Forecasting Agricultural Commodity Prices Using Multivariate Bayesian Machine Learning. Andres M. Ticlavilca, Dillon M. Feuz, and Mac McKee

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1 Foecasting Agicultual Commodity Pices Using Multivaiate Bayesian Machine Leaning Regession by Andes M. Ticlavilca, Dillon M. Feuz, and Mac McKee Suggested citation fomat: Ticlavilca, A. M., Dillon M. Feuz and Mac McKee Foecasting Agicultual Commodity Pices Using Multivaiate Bayesian Machine Leaning Regession. Poceedings of the NCCC-134 Confeence on Applied Commodity Pice Analysis, Foecasting, and Maket Risk Management. St. Louis, MO. [

2 Foecasting Agicultual Commodity Pices Using Multivaiate Bayesian Machine Leaning Regession Andes M. Ticlavilca, Dillon M. Feuz, and Mac McKee * Pape pesented at the NCCC-134 Confeence on Applied Commodity Pice Analysis, Foecasting, and Maket Risk Management St. Louis, Missoui, Apil 19-20,2010 Copyight 2010 by Andes M. Ticlavilca, Dillon M. Feuz and Mac McKee. All ights eseved. Reades may vebatim copies of this document fo non-commecial puposes by any means, povided that this copyight notice appeas on all such copies. * Andes M. Ticlavilca (andes.t@aggi .usu.edu) is a Gaduate Reseach Assistant at the Utah Wate Reseach Laboatoy, Depatment of Civil and Envionmental Engineeing, Utah State Univesity. Dillon M. Feuz (dillon.feuz@usu.edu) is a Pofesso at Utah State Univesity in the Applied Economics Depatment. Mac McKee (mac.mckee@usu.edu) is the Diecto of the Utah Wate Reseach Laboatoy and Pofesso at the Depatment of Civil and Envionmental Engineeing, Utah State Univesity.

3 Foecasting Agicultual Commodity Pices Using Multivaiate Bayesian Machine Leaning Regession The pupose of this pape is to pefom multiple pedictions fo agicultual commodity pices (one, two and thee month peiods ahead). In ode to obtain multiple-time-ahead pedictions, this pape applies the Multivaiate Relevance Vecto Machine (MVRVM) that is based on a Bayesian leaning machine appoach fo egession. The pefomance of the MVRVM model is compaed with the pefomance of anothe multiple output model such as Atificial Neual Netwok (ANN). Bootstapping methodology is applied to analyze obustness of the MVRVM and ANN. Keywods: Commodity pices, Foecasting, Machine leaning, Bayesian Intoduction The last few yeas thee has been an incease in the volatility of many agicultual commodity pices. This has inceased the isk faced by agicultual poduces. The main pupose of agicultual commodity pice foecasting is to allow poduces to make bette-infomed decisions and to manage pice isk. Simple pice foecast models such as naïve, o distibuted-lag models have pefomed quite well in pedicting agicultual commodity pices (Hudson, 2007). Othe models such as defeed futue plus histoical basis models (Kastens et al., 1998), autoegessive integated moving aveage (ARIMA) models and composite models (Tomek and Myes, 1993) lead to moe accuate estimates. Howeve, as the accuacy inceases, so does the statistical complexity (Hudson, 2007). Pactical applications of moe complex models ae limited by the lack of equied data and the expense of data acquisition. On the othe hand, the inceased volatility in agicultual commodity pices may incease the difficulty of foecasting accuately making the simple methods less eliable and even the moe complex foecast methods may not be obust in this new maket envionment. To ovecome these limitations, machine leaning (ML) models can be used as an altenative to complex foecast models. ML theoy is elated to patten ecognition and statistical infeence wheein a model is capable of leaning to impove its pefomance on the basis of its own pio expeience (Mjolsness and DeCoste, 2001). Examples of ML models include the atificial neual netwoks (ANNs), suppot vecto machines (SVMs) and elevance vecto machines (RVMs). ML models have been applied in financial economics modeling. Enke and Thawonwong (2005) used data mining and ANNs to foecast stock maket etuns. Co and Boosaawongse (2007) demonstated that ANNs outpefomed exponential smoothing and ARIMA models in foecasting ice expots. Also ML models have been applied in foecasting agicultual commodity pices. Shahwan and Odening (2007) used a hybid between ANNs and ARIMA model to pedict agicultual commodity pices. 1

