018_LinRegIntro(Scala)

Import Notebook

ScaDaMaLe Course site and book

Let us visit an interactive visual cognitive tool for the basics ideas in linear regression:

The following video is a very concise and thorough treatment of linear regression for those who have taken the 200-level linear algebra. Others can fully understand it with some effort and revisiting.

Linear Regression by Ameet Talwalkar in BerkeleyX: CS190.1x Scalable Machine Learning

(watch now 11:13):

Linear Regression by Ameet Talwalkar in BerkeleyX: CS190.1x Scalable Machine Learning

Ridge regression has a Bayesian interpretation where the weights have a zero-mean Gaussian prior. See 7.5 in Murphy's Machine Learning: A Probabilistic Perspective for details.

Please take notes in mark-down if you want.

For latex math within markdown you can do the following for in-line maths: Ai,j∈R1 \mathbf{A}_{i,j} \in \mathbb{R}^1 . And to write maths in display mode do the following:

A∈Rm×d\mathbf{A} \in \mathbb{R}^{m \times d}

You will need to write such notes for your final project presentation!

MillonSongs Ridge Regression by Ameet Talwalkar in BerkeleyX: CS190.1x Scalable Machine Learning

(watch later 7:47):

Linear Regression by Ameet Talwalkar in BerkeleyX: CS190.1x Scalable Machine Learning

Covers the training, test and validation and grid search... ridger regression...

Take your own notes if you like.

Gradient Descent by Ameet Talwalkar in BerkeleyX: CS190.1x Scalable Machine Learning

(watch later 11:19):

Gradient Descent by Ameet Talwalkar in BerkeleyX: CS190.1x Scalable Machine Learning

Please take notes if you want to. Most of the above should be review to most of you.

Getting our hands dirty with Darren Wilkinson's blog on regression

Let's follow the exposition in:

You need to scroll down the fitst link embedded in iframe below to get to the section on Analysis of quantitative data with Descriptive statistics and Linear regression sub-sections (you can skip the earlier part that shows you how to run everything locally in spark-shell - try this on your own later).

Show code

Let's do this live now...

