/usr/lib/R/site-library/DESeq2/script/vst.nb is in r-bioc-deseq2 1.18.1-1.
This file is owned by root:root, with mode 0o644.
The actual contents of the file can be viewed below.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 808 809 810 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843 844 845 846 847 848 849 850 851 852 853 854 855 856 857 858 859 860 861 862 863 864 865 866 867 868 869 870 871 872 873 874 875 876 877 878 879 880 881 882 883 884 885 886 887 888 889 890 891 892 893 894 895 896 897 898 899 900 901 902 903 904 905 906 907 908 909 910 911 912 913 914 915 916 917 918 919 920 921 922 923 924 925 926 927 928 929 930 931 932 933 934 935 936 937 938 939 940 941 942 943 944 945 946 947 948 949 950 951 952 953 954 955 956 957 958 959 960 961 962 963 964 965 966 967 968 969 970 971 972 973 974 975 976 977 978 979 980 981 982 983 984 985 986 987 988 989 990 991 992 993 994 995 996 997 998 999 1000 | (* Content-type: application/mathematica *)
(*** Wolfram Notebook File ***)
(* http://www.wolfram.com/nb *)
(* CreatedBy='Mathematica 7.0' *)
(*CacheID: 234*)
(* Internal cache information:
NotebookFileLineBreakTest
NotebookFileLineBreakTest
NotebookDataPosition[ 145, 7]
NotebookDataLength[ 35124, 991]
NotebookOptionsPosition[ 32281, 895]
NotebookOutlinePosition[ 32848, 915]
CellTagsIndexPosition[ 32805, 912]
WindowFrame->Normal*)
(* Beginning of Notebook Content *)
Notebook[{
Cell[CellGroupData[{
Cell["Variance-stabilizing transformation for DESeq", "Subtitle",
CellChangeTimes->{{3.540622683759346*^9, 3.5406226961574497`*^9}, {
3.540815669170343*^9, 3.540815673334532*^9}}],
Cell[BoxData[
StyleBox[
RowBox[{
RowBox[{"For", " ", "parametrized", " ", "dispersion", " ", "fit"}],
"\[IndentingNewLine]"}], "Subsubtitle"]], "Input",
CellChangeTimes->{{3.540815676143361*^9, 3.5408157093482103`*^9}}],
Cell[TextData[{
"This file describes the variance stabilizing transformation (VST) used by \
DESeq when parametric dispersion estimation is used.\nThis is a ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" notebook. The file ",
StyleBox["vst.pdf",
FontSlant->"Italic"],
" is produced from ",
StyleBox["vst.nb",
FontSlant->"Italic"],
"."
}], "Text",
CellChangeTimes->{{3.5407069553696613`*^9, 3.540707011225795*^9}}],
Cell[BoxData[""], "Input",
CellChangeTimes->{{3.540706950820628*^9, 3.540706952072029*^9}}],
Cell[TextData[{
"When using ",
StyleBox["estimateDispersions",
FontSlant->"Italic"],
" with ",
StyleBox["fitType=\"parametric\"",
FontSlant->"Italic"],
", we parametrize the relation between mean \[Mu] and dispersion \[Alpha] \
with two constants ",
Cell[BoxData[
FormBox[
SubscriptBox["a", "0"], TraditionalForm]],
FormatType->"TraditionalForm"],
" and ",
Cell[BoxData[
FormBox[
SubscriptBox["a", "1"], TraditionalForm]],
FormatType->"TraditionalForm"],
"as follows:"
}], "Text",
CellChangeTimes->{{3.540622754917987*^9, 3.5406228441955957`*^9}}],
Cell[CellGroupData[{
Cell[BoxData[
RowBox[{"\[Alpha]", " ", "=", " ",
RowBox[{
SubscriptBox["a", "0"], "+",
RowBox[{
SubscriptBox["a", "1"], "/", "\[Mu]"}]}]}]], "Input",
CellChangeTimes->{{3.540622709621419*^9, 3.540622751006365*^9}, {
3.5406228468149643`*^9, 3.540622847455463*^9}, {3.540622881311955*^9,
3.540622881653171*^9}}],
Cell[BoxData[
RowBox[{
SubscriptBox["a", "0"], "+",
FractionBox[
SubscriptBox["a", "1"], "\[Mu]"]}]], "Output",
CellChangeTimes->{3.5406228483207407`*^9, 3.540622882333549*^9,
3.5406235190432873`*^9, 3.54070632548785*^9}]
}, Open ]],
Cell[TextData[{
"In the package, ",
Cell[BoxData[
FormBox[
SubscriptBox["a", "0"], TraditionalForm]],
FormatType->"TraditionalForm"],
" is called the ",
StyleBox["asymptotic dispersion",
FontSlant->"Italic"],
" and ",
Cell[BoxData[
FormBox[
SubscriptBox["a", "1"], TraditionalForm]],
FormatType->"TraditionalForm"],
" the ",
StyleBox["extra-Poisson factor",
FontSlant->"Italic"],
"."
