11institutetext: Department of Computer Science and Engineering,
Southeast University, Nanjing 210096, China 22institutetext: Baidu Inc
22email<EMAIL_ADDRESS><EMAIL_ADDRESS>
# Learning Efficient Representations for Image-Based Patent Retrieval

Hongsong Wang*(🖂) 11 0000-0002-9464-1778 Yuqi Zhang* 22 0000-0001-7094-3838

###### Abstract

Patent retrieval has been attracting tremendous interest from researchers in
intellectual property and information retrieval communities in the past
decades. However, most existing approaches rely on textual and metadata
information of the patent, and content-based image-based patent retrieval is
rarely investigated. Based on traits of patent drawing images, we present a
simple and lightweight model for this task. Without bells and whistles, this
approach significantly outperforms other counterparts on a large-scale
benchmark and noticeably improves the state-of-the-art by 33.5% with the mean
average precision (mAP) score. Further experiments reveal that this model can
be elaborately scaled up to achieve a surprisingly high mAP of 93.5%. Our
method ranks first in the ECCV 2022 Patent Diagram Image Retrieval Challenge.

###### Keywords:

Image-Based Patent Retrieval Patent Search Sketch-Based Image Retrieval.

††footnotetext: *Equal Contribution. 🖂Corresponding author.

##  1 Introduction

With a large and ever-increasing number of scientific and technical documents
on the web every year, diagram image retrieval and analysis [33, 1] become
crucially important for intelligent document processing and understanding.
Different from natural images that contain rich information about color,
texture, and intensity, diagram images only involve line and shape. Although
existing computer vision methods have achieved significant progress on natural
image understanding (e.g., classification, detection and retrieval) [12, 26,
16, 31], diagram image understanding retains a challenging and less developed
area.

Image-based patent retrieval [11, 2, 37], one of the most typical research
topics of diagram image understanding, has drawn increasing interest from
intellectual property and information retrieval communities. The need for
effective and efficient systems of patent retrieval becomes inevitably crucial
due to the tremendous amounts of patent data. As visual image plays an
essential role in information retrieval, patent image search can help people
quickly understand the patent contents and check differences between patents.
However, accurate patent retrieval based on visual content remains an open
challenge. Commercial search engines such as Google and Baidu currently fail
to retrieve relevant patent images with the query of a patent drawing [15]. An
example of this task is illustrated in Fig. 1(a), where many unrelated drawing
images with a similar style to the query are retrieved.

(a) Image-based patent retrieval

(b) Different types of images

Figure 1: An illustration of image-based patent retrieval.

One of the well-concerned problems in computer vision most related to image-
based patent retrieval is sketch-based image retrieval [9]. Although both are
abstract drawings, there are subtle differences between sketch images and
patent drawings. As shown in Fig. 1(b), patent drawings contain more detailed
contours and patterns about the object when compared with sketch images.

Existing patent search systems mostly use textual and metadata information.
Only little published work studies content- and image-based patent retrieval.
These approaches represent the patent drawing with relational skeletons [13],
edge orientation autocorrelogram [27], contour description matrix [38],
adaptive hierarchical density histogram [30], etc. For example, the patent
retrieval system PATSEEK [27] applies shape-based retrieval and represents a
patent image with an edge orientation autocorrelogram. However, these
representations are low-level and handcrafted visual features, which are not
discriminative enough for large-scale applications. In addition, previous
public patent retrieval datasets such as the concept [29] only contains a
limited number of ID categories and images. The evaluation protocols of image-
based patent retrieval are also not consistent across most existing
approaches.

Recently, a large-scale dataset called DeepPatent [15] focused on patent
drawings is introduced. This dataset makes it possible for large-scale deep
learning-based image-based patent retrieval. The PatentNet [15] is currently
the only deep learning model designed for large-scale image-based patent
retrieval. However, this model is sophisticatedly trained in two stages with
different training losses. To the best of our knowledge, there exists no
single-stage training model that exploits the unique characteristics of patent
drawings and comprehensively studies how to train an effective patent
retrieval system from patent images.

To this end, we present a simple yet strong pipeline for image-based patent
retrieval which adopts a lightweight backbone and subsequently applies a neck
structure to obtain low-dimensional representation for efficient patent
retrieval. Training this network is straightforward by only using a
classification loss in a geometric angular space and data augmentation without
scaling, which is specifically designed for patent drawings according to their
characteristics. Experimental results demonstrate that the proposed method
significantly outperforms both the traditional and deep learning-based
counterparts. In addition, this baseline can be easily scaled up from the
current lightweight model to large models which achieve even better
performance.

##  2 Related Work

###  2.1 Image-Based Patent Retrieval

Patent retrieval [14] is the task of finding relevant patents for a given
image query, which can be a sketch, a photo or a patent image. An image and
text analysis approach automatically extracts concept information describing
the patent image content [29]. A retrieval system uses a hybrid approach that
combines feature extraction, indexing and similarity matching to retrieve
patent images based on text and shape features [18]. A combined method of
classification and indexing techniques retrieves patents for photo queries and
evaluates it on a large-scale dataset [10]. PatentNet [15] learns to embed
patent drawings into a common feature space using a contrastive loss function,
where similar drawings are close and dissimilar drawings are far apart. The
diversity and complexity of patent images which contain different objects and
the data scarcity and imbalance are main difficulties of image-based patent
retrieval.

###  2.2 Sketch-Based Image Retrieval

Sketch-based image retrieval [32] receives hand-drawn sketch as an input and
retrieves images relevant to the sketch. Existing deep learning approaches can
be roughly divided into three categories: generative models, cross-modal
learning and zero-shot learning. Generative models aim to generate synthetic
images or sketches from the input sketch or image using generative neural
networks [5, 3]. Cross-modal learning approaches learn a common feature space
for sketches and images using techniques such as metric learning, domain
adaptation and knowledge distillation [4, 23]. Zero-shot learning retrieves
images that belong to unseen categories using semantic information such as
attributes or word embeddings [35, 28]. This problem is extremely challenging
as the appearance gap between sketches and natural images makes it difficult
to learn effective cross-modal representations and style variations among
different sketchers introduce noise and ambiguity to the sketch
representations. In contrast, image-based patent retrieval does not involve
cross-modal learning and variations among patent drawings are smaller than
those among sketches.

##  3 Method

We present a simple yet effective training pipeline for image-based patent
retrieval, which is shown in Fig. 2. Similar to the strong baseline for person
re-identification (Re-ID) [17], the network mainly consists of the backbone,
the neck and the head. However, training strategies and structures of key
components are very different from those of Re-ID. Details are described as
follows.

Figure 2: The network structure of the proposed method. Dotted lines denote
modules that only appear in the training pipeline.

###  3.1 Backbone Network

EfficientNet-B0 [26] is an effective and efficient convolutional network
obtained by neural architecture search [39, 25]. Compared with ResNet-50 [12],
EfficientNet-B0 significantly reduces network parameters and FLOPS by an order
of magnitude. The main building block of this network is the mobile inverted
bottleneck called MBConv [24, 25], which uses the efficiency inverted
structure in the residual block. This architecture can be easily scaled up
from dimensions of depth, width, and resolution to obtain a family of models.

