# A Descriptive Method of Firm Size Transition Dynamics Using Markov Chain

###### Abstract

Social employment, which is mostly carried by firms of different types,
determines the prosperity and stability of a country. As time passing, the
fluctuations of firm employment can reflect the process of creating or
destroying jobs. Therefore, it is instructive to investigate the firm
employment (size) dynamics. Drawing on the firm-level panel data extracted
from the Chinese Industrial Enterprises Database 1998-2013, this paper
proposes a Markov-chain-based descriptive approach to clearly demonstrate the
firm size transfer dynamics between different size categories. With this
method, any firm size transition path in a short time period can be
intuitively demonstrated. Furthermore, by utilizing the properties of Markov
transfer matrices, the definition of transition trend and the transition
entropy are introduced and estimated. As a result, the tendency of firm size
transfer between small, medium and large can be exactly revealed, and the
uncertainty of size change can be quantified. Generally from the evidence of
this paper, it can be inferred that small and medium manufacturing firms in
China have greater job creation potentials compared to large firms over this
time period.

Boyang You111Ph.D. Student, Department of Economics, University of Bath.
Email<EMAIL_ADDRESS>Kerry Papps222Senior Lecturer, Department of
Economics, University of Bath. Centre for Analysis of Social Policy (CASP)
Labour, Education and Health Economics. Email<EMAIL_ADDRESS>
###### Key words:

Firm size, job creation, Markov chain, dynamics description.

###  1 Introduction

Firms are business organizations which carrying the vast majority of social
employment. When defining the size of firms as their number of employees
(Birch, 1979), the net job creation can be calculated by the firm size
variations. Early studies mostly focus on quantifying net job creation of
firms via cross-sectional data sets: Birch (1979, 1981, 1987), Davis &
Haltiwanger (1994), Neumark et al. (2011) all look into the job creation
abilities of different firm types including small, medium and large firms.

In practice, the process of firm job creation is continuous. The changes of
firm job creation in various time periods can be regarded as the dynamics of
firm size, making scholars examine this issue via time series approaches. One
of the classic method used is the panel vector autoregression (PVAR, Love &
Zicchino, 2006). By this model, the impact among firm characteristics can be
clearly estimated. Davis & Haltiwanger (1999, 2001), Koellinger & Roy (2012)
all deeply examine the firm employment or entrepreneurship determinants via
this method.

The PVAR method holds the advantages of expressing how the dynamic changes of
individual firms are influenced by its own. However, this approach cannot
clearly quantify the net job creation of firms in different size categories as
Birch’s studies do. Moreover, data stationary and data time length is strictly
required when using autoregressive models. Due to the existence of firm entry
& exit, existing firm-level data sets can hardly meet the requirement. This
inspires us to further propose a Markov-chain-based approach to describe the
dynamics of firm size.

Markov chain, which can describe the random transfer processes is widely used
in many aspects. Unlike autoregression models, any two-period transfer can be
described by establishing transition matrix. Using the properties of Markov
chain, multiple first-order transfer matrices can also describe the dynamics
over a continuous time period in an intuitive way. In the topic of firms,
there are currently not many relevant studies. Uyar (1972) forecasts the
replacement demand of employees using Markov chain. Horowitz & Horowitz (1968)
use a first-order Markov process to describe market change. Joining Markov
first-order transfer matrices with entropy, the industrial agglomeration of
the brewing industry is quantified. Kopecky & Suen (2010) propose a method
that can use Markov chain to approximate autoregression models.

In this paper, a method that describes firm size transition dynamics is
proposed based on Markov chain. Joint with a rich continuous panel data
exacted from the Chinese Industrial Enterprises Database 1998-2013, the first
order transfer matrices are firstly calculated, and the dynamic size transfer
path of Chinese manufacturing firms between 1998 to 2013 are revealed.
Furthermore, using the properties of Markov Chain, not only the job creation
amount but also the general firm growth trend of each can be presented by
calculating the probability of upward or downward transfer in the matrices.
Besides, following the idea of Horowitz & Horowitz (1968), entropy of firm
size are defined and measured. Through these results, the job creation
probability of firms in any time period or any size category can be directly
described, which may intuitively help policy maker recognize firms with which
size categories may create more jobs, and in turn formulate effective
employment-boosting polices for the target firms to alleviate the employment
pressure of the society in an effective way.

For the rest of this paper, section 2 illustrates the methodology of the
proposed method, in which the definition of firm size transition and the
Markov first-order transition matrix is given. Based on the transition
matrices, the transfer path, trend and entropy are defined. Section 3 shows
the results of using the proposed method over a firm-level data set, which
includes data set descriptions, calculations details, results analyses and
robustness check. Section 4 is the conclusion and discussion of this paper.

###  2 Methodology

In this section, the methodology of the transition analysis will be presented.
First, the size of the firm is divided into multiple states, which is judged
by the number of employees of firms from small to large. Next, existing panel
data will be rectangularized including firm entry and exit, and the Markov
state transfer matrices between different consecutive years will be
calculated. By analysing the obtained Markov matrices, a general trend of
inter-transition in terms of firm size may be concluded, while the average job
creation among all categories will be calculated based on transfer matrices.

####  2.1 Definition of firm size transition

The basic idea of using Markov chain is to illustrate how the size of firms,
which belongs to one of the $N$ categories, are relocated after $d$ years. The
entire process of the transfer constitutes multiple probability transition
matrices which can be considered as a Markov chain, while all these matrices
can be calculated from the sample data set.

Based on this idea, it is necessary to find that among all firms, how many of
them has transferred from size category $i$ to $j$ after $d$ years. To this
end, let $f_{ji}(D,D+d)$ be the possibility of the size of a firm transfers
from category $i$ to $j$, from base year $D$ to year $D+d$. Let the categories
in terms of firm size at year $d$ be
$S_{d},\>S_{d}\in\left\\{0,1,2,\text{\ldots},N\right\\}$. Then there is:

$f_{ji}(D,D+d)=Pr\left(S_{D+d}=j\mid
S_{D}=i\right)\;\;\;\;\;i,j\in\left\\{0,1,2,\text{\ldots},N\right\\}$ (1)

From equation (1), it is obvious to find that the firm size transition process
can be described as a Markov chain. Also, for notation simplicity, in this
paper, assuming base year $D=0$, which means the base year the firm size
transfer begins at year $0$. Hence, there is $f_{ji}(D,D+d)=f_{ji}(0,d)$.

####  2.2 Properties of Markov chain

According to Neumark, et al. (2011), firms are divided into 13 size categories
according to their employee numbers, in which category 0 represents the non-
existing firms at current year. Similar to You & Fan (2020), let $N=12$. Then,
the transfer matrix of the firm size can be considered as a $[13\times 13]$
matrix. Let $F(d-1,d)$ be a first order 13-state Markov transfer matrix. Based
on equation (2), there is:

$F(d-1,d)=\left[\begin{array}[]{cccc}f_{00}(d-1,d)&f_{01}(d-1,d)&\cdots&f_{0N}(d-1,d)\\\
f_{10}(d-1,d)&f_{11}(d-1,d)&\cdots&f_{1N}(d-1,d)\\\ \vdots&\vdots&&\vdots\\\
f_{N0}(d-1,d)&f_{N1}(d-1,d)&\cdots&f_{NN}(d-1,d)\end{array}\right],$ (2)

where elements in every column satisfies:

$\sum_{j=0}^{N}f_{ji}(d-1,d)=1$ (3)

Note that $F(d-1,d)$ tend to be a diagonally dominant matrix if firm entry and
exit (state 0) are excluded. This is because, for most companies, size change
between two consecutive years is relatively small. That is to say, for any
$i,j=0,1,2,\ldots,N,j\neq i$, roughly satisfying $f_{ii}(d-1,d)\geqslant
f_{ji}(d-1,d)$.

Also, according to the Law of total probability, there is:

$Pr\left(S_{d}=j\right)=\sum_{i=0}^{N}Pr\left(S_{d}=j\mid
S_{d-1}=i\right)Pr\left(S_{d-1}=i\right)$ (4)

Let $p_{j}(d)=Pr\left(S_{d}=j\right)$, then, there is:

$p_{j}(d)=\sum_{i=0}^{N}f_{ji}(d-1,d)p_{i}(d-1)$ (5)

##### Transition path

The Gibrat Law indicates that the evolution of the firm size is only related
to the size of its previous year. Following this idea, a transition path can
be described by a Markov chain. Let
$P(d)=\left[\begin{array}[]{cccc}p_{0}(d)&p_{1}(d)&\cdots&p_{N}(d)\end{array}\right]$${}^{{}^{\prime}}$.
For any $d$ there is:

$P(d)=F(d-1,d)P(d-1)$ (6)

Equation (6) can also be transformed as follows:

$\displaystyle P(d)$ $\displaystyle=F(d-1,d)P(d-1)$
$\displaystyle=F(d-1,d)F(d-2,d-1)P(d-2)$ $\displaystyle=\cdots$
$\displaystyle=F(d-1,d)F(d-2,d-1)\cdots F(1,2)F(0,1)P(0)$ (7)

In a special case when the Markov chain is homogeneous, define $F$ which
satisfies $F=F(d-1,d)=F(d-2,d-1)=\cdots=F(0,1)$, there is:

$P(d)=F^{d}P(0)$ (8)

It should be noted that the assumption given here is strong. In practice, the
transfer matrices obtained from sample data can hardly be homogeneous.

##### Transition trend

By analysing the transfer matrices, not only the probability of small firms
evolving into large firms, but also the transition trend can be discovered.
For example, when prosperity comes, firms may be willing to increase
recruitment, which will make the distribution of the transfer matrix elements
move to the bottom left corner; conversely, when recession comes, firms may
reduce recruitment or even layoffs, leading to the distribution of the
transfer matrix elements moving to the upper right corner.

In order to quantify this trend, let

$R(d)=\sum_{j<i}f_{ji}(d-1,d)p_{j}(d)$ (9)

$L(d)=\sum_{j>i}f_{ji}(d-1,d)p_{j}(d)$ (10)

Next, simply examine $Q(d)=\frac{L(d)}{R(d)}$, the transition trend at year
$d$ can be described.

##### Transition entropy

In the study of Horowitz & Horowitz (1968), the concept of entropy in
information theory, which may reveal the uncertainty, are used to quantify
industrial agglomeration. Following this idea, it is proper to introduce a
concept of transition entropy, which reveals the randomness of firm size
evolution into out topic. With this, the likelihood of firm size change in a
certain category can be quantified and compared.

