John ; Taylor

Sayim

Bilge

Henry Taylor

Bilge Sayim
email: bilge.sayim@psy.unibe.ch

One of the most important topics in current work on consciousness is what relationship it has to attention. Recently, one of the focuses of this debate has been on the phenomenon of identity crowding. Ned Block has claimed that identity crowding involves consciously perceiving an object that we are unable to pay attention to. Others have offered different interpretations, emphasising the role of cognitive inference over conscious perception. In this paper, we draw upon a range of empirical findings to argue against Block's interpretation of the data. We also argue that current empirical evidence strongly supports one particular version of the inference hypothesis. Finally, we consider the additional evidence Block gives in favour of his view, and argue that it fails to establish his position.

Identity Crowding

One of the most hotly debated questions in contemporary philosophy of mind and cognitive science is the relationship between attention and consciousness. 1 One of the focal points of this debate is the phenomenon of 'identity crowding'. Ned Block has argued that identity crowding provides an example of an object that is consciously seen but which we are unable to pay attention to. If Block is correct, this will be a decisive blow to many theorists (such as [START_REF] Mack | Inattentional Blindness[END_REF][START_REF] Prinz | The Conscious Brain: How Attention Engenders Experience[END_REF][START_REF] Montemayor | Consciousness, Attention and Conscious Attention[END_REF][START_REF] Cohen | The attentional requirements of consciousness[END_REF] who argue that attention is necessary for consciousness, and it would confirm the view of other thinkers (e.g. [START_REF] Mole | Attention and consciousness[END_REF][START_REF] Jennings | Consciousness Without Attention[END_REF][START_REF] Li | Rapid natural scene categorization in the near absence of attention[END_REF][START_REF] Koch | Consciousness and attention: two distinct brain processes[END_REF] who argue that consciousness is possible in the absence of attention. Furthermore, Block argues that we can have de re thoughts about an object that we are unable to attend to. 2 This runs counter to the majority view in philosophical discussion of de re thought (e.g. [START_REF] Campbell | Reference and Consciousness[END_REF][START_REF] Levine | Demonstrative thought[END_REF]and Tye 2009a, 2009b, 2010). For this reason, our assessment of identity crowding has important consequences throughout cognitive science, as well as philosophy of mind and language.

In this paper, we offer an analysis of this debate. We firstly examine the phenomenon of identity crowding and the different interpretations of it that have been given ( §1). Next, we draw upon a large body of empirical evidence to argue against Block's interpretation of identity crowding ( §2) and put forward a rival interpretation ( §3). Finally, we consider additional arguments put forward in favour of Block's view, and argue that they fail to establish Block's conclusion ( §4). Identity crowding is a particular variety of visual crowding (fig. 1). When you fixate your gaze on the upper '+' in fig. 1, you will probably find that you are able to identify the letter to the right of it. This is a normal instance of non-crowded peripheral vision. 'Crowding' is an effect that occurs when target items are 'flanked' by other items. For example, when you fixate on the middle '+' in fig. 1, you will most likely be unable to identify the middle letter in the array to the right of that '+'. This is a case of normal crowding. It occurs because the presence of the flanking letters disrupts the perception of the target letter (see [START_REF] Levi | Crowding -An essential bottleneck for object recognition: A minireview[END_REF] for a survey). Crowding is not constrained to letters but occurs with a large range of other stimuli. 3There are various proposed explanations of crowding, and a large empirical literature dedicated to it (e.g. [START_REF] Herzog | Crowding, grouping, and object recognition: A matter of appearance[END_REF], [START_REF] Levi | Crowding -An essential bottleneck for object recognition: A minireview[END_REF], [START_REF] Pelli | Crowding is Unlike Ordinary Masking: Distinguishing Feature Integration from Detection[END_REF], [START_REF] Parkes | Compulsory Averaging of Crowded Orientation Signals in Human Vision[END_REF][START_REF] Strasburger | Unfocused spatial attention underlies the crowding effect in indirect form vision[END_REF]). Ned Block follows the view that crowding is due to 'excessive feature integration', where the visual system is unable to differentiate which properties belong to the target, and which belong to the flankers, and the result is that in cases of normal crowding, one only sees a 'messy' texture (Block, 2013a, pp.171-2). The purpose of the present paper is not to contest this general approach to crowding, but to focus on a particular kind of crowding: identity crowding.

The crucial case for our purposes is the bottom row in fig. 1. When fixating the bottom '+', most observers find that they are able to identify the target letter. We follow Block (2013a) in referring to this phenomenon as 'identity crowding'. It occurs when the target is physically identical to the flankers. In this paper, our focus will be on the recent and highly controversial claim that identity crowding is a case where subjects are unable to direct attention to a target item, but can still consciously see that item (the item in question being the middle letter). We call this view the 'conscious perception' view. Block (2013aBlock ( , 2013b[START_REF] Block | The defective armchair: a reply to Tye[END_REF] defends this view. 4 Block's case can be divided into two main points (each of which will be more thoroughly explained later in the paper).

Firstly, he notes that subjects are usually able to detect identity crowded targets, discriminate them from the background (and from the flankers), identify them, and have de re thoughts about them. Block concludes from this that 'it is difficult to see a rationale for denying that one can consciously see [the target]' (2013a, p.175). Secondly, Block draws upon work based upon summary statistics, binding and 'singular elements' to argue that identity crowding is 'sufficiently perceptual to involve consciously seeing identity-crowded objects ' (2013b, p.31).

Views that oppose the conscious perception hypothesis offer a rival interpretation of identity crowding. These views all deny that subjects have a conscious experience of the middle item in identity crowding, and they explain subjects' abilities in terms of some kind of cognitive inference, rather than conscious perception without attention. Such interpretations fit into a family that we call 'inference views', but this is only a loose label, as different versions of the view emphasise inference to very different extents. Bradley [START_REF] Richards | Identity crowding and object seeing: a reply to Block[END_REF] claims that subjects unconsciously perceive the middle letter, but this unconscious perception is accompanied by a conscious judgement to the effect that there is an 'A' (see fig. 1), present. Henry [START_REF] Taylor | Is the grain of vision finer than the grain of attention? Response to Block[END_REF] suggests that the experience of the group of letters as a whole lacks the detail to represent the middle letter specifically, but that subjects are able to identify one of the flankers, and also identify that the group of letters as a whole is congruous and uniform. Taylor claims that subjects can use this information to infer that the target is the same as the flankers, and thus identify the target. Michael [START_REF] Tye | Does conscious seeing have a finer grain than attention[END_REF] claims that in the case of identity crowding, we only see the two flanking letters: '[w]hat your experience erroneously tells you is that there are two [letters] on the right, not three' (2014 p.156). Then, when asked what the 'middle' letter is, subjects infer (based on the fact that they can only see two similar letters) that any letter there must be the same.

Inference theorists and conscious perception theorists also differ on their interpretation of cases similar to identity crowding (e.g. fig. 2). When subjects fix their gaze on the cross in fig. 2, they typically find it difficult to direct attention toward each of the bars individually [START_REF] Intriligator | The Spatial Resolution of Visual Attention[END_REF]. Block (2013a, p.175) claims that each individual bar is consciously seen. Inference theorists reject this. Tye (2009aTye ( , 2009b[START_REF] Tye | Does conscious seeing have a finer grain than attention[END_REF] claims that one sees the collection of bars as a whole, without seeing each individual bar, and this experience allows subjects to infer properties of each individual bar. [START_REF] Taylor | Is the grain of vision finer than the grain of attention? Response to Block[END_REF] offers a similar view to Tye, though he emphasises the role of properties of the group such as congruity and uniformity in allowing subjects to infer properties of individual bars. Richards' (2013) view is again similar, though he emphasises the role of unconscious perception.

Importantly, Block agrees with these opponents that perception and inference play at least some role in identity crowding (Block, 2013b, p.31). The difference is that Block (unlike his opponents) thinks that there is conscious perception of the target in the absence of attention in cases of identity crowding and in cases like fig. 2. Block puts forth this claim with different strengths in different places, occasionally he seems to say that the target letter in identity crowding cases is always consciously seen, though it escapes attention (2013a, p.175), whilst at other times he claims that it is consciously seen 'at least in some cases' (2013b, p.31). Each version of the view raises its own distinctive questions. For example, if the claim is that the middle letter is only seen sometimes, then we require an account of what is going on in the cases where it is not seen; whilst no such need arises when the claim is that the letter is always seen. In any case, in this paper, we will argue that Block's arguments fall short of showing that the target is consciously seen, and that all of the data can be accommodated without needing to make this claim. We further claim that in cases of identity crowding, we have good reason to deny that the target is ever consciously seen.

Evidence Against Conscious Seeing in Identity Crowding.

One core part of Block's case against both Tye and Taylor is that they do not support their views with empirical evidence (Block, 2013b, pp.36-37 and2014). These complaints are fair, Block's opponents have by and large relied on intuition and introspection. In this section and the next, we remedy this: we review evidence related to the phenomenon of 'grouping' in crowding, and the relation of conscious target perception to the abilities to direct attention to the target. We argue that both kinds of evidence oppose Block's 'conscious perception' view.

Similarity and grouping in crowding

To Block, the conscious perception of the target in identity crowding is what explains subjects' abilities to identify, discriminate etc. the target. Of course, Block does not base his case on normal crowding (e.g. the middle row of fig. 1) because in these cases subjects are not able to identify, discriminate etc. the target (again, given that crowding is sufficiently strong), so there is no positive reason to claim that the target is consciously seen. In this section, we draw upon work on grouping to argue that we have more reason to doubt that the target is consciously perceived in the case of identity crowding than in the case of normal crowding! Given these findings, we have reason to doubt Block's view.

