky
et al., 2008).

We also searched for other features seen in IR-luminous systems, though these
are reliably measurable only with higher resolution data, so we do not present
flux measurements. Both SDSS 1214-0001 and SDSS 1427+2709 show strong
[NeIII]$\lambda$15.56, and weak but significant [NeV]$\lambda$14.32. SDSS
1214-0001 additionally has weak detections at the positions of
[ArII]$\lambda$6.99 and H2S(2)$\lambda$12.29. The other four objects show no
further features that we can identify. SDSS 1154+0300 has an apparent feature
at rest-frame $\sim 8\mu$m that we cannot find a reliable ID for, though as
the spectrum is low S/N it is possible this feature arises from the
juxtaposition of a declining PAH and a rising silicate feature. ISO 0056-2738
is too low S/N to infer the presence or otherwise of other features. We also
note that all six spectra likely contain residual structure due to fringing
that the defringing algorithm was unable to completely remove.

###  4.2. Qualitative Comparisons

####  4.2.1 Objects 3 & 4

We start by comparing the two objects with prominent PAHs, SDSS 1214-0001 and
SDSS 1427+2709, to well studied low-redshift objects (Figure 3). Both objects
closely resemble PAH dominated ULIRGs. It is superficially interesting that
SDSS 1214-0001 is a good match to IRAS 15206+3342, a local ULIRG with iron
absorption features in its UV spectrum, though not of great significance given
that many local ULIRGs have similar mid-IR spectral shapes, see e.g. the
position of IRAS 15206+3342 in the ‘network’ plot of Farrah et al. (2009b).
SDSS 1427+2709 on the other hand is well matched by ULIRGs with weaker PAHs
and a more pronounced continuum, such as Mrk 231. Neither system resembles
heavily absorbed ULIRG spectra such as Arp 220.

Moving on to comparisons with larger samples, we are hampered as there does
not exist a comprehensive mid-IR spectroscopic survey of HiBALs or LoBALs to
compare to. Even comparing to the general AGN population though, the
peculiarity of these two objects, in particular SDSS 1214-0001, is thrown into
sharp relief. Their PAHs are extraordinarily prominent, far more so than for
any QSO, radio loud or radio quiet, that we are aware of (Haas et al., 2005;
Hao et al., 2005; Shi et al., 2006; Maiolino et al., 2007), including those
selected to be far-IR luminous (Lutz et al., 2008; Martínez-Sansigre et al.,
2008), as well as the X-ray luminous ‘type 2 QSO’ objects (Sturm et al.,
2006). There are however a few systems with prominent PAHs among the Narrow
Line Seyfert 1 (NLS1, objects with H$\beta$ FWHMs of $<2000$km s-1, Osterbrock
& Pogge 1985) population (Sani et al., 2010), and several examples of Sy2
ULIRGs with strong PAHs (Armus et al. 2007; Imanishi et al. 2007; Farrah et
al. 2007b, see also the type 2 object LH901A in Sturm et al. 2006).

####  4.2.2 Objects 1, 2, 5 & 6

Turning to the other four sources; SDSS 2215-0045 is a good match to QSOs with
strong silicate emission and weak but detectable PAHs, of which several exist
(Hao et al., 2005). Interestingly, SDSS 2215-0045 is an excellent match to
PG1351+640, a IR-luminous, CO detected QSO with narrow BALs, FeII in emission
under H$\beta$, and slight morphological disturbance (Gelderman & Whittle,
1994; Falcke et al., 1995; Zheng et al., 2001; Evans et al., 2001).
Conversely, SDSS 1556+3517, with its weaker silicate emission feature and
negligible PAHs, is a good match both to some classical QSOs, and to NLS1
systems such as PG1211+143 (which has a high velocity outflow, Pounds & Page
2006; Shi et al. 2007) and IZw1 (which has optical and UV FeII emission and a
decaying starburst (Marziani et al., 1996; Schinnerer et al., 1998; Surace et
al., 1998)). SDSS 1154+0300 has a poorer quality IRS spectrum but is also
consistent with QSOs with weak silicate emission and negligible PAHs. ISO
0056-2738 is of too low signal-to-noise to draw any conclusions other than it
appears to be consistent with AGN generally.

