# Tropical cyclone intensity estimations over the Indian ocean using Machine
Learning

Koushik Biswas
Department of Computer Science, IIIT Delhi
New Delhi, India, 110020.
<EMAIL_ADDRESS>
& Sandeep Kumar
Department of Computer Science, IIIT Delhi
&
Department of Mathematics,
Shaheed Bhagat Singh College, University of Delhi
New Delhi, India, 110020.
<EMAIL_ADDRESS><EMAIL_ADDRESS>
Ashish Kumar Pandey
Department of Mathematics, IIIT Delhi
New Delhi, India, 110020.
<EMAIL_ADDRESS>

###### Abstract

Tropical cyclones are one of the most powerful and destructive natural
phenomena on earth. Tropical storms and heavy rains can cause floods, which
lead to human lives and economic loss. Devastating winds accompanying cyclones
heavily affect not only the coastal regions, even distant areas. Our study
focuses on the intensity estimation, particularly cyclone grade and maximum
sustained surface wind speed (MSWS) of a tropical cyclone over the North
Indian Ocean. We use various machine learning algorithms to estimate cyclone
grade and MSWS. We have used the basin of origin, date, time, latitude,
longitude, estimated central pressure, and pressure drop as attributes of our
models. We use multi-class classification models for the categorical outcome
variable, cyclone grade, and regression models for MSWS as it is a continuous
variable. Using the best track data of 28 years over the North Indian Ocean,
we estimate grade with an accuracy of 88% and MSWS with a root mean square
error (RMSE) of 2.3. For higher grade categories (5-7), accuracy improves to
an average of 98.84%. We tested our model with two recent tropical cyclones in
the North Indian Ocean, Vayu and Fani. For grade, we obtained an accuracy of
93.22% and 95.23% respectively, while for MSWS, we obtained RMSE of 2.2 and
3.4 and $R^{2}$ of 0.99 and 0.99, respectively.

_Keywords_ Machine Learning $\cdot$ Tropical Cyclone $\cdot$ Cyclone intensity
$\cdot$ Maximum sustained surface wind speed

##  1 Introduction

Tropical cyclones are rapidly rotating storm systems centered in a low-
pressure region. Tropical cyclones cause heavy rain, strong wind, large storm
surges near landfall, and tornadoes, which results in loss of property and
lives. About 1.9 million people have died because of tropical cyclones
worldwide during the last two centuries [1, 2]. The North Indian ocean (which
includes the Bay of Bengal and Arabian sea) alone has seen some of the most
devastating tropical cyclones. In 2019, both coasts of India experienced
substantial damages because of Vayu and Fani.

It is of high importance to estimate the intensity of a tropical cyclone. A
standard indicator of the intensity of the storm is the maximum sustained
surface wind speed (MSWS). The World Meteorological Organization categorizes
the low-pressure systems using the ranges of MSWS of the tropical cyclones
[3]. The categorization can be used to determine possible storm surges and
damage impact on land [4].

In most tropical cyclone basins, satellite-based Dvorak technique or
reconnaissance air-crafts are used to estimate MSWS [5]. These techniques
provide reasonable estimates but require advanced machinery. Therefore,
estimating MSWS from other tropical cyclone parameters is a significant
problem. Much work has been done towards this problem; see [6, 7] and
references therein for a complete history of the work relating to cyclone
intensity prediction.

We propose a method to estimate MSWS based on other characteristics of a
tropical cyclone like date, time, latitude, longitude, pressure drop and
estimated central pressure. We use machine learning algorithms to devise a
regression model to estimate MSWS from other characteristics. We further
employ machine learning classification algorithms to predict the grade of the
cyclone based on these characteristics.

##  2 Materials and methods

###  2.1 Data

The best track dataset of tropical cyclonic disturbances are collected from
the Regional Specialized Meteorological Centre, New Delhi
(http://www.rsmcnewdelhi.imd.gov.in/index.php?option=com_content&view=article&id=48&Itemid=194&lang=en)
has been used in this study for the period from 1990 to 2017 in the North
Indian ocean. The basin of origin, name (if there any), date and time of
occurrence, position (latitude and longitude), Class number (or T No.),
estimated central pressure, MSWS, pressure drop, grade, outermost closed
isobar and diameter of outermost closed isobar of tropical cyclones are
provided in the dataset. We define terms which we are going to use in the
analysis below [3]:

  * •

Basin of origin(BOO): The Arabian sea, Bay of Bengal, or land is the possible
basin of origins of any cyclone.

  * •

Date and Time: The date and time of the origin of the cyclone.

  * •

Latitude and Longitude: The latitude and longitude in degrees along the path
of the cyclone.

