The Use of Signal Filtering for Hog Inventory Estimation

Stephen Busselberg 

National Agricultural Statistics Service 

3251 Old Lee Hwy; Fairfax, VA 22030 

Abstract

The National Agricultural Statistics Service (NASS) uses probability surveys of hog owners to estimate quarterly 

hog inventories in the United States at the national and state levels. NASS receives data from external sources on the 

number of Canadian hog imports and exports; Canadian feeder pigs; and farm and commercial slaughter counts. A 

panel of commodity experts which forms the Agricultural Statistics Board (ASB) reviews the proprietary survey 

results and industry transaction data and compares them against a set of inter-inventory relationship constraints. 

Given the internal survey data, external transaction data, and the set of inventory relationship constraints, the ASB

establishes the NASS official published hog inventory estimates for the estimation quarter. The goal of this paper is

to propose the estimation of hog inventories by combining the NASS proprietary survey results, the non-proprietary 

hog transaction data, the ASB panel expert analysis, and the inter-inventory relationship constraints using 

statistically defensible methodology. In order to achieve this goal, this paper demonstrates the expression of hog

inventories in State-Space representation for use with an Extended Kalman Filter. Allocation of the U.S. level 

inventory estimates to the state level is formulated using Restricted Least Squares theory. 

Hog Estimation Overview

The current process used in hog inventory estimation has been stationary for many decades. The sheer length of this

probationary period leads to the question – why change the process now? The answer to this question requires a 

clear grasp of the scope of hog inventory estimation. To this end, the paper is structured to introduce fundamental 

concepts that provide a necessary foundation for understanding the current hog estimation process. This includes 

descriptions of the full spectrum of hog inventory items; background on the survey design and types of survey

estimates; details about the non-proprietary inventory transaction data and its sources, and some explanation as to

why the data provides a highly influential role in hog inventory estimation; a breakdown of the inter-inventory 

relationship constraints and their role in hog inventory estimation; information on the ASB and its origin and 

function in hog inventory estimation; and lastly, a brief introduction to key Office of Management and Budget 

(OMB) standards and guidelines for survey estimation. Once the fundamental details of hog inventory estimation 

have been conveyed, the paper will provide a brief overview of State-Space representation and the system equations

relevant to hog inventory estimation. Following the overview of the State-Space system equations, the paper will 

then derive the system equations that express hog inventories in State-Space form. Hog inventories expressed in

State-Space form will be used in conjunction with the Extended Kalman Filter in order to estimate those inventories 

given the survey results, non-proprietary inventory transaction data, inter-inventory relationship constraints, and the

ASB analysis. The paper will then cover hog inventory estimation at the U.S. and state levels followed by a 

comparison of empirical results calculated from three different parameterizations of the hog inventory system

equations. The three different parameterizations or “treatments” pertain to various ways of handling the ASB expert

analysis and its role in the estimation of hog inventories. 
 

                                                            

1 Published Hog Inventory Items 

This section provides an overview of the scope of published hog inventory items. It describes the level of detail at 

which inventories are published and how they relate.

The National Agricultural Statistics Service (NASS) publishes quarterly hog inventory estimates in terms of the 

number of hogs living on hog operations in a domain of reference at the end of that quarter. Hog owners, including 

contractors, are the target population. The quarters estimated are March, June, September, and December. The 

interpretation of a March inventory count means the number of hogs living on hog operations for the corresponding 

domain on March 1st. Likewise, a June inventory count refers to the number of hogs on June 1st. For a given domain 

and quarter of reference, hog inventories are provided for ten categories of inventory. The first of these categories is 

the sum total of all hogs and pigs. The total number of hogs and pigs is also partitioned into market weight group 

classes. The first of these market weight group classes is the number of market hogs weighing less than 50 lbs. The 

second group is those market hogs between 50 and 119 lbs. The third and fourth market weight groups are 

comprised of market hogs between 120 and 179 lbs, and market hogs over 180 lbs, respectively. The sum total of 

these four weight classes is reported as total market hogs. Additional categories cover hog reproduction and include 

the number births which survive weaning, the number of sows farrowed, and breeding herd size. The ratio of pig 

crop (weaned births) to sows farrowed is reported as the litter rate and can be interpreted as the mean number of pigs 

which survive past weaning born to a sow. The sum of the four weight classes equals the total number of market 

hogs, and the sum of total market hogs plus breeding herd equals the total number of hogs and pigs. Pig crop is 

contained within market hogs less than 50 lbs and market hogs 50-119 lbs. Sows farrowed are contained within 

breeding herd. In the estimation quarter of December 2009, the first two weight groups were redefined to the present 

day definitions. The first weight class of market hogs less than 50 lbs had been previously reported as market hogs 

less than 60 lbs, and the second weight class of market hogs between 50 and 119 lbs had been previously reported as 

market hogs between 60 and 119 lbs. Table 1 contains a summary list of the hog inventory items published at the 

U.S. and state levels in the National Agricultural Statistics Service’s quarterly Hog Report1

. 

Table 1 

 

 

 

 

 

 

Item Number Inventory Item Notation Relationship 

1 Total Hogs and Pigs H 

2 Pig Crop (weaned births) P 

3 Sows Farrowed S 

4 Market Hogs less than 50 lbs 罫怠

5 Market Hogs 50 – 119 lbs 罫態

6 Market Hogs 120 – 179 lbs 罫戴

7 Market hogs greater than 180 lbs 罫替

8 Market Hogs 警

替

布 罫賃

賃退怠

9 Breeding Herd 稽 茎伐警

10 Litter Rate T 鶏

鯨
 

 

 

 

 

2 Hog Survey Measurements 

This section covers the hog inventory survey including sampling frames, design, and types of estimators. 

Survey estimates of inventory are calculated from a stratified simple random sample design. The strata are 

partitioned according to size of hog operation with respect to the number of total hogs and pigs stored in NASS’s list 

frame. The sampling unit is any hog operation with the capacity to raise breeding or market hogs. In addition to the 

hog operation population list frame estimate referred to as the ADXX list frame survey estimate, there is also a 

multiframe estimate (ADMW). The ADMW estimate contains the inference for the hog operation population list 

frame plus an area frame component. The area frame component estimates the number of hogs belonging to owners 

who are Not On the List (NOL) frame. This NOL component is estimated using a separate area frame sampling 

design. The area frame survey is conducted on an annual frequency; however, the results are used to calculate 

quarterly estimates of the NOL component for total hogs and pigs, pig crop, sows farrowed, and breeding herd. The 

market weight group multiframe estimates are calculated on an annual basis in the quarter of December, and the 

other quarters are adjusted based on the December ADXX and NOL ratio. The NOL component adjusts for 

undercoverage of the list frame component. U.S. and state-level estimates of the variance of the survey estimates are 

also calculated according to the sampling design. Historical plots comparing NASS official ASB estimates and the 

ADXX and ADMW survey results are shown in Figure 1 through Figure 10. 

Figure 1 

 

 

Total Hogs Inventory

Number of Hogs Supressed

Official Published Estimate

List Frame Survey Result

Multiframe Survey Result

1990 1995 2000 2005 2010

Figure 2 

 

 

Pig Crop (Births) Inventory

Number of Hogs Supressed

Official Published Estimate

List Frame Survey Result

Multiframe Survey Result

1990 1995 2000 2005 2010

Figure 3 

 

 

Sows Farrowed Inventory

1990 1995 2000 2005 2010

Official Published Estimate

List Frame Survey Result

Multiframe Survey Result

Number of Hogs Supressed

Figure 4 
Figure 5 

 

 

Market Hogs 50-119 lbs Inventory

Official Published Estimate

List Frame Survey Result

Multiframe Survey Result

Number of Hogs Supressed

1990 1995 2000 2005 2010

Figure 6 

 

 

Market Hogs 120-179 lbs Inventory

Official Published Estimate

List Frame Survey Result

Multiframe Survey Result

Number of Hogs Supressed

1990 1995 2000 2005 2010

Figure 7 

 

 

Number of Hogs Supressed

Market Hogs 180+ lbs Inventory

Official Published Estimate

List Frame Survey Result

Multiframe Survey Result

1990 1995 2000 2005 2010

Figure 8 

 

 

 

Number of Hogs Supressed

Total Market Hogs Inventory

Official Published Estimate

List Frame Survey Result

Multiframe Survey Result

1990 1995 2000 2005 2010

Figure 9 

 

 

Number of Hogs Supressed

Breeding Herd Inventory

Official Published Estimate

List Frame Survey Result

Multiframe Survey Result

1990 1995 2000 2005 2010

Figure 10 

 

 

Pigs Per Sow Farrowed

Litter Rate

Official Published Estimate

List Frame Survey Result

Multiframe Survey Result

1995 2000 2005 2010

For each of the graphs, the differences between survey results and published inventory are attributed to the 

additional information provided by the non-proprietary inventory transaction data and a set of assumptions in the 

form of constraints on how inventory items relate one to another. These assumptions build the foundation for the 

system equations of the signal filter that will be derived in this paper. Actual survey results and their variances are 

never released to the public. 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3 Non-proprietary Inventory Transaction Data 

This section provides details on the inventory transaction data referred to in the previous section which NASS

obtains from external sources. This data is highly influential in the differences between the survey results and the 

NASS official published estimates determined by the ASB. The transaction data plays a key role in inter-inventory

relationship constraints that will be introduced in the next section.

The life of a hog from birth to slaughter is approximately six months. This implies that the reported pig crop in a 

given quarter is reflected in slaughter estimates two quarters later. Slaughter data is obtained from the Agricultural 

Marketing Service (AMS) and is partition into farm slaughter, federally inspected commercial slaughter, and nonfederally inspected commercial slaughter. According to AMS data, federally inspected commercial slaughter 

amounts to roughly 99% of total slaughter. This is significant because the majority of slaughter is federally 

inspected. Although NASS does not receive variance estimates for any of the hog commodity transaction data from 

outside sources, the slaughter estimates are assumed to have very low variance due to federal inspection. In addition 

to hog slaughter, NASS receives data on hog imports and exports to and from Canada from the Department of 

Commerce through the Foreign Agricultural Service. The transaction data is available at the U.S. level only. Table 2 

lists the inventory transaction data and the notation required to formulate the hog inventory constraints which will be 

given in section 4. 

Table 2 

 

 

 

Item Number Data Notation Function 

1 Slaughter L 

2 Imports I 

3 Exports E 

4 Canadian Feeder Pigs C 

5 Death Loss D 

6 Balance Sheet Net BSN 荊伐継伐経伐詣

4 Hog Inventory Relationship Constraints 

This section introduces the inter-inventory relationship constraints. These constraints are mathematical expressions

which relate inventory items to each other and to the external transaction data. The survey results are not published

because they do not satisfy these constraints.

The justification for publishing inventory estimates other than the survey results is the satisfaction of a set of 

assumed constraints. These constraints relate current inventory to past inventory, relate current and past inventory to 

the external transaction data, and reflect the hog growth cycle. The survey results do not satisfy the constraints. True 

hog and pig inventories are assumed to satisfy the constraints introduced in this section. The constraints will be 

given as mathematical expressions using the notation from Table 1 and Table 2. The subscript t will be used to index 

time, where the interval between 建 and 建髪倦 represents 倦 quarters, or 倦 consecutive three month intervals. The 

quarter of reference for 建噺な refers to the first quarter, and the quarter of reference 建噺券 refers to the last quarter in 

the time series. In general, 建噺券 refers to the most recent quarter of inventory estimates. All constraints pertain to 

the U.S. level of estimation. 

4.1 The Balance Sheet Equation 
 

 

 

Total hog inventory can be compared to a deposit account; an accounting system composed of deposits and 

withdrawals. Deposits are analogous to increases in inventory such as births and imports. Withdrawals are the 

decreases in inventory which consist of slaughter, exports, and death loss. This balance sheet concept forms the 

balance sheet equation where one quarter’s total hogs and pigs inventory is equal to the previous quarter’s total hogs 

and pigs inventory plus the quarter’s deposits minus the quarter’s withdrawals. The balance sheet equation is then 

茎痛 噺 茎痛貸怠 髪 鶏痛 髪 荊痛 伐 継痛 伐 経痛 伐 詣痛

噺 茎痛貸怠 髪 鶏痛 髪 稽鯨軽痛

Let 稽鯨迎痛

 be defined as the balance sheet residual at time 建 and 

稽鯨迎痛 噺 茎痛 伐 茎痛貸怠 伐 鶏痛 伐 稽鯨軽痛

If 稽鯨迎痛 噺 ど then 茎痛 噺 茎痛貸怠 髪 鶏痛 髪 稽鯨軽痛

 and the system is in balance. In setting published total hogs and pigs 

inventory, the balance sheet residual is allowed to vary at most by approximately one day’s slaughter (approximately 

500,000 hogs). It then follows that 

|稽鯨迎痛

| 判 のどど,どどど

|茎痛 伐 茎痛貸怠 伐 鶏痛 伐 稽鯨軽痛

| 判 のどど,どどど

We will now define the three month, six month, and twelve month balance sheet constraints as 

懲

保布茎痛貸賃 伐 茎痛貸怠貸賃 伐 鶏痛貸賃 伐 稽鯨軽痛貸賃保 判 のどど,どどど

賃退待

(1) 

with 計噺ど, 計噺な, and 計噺ぬ, respectively. 

Figure 11 

 

 

3-Month Balance Sheet Residuals

-10000 -5000 0 5000 10000

Official Published Estimate

List Frame Survey Result

Multiframe Survey Result

Constraint Bounds

Thousands of Hogs

1990 1995 2000 2005 2010

Year

Figure 12 
 

 

 

 

 

 

 

Figure 13 

 

 

12-Month Balance Sheet Residuals

-10000 -5000 0 5000 10000

Official Published Estimate

List Frame Survey Result

Multiframe Survey Result

Constraint Bounds

Thousands of Hogs

1990 1995 2000 2005 2010

Year

Figure 11, Figure 12, and Figure 13 plot the 3-month, 6-month, and 12-month balance sheet residuals for the 

ADXX (list frame) and ADMW (multiframe) survey results. The balance sheet residuals of the published ASB 

estimates are included for comparison with those of the survey results. The plots demonstrate the ASB’s attempt to 

contain the balance sheet residuals of the published estimates within the 500 thousand hogs bounds. 