####  4.2.3 The sample as a whole

Finally, we compare the ensemble properties of our sample to other classes of
active galaxies. We start by looking for matches with IR-selected AGN samples.
One sample where we might expect overlap is with the X-ray detected LIRGs in
Brand et al. (2008b), but the two samples do not resemble each other. None of
the 16 Brand et al. 2008b objects have detectable PAHs; though 9 have some
silicate absorption, the rest are featureless power laws. Our two PAH
dominated sources resemble some sources in optically faint 70$\mu$m selected
samples (Brand et al., 2008a; Farrah et al., 2009a), but 70$\mu$m selected
samples have no sources with silicates in emission. We do better if we instead
compare to IRS observations of high-redshift 15$\mu$m selected samples
(Hernán-Caballero et al., 2009), likely because we’re selecting on hotter
dust; 15$\mu$m samples contain objects comparable to our PAH strong objects,
but have few sources with silicates in emission. The one optically selected
sample that appears reasonably matched to ours is the NLS1 sample of Sani et
al. (2010), which contains both PAH dominated objects, and a few objects with
what appears to be weak silicate emission.

###  4.3. Spectral Diagnostics

We move on to quantitative diagnostics. As our spectra are of relatively low
S/N and do not have coverage in all the IRS modules, we use simple diagnostics
that allow for easy comparisons with other samples.

We start with the PAH features. The PAH flux ratios of our sample, considered
either as functions of each other or of PAH luminosity (Figures 4 and 5) are
slightly offset from those of low-redshift ULIRGs, but lie within the
dispersion of high-redshift 70$\mu$m or 24/r selected samples. This is
straighforward to understand. Local ULIRGs are selected without a bias towards
AGN, and have lower IR luminosities, on average, than our sample. Conversely,
the Sajina et al. (2007) & Dasyra et al. (2009) samples are (arguably) biased
towards AGN, and have comparable total IR luminosities to our objects. It is
thus not surprising that the PAH flux ratios and luminosities of our sample
resemble the high redshift comparison objects. It is also interesting that the
two systems with the strongest PAH detections (SDSS 1214-0001 & SDSS
1427+2709) are also the strongest 160$\mu$m detections, consistent with the
idea that starbursts are associated with colder dust, but the small size of
our sample means this consistency could simply be coincidence, so we do not
comment on it further.

We estimate star formation rates from the PAH features using:

$SFR[M_{\odot}yr^{-1}]=1.18\times 10^{-41}L_{P}[ergs\ s^{-1}]$ (2)

where $L_{P}$ is the combined luminosity of the 6.2$\mu$m and 11.2$\mu$m PAH
features (Farrah et al., 2007b). The values are listed in Table 2444Estimating
the star formation rates using the formula in Houck et al. (2007) gives
comparable results. Overall, they are comparable to those derived for other
high redshift IR-luminous sources, including ‘bump’ selected starbursts
(Farrah et al., 2008), sub-mm selected starbusts (Pope et al., 2008; Menéndez-
Delmestre et al., 2009), and far-IR bright QSOs (Lutz et al., 2008). The star
formation rate in SDSS 1214-0001 is extraordinarily high; assuming continuous
star formation then this star formation rate is capable of manufacturing
$10^{11}$M⊙ of stars in less than 100Myr, and $\lesssim 50$Myr if we assume a
late stage exponentially decaying burst.

Next, we employ the ‘Fork’ diagnostic of Spoon et al. 2007 (Figure 6), which
employs both the 6.2$\mu$m PAH feature and the 9.7$\mu$m silicate feature.
Here our sample is not distributed in a similar way to the 70$\mu$m or 24/r
selected sources, but instead lies along the lower branch of the fork. Spoon
et al. 2007 postulate that sources move around the Fork diagram as their power
source evolves, starting on the upper branch and then moving diagonally or
vertically downwards as starburst/AGN activity clears the obscuration from the
nuclear regions. Therefore, solely from this diagnostic, we would classify our
sample as late-stage ULIRGs.

