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279Figure 20.1: Various estimates of the underlying sinusoidal model, f = sinFigure 20.2: Training MSE and Test MSE as a function of model flexibility.
280However if we plot the test MSE, which is given by the blue curve, the situation is quite different.The test MSE initially decreases as we increase the flexibility of the model but eventuallystarts to increase again after we introduce more flexibility. Why is this? By increasing the modelflexibility we are letting it fit to unique "patterns" in the training data.However as soon as we introduce new test data the model cannot generalise well because these"patterns" are only random artifacts of the training data and are not an underlying property ofthe true sinusoidal form. We are in a situation of overfitting. Colloquially, the model is fittingitself to the noise and not the signal.In fact this property of a "u-shaped" test MSE as a function of model flexibility is an intrinsicproperty of supervised machine learning models known as the bias-variance tradeoff.It can be shown (see below in the Mathematical Explanation section) that the expected testMSE, where the expectation is taken across many test sets, is given by:E(y 0 − ˆf(x 0 )) 2 = Var( ˆf(x[0 )) + Bias ˆf(x20 )]+ Var(ɛ) (20.7)The first term on the right hand side is the variance of the estimate across many testing sets.It determines how much the average model estimation deviates as different testing data are used.In particular a model with high variance is suggestive that it is overfit to the training data.The middle term is the squared bias, which characterises the difference between the averagesof the estimate and the true values. A model with high bias is not capturing the underlying behaviourof the true functional form well. One can imagine the situation where a linear regressionis used to model a sine curve (as above). No matter how well "fit" to the data the model is, itwill never capture the non-linearity inherant in a sine curve.The final term is known as the irreducible error. It is the minimum lower bound for thetest MSE. Since we only ever have access to the training data points (including the randomnessassociated with the ɛ values) we cannot ever hope to get a "more accurate" fit than what thevariance of the residuals offer.Generally as flexibility increases we see an increase in variance and a decrease in bias. Howeverit is the relative rate of change between these two factors that determines whether the expectedtest MSE increases or decreases.As flexibility is increased the bias will tend to drop quickly (faster than the variance canincrease) and so we see a drop in test MSE. However, as flexibility increases further, there isless reduction in bias (because the flexibility of the model can fit the training data easily) andinstead the variance rapidly increases, due to the model being overfit.Our ultimate goal in supervised machine learning is to try and minimise theexpected test MSE, that is we must choose a supervised machine learning modelthat simultaneously has low variance and low bias.If you wish to gain a more mathematically precise definition of the bias-variance tradeoff thenyou can read the following section, otherwise it may be skipped.A More Mathematical ExplanationWe have now qualitatively outlined the issues surrounding model flexibility, bias and variance. Inthis section we are going to carry out a mathematical decomposition of the expected prediction
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ContentsI Introduction 11 Introduct
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38.2 Correlation . . . . . . . . .
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513.2 The Kalman Filter . . . . . .
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719 Support Vector Machines . . . .
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928 Kalman Filter-Based Pairs Tradi
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Limit of Liability/Disclaimer ofWar
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Part IIntroduction1
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4The aim of this book is to provide
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61.3 How Is The Book Laid Out?The b
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81.5 How Does This Book Differ From
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101.7.1 AlternativesThere are many
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Chapter 2Introduction to Bayesian S
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15Table 2.1: Comparison of Frequent
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172.2 Applying Bayes’ Rule for Ba
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19the next chapter. At this stage,
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21# Create and plot a Beta distribu
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Chapter 3Bayesian Inference of a Bi
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25• The fairness of the coin does
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273.4.3 Multiple Flips of the CoinN
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29x = np.linspace(0, 1, 100)params
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31Hence, all we need to do is re-ar
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33Figure 3.3: The prior and posteri
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Chapter 4Markov Chain Monte CarloIn
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374.2.1 Markov Chain Monte Carlo Al
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39We discussed the fact that we cou
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41Finally we specify the Metropolis
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43we do not need to compute 100,000
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45x, stats.beta.pdf(x, alpha, beta)
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Chapter 5Bayesian Linear Regression
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49non-informative priors.• Poster
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51# Use a linear model (y ~ beta_0
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53# Use the No-U-Turn Samplerstep =
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55plt.show()We can see the sampled
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57"""Calculates the Markov Chain Mo
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Chapter 6Bayesian Stochastic Volati
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61plt.show()Figure 6.1: Different r
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63( )yilog ∼ t(ν, 0, exp(−2s i
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656.3.2 Model Specification in PyMC
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67Figure 6.5: Overlay of samples fr
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69plt.xlabel(’Time’)plt.ylabel(
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Part IIITime Series Analysishttps:/
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74• Seasonal Variation - Many tim
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767.5 How Does This Relate to Other
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78Definition 8.1.1. Expectation. Th
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808.2 CorrelationCorrelation is a d
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82Definition 8.3.4. Stationary in t
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84Figure 8.2: Correlogram plotted i
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868.6 Next StepsNow that we’ve di
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88• Use our knowledge of time ser
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909.3.2 CorrelogramWe can also plot
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929.4.2 CorrelogramThe autocorrelat
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94created the difference series, we
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96there are a few that are marginal
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9810.1 How Will We Proceed?In this
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10010.4.1 RationaleIn the previous
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10210.4.4 Simulations and Correlogr
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104Figure 10.2: Realisation of AR(1
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10610.4.5 Financial DataAmazon Inc.
