12 Seeing and Psychophysics 12 1 Hubble telescope photograph Courtesy NASA Chapter 12 I magine you are with a friend watching the stars come out as twilight fades into night, 12.1. Your friend calls out that he can see a star in a certain location in the sky. You try hard but can't make it out, even though you are sure you are looking in th e right place. You conclude that your friend has a much better visual system than your own when it comes to detecting faint light sources. This kind of situation is not unusual. We are quite often called upon to detect faint stimuli , which relies on an ability to discriminate between the presence or absence of such stimuli. So detec- tion and discrimination are intimately related , although they are often treated as distinct abili- ties An example of discrimination is the ability to judge which of two stars is brighter. Psychologists have evolved some clever techniques for making careful measurements of people's ability to detect andlor discriminate stimuli. We explain these psy- chophysical methods and their theoretical under- pinnings in this chapter. Throughout this chapter we use the example of luminance, but these meth- ods can be applied to any perceived quantity, such as contrast, color, sound , and weight. Psychophysical methods have to be highly sensi- tive if they are to do justice to the sensitivity of the human visual system. For example, the fact that photoreceptors can signal the presence of a single Why read this chapter? What is the dimmest light that can be seen? What is the smallest difference in light that can be seen? These related questions are the subject of this chap- ter , which uses image light intensity (luminance) as an example, even though the ideas presented here apply to any stimulus property, such as motion or line orientation. We begin by introducing the idea of a threshold luminance, which, in principle, defines the dimmest light that can be seen. When we come to describe how to measure a threshold, we find that it varies from moment to moment, and between dif- ferent individuals, depending on their willingness to say yes, rather than on their intrinsic sensitivity to light. Two solutions to this dilemma are described. First, a two-alternative forced choice (2AFC) proce- dure measures sensitivity to light. Second, signal detection theory provides a probabilistic frame- work which explains sensitivity without the need to invoke the idea of thresholds. We discuss how these methods can be used to redefine the no- tion of a threshold in statistical terms. Finally, we give a brief historical overview of psychophysics. a 0.8 0.6 (/) Q) >- Q 0.4 0 2 0 0 2 3 4 5 6 Luminance b 0.8 0 6 W Q) >- Threshold = 3 Q 0 4 0.2 0 0 2 3 4 5 6 Luminance 12.2 Transition from not seeing to seeing a In an ideal world , the probability of seeing a brief flash of light would jump from zero to one as the light increased in luminance 282 b In practice , the probability of a yes response increases gradually , and defines a smooth S-shaped or sigmOidal psychometric function. The threshold is taken to be the mid-point of this curve : the luminance at which the probabil- ity of seeing the light is 0.5. Most quantities on the abscissa (horizontal axis) in plots of this kind are given in log units , so that each increment (e.g. , from 3 to 4) along this axis implies a multiplicative change in magnitude For example, if the log scale being used was to the base 10 then a single step on the abscissa would mean the luminance increased by a factor of ten. photon was first deduced using a simple psycho- physical method by Hecht, 5chlaer and Pirenne in 1942 (a photon is a discrete packet or quantum oflight). It was confirmed about 40 years later by measuring of the outputs of single photoreceptors. This high degree of visual sensitivity is why we can easily see a white piece of paper in starlight, even though each retinal photoreceptor receives only 1 photon of light every hundred seconds from the paper (with no moon light and no stray light from man-made sources). To put it another way, each se t of 100 photo receptors receives an average of a bout 1 photon per second in starlight. Whichever way we choose to put it, the eye is a remarkabl y efficient d e tector of light. (Note that only the rods, Ch 6, are active in the extremely low light level of starlight.) The smallest luminance that we can see is called the absolute threshold, whereas the small- est perceptible diffirence in luminance is called the difference threshold, difference limen, or just noticeable difference OND) for the historical reason that it was deemed to be just noticeable. But exactly what do we mean by "see" in terms of absolute thresholds and JNDs? Thi s i s a question that caused researchers in the 19th a nd early 20th centuries to develop what we will refer to as classi- cal psychophysics Classical Psychophysics: Absolute Thresholds Suppose you were presented with a series of brief light flashes of increasing