From ea04084cf94aa3eda38f770142ec3059cb1954c7 Mon Sep 17 00:00:00 2001 From: Markus Bilz Date: Mon, 26 Jun 2023 07:18:28 +0200 Subject: [PATCH] =?UTF-8?q?Cleanup=F0=9F=A7=B9=20(#417)?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- ...360\237\215\225Application study notes.md" | 19 -- ...2\244Embeddings for tabular data notes.md" | 61 ++-- .../\360\237\224\232Discussion notes.md" | 36 +++ .../\360\237\224\242Basic rules notes.md" | 4 +- .../\360\237\224\242EMO rule notes.md" | 2 +- .../\360\237\224\242Hybrid rules notes.md" | 10 - .../\360\237\217\205Comparsion.md" | 1 - ...\237\217\205Results of semi-supervised.md" | 37 +-- .../\360\237\217\205Results of supervised.md" | 8 - .../\360\237\217\205Results.md" | 7 - .../\360\237\247\223Discussion.md" | 65 +---- .../\360\237\252\246graveyard of ideas.md" | 269 ------------------ .../@caoInformationalContentOption2005.md" | 13 + .../accuracies_classical.png" | Bin 74905 -> 0 bytes .../classical-ise-results.png" | Bin 91530 -> 0 bytes .../eff-spread-finucane.png" | Bin 127556 -> 0 bytes .../effective-spead-options.png" | Bin 33347 -> 0 bytes .../feature-and-feature-set-definition.png" | Bin 163205 -> 0 bytes .../galatica.png" | Bin 155106 -> 0 bytes .../hyperparameter-ft-transformer.png" | Bin 58979 -> 0 bytes .../positional_encoding.png" | Bin 41610 -> 0 bytes .../presentation-of-effective-spread.png" | Bin 125663 -> 0 bytes .../pseudocode-of-algorithms.png" | Bin 84629 -> 0 bytes .../rankwise-correlations.png" | Bin 68577 -> 0 bytes .../summarized-results.png" | Bin 137082 -> 0 bytes .../viz-robustness.png" | Bin 256310 -> 0 bytes reports/Content/bibliography.bib | 43 --- reports/Content/results.tex | 5 +- reports/thesis.tex | 2 +- 29 files changed, 87 insertions(+), 495 deletions(-) create mode 100644 "references/obsidian/\360\237\223\221notes/\360\237\224\232Discussion notes.md" delete mode 100644 "references/obsidian/\360\237\223\226chapters/\360\237\217\205Comparsion.md" delete mode 100644 "references/obsidian/\360\237\223\226chapters/\360\237\252\246graveyard of ideas.md" create mode 100644 "references/obsidian/\360\237\223\245Inbox/@caoInformationalContentOption2005.md" delete mode 100644 "references/obsidian/\360\237\226\274\357\270\217Media/accuracies_classical.png" delete mode 100644 "references/obsidian/\360\237\226\274\357\270\217Media/classical-ise-results.png" delete mode 100644 "references/obsidian/\360\237\226\274\357\270\217Media/eff-spread-finucane.png" delete mode 100644 "references/obsidian/\360\237\226\274\357\270\217Media/effective-spead-options.png" delete mode 100644 "references/obsidian/\360\237\226\274\357\270\217Media/feature-and-feature-set-definition.png" delete mode 100644 "references/obsidian/\360\237\226\274\357\270\217Media/galatica.png" delete mode 100644 "references/obsidian/\360\237\226\274\357\270\217Media/hyperparameter-ft-transformer.png" delete mode 100644 "references/obsidian/\360\237\226\274\357\270\217Media/positional_encoding.png" delete mode 100644 "references/obsidian/\360\237\226\274\357\270\217Media/presentation-of-effective-spread.png" delete mode 100644 "references/obsidian/\360\237\226\274\357\270\217Media/pseudocode-of-algorithms.png" delete mode 100644 "references/obsidian/\360\237\226\274\357\270\217Media/rankwise-correlations.png" delete mode 100644 "references/obsidian/\360\237\226\274\357\270\217Media/summarized-results.png" delete mode 100644 "references/obsidian/\360\237\226\274\357\270\217Media/viz-robustness.png" diff --git "a/references/obsidian/\360\237\223\221notes/\360\237\215\225Application study notes.md" "b/references/obsidian/\360\237\223\221notes/\360\237\215\225Application study notes.md" index 7710dd53..8db09909 100644 --- "a/references/obsidian/\360\237\223\221notes/\360\237\215\225Application study notes.md" +++ "b/references/obsidian/\360\237\223\221notes/\360\237\215\225Application study notes.md" @@ -50,11 +50,6 @@ Found at: https://berkeley-defi.github.io/assets/material/Bid-Ask-Spreads-Measur - “Although it is clear that the classification errors associated with LR's algorithm and the tick test are significant, it is important to consider whether em? pirical research employing these methods might be biased by the errors. Since these methods are most commonly applied in estimating effective bid-ask spreads and signed volume, the possibility that imperfectly inferring trade direction might bias empirical inferences is tested by comparing true effective spreads to effective spreads estimated using LR's method and the tick test, and by comparing true signed volume to signed volume estimated using the two methods of inferring trade direction. The sample of 144 firms is first divided into dollar volume deciles, and mean effective percentage bid-ask spreads are estimated for each decile. Effective spreads are measured as twice the percentage half-spread for each observation, (3) ESl7 = 2(l00)(Dit)(Sit-Mit)/Mih where Du = 1 for purchases and ? 1 for sales, Sit is the trade price for stock i at time r, and Mit is the midpoint of the quoted spread. Du is measured in three ways: using the true order direction from the TORQ order file, using LR's algorithm, and using the tick test.13 The true effective spreads, estimated effective spreads, and quoted spreads are presented in (see screenshot below).” (([[@finucaneDirectTestMethods2000]], p. 569) - ${ }^{13}$ Since LR's algorithm always assigns a direction of one to trades above the midpoint and $-1$ to trades below the midpoint, and since the effective spread for trades at the midpoint is equal to zero, an equivalent expression for the effective spread under LR's method is: $\mathrm{ES}_{i t}=2(100)\left|S_{i t}-M_{i t}\right| / M_{i t}$. ([[@finucaneDirectTestMethods2000]]569) -![[eff-spread-finucane.png]] - -![[effective-spead-options.png]] -(copied from [[@savickasInferringDirectionOption2003]]897) - **Calculate differences:** Panel A indicates improvements can be made when using different quote lags and/ or trade direction algorithms. Staying with the $1 / 8$ ths pricing regime, we see the EMO algorithm with no lag for benchmark quote assignment reduces biases to $6.6 \%$ $(=12.78 / 11.99-1)$. For most categories, the EMO algorithm outperforms the LR algorithm and the Lee/Ready estimator. (For each row in the table we have bolded the cell with the least bias.) (found in [[@petersonEvaluationBiasesExecution2003]] 278) **Why we use the midpoint:** “In the presence of trading frictions, the transaction price P typically differs from the fundamental value X. The effective spread quantifies the difference, and may be viewed as a premium paid for the service of immediacy in securities trading. The nominal effective spread is defined as S = 2D(P − X ), (1) where D is a direction of trade indicator taking the value +1 for buyer-initiated trades, and -1 for seller-initiated trades. The multiplication by two is for consistency with the quoted bid-ask spread (defined below for a hypothetical roundtrip trade). For ease of exposition, I suppress stock and time subscripts for all variables in this section. Because the fundamental value at the time of transaction is unobservable, the effective spread is estimated relative to a proxy, which I denote ̃ X. An effective spread estimator can then be defined as ̃ S = 2D(P − ̃ X). (2) Various fundamental value proxies are distinguished with the superscript v, ̃ Xv. For example, I denote the midpoint ̃ Xmid. Similarly, an effective spread estimator utilising the fundamental value estimator v is denoted ̃ Sv. The midpoint effective spread, as defined by Blume and Goldstein (1992) and Lee (1993) as well as in the RegNMS Rule 605, is thus denoted ̃ Smid . An effective spread estimator is unbiased if the expected difference between the expressions in (1) and (2) is zero. The expected difference is E( ̃ S − S) = 2E(D(X − ̃ X)), (3) implying that the effective spread estimator is unbiased if and only if D and (X − ̃ X ) are uncorrelated. This can be expected either if investors are unable to assess the sign of (X − ̃ X ), or if the liquidity demand elasticity is zero. Consider again the example in the introduction. Whe” ([[@hagstromerBiasEffectiveBidask2021]], p. 317) @@ -75,20 +70,6 @@ The true transaction cost is necessarily unobservable in real data. In many econ ## Results -“We repeated this analysis with our dataset from the Frankfurt Stock Exchange. The results are presented in columns 2 and 3 of Table 5. The bias is even more dramatic. The traditional spread estimate is, on average, about twice as large as the “true” spread.8 A Wilcoxon test rejects the null hypothesis of equal medians (p < 0.01). Despite the large differences, the correlation between the two spread estimates is very high (ρ= 0.96). The magnitude of the relative bias (i.e., the traditional spread estimate divided by the “true” spread) is strongly negatively related to the classification accuracy. The correlation is –0.84.” ([[@theissenTestAccuracyLee2000]], p. 12) - -“Our algorithm improves classification accuracy by selecting the best combination of the tick and quote rules. To examine whether this improvement impacts the applications that rely on trade classifications we calculate effective spreads and price impact of trade.10 7.1. Effective spread and price impact of trade Using the trade direction provided by all algorithms, we compute effective spreads and price impacts and compare them to the effective spreads and price impacts using actual trade direction. Effective spread and price impact are computed as Effective spread ¼ 2 I ðTransaction Price MidpointÞ; ð6Þ % Effective spread ¼ Effective spread Transaction Price ; ð7Þ PIi ¼ I ðM i;tþ10 M i;tÞ; ð8Þ 9 While these coefficients are statistically significant, the magnitudes of the marginal effects are small and one should interpret the economic significance of these results accordingly. 10 The accuracy of trade direction estimation is not important for estimating effective spreads when trades are executed at quote midpoints at which points effective spreads are zero. where PIi is price impact for trade i at time t, I is trade indicator which equals 1 (1) for buy (sell) orders, Mi,t is quote midpoint at the time of trade, and Mi,t+10 is the quote midpoint 10 min after the trade. Table 6 Panel A shows that our algorithm provides the best estimate of effective spread. We conduct a t-test for difference in means to assess whether the effective spread of each algorithm is statistically significantly different from actual effective spread. Results indicate that the effective spread provided by our algorithm is a statistically significant unbiased estimate of the actual effective spread while the LR (EMO) rule provides upwardly (downwardly) biased estimates. Table 6 Panel B shows that the other algorithms provide biased estimates while our alternative algorithm provides statistically insignificant difference from the actual price impact. The results show that errors in trade side classification can result in substantial biased price impacts. The underestimations of price impacts are 5.26%, 29.47%, and 44.21% for the LR, EMO, and tick rule, respectively.“sing INET data, we assess the accuracy of the tick, LR, and EMO rules in identifying trade direction for NASDAQ stocks traded on an ECN. We show the limited success of these algorithms in classifying trade direction, especially for trades that occur inside the quotes. We propose an alternative algorithm that performs better than the current rules for classifying ECN trades, especially for trades inside the quotes.” ([[@chakrabartyTradeClassificationAlgorithms2007]], p. 3820) - -“The overall success rates of the LR, EMO, and tick rules are 74.42%, 75.80%, and 75.40%, respectively. Our Table 6 Application of trade classification algorithms: Effective spreads (May/June sample) Effective spread (in $) Mean differences t-Statistics % Effective spread Mean differences t-Statistics Panel A: Effective spreads ACTUAL 0.0343 0.0021 LR 0.0527 0.0184*** 21.45 0.0032 0.0011*** 27.92 EMO 0.0330 0.0014*** 7.74 0.0020 0.0001*** 26.4 TICK 0.0238 0.0105*** 11.87 0.0014 0.0006*** 10.1 Our alternative 0.0344 0.0001 0.49 0.0021 0.0000 0.37 Price impacts (in $) Mean differences t-Statistics Panel B: Price impacts ACTUAL 0.0095 LR 0.0090 0.0005*** 6.43 EMO 0.0067 0.0029*** 15.29 TICK 0.0053 0.0042*** 21.64 Our alternative 0.0091 0.0004 1.49 In this table we use the buy and sell directions proposed by each algorithm to compute the effective spreads and price impact of trade. We compare these measures to effective spreads and price impacts using the actual buy and sell directions. Mean differences are the effective spread of each algorithm minus the actual effective spread. t-statistics for mean differences are also provided for comparison. The effective spread is computed both in $ and percentage terms. *** 1% significance level. ([[@chakrabartyTradeClassificationAlgorithms2007]], p. 3820) - -![[presentation-of-effective-spread.png]] - -**Different take on effective spread, price impact** - -To analyse how the plus in classification accuracy translates into dollar values, I apply the different algorithms to the measurement of liquidity. The liquidity measures I consider are the dollar effective spread, the dollar price impact and the dollar realised spread for which the knowledge of the trade initiator plays a crucial role. The effective spread is defined as DESi = 2oi(pi − mi) where pi is the price per share of the i-th trade, mi is the spread mid-point associated with the i-th trade and oi is the direction of the trade initiator, that is +1 in case of buyer-initiated trades and −1 in case of seller-initiated trades. The effective spread measures the costs incurred by liquidity demanders relative to the ideal environment where trades execute at the mid-point. Opposite to the costs of liquidity demanders are the profits of liquidity suppliers. These gains are usually measured by the realised spread which subtracts the price impact from the effective spread, since the price impact is detrimental to the liquidity provider’s profits if prices move in the direction of the trade initiator. The price impact is given by DP Ii = 2oi(mi+∆ − mi) where mi+∆ is the mid-point at ∆ units, here chosen to be 10 minutes, after the i-th trade and the realised spread is, thus, DRSi = DESi − DP Ii = 2oi(pi − mi+∆). For each stock-day I compute the volume weighted averages L = P i viLi/V for 27 the respective liquidity measure Li ∈ {DESi , DP Ii , DRSi}. I compare the measures computed from the true trade initiator label and the knowledge of which mid-point belongs to which trade, with the measures computed using the estimated trade initiator label and the associated mid-point provided by the algorithms. The setups for the application of the algorithms to the data are as in the previous section. The timestamps of the trades are either not affected by noise or they are delayed by exponential noise with intensity β = 10−j with j = 2, 3. The precision of the timestamp is reduced to 10−i for i = 0, . . . , 3. Figure 9 shows the root-mean-square error and the estimated mean across all stock-days measured in cents.21 The red line in the graphs for the mean indicates the mean measured