  * •

Estimated central pressure (ECP): It is the surface pressure at the center of
the tropical cyclone as measured or estimated (in hPa (hectopascals)).

  * •

Pressure drop (PD): It is the drop in the pressure with respect to the
atmospheric pressure. It is also measured in hPa.

  * •

Maximum sustained surface wind (MSWS): The maximum sustained surface wind
speed is the highest average of 3 minutes surface wind speed occurring within
the circulation of the system. It is measured in knots (nautical miles per
hour), which is the same as 1.86 Kilometers per hour.

  * •

Grade: Any tropical cyclone that develops within the North Indian Ocean
between $100^{\circ}$E and $45^{\circ}$E is monitored by the India
Meteorological Department (IMD). Tropical cyclone intensity scale according to
cyclone category are given in the following table:

Grade | Low pressure system | MSWS (in knots)
---|---|---
1 | Low Pressure Area (LP) |  $<$17
2 | Depression (D) | 17-27
3 | Deep Depression (DD) | 28-33
4 | Cyclonic Storm (CS) | 34-47
5 | Severe Cyclonic Storm (SCS) | 48-63
6 | Very Severe Cyclonic Storm (VSCS) | 64-119
7 | Super Cyclonic Storm (SS) |  $\geq$120
Table 1: The classification of the low pressure systems by IMD.

Figure 1: Cyclones hitting India since 1990-2017.

There is a total of 4852 instances of cyclone measurements in the dataset
used, out of which we selected 4021 for our study, dropping all of those
instances which have any missing feature value. A pictorial description of
these cyclones, along with colour-coded grade, is shown in Figure 1. The date
is divided into three classes according to three seasons: Pre Monsoon - March
to May, Monsoon- June to September, Post Monsoon - October to February. Time
is divided into two categories according to day and night. We did not include
outermost closed isobar and diameter of outermost closed isobar as attributes
in our study, as very few data points were available in these columns. Table 2
describes the distribution of data in different categories.

Characteristics | Subdivisions | Number of data points
---|---|---
Basin of Origin |  | Arabian Sea
---
Bay of Bengal
Land
| 1149
---
2707
165
Season |  | Pre-Monsoon (March - May)
---
Monsoon (June to September)
Post-monsoon (October - February)
| 791
---
1208
2022
Grade |  | 1
---
2
3
4
5
6
7
| 1433
---
915
920
266
15
205
267
Table 2: Baseline Data.

###  2.2 Methodology

The MSWS is a continuous variable, while the grade is a categorical variable.
Therefore, we use various machine learning regression and classification
algorithms (XGBoost, Gradient Boosting Machine, Linear Regression, Decision
Tree, Random Forest, SVM, Naive Bayes, Logistic Regression) for the prediction
of MSWS and grade. In what follows, we briefly describe these algorithms.

####  2.2.1 Decision tree

Decision Tree [8] is one of the most popular supervised machine learning
algorithms used for both classification and regression techniques. The
algorithm can be represented by an inverted tree with a root node at the top
and other nodes connected to it through branches. Each node corresponds to a
feature and a value assigned to the feature, while each branch represents a
decision taken for the output variable based on the node it is emanating. To
decide which feature to be placed at a node, we use measures like the Gini
index, Entropy, or Information gain. For a given attribute $X$,

  * •

Entropy is defined as

$E(X)=\sum-P(X=x)\log_{2}(X=x),$

  * •

Information gain is defined as

$IG(X,x)=E(X)-E(X|x),$

  * •

Gini index is defined as

$1-\sum(P(X=x))^{2}$

where $P$ denotes the probability, $E(X)$ denotes the entropy and $E(X|x)$ is
the conditional entropy for a particular instance $x$ of $X$. We can determine
the importance of a given attribute of a feature vector by calculating one of
the above for that attribute.

####  2.2.2 Random Forest

Random forest [9] is an ensemble learning method that can be used for both
classification and regression. It generates multiple decision trees as part of
the training process and outputs the mode (average) of these trees as per the
classification (regression) problem. This approach solves the problem of
overfitting, which is prevalent in the case of Decision Trees.

####  2.2.3 Gradient Boosting Machine

Gradient Boosting Machine [10] is an ensemble machine learning technique that
is used for both classification and regression problems. It depends on the
boosting technique where each weak learner is assigned a large weight to
convert them to a strong learner in an iterative manner.

