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Statistical analysis of a time series relevant to passive systems of nuclear power plants

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Abstract

In this study, we present the methodology for fitting auto-regressive integrated moving average (ARIMA) model to one of the independent process parameter (atmospheric temperature) of passive safety systems. We also present how to generate the synthetic data of the modelled time series of this independent process parameter, which can be used for uncertainty propagation and reliability estimation of passive safety systems. As an illustration to the methodology of model fitting and synthetic data generation, a time series of monthly-maximum atmospheric temperature of district Chittaurgarh (Rajasthan, India) has been considered. With the help of methodology, a non-contiguous ARIMA model of AR (1,3,6,9,12), MA (1,3,6,9,12) has been found to represent the differenced (at lags 12) stationary series of monthly-maximum atmospheric temperature. A synthetic series of length 1224 months have been generated based on the finalized ARIMA model.

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Abbreviations

ACF:

Auto correlation function

AD:

Anderson–Darling

AR:

Auto-regression

ARCH:

Autoregressive conditional heteroskedasticity

ARMA:

Autoregressive–moving-average

ARIMA:

Autoregressive-integrated-moving-average

AIC:

Akaike information criterion

BIC:

Bayesian information criterion

C:

Contiguous ARMA model

NC:

Non-contiguous ARMA model

ECDF:

Empirical cumulative distribution function

MLE:

Maximum likelihood estimation

MA:

Moving-average

PACF:

Partial auto correlation function

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Author information

Authors and Affiliations

  1. Homi Bhabha National Institute, Anushaktinagar, Mumbai, 400094, India

    A. Chandrakar

  2. Radiological Physics and Advisory Division, Bhabha Atomic Research Centre, Trombay, Mumbai, 400085, India

    D. Datta

  3. Reactor Engineering Division, Reactor Design Development Group, Bhabha Atomic Research Centre, Trombay, Mumbai, 400085, India

    A. K. Nayak

  4. Reactor Safety Division, Reactor Design Development Group, Bhabha Atomic Research Centre, Trombay, Mumbai, 400085, India

    Gopika Vinod

Authors
  1. A. Chandrakar
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  2. D. Datta
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  3. A. K. Nayak
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  4. Gopika Vinod
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Corresponding author

Correspondence to A. Chandrakar.