from the true trade initiator label. (https://dauphine.psl.eu/fileadmin/mediatheque/chaires/fintech/articles/1_UPDATE_Simon_Jurkatis_YFS2019.pdf) - - -All of the standard output-based trade direction algorithms result in effective spread estimates that are systematically biased when compared to the LM method. The LR technique and the EOM modification both overstate effective spreads while the tick test understates effective spreads. Thus, existing trade direction algorithms do not accurately measure transaction costs when trade liquidity is used to classify trade direction. (file:///C:/Users/Markus/Downloads/V12N12p3.pdf) Classification of the direction of trading naturally developed from the availability of data. Since each order is either a buy or a sell, observing the trade would logically follow the order and a simple view of the order-trade sequence was employed by researchers. The complexity is apparent when examining the priority of the order. Market orders have time priority and they result in a trade quickly at whatever price, thus these are the trades typically observed at the extremes of the quoted bid-ask spread. On the other hand, limit orders have price priority and they do not necessarily result in a trade. A natural improvement in determining trade direction is the use of order data. Order data can be linked to each trade and the resultant classifications determined from the source of the trade rather than an inference. Using the source of the trade requires an underlying definition of trade origination that may not be readily apparent in trade direction algorithms that use trade and quote data. This issue was identified by Odders-White (2000) where the **timing of the order** is used as the basis for determining the trade initiator. In this approach, whoever places the last order (buyer or seller) is assumed to be the trade initiator. While choosing a chronological definition because it is clear-cut and easy to apply, Odders-White recognises that an alternative method would classify the trade initiator as the investor who demands liquidity. Lee (1992) and Peterson and Fialkowski (1994) identify the active side of the transaction as another theoretical basis to classify the trade initiator. While these approaches identify an underlying definition in determining trade direction, none provides a practical means to implement the theory. diff --git "a/references/obsidian/\360\237\223\221notes/\360\237\222\244Embeddings for tabular data notes.md" "b/references/obsidian/\360\237\223\221notes/\360\237\222\244Embeddings for tabular data notes.md" index 9a26e99d..42f2ffe2 100644 --- "a/references/obsidian/\360\237\223\221notes/\360\237\222\244Embeddings for tabular data notes.md" +++ "b/references/obsidian/\360\237\223\221notes/\360\237\222\244Embeddings for tabular data notes.md" @@ -2,73 +2,68 @@ ![[Pasted image 20230126132519.png]] (found here: https://images.ctfassets.net/dkgr2j75jrom/A6Xf1MfISZhiQWuyGFDpV/708d5fd82c14d8db6eb0d4af1c27d525/PillarPage-Qual-Quan-3.svg) -standard basis ($\mathbf{e}_{1}$ etc. / orthogonal and equidistant -> found these properties in [[@cerdaEncodingHighcardinalityString2022]]) +standard basis ($\mathbf{e}_{1}$ etc. / orthogonal and equidistant -> found these properties in [[@cerdaEncodingHighcardinalityString2022]]) Since its introduction it has become ubiquitous in natural language processing ([[@lampleLargeMemoryLayers2019]], p. 3; ...). Its success for language representations has also led to adaptions for image representations ([[@parmarImageTransformer2018]], [[@carionEndtoEndObjectDetection2020]] (Visionformer of Dosovitsky and Liu et. al)) (found in [[@tayEfficientTransformersSurvey2022]]), as well as tabular data representations ([[@huangTabTransformerTabularData2020]], [[@somepalliSaintImprovedNeural2021]], [[@gorishniyRevisitingDeepLearning2021]]). -In NLP, these embeddings represent words, but the same ideas have been used to process [image patches](https://arxiv.org/abs/2010.11929), [protein sequences](https://www.pnas.org/content/118/15/e2016239118), [graphs](https://arxiv.org/abs/1710.10903), [database schema](https://arxiv.org/abs/2012.15355), [speech](https://arxiv.org/abs/1910.09799), and [time series](https://arxiv.org/abs/2001.08317). (Found at borealis ai) +In NLP, these embeddings represent words, but the same ideas have been used to process [image patches](https://arxiv.org/abs/2010.11929), [protein sequences](https://www.pnas.org/content/118/15/e2016239118), [graphs](https://arxiv.org/abs/1710.10903), [database schema](https://arxiv.org/abs/2012.15355), [speech](https://arxiv.org/abs/1910.09799), and [time series](https://arxiv.org/abs/2001.08317). (Found at borealis ai) +- Explain how embeddings can be implemented. (querying matrix with one-hot-vector) +Embeddings are a memory efficient way of representing one-hot encoded features that can also capture additional relationships between factors (or levels) of that variable. (https://medium.com/@michi.jeremias/embeddings-in-tabular-data-990202daa59f). See also their explanation on the relation between one-hot-encoding and embeddings. +“Encoding the Data In language models, all tokens are embedded using the same procedure. However, in the tabular domain, different features can come from distinct distributions, necessitating a heterogeneous embedding approach. Note that tabular data can contain multiple categorical features which may use the same set of tokens. Unless it is known that common tokens possess identical relationships within multiple columns, it is important to embed these columns independently. Unlike the embedding of TabTransformer[18], which uses attention to embed only categorical features, we propose also projecting continuous features into a d−dimensional space before passing their embedding through the transformer encoder.” ([Somepalli et al., 2021, p. 4](zotero://select/library/items/PCV7XCHY)) ([pdf](zotero://open-pdf/library/items/N8H76CQW?page=4&annotation=Z6DBZJTE)) -~~Feature embeddings are flexible way to to represent tabular data. Though typically, models assume a static table structure which restricts the transferability across tables [[@wangTransTabLearningTransferable]]~~. -- Explain how embeddings can be implemented. (querying matrix with one-hot-vector) -Embeddings are a memory efficient way of representing one-hot encoded features that can also capture additional relationships between factors (or levels) of that variable. (https://medium.com/@michi.jeremias/embeddings-in-tabular-data-990202daa59f). See also their explanation on the relation between one-hot-encoding and embeddings. -“Encoding the Data In language models, all tokens are embedded using the same procedure. However, in the tabular domain, different features can come from distinct distributions, necessitating a heterogeneous embedding approach. Note that tabular data can contain multiple categorical features which may use the same set of tokens. Unless it is known that common tokens possess identical relationships within multiple columns, it is important to embed these columns independently. Unlike the embedding of TabTransformer[18], which uses attention to embed only categorical features, we propose also projecting continuous features into a d−dimensional space before passing their embedding through the transformer encoder.” ([Somepalli et al., 2021, p. 4](zotero://select/library/items/PCV7XCHY)) ([pdf](zotero://open-pdf/library/items/N8H76CQW?page=4&annotation=Z6DBZJTE)) - +“The most common variable types in structured data are continuous variables and discrete variables. Continuous variables such as temperature, price, weight can be represented by real numbers. Discrete variables such as age, colour, bus line number can be represented by integers. Often the integers are just used for convenience to label the different states and have no information in themselves. For example if we use 1, 2, 3 to represent red, blue and yellow, one can not assume that ”blue is bigger than red” or ”the average of red and yellow are blue” or anything that introduces additional information based on the properties of integers. These integers are called nominal numbers. Other times there is an intrinsic ordering in the integer index such as age or month of the year. These integers are called cardinal number or ordinal numbers. Note that the meaning or order may not be more useful for the problem than only considering the” ([Guo and Berkhahn, 2016, p. 3](zotero://select/library/items/5CUI2BTM)) ([pdf](zotero://open-pdf/library/items/TMTTKAZP?page=3&annotation=BW76G7AJ)) + +“Therefore, naively applying neural networks on structured data with integer representation for category variables does not work well. A common way to circumvent this problem is to use onehot encoding, but it has two shortcomings: First when we have many high cardinality features one-hot encoding often results in an unrealistic amount of computational resource requirement. Second, it treats different values of categorical variables completely independent of each other and often ignores the informative relations between them.” ([Guo and Berkhahn, 2016, p. 1](zotero://select/library/items/5CUI2BTM)) ([pdf](zotero://open-pdf/library/items/TMTTKAZP?page=1&annotation=XLJYFFSJ)) +“We map categorical variables in a function approximation problem into Euclidean spaces, which are the entity embeddings of the categorical variables. The mapping is learnt by a neural network during the standard supervised training process. Entity embedding not only reduces memory usage and speeds up neural networks compared with one-hot encoding, but more importantly by mapping similar values close to each other in the embedding space it reveals the intrinsic properties of the categorical variables.” ([Guo and Berkhahn, 2016, p. 1](zotero://select/library/items/5CUI2BTM)) ([pdf](zotero://open-pdf/library/items/TMTTKAZP?page=1&annotation=CYY7CN4R)) - -“The most common variable types in structured data are continuous variables and discrete variables. Continuous variables such as temperature, price, weight can be represented by real numbers. Discrete variables such as age, colour, bus line number can be represented by integers. Often the integers are just used for convenience to label the different states and have no information in themselves. For example if we use 1, 2, 3 to represent red, blue and yellow, one can not assume that ”blue is bigger than red” or ”the average of red and yellow are blue” or anything that introduces additional information based on the properties of integers. These integers are called nominal numbers. Other times there is an intrinsic ordering in the integer index such as age or month of the year. These integers are called cardinal number or ordinal numbers. Note that the meaning or order may not be more useful for the problem than only considering the” ([Guo and Berkhahn, 2016, p. 3](zotero://select/library/items/5CUI2BTM)) ([pdf](zotero://open-pdf/library/items/TMTTKAZP?page=3&annotation=BW76G7AJ)) +“With entity embedding we want to put similar values of a categorical variable closer to each other in the embedding space. If we use a real number to define similarity of the values then entity embedding is closely related to the embedding of finite metric space problem in topology” ([Guo and Berkhahn, 2016, p. 4](zotero://select/library/items/5CUI2BTM)) ([pdf](zotero://open-pdf/library/items/TMTTKAZP?page=4&annotation=EEMP9PII)) -“Therefore, naively applying neural networks on structured data with integer representation for category variables does not work well. A common way to circumvent this problem is to use onehot encoding, but it has two shortcomings: First when we have many high cardinality features one-hot encoding often results in an unrealistic amount of computational resource requirement. Second, it treats different values of categorical variables completely independent of each other and often ignores the informative relations between them.” ([Guo and Berkhahn, 2016, p. 1](zotero://select/library/items/5CUI2BTM)) ([pdf](zotero://open-pdf/library/items/TMTTKAZP?page=1&annotation=XLJYFFSJ)) +“The investigation in (Grinsztajn et al., 2022) pointed out three inherent characteristics of tabular data that impeded known neural networks from top-tier performances, including irregular patterns of the target function, the negative effects of uninformative features, and the nonrotationally-invariant features. Based on this, we furthermore identify two points that highly promote the capabilities of neural networks on tabular data. (i) An appropriate feature embedding approach. Though it was demonstrated (Rahaman et al., 2019; Grinsztajn et al., 2022) that neural networks are likely to predict overly smooth solutions on tabular data, a deep learning model was also observed to be capable of memorising random labels (Zhang et al., 2021). Since the target function patterns are irregular and spurious correlations between the targets and features exist, an appropriate feature embedding network should well fit the irregular patterns while maintaining generalizability. (ii) A careful feature interaction approach. Since features of tabular data are non-rotationally-variant and a considerable portion of them are uninformative, it harms the generalisation when a model incorporates needless feature interactions.” ([Chen et al., 2023, p. 1](zotero://select/library/items/UKRXZCJB)) ([pdf](zotero://open-pdf/library/items/Q8RGMXPL?page=1&annotation=49WB9QDJ)) (Do not cite paper, as it seems to be rather low-quality otherwise; cite Gristajin instead) -“We map categorical variables in a function approximation problem into Euclidean spaces, which are the entity embeddings of the categorical variables. The mapping is learnt by a neural network during the standard supervised training process. Entity embedding not only reduces memory usage and speeds up neural networks compared with one-hot encoding, but more importantly by mapping similar values close to each other in the embedding space it reveals the intrinsic properties of the categorical variables.” ([Guo and Berkhahn, 2016, p. 1](zotero://select/library/items/5CUI2BTM)) ([pdf](zotero://open-pdf/library/items/TMTTKAZP?page=1&annotation=CYY7CN4R)) +“The deep component is a feed-forward neural network, as shown in Figure 1 (right). For categorical features, the original inputs are feature strings (e.g., “language=en”). Each of these sparse, high-dimensional categorical features are first converted into a low-dimensional and dense real-valued vector, often referred to as an embedding vector. The dimensionality of the embeddings are usually on the order of O(10) to O(100). The embedding vectors are initialised randomly and then the values are trained to minimise the final loss function during model training. These low-dimensional dense embedding vectors are then fed into the hidden layers of a neural network in the forward pass.” ([Cheng et al., 2016, p. 2](zotero://select/library/items/EZUKWCIB)) ([pdf](zotero://open-pdf/library/items/QGHKZ7UB?page=2&annotation=KS6UM9FZ)) -“With entity embedding we want to put similar values of a categorical variable closer to each other in the embedding space. If we use a real number to