4.2 Death Loss Ratio 

The concept of the Death Loss Ratio is to acknowledge that there is a quantity of pig crop that dies and therefore 

cannot be counted in the market weight groups. These pigs survive past weaning and are within scope of the 

definition of pig crop. The weight of pigs born during a quarter is distributed between the first and a proportion of 

the second weight group. We will call that proportion 糠. If we look at these concepts in terms of annual increases, 

we have 

鶏痛 髪 系痛 罫怠禰 髪 糠痛罫態禰

鶏痛貸替 髪 系痛貸替

伴

罫怠禰貼填 髪 糠痛貸替罫態禰貼填

(2) 

Canadian Feeder Pigs are grouped with the births. Conceptually, this expression conveys that the annual increase in 

the number of pigs born and are weaned is greater than the annual increase in the first two market weight groups. 

The inequality implies disappearance from the weight group increase due to death loss after weaning that quarter. 

The value for 糠痛

 is time dependent due to a change in definition of the first two weight groups which happened in 

December 2009. Prior to December 2009, weight group 1 consisted of those market hogs weighing less than 60 lbs. 

Weight group 2 was composed of those market hogs weighing between 60 and 119 lbs. The parameter 糠痛

 is 

evaluated as follows: 

糠痛 噺 峽ど.ぬぬ 建 is prior to にどどぱ

ど.ねに otherwise

The values for alpha were determined by commodity analysts. The current commodity analysts enforce bounds 

which give the following Death Loss “Difference” constraint: 

鶏痛 髪 系痛 罫怠禰 髪 糠痛罫態禰

ど.どどねな 判 伐 判 ど.どどねぬ 鶏痛貸替 髪 系痛貸替 罫怠禰貼填 髪 糠痛貸替罫態禰貼填

(3) 

The Death Loss Ratio constraint in equation (4) is the true ratio version of the Death Loss Difference in equation (3). 

Figure 14 plots the historical Death Loss Ratio of the published inventory items. 
鶏痛 髪 系痛 罫怠禰 髪 糠痛罫態禰

な.どどねな 寄 峭 嶌

貸怠

寄 な.どどねぬ 鶏痛貸替 髪 系痛貸替 罫怠禰貼填 髪 糠痛貸替罫態禰貼填

 

 

 

 

 

(4) 

Figure 14 

Death Loss Ratio

0.99 1.00 1.01 1.02 1.03

Official Published

1.0037

1.0047

1990 1995 2000 2005 2010

Year

Figure 15 graphs the ASB death loss relationship by comparing the right side of equation (2) versus the left. Figure 

16 graphs the left and the right sides of equation (2) using the ADXX survey result. 

Figure 15 

Pt  Ct G1t

 tG2t



Pt4  Ct4 G1t4

 t4G2t4

0.95 1.00 1.05 1.10

Pt  Ct

Pt4  Ct4

G1t  tG2t

G1t4 t4G2t4

1990 1995 2000 2005 2010

Figure 16 

Pt  Ct G1t

 tG2t



Pt4  Ct4 G1t4

 t4G2t4

0.95 1.00 1.05 1.10

Pt  Ct

Pt4  Ct4

G1t  tG2t

G1t4 t4G2t4

1990 1995 2000 2005 2010

4.3 Weight Group Transition 

Where the Death Loss Ratio maps hog births to their corresponding weight classes during the quarter, the Weight 

Group Transition constraint maps those births and their weights to the heavier weight groups the following quarter. 

The Weight Group Transition constraint is an assumption about the growth of pigs within weight classes. It links the 

lighter two weight classes to the heavier two weight classes over the passing of a quarter. 

岫な伐糠痛

岻罫態禰 髪 罫戴禰 髪 罫替禰 半

罫怠禰貼迭 髪 糠痛貸怠罫態禰貼迭

岫な 伐 糠痛貸替岻罫態禰貼填 髪 罫戴禰貼填 髪 罫替禰貼填 罫怠禰貼天 髪 糠痛貸泰罫態禰貼天

(5) 

This constraint implies that the annual increase in weight groups three, four and a proportion of the second is the 

annual increase in weight group one and a proportion of the second one quarter in the past. 
 

 

 

 

 t4 t4 t4 t5  t5

1  tG2tG3tG4t

1  t4G2t4G3t4G4t4

G1t1 t1G2t1

G1t5 t5G2t5

0.95 1.00 1.05 1.10

1990 1995 2000 2005 2010

1 t4G2t4 G3t4G4t4 G1t5t5G2t5

1  tG2tG3tG4t

1  t4G2t4G3t4G4t4

G1t1 t1G2t1

G1t5 t5G2t5

0.96 0.98 1.00 1.02 1.04 1.06

1990 1995 2000 2005 2010

Figure 17 

1  tG2t G3tG4t G1t1t1G2t1



1 t 4G2  G3 G4 G1 t 5G2

Figure 18 

1  tG2t G3tG4t G1t1t1G2t1



Figure 17 plots the right side and left side of the inequality (5) using the ASB values. The right side of inequality (5) 

is the lagged version of the right side of equation (2). Figure 18 plots the right side and left side of the inequality (5) 

using the list frame ADXX values. The Death Loss Ratio constraint and the Weight Group Transition constraint 

represent the flow of hogs from pig crop births through the market weight groups until slaughtered. 

4.4 Pig Crop – Slaughter Ratio 

The time between birth and slaughter for a pig is approximately six months or two quarters. This implies that hogs 

born in quarter 建 are slaughtered in quarter 建髪に. The commodity analysts translate this concept into a ratio 

constraint where the annual increase in slaughter is equivalent to the increase in births two quarters in the past. This 

constraint is formulated as 

詣痛 噺

鶏痛貸態

詣痛貸替 鶏痛貸滞

(6) 

Figure 19 
Figure 19 graphs the left side slaughter ratio versus the right side of equation (6) with substituted ASB values, and 

Figure 20 adds substituted survey values for the right side of equation (6) for comparison of the survey performance 

versus the ASB published estimates. 

4.5 Market Hogs – Slaughter Ratio 

Constraint 4.4 is extended to include all market hogs by the annual increase in six months of slaughter. 

詣痛 髪 詣痛貸怠 噺

∑替

沈退怠 罫沈禰貼鉄 噺

警痛貸態

詣痛貸替 髪 詣痛貸泰 ∑替

珍退怠 罫珍禰貼展 警痛貸滞

(7) 

Equation (7) encompasses all four weight classes and conveys that the annual increase in total hogs with the 

exception of those hogs reserved for breeding is essentially the annual incease in two quarters of slaughter. Figure 

21 plots the left and right sides of equation (7) in terms of the ASB published values. Figure 22 substitutes the 

ADXX list frame survey results into the right side of equation (7)
 

 

 

 

Figure 23 

 

 

5WK

t1

5WK

L G4t



Lt3

G4t4

0.95 1.00 1.05 1.10

5 Week Slaughter Ratio (L)

Market Hogs 180+ Ratio

1995 2000 2005 2010

Figure 24 

 

 

 

5WK

t1

5WK

L G4t



Lt3

G4t4

0.95 1.00 1.05 1.10

5 Week Slaughter Ratio (L)

Market Hogs 180+ ASB Ratio

Market Hogs 180+ ADXX Ratio

1995 2000 2005 2010

Figure 23 compares the left and right sides of equation (8) substituting ASB values. Figure 24 substitutes the list 

frame ADXX survey result into the right side of equation (8)
 

 

 

 

 

 

 

 

 

 

 

 

 

survey results described in section
 

 

 

 

 

 

 

 

 

 

 

                                                            

     

NASS surveys its sampling frame of hog operations, obtains non-proprietary hog inventory transaction data, and 

submits these data to the ASB panel with the purpose of establishing quarterly hog inventory estimates that make 

sense from an historical standpoint, reflect congruency with inventory transaction data, and maintain inter-inventory 

consonance. This paper will demonstrate that the goals of NASS and the ASB, and the goals of the OMB Standards 

and Guidelines can be achieved using an Extended Kalman Filter. The Kalman Filter is a signal filtering tool for 

which there is an abundance of literature, supporting its use as statistically defensible methodology. Durbin and 

Koopman write, “the object of filtering is to update our knowledge of the system each time a new observation is 

brought in (Durbin and Koopman 2012).” Filtering is a methodology which can combine all observations of hog 

inventory; including the survey measurements from section 
 

 

 

 

 

8.2 The Observation Equation 

The observation equation relates a set of measurements or observations of the state to the state. In this paper, 

“observations” and “measurements” will be used interchangeably. The linear observations are modeled by the 

relationship 

検痛 噺 畦痛捲痛 髪 懸痛

(12) 

where 検 is a 倦抜な vector of measurements, 畦 is a 倦抜兼 measurement matrix which defines the linear relationship 

between the state 捲痛

 and the observations 検痛, and 懸痛 is a 倦抜な observation noise vector with a Gaussian distribution. 

迎 倹噺ど The first and second moments are 継岷懸痛

峅 噺 ど 褐 建 and 継範懸 懸

嫗

飯噺犯 ad 継範拳 嫗

痛 痛貸珍 . In dition, 痛貸沈懸痛貸珍飯噺ど 褐 件, 倹. If ど 倹塙ど

the relationship between the measurements and the state is nonlinear, equation (13) is used where 茎岫·岻 represents 

the system of nonlinear measurement relationships as a function of the state. 

検痛 噺 茎痛

岫捲痛岻 髪 懸痛

(13) 

9 Hog Inventory Transition Equations 

This section derives the hog inventory State-Space transition equations by putting equations (10) and (11) in terms 

of the published hog inventory items from Table 1 so that hog inventories can be estimated using the Extended 

Kalman Filter. This will be done using the constraints listed in section 4.

In order to estimate hog inventories using a Kalman Filter, they must be formulated in State-Space representation. 

This involves defining the state vector from the published items in Table 1, determining the parametric transition 

relationships for each state vector element that conform to the transition equations (10) and (11), and determining 

the parametric relationships of all measurements and observations to the state which conform to the observation 

equations (12) and (13). The constraints listed in section 4 define the behavior rules that can be adapted into StateSpace form which will be demonstrated in this section. Before establishing the state vector, we must define some 

weight group functions and a linear filter operator which will assist in formulating the transition relationships. We 

define the weight group functions 血怠

, 血態

血怠禰 噺 罫怠禰 髪 罫態禰

血態禰 噺 ln 磐罫怠禰 髪 糠痛罫態禰

卑

鶏痛 髪 系痛

血戴禰 噺 ln範岫な伐糠痛

岻罫態禰 髪 罫戴禰 髪 罫替禰

飯

血替痛 噺 ln盤罫替禰

匪

, 血戴, and 血替 as 

(14) 

(15) 

(16) 

(17) 

These functions should be familiar from the terms in the hog inventory constraints in section 4. We will use them to 

develop state transition relationships. In addition, we will use the linear filter operator of the first and fourth 

difference 

∆岫捲痛

岻 噺 捲痛 伐 捲痛貸怠 伐 捲痛貸替 髪 捲痛貸泰 (18) 

Given the weight group function equations (14) - (17) and the linear filter equation (18), the hog inventory state

vector elements and corresponding transitions are listed in Table 3 and Table 4, respectively. 
 

 

Table 3 

 

 

 

 

 

Element of 姉嗣

Item Values of k (x) 

1 ∆岫茎痛貸賃岻 倦 樺 岶ど,な,に,ぬ岼

2 ∆岫鶏痛貸賃岻 倦 樺 岶ど,な,に,ぬ岼

3 ∆岫鯨痛貸賃岻 倦 樺 岶ど,な,に,ぬ岼

4 ∆盤血怠禰貼入匪 倦 樺 岶ど,な,に,ぬ岼

5 茎痛貸賃 倦 樺 岶な,に,ぬ,ね,の岼

6 鶏痛貸賃 倦 樺 岶な,に,ぬ,ね,の岼

7 鯨痛貸賃 倦 樺 岶な,に,ぬ,ね,の岼

8 血怠痛貸賃 倦 樺 岶な,に,ぬ,ね,の岼

9 血態禰貼入 倦 樺 岶ど,な,に,ぬ,ね,の,は岼

10 血戴禰貼入 倦 樺 岶ど,な,に,ぬ,ね,の,は岼

11 血替禰貼入 倦 樺 岶ど,な,に,ぬ,ね岼

12 憲掴禰

捲 樺 岶茎, 鶏, 鯨, 血怠, 血態, 血戴, 血替

岼

Table 4 

 

Item Section Element Transition Function 

1 9.1 ∆岫茎痛

岻 ∆岫茎痛

岻 ∆岫茎 岻 痛貸沈 欣 禁 替 欣 禁 ∆岫鶏痛

岻 ∆岫鶏 岻 痛貸沈 噺 布漸

欽

欽

欽

∆ 兄 岫鯨痛

岻

禽

禽

禽

欽

欽

欽

岻

禽

禽

禽

∆岫鯨痛貸沈

辿退怠 琴∆盤血怠禰

匪筋 琴∆盤血怠禰貼 匪

日

筋

2 9.1 ∆岫鶏痛

岻

3 9.1 ∆岫鯨痛

岻

4 9.1 ∆盤血怠禰

匪

5 9.2 血態禰

血態禰 噺 血態禰貼填 伐 ln岫な.どどねに岻

6 9.3 

血戴禰

結

捗鉄禰貼迭 岫鶏痛貸怠 髪 系痛貸怠岻

血戴禰 噺 ln 峪

結

捗鉄禰貼天 岫鶏痛貸泰 髪 系痛貸泰岻

崋 髪 血戴禰貼填

7 9.4 血替禰 血替禰 噺 ln盤血怠禰貼迭匪 伐 ln盤血怠禰貼天匪 髪 血替禰貼填

8 10.2 憲掴禰

憲掴禰 噺 憲掴禰貼迭

The additive process noise term 拳痛 has been omitted for convenience from Table 4. In sections 9.1-9.4 we will 

develop justification for the transition relationships with the exception of Table 4 item 8 which is an estimate of the 

true survey bias which will be introduced in section 10.2. The true survey bias 憲 of inventory item 捲 defined as 憲掴

must be introduced here because it will be estimated and therefore must be defined as part of the state and given a 

transition. 