import breeze.stats.distributions._
def x = Gaussian(1.0,2.0).sample(10000)
val xRdd = sc.parallelize(x)
import breeze.stats.distributions._ x: IndexedSeq[Double] xRdd: org.apache.spark.rdd.RDD[Double] = ParallelCollectionRDD[752] at parallelize at command-2972105651606193:3 
println(xRdd.mean)
println(xRdd.sampleVariance)
1.0205021258164175 3.9857177279359335 
val xStats = xRdd.stats
xStats.mean
xStats.sampleVariance
xStats.sum
xStats: org.apache.spark.util.StatCounter = (count: 10000, mean: 1.020502, stdev: 1.996326, max: 8.089040, min: -6.775216) res1: Double = 10205.021258164175 
val x2 = Gaussian(0.0,1.0).sample(10000) // 10,000 Gaussian samples with mean 0.0 and std dev 1.0
val xx = x zip x2 // creating tuples { (x,x2)_1, ... , (x,x2)_10000}
val lp = xx map {p => 2.0*p._1 + 1.0*p._2 + 1.5} // { lp_i := (2.0*x + 1.0 * x2 + 1.5)_i, i=1,...,10000 }
val eps = Gaussian(0.0,1.0).sample(10000) // standrad normal errors
val y = (lp zip eps) map (p => p._1 + p._2)
val yx = (y zip xx) map (p => (p._1, p._2._1, p._2._2))
x2: IndexedSeq[Double] = Vector(-0.39294234164655323, 0.8555415681712383, 0.5047531440104407, -1.7447287479723848, 0.24074263385944478, -2.5060458696190544, -0.3625957248008947, -1.590842317493378, 0.6957126527559235, 0.4443439308434807, 1.0693145537223456, -1.2513129429033423, -0.35885347259074074, -0.4025589013089939, -0.5152842487123, 1.132054878042222, -0.5455845613656002, 0.047570983504133746, -0.49567564166101635, 1.0696261814411365, 0.05742293214978136, -0.34714185801134384, 1.8582008120255742, -0.6693454894738627, -0.24016699291536123, -0.12433010673208071, 0.862434789491131, -1.1316281054134634, -0.10575269488815854, 1.7745918273414025, -1.3061660094876109, -0.06572880649671899, 0.3873544789319996, -0.12210660688891804, -1.165943599251054, -0.36104975279667534, 0.42769260065418174, 2.5268542442818207, -0.4635835110880271, 0.5605742432439522, -0.18424533745017588, -0.6547651770934155, -0.3093631599212501, -1.6992194371388365, 1.0251808317424254, -0.49932778415411466, 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(1.7382278262107655,0.6911833508363038), (-1.9756499207390856,-1.1404580265857653), (2.2426105487932535,0.035760579673606574), (3.410303327117183,-0.043563376648401174), (-1.657936243193801,1.3953133669672373), (1.5904190641915654,-0.13154519383840338), (2.109846866051348,-0.5087661601151531), (1.7383262021843078,-0.13920955934160847), (0.5656777439959259,0.688437534855854), (0.5512630960372296,1.382020040244402), (1.6028860628432642,0.17362122242340142), (-1.796745067068136,-0.7252080007225248), (0.19781901056887174,0.05544934731138812), (2.5159873192784272,1.4275179195632797), (1.4464123272358083,-0.562485834087356), (2.814098282214936,-0.18222943553591694), (1.1740804299823382,-0.21308563190989152), (0.610475789529858,0.7798377783028813), (-0.42624197632233374,-1.2335920902335755), (3.625889016113135,0.6239593088850955), (0.2763769489768022,-0.38681536662494653), (0.9377840067167913,0.5153954440507872), (0.891404804340253,-0.4547634776863353), (-0.36535447057292614,-1.0675089357258334), (1.7797108170731182,0.20455738369516446), (0.7184782760676638,-0.4405023228090733), (-0.17923517056398053,-0.436098409565395), (-1.9223979428879878,-3.0914805600896087), (0.0314887574951036,-1.0384319108300781), (3.1915512652448115,0.16237362414927542), (0.07521161361985051,0.09173964941080463), (3.5309532938682597,1.0546072741601624), (0.3660623395309208,-1.4422890792230438), (2.5372590257109966,-1.3370778979747984), (1.6559484575999996,0.7517113206381191), (1.4924288421049785,0.3935092691102775), (1.5953453623043368,0.03811037742382486), (1.3617590586446204,0.530154778484631), (-2.534062257073355,0.3782070621461191), (0.9600591339372136,-1.023468891914467), (2.174999370984257,0.19316721687486774), (0.9971110428426018,0.6597527265457721), (-0.013924552549497538,-1.2236470241273742), (-0.14967707850183243,1.247514574470341), (2.044815209793586,2.2435222794640977), (2.4410440661546104,0.2995966594358739), (-2.349135440873335,-0.5291073022086537), (0.6790935771846645,-0.5571189095542917), (3.4139257418589426,0.11349223107935802), (-0.8626084002064136,1.1185669488897558), (-0.37391747446349344,-0.4086114125519275), (0.8215544444978548,-0.25112950363112646), (3.8618090128750024,-1.0608970223945158), (1.6800857686675532,0.17518246249620026), (1.0074005694996364,0.06651951321430995), (3.5318329860970725,-0.33034634898439175), (1.7950417717994815,0.6105745637327348), (-2.0336070695788977,0.5290626446720255), (-1.7054184477763195,2.018433101361855), (4.645023899050502,-0.9984024008645774), (1.3906629527780199,0.3260796702342513), (2.4740987497226086,-0.3250349484487747), (0.07847588428334362,-1.369769865042985), (2.475868347546835,0.9921216716815834), (0.6272709897847293,-0.3083526711709522), (-0.6858175920919738,1.0187493429512395), (0.45900544246416697,-0.24782056078650502), (-0.5745269203716783,-0.7712163035518746), (1.3701280631677617,-3.5604118223788594), (0.0948731510554367,0.32612291143440386), (-1.6852745529042497,0.4441259968243954), (1.6709633112857247,-1.4734210872528921), (2.3048068787912106,1.4195884324887544), (3.7027681434110313,-1.4331865020277441), (3.9197357289539516,0.5398675146254653), (3.4591727918555657,0.15162238606384537), (0.18141111286742795,0.43294396379213923), (1.1418538021916915,1.1004558318130622), (1.8458371307314194,-1.6225249551448202), (0.9244775944028731,0.890121110279592), (1.0864146130345944,-0.6294534363395626), (-0.3509024113650585,0.7003661935454487), (3.33728122430862,-0.1295402687875722), (0.2483092092697341,2.044822501564002), (3.9964209499931576,-0.8395262566314947), (5.041222565400819,-0.07787176492505386), (3.027508004726709,0.654084260252699), (-1.4004317476058188,-1.108634225084861), (1.9893187143877324,0.5376683045798573), (-1.510601790533327,-0.5998211693601727), (3.2336996361552006,1.0847220613718276), (1.3904890738297762,-1.2062551172960978), (-0.9985183459863718,-0.07704375636574601), (-0.42290918516396814,-0.22710371366501256), (0.557145424969743,0.6448982623891638), (0.6299424035592602,-1.3123779835257174), (1.2441200282402665,1.5005705164947913), (0.1416628957035737,-0.1530590588270949), (1.658439228273565,0.2968215409652417), (3.487027565153457,-0.17609073023938399), (0.057747803745601956,2.0723474186637874), (1.8158892220290406,0.18760101905530316), (0.47388630687051536,-1.1629786052155737), (-2.8455031410603127,-0.04644466219506257), (2.568676770051918,1.102684460944185), (-0.8084331842961674,-0.8597310131420156), (2.7731501328334534,1.378930732154014), (-2.2022646763305564,-0.6932188375333033), (-2.5747511745197507,-1.427569318035048), (0.045385950152493915,-0.4316528243072038), (0.48651128153317813,-2.438723481595101), (0.17211394730373364,0.7217274318503094), (2.4559329674426524,1.0155995742344546), (-1.949177472936245,0.052126413971228996), (3.151379032883542,-0.48445743390746254), (-2.008398402592464,0.4148805797424512), (2.1051344549387263,0.6133447153641186), (1.3539082426943376,0.8753910134615039), (-0.7773573700545062,-1.068076214407843), (3.136090265926976,-0.434556342004716), (0.8659619601166967,3.1269586338444797), (-2.756659224225406,-1.0898133189009078), (0.40970541978049146,-1.6792860321548917), (0.5481531611505823,1.1100130243189885), (1.4013179773665998,0.19785059877372296), (2.2133971545815765,1.1381515852738975), (-0.9491287644096225,1.6951931898940376), (0.5209829893925308,0.4849524249938774), (-2.9358313706560333,1.0375329570798115), (-2.1863280999191628,2.3962090984380273), (1.8973103339115684,0.10582200453150249), (-0.9438088917696854,0.12261906596813583), (-3.8161788015537486,-0.15685142307025152), (4.598243989073854,0.39785792430853506), (0.14157990229434658,-2.063728946132821), (0.7712012756880778,0.006718652661392625), (1.657954492689108,0.8811082917215027), (-1.9831854962565685,1.031451541304585), (-0.3480671912553863,1.03387374103542), (2.8738286816216583,-0.21407256559986743), (1.863514328094054,-0.5812016751411385), (-0.9428403558267164,-0.7583304518772646), (1.754785995282709,-1.7509940657503635), (0.013682513757415804,-0.6818646320163245), (-0.5248686453084859,0.03344337152195518), (6.071914898933014,0.07593147801064193), (-1.958247927497752,0.4280940512841111), (4.708222732187144,1.7068789507250506), (2.945611634833644,0.41457568923249366), (-2.873611006165206,0.7191845635928295), (-0.6308622794536811,-1.0212751896145356), (1.7810775590894088,0.5752383319361283), (1.3389808014307991,1.0226769514361524), (1.810527500961158,0.4821255564311748), (-0.6433026011303964,0.6925122151119577), (2.5056823298048334,-0.49731955088091157), (-2.920132259436649,-0.2964847770300842), (2.037990236118738,-1.7889384504694956), (1.4786574552223295,-1.7308205970757709), (-3.2626435921253814,-0.8978527372381635), (3.202111558516957,1.610161749388444), (3.0775529358503873,-2.3346966778146676), (-0.004858289779320346,-0.14005235243669392), (-0.2356611818482255,-0.2638362330334334), (1.0363597159426476,3.033939148221663), (0.5951880796739928,-0.4723921431007955), (-1.385157144701036,-1.9818612938644127), (3.168627913505496,-1.200180759439