define similarity of the values then entity embedding is closely related to the embedding of finite metric space problem in topology” ([Guo and Berkhahn, 2016, p. 4](zotero://select/library/items/5CUI2BTM)) ([pdf](zotero://open-pdf/library/items/TMTTKAZP?page=4&annotation=EEMP9PII)) +“Encoding the Data In language models, all tokens are embedded using the same procedure. However, in the tabular domain, different features can come from distinct distributions, necessitating a heterogeneous embedding approach. Note that tabular data can contain multiple categorical features which may use the same set of tokens. Unless it is known that common tokens possess identical relationships within multiple columns, it is important to embed these columns independently. Unlike the embedding of TabTransformer[18], which uses attention to embed only categorical features, we propose also projecting continuous features into a d−dimensional space before passing their embedding through the transformer encoder.” ([Somepalli et al., 2021, p. 4](zotero://select/library/items/PCV7XCHY)) ([pdf](zotero://open-pdf/library/items/N8H76CQW?page=4&annotation=Z6DBZJTE)) -“The investigation in (Grinsztajn et al., 2022) pointed out three inherent characteristics of tabular data that impeded known neural networks from top-tier performances, including irregular patterns of the target function, the negative effects of uninformative features, and the nonrotationally-invariant features. Based on this, we furthermore identify two points that highly promote the capabilities of neural networks on tabular data. (i) An appropriate feature embedding approach. Though it was demonstrated (Rahaman et al., 2019; Grinsztajn et al., 2022) that neural networks are likely to predict overly smooth solutions on tabular data, a deep learning model was also observed to be capable of memorising random labels (Zhang et al., 2021). Since the target function patterns are irregular and spurious correlations between the targets and features exist, an appropriate feature embedding network should well fit the irregular patterns while maintaining generalizability. (ii) A careful feature interaction approach. Since features of tabular data are non-rotationally-variant and a considerable portion of them are uninformative, it harms the generalisation when a model incorporates needless feature interactions.” ([Chen et al., 2023, p. 1](zotero://select/library/items/UKRXZCJB)) ([pdf](zotero://open-pdf/library/items/Q8RGMXPL?page=1&annotation=49WB9QDJ)) (Do not cite paper, as it seems to be rather low-quality otherwise; cite Gristajin instead) -“The deep component is a feed-forward neural network, as shown in Figure 1 (right). For categorical features, the original inputs are feature strings (e.g., “language=en”). Each of these sparse, high-dimensional categorical features are first converted into a low-dimensional and dense real-valued vector, often referred to as an embedding vector. The dimensionality of the embeddings are usually on the order of O(10) to O(100). The embedding vectors are initialised randomly and then the values are trained to minimise the final loss function during model training. These low-dimensional dense embedding vectors are then fed into the hidden layers of a neural network in the forward pass.” ([Cheng et al., 2016, p. 2](zotero://select/library/items/EZUKWCIB)) ([pdf](zotero://open-pdf/library/items/QGHKZ7UB?page=2&annotation=KS6UM9FZ)) +“Effect of embedding continuous features To understand the effect of learning embeddings for continuous data, we perform a simple experiment with TabTransformer. We modify TabTransformer by embedding continuous features into d dimensions using a single layer ReLU MLP, just as they use on categorical features, and we pass the embedded features through the transformer block. We keep the entire architecture and all training hyper-parameters the same for both TabTransformer and its modified version. The average AUROC of the original TabTransformer is 89.38. Just by embedding the continuous features, the performance jumps to 91.72. This experiment shows that embedding the continuous data is important and can boost the performance of the model significantly.” ([Somepalli et al., 2021, p. 8](zotero://select/library/items/PCV7XCHY)) ([pdf](zotero://open-pdf/library/items/N8H76CQW?page=8&annotation=JK3TUUXE)) -> better performance. -> leave open why it works. [[@grinsztajnWhyTreebasedModels2022]] -“Encoding the Data In language models, all tokens are embedded using the same procedure. However, in the tabular domain, different features can come from distinct distributions, necessitating a heterogeneous embedding approach. Note that tabular data can contain multiple categorical features which may use the same set of tokens. Unless it is known that common tokens possess identical relationships within multiple columns, it is important to embed these columns independently. Unlike the embedding of TabTransformer[18], which uses attention to embed only categorical features, we propose also projecting continuous features into a d−dimensional space before passing their embedding through the transformer encoder.” ([Somepalli et al., 2021, p. 4](zotero://select/library/items/PCV7XCHY)) ([pdf](zotero://open-pdf/library/items/N8H76CQW?page=4&annotation=Z6DBZJTE)) - -“Effect of embedding continuous features To understand the effect of learning embeddings for continuous data, we perform a simple experiment with TabTransformer. We modify TabTransformer by embedding continuous features into d dimensions using a single layer ReLU MLP, just as they use on categorical features, and we pass the embedded features through the transformer block. We keep the entire architecture and all training hyper-parameters the same for both TabTransformer and its modified version. The average AUROC of the original TabTransformer is 89.38. Just by embedding the continuous features, the performance jumps to 91.72. This experiment shows that embedding the continuous data is important and can boost the performance of the model significantly.” ([Somepalli et al., 2021, p. 8](zotero://select/library/items/PCV7XCHY)) ([pdf](zotero://open-pdf/library/items/N8H76CQW?page=8&annotation=JK3TUUXE)) -> better performance. -> leave open why it works. [[@grinsztajnWhyTreebasedModels2022]] - +“Apart from model designs, various data representation approaches, such as feature embedding (Gorishniy et al., 2022), discretization of continuous features (Guo et al., 2021; Wang et al., 2020), and rule search approaches (Wang et al., 2021), were proposed against the irregular target patterns (Tancik et al., 2020; Grinsztajn et al., 2022).” ([Chen et al., 2023, p. 2](zotero://select/library/items/UKRXZCJB)) ([pdf](zotero://open-pdf/library/items/Q8RGMXPL?page=2&annotation=65MMYTUW)) -“Apart from model designs, various data representation approaches, such as feature embedding (Gorishniy et al., 2022), discretization of continuous features (Guo et al., 2021; Wang et al., 2020), and rule search approaches (Wang et al., 2021), were proposed against the irregular target patterns (Tancik et al., 2020; Grinsztajn et al., 2022).” ([Chen et al., 2023, p. 2](zotero://select/library/items/UKRXZCJB)) ([pdf](zotero://open-pdf/library/items/Q8RGMXPL?page=2&annotation=65MMYTUW)) +“To answer all those questions, we propose to relax fixed table structures by introducing a Transferable Tabular Transformer (TransTab) for tables. The goal of TransTab is to convert each sample (a row in the table) to a generalizable embedding vector, and then apply stacked transformers for feature encoding.” ([Wang and Sun, p. 1](zotero://select/library/items/38EXIFQ9)) ([pdf](zotero://open-pdf/library/items/C9P6BQ9N?page=1&annotation=IRSLFH22)). “The other insight is to introduce supervised and self-supervised pretraining to improve model performance. We compare TransTab with multiple baseline methods on diverse benchmark datasets and five oncology clinical trial datasets. Overall, TransTab ranks 1.00, 1.00, 1.78 out of 12 methods in supervised learning, feature incremental learning, and transfer learning scenarios, respectively; and the proposed pretraining leads to 2.3% AUC lift on average over the supervised learning.” ([Wang and Sun, p. 1](zotero://select/library/items/38EXIFQ9)) ([pdf](zotero://open-pdf/library/items/C9P6BQ9N?page=1&annotation=E6CC35EC)) [[@wangLearningDeepTransformer2019]] -“To answer all those questions, we propose to relax fixed table structures by introducing a Transferable Tabular Transformer (TransTab) for tables. The goal of TransTab is to convert each sample (a row in the table) to a generalizable embedding vector, and then apply stacked transformers for feature encoding.” ([Wang and Sun, p. 1](zotero://select/library/items/38EXIFQ9)) ([pdf](zotero://open-pdf/library/items/C9P6BQ9N?page=1&annotation=IRSLFH22)). “The other insight is to introduce supervised and self-supervised pretraining to improve model performance. We compare TransTab with multiple baseline methods on diverse benchmark datasets and five oncology clinical trial datasets. Overall, TransTab ranks 1.00, 1.00, 1.78 out of 12 methods in supervised learning, feature incremental learning, and transfer learning scenarios, respectively; and the proposed pretraining leads to 2.3% AUC lift on average over the supervised learning.” ([Wang and Sun, p. 1](zotero://select/library/items/38EXIFQ9)) ([pdf](zotero://open-pdf/library/items/C9P6BQ9N?page=1&annotation=E6CC35EC)) [[@wangLearningDeepTransformer2019]] +“Some previous approaches either designed feature embedding approaches (Gorishniy et al., 2022) to alleviate overly smooth solutions inspired by (Tancik et al., 2020) or employed regularisation (Katzir et al., 2020) and shallow models (Cheng et al., 2016) to promote the model generalisation, while some neural networks were equipped with sophisticated feature interaction approaches (Yan et al., 2023; Chen et al., 2022; Gorishniy et al., 2021) for better selectively feature interactions.” ([Chen et al., 2023, p. 1](zotero://select/library/items/UKRXZCJB)) ([pdf](zotero://open-pdf/library/items/Q8RGMXPL?page=1&annotation=X3Z257LE)) (pretty intuitive) -“Some previous approaches either designed feature embedding approaches (Gorishniy et al., 2022) to alleviate overly smooth solutions inspired by (Tancik et al., 2020) or employed regularisation (Katzir et al., 2020) and shallow models (Cheng et al., 2016) to promote the model generalisation, while some neural networks were equipped with sophisticated feature interaction approaches (Yan et al., 2023; Chen et al., 2022; Gorishniy et al., 2021) for better selectively feature interactions.” ([Chen et al., 2023, p. 1](zotero://select/library/items/UKRXZCJB)) ([pdf](zotero://open-pdf/library/items/Q8RGMXPL?page=1&annotation=X3Z257LE)) (pretty intuitive) +“Recent works enhance tabular ML modelling using deep networks [5, 6, 7, 8] or designing self-supervision [2, 9, 10, 11]. Those existing works require the same table structure in training and testing data. However, there can be multiple tables sharing partially overlapped columns in the real world. Hence, learning across tables is inapplicable.” ([Wang and Sun, p. 1](zotero://select/library/items/38EXIFQ9)) ([pdf](zotero://open-pdf/library/items/C9P6BQ9N?page=1&annotation=3CDUAZDS)) -“Recent works enhance tabular ML modelling using deep networks [5, 6, 7, 8] or designing self-supervision [2, 9, 10, 11]. Those existing works require the same table structure in training and testing data. However, there can be multiple tables sharing partially overlapped columns in the real world. Hence, learning across tables is inapplicable.” ([Wang and Sun, p. 1](zotero://select/library/items/38EXIFQ9)) ([pdf](zotero://open-pdf/library/items/C9P6BQ9N?page=1&annotation=3CDUAZDS)) +Reason why embeddings could work. “Why are MLPs much more hindered by uninformative features, compared to other models? One answer is that this learner is rotationally invariant in the sense of Ng [2004]: the learning procedure which learns an MLP on a training set and evaluate it on a testing set is unchanged when applying a rotation (unitary matrix) to the features on both the training and testing set. On the contrary, tree-based models are not rotationally invariant, as they attend to each feature separately, and neither are FT Transformers, because of the initial FT Tokenizer, which implements a pointwise operation theoretical link between this concept and uninformative features is provided by Ng [2004], which shows that any rotationallly invariant learning procedure has a worst-case sample complexity that grows at least linearly in the number of irrelevant features. Intuitively, to remove uninformative features, a rotationaly invariant algorithm has to first find the original orientation of the features, and then select the least informative ones: the information contained in the orientation of the data is lost.” ([Grinsztajn et al., 2022, p. 8](zotero://select/library/items/G3KP2Z9W)) ([pdf](zotero://open-pdf/library/items/A3KU4A43?page=8&annotation=W6LGGVAC)) -Reason why embeddings could work. “Why are MLPs much more hindered by uninformative features, compared to other models? One answer is that this learner is rotationally invariant in the sense of Ng [2004]: the learning procedure which learns an MLP on a training set and evaluate it on a testing set is unchanged when applying a rotation (unitary matrix) to the features on both the training and testing set. On the contrary, tree-based models are not rotationally invariant, as they attend to each feature separately, and neither are FT Transformers, because of the initial FT Tokenizer, which implements a pointwise operation theoretical link between this concept and uninformative features is provided by Ng [2004], which shows that any rotationallly invariant learning procedure has a worst-case sample complexity that grows at least linearly in the number of irrelevant features. Intuitively, to remove uninformative features, a rotationaly invariant algorithm has to first find the original orientation of the features, and then select the least informative ones: the information contained in the orientation of the data is lost.” ([Grinsztajn et al., 2022, p. 8](zotero://select/library/items/G3KP2Z9W)) ([pdf](zotero://open-pdf/library/items/A3KU4A43?page=8&annotation=W6LGGVAC)) +“Our findings shed light on the results of Somepalli et al. [2021] and Gorishniy et al. [2022], which add an embedding layer, even for numerical features, before MLP or Transformer models this layer breaks rotation invariance. The fact that very different types of embedding seem to improve performance suggests that the sheer presence of an embedding which breaks the invariance is a key part of these improvements. We note that a promising avenue for further research would be to find other ways to break rotation invariance which might be less computationally costly than embeddings.” ([Grinsztajn et al., 2022, p. 9](zotero://select/library/items/G3KP2Z9W)) ([pdf](zotero://open-pdf/library/items/A3KU4A43?page=9&annotation=3DIWPNHH)) -“Our findings shed light on the results of Somepalli et al. [2021] and Gorishniy et al. [2022], which add an embedding layer, even