Appendix

Appendix

Time series of monthly maximum atmospheric temperature in Celsius (°C),

Data for the period of year 1901–2002

Location: Chittaurgarh, Rajasthan, India

Sampling frequency—Monthly

Year

Jan

Feb

Mar

Apr

May

Jun

Jul

Aug

Sept

Oct

Nov

Dec

1901

23.74

25.85

33.54

37.23

40.39

38.79

33.36

29.58

32.35

35.18

29.99

27.22

1902

26.57

28.62

35.39

38.59

39.92

37.50

33.14

31.32

31.32

33.37

28.98

25.53

1903

25.27

26.36

31.05

35.93

39.22

38.67

33.04

29.90

31.38

33.20

27.48

24.94

1904

25.21

27.62

31.78

38.23

39.78

36.47

31.05

29.65

31.98

33.64

29.28

26.16

1905

23.89

23.17

30.46

35.28

41.47

37.82

31.91

30.82

32.34

34.18

30.80

26.28

1906

23.70

25.48

31.77

37.05

40.90

36.56

31.53

30.33

30.58

33.08

30.01

26.33

1907

26.33

26.71

31.65

35.97

38.12

37.55

34.12

28.58

30.90

33.79

30.20

24.97

1908

24.88

27.08

31.58

37.86

39.36

37.29

30.21

28.72

30.60

32.68

28.86

24.94

1909

24.83

27.10

33.65

35.96

38.79

36.41

30.57

28.66

30.19

32.66

29.43

25.15

1910

24.75

27.68

32.73

36.12

39.16

35.53

30.42

29.41

31.20

31.87

27.54

25.70

1911

26.50

27.66

30.84

36.64

40.46

36.08

33.12

30.67

31.02

34.11

28.69

27.09

1912

26.39

29.29

32.25

37.60

40.32

38.46

31.98

29.18

30.50

33.13

28.22

25.66

1913

25.25

27.60

30.92

38.38

38.80

36.13

31.49

29.29

31.53

34.42

29.74

26.01

1914

27.15

27.20

31.97

37.39

40.80

36.36

31.34

29.76

32.02

33.60

30.63

25.48

1915

25.22

26.21

33.21

36.96

40.50

38.80

33.83

30.78

33.79

34.79

30.06

26.60

1916

26.21

26.59

34.88

38.20

39.78

35.39

32.81

29.59

31.62

32.27

27.58

25.14

1917

26.20

27.73

32.50

35.23

37.27

35.19

30.26

29.13

30.28

31.34

27.46

25.52

1918

23.82

28.15

32.48

35.44

40.06

36.12

32.88

30.54

31.48

33.97

30.52

25.20

1919

25.69

27.27

33.25

36.63

39.50

36.96

31.14

29.31

30.85

32.82

29.69

25.50

1920

25.69

27.11

33.42

36.70

36.73

35.48

30.51

29.18

32.69

34.52

30.44

26.36

1921

25.92

27.68

35.09

39.02

40.32

37.54

33.53

29.60

30.48

33.06

29.12

27.35

1922

24.62

29.19

33.68

38.07

39.61

36.75

31.64

29.48

30.72

32.88

29.02

25.22

1923

25.59

26.71

33.66

37.36

38.85

38.50

31.74

28.61

31.51

32.72

28.75

26.54

1924

25.04

28.16

34.85

38.20

38.27

37.72

32.19

29.86

30.59

31.78

27.60

25.77

1925

23.00

25.71

33.64

39.29

38.97

35.12

30.59

30.75

32.88

35.37

29.96

26.37

1926

26.11

30.04

32.50

34.72

38.66

39.18

32.30

29.71

30.82

32.93

27.59

25.16

1927

23.63

25.99

31.52

36.50

38.80

37.09

31.04

28.60

30.84

32.94

27.12

26.15

1928

24.93

27.78

33.08

36.99

39.75

36.78

31.45

29.53

30.84

33.20

29.62

25.01

1929

24.24

26.38

34.45

37.61

39.93

36.83

31.17

29.04

31.57

33.05

30.27

24.71

1930

25.14

26.97

33.43

36.59

39.37

37.07

30.44

29.30

32.93

34.60

28.62

26.52

1931

26.90

26.52

32.41

38.48

39.95

38.69

32.42

29.67

31.22

32.45

28.96

26.28

1932

27.89

27.03

32.85

36.64

39.38

37.23

31.69

29.45

32.09

34.25

28.44

26.40

1933

24.63

28.04

32.95

36.23

37.62

35.99

31.34

28.61

30.43

32.55

29.16

25.82

1934

23.48

28.85