}], "Text",
CellChangeTimes->{{3.540625116841147*^9, 3.54062515229095*^9}}],
Cell["The variance is hence", "Text",
CellChangeTimes->{{3.5406228589902277`*^9, 3.540622862149235*^9}}],
Cell[CellGroupData[{
Cell[BoxData[
RowBox[{"v", " ", "=", " ",
RowBox[{
RowBox[{"\[Mu]", " ", "+", " ",
RowBox[{"\[Alpha]", " ",
SuperscriptBox["\[Mu]", "2"]}]}], "//", "Expand"}]}]], "Input",
CellChangeTimes->{{3.540622864971992*^9, 3.540622905693181*^9}}],
Cell[BoxData[
RowBox[{"\[Mu]", "+",
RowBox[{
SuperscriptBox["\[Mu]", "2"], " ",
SubscriptBox["a", "0"]}], "+",
RowBox[{"\[Mu]", " ",
SubscriptBox["a", "1"]}]}]], "Output",
CellChangeTimes->{{3.5406228908884497`*^9, 3.540622906087739*^9},
3.540623520780364*^9, 3.540706328495348*^9}]
}, Open ]],
Cell[TextData[{
"A variance stabilizing transformation (VST) is a transformation ",
StyleBox["u",
FontSlant->"Italic"],
", such that, if ",
StyleBox["X", "InlineFormula",
FontSlant->"Italic"],
" is a random variable with variance-mean relation ",
StyleBox["v",
FontSlant->"Italic"],
", i.e.,",
Cell[BoxData[
FormBox[
RowBox[{
RowBox[{"Var", "(", "X", ")"}], "=",
RowBox[{"v", "(",
RowBox[{
StyleBox["E",
FontSlant->"Plain"], "(", "X", ")"}], ")"}]}], TraditionalForm]]],
", then ",
Cell[BoxData[
FormBox[
RowBox[{"u", "(", "X", ")"}], TraditionalForm]]],
" has stabilized variance, i.e., is homoskedastic.\[LineSeparator]\nA VST ",
StyleBox["u",
FontSlant->"Italic"],
" can be derived from a variance-mean relation ",
StyleBox["v",
FontSlant->"Italic"],
" by ",
Cell[BoxData[
FormBox[
RowBox[{
RowBox[{"u", "(", "x", ")"}], " ", "=",
RowBox[{
SuperscriptBox["\[Integral]", "x"],
FractionBox["d\[Mu]",
SqrtBox[
RowBox[{"v", "(", "\[Mu]", ")"}]]]}]}], TraditionalForm]]],
". \nHence, we can get a general VST with"
}], "Text",
CellChangeTimes->{{3.5406229237822933`*^9, 3.540622976488544*^9}, {
3.5406230186338882`*^9, 3.5406234172645473`*^9}, 3.540623709187056*^9}],
Cell[CellGroupData[{
Cell[BoxData[
RowBox[{
SubscriptBox["u", "0"], "=",
RowBox[{"Integrate", "[", " ",
RowBox[{
FractionBox["1",
SqrtBox["v"]], ",",
RowBox[{"{",
RowBox[{"\[Mu]", ",", "0", ",", "x"}], "}"}], ",", " ",
RowBox[{"Assumptions", "\[Rule]",
RowBox[{"{",
RowBox[{
RowBox[{
SubscriptBox["a", "0"], ">", "0"}], ",",
RowBox[{
SubscriptBox["a", "1"], ">", "0"}], ",",
RowBox[{"x", ">", "0"}]}], "}"}]}]}], "]"}]}]], "Input",
CellChangeTimes->{{3.540623530465404*^9, 3.5406235592399807`*^9}, {
3.540623599688093*^9, 3.5406236174438133`*^9}, {3.54062365174788*^9,
3.5406236985888433`*^9}, {3.5406237316160307`*^9, 3.540623763300437*^9}, {
3.5406239927503653`*^9, 3.5406240020590677`*^9}}],
Cell[BoxData[
FractionBox[
RowBox[{"Log", "[",
FractionBox[