9.1 Hogs, Pig Crop, Sows Farrowed, and Weight Group Function 讃層

We transition total hogs, pig crop, sows farrowed, and 血怠

 using a Vector Autoregressive (VAR) model. Figure 26 

illustrates the VAR transition model step-ahead predictions compared to ASB published numbers. 
Figure 26 

VAR Fit: H

55000 60000 65000

Published

1 Step-Ahead Prediction

95.45% CI

1990 1995 2000 2005 2010

VAR Fit: P

22000 24000 26000 28000 30000

Published

1 Step-Ahead Prediction

95.45% CI

1990 1995 2000 2005 2010

2700 2800 2900 3000 3100 3200 3300 3400

30000 32000 34000 36000 38000

VAR Fit: S VAR Fit: F1

Published Published

1 Step-Ahead Prediction

95.45% CI

1 Step-Ahead Prediction

95.45% CI

1990 1995 2000 2005 2010 1990 1995 2000 2005 2010

From visual inspection, the VAR model appears to be a reasonable transition. A 95.45% (two standard deviations of 

a Gaussian distribution) confidence interval is included to demonstrate the error term variance estimate. The VAR 

error term is analogous to the process noise in State-Space representation. For a simple comparison of the VAR 

prediction performance between inventory items, we can look at a bar plot of Coefficients of Variation calculated by 

the square-root of the VAR error term variance estimates divided by the series’ means. 

Figure 27 

e

N  100 xt

Percent

0.0 0.5 1.0 1.5 2.0

H P S F1

Figure 27 supports the general assumption that total hog and pigs is the “most stable” item to model at the U.S. 

level, meaning that the estimate of the standard error of the noise process is the smallest relative to the mean size for 

total hogs and pigs in comparison to the other inventory items’ process noise standard errors and mean sizes. It 
 

 

 

 

 

 

 

should also be noted that the Vector Autoregressive order is fixed at 4. The order that provides the minimum for 

canonical fit statistics such as Akaike’s Information Criterion (AIC) changes depending on the time window used to 

estimate the VAR model. Common AR orders of 3 and 4 as well as a couple outlier orders have been empirically 

produced. The order 4 was chosen because it was the mode and is more intuitive; an order of 4 will transition one 

year of data. 

9.2 Weight Group Function 讃匝

Weight group function 血態

 and its transition are derived from the Death Loss Ratio constraint in section 4.2. Defining 

膏痛

 as the death loss ratio at time 建, we can rewrite the Death Loss Ratio as a true ratio (commodity analysts calculate 

it as a difference) in terms of the weight group function 血態

. 

貸怠

膏痛 噺 磐 鶏痛 髪 系痛

卑 峭 罫怠禰 髪 糠痛罫態禰嶌

鶏痛貸替 髪 系痛貸替 罫怠禰貼填 髪 糠痛貸替罫態禰貼填

噺 結捗鉄禰貼填貸捗鉄禰

(19) 

The death loss ratio in equation (19) is expressed in terms of true hog and pig inventories. We can collect all terms 

of equation (19) at time 建 伐 倦|倦 噺 ど on the left side of the equation and all terms at time 建 伐 倦|倦 伴 ど on the right 

side of the equation which yields equation (20). 

罫怠禰 髪 糠痛罫態禰 噺

な 罫怠禰貼填 髪 糠痛貸替罫態禰貼填 結

栂禰

鶏痛 髪 系痛 膏痛 鶏痛貸替 髪 系痛貸替

(20) 

We have added multiplicative process noise term 結

栂禰

. By taking the natural log of both sides of equation (20), we 

can model a linear transition of 血態

 with the process noise error term 拳痛 in state space representation by 

ln岫膏痛

岻 噺 ln岫膏痛貸怠岻 髪 拳碇禰

血態禰 噺 血態禰貼填 伐 ln岫膏痛

岻 髪 拳痛 (21) 

(22) 

Equation (21) incorporates the natural log of the death loss ratio as part of the state vector with its own random walk 

transition and process noise in equation (22). The death loss ratio is essentially formulated here as a time variant 

level in the state vector. This parameterization is more reflective of the true historical death loss ratio in Figure 14. 

However, to conform to the assumptions of the current commodity analysts, we will assume the death loss ratio 膏痛

 is 

distributed symmetrically with constant mean value between its bounds of 1.0041 and 1.0043. Therefore, 継岷膏痛

峅 噺

な.どどねに. We replace 膏痛

 in the 血態 transition with the fixed value of its expectation 1.0042 and the 血態 transition 

becomes 

血態禰 噺 血態禰貼填 伐 ln岫な.どどねに岻 髪 拳痛 (23) 

Figure 28 plots the official published ASB values for 血態

 and the expectation of its transition in equation (23). It is 

more intuitive to examine the plot of 結

捗鉄禰 岫鶏痛 髪 系痛

岻 噺 罫怠禰 髪 糠痛罫態禰

 in Figure 29
 

Figure 28 

f2  1.00421f2t4wt

f2t

Ef2t

1990 1995 2000 2005 2010

Figure 29 

G1t tG2t 1.00421f2t4 wtPt Ct

f2tPt  Ct

Ef2tPt  Ct

1990 1995 2000 2005 2010

Canadian Feeder Pigs 系痛

 is treated as a fixed parameter. The value is known as it is supplied as external inventory 

transaction data. 

9.3 Weight Group Function 讃惣

The weight group function 血戴

 transition follows a similar derivation to血態. The constraint 4.3 can be written in terms 

of 血態

 and 血戴. 

結

捗典禰

半

結

捗鉄禰貼迭 岫鶏痛貸怠 髪 系痛貸怠岻

結

捗典禰貼填 結捗鉄禰貼天 岫鶏痛貸泰 髪 系痛貸泰岻

(24) 

Collection of terms at time 建 伐 倦|倦 噺 ど on the left and 建 伐 倦|倦 伴 ど on the right followed by taking the natural log 

of both sides of equation (24) yields the transition equation 

血戴禰 噺 ln 峪結

捗鉄禰貼迭 岫鶏痛貸怠 髪 系痛貸怠岻

結

捗鉄禰貼天 岫鶏痛貸泰 髪 系痛貸泰岻

崋 髪 血戴禰貼填 髪 拳痛

(25) 

The model noise process is multiplicative with 結

栂禰

. Figure 30 illustrates the official published ASB estimate of 血戴

and its expected transition. 

Figure 30 

 
 

 

 

 

 

 

 

 

 

9.4 Weight Group Function 讃想

The transition for weight group 4 from Table 4 is not derived from any of the constraints from section 4. 

罫替禰 噺

罫怠禰貼迭 髪 罫態禰貼迭 罫_ 罫怠禰貼天 髪 罫態禰貼天

ね痛貸替 (26) 

If we take the natural log of both sides of (26) with the multiplicative error term 結

栂禰

, we have the transition equation 

血替禰 噺 ln盤血怠禰貼迭匪 伐 ln盤血怠禰貼天匪 髪 血替禰貼填 髪 拳痛

(27) 

Its performance is illustrated below in Figure 31. 

Figure 31 

f1t1

G4t G4t4 wt

f1t5

G4

EG4

1990 1995 2000 2005 2010

10 The Hog Inventory Observation/Measurement Equations 

This section derives the hog inventory State-Space observation equations by putting equations (12) and (13) in terms 

of the published hog inventory items from Table 1 so that hog inventories can be estimated using the Extended 

Kalman Filter. This will be done using the constraints listed in section 4.

In order to put the measurements of inventory into the observation system of equations, we group the observations 

related to hog inventories into three categories according to source. We are using signal filtering methodology to 

estimate an unobserved signal for which we have “noisy” measurements. These measurements include the published 

ASB estimates, the survey results, and the non-proprietary hog inventory transaction data. We will categorize the 

ASB measurements of inventory as “expert analysis” measurements. The three categories of hog inventory 

observations are therefore expert analysis measurements, survey measurements, and external inventory transaction 

data. The categorization into these three groups helps distinctly separate how the types of measurements are treated. 

The relationships between inventory transaction data and hog inventories are defined in the observation equations 

through the relationship constraints introduced in section 4. Farm and commercial slaughter, hog imports and 

exports from Canada, and death loss estimates comprise this group. The survey results are treated as biased 

measurements of true inventories. The question remains how to treat the expert analysis measurements relative to 

true inventory. Sections 10.1 - 10.3 address the parameterization of each observation category. 
 

 

 

 

 

 

10.1 Expert Analysis Estimates and True Inventory 

The current hog inventory estimation process involves a panel of experts that take all given hog inventory data from 

internal surveys and external sources in order to find a solution that satisfies a set of assumptions in the form of 

constraints. This paper does not argue the validity or appropriateness of the constraints; rather, it proposes to 

incorporate the current process into a signal filtering model analog of the current hog estimation process. In order to 

maintain a smooth transition from the published board inventory estimates to published signal filter model estimates, 

the historical published ASB estimates must be included in the observation vector. Assuming that hog inventories 

will be estimated indefinitely, that particular observation or measurement must be continued. An example of 

methodology that eliminates ASB measurements and at the same time continues from where the ASB ends is to 

include a measurement in the observation vector that is the published measurement. The published inventory 

measurements are the ASB measurements before implementation of signal filtering methodology, and the model 

estimates become the published measurements post implementation of the Filter. This parameterization does require 

that certain complications be addressed, such as published revisions as new slaughter data and other data become 

available. In addition, model output becomes an observation, which is model input. An alternative approach is to 

continue using the ASB measurement as an “expert analysis” measurement. The previously published inventory 

becomes the past expert analysis measurements in the observation vector. Instead of publishing that ASB 

measurement, it is put into the signal filter with the other survey measurements and observation data. The Kalman 

Filter estimates are published together with their standard errors in the official release. Sections 10.1.1, 10.1.2, and 

10.1.3 discuss various treatments and parameterizations of the ASB expert analysis measurements. 

10.1.1 ASB Expert Analysis as an Unbiased Observation 

Incorporating Signal Filtering methodology as an analogue of the current estimation process implies continuing to 

supply an ASB measurement. There are many ways this can be done. The inclusion of ASB inventory measurements 

allows expert opinion the possibility to exert some influence in a statistically defensible model. One concern is the 

possibility that expert opinion will exert too much influence on the filter estimates relative to the other 

measurements so that filter estimates are simply “perturbed” ASB estimates. This concern will be addressed in later 

sections with model results for various proposed parameterizations. 

In order to establish a relationship between true inventory and measurements of true inventory, we must make some 

assumptions about true inventory. The ASB assumes that the survey results are biased and therefore publishes its 

own estimates of inventory rather than the survey results. Let us first assume that the ASB expert analysis estimates 

of true inventory are unbiased estimates of inventory. This is parameterized as

捲痛

凋聴喋 噺 捲痛 髪 懸痛

(28) 

where 捲 樺 岶茎, 鶏, 鯨岼. Another implication of this parameterization is 懸欠堅岫捲

凋聴喋

痛

岻 噺 迎. The variance estimate of the 

observation noise is parameterized as an estimate of the variance of the ASB estimate. Equation (28) is 

straightforward for total hogs, pig crop and sows farrowed. The four market weight groups add complexity because 

they are included as nonlinear functions in the state. If we use the ASB estimates in the weight group functions as 

the measurements for the weight groups, we can eliminate the nonlinearities in the system equations. 

For 血怠

, we have 

凋聴喋 髪 罫態禰凋聴喋 罫怠 噺 罫怠禰 髪 罫態禰 髪 懸痛 禰

凋聴喋 血怠 噺 血怠禰 髪 懸痛 禰

(29) 

For 血態

, we have 
 

 

 

 

 

 

 

 

凋聴喋 髪 糠痛罫態禰凋聴喋 n 磐罫怠禰 髪 糠痛罫態禰

ln 峭罫怠禰

凋聴喋 嶌噺l 卑髪懸痛

鶏痛 髪 系痛 鶏痛 髪 系痛

凋聴喋 血態 噺 血態禰 髪 懸痛 禰

(30) 

For 血戴

, we have 

凋聴喋 髪 罫戴禰凋聴喋 髪 罫替禰

ln範岫な伐糠痛

岻罫態禰

凋聴喋飯 噺 ln範岫な伐糠痛

岻罫態痛 髪 罫戴痛 髪 罫替禰

飯髪懸痛

凋聴喋 血戴 噺 血戴禰 髪 懸痛 禰

(31) 

For 血替

, we have 

ln盤罫替禰

凋聴喋匪 噺 ln盤罫替禰

匪髪懸痛

凋聴喋 血替 噺 血替禰 髪 懸痛 禰

(32) 

10.1.2 ASB Expert Analysis as a Biased Observation 

An argument can be made that the ASB estimates are also biased. If this is the case, we can also treat the bias as an 

unobserved signal and measure it in the state. Inclusion of the bias in the state requires a transition model for the 

bias terms in the system of transition equations. This parameterization is important in the analysis of the 

decomposition and the influence of ASB measurements in the observation vector which will be shown in section 13. 

We will be able to compare the influence of the ASB measurements on the estimates in the case that the ASB is 

treated as unbiased against the case in which the ASB measurements are allowed to contain possible bias. We 

express biased ASB measurements as 

捲痛

凋聴喋 噺 捲痛 髪 決掴禰 髪 懸痛

決掴禰 噺 決掴禰貼迭 髪 拳痛

(33) 

where 決掴禰

 is the bias term of item 捲 and is transitioned as a random walk. 