32.04

37.36

39.24

37.08

31.03

29.11

30.32

32.23

28.62

26.54

1935

23.15

27.65

32.35

34.61

39.62

37.66

30.85

29.15

30.56

32.49

29.47

26.07

1936

24.10

27.75

32.10

36.35

40.76

35.19

31.26

29.66

31.37

33.59

29.44

25.36

1937

24.22

28.26

31.41

36.63

39.50

37.55

30.58

29.70

31.51

32.84

30.08

24.70

1938

25.40

25.98

34.51

37.96

39.95

34.32

31.04

30.05

32.56

34.14

28.12

26.22

1939

26.35

27.83

30.64

35.65

39.14

36.34

31.67

31.30

31.37

34.11

29.35

26.52

1940

25.95

27.89

31.08

36.26

40.11

36.92

31.75

29.15

31.29

33.63

29.70

25.57

1941

24.63

28.55

34.42

38.41

39.99

37.42

31.69

30.05

32.45

35.67

30.16

27.55

1942

23.95

28.14

34.19

38.92

39.76

38.23

30.14

28.76

30.57

33.12

29.30

24.63

1943

25.07

27.61

33.69

36.68

40.23

36.05

30.57

28.79

30.84

32.76

29.91

26.45

1944

24.71

27.01

32.02

36.29

39.62

36.00

30.14

28.79

31.29

31.98

29.16

26.74

1945

23.05

26.88

33.49

36.18

38.72

37.01

30.15

29.52

30.53

32.20

28.36

24.24

1946

25.22

29.07

32.23

38.82

39.36

36.21

30.92

28.48

30.80

33.56

27.51

25.85

1947

23.94

27.41

33.61

37.24

40.14

37.88

31.82

30.28

30.25

31.90

29.62

26.23

1948

24.68

26.89

32.93

38.44

41.06

37.49

32.09

29.49

30.92

34.16

29.07

26.15

1949

27.24

27.92

33.69

38.76

41.14

36.81

31.70

30.26

31.92

33.58

28.44

25.34

1950

26.25

25.50

32.13

36.36

39.68

37.18

30.58

29.35

30.79

33.03

27.87

24.63

1951

24.10

26.90

32.80

35.50

39.31

36.27

33.05

29.87

33.59

36.39

31.11

25.93

1952

26.60

29.38

33.06

38.59

40.90

36.14

30.88

28.76

31.94

34.04

29.59

26.54

1953

24.80

30.01

35.36

37.69

39.69

37.31

32.10

29.51

31.66

33.19

29.41

28.15

1954

24.92

29.11

33.47

37.96

40.88

37.75

31.68

30.59

30.19

31.34

29.59

25.80

1955

25.36

28.05

35.18

35.49

39.10

36.81

32.44

29.43

30.64

31.89

27.79

25.93

1956

25.35

27.32

33.79

37.33

40.45

36.11

29.58

28.83

31.23

31.28

27.88

26.16

1957

24.46

26.06

31.60

36.51

38.33

37.58

31.72

30.29

31.31

33.68

31.34

26.65

1958

26.93

28.19

34.22

39.05

40.11

37.92

30.63

30.65

30.14

32.45

30.31

27.11

1959

25.24

27.34

34.61

37.61

39.60

37.01

30.44

28.84

30.42

33.01

29.26

26.59

1960

24.38

29.79

32.06

36.23

38.92

36.96

32.03

28.51

32.03

32.81

28.67

26.28

1961

24.27

25.45

33.56

36.70

39.58

36.24

31.12

29.89

30.03

31.78

29.26

24.74

1962

23.90

28.26

33.14

37.15

38.58

36.16

30.58

28.59

29.48

31.04

28.49

26.68

1963

25.75

30.26

33.73

37.42

39.56

37.25

32.51

28.72

31.03

33.53

31.56

27.06

1964

24.13

28.02

34.64

39.39

39.12

35.29

30.89

29.72

31.29

33.79

28.81

26.00

1965

27.35

28.65

33.21

36.90

39.20

37.91

31.99

30.63

31.40

34.68

31.40

26.46

1966

26.68

31.17

33.24

37.78

39.91

36.71

32.87

29.13

30.83

34.21

30.39

25.72

1967

23.49

29.74

31.85

35.83

38.44

35.71

31.05

28.02

29.42

32.35

30.17

27.24

1968

24.67

26.29

32.96

36.87

38.14

38.14

31.58

28.52

32.83

33.06

29.76

26.16

1969

25.26

28.33

35.64

37.51

38.81

37.40

31.84

29.20

31.58

34.34

31.52

27.23

1970

26.15

27.43

32.72

38.25

40.48

