Shrimp Data Modelling using Statistical Tools: State Space based Exponential Smoothing & Response Surface Methodology Ramasubramanian V., Martin Xavier, K. A. and Ananthan, P.S. CENTRAL INSTITUTE OF FISHERIES EDUCATION, MUMBAI
[email protected]
Applied Statistics - Method 1 of 2: State Space based Exponential Smoothing
Exponential smoothing • Weights - unequal - exponentially decreasing as we go into further past
• Simple exponential smoothing – if time series (TS) data has ‘horizontal’ component • Double exponential smoothing /Holt’s – if TS data has ‘trend’ component
• Triple exponential smoothing/ Winters – if TS data has ‘seasonal’ component as well
Exponential smoothing… Depending upon whether the components of time series data viz. Trend and Seasonality are i) Not present (N) ii) Additive (A) iii)Multiplicative (M) and in addition, considering the trend also as Damped(D) Seasonality N A M Trend N
N,N
N,A
N,M
A
A,N
A,A
A,M
M d
M,N d,N
M,A d,A
M,M d,M
Simple exponential smoothing • •
Let TS data be { Yt} The SES model F t+1 = α Y t + (1 – α) F t
• • • • •
Recursive model - F t+1 = f ( Y’s, F 1, α ) Choice of α and F 1 Flat horizon Adaptive Response Rate SES One-step-ahead forecasting
Simple exponential smoothing… •
If α = 0.2 then Yt
Yt
0.2
0.16
-1
Yt
-2
0.128
Yt-3
Yt-4
0.1024
0.0819
Exponential smoothing models (i)
Single Exponential Smoothing ( SES) yt (1)= yt -1 (1) + α [yt− yt-1 (1)]
(ii) Double Exponential Smoothing (DES) – Holt’s Level: lt = α yt +(1− α)(lt−1 + bt−1) Trend: bt = β(lt − lt−1)+(1− β)bt−1 Forecast: yt(h)= lt + bth (iii) Triple Exponential Smoothing (TES) – Multiplicative/Winters Level: Trend: Seasonal: Forecast:
yt lt = α s +(1− α)(lt−1 + bt−1) t -m bt = β (lt − lt−1)+(1− β)bt−1 st = γ yt / (lt−1 + bt−1)+(1− γ)st−m yt(h)=(lt + bt h)st−m+h
State Space modelling allow changes in the structure (read parameter estimates) of the system in a controlled manner as the pattern of data change over time whereas traditional models can be said to be time-invariant more general in the sense that they cover a wide range of models, the calculations needed to implement them can be put in recursive form leading to an unified framework whereas in traditional modelling it is not so each consecutive forecast is found by updating the previous forecast use additional information in the form of an assumed relationship between parameters of the models at different points of time forecasts based on state space models are likely to be more precise than that based on traditional models
State space model State transition equation zt+1= Fzt+Get+1 zt - state vector of dim. s
Let yt – obs. vector of dim. r (given variables) Note: First r components of zt consist of yt, s r Prediction of yt+k based on info. at time t. Then the last s-r elements of zt consist of elements of y t k | t , where k>0
In the state transition equation F - s x s coeff. transition matrix G - s x r coeff. input matrix with the first r rows and columns of G as an r x r identity matrix ee of et - indpt. normally distributed innovation vector dim. r with mean vector 0 and cov. matrix
Measurement or observation equation yt = I r 0z t
State Space based Exponential Smoothing Depending on the trend (additive) and seasonality (none, additive or multiplicative) components present in time series data, exponential smoothing methods on possible combinations of these components under a unified state space modelling framework has been employed for forecasting purposes