4 The pape epoted hee pesents a ML model to pefom monthly multi-step-ahead pedictions with pedictive confidence intevals fo agicultual commodity pices. Theefoe, the model ecognizes the pattens between multivaiate outputs (futue commodity pices) and multivaiate inputs (past data collected about commodity pices). This pape applies the Multivaiate Relevance Vecto Machine (MVRVM) (Thayananthan, 2005; Thayananthan et al., 2008). The MVRVM is an extension of the RVM algoithm developed by Tipping and Faul (2003). It can be used to poduce multivaiate outputs with confidence inteval, via its Bayesian appoach. The MVRVM has the same capabilities of the conventional RVM: high pediction accuacy, obustness, and chaacteization of uncetainty in the pedictions. The emainde of the pape descibes the MVRVM appoach, the model application fo agicultual commodity pice foecast, a compaison with an ANN model and conclusions. Model Desciption The MVRVM was developed by Thayananthan (2005) to povide an extension of the RVM algoithm fo egession (Tipping, 2001; Tipping and Faul, 2003) to multivaiate outputs. Given a set of taining examples of input-taget vecto pais {x (n), t (n) } N n 1, whee Ν is the numbe of patten obsevations, x Є R D is a input vecto, t Є R M is a output taget vecto, the model leans the dependency between input and output taget with the pupose of making accuate pedictions of t fo pevious values of x: t = W Φ(x) + ε (1) whee W is a M x P weight matix and P = N+1. The eo ε is assumed to be zeo-mean Gaussian with diagonal covaiance matix S=diag(σ 1 2,, σ M 2 ). Φ(x) is a vecto of basis functions of the fom Φ(x) = [1, K(x,x (1), K(x,x (N) )), whee K(x,x n ) is a kenel function (Tipping, 2001, Thayananthan, 2008). In this pape, we consideed a Gaussian kenel K(x,x n ) = exp(- -2 x- x (n) 2 ) whee is the kenel width paamete. A likelihood distibution of the weights is defined as a poduct of Gaussians of the weight vectos (w ) coesponding to each output taget (τ ) (Thayananthan, 2008): N n 1 N (n) (n) (n) 2 p({ t } W,S) ( t WΦ ( x ), S) ( τ w Φ,σ ) (2) n 1 whee Φ = [1, Φ(x 1 ), Φ(x 2 ),..., Φ(x N )]. To avoid ovefitting fom Equation (2), Tipping (2001) poposed constaining the selection of paametes by applying a Bayesian appoach and defining an explicit zeo-mean Gaussian pio pobability distibution ove the weights (Thayananthan, 2008): M 1 2

5 M P M 2 (w j 0,α j ) ( w 0, ) 1 j 1 1 p( W A) A (3) whee A = diag(α -2 1,, α -2 P ) T is a diagonal matix of hypepaametes α j, and w j is the (,j)th element of the weight matix W. Each α j contols the stength of the pio ove its associated weight (Tipping and Faul, 2003). The posteio distibution of the weights is popotional to the poduct of the likelihood and pio distibutions: N N p( W { t} n 1,S, A) p({ t} n 1 W,S) p( W A) (4) Then, this posteio paamete distibution can be defined as the poduct of Gaussians fo the weight vectos of each taget (Thayananthan, 2008): M N N n 1,S, A) p({ t} n 1 W,S) p( W A) ( w µ, ) 1 p( W { t} Σ (5) with covaiance and mean, Σ = (A + σ -2 Φ T Φ) -1 and µ = σ -2 Σ Φ T τ, espectively. An optimal weight matix can be obtained by estimating a set of hypepaametes that maximizes the data likelihood ove the weights in Equation (5) (Thayananthan, 2008). The maginal likelihood is then: N N p({ t } A,S) p({ t} W,S) p( W A)dW, n 1 n 1 M M 2 1 ( Φ,σ ) ( 0, 2 T -1 τ w w A) H exp( τ H τ ) (6) 2 1 whee H = σ 2 I + Φ A -1 Φ T. The optimal set of hypepaametes α opt = 1 1 opt P { α j } j 1 and noise paametes (σ opt ) 2 opt M = {σ } 1 ae obtained by maximizing the maginal likelihood using the fast maginal likelihood maximization algoithm poposed by Tipping and Faul (2003). Many elements of α go to infinity duing the optimization pocess, fo which the posteio pobability of the weight becomes zeo. These nonzeo weights ae called the elevance vectos (RVs) (Tipping and Faul, 2003). Then, we can obtain the optimal covaiance Σ opt = opt M { } 1 Σ and mean µ opt = opt M { } 1 µ. Given a new input x*, we can compute the pedictive distibution fo the coesponding output taget t* (Tipping, 2001) : p( t* t, α opt,( σ opt 2 opt 2 opt opt 2 ) ) = p( t* W,( σ ) ).p( W t, α,( σ ) ) dw (7) 3