RowBox[{"1", "+",
RowBox[{"2", " ", "x", " ",
SubscriptBox["a", "0"]}], "+",
SubscriptBox["a", "1"], "+",
RowBox[{"2", " ",
SqrtBox[
RowBox[{"x", " ",
SubscriptBox["a", "0"], " ",
RowBox[{"(",
RowBox[{"1", "+",
RowBox[{"x", " ",
SubscriptBox["a", "0"]}], "+",
SubscriptBox["a", "1"]}], ")"}]}]]}]}],
RowBox[{"1", "+",
SubscriptBox["a", "1"]}]], "]"}],
SqrtBox[
SubscriptBox["a", "0"]]]], "Output",
CellChangeTimes->{3.5406237656511507`*^9, 3.5406240086931467`*^9,
3.540706337835845*^9}]
}, Open ]],
Cell[TextData[{
"If ",
Cell[BoxData[
FormBox[
SubscriptBox["u", "0"], TraditionalForm]],
FormatType->"TraditionalForm"],
" is a VST, then so is ",
Cell[BoxData[
FormBox[
RowBox[{
RowBox[{"u", "(", "x", ")"}], "=",
RowBox[{
RowBox[{"\[Eta]", " ",
RowBox[{
SubscriptBox["u", "0"], "(", "x", ")"}]}], "+", "\[Xi]"}]}],
TraditionalForm]],
FormatType->"TraditionalForm"],
". Hence, this here is a VST, too:"
}], "Text",
CellChangeTimes->{{3.54062372243547*^9, 3.540623757039871*^9}, {
3.54062379375005*^9, 3.540623799888122*^9}}],
Cell[CellGroupData[{
Cell[BoxData[
RowBox[{"u", " ", "=", " ",
RowBox[{
RowBox[{"\[Eta]", " ",
SubscriptBox["u", "0"]}], " ", "+", " ", "\[Xi]"}]}]], "Input",
CellChangeTimes->{{3.540623420986374*^9, 3.540623478619273*^9}, {
3.540623773769244*^9, 3.540623774356125*^9}}],
Cell[BoxData[
RowBox[{"\[Xi]", "+",
FractionBox[
RowBox[{"\[Eta]", " ",
RowBox[{"Log", "[",
FractionBox[
RowBox[{"1", "+",
RowBox[{"2", " ", "x", " ",
SubscriptBox["a", "0"]}], "+",
SubscriptBox["a", "1"], "+",
RowBox[{"2", " ",
SqrtBox[
RowBox[{"x", " ",
SubscriptBox["a", "0"], " ",
RowBox[{"(",
RowBox[{"1", "+",
RowBox[{"x", " ",
SubscriptBox["a", "0"]}], "+",
SubscriptBox["a", "1"]}], ")"}]}]]}]}],
RowBox[{"1", "+",
SubscriptBox["a", "1"]}]], "]"}]}],
SqrtBox[
SubscriptBox["a", "0"]]]}]], "Output",
CellChangeTimes->{3.540623775240573*^9, 3.5406240154345617`*^9,
3.540706341376553*^9}]
}, Open ]],
Cell[TextData[{
"We will now choose the parameters \[Eta] and \[Xi] such that our VST \
behaves like ",
Cell[BoxData[
FormBox[
SubscriptBox["log", "2"], TraditionalForm]],
FormatType->"TraditionalForm"],
" for large values. Let us first look at the asymptotic ratio of the two \
transformations:"
}], "Text",
CellChangeTimes->{{3.5406237859608927`*^9, 3.540623835015697*^9}, {
3.540623912031002*^9, 3.5406239291035957`*^9}}],
Cell[CellGroupData[{
Cell[BoxData[
RowBox[{"Limit", "[",
RowBox[{
RowBox[{"u", "/",
RowBox[{"Log", "[",
RowBox[{"2", ",", "x"}], "]"}]}], ",",
RowBox[{"x", "\[Rule]", "\[Infinity]"}], ",",
RowBox[{"Assumptions", "\[Rule]",
RowBox[{"{",
RowBox[{
RowBox[{
SubscriptBox["a", "0"], ">", "0"}], ",",
RowBox[{
SubscriptBox["a", "1"], ">", "0"}], ",",