####  2.2.4 XGBoost

XGBoost [11] is one of the most popular recent supervised learning tree
boosting scalable machine learning algorithms, which is based on function
approximation and several regularization techniques. It is used for both
classification and regression problems. Let $\widehat{y_{i}}$ is the outcome
from the ensemble model defined as follows:

$\widehat{y_{i}}=\phi(x_{i})=\sum_{k=1}^{K}f_{k}(x_{i}),\quad
f_{k}\in\mathcal{F}$

where $\mathcal{F}=\\{f(x)=w_{q(x)}\\}$, $q:\mathbb{R}^{m}\rightarrow T$,
$w\in\mathbb{R}^{T}$ is the space of all regression trees and $T$ denotes the
total number of leaves in the tree. In the above equation, $f_{k}$ represents
a regression tree and $f_{k}(x_{i})$ is the outcome given by the $k$th tree to
the $i$th entries in the data. The goal in XGBoost is to minimize the
following regularized objective function:

$\mathcal{L}(\phi)=\sum_{i=1}^{n}l(y_{i},\widehat{y_{i}})+\sum_{k=1}^{K}\Omega(f_{k})$

where $l$ is the loss function. To avoid high complexity of the model, a
regularization term $\Omega$ is used which is given by

$\Omega(f_{k})=\gamma T+\frac{1}{2}\lambda||w||^{2}=\gamma
T+\frac{1}{2}\lambda\sum_{j=1}^{T}w_{j}^{2}$

Where $\gamma$ and $\lambda$ are regularization parameters, the best split at
any given node can be found from the following formula:

$\mathcal{L}_{split}=\frac{1}{2}\bigg{[}\frac{G_{L}^{2}}{H_{L}+\lambda}+\frac{G_{R}^{2}}{H_{R}+\lambda}-\frac{(G_{L}+G_{R})^{2}}{H_{L}+H_{R}+\lambda}\bigg{]}-\gamma$

Where $L$ stands for left-hand node and $R$ stands for right-hand node by
letting $I=I_{L}\cup I_{R}$. Figure 3 shows the XGBoost tree for the
estimation of MSWS.

####  2.2.5 Linear Regression

In Linear Regression [12], a hyperplane is estimated that gives best linear
relationship between independent variables (features) and dependent variable
(target). The prediction model (hypothesis) is given by :

$h_{\theta}(X)=\theta_{0}+\theta_{1}x_{1}+\theta_{2}x_{2}+\dots+\theta_{n}x_{n}$

where $X=(x_{1},x_{2},\dots,x_{n})$ represents the input vector and
$\theta=(\theta_{0},\theta_{1},\theta_{2},\dots,\theta_{n})$ are the
coefficients that determine the hyperplane. These coefficients are learned
through an iterative process called gradient descent by minimizing the
following loss function:

$J(\theta)=\sum_{i=1}^{m}(h_{\theta}(X_{i})-y_{i})^{2},$

where $X_{i}$ denotes the $i$th input vector and $y_{i}$ corresponding target
value.

####  2.2.6 Logistic Regression

Logistic regression [13] is a classifier that can be used to solve a
multiclass prediction problem. Its an extension of Linear Regression, where
the classification problem is converted into regression problem by estimating
the log(odds) of each class in place of probability itself. If $p_{i}$ denotes
the probability of $i$th class then the log(odds) for this class is defined as
$p_{i}/\sum_{j\not=i}p_{j}$.

####  2.2.7 Support Vector Machines (SVM)

SVM [14] can be used for both classification and regression problems. Like the
Linear regression, SVM tries to find a separating hyperplane, but with maximum
margin. The learning problem is converted into an objective (nonlinear)
maximization problem, subject to linear constraints. Using the tools of Linear
Programming Problem (LPP), few input vectors (called support vectors) are
selected that can be used for prediction. The nonlinear separating case of
input vectors can be handled with kernels techniques.

####  2.2.8 Naive Bayes

Naive Bayes [15] can be used for both classification and regression problems.
The Naive Bayes algorithm is based on Bayes’ theorem with an assumption that
the features are linearly independent. Suppose $X_{1},X_{2},\cdots,X_{n}$ are
real-valued attributes, and $Y$ is the set of all possible outcomes. Now
according to the Bayes’ theorem,

$P(Y=y_{i}|X_{1},X_{2},\cdots,X_{n})=\frac{P(Y=y_{i})P(X_{1},X_{2},\cdots,X_{n}|Y=y_{i})}{\sum_{k}P(Y=y_{k})P(X_{1},X_{2},\cdots,X_{n}|Y=y_{k})}.$

If we assume that $X_{i}$ are conditionally independent for given outcome set
$Y$, then the above equation can be written as

$P(Y=y_{i}|X_{1},X_{2},\cdots,X_{n})=\frac{P(Y=y_{i})\prod_{j}P(X_{j}|Y=y_{i})}{\sum_{k}P(Y=y_{k})\prod_{j}P(X_{j}|Y=y_{k})}$

The above equation is used for the classification problem. Similarly, we can
define Naive Bayes for regression problems, where the sum in the above
equation will be replaced by integration.