6 In Equation (7), both tems in the integand ae Gaussian. Then, this equation can be computed as: p( t* t, α opt,( σ opt 2 2 ) ) ( t* y*,( σ*) ) (8) whee y*=[ y 1 *,..., y *,... y M *] T is the pedictive mean with y * = (µ opt ) T Φ(x*); and (σ*) 2 = [(σ 1 *) 2,... (σ *) 2,..., (σ M *) 2 ] T is the pedictive vaiance with (σ *) 2 = (σ opt ) 2 + Φ(x*) T Σ opt Φ(x*) which contains the sum of two vaiance tems: the noise on the data and the uncetainty in the pediction of the weight paametes (Tipping, 2001). The standad deviation σ * of the pedictive distibution is defined as a pedictive eo ba of y * (Bishop 1995). Then, the width of the 90% Bayesian confidence inteval fo any y * can be ± 1.65.σ *. This Bayesian confidence inteval (which is based on pobabilistic appoach) should not be confused with a classical fequentist confidence inteval (which is based on the data). Data Monthly data fo cattle, hog and con pices wee obtained fo 21 yeas. The data wee obtained fom the Livestock Maketing Infomation Cente website and the data ae initially collected and epoted by the USDA-AMS. Con pices wee fom Omaha, NE maket; cattle pices wee fom Nebaska live fed cattle maket, and hog pices wee fom the Iowa/southen Minnesota maket. These ae all lage makets that ae fequently used as standads by which to judge othe makets. Pocedues Monthly data fom 1989 to 2003 wee used to tain each model and estimate the model paametes. Monthly data fom 2004 to 2009 wee used to test the models. The inputs used in the model to pedict monthly commodity pice ae expessed as x = [x tp-m ] T (9) whee, tp = time of pediction m = numbe of months pevious to the pediction time x1 tp-m = Commodity pice m months pevious to the pediction time The multiple output taget vecto of the model is expessed as t = [ t tp+1, t tp+2, t tp+3 ] T (10) whee, t tp+1 = pediction of commodity pice one month ahead t tp+2 = pediction of commodity pice two months ahead t tp+3 = pediction of commodity pice thee months ahead 4

7 Pefomance evaluation The kenel width is a smoothing paamete which defines a basis function to captue pattens in the data. This paamete cannot be estimated with the Bayesian appoach. Fo this pape, a sensitivity analysis is pefomed to estimate the kenel width that gives accuate test esults. The statistics used fo the selection of the model is the oot mean squae pecentage eo (RMSPE), and is given by: 2 N 1 t t * RMSPE.100 (11) N i 1 t whee t is the obseved output; t* is the pedicted output and N is the numbe of obsevations. The sensitivity analysis was done by building seveal MVRVM models with vaiation in the kenel width (fom 1 to 60) and the numbe of pevious months equied as input (fom 1 to 12 months). The selected model was the one with the minimum RMSPE of the aveage outputs coesponding to the testing phase. Bootstap analysis The bootstap method (Efon and Tibshiani,1998) was used to guaantee obustness of the MVRVM (Khalil et al., 2005a). The bootstap data set was ceated by andomly selecting fom the whole taining data set, with eplacement. This selection pocess was independently epeated 1,000 times to yield 1,000 bootstap taining data sets, which ae teated as independent sets (Duda et al., 2001). Fo each of the bootstap taining data sets, a model was built and evaluated ove the oiginal test data set. A obust model is one that shows a naow confidence bounds in the bootstap histogam (Khalil et al., 2005b). A naow confidence bounds implies low vaiability of the statistics with futue changes in the natue of the input data, which indicates that the model is obust. Compaison between MVRVM and ANN A compaative analysis between the developed MVRVM and ANNs is pefomed in tems of pefomance and obustness. Reades inteested in geate detail egading ANN and its taining functions ae efeed to Demuth et al. (2009). Seveal feed-fowad ANN models wee tained and tested with vaiation in the type of taining function, size of laye(fom 1 to 10) and the numbe of months pevious to the pediction time (fom 1 to 12 months). The selected model was the one with the minimum RMSPE of the aveage outputs coesponding to the testing phase. 5