The main feature of (normal) crowding is that it becomes harder to identify the target in the presence of flankers compared to unflanked targets (for reviews, see e.g., [START_REF] Herzog | Crowding, grouping, and object recognition: A matter of appearance[END_REF][START_REF] Levi | Crowding -An essential bottleneck for object recognition: A minireview[END_REF][START_REF] Whitney | Visual crowding: A fundamental limit on conscious perception and object recognition[END_REF]. One of the important factors determining the strength of this crowding effect is whether the target is 'grouped' with the flankers. Visual grouping occurs when certain distinct elements in the visual field are perceived as in some way 'connected' and belonging together (e.g., [START_REF] Köhler | Gestaltpsychology[END_REF][START_REF] Koffka | Principles of Gestalt Psychology[END_REF][START_REF] Wertheimer | Gestalt theory[END_REF]. An example is shown in fig. 3. Numerous studies have shown that the more the target groups with the flankers, the stronger is the effect of crowding, i.e. the more difficult it becomes to identify the target (for a review see, [START_REF] Herzog | Crowding, grouping, and object recognition: A matter of appearance[END_REF]. Grouping strength between the target and its (directly neighbouring) flankers is often determined by how similar/dissimilar their properties are. For example, a black target groups more strongly with black than with white flankers, and is harder to identify when it is surrounded by black flankers than by white flankers [START_REF] Kooi | The effect of similarity and duration on spatial interaction in peripheral vision[END_REF][START_REF] Sayim | Contrast polarity, chromaticity, and stereoscopic depth modulate contextual interactions in Vernier acuity[END_REF]. This similarity effect has been shown for a variety of features (e.g., colour: [START_REF] Kooi | The effect of similarity and duration on spatial interaction in peripheral vision[END_REF][START_REF] Sayim | Contrast polarity, chromaticity, and stereoscopic depth modulate contextual interactions in Vernier acuity[END_REF][START_REF] Manassi | Grouping, pooling, and when bigger is better in visual crowding[END_REF]shape: Nazir, 1992;depth: Sayim et al., 2008;[START_REF] Astle | Characterizing the role of disparity information in alleviating visual crowding[END_REF]. Importantly, target-flanker similarity was also correlated with the error rate in crowded letter discrimination. The upshot of these results is that the more the target and flanking letters were similar, the harder it was to identify the target, i.e., the higher was the error rate [START_REF] Bernard | The dependence of crowding on f!anker complexity and target-f!anker similarity[END_REF].

It is reasonable to take the increased difficulty of identifying targets as grouping strength increases to indicate increased disruption in the perception of the target. This is because crowding itself is due to a perceptual disruption of the target of one kind or another, so it is reasonable to assume that an increase in the effects of crowding indicate an increase in this disruption. Note that this need not commit us to any one particular theory of the mechanisms involved in crowding, we need only claim that the effect is perceptual in at least some way.

It might be argued that grouping changes the decision criterion, for example, by introducing a bias by the flankers. Systematic flanker bias has to be considered when the flankers contain features that are relevant for observers' responses. For example, in letter identification, flankers that strongly group with the target because of shape similarity could bias observers independent of perceptual changes of the target. In such cases, it is advisable to distinguish between disruption in the perception of the target and other factors, such as flanker bias, for example, by using signal detection theory (see, for example, [START_REF] Sayim | Grouping and crowding affect target appearance over different spatial scales[END_REF].

Additionally, there is reason to think that decrease in performance as grouping strength increases is mirrored in the phenomenology of the experience, as subjective estimates of how much the target 'stood out' (i.e., "ungrouped" from the flankers) are positively correlated with performance (e.g., [START_REF] Saarela | Global stimulus configuration modulates crowding[END_REF]. Furthermore, consider an alternative (non-perceptual) explanation of why subjects' abilities deteriorate as grouping increases. Such a view would have to claim that perception of the target remains approximately the same as grouping increases, but that subjects' judgements become steadily worse. Such a view is empirically unmotivated and inexplicable on current theories of crowding.

What's more, these effects are unsurprising on the view (that Block endorses) that crowding is due to 'excessive feature integration'. On this view, crowding is due to the visual system being unable to differentiate between the properties of the target and those of the flankers in peripheral vision, which leads to disruption in the conscious perception of the target. Given this, it is natural to suppose that the more similar the properties of the target and the flankers are, the harder it is for the visual system to differentiate them, and thus the more there will be excessive feature integration, and the more conscious perception of the target will be disrupted (note, however, that global grouping can trump effects of local similarity; e.g., [START_REF] Sayim | Contrast polarity, chromaticity, and stereoscopic depth modulate contextual interactions in Vernier acuity[END_REF][START_REF] Manassi | Grouping, pooling, and when bigger is better in visual crowding[END_REF].

The role of grouping in crowding is highly relevant for the discussion of identity crowding. In the standard case of identity crowding -one target letter with two flanking letters -target-flanker similarity is maximal (because, of course, the targets and the flankers look exactly the same). As a result, target-flanker grouping is extremely strong. Hence (since grouping is linked with disruption in the perception of the target) one would assume that conscious perception of the target will be maximally disrupted. By contrast, Block's claim (that of conscious seeing without attention) suggests that (in cases of identity crowding) perception of the target is superior compared to normal crowding. So, assuming that Block accepts that an increase in the strength of target-flanker grouping increases the effects of crowding, and thus increases disruption of the conscious perception of the target (and given the evidence explained above, it is very difficult to see how he could deny this), Block would have to claim that even though perception of the target is disrupted more as the target and the flankers get more and more similar, the exact reverse happens when they are identical: the disruptive effects of crowding in the visual system become weak, and the target can be consciously perceived and identified. Such a reversal of grouping and similarity effects in crowding at physical identity is highly implausible and not supported by empirical evidence. To the contrary, an accumulating body of studies shows that the relationship between grouping/similarity and crowding strength holds for a large range of stimuli, including almost identical targets and flankers close to physical similarity. We take these results to count strongly against Block's claim that the identity-crowded target is consciously perceived.

Attention in Crowding

Another difficulty for Block's general view is as follows: Block claims that in cases of identity crowding, subjects can consciously see and identify items that they are unable to attend to. But this is at odds with evidence suggesting that conscious perception and ease of identification of a target in crowding is correlated with how easy it is to pay attention to that target. To see this, consider that as grouping becomes stronger, the item in question 'blends in' more with its surroundings, whilst when grouping is weak, the item 'stands out' more.

The more that an item stands out, the easier it is to direct attention to that item, this is a well known phenomenon in visual search (e.g., Nothdurft, 2000;Treisman & Gelade, 1980).

Several studies showed that discrimination performance of a crowded target goes hand in hand with how much the target stands out (or "ungroups") from the flankers. This was shown with objective measures such as reaction time in visual search (e.g., [START_REF] Sayim | Quantifying target conspicuity in contextual modulation by visual search[END_REF][START_REF] Gheri | The relationship between search efficiency and crowding[END_REF]; but see [START_REF] Felisberti | The role of target salience in crowding[END_REF] and subjective measures (e.g., [START_REF] Saarela | Global stimulus configuration modulates crowding[END_REF][START_REF] Manassi | Grouping, pooling, and when bigger is better in visual crowding[END_REF]. For example, when the target stood out, i.e., was easily attended and efficiently searched, the disruptive effects of crowding were reduced compared to when the target did not stand out -the success of subjects at the discrimination task and directing attention to the target went hand in hand (e.g., [START_REF] Sayim | Quantifying target conspicuity in contextual modulation by visual search[END_REF].

Since Block claims that conscious seeing is possible without attention in crowding, he has difficulty explaining why performance is linked so closely to how easy it is to direct attention towards crowded items. Block's opponents have no such problems: they accept that there is an explanatory link between conscious seeing and attention, so their view would predict that there would be a link between how well subjects perform at these tasks and how easy it is to attend to the target, which is precisely what we see.

Evidence in Favour of the Inference View.

We have argued that grouping and attention effects in crowding tell against Block's interpretation. However, there is still something that remains to be explained: if in identity crowding, the target is not consciously seen, how is it that subjects can identify, discriminate and differentiate the target, as well as have de re thoughts about it? In this section, we turn to this question. We argue that subjects' (seeming) success can be explained in terms of grouping and congruity effects, flanker biases, and prior knowledge. Again, we supply evidence for these claims. We argue that the evidence supports Taylor's version of the inference view, and also supports some aspects of Tye's view.5

Grouping and Congruity

As we mentioned in §1, Taylor's view is that subjects identify a flanker, then infer (based on the fact that the group of letters as a whole appears congruous, i.e., no item "stands out") that the target is the same as the flankers. Taylor's claim that subjects can detect properties of the group such as 'congruity' neatly fits with our discussion of grouping above. As we pointed out, subjects are able to make accurate judgements about how much a target 'blends in' (i.e. groups) with its flankers. These judgements of how much a target 'blends in' with the group seem very similar to the kinds of judgements about 'congruity' that Taylor claims subjects can make. Since we know that subjects are capable of making these judgements, it is reasonable to assume that they feature in the inferences that allow them to correctly identify the crowded item. This will especially be the case when the targets are identical with the flankers, as grouping will be maximal, and the group of items will appear maximally congruous. Correlations between identification performance and grouping/ congruity support this view.

Errors and biases in crowding

In this section we argue that biases and errors to report the identity of the flankers are important factors in explaining subjects' performance in identity crowding. 6 Identity crowding is a special case of crowding because reporting a flanker is a correct response.