For the initial conditions of the filter, we set the bias parameters of all ASB measurements of inventory items to 

zero, allowing the filter to assess the biases starting with an initial assumption of zero bias. 

10.1.3 Published Inventory as an Observation 

Transitioning operationally from ASB published to Filter published is possible without the inclusion of ASB 

measurements of inventory when the published is treated as an unbiased measurement of inventory. The ASB sets 

an estimate for the current quarter and makes revisions to past estimates up to two quarters back in time. The 

backward revisions are due to the new slaughter data which give information on those hog inventories two quarters 

in the past. This can be implemented by the Filter operationally if the Kalman Filter estimates two quarters or more 

in the past become the observations. This is essentially the equivalent of using the fixed lag smoother 継岷捲痛貸態|検痛

峅 as 

a measurement in the observation vector. Given that 建噺倦 is the quarter of the last published ASB estimates before 

implementation of the Kalman Filter and 建噺券 is the most recent estimate quarter, the published inventory 

observation is defined below in equation (34). 
牒腸喋 捲痛 噺 捲痛 髪 懸痛

捲痛

凋聴喋 な判建判倦

捲痛

牒腸喋 噺 崔継岷捲痛

|検痛袋態峅 倦隼建隼券伐に

兼件嫌嫌件券訣 建 半 券 伐 に

 

(34) 

For any time point equal to or more recent than two quarters in the past relative to the target quarter of estimation 

建噺券, the published observation is coded as missing in the Kalman Filter. This is because a fixed lag smoother of 

lag 2 is not yet available for 建 樺 岶券 伐 な, 券岼 and the observation will be created for 建噺券伐に during the quarter. 

10.2 Survey Estimates and True Inventory 

Figure 1‐Figure 10 illustrate the degree of bias between the ASB estimate and the survey estimates for each 

inventory item. That bias demonstrated by the difference between survey results and the ASB published inventory 

estimates changes over time. In order to account for the bias in the survey measurements, we estimate it as part of 

the state. If 捲

凋帖諜諜

痛

 represents the ADXX survey result for inventory item 捲, we write 

捲痛

凋帖諜諜 噺 捲痛 髪 憲掴禰 髪 懸痛

(35) 

The above representation of the survey result for inventory item x shows the decomposition into the true inventory 

plus the bias term plus an observational noise process. This is a similar decomposition to the treatment of the biased 

ASB estimate in section 10.1.2 except that we use 憲 to represent the survey bias and 決 to represent the ASB bias. As 

the bias is part of the state, we also need a transition model which defines how the bias is correlated over time (if at 

all). If we assume that the ASB published estimate for total hogs and pigs is true inventory, the bias for the ADXX 

list frame survey result for total hogs and pigs can be modeled with an ARIMA(1,0,0)x(0,1,1)4. This bias model fit 

is demonstrated in Figure 32. 

Figure 32 

ADXX Total Hogs and Pigs Bias

ADXX Bias

Bias Prediction

1  z

4

1  zXt  1  z

4

Wt

1990 1995 2000 2005 2010

Time

Figure 33 

ACF Hog Bias Residuals PACF Hog Bias Residuals

-0.2 0.0 0.2 0.4 0.6 0.8 1.0

ACF

-0.2 -0.1 0.0 0.1 0.2

Partial ACF

0 1 2 3 4 5 1 2 3 4 5

Lag Lag

An ARIMA(0,1,0)x(0,1,1)4 could slightly reduce Akaike’s Information Criterion for the fit, however, does not fully 

remove autocorrelation in the residuals. Figure 33 gives the autocorrelation function and partial autocorrelation 

function of the residuals of the model fit from Figure 32, thereby providing evidence that this model removes 

significant autocorrelation structure. 
 

 

 

 

 

 

 

Pig crop and sows farrowed ADXX biases could be modeled similarly conditioned on the assumption that the ASB 

is unbiased. This ARIMA model does add significant complexity to the state-space model, particularly in 

dramatically increasing the dimension of the state in order to reflect this transition over multiple inventory items. 

Additionally it would require estimation of the measurement matrix in order to reflect the seasonal moving average 

terms. It is more reasonable to assume that the ASB estimates are in fact estimates of true inventory and may 

possibly contain biases of their own. In light of this assumption, we would not want to use a transition model that 

reflected biased bias terms. For the sake of model parsimony, we represent the survey bias term transitions for all 

survey inventory items with a random walk. Equation (36) is the transition of the survey bias term. 

憲掴禰 噺 憲掴禰貼迭 髪 懸痛 (36) 

For the survey litter rate, we base the observation equation on the ASB assumption 

凋帖諜諜 継 峪鶏

鯨

痛

痛

凋帖諜諜崋 蛤 鶏

鯨痛

痛

This implies that the ASB believes the survey litter rate is unbiased. The observation equation for the litter rate with 

multiplicative observational noise becomes 

鶏痛

凋帖諜諜

鯨痛

凋帖諜諜 噺

鯨

鶏

痛

痛

結

塚禰

ln岫劇痛

凋帖諜諜岻 噺 ln岫鶏痛

岻 伐 ln岫鯨痛岻 髪 懸痛

(37) 

The relationship between the ASB litter rate and the survey litter rates is shown in Figure 34. 

Figure 34 

Litter Rate

Published

ADMW

ADXX

1990 1995 2000 2005 2010

10.3 External Data and True Inventory 

Section 4 listed all of the hog inventory constraints. These constraints define how inventories relate to the various 

inventory items, and also how they relate to a number of external data items. The death loss ratio constraint from 4.2 

and the weight group transition constraint from 4.3 were included in the transition equation. These constraints 

involved transitional relationships of the multidimensional state over time. The remainder of the constraints from 

section 4 involves external transaction data and can be expressed in the observation equation. 
 

 

 

 

 

10.3.1 Balance Sheet Equation 

The balance sheet constraint was introduced in section 4.1 and the balance sheet net was defined in Table 2 of 

section 3. The balance sheet relationship is 

茎痛 噺 茎痛貸怠 髪 鶏痛 髪 稽鯨軽痛 髪 稽鯨迎痛

(38) 

If we define 稽鯨迎痛 噺 伐懸痛, the above can be written as 

稽鯨軽痛 噺 茎痛 伐 茎痛貸怠 伐 鶏痛 髪 懸痛

(39) 

The balance sheet equation is now in state-space form with the observation noise process equated to the balance 

sheet residual. In terms of elements of the state vector, the three month balance sheet measurement equation is 

稽鯨軽痛 噺 ∆岫茎痛

岻 髪 茎痛貸替 伐 茎痛貸泰 伐 ∆岫鶏痛岻 伐 鶏痛貸怠 伐 鶏痛貸替 髪 鶏痛貸泰 髪 懸痛

(40) 

The six month balance sheet measurement equation is 

怠

布 稽鯨軽痛貸賃 噺 ∆岫茎痛

岻 髪 茎痛貸怠 髪 茎痛貸替 伐 茎痛貸泰 伐 茎痛貸態 伐 ∆岫鶏痛岻 伐 に鶏痛貸怠 伐 鶏痛貸替 髪 鶏痛貸泰 髪 懸痛

賃退待

(41) 

The twelve month balance sheet measurement equation is 

戴

布 稽鯨軽痛貸賃 噺 ∆岫茎痛

岻 髪 茎痛貸怠 伐 茎痛貸泰 伐 ∆岫鶏痛岻 伐 に鶏痛貸怠 伐 鶏痛貸態 伐 鶏痛貸戴 伐 鶏痛貸替 髪 鶏痛貸泰 髪 懸痛

賃退待

(42) 

10.3.2 Breeding Herd “Smoother” 

Stricter constraints on the market weight groups and total hogs and pigs cause breeding herd to absorb some of the 

noise process due to the deterministic relationship they share. When uncontrolled, it is manifested in high frequency 

oscillations. Contrary to what these high frequency oscillations suggest, breeding herd should be a more stable 

inventory item, meaning that it does not change dramatically from quarter to quarter. This can be represented as an 

observation constraint where the change in breeding herd has “fixed” bounds. The measurement for this constraint is 

1, and the bounds can be set by a fixed diagonal element of the covariance matrix 迎噺継岷懸痛懸

嫗

痛

峅. The constraint is 

then 

な 噺

稽痛

結

塚禰

稽痛貸怠

ど 噺 ln岫稽痛

岻 伐 ln岫稽痛貸怠岻 髪 懸痛

(43) 

The remainder of the external constraints was introduced in section 4. Table 5 lists all observation equations as a 

function of the state specified in Table 3. The observation noise term is omitted for convenience. 
 

 

 

 

Table 5 

 

Item Section Obs Vector 

Element 

凋聴喋

Observation Equation 

1 0 隙痛

凋聴喋

隙痛

凋聴喋 噺 ∆岫隙痛

岻 髪 隙痛貸怠 髪 隙痛貸替 伐 隙痛貸泰 隙 樺 岶茎, 鶏, 鯨岼

2 0 血怠禰

凋聴喋

血怠禰

凋聴喋 噺 ∆盤血怠禰

匪 髪 血怠禰貼迭 髪 血怠禰貼填 伐 血怠禰貼天

3 0 血態禰

凋聴喋

凋聴喋 噺 血態禰

血態禰

4 0 血戴禰

凋聴喋

凋聴喋 噺 血戴禰

血戴禰

5 0 血替禰

凋聴喋 噺 血替禰

血替禰

6 10.2 凋帖暢調 隙痛

凋帖暢調 噺 隙痛 髪 憲諜禰

凋帖暢調 隙痛

凋帖諜諜

隙 樺 岶茎, 鶏, 鯨岼

7 10.2 凋帖諜諜 隙痛

諜

凋帖諜諜 噺 隙痛 髪 憲諜禰

隙痛

諜

隙 樺 岶茎, 鶏, 鯨岼

8 10.2 

ln 峭

鶏痛

諜

嶌

鯨痛

ln 峭

鶏痛

諜

嶌 噺 ln岫鶏痛

岻 伐 ln岫鯨痛岻

鯨痛

隙

樺 岶畦経警激, 畦経隙隙岼

10 0 稽鯨軽痛

稽鯨軽痛 噺 ∆岫茎痛

岻 髪 茎痛貸替 伐 茎痛貸泰 伐 ∆岫鶏痛岻 伐 鶏痛貸怠 伐 鶏痛貸替

髪 鶏痛貸泰

11 0 怠

布 稽鯨軽痛貸賃

賃退待

怠

布 稽鯨軽痛貸賃 噺 ∆岫茎痛

岻 髪 茎痛貸怠 髪 茎痛貸替 伐 茎痛貸泰 伐 茎痛貸態

賃退待

伐 ∆岫鶏痛

岻 伐 に鶏痛貸怠 伐 鶏痛貸替 髪 鶏痛貸泰

12 0 戴

布 稽鯨軽痛貸賃

賃退待

戴

布 稽鯨軽痛貸賃 噺 ∆岫茎痛

岻 髪 茎痛貸怠 伐 茎痛貸泰 伐 ∆岫鶏痛岻 伐 に鶏痛貸怠

賃退待

伐 鶏痛貸態 伐 鶏痛貸戴 伐 鶏痛貸替 髪 鶏痛貸泰

13 4.4 ln 磐

詣痛

卑

詣痛貸替

鶏痛貸態 ln 磐詣痛

卑 噺 ln 磐 卑

詣痛貸替 ∆岫鶏痛貸怠岻 伐 鶏痛貸怠 髪 鶏痛貸態 髪 鶏痛貸泰

14 4.5 ln 磐

詣痛 髪 詣痛貸怠 卑

詣痛貸替 髪 詣痛貸泰

泰調懲

See Market Slaughter* 

泰調懲 15 4.6 

ln 峭

詣痛袋怠

泰調懲嶌

詣痛貸戴

ln 峭

詣痛袋怠

泰調懲嶌 噺 血替禰 伐 血替禰貼填 詣痛貸戴

16 4.7 

ln岫. の岻

∆岫鯨痛

岻 髪 鯨痛貸怠 髪 鯨痛貸替 伐 鯨痛貸泰 ln岫. の岻 噺 ln 崕 崗

茎痛貸怠 伐 範岫鶏痛貸怠 髪 系痛貸怠岻結

捗鉄禰貼迭 髪 結捗典禰貼迭 飯

17 10.3.2 

0 See Breeding Herd Smoother** 

*Market Slaughter 

岫鶏痛貸態 髪 系痛貸態岻結

捗鉄禰貼鉄 髪 結捗典禰貼鉄 ln 磐 詣痛 髪 詣痛貸怠 卑 噺 ln 峪 崋

詣痛貸替 髪 詣痛貸泰 岷∆岫鶏痛貸怠岻 伐 鶏痛貸怠 髪 鶏痛貸態 髪 鶏痛貸泰 髪 系痛貸滞峅結

捗鉄禰貼展 髪 結捗典禰貼展

**Breeding Herd Smoother 

ど 噺 ln 崕∆岫茎痛

岻 髪 茎痛貸怠 髪 茎痛貸替 伐 茎痛貸泰 伐 範岫∆岫鶏痛

岻 髪 鶏痛貸怠 髪 鶏痛貸替 伐 鶏痛貸泰 髪 系痛岻結

捗鉄禰 髪 結捗典禰 飯

崗

茎痛貸怠 伐 範岫鶏痛貸怠 髪 系痛貸怠岻結

捗鉄禰貼迭 髪 結捗典禰貼迭 飯

11 Kalman Filter Estimates of U.S. Level Hog Inventories 

Given the hog inventory system equations in State-Space form from sections 9 and 10, the Extended Kalman Filter is 

used to estimate the state vector and its standard errors. The state vector contains functions of inventory items, and 

the standard errors estimated by the filter are standard errors of the functions of inventory items. This section

covers the transformations from state vector to U.S.-level inventory items. 