8 Results and Discussions Table 1 shows the selected kenel width and the numbe of input months fo each commodity pice foecasting. The selected models wee the ones with the lowest RMSPE of the aveage esults. Table 1 Selected MVRVM fo each commodity (testing phase) Numbe of Kenel RMSPE (%) Aveage RMSPE pevious months width 1-month 2-months 3-months (%) Con Cattle Hog Figues 1-3 show the pedicted outputs (full lines) of the MVRVM fo the testing phase fo con, cattle and hog pices espectively. These figues also show the 0.90 Bayesian confidence inteval (shaded egion) associated with the pedictive vaiance (σ *) 2 of the MVRVM in Equation (8). We can see that this Bayesian confidence inteval appeas to be unchanged duing the whole test peiod. As we mentioned in section 2, the pedictive vaiance is (σ *) 2 = (σ opt ) 2 + Φ(x*) T Σ opt Φ(x*). The fist tem depends on the noise on the taining data and the second tem depends on the pediction of the paamete when a new input x* is given. Fom ou esults, it was found that thee is a significant contibution fom the fist tem (the noise vaiance on the taining data) which made the contibution fom the second tem vey small (close to zeo). That is why the width of the confidence inteval fo the test esults appeas to be almost constant. The model leans the pattens fo one and two months ahead fo the thee commodities (Figues 1a, 1b, 2a, 2b, 3a and 3b). The pefomance accuacy is educed fo the thee-month ahead pice pediction of the thee commodities (Figues 1c, 2c and 3c). This accuacy eduction is found in most of the multiple-time-ahead pediction models, whee the fathe we pedict into the futue, the less accuate the pediction becomes. Also, we can see that the model pefomance deceases in ealy 2008 fo the con pice pedictions. As we mentioned in section 4, monthly data fom 1989 to 2003 wee used to tain the model and estimate the model paametes. The model pobably needs to be etained at this peiod (ealy 2008) and also it may need moe numbe of pevious months as inputs data duing this peiod. Howeve, we pefe not to povide moe definite conclusions, as they might not be sufficiently well suppoted. Moe detailed analysis egading stategies to impove model pefomance will be caied out fo futue eseach. 6

9 Figue 1. Obseved vesus pedicted monthly con pice of the MVRVM model with 0.90 Bayesian confidence intevals (shaded egion) fo the testing phase: (a) 1-month ahead, (b) 2- months ahead, (c) 3-months ahead 7

10 Figue 2. Obseved vesus pedicted monthly cattle pice of the MVRVM model with 0.90 Bayesian confidence intevals (shaded egion) fo the testing phase: (a) 1-month ahead, (b) 2- months ahead, (c) 3-months ahead 8

11 Figue 3. Obseved vesus pedicted monthly hog pice of the MVRVM model with 0.90 Bayesian confidence intevals (shaded egion) fo the testing phase: (a) 1-month ahead, (b) 2- months ahead, (c) 3-months ahead 9

12 Table 2 shows the selected ANN models fo two types of taining function. The model with conjugated-gadient-taining function shows the lowest RMSPE of the aveage esults fo con. The model with esilient-backpopagation-taining function shows the lowest RMSPE of the aveage esults fo cattle and hog. Theefoe they wee selected as the best type of taining function fo each model that descibes the input-output pattens. Table 2. Selected ANN models fo two types of taining function (testing phase) RMSPE (%) Con Cattle Hog Aveage RMSPE (%) Type of taining function 1-month 2-months 3-months 1-month 2-months 3-months 1-month 2-months 3-months Con Cattle Hog Resilient backpopagation Conjugate gadient with Powell-Beale estats Table 3 shows the selected size of laye and the numbe of input months fo each commodity pice foecasting fo the ANN models. Table 3. Selected ANN fo each commodity pice pediction (testing phase). Numbe of Size of RMSPE (%) Aveage RMSPE pevious months laye 1-month 2-months 3-months (%) Con Cattle Hog

13 Figue 4. Obseved vesus pedicted monthly con pice of the ANN model fo the testing phase: (a) 1-month ahead, (b) 2-months ahead, (c) 3-months ahead 11

14 Figue 5. Obseved vesus pedicted monthly cattle pice of the ANN model fo the testing phase: (a) 1-month ahead, (b) 2-months ahead, (c) 3-months ahead 12

15 Figue 6. Obseved vesus pedicted monthly hog pice of the ANN model fo the testing phase: (a) 1-month ahead, (b) 2-months ahead, (c) 3-months ahead 13