Therefore, it is essential to differentiate between responses to the flankers and responses to the target. This is particularly important because a common error in crowding is targetflanker substitution, i.e., reporting a flanker instead of the target (e.g., [START_REF] Strasburger | Unfocused spatial attention underlies the crowding effect in indirect form vision[END_REF].

Substitution errors may account for about 30% of the total error rate in crowding (e.g., [START_REF] Strasburger | Unfocused spatial attention underlies the crowding effect in indirect form vision[END_REF][START_REF] Bernard | The dependence of crowding on f!anker complexity and target-f!anker similarity[END_REF].

When target and flankers are different, substitution errors are noticed because observers do not correctly report the target. However, this is not the case in identity crowding where the target response is 'correct' even if the observer identified a flanker.

What's more, substitution errors in letter identification were shown to increase as targets and flankers get more similar [START_REF] Bernard | The dependence of crowding on f!anker complexity and target-f!anker similarity[END_REF]. Since the targets are identical in appearance to the flankers in cases of identity crowding, we would expect substitution errors to be rather high. This suggests that the (seemingly) correct identification of the target in identity crowding may be partly due to substitution errors, which go unnoticed because mistaking a flanker for the target still delivers the 'correct' answer. If this were so, it would tell against Block's interpretation that subjects' reports were due to conscious perception of the target.

In a recent study, [START_REF] Sayim | Grouping and crowding affect target appearance over different spatial scales[END_REF] used stimuli highly similar to the standard identity crowding stimulus and showed that subjects have a tendency to report the identity of (grouped) flankers. In these experiments, the stimuli were letters, just as they are in Block's central case. In the first experiment, a target letter T was presented in one of four orientations (Figure 4; [START_REF] Sayim | Grouping and crowding affect target appearance over different spatial scales[END_REF]. Observers indicated the orientation of the target. The target was flanked by two sets of flankers. One pair (the horizontal flankers) was presented to the sides of the target and always had different orientations from each other and the target. The other pair (the vertical flankers) was presented above and below the target, and always had the same orientation as each other (see Fig. 4A). When the target had the same orientation as the vertical flankers, the three letters were the same as in the standard case of identity crowding -three identical letters of the same orientation in a row (but vertically rather than horizontally arranged). The results showed that the proportion of correct responses was higher when the vertical flankers and the target had the same orientation, i.e., when they were identical. This is exactly what would be expected if target discrimination is better in identity crowding than in normal crowding, and it may initially seem to support Block's view. However, the experiment did not distinguish between responses to the flankers and responses to the target. For example, if observers always indicated the orientation of the vertical flankers, they would be 100% correct in the condition where the flankers and the target were the same and 0% correct in the condition where they were different. Similarly, in standard identity crowdingreporting a flanker is always correctly reporting the target.

To investigate if a bias to report the flanker orientation underlies the (seemingly) better performance when the vertical flankers and the target were identical, in the second experiment signal detection theory [START_REF] Macmillan | Detection Theory: A user's Guide[END_REF] was used to separate sensitivity (perceiving the target better) and bias (responses driven by non-target sources, e.g., the flankers). Observers indicated if the 'T' at the target location had an orientation that matched the orientation of a 'T' they were shown beforehand, the 'predefined orientation' (e.g., upright in Fig. 4B). If it did, they were instructed to respond 'present' and if it did not, subjects were to respond 'absent', so this is a straightforward Yes/No task. In half of the trials, the vertical flankers were of the same orientation as the predefined target orientation (the matched condition), in the other half not (the unmatched condition; for more details see [START_REF] Sayim | Grouping and crowding affect target appearance over different spatial scales[END_REF].

The important point is this: unlike in the previous experiment (and unlike the examples of identity crowding that Block cites) a bias to respond in line with the flankers does not improve performance in this case (in the matched condition compared to the unmatched condition), so if (contra Block) subjects tend to report "target present" when actually only the vertical flankers had the predefined orientation, this experiment would reveal that subjects were biased to report the orientation of the vertical flankers. Indeed, that is what was found. In the matched compared to the unmatched condition (see fig. 4B, bias results), observers frequently reported that the target was present even though it was not. In other words, when the vertical flankers had the same orientation as the predefined one, observers falsely reported that the target was of that same orientation. Hence, observers were biased to respond in line with the vertical flankers. The vital point is this: the results showed that identical flankers did not improve target perception, they merely made subjects more likely to report that the target was of the same orientation as the flankers.

The result of these two experiments are of major importance for the identity crowding debate. Block's view (conscious seeing without attention) depends on the claim that subjects' reports in identity crowding cases are based upon consciously "seeing" the target. This was not the case. Instead, the results showed that targets which are not the same as the flankers are often mistakenly seen to be the same as the flankers. So it is clear that subjects manifested a tendency to report the orientation of the flankers. Hence, (apparent) superior performance in identity crowding compared to normal crowding is likely due to this bias imposed by the flankers when they have the same orientation (as well as other factors, see below). This is bad news for Block, but what about his opponents' views? Firstly, return to Taylor's view. Taylor claims that subjects identify a flanker, and then infer (based on the fact that the group appears congruous) that the identity of the target is the same as that of the flanker. If this is so, we would expect subjects to be more likely to report the identity of a flanker the more congruous the group of letters is. Strikingly, this is precisely what was found. Sayim & Cavanagh (2013, p.9) performed a further experiment, which showed that the bias to report the flanker was not maintained when the vertical flankers were not aligned with the target (but with the innermost flanker). This suggests that subjects are only biased to report the identity of a flanker when the group of letters are better aligned, and thus are more congruous with the target in the centre. This strongly supports Taylor's view that congruity is a powerful factor in making subjects believe that the target letter is the same as the flanker letters. Importantly, it also shows that the effect is not a mere response bias, i.e., reporting the presence of a particular target orientation simply because that orientation is frequent in the entire stimulus. Now, compare this with Tye's view. We do not endorse Tye's claim that in all cases of identity crowding, subjects do only see two letters. 7 Tye's account is at odds with the fact that in crowding, subjects are typically able to tell whether a target is absent or not, which is difficult to explain if all subjects saw was two letters [START_REF] Levi | Suppressive and facilitatory spatial interactions in peripheral vision: Peripheral crowding is neither size invariant nor simple contrast masking[END_REF][START_REF] Livne | Configuration influence on crowding[END_REF][START_REF] Pelli | Crowding is Unlike Ordinary Masking: Distinguishing Feature Integration from Detection[END_REF]). 8 On Taylor's interpretation, there is no such problem, since Taylor allows that subjects have a conscious representation of a group of letters, subjects would be able to use information about the size and shape of this group to determine whether there were three items there or two, and thus detect the presence or absence of a target (cf. Taylor 2013, p.22) 9 . For this reason, we prefer Taylor's view to Tye's.

Tye is nonetheless correct that bias to report a flanker is an important factor. Block (2013a, p.175 and2014, p.162) argues that the identity crowding results cannot be explained in terms of subjects simply identifying the flankers and reporting them. He cites an experiment (Petrov and Popple, 2007) where three Gabor patches were presented 7 However, we will not debate the claim that subjects sometimes only perceive two letters, we only claim that this view has difficulty applying across the board and (as we shall show) advocates of the inference thesis do not need to make the extreme claim that Tye does, as the data can be accommodated for in other ways. 8 Block (2014, p.164) claims that Tye's view is also phenomenologically implausible. However, as such disputes are notoriously difficult to adjudicate, and because we do not need to take a stand on this, we will discuss it no further here. 9 Note that knowledge about the stimulus may strongly shape the outcome of an identity crowding experiment (see also 3.3.). For example, comparing the relative size of two stimuli when knowing that there are either two or three elements, is easier than indicating the absolute number of items without any comparisons.

peripherally, with each Gabor slanted left or right. Block notes that subjects performed with 96% accuracy when the Gabors were uniformly slanted (///) but only 53% accurately when the middle Gabor was differently slanted left (/\/). Block takes this to show that results in identity crowding cannot simply be based on subjects' identification of the flankers (2013a).

Furthermore, Block notes that other asymmetries, such as 72% correct responses in the (\/\) case vs only 53% correct responses in the (/\/) case in normal crowding 'can only be explained perceptually ' (2013b, p.31).

Start with the asymmetry between the (///) case and the case when the central Gabor is tilted to a different orientation from its flankers, such as (/\/). This is clearly the asymmetry that is more directly relevant to identity crowding, as in this case one of the sets of stimuli are identity crowded. Subjects' (apparently) superior performance in the (///) case can easily be explained in terms of the mechanisms we have already outlined. In the (///) case, subjects' accidentally reporting the orientation of one of the flankers ('flanker substitution') would go unnoticed, and be recorded as a 'correct' judgement, but this would not go unnoticed in the (/\/) case, so this will contribute to the apparently higher performance in the (///) case. Similarly, it is often assumed that in crowding, subjects can detect the presence of features of the stimuli, but cannot detect the location at which the features appear, which would again result in uncertainty about the orientation of the target and the flankers in the (/\/) case but not in the (///) case. Finally, we have already pointed out that subjects are more likely to report the identity of a flanker when the group appears congruous (as predicted by Taylor, see also Sayim and Cavanagh 2013, p.9). Since congruity is maximal in the (///) case, we would expect subjects to be most likely to report the identity of flankers in this case. These factors can easily account for subjects' apparently superior performance in the (///) case, as opposed to the (/\/) case.