According to the definition proposed by Shannon (1948), the entropy of the
$i-th$ column element of the transfer matrix, for each $f_{ji}(d-1,d)>0$, can
be expressed as:

$I_{i}(d)=-\sum_{j=0}^{N}f_{ji}(d-1,d)\cdot\ln f_{ji}(d-1,d),$ (11)

where $I_{i}(d)$ is the entropy of the $i-th$ firm category. Note that the
greater the entropy of the $i-th$ firm category, the greater the randomness of
its size change.

###  3 Evaluations over the data set

####  3.1 Data

The data set used in this paper is extracted from the Chinese Industrial
Enterprises Database 1998-2013. Variable SIZE is defined by the average number
of employees of firms in the year. The descriptive statistics of the data set
used in this paper is shown in Table 1.

Table 1 Descriptive statistics of Markov Transition section

Variable | Obs. | Variable Unit | Mean | Min | Max | Std. Dev.
---|---|---|---|---|---|---
SIZE | 3,747,157 | Person | 293.63 | 1 | 760,884 | 1,453.6
Data source: Chinese Industrial Enterprises Database.

The data set is selected unbalanced from 1998-2013 due to firm entry and exit.
The sample size in this section is 3,747,157, with its minimum value 1,
maximum value 760,884 and standard deviation 1453.6. For a purpose of
calculation, the unbalanced data set will be further rectangularized to a
balanced data set by filling 0 into missing values, which are considered as
non-existing firms.

####  3.2 Calculation details

First, referencing the firm size classification criterion used in our previous
study (You & Fan, 2020, Table 1), firms are divided into size categories of
0-13 with employee number boundaries of 0, 20, 50, 100, 250, 500, 1,000,
2,500, 5,000, 10,000, 25,000 and 50,000.

Second, according to Section 2.2, transfer matrices $F(d-1,d)$ are built based
on the probabilities
$f_{ji}(d-1,d),\>i,j\in\left\\{0,1,2,\text{\ldots},12\right\\}$. To calculate
each $f_{ji}(d-1,d)$, according to the Borel’s law of large numbers, when
sample size is sufficiently large, there is:

$\frac{Num_{ji}(d)}{Num_{i}(d-1)}\rightarrow^{p}Pr\left(S_{d}=j\mid
S_{d-1}=i\right)=f_{ji}(d-1,d),$ (12)

where $Num_{i}(d-1)$ is the number of the firms whose size is in category $i$
at year $d-1$, while $Num_{ji}(d)$ is the number of firms whose size category
changes from category $i$ at year $d-1$ to category $j$ at year $d$.

With this, a Markov chain that reflects dynamic transitions between firm size
categories can be established.

####  3.3 Results

As the sample data is collected between 1998 and 2013, there are 15 first-
order transfer matrices obtained, which are defined in equation (2) while
calculated by equation (12). Table 2 is the first-order transfer matrix from
1998 to 1999.

Table 2 The first-order transfer matrix from 1998 to 1999

Size | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12
---|---|---|---|---|---|---|---|---|---|---|---|---|---
0 | 0.0000 | 0.3688 | 0.2768 | 0.2173 | 0.1743 | 0.1366 | 0.1186 | 0.1219 | 0.1061 | 0.1024 | 0.0815 | 0.0926 | 0.0286
1 | 0.0546 | 0.3947 | 0.0526 | 0.0122 | 0.0055 | 0.0029 | 0.0014 | 0.0017 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000
2 | 0.1759 | 0.1099 | 0.4844 | 0.0914 | 0.0163 | 0.0068 | 0.0052 | 0.0018 | 0.0019 | 0.0000 | 0.0000 | 0.0000 | 0.0000
3 | 0.2425 | 0.0455 | 0.1223 | 0.5370 | 0.0921 | 0.0165 | 0.0077 | 0.0024 | 0.0019 | 0.0000 | 0.0043 | 0.0000 | 0.0000
4 | 0.2952 | 0.0566 | 0.0421 | 0.1235 | 0.6403 | 0.1501 | 0.0336 | 0.0091 | 0.0025 | 0.0017 | 0.0000 | 0.0000 | 0.0000
5 | 0.1279 | 0.0160 | 0.0168 | 0.0121 | 0.0628 | 0.6264 | 0.1542 | 0.0254 | 0.0025 | 0.0068 | 0.0000 | 0.0000 | 0.0000
6 | 0.0612 | 0.0029 | 0.0037 | 0.0054 | 0.0066 | 0.0550 | 0.6349 | 0.1404 | 0.0181 | 0.0017 | 0.0000 | 0.0000 | 0.0000
7 | 0.0313 | 0.0056 | 0.0010 | 0.0010 | 0.0018 | 0.0042 | 0.0418 | 0.6703 | 0.1685 | 0.0137 | 0.0000 | 0.0000 | 0.0000
8 | 0.0069 | 0.0000 | 0.0001 | 0.0001 | 0.0003 | 0.0013 | 0.0017 | 0.0247 | 0.6648 | 0.1758 | 0.0043 | 0.0185 | 0.0286
9 | 0.0028 | 0.0000 | 0.0001 | 0.0001 | 0.0000 | 0.0001 | 0.0005 | 0.0018 | 0.0293 | 0.6792 | 0.1631 | 0.0000 | 0.0000
10 | 0.0012 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0002 | 0.0006 | 0.0044 | 0.0188 | 0.7253 | 0.2222 | 0.0000
11 | 0.0003 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0002 | 0.0000 | 0.0000 | 0.0000 | 0.0215 | 0.6481 | 0.2571
12 | 0.0002 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0185 | 0.6857
Axis coordinates represent size categories, transfer probabilities in the
matrix. Full matrices see Appendix

As it is imagined before calculating, if firm entry and exit are excluded,
these matrices tend to be diagonally dominant matrices, with most of non-zero
values on the diagonal of the matrices. 333See Appendix for all 15 matrices
from 1998 to 2013.

Next, based on the equations in Section 2.3, transition trend and transition
entropy are calculated.444See Appendix for full results.

Figure 1 is the obtained transition trend based on equation (9) and (10). The
value of $Q(d)$ indicated a general transition trend at year $d$. It can be
observed that $Q(d)$ in 2008, 2009 and 2013 are relatively low. This indicates
that there is more job destruction happened in these two years. In contrast,
$Q(d)$ values in 2004, 2010 and 2012 are relatively high, which means job
creation is far more prevalent than job destruction in these years. The
massive job destruction in 2008 and 2009 might be attributed to the subprime
crisis, while the massive job creation 2004 and 2012 might be attributed to
national policy changes. Note that the job destruction trend in 2010 shall be
attributed to the criteria change of the database, which no longer include
firms with sales less than 20 million CNY instead of 5 million.

Figure 1 The transition trend from 1999 to 2013

Year means end year. e.g., year 1999 represents 1998 to 1999. $Q(d)$ is
transition trend. Full data see Appendix.

Figure 2 is the obtained empirical result based on equation (11). In the
figure firms in categories 1-3 are defined as small firms, firms in categories
4-6 are defined as mediums firms and firms in categories 7-12 are defined as
large firms. From different size categories it can be observed that small
firms tend to have greater transfer entropy while large firms tend to have
smaller transfer entropy. The entropy of small firms fluctuates around 1.2.
For the entropy of medium firms, there is a rough entropy difference of 0.1 in
average compared to small firms. The entropy of large firms have the lowest
value in average, which fluctuates around 1. From the perspective over time it
can be observed that in 2001, 2004, 2008 and 2010, the transition entropy
values increase significantly, which means the likelihood of size transfer
increase. This might be attributed to the instability of industrial economic
environment. Note that in 2012, due to missing data, the transition entropy
cannot be accurately estimated.

Figure 2 The transition entropy for small, medium and large firms from 1999 to
2013

Year in this figure means end year. $I_{i}(d)$ is transition entropy. Full
data see Appendix.

####  3.4 Robustness check of Markov transfer matrices

According to Neumark et al. (2011), the obtained results should be checked by
2-year periods instead of 1 in the consideration of robustness. To this end,
the 2nd-ordered Markov transfer matrices will be calculated and compared with
the previous results. According to equation (7), when $P(0)$ is given,

$P(d)=F(d-1,d)F(d-2,d-1)P(d-2)=F(d-2,d)P(d-2),$ (13)

where $d\geq 2$ and $F(d-2,d)$ is the second-ordered Markov transfer matrix
from $d-2$ to $d$. This means the second-ordered Markov transfer matrices
should completely equals to the product of two continuous first-order Markov
transfer matrices. Table 3 is the second-order transfer matrix 1998-2000
calculated from the product of two first-order transfer matrices 1998-1999 and
1999-2000. Table 4 is the second-order transfer matrix 1998-2000 calculated by
the cross-sectional data from 1998 to 2000.

Note that the later cannot be calculated by only the two cross-sectional data
sets of 1998 and 2000. This is because the firms that enter at 1999 while exit
at 2000 are ignored, which are considered as 0-to-0 state transfers from 1998
to 2000. To include all 0-to-0 state transfer in the second-order matrices,
the three cross-sectional data sets of 1998, 1999 and 2000 must be
rectangularized to a strongly balanced data set with missing values replaced
by 0. With this change, all firms that exist in any of the three years are
included, and it can be found that the second-order transfer matrix obtained
from data set equals to the product of two first-order transfer matrix, which
means the obtained first-order transfer matrices are correctly calculated.

Compare the second-order transfer matrix with the first-order transfer matrix,
it can be found that there are no significant differences between most of the
elements. The most obvious difference between the two kinds is that compared
to the second-order matrices, the diagonal elements of the first-order
matrices have larger values. This can be understood as when time interval
length increases, the probability that firms remain in the same categories may
gradually decrease. But in general, it can be considered that the results
obtained are not very sensitive to the time interval length.