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108Figure 10.6: Correlogram of Firs
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110> plot(gspcrt)Figure 10.8: First
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112model, as well as the conditiona
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114Figure 10.10: Realisation of MA(
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116Coefficients:ma1 intercept-0.729
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118> require(quantmod)> getSymbols(
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120> amznrt.maCall:arima(x = amznrt
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122Figure 10.16: Residuals of MA(1)
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124Figure 10.18: Residuals of MA(3)
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12610.6.3 RationaleTo date we have
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128Figure 10.20: Correlogram of an
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130Figure 10.22: Correlogram of an
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132Figure 10.23: Correlogram of the
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134Notice that there are some signi
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136I would like to thank you for be
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138Figure 11.1: Plot of simulated A
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140> amzn = diff(log(Cl(AMZN)))As i
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142We fit an ARIMA model by looping
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144being able to create strategy in
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14611.5.2 Why Does This Model Volat
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148ɛ t = σ t w t (11.17)σ 2 t =
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1502.5 % 97.5 %a0 0.1786255 0.21726
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152Figure 11.10: Residuals of an AR
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Figure 11.13: Squared residuals of
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156mean, but due to its stationarit
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158x t = pz t + w x,t (12.1)y t = q
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160Figure 12.3: Plot of x t and y t
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162> layout(1:2)> plot(badcomb, typ
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164We then created two subsequent s
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166Figure 12.6: Backward-adjusted c
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168Coefficients:(Intercept) ewaAdj3
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170> plot(rdsaAdj, rdsbAdj,xlab="RD
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172set.seed(123)## SIMULATED DATA##
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174plot(comb1$residuals, type="l",x
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176> q <- 0.6*z + rnorm(10000)> r <
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178Figure 12.12: Plot of s t , the
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180EWA.Adjusted.l2 EWC.Adjusted.l2
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182library("quantmod")library("tser
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184
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186• Smoothing - Estimating the p
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18813.2.1 A Bayesian ApproachRecall
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190E[y t+1 |D t ] = E[Ft+1θ T t +
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192of our linear regression. The ne
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194Figure 13.1: Scatterplot of the
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196Figure 13.2: Time-varying slope
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198delta = 1e-5trans_cov = delta /
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200
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202"risk manager". They will be use
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204Figure 14.1: Two-state Markov Ch
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206Figure 14.2: Hidden Markov Model
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208> bull_var <- 0.1> bear_mean <-
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210historical financial data.14.3.3
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212The same process will now be car
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214install.packages(’quantmod’)
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216
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Chapter 15Introduction to Machine L
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22115.3.1 Linear RegressionAn eleme
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223While seemingly not a traditiona
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Chapter 16Supervised LearningSuperv
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227ŷ = ˆf(x) = argmax z∈R p(y =
Page 242 and 243: Chapter 17Linear RegressionIn this
Page 244 and 245: 231if __name__ == "__main__":# Set
Page 246 and 247: 23317.3 Maximum Likelihood Estimati
Page 248 and 249: 235To simplify the notation the lat
Page 250 and 251: 237Once the full dataset is created
Page 252 and 253: 239if __name__ == "__main__":# Set
Page 254 and 255: 241))lr_model.coef_[0]# Create a sc
Page 256 and 257: Chapter 18Tree-Based MethodsIn this
Page 258 and 259: 245Figure 18.2: The resulting parti
Page 260 and 261: 24718.3 Decision Trees for Classifi
Page 262 and 263: 249In quantitative finance applicat
Page 264 and 265: 251(a) Grow a tree ˆf b with k spl
Page 266 and 267: 253tslag["Lag%s" % str(i+1)] = ts["
Page 268 and 269: 255The same approach is carried out
Page 270 and 271: 257seen whether it is feasible to p
Page 272 and 273: 259amzn = create_lagged_series("AMZ
Page 274 and 275: Chapter 19Support Vector MachinesIn
Page 276 and 277: 263• Non-Probabilistic - Since th
Page 278 and 279: 265The key point here is that it is
Page 280 and 281: 267classification. Overfitting mean
Page 282 and 283: 269Figure 19.6: Addition of a singl
Page 284 and 285: 27119.8 Support Vector MachinesThe
Page 286 and 287: 273p∑K(x i , x k ) = (1 + x ij x
Page 288 and 289: Chapter 20Model Selection andCross-
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Page 294 and 295: 281error for a particular model est
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Page 298 and 299: 28520.2.5 Python ImplementationWe a
Page 300 and 301: 287At this stage we have the necess
Page 302 and 303: 289..import pylab as plt....def plo
Page 304 and 305: 291fold = 0# Loop over the k-foldsf
Page 306 and 307: 293Figure 20.5: Test MSE curves for
Page 308 and 309: 295return tsretdef validation_set_p
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Page 314 and 315: Chapter 21Unsupervised LearningThe
Page 316 and 317: 303parameters of the model, θ.Conv
Page 318 and 319: Chapter 22Clustering MethodsIn this
Page 320 and 321: 3072. Iterate the following until c
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Page 324 and 325: 311clustered using K-Means and then
Page 326 and 327: 313ax.set_axis_bgcolor((1,1,0.9))ax
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Page 330 and 331: 317Figure 22.4: 3D scatterplot of n
Page 332 and 333: 319# Apply the K-Means Algorithm fo
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Page 336 and 337: 323sp500["ClusterMatrix"] = list(zi
Page 338 and 339: Chapter 23Natural Language Processi
