Monday, August 30, 2010

TempLS V2 Math basis

The Math basis of TempLS Ver 2

For Version 1 of TempLS I posted a PDF guide, TempLS.pdf, to the mathematical basis. It's a useful starting point here.

But this time I'm going to try to do it in HTML below. Colors help.

For more Version 2 information:


  • Index of posts by category
  • TempLS Version 2 Release
  • Ver 2 - Regional spatial variation.
  • Spatial Temperature distributions in TempLS V2
  • Plotting spatial trends in TempLS


  •  Notation

    I need to use a summation convention. I'm referring to various matrices etc as just indexed sets, and when they are multiplied together, every index that appears twice is understood to be summed over the range of that index. It's called a dummy index, because it doesn't matter what symbol you use (as long as it's not used for something else). Once the summation is done, that index doesn't connect with anything else.

    I'll be talking about weighted least squares, and an exception is needed for the weight function. It's indices are sometimes put in brackets, which means they should not be counted in this pairing. The index still varies with any summation that is going on.

    The reason for this is that summation signs are difficult in HTML. But also it gets easier with colors. I'll color indices subject to summation blue, standalone indices red, and exception indices lime green.

    So multiplying two matrices A and B (C=A*B) would be represented:
    Cik  =  Aij Bjk

    The model

    We have, on the GHCN and similar sets, a number of stations (s), with temperature readings (x) by year (y) and month (m). m varies 1:12. The model is that x can be represented by a local temperature L, with a value for each station and month (seasonal average), and a global temperature G depending on year alone.

    xsmy  ~  Lsm  +  Gy
    It's useful to maintain a convention that every term has to include every red index. So I'll introduce an identity Iy, for which every component is 1, and rewrite the model as
    xsmy  ~  IyLsm  +  IsmGy

    Weighted least squares.

    The L and G are parameters to be fitted. We minimise a weighted squares expression for the residuals. The weight function w has two important roles.
    • It is zero for missing values
    • It needs to ensure that stations make "fair" contributions to the sum, so that regions with many stations aren't over-represented. This means that the sum with weights should be like a numerical integral over the surface. w should be inversely proportional to station density.
    So we want to minimise w(smy)(xsmy - IyLsm - IsmGy)2

    Differentiation

    This is done by differentiating this expression wrt each parameter component. In the summation convention, when you differentiate wrt a paired index, it disappears. The remaining index goes from blue to red. The original scalar SS becomes a system of equations, as the red indices indicate:
    Minimise:w(smy)(xsmy - IyLsm - IsmGy)2
    Differentiate wrt L:   w(smy)Iy(xsmy - IyLsm - IsmGy) = 0
    Differentiate wrt G:w(smy)Ism(xsmy - IyLsm - IsmGy) = 0


    The identity operations have been useful, but can now be rationalised. The equations become:

    Differentiate wrt L:   w(sm)y IyLsm  +  wsmy Gy  =  w(sm)y xsmy
    Differentiate wrt G:wsmy Lsm  +  wsm(y) IsmGy  =  wsm(y) xsmy

    Equations to be solved

    These are the equations to be solved for L and G. It can be seen as a block system:
    A*L + B*G  = X1
    BT*L + C*G  = X2
    combining s and m as one index, and where T signifies transpose. Examining the various terms, one sees that A and C are diagonal matrices, so the overall system is symmetric.

    The first equation can be used to eliminate L, so
    (C -  BTA-1B)*G = X2  -  BTA-1X1
    This is not a large system to solve - G has of order 100 components (years). But the multiplications to create it are time-consuming. The alternative offered by TempLS is to use the conjugate gradient method on the whole system. This turns out to be fast and reliable.

    There is a rank deficiency in that a constant could be added to L and subtracted from G. This is resolved by forcing the first component of G to be zero. Later this can be adjusted to set the base period for G as an anomaly. The same issue arises with spatial models below.

    Spatial dependence

    That is what was done in Ver 1. In version 2, it is called model 1.The weight function w was estimated using a grid to get the station density locally.

    But we could try to put more parameters into G. In particular, we could try to let it vary spatially, and this is what is new in V2.

    We need a set of orthogonal functions, expressed as a matrix Psk, being a value at each station, where k indexes the functions. Actually, the functions needn't be strictly orthogonal, but should express the range of possible spatial variation.