Given the output feature maps of the backbone network, we use Generalized Mean
(GeM) pooling [22] instead of the commonly used max- or average-pooling to get
image features. The pooled feature dimension is the same as the number of
channels of feature maps. Let $f\in\mathbb{R}^{n}$ be the output feature
vector of dimension $n$. For the $k$-th component of vector $f$, the
formulation of GeM pooling is

${f_{k}}={(\frac{1}{{{\rm{|}}{{\rm
Z}_{\rm{k}}}{\rm{|}}}}{\sum\limits_{x\in{{\rm
Z}_{\rm{k}}}}x^{{p_{k}}}})^{{1\mathord{\left/{\vphantom{1{{p_{k}}}}}\right.\kern-1.2pt}{{p_{k}}}}}},$
(1)

where ${\rm Z}_{\rm{k}}$ is the $k$-th feature map, and $p_{k}$ is a learnable
pooling parameter, $p_{k}>0$. This operation is average-pooling when
$p_{k}=1$, and approximates max-pooling when $p_{k}\to\infty$. The GeM pooling
focuses on the salient and discriminative features from different spatial
positions of the image.

###  3.2 Neck Structure

The strong baseline for Re-ID presents a neck structure that consists of a
batch normalization (BN) layer and empirically analyzes the functions of this
BN layer while combining ID classification loss and triplet loss [17]. There
exists inconsistency between the classification loss and the triplet loss
while combing them in the same feature space, and the BN layer alleviates this
inconsistency by separating the two losses into different feature spaces.

Different from previous neck structures [17, 34] that involve solely a BN
layer for person re-identification, we design a neck structure that comprises
a fully connected (FC) layer and a consecutive BN layer. The FC layer maps the
pooled feature vector of an image into a compact and low-dimensional space,
and, as a result, reduces the parameters of the classifier. In addition, it
transforms feature vectors of various dimensions from different backbones to
fixed dimensions.

During training, the BN layer normalizes the feature distributions and
improves the intra-class compactness and inter-class discrimination, which
benefits the ID classification loss. In the inference phase, the output
feature of the FC layer is directly used for image retrieval.

###  3.3 Classification Head

The PatentNet [15] first trains the patent retrieval model with the
classification loss and then finetunes the network using either the triplet or
contrastive loss. The strong baseline [17] trains the person re-identification
network with both the classification and triplet losses. Our approach for
image-based patent retrieval is trained in a single stage and only uses
classification loss.

Let $x_{i}$ denote the retrieval feature vector of the $i$-th image sample,
and $y_{i}$ be the patent ID label. Suppose there are $m$ different ID
categories and $N$ samples, the widely used cross-entropy loss or the Softmax
loss $L_{\rm{c}}$ is

${L_{\rm{c}}}=-\frac{1}{N}\sum\limits_{i=1}^{N}{\log\frac{{{e^{W_{{y_{i}}}^{T}{x_{i}}+{b_{{y_{i}}}}}}}}{{\sum\nolimits_{j=1}^{m}{{e^{W_{j}^{T}{x_{i}}+{b_{j}}}}}}}},$
(2)

where $W$ and $b$ are the weight and bias parameters of the classifier that
consists of an FC layer.

The ArcFace loss [8] first performs L2 normalization upon the feature vector
and the weight of the classifier, then computes the cosine distance in a
geometric angular space. The loss $L_{a}$ is formulated as

${L_{a}}=-\frac{1}{N}\sum\limits_{i=1}^{N}{\log\frac{{{e^{s\cdot\cos({\theta_{{y_{i}}}}+m)}}}}{{{e^{s\cdot\cos({\theta_{{y_{i}}}}+m)}}+\sum\nolimits_{j=1,j\neq{y_{i}}}^{m}{{e^{s\cdot\cos{\theta_{j}}}}}}}},$
(3)

where $s$ and $m$ are scale and angular margin penalty hyperparameters,
respectively. As discussed in [8], the ArcFace loss could enhance similarity
for intra-class samples and increase diversity for inter-class samples, which
is particularly suitable for large-scale retrieval scenarios due to the huge
intra-class appearance variations.

Figure 3: Typical data augmentations without scaling transformation for image-
based patent retrieval. For Random Crop, the cropped image is not resized to
the original image resolution.

###  3.4 Data Augmentation Without Scaling

Scaling Transformation is a prevalent data augmentation technique for natural
image recognition and analysis. As the concerned object in images varies
considerably in different sizes, scaling transformation makes the model learn
scale-invariant features from images. For image understanding with deep
learning, scaling transformation is almost inevitably included in any data
augmentations.

Different from natural images, patent drawings span a relatively fixed line
width in the document, regardless of the content, and so objects in the
drawings vary little in size. For this reason, we present data augmentation
without scaling (AUG w/o S) for image-based patent retrieval. Typical AUG w/o
S transformations are shown in Fig. 3. Empirically studies of data
augmentations are provided in experiments.

##  4 Experiments

###  4.1 Datasets and Implementation Details

The DeepPatent [15] is a recent large-scale dataset for image-based patent
retrieval. It consists of a total of 45,000 different design patents and
350,000 drawing images. The object in a patent is presented in multiple
drawing images from different views. The training set has 254,787 images from
33,364 patents, and the validation set has 44,815 images from 5,888 patents.
This dataset is dedicated to visual patent retrieval based on detailed and
abstract drawings. The mean Average Precision (mAP) and ranked accuracy
(Rank-N) are used to evaluate the performance of retrieval methods.

The input image resolution is 256$\times$256 if not otherwise specified. The
output feature dimension of the FC layer in the neck structure is 512. The
scale and margin hyperparameters $s$ and $m$ in the ArcFace loss are 20 and
0.5, respectively. The network is trained using the AdamW optimizer with an
initial learning rate of 0.001 and weight decay of 0.0005. The batch size is
128, and the maximum number of iterations is 20,000.

###  4.2 Experimental Results

Table 1: Comparison with the state-of-the-art approaches for large-scale patent retrieval on the DeepPatent dataset. Method | mAP | Rank-1 | Rank-5 | Rank-20
---|---|---|---|---
HOG [7] | 8.3 | 27.2 | 31.7 | 35.9
SIFT FV [6] | 9.2 | 20.6 | 28.9 | 37.5
LBP [19] | 6.9 | 21.0 | 25.2 | 34.3
AHDH [30] | 9.5 | 28.8 | 34.3 | 39.9
VisHash [20] | 9.3 | 27.4 | 34.0 | 40.2
Sketch-a-Net [36] | 13.5 | 36.1 | 45.1 | 53.6
PatentNet [15] | 37.6 | 69.1 | 78.4 | 84.1
Ours | 71.2 | 88.9 | 95.8 | 98.1
Ours w/o ArcFace | 55.4 | 82.3 | 91.3 | 95.2

As image-based patent retrieval is a relatively new research topic and
traditional general techniques of computer vision are likely to perform well,
we compare our approach with both deep learning and traditional approaches.

Representative traditional methods are HOG [7], SIFT FV [6, 21], AHDH [30],
and VisHash [20]. Deep learning-based comparison methods are Sketch-a-Net [36]
and PatentNet [15]. The proposed method significantly outperforms both
traditional and deep learning approaches. It beats the recently proposed deep
learning-based PatentNet [15] by 33.5% and 19.8% for the scores of mAP and
Rank-1, respectively.