Table 3 The second-order transfer matrix 1998-2000 (calculated from product)

Size | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12
---|---|---|---|---|---|---|---|---|---|---|---|---|---
0 | 0.2165 | 0.2074 | 0.1968 | 0.1787 | 0.1588 | 0.1427 | 0.1319 | 0.1290 | 0.1261 | 0.1354 | 0.1441 | 0.1143 | 0.1987
1 | 0.0495 | 0.2219 | 0.0834 | 0.0485 | 0.0356 | 0.0261 | 0.0201 | 0.0177 | 0.0118 | 0.0111 | 0.0081 | 0.0088 | 0.0027
2 | 0.1251 | 0.1814 | 0.3274 | 0.1408 | 0.0661 | 0.0427 | 0.0335 | 0.0296 | 0.0246 | 0.0216 | 0.0173 | 0.0193 | 0.0060
3 | 0.1789 | 0.1488 | 0.1931 | 0.3704 | 0.1487 | 0.0678 | 0.0470 | 0.0378 | 0.0308 | 0.0282 | 0.0265 | 0.0248 | 0.0074
4 | 0.2345 | 0.1532 | 0.1298 | 0.1970 | 0.4676 | 0.2134 | 0.0929 | 0.0579 | 0.0388 | 0.0324 | 0.0258 | 0.0262 | 0.0080
5 | 0.1076 | 0.0542 | 0.0451 | 0.0428 | 0.0971 | 0.4145 | 0.1870 | 0.0584 | 0.0314 | 0.0200 | 0.0145 | 0.0116 | 0.0035
6 | 0.0527 | 0.0192 | 0.0158 | 0.0149 | 0.0187 | 0.0795 | 0.4210 | 0.1730 | 0.0484 | 0.0196 | 0.0141 | 0.0067 | 0.0017
7 | 0.0258 | 0.0115 | 0.0067 | 0.0056 | 0.0058 | 0.0109 | 0.0616 | 0.4567 | 0.2071 | 0.0493 | 0.0259 | 0.0208 | 0.0086
8 | 0.0057 | 0.0015 | 0.0011 | 0.0009 | 0.0009 | 0.0018 | 0.0037 | 0.0358 | 0.4271 | 0.1906 | 0.0335 | 0.0155 | 0.0181
9 | 0.0023 | 0.0005 | 0.0004 | 0.0004 | 0.0002 | 0.0004 | 0.0008 | 0.0032 | 0.0472 | 0.4547 | 0.2135 | 0.0335 | 0.0012
10 | 0.0009 | 0.0003 | 0.0002 | 0.0002 | 0.0001 | 0.0001 | 0.0003 | 0.0009 | 0.0062 | 0.0365 | 0.4479 | 0.2684 | 0.0739
11 | 0.0002 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0002 | 0.0000 | 0.0005 | 0.0005 | 0.0260 | 0.3548 | 0.1596
12 | 0.0002 | 0.0001 | 0.0001 | 0.0000 | 0.0000 | 0.0000 | 0.0001 | 0.0000 | 0.0000 | 0.0000 | 0.0027 | 0.0952 | 0.5106
Axis coordinates represent size categories, transfer probabilities in the
matrix.

Table 4 The second-order transfer matrix 1998-2000 (calculated from data)

Size | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12
---|---|---|---|---|---|---|---|---|---|---|---|---|---
0 | 0.2166 | 0.2074 | 0.1968 | 0.1786 | 0.1588 | 0.1426 | 0.1318 | 0.1290 | 0.1260 | 0.1354 | 0.1442 | 0.1144 | 0.1987
1 | 0.0496 | 0.2219 | 0.0835 | 0.0485 | 0.0356 | 0.0262 | 0.0200 | 0.0177 | 0.0118 | 0.0111 | 0.0081 | 0.0088 | 0.0027
2 | 0.1251 | 0.1813 | 0.3274 | 0.1409 | 0.0660 | 0.0426 | 0.0335 | 0.0296 | 0.0246 | 0.0217 | 0.0173 | 0.0193 | 0.0061
3 | 0.1789 | 0.1488 | 0.1931 | 0.3705 | 0.1488 | 0.0679 | 0.0471 | 0.0378 | 0.0309 | 0.0282 | 0.0265 | 0.0248 | 0.0074
4 | 0.2346 | 0.1532 | 0.1298 | 0.1970 | 0.4675 | 0.2134 | 0.0929 | 0.0580 | 0.0388 | 0.0324 | 0.0259 | 0.0263 | 0.0081
5 | 0.1076 | 0.0542 | 0.0452 | 0.0427 | 0.0970 | 0.4144 | 0.1871 | 0.0583 | 0.0315 | 0.0200 | 0.0145 | 0.0115 | 0.0035
6 | 0.0527 | 0.0192 | 0.0157 | 0.0150 | 0.0187 | 0.0795 | 0.4210 | 0.1729 | 0.0484 | 0.0196 | 0.0141 | 0.0068 | 0.0016
7 | 0.0259 | 0.0115 | 0.0066 | 0.0057 | 0.0058 | 0.0109 | 0.0616 | 0.4566 | 0.2071 | 0.0492 | 0.0259 | 0.0209 | 0.0086
8 | 0.0057 | 0.0016 | 0.0012 | 0.0008 | 0.0009 | 0.0018 | 0.0037 | 0.0358 | 0.4271 | 0.1906 | 0.0336 | 0.0156 | 0.0181
9 | 0.0022 | 0.0004 | 0.0004 | 0.0005 | 0.0002 | 0.0004 | 0.0008 | 0.0033 | 0.0472 | 0.4546 | 0.2136 | 0.0336 | 0.0013
10 | 0.0009 | 0.0002 | 0.0002 | 0.0002 | 0.0001 | 0.0002 | 0.0004 | 0.0008 | 0.0063 | 0.0366 | 0.4479 | 0.2685 | 0.0740
11 | 0.0002 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0002 | 0.0000 | 0.0004 | 0.0004 | 0.0260 | 0.3548 | 0.1596
12 | 0.0002 | 0.0001 | 0.0001 | 0.0000 | 0.0000 | 0.0000 | 0.0001 | 0.0000 | 0.0001 | 0.0000 | 0.0027 | 0.0951 | 0.5106
Axis coordinates represent size categories, transfer probabilities in the
matrix.

###  4 Conclusion and Discussion

In this paper, a Markov-chain-based descriptive method is proposed for
presenting firm size transition dynamics. By dividing firms into multiple
states according to their size categories, a Markov chain that reveals the
internal transfer between size categories is established. Using the properties
of Markov chain, the definition of transition path, transition trend and
transition entropy are introduced based on first-order transition matrices. By
calculating the transition path, the growth probabilities of firms in terms of
size for a certain time period can be clearly demonstrated; by evaluating the
transition trend, the tendency of firm size transition towards large firms or
small and medium firms in any time period can be revealed; by calculating
transition entropy of a certain firm size category, the uncertainty of firm
size transition, or in other words, the likelihood of firm size change can be
quantified.

Furthermore, the proposed descriptive method is used on the rich firm-level
data set extracted from the Chinese Industrial Enterprises Database 1998-2013.
From the transition matrices it can be observed that most firms have minor
size changes, while a small number of firms may grow or deteriorate
significantly. From the result of transition trend, it can be found that in
2004, 2010 and 2012 Chinese manufacturing firms tend to transfer into larger
firms, which implies economic growth; in 2008, 2009 and 2013 however, firms
tend to transfer into smaller firms, which implies economic recession. From
the result of transition entropy, it can be observed that compared to large
firms, small and medium firms in China are much more likely to change in terms
of size. In general, evidence concludes that small and medium manufacturing
firms in China have greater job creation potentials compared to large firms
over the time period.

Nevertheless, the research remains imperfect. First, the proposed Markov-
chain-based method is only applicable to describe the fact that has occurred.
There is no function of prediction so far. In addition, currently there is no
sufficient precedent paper fully investigate job creation issue via Markov
chain. So, whether the property of Markov chains is applicable to empirical
analysis and forecasting under all circumstances needs further discussion.
Finally, this paper follows the classification standard proposed by Neumark et
al. (2011) to classify firm size, which is extensively used but not linear.
According to previous research, it can be roughly considered that this
classification conforms to a log-normal function (You & Fan, 2020), yet this
standard might not be the best on describing categories of small firms.

### References

  * [1] Birch, D. G. W. (1979). The Job-Generation Process. Cambridge, MA: MIT Program on Neighborhood and Regional Change.

  * [2] Birch, D. G. W. (1981). Who Creates Jobs? The Public Interest, 65, 3-14.
  * [3] Birch, D. G. (1987). Job creation in America: How our smallest companies put the most people to work. University of Illinois at Urbana-Champaign’s Academy for Entrepreneurial Leadership Historical Research Reference in Entrepreneurship.
  * [4] Davis, S. J. , Haltiwanger, J. , & Schuh, S. D. (1994). Small business and job creation: dissecting the myth and reassessing thefacts. Nber Working Papers, 29(3), 13-21.
  * [5] Davis, S. J., & Haltiwanger, J. (1999). On the driving forces behind cyclical movements in employment and job reallocation. American Economic Review, 89(5), 1234-1258.
  * [6] Davis, S. J., & Haltiwanger, J. (2001). Sectoral job creation and destruction responses to oil price changes. Journal of monetary economics, 48(3), 465-512.
  * [7] Horowitz, A., & Horowitz, I. (1968). Entropy, Markov processes and competition in the brewing industry. The Journal of Industrial Economics, 196-211.
  * [8] Koellinger, P. D., & Roy Thurik, A. (2012). Entrepreneurship and the business cycle. Review of Economics and Statistics, 94(4), 1143-1156.
  * [9] Kopecky, K. A., & Suen, R. M. (2010). Finite state Markov-chain approximations to highly persistent processes. Review of Economic Dynamics, 13(3), 701-714.
  * [10] Love, I., & Zicchino, L. (2006). Financial development and dynamic investment behavior: Evidence from panel VAR. The Quarterly Review of Economics and Finance, 46(2), 190-210.
  * [11] Neumark, D., Wall, B., & Zhang, J. (2011). Do small businesses create more jobs? New evidence for the United States from the National Establishment Time Series. The Review of Economics and Statistics, 93(1), 16-29.
  * [12] Shannon, C. E. (1948). A mathematical theory of communication. Bell system technical journal, 27(3), 379-423.
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  * [14] You, B., & Fan, C. (2020). An empirical study of net job creation, firm size and firm age in China. Journal of Entrepreneurship and Innovation in Emerging Economies, 6(1), 220-231.