We move on to consider mid-IR continuum diagnostics. Given the limited
wavelength coverage, we use the rest-frame 6$\mu$m continuum luminosity as a
proxy for the bolometric AGN luminosity (Nardini et al., 2008; Watabe et al.,
2009). Following Nardini et al. 2008, we define:

$R=\frac{L_{6}}{L_{IR}}$ (3)

where $L_{6}$ is the 6$\mu$m luminosity as measured from the IRS spectra and
$L_{IR}$ is the 1-1000$\mu$m luminosity from F07. The fractional contribution
of an AGN to the 6$\mu$m luminosity, $\alpha_{6}$, is then:

$\alpha_{6}=\frac{1}{R_{S}-R_{A}}\left(\frac{R_{A}R_{S}}{R}-R_{A}\right)$ (4)

where $R_{S}$ and $R_{A}$ are the equivalents of $R$ for ‘pure’ starbursts and
AGN, with values of $(117\pm 8)\times 10^{-4}$ and $0.32\pm 0.1$ respectively
(Nardini et al., 2008). Therefore, the fractional contribution of the AGN to
the total IR luminosity, $\alpha_{bol}$, is:

$\alpha_{bol}=\frac{\alpha_{6}}{\alpha_{6}+\left(\frac{R_{A}}{R_{S}}\right)(1-\alpha_{6})}$
(5)

The resulting $\alpha_{bol}$ values are listed in Table 1. For comparison, we
also list the $\alpha_{bol}$ values obtained from the SEDs in F07. In four
cases the values are consistent, but for two (SDSS 1556+3517 & SDSS 2215-0045)
they are not. We think it likely that the SED derived values are more
reliable, as they are direct measurements from the (albeit sparsely sampled)
full SEDs, while the values computed using Equation 5 are calibrated using a
wide range of sources. It is interesting that these two objects are the only
two with detected silicate emission, but we do not know if this is the cause
of the discrepancy, or coincidence. Considering either method though, the
sources with strong PAHs have weak AGN contributions to the total IR
luminosity, and the range in $\alpha_{bol}$ values for the whole sample is
similar to that seen in local ULIRGs.

Finally, we fit the source with the most prominent PAHs, SDSS 1214-0001, with
PAHFIT (Smith et al., 2007). This is not an attempt to reconstruct star
formation parameters as PAHFIT is intended for systems where emission from
stars provides at least most of the flux across the mid-IR, but it does serve
as a test of how much of the mid-IR flux in SDSS 1214-0001 is attributable to
star formation. The result is shown in Figure 7. The fit is good, with
$\chi^{2}_{red}=1.4$. The fit is poor at $\lesssim 6\mu$m, with a steeply
rising contribution from ‘starlight’ with decreasing wavelength that is
probably an attempt to fit the AGN continuum. At $\lambda\gtrsim 6\mu$m
though, we obtain an excellent fit. The PAHs are well fitted by the Drude
profiles, and the continuum is predicted to mostly come from dust at 50k, with
small contributions from the 135k and 300k components.

##  5\. Discussion & Conclusions

A general caveat to all of what follows is our small and heterogeneous sample.
Our conclusions should thus be regarded as tentative.

The mid-IR spectra of our sample span a wide range of shapes. We see classical
QSO spectra with hot silicate dust, together with classical starburst spectra
with strong PAHs. The ‘starburst’ spectra have more prominent PAHs than those
seen in any other QSO so far observed, while the hot silicate dust sources do
not have a close match in any purely mid/far-IR selected sample that we are
aware of. It is possible that our heterogeneous selection leads to the
heterogeneity of the spectra, but if this were true then we might expect the
ISO selected object to have the strongest PAHs, which is not the case. Indeed,
the two spectra at the extreme ends of the range of spectral shapes are both
SDSS objects. The star formation rates of our sample span levels comparable to
those in the most luminous starbursts found in any survey at any wavelength,
to those seen in moderately IR-luminous AGN. The spectral diagnostics mark our
sample as late-stage ULIRGs with a wide range of fractional AGN contributions.