###  4.3 Analysis and Visualization

(a) Ablation studies

(b) Effects of augmentations

Figure 4: Effects of key components of the proposed method.

(a) Ablation studies

(b) Effects of augmentations

Figure 5: Scalability analysis of the proposed method. For the sake of
simplicity, B0 and B5 denote EfficientNet-B0 and EfficientNet-B5,
respectively, and $m\times m$ means that the input resolution is $m\times m$.

Ablation studies of key components of the proposed method are shown in Fig.
4(a). ArcFace loss significantly improves the results over Softmax loss for
large-scale image-based patent retrieval, as it increases the value of mAP
from 55.4% to 71.2%. Data augmentation without scaling (AUG w/o S) further
boosts the performance over the proposed strong baseline B0-ArcFace which
adopts EfficientNet-B0 as the backbone and uses the ArcFace classification
loss.

In Fig. 2, AUG w/o S are specifically designed in the proposed method. Here we
experimentally evaluate different augmentations techniques and provide results
in Fig. 4(b). Surprisingly, we find that the commonly used Random Resized Crop
severely hurts the performance while Random Crop without scaling could improve
performance. The results confirm our hypothesis in Subsection 3.4 that scaling
transformation is not suitable for image-based patent retrieval. Another
observation is that Random Rotation also dramatically damages the performance.
One possible reason is that drawing images are horizontally placed in the
document without any geometric distortion.

Although state-of-the-art performance has been reached, the proposed strong
baseline can be scaled up to achieve even stronger performance. As a trial
experiment, we extend the model from three aspects: increasing input
resolution, using a large backbone, and post-processing with re-ranking. The
results are summarized in Fig. 5. Notably, the mAP score is first boosted from
71.2% of the baseline to 87.7% with the input image resolution of $512\times
512$, then gets 89.9% using EfficientNet-B5, and finally reaches 93.5% by
combining these three expanding techniques. Notably, the mAP score is
astonishingly boosted from 71.2% of the baseline to 93.5% by combining these
three expanding techniques. The final extended approach wins the first place
in the Image Retrieval Challenge of an International Conference.

Figure 6: Comparison of image-based patent retrieval results of different
approaches of ours and the state-of-the-art PatentNet [15]. Given a query
image, we only show the top-5 retrieved images. Symbols ✓ and ✗ denote correct
and wrong hits, respectively.

To further investigate the retrieval performance, we visualize the search
results of different approaches in Fig. 6. We find that our approach is robust
to viewpoints. For the query patent (b) which is an aerial view of a patent
object, our approach successfully retrieves the patent images from other
different views while the recent state-of-the-art PatentNet [15] fails to find
this patent for the top-4 retrieved results. For the query patent (c) which
contains apparent texture patterns, the PatentNet [15] find other patents with
similar shapes by mistake while ours correctly retrieves the right patents
from other views.

##  5 Conclusion

In this paper, we empirically study image-based patent retrieval and present a
simple, lightweight yet strong model which consists of a backbone network,
neck structure, and training head. We find that the loss function plays an
important role in this task, and deliberately choose the ArcFace
classification loss. We also find that the classic crop and resize data
augmentation greatly hurts the performance, and accordingly devise data
augmentation without scaling which benefits the performance. To study the
scalability of the model, we scale this baseline to large models by several
expanding techniques such as increasing input resolution and post-processing
with re-ranking. Astonishingly, such techniques significantly boost the
retrieval performance, which demonstrates that the proposed model has good
scalability. In the future, we will investigate more robust losses for image-
based patent retrieval.

## Acknowledgement

This work was supported by the Start-up Research Fund of Southeast University
under Grant RF1028623063. This work is also supported by the Big Data
Computing Center of Southeast University.

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# Data Analytics Driven Controlling: bridging statistical modeling and
managerial intuition ††thanks: This research was supported by the Deutsche
Forschungsgesellschaft through the International Research Training Group 1792
”High Dimensional Nonstationary Time Series”. This research is based upon work
from COST Action 19130, supported by COST (European Cooperation in Science and
Technology). Danial Saef acknowledges the support of PwC.

Kainat Khowaja Humboldt-Universität zu Berlin, International Research Training
Group 1792, Spandauer Str. 1, 10178 Berlin, Germany. Email: kainat.khowaja@hu-
berlin.de Danial Saef Humboldt-Universität zu Berlin, International Research
Training Group 1792, Spandauer Str. 1, 10178 Berlin, Germany. PwC, Germany.
Email<EMAIL_ADDRESS>Sergej Sizov PricewaterhouseCoopers, Germany.
Email<EMAIL_ADDRESS>Wolfgang Karl Härdle Humboldt-Universität zu
Berlin, BRC Blockchain Research Center, Berlin; Sim Kee Boon Institute,
Singapore Management University, Singapore; WISE Wang Yanan Institute for
Studies in Economics, Xiamen University, Xiamen, China; National Chiao Tung
University, Dept Information Science and Finance, Hsinchu, Taiwan, ROC;
Charles University, Dept Mathematics and Physics, Prague, Czech Republic;
Grant CAS: XDA 23020303 and DFG IRTG 1792 gratefully acknowledged. Email:
<EMAIL_ADDRESS>
###### Abstract

Strategic planning in a corporate environment is often based on experience and
intuition, although internal data is usually available and can be a valuable
source of information. Predicting merger & acquisition (M&A) events is at the
heart of strategic management, yet not sufficiently motivated by data
analytics driven controlling. One of the main obstacles in using e.g. count
data time series for M&A seems to be the fact that the intensity of M&A is
time varying at least in certain business sectors, e.g. communications. We
propose a new automatic procedure to bridge this obstacle using novel
statistical methods. The proposed approach allows for a selection of adaptive
windows in count data sets by detecting significant changes in the intensity
of events. We test the efficacy of the proposed method on a simulated count
data set and put it into action on various M&A data sets. It is robust to
aberrant behaviour and generates accurate forecasts for the evaluated business
sectors. It also provides guidance for an a-priori selection of fixed windows
for forecasting. Furthermore, it can be generalized to other business lines,
e.g. for managing supply chains, sales forecasts, or call center arrivals,
thus giving managers new ways for incorporating statistical modeling in
strategic planning decisions.

##  1 Introduction

Data driven insights can improve corporate decision making. In large
organizations, a variety of financial data is available due to reporting
requirements and organizational purposes. However, expertise in the field of
data analytics is scarce. New methods for robotic data evaluation can help
organizations cut costs by shifting resources away from manual tasks and
towards tasks that require supervision. Managers can derive valuable insights
from forecasts based on internal company data to deal with common problems in
a corporate environment such as demand forecasting for supply chain planning
(Yelland et al., 2010), call center arrival times (Taylor, 2007, 2011) and
Oreshkin et al. (2016), sales forecasting (Kolsarici and Vakratsas, 2015), or
mergers & acquistions (M&A) forecasting (Very et al., 2012). However, the
available datasets are usually subject to non-stationarity and structural
breaks, and they usually make manual efforts to fit a meaningful model
necessary. To deal with such problems, experts employ techniques on change
point detection or finding stable parameter windows. To the best of our
knowledge, no combination of such techniques was explored for a framework
incorporating count data. To address this gap, we propose a method that
automatically detects locally homogeneous time windows and corresponding
parameters in an automated way that can be used to generate point or density
forecasts.