luminance , starting from a luminance that lies below your absolute threshold. As the luminance is increased you might expect that there comes a point at which you are s uddenly able to see the light flashes, so that the probability of seeing light flashes increases from zero (no flash is seen) to certainty (probability= 1 ; every flash is seen), 12.2a. However, in practice this isn't what happens. Instead , the probability of seeing light flashes gradually increases as luminance increases , as depicted by the psychometric func- tion in 12.2b. This psychometric function implies that there comes a point, as the light flash luminance increa s - es above zero , at which it starts to become visible, but it isn ' t seen reliably on every presentation. For many presentations of the same (low) luminance, sometimes you see the flash , and sometimes you don ' t. As the luminance is increa se d, the probabil- ity of seeing the flash increases gradually from zero to one. If you respond yes when the light is seen, and no when it is not, then your absolute thresh- old is taken to be the luminance which yields a yes response 50% of the time A major problem in measuring absolute thresholds plagued ps yc hophysics for many years: Seeing and Psychophysics different people have different criteria for saying yes. Suppose two people see the same thing when exposed to a very dim light One of them may be willing to say yes, whereas the other may say no because s/he requires a higher degr ee of certainty. (Remember, we are talking here about very low luminances, for which percepts are anything but vivid.) Let's call these two people, which we assume have identical visual systems, Cautious and Risky , respectively. Both Cautious and Risky see the same thing, but Cautious demands a great deal of evi- dence before being willing to commit. In contrast, Risky is a more flamboyant individual who does not need to be so certain before saying yes. Let's put Cautious and Risky in a laboratory , and show them a set of, say, 10 lights with lumi- nance values ranging from zero, through very dim, to clearly visible. We present each of these lights 100 times in a random order, so there will be 1000 trials in this experiment (i.e., 100 trials per lumi- nance level x 10 luminance levels). As shown in 12.2b, at very low luminance levels, the light can- not be seen , and so the proportion of yes responses is close to zero. As luminance increases , the pro- portion of yes responses incre ases, until at very high luminance values the proportion of yes responses is close ro unity. If the proportion of yes responses for each if the 10 luminance levels is plotted against luminance then the data would look rather like the psychometric function in 12.2b However, if we compare the corresponding curves for Cautious and Risky, 12.3 , we find that both psychometric functions are identical in shape, but that Cautious' curve is to the right of Risky's curve. This graph implies that a light which induces a yes response of say 20% of the time from Risky might induce a yes re spo nse 5% of the time Cautious. Thus, all of Cautious' responses " lag behind ' Risky's responses in terms of luminance. Recall that both Risky and Cautious see the same thing , but Risky is just more willing to say " Yes - that almost invisible dim percept was caused by a light flash rather than being an artefact of my visual system. " To sum up : the absolute threshold is the lumi- nance which evokes a yes response on 50% of trials, but Risky's estimated threshold is much lower than Cautious', even though they both see the same thing The differenc e between the estimated thresholds 283 en Q) >- Q Chapter 12 thre s hold and the JND are independent quanti- ties: knowing one tells us nothing about the value , , of the other one, in principle, at least. In practice , , a low absolute threshold is usually associated with I 0.8 a small JND, but it is possible for an observer to have a l ow absolute threshold and a large JND, or 0.6 vice versa To illustrate the ind ependence between absolute 0.4 thre s holds and JNDs , notice that the curves in 12.3 have the same slopes. This means that Cau- 0 .2 tious and Risky have the same JND despite their , different absolute thresholds The point is that , 0 the JND is defined as the difference in luminance 0 2 3 4 5 6 7 8 9 10 Luminance a 12.3 Individual differences in responding The proportion of trials on which two observers , Cautious (red dashed curve) and Risky (black solid curve) are willing to respond yes to the question : " Did you see a light?" of Cautious and Risky arises only because Risky is predisposed to say yes more often than Cautious. This is clearl y nonsensical because we are inter- ested in mea s uring the sensitivities of Cautious and Risky 's visual systems, not their personalities. Fortunately, it can be fixed using signal detection theory (SD1) and a procedure known as two-al- ternative forced choice (2AFC), both of w hi ch are described l a ter. b Just Noticeable Differences The difference threshold or JND is usually defined as the change in luminance required to increase the proportion of yes re s ponses from 50 % to 7 5%. Consider two observers called Sensitive a nd Insen- sitive, whose ps yc hometric function s are shown in 12.4a: Sensitive (solid black curve) and Insensitive (dashed red curve). A small increment in lumi- nance has a dramatic effect on the proportion of yes responses from Sensitive, but the same luminance change has a relatively small effect on the propor- tion of yes responses from Insensitive. 