If 建噺券 represents the most recent target survey period of reference, we are interested in estimating the vector 
圏津 噺 岷茎津 鶏津 鯨津 罫怠韮

罫態韮罫戴韮

罫替韮 警津 稽津 劇津峅

嫗

 

 

 

 

and its variance. The vector 圏津 represents all published inventory items from Table 1 at the U.S. level. The vector of 

state elements at time 建噺券 from Table 3 is however 

捲津 噺 範∆岫茎津

岻 ∆岫鶏津岻 ∆岫鯨津岻 ∆盤血怠津

匪 血態韮

血戴韮血替韮 ... 飯嫗

We express the vector of inventory items to be published as a vector of functions of the state vector. 

圏津 噺 繋岫捲津

岻

The vector of functions 繋岫捲津

岻 contains both linear and nonlinear functions. The Extended Kalman Filter (EKF) uses 

a first order Taylor Series approximation of the first and second central moments. For linear functions, this will be 

the exact mean and variance. A first order Taylor Series expansion of hog inventories 圏津 噺 繋岫捲津

岻 about 継岷捲津峅 is 

繋岫捲津

岻 噺 繋岫継岷捲津峅岻 髪 繋椛岫継岷捲津

峅岻岫捲津 伐 継岷捲津

峅岻

where 繋

椛

岫·岻 represents the Jacobian of 繋岫·岻. Taking the first order approximation and subtracting its expectation 

yields 

繋岫捲津

岻 伐 継岷繋岫捲津岻峅 噺 繋椛岫継岷捲津峅岻岫捲津 伐 継岷捲津峅岻

嫗 懸欠堅岷繋岫捲津

岻峅 噺 Σ槌韮 噺 繋椛

岫継岷捲津峅岻鶏津繋

椛

岫継岷捲津峅岻

The first order Taylor Series approximation of 圏津 and its covariance matrix are therefore 

圏津 噺 繋岫継岷捲

嫗

津

峅岻

Σ槌韮 噺 蛍津鶏津蛍津

(44) 

(45) 

 

嫗 where 蛍津 噺 繋椛

岫継岷捲津峅岻, and 鶏津 噺 継岷岫捲津 伐 継岷捲津峅岻岫捲津 伐 継岷捲津峅岻 峅.

The vector of functions of Kalman Filter estimates of hog inventories 圏痛 噺 繋岫捲痛

岻 is as follows: 
 

 

 

 

 

 

 

 

 

 

 

 

 

12 State-Level Inventories 

This section covers Restricted Least Squares allocation of the Kalman Filter estimates of U.S.-level inventories to

the states. State-level survey results must be calibrated to sum to the U.S. estimates for each of the inventory items 

with the exception of the litter rate. 

At the state level, there is no formal ASB panel analogous to the one at the U.S. level. However, the state 

commodity analysts set a “state recommendation” which most often differs from the survey results. When a state 

believes its survey results to be biased, they choose to recommend inventory estimates differing from the survey 

results. This is often due to significant undercoverage or nonresponse from extreme operators – those farms which 

are so large, they form a majority of the hog production in a given state. The constraints that we have formulated in 

State-Space representation for the Kalman Filter estimates of U.S.-level hog inventories are U.S.-level constraints. 

This is attributed to the fact that the external data is available at the U.S. level only. The states’ inventory estimates 

must sum to the U.S.-level inventory estimates for total hogs, pig crop, sows farrowed, the market weight groups, 

and breeding herd. The state-level estimates must be adjusted for this constraint to hold. This is accomplished using 

Restricted Least Squares techniques. 

We now establish notations and definitions in order to derive the Restricted Least Squares methodology for 

estimating state-level inventory. The time subscript t may be omitted for convenience unless 建塙券. 

Notation Definition 

捲津|津

The expected value of the state at time 建 噺 券 given measurements of the state at time 建 樺 岶な,に, ... , 券岼 or

継岷捲津

|検津峅. This is the vector of Kalman Filter estimates of the state for the most recent estimation quarter 

for hog inventory items. 

鶏津|津

The covariance matrix of 捲津|津 from the Kalman Filter. 

嫗

鶏津|津 噺 継峙盤捲津 伐 捲津|津匪盤捲津 伐 捲津|津匪 |検津峩

検津 Vector of measurements of inventory for time 建 噺 券.

紅聴蝶 Vector of list frame survey results ordered by inventory item and state at time 建 噺 券. 

紅聴蝶

茅

Vector of list frame survey results ordered by subset of inventory items and state at time 建 噺 券. The 

subset of inventory items consists of pig crop (P), sows farrowed (S), market hogs <50 lbs (G1), market 

hogs 50-119 lbs (G2), market hogs 120-179 lbs (G4), market hogs over 180 lbs (G4), and breeding herd 

(B). 

紅博博博博

聴蝶

茅

Vector of list frame survey results ordered by subset of inventory items and state at time 建 噺 券. The 

subset of inventory items consists of total hogs and pigs (H), total market hogs (M), and litter rate (T). 

These are the inventory items that can be derived from 紅聴蝶

茅

. 

紅聴眺

茅

Vector of state recommendations ordered by subset of inventory item and state at time 建 噺 券. The subset 

of inventory items consists of pig crop (P), sows farrowed (S), market hogs <50 lbs (G1), market hogs 

50-119 lbs (G2), market hogs 120-179 lbs (G3), market hogs over 180 lbs (G4), and breeding herd (B). 

紅懲庁

Vector of state-level estimates of inventory derived from the U.S.-level Kalman Filter estimates of 

inventory using Restricted Least Squares allocation. The vector elements are ordered by inventory item 

and state. 

紅懲庁

茅

Vector of state-level estimates of a subset of inventory items derived from the U.S.-level Kalman Filter 

estimates of inventory using Restricted Least Squares allocation. The vector elements are ordered by 

inventory item and state. The subset of inventory items consists of pig crop (P), sows farrowed (S), 

market hogs <50 lbs (G1), market hogs 50-119 lbs (G2), market hogs 120-179 lbs (G4), market hogs 

over 180 lbs (G4), and breeding herd (B). 

紅博博博博博

懲庁

茅

Vector of state-level estimates of a subset of inventory items derived from the U.S.-level Kalman Filter 

estimates of inventory using Restricted Least Squares allocation. The vector elements are ordered by 

inventory item and state. The subset of inventory items consists of total hogs and pigs (H), total market 

hogs (M), and litter rate (T). 
 

 

 

 

 

 

絞岫紅懲庁

茅

岻

The vector of functions that calculates 紅博博

懲庁

博博博茅

 from 紅懲庁

茅

.

稽痛 髪 罫怠禰 髪 罫態禰 髪 罫戴痛 髪 罫替禰

欣 禁 罫怠禰 髪 罫態禰 髪 罫戴痛 髪 罫替禰

絞岫紅懲庁

茅

岻 噺 噺 煩警

茎

劇痛

痛

痛晩

欽

欽

欽

禽

禽

禽

鶏痛

琴 鯨痛

筋

絞

椛

岫紅懲庁

茅

岻 The Jacobian of 絞岫紅懲庁

茅

岻.

経 Diagonal weight matrix containing the difference 紅聴眺

茅 伐 紅聴蝶茅

圏聴蝶 Vector of list frame survey results at the U.S. level at time 建 噺 券. 

圏聴蝶

茅

Vector of a subset of list frame survey results at the U.S. level at time 建 噺 券. The subset of inventory 

items consists of pig crop (P), sows farrowed (S), market hogs <50 lbs (G1), market hogs 50-119 lbs 

(G2), market hogs 120-179 lbs (G4), market hogs over 180 lbs (G4), and breeding herd (B). 

圏博博

聴蝶

茅博博

Vector of a subset of list frame survey results at the U.S. level at time 建 噺 券. The subset of inventory 

items consists of total hogs and pigs (H), total market hogs (M), and litter rate (T). 

圏懲庁

Vector of Kalman Filter estimates of hog inventory at the U.S. level at time 建 噺 券.

圏懲庁 噺 繋盤捲津|津匪. 

圏懲庁

茅

Vector of a subset of Kalman Filter estimates of hog inventory at the U.S. level at time 建 噺 券. The subset 

of inventory items consists of pig crop (P), sows farrowed (S), market hogs <50 lbs (G1), market hogs 

50-119 lbs (G2), market hogs 120-179 lbs (G4), market hogs over 180 lbs (G4), and breeding herd (B).

圏懲庁

茅 噺 繋茅

盤捲津|津匪. 

圏博博博博博

懲庁

茅

Vector of a subset of Kalman Filter estimates of hog inventory at the U.S. level at time 建 噺 券. The subset 

of inventory items consists of total hogs and pigs 博博博博 (H), total market hogs (M), and litter rate (T). 博圏懲庁

茅 噺

繋博博博茅盤捲津|津匪

戟

嫗 A linear operator such that 圏聴蝶

茅 噺 戟紅聴蝶茅

 and 圏懲庁

茅 噺 戟紅懲庁茅

. 戟 噺 岷薩愚層峅 where 1 is a のど 抜 な vector of 1s 

and I is a ば抜ば identity matrix representing the seven inventory items 鶏, 鯨, 罫怠

, 罫態, 罫戴, 罫替, 稽.

Σ掴

嫗 The covariance matrix of 捲, or 継岷岫捲 伐 継岷捲峅岻岫捲 伐 継岷捲峅岻 峅

Σ掴槻

嫗 The covariance matrix of 捲 and 検, or 継岷岫捲 伐 継岷捲峅岻岫検 伐 継岷検峅岻 峅

計津 The Kalman Gain from the Kalman Filter at time 建 噺 券. 

蛍津 The Jacobian of the vector of functions 繋盤捲津|津匪. 

蛍津

茅 The Jacobian of the vector of functions 繋

茅

盤捲津|津匪

喧珍 Pig crop for state j.

嫌珍 Sows farrowed for state j.

布 喧

珍

Pig crop at the U.S. level summed over all states. 

布 嫌

珍

Sows farrowed at the U.S. level summed over all states. 

購椎

態

乳

Variance of list frame survey result for pig crop for state j.

購鎚

態

乳

Variance of list frame survey result for sows farrowed for state j. 

購椎乳鎚乳

Covariance of list frame survey results for pig crop and sows farrowed for state j.

購椎乳鎚乳 噺 購鎚乳椎乳

嫗 We define 紅沈珍 to be the state-level inventory for inventory item i from Table 1 state j. 紅噺範紅沈珍飯 for 件 樺 岶な,に, ... ,など岼

嫗

and 倹 樺 岶な,に, ... ,のど岼 ordered by 件 then 倹. 紅

茅 噺 範紅沈珍飯 for 件 樺 岷に,ぬ,ね,の,は,ば,ひ峅. 

博紅博博茅

 will be used to denote the 

complement of 紅

茅

. 紅

茅博 茅博博 聴眺 噺 紅茅

聴眺 姦 紅博博 博 茅 博博

聴眺 represents the vector of state recommendations, and 紅聴蝶 噺 紅聴蝶 姦 紅聴蝶 represents 

the vector of state survey results. We define 戟 噺 岷荊愚な峅

嫗

 where 1 is a のど 抜 な vector of 1s and I is a ば抜ば identity 

matrix. 圏

茅

聴眺 噺 戟紅茅聴眺 is the U.S.-level summed state recommendation vector for the appropriate subset of inventory 

items and 圏

茅 茅

聴蝶 噺 戟紅聴蝶 is the U.S.-level summed survey results vector of the subset of inventory items. The vector 

圏懲庁 contains all hog inventory items estimated by the Kalman Filter at the U.S. level. The Restricted Least Squares 

Notation Definition 
(RLS) estimate for the Ordinary Least Squares (OLS) regression parameter in the model 検 噺 隙紅 髪 継 with

継~軽岫ど, Σ帳

岻 is 

紅眺挑聴 噺 紅潮挑聴 髪 岫隙

嫗隙岻貸怠戟嫗

岷戟岫隙

嫗隙岻貸怠戟嫗

峅

貸怠岫圏 伐 戟紅潮挑聴岻 (46) 

A derivation can be found in Green (2000). Equation (46) can be rewritten as 

嫗 嫗 紅懲庁

茅 噺 紅聴蝶

茅 髪 経戟 岷戟経戟 峅

貸怠岫圏懲庁

茅 伐 戟紅聴蝶

茅

岻 (47) 

where 経噺穴件欠訣岫紅

茅

聴眺 伐 紅茅聴蝶岻 is a diagonal weight matrix. This formulation creates the adjustments to the survey 

results according to the assumed degree of bias. If the state recommendations are the state survey results, the 

diagonal element or weight is equal to zero and the RLS adjustment is zero. The survey results are published. The 

degree of adjustment to the survey results is therefore correlated with the assumed bias. The Kalman Filter 

Restricted Least Squares state-level hog inventory estimates are 

紅懲庁

茅 噺 岫荊 伐 傑戟岻紅聴蝶

茅 髪 傑圏懲庁

茅

博紅博

懲庁

博博博茅 噺 絞岫紅懲庁

茅

岻

紅懲庁 噺 懲庁 懲庁 岷紅

茅 紅博博博博博茅

峅

嫗

where 傑噺経戟

嫗

岷戟経戟嫗峅

貸怠 and 絞岫·岻 is the vector of functions that gives total hogs (summation), total market hogs 

(summation), and litter rate (ratio of sums). The covariance matrix is given by 

Σ庭

茅 噺 岫荊 伐 傑戟岻Σ庭

茅 岫荊 伐 傑戟岻

嫗 髪 傑蛍茅鶏 蛍茅 嫗

津 津 津 傑

嫗 髪 岫荊 伐 傑戟岻Σ庭

茅 超韮

計

嫗

津蛍

茅 嫗

傑

嫗 髪 傑蛍茅計 Σ 岫荊 伐 傑戟岻

嫗

凪鈍 庭

茅

縄楠 縄楠 津 津 津 超韮 縄楠

Σ 噺 絞椛

岫紅

茅

岻Σ 絞

椛

岫紅

茅 嫗

博博博茅博博博

茅

庭 懲庁 庭

岻

凪鈍 懲庁 凪鈍

Σ庭

茅 博博博 庭

茅博博博 噺 岫荊伐傑戟岻範Σ 岫荊伐傑戟岻

嫗 髪 Σ 計 紅茅

岻

嫗

嫦

嫗

蛍

茅 嫗

傑

嫗

飯絞椛

岫

茅 嫗 茅 茅

庭

茅 庭茅 超韮 津 津 懲庁 髪 傑範蛍津計津Σ超 庭茅

韮

岫荊伐傑戟岻 髪 蛍津鶏津蛍津 傑

嫗

飯絞椛

岫紅

茅 嫗

凪鈍 凪鈍 縄楠 縄楠 縄楠 懲庁岻

Σ庭

茅 Σ庭

茅 庭博博博茅博博博

Σ 噺 峪 凪鈍 凪鈍 凪鈍

庭 崋

懲庁 Σ庭 庭 庭 博博博茅博博博 茅 Σ博博博茅博博博

凪鈍 凪鈍 凪鈍

In order to derive the covariance matrix of the state allocation, we first need to derive the covariance of the survey 

results with the estimated state Σ 懸盤紅茅

痴

茅

SV掴韮|韮 噺 潔剣 聴蝶, 捲津|津匪. 