These results, combined with their optical classification as reddened QSOs
with a strong outflow, are in principle compatible with a continuum of ‘end-
of-ULIRG’ vs ‘peculiar QSO’ contributions to the FeLoBAL QSO population. To
frame the discussion, we describe three points on this continuum in detail:

1: FeLoBAL QSOs are a transition stage between starburst dominated ULIRGs and
QSOs. As our sample span the entirety of this transition, and because FeLoBALs
are observed to be rare555A further reason behind the rarity of FeLoBALs is
that they they are hard to detect; FeLoBAL QSOs are red, have few or no broad
emission lines and no UV excess, and are thus hard to find in standard QSO
searches (see e.g. Appenzeller et al. 2005). The iron absorption features
themselves are weak., we can set constraints on the timescale of this
transition depending on how intrinsically common the Fe absorption features
are. If Fe absorption is ubiquitous in such a transition, then the transition
must be short in comparison to the ULIRG or QSO lifetime. Taking the ULIRG and
QSO lifetimes to be $\sim$108 years (Tacconi et al., 2008), then the
transition must take $\lesssim$107 years (see also Gregg et al. 2002). If on
the other hand the iron absorption features are intrinsically rare, then the
transition can take longer.

2: FeLoBALs are comprised of comparable fractions of two galaxy populations,
one that is transitioning from a ULIRG to a QSO, and another that is an
unusual phase that QSOs go through, unconnected to mergers. Here the
heterogeneity of our sample arises from observing two unconnected populations.

3: FeLoBAL QSOs are entirely an ‘unusual QSO’ class, and have nothing to do
with mergers or otherwise ‘transitioning’ systems. Instead, the iron
absorption features mark QSOs with both an outflow and atypically large
amounts of iron in their ISM.

Formally, we cannot rule any of these scenarios out. The third though seems
unlikely. It is hard to see how such a heterogeneous set of mid-IR spectra
could arise from observing a peculiar class of QSO in which nothing more
interesting than an outflow with large quantities of iron is going on,
especially if we assume that the restrictive selection on rest-frame UV
properties means we are viewing them over a restricted range in line of sight.
Furthermore, most systems with star formation rates in the range seen in our
sample are ULIRGs, which (at low redshift, at least) are nearly all mergers.

Distinguishing between the first and second scenarios is however more
difficult. On one hand, the first scenario uses a single origin for which
there is independent evidence from studying ULIRGs. The second scenario thus
seems contrived, as it proposes two origins where one will do. On the other
hand, our sample is small, and it is tempting to place undue emphasis on the
properties of SDSS 1214-0001, as it is so peculiar. If this object were
removed then the ‘unusual QSO’ scenario would become more attractive. That
said, the counter-argument also holds; remove (say) SDSS 1556+3517, and the
‘end-of-ULIRG’ scenario becomes more attractive. We are mindful though that it
is only the properties of SDSS 1214-0001 (and to an extent SDSS 1427+2709)
that make us seriously consider the ‘end-of-ULIRG’ scenario.

Recent work that may shed light on this is the study of NLS1’s by Sani et al.
(2010). Their spectra show that at least some NLS1’s harbor intense star
formation and a weak mid-IR AGN continuum. NLS1’s also have strong optical Fe
emission lines, a large soft X-ray excess, and exhibit rapid, large amplitude
X-ray variability. Furthermore, their black hole masses appear to be smaller
than those in BLS1’s of comparable luminosity, though there is controversy
over whether NLS1’s lie below (Peterson et al., 2000; Grupe & Mathur, 2004) or
on (Botte et al., 2005; Komossa & Xu, 2007; Decarli et al., 2008) the
$M_{BH}-\sigma_{*}$ relation (Tremaine et al., 2002). On one level this hints
at interesting links between NLS1s and FeLoBAL QSOs; for example, FeLoBAL QSOs
may be analogues of NLS1s seen more pole-on than edge-on, and in which a
strong outflow has formed. Exploring this idea in detail requires
significantly larger samples, so we do not pursue it here. More generally
however, it implies that relative orientation may also play a role in the
FeLoBAL QSO phenomenon.

We think it likely that the simplest solution that is consistent with all the
observations, and the apparent rarity of FeLoBALs, is the correct one. We
therefore propose that FeLoBAL QSOs as a class are rapidly evolving, youthful
QSOs in which a starburst is coming to an end, and the AGN is in the last
stages of burning through its surrounding dust. We also propose, with more
reserve, that (1) we view FeLoBAL QSOs over a restricted line of sight range,
where we see them more pole-on than edge on, and (2) the outflow has in some
cases partially cleared the dust from around the AGN, leaving the IR emission
from the starburst to dominate.