for numerical features, before MLP or Transformer models this layer breaks rotation invariance. The fact that very different types of embedding seem to improve performance suggests that the sheer presence of an embedding which breaks the invariance is a key part of these improvements. We note that a promising avenue for further research would be to find other ways to break rotation invariance which might be less computationally costly than embeddings.” ([Grinsztajn et al., 2022, p. 9](zotero://select/library/items/G3KP2Z9W)) ([pdf](zotero://open-pdf/library/items/A3KU4A43?page=9&annotation=3DIWPNHH)) +“4.3 Embedding Layer Since the feature representations of the categorical features are very sparse and high-dimensional, a common way is to represent them into low-dimensional spaces (e.g., word embeddings). Specifically, we represent each categorical feature with a low-dimensional vector, i.e., ei = Vixi, (2) where Vi is an embedding matrix for field i, and xi is an one-hot vector. Often times categorical features can be multi-valued, i.e., xi is a multi-hot vector. Take film watching prediction as an example, there could be a feature field Genre which describes the types of a film and it may be multi-valued (e.g., Drama and Romance for film “Titanic”). To be compatible with multi-valued inputs, we further modify the Equation 2 and represent the multi-valued feature field as the average of corresponding feature embedding vectors: ei = 1 q Vixi, (3) where q is the number of values that a sample has for i-th field and xi is the multi-hot vector representation for this field. To allow the interaction between categorical and numerical features, we also represent the numerical features in the same lowdimensional feature space. Specifically, we represent the numerical feature as em = vmxm, (4) where vm is an embedding vector for field m, and xm is a scalar value. By doing this, the output of the embedding layer would be a concatenation of multiple embedding vectors, as presented in Figure 2.” ([Song et al., 2019, p. 4](zotero://select/library/items/2PWVWL5T)) ([pdf](zotero://open-pdf/library/items/HBV6667L?page=4&annotation=UVTMKE9G)) -“4.3 Embedding Layer Since the feature representations of the categorical features are very sparse and high-dimensional, a common way is to represent them into low-dimensional spaces (e.g., word embeddings). Specifically, we represent each categorical feature with a low-dimensional vector, i.e., ei = Vixi, (2) where Vi is an embedding matrix for field i, and xi is an one-hot vector. Often times categorical features can be multi-valued, i.e., xi is a multi-hot vector. Take film watching prediction as an example, there could be a feature field Genre which describes the types of a film and it may be multi-valued (e.g., Drama and Romance for film “Titanic”). To be compatible with multi-valued inputs, we further modify the Equation 2 and represent the multi-valued feature field as the average of corresponding feature embedding vectors: ei = 1 q Vixi, (3) where q is the number of values that a sample has for i-th field and xi is the multi-hot vector representation for this field. To allow the interaction between categorical and numerical features, we also represent the numerical features in the same lowdimensional feature space. Specifically, we represent the numerical feature as em = vmxm, (4) where vm is an embedding vector for field m, and xm is a scalar value. By doing this, the output of the embedding layer would be a concatenation of multiple embedding vectors, as presented in Figure 2.” ([Song et al., 2019, p. 4](zotero://select/library/items/2PWVWL5T)) ([pdf](zotero://open-pdf/library/items/HBV6667L?page=4&annotation=UVTMKE9G)) - “Transformer-like architectures have a specific way to handle numerical features of the data. Namely, they map scalar values of numerical features to high-dimensional embedding vectors, which are then mixed by the self-attention modules.” ([Gorishniy et al., 2022, p. 1](zotero://select/library/items/V9AJAB5T)) ([pdf](zotero://open-pdf/library/items/YMZCLKEQ?page=1&annotation=NINXYJZY)) -“As another important finding, we demonstrate that the step of embedding the numerical features is universally beneficial for different deep architectures, not only for arXiv:2203.05556v2 [cs.LG] 15 Mar 202” ([Gorishniy et al., 2022, p. 1](zotero://select/library/items/V9AJAB5T)) ([pdf](zotero://open-pdf/library/items/YMZCLKEQ?page=1&annotation=7CKCYKGR)) +“As another important finding, we demonstrate that the step of embedding the numerical features is universally beneficial for different deep architectures, not only for arXiv:2203.05556v2 [cs.LG] 15 Mar 202” ([Gorishniy et al., 2022, p. 1](zotero://select/library/items/V9AJAB5T)) ([pdf](zotero://open-pdf/library/items/YMZCLKEQ?page=1&annotation=7CKCYKGR)) + +“To sum up, our contributions are as follows: 1. We show that embedding schemes for numerical features are an underexplored research question in tabular DL. Namely, we show that more expressive embedding schemes can provide substantial performance improvements over prior models. 2. We demonstrate that the profit from embedding numerical features is not specific for Transformer-like architectures, and proper embedding schemes benefit traditional models as well. 3. On a number of public benchmarks, we achieve the new state-of-the-art of tabular DL.” ([Gorishniy et al., 2022, p. 2](zotero://select/library/items/V9AJAB5T)) ([pdf](zotero://open-pdf/library/items/YMZCLKEQ?page=2&annotation=UI5Y5QHM)) -“To sum up, our contributions are as follows: 1. We show that embedding schemes for numerical features are an underexplored research question in tabular DL. Namely, we show that more expressive embedding schemes can provide substantial performance improvements over prior models. 2. We demonstrate that the profit from embedding numerical features is not specific for Transformer-like architectures, and proper embedding schemes benefit traditional models as well. 3. On a number of public benchmarks, we achieve the new state-of-the-art of tabular DL.” ([Gorishniy et al., 2022, p. 2](zotero://select/library/items/V9AJAB5T)) ([pdf](zotero://open-pdf/library/items/YMZCLKEQ?page=2&annotation=UI5Y5QHM)) +“the existing architectures (Gorishniy et al., 2021; Somepalli et al., 2021; Kossen et al., 2021; Song et al., 2019; Guo et al., 2021) construct embeddings for numerical features using quite restrictive parametric mappings, e.g., linear functions, which can lead to suboptimal performance.” ([Gorishniy et al., 2022, p. 1](zotero://select/library/items/V9AJAB5T)) ([pdf](zotero://open-pdf/library/items/YMZCLKEQ?page=1&annotation=MFFC6S4Z)) -“the existing architectures (Gorishniy et al., 2021; Somepalli et al., 2021; Kossen et al., 2021; Song et al., 2019; Guo et al., 2021) construct embeddings for numerical features using quite restrictive parametric mappings, e.g., linear functions, which can lead to suboptimal performance.” ([Gorishniy et al., 2022, p. 1](zotero://select/library/items/V9AJAB5T)) ([pdf](zotero://open-pdf/library/items/YMZCLKEQ?page=1&annotation=MFFC6S4Z)) - -“We stack these and apply an identical linear embedding to each of n datapoints, obtaining an input representation H(0) ∈ Rn×d×e (Fig. 2b).” ([Kossen et al., 2021, p. 3](zotero://select/library/items/WCUZUJHD)) ([pdf](zotero://open-pdf/library/items/DPPA9MFF?page=3&annotation=VKPT3UA2)) +“We stack these and apply an identical linear embedding to each of n datapoints, obtaining an input representation H(0) ∈ Rn×d×e (Fig. 2b).” ([Kossen et al., 2021, p. 3](zotero://select/library/items/WCUZUJHD)) ([pdf](zotero://open-pdf/library/items/DPPA9MFF?page=3&annotation=VKPT3UA2)) We are unable to find a seminal work that explicitly documents a theoretical justification for when encoding categorical variables with embedding layers can improve a model’s performance. Therefore, we see an opportunity for future work to provide this theoretical justification. ([[@hancockSurveyCategoricalData2020]]) -“Guo and Berkhahn employed an automatic technique to win third place in a 2015 Kaggle competition. The technique is known as entity embedding. Entity embedding is a method to embed categorical variables in real-valued vector spaces [3]. Guo and Berkhan’s success gives us an indication that neural networks are perhaps capable of operating on all sorts of qualitative data, if we can map that data to vectors we can use as input to neural networks effectively. In [3] Guo and Berkhahn do not reveal a new technique. In fact, we find that in the source code for their Kaggle competition entry [35] Guo and Berkhahn use the Keras library Embedding Layer function.” (Hancock and Khoshgoftaar, 2020, p. 27) (? Not sure if they deserve all the credit?) +“Guo and Berkhahn employed an automatic technique to win third place in a 2015 Kaggle competition. The technique is known as entity embedding. Entity embedding is a method to embed categorical variables in real-valued vector spaces [3]. Guo and Berkhan’s success gives us an indication that neural networks are perhaps capable of operating on all sorts of qualitative data, if we can map that data to vectors we can use as input to neural networks effectively. In [3] Guo and Berkhahn do not reveal a new technique. In fact, we find that in the source code for their Kaggle competition entry [35] Guo and Berkhahn use the Keras library Embedding Layer function.” (Hancock and Khoshgoftaar, 2020, p. 27) (? Not sure if they deserve all the credit?) -Unlike unstructured data found in nature, structured data with categorical features may not have continuity at all and even if it has it may not be so obvious. The continuous nature of neural networks limits their applicability to categorical variables. Therefore, naively applying neural networks on structured data with integer representation for category variables does not work well. A common way to circumvent this problem is to use onehot encoding, but it has two shortcomings: First when we have many high cardinality features one-hot encoding often results in an unrealistic amount of computational resource requirement. Second, it treats different values of categorical variables completely independent of each other and often ignores the informative relations between them. (Guo Berkhahn) \ No newline at end of file +Unlike unstructured data found in nature, structured data with categorical features may not have continuity at all and even if it has it may not be so obvious. The continuous nature of neural networks limits their applicability to categorical variables. Therefore, naively applying neural networks on structured data with integer representation for category variables does not work well. A common way to circumvent this problem is to use onehot encoding, but it has two shortcomings: First when we have many high cardinality features one-hot encoding often results in an unrealistic amount of computational resource requirement. Second, it treats different values of categorical variables completely independent of each other and often ignores the informative relations between them. (Guo Berkhahn) \ No newline at end of file diff --git "a/references/obsidian/\360\237\223\221notes/\360\237\224\232Discussion notes.md" "b/references/obsidian/\360\237\223\221notes/\360\237\224\232Discussion notes.md" new file mode 100644 index 00000000..8183d4b5 --- /dev/null +++ "b/references/obsidian/\360\237\223\221notes/\360\237\224\232Discussion notes.md" @@ -0,0 +1,36 @@ + +- moneyness / time-to-maturity / how do both relate with trade classification / motives +- low accuracy for index options + - Study sources of missclassification. See e. g., [[@savickasInferringDirectionOption2003]] + - The extent to which inaccurate trade classification biases empirical research dependes on whether misclassifications occur randomly or systematically [[@theissenTestAccuracyLee2000]]. This document also contains ideas how to study the impact of wrong classifications in stock markets. Might different in option markets. +- low accuracy for trades outside the quotes + - see also [[@ellisAccuracyTradeClassification2000]] for trades inside and outside the spread +- high gains for options for otm options and options with long maturity + - Accuracy is not the sole criterion. Depends on whether error is systematic or not. Thus, we do application study. See reasoning in ([[@theissenTestAccuracyLee2000]]) +- performance gap in classical rules +- strong performance of neural networks / tree-based ensembles + - We identify missingess in data to be down-ward biasing the results of classical estimators. ML predictors are robust to this missingness, as they can handle missing values and potentially substitute. +- methodology + - our study puts special emphasises on thoughtful tuning, data pre-processing. +- the elephant in the room: + - labelled data and cmputational data. + - Finetune. Low cost of inference +- which algorithm is no preferable? Do Friedman rank test + + +## Algorithm +2.3.7 How to Write the Discussion  Assessment of the results  Comparison of your own results with the results of other studies = Citation of already published literature!  