####  2.2.9 Metrics

To evaluate the performance of regression models for MSWS, we use the Root
Mean square error (RMSE) and Coefficient of determination ($R^{2}$).

  * •

RMSE: If there are $m$ sample points with $y_{i}$ as actual value and
$\widehat{y_{i}}$ as predicted value evaluated from the model, then RMSE is
defined as

$\sqrt{\frac{1}{m}\sum_{i=1}^{m}(y_{i}-\widehat{y_{i}})^{2}}.$

RMSE is always non-negative and should be close to $0$.

  * •

$R^{2}$: The coefficient of determination ($R^{2}$) is defined as

$R^{2}=1-\frac{\text{SE}_{\hat{y}}}{\text{SE}_{\bar{y}}}$

where total sum of squares, $\text{SE}_{\bar{y}}$, is defined as
$\text{SE}_{\bar{y}}=\sum_{i=1}^{m}(y_{i}-\bar{y_{i}})^{2}$ and residual sum
of squares, $\text{SE}_{\hat{y}}$ is defined as
$\text{SE}_{\hat{y}}=\sum_{i=1}^{m}(y_{i}-\widehat{y_{i}})^{2}$. Here,
$\bar{y}$ is the mean of the data, $\bar{y}=\frac{1}{m}\sum_{i=1}^{m}y_{i}$.

The confusion matrix is used to determine the performance of the
classification model on the test data. For classification models, multi-class
classification accuracy has been measured using the confusion matrix [16].
Accuracy is the ratio between all correctly predicted samples to all possible
samples.

$\text{Accuracy}=\frac{\text{correctly predicted samples}}{\text{total number
of test samples}}$

##  3 Results and Discussions

###  3.1 Correlation analysis

Figure 2: Correlation between variables.

The correlation matrix of all variables is given in Figure 2. The grade is
weakly correlated with all the variables except ECP and PD. Also, the
correlation of grade with ECP is negative, suggesting that if central pressure
is low, the intensity of the cyclone is high. The MSWS shares a similar
correlation with ECP as the grade. This is not surprising as grade is directly
evaluated from MSWS; see Table 1. PD has a strong positive correlation with
MSWS. A linear regression suggests the following relationship between MSWS and
PD

$\operatorname{MSWS}\approx 1.6\operatorname{PD}+22.3$

in the North Indian Ocean. Notice that in [17], a similar relationship between
MSWS and PD ($\text{MSWS}\approx 1.176\text{PD}+30$) was reported for tropical
cyclones in Central North Pacific Ocean.

###  3.2 Model selection and validation

We use 10-fold cross-validation for each of the models. In each fold, we split
the data into training and validation sets in the ratio of 4:1. Then, each ML
algorithm is applied to the training set to train the model. At every step,
the performances (RMSE, $R^{2}$, or accuracy) of the model are recorded, and
the average of each of these performances is reported in Tables 3(a) and 3(b).

Model | RMSE | $R^{2}$
---|---|---
XGBoost | 2.30 | .99
Gradient Boosting Machine | 2.80 | 0.97
Decision Tree | 3.91 | 0.94
Random Forest | 3.12 | 0.96
Linear Regression | 5.07 | 0.92
SVM Kernel:-RBF Kernel:Linear Kernel: Polynomial (4th-degree) |  6.11 5.69 3.93 |  0.90 0.91 0.95
Naive Bayes | 3.38 | 0.97
(a) Regression Analysis on MSWS.

Model | Accuracy
---|---
XGBoost | 87.15
GBM | 85.73
Decision Tree Entropy(Depth-4) Gini(Depth-4) |  87.91 84.76
Random forest | 85.95
Naive Bayes | 86.39
Logistic | 71.28
SVM Kernel:-RBF Kernel: Linear Kernel: Polynomial (degree 4) |  78.48 86.90 84.36
(b) Classification(Multi-class) Analysis on Cyclone grade.

It is evident from Table 3(a) that XGBoost is outperforming other models with
an RMSE of 2.3 and $R^{2}$ of 0.99. Notice from Table 1 that the range of
values of MSWS for a particular grade is always greater than or equal to 5,
and since XGBoost is predicting MSWS with an RMSE of 2.3, we expect that
XGBoost will also predict grade with very high accuracy. That is definitely
the case, as from Table 3(b), XGboost has an accuracy of 87.15% in predicting
the grade. However, the Decision Tree with Entropy of depth 4 outperforms
XGBoost in predicting the grade with an accuracy of 87.91%.

