(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 6.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 2601302, 43998] NotebookOptionsPosition[ 2589401, 43626] NotebookOutlinePosition[ 2590154, 43654] CellTagsIndexPosition[ 2590111, 43651] WindowFrame->Normal ContainsDynamic->True *) (* Beginning of Notebook Content *) Notebook[{ Cell["\<\ Computational Vision U. Minn. Psy 5036 Daniel Kersten Lecture 5\ \>", "Subtitle", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["Initialize standard library files:", "Subsubsection"], Cell[BoxData[ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"<<", "\"\\""}], ";", RowBox[{"<<", "\"\\""}], ";", RowBox[{"<<", "\"\\""}]}], ")"}], " ", ";", RowBox[{"Off", "[", RowBox[{"General", "::", "\"\\""}], "]"}], ";"}]], "Input", CellGroupingRules->"TitleGrouping", AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell["Goals", "Section"], Cell[CellGroupData[{ Cell["Last time", "Subsection"], Cell[CellGroupData[{ Cell[TextData[StyleBox["Ideal Observer Analysis: Essential idea", FontSize->12, FontColor->RGBColor[0.2, 0.2, 0.8]]], "Subsubsection"], Cell[TextData[{ StyleBox["Ideal observer", FontWeight->"Bold"], "\n\tModel the data (image) generation process\n\tDefine the inference task\n\ \tDetermine optimal performance" }], "Text"], Cell[TextData[{ StyleBox["Compare human performance to the ideal", FontWeight->"Bold"], "\n\tIdeal normalizes for information available" }], "Text"], Cell[TextData[{ StyleBox["Explain discrepancies in terms of:", FontWeight->"Bold"], "\n\tfunctional adaptation\n\tmechanism" }], "Text"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Psychophysical tasks & techniques (from the previous lecture)\ \>", "Subsection"], Cell[CellGroupData[{ Cell["The Receiver Operating Characteristic (ROC)", "Subsection"], Cell[TextData[{ "Although we can't directly measure the internal distributions of a human \ observer's decision variable, we've seen that we can measure hit and false \ alarm rates, and thus d'. \nBut one can do more, and actually test to see if \ an observer's decisions are consistent with Gaussian distributions with equal \ variance. If the criterion is varied, we can obtain a set of n data points: \n\ ", StyleBox["{(hit rate 1, false alarm rate 1), (hit rate 2, false alarm rate \ 2), ..., (hit rate n, false alarm rate n)} \n", FontSize->10], "all from one experimental condition (i.e. from one signal-to-noise ratio, \ call it ", Cell[BoxData[ FormBox[ SuperscriptBox[ SubscriptBox["d", "ideal"], "'"], TraditionalForm]]], "). This is because as the hit rate varies, so does the false alarm rate \ (see the above figures showing how hit and false alarm rates relate to area \ under the signal and noise distributions.). One could compute the d' for each \ pair and they should all be equal for the ideal observer. Of course, we would \ have to make a large number of measurements for each one--but on average, \ they should all be equal. \n\tTo get meaningful and equal d's for each pair \ of hit and false alarm rates assumes that the underlying relative separation \ of the signal and noise distributions remain unchanged and that the \ distributions are Gaussian, with equal standard deviation. We might know \ this is true (or true to a good approximation) for the ideal, but we have no \ guarantee for the human observer. Is there a way to check? Suppose the signal \ and noise distributions look like:" }], "Text"], Cell[GraphicsData["CompressedBitmap", "\<\ eJzNXNmzlcURP55z7s6FexGRJWwCKquAC26ILIKKIIi4RhERRBOXoCkSk8qr b8mjVPKWtchWMY8klapUnlLlY1JZqkyqsm/Fv3Bzumf695vpr7/LJU85VdzT X0/Pr3t6emZ65szH4VPvnjvzxql3Xzt9avXB86fePvfa6XdWH3jr/IDVu67T ue7LnU7nV6s7Qs8MyPxHP1+RP/kh0z0t7s1cvnx5Jn86R/TvkPIygP4bPKe6 d6W6g2qXLl2a2b9/vz4/oX9rnorvhPiVK1dmpqamKvGSV4kP6ZMUCZoYcPHi xU5XS/tCK2/Hjh0zH330UWKP6KOwByytXjx3TGbNmjXKF80iJypE5sKFC4A/ ceKEyonMQFU3u0ksya4CXGmJiJvVA3pQI7myVCkYg/JBfQD3AZwr9gJTXYuz /mGt8vHHH6tekcnPCWKhd6CKla2Tz9mzZ+E1gZc6Z86cKZuVtM2HFfKvNNos KGkpl2hAOyfCKoXR5hztai0cz9EwCsMFUQwXo61BA4GuOmW86gZBM5e759Se adgoaCW60MKzaqLJ3CQy8hF3vf/++x3rLTFcGme9Vfggy0xW+gr/RLiubr+q ixaPNVpsqs3kQVM6Fg2RA8vmZWfCBRIlIj8+Pl42LXlveWi1qSj7y0wbDLHW lgZmlx6qw0Hbo1UHonkopOEpkB31zuDTFY02UMR2zIWDT362ygIm9qWxPXiy euJfabsUZgGpN5LrFZAVltg/U8wTBX5FpwmgrV7fy7JdyQL4Tz4yhKTLBt+I yuwkNFtkT548WUx3w2iD1Jcoljo2BYdNHCtNrVSIqTmgc7NG20y0puUmmZaR 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The main point of this plot is to see if the data tend to fall on a \ straight line with slope of one. If a straight line, this would support the \ Gaussian assumption. A slope = 1 supports the assumption of equal variance \ Gaussian distributions.\n\tIn practice, there are several ways of obtaining \ an ROC curve in human psychophysical experiments. One can vary the criterion \ that an observer adopts by varying the proportion of times the signal is \ presented. As observers get used to the signal being presented, for example, \ 80% of the time, they become biased to assume the signal is present. One \ needs to block trials in groups of, say 400 trials per block, where the \ signal and noise priors are fixed for a given block.\n\tOne can also use a ", StyleBox["rating scale", FontSlant->"Italic"], " method in which the observer is asked to say how confident she/he was \ (e.g. 5 definitely, 4 quite probable, 3 don't know for sure, 2, unlikely, 1 \ definitely not). Then we can bin the proportion of \"5's\" when the signal \ vs. noise was present to calculate hit and false alarm rates for that rating, \ do the same for the \"4's\", and so forth. The assumption is that an observer \ can maintain not just one stable criterion, but four---the observer has in \ effect divided up the decision variable (x) domain into 5 regions. An \ advantage of the rating scale method is efficiency--relatively few trials are \ required to get an ROC curve. Further, in some experiments, ratings seem \ psychologically natural to make. But if there is any \"noise\" in the \ decision criterion itself, e.g. due to memory drift, or whatever, this will \ act to decrease the estimate of d' in both yes/no and rating methods." }], "Text"] }, Closed]], Cell[CellGroupData[{ Cell["Applications of ROC to neural measures", "Subsection"], Cell["\<\ The area under the ROC curve provides a useful measure of sensitivity even if \ the additive gaussian model isn't known to be correct. It can also be thought \ of as a measure of how much information about signal vs. no signal can be \ extracted from the data. ROC curves can be used characterize the sensitivity \ of single neurons, as well as gross overall measures of activity such as \ comes from brain imaging data. In the figure below, the gray lines represent a behavioral response by a \ human observer--i.e. when the signal is high, the observer is indicating \ subjective \"detection\". 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Is there a way to reduce \ the of a fluctuating criterion?problem of a fluctuating criterion?\ \>", "Text"], Cell[CellGroupData[{ Cell["\<\ Relating performance (proportion correct) to signal-to-noise ratio, d'. \ \>", "Subsubsection"], Cell["\<\ In psychophysics, the most common way to minimize the problem of a varying \ criterion is to use a two-alternative forced-choice procedure (2AFC). In a \ 2AFC task the observer is presented on each trial a pair of stimuli. One \ stimulus has the signal (e.g. high flash), and the other the noise (e.g. low \ flash). The order, however, is randomized. So if they are presented \ temporally, the signal or the noise might come first, but the observer \ doesn't know which from trial to trial. In the spatial version, the signal \ could be on the left of the computer screen with the