111erefore, Sensitive requires a s maller increase in luminance than Insensitive to raise the proportion of yes responses from 50 % to 75% It follow s that Sensi- tive must have a smaller JND than Insensitive , even though both observers may have the same absolute threshold , as in 12.4b. Thus , the abso lut e 284 0.8 en 0.6 Q) >- Q 0.4 0 2 o --/ ) I I I I I ----:-: - , " , , , , I I o 2 3 4 5 6 7 8 9 10 Luminance 12.4 Individual differences in psychometric function a Two observers with different JNDs and different absolute thresholds. b Two observers with different JNDs and the same absolute threshold Note: 12.3 shows two observers with the same JNDs , but different absolute thresholds required to increase the proportion of yes responses from 50% to 75%, a change which i s unaffected by th e Lo cation of the curve. Thu s, the data from Cautious and Risky give biased estimates of their absolute thresholds , but accurate estimates of their JND s Caveat: The definition for the JND given above is: the change in lumin a n ce required to increase the proportion of yes responses from 50% to 7 5% However, the value of75% i s quite arbitrary, and i s used for hi sto rical rea so n s; it could just as easily have been 80% or 60%. The main thing for mean- ingful comparisons is to have a consistent measure of the steepness of the psychometric function In fact, for reasons that will becom e clear , we wi ll use 76% as the upper limit for the JND later in this c hapter. Signal Detection Theory: The Problem of Noise The single most important impediment to percep- tions around thresholds is noise. Indeed, as we will see, the reason why psychometric functions do not look like 12.2a is due to the effects of noise. This is not the noise of cat's howling , tires screeching, or com puter fans humming, but the random fluctua- tions that plague every receptor in the eye, a nd every neuron in the brain. The unwanted effects of noise can be reduced , but they cannot be elimi- nated. It is therefore imperative to have a theoreti- ca l framework which a llows u s (and the brain) to d eal rational l y with this unavoidabl e obstacle to perception. Such a framework i s signal detection theory. We will explain its details in du e course but for the moment we give a non-techni cal overview of the main ideas. Think about what the brain has to contend with in trying to decide whether or not a light is pre se nt. The eye is full of receptor s with fluctuating membrane potentials (receptors do not have firing rates), which feed into bipolar cel l s with fluctuat- ing membrane potentials , and these in turn feed r e tinal ganglion cells with fluctuating firing rates , whose axo n s connect to brain neuron s with their own fluctuating firing rates. All these fluctuations are u s uall y smal l , but then so i s the change induced b y a very dim stimu lu s. More importantly, these fluctuations happen whether or not a stimulus is pr ese nt. Th ey are examples of noise. What chance d oes the brain s tand of det ec ting a very dim light 285 Seeing and Psychophysics in the face of this noise ? Sutpr i singly, th e answer is, "a good chance." However, certainty i s rul e d out because of the effects of noi se. The amount of noise varies randomly from moment to moment. To illustrate thi s, l e t 's con- centrate first on the receptors. They hav e a resting potential of about -35 m V (milliVolts), and their response to light is to hyperpoLar ize to about -55 mY. However , to keep things arithmetically s imple (and to help later in generalizing from this exam- ple) we will define th e ir me a n resting potential as o mV (just imagine that +35 mV i s a dd ed to eac h measurement of receptor output). It will help to have a symbol for a receptor 's membrane pot e n- tial a nd we w ill use r for this purpose (think of r standing for response). The variable r i s known as a random variable because it can take on a different value every time it is observed due to the effects of noise. We will use u to denote the mean value of r The key point is that variations in r occur due solely to noise, as shown in 12.5a. If we construct a histogram of r values (as in l2.5b) then we see that th e mean value of r is indeed equal to zero For simp licity , we assume that thi s distribution of r values is gaussian or normal with mean u = 0 m V, 12.5c. In order to help und e rstand how