Components  Principles, relationships, generalizations shown by the results = Discussion, not recapitulation of the results  Exceptions, lack of correlation, open points  Referring to published work: = Results and interpretations in agreement with or in contrast to your results  Our Recommendations: The writing of the chapter “Discussion” is the most difficult one. Compare your own data/results with the results from other already published papers (and cite them!). Outline the discussion part in a similar way to that in the Results section = consistency. Evaluate whether your results are in agreement with or in contrast to existing knowledge to date. You can describe why or where the differences occur, e.g. in methods, in sites, in special conditions, etc. Sometimes it is difficult to discuss results without repetition from the chapter “Results”. Then, there is the possibility to combine the “Results” and “Discussion” sections into one chapter. However, in your presentation you have to classify clearly which are your own results and which are taken from other studies. For beginners, it is often easier to separate these sections. + + +## Other +- calculate $z$-scores / $z$-statistic of classification accuracies to assess if the results are significant. (see e. g., [[@theissenTestAccuracyLee2000]]) +- provide $p$-values. Compare twitter / linkedin posting of S. Raschka on deep learning paper. +- When ranking algorithms think about using the onesided Wilcoxon signed-rank test and the Friedman test. (see e. g. , code or practical application in [[@perez-lebelBenchmarkingMissingvaluesApproaches2022]]) +- To test these hypotheses it would be best if we had the precise motivation behind the trades. While such analysis is not feasible here, using trade classification algorithms, we are able to assign stock and option volume as buyer or seller initiated. Easley et al. (1998) show how this directional volume is more informative than raw volume, because signed volume provides important information about the motivation of the trade (bullish or bearish). ([[@caoInformationalContentOption2005]]) +- [https://doi.org/10.1287/mnsc.2019.3398](https://doi.org/10.1287/mnsc.2019.3398) +- https://pdf.sciencedirectassets.com/271671/1-s2.0-S0304405X20X00067/1-s2.0-S0304405X19302831/am.pdf?X-Amz-Security-Token=IQoJb3JpZ2luX2VjEJT%2F%2F%2F%2F%2F%2F%2F%2F%2F%2FwEaCXVzLWVhc3QtMSJHMEUCIDixxfTiKliJIuzoOXxlII71RLwniTDskEPKeGqAyItEAiEA4%2Fytxevo9ZXJNkxW1jrTnKzaaobWySQgbq68siGmgwQquwUI7f%2F%2F%2F%2F%2F%2F%2F%2F%2F%2FARAFGgwwNTkwMDM1NDY4NjUiDLKVDK%2Bh4Dg%2B7qRvKCqPBUP%2BVpVVJhJWgxEXvneeMcgDHwz3Q%2BFj4BacIV3D2cLphWRHZirMPOW1Scz6VIzOfzGnUYgdZXRNb0yT8KQur9GvN%2B1TwgULtgOLUlII49PNpfnhgo%2B5TFji2%2BRpB4Bs7BoBu6JZH6x2vrhjfqFGSsl19%2Bsyxe3zfS%2BYZzLkEUBwXTVS0Omt3AWowOaltN5qRbzjH0M16ijT3HTA3BTtQJLZe%2BNqqKsohziXZJ2GIC0I%2BswnrB9qpx8TplWGO62ITP0I4Xa4F2GhzByCl2nrGKeHUdJ03VUa3dYpyw4ml8n3E7ADheEZh4yhh8W3GS%2Btc2AkrpJkl9JpInWeTwijmC5rsQVtRZfYLCNFXdSZkPtWFWOBYM0WVIiRHMx8urSTYs%2FQ5XiP61nmWn%2BlIdyeLDYgg8uYcBCwciMCBdfBKu86mAK42snqIIJC8fHQ6RjZ0HkTxXK3ecfWG9ZD5LYrwOig7B30VufNzSvG%2FnJ8UxeUOPfXcX9Ob8OEUuaWzvTCSU3%2BIsw8vx4%2BpScmof5EwYvWDb4ndAD02RdDsAps9DjoZT0Fo6ezxpGZYsSzJ%2FRvzXoxrKIhTVugO2%2BDJubQ9sHIex7HBmGf1dM3j56ypwqghzdFmDohh4bPT3oYbBkQkIeojKcuifG6RPtAROKxHdSv0Htm9LZrdehnkehKyeFESJ8pcZ9IrTP5sejH9%2BHrVo8m7gjUaYTw6vWdsQxw1dCZ96jSuoANt2O8QvxR5S%2BG078zV0yhx76Y1nuhz3Dzgk%2FJCwLUwklQcsGDNZzXKhuXoZkpyE1sHDgqeSwDU7xYdJEZnQy60exHcdnjh5qQt1cY3ZCb4EMH8Y4yUtDwJgf5YOlxHJo8rElAkn5T3e0w2YDWpAY6sQGhOGrkIFN%2FaJBDi7pCA1DemSn4hZ5FnHX1%2Fh492NDWmIx5ojFpVIHW5WftmRayIG6R4hzuIank8S0YgtSwnLUQpKMXKQEBeTmLwzHqVg7sHjO8dIJ0dtTywPmoP%2Fz7Op4wRbImbrCavUEUdf9bIt8ywHhHPJowJ9OnPNLByKti3DmdKLe%2FjO1iMpJGKTCwRu4WN9EDOsW3Jxo1xPUe3l2EUBbOB5oy6z3j8m1qo4bMbDE%3D&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Date=20230623T124336Z&X-Amz-SignedHeaders=host&X-Amz-Expires=300&X-Amz-Credential=ASIAQ3PHCVTYUUDJI6EP%2F20230623%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Signature=f0e2119f981b01f0c6f371ba9815e791e0ff23c077563ba5b6e75152e2e77385&hash=97254ce55dfdd1c8959a1ac605bad32a3be5f6fff1c70209a668432c2a4bbeff&host=68042c943591013ac2b2430a89b270f6af2c76d8dfd086a07176afe7c76c2c61&pii=S0304405X19302831&tid=pdf-81ea8b48-352c-43fe-bfe6-12fecbdb988f&sid=d1961e3d9967514b7918a74-b56e3090c4eagxrqb&type=client +- https://doi.org/10.1287/mnsc.2019.3529 +- https://www.dropbox.com/s/1i4zxc23qm00bv9/OptionMarketMakers.032623.pdf?dl=0 +- https://dmurav.com/CV_Dmitry_Muravyev_202305.pdf +- for index options see [[@chordiaIndexOptionTrading2021]] +- To test these hypotheses it would be best if we had the precise motivation behind the trades. While such analysis is not feasible here, using trade classification algorithms, we are able to assign stock and option volume as buyer or seller initiated. Easley et al. (1998) show how this directional volume is more informative than raw volume, because signed volume provides important information about the motivation of the trade (bullish or bearish). ([[@caoInformationalContentOption2005]]) \ No newline at end of file diff --git "a/references/obsidian/\360\237\223\221notes/\360\237\224\242Basic rules notes.md" "b/references/obsidian/\360\237\223\221notes/\360\237\224\242Basic rules notes.md" index 757ddd26..300aed85 100644 --- "a/references/obsidian/\360\237\223\221notes/\360\237\224\242Basic rules notes.md" +++ "b/references/obsidian/\360\237\223\221notes/\360\237\224\242Basic rules notes.md" @@ -30,11 +30,9 @@ While the information about the initator of a trade is missing in public data se “Methods of inferring trade direction can be classified as: tick tests, which use changes in trade prices; the quote method, which compares trade prices to quotes;” (Finucane, 2000, p. 557) +“Methods of inferring trade direction can be classified as: tick tests, which use changes in trade prices; the quote method, which compares trade prices to quotes;” (Finucane, 2000, p. 557) -“Methods of inferring trade direction can be classified as: tick tests, which use changes in trade prices; the quote method, which compares trade prices to quotes;” (Finucane, 2000, p. 557) - - **Synonyms** 1. Broader term is **trade site classification** = assign the side to a to a transaction and differentiate between buyer- and seller-initiated transactions diff --git "a/references/obsidian/\360\237\223\221notes/\360\237\224\242EMO rule notes.md" "b/references/obsidian/\360\237\223\221notes/\360\237\224\242EMO rule notes.md" index 4107befe..1306c245 100644 --- "a/references/obsidian/\360\237\223\221notes/\360\237\224\242EMO rule notes.md" +++ "b/references/obsidian/\360\237\223\221notes/\360\237\224\242EMO rule notes.md" @@ -16,7 +16,7 @@ Tags: #trade-classification #emo - **Effective spreads:** “However, the results in the previous sections indicate that the correct classification rate of the EMO technique is lower than that of the quote and the LR methods. Thus, the EMO method, like all the other approaches, must be used with care. It combines spread underestimation of the tick method with the overestimation of the quote method in such proportions as to yield spread estimates that happen to be close to the actual values.” ([[@savickasInferringDirectionOption2003]], 2003, p. 897) - **Effective spreads:** “A more substantial improvement arises when our algorithm is used for calculating effective spreads. In particular, we find that using the LR algorithm significantly overstates the size of the effective spread in the Nasdaq market.2 Using the correct buy/sell indicator from the Nasdaq audit data, we calculate an average effective spread of 1.34%, whereas using the LR classifications results in an effective spread of 1.96%.” ([Ellis et al., 2000, p. 531](zotero://select/library/items/54BPHWMV)) ([pdf](zotero://open-pdf/library/items/TTB4YUW6?page=4&annotation=9VPBNHNN)) - **Limitations:** “If the second trade of the day takes place at the same price as the first one, both trades cannot be classified by the tick rule. The LR and EMO rules also lose some trades since they are combinations of the tick and the quote rules.” (Savickas and Wilson, 2003, p. 886) -> Wouldn't this be filled by the prev. trade that is different, even if it is from long ago? -- (for a short discussion see [[@carrionTradeSigningFast2020]]) +- (for a short discussion see [[@carrionTradeSigningFast2020]]) **Notation:** Turn it into simple formula, instead of lengthy algorithm diff --git "a/references/obsidian/\360\237\223\221notes/\360\237\224\242Hybrid rules notes.md" "b/references/obsidian/\360\237\223\221notes/\360\237\224\242Hybrid rules notes.md" index 1e079f6e..fa04dca7 100644 --- "a/references/obsidian/\360\237\223\221notes/\360\237\224\242Hybrid rules notes.md" +++ "b/references/obsidian/\360\237\223\221notes/\360\237\224\242Hybrid rules notes.md" @@ -15,16 +15,6 @@ Tags: #trade-classification “Sophisticated algorithms combine the quote and tick rule: Thus, trades at the midpoint are always classified by the tick rule, and trades at the best bid or ask are classified by the quote rule. The three most common algorithms differ in how they divide the remaining trades between quote and tick rule, as illustrated in Fig. 1.” ([Pöppe et al., 2016, p. 166](zotero://select/library/items/5A83SDDB)) ([pdf](zotero://open-pdf/library/items/4XIK47X6?page=2&annotation=4A3YAHN2)) - - - - - **Bridge to ML:** 🌉 Interestingly, researchers gradually segment the decision surface starting with quote and tick rule, continuing with LR, EMO and CLNV. This is very similar to what is done in a decision tree. Could be used to motivate decision trees. All the hybrid methods could be considered as an ensemble with some sophisticated weighting scheme (look up the correct term) -> In recommender the hybrid recommender is called switching. - Current hybrid approaches use stacking ([[@grauerOptionTradeClassification2022]] p. 11). Also, due to technical limitations. Why not try out the majority vote/voting classifier with a final estimator? Show how this relates to ML. - In stock markets applying those philtres i. e. going from tick and quote rule did not always improve classification accuracies. The work of [[@finucaneDirectTestMethods2000]] raises critique about it in the stock market. - -**Algorithms:** -![[pseudocode-of-algorithms.png]] -(found in [[@jurkatisInferringTradeDirections2022]]). Overly complex description but helpful for implementation? - -![[Pasted image 20230421141343.png]] \ No newline at end of file diff --git "a/references/obsidian/\360\237\223\226chapters/\360\237\217\205Comparsion.md" "b/references/obsidian/\360\237\223\226chapters/\360\237\217\205Comparsion.md" deleted file mode 100644 index fd9e641f..00000000 --- "a/references/obsidian/\360\237\223\226chapters/\360\237\217\205Comparsion.md" +++ /dev/null @@ -1 +0,0 @@ -tbd \ No newline at end of file diff --git "a/references/obsidian/\360\237\223\226chapters/\360\237\217\205Results of semi-supervised.md" "b/references/obsidian/\360\237\223\226chapters/\360\237\217\205Results of semi-supervised.md" index 850338e9..270f4df7 100644 --- "a/references/obsidian/\360\237\223\226chapters/\360\237\217\205Results of semi-supervised.md" +++ "b/references/obsidian/\360\237\223\226chapters/\360\237\217\205Results of semi-supervised.md" @@ -1,7 +1,5 @@ We compare the performance of pre-trained Transformers and self-trained gradient-boosting on the gls-ise and gls-cboe test set. Results are reported in cref-tab-semi-supervised-results. -![[results-semis-supervised.png]] - Identical to the supervised case, our models consistently outperform their respective benchmarks. Gradient boosting with self-training surpasses $\operatorname{gsu}_{\mathrm{small}}$ by percentage-3.35 on gls-ise and percentage-5.44 on gls-cboe in accuracy. Improvements for larger feature sets over $\operatorname{gsu}_{\mathrm{large}}$ are marginally lower to the supervised model and range between percentage-4.55 and percentage-7.44. The results do not support the hypothesis, that incorporating unlabelled trades into the training corpus improves the performance of the classifier. We explore this finding in detail. @@ -9,37 +7,4 @@ The results do not support the hypothesis, that incorporating unlabelled trades **Finding 5: Unlabelled Trades Provide Poor Guidance** todo() -To summarize, despite the significantly higher training costs, semi-supervised variants do not provide better generalisation performance than supervised approaches. We subsequently evaluate if semi-supervised learning improves robustness, if not performance. - - - -While the performance of semi-supervised classifiers is competitive to the of supervised - -Despite the strong performance of our classifiers, semi-supervised methods do not deliver - - -Overall, out-of-sample performance is lower than for the supervised variants. Th - -Accuracy is not the sole criterion. Depends on whether error is systematic or not. Thus, we do application study. See reasoning in \textcite{theissenTestAccuracyLee2000} - - -Use $t$-SNE to assess the output of the supervised vs. the semi-supervised train models. See [[@leePseudolabelSimpleEfficient]] and [[@banachewiczKaggleBookData2022]] for how to use it. -See [[@vandermaatenVisualizingDataUsing2008]] for original paper. -![[t-sne-map.png]] - - - - -- Results for random classifier -- What would happen if the classical rules weren't stacked? -- Confusion matrix -- ROC curve. See e. g., [this thread](https://stackoverflow.com/a/38467407) for drawing ROC curves - -![[visualise-classical-rules-vs-ml.png]] -(print heatmap with $y$ axis with ask, bid and mid, $x$-axis could be some other criteria e. g. the trade size or none. If LR rule was good fit for options, accuracy should be evenly distributed and green. Visualise accuracy a hue / colour) -- calculate $z$-scores / $z$-statistic of classification accuracies to assess if the results are significant. (see e. g., [[@theissenTestAccuracyLee2000]]) -- provide $p$-values. Compare twitter / linkedin posting of S. Raschka on deep learning paper. -- When ranking algorithms think about using the onesided Wilcoxon signed-rank test and the Friedman test. (see e. g. , code or practical application in [[@perez-lebelBenchmarkingMissingvaluesApproaches2022]]) -- Study removal of features with high degree of missing values with feature permutation. (see idea / code done in [[@perez-lebelBenchmarkingMissingvaluesApproaches2022]]) -- How do classical rules compare to a zero rule baseline? Zero rule baseline predicts majority class. (variant of the simple heuristic). The later uses simple heuristics to perform a heuristic. -- Compare against "existing solutions" e. g., LR algorithm, depth rule etc. \ No newline at end of file +To summarize, despite the significantly higher training costs, semi-supervised variants do not provide better generalisation performance than supervised approaches. We subsequently evaluate if semi-supervised learning improves robustness, if not performance. \ No newline at end of file diff --git "a/references/obsidian/\360\237\223\226chapters/\360\237\217\205Results of supervised.md" "b/references/obsidian/\360\237\223\226chapters/\360\237\217\205Results of supervised.md" index c6764e98..fe496ef2 100644 --- "a/references/obsidian/\360\237\223\226chapters/\360\237\217\205Results of supervised.md" +++ "b/references/obsidian/\360\237\223\226chapters/\360\237\217\205Results of supervised.md" @@ -20,8 +20,6 @@ footnote-what is link to accuracy todo-Discuss results Null hypothesis: A and B have the same error rate. -![[mcnemar-ise.png]] - Statistical significance is used to compare learning algorithms on achieved error rates. The three most frequently used statistical tests for comparing two learning algorithms [96], [97] are: 1) McNemar test [98]: denote the number of data instances misclassified by the first classifier and correctly classified by the second classifier by a, and denote b in the opposite way. The McNemar statistic is computed as M ¼ signða  bÞða  bÞ 2 =ða þ bÞ to test whether two classifiers perform equally well. The test follows the x2 distribution; 2) https://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=8496795 McNemar’s chi-squared (χ2 ) test (McNemar 1947) was used to statistically compare error matrices by testing for the marginal homogeneity between two classifiers. Marginal homogeneity refers to the equality (i.e. lack of statistically significant difference) in the overall distributions of row or column variables predicted by one algorithm compared to another. It is a simple yet powerful method to compare class-wise predictions between algorithms. The test is parametric, has a low type I error and consists of a straightforward formulation (Dietterich 1998; de Leeuw et al. 2006). https://www.tandfonline.com/doi/pdf/10.1080/15481603.2019.1650447?needAccess=true&role=button @@ -97,12 +95,6 @@ and estimate McNemar's This allows more detailed analysis than simply observing the proportion of correct classifications (accuracy). Accuracy will yield misleading results if the data set is unbalanced; that is, when the numbers of observations in different classes vary greatly. -![[confusion-matrix-ise.png]] -(ise) -![[confusion-matrix-cboe.png]] -(cboe) - - - The recommendation of the McNemar’s test for models that are expensive to train, which suits large deep learning models. - How