RowBox[{"x", ">", "0"}]}], "}"}]}]}], "]"}]], "Input",
CellChangeTimes->{{3.540623840409006*^9, 3.540623897293412*^9}}],
Cell[BoxData[
FractionBox[
RowBox[{"\[Eta]", " ",
RowBox[{"Log", "[", "2", "]"}]}],
SqrtBox[
SubscriptBox["a", "0"]]]], "Output",
CellChangeTimes->{
3.5406238532935*^9, {3.540623884009026*^9, 3.540623898620083*^9},
3.540623932367114*^9, 3.540624018553113*^9, 3.540706350438925*^9}]
}, Open ]],
Cell["\<\
Hence, if we set \[Eta] as follows, both tranformations have asymptotically \
the ratio 1.\
\>", "Text",
CellChangeTimes->{{3.540623941645524*^9, 3.540623964772687*^9}}],
Cell[CellGroupData[{
Cell[BoxData[
RowBox[{"\[Eta]", "=",
FractionBox[
SqrtBox[
SubscriptBox["a", "0"]],
RowBox[{"Log", "[", "2", "]"}]]}]], "Input",
CellChangeTimes->{{3.5406239683083763`*^9, 3.540623985663389*^9}}],
Cell[BoxData[
FractionBox[
SqrtBox[
SubscriptBox["a", "0"]],
RowBox[{"Log", "[", "2", "]"}]]], "Output",
CellChangeTimes->{3.54062402054701*^9, 3.5407063538642073`*^9}]
}, Open ]],
Cell["We also want the difference to vanish for large values:", "Text",
CellChangeTimes->{{3.5406240570203876`*^9, 3.540624066501199*^9}}],
Cell[CellGroupData[{
Cell[BoxData[
RowBox[{"Limit", "[",
RowBox[{
RowBox[{"u", "-",
RowBox[{"Log", "[",
RowBox[{"2", ",", "x"}], "]"}]}], ",",
RowBox[{"x", "\[Rule]", "\[Infinity]"}], ",",
RowBox[{"Assumptions", "\[Rule]",
RowBox[{"{",
RowBox[{
RowBox[{
SubscriptBox["a", "0"], ">", "0"}], ",",
RowBox[{
SubscriptBox["a", "1"], ">", "0"}], ",",
RowBox[{"x", ">", "0"}]}], "}"}]}]}], "]"}]], "Input",
CellChangeTimes->{{3.540624078076254*^9, 3.5406240782935953`*^9}}],
Cell[BoxData[
RowBox[{"\[Xi]", "+",
FractionBox[
RowBox[{"Log", "[",
FractionBox[
RowBox[{"4", " ",
SubscriptBox["a", "0"]}],
RowBox[{"1", "+",
SubscriptBox["a", "1"]}]], "]"}],
RowBox[{"Log", "[", "2", "]"}]]}]], "Output",
CellChangeTimes->{3.540624079366681*^9, 3.5407063578189287`*^9}]
}, Open ]],
Cell["So, we set", "Text",
CellChangeTimes->{{3.540624088556891*^9, 3.540624089891953*^9}}],
Cell[CellGroupData[{
Cell[BoxData[
RowBox[{"\[Xi]", "=",
RowBox[{"-",
FractionBox[
RowBox[{"Log", "[",
FractionBox[
RowBox[{"4", " ",
SubscriptBox["a", "0"]}],
RowBox[{"1", "+",
SubscriptBox["a", "1"]}]], "]"}],
RowBox[{"Log", "[", "2", "]"}]]}]}]], "Input",
CellChangeTimes->{{3.540624101066662*^9, 3.54062410215302*^9}}],
Cell[BoxData[
RowBox[{"-",
FractionBox[
RowBox[{"Log", "[",
FractionBox[
RowBox[{"4", " ",
SubscriptBox["a", "0"]}],
RowBox[{"1", "+",
SubscriptBox["a", "1"]}]], "]"}],
RowBox[{"Log", "[", "2", "]"}]]}]], "Output",
CellChangeTimes->{3.5406241034473743`*^9, 3.5407063616117477`*^9}]
}, Open ]],
Cell["Check that both limits are now correct:", "Text",