## Appendix

#### Obtained First-order Markov Transfer Matrices

The obtained first-order Markov transfer matrices from 1998 to 2013 are
presented as follows:

Table 5 The first-order transfer matrix from 1998 to 1999

Size | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12
---|---|---|---|---|---|---|---|---|---|---|---|---|---
0 | 0.0000 | 0.3688 | 0.2768 | 0.2173 | 0.1743 | 0.1366 | 0.1186 | 0.1219 | 0.1061 | 0.1024 | 0.0815 | 0.0926 | 0.0286
1 | 0.0546 | 0.3947 | 0.0526 | 0.0122 | 0.0055 | 0.0029 | 0.0014 | 0.0017 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000
2 | 0.1759 | 0.1099 | 0.4844 | 0.0914 | 0.0163 | 0.0068 | 0.0052 | 0.0018 | 0.0019 | 0.0000 | 0.0000 | 0.0000 | 0.0000
3 | 0.2425 | 0.0455 | 0.1223 | 0.5370 | 0.0921 | 0.0165 | 0.0077 | 0.0024 | 0.0019 | 0.0000 | 0.0043 | 0.0000 | 0.0000
4 | 0.2952 | 0.0566 | 0.0421 | 0.1235 | 0.6403 | 0.1501 | 0.0336 | 0.0091 | 0.0025 | 0.0017 | 0.0000 | 0.0000 | 0.0000
5 | 0.1279 | 0.0160 | 0.0168 | 0.0121 | 0.0628 | 0.6264 | 0.1542 | 0.0254 | 0.0025 | 0.0068 | 0.0000 | 0.0000 | 0.0000
6 | 0.0612 | 0.0029 | 0.0037 | 0.0054 | 0.0066 | 0.0550 | 0.6349 | 0.1404 | 0.0181 | 0.0017 | 0.0000 | 0.0000 | 0.0000
7 | 0.0313 | 0.0056 | 0.0010 | 0.0010 | 0.0018 | 0.0042 | 0.0418 | 0.6703 | 0.1685 | 0.0137 | 0.0000 | 0.0000 | 0.0000
8 | 0.0069 | 0.0000 | 0.0001 | 0.0001 | 0.0003 | 0.0013 | 0.0017 | 0.0247 | 0.6648 | 0.1758 | 0.0043 | 0.0185 | 0.0286
9 | 0.0028 | 0.0000 | 0.0001 | 0.0001 | 0.0000 | 0.0001 | 0.0005 | 0.0018 | 0.0293 | 0.6792 | 0.1631 | 0.0000 | 0.0000
10 | 0.0012 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0002 | 0.0006 | 0.0044 | 0.0188 | 0.7253 | 0.2222 | 0.0000
11 | 0.0003 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0002 | 0.0000 | 0.0000 | 0.0000 | 0.0215 | 0.6481 | 0.2571
12 | 0.0002 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0185 | 0.6857

Table 6 The first-order transfer matrix from 1999 to 2000

Size | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12
---|---|---|---|---|---|---|---|---|---|---|---|---|---
0 | 0.0000 | 0.3837 | 0.2840 | 0.2272 | 0.1874 | 0.1582 | 0.1433 | 0.1457 | 0.1382 | 0.1538 | 0.1587 | 0.1111 | 0.2424
1 | 0.0951 | 0.4501 | 0.0601 | 0.0266 | 0.0192 | 0.0121 | 0.0077 | 0.0054 | 0.0006 | 0.0016 | 0.0000 | 0.0000 | 0.0000
2 | 0.2079 | 0.1071 | 0.5227 | 0.0819 | 0.0197 | 0.0093 | 0.0050 | 0.0026 | 0.0012 | 0.0000 | 0.0000 | 0.0000 | 0.0000
3 | 0.2575 | 0.0326 | 0.1018 | 0.5490 | 0.0786 | 0.0166 | 0.0100 | 0.0034 | 0.0018 | 0.0016 | 0.0040 | 0.0000 | 0.0000
4 | 0.2718 | 0.0196 | 0.0250 | 0.1046 | 0.6278 | 0.1252 | 0.0292 | 0.0168 | 0.0066 | 0.0016 | 0.0040 | 0.0000 | 0.0000
5 | 0.1026 | 0.0046 | 0.0044 | 0.0083 | 0.0608 | 0.6136 | 0.1218 | 0.0181 | 0.0203 | 0.0016 | 0.0079 | 0.0000 | 0.0000
6 | 0.0398 | 0.0013 | 0.0015 | 0.0021 | 0.0053 | 0.0605 | 0.6330 | 0.1152 | 0.0191 | 0.0131 | 0.0119 | 0.0000 | 0.0000
7 | 0.0192 | 0.0010 | 0.0006 | 0.0004 | 0.0010 | 0.0039 | 0.0484 | 0.6623 | 0.1382 | 0.0196 | 0.0278 | 0.0159 | 0.0000
8 | 0.0037 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0004 | 0.0013 | 0.0290 | 0.6292 | 0.1162 | 0.0159 | 0.0000 | 0.0000
9 | 0.0013 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0001 | 0.0002 | 0.0011 | 0.0407 | 0.6547 | 0.1468 | 0.0000 | 0.0000
10 | 0.0007 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0004 | 0.0036 | 0.0360 | 0.6032 | 0.2063 | 0.0303
11 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0001 | 0.0000 | 0.0006 | 0.0000 | 0.0198 | 0.5397 | 0.0303
12 | 0.0002 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0001 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.1270 | 0.6970

Table 7 The first-order transfer matrix from 2000 to 2001

Size | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12
---|---|---|---|---|---|---|---|---|---|---|---|---|---
0 | 0.0000 | 0.4047 | 0.3289 | 0.2718 | 0.2386 | 0.2143 | 0.1967 | 0.1857 | 0.1828 | 0.1813 | 0.1187 | 0.0465 | 0.1795
1 | 0.0586 | 0.2635 | 0.0502 | 0.0115 | 0.0056 | 0.0037 | 0.0023 | 0.0018 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000
2 | 0.1947 | 0.1276 | 0.4332 | 0.0879 | 0.0184 | 0.0074 | 0.0037 | 0.0018 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000
3 | 0.2659 | 0.0841 | 0.1323 | 0.4812 | 0.0925 | 0.0158 | 0.0061 | 0.0030 | 0.0020 | 0.0000 | 0.0000 | 0.0000 | 0.0000
4 | 0.2891 | 0.0804 | 0.0441 | 0.1292 | 0.5658 | 0.1551 | 0.0332 | 0.0119 | 0.0055 | 0.0000 | 0.0046 | 0.0233 | 0.0000
5 | 0.1143 | 0.0249 | 0.0092 | 0.0143 | 0.0690 | 0.5387 | 0.1558 | 0.0242 | 0.0130 | 0.0072 | 0.0000 | 0.0000 | 0.0000
6 | 0.0476 | 0.0097 | 0.0014 | 0.0030 | 0.0081 | 0.0589 | 0.5507 | 0.1404 | 0.0157 | 0.0018 | 0.0000 | 0.0233 | 0.0256
7 | 0.0218 | 0.0049 | 0.0006 | 0.0008 | 0.0016 | 0.0052 | 0.0491 | 0.6038 | 0.1978 | 0.0287 | 0.0137 | 0.0000 | 0.0256
8 | 0.0051 | 0.0002 | 0.0000 | 0.0002 | 0.0002 | 0.0006 | 0.0019 | 0.0255 | 0.5553 | 0.1813 | 0.0365 | 0.0233 | 0.0000
9 | 0.0020 | 0.0000 | 0.0000 | 0.0001 | 0.0000 | 0.0001 | 0.0004 | 0.0014 | 0.0266 | 0.5781 | 0.2237 | 0.0000 | 0.0000
10 | 0.0006 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0001 | 0.0005 | 0.0014 | 0.0215 | 0.5936 | 0.1628 | 0.0513
11 | 0.0002 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0091 | 0.7209 | 0.1795
12 | 0.0001 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0002 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.5385

Table 8 The first-order transfer matrix from 2001 to 2002

Size | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12
---|---|---|---|---|---|---|---|---|---|---|---|---|---
0 | 0.0000 | 0.3372 | 0.2247 | 0.1651 | 0.1272 | 0.1014 | 0.0869 | 0.0744 | 0.0724 | 0.0633 | 0.0842 | 0.0800 | 0.0000
1 | 0.0548 | 0.4986 | 0.0503 | 0.0110 | 0.0053 | 0.0029 | 0.0020 | 0.0023 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000
2 | 0.2198 | 0.1140 | 0.5684 | 0.0825 | 0.0164 | 0.0064 | 0.0029 | 0.0024 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000
3 | 0.2800 | 0.0257 | 0.1239 | 0.6103 | 0.0842 | 0.0129 | 0.0062 | 0.0016 | 0.0007 | 0.0000 | 0.0000 | 0.0000 | 0.0000
4 | 0.2853 | 0.0194 | 0.0272 | 0.1208 | 0.6937 | 0.1354 | 0.0222 | 0.0104 | 0.0043 | 0.0037 | 0.0105 | 0.0000 | 0.0385
5 | 0.0960 | 0.0043 | 0.0047 | 0.0082 | 0.0659 | 0.6721 | 0.1329 | 0.0163 | 0.0135 | 0.0074 | 0.0000 | 0.0000 | 0.0000
6 | 0.0389 | 0.0005 | 0.0006 | 0.0017 | 0.0060 | 0.0648 | 0.6919 | 0.1200 | 0.0121 | 0.0149 | 0.0000 | 0.0000 | 0.0000
7 | 0.0176 | 0.0001 | 0.0001 | 0.0004 | 0.0012 | 0.0035 | 0.0531 | 0.7400 | 0.1412 | 0.0093 | 0.0053 | 0.0000 | 0.0000
8 | 0.0049 | 0.0000 | 0.0001 | 0.0000 | 0.0001 | 0.0004 | 0.0013 | 0.0306 | 0.7204 | 0.1564 | 0.0105 | 0.0000 | 0.0000
9 | 0.0020 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0001 | 0.0005 | 0.0013 | 0.0348 | 0.7151 | 0.1421 | 0.0000 | 0.0000
10 | 0.0007 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0001 | 0.0007 | 0.0007 | 0.0298 | 0.7316 | 0.1400 | 0.0000
11 | 0.0001 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0105 | 0.7600 | 0.0385
12 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0053 | 0.0200 | 0.9231

Table 9 The first-order transfer matrix from 2002 to 2003

Size | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12
---|---|---|---|---|---|---|---|---|---|---|---|---|---
0 | 0.0000 | 0.4219 | 0.2423 | 0.1800 | 0.1440 | 0.1164 | 0.1014 | 0.0919 | 0.1042 | 0.1612 | 0.0821 | 0.0652 | 0.1111
1 | 0.0469 | 0.4257 | 0.0398 | 0.0084 | 0.0035 | 0.0024 | 0.0019 | 0.0008 | 0.0013 | 0.0000 | 0.0000 | 0.0000 | 0.0000
2 | 0.2315 | 0.1105 | 0.5546 | 0.0754 | 0.0133 | 0.0044 | 0.0016 | 0.0011 | 0.0007 | 0.0018 | 0.0000 | 0.0000 | 0.0000
3 | 0.2947 | 0.0255 | 0.1325 | 0.6030 | 0.0803 | 0.0108 | 0.0061 | 0.0026 | 0.0007 | 0.0036 | 0.0000 | 0.0000 | 0.0000
4 | 0.2796 | 0.0125 | 0.0259 | 0.1230 | 0.6794 | 0.1220 | 0.0198 | 0.0099 | 0.0046 | 0.0036 | 0.0051 | 0.0000 | 0.0000
5 | 0.0933 | 0.0022 | 0.0037 | 0.0078 | 0.0724 | 0.6620 | 0.1222 | 0.0124 | 0.0073 | 0.0054 | 0.0051 | 0.0000 | 0.0000
6 | 0.0346 | 0.0013 | 0.0009 | 0.0020 | 0.0058 | 0.0760 | 0.6814 | 0.1016 | 0.0139 | 0.0163 | 0.0051 | 0.0217 | 0.0000
7 | 0.0147 | 0.0003 | 0.0001 | 0.0003 | 0.0010 | 0.0053 | 0.0646 | 0.7341 | 0.1228 | 0.0199 | 0.0205 | 0.0000 | 0.0000
8 | 0.0033 | 0.0000 | 0.0001 | 0.0001 | 0.0003 | 0.0006 | 0.0008 | 0.0444 | 0.6899 | 0.1014 | 0.0051 | 0.0000 | 0.0000
9 | 0.0009 | 0.0001 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0002 | 0.0011 | 0.0525 | 0.6522 | 0.1128 | 0.0000 | 0.0000
10 | 0.0003 | 0.0001 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0020 | 0.0326 | 0.7179 | 0.0217 | 0.0370
11 | 0.0001 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0018 | 0.0410 | 0.8696 | 0.1481
12 | 0.0001 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0051 | 0.0217 | 0.7037