Derivation of Σ庭

茅

縄楠掴韮|韮

嫗

Σ

茅 茅

痴SV掴韮|韮 噺 継峙岫紅聴蝶 伐 継岷紅聴蝶峅岻盤捲津|津 伐 継範捲津|津飯匪 峩

捲津|津 噺 捲津|津貸怠 伐 計津訣盤捲津|津貸怠匪 髪 計津桁津

継範捲津|津飯 噺 継範捲津|津貸怠 伐 計津訣盤捲津|津貸怠匪飯 髪 計津継岷桁津

峅

捲津|津 伐 継範捲津|津飯噺捲津|津貸怠 伐 計津訣盤捲津|津貸怠匪 伐 継範捲津|津貸怠 伐 計津訣盤捲津|津貸怠匪飯 髪 計津

岫桁津 伐 継岷桁津

峅岻

Σ庭

茅 噺 Σ庭

嫗

掴

茅 茅

縄楠 韮|韮 廷盤掴 匪 縄楠 韮|韮貼迭 髪 Σ庭縄楠超韮

計津

噺 Σ庭

茅

縄楠超韮

計

嫗

津

The final result is because Σ庭縄楠廷盤掴 匪 噺 ど 韮|韮貼迭 as the survey results at time n are independent and therefore not 

correlated with the prediction of the state at time n; nor any function thereof. All terms with 捲津|津貸怠 are collapsed 

into the function 紘盤捲津|津貸怠匪. The covariance matrix Σ庭縄楠超韮

 is nonzero for the U.S. survey result observations (sum 

and litter rate). Otherwise it is zero. 
庭

庭 庭

庭

嫗 Σ

縄楠

茅 槌凪鈍茅 噺 継岷岫紅聴蝶

茅 伐 継岷紅聴蝶茅

峅岻岫圏懲庁

茅 伐 継岷圏懲庁茅

峅岻 峅

圏懲庁

茅 噺 繋茅

盤捲津|津匪

圏懲庁

茅 噺 繋茅

盤継範捲津|津飯匪 髪 繋

茅椛

盤継範捲津|津飯匪盤捲津|津 伐 継範捲津|津飯匪

継岷圏懲庁

茅

峅 噺 繋茅

盤継範捲津|津飯匪

圏懲庁

茅 伐 継岷圏懲庁茅

峅 噺 繋茅椛

盤継範捲津|津飯匪盤捲津|津 伐 継範捲津|津飯匪

嫗

Σ

縄楠

茅 槌凪鈍茅 噺 Σ

縄楠

茅 掴韮|韮繋

茅椛

盤捲津|津匪

噺 Σ

縄楠

茅 超韮

計津

嫗

蛍津

茅 嫗

Derivation of Σ庭

茅

凪鈍

庭

庭 庭 庭 庭

庭 庭 庭

嫗 Σ

凪鈍

茅 噺 継岷岫紅懲庁

茅 伐 継岷紅懲庁茅

峅岻岫紅懲庁

茅 伐 継岷紅懲庁

茅

峅岻 峅

紅懲庁

茅 噺 岫荊 伐 傑戟岻紅聴蝶

茅 髪 傑圏懲庁

茅

継岷紅懲庁

茅

峅 噺 岫荊 伐 傑戟岻継岷紅聴蝶

茅

峅 髪 傑継岷圏懲庁

茅

峅

紅懲庁

茅 伐 継岷紅懲庁茅

峅 噺 岫荊 伐 傑戟岻岫紅聴蝶

茅 伐 継岷紅聴蝶

茅

峅岻 髪 傑岫圏懲庁

茅 伐 継岷圏懲庁茅

峅岻

嫗 嫗 嫗 嫗 Σ

凪鈍

茅 噺 岫荊 伐 傑戟岻Σ

縄楠

茅 岫荊 伐 傑戟岻 髪 傑Σ槌凪鈍

茅 傑 髪 岫荊 伐 傑戟岻Σ

縄楠

茅 槌凪鈍

茅 傑 髪 傑Σ

縄楠

茅 槌凪鈍茅 岫荊 伐 傑戟岻

噺 嫗 嫗 嫗 嫗 嫗 嫗 嫗

岫荊 伐 傑戟岻Σ

縄楠

茅 岫荊 伐 傑戟岻 髪 傑蛍津

茅鶏津蛍津茅 傑 髪 岫荊 伐 傑戟岻Σ

縄楠

茅 超韮

計津蛍津

茅 傑 髪 傑蛍津茅計津Σ超韮 縄楠

茅 岫荊 伐 傑戟岻

Derivation of Σ庭

茅

縄楠庭

茅

凪鈍

庭 庭

庭 庭

庭 庭

庭 庭

嫗 Σ

縄楠

茅

凪鈍

茅 噺 継岷岫紅聴蝶

茅 伐 継岷紅聴蝶茅

峅岻岫紅懲庁

茅 伐 継岷紅懲庁茅

峅岻 峅

岫紅懲庁

茅 伐 継岷紅懲庁茅

峅岻嫗 噺 岫紅聴蝶

茅 伐 継岷紅聴蝶茅

峅岻嫗岫荊 伐 傑戟岻

嫗 髪 岫圏懲庁

茅 伐 継岷圏懲庁茅

峅岻嫗傑

嫗

嫗 嫗 嫗 嫗 Σ

縄楠

茅

凪鈍

茅 噺 岫紅聴蝶

茅 伐 継岷紅聴蝶茅

峅岻岫紅聴蝶

茅 伐 継岷紅聴蝶茅

峅岻 岫荊 伐 傑戟岻 髪 岫紅聴蝶

茅 伐 継岷紅聴蝶茅

峅岻岫圏懲庁

茅 伐 継岷圏懲庁茅

峅岻 傑

噺 Σ

縄楠

茅 岫荊 伐 傑戟岻

嫗 髪 Σ

縄楠

茅 槌凪鈍茅 傑

嫗

噺 Σ

縄楠

茅 岫荊 伐 傑戟岻

嫗 髪 Σ

縄楠

茅 超韮

計津

嫗

蛍津

茅 嫗

傑

嫗

Derivation of Σ槌

茅

凪鈍庭

茅

凪鈍

庭 庭

庭

圏懲庁

茅 伐 継岷圏懲庁茅

峅 噺 蛍津

茅

盤捲津|津 伐 継範捲津|津飯匪

嫗 嫗 嫗 嫗 嫗 岫紅懲庁

茅 伐 継岷紅懲庁茅

峅岻 噺 岫紅聴蝶

茅 伐 継岷紅聴蝶茅

峅岻 岫荊 伐 傑戟岻 髪 岫圏懲庁

茅 伐 継岷圏懲庁茅

峅岻 傑

嫗 嫗 Σ槌凪鈍

茅

凪鈍

茅 噺 蛍津

茅Σ掴韮|韮 縄楠

茅 岫荊 伐 傑戟岻 髪 蛍津

茅Σ掴韮|韮槌凪鈍

茅 傑

噺 蛍 嫗 嫗

津

茅計津Σ超韮 縄楠

茅 岫荊 伐 傑戟岻 髪 蛍津

茅鶏津蛍津

茅

嫦

傑

Derivation of Σ庭

茅 博博博博博博 庭

茅

凪鈍 凪鈍

庭 庭

庭 庭 庭 庭 庭

嫗

Σ

凪鈍

茅 博博

凪鈍

博博博博 茅 噺 継峙岫紅懲庁

茅 伐 継岷紅懲庁茅

峅岻盤紅博博

懲庁

博博博茅 伐 継範紅博博

懲庁

博博博茅

飯匪 峩

紅懲庁

茅 伐 継岷紅懲庁茅

峅 噺 岫荊 伐 傑戟岻岫紅聴蝶

茅 伐 継岷紅聴蝶茅

峅岻 髪 傑岫圏懲庁

茅 伐 継岷圏懲庁茅

峅岻

嫗

嫗 嫗 盤紅博博

懲庁

博博博茅 伐 継範紅博博

懲庁

博博博茅

飯匪 噺 岫紅懲庁

茅 伐 継岷紅懲庁茅

峅岻 絞

椛

岫紅懲庁

茅

岻

嫗 嫗 Σ

凪鈍

茅 博博

凪鈍

博博博博 茅 噺 岫荊 伐 傑戟岻Σ

縄楠

茅

凪鈍

茅 絞

椛

岫紅懲庁

茅

岻 髪 傑Σ槌凪鈍

茅

凪鈍

茅 絞

椛

岫紅懲庁

茅

岻

Derivation of Σ庭

茅

縄楠槌

茅

凪鈍

態

な な ど ど 欣

購椎迭

ど

態

購椎迭鎚迭ど

禁

ど ど な な 欽 ど 購椎鉄

ど 購椎鉄鎚鉄禽

繋

椛

岫E岷X峅岻ΣX 噺 頒 な な

伐

な

伐

な 番

欽 態 禽 噺

購鎚迭椎迭

ど 購鎚迭ど

∑珍 喧 ∑珍 喧 ∑珍 嫌 ∑珍 嫌

欽

琴 ど 購鎚鉄椎鉄

ど 購鎚

態

鉄

禽

筋

態 態 欣

購椎迭

購椎鉄購椎迭

態

鎚迭

購椎鉄

態

鎚鉄 禁

購鎚迭椎迭

購鎚鉄椎鉄

購鎚迭購鎚鉄

欽

欽

欽

な

購椎

態

迭 伐

な

購鎚迭椎迭

な

購椎

態

鉄 伐

∑

な

珍 嫌

購鎚鉄椎鉄

な

購椎迭鎚迭 伐

∑

な

珍 嫌

購鎚

態

迭

な

購椎鉄鎚鉄 伐

な

購鎚

態

鉄

禽

禽

禽

琴∑珍 喧 ∑珍 嫌 ∑珍 喧 ∑珍 喧 ∑珍 喧 ∑珍 嫌 筋

 

 

 

These results can be expanded to any number of states. 

13 Filter Results 

This section provides results for three scenarios of Kalman Filter results. The results are from three different

treatments or parameterizations of the ASB estimates in the Filter. For the first, we omit an ASB measurement. For 

the second treatment, we assume the ASB is biased. For the third treatment, we assume the ASB is unbiased. Details 

are provided on specific parameterizations of the observations so that it is clear how the treatments differ. 

Decomposition of the estimates into relative absolute net contributions is explained.

In this paper, we have defined the hog inventory system equations. Their derivations have been presented without 

any evidence of the performance of Filter inventory estimates by the various parameterizations. Performance can be 

evaluated more easily when there exist some unbiased measures of the true signal to which we can compare the filter 

results. Signal Filtering is used in situations where one or more measurements of an unobservable signal are 

collected, and by the very nature of the problem, the true signal is not available, so a comparison to truth cannot be 

made. This is the case with hog inventory estimation. As true inventories are not available for comparison, we will 

compare the Filter estimates to the ASB published, and to each other across treatments; and examine the inventory 

constraints to judge whether the Filter estimates follow the rules and “make sense”. For each of the ten published 

inventory items; namely total hogs and pigs (H), pig crop (P), sows farrowed (S), market hogs less than 50 lbs (G1), 

market hogs between 50 and 119 lbs (G2), market hogs between 120 and 179 lbs (G3), market hogs greater than 180 

lbs (G4), total market hogs (M), breeding herd (B), and litter rate (T); we will present graphically the results of a 

fixed interval Kalman Smoother estimate of inventory for three different parameterizations or treatments of the 

observations. We will compare the estimates of the variances of the Kalman Smoother estimates between the three 

treatments. We will also decompose the Smoother at the last point in time 継岷捲痛

|検津

峅|建 噺 券 into what will be defined 

as absolute relative net contributions from categories of observations. This will provide understanding as to which 

data items are influential in each treatment of the measurements. Additionally, we will include the estimates of the 

variances of the process noise and observation noise obtained by the Expectation Maximization algorithm for each 

of the three treatments. These are the diagonals of the Q and R matrices from sections 8.1 and 
 

 

 

 

 

 

 

 

sheet residuals are bounded by 罰のどど thousand hogs. It is also essential to note that the ASB gives heavy weight to 

the survey litter rate. Omitting an ASB measurement and estimating the State-Space model parameters without a 

fixed constraint on the litter rate results in an observation noise estimate for the survey litter rate that is not 

consistent with ASB behavior. ASB behavior can be reflected in the Filter without ASB measurements by fixing the 

variance parameter associated with the survey litter rate observation equations. Fixing the variance in this way puts 

strict bounds on how far the filter estimate for the litter rate will deviate from the survey litter rate. For example, if 

the variance of the observation noise associated with the survey litter rate were fixed at zero, the filter estimate for 

the litter rate would be exactly the survey litter rate, and the filter estimates for pig crop and sows farrowed would 

reflect this. The farther that variance is fixed from zero, the more the filter estimate for litter rate can deviate from 

the survey litter rate. The breeding herd smoother is an imposed constraint and therefore also fixed. We treat the 

ratio between survey sows farrowed and previous quarter’s breeding herd as unbiased. Although the sows 

farrowed/breeding herd constraint introduced in section 4.7 equation (9) defines sows farrowed as one half of the 

previous quarter’s breeding herd, the 

Figure 25 graph demonstrates that the historical ASB estimates do not agree. As the ASB treats the survey ratio of 

pig crop to sows farrowed as unbiased, we assume the survey ratio of sows farrowed to previous quarter’s breeding 

herd is also unbiased. It is understood that the breeding herd in the denominator is from a different survey quarter 

and hence survey result than the numerator since it is lagged one quarter; however, the survey results are similar for 

the same survey quarter as for the lagged quarter as shown in Figure 35. 