As a motivating example we apply this newly developed algorithm to forecast
M&A intensity in different industries. M&A are especially interesting as they
frequently occur in different markets and industries, and are relevant both to
the financial industry that generates revenue by supporting their execution,
as well as those companies that are observing their own industry and their
competitors. Recent approaches are using time series models, as in Very et al.
(2012) or a revealed preference model as in Akkus et al. (2015). Figure 1
shows an example data set of mergers and acquisitions of German energy market
that illustrates the presence of non-stationarity and structural breaks in
such time series.

Figure 1:  Time series of count of mergers and acquisitions per month, with
moving average curve of  1 year and 3 years.  LPA_Empiricalstudy

Empirical evidence strongly suggests that mergers are often clustered in time
as waves, see Martynova and Renneboog (2005), Harford (2005) and Maksimovic et
al. (2013). Ahern and Harford (2014) find that their activity is subject to
network effects and that these waves largely occur within industries, but can
also be transmitted to connected industries. Furthermore, shocks of any kind,
even if they lead to merger waves, are difficult to predict. Wave patterns
seem to be heterogeneous and differ both in time and in industries. Following
their argumentation, we conclude that data on M&A should be evaluated per
industry and geographic location due to differences in regulation, innovation
power, technology, and stock markets.

Predictive models for M&A intensity could be identified through aggregating
acquired knowledge and tailored to specific industries and markets.
Alternatively, time-series models, e.g. ARMA can be used. While a co-variate
based model provides explainability to the user, it requires manual efforts to
gather data and incorporate expert knowledge to define relevant variables and
calibrate their impact on a predictive model. Data gathering can be time-
consuming and expensive, and knowledge about modeling heterogeneous industries
in a specific application requires domain knowledge, which is often scarce and
narrowed to said application. Time series models are an alternative, as they
can be adapted to any other data set with comparable structure. However, such
models need to be robust to non-stationarity, structural breaks and wave
patterns that limit their predictive power.

There is no doubt that only time varying time series models approximate the
dynamics of the underlying series better than any homogeneous parameter
approach, e.g. a fixed ARMA($p,q$) model. Therefore, we employ an adaptive
estimation method called Local Parametric Approach (LPA). The quantitative
implementation was first proposed in Spokoiny (1998), advances are made in
Mercurio and Spokoiny (2004) and Spokoiny (2009). It helps us to find locally
homogeneous time intervals with stable parameters and guarantees a trade-off
between parameter variance and modeling bias. The technique is based on a
series of likelihood ratio tests to determine assumed but unknown change
points in the underlying series. As a result one finds local intervals of
homogeneity and efficient estimates at each point in time.

Since we are often dealing with small sample sizes, and the test statistic
distribution is unknown, we need a method to approximate this distribution.
Recent advances in bootstrapping methodology allow us to generate confidence
sets and critical values that non-asymptotically approximate the true
distribution. Here, we couple LPA with multiplier bootstrap (MBS) (Spokoiny
and Zhilova, 2015) for approximating a critical value for the testing
procedure. MBS builds up on wild bootstrap, that originates from Wu (1986) and
Beran (1986). An important application is reported in Härdle and Mammen
(1993), and Mammen (1993). Further advancements are made in Chatterjee and
Bose (2005) and Arlot et al. (2010). Notable publications that precede
Spokoiny and Zhilova (2015) are Bücher and Dette (2013) and Chernozhukov et
al. (2013). Klochkov et al. (2019) present an application in the context of a
conditional autoregressive Value at Risk model. We generalize this LPA idea to
any data that is Poisson jump distributed, although our simulation study
indicates that these assumptions could be relaxed to the general membership to
any of the exponential families. This indication is useful in bridging
business requirements with the robustness of novel statistical methods through
flexible automated estimations, since many problems in strategic management,
e.g. forecasting M&A intensity, are subject to data sets for which strict
assumptions on underlying distribution, stationarity, or absence of structural
breaks may not be fulfilled.

We detect locally homogeneous windows by computing a non-parametric likelihood
ratio statistics. This approach is related to the branch of change point
detection methods. Chen and Gupta (2011), Eckley et al. (2011), and
Aminikhanghahi and Cook (2017) summarize and evaluate diverse methods of
change point detection. Notable approaches are Hinkley and Hinkley (1970),
(Hsu, 1979), (Haccou et al., 1987), and (Chen and Gupta, 1999) that propose
change point detection methods for gamma, exponentially, and normally
distributed data respectively. Kutoyants and Spokoiny (1999) propose an
adaptive procedure as well as theoretical properties for Poisson distributed
data. Chen and Gupta (2011) provides the null distribution of a likelihood
ratio test for Poisson distributed random variables, but they do not evaluate
the efficiency of the procedure on real data.

Both change point detection and homogeneous window approaches can serve as
determinator for an optimal forecasting window. Recent approaches for optimal
window selection are Giraitis et al. (2013), Pesaran et al. (2013), and Inoue
et al. (2017). They address parameter instability and frequent structural
breaks and indicate that adaptive window selection is favorable over choosing
fixed window sizes, such as a 1 year or 3 year moving average. Since we aim at
developing a generally applicable method that is robust to different data
characteristics, we need an adaptive window. Hence, we pursue a nonparametric
approach that is independent of knowledge about or assumptions on the dataset,
except that the values are generated by a Poisson process with smooth but time
varying intensity over some unknown time window.

Our approach extends the previous literature as it serves as a generic toolbox
that could easily be adapted to other applications and gives density forecasts
that can (but do not have to) be adjusted by incorporating knowledge from
industry experts and is adaptable to arbitrary frequencies. Although we show
how to forecast the density of M&A, our methodology could also extend research
in other areas that typically use Poisson processes.

The remainder of this paper is structured as follows: Section 2 describes the
algorithm that is based on a combination of LPA, MBS and put into action in an
iterative procedure. Section 3 contains an experimental study. We describe the
evaluation method, verify the robustness of the presented algorithm in
simulation scenarios, apply it empirically on a dataset of mergers &
acquisitions and show how it can be used to generate forecasts. Section 4
presents key results, such as robustness to diverse data inputs and
adaptability to other applications. Section 5 discusses limitations, such as
in the evaluation approach or computational costs and suggests next steps like
density forecasting, introducing a judgemental component and extending the
test-statistic. All numerical algorithms can be found on Quantlet.de .

##  2 Methodology

###  2.1 Basic Idea

Planning processes in corporate environments are based on internal financial
data of different kinds. Traditionally, these problems have been solved using
diverse time series models. However, it is difficult to use them since real
time series often are non stationary and have structural breaks. Automated
analyses can be beneficial as they make modelling easier. To contribute to
solving this problem, we focus on detecting locally homogeneous intervals with
stable parameters. To be more specific, we focus on a count data model where
the time varying intensity determines a Poisson process. Take again figure 1
as an example. We aim to detect the years of ’95 - ’97 as a structural break
and capture therefore the non stationary component (a slight uptrend is
observable). We find locally homogeneous windows, verifying that the procedure
is working and show how it can be used to obtain density forecasts.