Table 10 The first-order transfer matrix from 2003 to 2004

Size | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12
---|---|---|---|---|---|---|---|---|---|---|---|---|---
0 | 0.0000 | 0.5122 | 0.3566 | 0.2821 | 0.2334 | 0.1984 | 0.1758 | 0.1692 | 0.1691 | 0.1715 | 0.1611 | 0.1207 | 0.0400
1 | 0.0808 | 0.3336 | 0.0361 | 0.0062 | 0.0034 | 0.0022 | 0.0016 | 0.0008 | 0.0006 | 0.0019 | 0.0000 | 0.0000 | 0.0000
2 | 0.2631 | 0.1118 | 0.4433 | 0.0771 | 0.0156 | 0.0070 | 0.0036 | 0.0034 | 0.0057 | 0.0000 | 0.0000 | 0.0000 | 0.0000
3 | 0.2893 | 0.0234 | 0.1294 | 0.5001 | 0.0859 | 0.0161 | 0.0086 | 0.0035 | 0.0051 | 0.0039 | 0.0000 | 0.0000 | 0.0000
4 | 0.2469 | 0.0142 | 0.0300 | 0.1237 | 0.5819 | 0.1376 | 0.0310 | 0.0143 | 0.0102 | 0.0039 | 0.0000 | 0.0000 | 0.0000
5 | 0.0776 | 0.0018 | 0.0039 | 0.0093 | 0.0725 | 0.5615 | 0.1318 | 0.0247 | 0.0153 | 0.0058 | 0.0111 | 0.0000 | 0.0000
6 | 0.0277 | 0.0018 | 0.0006 | 0.0012 | 0.0061 | 0.0724 | 0.5795 | 0.1140 | 0.0197 | 0.0234 | 0.0167 | 0.0172 | 0.0000
7 | 0.0110 | 0.0010 | 0.0001 | 0.0002 | 0.0010 | 0.0046 | 0.0661 | 0.6247 | 0.1360 | 0.0312 | 0.0222 | 0.0172 | 0.0000
8 | 0.0024 | 0.0003 | 0.0000 | 0.0000 | 0.0001 | 0.0003 | 0.0017 | 0.0429 | 0.5900 | 0.1267 | 0.0444 | 0.0000 | 0.0000
9 | 0.0007 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0003 | 0.0019 | 0.0451 | 0.5828 | 0.0889 | 0.0345 | 0.0000
10 | 0.0003 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0001 | 0.0005 | 0.0032 | 0.0468 | 0.6333 | 0.0862 | 0.0400
11 | 0.0001 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0002 | 0.0000 | 0.0019 | 0.0222 | 0.6379 | 0.0800
12 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0862 | 0.8400

Table 11 The first-order transfer matrix from 2004 to 2005

Size | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12
---|---|---|---|---|---|---|---|---|---|---|---|---|---
0 | 0.0000 | 0.3970 | 0.2336 | 0.1675 | 0.1227 | 0.0959 | 0.0902 | 0.0884 | 0.0997 | 0.0944 | 0.1020 | 0.0357 | 0.0370
1 | 0.0726 | 0.3971 | 0.0323 | 0.0064 | 0.0033 | 0.0016 | 0.0019 | 0.0012 | 0.0012 | 0.0000 | 0.0000 | 0.0179 | 0.0000
2 | 0.2776 | 0.1538 | 0.5566 | 0.0705 | 0.0124 | 0.0044 | 0.0017 | 0.0010 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000
3 | 0.3012 | 0.0330 | 0.1407 | 0.6128 | 0.0801 | 0.0112 | 0.0054 | 0.0021 | 0.0012 | 0.0020 | 0.0000 | 0.0000 | 0.0000
4 | 0.2344 | 0.0151 | 0.0305 | 0.1312 | 0.6958 | 0.1251 | 0.0193 | 0.0082 | 0.0031 | 0.0000 | 0.0000 | 0.0000 | 0.0000
5 | 0.0713 | 0.0024 | 0.0049 | 0.0094 | 0.0770 | 0.6744 | 0.1155 | 0.0107 | 0.0055 | 0.0000 | 0.0102 | 0.0000 | 0.0000
6 | 0.0266 | 0.0010 | 0.0010 | 0.0019 | 0.0070 | 0.0804 | 0.6831 | 0.0928 | 0.0098 | 0.0120 | 0.0051 | 0.0000 | 0.0000
7 | 0.0119 | 0.0003 | 0.0003 | 0.0004 | 0.0015 | 0.0059 | 0.0785 | 0.7419 | 0.1058 | 0.0080 | 0.0204 | 0.0000 | 0.0000
8 | 0.0027 | 0.0002 | 0.0000 | 0.0000 | 0.0001 | 0.0008 | 0.0029 | 0.0493 | 0.6936 | 0.1285 | 0.0051 | 0.0000 | 0.0000
9 | 0.0011 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0002 | 0.0010 | 0.0032 | 0.0765 | 0.6948 | 0.0714 | 0.0000 | 0.0000
10 | 0.0004 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0003 | 0.0010 | 0.0031 | 0.0602 | 0.7398 | 0.1250 | 0.0370
11 | 0.0001 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0001 | 0.0000 | 0.0006 | 0.0000 | 0.0408 | 0.7857 | 0.0741
12 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0001 | 0.0001 | 0.0000 | 0.0000 | 0.0051 | 0.0357 | 0.8519

Table 12 The first-order transfer matrix from 2005 to 2006

Size | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12
---|---|---|---|---|---|---|---|---|---|---|---|---|---
0 | 0.0000 | 0.2534 | 0.1374 | 0.1052 | 0.0811 | 0.0648 | 0.0505 | 0.0533 | 0.0584 | 0.0573 | 0.0183 | 0.0500 | 0.0000
1 | 0.0765 | 0.5485 | 0.0343 | 0.0070 | 0.0033 | 0.0015 | 0.0011 | 0.0011 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000
2 | 0.3049 | 0.1509 | 0.6777 | 0.0722 | 0.0131 | 0.0043 | 0.0021 | 0.0010 | 0.0000 | 0.0017 | 0.0000 | 0.0000 | 0.0000
3 | 0.3046 | 0.0295 | 0.1247 | 0.7001 | 0.0792 | 0.0110 | 0.0054 | 0.0023 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000
4 | 0.2226 | 0.0143 | 0.0227 | 0.1076 | 0.7593 | 0.1175 | 0.0205 | 0.0073 | 0.0017 | 0.0017 | 0.0046 | 0.0000 | 0.0000
5 | 0.0610 | 0.0026 | 0.0027 | 0.0066 | 0.0592 | 0.7370 | 0.1108 | 0.0146 | 0.0040 | 0.0035 | 0.0000 | 0.0000 | 0.0000
6 | 0.0193 | 0.0007 | 0.0005 | 0.0012 | 0.0040 | 0.0601 | 0.7517 | 0.0903 | 0.0081 | 0.0122 | 0.0046 | 0.0000 | 0.0333
7 | 0.0081 | 0.0001 | 0.0001 | 0.0002 | 0.0008 | 0.0035 | 0.0559 | 0.7909 | 0.0960 | 0.0087 | 0.0000 | 0.0000 | 0.0000
8 | 0.0020 | 0.0001 | 0.0000 | 0.0000 | 0.0001 | 0.0003 | 0.0016 | 0.0369 | 0.7763 | 0.1285 | 0.0046 | 0.0000 | 0.0000
9 | 0.0007 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0003 | 0.0017 | 0.0538 | 0.7257 | 0.0734 | 0.0000 | 0.0333
10 | 0.0002 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0001 | 0.0006 | 0.0017 | 0.0590 | 0.8624 | 0.0833 | 0.0000
11 | 0.0001 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0017 | 0.0275 | 0.7667 | 0.0000
12 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0046 | 0.1000 | 0.9333

Table 13 The first-order transfer matrix from 2006 to 2007

Size | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12
---|---|---|---|---|---|---|---|---|---|---|---|---|---
0 | 0.0000 | 0.2567 | 0.1320 | 0.0962 | 0.0749 | 0.0557 | 0.0486 | 0.0445 | 0.0429 | 0.0475 | 0.0364 | 0.0175 | 0.0000
1 | 0.0784 | 0.5182 | 0.0294 | 0.0059 | 0.0029 | 0.0013 | 0.0011 | 0.0005 | 0.0000 | 0.0017 | 0.0000 | 0.0000 | 0.0000
2 | 0.3235 | 0.1669 | 0.6846 | 0.0694 | 0.0124 | 0.0036 | 0.0015 | 0.0003 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000
3 | 0.2972 | 0.0377 | 0.1247 | 0.7113 | 0.0774 | 0.0113 | 0.0061 | 0.0011 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000
4 | 0.2142 | 0.0175 | 0.0249 | 0.1074 | 0.7642 | 0.1171 | 0.0198 | 0.0085 | 0.0016 | 0.0017 | 0.0000 | 0.0000 | 0.0000
5 | 0.0578 | 0.0026 | 0.0038 | 0.0084 | 0.0630 | 0.7416 | 0.1119 | 0.0143 | 0.0071 | 0.0034 | 0.0000 | 0.0000 | 0.0000
6 | 0.0190 | 0.0002 | 0.0006 | 0.0011 | 0.0044 | 0.0639 | 0.7457 | 0.1089 | 0.0087 | 0.0051 | 0.0000 | 0.0000 | 0.0000
7 | 0.0077 | 0.0001 | 0.0001 | 0.0003 | 0.0007 | 0.0051 | 0.0638 | 0.7791 | 0.1505 | 0.0136 | 0.0162 | 0.0000 | 0.0000
8 | 0.0014 | 0.0000 | 0.0000 | 0.0000 | 0.0001 | 0.0004 | 0.0011 | 0.0411 | 0.7277 | 0.1104 | 0.0081 | 0.0351 | 0.0000
9 | 0.0005 | 0.0001 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0003 | 0.0015 | 0.0576 | 0.7589 | 0.0850 | 0.0000 | 0.0000
10 | 0.0001 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0001 | 0.0001 | 0.0027 | 0.0560 | 0.8462 | 0.0877 | 0.0286
11 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0011 | 0.0017 | 0.0081 | 0.8421 | 0.0857
12 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0001 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0175 | 0.8857