Exponential Smoothing Models via State Space Depending on trend and seasonality components in time series data, the common exp. smoothing methods, viz. simple, Holt and Winters collectively written in their error-correction form as (Hyndman et al. , 2002; 2008) lt = Qt + α(Pt − Qt) bt = bt−1 + β(Rt − bt−1) st = st−m + γ(Tt − st−m) on possible combinations of these components can fitted in state space form
lt - series level at time t bt - additive trend at t st - seasonal component of series at t m – no. of seasons in a year • The values of Pt, Qt, Rt, and Tt vary according to which of the trend-seasonality combination the method belongs to (see Table for no trend; similarly additive trend a separate table is there), and α, β and γ are smoothing constants • Exponential smoothing model via state space is z t (l t , b t , s t , s t 1 ,..., s t m1 ) yt =μt + εt and, with z t f z t 1 g z t 1 t where εt is Gaussian WN (0, σ2 )
Seasonality
Form of the model (No trend)
Error correction
None
Pt = Yt Qt = lt−1 Yt(h) = lt
μ =l State space t t-1 lt = lt-1+αεt
Additive
Multiplicative
Pt = Yt − st−m Qt = lt−1 Tt = Yt − Qt Yt(h) = lt + st+h−m
Pt = Yt/st−m Qt = lt−1 Tt = Yt/Qt Yt(h) = l t s t+h−m
μt = lt-1+st-m lt = lt-1+αεt st = st-m+γεt
μt = lt-1st-m lt = lt-1+αεt/ st-m st = st-m+γεt/ lt-1
Application • Weekly data of price indicators of marine product exports on specific grades of Litopenaeus vannamei shrimp have been considered • Source: PRIME (MPEDA, Kochi) • Export prices (US $ per kg) from India to USA • Product form - HLSO –> Head-Less Shell-On • Grade: 25/30 i.e. around 25-30 shrimp counts per kg • Origin:- Mostly Vizag/ Orissa/ Chennai • Period: 02Dec2011 to 17Jan2014 (112 weeks)
Application • Data points - 80% available during period in question • Conversion to fortnightly data / also imputation done for missing data • Data fitting – 25th Fortnight of 2011 to 22nd Fortnight of 2013 (48 data points) • Data validation – subsequent 8 data points • Models – Holt Exp. Smoothing – State Space based Exp. Smoothing
Weekly export prices of Vannamei shrimp
Fortnightly export prices of Vannamei shrimp
Results Exp. Smoothing Model → Error Alpha Beta Sigma AIC
Holt
State space based
Additive 0.9500 0.0001 0.5416 136.3482
Multiplicative 0.8500 0.0721 0.0605 132.0862
Forecast performance measure Yt Ft 1 MAPE x 100 n Yt • n – no. of time points in forecast period • Yt - actual value in time t • Ft - forecast at time t
Results Fortnight No. 49 50 51 52 53 54 55 56
Actual
Forecast Holt
13.92 14.53 14.74 13.29 13.69 13.30 14.46 14.69 MAPE
13.27 13.39 13.51 13.63 13.75 13.88 14.00 14.12 3.80
State space based 13.47 13.74 13.99 14.25 14.50 14.56 15.01 15.26 3.43
Applied Statistics - Method 2 of 2: Response Surface Methodology
Response Surface Methodology • Optimization of glucosamine production • Glucosamine is a neutraceutical from chitin extracted from Metapenaeus Dobsoni shrimp shell waste
• By studying the effect of factors viz., temperature, reaction time, acid to Chitin ratio and acid strength • Response variable – Glucosamine production in grams percentage i.e. how much grams of it in 100 grams of Chitin
Values of the factors in CCD Levels Factor
Units
-2
-1
0
1
2
Temperature
o
80
85
90
95