Turn now to the other asymmetry: subjects give 72% correct responses in the (\/\) case as opposed to 53% in the (/\/) case. Neither of these are cases of identity crowding, so it is difficult to see why this asymmetry should support Block's case that the middle item is consciously seen in the case of identity crowding. Block does not connect this case with identity crowding specifically, rather he seems to use it only to establish a more modest claim, which is that crowding effects in general can only be explained in terms of perception (2013b, p.31). Furthermore, in response to Taylor, Block concedes that 'Taylor is right that these asymmetries show only that the abilities I appealed to must be at least partially perceptual' (2013b, p.31. Our emphasis. See also his 2014 p.162).

Block's claim that these asymmetries are to be explained partly perceptually is quite modest, and we can agree with it. After all, our explanation of the asymmetries between identity crowding and normal crowding rely explicitly on subjects' conscious perception of the group of letters as a whole, and perceptual experience of the flankers, as well as congruity and uniformity properties. Furthermore, these conscious perceptual experiences will of course be underwritten by some unconscious perceptual processes. 10 So, we are happy to agree that the asymmetry between these two cases of normal crowding can be explained partially perceptually as well.

As Block notes, the mechanism underlying the (\/\) vs (/\/) asymmetry is not well understood (2013b, p.31) so we will not take a stand on this issue here. However, our view is compatible with a wide range of proposed explanations of this phenomenon. For example, Petrov and Popple speculate that we may be able to explain the asymmetry in terms of optic 10 We will expand on this point about unconscious perception in §4.

flow and saccadic eye movements (2007, p.5). Block himself notes this as a possible explanation (2013b, p.31). We do not commit to this view here, but nothing we have said is inconsistent with it.

Effects of knowledge

In identity crowding illustrations, observers know that they are looking at an example of identity crowding. Hence, they know in advance that, for example, three letters are presented. Without gaze contingent presentation, (presentation of the stimulus only when the observer keeps fixation at a particular point away from the stimulus), or very short presentation times, it is difficult to prevent observers from directly looking at the stimulus.

Hence, prior knowledge either about the entire stimulus or the flanker identity is likely to occur and have an effect in identity crowding. This has been shown in a variety of experiments. [START_REF] Zhang | Legibility of Chinese characters in peripheral vision and the top-down influences on crowding[END_REF] showed that if subjects have prior knowledge of the set of items the target and flankers were drawn from, then superior performance was observed compared to conditions in which the set was not known before the experiment. Similarly, if subjects are cued with an item in the fovea, and then are made to scrutinize a crowded target for that item, success increased when the cued item was the same as the target (Sayim et al., 2014) 11 . This effect of prior knowledge even occurs with extremely complex crowded stimuli, such as abstract paintings (Sayim, Myin, & Van Uytven, 2015).

It is likely that subjects can reach conclusions about the identity crowded stimuli based upon these knowledge effects. As observers know the target in identity crowding because they can directly look at them, conclusions about the identity of the target are influenced by this prior knowledge. Having prior knowledge will obviously make the question of what the target letter is very easy. Again, this suggests another explanation of subjects' abilities, other than conscious perception of the target 3.4 Subjects' abilities: conclusions.

We have discussed empirical results which indicate that the (seemingly) better performance in identity crowding can be attributed to several factors without invoking the conscious perception view. These factors include the regularity of the group of letters, i.e., the lack of "standing out" of the target (leading subjects to make inferences based upon 'congruity' properties of the group), in conjunction with identification of a flanker, biases to report the identity of the flankers and knowledge of the stimulus. We claim that it is some variety of these factors that accounts for subjects' performance in identity crowding. What's more, unlike Block's other opponents, we have supported this claim with a large range of empirical evidence.

Summary statistics and the phenomenology of crowding.

In this section we consider some additional arguments that Block uses to support the conscious perception hypothesis. Block (2013b, pp.32-36) refers to an experiment that applied a texture synthesis algorithm to images to produce new 'scrambled' images which should be indiscernible from the original, when both images are viewed in peripheral vision [START_REF] Balas | A summary Statistic Representation in Peripheral Vision Explains Visual Crowding[END_REF]). 12 These scrambled images are called 'mongrels'. Balas et al. suggest (albeit tentatively) that the perceptual system may represent stimuli viewed in the periphery using a process similar to this mongrelisation algorithm (2009, pp.11-14).

When discussing the summary statistics data, it is important to separate the different points that Block makes. The first is to do with binding. Block notes that in many of the mongrels, the middle letter is recognizable as an upright 'T'. He then concludes that: '[t]hese figures suggest that the visual system can sometimes bind features to some objects in peripheral vision so long as the objects are all the same ' (2013b, pp.35-36).

We are happy to accept that the visual system can bind features in peripheral vision. However, it is well known that perceptual binding can occur in the absence of consciousness [START_REF] Dehaene | Letter Binding and Invariant Recognition of Masked Words: Behavioral and Neuroimaging Evidence[END_REF], so the presence of binding does not tell us whether perception of the object is conscious. In crowding generally, the view that crowded targets which cannot be discerned are nevertheless processed by the visual system and affect perception and behavior, is well supported by empirical results [START_REF] He | Attentional resolution and the locus of visual awareness[END_REF][START_REF] Yeh | Semantic priming from crowded words[END_REF][START_REF] Kouider | Preference is biased by crowded facial expressions[END_REF], Faivre & Kouider, 2011). Though these studies did not concern identity crowding, they do show that unconscious perception of crowded items occurs, and so we see no reason to resist the claim that it may occur in the case of identity crowding. Similar things go for Block's claim (based on [START_REF] Burge | Origins of Objectivity[END_REF] that unconscious seeing and conscious seeing form a natural kind (2013a, p.180 and 2014, p.161). 13 We are happy to accept all of these points, we simply deny that the target is consciously perceived. 14 Crucially for our purposes, Block extends this discussion in an attempt to demonstrate that the middle letter is consciously represented, at least sometimes. Block argues that we have good reason to think that the phenomenology of seeing the 'TTT' stimulus in peripheral vision is accurately represented by the different mongrels in fig. 2, saying that: 13 Block further claims that in identity crowding, our perception of the target involves a 'singular element ' (2013b, p.36). A singular element is part of the content of a perceptual episode that represents the individuality of the percept, the fact that the percept is a particular (Burge 2009, p.287). Again, we are happy to accept this, but we claim that singular elements are unconscious in the identity crowding case. Burge himself agrees that singular elements can be present in unconscious perceptual episodes (2010). 14 This point also applies to an argument for a similar conclusion that Block bases on Freeman and Pelli (2007), where subjects were just as good in performance at a change-detection task when the cued items were crowded letters, as when they were uncrowded ones. Block takes this to imply that perception in crowding is object-perception (2013b, p.36) and that a Burgean 'singular element' is present in crowded perception (2014, p.161. See also Block 2013a, pp.178-182). As above, we will not dispute these claims, we simply claim that all of this is unconscious perception.

'the mongrelisation algorithms are designed to reproduce the conscious qualities of peripheral vision… there is reason to think that the mongrelisation process is telling us about conscious vision ' (2013b' ( , p.37. Cf. 2014, p.162), p.162). 15 If we accept Block's claim that the phenomenology of identity crowding is accurately captured by the mongrels, then (since some of the mongrels contain a clear middle letter 'T') we have good reason to accept that the middle 'T' is represented in consciousness, at least sometimes.

This argument hinges on accepting that the mongrels accurately represent the phenomenology of peripheral vision. However, we do not think that Block is warranted in drawing this conclusion from the Balas et al. data, for two reasons: firstly, the claim that mongrels are a good representation of what the original would look like when viewed peripherally is based on an experiment that involved exposing two different groups of subjects to different stimuli, and asking them to identify the stimuli. One group viewed the original images in peripheral vision, whilst the other viewed mongrels in foveal vision. The correlation of performance between the tasks was taken as the indication that the mongrel accurately 'matched' what the original would look like in peripheral vision. However, it simply doesn't follow from the fact that performance at an identification task was the same in both cases that the phenomenology of the original in peripheral vision is the same as that of the mongrel in foveal vision. More generally, the experiment that Block describes is put forward extremely tentatively by Rosenholtz and her colleagues, as is the method outlined above for 'matching' mongrels with original images. We would require far more converging evidence before we accepted that mongrels are an accurate portrayal of the phenomenology 15 Simoncelli and Portilla themselves note that whether the algorithm is reproducing the phenomenology of conscious vision is in need of experimental verification using 'subjective measurements ' (1998, p.4). of peripheral vision. Balas et al. themselves typically claim that the mongrels accurately capture 'the information available to the visual system under conditions of crowding' (2009, p.10), so it is unclear whether they are making a claim about phenomenology, as Block does.

Secondly, (and, we think, devastatingly) for any one original image, there are infinite possible mongrels. As you can see from fig. 5, some of them involve a central letter, and others do not. So even if we do assume that some of the mongrels accurately represent how the original image appears in peripheral vision, we have no reason to think that this includes the mongrels that contain the middle letter. Block could argue that a high proportion of the set of possible mongrels include a middle letter. However, the problem is that whether or not this is true, there is still nothing forcing Block's opponents into accepting that the phenomenology of identity crowding resembles any of these particular mongrels (or even any mongrels at all). Moreover, it is not sufficient to claim that a high proportion of the stimuli contain the (correct) middle letter because the important aspect is that the proportion should be higher than that for "normally" crowded targets under similar conditions. Drawing on suggestions by [START_REF] Balas | A summary Statistic Representation in Peripheral Vision Explains Visual Crowding[END_REF], p.5), Block (2013b, p.35) notes that when we view items in conditions of crowding, our perception of crowded items may shift between the different possible mongrels in figure 5. He agrees with Balas et al. when they say: 'this may be the explanation for the shifting and dynamic percept many observers experience when attending to their peripheral vision ' (2009, p.5).