    The continuous function that this produces is Pk(lat,lon)Hk, where H is a set of coefficients to be found.

    Then it's simpler if the dependence of G on years is fixed in advance. A model expressing these dependences is:
    xsmy ~ IyLsm +  ImJyPskHk
    For J, there are two forms used:
    • J=y, yr0≤y≤yr1 gives a trend in C/yr between those years. This is model 2
    • J=1, yr0≤y≤yr1 gives an average C between those years. In Ver 2.0, this is model 3, and is restricted to a single year.

    Differentiation

    This time the sum of squares  of residuals is differentiated wrt L and H:
    Minimise:w(smy)(xsmy - IyLsm -  ImJyPskHk)2
    Differentiate wrt L:    w(smy)Iy(xsmy -  IyLsm -  ImJyPskHk) = 0
    Differentiate wrt H:w(smy) ImJy Psk (xsmy -  IyLsm -  ImJyPsjHj ) = 0


    Again it can be re-expressed as a symmetric system:
    Differentiate wrt L:   w(sm)y IyLsm  +  w(s)my JyPskHk  =  w(sm)y xsmy
    Differentiate wrt H:w(s)my Jy Psk Lsm  +  w(smy) ImIm JyJy Psk Psj Hj  =  w(s)m(y) Jy Psk xsmy

    Solving


    Solving is by the same elimination method - the multiplier of L in the first equation is again a diagonal matrix. The conjugate gradient method is less attractive here.

    Spatial methods can be used over the whole globe or over sub-regions. There is a tension between the resolution you would like to get by using many orthogonal functions, and the information available. If there are too few stations for the resolution, the final matrix will be ill-conditioned, and spurious results will occur.

    TempLS deals with this by doing a diagonalization of that final symmetric matrix, and inverts a subsystem based on the larger eigenvalues - a kind of PCA. The user can specify where the cutoff occurs, using a variable eigcut. This should be a small number, between 0 and 1, but closer to 0. I often use 0.01 - higher values are more conservative. A strategy is to try different values until the results do not depend on your choice.

    Orthogonal functions

    The types currently available are
    • Type 1 - spherical harmonics, for the whole sphere
    • Type 0 - trig functions (of lat-lon) for use on approximately rectangular regions
    The number num_orthogs in the user file gives the approximate resolution (in half-periods)in one dimension. The total number of functions generated is approximately the square of this. Values in the range 8-12 are common - maybe up to 16 on the globe. Remember that it will generally be over-ridden by the eigcut facility - ie if you set num_orthogs too high, you'll just do some extra work for the same result, while if you set it low, you won't get the resolution that you could.

    2 comments:

    1. OK, I'm confused on spatial dependence. You're using Psk Hk to account for a geographical dependence. But that term is time-independent, so should look the same in every map. And you're scaling it by a factor which varies with year.
      So what the model is saying is that any change over time happens according to a geographical function scaled by a time function, plus the local adjustment. Is that right?
      I don't understand the bullet points describing J at all, I'm afraid.
      Kevin C

      ReplyDelete
    2. Kevin,
      It has the form of a general expansion in orthogonal functions. If J_y is linear, say, then it is effectively a spherical harmonics expansion of trend. The coefficients are fitted jointly with the station offsets.

      The way I do it by month means that really you just couple a SH series for a single month temp with the local offsets. Although fitting jointly has some theoretical advantages, I think in practice the coupling is weak, and as I said at SKS, there isn't much difference to just calculating the anomalies without space dependence and fitting the SH coefficients afterwards.

      In principle J_y could be any function of time. But because you have only one vs a whole family of space functions, it should be something fundamental. I've used just linear, so it's just fitting a spatially varying trend, and a sort of pulse function. The pulse function isn't saying that that is how the temp actually progressed. It's just saying that you weight the current month or year differently. You only look at past values to get local averages.

      The main practical advantage of the joint fitting is that you avoid the troubles of GISS etc in dealing with an anomaly base period where some stations don't have data.

      Let me try to put it another way. If you were doing a 2D fourier series on a square, say, you would make a series a_jk sin(jx) sin(ky) (plus cos terms etc). j and k would vary over a similar range. Here I'm in effect reducing one range to a single value. That is time, and the (partial) justification of the lopsidedness is that we have more data in space than in time (measured in years).

      ReplyDelete