Moreover, if we fix the classification model for the grade to be the Decision
Tree with Entropy of depth 4, Table 4 represents the accuracy in predicting a
particular category for the grade. The model predicts the top three high-
intensity categories (SCS, VSCS, and SS) of grade with an average accuracy of
98.84%.

Category | Accuracy
---|---
LP | 98.33
D | 77.92
DD | 78.37
CS | 88.72
SCS | 100
VSCS | 99.51
SS | 97
Table 4: Classification accuracy of different Cyclone Grade. Figure 3: XGBoost
tree for MSWS.

###  3.3 Testing on Vayu and Fani

Figure 4: Scatter plot of actual and model-predicted MSWS for Fani and Vayu.

Figure 5: Actual and model predicted grade along track of Fani.

Figure 6: Actual and model predicted grade along track of Vayu.

We test our model on two recent tropical cyclones, Vayu and Fani. Vayu was a
grade 7 tropical cyclone which hit the Indian west coast in June 2019. Around
6.6 million people were affected in northwestern India by the cyclone [18].
Fani was also a grade 7 tropical cyclone that hit the Indian state of Odisha
in April-May 2019. Due to Fani, India and Bangladesh faced heavy damages. At
least 89 people have been reported died, and damages caused estimated around
US$8.1 billion [19].

We checked the performance of the best model to predict MSWS, XGBoost, on Vayu
and Fani. The RMSE is 2.2 and 3.4, while $R^{2}$ is 0.99 and 0.99 for Vayu and
Fani, respectively. Figure 4 depicts the actual values of MSWS and values
predicted by the XGBoost model during the course of Vayu and Fani.

The best model to predict grade, Decision Tree with Entropy with depth 4,
predicts different grades during the course of Vayu and Fani with an accuracy
of 93.22% and 95.23%, respectively. The actual and predicted grades along the
track of Vayu and Fani is shown in Figures 5 and 6.

##  4 Conclusion

Estimating the intensity of tropical cyclones on a real-time basis is a
problem worth studying, considering the human life and economic loss involved.
In this study, we explored various machine learning techniques and reported
their performance to estimate the Maximum Surface Sustained Wind Speed and
intensity of the tropical cyclone. Our research finds that the ML model
XGBoost and Decision Tree can be used for the estimation of MSWS and intensity
with excellent performance over the North Indian ocean.

## Acknowledgement

Authors are thankful to the Indian Meteorological Department (IMD) for
providing the data archives.

## Conflict of Interest

All the authors declare that they have no conflict of interest.

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# Analysis and discussion of the recent $W$ mass measurements

GIGI ROLANDI  CERN Ginevra and Scuola Normale Superiore Pisa

###### Abstract

The ATLAS and CDF measurements of the W mass are compared discussing some
similarities and differences in the quoted systematic uncertainties.

##  1 Introduction

The Organizing Committee asked me to prepare a short presentation to stimulate
the discussion on the recent $W$ mass measurement [1] of the ATLAS
collaboration nicely presented by Nansi Andari [2]. I decided to show a
comparison of this measurement with the old CDF measurement [3] that has the
same precision of the ATLAS one: 19 MeV. Since I refer the reader quite often
to figures published in the two papers it is convenient to read this note
having the two papers [1, 3] at hand.

Both measurements are done using electronic and muonic decays of the W. The
CDF sample consists of 1.1 Millions $W$, evenly distributed between $W^{+}$
and $W^{-}$, produced in $p\bar{p}$ collisions at $\sqrt{s}=2\;{\rm TeV}$ ;
the ATLAS sample comprises 7.8 Millions $W^{+}$ and 5.9 Millions $W^{-}$
produced in $pp$ collisions at $\sqrt{s}=7\;{\rm TeV}$.

The $W$ mass is measured in $W\rightarrow\ell\nu$ decays from the transverse
momentum distribution of the charged lepton111In the following I will use the
word lepton to indicate the charged lepton ($p_{T}$) and from the transverse
mass distribution ($M_{T}$), where the information of transverse momentum of
the lepton is combined with the transverse momentum of the hadronic system
recoiling to the W ($\vec{h}$):
$M_{T}^{2}=2\left(p_{T}|\vec{p_{T}}+\vec{h}|+p_{T}^{2}+\vec{p_{T}}\vec{h}\right)$.
The CDF collaboration uses also the missing transverse energy distribution
$p_{T}^{\nu}=|\vec{p_{T}}+\vec{h}|$. For each experimental distribution
($p_{T}$,$M_{T}$,$p_{T}^{\nu}$) a value of the W mass is determined comparing
the measured distribution to templates generated with different W mass
hypotheses using a carefully calibrated MonteCarlo. The correlated values of
the W mass are combined in order to minimize the total uncertainty.