Figure 35 

ADXX List Only Survey Sows Farrowed / Breeding Herd

0.40 0.42 0.44 0.46 0.48 0.50 0.52

St Bt1

St Bt

1990 1995 2000 2005 2010

Table 7 below lists the hog inventory measurement notation, description, and how it is parameterized in the 

treatment. For these three treatment scenarios, we use only the ADXX list frame survey result with the exception of 

the litter rate. The ADMW multiframe survey result is not computed quarterly for all inventory items and it is 

sufficient to use the list frame survey result in the treatment scenarios. 

Table 7 
 

 

 

 

 

Notation Description 

Parameterization 

Of Observation 

Noise 

ADXX.F4 List frame survey result for f4 Estimated 

BSN3 3 Month Balance Sheet Net Fixed 

BSN6 6 Month Balance Sheet Net Fixed 

BSN12 12 Month Balance Sheet Net Fixed 

RATIO.P Slaughter Ratio – Pig Crop (section 4.4) Estimated 

RATIO.M Slaughter Ratio – Market Hogs (section 4.5) Estimated 

SL5WKS 5 Week Slaughter Ratio (section 4.6) Estimated 

ADMW.SL Multiframe Survey result for litter rate Fixed 

ADXX.SL List frame survey result for litter rate Fixed 

.5 List frame survey result for St

/Bt-1 Fixed 

1 Breeding Herd Smoother (section 10.3.2) Fixed 

13.2 Treatment 2 – ASB Measurements as Biased Estimates 

For the second scenario, Treatment 2, we add the ASB measurements to the observations and use the 

parameterization introduced in 10.1.2 which defines the ASB measurements as biased. The other items remain the 

same. 

Table 8 

 

 

 

 

 

 

 

 

Notation Description 

Parameterization 

Of Observation 

Noise 

HP ASB estimate for total hogs and pigs Estimated 

PP ASB estimate for pig crop Estimated 

SP ASB estimate for sows farrowed Estimated 

F1.P ASB estimate for f1 Estimated 

F2.P ASB estimate for f2 Estimated 

F3.P ASB estimate for f3 Estimated 

F4.P ASB estimate for f4 Estimated 

ADXX.H List frame survey result for total hogs and pigs Estimated 

ADXX.P List frame survey result for pig crop Estimated 

ADXX.S List frame survey result for sows farrowed Estimated 

ADXX.F1 List frame survey result for f1 Estimated

ADXX.F2 List frame survey result for f2 Estimated 

ADXX.F3 List frame survey result for f3 Estimated 

ADXX.F4 List frame survey result for f4 Estimated 

BSN3 3 Month Balance Sheet Net Fixed 

BSN6 6 Month Balance Sheet Net Fixed 

BSN12 12 Month Balance Sheet Net Fixed 

RATIO.P Slaughter Ratio – Pig Crop (section 4.4) Estimated 

RATIO.M Slaughter Ratio – Market Hogs (section 0) Estimated 

SL5WKS 5 Week Slaughter Ratio (section 4.6) Estimated 

ADMW.SL Multiframe Survey result for litter rate Fixed 

ADXX.SL List frame survey result for litter rate Fixed 

.5 List frame survey result for St

/Bt-1 Fixed 

1 Breeding Herd Smoother (section 10.3.2) Fixed 

13.3 Treatment 3 – ASB Measurements as Unbiased Estimates 
 

 

 

For Treatment 3, we remove the ASB bias component from the system equations. The filter treats the ASB 

measurements as unbiased estimates of true inventory. As the ASB estimates are parameterized as unbiased and they 

give heavy weight to the survey litter rate, there is no longer a need to fix the variances associated with the litter 

rates noise processes and they are therefore estimated. Lastly, the ratio observation of survey sows farrowed to 

breeding herd is replaced by the ASB values and the variances is estimated instead of fixed. These changes are 

bolded in Table 9. All other measurements remain the same. 

Table 9 

 

 

 

 

 

 

 

 

 

Notation Description 

Parameterization 

Of Observation 

Noise 

HP ASB estimate for total hogs and pigs Estimated 

PP ASB estimate for pig crop Estimated 

SP ASB estimate for sows farrowed Estimated 

F1.P ASB estimate for f1 Estimated 

F2.P ASB estimate for f2 Estimated 

F3.P ASB estimate for f3 Estimated 

F4.P ASB estimate for f4 Estimated 

ADXX.H List frame survey result for total hogs and pigs Estimated 

ADXX.P List frame survey result for pig crop Estimated 

ADXX.S List frame survey result for sows farrowed Estimated 

ADXX.F1 List frame survey result for f1 Estimated

ADXX.F2 List frame survey result for f2 Estimated 

ADXX.F3 List frame survey result for f3 Estimated 

ADXX.F4 List frame survey result for f4 Estimated 

BSN3 3 Month Balance Sheet Net Fixed 

BSN6 6 Month Balance Sheet Net Fixed 

BSN12 12 Month Balance Sheet Net Fixed 

RATIO.P Slaughter Ratio – Pig Crop (section 4.4) Estimated 

RATIO.M Slaughter Ratio – Market Hogs (section 0) Estimated 

SL5WKS 5 Week Slaughter Ratio (section 4.6) Estimated 

ADMW.SL Multiframe Survey result for litter rate Estimated 

ADXX.SL List frame survey result for litter rate Estimated 

.5 ASB estimate for St

/Bt-1 Estimated 

1 Breeding Herd Smoother (section 10.3.2) Fixed 

13.4 Decomposition of 撮岷姉仔|姿仔

峅 into Absolute Relative Net Contributions 

In addition to comparing the fixed interval Kalman Smoother estimates of the state and their associated variances, 

we also examine the decomposition of the estimates. This is a way of comparing between the treatments which 

observations are most influential. For the sake of simplicity and relevance, we will look at the decomposition of the 

measurement contributions for the most recent measurement vector in time (建噺券) for all three treatments and 

inventory items which are linear functions of the state. Specifically these are total hogs and pigs, pig crop, sows 

farrowed, and the weight group functions (the weight groups themselves are nonlinear functions of the state). As a 

measurement of inventory can have a positive or negative contribution, we will compare the absolute relative net 

contributions. We categorize the measurements according to measurement type. The six categories are ASB 

measurements, SURVEY measurements, BALANCE sheet measurements, SLAUGHTER ratio measurements, the 

cumulative historical contribution which will be explained in this section and which we will call “MODEL”, and 

OTHER measurements. 

Table 10 shows the breakdown of observations into categories. 
 

 

 

 

 

Table 10 

Observation Category 

HP 

ASB 

PP 

SP 

F1.P 

F2.P 

F3.P 

F4.P 

ADXX.H 

SURVEY 

ADXX.P 

ADXX.S 

ADXX.F1 

ADXX.F2 

ADXX.F3 

ADXX.F4 

ADXX.SL 

ADMW.SL 

BSN3 

BSN6 BALANCE

BSN12 

RATIO.P 

RATIO.M SLAUGHTER 

SL5WKS 

.5 OTHER 1 

 MODEL 

The “other” category consists of the sows farrowed/breeding herd ratio and the breeding herd constraint. We define 

a matrix 警 where the columns are the six categories and the rows are the observations. The values of M are 1 if the 

observation belongs to the category and zero otherwise. Define a matrix 系 where the columns are the seven 

inventory items H, P, S, F1, F2, F3, and F4; and the rows are the observations. The values of 系 contain the 

contribution of the observations such that the column sums equal the fixed interval Kalman Smoother estimates. Let 

1 be a vector of ones with dimensions は抜な . The absolute relative net contribution by category (ARNC) is then 

calculated as 

嫗

畦迎軽系 噺

|警 系|

嫗 嫗 層層 |警 系|

(48) 

The 畦迎軽系 matrix contains the absolute relative net contributions by inventory item (columns) and category (rows). 

Note that the brackets in this case stand for the absolute value and not the matrix determinate. 

We now derive the matrix 系 containing the net contribution by inventory item (columns) and observation (rows). 

For a linear system of transition and observation equations, the Kalman Filter and Kalman smoother at 建噺券

provided in Shumway & Stoffer (2006) is 

捲津|津 噺 捲津|津貸怠 髪 計津盤検津 伐 畦津捲津|津貸怠匪 (49) 
貸怠 where 計津 噺 鶏津|津貸怠畦

嫗

盤畦 鶏 畦嫗

津 津 津|津貸怠 津 髪 迎匪 is the Kalman Gain and 

嫗

鶏津|津貸怠 噺 継峙盤捲津 伐 捲津|津貸怠匪盤捲津 伐 捲津|津貸怠匪 |検津貸怠峩 is the variance of the state prediction 捲津|津貸怠. We can rewrite 

equation (49) as 

捲津|津 噺 岫荊伐計津畦津

岻捲津|津貸怠 髪 計津検津 (50) 

From the State-Space transition equation 捲津|津貸怠 噺 Φ捲津貸怠|津貸怠, we can write equation (50) which is the Kalman 

Filter and Smoother at 建噺券 as the recursive relationship 

捲津|津 噺 岫荊伐計津畦津

岻Φ捲津貸怠|津貸怠 髪 計津検津 (51) 

The contribution 岫荊伐計津畦津

岻Φ in the first term of equation (51) weights the previous quarter’s state 捲津貸怠|津貸怠, and 

the contribution 計津 from the second term weights the most recent measurements of inventory 検津. As this is a 

recursive equation, we will show that we can write this equation in terms of all measurements and the initial state

捲待|待. We start with the recursive equation (51) and calculate 捲津貸怠|津貸怠 as 

捲津貸怠|津貸怠 噺 岫荊伐計津貸怠畦津貸怠岻Φ捲津貸態|津貸態 髪 計津貸怠検津貸怠 (52) 

Substitution of (52) into (51) yields 

捲津|津 噺 岫荊伐計津畦津

岻Φ岫荊伐計津貸怠畦津貸怠岻Φ捲津貸態|津貸態 髪 岫荊伐計津畦津岻Φ計津貸怠検津貸怠 髪 計津検津 (53) 

With an additional third recursion, the pattern is recognizable 

捲津|津 噺 岫荊伐計津畦津

岻Φ岫荊伐計津貸怠畦津貸怠岻Φ岫荊伐計津貸態畦津貸態岻Φ捲津貸戴|津貸戴

髪 岫荊伐計津畦津

岻Φ岫荊伐計津貸怠畦津貸怠岻Φ計津貸態検津貸態 髪 岫荊伐計津畦津

岻Φ計津貸怠検津貸怠 髪 計津検津

(54) 

and it becomes evident that 捲津|津 can be written 

津貸怠 津貸怠 陳

捲津|津 噺 煩敷岫荊伐計津貸賃畦津貸賃岻Φ晩 捲待|待 髪 布 崛敷盤荊 伐 計津貸珍袋怠畦津貸珍袋怠匪Φ崑 計津貸陳検津貸陳

賃退待 陳退待 珍退怠

(55) 

For the initial state 捲待|待 in equation (55) we are using published data with absolute certainty (initial variance of 

zero). We will call this 検待 噺 捲待|待. The result is that we can write the final Kalman Filter/Fixed interval Smoother 

estimate at 建噺券 as a “weighted average” of the initial state and all measurements. 

津

捲津|津 噺 布 膏陳検津貸陳

緊

陳退待

陳

計津貸陳 兼噺ど

衿

衿崛敷盤荊 伐 計津貸珍袋怠畦津貸珍袋怠匪Φ崑 計津貸陳 ど隼兼 隼 券

膏陳 噺 珍退怠 芹 津貸怠 衿

衿 敷岫荊 伐 計津貸賃畦津貸賃岻Φ 兼 噺 券

菌 賃退待

(56) 

(57) 
The interpretation is that the estimate 捲津|津 is a composite of the initial state and all measurements in time weighted 

by the state-space model parameters. For an analysis of the decomposition, we will partition out from the summation 

in equation (56) the most recent data vector at 兼噺ど

津

捲津|津 噺 計津検津 髪 布 膏陳検津貸陳 (58) 

陳退怠

The first term of equation (58) is the contribution of the most recent measurement vector 検津; the second term is the 

composite contribution from the initial state and historical measurement vectors. This aggregate contribution of the 

historical observations is the “Model” category for the absolute relative net contributions. Each term’s contribution 

in the summation could be calculated using the appropriate 膏陳. For the nonlinear version of the decomposition, we 

have the Filter equations 

捲津|津 噺 捲津|津貸怠 髪 計津範検津 伐 茎盤捲津|津貸怠匪飯

捲津|津貸怠 噺 罫盤捲津貸怠|津貸怠匪

(59) 

The result is the recursion equation 

捲津|津 噺 罫盤捲津貸怠|津貸怠匪 髪 計津版検津 伐 茎範罫盤捲津貸怠|津貸怠匪飯繁

噺 罫盤捲津貸怠|津貸怠匪 伐 計津茎範罫盤捲津貸怠|津貸怠匪飯 髪 計津検津

噺 蛍盤捲津貸怠|津貸怠匪 髪 計津検津

(60) 

where 計津検津 contains the contribution of the most recent data and 蛍盤捲津貸怠|津貸怠匪 contains the contributions of 

cumulative past measurements analogous with the linear weighted average component ∑津

陳退怠 膏陳検津貸陳. The 

contribution matrix is calculated as 

嫗

系噺範岫計 抜 な検嫗

津 津

岻||岫荊伐計 畦

茅

津

岻捲

茅

嫦

津|津貸怠飯 畦 (61) 

where 抜 is the elementwise multiplication operator, || appends matrix columns, 1 is an 兼抜な vector of ones in 

which 兼 is the dimension of the state, and 畦

茅

 represents the linear transformation for which 

嫗

畦

茅捲津|津 噺 範茎津 鶏津 鯨津 血怠韮

 血態韮 血戴韮

 血替韮

飯 . Substitution of (61) into (48) gives the Absolute Relative Net Contribution 

matrix. 