However, this point cannot establish that the mongrels accurately represent the phenomenology of crowding. Firstly, it is unclear whether the phenomenology of crowding really is 'shifting and dynamic'. If we know anything about crowding, it is that its phenomenology is extremely hard to put one's finger on. Secondly, even if we accept this description of the phenomenology, we need not accept that it is due to changes which correlate with different images akin to the different mongrels. Such 'dynamic' phenomenology may be as a result of subtle shifts in attention to the group of items: it is well known that attention to items can change their perceived appearance (e.g. [START_REF] Carrasco | Attention alters appearance[END_REF][START_REF] Tse | Voluntary attention modulates the brightness of overlapping transparent surfaces[END_REF].

We must now discuss a pilot study that Block mentions (2013b, pp.35-36). Some applications of the algorithm result in crowded stimuli that contain letters that are inverted relative to the original image. Block notes that if we take seriously the claim that the mongrelization algorithms accurately represent what is happening in cases of crowding then this will predict that subjects should find it difficult to discern mongrel stimuli with inverted letters from original stimuli without such inversions. Balas et al. performed a pilot experiment which shows that subjects have difficulty telling apart a set of stimuli containing upright A's from one containing an inverted A under crowding (2009, p.11).

However, the pilot study cannot support any substantial claims about whether the mongrels accurately represent the phenomenology of peripheral vision. 17 The first important point is that the study was only a pilot study. For example, there were no baseline measurements, no control conditions, and it was a null finding (chance performance). To our knowledge, a full version of this experiment has not been performed, so we cannot draw any firm conclusions from it.

In any case, this is only a discrimination task, and as a result is subject to certain limitations. Even if a full experiment were carried out, the most we could conclude from a discrimination task such as this would be that (whatever the phenomenology of the experience of the original and the experience of the mongrel were like) it was hard for subjects to tell them apart. This is, of course, what would be expected if the mongrelisation algorithm was working properly. However, the inability of subjects to discern a set of upright 'A's from a set containing an inverted one can only tell you that subjects find the two sets of stimuli difficult to discern. We cannot draw from this any conclusions about what the phenomenology of either set of stimuli actually was like. It tells us nothing about what the phenomenological appearance of each one was to the subject. So we are not entitled to draw any firm conclusions about the phenomenal appearance of either set of letters, which is what would be required for Block's argument to hold up. Of course, we are not expressing scepticism about the general possibility of using introspective measures to establish the phenomenology of crowding, 18 we are simply pointing out the limitations of discrimination tasks such as this one.

Conclusion

Before drawing any firm conclusions about identity crowding, it is important to examine all of the available evidence. In this paper, we have argued that Block's interpretation of the crowding data is untenable. Though we have disagreed with Block's view, we are in complete agreement with him that crowding generally (and identity crowding particularly) provides a rich source of insights about attention and its relationship to consciousness, and 18 For example, [START_REF] Sayim | Drawings of the visual periphery reveal appearance changes in crowding Perception 42[END_REF] use a paradigm on which subjects are asked to draw the crowded items (see also Coates, Wagemans, & Sayim, in press). There is good reason to think that this gives us a better insight into the phenomenology of crowding than simple discrimination tasks.

Figure 1 :

 1 Figure 1: An example of identity crowding. In the first row, an unflanked target is shown. The second row shows a 'normal' case of crowding and the third row the 'identity' crowded target (details see text).

Figure 2 :

 2 Figure 2: Identity crowded lines. Adapted from Intriligator and Cavanagh (2001). (Details see text).

Figure 3 :

 3 Figure 3: an example of grouping. On the left, the central A strongly 'groups' with the two close-by letters to

Figure 4 :

 4 Figure 4: A) Observers indicated the orientation of the target T in the centre. When the target had the same

Figure 5 :

 5 Figure 5: From Block (2013b, p.35): John Wiley and Sons, Inc. Reproduced with Permission. Block credits Ruth Rosenholtz with creation of the image.

Fig. 5

 5 Fig.5 shows the results of applying the mongrelisation algorithm to stimuli used in

See Carrasco (2011[START_REF] Van Boxtel | Consciousness and attention: on sufficiency and necessity[END_REF] and[START_REF] Kentridge | Attention without awareness: a brief review[END_REF] for surveys of work in cognitive science.See[START_REF] Watzl | The philosophical significance of attention[END_REF] and[START_REF] Wu | Attention[END_REF], for philosophical surveys.

By 'de re thoughts' we mean demonstrative thoughts, thoughts that 'mentally point' to objects.

Following convention, we refer to the middle letter as the 'target' and the outer two letters as 'flankers'.

Though Block grants that there may be spatial (as opposed to object-based) attention to it (2013a, p.173).

For reasons of space, we will not be able to discuss Richards' view much more, nor will we have the space to dispute Block's criticisms of Richards(Block, 2013b, p. 31). Of course, the fact that the evidence we supply does not support Richards' view does not imply that his view is wrong.

Again, though Taylor and Tye both suggest possibilities similar to these, they do not discuss any evidence related to them, as we do here.

Note that simultaneous presentation of the foveal item and the stimulus showed a stronger effect than presenting the foveal item before the stimulus, indicating that the effect of target-flanker grouping was stronger than cueing.

See Portilla and Simoncelli (2000),[START_REF] Simoncelli | Texture characterization via joint statistics of wavelet coefficient magnitudes[END_REF] and[START_REF] Freeman | Metamers of the Ventral Stream[END_REF] for more on these algorithms.

Importantly, Balas et al. themselves do not claim that it does.

The authors would like to thank Chris Mole, Bence Nanay, an anonymous referee and the editor for comments previous drafts of the paper. Thanks also to Ned Block for helpful conversation. Henry Taylor would like to thank the Leverhulme Trust and Isaac Newton Trust for an Early Career Fellowship (ECF-2015-088) that provided support while the paper was being written. Thanks also to the Royal Institute of Philosophy for providing a Jacobsen Fellowship whilst research for the paper was carried out. Bilge Sayim was supported by the Swiss National Science Foundation (PP00P1_163723, BS) and an FWO Pegasus Marie Curie fellowship.


domain: sde math

text:
Carole Delenne

Vincent Guinot

Bernard Cappelaere

Weak form-based objective functions in hydrology Application to a 1D transport/degradation model

Hypotheses:

-Water elevation controlled so that A independent of time => Q independent of space k(t) piecewise linear Conclusions -Weak form-based OF proves easier to use than distancebased OF in practice -It is well-admitted that calibration and validation should be carried out using different data sets; we may also admit that they should be carried out using different types of OF! -Sensitivity analysis can help select measurements to be used for calibration of any given parameter

Application

Each parameter calibrated using all the measurements BUT a parameter should not be calibrated using non-sensitive output for that parameter

1-

  Context and ObjectiveWeak form-based objective functions (WOF) can be a good alternative to the classical distance-based functions (DOF) for model calibration[1]. WOF are monotone with respect to a given parameter when the sensitivity of the model output keeps a constant sign; then: -The calibration problem is changed from an optimization problem into a root finding problem -WOF are less prone to local extrema (when one parameter is to be calibrated, they have a single optimum) -When several parameters are calibrated, their redundancy can be assessed by superimposing the contour lines of different WOF. The objective is to apply WOF to the calibration of the degradation coefficient (k(t) depending on time) of a 1D pollutant transport/degradation model.Generationof «observed» values with all parameters constant. The primary variable C(x,t) cannot be measured easily -Use of a secondary variable: C(L,t) at the outlet -Model output «as close as possible» to the data -Classical measures of closeness (DOF): -Weak form-based (monotone wrt k): : several values of (k1,k3) give acceptable values of Wp (error compensation): e Model error: Generation of «observed» values with variable, noisy input and «measurement» uncertainty V. Guinot, B. Cappelaere, C. Delenne and D. Ruelland, Towards improved criteria for hydrological model calibration: theoretical analysis of distance-and weak form-based functions, Journal of Hydrology, in press. with and Calibration of 1 parameter n with WOF: -Choose a value for p and solve W p (k n )=0 using e.g. Newton's algorithm : With obtained by solving the sensitivity equation: Calibration of n parameters k n with WOF -Find the intersection of n WOF with n values for p.

  error by the sensitivity gives more weight to the error when the solution is sensitive to the parameter and avoid compensations.

variable & noisy input


Oriol Pont

Antonio Turiel
email: turiel@icm.csic.es

Conrad J Perez-Vicente
email: conrad@ffn.ub.es

Description, modelling and forecasting of data with optimal wavelets

Keywords: Optimal wavelet, Cascade processes, Microcanonical multifractal formalism, Time series forecasting

Cascade processes have been used to model many different self-similar systems, as they are able to accurately describe most of their global statistical properties. The so-called optimal wavelet basis allows to achieve a geometrical representation of the cascade process-named microcanonical cascade-that describes the behavior of local quantities and thus it helps to reveal the underlying dynamics of the system. In this context, we study the benefits of using the optimal wavelet in contrast to other wavelets when used to define cascade variables, and we provide an optimality degree estimator that is appropriate to determine the closest-to-optimal wavelet in real data. Particularizing the analysis to stock market series, we show that they can be represented by microcanonical cascades in both the logarithm of the price and the volatility. Also, as a promising application in forecasting, we derive the distribution of the value of next point of the series conditioned to the knowledge of past points and the cascade structure, i.e., the stochastic kernel of the cascade process.