to transform prediction results from two classifiers into a contingency table and how the table is used to calculate the statistic in the McNemar’s test. - How to calculate the McNemar’s test in Python and interpret and report the result. diff --git "a/references/obsidian/\360\237\223\226chapters/\360\237\217\205Results.md" "b/references/obsidian/\360\237\223\226chapters/\360\237\217\205Results.md" index 11209b59..15051e29 100644 --- "a/references/obsidian/\360\237\223\226chapters/\360\237\217\205Results.md" +++ "b/references/obsidian/\360\237\223\226chapters/\360\237\217\205Results.md" @@ -8,16 +8,12 @@ Our results are approximately similar to ([[@grauerOptionTradeClassification2022 From all rules, tick-based algorithms perform worst when applied to trade prices at the trading venue with accuracies of a random guess, percentage-49.67 or percentage-51.47. For comparison, a simple majority vote would achieve percentage-51.40 accuracy. The application to trade prices at the inter-exchange level marginally improves over a random / dummy classification, achieving accuracies of percentage-55.25 for the reversed tick test. Due to the poor performance, of tick-based algorithms at the exchange level, we estimate all hybrids with $\operatorname{tick}_{\mathrm{all}}$ or $\operatorname{rtick}_{\mathrm{all}}$. -![[classical-ise-results.png]] - Quote-based algorithms, outperform tick-based algorithms delivering accuracy up to percentage-63.71, when estimated on the gls-NBBO. The superiority of quote-based algorithms in option trade classification has previously been documented in ([[@savickasInferringDirectionOption2003]]) and ([[@grauerOptionTradeClassification2022]]). The performance of hybrids, such as the gls-LR algorithm, hinges with the reliance on the tick test. Thus, the gls-emo rules and to a lesser extent the gls-clnv rules perform worst, achieving accuracies between percentage-55.42 and percentage-57.57. In turn, variants of the gls-LR, which uses the quote rule for most trades, is among the best performing algorithms. By extension, $\operatorname{gsu}_{\mathrm{small}}$ further reduces the dependence on tick-based methods through the successive applications of quote rules, here $\operatorname{quote}_{\mathrm{nbbo}} \to \operatorname{quote}_{\mathrm{ex}}$. Notably, the combination of ([[@grauerOptionTradeClassification2022]]33) including overrides from the trade size and depth rules performs best, achieving percentage-67.20 accuracy on the gls-ise test set and percentage-75.03 on the entire dataset. Yet, the performance deteriorates most sharply between sets. -![[accuracies_classical.png]] - **Finding 1: Accuracy of tick-based algorithms is downward-biased by missingness** - grauer et al trace back low accuracy of tick-based algorithms to illiquidity in option markets. - We do not have the time to previous trades. One would expect higher performance for more frequently traded options. Results of Grauer doe not indicate such a behaviour. @@ -48,9 +44,6 @@ We repeat the analysis on the gls-cboe dataset in cref-table-cboe and observe a For example, if the 50,000 transactions misclassi"ed by the Lee and Ready method constitute a representative cross-section of the entire sample, then the misclassi"cation will simply add noise to the data. In this case, the 85% accuracy rate is quite good. If, on the other hand, the Lee and Ready method systematically misclassi"es certain types of transactions, a bias could result. - -![[summarized-results.png]] - Our remaining analysis is focused on the test set. By extension, we also estimate rules combinations involving overrides from the tradesize rule ($\operatorname{tsize}$) and the depth rule ($\operatorname{depth}$) on the top-performing baselines of FS1. Consistent with the recommendation of ([[@grauerOptionTradeClassification2022]]14), we find that a deep combination of the $\operatorname{tsize}_{\text{ex}} \to \operatorname{quote}_{\text{nbbo}} \to \operatorname{quote}_{\text{ex}} \to \operatorname{depth}_{\text{nbbo}} \to \operatorname{depth}_{\text{ex}} \to \operatorname{rtick}_{\text{all}}$ achieves the highest validation. For brevity, we refer to this hybrid as the gls-GSU method. Much of the performance improvements is owed to the trade size and depth rules, which reduce the dependence on the reverse tick test as a last resort and provide overrides for trades at the quotes, improving validation accuracy to percent-68.8359. diff --git "a/references/obsidian/\360\237\223\226chapters/\360\237\247\223Discussion.md" "b/references/obsidian/\360\237\223\226chapters/\360\237\247\223Discussion.md" index a8fb86c7..0f17ee7b 100644 --- "a/references/obsidian/\360\237\223\226chapters/\360\237\247\223Discussion.md" +++ "b/references/obsidian/\360\237\223\226chapters/\360\237\247\223Discussion.md" @@ -1,64 +1,3 @@ -- [https://doi.org/10.1287/mnsc.2019.3398](https://doi.org/10.1287/mnsc.2019.3398) -- https://pdf.sciencedirectassets.com/271671/1-s2.0-S0304405X20X00067/1-s2.0-S0304405X19302831/am.pdf?X-Amz-Security-Token=IQoJb3JpZ2luX2VjEJT%2F%2F%2F%2F%2F%2F%2F%2F%2F%2FwEaCXVzLWVhc3QtMSJHMEUCIDixxfTiKliJIuzoOXxlII71RLwniTDskEPKeGqAyItEAiEA4%2Fytxevo9ZXJNkxW1jrTnKzaaobWySQgbq68siGmgwQquwUI7f%2F%2F%2F%2F%2F%2F%2F%2F%2F%2FARAFGgwwNTkwMDM1NDY4NjUiDLKVDK%2Bh4Dg%2B7qRvKCqPBUP%2BVpVVJhJWgxEXvneeMcgDHwz3Q%2BFj4BacIV3D2cLphWRHZirMPOW1Scz6VIzOfzGnUYgdZXRNb0yT8KQur9GvN%2B1TwgULtgOLUlII49PNpfnhgo%2B5TFji2%2BRpB4Bs7BoBu6JZH6x2vrhjfqFGSsl19%2Bsyxe3zfS%2BYZzLkEUBwXTVS0Omt3AWowOaltN5qRbzjH0M16ijT3HTA3BTtQJLZe%2BNqqKsohziXZJ2GIC0I%2BswnrB9qpx8TplWGO62ITP0I4Xa4F2GhzByCl2nrGKeHUdJ03VUa3dYpyw4ml8n3E7ADheEZh4yhh8W3GS%2Btc2AkrpJkl9JpInWeTwijmC5rsQVtRZfYLCNFXdSZkPtWFWOBYM0WVIiRHMx8urSTYs%2FQ5XiP61nmWn%2BlIdyeLDYgg8uYcBCwciMCBdfBKu86mAK42snqIIJC8fHQ6RjZ0HkTxXK3ecfWG9ZD5LYrwOig7B30VufNzSvG%2FnJ8UxeUOPfXcX9Ob8OEUuaWzvTCSU3%2BIsw8vx4%2BpScmof5EwYvWDb4ndAD02RdDsAps9DjoZT0Fo6ezxpGZYsSzJ%2FRvzXoxrKIhTVugO2%2BDJubQ9sHIex7HBmGf1dM3j56ypwqghzdFmDohh4bPT3oYbBkQkIeojKcuifG6RPtAROKxHdSv0Htm9LZrdehnkehKyeFESJ8pcZ9IrTP5sejH9%2BHrVo8m7gjUaYTw6vWdsQxw1dCZ96jSuoANt2O8QvxR5S%2BG078zV0yhx76Y1nuhz3Dzgk%2FJCwLUwklQcsGDNZzXKhuXoZkpyE1sHDgqeSwDU7xYdJEZnQy60exHcdnjh5qQt1cY3ZCb4EMH8Y4yUtDwJgf5YOlxHJo8rElAkn5T3e0w2YDWpAY6sQGhOGrkIFN%2FaJBDi7pCA1DemSn4hZ5FnHX1%2Fh492NDWmIx5ojFpVIHW5WftmRayIG6R4hzuIank8S0YgtSwnLUQpKMXKQEBeTmLwzHqVg7sHjO8dIJ0dtTywPmoP%2Fz7Op4wRbImbrCavUEUdf9bIt8ywHhHPJowJ9OnPNLByKti3DmdKLe%2FjO1iMpJGKTCwRu4WN9EDOsW3Jxo1xPUe3l2EUBbOB5oy6z3j8m1qo4bMbDE%3D&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Date=20230623T124336Z&X-Amz-SignedHeaders=host&X-Amz-Expires=300&X-Amz-Credential=ASIAQ3PHCVTYUUDJI6EP%2F20230623%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Signature=f0e2119f981b01f0c6f371ba9815e791e0ff23c077563ba5b6e75152e2e77385&hash=97254ce55dfdd1c8959a1ac605bad32a3be5f6fff1c70209a668432c2a4bbeff&host=68042c943591013ac2b2430a89b270f6af2c76d8dfd086a07176afe7c76c2c61&pii=S0304405X19302831&tid=pdf-81ea8b48-352c-43fe-bfe6-12fecbdb988f&sid=d1961e3d9967514b7918a74-b56e3090c4eagxrqb&type=client -- https://doi.org/10.1287/mnsc.2019.3529 -- https://www.dropbox.com/s/1i4zxc23qm00bv9/OptionMarketMakers.032623.pdf?dl=0 -- https://dmurav.com/CV_Dmitry_Muravyev_202305.pdf -- for index options see [[@chordiaIndexOptionTrading2021]] -- To test these hypotheses it would be best if we had the precise motivation behind the trades. While such analysis is not feasible here, using trade classification algorithms, we are able to assign stock and option volume as buyer or seller initiated. Easley et al. (1998) show how this directional volume is more informative than raw volume, because signed volume provides important information about the motivation of the trade (bullish or bearish). (cao paper) - -- see also [[@ellisAccuracyTradeClassification2000]] for trades inside and outside the spread - -- Whilst we reach the same conclusion, we estimate that large models should be trained for many more training tokens than recommended by the authors. - -- results for classical rules demonstrate that classical choices for option trade classification. -- We identify missingess in data to be down-ward biasing the results of classical estimators. ML predictors are robust to this missingness, as they can handle missing values and potentially substitute. - -- our study puts special emphasise on thoughtful tuning, data pre-processing. -- - -- the elephant in the room, labelled data and cmputational data. Finetune. Low cost of inference -- our results contradict ronen et al. neural networks can achieve sota performance if well tuned. - - -es it mean? Point out limitations and e. g., managerial implications or future impact. -- How do wide models compare to deep models -- Study sources of missclassification. See e. g., [[@savickasInferringDirectionOption2003]] -- Would assembeling help here? As discussed in [[@huangSnapshotEnsemblesTrain2017]] ensembles can only improve the model, if individual models have a low test error and if models do not overlap in the samples they missclassify. -- The extent to which inaccurate trade classification biases empirical research dependes on whether misclassifications occur randomly or systematically [[@theissenTestAccuracyLee2000]]. This document also contains ideas how to study the impact of wrong classifications in stock markets. Might different in option markets. -- Ceveat is that we don't know the true labels, but rather subsets. Could be biased? - -“The established methods, most notably the algorithms of Lee and Ready (1991) (LR), Ellis et al. (2000) (EMO), and Chakrabarty et al. (2007) (CLNV), classify trades based on the proximity of the transaction price to the quotes in effect at the time of the trade. This is problematic due to the increased frequency of order submission and cancellation. With several quote changes taking place at the time of the trade, it is not clear which quotes to select for the decision rule of the algorithm.” (Jurkatis, 2022, p. 6) - -To put these results in perspective, our best model using additional trade size and option features improves over the frequently employed tick rule, quote rule, and gls-lr algorithm by more than (74.12 - 57.10) on the gls-ISE sample. - -Cost of inference is low. Good practical use. - - - -We have proposed a series of analysis methods for understanding the attention mechanisms of models and applied them to BERT. While most recent work on model analysis for NLP has focused on probing vector representations or model outputs, we have shown that a substantial amount of linguistic knowledge can be found not only in the hidden states, but also in the attention maps. We think probing attention maps complements these other model analysis techniques, and should be part of the toolkit used by researchers to understand what neural networks learn about language. - -In this work, we have systematically evaluated typical pretraining objectives for tabular deep learning. We have revealed several important recipes for optimal pretraining performance that can be universally beneficial across various problems and models. Our findings confirm that pretraining can significantly improve the performance of tabular deep models and provide additional evidence that tabular DL can become a strong alternative to GBDT. - - -Discussion AlphaDev discovers new, state-of-the-art sorting algorithms from scratch that have been incorporated into the LLVM C++ library, used by millions of developers and applications around the world23–25. Both AlphaDev and stochastic search are powerful algorithms. An interesting direction for future research is to investigate combining these algorithms together to realize the complementary advantages of both approaches. It is important to note that AlphaDev can, in theory, generalize to functions that do not require exhaustive verification of test cases. For example, hashing functions48 as well as cryptographic hashing functions49 define function correctness by the number of hashing collisions. Therefore, in this case, AlphaDev can optimize for minimizing collisions as well as latency. AlphaDev can also, in theory, optimize complicated logic components within the body of large, impressive functions. We hope that AlphaDev can provide interesting insights and inspire new approaches in both the artificial intelligence and program synthesis communities. - -In this paper, we presented a series of language models that are released openly, and competitive with state-of-the-art foundation models. Most notably, LLaMA-13B outperforms GPT-3 while being more than 10× smaller, and LLaMA-65B is competitive with Chinchilla-70B and PaLM-540B. Unlike previous studies, we show that it is possible to achieve state-of-the-art performance by training exclusively on publicly available data, without resorting to proprietary datasets. We hope that releasing these models to the research community will accelerate the development of large language models, and help efforts to improve their robustness and mitigate known issues such as toxicity and bias. Additionally, we observed like Chung et al. (2022) that finetuning these models on instructions lead to promising results, and we plan to further investigate this in future work. Finally, we plan to release larger models trained on larger pretraining corpora in the future, since we have seen a constant improvement in performance as we were scaling. - -We are excited about the future of attention-based models and plan to apply them to other tasks. We plan to extend the Transformer to problems involving input and output modalities other than text and to investigate local, restricted attention mechanisms to efficiently handle large inputs and outputs such as images, audio and video. Making generation less sequential is another research goals of ours. - -The challenges encountered throughout this work can be revisited in future work. T - -n this thesis, we proposed different strategies to estimate the contrast of subspaces on a data stream in real-time. We introduced a new index structure allowing efficient insert- and delete-operations to improve the runtime of the contrast estimator. Building on this, we successfully extended GMD to the streaming setting. Our empirical results show that the contrast estimated by the Exponential Weighting Strategy is close to the contrast reported by the baseline algorithm while running three times faster. Our subspace search algorithm runs up to 30 times faster than the baseline algorithm and produces equally-good or better results than competitors when applied to downstream data analysis tasks, such as outlier detection. The results of our comprehensive experiments on real-world data sets indicate that outlier detection results in a streaming setting profit from combining the results obtained in individual high-contrast subspaces. In future work, we will investigate the effect of different update schemes running the subspace search only at every k-th time-step, which reduces the runtime further. - -In this work, we have systematically evaluated typical pretraining objectives for tabular deep learning. We have revealed several important recipes for optimal pretraining performance that can be universally beneficial across various problems and models. Our findings confirm that pretraining can significantly improve the performance of tabular deep models and provide additional evidence that tabular DL can become a strong alternative to GBDT. - - -2.3.8 How to Write the Conclusion  Answers to the questions formulated in the chapter Introduction  Same order as the questions  Recommendations  Theoretical implications  Possible practical applications = Including the significance of the work = Defining scientific truth - - -The challenges encountered throughout this work can be revisited in future work. This includes the Kafka configuration, Jaeger code instrumentation as well as the reproduction of the tracing data. To what extent different distributed tracing tools create different or the same results is another point for further investigation. The focus of the architecture extraction script introduced in section 6.2.1 are the span ids in order to make all communication relations visible. Other approaches taking single traces into account or considering the internal communications inside the microservices are possible avenues for future work. The use case project used in this work is written in Java and the MOM used is Kafka. The architecture extraction is based on JSON data as output from the distributed tracing tool and can be reused for other projects written in other programming languages. To which extent the instrumentation of the source code is suitable in other programming languages is also an open issue. The usage of other distributed tracing tools or projects utilizing a different type of MOM is another aspect that can be considered in future work. - - -2.3.7 How to Write the Discussion  Assessment of the results  Comparison of your own results with the results of other studies = Citation of already published literature!  