A Bayesian Approach to Demand Estimation

Available online at www.sciencedirect.com

ScienceDirect Procedia Economics and Finance 26 (2015) 777 – 784

4th World Conference on Business, Economics and Management, WCBEM

A Bayesian Approach to Demand Estimation H.Kıvanc Aksoya, Asli Gunera * a

Eskisehir Osmangazi University, Eskisehir, 26100, TURKEY

Abstract Companies have been using various methods to ensure their sustainability and to increase their profit. Demand estimation can be defined as a process that involves coming up with an estimate of the amount of end use demand for a product or service. Companies use demand estimation techniques for various reasons. One of the reasons that businesses use demand estimation is to determining the production amount in a certain period of time and accordingly keep the inventory level under control. An additional consideration is to assist with pricing. Decision maker’s uses various techniques for demand estimation. One of the techniques is using the earlier period data and the other is utilizing the test markets which are similar to larger targeted markets.In this paper, we resort to Bayesian approach for demand estimation and utilized the textile products export data which belongs to years between 2002 and 2014. In Bayesian analysis the new information is combined with the previously available information. At this point the prior information (distribution) corresponds to the historical data or the subjective thought of the decision maker about the unknown parameter of the involved process. The performance of the following updates depends on the prior information therefore the determination of prior information is significant. In this research to obtain the future demand level for the previously mentioned data we utilize the conjugate prior families to obtain the posterior distribution. © B.V. This is an open access article under the CC BY-NC-ND license © 2015 2015The TheAuthors. Authors.Published PublishedbybyElsevier Elsevier B.V. (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of Academic World Research and Education Center. Peer-review under responsibility of Academic World Research and Education Center Keywords:demand estimation; bayesian approach; conjugate families; prior distribution; posterior distribution.

1. Introduction Demand estimation can be defined as a process that involves coming up with an estimate of the amount of end use demand for a product or service. Companies use demand estimation techniques for various reasons. One of the reasons that businesses use demand estimation is to determining the production amount in a certain period of time and accordingly keep the inventory level under control. An additional consideration is to assist with pricing. The

* Asli Guner. Tel.: +90-222-2393750/2251 E-Mail address: [email protected]

2212-5671 © 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Peer-review under responsibility of Academic World Research and Education Center doi:10.1016/S2212-5671(15)00844-8

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demand level for an item usually follows a random pattern and most of the cases its distribution function cannot be obtained completely. One of the reasons of this situation is if the product or service is freshly introduced to the market or its demand level highly fluctuating with season as its instance in the fashion industry. To cope with the demand level uncertainty in a dynamic environment it is appropriate to start with an observed earlier period data and update the demand level once the new figures are available. In this paper we utilized Bayesian approach for successful demand estimation process through continuously revising the demand distribution combining the previous information with a most recent data. There is an expanding body of literature in the area of demand estimation and managers or business owners from various sectors make use of numerous techniques for demand estimation. Bayesian approach is various applications in diverse areas such as stock price estimation, population modeling, marketing and estimation of expansion parameter(s). Berk, Guler and Levine (2001), used Bayesian approach for demand estimation for newsboy type inventory model where lost sales circumstances occur. Authors first considered the completely observed demand case and derived the cost function according to the Gamma distributed demand and obtain the optimal conditions accordingly. In this research authors also examined censored data and completely unobserved cases for both Poisson and Normal distributed demand data. In a later article, Berk, Gurler and Levine (2007), remarked that Lovejoy (1990) single period inventory approach by myopic control principle and employ first two moment approximation approach for the posterior distribution. Albert and Chib (1993) and Tektas (2006) used both maximum likelihood method and Bayesian method to obtain parameter estimation in binary logit and probit models. Authors concluded that in small samples case Bayesian methods are more useful than maximum likelihood methods which are not asymptotical property for small samples. In Bayesian approach prior probability distribution states that one’s belief about the process before some additional information obtained. Conjugate prior families were first discussed and formalized by Raiffa and Schlaifer (1961). The definition and construction of conjugate prior distributions depends on the existence and identification of sufficient statistics of fixed dimension for the given likelihood function. Raiffa and Schlaifer (1961) also show that the posterior distribution arising from the conjugate prior is itself a member of the same family as the conjugate prior. When the prior and posterior distributions both belong to the same family, the prior is said to be closed under sampling. Raiffa & Schlaifer (1961) and DeGroot (1970) note that if, and only if, the data generating process is a member of the exponential family, and meet certain regularity conditions, must there exist a set of sufficient statistics of fixed dimension. It is only for these processes that there must exist a set of simple operations for updating the prior into the posterior. In this paper, we resort to Bayesian approach for demand estimation and utilized the textile products export data which belongs to years between 2002 and 2014. In Bayesian analysis the new information is combined with the previously available information. At this point the prior information (distribution) corresponds to the historical data or the subjective thought of the decision maker about the unknown parameter of the involved process. The performance of the following updates depends on the prior information therefore the determination of prior information is significant. In this research to obtain the future demand level for the previously mentioned data we utilize the conjugate prior families to obtain the posterior distribution. 2. Bayesian Approach In Bayesian analysis the new information is combined with the previously available information. At this point the prior information (distribution) corresponds to the historical data or the subjective thought of the decision maker about the unknown parameter of the involved process. The consequential decision or inferential statement (posterior distribution) combined all available information about the uncertain parameter of the process. The performance of the succeeding updates depends on the prior information therefore the determination of prior information is significant. If there isn’t any basis of the prior information then the decision maker may consider the noninformative priors about the random variable which represents the unknown parameter may obtain any value in its domain evenly likely (Hill 1997, 1999; Shih 2001; Winkler 2003). Bayesian update process can be described using three distinct probability distributions; prior, likelihood and posterior probabilities: x Prior probability represents our knowledge before we observe evidence. The prior probability of an event A is expressed as P (A).