There are three ways to test this conclusion. First, FeLoBAL QSO hosts should
be in the final stages of merging, so their host galaxies should show slight
signs of morphological disturbance. Second, larger mid-IR spectroscopic
surveys of FeLoBAL QSOs should show a similar range of spectral shapes to
ours. Third, far-IR photometric surveys of FeLoBALs should find both a
moderate enhancement of the fraction of far-IR bright sources compared to the
classical QSO population, corresponding to those FeLoBAL QSOs that still
harbor high star formation rates, and a correlation between far-IR luminosity
and PAH equivalent width. Moreover, it would be useful to perform radiative
transfer modelling of a BAL wind through a starburst to determine the
conditions under which iron absorption is observed in such systems, and so
provide insight into whether our conclusions are feasible.

We close with a caveat and a speculative comment. First, there is an important
parameter that we cannot address; the way in which iron absorption features in
FeLoBALs signpost starburst and AGN activity. For example, if the Fe
absorption only occurs when a starburst is ending and an AGN is blowing away
its surrounding dust, but can occur throughout such a transition, then our
first scenario is likely correct. As a contrived counter-example, if Fe
absorption can only occur during the early phases of such a transition, but
can occur randomly in reddened AGN in which there is no significant star
formation, then in a small sample such as ours we would conclude, incorrectly,
that all FeLoBAL QSOs were a ULIRG-to-QSO transition where the Fe absorption
is present throughout. Resolving this issue is beyond the scope of this paper,
so we note it as a caveat to our conclusions.

Second, Hall et al. (2002) note that two of our sample, SDSS 1214-0001 and
SDSS 2215-0045 have [FeIII]/[FeII] ratios greater than unity, implying the
BALs arise in unusually dense, hot gas. The two ratios however are different;
SDSS 2215-0045 has a [FeIII]/[FeII] ratio much greater than unity, while SDSS
1214-0001 has an [FeIII]/[FeII] ratio only slightly greater than unity. While
conclusions drawn on only two objects are not trustworthy, this difference in
ratio is consistent with our proposed evolutionary sequence. SDSS 1214-0001
has strong PAHs and silicates in absorption, while SDSS 2215-0045 has weak
PAHs and silicates in emission, implying that SDSS 1214-0001 is at an earlier
evolutionary phase than SDSS 2215-0045. Moreover, the enhanced [FeIII]/[FeII]
ratio in SDSS 2215-0045 compared to SDSS 1214-0001 could be interpreted as the
BAL outflow in SDSS 2215-0045 being more developed666This is consistent with
the ‘youth’ hypothesis that Hall et al. (2002) give for SDSS 2215-0045.
Further support for this idea would be an [FeIII]/[FeII] ratio greater than
unity in SDSS 1427+2709, but the only published spectrum we are aware of is in
Becker et al. (2000), in which the FeIII lines lie in a noisy region at the
extreme blue end of the bandpass. They do not however appear to be
significantly stronger than the FeII lines, so we simply note this as an
interesting avenue for future work.

We thank the referee for a helpful report. This work is based on observations
made with the Spitzer Space Telescope, which is operated by the Jet Propulsion
Laboratory, California Institute of Technology under a contract with NASA.
Support for this work was provided by NASA. This research has made extensive
use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet
Propulsion Laboratory, California Institute of Technology, under contract with
NASA. DF thanks STFC for support via an Advanced Fellowship. NC gratefully
acknowledges support from a Cottrell College Science Award from the Research
Corporation and from an NSF RUI grant.