612), (1.7659099443795578,-0.1958021131712377), (-2.2099743662353757,1.8451977668782862), (-2.030174316332396,-1.5854513074199188), (1.7759112730502544,-0.6861915752981226), (-1.4724017242345253,-0.8314228441600172), (-3.090438319803951,-0.5338604341531643), (-1.002042900848683,1.781652090388434), (2.6306174515890004,-0.9076613846556846), (-1.908448678609222,-0.6071641904585964), (1.4736263162987941,0.25535155484588434), (0.7736703824491263,1.5909788064424866), (1.7498656979196905,1.2231795468357605), (0.9366265907877988,-1.5369768230738219), (-0.09536435134844679,0.9778022047574955), (0.8177132188563772,-0.11568918189852785), (-3.4121939733497646,-0.43046323075751886), (3.4314196035854083,0.4802026430220832), (1.2692425660828197,2.1631891668981025), (0.9537947988635598,0.4690583697476523), (1.9696982561485377,1.0674375424842637), (3.9301829681795217,0.9208080500687618), (-0.4472709746146637,0.6843434412137375), (-2.253286341939513,-1.6794382154599943), (0.026135514256517056,1.0870090254688949), (2.4177497341772787,1.9248747606955043), (-0.835370698709712,0.6934671953307943), (4.135177415806311,0.627881336513211), (-1.3712161103055798,0.3744662154421754), (1.410688895640302,1.763784024349061), (-1.942154218131955,-0.8493808371900399), (0.38330995987308936,-0.567609188833336), (-1.0334804970971412,-0.2201046369367792), (-1.832625134688632,-1.6873491715309796), (2.5310364388620794,0.2371190560666023), (3.452713263233051,1.075251848522166), (-0.6837517757525475,1.4368687592287923), (-0.2942544269504741,0.8762404262364163), (1.4256959022956015,-0.07711565494502963), (2.7974786021190114,0.17698197228232837), (1.323363871505369,-0.3600776896107076), (2.7466288331322652,1.150261456652663), (-1.413730211824194,-0.5002... 
val rddLR = sc.parallelize(yx)
rddLR.take(5)
rddLR: org.apache.spark.rdd.RDD[(Double, Double, Double)] = ParallelCollectionRDD[756] at parallelize at command-2972105651606197:1 res2: Array[(Double, Double, Double)] = Array((-0.6846711453351213,-1.5289467818444833,-0.39294234164655323), (5.497385056408215,1.699419230239045,0.8555415681712383), (3.192289974143172,0.5678191719510146,0.5047531440104407), (9.62629002945858,4.451573214358624,-1.7447287479723848), (2.3798191201086443,-0.1120835302184493,0.24074263385944478)) 
val dfLR = rddLR.toDF("y","x1","x2")
dfLR.show
dfLR.show(5)
+--------------------+--------------------+--------------------+ | y| x1| x2| +--------------------+--------------------+--------------------+ | -0.6846711453351213| -1.5289467818444833|-0.39294234164655323| | 5.497385056408215| 1.699419230239045| 0.8555415681712383| | 3.192289974143172| 0.5678191719510146| 0.5047531440104407| | 9.62629002945858| 4.451573214358624| -1.7447287479723848| | 2.3798191201086443| -0.1120835302184493| 0.24074263385944478| | 2.8219504009739422| 2.593362084700087| -2.5060458696190544| | 3.612693960033294| 0.5534949593776934| -0.3625957248008947| |-0.43223384405620746| 0.610162799267687| -1.590842317493378| | 2.6768474073335398| 0.6079959554264716| 0.6957126527559235| | 7.1521815633527535| 3.5570523071163964| 0.4443439308434807| | -0.9227166193659311| -1.6847794191419774| 1.0693145537223456| | -2.468553589310911| -0.9209298511507433| -1.2513129429033423| | 3.898027197661611| 1.7696585529469604|-0.35885347259074074| | 6.732593845464607| 2.513719756856524| -0.4025589013089939| | 0.43416427898655346| -0.5101432900051885| -0.5152842487123| | 4.214978206657414| 0.9531168056474683| 1.132054878042222| | 1.4665466181948976|-0.18160386853639432| -0.5455845613656002| | -3.7361996916879003| -1.6720534639191835|0.047570983504133746| | 0.9602040797406759| -0.1128951628965178|-0.49567564166101635| | 12.928871086510929| 5.053202646224175| 1.0696261814411365| +--------------------+--------------------+--------------------+ only showing top 20 rows +-------------------+-------------------+--------------------+ | y| x1| x2| +-------------------+-------------------+--------------------+ |-0.6846711453351213|-1.5289467818444833|-0.39294234164655323| | 5.497385056408215| 1.699419230239045| 0.8555415681712383| | 3.192289974143172| 0.5678191719510146| 0.5047531440104407| | 9.62629002945858| 4.451573214358624| -1.7447287479723848| | 2.3798191201086443|-0.1120835302184493| 0.24074263385944478| +-------------------+-------------------+--------------------+ only showing top 5 rows dfLR: org.apache.spark.sql.DataFrame = [y: double, x1: double ... 1 more field] 
// importing for regression
import org.apache.spark.ml.regression.LinearRegression
import org.apache.spark.ml.linalg._
 