CellChangeTimes->{{3.540624108213401*^9, 3.5406241294046183`*^9}}],
Cell[CellGroupData[{
Cell[BoxData[
RowBox[{"Limit", "[",
RowBox[{
RowBox[{"u", "/",
RowBox[{"Log", "[",
RowBox[{"2", ",", "x"}], "]"}]}], ",",
RowBox[{"x", "\[Rule]", "\[Infinity]"}], ",",
RowBox[{"Assumptions", "\[Rule]",
RowBox[{"{",
RowBox[{
RowBox[{
SubscriptBox["a", "0"], ">", "0"}], ",",
RowBox[{
SubscriptBox["a", "1"], ">", "0"}], ",",
RowBox[{"x", ">", "0"}]}], "}"}]}]}], "]"}]], "Input"],
Cell[BoxData["1"], "Output",
CellChangeTimes->{3.540624144767686*^9, 3.540706364776153*^9}]
}, Open ]],
Cell[CellGroupData[{
Cell[BoxData[
RowBox[{"Limit", "[",
RowBox[{
RowBox[{"u", "-",
RowBox[{"Log", "[",
RowBox[{"2", ",", "x"}], "]"}]}], ",",
RowBox[{"x", "\[Rule]", "\[Infinity]"}], ",",
RowBox[{"Assumptions", "\[Rule]",
RowBox[{"{",
RowBox[{
RowBox[{
SubscriptBox["a", "0"], ">", "0"}], ",",
RowBox[{
SubscriptBox["a", "1"], ">", "0"}], ",",
RowBox[{"x", ">", "0"}]}], "}"}]}]}], "]"}]], "Input",
CellChangeTimes->{{3.5406241492285852`*^9, 3.540624149396823*^9}}],
Cell[BoxData["0"], "Output",
CellChangeTimes->{3.5406241503157988`*^9, 3.5407063658855057`*^9}]
}, Open ]],
Cell["Hence, we arrive at this VST:", "Text",
CellChangeTimes->{{3.540624156886202*^9, 3.5406241624452667`*^9}}],
Cell[CellGroupData[{
Cell[BoxData[
RowBox[{"FullSimplify", "[",
RowBox[{"u", ",",
RowBox[{"Assumptions", "->",
RowBox[{"{",
RowBox[{
RowBox[{
SubscriptBox["a", "0"], ">", "0"}], ",",
RowBox[{
SubscriptBox["a", "1"], ">", "0"}], ",",
RowBox[{"x", ">", "0"}]}], "}"}]}]}], "]"}]], "Input",
CellChangeTimes->{{3.54062416605935*^9, 3.540624190033182*^9}, {
3.54062534911759*^9, 3.5406253565459948`*^9}}],
Cell[BoxData[
FractionBox[
RowBox[{"Log", "[",
FractionBox[
RowBox[{"1", "+",
RowBox[{"2", " ", "x", " ",
SubscriptBox["a", "0"]}], "+",
SubscriptBox["a", "1"], "+",
RowBox[{"2", " ",
SqrtBox[
RowBox[{"x", " ",
SubscriptBox["a", "0"], " ",
RowBox[{"(",
RowBox[{"1", "+",
RowBox[{"x", " ",
SubscriptBox["a", "0"]}], "+",
SubscriptBox["a", "1"]}], ")"}]}]]}]}],
RowBox[{"4", " ",
SubscriptBox["a", "0"]}]], "]"}],
RowBox[{"Log", "[", "2", "]"}]]], "Output",
CellChangeTimes->{{3.5406241686802197`*^9, 3.54062419149958*^9},
3.54062535102468*^9, 3.540706368929482*^9}]
}, Open ]],
Cell[TextData[{
"This VST (red) now behaves asymptotically as ",
Cell[BoxData[
FormBox[
SubscriptBox["log", "2"], TraditionalForm]],
FormatType->"TraditionalForm"],
" (blue), shown here for typical values for ",
Cell[BoxData[
FormBox[
SubscriptBox["a", "0"], TraditionalForm]],
FormatType->"TraditionalForm"],
" and ",
Cell[BoxData[
FormBox[
SubscriptBox["a", "1"], TraditionalForm]],
FormatType->"TraditionalForm"],
"."