this histogram is constructed, note that the horizontaL dashed (blue) line in 12.5a which mark s r = 10m V, gets transformed in to a vertica L d as hed (blue) lin e in the histogram of r values in l2.5b,c As thi s di s tribution consists entirely of noise, it is known as th e noise distribution. So mu c h for the nois e di s tribution of response s when no stimu lu s is pres e nt What happens when a light flash is shown? The flash respon se is added to the noise to create another distribution , called the signal distribution. Obviously, if the lumi - nan ce of the flash is too l ow to create any effect in the visual sys tem then the signal di s tribution is identical to the noise distribution. How eve r , as the luminance of the flash increases, it begins to create a response, and so the signal di s tribution becomes shifted, 12.6a,b. Let's now apply these ba sic idea s about noi se by considering the output of a single r ece ptor in the eye to a l ow intensity light. The brain is being asked to solve a difficult statistica l problem: given that a certain receptor output r was observed dur- ing the last trial (which ma y la s t a seco nd or two), a Chapter 12 20 Trial number 12.5 Noise in receptors a If we measured the output r of a single photoreceptor over 1000 trials then we would obtain the values plotted her e, because r varies randomly from trial to trial. Note that thi s r ecep tor is assumed to be in total darkness here The probability that r is greater than some criterion value c (set to 10 mV here) is given by the proportion of dots above the blue dashed line , and is written as p(r > c). b Hi stogram of r values measured over 1000 trials shows that the mean receptor output is u = 10 mV and that the variation around this mean value has a standard devia- tion of 10 mV. The probability p(r > c) is given by the pro- portion o f histogram area to the right of the c riterion c indi- cated h ere by the blue dashed vertical line. c The histogram in b is a good approximation to a gaus- sian or normal distribution of r values , as indicated by th e solid (b la ck) curve. Notice that this distribution has been " normalized to unit area ." This means that the values plotted o n the ordinate ( v ertica l axis) have been adjusted to that the area under the curves adds up to unity The r esu lting distribution is ca lled a probability density func- tion or pdf. was a li ght presenr or not? Remember that r varies randoml y from second to seco nd whether or not a s timulu s is presenr, due ro noise. So some trials are assoc iated with a recepror ourput caused by the dim li ght , a nd other trials have a l arger recepror ourput eve n if no li ght is presenr, again, due to noi se. Of course, on average, the rec e pror outpur tend s ro b e larger when the dim light is on man when it is off. However, on a sma ll proportion of tria l s, th e presence of noise reverses this situation, and th e recepror output associated with no li ght is l arger than it i s with me dim light on. The up s h ot is that the observer h as to se t a cri- terion re cepro r output, which we denote as c, for 286 b c C :::J o U Re cepto r output ( mV) Receptor output ( mV) sayingyes. As thi s recepror output correspo nd s ro a specific luminan ce, the observer effectively chooses a criter i on luminance, 12.7. So m etimes this c rit eri- on yields a correct yes On other trials , the fluctuat- in g act i v i ty l evels l eads ro an incorrect yes. Th ese ideas a ll ow u s ro see why the ps yc h o m et - ric function has its c h aracteristic S-shape. Whether or not these responses are correct, th e proportion of trials on which the criterion is exceeded, and therefore the proportion of yes responses , gradua ll y in creases as the luminance increases, 12.8. Having given an outline of SOT, we are now read y ro examine it in mor e detail. Seeing and Psychophysics a b 50 ...... : '. :', ; .::. 40 ..... ,," . ' .... '." " .... : .. ' :. : ,. " 1 1" C ' •• ' ,c -cc '-i,', i ,\' C,,' : c o () -20 .; o 200 400 600 800 1000 Trial number -20 o 20 Receptor output (mV) 60 40 12.6 Signal added to noise creates the signal distribution a Measured values of receptor output rwith the light off (lower red dots) and on (upper green dots), with the mean of each set of rvalues given by a horizontal black line. b Histograms of the two sets of dots shown in a Signal Detection Theory: The Nuts and Bolts In essence , SOT augments classical psychophysi- cal methods with the addition of catch trials. If an absolute threshold is being measured then each catch trial contains a stimulus with zero lumi- nance. If a JNO is being measured then each trial contains two stimu li , and the observer has to state if they are different (with a yes response) or not (no response), and a catch trial has zero difference be- tween two stimuli. The effect of this subtle change is dramatic, and l eads to an objective measure of abso lut e thresholds, JNOs, and a related measure of sensitivity, known