###  2.2 Stochastics

Let $Y_{t}\in\mathbb{N},t=0,...,T$ be a count data time series such as the
count of M&A series, see in figure 1. Think of $Y_{t}\sim Poisson(\theta)$,
where $\theta$ represents the rate or average number of occurrences in a fixed
interval. Since we allow for time variation in our model, for any interval
$I=[a,b]$ with $a<b$ and $a,b\in\\{0,...,T\\}$, we write $(Y_{t})_{t\in I}\sim
Poisson(\theta)$

The log likelihood function on $I$ is:

$\displaystyle L_{I}(\theta)$ $\displaystyle=\sum_{t\in
I}\log(\theta^{Y_{t}}e^{-\theta}/Y_{t}!)$ $\displaystyle=\log\theta\sum_{t\in
I}Y_{t}-\sum_{t\in I}\theta-\sum_{t\in I}\log(Y_{t}!)$ (1)

the MLE ${\tilde{\theta}_{I}}$ based on observations in ${i\in I}$ is:

$\tilde{\theta}_{I}\mathrel{\overset{\makebox[0.0pt]{\mbox{\tiny
def}}}{=}}\underset{\theta\in\Theta}{\text{argmax }}L_{I}(\theta)$ (2)

which for a Poisson model is the sample mean.

###  2.3 Local Parametric Approach

LPA, first introduced by Spokoiny (1998) is based on the phenomenon that a
series of locally parametric models can describe the features of a time series
better than a global parametric model. The basic idea is that given a time
series and a model for its dynamics, one finds locally stationary intervals of
the time series in an online fashion. This is done by finding the set of most
recent observations, such that the model parameters are approximately stable
in that interval. This set of time points is called interval of homogeneity.
Employing the same procedure at each point in time, one locally estimates the
parameter (Härdle et al., 2015). The merit of LPA is that it does not require
an explicit expression of the law of the dynamics of the parameter, but only
assumes that the parameter is constant on some unknown time interval in the
past (Spokoiny, 2009).

In order to check the homogeneity of an interval $I=[a,b]$, LPA looks for some
break point $\tau\in(a,b)$ such that $A_{\tau}=[a,\tau)$ has one parameter and
$B_{\tau}=[\tau,b]$ has another parameter. If at least one break point exists
in the interval $I$, we conclude that the interval is non-homogeneous
(Klochkov et al., 2019).

The testing hypotheses are therefore:

$\displaystyle H_{0}(I)$ $\displaystyle:(Y_{t})_{t\in I}\sim
Poisson(\theta^{*}_{I}),\theta^{*}_{I}\in\Theta,$ (3) vs $\displaystyle
H_{1}(I)$ $\displaystyle:(Y_{t})_{t\in A_{\tau}}\sim
Poisson(\theta^{*}_{A_{\tau}}),\theta^{*}_{A_{\tau}}\in\Theta,$
$\displaystyle(Y_{t})_{t\in B_{\tau}}\sim
Poisson(\theta^{*}_{B_{\tau}}),\theta^{*}_{B_{\tau}}\in\Theta,$
$\displaystyle\text{with some }\theta^{*}_{A_{\tau}}\neq\theta^{*}_{B_{\tau}}$

The LR test statistic for a breakpoint $\tau$ is:

$\mathcal{T}_{I,\tau}=L_{A_{\tau}}(\tilde{\theta}_{A_{\tau}})+L_{B_{\tau}}(\tilde{\theta}_{B_{{\tau}}})-L_{I}(\tilde{\theta}_{I})$
(4)

Since one has many candidates $\tau\in J$, one arrives at:

$\mathcal{T}_{I}=\max_{\tau\in J}\mathcal{T}_{I,\tau}$ (5)

This unfortunately has a very intractable distribution, hence the critical
values $\mathfrak{z}_{I}(\alpha)$, indicating that the test indicates that the
test statistic rejects $H_{0}$ in equation (3) i.e. when

$\mathcal{T}_{I}\geq\mathfrak{z}_{I}(\alpha)$ (6)

is hard to calculate. Indeed the limiting distribution of $\mathcal{T}_{I}$ is
different from general likelihood ratio tests due to the presence of nuisance
parameters (breakpoints) in the alternative hypothesis which are not
identified under the null hypothesis. Hence, convergence of the generalized LR
statistics to a $\chi^{2}$ distribution according to Wilk’s phenomenon can not
be put into action. While the asymptotic distribution of the sup-LR test in
equation (5) can still be derived (Andrews and Ploberger, 1994), a large
enough sample size is required for its asymptotic critical values to be
applicable. Certainly, that is not the case in most of the practical
situations where only small samples of data are available.

Spokoiny and Zhilova (2015) provide a non-asymptotic result for mis-specified
models with small sampling sizes. The technique is called multiplier bootstrap
(MBS) which is discussed in detail in the next section.

###  2.4 Multiplier bootstrap

Since the asymptotic distribution for the LR test statistic is not available
for small samples, we approximate the unknown log-likelihood distribution
using the bootstrap. First, introduce random weights to the previously defined
likelihood function:

$\displaystyle L_{I}^{\circ}(\theta)=\sum_{t\in I}w_{t}l_{t}(\theta)$

where the weights $w_{t}\text{ are with }\operatorname{E}(w_{t})=1\text{ and
}\mathrm{Var}(w_{t})=1\text{ }iid$. The bootstrap version of (1) is given by:

$\displaystyle L_{I}^{\circ}(\theta)=\log\text{ }\theta\sum_{t\in
I}Y_{t}w_{t}-\sum_{t\in I}w_{t}\theta-\sum_{t\in I}\log(Y_{t}!)w_{t}.$ (7)

The bootstrap MLE is then defined as:

$\displaystyle\tilde{\theta}_{I}^{\circ}=\arg\max L_{I}^{\circ}(\theta),$

which is:

$\displaystyle\tilde{\theta}^{\circ}_{I}=\frac{\sum_{t\in
I}Y_{t}w_{t}}{\sum_{t\in I}w_{t}}.$

It follows that the corresponding bootstrap of (10) is:

$\mathcal{T}_{I,\tau}^{\circ}={}L_{A,\tau}^{\circ}(\tilde{\theta}_{A,\tau}^{\circ})+L_{B,\tau}^{\circ}(\tilde{\theta}_{B,\tau}^{\circ})-\sup_{\theta}\left\\{L_{A,\tau}^{\circ}(\theta)+L_{B,\tau}^{\circ}(\theta+\tilde{\theta}_{B,\tau}-\tilde{\theta}_{A,\tau})\right\\},$
(8)

A penalty term $\tilde{\theta}_{B,\tau}-\tilde{\theta}_{A,\tau}$ is introduced
to compensate for model misspecification bias. Klochkov et al. (2019) shows
that the distribution of this test conditional on the data mimics the ”true”
distribution of $\mathcal{T}_{I}$ with high probability. Using (8) we can
obtain the critical value through simulations. Indeed, the critical value
$z_{I}^{\circ}(\alpha)$ is defined as:

$\displaystyle z_{I}^{\circ}(\alpha)=z_{I}^{\circ}(\alpha;Y)=\inf\left\\{z\geq
0:\textup{P}(\mathcal{T}_{I}^{\circ}>z^{2}/2)\leq\alpha\right\\}.$ (9)

###  2.5 Algorithm

The algorithm for an adaptive window length selection at each point in time is
now straightforward. It is based on sequential testing of the hypotheses on a
nested set of intervals $\\{I_{k}\\}_{k=0,1,\ldots,K}$, where $I_{0}\subset
I_{1}\subset\ldots\subset I_{K}$. Let $n_{k}=|I_{k}|$ be the number of
observations in each interval. The first interval $I_{0}$ is assumed to be
homogeneous with length $n_{0}$. Then, for each interval $I_{k}$, the null
hypothesis of parameter homogeneity is tested against the alternative of a
change point at an unknown location $\tau$ within $I_{k}$, as in (3).