H. Kıvanc Aksoy and Asli Guner / Procedia Economics and Finance 26 (2015) 777 – 784

x x

779

Likelihood represents a factor that is used to update our prior knowledge. The likelihood for an event A and an evidence B is expressed in terms of a conditional probability P( B | A) . Posterior represents combined probability of initial probability and additional information from the process. The posterior probability of an event A given the evidence B is expressed in terms of a conditional probability P( A | B) .

In Bayesian approach forecasters update their knowledge in response to an observed event iteratively; this process is depicted in Figure 1.

Fig. 1: Bayesian update model

In summary, Bayes’ Theorem can be expressed as follows:

P( A | B)

P( B | A) P( A) P( B)

(1)

It states that the posterior is proportional to the product of the prior and the likelihood. In other words, we can obtain our posterior knowledge by multiplying the prior and the likelihood and scaling the product. Figure 1 graphically represents how the forecasters update their knowledge using the prior, likelihood, and posterior distributions. As seen in the diagram, the update process is iterative: The current posterior becomes the prior of the next step. The process iterates when a new event is observed. Bayesian approach is an excellent "procedure generator," even if one's evaluation criteria are frequentist provided that the prior distributions introduce only a small amount of information. This agnostic view considers features of the prior (possibly the entire prior) as "tuning parameters" that can be used to produce a decision rule with broad validity. The Bayesian approach will be even more effective if one desires to structure an analysis using either personal opinion or objective information external to the current data set (Carlin and Louis, 2000). In Bayesian approach, inference regarding to posterior distribution is given by for both discrete and continuous cases as follows;

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p(T | x)

posterior

P(T i | x1 , x2 ,..., xn )

­ P( x n | T i , x1 , x2 ,..., xn 1 ) P(T i | x1 , x2 ,..., xn 1 ) , discrete ° I ° ¦ P( x n | T k , x1 , x2 ,..., xn 1 ) P(T k | x1 , x2 ,..., xn 1 ) °k 1 ® P( x | T , x , x ,..., x ) P(T | x , x ,..., x ) n i n 1 i 1 2 n 1 1 2 ° , continious ° I x ,..., x ) d T ( | , , ,..., ) ( | , P x T x x x P T x °³ n 1 2 n k 1 2 n 1 k 1 ¯k 1

­ likelihood u prior , discrete ° f ° ¦ likelihood u prior ° f ® likelihood u prior °f , continious ° u likelihood prior °³ ¯ f

(2)

(3)