as d' (pronounced d-prime). The question the observer has to answer on each trial is the same as before: did I see a light (abso- lut e thresholds) , or a difference between two lights (JNOs)? If an observer responds yes on a catch trial then this implies that internal noise has exceeded the observer ' s criterion for deciding whether or not a light is present (abso lut e threshold), or whether two li ghts have different luminances (JNO). Thus , the use of catch trials effectively permits the amount of noise to be est im ated. Histograms and pdfs Understanding SOT requires knowing some tech- nical details about distributions. Readers familiar with standard deviations, and the relationship between histograms and probability density func- tions (pdfs) may want to skip this section. We can see from 12.5a,b that values of the receptor output r around the mean (zero in o ur example) are most common. If we wanted to know precisely how common they are , we can use 12.5b. For examp l e, each co lumn or bin in 12.5b h as a width of 3 m V, so that values of r between zero and 3 m V contribute to the height of the first co lumn to the right of zero This bin has a height of 119, indicating that there were a total of 119 values or r which fell between zero and 3 m V. As we measured a total of 1000 values of r, it fol- lo ws that 119/1000 or 0.119 (i.e. , 11.9%) of all recorded va lu es of r fell between zero and 3 m V. 287 Noise Sensor r Signal More noise Neural response "yes" "no" 12.7 Overview of a perceptual system faced with noise 0.4 0 35 0 .3 0.25 'C' Q 0 2 0.15 0.1 Chapter 12 2 4 6 Receptor output mV) 0.4 0 35 0 3 0 25 'C' Q 0.2 0 15 0 1 0 05 10 0.8 Ul 0 6 (l) >- Q 0.4 a a 2 4 6 8 10 Receptor output (mV) 12.8 The psychometric function and SOT 0.4 0.35 0 3 0 25 'C' Q 0.2 0.15 0 1 'C' Q 10 0.4 0.35 0 3 0.25 0 2 0 15 0 1 0 05 10 As the receptor output increases the signal distribution (green) moves to the right , increasing the probability of a yes reSDonse for a fixed criterion of c = 7 mV. Thus, the probability that r is between zero and 3 mV is 0.119 Now suppose we wanted ro know the probability that r is greater than, say, 10m V, given that the light is off. At this stage, it will abbreviate matters if we define some simple notation. The prob a bility that r is greater than 10 mV is written as per > l0loffi: the vertical bar stands for " given that. " Thus, per > 10 loffi is read as " the probability that r> 10 given that the light is off." This is a con- ditional probability because the probability that r> lOis conditional on the state of the light. From 12.5a, it is clear that per > 10 loffi is related to th e number of dots above the r = 10m V dashed line. In fact, per > 10 loffi is given by the proportion of dot s above the dashed line. Now, each of these dots contributes to one of the bins above the verti- cal blue line in 12.5b. It follows that per> 10loffi is given b y the s ummed heights of these bins to the right of 0.0 m V, expressed as a proportion of the summed heights of all bins (which must be 1000 b eca use we measured 1000 r values). The summed heights of these bins comes to 150, so it follows that there are 150 r values that exceed 10m V, and therefore thatp(r > 10loffi = 150/1000 = 0.15. This procedure can be applied to any value or range of r values. For example, if we wanted to know the probability that r is greater than 0 m V then we would add up all the bin heights to the right of 0 m V On average , we would find that this accounts for half of the measured r values, so that p(r>Oloffi = OS At this stage we can simply note that each bin has a finite width (00 m V), so that per > 1Oioffi i s actually the area of the bins for which r> 10, ex- pressed as a proportion of the total area of all bins. This will become important very soon. If we overlay the curve which corresponds to a gaussian curve then we see that this is a good fit to our histogram , as in 12.5c. In fact , the histogram in 12.5c has the same shape as that in 12.5b, but it has been set to have an area of 1.0 (technically, it is said to have been " normalized to have unit area, " as explained in the figure legend). Eac h column in 12.5b has an area given by its height multiplied by 288 10 its width (3 m V in this case). If we add up all the column heights (of the 1000 bins) and multiply by the column widths (3 m V) then we obtain 3000 = 1000 x 3, which is the total area of the histogram. If we divide the area of each column by 3000 then the total area of the new histogram is one (because 3000/3000 = 1). If you look closely at 12.5c then you will see that this has been done already. Instead of a maximum bin height of 125 in 12.5b, the maximum height in 12.5c is around 0.4; and, as we have noted, instead of an area of 3000 in 12.5b, the area of 12.5c is exactly one. This transformation from an ordinary histogram to a histogram with unit area is useful because it allows us to compare the histogram to stand- ard curves, such as