negative values kernel density estimation

I have obtained the monthly temperature distribution using kernel density estimate. And using SVD(Singular Value Decomposition) and regression model, I forecast the monthly temperature distribution. But I found that some estimated kernel density values are negative. How to deal with these negative values?

[Bjorn Gustavsson]

It might be easier to have opinions if you could share a figure illustrating this.

[Adam Danz]

Code snippets are helpful, too, so we can see exactly what you're doing.

[ZHIMIN YAN]

[U,S,V] = svd(z);

U1 = U(:,1);

Mdl = arima('Constant',0,'ARLags',1,'SARLags',12,'D',1,...

'Seasonality',12,'MALags',1,'SMALags',12);

EstMdl = estimate(Mdl,U1);

numperiods = 11;

[Forecast,ForecastMSE] = forecast(EstMdl,numperiods,U(:,1));

Tempdiff_forecast = Forecast*S(1,1)*V(:,1)';

z is kernel density value, and Tempdiff_forecast is kernel density forecast value. And some values are negative. How to deal with these negative values?

[Adam Danz]

Could you provide the z values?

[ZHIMIN YAN]

This is z values. Thank you very much!!

zmat = load("z.mat");
z = zmat.z;

[U,S,V] = svd(z);
U1 = U(:,1);
Mdl = arima('Constant',0,'ARLags',1,'SARLags',12,'D',1,...
            'Seasonality',12,'MALags',1,'SMALags',12);
EstMdl = estimate(Mdl,U1);

 
    ARIMA(1,1,1) Model Seasonally Integrated with Seasonal AR(12) and MA(12) (Gaussian Distribution):
 
                 Value      StandardError    TStatistic      PValue   
                ________    _____________    __________    ___________

    Constant           0              0           NaN              NaN
    AR{1}        0.04126       0.054551       0.75636          0.44944
    SAR{12}      -0.1694       0.049563       -3.4179       0.00063099
    MA{1}       -0.81241       0.036031       -22.548      1.4186e-112
    SMA{12}     -0.78822       0.035602        -22.14      1.3097e-108
    Variance    0.003309     0.00019167        17.264        8.771e-67

numperiods = 11;
[Forecast,ForecastMSE] = forecast(EstMdl,numperiods,U(:,1));
Tempdiff_forecast = Forecast*S(1,1)*V(:,1)';

[Adam Danz]

You start with negative z values.

data = load('z.mat')

data = struct with fields:
    z: [372×401 double]

[zmin, zmax] = bounds(data.z, 'all') % show range of z values

zmin = -0.0924

zmax = 0.0993

And you're training your model using negative values,

[U,S,V] = svd(data.z);
U1 = U(:,1)

U1 = 372×1

-0.0222 -0.0145 -0.0579 0.0195 -0.0293 0.0161 -0.0491 -0.1093 -0.0165 -0.0512

Your new model EstMdl contains estimated parameters based on the training data U1. So, when you forecast that fitted model, the forecast also has negative values.

[ZHIMIN YAN]

So you mean that I should delete the negative values in z, ensure that the predicted results are all positive?

[ZHIMIN YAN]

Can I delete the negative values from the predicted kernel density values and rescale the kernel density values so that the bandwidth to be 1?

[Adam Danz]

> So you mean that I should delete the negative values in z

No, I would take a step back and investigate. Do you expect there to be negative values in z? If not, then how did they get there? Perhaps something went wrong with your calculations of z or perhaps your expectations of what z should be aren't correct expectations. If you do expect there to be negative values in z or that negative values are possible, then I would re-think whether it is a problem that the forecast produces negative values.

If z isn't meaningful data and you're using z to poke around at the model, then it's completely fine to replace the negative values or use an entirely different set of data. But if z is meaningful data, you can't just delete some values because they are causing problems.

I don't know enough about what the data are or about the forecasting you're using to suggest the next steps.

[ZHIMIN YAN]

z in my data is the difference between actual temperature kernel density distribution and average temperature kernel density distribution. So negative values in z are also meaningful data. And my purpose is to predict the temperature kernel distribution. Thank you very much!