Table 14 The first-order transfer matrix from 2007 to 2008

Size | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12
---|---|---|---|---|---|---|---|---|---|---|---|---|---
0 | 0.0000 | 0.3513 | 0.2895 | 0.2654 | 0.2463 | 0.2314 | 0.2254 | 0.2339 | 0.2406 | 0.2324 | 0.2444 | 0.1525 | 0.2121
1 | 0.0830 | 0.3699 | 0.0379 | 0.0058 | 0.0030 | 0.0011 | 0.0010 | 0.0004 | 0.0011 | 0.0000 | 0.0000 | 0.0000 | 0.0000
2 | 0.2798 | 0.1998 | 0.4840 | 0.0956 | 0.0181 | 0.0053 | 0.0024 | 0.0012 | 0.0005 | 0.0016 | 0.0000 | 0.0000 | 0.0000
3 | 0.2644 | 0.0508 | 0.1405 | 0.4989 | 0.1061 | 0.0150 | 0.0051 | 0.0024 | 0.0005 | 0.0000 | 0.0000 | 0.0000 | 0.0000
4 | 0.2287 | 0.0256 | 0.0420 | 0.1205 | 0.5503 | 0.1566 | 0.0254 | 0.0095 | 0.0033 | 0.0016 | 0.0000 | 0.0000 | 0.0000
5 | 0.0939 | 0.0021 | 0.0056 | 0.0122 | 0.0688 | 0.5241 | 0.1425 | 0.0185 | 0.0087 | 0.0000 | 0.0038 | 0.0000 | 0.0000
6 | 0.0315 | 0.0004 | 0.0004 | 0.0014 | 0.0062 | 0.0604 | 0.5274 | 0.1214 | 0.0087 | 0.0032 | 0.0038 | 0.0000 | 0.0000
7 | 0.0128 | 0.0001 | 0.0001 | 0.0003 | 0.0011 | 0.0055 | 0.0684 | 0.5651 | 0.1592 | 0.0192 | 0.0038 | 0.0000 | 0.0303
8 | 0.0045 | 0.0000 | 0.0000 | 0.0000 | 0.0001 | 0.0006 | 0.0021 | 0.0440 | 0.5160 | 0.1378 | 0.0226 | 0.0169 | 0.0000
9 | 0.0011 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0001 | 0.0004 | 0.0032 | 0.0581 | 0.5513 | 0.0865 | 0.0000 | 0.0303
10 | 0.0004 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0005 | 0.0033 | 0.0529 | 0.6015 | 0.1017 | 0.0606
11 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0263 | 0.6441 | 0.0909
12 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0075 | 0.0847 | 0.5758

Table 15 The first-order transfer matrix from 2008 to 2009

Size | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12
---|---|---|---|---|---|---|---|---|---|---|---|---|---
0 | 0.0000 | 0.3340 | 0.2711 | 0.2476 | 0.2365 | 0.2324 | 0.2216 | 0.2147 | 0.2391 | 0.2188 | 0.1690 | 0.1250 | 0.1923
1 | 0.0843 | 0.4809 | 0.0397 | 0.0067 | 0.0034 | 0.0014 | 0.0007 | 0.0013 | 0.0000 | 0.0020 | 0.0000 | 0.0000 | 0.0000
2 | 0.3238 | 0.1486 | 0.5699 | 0.0984 | 0.0196 | 0.0062 | 0.0033 | 0.0018 | 0.0020 | 0.0000 | 0.0000 | 0.0000 | 0.0000
3 | 0.2821 | 0.0230 | 0.0960 | 0.5582 | 0.1064 | 0.0173 | 0.0057 | 0.0050 | 0.0020 | 0.0000 | 0.0000 | 0.0000 | 0.0000
4 | 0.1985 | 0.0106 | 0.0184 | 0.0787 | 0.5769 | 0.1514 | 0.0245 | 0.0116 | 0.0048 | 0.0000 | 0.0000 | 0.0000 | 0.0000
5 | 0.0692 | 0.0021 | 0.0040 | 0.0089 | 0.0526 | 0.5406 | 0.1488 | 0.0164 | 0.0041 | 0.0000 | 0.0047 | 0.0000 | 0.0000
6 | 0.0239 | 0.0006 | 0.0006 | 0.0011 | 0.0037 | 0.0461 | 0.5513 | 0.1244 | 0.0109 | 0.0039 | 0.0047 | 0.0208 | 0.0000
7 | 0.0127 | 0.0002 | 0.0002 | 0.0002 | 0.0008 | 0.0041 | 0.0422 | 0.5930 | 0.1653 | 0.0156 | 0.0141 | 0.0000 | 0.0000
8 | 0.0033 | 0.0001 | 0.0000 | 0.0000 | 0.0000 | 0.0004 | 0.0016 | 0.0298 | 0.5171 | 0.1523 | 0.0047 | 0.0000 | 0.0000
9 | 0.0014 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0001 | 0.0004 | 0.0019 | 0.0512 | 0.5645 | 0.1127 | 0.0208 | 0.0000
10 | 0.0005 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0027 | 0.0391 | 0.6573 | 0.0625 | 0.0000
11 | 0.0002 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0002 | 0.0007 | 0.0020 | 0.0235 | 0.7292 | 0.0385
12 | 0.0001 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0020 | 0.0094 | 0.0417 | 0.7692

Table 16 The first-order transfer matrix from 2009 to 2010

Size | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12
---|---|---|---|---|---|---|---|---|---|---|---|---|---
0 | 0.0000 | 0.3499 | 0.3060 | 0.2341 | 0.1740 | 0.1368 | 0.1041 | 0.0926 | 0.0877 | 0.0863 | 0.0635 | 0.0392 | 0.1379
1 | 0.0410 | 0.3177 | 0.0450 | 0.0089 | 0.0043 | 0.0015 | 0.0007 | 0.0004 | 0.0008 | 0.0000 | 0.0000 | 0.0000 | 0.0000
2 | 0.1737 | 0.2284 | 0.4389 | 0.1204 | 0.0360 | 0.0171 | 0.0039 | 0.0023 | 0.0017 | 0.0000 | 0.0000 | 0.0000 | 0.0000
3 | 0.2119 | 0.0612 | 0.1641 | 0.4674 | 0.1351 | 0.0435 | 0.0130 | 0.0069 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000
4 | 0.2132 | 0.0324 | 0.0393 | 0.1538 | 0.5467 | 0.2106 | 0.0553 | 0.0200 | 0.0066 | 0.0126 | 0.0053 | 0.0000 | 0.0000
5 | 0.1890 | 0.0081 | 0.0051 | 0.0124 | 0.0934 | 0.4929 | 0.1872 | 0.0438 | 0.0215 | 0.0147 | 0.0053 | 0.0000 | 0.0000
6 | 0.1041 | 0.0017 | 0.0011 | 0.0021 | 0.0088 | 0.0897 | 0.5362 | 0.1790 | 0.0373 | 0.0126 | 0.0106 | 0.0000 | 0.0000
7 | 0.0485 | 0.0006 | 0.0003 | 0.0007 | 0.0014 | 0.0070 | 0.0968 | 0.5942 | 0.2467 | 0.0674 | 0.0053 | 0.0196 | 0.0000
8 | 0.0120 | 0.0001 | 0.0001 | 0.0001 | 0.0003 | 0.0007 | 0.0021 | 0.0574 | 0.5265 | 0.2253 | 0.0370 | 0.0196 | 0.0000
9 | 0.0042 | 0.0001 | 0.0000 | 0.0000 | 0.0000 | 0.0001 | 0.0005 | 0.0025 | 0.0679 | 0.5095 | 0.1852 | 0.0196 | 0.0000
10 | 0.0017 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0001 | 0.0010 | 0.0025 | 0.0716 | 0.6614 | 0.1961 | 0.0690
11 | 0.0003 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0001 | 0.0000 | 0.0008 | 0.0000 | 0.0212 | 0.6667 | 0.1379
12 | 0.0003 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0053 | 0.0392 | 0.6552

Table 17 The first-order transfer matrix from 2010 to 2011

Size | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12
---|---|---|---|---|---|---|---|---|---|---|---|---|---
0 | 0.0000 | 0.7743 | 0.6891 | 0.6076 | 0.4969 | 0.5833 | 0.5908 | 0.5649 | 0.5683 | 0.5741 | 0.5281 | 0.4752 | 0.6143
1 | 0.0167 | 0.0057 | 0.0060 | 0.0065 | 0.0075 | 0.0062 | 0.0063 | 0.0059 | 0.0055 | 0.0066 | 0.0065 | 0.0099 | 0.0000
2 | 0.0386 | 0.0191 | 0.0157 | 0.0119 | 0.0123 | 0.0097 | 0.0087 | 0.0094 | 0.0091 | 0.0094 | 0.0022 | 0.0099 | 0.0000
3 | 0.1150 | 0.0438 | 0.0546 | 0.0419 | 0.0270 | 0.0169 | 0.0163 | 0.0160 | 0.0121 | 0.0113 | 0.0411 | 0.0297 | 0.0286
4 | 0.4993 | 0.1354 | 0.1891 | 0.2277 | 0.1966 | 0.0696 | 0.0394 | 0.0372 | 0.0414 | 0.0415 | 0.0368 | 0.0594 | 0.0429
5 | 0.2544 | 0.0203 | 0.0422 | 0.0961 | 0.2263 | 0.1950 | 0.0636 | 0.0145 | 0.0033 | 0.0009 | 0.0022 | 0.0000 | 0.0000
6 | 0.0538 | 0.0012 | 0.0029 | 0.0074 | 0.0294 | 0.1062 | 0.1969 | 0.0825 | 0.0104 | 0.0057 | 0.0022 | 0.0000 | 0.0000
7 | 0.0172 | 0.0002 | 0.0004 | 0.0009 | 0.0037 | 0.0121 | 0.0729 | 0.2212 | 0.1059 | 0.0179 | 0.0022 | 0.0000 | 0.0000
8 | 0.0035 | 0.0000 | 0.0000 | 0.0001 | 0.0003 | 0.0008 | 0.0043 | 0.0427 | 0.1893 | 0.0812 | 0.0130 | 0.0099 | 0.0000
9 | 0.0010 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0001 | 0.0007 | 0.0049 | 0.0463 | 0.1936 | 0.0823 | 0.0198 | 0.0143
10 | 0.0004 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0001 | 0.0007 | 0.0075 | 0.0557 | 0.2165 | 0.0990 | 0.0429
11 | 0.0001 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0001 | 0.0010 | 0.0019 | 0.0584 | 0.2178 | 0.0429
12 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0087 | 0.0693 | 0.2143