Components  Principles, relationships, generalizations shown by the results = Discussion, not recapitulation of the results  Exceptions, lack of correlation, open points  Referring to published work: = Results and interpretations in agreement with or in contrast to your results  Our Recommendations: The writing of the chapter “Discussion” is the most difficult one. Compare your own data/results with the results from other already published papers (and cite them!). Outline the discussion part in a similar way to that in the Results section = consistency. Evaluate whether your results are in agreement with or in contrast to existing knowledge to date. You can describe why or where the differences occur, e.g. in methods, in sites, in special conditions, etc. Sometimes it is difficult to discuss results without repetition from the chapter “Results”. Then, there is the possibility to combine the “Results” and “Discussion” sections into one chapter. However, in your presentation you have to classify clearly which are your own results and which are taken from other studies. For beginners, it is often easier to separate these sections. - +**Notes:** +[[🔚Discussion notes]] diff --git "a/references/obsidian/\360\237\223\226chapters/\360\237\252\246graveyard of ideas.md" "b/references/obsidian/\360\237\223\226chapters/\360\237\252\246graveyard of ideas.md" deleted file mode 100644 index 3960b42a..00000000 --- "a/references/obsidian/\360\237\223\226chapters/\360\237\252\246graveyard of ideas.md" +++ /dev/null @@ -1,269 +0,0 @@ -## Feature Engineering -Our largest feature set als contains dates and times of the trade. In contrast to other continuous features, the features are inherently cyclic. We exploit this property for hours, days, and months and apply a fourier transform to convert the features into a smooth variable using formula [[#^773161]]: - -$$ -\begin{aligned} -x_{\sin} &= \sin\left(\frac{2\pi x}{\max(x)} \right), \text{and}\\ -c_{\cos} &= \cos\left(\frac{2\pi x}{\max(x)} \right), -\end{aligned} -$$ -^773161 -(There is close link to [[🧵Positional Embedding]]) and [[@tancikFourierFeaturesLet2020]] and [[@gorishniyEmbeddingsNumericalFeatures2022]] -%% -https://en.wikipedia.org/wiki/Sine_wave -Its most basic form as a function of time $(t)$ is: -$$ -y(t)=A \sin (2 \pi f t+\varphi)=A \sin (\omega t+\varphi) -$$ -where: -- A, amplitude, the peak deviation of the function from zero. -- $f$, ordinary frequency, the number of oscillations (cycles) that occur each second of time. -- $\omega=2 \pi f$, angular frequency, the rate of change of the function argument in units of radians per second. -- $\varphi$, phase, specifies (in radians) where in its cycle the oscillation is at $t=0$. -When $\varphi$ is non-zero, the entire waveform appears to be shifted in time by the amount $\varphi / \omega$ seconds. A negative value represents a delay, and a positive value represents an advance. -%% - -where $x$ is the raw input and $x_{\sin}$ and $x_{\cos}$ are the cyclical features. This cyclic continuous encoding, has the effect of preserving temporal proximity, as shown in Figure [[#^278944]]. As visualised for dates, the month's ultimo and the next month's first are close to each other in the individual features and on the unit circle. [^3] - -![[positional_encoding.png]] -(found here similarly: https://www.researchgate.net/figure/A-unit-circle-example-of-frequency-encoding-of-spatial-data-using-the-Fourier-series-a_fig2_313829438) ^278944 - - - - - -## Embeddings -This is similar to the work of ([[@easleyDiscerningInformationTrade2016]] 272), who modify the tick rule and GLSC-BVC algorithm to obtain probability estimates of a buy from individual or aggregated trades, but with a sole focus on trade signing on a trade-by-trade basis. - -## Applicaton study -See [[@jurkatisInferringTradeDirections2022]]. -Effective spread calculation. See e. g., [[@ellisAccuracyTradeClassification2000]]. - -“To give the improvement in classification accuracy more economic meaning, I apply the trade classification methods to the estimation of transaction costs. The transaction costs in turn are used in a portfolio optimisation exercise. The results show that an investor with a mean-variance utility function would be willing to forgo up to 33 bps on yearly returns to use the proposed algorithm to estimate transaction costs instead of the LR algorithm.” ([[@jurkatisInferringTradeDirections2022]], 2022, p. 7) - -## Transformer - -At times we fall back to the Transformer for machine translations, to develop a deeper understanding of the architecture and its components. - -## TabTransformer -Motivated by the success of contextual embeddings in natural language processing ([[@devlinBERTPretrainingDeep2019]]) ([[@liuRoBERTaRobustlyOptimized2019]]), ([[@huangTabTransformerTabularData2020]]4) propose with *TabTransformer* an adaption of the classical Transformer for the tabular domain. - -In large-scale experiments ([[@huangTabTransformerTabularData2020]]5--6) can show, that the use of contextual embeddings elevates both the robustness to noise and missing data of the model. For various binary classification tasks, the TabTransformer outperforms other deep learning models e. g., vanilla multi-layer perceptrons in terms of *area under the curve* (AUC) and can compete with [[🐈Gradient Boosting]]. - - - -# Feature Sets - -| Feature | Feature Category | Why? | FS 1 (Classical) | FS 2 (F1 + Grauer) | FS 4 (F3 + Others) | Transform | -|-----------------------|------------------------------|-----------------------------------------------------------------------------------------------------------------------------|------------------|--------------------|--------------------|-------------| -| TRADE_PRICE | tick rule | See [[@leeInferringTradeDirection1991]] | x | x | x | log | -| price_ex_lag | tick rule | See above. | x | x | x | log | -| price_all_lag | tick rule | See above. | x | x | x | log | -| chg_ex_lag | tick rule | See above. | x | x | x | standardise | -| chg_all_lag | tick rule | See above. | x | x | x | standardise | -| price_ex_lead | reverse tick rule | See above. | x | x | x | log | -| price_all_lead | reverse tick rule | See above. | x | x | x | log | -| chg_ex_lead | reverse tick rule | See above. | x | x | x | standardise | -| chg_all_lead | reverse tick rule | See above. | x | x | x | standardise | -| BEST_BID | quote rule | See above. | x | x | x | log | -| bid_ex | quote rule | See above. | x | x | x | log | -| BEST_ASK | quote rule | See above. | x | x | x | log | -| mid_ex | mid quote 🆕 | See above. | | | | log | -| mid_best | mid quote 🆕 | See above. | | | | log | -| ask_ex | quote rule | See [[@leeInferringTradeDirection1991]] | x | x | x | log | -| bid_ask_ratio_ex | Ratio of ask and bid 🆕 | ? | | x | x | standardise | -| spread_ex | Absolute spread 🆕 | ? | | | | standardise | -| spread_best | Absolute spread 🆕 | ? | | | | standardise | -| price_rel_nbb | Tradeprice rel to nbb 🆕 | Relates trade exchange with nation-wide best. | | x | x | standardise | -| price_rel_nbo | Tradeprice rel to nbo 🆕 | See above. | | x | x | standardise | -| prox_ex | EMO / CLNV | Most important predictor in [[@ellisAccuracyTradeClassification2000]] and [[@chakrabartyTradeClassificationAlgorithms2012]] | x | x | x | standardise | -| prox_best | EMO / CLNV | See above. | x | x | x | standardise | -| bid_ask_size_ratio_ex | Depth rule | See [[@grauerOptionTradeClassification2022]] | | x | x | standardise | -| bid_size_ex | Depth rule / Trade size rule | See above. | | x | x | standardise | -| ask_size_ex | Depth rule / Trade size rule | See above. | | x | x | standardise | -| rel_bid_size_ex | Trade size rule | See above. | | x | x | standardise | -| rel_ask_size_ex | Trade size rule | See above. | | x | x | standardise | -| TRADE_SIZE | Trade size rule | See above. | | x | x | standardise | -| STR_PRC | option | ? | | | x | log | -| day_vol | option | ? | | | x | log | -| bin_root | option 🦺(many `UNKWN`) | ? | | | x | binarize | -| time_to_maturity | option | ? | | | x | standardise | -| moneyness | option | ? | | | x | standardise | -| bin_option_type | option | ? | | | x | binarize | -| bin_issue_type | option | See [[@ronenMachineLearningTrade2022]]. Learn temporal patterns. Data is ordered by time. | | | x | binarize | -| date_month_sin | date | See above. | | | x | pos enc | -| date_month_cos | date | See above. | | | x | pos enc | -| date_day_sin | date | See above. | | | x | pos enc | -| date_day_cos | date | See above. | | | x | pos enc | -| date_weekday_sin | date | See above. | | | x | pos enc | -| date_weekday_cos | date | See above. | | | x | pos enc | -| date_time_sin | date | See above. | | | x | pos enc | -| date_time_cos | date | See above. | | | x | pos enc | -| date_year | date 🦺(uniformative) | See above. | | | | None | - -## LR algorithm -The algorithm is derived from an analysis of stock trades inside the quotes ([[@leeInferringTradeDirection1991]] 742). - -## Residual connections - - -## Positional Embedding -Positional embeddings are not the only way to fix the location, however. Later works, like ([[@daiTransformerXLAttentiveLanguage2019]]4--5), remove the positional encoding in favour of a *relative position encoding*, which is only considered during computation. - - -## Point-wise FFN -and ([[@gevaTransformerFeedforwardLayers2021]]). - -Later variants (see e. g., [[@devlinBERTPretrainingDeep2019]] or [[@radfordImprovingLanguageUnderstanding]]) commonly replace the $\operatorname{ReLU}$ with the *Gaussian Error Linear Units* $\operatorname{GELU}$ ([[@hendrycksGaussianErrorLinear2020]], p. 2) activation, which has empirically proven to improve the performance and convergence behaviour of Transformers ([[@narangTransformerModificationsTransfer2021]], p. 16; and [[@shazeerGLUVariantsImprove2020]] p. 4). - -Like the [[🅰️Attention]] sub-layer, the feed-forward sub-layer is surrounded by residual connections ([[@heDeepResidualLearning2015]]) and followed by a layer-normalisation ([[@baLayerNormalization2016]] (p. 4)) layer. - - -### Normality - -Test log-normality visually with qq-plots (https://stackoverflow.com/questions/46935289/quantile-quantile-plot-using-seaborn-and-scipy) or using statistical tests e. g., log-transform + normality test. https://stats.stackexchange.com/questions/134924/tests-for-lognormal-distribution - -- min-max scaling and $z$ scaling preserve the distribution of the variables (see [here.](https://stats.stackexchange.com/a/562204/351242)). Applying both cancels out each other (see proof [here.](https://stats.stackexchange.com/a/562204/351242)). - -There are controversies(Note zero imputation can be problematic for neural nets, as shown in [[@yiWhyNotUse2020]] paper) -- For imputation look into [[@perez-lebelBenchmarkingMissingvaluesApproaches2022]] -- [[@josseConsistencySupervisedLearning2020]] also compare different imputation methods and handling approaches of missing values in tree-based methods. -- for visualisations and approaches see [[@zhengFeatureEngineeringMachine]] and [[@butcherFeatureEngineeringSelection2020]] -- [[@yiWhyNotUse2020]] and [[@smiejaProcessingMissingData2018]] contain various references to papers to impute missing data in neural networks. -- add no missing indicator to keep the number of parameters small. - -- [[@lemorvanWhatGoodImputation2021]] for theoretical work on imputation. -- For patterns and analysis of imputed data see https://stefvanbuuren.name/fimd/ch-analysis.html - - - - %%we normalise all continous features into a range of $[-1,1]$ using formula [[#^5d5445]]: - -$$ -x^{\prime}=-1+\frac{2(x-\min (x))}{\max (x)-\min (x)} \tag{1} -$$ -$$ -X_{n o r m}=\frac{X-X_{\min }}{X_{\max }-X_{\min }} -$$ - -%% - - -- TODO: Why do we perform feature scaling at all? -- TODO: Try out robust scaler, as data contains outliers. Robust scaler uses the median which is robust to outliers and iqr for scaling. -- TODO: Try out different IQR thresholds and report impact. Similarily done here: https://machinelearningmastery.com/robust-scaler-transforms-for-machine-learning/ -- We scale / normalise features to a $\left[-1,1\right]$ scale using statistics estimated on the training set to avoid data leakage. This is also recommended in [[@huyenDesigningMachineLearning]]. Interestingly, she also writes that empirically the interval $\left[-1,1\right]$ works better than $\left[0,1\right]$. Also read about this on stackoverflow for neural networks, which has to do with gradient calculation. -- Scale to an arbitrary range $\left[a,b\right]$ using the formula from [[@huyenDesigningMachineLearning]]: -$$ -x^{\prime}=a+\frac{(x-\min (x))(b-a)}{\max (x)-\min (x)} -$$ -- Feature scaling theoretically shouldn't be relevant for gradient boosting due to the way gbms select split points / not based on distributions. Also in my tests it didn't make much of a difference for gbms but for transformers. (see https://github.com/KarelZe/thesis/blob/main/notebooks/3.0b-mb-comparsion-transformations.ipynb) -- [[@ronenMachineLearningTrade2022]] performed no feature scaling. -- [[@borisovDeepNeuralNetworks2022]] standardise numerical features and apply ordinal encoding to categorical features, but pass to the model which ones are categorical features. -- [[@gorishniyRevisitingDeepLearning2021]] (p. 6) use quantile transformation, which is similar to the robust scaler, see https://scikit-learn.org/stable/auto_examples/preprocessing/plot_all_scaling.html#sphx-glr-auto-examples-preprocessing-plot-all-scaling-pyf) Note