the gaussian curve overlaid in 12.5c. As we reduce the bin width, and as we increase the number of measured values of r, this histogram becomes an increasingly good approxi- mation to the gaussian curve in 12.5c. In the limit, as the number of samples of r tends to infinity, and as the bin size tends to zero, the histogram would be an exact replica of the gaussian curve, and in this limit the histogram is called a probability density function , or pdf Just as the histogram allowed us to work out the probability of r being within any given range of values, so does the pdf, but without having to count column heights. For example, the prob- ability that rdO mV is given by the area under the pdf curve to the left of the dashed blue line in 12.5c. If we start at the left hand end of the curve and work out areas to the left of increasing values of r, we end up with a curve shaped like the psychom e tric function. Because this curve gives the cumulative total of areas to the left of any given point it is known as a cumulative density function or cdf This area is can be obtained from a standard tabl e of values relevant to the gaussian distribution found in most textbooks on statistics. One aspect that we have not yet discussed is the amount of random variability in values of r. This is revealed by the width of the histogram of r values. A standard measure of variability is the standard deviation, which is denoted by the Greek letter a (sigma). Given a set of n (where n = 1000 here) values of r, if u is the mean then the standard deviation is a = VO/n IV - U) 2 ), , , Seeing and Psychophysics where the symbol I stands for summation. In words, if we take the difference between each measured value of r and the mean u, and then square all the se differences, and then add them all up, and then take their mean (by dividing by n), and finally take the square root of this mean, then we obtain the standard deviation. The equation for a gaussian distribution is per) = kexp(-(u - r)2/(2a 2 )), where k = 1/[aV(2rr)] ensures that the area under the gaussian curve sums to unity. But if we define a new variable z for which the mean is zero and the standard deviation is one z = (u - r)/a then we can express the gaussian in its standard form as p(z) = k z exp(-:i/2), where k = 1I[vi(2rr)]. z The standard form of the gaussian distribution has a mean of zero, a standard deviation of a = 1 , and an area of unity; this is the form used in most statistical tables Any data set of n values of r can be transformed into this normalized form by subtracting its mean u from all n values of r, and dividing each r value by the standard deviation a of the n values. The resultant data have a mean of zero and a standard deviation of unity. This was done in order to transform our raw values of r in 12.5a,b to the normalized values shown in 12.5c. Thus, when a normalized gaussian curve is overlaid on 12.5c, the fit is pretty good. Conversely, we can go the other way and scale a normalized gaussian curve to get a rough fit to our raw data. This is achieved by adding the data mean u to the normal- ized gaussian mean of zero, and by multiplying the standard deviation of the normalized gaussian (which is unity) by the standard deviation of our data, as in 12.5b. In order to give an impression of what happens to a gaussian curve as we vary the standard devia- tion , two gaussians with standard deviations of a = 5 mV and a = 10 mV are shown in 12.9 Note that the heights of these two curves are dif- ferent because they both have unit area. Forcing 289 Chapter 12 0.8 0 7 0.6 0.5 'C" 0.4 Q 0.3 0.2 0.1 ity that a given value r is less than or equal to some reference value x is the area und er the c ur ve to the left of x p(r 5 x) = <I>((x - u)/cr), where the function <I> (the Greek l etter, phi) returns the area under the c urve to the l eft of x for a gaus- sian with mean u and stan dard deviation cr. Note that the quantity z = (x - u)/cr expresses the difFer- ence (x - u) in units of cr, is known as a z-score, and was defined on the previous page The function <I> is a called a cumulat i ve density function , because it returns the cumulative total -20 -10 0 10 Receptor output (mV) 30 area under the gaussian pdf to the left of z For 12.9 Gaussian curves with different standard deviations The narrow gaussian has a standard deviation of a = 5 mV , and the wide gaussian has a = 10 mV , as indicated by the horizontal dashed line attached to each curve. The curves have different heights because they have different stand- ard deviations, but the same area (unity) , which means we can treat each distribution as a probability density function (pdf). The abscissa defines values of r, and the ordinate indicates the probability density p(r) for each value of r a distribution or a histogram to have unit area is useful if we wish to interpret areas as probabilities, and it a l so allows us to treat both gaussian curves as pdf s with different standard deviations. Before