Table 18 The first-order transfer matrix from 2011 to 2012

Size | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12
---|---|---|---|---|---|---|---|---|---|---|---|---|---
0 | 0.0000 | 0.0014 | 0.0016 | 0.0020 | 0.0017 | 0.0013 | 0.0007 | 0.0006 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000
1 | 0.0439 | 0.9986 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000
2 | 0.0995 | 0.0000 | 0.9984 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000
3 | 0.1990 | 0.0000 | 0.0000 | 0.9896 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000
4 | 0.5573 | 0.0000 | 0.0000 | 0.0084 | 0.9938 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000
5 | 0.0816 | 0.0000 | 0.0000 | 0.0000 | 0.0045 | 0.9958 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000
6 | 0.0140 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0029 | 0.9966 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000
7 | 0.0036 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0027 | 0.9980 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000
8 | 0.0008 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0014 | 0.9952 | 0.0000 | 0.0000 | 0.0000 | 0.0000
9 | 0.0002 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0048 | 0.9984 | 0.0000 | 0.0000 | 0.0000
10 | 0.0001 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0016 | 0.9961 | 0.0000 | 0.0000
11 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0039 | 1.0000 | 0.0000
12 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 1.0000

Table 19 The first-order transfer matrix from 2012 to 2013

Size | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12
---|---|---|---|---|---|---|---|---|---|---|---|---|---
0 | 0.0000 | 0.6480 | 0.5886 | 0.5510 | 0.5192 | 0.4873 | 0.4447 | 0.3886 | 0.3427 | 0.3101 | 0.2605 | 0.2055 | 0.1250
1 | 0.0255 | 0.0150 | 0.0121 | 0.0079 | 0.0038 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000
2 | 0.0517 | 0.0257 | 0.0286 | 0.0192 | 0.0110 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000
3 | 0.0997 | 0.0402 | 0.0492 | 0.0511 | 0.0338 | 0.0023 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000
4 | 0.3372 | 0.1272 | 0.1491 | 0.1637 | 0.1636 | 0.0722 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000
5 | 0.4385 | 0.1438 | 0.1723 | 0.2071 | 0.2535 | 0.2837 | 0.0443 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000
6 | 0.0398 | 0.0000 | 0.0000 | 0.0000 | 0.0151 | 0.1544 | 0.4402 | 0.0211 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000
7 | 0.0060 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0709 | 0.5741 | 0.0106 | 0.0000 | 0.0000 | 0.0000 | 0.0000
8 | 0.0012 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0161 | 0.6362 | 0.0063 | 0.0000 | 0.0000 | 0.0000
9 | 0.0002 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0106 | 0.6709 | 0.0038 | 0.0000 | 0.0000
10 | 0.0002 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0127 | 0.7356 | 0.0000 | 0.0000
11 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.7945 | 0.0000
12 | 0.0001 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.8750

#### Obtained Transition Trend and Entropy Results

The calculated transition trend and transition entropy of each year/category
are presented as follows:

Table 20 The transition trend from 1999 to 2013

Year* | 1999 | 2000 | 2001 | 2002 | 2003 | 2004 | 2005 | 2006
---|---|---|---|---|---|---|---|---
L(d) | 0.2825 | 0.2268 | 0.3446 | 0.2661 | 0.2889 | 0.4667 | 0.2396 | 0.2534
R(d) | 0.2500 | 0.2744 | 0.2791 | 0.2077 | 0.2092 | 0.2173 | 0.2144 | 0.1526
Q(d) | 1.1300 | 0.8265 | 1.2347 | 1.2814 | 1.3811 | 2.1482 | 1.1179 | 1.6603
Year* | 2007 | 2008 | 2009 | 2010 | 2011 | 2012 | 2013 |
L(d) | 0.2609 | 0.1289 | 0.2185 | 0.5030 | 0.4439 | 0.1344 | 0.3185 |
R(d) | 0.1448 | 0.3643 | 0.3045 | 0.2122 | 0.4695 | 0.0313 | 0.4952 |
Q(d) | 1.8020 | 0.3538 | 0.7174 | 2.3700 | 0.9455 | 4.2973 | 0.6432 |
*For simplicity, year in this table means end year. e.g., year 1999 represents 1998 to 1999.

Table 21 The transition entropy from 1999 to 2013

Category | 1999 | 2000 | 2001 | 2002 | 2003 | 2004 | 2005 | 2006
---|---|---|---|---|---|---|---|---
0 | 1.7736 | 1.7277 | 1.7185 | 1.6733 | 1.6278 | 1.6290 | 1.6127 | 1.5613
1 | 1.3928 | 1.1951 | 1.5558 | 1.1599 | 1.1458 | 1.1342 | 1.2228 | 1.1497
2 | 1.3500 | 1.2285 | 1.3404 | 1.1952 | 1.1900 | 1.2450 | 1.1945 | 1.0183
3 | 1.2867 | 1.2593 | 1.3220 | 1.1629 | 1.1601 | 1.2444 | 1.1441 | 0.9938
4 | 1.1261 | 1.1642 | 1.2224 | 1.0384 | 1.0507 | 1.1789 | 1.0281 | 0.8853
5 | 1.1603 | 1.2119 | 1.2745 | 1.0764 | 1.0954 | 1.2498 | 1.0754 | 0.9278
6 | 1.1683 | 1.1962 | 1.2808 | 1.0500 | 1.0657 | 1.2618 | 1.0817 | 0.9123
7 | 1.0803 | 1.1201 | 1.1841 | 0.9443 | 0.9532 | 1.1916 | 0.9470 | 0.8220
8 | 1.0632 | 1.2053 | 1.2262 | 0.9648 | 1.0553 | 1.2902 | 1.0483 | 0.8274
9 | 0.9908 | 1.1101 | 1.1673 | 0.9726 | 1.1531 | 1.3151 | 1.0128 | 0.9745
10 | 0.8625 | 1.2570 | 1.1447 | 0.9133 | 1.0346 | 1.2242 | 0.9817 | 0.5904
11 | 0.9833 | 1.2305 | 0.9369 | 0.7641 | 0.5489 | 1.2205 | 0.7594 | 0.7908
12 | 0.8113 | 0.8070 | 1.2899 | 0.3247 | 0.8962 | 0.6060 | 0.5733 | 0.2910
Category | 2007 | 2008 | 2009 | 2010 | 2011 | 2012 | 2013 | AVG
0 | 1.5468 | 1.6739 | 1.6156 | 1.8822 | 1.3956 | 1.3068 | 1.3760 | 1.6081
1 | 1.2019 | 1.3193 | 1.1566 | 1.4072 | 0.7997 | 1.0106 | 1.1086 | 1.1307
2 | 1.0085 | 1.2761 | 1.1291 | 1.3249 | 0.9799 | 1.0119 | 1.2020 | 1.1130
3 | 0.9727 | 1.2736 | 1.1841 | 1.3536 | 1.1267 | 1.0629 | 1.2169 | 1.1176
4 | 0.8731 | 1.2259 | 1.1745 | 1.3227 | 1.3187 | 1.2413 | 1.2330 | 1.0589
5 | 0.9191 | 1.2694 | 1.2368 | 1.4216 | 1.2623 | 1.2298 | 1.1999 | 1.0940
6 | 0.9242 | 1.2920 | 1.2370 | 1.3713 | 1.2942 | 1.1244 | 1.0473 | 1.0805
7 | 0.8313 | 1.2217 | 1.1834 | 1.2902 | 1.3530 | 1.1156 | 0.8338 | 0.9981
8 | 0.9260 | 1.2774 | 1.2762 | 1.3547 | 1.3828 | 1.0304 | 0.7511 | 1.0453
9 | 0.8960 | 1.2109 | 1.1925 | 1.4336 | 1.3806 | 1.0119 | 0.7182 | 1.0360
10 | 0.6163 | 1.1434 | 1.0900 | 1.1237 | 1.4769 | 1.0255 | 0.5975 | 0.9388
11 | 0.6173 | 1.0806 | 0.9571 | 1.0749 | 1.5863 | 1.0000 | 0.5079 | 0.8706
12 | 0.4197 | 1.2465 | 0.6443 | 1.0079 | 1.1971 | 0 | 0.3768 | 0.6994


?!

#  Weakly-supervised continual learning
for class-incremental segmentation

###### Abstract

Transfer learning is a powerful way to adapt existing deep learning models to
new emerging use-cases in remote sensing. Starting from a neural network
already trained for semantic segmentation, we propose to modify its label
space to swiftly adapt it to new classes under weak supervision. To alleviate
the background shift and the catastrophic forgetting problems inherent to this
form of continual learning, we compare different regularization terms and
leverage a pseudo-label strategy. We experimentally show the relevance of our
approach on three public remote sensing datasets. Code is open-source and
released in this repository: https://github.com/alteia-ai/ICSS.

(a) Initial prediction

(b) New class annotations

(c) New prediction

(d) Ground-truth

Fig. 1: Add building in the {background, road} label space with clicked
annotations. Initial prediction (a) leads to new-class annotations (b) to
collect the new prediction (c).

##  1 Introduction

Semantic segmentation, or pixel-wise classification of an image, is a
challenging computer vision task with various remote sensing applications such
as land-cover mapping, building recognition or change detection. Nowadays,
deep neural networks (DNNs) are commonly applied for such purposes. They are
usually trained once on a given training dataset and then deployed to
automatically process potentially large volumes of data.

This approach outcomes frozen algorithms that can lead to excellent outputs in
ideal conditions. However, there are still many open questions, such as domain
adaptation or limited training data, for which such a static approach is often
not suitable. Another important issue of these algorithms is their lack of
flexibility due to their frozen output space fixed before training. Many works
now use semi-supervised learning [1] and weakly supervised learning [2] to
address limited training data issues and domain adaptation [3, 4]. Another
promising approach to handle these problems comes with interactive learning
[5], where the algorithm learns to adapt to user inputs. To improve the neural
networks plasticity for semantic segmentation, recent works propose customized
losses for class-incremental segmentation.