}], "Text",
CellChangeTimes->{{3.540624324957206*^9, 3.5406243429895267`*^9}, {
3.540624623453052*^9, 3.5406246604775143`*^9}}],
Cell[CellGroupData[{
Cell[BoxData[
RowBox[{"Plot", "[", " ",
RowBox[{
RowBox[{"{",
RowBox[{
RowBox[{"u", "/.",
RowBox[{"{",
RowBox[{
RowBox[{
SubscriptBox["a", "0"], "\[Rule]", ".01"}], ",",
RowBox[{
SubscriptBox["a", "1"], "->", "3"}]}], "}"}]}], ",",
RowBox[{"Log", "[",
RowBox[{"2", ",", "x"}], "]"}]}], "}"}], ",",
RowBox[{"{",
RowBox[{"x", ",", "0", ",", "10000"}], "}"}], ",",
RowBox[{"PlotStyle", "\[Rule]",
RowBox[{"{",
RowBox[{"Red", ",", "Blue"}], "}"}]}]}], "]"}]], "Input",
CellChangeTimes->{{3.540624219476747*^9, 3.540624313819049*^9}, {
3.540624609588531*^9, 3.5406246195269203`*^9}}],
Cell[BoxData[
GraphicsBox[{{}, {},
{RGBColor[1, 0, 0], LineBox[CompressedData["
1:eJwVz3k4lAsbBnBRoUJIpTRmsc1LC4noq25ZsuV0hJxsZSvLZxtpbNlphsiS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"]]},
{RGBColor[0, 0, 1], LineBox[CompressedData["
1:eJwVz3k01QkfBnCULGPJSCGuu+Hen8qMsnSN8djaeCNJvMlOtsle1ixXVxfZ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"]]}},
AspectRatio->NCache[GoldenRatio^(-1), 0.6180339887498948],
Axes->True,
AxesOrigin->{0, 8.},
PlotRange->{{0, 10000}, {7.603101236704731, 13.316142815535127`}},
PlotRangeClipping->True,
PlotRangePadding->{
Scaled[0.02],
Scaled[0.02]}]], "Output",
CellChangeTimes->{
3.5406242472538157`*^9, {3.540624281049045*^9, 3.540624314895378*^9},
3.5406246200625257`*^9, 3.540706377585573*^9}]
}, Open ]],
Cell["\<\
For small values, however, the VST (red) compresses the dynamics much more \
dramatically than the logarithm (blue) and the identity (green). This \
reflects that the strong Poisson noise makes differences uninformative for \
small values.\
\>", "Text",
CellChangeTimes->{{3.540624693085382*^9, 3.5406247289244823`*^9}, {
3.5406248163017282`*^9, 3.540624917261745*^9}}],
Cell[CellGroupData[{
Cell[BoxData[
RowBox[{"Plot", "[", " ",
RowBox[{
RowBox[{"{",
RowBox[{
RowBox[{"u", "/.",
RowBox[{"{",
RowBox[{
RowBox[{
SubscriptBox["a", "0"], "\[Rule]", ".01"}], ",",
RowBox[{
SubscriptBox["a", "1"], "->", "3"}]}], "}"}]}], ",",
RowBox[{"Log", "[",
RowBox[{"2", ",", "x"}], "]"}], ",", " ", "x"}], "}"}], ",",
RowBox[{"{",
RowBox[{"x", ",", "0", ",", "100"}], "}"}], ",",
RowBox[{"PlotStyle", "\[Rule]",
RowBox[{"{",
RowBox[{"Red", ",", "Blue", ",", "Green"}], "}"}]}], ",",
RowBox[{"PlotRange", "\[Rule]",
RowBox[{"{",
RowBox[{"0", ",", "20"}], "}"}]}]}], "]"}]], "Input",
CellChangeTimes->{{3.5406243535636806`*^9, 3.5406244096271353`*^9}, {
3.540624670184353*^9, 3.540624671231537*^9}, {3.540624734125123*^9,
3.540624734432403*^9}, {3.540624796200985*^9, 3.540624806534687*^9}, {
3.540624844102325*^9, 3.540624845507819*^9}}],
Cell[BoxData[
GraphicsBox[{{}, {},
{RGBColor[1, 0, 0], LineBox[CompressedData["
1:eJwV1Gs4lAkbB3DEOCwlKZlnZp4ph7AqZCni+dc6FDkuedLqoJhRyXlzjthe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"]]},
{RGBColor[0, 0, 1], LineBox[CompressedData["
1:eJwt0nk01PsbB/DBYBBZBjPfpCmafsnFRJLo81ZJ1pD4Ii2iRiR73VKkiEbW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"]]},
{RGBColor[0, 1, 0],
LineBox[{{2.040816326530612*^-6, 2.040816326530612*^-6}, {
0.03067179205596268, 0.03067179205596268}, {0.06134154329559883,