noise on the right, or \ vice versa. One can show that for 2AFC:\ \>", "Text"], Cell[BoxData[ RowBox[{ SuperscriptBox["d", "'"], "=", " ", RowBox[{ RowBox[{"-", " ", SqrtBox[ RowBox[{"2", " "}]]}], "z", RowBox[{"(", RowBox[{"proportion", " ", "correct"}], ")"}]}]}]], "NumberedEquation"], Cell[TextData[{ "Exercise: Prove ", Cell[BoxData[ RowBox[{ SuperscriptBox["d", "'"], "=", " ", RowBox[{ RowBox[{"-", " ", SqrtBox[ RowBox[{"2", " "}]]}], "z", RowBox[{"(", RowBox[{"proportion", " ", "correct"}], ")"}]}]}]]] }], "Exercise"] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Calculating the Pattern Ideal's d' for a two-alternative forced-choice \ experiment from a z-score of the proportion correct. (see Homework Assignment \ #1)\ \>", "Subsubsection", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ For our 2AFC experiment, the observer gets two images to compare. One has the \ signal plus noise, and the other just noise. But the observer doesn't know \ which one is which. 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This decision requires some measure of comparison between the input and \ the stored \"template\" ", StyleBox["s", FontWeight->"Bold"], ". \nGiven two patterns represented by vectors ", StyleBox["x", FontWeight->"Bold"], " and ", StyleBox["s", FontWeight->"Bold"], ", how can we measure how close or similar they are? \nSome possibilities \ are: ", StyleBox["Abs[x-s]", FontWeight->"Bold"], ", ", StyleBox["Cos[x,s]", FontWeight->"Bold"], ", or ", StyleBox["Dot[x,s]", FontWeight->"Bold"], ". \nWe will see below that the ideal strategy is to compute the cross - \ correlation decision variable for each image (i.e. the dot product between \ each image data vector, say ", StyleBox["x", FontWeight->"Bold"], ", and an exact template of the signal, ", StyleBox["s", FontWeight->"Bold"], ", one is looking for), and pick the image which gives the larger cross - \ correlation." }], "Text"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Probability Overview", "Section"], Cell[TextData[{ "For terminology, a fairly comprehensive outline, and overview, see \ notebook:", Cell[BoxData[ FormBox[ ButtonBox[ RowBox[{" ", RowBox[{"ProbabilityOverview", ".", "nb", " "}]}], BaseStyle->"Hyperlink", ButtonData->{ URL[ "http://gandalf.psych.umn.edu/users/kersten/kersten-lab/courses/\ Psy5036W2008/Lectures/3_TheIdealObserver/ProbabilityOverview.nb"], None}], TraditionalForm]], FormatType->"TraditionalForm"], "in the syllabus web page, and for a general introduction,", ButtonBox[" Griffiths and Yuille (2008)", BaseStyle->"Hyperlink", ButtonData->{ URL["http://gandalf.psych.umn.edu/users/kersten/kersten-lab/coursepapers/\ GriffithsYuille2008.pdf"], None}], "." }], "Text"], Cell["\<\ For the section below, we'll use the properties of independence. 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Show that a simple decision variable for \ detecting a known fixed pattern in white gaussian noise is the dot product, \ or cross-correlation, of the observation image ", StyleBox["x", FontWeight->"Bold"], " with the known signal image ", StyleBox["s", FontWeight->"Bold"], ".\n\tr = ", StyleBox["x", FontWeight->"Bold"], "\[Bullet]", StyleBox["s ", FontWeight->"Bold"], ", or alternatively written as", StyleBox["\n\t", FontWeight->"Bold"], Cell[BoxData[ FormBox[ RowBox[{"r", "=", RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"i", "=", "1"}], "N"], " ", RowBox[{ RowBox[{"x", "(", "i", ")"}], " ", RowBox[{"s", "(", "i", ")"}]}]}]}], TraditionalForm]], "NumberedEquation"], "\n2. Show that d' is given by:" }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[GraphicsData["CompressedBitmap", "\<\ eJyVV0tu1EAQ7cx4Pp4PM7BGykwkho+QQNnCIsAQ2PBPtsAQEiULBIRI2bEg QG4Q5Qq5Qs6QK+QKuYKp113VLnvajrE0Y7u6Xv2ruv1itre9+WW2t7MxGz3b nX3b3tn4MVr9ukuk+oIx5oh+ayOD54Qe+e8ooYsWS24ze6snp6enyfr6ejIe