In both analyses the events are selected requiring one identified lepton
matched to the trigger and $30\,{\rm GeV}<p_{T}$ , $30\,{\rm GeV}<p_{T}^{\nu}$
and $60\,{\rm GeV}<M_{T}$. The cut of the recoil is $h<15$ GeV in CDF and
$h<30$ GeV in ATLAS. The comparison between the measured distribution and the
template distributions is done in a limited region (fit range) of the variable
of interest. The fit ranges in CDF are $32<p_{T}<48$ GeV and $65<M_{T}<90$ GeV
in CDF and $32<p_{T}<45$ GeV and $66<M_{T}<99$ GeV in ATLAS.

The result of the CDF analysis is $M_{W}=80\,387\pm 12({\rm stat.})\pm 15({\rm
syst.})$ MeV; the result of the ATLAS analysis is $M_{W}=80\,370\pm 7({\rm
stat.})\pm 11({\rm exp.\,syst.})\pm 14({\rm mod.\,syst.})$ MeV. The total
uncertainty is 19 MeV in both analyses. The weights of the muon and electron
samples are similar in the two analyses : 0.62/0.38 in CDF and 0.57/0.43 in
ATLAS. The weights of the $M_{T}$ / $p_{T}$ fits in ATLAS are 0.14/0.86; the
weights of the $M_{T}$/ $p_{T}$ / $p_{T}^{\nu}$ fits in CDF are
0.53/0.31/0.16. The ATLAS measurement is mainly done using the $p_{T}$
observable while the CDF one is mainly done using the $M_{T}$ observable that
is less sensitive to prior knowledge of the transverse momentum distribution
of the $W$.

Table 1 shows a breakdown of the uncertainties in the two analyses using
separately the $p_{T}$ and $M_{T}$ observables. I have followed the
uncertainties breakdown structure presented in the CDF paper and I have
adapted the uncertainties quoted in the ATLAS paper to this somewhat different
list.

Table 1: Break down of the uncertainties in the $W$ mass measurement in CDF and ATLAS analyses. All figures in MeV | CDF/$p_{T}$ | CDF/$M_{T}$ | ATLAS/$p_{T}$ | ATLAS/$M_{T}$
---|---|---|---|---
Statistical | 16 | 15 | 7.2 | 9.6
Lepton Scale and Resolution | 7 | 7 | 6.5 | 6.5
Recoil Scale and Resolution | 5.5 | 6 | 2.5 | 13
Backgrounds | 4 | 3.5 | 4.6 | 8.3
PDFs | 9 | 10 | 9 | 10.2
W transverse momentum model / QCD | 9 | 3 | 8.3 | 9.6
Photon Radiation/EWK | 4 | 4 | 5.7 | 3.4

##  2 Statistical uncertainty

The ATLAS sample is about 12 times the CDF sample and one would naively expect
a reduction of the statistical uncertainty by a a factor $\sqrt{12}\simeq
3.5$. Table 1 shows ratios of 2.2 and 1.5 for the statistical uncertainties of
the $p_{T}$ and $M_{T}$ analyses. The $p_{T}$ spectra in the two experiments
are quite similar and the smaller fit range in ATLAS accounts for less than
20% reduction of the statistics. The $M_{T}$ spectrum in ATLAS is some 20%
larger due to the worse recoil resolution. These effects do not explain the
observed difference between the statistical uncertainties.

The ATLAS sample is divided in categories that are also fitted separately. The
sample of $W^{+}\rightarrow\mu\nu$ with $\eta<0.8$ has a roughly 1.2 Million
events as can be estimated from figure 20 of reference [1]. The statistical
uncertainty on the $p_{T}$ and $M_{T}$ fits done on this sample are 22 and 29
MeV (see table 10 of reference [1]), a factor 1.3 and 1.8 larger than the CDF
quoted statistical uncertainty.

##  3 Uncertainty due to the recoil

The systematic uncertainty in the $p_{T}$ fit due to the recoil is caused by
the event selection that depends on the recoil [4]. This uncertainty is
evaluated in CDF modeling the recoil distribution using Z data and assessing
the uncertainty on $M_{w}$ from the difference in the measured mass obtained
repeating the $W$ mass fit with a 1$\sigma$ variation of the recoil model. In
CDF this results in an uncertainty of about 6 MeV in the $p_{T}$ fit and is
caused by events that statistically are accepted or rejected by the selection.
The worse the recoil resolution the larger is this effect [4] .