val lm = new LinearRegression
lm.explainParams
lm.getStandardization
lm.setStandardization(false)
lm.getStandardization
lm.explainParams
import org.apache.spark.ml.regression.LinearRegression import org.apache.spark.ml.linalg._ lm: org.apache.spark.ml.regression.LinearRegression = linReg_6bc65ac2a454 res5: String = aggregationDepth: suggested depth for treeAggregate (>= 2) (default: 2) elasticNetParam: the ElasticNet mixing parameter, in range [0, 1]. For alpha = 0, the penalty is an L2 penalty. For alpha = 1, it is an L1 penalty (default: 0.0) epsilon: The shape parameter to control the amount of robustness. Must be > 1.0. (default: 1.35) featuresCol: features column name (default: features) fitIntercept: whether to fit an intercept term (default: true) labelCol: label column name (default: label) loss: The loss function to be optimized. Supported options: squaredError, huber. (Default squaredError) (default: squaredError) maxIter: maximum number of iterations (>= 0) (default: 100) predictionCol: prediction column name (default: prediction) regParam: regularization parameter (>= 0) (default: 0.0) solver: The solver algorithm for optimization. Supported options: auto, normal, l-bfgs. (Default auto) (default: auto) standardization: whether to standardize the training features before fitting the model (default: true, current: false) tol: the convergence tolerance for iterative algorithms (>= 0) (default: 1.0E-6) weightCol: weight column name. If this is not set or empty, we treat all instance weights as 1.0 (undefined) 
// Transform data frame to required format
val dflr = (dfLR map {row => (row.getDouble(0), 
           Vectors.dense(row.getDouble(1),row.getDouble(2)))}).
           toDF("label","features")
dflr.show(5)
+-------------------+--------------------+ | label| features| +-------------------+--------------------+ |-0.6846711453351213|[-1.5289467818444...| | 5.497385056408215|[1.69941923023904...| | 3.192289974143172|[0.56781917195101...| | 9.62629002945858|[4.45157321435862...| | 2.3798191201086443|[-0.1120835302184...| +-------------------+--------------------+ only showing top 5 rows dflr: org.apache.spark.sql.DataFrame = [label: double, features: vector]