moving o n , a few facts about pdfs are worth noting. First, the total area under a pdf is unity (one). Thi s corresponds to the fact that if we add up the probabilities of all of the observed values of r then thi s must come to 1.0. Second, as with the histogram example above, any area under the curve defines a ptobability. Because the area of a bin eq ual s its width times it s h eight, it follows that the h eig ht a lon e of the c urv e cannot be a probability. The height of the curve is called a probability density , and must be multiplied by a bin width to obta in a probability (correspondi ng to an area under the pdf) Third , for a gaussian di s tribution with mean u and sta ndard deviation cr , the area under the curve between u and u + cr occupies 34% of the total area. This implies that the probability that r i s between u and U + cr is 0.34. As half of the area under the curve lie s to the left of u, this impli es that the probability that r is less than u + cr is 0.84 = (0.50 + 0.34). More generally, the probabil- example, p(r 5 10) = <I>(z = 1 ) = 0.8 4 Signal and Noise Distributions Returning to our li g ht example, consider what happens if the li ght is on. This situation is s hown by the upper (green) set of dots in 12.6a, and the corresponding histogram of r values on the right hand side of 12.6a Let 's assume that turning the light on in creases the mean output to u = 30 m V For simplicity, we w ill assume that the standard deviation remains constant at cr = 10 m V We refer to this " li ght on " distribution of r values as the signal distribution. In order to distinguish between the signal and noise distributions, we refer the their means as U s and u ,, ' respectively, and to their standard devia- tions as cr s a nd cr " , respectively. However , as we assume that cr s = cr " , the standard deviation will usuall y b e referred to without a subscript. Now, we know that if the light is off then th e mean output is U II = 0 m V, but at any g i ven mo- ment the observed value of the o utput r fluctuates around 0 m V. Let ' s assume that we observe a value of r = 10m V Does this imply that the light is on or off? Before we answer this , consider the values that could be observed if the light is off in com- pari s on to values that could be observed if the li ght is on , as shown in 12.6a A value of r = 10m V i s one standard deviation above the mean va lu e U II = 0 m V associated with the light being off (because cr = 10 mV here) , but it is two standard deviations below the mean value of u = 30 m V s 290 associated with the light being on. So , even though an observed value of r = 10m V is unlikely if the light is off, such a value is even more unlikely if the light is on. Given that the observer i s required to respond yes or no for each trial, these considera- tions mean that an ideal observer should respond no. But as we shall see, most observers are not ideal, or at least not ideal in the sense of minimiz- ing the proportion of incorrect responses. The Criterion As explained above, given an observed value of the receptor output r, deciding whether or not this output means the light is on amounts to choos- ing a criterion, which we denote as c. If r is greater than c (i.e., r> c) then the observer decides that the light is on, and responds with a yes. Conversely, if r is less than c (i.e., r<c) then the observer decides that the light is off, and responds with a no. As a reminder, over a large of number trials , say 1000 , we pr ese nt a light or no light to an observer. On each trial, the observer has to indicate whether or not a light was seen. As the light is either on or off, and as the observer can respond either yes or no, there are four possible outcomes to each trial: 1) light on, observer responds yes, a hit, H 2) light on, observer responds no, a miss, M 3) light off, observer responds yes, afalse-alarm, FA 4) light off, observer responds no, a correct rejection, CR. Response Stimulus Catch Trial present Stimulus not present " Yes" Hit False alarm " No " Miss Correct rejection If we measure the observer's responses over a large number of trials then we can obtain estimates of each of these quantities. Each quantiry corresponds to a region of one of the histograms in 12.Gb , which ha s been redrawn in terms of gaussian pdf s in 12.10. Here, the light-off or noise pdflies to the left of the light-on or s ignal pdf. Following the line of reasoning outlined in the previous section: H: the hit rate equals the large (red) area of the signal pdf to the right of the vertical blue criterion, 291 "c- 0.4 0.35 0.3 0.25 Seeing and Psychophysics Criterion o: 0.2 0 .1 5 0.1 o 20 40 Receptor output (mV) 12.10 Estimating d' The distance d ' (d-prime) between the peaks of the noise (left) and signal (right) distributions can be estimated from a knowledge of two quantities: the hit rate H , and the false alarm rate FA. The dashed (blue) line is the criterion c, and the observer responds yes only