Introduction

Stock markets, as well as many other trading markets, are formed by a great diversity of interacting agents, each with their own characteristics, such as reacting times, budgetary constraints and so on. As a consequence, and due to the large amount of agents taking part in a typical market, many econometric indicators behave in a complex, scale-invariant fashion, a feature that has been taken into account in many different models [START_REF] Arneodo | direct" causal cascade in the stock market[END_REF][START_REF] Calvet | Forecasting multifractal volatility[END_REF][START_REF] Perelló | Multiple time scales in volatility and leverage correlations: a stochastic volatility model[END_REF]). However, scale invariance can be exploited not only in the design of models, but also in analysis tools capable of extracting new information from time series of dynamical systems. One of the most promising theories for the description of scale-invariant data concerns multifractal systems [START_REF] Muzy | Multifractal returns and hierarchical portfolio theory[END_REF][START_REF] Turiel | Multifractal geometry in stock market time series[END_REF], and more particularly multifractal systems in the microcanonical approach (Turiel andPérez-Vicente 2003, 2005;Turiel et al. 2008b). A full discussion of the microcanonical multifractal formalism (MMF) exceeds the scope of this paper; here we will just make use of those facts relevant for our study, namely the existence of adequate wavelet representations for which data can be represented as an explicit cascade process across scales. With the aid of microcanonical cascades one can maximize the amount of information that some scales convey about the others. Besides, it is possible to provide an analytical model describing the evolution of the series, and produce high quality forecasts for both the returns and volatility evolution of stock market data.

The paper is structured as follows: next section introduces the data that will be used to illustrate the theory. Then, in Sect. 3 the multiscale cascade model is introduced. In Sect. 4 the scale invariant properties of cascade are discussed, and Sect. 5 is devoted to introduce the optimal wavelet, for which the cascade is directly verified; as an application to econometric data, the best wavelet from a bank of standard wavelet bases is obtained. Then, the direct cascade model furnished by the optimal wavelet is used to derive analytically the stochastic kernel, namely the distribution of future values conditioned by the known past values, and the entire Sect. 6 is devoted to this goal. Finally, we present in Sect. 7 our conclusions.

Description of the data and notation

We have processed a group of data belonging to the Spanish stock market (IBEX). The group is formed by daily series of 35 different assets (those with the largest liquidity in the Spanish market) during approximately ten years (from June 1996 to June 2006, although some series are slightly shorter) containing 76,663 points. We do not try to correct systematic deviations by any mean. In that sense, we always identify the ending of a session as the instant just preceding the opening of the following, no matter the actual time interval between them (sometimes several non-working days). An example of this series is shown in Fig. 1.

We are interested in relative variations of the price, i.e., the ratio of the absolute value to the absolute variation. For that reason, we will work on series formed by logarithms of prices. In this way the absolute variation between two consecutive instants (approximately, the derivative with respect to time) approximates the relative variation for the original stock series. These series have been shown to have multifractal properties in the sense of MMF (Turiel andPérez-Vicente 2003, 2005;Turiel et al. 2008b), so they are appropriate for the present study. Throughout the paper, time series will be represented by a discrete, time-ordered collection of values x 1 , x 2 , . . ., where x n = x(t n ) and the sampling times t n are equally spaced, t i = t 0 + n∆t. For description and forecast, we will take a finite number N of past, known values, forming a vector x = (x 1 , . . . , x N ). The unknown value at the following time instant (which would correspond to x N +1 ) will be denoted with a different letter, y, to emphasize that this is the forecast.

Wavelet representation of cascade processes

Scale-invariant systems cannot be described appropriately by any statistical quantity covering only one or few scales; on the contrary, to gain full understanding of those systems the influence of all scales must be taken into account. Usual data representations are based on a single observation scale, the most natural one being the rate at which the series has been sampled. For a scale-invariant signal to focus exclusively on a single scale leads to an inefficient information processing, which motivates the use of other representations where the impact of the different scales can be explicitly decomposed. This is the case of wavelet bases [START_REF] Daubechies | Ten lectures on wavelets[END_REF][START_REF] Mallat | A wavelet tour of signal processing[END_REF]. For the sake of simplicity, we will just consider orthonormal, dyadic wavelet bases.

Given a discrete or continuous time series x(t), we can represent it in a dyadic wavelet basis of a mother wavelet Ψ as follows:

x(t) = j 0 j=-∞ k α j,k Ψ j,k (t) + k β j 0 ,k Φ j 0 ,k (t) (1)
where Φ is the unit function associated to Ψ and Ψ j,k (t) = 2 -j/2 Ψ \ ( 2 -j tk) [START_REF] Mallat | A wavelet tour of signal processing[END_REF]. In this representation, the first term accounts for the details, i.e., the variations that contribute at different scales, starting from the infinitely resolved one ( j = -∞) and including all of the coarser scales up to a conventionally fixed one ( j 0 ). The second term represents the approximation at that scale j 0 , which accounts for all of the remaining variations smoother than that of j 0 . The wavelet coefficients α j,k and the approximation coefficients β j 0 k can be easily obtained when the wavelet basis is orthonormal by simple projection:

α j,k = Ψ j,k , x = dt Ψ j,k (t) x(t)
(2) and

β j,k = Φ j,k , x = dt Φ j,k (t) x(t) (3)
For a signal defined on an unbounded domain, the separation scale j 0 is chosen conventionally (eventually j = +∞, what makes the second contribution to vanish). A different situation arises when the signal has a bounded domain. First, all the sums on the position k are limited to positions included in the domain. Second, j 0 cannot be greater than the integral scale J (N = 2 J ), and for that reason it is normally chosen as it. On the other hand, if the signal is discretised, this imposes a minimum possible scale (so truncating the lower end of the sum in the first term) which can conventionally set to j = 0. Hence, in practice Eq. ( 1) is reduced to:

x n = J j=0 k∈Z 2 j α j,k Ψ j,k (t n ) + β J,0 Φ J,0 (t n ) (4)
Notice that as wavelets form a basis, the wavelet coefficients in Eqs. ( 1) and (4) are algebraically independent. But when signals have a multifractal structure there is an additional constraint evidenced by wavelet coefficients. This is the cascade relation (Turiel and Parga 2000b;Turiel et al. 2003;[START_REF] Pont | On optimal wavelet bases for the realization of microcanonical cascade processes[END_REF], which relates a wavelet coefficient at a coarse scale (the Parent coefficient) with the wavelet coefficients at the next finer scale and with the same position at the coarse level (the Children coefficients). The cascade relation is expressed as follows:

α j,k . = η j,k α j-1, k 2 (5)
where the symbol " .

=" means that both sides are equally distributed and the variables η j,k are equally distributed for any j, k, and are independent from α j-1, k 2 . The variables η j,k are known as cascade variables and are related to multifractal invariants in the MMF (Turiel et al. 2008b;[START_REF] Pont | On optimal wavelet bases for the realization of microcanonical cascade processes[END_REF]. The notation k 2 means the integer part, rounding down, of k 2 ; this implies that one parent coefficient has exactly two children coefficients (e.g. k = 2k and k = 2k + 1). Notice that for any wavelet we could always define the child/parent ratio η j,k , namely:

η j,k = α j,k α j-1, k 2 (6)
In general η j,k is not independent of α j-1, k 2 except for a particular choice of wavelet basis, called optimal wavelet basis, for which the child/parent ratio η j,k is independent of the parent coefficient α j-1, k 2 and hence it can be considered a cascade variable.

This kind of wavelet is discussed in Sect. 5.

Scale invariant properties of the cascade processes

A dyadic cascade variable, i.e., a variable verifying Eq. ( 5), cannot follow an arbitrary distribution. In fact, the variables η j,k must follow an infinitely divisible distribution [START_REF] Novikov | Infinitely divisible distributions in turbulence[END_REF][START_REF] She | Universal scaling laws in fully developed turbulence[END_REF]Turiel and Parga 2000a), for a scale ratio of 1

2 . An infinitely divisible distribution implies that the product of two cascade variables is also a cascade variable, with a scale ratio equal to the product of the scale ratios (Turiel and Parga 2000a). Cascade variables η are not scale invariant, and neither is their distribution. However, they can be related to scale invariant quantities through the introduction of the MMF (Turiel et al. 2008b). We will not present a full discussion of MMF here, just a simplified, more qualitative introduction which would serve to support our experiences.

According to Eq. ( 5), any wavelet coefficient α j,k is distributed as the product of j independent dyadic cascade variables, in the way:

α j,k . = j-1 j =0 η j-j , k 2 j α 0.0 (7)
If the variables η j,k are infinitely divisible, they verify:

η j,k = 1 2 h j,k (8)
where h j,k is a scale-invariant quantity, the transition singularity exponent between the scales j and j -1 at the point k [START_REF] Pont | On optimal wavelet bases for the realization of microcanonical cascade processes[END_REF]Turiel et al. 2008b). We can thus define a punctual estimate of the singularity exponent associated to the whole cascade at the point 2 -j k by taking logarithms in Eq. ( 7) and normalizing by the scale factor j log 2, namely:

ĥ j,k = - log α j,k j log 2 . = h j,k - log α 0,0 j log 2 (9)
where h j,k is a dimensionless, scale invariant field known as singularity exponent, which is the average of the transition singularity exponents,

h j,k = 1 j j-1 j =0 h j-j , k 2 j (10)
Singularity exponents characterize the degree of regularity or irregularity of a function at a point, and serve to characterize the multifractal geometry of scale invariant fields in MMF (Turiel et al. 2008b). The multifractal structure of the signal is intimately linked to its dynamic properties (Turiel andPérez-Vicente 2003, 2005;Turiel et al. 2005aTuriel et al. ,b, 2008a)). This fact explains why it is so much important to obtain the singularity exponents for tasks such as reconstruction or forecasting. Punctual estimates as the one given by Eq. ( 9) are good approximations to the actual value of the singularity exponent when the scale ratio is large enough, namely j is large enough and so the second factor in Eq. ( 9) can be neglected [START_REF] Turiel | Numerical methods for the estimation of multifractal singularity spectra on sampled data: a comparative study[END_REF][START_REF] Pont | Application of the microcanonical multifractal formalism to monofractal systems[END_REF]).