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329for export as shippers are now e
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331tungtung-oilveg-oilwheatwoolwpiy
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333in order to minimise memory usag
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335’The Tokyo Grain Exchange said
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337for t in d[0]:if t in topics:d_t
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339This would mean that we are givi
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341partitions into the training and
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3430.660194174757[[21 0 0 0 2 3 0 0
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345self.encoding = encodingdef _res
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347"""topics = open("data/all-topic
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Part VQuantitative Trading Techniqu
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352While vectorised backtests are s
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354• Backtest - Encapsulates the
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356The second strategy provides a 3
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35825.4.1 MonthlyLiquidateRebalance
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360"""Size the order to reflect the
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362Figure 25.1: US Equities/Bonds 6
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364five years relative to US equiti
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366# Use the monthly liquidate and
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368import datetimefrom qstrader imp
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37026.2 Strategy ImplementationTo i
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372final.order[1],final.order[3]),
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374> spIntersect = merge( spArimaGa
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376enters what looks to be more a s
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378# Output the CSV file to "foreca
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380
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382natural gas.The hypothesis prese
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384## Carry out linear regression o
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38627.5 Python QSTrader Implementat
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388that arrive out of order in the
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390if self.invested is not None:if
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392take into account commission dif
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394adf.test(comb$residuals, k=1)# c
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396def go_short_units(self):"""Go s
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398from qstrader.trading_session.ba
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400default=’’,help=’Pickle (.
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402Hence we can "long the spread" i
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404class KalmanPairsTradingStrategy
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406if all(self.latest_prices > -1.0
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408It also changes the FixedPositio
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410’--testing/--no-testing’, de
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412• Asset Selection - Choosing a
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414if self.R is not None:self.R = s
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416)self.events_queue.put(SignalEve
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418@click.option(’--testing/--no-
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420• Short-Term Trend - Predict m
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422It takes the CSV file path as an
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424produce a final truth column whe
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426bias-variance trade-off of the m
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428used by the machine learning mod
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430IQFeedIntradayCsvBarPriceHandler
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432’--filename’,default=’’,
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Figure 29.2: Tearsheet for Random F
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436):lookforward_minutes=5,up_down_
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438#ts["UpDown"] = down_tot & up_to
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440events_queue - A handle to the s
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442from intraday_ml_strategy import
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444if __name__ == "__main__":main()
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446sentiment.In recent years there
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448Ticker Name Period LinkMSFT Micr
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450self.statistics.update(event.tim
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452if self.end_date is not None:sen
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454a sent_sell corresponding exit t
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456sentiment_handler = SentdexSenti
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458Figure 30.2:volatile, posting mo
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460fication by adding many more sto
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462events_queue = queue.Queue()csv_
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464def main(config, testing, ticker
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466In quantitative trading this pro
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468self.tickers[ticker]["timestamp"
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470df["Adj Close"][mask],".", lines
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472from qstrader.event import (Sign
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474class RegimeHMMRiskManager(Abstr
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476# If in the undesirable regime,
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47831.5 Strategy Results31.5.1 Tran
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480Figure 31.3: Trend Following Reg
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482ax.grid(True)plt.show()if __name
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484elif short_sma < long_sma and se
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486elif action == "SLD":if self.inv
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488backtest = Backtest(price_handle
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490holding an instrument representi
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492rolling_sharpe_s = np.sqrt(self.
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494In order to use the annualised r
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496Figure 32.2: Aluminum Smelting S
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498
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500[18] Wikipedia: Viterbi algorith
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502[56] Investor, S. Regime detecti
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504[92] Sinclair, E. Volatility Tra
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