Even though fine-tuning a deep network on a small set of samples of new
classes seems quite risky due to both catastrophic forgetting and lack of
convergence threats, we show in this work that this is possible within the
correct framework. Precisely, we combine weak and class-incremental
segmentation works to propose to increment neural network label space using
sparse clicked annotations, as showed by Figure 1. This set-up is suited for
the interactive learning of new classes since these annotations are
particularly handy for an annotator [2]. Currently, it might be easier to ask
an operator to annotate training data with the new segmentation classes to
fully retrain the algorithm. Yet, the purpose of the proposed framework is
precisely to be able to learn a new class on new data. This method is
therefore suitable for use-cases where initial training data is unavailable
due to privacy concerns or where accurate predictions are needed quickly on
small datasets.

To this purpose, we first compare different regularizations to fully benefit
from the sparse annotations. We also explore a pseudo-label strategy to
alleviate catastrophic forgetting. To summarize, the contributions of this
paper are: (1) A framework to increase the label space of a DNN for semantic
segmentation with sparse annotations; (2) A pseudo-label strategy and a
comparison of different regularizations for this weakly supervised fine-
tuning; (3) An experimental validation of the proposed approach on three
public remote sensing datasets.

##  2 Related work

###  2.1 Semi and weak supervision in semantic segmentation

When labels are scarce or flawed, the common training paradigm is to learn to
perform the segmentation task with the available labels while leveraging
unlabeled data to learn a better inner representation as support. In semantic
segmentation, weak labels can take various forms like incomplete ground-truth
maps or bounding boxes. Recently, different works in remote sensing explore
these different aspects. In [1], the authors learn from a mix of fully
annotated images and unannotated ones with auxiliary tasks while [3] designs a
custom regularization weighted by the proportion of labeled data to address
semi-supervised domain adaptation. In weakly supervised learning, semantic
segmentation can be achieved from bounding-boxes [4], rough polygons [6] or
image-level labels [7]. Closely related to our kind of supervision, DISCA [5]
and FESTA [2] leverage point annotations either as user input clicks or
partial labels. We here use similar point annotations to add a new
segmentation class to a pretrained DNN.

###  2.2 Class-incremental semantic segmentation

The goal of class incremental learning is to modify the output space of DNNs
to add label classes. This problem was already considered two decades ago [8]
and two main pitfalls are identified. First, like in other class-incremental
tasks, it is important to prevent the catastrophic forgetting of the
previously learned knowledge. Second, specific to semantic segmentation, the
background-shift first tackled in [9] has to be dealt with. Indeed, the new
class comes from the background, which causes a discrepancy with the
previously learned background class. Many methods addressed these two problems
by simply storing previous examples [10] but more recent works consider this
constraint to be too restrictive due to limited storage or security reasons.
Hence, we draw inspiration from [11, 12] which design customized
regularization to address the two identified pitfalls. [11] proposes
distillation losses which enforce statistical matches between the networks.
Inspired by few-shot and contrastive learning, [12] relies on prototypes
representing the different semantic classes to distinguish them.

##  3 Methodology

###  3.1 Scenario, baseline and constraints

A DNN pre-trained for semantic segmentation to detect $N-1$ classes is applied
on a new image $I$. The goal is to add a new possible segmentation class and
to provide an accurate segmentation map with respect to these $N$ classes. To
this aim, $M$ annotations are provided to make the network interactively learn
the $N^{\textit{th}}$ segmentation class. In this work, we follow findings
from [2] and use point annotations as they appear to be particularly handy for
a potential user. In practice, the last layer of the DNN is modified to
increment its output space. As we observed that freezing parts of the network
lowered its accuracy, the entire network is then retrained with a cross-
entropy (CE) loss on the annotated pixels. However, three problems arise with
this simple baseline: (1) Catastrophic forgetting since the new provided
labels only belong to the new class; (2) Background shift since the new class
was previously classified as background. So, the network has to both learn to
detect a new class and to modify the learned representation of the background
class; (3) Poor class representation since the network has to learn the new
class representation from sparse annotations (i.e. highly incomplete ground-
truth).

To address these problems, we notably rely on a pseudo-label strategy and
compare different regularizations added to the cross-entropy loss.

Following notation from [10], we refer through the rest of this paper to the
previous network with the $N-1$ class prediction as the memory network and to
the new one as the updated network. We also refer to the set of semantic
classes except the background as the classes of interest.

Ground-truth

Control prediction

Memory prediction

Baseline prediction

SDR prediction

Fig. 2: Building and road segmentation with 300 building annotations and road
pseudo-labels on an example from Vaihingen compared to ground-truth and
control prediction.

###  3.2 Learning from pseudo-labelling

The prediction of the memory network is used to sample old classes labels.
This ensures that the user does not have to provide such old-class annotations
while preventing the network from forgetting them. In addition, the
interactive user inputs provide both the new class and the background labels.
These background labels avoid the aforementioned background-shift issue since
background predictions from the memory network could actually belong to the
new class.
To emphasize the N annotations of the new class, only a maximum of N of the
most confident old-classes pixels are considered as annotations.

###  3.3 Regularization

In order to make the most of the sparse annotations, we consider the following
regularization schemes.

DISCA. Since the new class was previously part of the background, the
predictions over the other classes of interest should remain similar. To
enforce this property, we follow [5] and consider adding a cross entropy
regularization term over the pixels which are predicted as belonging to
classes of interest by the memory network. This can be seen as an output-level
knowledge distillation loss.

PodNet. Following PodNet [11], we enforce a statistic match at the encoder
level over the channels between the memory network and the updated one with a
$L^{2}$ intermediate-level knowledge distillation loss.

SDR. Inspired by few-shots and contrastive learning, we follow [12] to
regularize the DNN latent space. This aims to reduce forgetting whilst
improving the recognition of the new class. Concretely, we add a prototype-
based regularization at the encoder level of the DNN. Prototypes are vectors
representative of each segmentation class and are computed on the fly at each
new learning step. This regularization is composed of three terms: a matching
term enforces that the updated prototypes stay close to the previous ones, a
repulsive term ensures that the prototypes are far from each other and an
attracting term pushes the pixels to remain close to their associated
prototype.

FESTA. To deal with sparse annotations for segmentation in remote sensing, [2]
introduces an unsupervised loss that accounts for neighbourhood structures
both in spatial and feature domains as it assumes that nearby pixels share
labels.

##  4 Experiments

###  4.1 Experimental set-up

We evaluate our approach on the ISPRS Potsdam & Vaihingen datasets [13]
respectively composed of 38 ($6000\times 6000$ pxls, 5cm res.) and 33 (various
sizes, 9cm res.) images and on SemCity Toulouse [14] composed of 4 images
($3504\times 3452$ pxls, 50 cm res.). We pretrain the network for old class
(road\star\star$\star$Except when specified otherwise.) segmentation using
dense ground-truth maps on 30 (Potsdam), 26 (Vaihingen) and 2 (Toulouse)
images. We fine-tune for new class (building11footnotemark: 1) using
30011footnotemark: 1 clicked new-class and background annotations simulated
from the ground-truth on the remaining data. To compare with a potential upper
bound, we consider a control network directly pretrained on dense {road,
building} ground-truth maps (and not fine-tuned). For simplicity, all
experiments are performed separately within each dataset (i.e. not in domain
adaptation).

The considered metric is the Intersection over Union (IoU) averaged image-
wise. Except when we analyze the impact of the number of annotations, we
simulate 300 new-class and background annotations.

For all experiments, we use a LinkNet [15] architecture trained using Adam
optimizer with a learning rate of $10^{-4}$ during 10 pseudo-epochs. Each
pseudo-epoch consists in $10\,000$ $256\times 256$ labeled samples randomly
chosen from training data. We infer using a $256\times 256$ sliding window
with an overlap of $50\%$. Due to the stochastic nature of the optimization
process and the simulation of the annotations, all experiments are averaged on
3 runs to obtain statistically significant results.

During fine-tuning, we use an Adam optimizer with a learning rate of $2\cdot
10^{-5}$. We fine-tune for 30 steps and select the best performances obtained
in the last 15 steps. Each step consists of 10 back-propagation iterations.

###  4.2 Approach assessment

Table 1: Comparison of the different regularizations. Contr. stands for Control and Bas. for Baseline. | Contr. | Bas. | FESTA | DISCA | PodNet | SDR
---|---|---|---|---|---|---
Pot. | 76.4 |  68.7+- 2.7 |  66.5+- 8.4 |  68.2+- 4.1 |  71.7+- 2.3 | 72.0+- 0.3
Vai. | 84 |  76.2+- 2.3 |  75.1+- 3.4 |  74.8+- 1.8 |  74.1+- 5.4 | 79.7+- 1.6
Toul. | 74 |  63.2+- 3.2 |  62.4+- 1.5 |  54.0+- 0.9 |  65.4+- 1.0 | 67.7+- 0.6

As indicates Table 1, the DNN is able to learn a new class with clicked
annotations. Even without additional regularizations, it reaches an IoU over
the three classes of 68.7% on Potsdam, 76.2% on Vaihingen and 63.2% on
Toulouse. Appropriate regularization further improves the performances up to
4%. Indeed, SDR consistently improves the results on the three datasets: 3.3%
on Potsdam, 3.5% on Vaihingen and 4.5% on Toulouse. Moreover, SDR tends to
stabilize the results compared to the baseline, as testifies the standard
deviation measures (e.g. the std on Toulouse is of 0.6 for SDR and of 3.2 for
the baseline). As shows Figure 2, it visually translate into sharper contours,
even though the baseline already produces visually accurate results. However,
the other regularizations under-perform in this setting in regards to SDR and
can even lead to worse results than the baseline on Vaihingen and Toulouse.
Hence, the distillation losses and FESTA seem to be less relevant for this
problem than the latent space regularization proposed by SDR.

###  4.3 Influence of the number of annotations

Fig. 3: Metric evolution on the three datasets w.r.t. the number of
annotations. Control network performances are in dots.

To better apprehend our approach, we analyze the influence of the number of
annotations over the performances. According to the results of the previous
section, we fine-tune the network using SDR regularization. As we can observe
on Figure 3, less than one hundred annotations leads to sub-optimal results
and the performances then greatly improve with respect to the number of
annotations. After 100 annotations for the two ISPRS datasets and 300
annotations for the SemCity dataset, the DNN almost catches up with the
control network performances and then stabilizes on a plateau.

##  5 Conclusion

To conclude, we have shown that it is possible to make a DNN learn on the fly
new segmentation classes after the initial training with point supervision
using pseudo-labelling and relevant regularizations. We have assessed the
method efficiency with experiments on three public remote sensing datasets.
Moreover, with a sufficient amount of annotations, the performances even
almost catch up with a DNN trained on the classes of interest in a fully
supervised way.

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