0.06134154329559883}, {0.12268104577487113`, 0.12268104577487113`}, {
0.2453600507334157, 0.2453600507334157}, {0.4907180606505049,
0.4907180606505049}, {0.9814340804846833, 0.9814340804846833}, {
1.96286612015304, 1.96286612015304}, {4.090835708545865,
4.090835708545865}, {6.07778835701521, 6.07778835701521}, {
8.025764881887605, 8.025764881887605}, {10.138846915816112`,
10.138846915816112`}, {12.110912009821138`, 12.110912009821138`}, {
14.248082612882277`, 14.248082612882277`}, {16.346277092346465`,
16.346277092346465`}, {18.303454631887174`, 18.303454631887174`}, {20.,
20.}}]}},
AspectRatio->NCache[GoldenRatio^(-1), 0.6180339887498948],
Axes->True,
AxesOrigin->{0, 0},
PlotRange->{{0, 100}, {0, 20}},
PlotRangeClipping->True,
PlotRangePadding->{
Scaled[0.02], Automatic}]], "Output",
CellChangeTimes->{{3.540624354626749*^9, 3.540624410273096*^9},
3.540624672639637*^9, 3.540624735335573*^9, {3.540624800646563*^9,
3.540624806947823*^9}, 3.5406248461654253`*^9, 3.5407063812065363`*^9}]
}, Open ]],
Cell["A template for the R code in the function:", "Text",
CellChangeTimes->{{3.5407065548336563`*^9, 3.540706563191538*^9}}],
Cell[CellGroupData[{
Cell[BoxData[
RowBox[{
RowBox[{"CForm", "[",
RowBox[{"FullSimplify", "[",
RowBox[{"u", ",",
RowBox[{"Assumptions", "->",
RowBox[{"{",
RowBox[{
RowBox[{
SubscriptBox["a", "0"], ">", "0"}], ",",
RowBox[{
SubscriptBox["a", "1"], ">", "0"}], ",",
RowBox[{"x", ">", "0"}]}], "}"}]}]}], "]"}], "]"}], "/.",
RowBox[{"{",
RowBox[{
RowBox[{
SubscriptBox["a", "0"], "\[Rule]", "asymptDisp"}], ",",
RowBox[{
SubscriptBox["a", "1"], "\[Rule]", "extraPois"}], ",",
RowBox[{"x", "\[Rule]", "q"}]}], "}"}]}]], "Input",
CellChangeTimes->{{3.540706440235935*^9, 3.54070654712416*^9}}],
Cell["\<\
Log((1 + extraPois + 2*asymptDisp*q +
2*Sqrt(asymptDisp*q*(1 + extraPois + asymptDisp*q)))/
(4.*asymptDisp))/Log(2)\
\>", "Output",
CellChangeTimes->{{3.5407064886833467`*^9, 3.540706495739716*^9}, {
3.540706529058442*^9, 3.5407065480525093`*^9}}]
}, Open ]],
Cell[BoxData["\[IndentingNewLine]"], "Input",
CellChangeTimes->{3.5408157360105877`*^9}],
Cell[BoxData[
StyleBox[
RowBox[{"For", " ", "local", " ", "dispersion", " ", "fit"}],
"Subsubtitle"]], "Input",
CellChangeTimes->{{3.540815731390847*^9, 3.540815731815278*^9}}],
Cell[TextData[{
"In case of a local dispersion fit, the variance-stabilizing transformation ",
Cell[BoxData[
FormBox[
RowBox[{
RowBox[{"u", "(", "x", ")"}], " ", "=",
RowBox[{
SuperscriptBox["\[Integral]", "x"],
FractionBox["d\[Mu]",
SqrtBox[
RowBox[{"v", "(", "\[Mu]", ")"}]]]}]}], TraditionalForm]]],
"is obtained by numerical integration of the fitted mean-dispersion relation \
",
Cell[BoxData[
FormBox[
RowBox[{"v", "(", "\[Mu]", ")"}], TraditionalForm]],
FormatType->"TraditionalForm"],
" (by adding up along a asinh-spaced grid and a fitting a spline). Then, the \
scaling parameters \[Eta] and \[Xi] (see above) are chosen such that the VST \
is equal to ",
Cell[BoxData[
FormBox[
SubscriptBox["log", "2"], TraditionalForm]],
FormatType->"TraditionalForm"],
" for two large normalized count values (for which the 95- and the \
99.9-percentile of the sample-averaged normalized count values are used.)"