The measurement of the recoil in CDF has a response of 65% and a resolution on
the true recoil (eg corrected for the response) of 7 GeV. In ATLAS the
response is larger, close to 90% and the resolution is about 10 GeV. The
systematic uncertainty due to recoil on the $p_{T}$ fit in ATLAS is 2.5 MeV
that is small compared to the 6 MeV of CDF. I wonder if this is due to the
optimization procedure of ATLAS that is choosing the fit range that minimizes
the total uncertainty. In CDF the fit range is decided a priori.

##  4 Uncertainty due to the PDFs

In CDF the uncertainty due to the Parton Distribution Functions (PDF) is
computed repeating the $M_{w}$ fits with the different independent variations
of the PDFs 222 The PDF distributions are given as a central set of
distributions for each relevant parton and a number of replicas each
corresponding to 1$\sigma$ of the uncertainty of the PDF fit. These replicas
correspond to independent variations in the PDF fit. and assessing the
uncertainty as the sum in quadrature of the variations of $M_{w}$ resulting
from each fit. They use CTEQ6.6 and MSTW2008 PDFs. The result is a systematic
uncertainty of 10 MeV.

ATLAS uses CT10nnlo PDFs and has a procedure similar to the one adopted by
CDF. In ATLAS the fit is done in categories dividing the sample in four lepton
rapidity bins, separately for $W^{+}$ and $W^{-}$. The result of the fits
shows large beneficial correlation when the PDF distributions are varied. The
PDF uncertainty in the fits of $W^{+}$ and $W^{-}$ separate samples is 13.5
and 14.5 MeV respectively for the $p_{T}$ fit and 16.9 and 16.2 MeV for the
$M_{T}$ fit. The uncertainty on the fits combining $W^{+}$ and $W^{-}$ are 9
MeV and 10 MeV respectively , i.e. a reduction of 40% of the uncertainty ! One
of the reasons is that the $c\bar{s}$ contribution to the $W^{+}$ sample is
almost identical (bin by bin in the variable of interest) to the $s\bar{c}$
contribution to the $W^{-}$. When combining the two samples the contribution
to the PDF uncertainty of the less known $s$ and $c$ quarks parton
distribution functions is reduced.

Another interesting effect shown by the ATLAS analysis is that the PDF
uncertainties in the different $\eta$ categories have also beneficial
correlations. As shown in table 2 the PDF uncertainty is reduced when
combining the different $\eta$ bins.

Table 2: Uncertainty due to Parton Distribution Functions in the fit done in different $\eta$ categories in the ATLAS analysis. Figures from table 10 and table 11 of reference [1]. All figures are in MeV. $\eta$ range | 0-0.8 | 0.8-1.4 | 1.4-2.0 | 2.0-2.4 |  | combined
---|---|---|---|---|---|---
PDF $W^{+}$/$p_{T}$ fit | 24.7 | 20.6 | 25.2 | 31.8 |  | 14.5
PDF $W^{+}$/$M_{T}$ fit | 28.4 | 23.3 | 27.2 | 32.8 |  | 16.9

##  5 Uncertainty due to the modeling of the W transverse momentum

The W transverse momentum (pt) distribution is a very important ingredient of
the W mass analysis. It must be known to very good precision when fitting the
lepton $p_{T}$ distribution and its effect is somewhat mitigated in the fit of
the $M_{T}$ distribution depending on the available statistics and on the
response and resolution on the measurement of the recoil.

The strategy adopted by ATLAS and CDF is to measure and model precisely the Z
pt distribution and use the model to predict the W pt distribution.

CDF fits the measured Z pt distribution using RESBOS [5] determining the two
parameter of interest ($g_{2}$ and $\alpha_{s}$) and gets an agreement at the
2% level between the fitted function and the measured distributions as shown
in figure 5 of reference [3]. They use the same function to predict the W pt
distribution and they quote an uncertainty on the $W$ mass of 9 and 3 MeV for
the $p_{T}$ and $M_{T}$ fits computed from the variations in 1$\sigma$ of the
fitted parameters of the model. The smaller uncertainty in the $M_{T}$ fit is
what one expects because of the reduced dependence of the $W$ transverse mass
on the boson pt distribution.