Since the setup deals with nested intervals, but the existence and location of
a change point are unknown, only additional points in each new interval are
considered as possible change points. The candidate set for change points in
each interval is defined as $J_{k}=I_{k}\backslash I_{k-1}$. Using each point
$\tau\in J_{k}$, the left and right intervals are constructed as
$A_{k,\tau}=[i_{0}-n_{k+1},\tau]$ and $B_{k,\tau}=(\tau,i_{0}]$ respectively
(see figure (2)). The test statistic is calculated similarly to equation (5)
as

$\mathcal{T}_{I_{k},\tau}=L_{A_{{k},\tau}}(\tilde{\theta}_{A_{{k},\tau}})+L_{B_{{k},\tau}}(\tilde{\theta}_{B_{{k},\tau}})-L_{{}_{I_{k+1}}}(\tilde{\theta}_{I_{k+1}})$
(10)

where $A_{{k},\tau}$ and $B_{{k},\tau}$ are as previously specified and we
test at every point $\tau\in J_{k}$ for a change point. The $k$th interval is
rejected if

$\max_{j\in
J_{k}}\mathcal{T}_{I_{k},\tau}\leq\mathfrak{z}_{I_{k}}^{\circ}(\alpha)$ (11)

and $\mathfrak{z}_{I_{k}}^{\circ}$ is generated via multiplier bootstrap as
explained in the previous section. If the interval $I$ is not rejected, i.e.
there exists no change point and it is homogeneous, we continue the testing
procedure by choosing a bigger interval. Otherwise, the length of the last
non-rejected interval $\widehat{I}$ is the interval of homogeneity and
$\widehat{\theta}_{i_{0}}=\widehat{\theta}_{\widehat{I}}$ is the respective
adaptive estimate of $\widehat{I}$.

Homogeneity testing for $I_{k}$ utilizes also part of observations of
$I_{k+1}$. Hence, the pre-definition of intervals is crucial. Following that,
the choice of interval lengths affects the test results, and therefore
requires careful selection. We employ a geometric increase of intervals like
Härdle et al. (2015) and Klochkov et al. (2019). Based on the initial length
$n_{0}$, the intervals lengths are defined by

$n_{k}=\bigg{\lceil}n_{0}c^{k}\bigg{\rceil}$ (12)

where $c$ is a geometric multiplier, chosen slightly above 1 to ensure a
monotonic increase of interval lengths, but not by a big margin. Furthermore,
instead of taking a constant number of intervals $K$ for testing as proposed
by Härdle et al. (2014) and Klochkov et al. (2019), we select $K$ to be the
smallest integer such that the whole time series is covered under a
geometrically increasing length.

Figure 2: Iterative algorithm

In summary, the LPA algorithm for the adaptive choice of an interval of
homogeneity and the corresponding MLE is given by the following iterative
procedure:

  1. 1.

Initialization: Select $I_{0},I_{1},I_{2},$ and define $J_{1}=I_{1}\backslash
I_{0},(\forall\tau\in J_{1}),$ $A_{1,\tau}=[i_{0}-n_{2},\tau]$,
$B_{1,\tau}=(\tau,i_{0}]$.

  2. 2.

Iteration: For each iteration, select $I_{k-1},I_{k},I_{k+1},$,
$J_{k}=I_{k}\backslash I_{k-1},(\forall\tau\in J_{k})$
$A_{k,\tau}=[i_{0}-n_{k+1},\tau],$ $B_{k,\tau}=(\tau,i_{0}]$.

  3. 3.

Testing homogeneity: Calculate test statistics in equation (10) and select
critical value with multiplier bootstrap. Test hypothesis in equation (3)
using equation (11).

  4. 4.

Loop: If $I_{k}$ is accepted, take the next interval $I_{k+1}$. Otherwise set
$\widehat{I}$ to the latest non rejected $I_{k}$.

  5. 5.

Adaptive estimator: Take interval $\widehat{I}$ as interval of homogeneity and
$\widehat{\theta}_{i_{0}}=\widehat{\theta}_{\widehat{I}}$ as adaptive estimate
of $\widehat{I}$. Repeat the procedure for each point in time (different
$i_{0}$)

##  3 Experimental results

The following lines seek to answer the question whether or not we can generate
better forecasts by using the described adaptive methodology, and whether or
not the methodology is robust with respect to previously unseen and thus
unpredictable patterns. We compare one-step-ahead and multi-period point
parameter forecasts of the proposed locally adaptive procedure to a baseline
of one year and three years moving averages in a pseudo-out-of-sample
approach. We acknowledge that this is a fairly simple approach. A more
sophisticated evaluation approach would be to generate multi-period density
forecasts that could be evaluated using a tailored loss-function as in Diebold
et al. (1998), Diebold (2015), and Gonzalez-Rivera and Sun (2014). However, as
this is beyond the scope of this paper (or altogether another paper), we leave
it open for future work.

###  3.1 Simulation study

This section evaluates the performance of the proposed technique using
simulated datasets as shown in figure 3. We create scenarios that mimic common
patterns in financial datasets. The simulations focus both on short-term
shocks and regime shifts. Starting with the simplest piece-wise constant
model, we gradually increase the complexity of the simulations by generating
Poisson distributed data and finally test the robustness of the methodology by
changing the underlying model to follow an exponential distribution. For each
scenario, we consider a time series $(Y_{t})_{t=1}^{300}$ with the following
specifications:

  1. (a)

Regime shifts with piece-wise constant model: $(Y_{t})_{t=1}^{100}=1$,
$(Y_{t})_{t=101}^{200}=10$ and $(Y_{t})_{t=201}^{300}=20$

  2. (b)

Regime shifts with Poisson model: $(Y_{1t})_{t=1}^{100}$ with $\theta_{1}=1$,
$(Y_{2t})_{t=101}^{200}$ with $\theta_{2}=10$ and $(Y_{3t})_{t=201}^{300}$
with $\theta_{3}=20$

  3. (c)

Short term shock with piece-wise constant model: $(Y_{t})_{t=1}^{199}=1$,
$(Y_{t})_{t=200}^{200}=10$ and $(Y_{t})_{t=201}^{300}=1$

  4. (d)