3. Model Application The drive of demand estimation is to find a business’s potential end use demand level consequently decision makers can make accurate judgments about product pricing, production control and inventory organization and market potential. The demand level for an item is generally a random variable, and its distribution is not known completely. The demand for an item is generally random, and its distribution is not known completely. One of the reasons of this situation is if the product is recently introduced to the market or its demand level highly fluctuating with season as it’s an instance in the fashion industry. In such situations, it is efficient those subjectively assign a particular form for the demand distribution, and update it as additional information is obtained. Bayesian approach for updating the demand distribution is an efficient tool for continuously improves the probability distribution and provides the most updated demand level information for a given period. With any problem of statistical inference it is the definition of the problem itself that determines the type of data involved. And the analyst’s understanding of the process from which the data arise determines the appropriate likelihood function to be used with Bayes’ theorem. Thus, the only way to produce an analytically tractable solution to the integral is through the choice of the prior given the likelihood function. In the search for a family of conjugate prior distributions, it must be remembered that no one is at liberty to simply choose any prior distribution that works mathematically; a conjugate prior distribution must be specified that adequately describes the experimenter’s knowledge of the unknown parameter(s) before the experiment is executed. Consequently, to be of practical use, a conjugate prior family of distributions must produce an analytically tractable solution to the integration problem and it must be flexible enough to model the prior degree of belief in the parameters of interest (Fink, 1997). A conjugate prior is constructed by first factoring the likelihood function into two parts. One factor must be independent of the parameter(s) of interest but may be dependent on the data. The second factor is a function dependent on the parameter(s) of interest and dependent on the data only through the sufficient statistics. The conjugate prior family is defined to be proportional to this second factor. In this paper, we resort to Bayesian approach for demand estimation and utilized the textile products export data which belongs to years between 2002 and 2014. We use the conjugate family for normal distribution. Firstly, we use export data between years 2002 and 2006 for prior information, data of 2007 for likelihood and we estimate the average export figure of 2008. This process is depicted in Table 1 and Figure 2.

H. Kıvanc Aksoy and Asli Guner / Procedia Economics and Finance 26 (2015) 777 – 784

781

Table 1: Estimation results of conjugate family

Mean Std dev.

2002-2006

2007

Prior

Likelihood

Conjugate posterior

1.024.740,483 1.337.421,250 181.541,616

Percentage Error Real (%) Conjugate 1.311.671,000 Family

2008

136.911,589

1.323.271,890 38.618,377

153.358,572

0,88

Prior

Likelihood

Conjugate Posterior

Real

Fig. 2: Export data: Prior: between years 2002 and 2006, Likelihood: 2007, estimated value (posterior) and real value of 2008.

Then, we used this posterior information as prior information for the subsequent iteration and by using the data of 2009 for as additional information for the process (likelihood) we obtained updated posterior information for 2010. This process is depicted in Table 2 and Figure3. Table 2: Estimation results of conjugate family

Mean Std dev.

2008

2009

Prior

Likelihood

1.323.271,890 1.108.159,083 38.618,377

122.724,191

2010 Conjugate posterior

Real

Percentage Error (%)

1.206.462,527

1.218.549,219

Conjugate Family

26.106,295

117.118,473

0,99

782

H. Kıvanc Aksoy and Asli Guner / Procedia Economics and Finance 26 (2015) 777 – 784 Prior

Likelihood

Conjugate Posterior

Real

Fig. 3: Prior: estimated value (posterior) of 2008, Likelihood: 2009, estimated value (posterior) and real value of 2010.

In a similar manner, we considered the year 2010 data as prior information and update it via 2011, 2012, 2013 and first 11 months of 2014. Hence we obtain the posterior estimation of average export figure for December 2014. This computation process and parameters of the distribution is given in Table 3 and Figure 4. Table 3: Estimation results of conjugate family

Mean Std dev.

2010

2011-2013& first 11 months of 2014

Prior

Likelihood

Conjugate posterior

Real

1.206.462,527

1.425.114,495

1.341.609,526

1.372.215,00

Conjugate Family

26.106,295

140.684,786

16.133,356

-

2,23

December 2014

Percentage Error (%)

H. Kıvanc Aksoy and Asli Guner / Procedia Economics and Finance 26 (2015) 777 – 784 Prior

783

Likelihood

Conjugate Posterior

Fig. 4: Prior: estimated value (posterior) of 2010, Likelihood: between years 2011 and 2013, first 11 months of 2014, estimated value (posterior) of 2014

4. Conclusion In this paper, we resort to Bayesian approach for demand estimation and utilized the textile products export data which belongs to years between 2002 and 2014. We use the conjugate family for normal distribution. Firstly, we use export data between years 2002 and 2006 for prior information, data of 2007 for likelihood and we estimate the average export figure of 2008. Secondly we used this posterior information as prior information for the subsequent iteration and by using the data of 2009 for as additional information for the process (likelihood) we obtained updated posterior information for 2010. Finally we considered the year 2010 data as prior information and update it via 2011, 2012, 2013 and first 11 months of 2014. Hence we obtain the posterior estimation of average export figure for December 2014. From these processes we observe that posterior distribution’s standard deviation is decreasing. Also, Bayesian approach estimates the real demand data within a 3% error after the third updating process. Acknowledgement The authors acknowledge the support from Eskisehir Osmangazi University Scientific Research Projects (contract no: 201419A102).

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