16 Figues 4-6 show the obseved (dots) and pedicted (full lines) outputs of the ANN fo the testing phase fo con, cattle and hog pices espectively. RMSPE and RMSE statistics fo the MVRVM and ANN pediction pefomance ae displayed in Table 4. We can see that the MVRVM outpefoms, has a smalle foecast eo, the ANN fo con pice pediction one month ahead, cattle pice pediction one, two and thee months ahead, and hog pice pediction one and thee months ahead. On the othe hand, ANN outpefoms MVRVM fo con pice pediction two and thee months ahead. Hog pice pedictions fo two months ahead ae simila fo both models. The ANN cattle model (Figue 5) appeas to be shifted with a 1-3 month lag. This ANN model may need moe numbe of pevious months as inputs data duing this peiod. On the othe hand, the MVRVM cattle model (Figue 2) can ovecome the pefomance lag poblems since its Bayesian appoach allows us to calculate pedictive confidence intevals, instead of just poviding a single taget output (Bishop 1995) such as is the ANN model esults. Table 4. Model Pefomance using RMSPE and RMSE (testing phase) Con Cattle Hog Model Statistics 1-month 2-months 3-months 1-month 2-months 3-months 1-month 2-months 3-months MVRVM RMSPE (%) RMSE ($) ANN RMSPE (%) RMSE ($) Figues 7-12 show the bootstap histogams fo the RMSE test based on 1,000 bootstap taining data sets of the MVRVM and ANN models fo con, cattle and hog pices espectively. The bootstapped histogams of the MVRVM models (Figues 7, 9 and 11) show naow confidence bounds in compaison to the histogams of the ANN models (Figues 8, 10 and 12). Theefoe, the MVRVM is moe obust. 14

17 Figue 7. Bootstap histogams of the MVRVM model of con pice pedictions fo the RMSPE: a) one month ahead, b) two months ahead, c) thee months ahead Figue 8. Bootstap histogams of the ANN model of con pice pedictions fo the RMSPE: a) one month ahead, b) two months ahead, c) thee months ahead 15

18 Figue 9. Bootstap histogams of the MVRVM model of cattle pice pedictions fo the RMSPE: a) one month ahead, b) two months ahead, c) thee months ahead Figue 10. Bootstap histogams of the ANN model of cattle pice pedictions fo the RMSPE: a) one month ahead, b) two months ahead, c) thee months ahead 16

19 Figue 11. Bootstap histogams of the MVRVM model of hog pice pedictions fo the RMSPE: a) one month ahead, b) two months ahead, c) thee months ahead Figue 12. Bootstap histogams of the ANN model of hog pice pedictions fo the RMSPE: a) one month ahead, b) two months ahead, c) thee months ahead 17

20 Summay and Conclusions This pape applies a MVRVM model to develop multiple-time-ahead pedictions with confidence intevals of monthly agicultual commodity pices. The pedictions ae one, two and thee months ahead of pices of cattle, hogs and con. The MVRVM is a egession tool extension of the RVM model to poduce multivaiate outputs. The statistical test esults indicate an oveall good pefomance of the model fo one and two month s pediction fo all the commodity pices. The model pefomance deceased in ealy 2008 fo the con pice pedictions. The pefomance also deceased fo the thee-month pediction of the thee commodity pices. The MVRVM model outpefoms the ANN most of the time with the exception of con pice pediction two and thee months ahead. Howeve, the bootstap histogams of the MVRVM model show naow confidence bounds in compaison to the histogams of the ANN model fo the thee commodity pice foecasts. Theefoe, the MVRVM is moe obust. The esults pesented in this pape have demonstated the oveall good pefomance and obustness of MVRVM fo simultaneous multiple-time-ahead pedictions of agicultual commodity pices. The potential benefit of these pedictions lies in assisting poduces in making bette-infomed decisions and managing pice isk. Futue wok In this eseach, we have not analyzed the spase popety (low complexity) of the MVRVM since we have woked with elatively small data set (166 monthly obsevations) to tain the model. Futue eseach will be pefomed by analyzing weekly pice (moe than 1000 obsevations) and fully exploit the spase chaacteistics of the Bayesian appoach when dealing with lage dataset. Also, the elevance vectos (RVs) (which ae elated to the spase popety) ae the summay of the most essential obsevations of the taining data set to build the MVRVM. In this pape we have not analyzed the RVs since we ae dealing with small data set. Futue eseach will be pefomed by analyzing with moe details the statistical meaning of RVs with espect to a lage taining data set. Fo example, this analysis can also be elated to whethe we would ecommend educing the numbe of histoical data obsevations fo etaining the model with new data in the futue. The kenel width and the numbe of pevious months equied as input cannot be estimated with the Bayesian appoach. Fo this pape, a sensitivity analysis (by tial and eo) was pefomed to estimate these paametes that gave accuate test esults. We could see that the oveall test esults ae good. Howeve, the model pefomance deceases fo some peiods ( i.e. ealy 2008 fo the con pice pedictions). Theefoe, moe analysis egading the selection of these paametes (kenel width an numbe of pevious months as inputs) will be caied out in a follow-up pape. 18

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