ATLAS uses PYTIA8 to fit the Z pt distribution varying the intrinsic
transverse momentum of the incoming partons, the value of $\alpha_{s}$ used
for the QCD ISR, and the value of the ISR infrared cut-off. The resulting tune
(AZ tune) reproduces the Z pt spectrum with an agreement at 1-2% level. When
exporting the model to the W pt distribution the variation of the AZ tune fit
parameters produce a systematic uncertainty of about 3 MeV. In addition ATLAS
consider systematic uncertainties due to other contribution to the W
production model including the charm mass, the variation of the factorization
scale of the parton shower with heavy-flavour decorrelation, the parton shower
PDF uncertainty and the uncertainty of the angular coefficients. The resulting
uncertainties on the $W$ mass in the $p_{T}$ and $M_{T}$ fits from these
sources are quite similar as shown in table 3 of reference [1], resulting in
an uncertainty of 8 MeV and 9 MeV on the $P_{T}$ and $M_{T}$ fits.

Figure 1: Plots from the talk of N. Andari [2]. Left: ratio between the Z and
W pt spectra of different calculations/generators. The blue band shows the
systematic uncertainty in the ATLAS analysis. Right: $u_{||}$ is the component
of the W recoil parallel to direction of the lepton in W events. The plot
shows the ratio between the predicted and measured distributions

Figure 1 left shows the ratio between the pt distribution of the W and Z
bosons for the model used by the ATLAS analysis and other predictions of the
boson pt spectrum. Resummed calculations like DYRES [6] predict an harder
spectrum compared to the one used in the analysis and the difference is not
covered by the quoted systematic uncertainty band. In order to validate the
model ATLAS measures in W events the recoil component parallel to the lepton
direction $u_{||}$ (both vectors in the transverse plane). Due to events
selection cuts, when $u_{||}$ is close to 30 GeV in absolute value the recoil
is forced to be parallel or antiparallel to the lepton direction. The rate of
goes to zero approaching $u_{||}$ of 30 GeV from below and -30 GeV from above
as can be seen in figure 19 of reference [1]. Events where the W pt is
parallel to the lepton appear near -30 GeV in figure 1 b. These are events
where the lepton transverse momentum exceeds 60 GeV. The events near +30 GeV
in the same plot are events were the W pt is antiparallel to the lepton
direction: the neutrino momentum exceeds 60 GeV and the lepton momentum is
larger than 30 GeV. Because of the resolution and of the response of the
recoil, the rate of events near $u_{||}\sim+30$ GeV is larger than the rate
near -30 GeV. The plot shows that near $u_{||}\sim-30$ GeV the data are not in
agreement with the resummed calculations that predict a larger cross section
than what measured in data by some 5%. This region is characterized by leptons
with large transverse momentum that are produced in events where the boson pt
is large. This is not the case in the region near $u_{||}\sim+30$ GeV. Here
the boson pt can also be quite small because of the resolution in the
measurement of the recoil.

The question why resummed calculations do describe the Z pt distribution in
$p\bar{p}$ at 2 TeV and do not describe the $pp$ data at 7 TeV remains open.

##  6 Conclusions

ATLAS has produced the first W mass measurement at LHC with the data collected
at $\sqrt{s}=$ 7 TeV. This is a very welcome event and opens the road to
future measurements using even larger data samples. This analysis shows that
the systematic uncertainty due to PDFs can be kept under control and possibly
brought below the 10 MeV level. There are still few aspects to be understood
among which why the measured Z pt distribution is not in agreement with
prediction using resummation calculations.

## Acknowledgments

I would like to thank Patrick Janot and the organizing committee for inviting
me to give this presentation. Moriond Electroweak is in my opinion the most
stimulating conference in HEP and it is always a pleasure to participate
actively to the works.

## References

## References

  * [1] M. Aaboud et al. [ATLAS Collaboration], “Measurement of the $W$-boson mass in pp collisions at $\sqrt{s}=7$ TeV with the ATLAS detector,” arXiv:1701.07240 [hep-ex].
  * [2] Nansi Andari “W mass measurement in these proceedings
  * [3] T. A. Aaltonen et al. [CDF Collaboration], “Precise measurement of the W -boson mass with the Collider Detector at Fermilab,” Phys. Rev. D 89, no. 7, 072003 (2014) doi:10.1103/PhysRevD.89.072003 [arXiv:1311.0894 [hep-ex]].
  * [4] A. Kotwal, CDF collaboration, private communication.
  * [5] C. Balazs, J. w. Qiu and C. P. Yuan, Phys. Lett. B 355 (1995) 548 doi:10.1016/0370-2693(95)00726-2 [hep-ph/9505203].
  * [6] S. Catani, D. de Florian, G. Ferrera and M. Grazzini, JHEP 1512 (2015) 047 doi:10.1007/JHEP12(2015)047 [arXiv:1507.06937 [hep-ph]].