Structural break with piece-wise constant model: $(Y_{t})_{t=1}^{180}=10$,
$(Y_{t})_{t=181}^{200}=7$ and $(Y_{t})_{t=201}^{300}=10$

  5. (e)

Structural break with Poisson model: $(Y_{1t})_{t=1}^{180}$ with
$\theta_{1}=5$, $(Y_{2t})_{t=181}^{200}$ with $\theta_{2}=1$ and
$(Y_{3t})_{t=201}^{300}$ with $\theta_{3}=5$

  6. (f)

Regime shifts with exponential model: $(Y_{1t})_{t=1}^{100}$ with
$\theta_{1}=0.1$, $(Y_{2t})_{t=101}^{200}$ with $\theta_{2}=1$ and
$(Y_{3t})_{t=201}^{300}$ with $\theta_{3}=10$

(a)

(b)

(c)

(d)

(e)

(f)

Figure 3: Simulated series (left), homogeneous windows (middle) and MLE
(right).  LPA_Simulations

The simulated data of scenarios (a)-(f) are shown in the time series plots on
the left hand side in figure (3). Since the algorithm requires pre-selection
of the intervals, we fix c in equation (12) at 1.35. This seemingly arbitrary
choice ensures that we neglect only few unknown homogeneous intervals (if
any), while gaining computational efficiency. Assuming a minimal homogeneous
window ($n_{0}$) of 5 months, we compute $K$ as described in section (2). For
example, the candidate homogeneous windows for the latest period in the time
series are $[6,9,12,16,22,30,40,55,74,100,135,183,247,300]$. Note that since
we allow the number of intervals to vary with the time point for which the
homogeneous windows has to be determined, the number of intervals being tested
gradually decrease as the number of historical data points in the time series
decrease. The algorithm is applied to find the homogeneous window and
corresponding MLE estimate for each point in time in each simulation and the
results are presented in figure (3). Also, for each scenario, we run 100
simulations to smoothen the estimated window sizes and construct confidence
intervals that are depicted by dotted lines in all plots in figure (3).

Plot (a) in figure (3) shows that the procedure detects intervals of
homogeneity correctly in all cases of this fairly simple scenario. It is
robust to small and big values and gives a good approximation of the true
parameters. The step-wise increase of homogeneous windows and the fluctuation
of MLEs is a consequence of the limited number of possible time windows $K$
that the algorithm tests for. If $K$ were to be increased, for example
arithmetically, we would expect the algorithm to generate a straight downward
slopping line instead. Due to computational limitations, we illustrate this
only on scenario (a) in figure (4) and avoid such an experiment for other
scenarios.

(a)  Figure 4: Simulated series (left), homogeneous windows (middle) and MLE
(right) using arithmetically increasing intervals in LPA.  LPA_Simulations

Next, a more realistic scenario with values generated from a Poisson process
in (b) shows that the procedure is robust to noise even when the sample size
is small. On the other hand, when $n$ is small, the problem of multiple
testing causes many false negatives. This becomes evident in the first third
of (b). This can be easily dealt with by changing the number of tested
intervals in the algorithm. Regardless of this issue, the MLE remains close to
the true parameter value.

Further, some temporary shocks and small structural breaks are mimicked in
simulations (c)-(e). (c) shows that a large shock is detected accurately, with
an alarm to select smaller window size when the time is close to the change in
mean. The simulation in (d) shows that temporary changes in mean are not
always recognised by the algorithm, but the MLE after such short-term shocks
is still affected. The procedure also detects structural breaks within a
Poisson framework accurately as depicted in (e), but the estimated time
windows are inaccurate shortly after the break. With increasing $n$, the
accuracy is restored and the approximated MLE remains accurate.

Finally, to check the robustness of the algorithm, values are generated from
an exponential distribution. The simulation in (f) shows that the algorithm
detects breakpoints correctly even with exponentially distributed data.
Accordingly, the assumption of Poisson distributed values can be easily
relaxed towards models from any exponential family. This robustness allows the
algorithm to be used even with misspecified models.

For each simulation scenario, we also compare the results with fixed window
estimates of 12 months (high variance) and 36 months (high bias) and show the
results in figure (5). All subplots show that LPA finds a balance between
variance and bias by selecting a smaller or bigger window size when necessary.
Plot (a) shows that while the estimates of LPA fluctuate (as described above),
the shift in regime is detected earlier by LPA than by fixed window estimates.
Plots (b) and (e) show that for the simulated intervals with constant mean,
fixed window estimates indicate a highly fluctuating mean, while LPA
recognizes intervals of time homogeneity correctly. Moreover, the impact of
the short term shock disappears faster in the LPA estimates in (c) and (d).
However, LPA underestimates the magnitude of shocks as visible in (d).

In conclusion, the algorithm accurately detects small shocks, regime shifts
and temporary mean changes in time series without strict assumptions on the
underlying model. The results could be improved further by correct selection
of interval lengths and numbers of intervals being tested, as they have a
significant impact on the precision and accuracy of the results.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 5:  Time series of simulated data and one step ahead prediction with
estimates from  LPA, 1 year fixed window and  3 year fixed window.
LPA_Simulations

###  3.2 Use case study

In this section, we apply LPA to a working example of time series that are
relevant to businesses in the financial industry. We use a data set of mergers
& acquisitions per month that we acquired from the database Refinitiv Eikon
Deals (restricted access through subscription). We consider different
industries according to the Thomson Reuters Business Classification scheme in
Germany. The dataset consists of a total of 9,969 observations in ten
industries in the US and Germany. We put the procedure into action on the
following three industries in Germany:

  1. 1.

Financials

  2. 2.

Telecommunication

  3. 3.

Energy

Due to seemingly inconsistent recordings of transactions, we considered only
424 observations from 01-1984 to 04-2020 in Germany for each industry. We do
not make any assumptions regarding correlations of these industries or between
countries.

Following previous literature, we consider different industries separately.
Without insider knowledge, the frequency of M&As could be assumed to be random
and i.i.d. over certain time windows, which is based on the intuition that we
usually learn about mergers only after they are announced. The i.i.d.
assumption states in this context that the observed industries are of a
specific, somewhat stable structure in the short run, but can be interrupted
by exogenous factors that change industry dynamics. Given these
characteristics, we model the number of M&A with a Poisson process and apply
LPA to forecast the next period given the MLE of the last observed homogeneous
time window. We do not preprocess the data and let the procedure account for
structural breaks and non-stationarity.

Similar to the simulation study in previous section, we fix $n_{0}$ at 5
months, c at 1.35, compute K as in section (2), and run 100 simulations to
smoothen the estimated window sizes. We keep $n_{0}$ and c for the interval
selection through equation (12) constant as an aim of the proposed algorithm
is to evaluate unseen data even if no knowledge about it is available. Some
example cases are illustrated to verify the robustness of the algorithm with
respect to different types of time series in figure (6). The plot on the right
hand side shows the homogeneous windows for each point in time while the
corresponding LPA estimate, 12 month estimate and 36 month estimate as one
period ahead forecast are shown in the left plot.