}], "Text",
CellChangeTimes->{{3.540815740854124*^9, 3.540815862626584*^9}, {
3.540815907181427*^9, 3.540816034208343*^9}, {3.540816089844325*^9,
3.5408161426064177`*^9}, {3.5408161836470623`*^9, 3.540816250332963*^9}}]
}, Open ]]
},
WindowSize->{640, 750},
WindowMargins->{{148, Automatic}, {Automatic, 24}},
PrintingPageRange->{Automatic, Automatic},
PrintingOptions->{"Magnification"->1.,
"PaperOrientation"->"Portrait",
"PaperSize"->{594.3000000000001, 840.51},
"PostScriptOutputFile"->"/home/anders/work/SVN/DESeq/inst/doc/vst.pdf"},
FrontEndVersion->"7.0 for Linux x86 (64-bit) (February 25, 2009)",
StyleDefinitions->"Default.nb"
]
(* End of Notebook Content *)
(* Internal cache information *)
(*CellTagsOutline
CellTagsIndex->{}
*)
(*CellTagsIndex
CellTagsIndex->{}
*)
(*NotebookFileOutline
Notebook[{
Cell[CellGroupData[{
Cell[567, 22, 182, 2, 85, "Subtitle"],
Cell[752, 26, 230, 5, 66, "Input"],
Cell[985, 33, 431, 13, 71, "Text"],
Cell[1419, 48, 92, 1, 32, "Input"],
Cell[1514, 51, 575, 20, 51, "Text"],
Cell[CellGroupData[{
Cell[2114, 75, 330, 8, 32, "Input"],
Cell[2447, 85, 234, 6, 47, "Output"]
}, Open ]],
Cell[2696, 94, 487, 19, 31, "Text"],
Cell[3186, 115, 105, 1, 31, "Text"],
Cell[CellGroupData[{
Cell[3316, 120, 255, 6, 32, "Input"],
Cell[3574, 128, 305, 8, 33, "Output"]
}, Open ]],
Cell[3894, 139, 1272, 42, 145, "Text"],
Cell[CellGroupData[{
Cell[5191, 185, 768, 20, 61, "Input"],
Cell[5962, 207, 679, 22, 72, "Output"]
}, Open ]],
Cell[6656, 232, 578, 20, 31, "Text"],
Cell[CellGroupData[{
Cell[7259, 256, 264, 6, 32, "Input"],
Cell[7526, 264, 769, 24, 72, "Output"]
}, Open ]],
Cell[8310, 291, 437, 11, 51, "Text"],
Cell[CellGroupData[{
Cell[8772, 306, 515, 15, 32, "Input"],
Cell[9290, 323, 301, 8, 52, "Output"]
}, Open ]],
Cell[9606, 334, 180, 4, 31, "Text"],
Cell[CellGroupData[{
Cell[9811, 342, 211, 6, 63, "Input"],
Cell[10025, 350, 178, 5, 54, "Output"]
}, Open ]],
Cell[10218, 358, 139, 1, 31, "Text"],
Cell[CellGroupData[{
Cell[10382, 363, 517, 15, 32, "Input"],
Cell[10902, 380, 330, 10, 63, "Output"]
}, Open ]],
Cell[11247, 393, 92, 1, 31, "Text"],
Cell[CellGroupData[{
Cell[11364, 398, 354, 11, 70, "Input"],
Cell[11721, 411, 323, 10, 63, "Output"]
}, Open ]],
Cell[12059, 424, 123, 1, 31, "Text"],
Cell[CellGroupData[{
Cell[12207, 429, 449, 14, 32, "Input"],
Cell[12659, 445, 92, 1, 31, "Output"]
}, Open ]],
Cell[CellGroupData[{
Cell[12788, 451, 517, 15, 32, "Input"],
Cell[13308, 468, 96, 1, 31, "Output"]
}, Open ]],
Cell[13419, 472, 113, 1, 31, "Text"],
Cell[CellGroupData[{
Cell[13557, 477, 434, 12, 32, "Input"],
Cell[13994, 491, 695, 21, 68, "Output"]
}, Open ]],
Cell[14704, 515, 579, 19, 51, "Text"],
Cell[CellGroupData[{
Cell[15308, 538, 685, 20, 55, "Input"],
Cell[15996, 560, 5767, 102, 227, "Output"]
}, Open ]],
Cell[21778, 665, 382, 7, 71, "Text"],
Cell[CellGroupData[{
Cell[22185, 676, 958, 25, 55, "Input"],
Cell[23146, 703, 6513, 112, 256, "Output"]
}, Open ]],
Cell[29674, 818, 126, 1, 31, "Text"],
Cell[CellGroupData[{
Cell[29825, 823, 676, 20, 55, "Input"],
Cell[30504, 845, 273, 6, 65, "Output"]
}, Open ]],
Cell[30792, 854, 89, 1, 55, "Input"],
Cell[30884, 857, 183, 4, 37, "Input"],
Cell[31070, 863, 1195, 29, 159, "Text"]
}, Open ]]
}
]
*)
(* End of internal cache information *)
|