35.70

31.91

29.54

30.31

33.67

28.86

26.10

1971

24.45

27.56

32.58

38.33

38.14

34.49

29.24

29.02

31.11

32.21

29.22

25.85

1972

25.61

25.47

34.29

36.35

39.26

37.40

32.35

30.63

31.06

33.37

29.94

26.72

1973

24.45

28.84

33.01

39.14

40.36

36.44

30.58

28.95

30.06

33.02

29.31

25.99

1974

24.20

26.53

35.04

38.04

39.91

36.64

32.22

30.64

33.17

33.27

29.00

25.28

1975

24.88

26.96

30.81

36.82

40.51

36.10

30.08

29.49

29.99

32.60

28.18

26.52

1976

26.06

27.99

33.55

36.53

39.13

35.43

31.48

28.92

30.05

33.79

31.69

26.67

1977

24.80

29.35

35.51

37.65

39.00

36.05

29.99

28.75

30.19

34.32

31.21

29.17

1978

24.89

26.64

30.91

36.48

40.34

35.35

29.58

28.40

30.62

33.48

30.95

24.92

1979

26.22

26.19

31.30

37.87

37.32

37.22

31.74

29.52

32.80

34.61

30.89

26.84

1980

25.88

30.06

32.75

38.61

40.69

35.67

31.48

30.18

32.34

35.11

31.02

25.28

1981

25.59

29.11

32.62

38.49

40.00

37.80

31.13

29.07

32.17

34.05

28.32

25.13

1982

25.69

26.17

30.59

35.84

36.90

37.53

32.86

29.44

32.57

34.75

29.33

27.62

1983

24.96

26.59

32.52

34.52

38.64

36.83

32.12

29.96

31.65

32.36

28.44

26.24

1984

24.52

24.80

34.83

38.80

41.12

36.82

31.50

28.40

30.29

33.24

29.39

26.55

1985

24.95

29.33

35.56

37.80

40.30

37.29

31.67

28.88

31.79

31.71

30.41

28.75

1986

25.43

27.25

33.21

38.24

39.14

37.21

30.84

28.96

32.76

34.11

31.19

25.86

1987

26.40

29.17

34.11

38.30

37.79

37.52

33.39

31.92

33.71

34.90

31.36

27.22

1988

26.58

29.77

33.45

38.71

41.31

36.90

30.59

29.82

32.37

33.68

30.05

27.41

1989

24.64

27.75

32.72

37.24

40.13

35.80

31.20

28.74

32.47

34.04

31.42

26.76

1990

27.81

27.63

32.08

37.70

39.44

36.79

30.39

29.74

30.71

33.10

30.73

26.63

1991

24.57

28.33

33.59

36.73

37.51

38.29

32.30

29.38

31.41

33.49

29.74

27.06

1992

26.50

26.83

33.23

37.05

39.01

38.83

32.78

29.43

30.42

32.80

29.47

27.56

1993

26.56

29.17

32.21

37.28

40.42

37.05

31.16

30.58

30.98

34.46

31.07

27.73

1994

26.75

27.54

34.94

36.92

40.81

36.18

29.89

29.17

30.61

33.23

30.19

26.96

1995

24.44

28.19

31.82

37.04

39.83

39.71

31.98

29.62

32.12

35.15

30.29

27.60

1996

25.95

28.87

35.14

37.83

39.73

37.55

31.96

28.60

31.16

33.21

29.99

26.83

1997

25.02

27.86

33.55

35.94

37.63

35.60

31.66

29.65

31.60

31.72

30.40

25.37

1998

25.50

28.25

32.33

38.33

41.05

37.90

31.83

31.24

32.04

34.79

30.71

27.82

1999

25.07

28.81

33.85

39.50

40.04

36.36

31.53

30.03

31.73

33.43

30.96

26.56

2000

26.62

26.70

32.93

39.58

39.19

37.08

30.90

30.81

32.24

34.57

31.75

28.03

2001

25.31

28.48

33.70

37.39

39.11

34.60

29.85

30.19

33.33

34.52

30.85

27.70

2002

25.30

27.60

33.98

39.30

41.05

37.45

33.19

29.89

32.48

34.57

31.32

28.99

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Chandrakar, A., Datta, D., Nayak, A.K. et al. Statistical analysis of a time series relevant to passive systems of nuclear power plants. Int J Syst Assur Eng Manag 8, 89–108 (2017). https://doi.org/10.1007/s13198-016-0570-8

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