j+kN6DqeLe3k5MTcdewf7a2RbG1tJawmWV5etiw1Xjs/P0+Oj48tfDgcQrS5 4+Af7C3KwKfTaYuNgDKQaNncdoj3LPTs7Mwj8Lu4uDBthsEA0AhubmlYlOzv 72tFMZPhrZDJ9JtZkMjDjwR0WA18AYnuJocQwzOICBq1mkkWdHh46FfJ407A ZHrOgRBcFfgQiMiTbLi1QxS4uBJIR5zSGQdcJQ8mui4im3elqcXZo8dM9sgL M9EFmCJRNcTeDKgjizjun1RViHMIJ+TWVEVzAojTVTTegQHHkhP1mdlBRlmS ONNldojmy4wd+3fFDlOhcenyNqtwi9gnBEKVGt59ayGOsgbj0Fp/HbzOhlGd ScB8IyOIuMAeMQ3JF98OtIjY+iUxBxRaUXcID94hpmbFRJIumRW/lSSnvWmh +oKDpL3OEnCJBFo6yEtoWEIzUwn4Ib901SA/40DkQ8RVKJHNVyHGzDWsDfC6 mZWh1VFMG4GeIYsL4XrCkKhGgQkYY390/rPNSq6JYj0yKUx2Hq+l8fFw3bZe 8Zw9V73dvbwgQejxREIbLtiueBBgKZ5Bsagm16TEEptLIy2NuvSiZfvlMPm9 AEM9SjEtXbi+diSUP50MtzG0fLIQUlEFt9DjNRaHstOXBApFzuJiFodC1mXB /W83hFqg2CWClC6WNAgUF63KyJIck90mh1BtjSkmCCkWIgli6IMlTNIOCL9m olzkdigG0Nt9PDbx9DyL0A0Ae4sRV3hNIThMDV9Aoh/23wOhpQQ435s2dTAO ggAAmGh2NrcTIzXjWgdMPMelyoDhaWFG1Vl7zopV3OJUTSd/PrAVKKUjs5My PPo/dlb3FLdOqi6NH5VTsMtpfbE6a7dIjVSYS7PU0GLpKgub+sy5+ZdNt0wB aR9nVjJ3nFHaKrBWUt1nbO68OQmTWeSKD026CfXYcd2GMiVulKyFRdo9uMOw 0K4klS/HaD1xiDYsgfaURlu10ZxG3fToJUxOaVY1mfxRQY4P4OiXyGDVL33U 4oLIUEkGp0OTFeqzSbdERj+sMHsAsEeNxM9XHtDMWtesXr/e3rrVRbaVOTZ2 bW9Y4BjEn00dXxr9CtxxUAV/q4kbYpt0DLaT/EdMubuXy2NDXnmb2xxVnSfJ I4AiEHuvGFMt99VkzhkUc04wZvUIBghZgyCii8Ohwtfn/341cdoMK+a6JRS2 8jjP/8QRXns/SgkW8riA8MZDNMFyPCogvPWQUoKFrCjCw7QEzDvcHpQRgt8/ ZuEf4M/F/g==\ \>"], "Graphics", Evaluatable->False, AspectRatioFixed->True, ImageSize->{92, 51.75}, ImageMargins->{{34, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}] }, Closed]], Cell[TextData[{ StyleBox["s", FontWeight->"Bold"], " and ", StyleBox["x", FontWeight->"Bold"], " are a vectors, i.e. lists, of the image intensities, and \[Sigma] is the \ standard deviation of the added gaussian noise." }], "Text"], Cell[CellGroupData[{ Cell["\<\ 1. Cross correlation produces an ideal decision variable: Proof\ \>", "Subsubsection"], Cell["\<\ What is the optimal decision variable? Starting from the maximum a posteriori \ rule, we saw that basing decisions on the likelihood ratio is ideal, in the \ sense of minimizing the probability of error. So the likelihood ratio is a \ decision variable. But it isn't the only one, because any monotonic function \ is still optimal. So our goal is to pick a decision variable which is simple, \ intuitive, and easy to compute. But first, we need an expression for the \ likelihood ratio:\ \>", "Text"], Cell[BoxData[ FormBox[ StyleBox[ FractionBox[ RowBox[{"p", " ", RowBox[{"(", RowBox[{"x", "|", RowBox[{"signal", " ", "plus", " ", "noise"}]}], ")"}]}], RowBox[{"p", " ", RowBox[{"(", RowBox[{"x", " ", "|", " ", RowBox[{"noise", " ", "only"}]}], ")"}]}]], FontWeight->"Bold"], TraditionalForm]], "NumberedEquation"], Cell[TextData[{ "where ", StyleBox["x", FontWeight->"Bold"], " is the vector representing the image measurements actually observed" }], "Text"], Cell[TextData[{ "\t", StyleBox["x = s + n, under signal plus gaussian noise condition", FontWeight->"Bold"], "\n\t", StyleBox["x = n, under gaussian noise only condition", FontWeight->"Bold"] }], "Text"], Cell[TextData[{ "\nFirst let's consider just one pixel of intensity x. Under the signal plus \ noise condition, the values of x fluctuate about the average signal intensity \ s with a Gaussian distribution (", StyleBox["gp[ ]", FontWeight->"Bold"], ") with mean s and standard deviation \[Sigma].