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Table 1FeLoBAL QSO Sample

Object | RA (2000) | Dec |  $z_{sys}$aaSystemic redshift from narrow optical emission lines |  $z_{abs}$bbRedshift of peak absorption from the broad UV absorption lines | $f_{24}$ | $f_{70}$ | $f_{160}$ |  $\alpha_{bol}|_{6\mu m}$ddFractional AGN luminosity, computed using the prescription in §4.3 (Nardini et al., 2008). |  $\alpha_{bol}|_{sed}$eeFractional AGN luminosity, derived from the SED fits in F07.  | $L_{IR}$
---|---|---|---|---|---|---|---|---|---|---
(1) ISO J005645.1-273816 | 00 56 45.2 | -27 38 15.6 | 1.78 | 1.75 | 1.6 |  $<$7.0 |  $<$50 | 0.09 - 0.40 | $>0.26$ | 12.5-13.0
(2) SDSS J115436.60+030006.3 | 11 54 36.6 | 03 00 06.4 | 1.46ccHall et al. (2002) | 1.36 | 7.6 | 17.9 |  $<$50 | 0.35 - 0.60 | $>0.40$ | 12.9-13.1
(3) SDSS J121441.42-000137.8 | 12 14 41.4 | -00 01 37.9 | 1.05 | 0.99 | 4.7 | 38.3 | 78.9 | 0.07 - 0.15 | $<0.37$ | 12.7-12.9
(4) SDSS J142703.62+270940.3 | 14 27 03.6 | 27 09 40.3 | 1.17 | ? | 4.8 | 32.6 | 68.1 | 0.09 - 0.17 | $<0.61$ | 12.8-13.0
(5) SDSS J155633.78+351757.3 | 15 56 33.8 | 35 17 58.0 | 1.50 | 1.48 | 13.9 | 24.7 |  $<$50 | 0.40 - 0.70 | $>0.75$ | 13.1-13.3
(6) SDSS J221511.93-004549.9 | 22 15 11.9 | -00 45 49.9 | 1.48 | 1.36 | 10.4 | 27.2 |  $<$50 | 0.09 - 0.30 | $>0.36$ | 13.0-13.4

Note. — All flux densities are quoted in mJy. Errors are typically 10% at
24$\mu$m, 20% at 70$\mu$m, and 25% at 160$\mu$m. Luminosities are the
logarithm of the rest-frame 1-1000$\mu$m luminosity, in units of solar
luminosities (3.826$\times 10^{26}$ Watts), taken from F07. Limits and ranges
are 3$\sigma$.

Table 2Spectral Measurements

Object | PAH 6.2$\mu$m | PAH 7.7$\mu$m | PAH 11.2$\mu$m | $S_{sil}$ | SFRaaStar formation rate, determined from Equation 2. Error is solely that derived from the uncertainty in the fluxes.
---|---|---|---|---|---
| Flux | EW | Flux | EW | Flux | EW |  | M⊙ yr-1
(1) ISO 0056-2738 | 6.1$\pm$8.9 | 0.06$\pm$0.09 | 6.0$\pm$8.0 | 0.08$\pm$0.12 | 4.1$\pm$3.9 | 0.08$\pm$0.08 | 0.25$\pm$0.11 | $2600\pm 2500$
(2) SDSS 1154+0300 | 5.2$\pm$5.7 | 0.01$\pm$0.01 | 15.1$\pm$9.2 | 0.04$\pm$0.02 | 0.3$\pm$4.6 | 0.01$\pm$0.02 | 0.15$\pm$0.08 | $900\pm 1100$
(3) SDSS 1214-0001 | 16.3$\pm$3.3 | 0.06$\pm$0.01 | 56.5$\pm$7.0 | 0.25$\pm$0.05 | 22.5$\pm$5.6 | 0.24$\pm$0.06 | -0.25$\pm$0.05 | $2700\pm 500$
(4) SDSS 1427+2709 | 7.3$\pm$2.6 | 0.03$\pm$0.01 | 25.0$\pm$6.4 | 0.10$\pm$0.03 | 2.4$\pm$2.1 | 0.02$\pm$0.01 | -0.10$\pm$0.05 | $900\pm 350$
(5) SDSS 1556+3517 | 1.0$\pm$3.0 | 0.01$\pm$0.01 | 11.2$\pm$6.6 | 0.02$\pm$0.01 | 3.0$\pm$3.9 | 0.01$\pm$0.02 | 0.22$\pm$0.04 | $700\pm 800$
(6) SDSS 2215-0045 | 10.5$\pm$5.3 | 0.03$\pm$0.01 | 28.1$\pm$6.8 | 0.09$\pm$0.02 | 7.7$\pm$3.7 | 0.02$\pm$0.01 | 0.62$\pm$0.10 | $2700\pm 1000$

Note. — PAH fluxes are in units of $10^{-22}$W cm-2 and equivalent widths in
$\mu$m.