if the receptor output r is greater than c (i.e , if r>c). The hit rate H is equal to the area of the (red) region of the signal pdf to the right of the criterion, and FA is equal to the area of the (yellow) region of the noise pdf to the right of the criterion. M: the miss rate equals the small (green) area of the signal pdf to the left of the criterion, FA: the false alarm rate equals the (yellow) area of the signal pdf to the right of crite- rion, CR: the correct rejection rate equals the large (light blue) area of the noise pdf to the left of the criterion If we choose c = 10m V then an observed value of r = 20 m V would allow us to respond yes, because r> c. I f we adopted a cri terion of c = 10m V, how often would we be correct in responding yes given that the light is on? This is given by the conditional probabiliry p(yeslon), and is equal to the proportion of the signal pdf shaded red in 12.11. Note that the hit rate H = p(yeslon) can be made as large as we like simply by decreasing the value of the criterion c, which moves the vertical dashed line leftward in 12 10 For example, if c is set to -20 m V then almost all observed valu es of rare above c, so we respond yes for almost any observed value of r, 12 1Ia. It 's as if we adopt an extremely laissez foire or risky approach, and treat almost anything as a sign that the light is on. This implies Chapter 12 a .,--- Criter i on 0 .4 0.3 5 0 .3 S 0. 2 5 Q. 0.2 0. 15 0 1 0.05 0 - 20 o 20 4 0 Re c eptor output (mV ) 12.11 Effect of criterion The criterion c is given by the position of the blue dashed line. b 0.4 0 .3 5 0 .3 S 0 25 Q. 0 2 0 15 0 1 Crit e rion 0. 05 R ec eptor output ( mV ) a A low criterion of c = - 20 mV yields a large hit rate H (the red area of signal pdf to the right of c ) , but also yields a large false alarm rate FA (the yellow area of the noise pdf to the right of c) b A high criterion criterion of c = 50 mV yields a low FA (the area of the noise pdf to the right of c, which is so small it is not visibl e h e r e), but al s o a low H ( the red ar e a of the signal pdf to the right of c ) t h a t i f m e li g ht is o n th e n we r es p o nd yes, so th a t o ur hit rate b eco m es close to 100%, f o r exa mpl e, p(yes l on) = 0.9 10 Thi s m ay seem l i k e goo d n ews b ut it i s acco mp a ni e d b y so m e b a d n ews. Se nin g c = -2 0 m V g u a r a nt ees a hi g h hit r a t e, but it a l so e n s ur es th a t we a lm os t a l ways r es p o nd yes eve n w h e n th e li g ht is off, r es ultin g in a hi g h false a l a rm ra t e FA = p(yes l o./fJ. For exa mpl e, if r = - 1 0 m V th e n it i s n o t lik ely t hat th e l ig ht i s o n Bu t we wo uld respond yes i f r = -10 m V a nd i f o ur cr i ter i o n (c = - 2 0 m V) is set to yi eld a hi g h hit rate In ot h er wo rd s, th e p roba b i l ity p(yes I ojf) of r es p o ndin gyes g iv e n th a t th e li g h t i s o ff i s clo se to uni ty (g i ve n b y th e ye ll ow a r ea i n 12.11a ). Thu s, se nin g th e cri t er ion c to a very l ow valu e incr eases t h e hit r a t e, th e y ell ow r eg i o n in 12.11a , bu t it a l so i n creases t h e fal se a l a rm r a t e. I f we n ow reve r se th is st r a t egy a nd se t c to b e ve r y hi g h (say, c = 5 0 m V) th e n lif e d oes n o t ge t m u c h b et t er, 12.11b. In t h is case, it i s as i f we adopt a n ex tr e m ely ca u t i o u s ap p roac h , an d w ill n o t i nte r p r et eve n l arge val u es of r as i ndi ca t i n g t h e li g ht is o n Co n se qu e ntl y, we ra rel y r es p o nd yes w h e n th e li g ht i s o n , so th e fa l se al a rm r a t e FA i s a lm ost ze ro (w hi c h i s goo d ). H o w e v e r , a hi g h cr it e ri o n m ea n s th a t w e r a rel y r es p o nd yes eve n w h e n t h e li g ht i s o n , y i e ldin g a hit r a t e H cl ose to zero (w h ic h is b a d ). Th i s s i t u a ti o n is s h ow n 29 2 g r a ph ic all y in 12.11a , b Cr u c i al l y, th ere is no v alu e for c w h ic h g u a r a nt ees t h a t o ur d ec i s i o n s a r e al ways corre ct. H oweve r , th e r e i s a val u e of c w h ic h g u a r a nt ees t h a t we are r i g ht as of t e n as p oss ibl e. I f th e li g h t i s o n durin g half th e tri als m e n mi s v alu e i s exac tl y m i d -way b e tw ee n th e n oise a nd s i g n a l di s tributi o n s In s umm ary, a l ow c rit e ri o n (Laissezjaire o r ve r y r i sky) yiel d s a l a r ge hit rate but a l a r ge false alarm ra t e, w h ereas a hi g h (very ca uti o u s) c rit e r io n y i el d s a l ow hi t rate but a l ow false a l a rm r ate. I deal l y, we wo uld li ke to h ave a l a r ge hit r a t e a nd a l ow false a l ar m rate. G i ve n th a t n e ith e r a ve r y hi g h n o r a ve r y l ow val u e f or th e c rit e ri o n see m sa ti sfacto r y, it f o ll ows th a t m e r e mu st b e a va lu e so m ew h ere b etween th ese ex tr e m es w hi c h y ield s a se n s ibl e co mp ro mi se, w hi c h turn s o ut to b e th e midp o int b etwee n t h e m ea n s o f th e sig n al a nd n