The distribution of singularity exponents as obtained at a scale j is not scale invariant. However, it can be related to the singularity spectrum of the underlying multifractal hierarchy (Frisch 1995;[START_REF] Arneodo | direct" causal cascade in the stock market[END_REF]. Let r = 2 -j be the scale ratio associated to h j,k , according to Eq. ( 10). The distribution of values of h j,k , ρ\( h j,k ) verifies:

ρ\( h j,k ) = A 0 r d-D(h j,k ) (11)
where d is the dimension of the embedding space (d = 1 for time series) and D(h) is the function relating the value of the singularity exponent, h, with the fractal dimension of the associated singularity component [START_REF] Falconer | Fractal geometry: mathematical foundations and applications[END_REF]Turiel et al. 2008b). If the mode (i.e., most probable value) of h j,k has a known dimension [which must be d if the support of the multifractal is the whole space [START_REF] Turiel | Numerical methods for the estimation of multifractal singularity spectra on sampled data: a comparative study[END_REF](Turiel et al. , 2008b))], then the singularity spectrum can be directly retrieved from Eq. ( 11) by a log-log transformation [START_REF] Turiel | Numerical methods for the estimation of multifractal singularity spectra on sampled data: a comparative study[END_REF]:

D(h j,k ) = d + ρ\( h j,k \)/ρ 0 j log 2 (12)
where ρ 0 is the maximum value of the function ρ\( h j,k ). The singularity spectrum is a global scale invariant quantity, so for any value h 0 and any j, k, j , k we should observe D(h j,k = h 0 ) = D(h j k = h 0 ). In fact, the equality of these factors at different scales is one of the conditions for MMF to be valid (Turiel et al. 2008b). In Fig. 2 we show that the experimental singularity spectra obtained from the series of returns and from the series of volatilities (derived from our IBEX 35 dataset) are coincident within the experimental uncertainty at three different scales [notice that the right tail is always worse determined; see discussion in [START_REF] Turiel | Numerical methods for the estimation of multifractal singularity spectra on sampled data: a comparative study[END_REF]]. Remarkably enough, return and volatility singularity spectra are very similar. This should be expected because volatility is a measure of the amplitude of returns. As singularity exponents describe function regularity (Turiel et al. 2008b), those of returns approximately coincide with those of volatilities. Let us finally remark that the knowledge of the singularity spectrum D(h) provides information of the distribution of the cascade variables at any scale. Simply, by the application of Eqs. ( 9) and ( 11) we can know the distribution of α j,k at any scale j.

Optimal wavelet

Optimality and the Q-parameter

As commented before, an optimal wavelet is such that the variable η j,k is optimal, namely it is independent of the parent wavelet coefficient α j-1, k 2 and is distributed according a multiplicative cascade. This choice of wavelet basis is very convenient, as it implies extracting the independent factors relating scales thus minimizing redundancy, what can be used for tasks such as coding or inference. The existence of an optimal wavelet basis was first hypothesized in (Turiel and Parga 2000b) for the case of natural images, and in the same reference a constructive formula able to retrieve the optimal wavelet from a set of data was given. However, that formula requires to have a very large training dataset with many independent members, something which is rarely the case. For that reason, some attempts to design less data-demanding algorithms to retrieve the optimal wavelet from data have recently been enterprised. In [START_REF] Pont | On optimal wavelet bases for the realization of microcanonical cascade processes[END_REF] a criterion to assess the degree of optimality of a wavelet basis was presented. It is based on a parameter named Q, which compares the expectation value of the ratio child/parent with the expectation value of a cascade variable (which is a priori known from some constraints). More precisely, Q is defined as follows:

Q = | η| |η| ( 13
)
where the expectation values • are taken on an ensemble of realizations (including different scales j and positions k). A wavelet is optimal if and only if Q = 1, and any non-optimal wavelet will have Q > 1 [START_REF] Pont | On optimal wavelet bases for the realization of microcanonical cascade processes[END_REF]. The Q parameter has many advantages with respect to other criteria on wavelet optimality, as for instance the parameter I measuring the mutual information between η and its parent coefficient (I ≥ 0 and I = 0 if and only if the optimal wavelet). First, Q varies continuously under perturbations on the optimal wavelet, what makes it more appropriate for continuous strategies of optimization. Second, Q is less demanding in data and more stable, giving a more accurate description of the wavelet properties than I . Another, more visual criterion to assess the optimality of a wavelet basis consists in plotting joint histograms of log |α C | versus log |α P | (where the subindexes C and P denote "Children" and "Parent", respectively, used here to alleviate notation). This kind of histogram plots were introduced by Simoncelli and co-workers [START_REF] Buccigrossi | Image compression via joint statistical characterization in the wavelet domain[END_REF][START_REF] Wainwright | Random cascades on wavelet trees and their use in modeling and analyzing natural images[END_REF]. If the wavelet is optimal, the equality in Eq. 5 implies that:

log |α C | = log |η| + log |α P | (14)
Hence, the joint histogram of log |α C | versus log |α P | for an optimal wavelet must have a shape of straight line with slope 1, with the values of log |η| distributed around it. If the wavelet is not optimal, for small values of log |α P | a deviation from this linear behavior will be observed and the predictability of α P about α C diminishes. As a consequence, α C becomes independent of α P and the histogram displays a horizontal straight line. It is thus possible to roughly assess the degree of optimality of a wavelet by knowing the size of the range on which the optimal linear behavior, Eq. 14, holds. In general, however, it is better to work with more quantitative measurements such as Q, using joint histograms just for illustration purposes. We show examples of joint histograms derived from real data in the following.

Optimal wavelet for Spanish stock market series

We have constructed a bank of 24 orthonormal dyadic wavelet bases. The bases included in this bank are some of the most frequently used in the wavelet literature: Haar, Daubechies of orders 2-10 (signatures Dau2-9 and DauA), Symlets of orders 4-9 (signatures Sym4-9), Coiflets of orders 1 to 5 (signatures Coi1-5), and spline or Battle-Lemarié of orders 1,2,3 and 6 (signatures BL1-3 and BL6). We have searched which wavelet in this bank is closest to optimality for our dataset according to the Q criterion. Results are summarized in Table 1.

It is observed that the best wavelet (in terms of optimality) in the bank is the Battle-Lemarié wavelet of order 3. A similar experience conducted on volatility series throws that the optimal wavelet is a different one, Symlet of order 7 (in that case, Sym7 attains a Q = 2.22 while for other wavelets of the bank the average value of Q is 2.95 and the farthest from optimality is Haar with Q = 5.71). Should one expect to have a relation between the optimal wavelet for log-prices and that of volatility? In fact the answer is yes. Volatility is the amplitude of return variations, i.e., it represents some kind of modulus of the derivative of log-prices, and the optimal wavelet of the derivative series is the derivative of the optimal wavelet of the series, as explained in Table 1 Histograms of child wavelet coefficient α C conditioned by the value of its parent coefficient α P (in logarithmic scale) and values of Q for different wavelet bases They are sorted from closest to farthest from optimality, according to the value of Q. All histograms range from -6 to 3.5 in both axes. The elbow shape is a direct consequence of loss of optimality, so its position depends on the degree of optimality (for Battle-Lemarié 3, the best wavelet, it is located the leftmost, at about 1 3 of the histogram, while for Haar, the worst wavelet, it is the rightmost, at about 1 2 of the histogram) Fig. 3 Comparison of the optimal wavelet obtained for volatility series (Symlet of order 7) and the time derivative of optimal wavelet obtained for the logarithm of the quotation price \(Battle-Lemarié of order 3) Turiel and Parga (2000b). As shown in Fig. 3, Symlet of order 7 is very similar to the derivative of Battle-Lemarié of order 3. This confirms the validity of the cascade description in terms of optimal wavelets for both series.

Conditioned distribution of quotation values according the cascade model

Settings

We will try now to determine the distribution of the series value y conditioned to the knowledge of the vector x of N previous events, that we will denote by ρ y (y|x). Due to the high dimensionality of this function, it cannot be estimated in general and it rather needs to be modeled. We will make use of the cascade properties that we have introduced in the previous sections to model this conditioned PDF. First, we assume that our wavelet basis is optimal, so:

α j,k = η j,k α j-1, k 2 (15)
where the variables η j,k are all identically distributed according to a known PDF ρ η and each variable η j,k is independent of

α j-1, k 2 . As α j-1, k 2 can be further decom- posed as η j-1, k 2 α j-2, k 4
, we conclude that η j,k is independent of its grand-parent

η j-2, k 4
and by induction, it is independent of all its ancestors. Notice however that this does not imply that η j,k is independent of η j,k ; the horizontal correlations must be studied and implemented. Due to the linearity in the definition of the wavelet coefficients, Eq. ( 2), α j,k can be expressed as a linear function of y, with coefficients depending on x, namely:

α j,k = a j,k y + b j,k (16)
Hence, the cascade variables are expressed as:

η j,k = a j,k y + b j,k a j-1, k 2 y + b j-1, k 2 (17)
As there is a deterministic relation between each cascade variable and the variable y, the conditioned distribution of the cascade variables is expressed as a combination of delta functions, namely:

ρ\( {η j,k }|y, x) = j,k δ ⎛ ⎝ η j,k - a j,k y + b j,k a j-1, k 2 y + b j-1, k 2 ⎞ ⎠ (18)
If we integrate this distribution with ρ y (y|x) we will obtain the distribution of cascade variables conditioned by x only, namely:

ρ\( {η j,k }|x) = dy ρ y (y|x\ ) ρ( {η j,k }|y, x) (19)
We need to propose a model for ρ\( {η j,k }|x) so we can solve for ρ y (y|x).