Figure 1.— Spitzer-IRS low-resolution spectra of our sample, plotted in the
rest-frame using the optical emission line redshifts. The data are plotted in
black, with 1$\sigma$ errors in grey. Flux densities are in mJy and
wavelengths are in $\mu$m. The number in the top left of each panel is the ID
number in Table 1. The green solid (dashed) lines mark the 6.2$\mu$m,
7.7$\mu$m, 8.6$\mu$m, 11.2$\mu$m and 12.7$\mu$m PAH features assuming the
systemic (peak absorption, if available) redshift. The red lines perform the
same function for the [Ne II]$\lambda$12.81, [Ne V]$\lambda$14.32 and [Ne
III]$\lambda$15.56 lines. The yellow shaded region shows the approximate
extent of the 9.7$\mu$m silicate absorption feature, but see Bowey et al.
1998; Draine 2003; Nikutta et al. 2009.

Figure 2.— An example of the spline fits used to determine the shape of the
underlying continuum for the PAH fluxes, in this case for the 6.2$\mu$m PAH
feature in SDSS 1214-0001. The (rest-frame) spectrum is plotted in black. The
points used to define the continuum are shown in green, while the spline fit
and its residuals are shown in blue. The errors on the spectrum have been
omitted for clarity. The dashed line shows the adopted peak wavelength of the
PAH feature, while the dotted lines show the lower and upper limits of the
integration. This plot also demonstrates the excellent agreement between the
two nods; the ‘double’ blue error bars show separately the error bars for the
two nod positions, which have slightly offset wavelength grids.

Figure 3.— Comparison of our spectra to well studied low-redshift objects.
Objects showing silicate absorption in the left panel, and the others are in
the right panel. Comparison spectra are taken from Brandl et al. (2006); Armus
et al. (2007); Shi et al. (2007) and Imanishi et al. (2007). Left: Red - Mrk
231. Orange - composite of local $\simeq 10^{11}$L⊙ starbursts. Green - Arp
220. Blue - IRAS 15206+3342 (a local ULIRG with iron absorption features in
its UV spectrum). Right: Red - 3C 273. Orange - I Zw 1 (a low-z NLS1). Green -
PG 1211+143. Blue - PG 1351+640 (both far-IR luminous QSOs).

Figure 4.— PAH ratio diagnostic plots, divided by comparisons to (top row)
low-redshift and (bottom row) high redshift samples. Black: FeLoBALs. Green:
low-z ULIRGs with measurable 7.7$\mu$m PAHs (Spoon et al., 2007; Desai et al.,
2007). Red: low-z AGN (Weedman et al., 2005). Blue: low-z starbursts with
L${}_{IR}<10^{11}$L⊙ (Brandl et al., 2006). Purple: high-z $70\mu$m selected
(Brand et al., 2008a; Farrah et al., 2009a). Grey: high-z ‘bump’ selected
(Farrah et al., 2008). Orange: high-$24\mu$m/0.7$\mu$m selected Dasyra et al.
(2009). Cyan: high-z $24\mu$m/8$\mu$m and $24\mu$m/0.7$\mu$m selected Sajina
et al. (2007). Yellow: high-z sub-mm selected (Pope et al., 2008; Menéndez-
Delmestre et al., 2009).

Figure 5.— PAH luminosity diagnostic plots. Color coding is the same as in
Figure 4.

Figure 6.— ‘Fork’ diagnostic diagram (Spoon et al., 2007). The FeLoBALs are
plotted in black, while the other points are color-coded as in Figure 4. Plots
of the composite spectra for each region are in figure 2 of Spoon et al.
(2007). With respect to the comparison objects plotted in Figure 3; Mrk 231 is
class 1A, IRAS 15206+3342 is class 1B, Arp 220 is class 3B, the starburst
composite is class 1C, and the remaining objects are all class 1A. The ‘bump’
sources lie close together on this plot, so, for clarity, we plot their
average as a single point.

Figure 7.— PAHfit results for SDSS 1214-0001. Details of the model parameters
can be found in Smith et al. (2007). Vertical black lines: observed data.
Green: combined best-fit model. Red solid lines: thermal dust continuum
components. Magenta: starlight continuum. Black solid: combined
starlight+thermal dust continuum. Violet: fine structure+molecular features.
Blue: PAH features. Black dotted: extinction curve.