\nSo under the signal plus \ noise condition, the likelihood ", StyleBox["p[x|s] ", FontWeight->"Bold"], "is the ", StyleBox["gp[x-s; \[Sigma]]", FontWeight->"Bold"], ":" }], "Text"], Cell["\<\ gp[x_,s_,\[Sigma]_]:= (1/(\[Sigma]*Sqrt[2 Pi])) Exp[-(x-s)^2/(2 \[Sigma]^2)]\ \>", "Input"], Cell[CellGroupData[{ Cell["gp[x,s,\[Sigma]]", "Input"], Cell[BoxData[ FormBox[ FractionBox[ SuperscriptBox["\[ExponentialE]", RowBox[{"-", FractionBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"x", "-", "s"}], ")"}], "2"], RowBox[{"2", " ", SuperscriptBox["\[Sigma]", "2"]}]]}]], RowBox[{ SqrtBox[ RowBox[{"2", " ", "\[Pi]"}]], " ", "\[Sigma]"}]], TraditionalForm]], "Output"] }, Closed]], Cell["\<\ Again, consider just one pixel of intensity x. Under the noise only \ condition, the values of x fluctuate about the average intensity \ corresponding to the mean of the noise, which we assume is zero. So under the noise only condition, the likelihood p[x|n] is:\ \>", "Text"], Cell[CellGroupData[{ Cell["gp[x,0,\[Sigma]]", "Input"], Cell[BoxData[ FormBox[ FractionBox[ SuperscriptBox["\[ExponentialE]", RowBox[{"-", FractionBox[ SuperscriptBox["x", "2"], RowBox[{"2", " ", SuperscriptBox["\[Sigma]", "2"]}]]}]], RowBox[{ SqrtBox[ RowBox[{"2", " ", "\[Pi]"}]], " ", "\[Sigma]"}]], TraditionalForm]], "Output"] }, Closed]], Cell[TextData[{ "But we actually have a whole pattern of values of x, which make up an image \ vector ", StyleBox["x", FontWeight->"Bold"], ". So consider a pattern of image intensities represented now by a vector ", StyleBox["x = {x[1],x[2],...x[N]}", FontWeight->"Bold"], ". Let the mean values of each pixel under the signal plus noise condition \ be given by vector ", StyleBox["s = {s[1],s[2],...,s[N]}", FontWeight->"Bold"], ". The joint probability of an image observation ", StyleBox["x", FontWeight->"Bold"], ", under the signal hypothesis is:" }], "Text"], Cell[CellGroupData[{ Cell["\<\ Product[gp[x[i],s[i],\[Sigma]],{i,1,N}]\ \>", "Input"], Cell[BoxData[ FormBox[ RowBox[{ UnderoverscriptBox["\[Product]", RowBox[{"i", "=", "1"}], "N"], FractionBox[ SuperscriptBox["\[ExponentialE]", RowBox[{"-", FractionBox[ SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"x", "(", "i", ")"}], "-", RowBox[{"s", "(", "i", ")"}]}], ")"}], "2"], RowBox[{"2", " ", SuperscriptBox["\[Sigma]", "2"]}]]}]], RowBox[{ SqrtBox[ RowBox[{"2", " ", "\[Pi]"}]], " ", "\[Sigma]"}]]}], TraditionalForm]], "Output"] }, Closed]], Cell[TextData[{ "\nThis is because we are assuming independence. In general, whether we can \ assume independence depends on the problem. In our case, the samples are \ independent by definition--as \"experimenters\" we generate the noise as \ independent samples. \nIndependence between pixels means we can multiply the \ individual probabilities to get the global joint image probability. (See \ above and ", StyleBox["ProbabilityOverview.nb", FontWeight->"Bold"], ")\nThe joint probability of an image observation ", StyleBox["x", FontWeight->"Bold"], ", under the noise only hypothesis is:" }], "Text"], Cell[CellGroupData[{ Cell["Product[gp[x[i],0,\[Sigma]],{i,1,N}]", "Input"], Cell[BoxData[ FormBox[ RowBox[{ UnderoverscriptBox["\[Product]", RowBox[{"i", "=", "1"}], "N"], FractionBox[ SuperscriptBox["\[ExponentialE]", RowBox[{"-", FractionBox[ SuperscriptBox[ RowBox[{"x", "(", "i", ")"}], "2"], RowBox[{"2", " ", SuperscriptBox["\[Sigma]", "2"]}]]}]], RowBox[{ SqrtBox[ RowBox[{"2", " ", "\[Pi]"}]], " ", "\[Sigma]"}]]}], TraditionalForm]], "Output"] }, Closed]], Cell["Now we have what we need for the likelihood ratio:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"Product", "[", RowBox[{ RowBox[{"gp", "[", RowBox[{ RowBox[{"x", "[", "i", "]"}], ",", RowBox[{"s", "[", "i", "]"}], ",", "\[Sigma]"}], "]"}], ",", RowBox[{"{", RowBox[{"i", ",", "1", ",", "N"}], "}"}]}], "]"}], "/", RowBox[{"Product", "[", RowBox[{ RowBox[{"gp", "[", RowBox[{ RowBox[{"x", "[", "i", "]"}], ",", "0", ",", "\[Sigma]"}], "]"}], ",", RowBox[{"{", RowBox[{"i", ",", "1", ",", "N"}], "}"}]}], "]"}]}]], "Input"], Cell[BoxData[ FormBox[ FractionBox[ RowBox[{ UnderoverscriptBox["\[Product]", RowBox[{"i", "=", "1"}], "N"], FractionBox[ SuperscriptBox["\[ExponentialE]", RowBox[{"-", FractionBox[ SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"x", "(", "i", ")"}], "-", RowBox[{"s", "(", "i", ")"}]}], ")"}], "2"], RowBox[{"2", " ", SuperscriptBox["\[Sigma]", "2"]}]]}]], RowBox[{ SqrtBox[ RowBox[{"2", " ", "\[Pi]"}]], " ", "\[Sigma]"}]]}], RowBox[{ UnderoverscriptBox["\[Product]", RowBox[{"i", "=", "1"}], "N"], FractionBox[ SuperscriptBox["\[ExponentialE]", RowBox[{"-", FractionBox[ SuperscriptBox[ RowBox[{"x", "(", "i", ")"}], "2"], RowBox[{"2", " ", SuperscriptBox["\[Sigma]", "2"]}]]}]], RowBox[{ SqrtBox[ RowBox[{"2", " ", "\[Pi]"}]], " ", "\[Sigma]"}]]}]], TraditionalForm]], "Output"] }, Closed]], Cell[TextData[{ "So at this point, we could just stop and write a program to use this \ product to make ideal decisions. E.g. if the product is bigger than 1, choose \ the signal hypothesis, and if less than 1 choose the noise hypothesis. \nBut \ we can get a much simpler rule with a little more work. \nThis is because any \ monotonic function, ", StyleBox["f()", FontWeight->"Bold"], " of the likelihood ratio would give the same performance (i.e. choose \ signal if ", StyleBox["f(likelihood ratio)>f(1)", FontWeight->"Bold"], ", and noise otherwise), let's try one--the natural logarithm will turn the \ product into a sum:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Log", "[", FractionBox[ RowBox[{ UnderoverscriptBox["\[Product]", RowBox[{"i", "=", "1"}], "N"], RowBox[{"gp", "[", RowBox[{ RowBox[{"x", "[", "i", "]"}], ",", RowBox[{"s", "[", "i", "]"}], ",", "\[Sigma]"}], "]"}]}], RowBox[{ UnderoverscriptBox["\[Product]", RowBox[{"i", "=", "1"}], "N"], RowBox[{"gp", "[", RowBox[{ RowBox[{"x", "[", "i", "]"}], ",", "0", ",", "\[Sigma]"}], "]"}]}]], "]"}]], "Input"], Cell[BoxData[ FormBox[ RowBox[{"log", "(", FractionBox[ RowBox[{ UnderoverscriptBox["\[Product]", RowBox[{"i", "=", "1"}], "N"], FractionBox[ SuperscriptBox["\[ExponentialE]", RowBox[{"-", FractionBox[ SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"x", "(", "i", ")"}], "-", RowBox[{"s", "(", "i", ")"}]}], ")"}], "2"], RowBox[{"2", " ", SuperscriptBox["\[Sigma]", "2"]}]]}]], RowBox[{ SqrtBox[ RowBox[{"2", " ", "\[Pi]"}]], " ", "\[Sigma]"}]]}], RowBox[{ UnderoverscriptBox["\[Product]", RowBox[{"i", "=", "1"}], "N"], FractionBox[ SuperscriptBox["\[ExponentialE]", RowBox[{"-", FractionBox[ SuperscriptBox[ RowBox[{"x", "(", "i", ")"}], "2"], RowBox[{"2", " ", SuperscriptBox["\[Sigma]", "2"]}]]}]], RowBox[{ SqrtBox[ RowBox[{"2", " ", "\[Pi]"}]], " ", "\[Sigma]"}]]}]], ")"}], TraditionalForm]], "Output"] }, Closed]], Cell["which is equal to:", "Text"], Cell[BoxData[ FormBox[ RowBox[{"Log", "(", RowBox[{ UnderoverscriptBox["\[Product]", RowBox[{"i", "=", "1"}], "N"], FractionBox[ SuperscriptBox["\[ExponentialE]", RowBox[{"-", FractionBox[ RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{ RowBox[{"x", "(", "i", ")"}], "-", RowBox[{"s", "(", "i", ")"}]}], ")"}], "2"], "-", SuperscriptBox[ RowBox[{"x", "(", "i", ")"}], "2"]}], RowBox[{"2", " ", SuperscriptBox["\[Sigma]", "2"]}]]}]], RowBox[{ SqrtBox[ RowBox[{"2", " ", "\[Pi]"}]], " ", "\[Sigma]"}]]}], ")"}], TraditionalForm]], "NumberedEquation"], Cell["which is monotonic with:", "Text"], Cell[BoxData[ FormBox[ RowBox[{"Log", "[", RowBox[{ UnderoverscriptBox["\[Product]", RowBox[{"i", "=", "1"}], "N"], SuperscriptBox["\[ExponentialE]", FractionBox[ RowBox[{"2", RowBox[{"x", "(", "i", ")"}], RowBox[{"s", "(", "i", ")"}]}], RowBox[{"2", " ", SuperscriptBox["\[Sigma]", "2"]}]]]}], "]"}], TraditionalForm]], "NumberedEquation"], Cell["which simplifies to", "Text"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"(", RowBox[{"1", "/", " ", SuperscriptBox["\[Sigma]", "2"]}], ")"}], RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"i", "=", "1"}], "N"], " ", RowBox[{ RowBox[{"x", "(", "i", ")"}], " ", RowBox[{"s", "(", "i", ")"}]}]}]}], TraditionalForm]], "NumberedEquation"], Cell["which is monotonic with:", "Text"], Cell[CellGroupData[{ Cell["Sum[x[i] s[i],{i,1,N}]", "Input"], Cell[BoxData[ FormBox[ RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"i", "=", "1"}], "N"], RowBox[{ RowBox[{"s", "(", "i", ")"}], " ", RowBox[{"x", "(", "i", ")"}]}]}], TraditionalForm]], "Output"] }, Closed]], Cell[BoxData[ FormBox[ RowBox[{"r", "=", RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"i", "=", "1"}], "N"], " ", RowBox[{ RowBox[{"x", "(", "i", ")"}], " ", RowBox[{"s", "(", "i", ")"}]}]}]}], TraditionalForm]], "NumberedEquation"], Cell["\<\ In other words, we've proven that the dot product, r, (or cross-correlation \ or matched filter) provides a decision variable which is optimal--in the \ sense that if we use the rule, the probability of error will be least. Now, \ let's calculate d'.\ \>", "Text"] }, Closed]], Cell[CellGroupData[{ Cell["2. Derive formula for d'", "Subsubsection"], Cell["By definition", "Text"], Cell[CellGroupData[{ Cell["d'=(\[Mu]2 - \[Mu]1)/\[Sigma]", "Input"], Cell[BoxData[ FormBox[ FractionBox[ RowBox[{"\[Mu]2", "-", "\[Mu]1"}], "\[Sigma]"], TraditionalForm]], "Output"] }, Closed]], Cell["\<\ where u2 is the mean of the decision variable, r ,under the signal hypothesis \ (i.e. \"switch set to send signal\"), and u1 is the mean under the noise-only \ hypothesis (i.e. switch set to not send signal). (For our light \ discrimination example, \[Mu]2 = b, and \[Mu]1 =d)\ \>", "Text"], Cell[TextData[{ "To get d', we need formulas for the means and standard deviation for the \ decision variable, r under the two hypotheses, \"signal plus noise\" vs. \ \"noise\" only.\n", StyleBox["First", FontWeight->"Bold"], ", ", StyleBox["suppose the switch is set for signal trials", FontSlant->"Italic"], ". What is the average and standard deviation of r? I.e. \[Mu]2 and ", "\[Sigma]?" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"\[Mu]2", " ", "=", " ", RowBox[{ RowBox[{"Average", "[", "r", "]"}], " ", "=", " ", RowBox[{ RowBox[{"Average", "[", FormBox[ RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"i", "=", "1"}], "N"], " ", RowBox[{ RowBox[{"x", "(", "i", ")"}], " ", RowBox[{"s", "(", "i", ")"}]}]}], TraditionalForm], "]"}], "=", RowBox[{ FormBox[ RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"i", "=", "1"}], "N"], " ", RowBox[{ RowBox[{"Average", "[", RowBox[{"x", "(", "i", ")"}], "]"}], RowBox[{"s", "(", "i", ")"}]}]}], TraditionalForm], "=", RowBox[{ FormBox[ RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"i", "=", "1"}], "N"], " ", RowBox[{ RowBox[{"s", "(", "i", ")"}], " ", RowBox[{"s", "(", "i", ")"}]}]}], TraditionalForm], "=", FormBox[ RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"i", "=", "1"}], "N"], " ", SuperscriptBox[ RowBox[{"s", "(", "i", ")"}], "2"]}], TraditionalForm]}]}]}]}]}]], "NumberedEquation"], Cell[BoxData[ FormBox[ RowBox[{"\[Mu]2", " ", "=", RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"i", "=", "1"}], "N"], SuperscriptBox[ RowBox[{"s", "(", "i", ")"}], "2"]}]}], TraditionalForm]], "NumberedEquation"] }, Closed]], Cell["(Because x(i)=s(i)+n(i), Average[x(i)]=s(i).)", "Text"], Cell["And the variance is:", "Text"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"Var", "(", RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"i", "=", "1"}], "N"], RowBox[{ RowBox[{"x", "(", "i", ")"}], " ", RowBox[{"s", "(", "i", ")"}]}]}], ")"}], "=", RowBox[{ RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"i", "=", "1"}], "N"], " ", RowBox[{ SuperscriptBox[ RowBox[{"s", "(", "i", ")"}], "2"], RowBox[{"Var", "[", RowBox[{"x", "(", "i", ")"}], "]"}]}]}], "=", RowBox[{ UnderoverscriptBox[ RowBox[{ SuperscriptBox["\[Sigma]", "2"], "\[Sum]"}], RowBox[{"i", "=", "1"}], "N"], " ", SuperscriptBox[ RowBox[{"s", "(", "i", ")"}], "2"]}]}]}], TraditionalForm]], "NumberedEquation"], Cell["\<\ (We've used to rules from above: Var[Y + Z] = Var[Y] + Var[Z], but one is a \ constant, so because Var[constant + n]=Var[n]. And, recall that Var[c Y] = c^2 Var[Y])\ \>", "Text"], Cell[TextData[{ StyleBox["Second", FontWeight->"Bold"], ", suppose the switch is set for noise only trials. The average of the dot \ product is:" }], "Text"], Cell[BoxData[ RowBox[{"\[Mu]1", " ", "=", " ", RowBox[{ RowBox[{"Average", "[", "r", "]"}], " ", "=", " ", RowBox[{ RowBox[{"Average", "[", FormBox[ RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"i", "=", "1"}], "N"], " ", RowBox[{ RowBox[{"x", "(", "i", ")"}], " ", RowBox[{"s", "(", "i", ")"}]}]}], TraditionalForm], "]"}], "=", RowBox[{ FormBox[ RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"i", "=", "1"}], "N"], " ", RowBox[{ RowBox[{"Average", "[", RowBox[{"x", "(", "i", ")"}], "]"}], RowBox[{"s", "(", "i", ")"}]}]}], TraditionalForm], "=", RowBox[{ FormBox[ RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"i", "=", "1"}], "N"], " ", RowBox[{"0", " ", RowBox[{"s", "(", "i", ")"}]}]}], TraditionalForm], "=", FormBox["0", TraditionalForm]}]}]}]}]}]], "NumberedEquation"], Cell["The variance is the same as for the signal case:", "Text"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"Var", "(", RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"i", "=", "1"}], "N"], RowBox[{ RowBox[{"x", "(", "i", ")"}], " ", RowBox[{"s", "(", "i", ")"}]}]}], ")"}], "=", RowBox[{ RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"i", "=", "1"}], "N"], " ", RowBox[{ SuperscriptBox[ RowBox[{"s", "(", "i", ")"}], "2"], RowBox[{"Var", "[", RowBox[{"x", "(", "i", ")"}], "]"}]}]}], "=", RowBox[{ UnderoverscriptBox[ RowBox[{ SuperscriptBox["\[Sigma]", "2"], "\[Sum]"}], RowBox[{"i", "=", "1"}], "N"], " ", SuperscriptBox[ RowBox[{"s", "(", "i", ")"}], "2"]}]}]}], TraditionalForm]], "NumberedEquation"], Cell["", "Text"], Cell["So d' is:", "Text"], Cell[CellGroupData[{ Cell["\<\ Sum[s[i]^2, {i, 1, N}]/Sqrt[(\[Sigma]^2 Sum[s[i]^2,{i,1,N}])]\ \>", "Input"], Cell[BoxData[ FormBox[ FractionBox[ RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"i", "=", "1"}], "N"], SuperscriptBox[ RowBox[{"s", "(", "i", ")"}], "2"]}], SqrtBox[ RowBox[{ SuperscriptBox["\[Sigma]", "2"], " ", RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"i", "=", "1"}], "N"], SuperscriptBox[ RowBox[{"s", "(", "i", ")"}], "2"]}]}]]], TraditionalForm]], "Output"] }, Closed]], Cell[CellGroupData[{ Cell["\<\ FullSimplify[Sum[s[i]^2, {i, 1, N}]/Sqrt[(\[Sigma]^2 Sum[s[i]^2,{i,1,N}])],\ \[Sigma]>0]\ \>", "Input"], Cell[BoxData[ FormBox[ FractionBox[ SqrtBox[ RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"i", "=", "1"}], "N"], SuperscriptBox[ RowBox[{"s", "(", "i", ")"}], "2"]}]], "\[Sigma]"], TraditionalForm]], "Output"] }, Closed]], Cell["Or:", "Text"], Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"d", "'"}], "=", RowBox[{ FractionBox[ SqrtBox[ RowBox[{ UnderoverscriptBox["\[Sum]", RowBox[{"i", "=", "1"}], "N"], SuperscriptBox[ RowBox[{ StyleBox["s", FontWeight->"Plain"], "(", "i", ")"}], "2"]}]], "\[Sigma]"], "=", FractionBox[ SqrtBox[ RowBox[{"s", ".", "s"}]], "\[Sigma]"]}]}], TraditionalForm]], "NumberedEquation"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Calculating the Pattern Ideal's d' for a two-alternative forced-choice \ experiment from a z-score of the proportion correct.\ \>", "Subsection", Evaluatable->False, AspectRatioFixed->True], Cell["\<\ Recall that we had an expression for d' for a yes/no experiment in which we \ measured hit and false alarm rates. We've seen the expression for d' for a 2AFC experiment earlier lecture, but \ let's review it. For a 2AFC experiment, the observer gets two images to compare. One has the \ signal plus noise, and the other just noise. But the observer doesn't know \ which one is which. An ideal strategy is to compute the cross-correlation \ decision variable for each image, and pick the image which gives the larger \ cross-correlation. 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