Multidimensional_linear_regression.ipynb

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    "\"\"\" THIS FUNCTION CALCULATES A LINEAR REGRESSION OF 2 VARIABLES WITH MORE THAN 1 DIMENSION\n",
    "\n",
    "Code written by: Hrishi (https://hrishichandanpurkar.blogspot.com/2017/09/vectorized-functions-for-correlation.html)\n",
    "\n",
    "Uploaded by Paloma Trascasa-Castro\n",
    "\n",
    "\"\"\"\n",
    "\n",
    "\n",
    "# First import some modules to make this function work\n",
    "\n",
    "import xarray as xr \n",
    "from  scipy import stats, signal #Required for detrending data and computing regression\n",
    "\n",
    "# Define the function \"lag_linregress_3D\"\n",
    "\n",
    "def lag_linregress_3D(x, y, lagx=0, lagy=0):\n",
    "    \"\"\"\n",
    "    Input: Two xr.Datarrays of any dimensions with the first dim being time. \n",
    "    Thus the input data could be a 1D time series, or for example, have three dimensions (time,lat,lon). \n",
    "    Datasets can be provied in any order, but note that the regression slope and intercept will be calculated\n",
    "    for y with respect to x.\n",
    "    Output: Covariance, correlation, regression slope and intercept, p-value, and standard error on regression\n",
    "    between the two datasets along their aligned time dimension.  \n",
    "    Lag values can be assigned to either of the data, with lagx shifting x, and lagy shifting y, with the specified lag amount. \n",
    "    \"\"\" \n",
    "    #1. Ensure that the data are properly alinged to each other. \n",
    "    \n",
    "    x,y = xr.align(x,y)\n",
    "    \n",
    "    \n",
    "    #2. Add lag information if any, and shift the data accordingly. If not needed, jump to step 3\n",
    "    if lagx!=0:\n",
    "        #If x lags y by 1, x must be shifted 1 step backwards. \n",
    "        #But as the 'zero-th' value is nonexistant, xr assigns it as invalid (nan). Hence it needs to be dropped\n",
    "        x   = x.shift(time = -lagx).dropna(dim='time')\n",
    "        #Next important step is to re-align the two datasets so that y adjusts to the changed coordinates of x\n",
    "        x,y = xr.align(x,y)\n",
    "\n",
    "    if lagy!=0:\n",
    "        y   = y.shift(time = -lagy).dropna(dim='time')\n",
    "        x,y = xr.align(x,y)\n",
    " \n",
    "    #3. Compute data length, mean and standard deviation along time axis for further use: \n",
    "    n     = x.shape[0]\n",
    "    xmean = x.mean(axis=0)\n",
    "    ymean = y.mean(axis=0)\n",
    "    xstd  = x.std(axis=0)\n",
    "    ystd  = y.std(axis=0)\n",
    "    \n",
    "    #4. Compute covariance along time axis\n",
    "    cov   =  np.sum((x - xmean)*(y - ymean), axis=0)/(n)\n",
    "    \n",
    "    #5. Compute correlation along time axis\n",
    "    cor   = cov/(xstd*ystd)\n",
    "    \n",
    "    #6. Compute regression slope and intercept:\n",
    "    slope     = cov/(xstd**2)\n",
    "    intercept = ymean - xmean*slope  \n",
    "    \n",
    "    #7. Compute P-value and standard error\n",
    "    #Compute t-statistics\n",
    "    tstats = cor*np.sqrt(n-2)/np.sqrt(1-cor**2)\n",
    "    stderr = slope/tstats\n",
    "    \n",
    "    from scipy.stats import t\n",
    "    pval   = t.sf(tstats, n-2)*2\n",
    "    pval   = xr.DataArray(pval, dims=cor.dims, coords=cor.coords)\n",
    "\n",
    "    return cov,cor,slope,intercept,pval,stderr"
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    "# Now all you need to do is to prepare your data and run the funcion:\n",
    "\n",
    "# Convert your data to an xr.Darray. For example I used 2 3-D cubes (time,latitude,longitude)\n",
    "# if I want to regress wind onto sst, then:\n",
    "\n",
    "y=xr.DataArray(wind.data)\n",
    "x=xr.DataArray(sst.data)\n",
    "\n",
    "# Calculate covariance, correlation, slope, intercept, pvalue and standard error\n",
    "# of your linear regression\n",
    "\n",
    "cov,cor,slope,intercept,pval,stderr = lag_linregress_3D(x,y)\n",
    "\n",
    "print (slope)"
   ]
  }
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