嫗

嵳警嫗範岫計

嫗

津 抜 な検津

岻||岫荊伐計津畦

茅

岻捲津|津貸怠飯 畦茅

嫦

嵳

畦迎軽系 噺 嫗

層層嫗

嵳警嫗範岫計津 抜 な検嫗

津

岻||岫荊伐計津畦茅岻捲津|津貸怠飯 畦茅

嫦

嵳

(62) 
 

 

 

13.5 Contribution Tables 

The remainder of this paper contains the results of hog inventory estimation through Signal Filtering based on the 

three treatments defined in sections 13.1 - 13.3. Table 11 contains the relative absolute net contribution tables for 

each of the three treatments by categorized measurements; ASB, survey, balance sheet data, slaughter data, other 

(composed of sows to breeding herd ratio and breeding herd smoother), and historical observations (called “Model”) 

. Following these tables are graphs of the estimates and charts of the contribution allocations organized by inventory 

item, as well as graphs of the constraints. It should be noted that because the natural log of 1 is zero, the contribution 

of the breeding herd smoother is incalculable. 

Table 11 
Table 12: Summary Contribution Equally Weighted over all Inventory Intentory Items 
Figure 36: Total Hogs and Pigs 
Figure 37: Pig Crop 
 

Figure 38: Sows Farrowed 
Figure 39: Weight Group Function 1 
Figure 40: Weight Group Function 2 
 

Figure 41: Weight Group Function 3 
Figure 42: Weight Group Function 4 


Figure 47: Total Market Hogs 
Figure 49: Litter Rate 
Figure 51: Weight Group Ratio Transition 
Figure 53: Market Hogs and Slaughter Ratio 
Figure 55: Three Month Balance Sheet Residual 
Figure 57: Twelve Month Balance Sheet Residual 
 

 

 

 

 

 

 

 

13.6 Summary of Results 

Treatments 1, 2, and 3 demonstrate that it is possible to start at the same level of inventory at a particular point in 

time and generate many different solutions, given the same constraints that the ASB uses to set its published 

numbers. The variance parameters 芸 and 迎 were estimated to maximize the likelihood of the realization of the 

observation vectors, given the constraints and assumptions discussed in this paper. If a particular constraint - for 

example, a slaughter ratio - is not deemed adequate, the corresponding noise process in these covariance matrices 

could be fixed, as was done with the survey litter rates in order to enforce ASB behavior in the treatments in which 

there were no ASB measurements, or in which the ASB measurements were parameterized to contain possible bias. 

Strictly enforcing too many constraints can have computational complications, as a solution may not in fact exist 

that satisfies all constraints. 

13.6.1 Treatment 1 – No Expert Measurements 

The plots of the fixed interval Kalman Smoother estimates in Figure 36 through Figure 49 show that the results for 

pig crop, sows farrowed, weight groups 1 and 2, and the litter rate appear to agree for the most part with the ASB 

published inventories. Total hog inventory, total market hogs, and consequently breeding herd appear significantly 

different. Further inspection shows that these differences are attributed to weight groups 3 and 4 (Figure 45 and 

Figure 46). These weight groups are encapsulated in weight group function 3 (Figure 41), which also deviates 

considerably from the ASB published. Both the survey results and ASB measurements for 血戴

 and 血替 will prove to be 

highly influential observations in treatments 2 and 3 relative to the other measurements. 

One notable result is the slaughter ratios’ lack of contribution to the estimates in all three treatments. We are 

examining the influence of the measurements of inventory at the last point in time on the estimates of inventory at 

that time. In section 4.4 it was established that slaughter has a lagged effect on inventory by two quarters. One 

would expect that the slaughter ratios at time 建噺券 would have some influence on the inventory estimates at some 

point 券伐に判建判券. However this is not the case. The maximum relative absolute net contribution of the slaughter 

ratios in treatment 1 is 0.38% in the Kalman Smoother for total hogs and pigs at time 建噺券. In treatment 2 the 

maximum relative absolute net contribution of the slaughter ratios is 0.05% for 血替

 at time 建噺券. In treatment 3 the 

maximum relative absolute net contribution of the slaughter ratios is 0.25% for the total hogs and pigs estimate at 

建噺券. 

13.6.2 Treatment 2 – Biased Expert Measurements 

The initial state of the Filter at 捲待|待 噺 継岷捲待

|検待峅 for all three treatments is set at the published ASB values and is 

parameterized with zero uncertainty i.e. 懸欠堅盤捲待|待匪 噺 鶏待|待 噺 ど. This means that for the initial starting position of 

each filter scenario, the ASB values are treated as absolute truth. The initial values of the bias parameters 

corresponding to each ASB measurement in treatment 2 are initialized at zero. The visually apparent biases in 

weight group 3 and weight group 4 seen in treatment 1 are also observable in the filter results of treatment 2. As 

treatment 2 estimates the ASB bias as part of the state and we estimate the variance of the state within the filter, we 

can test the null hypothesis that the bias is equal to zero versus the alternative that there exists a nonzero bias in the 

ASB estimates. Graphs of the p-values of this hypothesis test for each point in time are given in Figure 58 through 

Figure 64. P-values that fall below the dashed red line in the graphs indicate there is enough evidence to reject the 

null hypothesis that the bias is zero at that particular point in time at a significance level of 糠 噺 ど.どの. This analysis 

does not take into account simultaneous inference across time and inventory items. Independent hypothesis testing 

of the Treatment 2 bias parameters for pig crop, Figure 59, and sows farrowed, Figure 60, shows greatest lack of 

evidence of bias for those two inventory items. There is sufficient evidence to support the existence of bias in the 

ASB measurements as time progresses for total hogs and pigs and the four weight group functions. 
Figure 58 

P-Values for ASB Bias Parameter: H

0.0 0.2 0.4 0.6 0.8 1.0

1990 1995 2000 2005 2010

Year

Figure 59 

P-Values for ASB Bias Parameter: P

0.0 0.2 0.4 0.6 0.8 1.0

1990 1995 2000 2005 2010

Year

Figure 60 

 

0.0 0.2 0.4 0.6 0.8 1.0

P-Values for ASB Bias Parameter: S

1990 1995 2000 2005 2010

Year

Figure 61 

 

0.0 0.2 0.4 0.6 0.8 1.0

P-Values for ASB Bias Parameter: F1

1990 1995 2000 2005 2010

Year

Figure 62 

 

0.0 0.2 0.4 0.6 0.8 1.0

P-Values for ASB Bias Parameter: F2

1990 1995 2000 2005 2010

Year

Figure 63 

 

0.0 0.2 0.4 0.6 0.8 1.0

P-Values for ASB Bias Parameter: F3

1990 1995 2000 2005 2010

Year

Figure 64 
 

 

 

 

 

 

 

 

 

P-Values for ASB Bias Parameter: F4

0.0 0.2 0.4 0.6 0.8 1.0

1990 1995 2000 2005 2010

Year

The mean relative absolute net contribution by category for Treatment 2 in Table 12 on page 40 shows that both the 

survey results and the ASB measurements are highly influential measurements in the inventory estimates. The 

survey results contribute on average 42.28% of the estimates and the ASB measurements contribute 44.84% of the 

estimates. Of these percentages, the survey results and ASB estimates for weight group functions 3 and 4 make up 

over 75% of the survey and ASB relative absolute net contributions to the inventory item estimates on average (see 

Figure 65 and Figure 66). These functions contain weight groups 3 and 4 which by visual inspection contain the 

most relative bias. Intuitively it could be hypothesized that weight group 3 is not restricted by as many constraints as 

the other inventory items, and therefore the filter relies heavily on observations containing information on weight 

group 3. It is also worth noting the presence of bias in the ASB measurement for weight group 4 and weight group 

function 4 (Figure 46, middle) despite the “good behavior” of the five week slaughter ratio relative to the annual 

ratio of weight group 4 (Figure 54, middle). 

Figure 65 

Mean Survey Contribution for Treatment 2

ADXX.F4

ADXX.P

ADXX.F3

ADXX.S

ADXX.H

ADXX.F1

ADXX.F2

Figure 66 
 

 

 

 

 

 

13.6.3 Treatment 3 – Unbiased Expert Measurements 

The results for the final parameterization of ASB measurements as unbiased estimates of true inventory demonstrate 

that congruency can be achieved with external data and the inventory estimates, in addition to a smooth transition 

from the publishing of ASB measurements to Kalman Smoother measurements. Treatment 3 in Table 12 on page 40 

shows the categorized mean relative absolute net contribution of each measurement type to hog inventories. In 

comparing Treatment 3 to Treatment 2 with regard to the measurement contributions, it appears that treating the 

ASB measurements as unbiased resulted in a more uniform overall contribution of measurements on the final 

inventory estimates than in the case of the biased ASB measurements (Treatment 2, Table 12). The absolute relative 

contribution of the historical estimates (labeled “model”) on the last inventory measurements increased from the 

Treatment 2 level. In Figure 67 and Figure 68 we see that the most influential survey results and ASB measurements 

are weight group functions 3 and 4, similar to the contribution profile of Treatment 2. When the ASB measurements 

are treated as unbiased, the ASB measurement for weight group function 4 shows comparable influence to that of 

weight group function 3 (Figure 68). 

Figure 67 

Mean Survey Contribution for Treatment 3

ADXX.H

ADXX.F4

ADXX.P

ADXX.S

ADXX.F1

ADXX.F2

ADXX.F3

Figure 68 

Mean ASB Contribution for Treatment 3

HP

PP

F4.P

SP

F1.P

F2.P

F3.P

13.7 Standard Errors 

Graphs of the standard errors that follow demonstrate that when the ASB measurement is included and treated as an 

unbiased estimate, the fixed interval Kalman Smoother variance estimates are minimized between all three 

treatments. For Sows Farrowed, the standard errors are very close between Treatments 2 and 3. For all other 

inventory items, Treatment 3 resulted in the smallest standard errors. Excluding expert opinion resulted in the 

highest standard errors of the fixed interval Kalman Smoother estimates. 
 

 

 

Kalman Filter Standard Errors: Total Hogs

200 400 600 800

Treatment 1 - No ASB

Treatment 2 - ASB Biased

Treatment 3 - ASB Unbiased

1990 1995 2000 2005 2010

Time

 

 

 

Kalman Filter Standard Errors: Pig Crop

50 100 150 200 250

Treatment 1 - No ASB

Treatment 2 - ASB Biased

Treatment 3 - ASB Unbiased

1990 1995 2000 2005 2010

Time

 

 

 

 

10 15 20 25

Kalman Filter Standard Errors: Sows Farrowed

Treatment 1 - No ASB

Treatment 2 - ASB Biased

Treatment 3 - ASB Unbiased

1990 1995 2000 2005 2010

Time

 

 

 

 

100 200 300 400

Kalman Filter Standard Errors: Mkt <50 lbs

Treatment 1 - No ASB

Treatment 2 - ASB Biased

Treatment 3 - ASB Unbiased

1990 1995 2000 2005 2010

Time

 

 

 

 

200 400 600 800

Kalman Filter Standard Errors: Mkt 50-119 lbs

Treatment 1 - No ASB

Treatment 2 - ASB Biased

Treatment 3 - ASB Unbiased

1990 1995 2000 2005 2010

Time

 

 

 

 

100 200 300 400 500

Kalman Filter Standard Errors: Mkt 120-179 lbs

Treatment 1 - No ASB

Treatment 2 - ASB Biased

Treatment 3 - ASB Unbiased

1990 1995 2000 2005 2010

Time

 

 

 

 

50 100 150 200 250

Kalman Filter Standard Errors: Mkt >180 lbs

Treatment 1 - No ASB

Treatment 2 - ASB Biased

Treatment 3 - ASB Unbiased

1990 1995 2000 2005 2010

Time

 

 

 

 

100 200 300 400 500 600

Kalman Filter Standard Errors: F1

Treatment 1 - No ASB

Treatment 2 - ASB Biased

Treatment 3 - ASB Unbiased

1990 1995 2000 2005 2010

Time
 

 

 

 

Kalman Filter Standard Errors: F2

0.002 0.003 0.004 0.005 0.006 0.007

Treatment 1 - No ASB

Treatment 2 - ASB Biased

Treatment 3 - ASB Unbiased

1990 1995 2000 2005 2010

Time

 

 

 

Kalman Filter Standard Errors: F3

0.005 0.010 0.015 0.020

Treatment 1 - No ASB

Treatment 2 - ASB Biased

Treatment 3 - ASB Unbiased

1990 1995 2000 2005 2010

Time

 

 

 

 

0.005 0.010 0.015 0.020 0.025 0.030

Kalman Filter Standard Errors: F4

Treatment 1 - No ASB

Treatment 2 - ASB Biased

Treatment 3 - ASB Unbiased

1990 1995 2000 2005 2010

Time
References 

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Durbin, J. and Koopman, S.J. Time Series Analysis by State Space Methods. Oxford: Oxford University Press, 2012. 

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