100
Reaction time
minutes
15
45
75
105
145
-
1:1
2:1
3:1
4:1
5:1
Percentage
30
32
34
36
38
Acid : Chitin ratio
Acid strength
Celsius
RSM model The model fitted was Y = f( A, B, C, D, AB, BC, BD, CD, AA, BB, CC, DD) AC & AD not included Software: The Unscrambler
A
B -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1
C
D
RSM model -1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1
-1 -1 -1 1 1 1 1 -1 -1 -1 -1 1 1 1 1
Y -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 1 1
64.46 53.61 70.43 69.24 68.45 69.97 70.29 66.95 66.60 67.90 69.32 68.39 66.6 69.41 70.29 69.31
RSM data Points
Axial
Central
A
B
C
D
Y
-2
0
0
0
66.52
2
0
0
0
68.72
0
-2
0
0
69.75
0
2
0
0
70.05
0
0
-2
0
53.82
0
0
2
0
69.57
0
0
0
-2
64.01
0
0
0
2
70.56
0
0
0
0
70.17
0
0
0
0
71.56
0
0
0
0
68.44
0
0
0
0
70.51
0
0
0
0
70.17
RSM ANOVA
RSM model coefficients
Generation of Response Surfaces by Fixing Values of the factors in CCD Levels Factor
Units
-2
-1
0
1
2
Temperature
o
80
85
90
95
100
Reaction time
minutes
15
45
75
105
145
-
1:1
2:1
3:1
4:1
5:1
Percentage
30
32
34
36
38
Acid : Chitin ratio
Acid strength
Celsius
Optimal Glucosamine production Fact Temp. ors A Deg. Celsius AB 87.63 AC 88.34 AD 88.24 BC BD CD
Reaction Acid:Chitin Acid time Strength B C D Percentage Minutes Ratio 134.93 3.45 35.68 134.85 134.27
2.80 3.33
32.89 35.10
Maximum Glucosamine Production Y Grams % 72.96 70.84 70.80 72.90 72.99 71.01
Optimal Glucosamine production- SAS output Estimated Ridge of Maximum Response for Variable Yield Factor Values
Radius 0.0
Estimated Response 70.170000
Standard Error 1.434628
0.1
70.422118
1.432548 -0.008634 0.044656 0.078070 0.042853
0.2
70.628470
1.426457 -0.010912 0.096751 0.150524 0.088672
0.3
70.791999
1.416808 -0.003864 0.161794 0.211996 0.137352
0.4
70.918603
1.404373 0.012709 0.249282 0.252847 0.183753
0.5
71.018852
1.390278 0.025355 0.369703 0.262662 0.209011
0.6
71.107231
1.376028 0.013938 0.511358 0.245415 0.195168
0.7
71.195649
1.363540 -0.014479 0.645484 0.219085 0.158566
0.8
71.289612
1.355148 -0.047903 0.766382 0.192168 0.115910
0.9
71.391198
1.353567 -0.082382 0.877731 0.166104 0.072193
1.0
71.501295
1.361795 -0.116899 0.982681 0.140907 0.028604
A 0
B 0
C 0
D 0
Thank you
References follow…
References • Hyndman, R. J., Koehler, A.B., Snyder, R.D. and Grose, S. (2002) A state space framework for automatic forecasting using exponential smoothing methods, International Journal of Forecasting, 18, 439-54. • Hyndman, R. J., Koehler, A.B., Ord, J.K. and Snyder, R.D. (2005). Prediction intervals for exponential smoothing using two new classes of state space models, Journal of Forecasting, 24, 17-37. • Hyndman, R.J., Akram, Md., and Archibald, B. (2008) "The admissible parameter space for exponential smoothing models". Annals of Statistical Mathematics, 60(2), 407–426. • Hyndman, R.J., Koehler, A.B., Ord, J.K., and Snyder, R.D. (2008) Forecasting with exponential smoothing: the state space approach, Springer-Verlag. http://www.exponentialsmoothing.net.ht
References… • Box, G.E.P., Hunter, W.G. and Hunter, J.S. (1978). Statistics for experimenters: An introduction to design, data analysis and model building, John Wiley & Sons, New york. • The Unscrambler X, CAMO Software India Pvt. Ltd., Bangalore (Norway based product) • R version 3.0.2 (2013-09-25) The R Foundation for Statistical Computing.
• SAS software
