Meteo

Meteorology & Markov Chains for the HP-41

Overview

-We assume that the weather on day j+1 only depends on the weather on day j ( Markov property )
-This is perhaps not completely true but it's a good approximation.
-The following routine simply computes the Transition Matrix from observations of N consecutive days

-The weather is defined by n states, for example 1 = sunny day , 2 = cloudy ( without rain ) , 3 = rainy day
( do not use 0 )

Program Listing

Data Registers: • R00 = n = number of states ( Registers R00 and Rbb thru Ree are to be initialized before executing "METEO" )

• Rbb = x₁ ...................... • Ree = x_N bb > n²+n Rn²+1 to Rn²+n: temp

>>>> When the program stops, R01 thru Rn² contain the elements of the Transition Matrix

Flags: /
Subroutines: /

-Lines 02 and 53-54 are only useful to save bbb.eee otherwise, they can be deleted
-If you don't have an HP-41CX, replace line 10 by SIGN CLX LBL 00 STO IND L ISG L GTO 00

01 LBL "METEO"
02 STO M
03 RCL 00
04 X^2
05 LASTX
06 +
07 E3
08 /
09 ISG X
10 CLRGX
11 X<>Y

12 LBL 01
13 RCL IND X
14 ENTER^
15 ISG Z
16 X=0?
17 GTO 02
18 RCL 00
19 ST* Y
20 -
21 RCL IND Z
22 +

23 X<>Y
24 RCL 00
25 X^2
26 +
27 SIGN
28 ST+ IND Y
29 ST+ IND L
30 R^
31 GTO 01
32 LBL 02
33 RCL 00

34 ENTER^
35 X^2
36 .1
37 %
38 +
39 ST+ Y
40 LBL 03
41 RCL 00
42 E3
43 /
44 -

45 LBL 04
46 RCL IND Z
47 ST/ IND Y
48 RDN
49 DSE X
50 GTO 04
51 DSE Y
52 GTO 03
53 CLX
54 X<> M
55 END

( 87 bytes )

STACK	INPUT	OUTPUT
X	bbb.eee	bbb.eee

with bbb > n²+n

Example: You observe the weather during 21 consecutive days, it yields: 112212333231122322121 ( 1 = Sun , 2 = Clouds without Rain , 3 = Rain )

n = 3 STO 00 and store these 21 integers into registers R13 R14 .............. R33 control number = 13.033

13.033 XEQ "METEO" >>>> 13.035 --- Execution time = 15s ---

And we get:

      R01   R04   R07        1/3   1/3   1/5
      R02   R05   R08   =   2/3   1/3   2/5   =   Transition Matrix = T
      R03   R06   R09         0     1/3   2/5

-What will be tomorrow's weather ?

1- If today is a sunny day, look at the 1st column:

         the probability that tomorrow will be sunny is 1/3
         the probability that tomorrow will be cloudy is 2/3
         the probability that tomorrow will be rainy is 0

2- If today is a cloudy day, look at the 2nd column:

        the probability that tomorrow will be sunny is 1/3
        the probability that tomorrow will be cloudy is 1/3
        the probability that tomorrow will be rainy is 1/3

3- If today is a rainy day, look at the 3rd column:

        the probability that tomorrow will be sunny is 1/5
        the probability that tomorrow will be cloudy is 2/5
        the probability that tomorrow will be rainy is 2/5

Notes:

-The sum of the elements of each column is always 1

-Limit of T^k as k tends to infinity =

       0.3    0.3    0.3
      0.45 0.45 0.45
      0.25 0.25 0.25

-Using one register to store just a one-digit number is of course wasteful.
-With the following variant, you can store the data by groups of up to 10 states, provided that n < 10

-If you don't have an HP-41CX, replace line 16 by SIGN CLX LBL 00 STO IND L ISG L GTO 00

01 LBL "METEO"
02 STO N
03 INT
04 DSE X
05 LASTX
06 FRC
07 RCL 00
08 X^2
09 LASTX
10 +
11 E3
12 ST* Z
13 ST/ T
14 /
15 ISG X
16 CLRGX

17 RDN
18 +
19 STO M
20 RCL IND X
21 STO Y
22 10
23 ST/ Z
24 MOD
25 X<>Y
26 LBL 01
27 INT
28 X=0?
29 GTO 02
30 RCL X
31 10
32 ST/ Z

33 MOD
34 STO O
35 RCL 00
36 ST* Y
37 -
38 ST+ Z
39 SIGN
40 ST+ IND Z
41 RCL 00
42 X^2
43 RCL O
44 +
45 X<>Y
46 ST+ IND Y
47 LASTX
48 R^

49 GTO 01
50 LBL 02
51 DSE M
52 X<0?
53 GTO 02
54 CLX
55 RCL IND M
56 GTO 01
57 LBL 02
58 RCL 00
59 ENTER^
60 X^2
61 .1
62 %
63 +
64 ST+ Y

65 LBL 03
66 RCL 00
67 E3
68 /
69 -
70 LBL 04
71 RCL IND Z
72 ST/ IND Y
73 RDN
74 DSE X
75 GTO 04
76 DSE Y
77 GTO 03
78 RCL N
79 CLA
80 END

( 127 bytes )

STACK	INPUT	OUTPUT
X	bbb.eee	bbb.eee

with bbb > n²+n

Example: With the same observations, 112212333231122322121 may be stored as follows:

    112212333   STO 13                    1122123332   STO 13
    231122322   STO 14         or        3112232212   STO 14        and many other ways
          121         STO 15                             1            STO 15

-In both cases, 13.015 XEQ "METEO" gives 13.015 ( in 18 seconds ) and the same transition matrix.

Note:

The weather model may of course be improved, for instance:

   1 = Sun without wind
   2 = Sun & wind
   3 = Cloud ( without rain ) without wind
   4 = Cloud & wind ( without rain )
   5 = Rain without wind
   6 = Rain & wind

... and so on ...

hp41programs

Meteorology & Markov Chains for the HP-41