that [[@grinsztajnWhyTreebasedModels2022]] only applied quantile transformations to all features, thus not utilise special implementations for categorical variables. - -# Random Forests - -As stated previously, decision trees suffer from a number of drawbacks. One of them being prone to overfitting. To mitigate the high variance, several trees can be combined to form an *ensemble* of trees. The final prediction is then jointly estimated from all member within the ensemble. - -Popular ensemble methods for trees include bagging and random forests [[@breimanRandomForests2001]]. Also, Boosting is another approach that learns a sequence base learners such as simplified decision trees. [[@hastietrevorElementsStatisticalLearning2009]] (p. 587) We present boosting as part of section [[🐈Gradient Boosting]]. - -Bagged trees and Random Forests have in common to learn $B$ independent trees. However, they differ in whether learning is performed on a random subset of data or whether splits consider only a portion of all features. - -Let's consider the regression case only. With bagging each tree is trained on a random subset of data, drawn with replacement from the training set. Learning a predictor on a so-called bootstraped sample, still causes the single tree to overfit. Especially, if trees are deep. Pruning the trees, selecting the best performing ones and averaging their estimates to a bagged predictor, helps to improve the accuracy [[@breimanBaggingPredictors1996]]. - -Besides this, averaging the estimates of several trees, bagging maintaines the desirable low-bias property of a single tree, assuming trees are grown large enough to capture subtleties of the data, while also improving on the variance. [[@hastietrevorElementsStatisticalLearning2009]] - -Yet, one issue bagging can not resolve is, that bagged trees are not independent [[@hastietrevorElementsStatisticalLearning2009]]. This is due to the limitation, that all trees select their best split attributes from the same set of features. If features dominate in the bootstrap samples, they will yield similiar splitting sequences and thus highly correlated trees. - -The variant of Bagging named *Random Forests* addresses the high correlation among trees, by considering only a random subset of all features for splitting. Random forests for regression, as introduced by [[@breimanRandomForests2001]] consist of $B$ trees, that are grown in parallel to form a forest. At each split only a random subset of $m$ features is considered for splitting. Typically, $m$ is chosen to be the $\sqrt{p}$ of all $p$ input variables [[@hastietrevorElementsStatisticalLearning2009]]. - -The random forest predictor is then estimated as the average overall the set of $\left\{T\left(x ; \Theta_{b}\right)\right\}_{1}^{B}$ trees: -$$ -\hat{f}_{\mathrm{rf}}^{B}(x)=\frac{1}{B} \sum_{b=1}^{B} T\left(x ; \Theta_{b}\right), -$$ - -with $\Theta_{b}$ being a parameter vector of the $b$-th tree [[@hastietrevorElementsStatisticalLearning2009]]. - -As the variables considered for splitting differ from one split and one tree to another, the trees are less similar and hence correlated. Random Forests achieve a comparable accuracy to Boosting or even outperform them. As trees do not depend on previously built trees, they can be trained in parallel. These advantages come at the cost of of lower interpretability compared to decision trees. [[@breimanRandomForests2001]] - -In the next section we discuss Boosting approaches, that grow trees in an adaptive manner. - - - - -A FFN tries to approximate an arbitrary function $f^{*}$. To do so, it defines a mapping $\boldsymbol{y}=f(\boldsymbol{x} ; \boldsymbol{\theta})$ from some input $\boldsymbol{x}$ to some output $\boldsymbol{y}$ and learns the parameters $\boldsymbol{\theta}$, that approximate the true output best. - -Structurally, a FFN consists of an input layer, one or more hidden layer and output layer. Thereby, each layer is made up of neurons and relies on input from the previous layer. In the most trivial case, the network consists of only a single hidden layer and the output layer. Formally, the output is calculated as shown in equation (...). $\mathbf{X} \in \mathbb{R}^{n \times d}$ denotes the input consisting of $d$ features and $n$ samples, $\mathbf{H} \in \mathbb{R}^{n \times h}$ the output of the hidden layer with $h$ hidden units and $\mathbf{O} \in \mathbb{R}^{n \times q}$ the final output. The weights and bias for the hidden layer and output layer are denoted by $\mathbf{W}^{(1)} \in \mathbb{R}^{d \times h}$ and biases $\mathbf{b}^{(1)} \in \mathbb{R}^{1 \times h}$ and output-layer weights $\mathbf{W}^{(2)} \in \mathbb{R}^{h \times q}$ and biases $\mathbf{b}^{(2)} \in \mathbb{R}^{1 \times q} .$ - -$$ -\begin{aligned} \mathbf{H} &=\sigma\left(\mathbf{X} \mathbf{W}^{(1)}+\mathbf{b}^{(1)}\right) \\ \mathbf{O} &=\mathbf{H} \mathbf{W}^{(2)}+\mathbf{b}^{(2)} \end{aligned} -$$ -As seen above, an affine transformation is applied to the input, followed activation function $\sigma(\cdot)$, that decides whether a neurone in the hidden layer is activated. The final prediction is then obtained after another affine transformation the output layer. Here, the parameter set consists of $\boldsymbol{\theta} = \left \{\mathbf{W}^{(1)}, \mathbf{b}^{(1)},\mathbf{W}^{(2)}, \mathbf{b}^{(2)} \right\}$. - -To learn the function approximation, FFNs are trained using backpropagation by adjusting the parameters $\boldsymbol{\theta}$ of each layer to minimise a loss function $\mathcal{L}(\cdot)$. As backpropagation requires the calculation of the gradient, both the activation and loss functions have to be differentiable. - -ReLU is a common choice. It's non-linear and defined as the element-wise maximum between the input $\boldsymbol{x}$ and $0$: - -$$ -\operatorname{ReLU}(\boldsymbol{x})=\max (\boldsymbol{x}, 0). -$$ - -The usage of ReLU as activation function is desirable for a number of reasons. First, it can be computated efficiently as no exponential function is required. Secondly, it solves the vanishing gradient problem present in other activation functions [[@glorotDeepSparseRectifier2011]]. - -Networks with a single hidden layer can approximate any arbitrary function given enough data and network capacity [[@hornikMultilayerFeedforwardNetworks1989]]. -In practise, similiar effects can be achieved by stacking several hidden layers and thereby deepening the network, while being more compact [[@zhangDiveDeepLearning2021]]. - -Deep neural nets combine several hidden layers by feeding the previous hidden layer's output into the subsequent hidden layer. Assuming a $\operatorname{ReLU}(\cdot)$ activation function, the stacking for a network with two hidden layers can be formalised as: $\boldsymbol{H}^{(1)}=\operatorname{ReLU}_{1}\left(\boldsymbol{X W}^{(1)}+\boldsymbol{b}^{(1)}\right)$ and $\boldsymbol{H}^{(2)}=\operatorname{ReLU}_{2}\left(\mathbf{H}^{(1)} \mathbf{W}^{(2)}+\mathbf{b}^{(2)}\right)$. - -Feed forward networks are restricted to information flowing through the network in a forward manner. To also incorporate feedback from the output, we introduce Recursive Neural Nets as part of section (...). - -## Intro -Moved to [[🥬SAGE values]]. - -Many model-agnostic methods are based on the randomly permuting features values. In this work, we specifically consider the variants *permutation feature importance* ([[@breimanRandomForests2001]]23--24) and partial-dependence plots ([[@friedmanGreedyFunctionApproximation2001]]26--28). Both serve a complementary purpose. Permutation feature importance derives the feature importance from the change in predictive accuracy before and after permuting a feature randomly, whereas partial dependence plots visualise the average change in prediction, if feature values are altered. These are widely adopted and computationally efficient. - -### Permutation feature importance -Permutation feature importance derives the importance from the mean decrease in accuracy before and after permuting a feature randomly. Expectedly, permuting features breaks the association with the target. Thus, the permutation of important features leads to a sharp decrease in accuracy, whereas unimportant features leave the accuracy unaffected. -The importance measure was originally proposed ([[@breimanRandomForests2001]]23--24) for random forests, and has later been extended by ([[@fisherAllModelsAre]]??) into a model-agnostic feature importance measure. - -Given our feature matrix $\mathbf{X}$, we can define a second permuted version, $\mathbf{X}^{\pi,j}$, where the $j$-th feature is randomly permuted by $\pi$. Using $L(y_i, h(\mathbf{x}_i))$ for predicting $y_i$ from $h(\mathbf{x}_{i})$, the importance of $j$-th feature is given by: -$$ -\operatorname{VI}^{\pi}_{j} = \sum_{i=1}^{N} L(y_{i}, h(\mathbf{x}_{i}^{\pi,j})) - L(y_{i}, h(\mathbf{x}_{i})), -$$ -which is the increase in loss, i.e., accuracy, before and after permutation ([[@hookerUnrestrictedPermutationForces2021]]82). While ([[@breimanRandomForests2001]]23--24) uses a single permutation, ([[@fisherAllModelsAre]]??) consider multiple, random permutations. By definition, random feature importance only yields global feature importances, as the measure is aggregated from all $N$ samples. - -### Extending permutation feature importance ✅ -Random feature permutation has the desirable properties of being easy to interpret, computationally efficient and model-agnostic. Like other feature importance measures, including SHAP or LIME, it assumes independence between features ([[@aasExplainingIndividualPredictions2021]]2). - -As defined in cref-eq-random-feature-permutation, every feature is permuted independently from other features which artificially breaks correlations between features and creates unrealistic feature combinations. Consider, for example, the apparent correlation between the ask, bid price and trade price. Permuting only the ask, could result in strongly negative or extremely large spreads, whereas bid and trade price remain unchanged. In effect, the presence of correlated features, leads to an overestimate of the importance of correlated features ([[@stroblConditionalVariableImportance2008]]3). - -Vice versa, can the presence of a correlated features decrease the importance of the associated feature, as the feature importance now distributed across the features, thereby underestimating the true importance of the features. This effects all features, where information is encoded redundantly, such as the bid-ask ratio. (footnote-for an extended discussion of substitution effects on feature importance in the financial domain see ([[@lopezdepradoAdvancesFinancialMachine2018]]114--118). - -To alleviate the bias from correlated / depenedent, we group dependent features and estimate the feature importance on the group-level. Arranging all features in a tree-like hierarchy gives us the freedom to derive feature importances at different levels, enabling cross-comparisons between classical rules and machine learning based classifiers, as grouping of raw and derived features makes the implementation of classical rules transparent. (footnote: Consider the implementation of the tick rule. Here, the implementation could use the feature price lag (ex) or calculate the price change from the trade price and price lag (ex). If not grouped, feature importances would be attributed to either the derived feature or raw features causing difficulties in comparison with machine learning classifiers, which have access to all three features simultaneously. Grouping all three features resolves this issue at the cost of interpretability.). Other than the classical permutation importance from cref-eq-random-feature-permutation, all features sharing the same parent node are permuted together. We define the following dependency structure: - -```mermaid - -graph TB -A((1))-->B((2)) -A-->C((3)) -A-->D((4)) -B-->E((5)) -B-->F((6)) -B-->G((7)) -C-->H((8)) -D-->I((9)) -D-->J((10)) -``` -Groupings are created to be mutually exclusive and based on the dependency structure of classical trade classification algorithms. The computational demand is comparable to classical feature permutation, as grouping results in fewer permutations, but the analysis may be repeated on several sub-levels. - -To this end, we want to emphasise, that our approach is different from ([[@ronenMachineLearningTrade2022]]52) as we do not estimate the improvement from adding new features, but keep the feature sets fixed-sized and permute them. - -### Partial dependence plots -Related to the concept of random feature permutation are partial dependence plots by ([[@friedmanGreedyFunctionApproximation2001]] 26--28) visualise the dependency between a single (or multiple) feature and the effect on the traget value as the feature value is adjusted / after marginalising out all other features. Let $\mathbf{X}^{x,j}$ be a matrix, where the $j$-th feature value is replaced by $x$ and all other features are unaltered, partial feature dependence function: -$$ -\operatorname{PD}_{j}(x) = \frac{1}{N} \sum_{i=1}^{N} f(\mathbf{x}^{x,j}_{i}), -$$ -now gives the marginal average over all other features ([[@hookerUnrestrictedPermutationForces2021]]81). By iterating over a grid of (observed) $x$, we obtain the partial dependence plot for the feature. - -Like random feature permutation, partial dependence plots are a global feature importance measure, unable to capture dependencies between features and visualisation is constrained to two dimensions or features at once ([[@hastietrevorElementsStatisticalLearning2009]] p. 388). Despite these limitation, partial dependence plots to help us verify the assumed relationships in classical rules, such as the the linear relationship in the tick rule, with the learnt relationships in our classifier. diff --git "a/references/obsidian/\360\237\223\245Inbox/@caoInformationalContentOption2005.md" "b/references/obsidian/\360\237\223\245Inbox/@caoInformationalContentOption2005.md" new file mode 100644 index 00000000..deee3a97 --- /dev/null +++ "b/references/obsidian/\360\237\223\245Inbox/@caoInformationalContentOption2005.md" @@ -0,0 +1,13 @@ +*title:* Informational Content of Option Volume Prior to Takeovers +*authors:* Charles Cao, Zhiwu Chen, John M. Griffin +*year:* 2005 +*tags:* +*status:* #📥 +*related:* +*code:* +*review:* + +## Notes 📍 + +## Annotations 📖 +Note: \ No newline at end of file diff --git "a/references/obsidian/\360\237\226\274\357\270\217Media/accuracies_classical.png" "b/references/obsidian/\360\237\226\274\357\270\217Media/accuracies_classical.png" deleted file mode 100644 index 91746dec49b34a16506117acf21e186ed9a698aa..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 74905 zcmYiM19T-_*98ivJ9ct9?AW&5v2EMNiP5ob=ft+zu{yRpw(UR9`~7#^J4V$g?OkK8 zT652}_O2DKASaFp_Y)2b3=C0HLPQA+?0WMPvRbb zT|k%%$_j#k)y2ZU8A5(t!#YT4I)i~B_5Zhj8@4Ys0R#Iik`xhCao0P`hIU66T^jxz zmA&jz{|@UP1O^Jy6a99a|2^NJcN5;VF7n9wb0DCGff%E*6h`W@D@2ZxDk$h~^Kp&E z?a}OVLYAZ#EJr`b!_L+0^z7oV-&9t5U08T{JmANVUPgre|28e!Wc?6Btp7HqfMDYP z-9s?Oz=-(2$5EOZl+6EkYGw!;`M*wS!T)bVeEd*DtOG#b?Cf_p8GPu}e@IeMIX_2# z=K4L--6+dP0L@~~BZ-#yg7}~x|MV9_J!5=4oi!X@9d-&Zo%LQ*PE0a)Y`n4-bDF?NyVE?|q6hs#L-< 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