(1)

(2)

(3)

Figure 6:  Time series of original data and one step ahead prediction with
estimates from  LPA, 1 year fixed window and  3 year fixed window (left)
Homogeneous windows (right) for (1) Financials, (2) Telecommunication and (3)
Energy.  LPA_Empiricalstudy

The first plot in figure (6) shows the German financial industry, where an
upward trend in the number of mergers can be observed over time. The plot
shows that LPA estimates closely mimic this trend, and is much more responsive
to shocks as in the mid-90s. Moreover, contrary to the green 1-year moving
average curve, which shows high variance in the MLE estimate, LPA recognizes
the regions within time series where the average number of M&A is
approximately constant, such as from 2010 to 2020. The LPA estimates on the
left suggest that the selected window size differs over time (sometimes up to
200, sometimes only a few observations) and using a fixed time window for
generating forecasts is not recommended according to the technique. Comparing
the window plot of the same industry with the simulation plots (a) and (b)
from the previous section, we identify six regimes in the German financial
industry (1984-1988, 1988-1990, 1990-1994, 1994-1999, 1999-2006 and
2006-2020). These regimes could be associated with external events, such as: a
merger wave in 1984, globalization in late 1980s, the German bank merger wave
in 1990s, German market liberalization in mid-90s, and the technological
innovations in late 20th century.

The second part of figure (6) shows the German telecommunication industry,
where the number of M&As per month seems stable and stationary, except for a
short merger wave in the mid-90s due to the liberalization of the market and
another short wave around 2001 (possibly related to the dotcom-bubble). LPA
suggests a stable time series as the algorithm recommends to select large
windows, and sometimes the whole time series. Only during the interval where a
shock can be seen on the left, LPA’s choice of homogeneous window on the right
restricts the time window to a few years (for example between 1994 and 2001).
We see that LPA can handle time-varying parameters while facing a trade-off
between parameter variability and modelling bias. A similar conclusion can be
drawn from the energy markets plots in (3) of figure (6).

Next, we forecast the number of M&As for 02-2020 to 04-2020 using the LPA
estimate and the fixed 1-year and 3-year window estimates of 01-2020. The
estimates and mean squared errors (MSE) of estimation are summarized in the
table (1). Moreover, apart from the selection time-varying window lengths, we
propose that LPA can also provide guidance on a priori selection of fixed
windows. Looking at the distribution plot of window lengths (as a proportion
of data points in the time series before the time of evaluation) in figure (7)
and selecting the window with highest frequency $(w)$ as the fixed time window
for each time series, we also make a forecast and report the MSE in the same
table.

| Financials | Telecom. | Energy
---|---|---|---
LPA estimate | 52.99 | 3.43 | 4.23
12 month MA estimate | 55.58 | 2.42 | 3.08
36 month MA estimate | 55.50 | 2.94 | 4.17
Most recurring window(w) | 160 | 184 | 309
w month MA estimate | 55.25 | 2.90 | 4.62
LPA MSE | 1008.24 | 3.11 | 8.61
12 month MSE | 1159.29 | 7.34 | 3.29
36 month MSE | 1154.25 | 4.89 | 8.25
w month MSE | 1139.23 | 5.06 | 11.03
Table 1: Forecast results for 02-2020 to 04-2020 based on the adaptively
selected MLE, fixed windows and most recurring window proportionally $(w)$ in
the distribution plots in figure (7)

(1)

(2)

(3)

Figure 7: Distribution of estimated interval length as a proportion of data
points for (1) Financials, (2) Telecommunication and (3) Energy.
LPA_Empiricalstudy

The table shows that LPA produces the smallest MSE in the financials and
telecommunication industry. However, in the energy sector the 1 year fixed
window outperforms LPA, perhaps due to its high fluctuations. Similarly, the
homogeneous time window recommended by LPA for the financial industry was
roughly 160 months. Choosing this time window resulted in a lower MSE than
using 12 and 36 months fixed window estimates. For the energy sector, however,
LPA recommends the selection of a very long window, for which the estimate
deviates significantly from the true value, but still captures the stationary
aspect of the time series. As a whole, the results show that wave-like
patterns and level shifts are accurately detected, and that a plausible fit is
achieved under the assumption that M&A follow a Poisson distribution with
time-varying parameter.

The analysis shows that the proposed technique can be applied to real-world
financial data and gives additional insights based merely on an automatic
procedure. It can be used as a baseline for forecasting or simulation
approaches. The MLE series could be forecasted using e.g. autoregressive
models that generate different scenarios. Business experts could then make a
judgement about the likeliness of these scenarios. Moreover, the availability
of a time-varying parameter allows analysts to use more sophisticated
approaches, e.g. forecasting densities of various time-series that are assumed
to follow a Poisson distribution, such as the examples mentioned in the
introduction (call centers, sales, supply chains). Businesses are often not
interested in exact values, but in having an overview over the range of
expected figures, and forecasts of low quality can lead to costly
miscalculations. The proposed procedure could help increase forecast quality
and availability and thus generate business value if employed.

##  4 Conclusion

A new algorithm for automatically finding homogeneous time intervals in a non
stationary context is proposed. Trends, cyclical components, and even black
swan events can be analysed with this technique. The algorithm is based on a
fruitful combination of a local parametric model (here a Poisson process) and
MBS. We conduct a simulation study that indicates that the procedure is indeed
robust, even if the input data is not Poisson distributed. These results are
then used to evaluate a data set of M&A industry-based in Germany. The results
show that in this non-parametric model-free context we obtain interpretable
structural breaks and changes in trend. In conclusion, we provide an easy-to
use solution for analyzing large data collections automatically that addresses
the most common problems in time series modeling and can be adapted to diverse
applications, including call center arrivals, sales forecasting or supply
chain decisions.

##  5 Future Work

This work is subject to several limitations that are caused either by
computational cost or additional complexity that exceeds the scope of this
work. As the proposed algorithm is computationally expensive, we evaluated
only three German industries and admit that this choice is, to an extend,
arbitrarily taken. However, data for all industries in the US and Germany is
available and could be used. The simulations showed the the proposed method
had difficulties with detecting small changes. We also saw that the method is
not in all cases better than choosing a simple approach using moving averages.
An evaluation based on all 20 datasets could give further insights on the
robustness of the method. Furthermore, the evaluation using a pseudo-out-of-
sample point forecast is fairly limited and we justify the superiority of our
approach based on the simple measure of a mean-squared-error. However, as
Diebold (2015) outlines, this approach provides no insurance against over-
fitting, are costly as they discard data and it is questionable whether they
provide any benefit. Another questionable point is whether the occurrence of
M&A per month strictly follows a time-varying Poisson process, as assumed in
this paper.

Answering this question using a sophisticated evaluation approach through
generating (multi-period) density forecasts would give more insights and could
be an interesting area for future research. Future research could also
evaluate the fit to a Poisson distribution with time-varying MLE, but with
respect to different values of $n_{0}$ and c. Alternatively, other methods for
increasing the tested homogeneous interval could be compared to a geometric
approach, e.g. random increases that follow a Poisson process or arithmetic
increases. Furthermore, the simulation study indicated that the procedure
could be generalized to work with any data that is assumed to follow a
distribution of the exponential family. Finally, a possible extension could
include a more general version of the current test statistic that looks into
both directions.

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