The model

We propose the following model for ρ y (y|x):

ρ\( {η j,k }|x) = κ ⎛ ⎝ j,k ρ η \(η j,k ) ⎞ ⎠ χ x ({η j,k }) (20)
where χ x ({η j,k }) is a set function (it takes only the values 0 and 1) that restricts the values of the variables η j,k to a subset of really accessible values. That is, we assume that the probabilities are independent but not every possible value of η j,k will be visited once x is fixed, which explains the normalization prefactor κ. Nevertheless, we assume that the pyramid is large enough not to modify the shape of the marginal distributions.

The solution

Recalling Eq. ( 19) and using Eq. ( 18) we have:

ρ\( {η j,k }|x) = dy ρ y (y|x) j,k δ ⎛ ⎝ η j,k - a j,k y + b j,k a j-1, k 2 y + b j-1, k 2 ⎞ ⎠ (21)
We will next show that the following function:

ρ y (y|x) = κ j,k ρ η ⎛ ⎝ a j,k y + b j,k a j-1, k 2 y + b j-1, k 2 ⎞ ⎠ (22)
verifies the proposed model, Equation ( 20). The values of η j,k actually visited are those of the shape

η j,k = a j,k y+b j,k a j-1, k 2 y+b j-1, k 2 j,k
where y can vary but must be the same for all η j,k in the same realization of the pyramid. Substituting Eq. ( 22) in Eq. ( 21) we have:

ρ\( {η j,k }|x) = κ dy j,k ρ η \(η j,k \ ) δ ⎛ ⎝ η j,k - a j,k y + b j,k a j-1, k 2 y + b j-1, k 2 ⎞ ⎠ (23)
If we assume that the pyramid is large enough so that the distribution of values η j,k across the pyramid is not dependent on the particular value of y considered, we can take the product of PDFs outside the integral and hence:

ρ\( {η j,k }|x) = κ ⎛ ⎝ j,k ρ η \(η j,k ) ⎞ ⎠ dy j,k δ ⎛ ⎝ η j,k - a j,k y + b j,k a j-1, k 2 y + b j-1, k 2 ⎞ ⎠ (24)
Notice that the remaining integral is a set function with support on the values of η j,k which are accessible only. Hence,

χ x ({η j,k }) = dy j,k δ ⎛ ⎝ η j,k - a j,k y + b j,k a j-1, k 2 y + b j-1, k 2 ⎞ ⎠ (25)
and so Eq. ( 20) follows. We conclude that Eq. ( 22) is the expression of the distribution of y conditioned by the past values x according to the cascade model.

Asymptotic limits and stability of the solution

The stochastic model given by Eq. ( 22) is very appealing, but its application to real data requires some care. First of all, let us realize that the model is based on the assumption that the wavelet used is optimal. This is a bit delicate for the case in which the denominators a j-1, k 2 y + b j-1, k 2 vanish, as for these, as shown by the conditioned histograms in Table 1, the cascade model is no longer valid. In fact, for those values of y making the denominator to vanish the numerator should also vanish, what implies

a j,k /a j-1, k 2 = b j,k /b j-1, k 2
. Hence, such a constraint should be implemented in an appropriate numerical scheme in order to obtain stable results.

Another interesting limit is when |y| → ∞. In that case, the solution collapses to a fixed value,

ρ y (y|x) → |y|→∞ κ j,k ρ η ⎛ ⎝ a j,k a j-1, k 2 ⎞ ⎠ (26)
In order to define an integrable distribution, ρ y (y|x) → |y|→∞ 0, so this has two consequences. First, there exists a finite maximum value η ∞ for the variable η, so ρ η \(η > η ∞ ) = 0. Second, for at least one j, k, we must have a j,k /a j-1, k 2 > η ∞ . The first condition is in fact trivially verified, as discussed in [START_REF] Turiel | Multifractal geometry in stock market time series[END_REF]Turiel et al. 2008b): a physical signal has always this finite maximum. The second property implies that a well-realized cascade must have one of the ratios a j,k /a j-1, k 2 large enough. This property can be used as a control check on the validity of the cascade model to a given case. Let us finally remark that if the two conditions are fulfilled hence the range of valid values of |y| is bounded, so there is a minimum and a maximum possible value of y.

Conditioned distribution and maximum likelihood forecasting

Since ρ y (y|x) takes into account the presence of the cascade, it should be a better predictor than just considering the marginal distribution of returns ρ r (r ) centered on the last point: ρ ŷ ( ŷ) = ρ r ( ŷx N ). 1 In Fig. 4 we show the distributions ρ y (y|x) and ρ ŷ ( ŷ) for a randomly chosen point of the TEF series. We can see that our modeled distribution, ρ y (y|x), is visibly skewed, meaning that the cascade at this point is making negative returns more likely than positive ones. We also see that this distribution is narrower than the return-derived distribution, i.e, for this point the cascade structure is implying a reduction of volatility. Similar results are seen in other points of all the series.

The presented model is not rigid in the sense that it does not forecast an exact future value, but a distribution of possible future values. This allows to forecast not only the most likely future value but also the instant volatility and higher-moment information conditioned by a given past. If we concentrate on the most probable value only, this is a maximum likelihood estimation of the future value. We have calculated these most likely future values sequentially and compared them to the actual observed values, all along the series and for all the series. Since volatility changes over time and also between series, we have calculated the difference between actual and predicted values, divided by the estimated instantaneous 1-day volatility. This quotient is around 0.85, similar for all the series (between 0.84 and 0.87). This means that the knowledge furnished by our model allows reducing expected risk in about 15% from an estimation of volatility at the same point.

Conclusions

The development of new tools designed to improve signal processing for self-similar scale-invariant systems such as multifractals is in continuous evolution. In this context, wavelets play a key role. An important concept in the description of multifractal systems is to know whether dynamical variables follow a cascade process in analogy to the case of Fully Developed Turbulence. Wavelet projections are appropriate to find and characterize such a cascade, because of their ability to tune the scale of changes in the variables. In this way, the wavelet coefficients of a signal governed by a cascade follow multiplicative relations able to characterize the cascade process and provide relevant information about its dynamics. The multiplicative factors that relate wavelet coefficients at consecutive scales are called cascade variables and are a representation basis of the signal. Let us remark that to deal appropriately with the cascade not any wavelet is efficient. In fact, only the optimal wavelet of the system ensures independence of the cascade variables across scales. Therefore, to find such a wavelet is extremely important due to its potential applications in encoding, forecasting and, in general, signal processing. Any other wavelet systematically induces a certain level of noise that perturbs the cascade relations. Unfortunately, the search for such a representation basis is a complex issue for real data.

In this paper we have addressed some of these important problems. First, we have shown that stock market time series can be described in terms of cascade processes. We have shown that both log-prices and volatility are multifractal observables following a multiplicative process across scales. This result reveals the existence of a hierarchical structure whose analysis can be relevant not only to improve the statistical characterization of variables but also to forecast them. To complete the analysis, we have provided a new criterion to measure the optimality degree of a wavelet in an accurate, robust and little data-demanding way, specially appropriate for real data consisting of limited datasets. With this optimality degree estimator we have found the closest to optimal wavelet among a bank of standard wavelets for the logarithm of quotation price (Battle-Lemarié 3) and the volatility series (Symlet 7).

We have also faced explicitly the problem of forecasting. In this context, we have provided a theoretical model able to compute the probability distribution of an unknown point of the series conditioned to the knowledge of previous events making use of the cascade properties and knowing the optimal wavelet basis. As far as we know, this is the first theoretical approach in these terms and opens a new door to the problem. We have shown that in practice the distribution evolves and changes width and skewness, i.e., at some points the cascade favors positive returns while in others it favors negative returns, or similarly it implies a volatility increase at some points and decrease in others. Additionally, a simple maximum likelihood estimator shows a discrepancy smaller than the volatility at the same point.

Fig. 1

 1 Fig. 1 Daily series of the logarithm of quotation price for Telefónica (TEF) between June 3rd 1996 and May 30th 2006

Fig. 2

 2 Fig.2Singularity spectra derived from returns (left) and volatilities (right). The absolute values of returns and the squares of volatilities have been averaged at three different time windows, i.e., they have been projected with a box wavelet. The scales (sizes of the windows) are as follows: +: 2 days; ×: 4 days; * :8 days. For simplicity, we use a box wavelet, as we have seen that the marginal distributions of singularity exponents, ρ h j,k , are the same with almost any wavelet (the optimal wavelet basis is relevant only for calculation of the parent/child joint distribution)

Fig. 4

 4 Fig. 4 Distribution ρ y (y|x) for the TEF series on February 1st 2002 (light/red), compared to the marginal distribution of returns added to the last point ρ ŷ ( ŷ) (dark/blue). The actual logarithm of price that day was 2.71 (circle)

With the marginal distribution of returns ρ r (r ) we can construct a simple predictor of y from the last price x N as follows: ŷ = x N + r , with a random return r chosen according to ρ r (r ). x N is the last element of vector x and hence the element just preceding y. Obviously, the distribution of ŷ is just the distribution ρ r shifted in its argument by -x N : ρ ŷ ( ŷ) = ρ r ( ŷx N ).

Acknowledgments This work is a contribution to OCEANTECH (PIF 2006 Project) and FIS2006-13321-CO2-01. O. Pont is funded by a Ph.D. contract from Generalitat de Catalunya.